Diffusion in Minerals and Melts 9781501508394, 9780939950867

Citation preview

REVIEWS IN MINERALOGY AND GEOCHEMISTRY Volume 72

2010

Diffusion in Minerals and Melts EDITORS Youxue Zhang

University of Michigan Ann Arbor, Michigan, U.S.A.

Daniele J. Cherniak

Rensselaer Polytechnic Institute Troy, New York, U.S.A.

ON THE COVER: Top Left: A BSE image showing zonation of zircon (Zhang 2008, Geochemical Kinetics). Lower Right: Ar diffusivity in air, water, melts and hornblende, and heat diffusivity as a function of temperature (data are from various sources).

Series Editor: Jodi J. Rosso MINERALOGICAL SOCIETY OF AMERICA GEOCHEMICAL SOCIETY

Reviews in Mineralogy and Geochemistry, Volume 72 Diffusion in Minerals and Melts ISSN ISBN

1529-6466

978-0-939950-86-7

COPYRIGHT 2 0 1 0 T H E M I N E R A L O G I C A L S O C I E T Y OF A M E R I C A 3 6 3 5 CONCORDE PARKWAY, SUITE 5 0 0 CHANTILLY, VIRGINIA, 2 0 1 5 1 - 1 1 2 5 , U . S . A . WWW.MINSOCAM.ORG The appearance of the code at the bottom of the first page of each chapter in this volume indicates the copyright owner's consent that copies of the article can be made for personal use or internal use or for the personal use or internal use of specific clients, provided the original publication is cited. The consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other types of copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. For permission to reprint entire articles in these cases and the like, consult the Administrator of the Mineralogical Society of America as to the royalty due to the Society.

Diffusion in Minerals and Melts 72

Reviews in Mineralogy and Geochemistry

72

FROM THE SERIES EDITOR The chapters in this volume represent an extensive compilation of the material presented by the invited speakers at a short course on Diffusion in Minerals and Melts held prior (December 11-12,2010) to the Annual fall meeting of the American Geophysical Union in San Francisco, California. The short course was held at the Napa Valley Marriott Hotel and Spa in Napa, California and was sponsored by the Mineralogical Society of America and the Geochemical Society. At the MSA website, www.minsocam.org/MSA/RIM, the supplemental material associated with this volume can be found and the reader is encouraged to have a look at it. Any errata will also be posted there. The reader will also be able to find links to the electronic copies of this and other RiMG volumes. Jodi 3*. P-osso. Series Editor West Richland, Washington October 2010

PREFACE Geologists often need to apply diffusion theory and data to understand the degree of mass transfer, infer temperature-time histories, and address a wide range of geological problems. The aim of this volume is to provide practitioners the necessary background and data for such applications. We have made efforts to present a comprehensive overview, with discussion and assessment of diffusion data in a broad range of rock-forming minerals and all geologically relevant melts. Extensive data tables are provided as online supplements (as well as at websites maintained by individual authors), both for general usage by readers, and for experimentalists and theoreticians in the field to develop greater understanding of diffusion and plan future research directions. We would like to take this opportunity to thank the authors of individual chapters, and those who reviewed the chapters. The reviewers are: Don Baker, Harald Behrens, Bill Carlson, Michael Carroll, Fidel Costa, John Farver, John Ferry, Jiba Ganguly, Matt Heizler, Jannick Ingrin, Motoo Ito, David Kohlstedt, Ted Labotka, Chip Lesher, Yan Liang, Thomas Mueller, Jim Mungall, Martin Reich, Rick Ryerson, Jim Shelby, Frank Spera, Jim Van Orman, Yong-Fei Zheng, and anonymous reviewers. This volume and the accompanying short course in Napa Valley were made possible by generous support for student participants from the US National Science Foundation. The preparation of this volume and the short course benefited tremendously from the efforts of Jodi Rosso and Alex Speer. Youxue Zhang Ann Arbor, Michigan

1529-6466/10/0072-0000$05.00

Daniele Cherniak Troy, New York

DOT: 10.2138/rmg.2010.72.0

TABLE OF CONTENTS 1

Diffusion in Minerals and Melts: Introduction Y. Zhang, D.J. Cherniak

INTRODUCTION: RATIONALE FOR THIS VOLUME SCOPE AND CONTENT OF THIS VOLUME REFERENCES

Z.

1 2 3

Diffusion in Minerals and Melts: Theoretical Background Y. Zhang

INTRODUCTION FUNDAMENTALS OF DIFFUSION Basic concepts Microscopic view of diffusion Various kinds of diffusion General mass conservation and various forms of the diffusion equation Diffusion in three dimensions (isotropic media) SOLUTIONS TO BINARY AND ISOTROPIC DIFFUSION PROBLEMS Thin-source diffusion Comments about fitting data Sorption or desorption Diffusion couple or triple Diffusive crystal dissolution Variable diffusivity along a profile Homogenization of a crystal with oscillatory zoning One dimensional diffusional exchange between two phases at constant temperature Spinodal decomposition Diffusive loss of radiogenic nuclides and closure temperature DIFFUSION IN ANISOTROPIC MEDIA MULTICOMPONENT DIFFUSION Effective binary approach, FEBD and SEBD Modified effective binary approach (activity-based effective binary approach) Diffusivity matrix approach Activity-based diffusivity matrix approach Origin of the cross-diffusivity terms DIFFUSION COEFFICIENTS Temperature dependence of diffusivities; Arrhenius relation Pressure dependence of diffusivities Diffusion in crystalline phases and defects Diffusivities and oxygen fugacity Compositional dependence of diffusivities iv

5 6 6 9 10 14 17 18 18 19 20 22 23 25 26 27 28 29 32 35 36 39 40 42 42 43 43 43 45 47 47

Diffusion in Minerals and Melts - Table of Contents Relation between diffusivity, particle size, particle charge, and viscosity Diffusivity and ionic porosity Compensation "law" Interdiffusivity and self diffusivity CONCLUSIONS ACKNOWLEDGMENTS REFERENCES APPENDIX 1. EXPRESSION OF DIFFUSION TENSOR IN CRYSTALS WITH DIFFERENT SYMMETRY

3

48 50 50 50 53 53 53 58

Non-traditional and Emerging Methods for Characterizing Diffusion in Minerals and Mineral Aggregates E.B. Watson, R. Dohmen

INTRODUCTION THE THIN-FILM METHOD AND PULSED LASER DEPOSITION (PLD): PRINCIPLES AND RECENT DEVELOPMENTS Definition of a thin film Why use thin films? Fitting of diffusion profiles from thin-film diffusion couples Analytical solutions - examples Fitting uncertainties Pulsed laser ablation: a versatile method for thin film deposition Application of PLD to diffusion studies - examples Single layer configurations Double layer configurations THE POWDER-SOURCE TECHNIQUE Overview and history Rationale and details Analytical considerations, advantages and drawbacks ION IMPLANTATION AND DIFFUSION EXPERIMENTS Introduction Interactions between energetic ions and solids Ion implantation Mathematical aspects of implantation and diffusion Complications and examples THE DETECTOR-PARTICLE METHOD FOR STUDIES OF GRAIN-BOUNDARY DIFFUSION Context and history The detector-particle approach: general considerations and examples Numerical simulation: constant-surface model A simple analysis of the detector-particle method Concluding remarks on detector particles ACKNOWLEDGMENTS REFERENCES

v

61 63 63 64 65 65 67 68 70 71 74 78 78 79 80 82 82 83 84 85 87 90 90 91 94 99 100 101 101

Diffusion in Minerals and Melts - Table of Contents Analytical Methods in Diffusion Studies D.J. Cherniak, R. Hervig, J. Koepke, Y. Zhang, D. Zhao INTRODUCTION "CLASSICAL" METHODS FOR MEASURING DIFFUSION PROFILES USING RADIOACTIVE TRACERS Serial sectioning Autoradiography ELECTRON MICROPROBE ANALYSIS Principles of EMPA Instrumentation for EMPA Applications and limitations of EMPA Summary SECONDARY ION MASS SPECTROMETRY (SIMS) Basic principles of SIMS Using SIMS to measure diffusion profiles Depth profile analyses Ion implantation and SIMS Summary comments LASER ABLATION ICP-MS (LA ICP-MS) RUTHERFORD BACKSCATTERING SPECTROMETRY (RBS) Basic principles of RBS Depth and mass resolution Example applications of R B S in diffusion studies NUCLEAR REACTION ANALYSIS (NRA) ELASTIC RECOIL DETECTION (ERD) FOURIER TRANSFORM INFRARED SPECTROSCOPY Vibrational modes and infrared absorption Instrumentation for Infrared Spectroscopy Different types of IR spectra Calibration Applications to geology SYNCHROTRON X-RAY FLUORESCENCE MICROANALYSIS (li-SRXRF) Instrumental setup, spectra acquisition and data processing Sample preparation Applications of |i-SRXRF for measuring trace element diffusivities in silicate melts ACKNOWLEDGMENTS REFERENCES

D

107 109 109 110 Ill Ill 113 120 123 123 123 125 129 134 134 134 137 137 140 141 143 147 148 148 152 152 153 155 156 156 158 158 160 160

Diffusion of H, C, and O Components in Silicate Melts Y. Zhang, H. Ni

INTRODUCTION

171 vi

Diffusion in Minerals and Melts - Table of Contents DIFFUSION OF THE H 2 0 COMPONENT H 2 0 speciation: equilibrium and kinetics H 2 0 diffusion literature H 2 0 diffusion, theory and data summary MOLECULAR H2 DIFFUSION DIFFUSION OF THE CO, COMPONENT OXYGEN DIFFUSION Self-diffusion of oxygen in silicate melts under dry conditions Chemical diffusion of oxygen under dry conditions " S e l f ' diffusion of oxygen in the presence of H 2 0 " S e l f ' diffusion of oxygen in natural silicate melts in natural environments Contribution of C 0 2 diffusion to 1 8 0 transport in C0 2 -bearing melts Oxygen diffusion and viscosity: applicability of the Eyring equation 0 2 DIFFUSION IN PURE SILICA MELT SUMMARY AND CONCLUSIONS ACKNOWLEDGMENTS REFERENCES

O

172 172 178 180 191 197 199 200 207 209 211 213 216 217 219 219 219

Noble Gas Diffusion in Silicate Glasses and Melts H. Behrens

INTRODUCTION EXPERIMENTAL AND ANALYTICAL METHODS Studies at atmospheric and sub-atmospheric pressure Studies at high-pressure DIFFUSION SYSTEMATICS Temperature dependence of diffusivity Pressure dependence of diffusivity Comparison of different noble gases in the same matrix glass COMPOSITIONAL EFFECTS ON NOBLE GAS DIFFUSION He diffusion Ne diffusion Ar diffusion Kr, Xe and Rn diffusion COMPARISON OF NOBLE GASES AND MOLECULAR SPECIES H2 diffusion H 2 0 diffusion O, diffusion N2 diffusion CO, diffusion ACKNOWLEDGMENTS RERERENCES APPENDIX

vii

227 228 228 230 232 232 233 236 238 238 240 241 248 249 249 250 250 251 252 252 253 257

Diffusion

in Minerals

and Melts

- Table of

Contents

Observations and Applications to Magmatic Systems C.E. Lesher INTRODUCTION ADDITIONAL TERMINOLOGY THEORETICAL CONSIDERATIONS Self and tracer diffusion Intradiffusion Polyanionic diffusion EXPERIMENTAL METHODS AND DATA Thin source method Diffusion couple method Capillary-reservoir method Gas exchange method DISCUSSION Background Ionic charge and size Temperature Viscosity and the Eyring diffusivity Pressure CONCLUDING REMARKS ACKNOWLEDGMENTS REFERENCES

O

269 270 271 271 276 280 283 283 284 284 285 285 285 286 288 291 296 303 305 305

Diffusion Data in Silicate Melts Y. Zhang, H. Ni, Y. Chen

INTRODUCTION Terminology General comments about experimental methods to extract diffusivities Grouping of the elements Data compilation Quantification of D as a function of '/ . H 2 0, P,f0l and melt composition DIFFUSION OF INDIVIDUAL ELEMENTS Diffusion of major elements versus minor and trace elements H diffusion The alkalis (Li, Na, K, Rb, Cs, Fr) The alkali earths (Be, Mg, Ca, Sr, Ba, Ra) B, Al, Ga, In, and T1 C, Si, Ge, Sn and Pb N, P,As, Sb, Bi O, S, Se, Te, Po F, CI, Br, I, At He, Ne, Ar, Kr, Xe, Rn viii

311 312 313 315 315 317 317 317 320 320 330 340 345 352 354 356 360

Diffusion in Minerals and Melts - Table of Contents Sc, Y, REE Ti, Zr, Hf V, Nb, Ta Cr, Mo, W Mn, Fe, Co, Ni, Cu, Zn Tc, Ru, Rh, Pd, Ag, Cd Re, Os, Ir, Pt, Au, Hg Ac,Th, Pa, U DISCUSSION The empirical model by Mungall (2002) Effect of ionic size on diffusivities of isovalent ions Dependence of diffusivities on melt composition Diffusivity sequence in various melts CONCLUDING REMARKS ACKNOWLEDGMENTS REFERENCES

s

360 375 380 383 383 389 389 391 393 393 395 397 398 402 404 404

Multicomponent Diffusion in Molten Silicates: Theory, Experiments, and Geological Applications Y. Liang

INTRODUCTION IRREVERSIBLE THERMODYNAMICS AND MULTICOMPONENT DIFFUSION The rate of entropy production Diffusing species and choice of endmember component GENERAL FEATURES OF MULTICOMPONENT DIFFUSION Solutions to multicomponent diffusion equations Essential features of multicomponent diffusion EXPERIMENTAL STUDIES OF MULTICOMPONENT DIFFUSION Experimental design and strategy Inversion methods Experimental results EMPIRICAL MODELS FOR MULTICOMPONENT DIFFUSION Empirical models Experimental tests of the empirical models GEOLOGICAL APPLICATIONS Modeling isotopic ratios during chemical diffusion in multicomponent melts Convective crystal dissolution in a multicomponent melt Crystal growth and dissolution in a multicomponent melt FUTURE DIRECTIONS ACKNOWLEDGMENTS REFERENCES

ix

409 411 411 412 414 414 415 423 423 425 428 434 434 436 437 437 438 441 442 443 443

Diffusion in Minerals and Melts - Table of Contents IU

Oxygen and Hydrogen Diffusion in Minerals J.R. Farver

INTRODUCTION EXPERIMENTAL METHODS Bulk exchange experiments Single crystal experiments ANALYTICAL METHODS Mass Spectrometry Nuclear Reaction Analysis Fourier Transform Infrared Spectroscopy Other methods RESULTS Quartz Feldspars Olivine Pyroxene Amphiboles Sheet silicates Garnet Zircons Titanite Melilite Tourmaline and beryl Oxides Carbonates Phosphates DISCUSSION Effect of temperature Effect of mineral structure Empirical methods Anisotropy Pressure dependence Effect of water Hydrogen chemical diffusion and the role of defects ACKNOWLEGMENTS REFERENCES

I I

447 447 447 448 449 449 450 450 450 451 451 455 461 465 470 471 472 474 474 475 476 477 480 482 483 483 485 486 486 488 488 489 490 490

Diffusion of Noble Gases in Minerals E.F. Baxter

INTRODUCTION The interpretive challenge of bulk-degassing experiments HELIUM He in apatite x

509 510 513 514

Diffusion in Minerals and Melts - Table of Contents He in titanite He in zircon and zircon-structure rare earth element orthophosphates He in monazite and monazite-structure rare earth element orthophosphates He diffusion in other minerals ARGON Ar in micas Arin amphibole Ar in feldspar Ar diffusion in other minerals THE OTHER NOBLE GASES: NEON, KRYPTON, XENON, RADON THEMES IN NOBLE GAS DIFFUSION IN MINERALS Effect of radiation damage Effect of deformation Multi-domain diffusion Multi-path diffusion Synthesis: relative diffusivities of the noble gases in minerals CHOOSING THE "RIGHT" DIFFUSION DATA Role of noble gas diffusion data in Ar/Ar and (U-Th)/He thermochronology SUGGESTIONS FOR FUTURE STUDY Diffusion at high pressures and temperatures Diffusion of Ar and He in common mantle minerals In situ depth profile analysis Quantification of noble gas diffusion within "fast paths" Integrated studies with multiple noble gases Quantification of effects of radiation damage, defects, and deformation ACKNOWLEDGMENTS REFERENCES

I Zi

520 520 523 523 527 528 529 529 530 532 534 534 535 536 537 539 542 545 548 548 548 551 551 551 551 552 552

Cation Diffusion Kinetics in Aluminosilicate Garnets and Geological Applications ./. Ganguly

INTRODUCTION NOMENCLATURE OF DIFFUSION COEFFICIENTS EXPERIMENTAL DETERMINATION OF DIFFUSION COEFFICIENTS Experimental methods Modeling of experimental data EXPERIMENTAL DATA AND DISCUSSION Self/tracer diffusion coefficients of Mn, Fe 2+ and Mg Diffusion properties of Ca Tracer diffusion coefficients of tri valent rare earth ions D-MATRIX, UPHILL DIFFUSION AND CHEMICAL WAVES COMMENTS ON EXTRAPOLATION AND GEOLOGICAL APPLICATION OF EXPERIMENTAL DIFFUSION DATA Change of diffusion mechanism and extrapolation of diffusion data Modeling prograde vs. retrograde profiles xi

559 560 561 561 564 566 566 573 575 578 580 580 580

Diffusion in Minerals and Melts - Table of Contents Treatment of diffusion data A SEMI-EMPIRICAL MODEL OF DIVALENT CATION DIFFUSION Carlson model Discussion GEOLOGICAL APPLICATIONS Modeling multicomponent diffusion profiles using effective binary diffusion formulation Cooling rates of metamorphic rocks: diffusion modeling of garnet vs. geochronological constraints Subduction and exhumation rates Modeling partially modified growth zoning of garnets in metamorphic rocks Interpretation of REE patterns of basaltic magma Sm-Nd and Lu-Hf geochronology of garnets in metamorphic rocks CONCLUDING REMARKS " ACKNOWLEDGMENTS REFERENCES APPENDIX: COMBINED ANALYTICAL AND NUMERICAL METHOD FOR MODELING MULTICOMPONENT DIFFUSION PROFILES

1 3

580 581 581 582 585 586 587 587 589 592 594 596 598 598 600

Diffusion Coefficients in Olivine, Wadsleyite and Ringwoodite S. Chakraborty

INTRODUCTION OLIVINE Structure of olivine and types of diffusion coefficients Diffusion mechanisms in olivine Diffusion of divalent cations Diffusion of Si and oxygen Diffusion of ions that enter olivine via heterovalent substitutions INFORMATION FROM OLIVINES OTHER THAN Fe-Mg BINARY SOLID SOLUTIONS SPECTROSCOPIC MEASUREMENTS COMPUTER CALCULATIONS WADSLEYITE AND RINGWOODITE Diffusion of divalent cations Diffusion of silicon and oxygen Diffusion of ions that are incorporated by heterovalent substitutions A SUMMARY, AND APPLICATIONS OF DIFFUSION DATA IN OLIVINE, WADSLEYITE AND RINGWOODITE ACKNOWLEDGMENTS REFERENCES

xii

603 603 603 605 608 620 623 627 628 628 629 630 631 633 633 635 635

Diffusion in Minerals and Melts - Table of Contents Diffusion in Pyroxene, Mica and Amphibole D.J. Chemiak, A. INTRODUCTION CATION DIFFUSION IN PYROXENES Pioneering approaches More recent investigations of major element diffusion Diffusion of major element cations in clinopyroxenes Diffusion in synthetic versus natural crystals Major element cation diffusion in orthopyroxenes Pyroxene point defect chemistry Diffusion of minor and trace elements in pyroxene Comparison of diffusion of cations in pyroxene DIFFUSION IN AMPHIBOLES AND MICAS F-OH interdiffusion in tremolite Sr diffusion in tremolite and hornblende Sr diffusion in fluorphlogopite K and Rb diffusion in biotite ACKNOWLEDGMENTS REFERENCES APPENDIX

I 3

Dimanov 641 641 643 644 645 656 656 658 661 672 676 677 678 679 679 680 680 685

Cation Diffusion in Feldspars D.J.

INTRODUCTION DIFFUSION OF MAI OR CONSTITUENTS Sodium Potassium K-Na interdiffusion Calcium Barium CaAl-NaSi interdiffusion Silicon DIFFUSION OF MINOR AND TRACE ELEMENTS Lithium Rubidium Magnesium Iron Strontium Lead Radium Rare Earth Elements COMPARISON OF RELATIVE DIFFUSIVITIES OF CATIONS IN VARIOUS FELDSPAR COMPOSITIONS xiii

Cherniak 691 692 692 695 696 698 699 700 703 705 705 705 707 708 708 717 721 721 723

Diffusion in Minerals and Melts - Table of Contents Albite K-feldspar Intermediate alkali feldspars Anorthite Labradorite Oligoclase ACKNOWLEDGMENTS REFERENCES

1 6

723 723 725 725 726 728 728 728

Diffusion in Quartz, Melilite, Silicate Perovskite, and Mullite D.J.

INTRODUCTION DIFFUSION IN QUARTZ Silicon Aluminum and gallium Alkali elements - Li, Na, K Calcium Titanium Diffusion in quartz - a summary DIFFUSION IN MELILITE Al+Al Mg + Si interdiffusion Mg Mn, Fe, Co, and Ni Ca, Sr, andBa Potassium Diffusion in melilite - a summary DIFFUSION IN SILICATE PEROVSKITE Silicon Fe-Mg interdiffusion DIFFUSION IN MULLITE ACKNOWLEDGMENTS REFERENCES

I /

Cherniak 735 735 736 738 739 741 741 742 743 743 743 746 748 749 750 751 751 753 753 754 754

Diffusion in Oxides J.A. Van Orman, K.L. Crispin

INTRODUCTION PERICLASE General considerations Oxygen Magnesium

757 758 758 759 763 xiv

Diffusion in Minerals and Melts - Table of Contents Other group IIA divalent cations Group IIIA and IIIB trivalent cations Tetravalent cations Transition metals Hydrogen SPINEL Oxygen Magnesium Fe-Mg interdiffusion Mg-Al interdiffusion Cr-Al interdiffusion Hydrogen MAGNETITE Oxygen Iron Other cations RUTILE Oxygen Tetravalent and pentavalent cations Divalent and trivalent cations Monovalent cations ACKNOWLEDGMENTS REFERENCES APPENDIX

IO

766 769 771 771 783 783 784 785 786 787 787 788 788 789 791 794 796 797 799 801 803 804 804 810

Diffusion in Accessory Minerals: Zircon, Titanite, Apatite, Monazite and Xenotime D.J. Cherniak

INTRODUCTION DIFFUSION IN ZIRCON Lead Rare Earth Elements (REE) Tetravalent cations Cation diffusion in zircon - a summary DIFFUSION IN TITANITE Strontium and Lead Neodynium Zirconium Summary of diffusion data for titanite DIFFUSION IN MONAZITE Calcium and Lead Thorium DIFFUSION IN XENOTIME DIFFUSION IN APATITE Lead and Calcium xv

827 827 828 832 835 838 841 841 843 843 844 844 845 847 848 850 850

Diffusion in Minerals and Melts - Table of Contents Strontium Manganese Rare Earth Elements (REE) Phosphorus Uranium and Thorium F-OH-C1 Comparison of diffusivities of cations and anions in apatite COMPARISON OF DIFFUSIVITIES AMONG ACCESSORY MINERALS Lead Rare Earth Elements (REE) Thorium and Uranium ACKNOWLEDGMENTS REFERENCES

I 7

852 853 854 858 858 859 860 861 861 862 863 864 864

Diffusion in Carbonates, Fluorite, Sulfide Minerals, and Diamond D.J.

INTRODUCTION CARBONATES Carbon Calcium Magnesium Strontium and Lead Rare Earth Elements Diffusion in calcite - an overview FLUORITE Fluorine Calcium Strontium, Yttrium and Rare Earth Elements DIAMOND SULFIDE MINERALS Pyrite Pyrrhotite Sphalerite Chalcopyrite Galena Summary of diffusion findings for the sulfides ACKNOWLEDGMENTS REFERENCES

xvi

Cherniak 871 871 872 875 876 877 878 879 880 881 883 883 885 885 886 888 889 891 892 892 893 894

Diffusion in Minerals and Melts - Table of Contents z u

Diffusion in Minerals: An Overview of Published Experimental Diffusion Data J.B. Brady, D.J.

Cherniak

INTRODUCTION ARRHENIUS RELATIONS DIFFUSION COMPENSATION DIAGRAMS IONIC POROSITY DIFFUSION ANISOTROPY CONCLUDING REMARKS ACKNOWLEDGMENTS REFERENCES

L\

899 900 904 911 913 917 917 917

Diffusion in Poly crystalline Materials: Grain Boundaries, Mathematical Models, and Experimental Data R. Dohmen, R. Milke

INTRODUCTION Geological relevance of grain boundary diffusion Physical nature of a grain/interphase boundary Thermodynamic model for interfaces THE ISOLATED GRAIN BOUNDARY Basic mathematical description Kinetic regimes and diffusion penetration distances THE MONOPHASE POLYCRYSTALLINE AGGREGATE Models and kinetic regimes Bulk diffusion coefficients A geological example Profile analysis - the Le Claire approach Complexities of real and polyphase systems Asymmetric grain boundaries/interphase boundaries The migrating isolated grain boundary Presence of dislocations/sub-grain boundaries Element/isotope exchange mediated by grain boundary diffusion EXPERIMENTAL METHODS Setup with bi-crystals Setup with a polycrystalline aggregate Source-sink studies EXPERIMENTAL DATA Parameters affecting grain boundary diffusion coefficients Direct measurement of tracer diffusion in polycrystals of geological relevance Concluding remarks ACKNOWLEDGMENTS REFERENCES xvii

921 921 922 925 927 927 928 932 932 934 936 937 940 941 942 944 946 947 948 949 950 950 950 955 964 966 966

Diffusion in Minerals and Melts - Table of Contents

LL

Theoretical Computation of Diffusion in Minerals and Melts N. de Koker, L. Stixrude

INTRODUCTION THEORETICAL FOUNDATIONS Thermodynamic description Statistical mechanical description COMPUTATIONAL APPROACHES Characterization of bonding Adding temperature Computation of diffusion SELECTED APPLICATIONS Liquids and melts Solids A VIEW TO THE FUTURE ACKNOWLEDGMENTS REFERENCES

971 972 972 974 976 977 978 980 981 981 988 991 991 991

Applications of Diffusion Data to High-Temperature Earth Systems T. Mueller, E.B. Watson, T.M. Harrison INTRODUCTION DECIPHERING KINETICALLY CONTROLLED PROCESSES USING DIFFUSION Mass transport in geological systems Diffusion in minerals Control of solid-state reaction rates and compositions of reaction products by diffusion Metamorphic example of diffusion-limited uptake: REE behavior during garnet growth Chemical diffusive fractionation Diffusive fractionation in a thermal gradient THERMOCHRONOLOGY Background Bulk closure Continuous histories Dating metamorphic events GEOSPEEDOMETRY The concept of geospeedometry Deciphering timescales from kinetic modeling Diffusion in two or three dimentions and the effect of geometry Example: Deciphering short-term metamorphic events and timescales ACKNOWLEDGMENTS...." REFERENCES xviii

997 ...999 999 1002 1005 1011 1014 1017 1018 1018 1019 1021 1024 1025 1025 1026 1027 1029 1032 1032

1

Reviews in Mineralogy & Geochemistry Vol. 72 pp. 1-4, 2010 Copyright © Mineralogical Society of America

Diffusion in Minerals and Melts: Introduction Youxue Zhang Department of Geological Sciences The University of Michigan Ann Arbor, Michigan, 48109-1005, U.S.A. youxue @ umich. edit

Daniele J. Cherniak Department of Earth & Environmental Sciences Rensselaer Polytechnic Institute Troy, New York, 12180, U.S.A. chernd @ rpi.edu

INTRODUCTION: RATIONALE FOR THIS VOLUME Because diffusion plays a critical role in numerous geological processes, petrologists and geochemists (as well as other geologists and geophysicists) often apply diffusion data and models in a range of problems, including interpretation of the age of rocks and thermal histories, conditions for formation and retention of chemical compositional and isotopic zoning in minerals, controls on bubble sizes in volcanic rocks, and processes influencing volcanic eruptions. A major challenge in the many applications of diffusion data is for researchers to find relevant and reliable data. For example, diffusivities determined in different labs may differ by orders of magnitude. Sometimes the differences are a result of limitations not recognized in certain diffusion studies due to the materials or methodologies used. For example, diffusivities determined through bulk analyses are often orders of magnitude greater than those obtained from directly measured diffusion profiles; the former are often affected by cracks, extended defects and/or other additional diffusion paths whose influence may not be recognized without direct profiling. Differences in depth resolution of analytical techniques may also contribute to discrepancies among measured diffusivities, as can the occurrence of non-diffusional processes (e.g., convection, crystal dissolution or surface reaction) that may compromise or complicate diffusion experiments and interpretations of results. Sometimes the discrepancies among datasets may be due to subtle variations in experimental conditions (such as differing oxygen fugacities, pressures, or variations in H 2 0 content of minerals and melts used in respective experimental studies). Experts in the field may be able to understand and evaluate these differences, but those unfamiliar with the field, and even some experimental practitioners and experienced users of diffusion data, may have difficulty discerning and interpreting dissagreements among diffusion findings. For those who want to investigate diffusion through experiments, it is critical to understand the advantages and limitations of various experimental approaches and analytical methods in order to optimize future studies, and to obtain a clear sense of the "state of the art" to put their own findings in perspective with earlier work. Two early books were important landmarks in diffusion studies in geology. One was a special publication by Carnegie Institution of Washington edited by Hofmann et al. (1974) titled "Geochemical Transport and Kinetics." The other was a Reviews of Mineralogy volume edited by Lasaga and Kirkpatrick (1981) titled "Kinetics of Geochemical Processes." Various recent tomes are available on diffusion theory in metallurgy, chemical engineering, materials 1529-6466/10/0072-0001 $05.00

DOI: 10.2138/rmg.2010.72.1

2

Zhang & Cherniak

science, and geology (e.g., Kirkaldy and Young 1987; Shewmon 1989; Cussler 1997; Lasaga 1998; Glicksman 2000; Balluffi et al. 2005; Mehrer 2007; Zhang 2008) and the mathematics of solving diffusion problems (e.g., Carslaw and Jaeger 1959; Crank 1975). There have also been summaries of geologically relevant diffusion data (e.g., Freer 1981; Brady 1995), review articles and book chapters presenting diffusion data for specific mineral phases (e.g., Yund 1983; Giletti 1994; Cherniak and Watson 2003) and for specific species in minerals and melts (e.g., Chakraborty 1995; Cole and Chakraborty 2001; Watson 1994) and applications of diffusion in geology (e.g., Ganguly 1991; Watson and Baxter 2007; Chakraborty 2008). However, there is no single resource that reviews and evaluates a comprehensive collection of diffusion data for minerals and melts, and previously published summaries of geologically-relevant diffusion data predate the period in which a large proportion of the existing reliable diffusion data have been generated. This volume of Reviews in Mineralogy and Geochemistry attempts to fill this void. The goal is to compile, compare, evaluate and assess diffusion data from the literature for all elements in minerals and natural melts (including glasses). Summaries of these diffusion data, as well as equations to calculate diffusivities, are provided in the chapters themselves and/or in online supplements. Suggested or assessed equations to evaluate diffusivities under a range of conditions can be found in the individual chapters. The aim of this volume is to help students and practitioners to understand the basics of diffusion and applications to geological problems, and to provide a reference for and guide to available experimental diffusion data in minerals and natural melts. It is hoped that with this volume students and practitioners will engage in the study of diffusion and the application of diffusion findings to geological processes with greater interest, comprehension, insight, and appreciation.

SCOPE AND CONTENT OF THIS VOLUME This volume begins with three general chapters. One chapter presents the basic theoretical background of diffusion (Zhang 2010), including definitions and concepts encountered in later chapters. This chapter is not meant to be comprehensive, as detailed, book-length treatments of diffusion theory can be found in other sources. Some discussion of advanced topics of diffusion theory and mechanisms can be found in individual chapters throughout the volume, including models for diffusion in melts (Lesher 2010), multi-species diffusion (Zhang and Ni 2010), multicomponent diffusion (Liang 2010; Ganguly 2010), and defect chemistry (Chakraborty 2010; Cherniak and Dimanov 2010; Van Orman and Crispin 2010). Diffusion data for minerals and melts are most commonly obtained through experimental studies which require analyses of the experimental products; these considerations are reflected in the topics of the next two chapters. For readers who are interested in carrying out experimental research or understanding experimental results and diffusion data, the second general chapter (Watson and Dohmen 2010) covers experimental methods in diffusion studies, with focus on nontraditional and emerging methods. Additional discussion of experimental methods in diffusion studies is provided in Ganguly (2010) and Farver (2010). The third general chapter reviews a range of analytical techniques applied in analyses of diffusion experiments (Cherniak et al. 2010). Experimental methods and analytical techniques are also described in other chapters in the context of discussion of specific diffusion studies. The next five chapters are on diffusion in melts (including glasses), focusing on natural melts relevant in geological systems. Zhang and Ni (2010) discuss the diffusion of H, C and O in silicate melts, which involves multi-species diffusion, where one species (such as molecular H 2 0 ) may contribute to the diffusion of two elements (such as H and O in this case). They also assess the relative importance of various diffusing species, and extract oxygen diffusion data in hydrous silicate melts from diffusion data for water. Behrens (2010) offers a thorough review and evaluation of noble gas diffusion data for natural silicate melts and industrial glasses. Lesher (2010) elaborates on the various diffusion models for self diffusion, tracer diffusion, isotopic

Introduction to Diffusion in Minerals and Melts

3

diffusion and trace element diffusion. Zhang et al. (2010) summarize available diffusion data (focusing on effective binary diffusivities) of all elements in silicate melts. Liang (2010) presents a systematic assessment of multicomponent diffusion studies for silicate melts. The next eleven chapters review and evaluate diffusion data for minerals. Farver (2010) reviews H and O diffusion data for a range of mineral phases and examines the effect of oxygen, hydrogen and water fugacities on diffusion. Noble gas diffusion in minerals, notably diffusion of the important radiogenic nuclides 40 Ar and 4 He for application in closure temperature determinations and thermochronometry, is reviewed by Baxter (2010). Ganguly (2010) assesses cation diffusion data in garnet, with discussion of multicomponent diffusion in garnet and its geological applications. Chakraborty (2010) focuses on diffusion in (Fe,Mg) 2 Si0 4 polymorphs (olivine, wadsleyite and ringwoodite) with a discussion of the role of defects in diffusion and the effects of pressure on diffusion in these phases. Diffusion of major and trace elements in pyroxenes, amphibole, and mica is discussed by Cherniak and Dimanov (2010). Cherniak (2010a) reviews diffusion data for feldspars, examining the effects of feldspar composition on diffusion in this common crustal mineral. Cherniak (2010d) summarizes diffusion data for the silicate phases quartz, melilite, silicate perovskite, and mullite. Van Orman and Crispin (2010) discuss diffusion in oxide minerals including periclase, magnesium aluminate spinel, magnetite, and rutile, and explore the intricacies of defect chemistry and its effects on diffusion in these deceptively simple compounds. Cherniak (2010b) reviews diffusion in the accessory minerals zircon, monazite, apatite, and xenotime, phases important in geochronologic studies. Diffusion in other minerals, including carbonates, sulfide minerals, fluorite and diamond, is reviewed by Cherniak (2010c). Brady and Cherniak (2010) take a broad overview of extant diffusion data for minerals, examining possible relations among diffusivities for various mineral phases and diffusants to assess trends and correlations that may be of value in developing or refining predictive models and empirical relations. The next two chapters discuss the specialized topics of grain-boundary diffusion and computational methods for determining diffusion coefficients. Dohmen and Milke (2010) present existing data for grain boundary diffusion in polycrystalline materials, discuss theoretical underpinnings and the different types of grain-boundary diffusion regimes, and outline mathematical treatments and experimental approaches for quantifying grain-boundary diffusion. Computation of diffusion coefficients using ab initio methods and molecular dynamics simulations are reviewed by De Koker and Stixrude (2010) with focus on recent progress and what the future may bring for these rapidly-developing techniques. The final chapter is devoted to geological applications of diffusion data (Mueller et al. 2010). The applications outlined include not only forward problems of applying diffusion theory and data to infer rates and extents of diffusion-related processes, but also inverse problems of thermochronology and geospeedometry.

REFERENCES Balluffi RW, Allen SM, Carter WC, Kemper RA (2005) Kinetics of Materials. Wiley-Interscience, Hoboken, N.J Baxter EF (2010) Diffusion of noble gases in minerals. Rev Mineral Geochem 72:509-557 Behrens H (2010) Noble gas diffusion in silicate glasses and melts. Rev Mineral Geochem 72:227-267 Brady JB (1995) Diffusion data for silicate minerals, glasses, and liquids. In: Mineral Physics and Crystallography, A Handbook of Physical Constants, Reference Shelf 2. Ahrens TJ (ed) AGU, Washington, D.C., p 269-290 Brady JB, Cherniak DJ (2010) Diffusion in minerals: an overview of published experimental diffusion data. Rev Mineral Geochem 72:899-920 Carslaw HS, Jaeger JC (1959) Conduction of Heat in Solids. Clarendon Press, Oxford Chakraborty S (1995) Diffusion in silicate melts. Rev Mineral Geochem 32:411-504 Chakraborty S (2008) Diffusion in solid silicates; a tool to track timescales of processes comes of age. Ann Rev Earth Planet Sci 36: 153-190

4

Zhang & Cherniak

Chakraborty S (2010) Diffusion coefficients in olivine, wadsleyite and ringwoodite. Rev Mineral Geochem 72:603-639 Cherniak DJ (2010a) Cation diffusion in feldspars. Rev Mineral Geochem 72:691-734 Cherniak DJ (2010b) Diffusion in accessory minerals: zircon, titanite, apatite, monazite and xenotime. Rev Mineral Geochem 72:827-870 Cherniak DJ (2010c) Diffusion in carbonates, fluorite, sulfide minerals, and diamond. Rev Mineral Geochem 72:871-897 Cherniak DJ (2010d) Diffusion in quartz, melilite, silicate perovskite, and mullite. Rev Mineral Geochem 72:735-756 Cherniak DJ, Dimanov A (2010) Diffusion in pyroxene, mica and amphibole. Rev Mineral Geochem 72:641690 Cherniak DJ, Hervig R, Koepke J, Zhang Y, Zhao D (2010) Analytical methods in diffusion studies. Rev Mineral Geochem 72:107-169 Cherniak DJ, Watson EB (2003) Diffusion in zircon. Rev Mineral Geochem 53:113-143 Cole DR, Chakraborty S (2001) Rates and mechanisms of isotopic exchange. Rev Mineral Geochem 43:83-223 Crank J (1975) The Mathematics of Diffusion. Clarendon Press, Oxford Cussler EL (1997) Diffusion: Mass Transfer in Fluid Systems. Cambridge Univ. Press, Cambridge, England de Koker N, Stixrude L (2010) Theoretical computation of diffusion in minerals and melts. Rev Mineral Geochem 72:971-996 Dohmen R, Milke R (2010) Diffusion in polycrystalline materials: grain boundaries, mathematical models, and experimental data. Rev Mineral Geochem 72:921-970 Farver JR (2010) Oxygen and hydrogen diffusion in minerals. Rev Mineral Geochem 72:447-507 Freer R (1981) Diffusion in silicate minerals and glasses: a data digest and guide to the literature. Contrib Mineral Petrol 76:440-454 Ganguly J (2010) Cation diffusion kinetics in aluminosilicate garnets and geological applications. Rev Mineral Geochem 72:559-601 Ganguly J (ed) (1991) Diffusion, Atomic Ordering, and Mass Transport: Selected Topics in Geochemistry. Advances in Physical Geochemistry, Vol. 8. Springer Giletti BJ (1994) Isotopic equilibrium/disequilibrium and diffusion kinetics in feldspars. In: Feldspars and their reactions. NATO Advanced Study Institutes Series. Series C: Mathematical and Physical Sciences. Parsons I (ed) D. Reidel Publishing, Dordrecht-Boston, 421:351-382 Glicksman M E (2000) Diffusion in Solids: Field Theory, Solid-State Principles, and Applications. Wiley, New York Hofmann AW, Giletti BJ, Yoder HS, Yund RA (1974) Geochemical Transport and Kinetics. Carnegie Institution of Washington Publ., Vol 634. Washington, DC Kirkaldy JS, Young DJ (1987) Diffusion in the Condensed State. The Institute of Metals, London Lasaga AC (1998) Kinetic Theory in the Earth Sciences. Princeton University Press, Princeton, NJ Lasaga AC, Kirkpatrick RJ (eds) (1981) Kinetics of Geochemical Processes. Reviews in Mineralogy, Vol 8. Mineralogical Society of America, Washington DC Lesher CE (2010) Self-diffusion in silicate melts: theory, observations and applications to magmatic systems. Rev Mineral Geochem 72:269-309 Liang Y (2010) Multicomponent diffusion in molten silicates: theory, experiments, and geological applications. Rev Mineral Geochem 72:409-446 Mehrer H (2007) Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-Controlled Processes. Springer, Berlin Mueller T, Watson EB, Harrison TM (2010) Applications of diffusion data to high-temperature earth systems. Rev Mineral Geochem 72:997-1038 Shewmon PG (1989) Diffusion in Solids. Minerals. Metals & Materials Society, Warrendale, PA Van Orman JA, Crispin KL (2010) Diffusion in oxides. Rev Mineral Geochem 72:757-825 Watson EB (1994) Diffusion in volatile-bearing magmas. Rev Mineral 30:371-411 Watson EB, Baxter EF (2007) Diffusion in solid-Earth systems. Earth Planet Sci Lett 253:307-327 Watson EB, Dohmen R (2010) Non-traditional and emerging methods for characterizing diffusion in minerals and mineral aggregates. Rev Mineral Geochem 72:61-105 Yund RA (1983) Diffusion in feldspars. In: Feldspar Mineralogy, Short Course Notes 2. Ribbe P (ed) Mineralogical Society of America, p 203-222 Zhang Y (2008) Geochemical Kinetics. Princeton University Press, Princeton, NJ Zhang Y (2010) Diffusion in minerals and melts: theoretical background. Rev Mineral Geochem 72:5-59 Zhang Y, Ni H (2010) Diffusion of H, C, and O components in silicate melts. Rev Mineral Geochem 72:171225 Zhang Y, Ni H, Chen Y (2010) Diffusion data in silicate melts. Rev Mineral Geochem 72:311-408

Reviews in Mineralogy & Geochemistry Vol. 72 pp. 5-59, 2010 Copyright © Mineralogical Society of America

Diffusion in Minerals and Melts: Theoretical Background Youxue Zhang Department of Geological Sciences The University of Michigan Ann Arbor, Michigan, 48109-1005, U.S.A youxue @ umich. edu

INTRODUCTION Diffusion is due to thermally activated atomic-scale random motion of particles (atoms, ions and molecules) in minerals, glasses, melts, fluids, and gases (Fig. 1). The random motion leads to a net flux when the concentration (more strictly speaking, the chemical potential) of a component is not uniform. Even though diffusion is a microscopic process, it can lead to macroscopic effects. For example, the initial phase of explosive volcanic eruptions (or more commonly encountered champagne eruptions) is powered by bubble growth, which in turn is controlled by diffusion that brings gas molecules into bubbles. This chapter provides a brief review of the theory of diffusion in minerals and melts (including glasses). More complete coverage of diffusion theory can be found in Crank (1975), Kirkaldy and Young (1987), Shewmon (1989), Cussler (1997), Lasaga (1998), Glicksman (2000), Balluffi et al. (2005), Mehrer (2007), and Zhang (2008). In minerals, diffusive transport is the only mechanism for particles to move from one location to another. For example, homogenization of a zoned crystal and loss of radiogenic

OOOOO o

O O o O O o

f p ^ o o o O ° n ° 0 Ou 0 0

oQooo 0

0

0

o

oo o oo°o

pOOQ

0

OqO

o

° o i o Ju> o ° o

°o5 l O oo oo®d t o •

n o t o

Figure 1. An example of random motion of particles. Initially (the left panel), all A particles (such as Fe 2+ ions in garnet) represented by filled circles are in the lower side, and all B particles (such as Mg 2 + ions in garnet) represented by open circles are in the upper side. Due to random motion, there is a net flux of A from the lower side to the upper side, and a net flux of B from the upper side to the lower side (the middle and right panels). As time increases, A and B will eventually become randomly and uniformly distributed in the whole system. This situation for diffusion is often encountered in diffusion experiments and is referred to as a diffusion couple. 1529-6466/10/0072-0002$ 10.00

DOI: 10.2138/rmg.2010.72.2

6

Zhang

nuclides (such as 40Ar from the decay of 40K) from a mineral are through diffusion. In silicate melts, mass transport can be through either diffusion or flow (or convection). Only diffusion is covered in this chapter. Even when convection is present, it is still necessary to understand diffusion because in the boundary layer mass transport is through diffusion. Diffusion also plays a role during crystal growth and dissolution in a melt, key processes in magma solidification and evolution. One of the most important geological applications of diffusion is the inverse problem, to infer the details of thermal histories and factors such as closure temperature, apparent equilibrium temperature, and cooling rates from diffusion properties (Zhang 2008). Thermochronology and its application to the understanding of tectonic uplift and erosion rates, require a thorough understanding of diffusion in minerals. The mathematics of diffusion is complicated. An excellent reference book is by Crank (1975), which provides analytical solutions to many diffusion problems. The mathematical description of diffusion is similar to that of heat conduction. Hence, analytical solutions to heat conduction problems (e.g., Carslaw and Jaeger, 1959) can also be applied to diffusion. Because the mathematical treatment is in itself specialized and can be found in the aforementioned treatises, in this chapter, I focus on concepts of diffusion relevant to geological and experimental diffusion studies, rather than the mathematical solutions. Solutions for specific diffusion problems will be given without derivations. FUNDAMENTALS OF DIFFUSION Basic concepts The German physiologist Adolf Fick (1829-1901) investigated diffusive mass transport and proposed the following phenomenological law that describes diffusion by analogy to Fourier's law of heat conduction (1)

where J is the diffusive mass flux (a vector), D is the diffusion coefficient (also referred to as the diffusivity), C is the concentration of the component under consideration (in mass per unit volume, such as kg/ml or number of atoms per m3, or mol/m3), x is distance, dC/dx is the concentration gradient (a vector), and the negative sign means that the direction of the diffusive flux is opposite to the direction of the concentration gradient (i.e., diffusive flux goes from high to low concentration, but the gradient is from low to high concentration). Hence, when the concentration gradient is large (i.e., the concentration profile is steep), the diffusive flux is also large. The unit of D is length2/time, such as m2/s, mm2/s, and |_im2/s (1 m2/s = 106 mm2/s = 1012 |am2/s). The value of the diffusivity is an indication of the "rate" of diffusion and, hence, is essential in quantifying diffusion. Diffusivities depend on several factors, including temperature, pressure, composition, and physical state and structure of the phase, and sometimes oxygen fugacity. Some general relations between diffusivities and other parameters will be presented later in this chapter. Diffusivity values in various systems are the main focus of this volume, of which this chapter is a part. When diffusion is mentioned without special qualification, it refers to volume diffusion occurring inside a phase due to thermally activated random motion (in contrast to grainboundary diffusion or eddy diffusion in natural waters). Typical values of diffusion coefficients are (see Fig. 2 for diffusivity of a neutral gas species as a function of temperature; see also Watson and Baxter 2007 for generalized diffusion behaviors in geological materials): •

In gas, D is large, about 10 ^ m2/s in air at 300 K;

Theoretical

Background

of Diffusion

in Minerals

and

Melts

7

qC£

1

1.5

2

2.5

3

3-5

1000/7 (T in K) Figure 2. Ar diffusion data in air (gas) (calculated using relations in Cussler 1997). water (liquid) (Wise and Houghton 1966). basalt melt (Nowak et al. 2004), rhyolite melt (Behrens and Zhang 2001) and the mineral hornblende (Harrison 1981).

•

In aqueous solution. D is intermediate, about 10 '' m2/s in water at 300 K;

•

In silicate melts. D is small, about 10~u m2/s at 1600 K for divalent cations;

•

In minerals, D is extremely small, about 10

17

m 2 /s at 1600 K for divalent cations.

Grain-boundary diffusion is diffusion along interphase interfaces, including mineralfluid interfaces (or surfaces) or mineral-mineral interfaces. Eddy (or turbulent) diffusion in fluid phases is due to non-thermal random disturbances such as waves, fish swimming, boats cruising, etc. Hence, eddy diffusion is fundamentally different from thermally activated volume diffusion. Both grain-boundary diffusivities and eddy diffusivities are often several orders of magnitude higher than the respective volume diffusivities listed above. In Fick's first law. the diffusive flux is related to the concentration gradient. In diffusion studies, often we need to determine how a concentration profile would evolve with time given the initial concentration distribution. For this purpose, we need an equation (referred to as the diffusion equation) to describe how the concentration is related to space and time, such as C(x.r) for the one-dimensional case. The one-dimensional diffusion equation often takes the following form

dt

3.r

where D is independent of C and x. Equation (2) is also referred to as Fick's second law. Below is a derivation of Equation (2) from Equation (1) and the mass balance condition. Consider diffusion across a thin sheet with the left side at x and the right side at x+dx (thickness of dx). Assume that the flux is one-dimensional along the x direction (Fig. 3). Then the total mass variation in the volume defined by thickness dx and an arbitrary area S and equals the flux into the sheet from the left side (x), JXS, minus the flux out of the sheet from the right side (x+dx). / v+dv S:

8

Zhang

T "x+dx

Figure 3. Sketch of fluxes into and out of an element volume. The flux along the x-axis points to the right (the x-axis also points to the right). The flux at x is Jx, and that at x+dx is -Ai+dA- The net flux into the small volume is (Jx - /j+dr), which causes the mass and density in the volume to vary.

£

x

x+dx Sdx = JXS - Jx+dxS =

^ Sdx

(3)

where Jx (a scalar) is the flux along increasing x direction (the vector flux J = JJ, where i is the unit vector along the x axis). Hence,

dJx(x) dx

dt

Combining the above with Fick's first law (Eqn. 1) leads to:

at

ox

ox

If D is independent of C and x, the above is simplified to Equation (2). In three dimensions, the diffusive flux for a component (Eqn. 1) takes the following form:

J= - DVC

(6)

the mass balance equation (Eqn. 4) becomes: £

dt

= -V-J

(7)

and the diffusion equation (Eqn. 5) becomes:

riC

— = V • (DVC)

dt

(8)

where V is the gradient operator when it is applied to a scalar C, and the divergent operator when it is applied to a vector VC (i.e., V turns a scalar to a vector and a vector to a scalar). From Equation (2), it can be seen that if d2C/dx2 = 0 at a position (e.g., point 1 in Fig. 4), i.e., if C is locally a linear function of x (including the case of constant concentration), then dC/dt = 0, meaning that the concentration at the position would not vary with time. If d2C/dx2 > 0 at the position (point 2 in Fig. 4; concave up), then dC/dt > 0, meaning that the concentration at the position would increase with time. If d2C/dx2 < 0 at the position (point 3 in Fig. 4; concave down), then the concentration at the position would decrease with time. Although we often talk about diffusion "rate", and the rate is related to the diffusion coefficient, diffusion is a peculiar process in which there is no single diffusion "rate". From solutions

Theoretical Background of Diffusion in Minerals and Melts

9

Figure 4. Concentration profile C versus x, and the corresponding d2C/dx2 versus x (arbitrary units) to illustrate whether C increases, decreases or stays the same with time. At point 1, 9 2 C/9.r = 0 and hence dC/dt = 0. At point 2 (concave up), d2C/dx2 > 0 and hence dC/dt > 0. At point 3 (concave down), d2C/dx2 < 0 and hence dC/dt < 0.

of the diffusion equation, the diffusion distance is proportional not to duration, but to the square root of duration: the relation is often written as X = yfDt

(9)

This distance may also be referred to as the mid-concentration distance, or half distance of diffusion (Zhang 2008, p. 201-204), which will become clearer later. Defining the diffusion "rate" as how rapidly the diffusion distance advances with time (cLr/di), then the "rate" equals 0.5(D/t) m , and is infinity at t = 0 and then decreases gradually with time. The diffusivity increases rapidly with temperature, following the Arrhenius relation (Fig. 2), D = Dae~EIRT

(10)

where 7 is the absolute temperature in K. D{) is the pre-exponential factor and equals the value of D at T = E is the activation energy and is a positive number, and R is the universal gas constant. The pressure dependence of diffusivities can be either positive or negative. The following equation is often used to describe both the temperature and pressure dependence of diffusivity D = Dne-{E+PAV)IRT

(11)

where AV is referred to as the activation volume, which can be either positive (leading to a decrease of D with increasing P) or negative (leading to an increase of D with increasing P). Negative AVis not rare. From Equation (11), the activation energy depends on pressure (when AV * 0). Similarly, the activation volume AVmay also depend on temperature, which would change the form of the above equation (see later discussion). Microscopic view of diffusion Microscopically and statistically, diffusion can be quantified using random walk of particles (atoms, ions, or molecules). Consider, for example, diffusion of Mg 2+ (counterbalanced by Fe 2 + in the opposite direction) in garnet along any direction, labeled as the x direction. (A cubic crystal is used here so that the effect of diffusional anisotropy does not have to be considered.) Consider two adjacent lattice planes (left and right) at distance / apart. If the jumping distance of Mg 2+ is / and the frequency of Mg 2 + ions jumping away from the original position is/, then the number of Mg 2 + ions jumping from left to right is Ym^f&t and that from right to left is Vm^fdt, where n L and nR are the number of Mg 2 + ions per unit area on the left and

10

Zhang

right planes. The factor Vi in the expressions is due to the fact that the ions in each plane can jump to both sides, but we consider only one direction. The jumping frequency/is assumed to be the same from left to right or from right to left, i.e., random walk is assumed. Therefore, the net flux from the left plane to the right plane is J = \(nL~nl{)f

(12)

Since nL = ICL and nR = ICR where CL and CR are the concentrations of Mg 2+ on the left and right planes, then J = \l{CL-CR)f

(13)

Because C, - CR = -IdC/dx, we have

J=

2

ox

(14)

Comparing this with Fick's law (Eqn. 1), we have D = U2f

(15)

Thus, microscopically, in one-dimensional diffusion, the diffusion coefficient may be interpreted as one-half of the jumping distance squared times the overall jumping frequency. Since I is of the order 3xlO~10 m (the interatomic distance in a lattice), the jumping frequency can be roughly estimated from D. For D ~ 10 17 m2/s, as in a typical mineral at high temperature, the jumping frequency is 2D/l2 ~ 220 per second. Because ion jumping requires a site to accept the ion, the jumping frequency in minerals depends on the concentration of vacancies and other defects. Hence, high defect concentrations lead to high diffusivities. In melts, the jumping frequency is much higher (about 108 per second), depending on the flexibility of the liquid structure, and may also be related to viscosity. The above analysis can be carried forward to the full statistical treatment of random walk using either theoretical analysis (Gamow 1961) or computer simulations (Kleinhans and Friedrich 2007). If initially a large number (trillions) of particles were at a single position (defined as x = 0), after more than 100 jumping steps, the distribution can be approximated well by a continuous function. For one-dimensional diffusion, the concentration of particles at x (or the probability of finding a particle at x) follows the Gaussian distribution: C(x,0=

M

(4nDt)

U2 e-

z2,(4Di>

(16)

where M is the total number of particles, all of which were initially at x = 0. This diffusion problem is known as diffusion from an instantaneous plane source. Various kinds of diffusion There are many kinds of diffusion encountered in nature and experimental studies. The definitions may differ somewhat in the literature, making it less straightforward to deal with the terms. Below is a summary of the various kinds of diffusion described in most of the geological literature. Because diffusion involves a diffusing species in a diffusion medium, it can be classified based on either the diffusion medium or the diffusing species. When considering the diffusion medium, thermally activated diffusion may be classified as volume diffusion and grain-boundary diffusion. Volume diffusion is diffusion in the interior of a phase; an example is the diffusion of Mg and Fe2+ in a garnet crystal, leading to the homogenization of

Theoretical Background of Diffusion in Minerals and Melts

11

a garnet crystal initially zoned in Fe 2+ and Mg (Ganguly 2010, this volume). Volume diffusion is what is typically referred to when we simply say "diffusion" without further qualifiers. In volume diffusion, the diffusion medium can be either isotropic or anisotropic. In an isotropic diffusion medium, diffusion properties do not depend on direction. Both melts (and glasses) and isometric minerals are isotropic diffusion media, but non-isometric minerals are in general anisotropic diffusion media (although in some cases, the dependence of diffusivities on the direction is weak). Anisotropic diffusion will be treated later in this chapter. Grain-boundary diffusion is diffusion along interphase interfaces, including mineral-fluid interfaces (or surfaces), interfaces between the same minerals, and those between different minerals. Because many bonds are not satisfied for atoms on the interface, there are generally very high concentrations of defects, leading to very high grain-boundary diffusivities compared with volume diffusivities. For example, at 1473 K, the grain-boundary diffusivity of Si at forsterite-forsterite boundaries is about 9 orders of magnitude greater than the volume diffusivity of Si in forsterite (Farver and Yund 2000). Grain-boundary diffusion will be the subject of a chapter in this volume (Dohmen and Milke 2010, this volume). Considering differences in the diffusing species, diffusion can be classified as self diffusion, tracer diffusion, or chemical diffusion that can be further distinguished as trace element diffusion, binary diffusion, multispecies diffusion, multicomponent diffusion, and effective binary diffusion. Below is a discussion of these terms; first the definition used in this work is shown, then alternative definitions are also mentioned. Self diffusion. There is no chemical potential gradient in the system in terms of elemental composition but there is difference in the isotopic ratios (or chemical potential gradients are present only in isotopes) (Lasaga 1998; Zhang 2008). The diffusion is monitored through difference in isotopic fractions. For example, in a diffusion couple made of basalt melt, one side may have a high 44 Ca/Z'Ca ratio and the other side has normal Ca isotope ratios, but the elemental composition of the melts in both sides of the couple is uniform (e.g., in the experiments of LaTourrette et al. 1996; LaTourrette and Wasserburg 1997). In Figure 1, one may view the solid circles as 44 Ca-enriched Ca, and the open circles as normal Ca, and the matrix is the haplobasalt melt. Because there are no chemical (or elemental) gradients, the diffusivity, which often depends on chemical composition of the system, is assumed to be constant. This works well for self diffusion without exceptions. Small differences, 10%, such as from a mineral to a melt), concentrations in mol/m3 or kg/m 3 should be used. Equation (8) is the general equation for diffusion in a binary system without reaction terms or multiple species. If D is constant, Equation (8) becomes — = DV2C dt

(28)

If diffusion is one-dimensional, Equation (8) becomes Equation (5). If diffusion is onedimensional and D is independent of C and x, then Equation (8) becomes Equation (2). All the above equations are for binary systems and isotropic diffusion media. Diffusion in multicomponent systems or anisotropic diffusion media is more complex and will be discussed in later sections (and chapters). Diffusion equations in three dimensions in isotropic media are discussed below.

Theoretical Background of Diffusion in Minerals and Melts

17

Diffusion in three dimensions (isotropic media) In general, three-dimensional diffusion is much more complicated unless the boundary shape is simple (such as spherical surfaces) and there is high symmetry (such as spherical symmetry). The forms of the diffusion equations are summarized below (for details, see Crank 1975; Carslaw and Jaeger 1959). The three-dimensional diffusion equation takes the following form in Cartesian coordinates:

d_ D* dz dz

ac__a_rDac

dt

dy

dx ^ dx

3>
0. There is no additional flux from either side of the sample, which means that dC/dx = 0 at both ends (the end with the thin film is x = 0, and the other end is x = °° if diffusion has not reached this end). This mathematical problem is similar to that of random walk in one dimension (Eqn. 16), except that in the thin source problem, diffusion goes in only one direction instead of both directions. Hence, the resulting concentration profile (i.e., the solution to this diffusion problem) is two times that in Equation (16): C

(

x

' °

=

c

°

e

^ '

i

4

m

(37)

where x is distance measured from the surface on which the tracer was applied, C is the concentration of the diffusant (e.g., measured by counting the number of decays in the case of a radiotracer), M i s the initial mass of the diffusant in the thin film per applied area, C 0 is the concentration of the diffusant at the surface (x = 0), which decreases by half as time is quadrupled. Defining the mid-concentration distance (x1/2) as the distance at which C = C ( /2, then x, /2 = 1.6651(£>f)i/2

(38)

Theoretical Background of Diffusion in Minerals and Melts

19

The above is similar to the general form of Equation (9). If the thin film thickness is < 0. lx ]/2 , then the solution (Eqn. 37) applies well. Otherwise, the solution may not be accurate. If the "thin" film thickness is > 0.2xm, the source is not thin any more, and the problem should be treated as extended source diffusion or finite-medium diffusion (e.g., Zhang 2008). If the "thin" film thickness is > 2x,/2, then the tracer diffusion is almost equivalent to a diffusion couple (discussed in a later section), with one half being the "thin" film, and the other half the diffusion medium of interest. In this case, the tracer diffusion becomes chemical diffusion across two very different compositions (effective binary diffusion in a diffusion couple). When a radiotracer is used as the diffusing species, the integrated concentrations are often measured using the residual activity method (e.g., Jambon and Carron 1976; Behrens 1992). After the experiment, the radioactive nuclide on the surface is washed away, and the radioactivity in the whole sample is measured. Then a thin layer of the sample (e.g., 0.005 mm) is polished off, and the total residual radioactivity of the remaining sample is measured. And another layer is polished off, and the residual activity measured, and so on. Hence, every measurement is total radioactivity from x to °° where x starts at zero (the first measurement) and gradually increases. Hence, the solution is the integration of Equation (37):

A(x,t) = ]c(x,t)dx = C~je-x!'{4D,)dx = A0 e

r f c

^=

(39>

where A is defined as the measured residual radioactivity, and erfc is the complementary error function. To the uninitiated, the shapes of the two profiles (Eqns. 37 and 39) may appear similar, but there are important differences between the two profiles. For example, the slope is zero at x = 0 for Equation (37), but the slope is the steepest at x = 0 for Equation (39). Comments about fitting data When analytical data are fit by Equation (37) or (39), one may choose to carry out nonlinear fit using the equations directly. In the past, this was difficult because one would have to write a software program to do so (e.g., Press et al. 1992). More recently, nonlinear fitting has become easier because many commercially available programs can carry out such fitting. Another approach is to linearize the relations and do a linear fitting; an advantage is that it is simple and visually easy to verify such relations. Hence, many authors have used linearized fitting. Equation (37) is linearized as follows: In C = In Cn — - — 0 4 Dt

(40)

A plot of InC versus x 2 would be a straight line and D can be found from the slope. Equation (39) is linearized as follows:

-jADt

= erfc

A_

f

A,

(41)

where erfc 1 means the inverse of the complementary error function. A plot of erfc '(A!A0) versus x would be a straight line and D can also be found from the slope. If analytical errors are much smaller than every measured concentration (e.g., 1% relative precision for all measured concentrations), linearized fitting will work well. However, the relative uncertainty of measurements at low concentrations is often large. Therefore, the error in InC (which is the relative error for C) increases as x increases in Equation (40), and error in erfc~'(/4M0) also increases as x increases. One must be careful either to do an errorweighted fitting, or only use data with high relative precision (e.g., Fig. 3-29b in Zhang 2008). Otherwise, the fit might be dominated by data with large errors and D from the fitting would

Zhang

20

not be accurate. Hence, nonlinear fitting has the advantage of handling errors much better (the data with small concentrations and consequently large errors are not emphasized in nonlinear fitting) and is the preferred method, especially since nonlinear fitting programs are now more readily available. The above comments about fitting data also apply to fitting other kinds of diffusion profiles discussed below.

Sorption or desorption Sorption or desorption of gases into or from a mineral occurs often in nature. For example, loss of radiogenic Ar and He (important for thermochronology) as well as other volatiles from minerals can be considered desorption. Sorption of water into minerals and glasses occurs in nature and can change the properties of the mineral and glasses. In diffusion studies, sorption and desorption experiments are often undertaken to obtain effective binary diffusivities of volatile components in melts and minerals (e.g., Dingwell and Scarfe 1985). The method has also been applied to determine , 8 0 diffusivities in melts and minerals under hydrous conditions (e.g., Giletti et al. 1978). In desorption experiments, a mineral or glass initially containing volatiles is heated in a gas medium that is devoid of the volatile component of interest. The surface condition is hence a zero concentration (or some low equilibrium concentration). In sorption experiments, a mineral or glass initially free (or almost free) of the volatile component of interest is heated in a gas or fluid containing the component of interest in the diffusion study. The surface boundary condition is a fixed concentration of the volatile component. Mathematically, the two problems (sorption and desorption) are similar, with the only difference being the initial and surface concentrations. This diffusion problem is known as the half-space diffusion problem with constant initial and surface concentrations. If the diffusivity D is constant, and diffusion from one surface has not reached the center of the sample (hence a semi-infinite medium), the resulting diffusion profile is as follows: C = C+ (C, - C t )erf

V4Z)f

(42)

where erf is the error function, C, is the initial concentration of the volatile component in the sample, and Cs is the surface concentration. Figure 6 shows a diffusion profile during sorption of Ar into a rhyolite melt. For desorption experiments, if the surface concentration is zero, the solution becomes: C = C erf—¡=^=

V4Dt

(43)

For sorption experiments, if the initial concentration is zero, the solution becomes: C = C eric —¡==

y/4Dt

(44)

If concentration profiles can be measured, the above equations can be used to fit data and D can be obtained. The mid-concentration distance for sorption and desorption is: Xi/2 = 0.9539(D0, /2

(45)

In sorption or desorption experiments, the concentration of the volatile component often only changes by hundreds or thousands of ppm, meaning the concentration gradients of major components are small. Hence, the diffusivity is often constant across the profile and the above solutions can be applied. For some diffusant such as H 2 0 in glass or minerals, even when the concentration is low (thousands of ppm, even down to tens of ppm), the concentration profiles

Theoretical

Background

of Diffusion

in Minerals

and

Melts

21

.v ( m m ) Figure 6. Diffusion profile for Ar sorption into a rhyolite melt (experiment RhyAr4-0 at 1375 K and 0.5 GPa of Ar pressure; Behrens and Zhang (2001).

cannot be fit by the above equations (e.g., Dmry and Roberts 1963; Delaney and Karsten 1981; Zhang et al. 1991a; Wang et al. 1996), signifying that D must depend on the concentration itself. After sorption or desorption experiments, sometimes the concentration profiles cannot be measured, but only the total mass gain or loss as a function of time is measured. If D is constant, the total mass gain or loss from both surfaces of a parallel plate (if loss from other surfaces is negligible) can be described by the following equation (Crank 1975); M, _ 4-v/D?

l + 2 ^ y (-l)"ierfc-^= ti 2 -jDt

(46)

where M, and M„ are the amount of the volatile component of interest entering (or exiting) the plate of thickness L at time t and time and ierfc is the integrated complementary error function. For small times (more specifically, when MJM„ < 0.6), diffusion has not reached the center yet and the above equation can be simplified as (Crank 1975);

M„

y/nL

That is, a plot of Mt versus tyl is a straight line. If D depends on concentration, the linearity between Mt and tm still holds, but the diffusivity derived from such data is an average diffusivity, and depends on whether sorption data are averaged (from which one obtains the diffusion-in diffusivity, Dm), or desorption data are averaged (from which one obtains the diffusion-out diffusivity, D0ut). Din and Z>lUI can be different, depending on how D depends on concentration. In some experiments, one single sphere, or more often, many spheres of roughly equal radius a, are investigated for mass gain or loss to obtain diffusivities. The equation to describe such results is (Crank 1975); M i rrc v> i•e rff c -n 7a = 1 — ,L = 6 - ^ < H + 2 V Vrc« [ ti -jDt J

iDt a

nos (48)

22

Zhang

where M, and are the amount of diffusant (for example, 1 s O in the case of oxygen diffusion ,s studied using an O tracer) entering (or exiting) the sphere of radius a at time t and time The above equation converges rapidly for small times. Furthermore, if MJM„ < 0.9, the above equation can be simplified as (Zhang 2008, p 291): M^

6

(49)

yjn

a

In the literature (e.g., Muehlenbachs and Kushiro 1974), the following equation is also used to fit experimental data for spheres, which converges rapidly at large diffusion times (Crank 1975): M
0 in the upper half. The combined solution of both halves is: C=C"

+ C

2

' + Cu ~C' erf-pL= 2 V4D?

(51)

where the Cv and CL are the initial concentrations in the upper and lower halves. Measured concentration profiles can be fit to the above equation to obtain D. In such fitting, Cv and CL can often be obtained from measured concentrations at the two ends (each can be obtained by averaging many points) and can be fixed in the fitting. Hence, there is essentially only one unknown parameter, D, to be obtained from the fitting. However, often the interface position is not known accurately, although it may be roughly estimated. Hence, the fitting often takes the following form: C=

C

U+CL 2

+

C

U~ 2

C

L

e v i

^

=

(52)

Dt

where za (the position of the Matano interface, defined by mass balance so that the diffusant loss from one side is equal to the diffusant gain on the other side; see Eqn. 54 below) is also

Theoretical

Background

of Diffusion

in Minerals

and

Melts

23

Figure 7. The diffusion couple setup and the resulting concentration profiles. On the left is a drawing of the diffusion couple configuration with high concentration of the component of interest in the upper half, and low concentration in the lower half (see also Fig. 1). The evolution of the concentration with time is shown on the right for three different times (arbitrary unit).

a fitting parameter and allowed to vary to optimize the fitting. The value of Zu does not have much meaning; it only indicates how well one estimated the interface position before the fitting. The definition of the mid-concentration distance takes some thought for a diffusion couple. If it were defined as the mid-concentration between the two halves, then it would not move at all, inconsistent with diffusive flux into a medium. The adopted definition is to consider diffusion in each half as having a constant surface concentration. Then the mid-concentration distance is the same as in sorption or desorption experiments, with x\a = 0.9539(Dt) m . Some authors use diffusion triples (e.g., Behrens and Hahn 2009), which are essentially two diffusion couples sharing one common half, in one experiment. In a diffusion triple, three glass or mineral cylinders are stacked together as upper, middle and lower thirds, making two diffusion couples. In nature, diffusion between two layers of a crystal differing in elemental or isotopic compositions may be viewed as a diffusion couple, as can diffusion between two layers of melts (though it is difficult to avoid convection in natural systems). For the complete homogenization of a diffusion couple, the initial concentration evolution is similar to Equation (51), but the concentration evolution after the diffusant has reached at least one end of the material depends on the boundary conditions at the ends (e.g., whether the ends are kept at constant concentration or there is no flux from the outside) as well as the dimensions of the initial two layers. Diffusive crystal dissolution Crystal dissolution and growth are common in magma chambers. Diffusive crystal dissolution has been applied to obtain chemical diffusivities and to treat multicomponent diffusion (Harrison and Watson 1983; Zhang et al. 1989; Liang 1999). Crystal dissolution rather than crystal growth is adopted in diffusion studies because crystal dissolution can be controlled well; for crystal growth experiments, where new crystals form cannot be wellcontrolled. The modifier "diffusive" is also important: it means that convection needs to be avoided to study diffusion.

24

Zhang

In the design of diffusive crystal dissolution, a gem-quality crystal disk and aglass cylinder are joined vertically with a horizontal interface to minimize convection (Fig. 8). If the melt due to the dissolution of the crystal has a higher density than the ambient (or initial) melt, the crystal is placed at the bottom: otherwise the crystal is placed on the top to minimize convection. Thus mass transport is entirely controlled by diffusion. At a fixed high temperature, the dissolution of the crystal often rapidly establishes a constant melt composition at the interface (Zhang et al. 1989: Chen and Zhang 2008. 2009), and diffusion carries the flux into the melt interior. The diffusion is often complicated due to (i) multicomponent effects and (ii) major compositional variation in the melt.

Figure 8. Setup of an olivine dissolution experiment. Because the dissolution of olivine produces a melt (interface melt) with greater density than the initial melt, olivine is placed at the bottom of a melt to minimize convection in the melt. In this case, the olivine crystal is larger in diameter than the melt so that the edge of olivine is preserved for the accurate determination of the olivine dissolution distance (Chen and Zhang 2008).

For the dissolution of low-solubility minerals such as zircon, the concentration gradients in major oxides are often negligible, and the diffusivity of the main mineral component is roughly constant along a profile. The solution to this diffusion problem is (Zhang et al. 1989): e r f c ^

C = C. + ( C - C . )

(53) erfc^ii V4Dt

where C, is the initial concentration of the main mineral component (such as Zr0 2 ) in the melt, Cs is the concentration of the component in the interface melt, which is a fitting parameter, and L is the melt growth distance, which is often negligible for dissolution of low-solubility minerals (which are also slowly dissolving minerals) such as zircon. For the dissolution of high-solubility minerals such as pyroxenes and olivine, the concentration gradients in major oxides are significant and the above equation does not work well for most components because of the multicomponent effects. However, for the major mineral component (the component whose concentration in the mineral is much higher than that in the melt, such as MgO during olivine dissolution), it is often possible to treat its diffusion as effective binary diffusion. In such cases. Equation (53) may be applied to fit the data to estimate the effective binary diffusivity. Furthermore, for high-solubility minerals (which are also rapidly dissolving minerals), the melt growth distance L must be determined independently (often from the mineral dissolution distance multiplied by the ratio of the mineral density over the melt density) to apply Equation (53) to fit data. In earlier experimental studies of crystal dissolution, convection was often present (e.g.. Brearley and Scarfe 1986), but was either not considered or incorrectly treated (see Zhang et

Theoretical Background of Diffusion in Minerals and Melts

25

al. 1989 for more discussion). Hence, the extracted diffusivities based on crystal dissolution experiments in these studies were often incorrect. Theoretically, there is also a short diffusion profile in the crystal, which is too short to be measured. Furthermore, the dissolution of the crystal shortens the diffusion profile in the crystal (Zhang 2008, p 378-389). Variable diffusivity along a profile In some diffusion experiments, the diffusivity may vary along a concentration profile. This can happen in at least two scenarios. One is when the major element composition changes significantly along a diffusion profile, such as in the case of Fe-Mg interdiffusion in olivine (Chakraborty 2010, this volume), in which diffusion has a strong compositional dependence. The other is in the case of components such as H 2 0 , where the diffusivity varies with its own concentration due to the effects of speciation even when the compositional variation of major components is negligible. To solve the diffusion equation with concentration-dependent diffusivity, numerical methods are necessary (e.g., Crank 1975; Press et al. 1994), which often is only slightly more difficult than working on complicated analytical solutions to a diffusion problem. In experimental studies, however, the interest is in obtaining the diffusivities from the measured concentration profiles, which is an inverse problem. There are two methods to extract diffusion coefficients if the diffusivity varies along a concentration profile. In one method, the functional form of the variation of the diffusivity with concentration is known, even though some parameters in the function are not known. For example, the diffusivity might be proportional to the concentration: D = aC, where a is the value of D when C = 1. Or the diffusivity may be linear in C: D = aC+b. Or the diffusivity might be an exponential function of concentration: D = b exp(aC) (i.e. InD is linear in C), where b is the value of D when C = 0. If the functional form is known but not the parameters a and b, the diffusion equation can be solved for given values of a and b, and the solution is compared with the experimental profile. By adjusting a and b to fit the concentration profile, the parameters can be found, so that the way in which D varies with C can be determined. The fitting can be complicated but specific programs have been written to accomplish this task (e.g., Zhang et al. 1991a; Zhang and Behrens 2000; Ni and Zhang 2008). If the functional form of the dependence of D on C is not known and cannot be guessed, then Boltzmann-Matano method, based on an application of the Boltzmann analysis by Matano (1933), can be applied to obtain diffusivities at every point along a profile. This method is most often applied to diffusion couples. In the original method, it is necessary to first find the Matano interface between the two halves of the diffusion couple (which may or may not be the physically marked initial interface between the two halves), so that x defined relative to the Matano interface (i.e., x = 0 at the Matano interface) satisfies: •c.

(54)

where CL and Cv are the concentrations at the two ends, x < 0 in the lower half of the couple, and x > 0 in the upper half of the couple. After obtaining the Matano interface, then the diffusivity at any x = x 0 (which also means at a C corresponding to xa) can be found (Crank 1975):

D

Co)

2t(dC!dx)

(55)

where t is the experimental duration. The key in minimizing the errors in extracting D using the above expression is to obtain accurate integrals and slopes, which requires smooth concentration profiles. Often the experimental data are smoothed objectively, either manually

26

Zhang

or by some kind of piecewise fitting (because it is not known what function can fit the whole profile). Furthermore, D values obtained using the above method near the two ends often have large errors. If the method is applied carefully, the general trend of D versus C is often acceptable, but small undulations may be artifacts of inaccurate slopes and integrations. A trivial variation of Equation (55) is

-J D cw

"

=

° xdC (56)

2t(dC/dx)Mo

A modified approach based on the Boltzmann analysis is provided by Sauer and Freise (1962). The advantage of this method is that there is no need to find the Matano interface. Define

(C-C.)

y= 7

K-CL)

^

(57)

which may be referred to as the normalized concentration. D can be found as follows: 1

D.a,c
lj (1 - y)dx + (1 - y\Xa

ydx

(58)

Again, the key in obtaining reliable D is to obtain accurate integrals and slopes, which requires smooth concentration profiles. The Boltzmann analysis can also be adapted for use in studies of diffusive crystal dissolution in order to extract diffusivities. The equation is (Zhang 2008): D,f(„=

c

1

" f (x-L)dC

(59)

where the upper limit of the integration C, is the initial concentration in the melt, x is the distance from the crystal-melt interface, x 0 is the position at which the diffusivity is obtained, and L is the melt growth distance. Homogenization of a crystal with oscillatory zoning In nature, a crystal (such as plagioclase) may be oscillatorily zoned. Idealize the initial oscillatory zones as follows: the concentration in the zones can be described by a sine or cosine function (which also implies constant width of every zone): R

C

AULo=a

+

2TIX

^

teÌn

(60)E

v P /

where a is the average An content in a plagioclase crystal, b is the peak amplitude (or half of peak-to-peak amplitude), and p is the width of each zone (or period of the oscillation), e.g., from one maximum to the neighboring maximum in Figure 9. As diffusion proceeds in a closed system (nothing entering or leaving the system), the concentration profile would evolve asr 2nx ^ C.„ =a + be~i% r:"lLl sin v P /

(61)

where D is the diffusivity of the coupled cation exchange Ca+Al Na+Si, which changes the concentration of the An component in plagioclase. That is, both the average An content and the period of the zoning stay the same, but the compositional amplitude of the zoning decreases

Theoretical

Background

of Diffusion

in Minerals

and

27

Melts

x (mm) Figure 9. Homogenization of an oscillatorily zoned crystal with time.

exponentially with time (Fig. 9). If the initial oscillatory zoning is periodic but has sharp boundaries, the solution would be an infinite series of sine or cosine functions. One dimensional diffusional exchange between two phases at constant temperature Often, two minerals in contact have common components that may be exchanged. For example, garnet and olivine both contain Fe2+ and Mg2+. and the two cations can exchange through diffusion (Fig. 10). Garnet and spinel may exchange Mn-Fe 2+ -Mg in divalent sites, and Al-Cr-Fe1+ in the trivalent sites. The following results are from Zhang (2008. p 426-430). Assume (i) each phase is initially uniform in composition, (ii) the exchange is between only two components (binary diffusion), (iii) the contact interface between the two minerals is flat (planar), (iv) either the mineral is isotropic or diffusion in an anisotropic mineral is along a principal axis of diffusion (see below on diffusion in anisotropic medium), (v) diffusion has not proceeded to the center of either mineral yet. and (vi) D in each mineral phase is constant. Then, the problem is one dimensional and has analytical solutions. Furthermore, assume that there is instantaneous equilibrium at the contact between the surfaces of two minerals and that the equilibrium condition is described by a constant exchange coefficient K a (which depends on temperature): K,=

[xt/xt)

(62)

where X means mole fractions, superscripts A and B are the two mineral phases (e.g., A = olivine and B = garnet), subscripts 1 and 2 are the two components (e.g., 1 = Mg and 2 = Fe), X* is the mole fraction of component 1 in mineral A, the interface is at x = 0, mineral B is on the side of a: > 0, mineral A is on the side of a: < 0, "a: = +0" means the surface of mineral B (x approaches zero (interface) from x > 0 side), and "x = - 0 " means the surface of mineral A (x approaches zero (interface) from x < 0 side). The solution for the concentration evolution as a function of time is: (63a) (63b) where Xy and X" are the initial mole fractions of component 1 in minerals A and B, Xy_u and

Zhang

28

Dol/Dct=

16 Gurnet

Olivine

0 jr

0.5

Figure 10. F e - M g exchange between olivine and garnet. The figure on the left shows the geometry of the diffusional exchange, with the horizontal direction being along c-axis of olivine. The figure on the right shows the calculated diffusion profile in the two phases using Equations (63a) and (63b). KD = (Fe/Mg) G l / (Fe/Mg) 0 i = 1.7.

Xf+U are the mole fractions of component 1 at the interfaces of minerals A and B. DA and D B are interdiffusivities between components 1 and 2 in minerals A and B. The mole faction of component 2 in each mineral can be found by stoichiometry (e.g., the sum of mole fractions of 1 and 2 in every mineral is 1). Given initial conditions X* and and diffusivities. there are still two unknowns ( X , A a n d X,B+0) in the above two equations, which can be solved from two equations: Equation (62) (surface equilibrium) and the following (mass balance):

{xtu-x*y^=(x?M)-x»y^

(64)

where pA and p B are the molar densities of components 1 and 2 in minerals A and B. For example, = 43.48 mol/L and = 25.64 mol/L if there are no other divalent cations. Calculated profiles are shown in Figure 10b. Spinodal decomposition Spinodal decomposition is the spontaneous decomposition of a single phase to two phases. For example, alkali feldspar at high temperature can be a single phase. As the temperature becomes lower, it may spontaneously separate into two phases, albite and orthoclase. The intergrowth of the two phases is called perthite. The separation of a single uniform phase into two phases of similar structure is called spinodal decomposition. It is accomplished by diffusion and thermal fluctuation. In the process, diffusion may transport elements from low concentration to high concentration (referred to as uphill diffusion), opposite to the transport direction during normal diffusion. Spinodal decomposition in a binary system illustrates that diffusion is not simply responding to concentration differences to homogenize the system, but is a response to the chemical potential (or chemical activity) difference. Diffusion reduces the Gibbs free energy of the system. In ideal or close to ideal binary mixtures, the entropy portion of the Gibbs free energy dominates the total Gibbs free energy of mixing. The chemical potential of a component increases as the concentration of the component increases. Hence, diffusion homogenizes the system, which minimizes the Gibbs free energy of the system. In highly non-ideal binary mixtures, the chemical potential (or activity) may decrease as concentration increases when the enthalpy part of the Gibbs free energy dominates the total mixing energy.

Theoretical Background of Diffusion in Minerals and Melts

29

Then, diffusion would still be downhill in terms of the chemical potential (or activity) gradient, but can be uphill in terms of concentration gradient. Hence, Fick's law is an approximation of the following more accurate diffusion law in a binary system (Zhang 1993): J=

, 3 — + — D p dt dx — dx —- dy —- dz I dy —- dx —— dy —~ dz d

f

n

d C

r,

d C

r,

(79)

d C

+— Dn — + D, 3 — + £>33 — dz I — - dx dy — dz. If all the tensor components are constant, then

dC d2C d2C d2C d2C d2C d2C = D, D D 2D, -2D, -2D, (80) dt —— dx2 —— dy2 —— dz2 —— 3x3)' —— dxdz —— dydz

Zhang

34

The above equation can be simplified using coordinate transformation to become:

dC

n

d2C

n

d2C

— = D„—~+D„—t a 2 P

dt

da

3p2

n

d2C

+ D„—t-

'df

(81)

where a , (5, and y are the principal axes of diffusion, and Da, Dp and D., are the principal diffusivities (diffusivities along the principal axes). For the above equation to be applicable, the boundary conditions must also be transformable. The fluxes along a principal axis of diffusion can be written as:

Ja=-DadC/d

a

(82a)

yp=-L»p3C/ap

(82b)

Jy = -DydC / dy

(82c)

One-dimensional diffusion equation along each principal axis of diffusion can be written as:

ac_D yc — = Dp , —2 dt ~ ap

(83b)

f-.f

Diffusion along each principal axes is hence unaffected by diffusion in other directions. In melts, glasses and isometric minerals, any direction can be taken as the principal diffusion direction. In non-isometric minerals, the crystallographic axes can be taken as the principal diffusion axes if the symmetry is at least orthorhombic. In addition, in hexagonal, tetragonal, and trigonal minerals, diffusion along any direction in the plane perpendicular to the c-axis can be taken as a principal axis and can be treated simply as one-dimensional diffusion. Diffusion along a direction that is not a principal axis cannot be treated as one-dimensional diffusion, because diffusion along other directions would also contribute to the flux along the direction of consideration; i.e., the diffusive flux J is not parallel to -VC. The three-dimensional diffusion equation (Eqn. 81) can be simplified further with the following axis transformation: a = a/JIT

(84a)

p' = p / ^

(84b)

y =y

(84c)

The transformed diffusion equation becomes

dC _ d2C

d2C

d2C

which is identical to the diffusion equation in isotropic media with D = 1. Hence, after convoluted transformations, the complicated diffusion equation in anisotropic media with a constant diffusivity tensor (Eqn. 80) is simplified to the diffusion equation in isotropic media. Hence, in theory, analytical solutions for diffusion in anisotropic media can be obtained using these transformations. However, the three-dimensional initial and boundary conditions must also be transformed, which is not always easy. Furthermore, the transformations may have drastically

Theoretical

Background

of Diffusion

in Minerals

and

Melts

35

changed the shape of a crystal, from the physical shape in natural coordinates to the effective shape in the transformed coordinates. The effective shape might be highly unintuitive. Figure 12 gives an example. Diffusion in anisotropic media is often treated with major simpliFigure 12. Comparison of the physical shape and effective shape in terms of fications. If total loss or gain of the diffusion. The upper diagram shows the diffusant is of interest, as in the difphysical shape of mica (plane sheet), and fusive loss of radiogenic nuclides in the shape on the right hand side is the understanding closure temperature shape after transformation to lengthen the vertical axis by a factor of 100 (because the and closure age. the shape of the difference in oxygen diffusivities is 4 orders mineral grains (necessary for deterof magnitude). Hence, diffusion along the mining the shape factor) is the efz direction can be ignored, and the mica fective shape. For example, Fortier can be treated as a cylinder in terms of l s O and Giletti (1991) showed that under diffusion. hydrothermal conditions the 1 8 0 diffusivity in mica parallel to the c-axis is about 4 orders of magnitude slower than diffusion in the plane perpendicular to the c-axis (even though mica is monoclinic, it is approximately hexagonal in terms of many of its properties). Hence, the physical shape of mica is platy, but the effective shape of mica in terms of l s O diffusion under hydrothermal conditions is an "infinitely" long cylinder, meaning that bulk mass loss or gain is through diffusion in the plane perpendicular to the c-axis (Fig. 12). MULTICOMPONENT DIFFUSION Diffusion in multicomponent systems (having three or more components) is also complicated. In-depth treatment of multicomponent diffusion will be covered in other chapters (Liang 2010: Ganguly 2010) in this volume. The general aspects and some simple treatments are covered here. Fick's law (Eqn. 1) is for binary systems only. When three of more components (such as A, B, C and D) are present, experiments show that the diffusion of a given component A depends not only on the concentration gradient of A, but also on the concentration gradients of B and C. Consider a melt with N components. Because the summation of concentrations (such as mole fractions or mass fractions) of all components is 100%, there are only N-1 independent components in an /V-componenl system. (For example, in a binary system, only the concentration of one component is independent.) Let n = N-1. If the iV-component system is a stable phase (i.e., no spinodal decomposition), the diffusive flux of components can be written as (De Groot and Mazur 1962): J, = -DuVCt

- D12VC2... - DlflVC„

J, = - A . V C , - D2yC2...

- A„VC„

J„ = - Z ^ V C , - DirVC,... - DmVCn

(86a) (86b) (86n)

where D,-,'s are diffusion coefficients for component i due to concentration gradients of component j. D,,'s are referred to as the main or on-diagonal diffusivities. and D,-,'s when i±j are referred to as the off-diagonal or cross diffusivities. In matrix notation, the above becomes:

Zhang

36

Dn D,

D, D.

A.,

YC

D,

D,

A.

vc.

vc, (87)

The diffusivity matrix is not to be confused with the diffusivity tensor; they are different in at least two aspects: (i) the meanings are different, one refers to diffusion in a multicomponent system but isotropic medium, and the other refers to binary diffusion in an anisotropic medium; (ii) the diffusivity tensor in anisotropic system is always represented by a 3x3 symmetric matrix, but the diffusivity matrix (always a square matrix) is n by n where n > 2 and is nonsymmetrical. For diffusion in a multicomponent system in an anisotropic medium, the full rigorous description would require a diffusivity matrix in which every element is a tensor. Such a diffusion problem has not yet been solved. Even if the mathematical complexity can be overcome in the foreseeable future, it would be impossible to obtain all the coefficients in the tensor matrix experimentally. Because multicomponent and anisotropic minerals are common (e.g., mica, hornblende, pyroxenes, etc), diffusion in nature is truly complicated. Hence, simplifications are usually made out of necessity because requiring theoretical rigor would simply mean getting nowhere in understanding geological problems which otherwise can be approximately quantified. Even for diffusionally isotropic natural silicate melts with typically four or more major components and numerous minor and trace components, no reliable diffusivity matrix has been obtained. In the near future, it may be possible to extract diffusivity matrices of the major components. However, unless theories can be developed to calculate the diffusivity matrix for trace elements in natural melts, the diffusivity matrix approach is unlikely workable for trace elements because it is impractical to experimentally obtain diffusivity matrices involving both major and trace elements (about 80 components in total). Hence, in the foreseeable future, the simple effective binary approach, or some modified simple approach, will be necessary. In this section, the various approaches and their advantages and disadvantages are briefly outlined. Effective binary approach, FEBD and SEBD The effective binary diffusion approach (Cooper 1968) is the most widely used simple treatment of diffusion in a multicomponent system. In this approach, the diffusion of a component in a multicomponent system is treated as diffusion in a binary system, of which one is the component of interest and the other is all the other components combined. That is, the flux of component i in one dimension is simply expressed as (Eqn. 6): J, = "

dx

(88)

The diffusion equation is hence Equation (5). Compared to the multicomponent diffusivity matrix approach (one dimensional form of Eqn. 87) in which J, = -ZD,-;3C/3jc, it can be seen that

JL, dC / dx Di = y Dij—I = D - x D 0(. 5 dC,. / dx i=i,j*i -f

(89)

Hence, the effective binary diffusivity depends on the on-diagonal and off-diagonal diffusivities involving the component of interest, as well as the concentration gradients of all components. With the effective binary approach, the solutions for binary diffusion are used for diffusion of a component in a multicomponent system and to extract effective binary diffusivities. Almost all chemical diffusion data in minerals and melts are obtained using this approach.

Theoretical

Background

of Diffusion

in Minerals

and

Melts

37

These chemical diffusivities are called effective binary diffusivities (EBD, which can also mean effective binary diffusion). The approach only works when there is no uphill diffusion (e.g.. Fig. 13): otherwise, the effective binary diffusivity would change from positive to negative in a single profile. 1.8 ^

1-7 • c

t:

i.6

•

•

• •

•'ft i

1.5 -

* t

.

•

••• •

T5 0 1.4 -0.6

-0.4

-0.2

0

x (mm)

0.2

0.4

0.6

Figure 13. Uphill diffusion in a diffusion couple experiment (Van D e r L a a n e t al. 1994).

Because the EBD approach does not work for uphill diffusion profiles, authors typically ignore uphill diffusion profiles in data treatment, often simply marking them as "uphill". In binary systems, the presence or absence of uphill diffusion can be predicted from thermodynamics: uphill diffusion would occur when the phase is not stable and undergoes spinodal decomposition. However, in multicomponent systems, whether uphill diffusion would occur for a given element (such as Ca or Ce) in a diffusion couple or during crystal dissolution (Zhang et al. 1989) cannot be predicted yet. Even in the absence of uphill diffusion, the EBD approach is still prone to many limitations. The most serious one is that effective binary diffusivity of an element may depend not only on the major oxide concentrations, but also on their gradients (Liang 2010, this volume). This may sound strange but can be understood in the following example. The Si0 2 diffusivity in CaO-AI 2 O r Si0 2 melts along constant CaO concentration (the CaO concentration gradient is nearly zero, and the A1 2 0 3 concentration gradient roughly compensates the Si02 concentration gradient) is essentially the SiOo-AW h interdiffusivity, but is essentially the Si0 2 -Ca0 interdiffusivity along a constant A1 2 0 3 concentration. It is hence understandable that the two are different even at the same bulk composition, as shown by Liang et al. (1996). Along other compositional directions, even though the diffusivity cannot be simply viewed as an interdiffusivity, the effective binary diffusivity is expected to also depend on other compositional gradients. Because compositional gradients can change as diffusion proceeds, especially in a finite system, the EBD can also change as diffusion proceeds (Liang 2010, this volume). Even knowing that it has limitations, EBD approach is still the most widely used approach in treating diffusion in multicomponent systems because other approaches are too complicated. In the foreseeable future, the EBD approach will still likely be the method of choice for trace element diffusion even if the diffusivity matrix for major components could be obtained. Hence, it is necessary to understand under what conditions the EBD approach is more reliable than others. Cooper (1968) summarized that the EBD approach is meaningful when (i) the concentration gradients of all components are in the same direction; and (ii) either a steady state exists or the diffusion media is infinite or semi-infinite. However, there could still be uphill diffusion even when the two conditions are satisfied, and uphill diffusion is difficult to treat using the EBD approach. Below, some specific situations are discussed:

38

Zhang (a) The EBD approach works well if the base composition of a diffusion system is the same, but there is one relatively minor component (or a few minor to trace components) up to several percent that diffuses in or out, such as in cases of sorption or desorption, hydration or dehydration, bubble growth, or a diffusion couple between two rhyolite melts with only small difference in Sr concentration. That is, the compositional difference in the system is due to the dilution effect of the presence (or absence) of this component under consideration. If the component diffuses more rapidly than other components, the effective binary diffusion approach works even better. Diffusivities from such experiments can be applied accurately to similar situations when the base compositions are the same. (b) In a multicomponent system, if the initial compositional difference is only between two components such as N a 2 0 and K 2 0, the diffusion of each of the two components can be treated as effective binary diffusion. In fact, this case is similar to interdiffusion between two components, and hence the two components should have the same interdiffusivity. Diffusion of other components in the system may not be treated as EBD. (c) Even if there are major concentration gradients in multiple components, as long as they are consistent in the direction and relative magnitude, the EBD approach can be applied to the component with the largest concentration gradient. Examples include dissolution of a specific mineral in a specific melt (such as olivine dissolution in a basalt melt, or quartz dissolution in an andesite melt). In this case, the component with the largest concentration gradient (such as MgO during olivine dissolution in a basalt melt, or Si0 2 during quartz dissolution in an andesite melt, referred to as the principal equilibrium determining component by Zhang et al. 1989) can be treated as EBD. Such effective binary diffusivities can be applied to the identical situations in nature under similar boundary conditions such as semi-infinite diffusion media, meaning MgO EBD extracted from olivine dissolution experiments in basalt melt can be applied to predict MgO diffusion in nature during olivine dissolution in a basalt melt, but it may not be applicable to MgO diffusion during clinopyroxene dissolution in a similar melt, or olivine dissolution in a different melt. Diffusion of other components may or may not be treated as EBD. (d) In diffusion couple studies when two different melts are placed together, effective binary diffusivities may be extracted for components showing large concentration differences between the two halves. These EBD values may be applied to diffusion couples with similar compositions in the two halves. However, if the concentration gradient is switched for some major component (e.g., in one diffusion couple, A1 2 0 3 concentration is 13 wt% in the basalt half and 17 wt% in the andesite half, whereas in the other diffusion couple, A1 2 0 3 is 17 wt% in the basalt half and 13 wt% in the andesite half), EBD from one case may not be applied to the other. Liang et al. (1996) (their Fig. 5) showed an example of this.

As can be seen from the above, effective binary diffusivities are a large category and cover many different situations. Some EBD values are more reliable than others. For easy reference, I propose three types of effective binary diffusivities based on their consistency and reliability: (1) Interdiffusivity or interdiffusion (ID). Binary diffusion is interdiffusion. In multicomponent systems, if the concentration gradients are primarily in two components, and concentrations of all other components are roughly uniform (case (b) above), then it can be referred to as multicomponent interdiffusion. In this case, the diffusivities of the two components are roughly the same, and can be treated well and consistently using the effective binary approach. The interdiffusivity would depend on the bulk composition but not on the concentration gradients of other components (these are es-

Theoretical

Background

of Diffusion

in Minerals

and

Melts

39

sentially zero). Diffusion of other components may not be treated well by the effective binary approach (e.g.. often there can be uphill diffusion for other components). (2) FEBD (first type of effective binary diffusion, or first type of effective binary diffusivity). FEBD corresponds to the diffusion situation of case (a) above. Because this situation is often encountered in experiments and nature (especially sorption/ desorption. bubble growth, and explosive volcanic eruptions), and because of the high degree of consistency of this type of effective binary diffusivity, it is considered to be a special type of EBD, called the first type of effective binary diffusion, with the acronym FEBD. In principle, when the concentration of the component of interest becomes low enough, FEBD approaches the tracer diffusivity (or trace element diffusivity in the absence of major concentration gradients). (3) SEBD (second type of effective binary diffusion, or second type of effective binary diffusivity): All other types of effective binary diffusivities are less reliable and are grouped as SEBD, even though some may be more consistent than others. SEBD values can be applied to systems very similar to experimental systems in terms of bulk composition as well as the direction and size of concentration differences. In studies of diffusion in multicomponent systems, often all monotonic concentration profiles (no uphill diffusion) are treated using the effective binary diffusion approach, assuming a constant SEBD. The fits may not be perfect (e.g., the SEBD of MgO during olivine dissolution in an andesite melt seems to increase with increasing Si0 2 content: Fig. 14). In such a case, one is tempted to make efforts to determine how the SEBD varies along the profile and associate the variation with Si0 2 or other concentration changes. However, this may not be correct, because the SEBD variation might be due to concentration gradient variations, rather than the compositional variations. Hence, in treating SEBD profiles, the simplest approach is to fit the profile with a constant SEBD and ignore the small misfits because of the complexity of SEBD.

M g O d i f f u s i o n in andesite melt d u r i n g olivine dissolution 1558 K, 0.55 GPa, 5 hrs st 51

8

Figure 14. MgO diffusion profile in the melt during olivine dissolution in andesite melt (Zhang et al. 1989). The fit curve, assuming constant SEBD. does not match the data well, e.g., at the region near x = 1 mm. The slower decrease of the concentration implies higher diffusivity in this region even though the Si0 2 concentration here is high (about 51 wt%) compared to Si0 2 near x = 0 (about 52 wt%). This misfit is most likely due to the dependence of the SEBD of MgO on concentration gradients rather than on the bulk composition itself.

Modified effective binary approach (activity-based effective binary approach) The modified effective binary approach (also called the activity-based effective binary approach) was proposed by Zhang (1993) and based on rough chemical activity estimation in silicate melts. It is assumed that the diffusive flux of a component i is proportional to the

Zhang

40

activity gradient of the component alone (Eqn. 65): J. = ——Va.

(90)

y,

where a, and y, are the chemical activity and activity coefficient of component i with a, = y,C and (D, is the "intrinsic" effective binary diffusivity, different from the normal effective binary diffusivity. The above equation reduces to Equation (88) when y, is constant. The difference between the effective binary approach and the modified effective binary approach is that the concentration gradient is replaced by the activity gradient. From Equation (66) (or comparing Eqns. 88 and 90). the "intrinsic" effective binary diffusivity and the normal effective binary diffusivity are related as:

D, =

1 +

d lny, d I ii C.

(91)

Zhang (1993) showed that the approach could fit and predict uphill diffusion profiles during crystal dissolution experiments (Fig. 15). which cannot be treated by effective binary diffusion. Lesher (1994) developed a similar model to treat uphill diffusion of trace elements in a diffusion couple. Even though the model can handle uphill diffusion profiles and may also fit monotonic diffusion profiles better, the approach has two disadvantages: (i) it is complicated, and (ii) the activity model for silicate melts is uncertain. The applicability (or inapplicability) of the approach needs to be explored further.

i

1

1

r

O l i v i n e d i s s o l u t i o n in a n d e s i t e # 2 2 9 . 1458 K , 0 . 5 G P a

Figure 15. Fitting an uphill diffusion profile of FeO in an andesite melt during olivine dissolution using the modified effective binary diffusion model (from Zhang 1993).

0.2

0.4

0.6

0.8

x (mm) Diffusivity matrix approach The diffusivity matrix approach is the classical and rigorous method to describe multicomponent diffusion (Eqns. 86 and 87; see in-depth discussion by Liang 2010. this volume). This approach works well if the ^-component mixture is not very non-ideal. The approach fails when the mixture is unstable, leading to spinodal decomposition because some eigenvalues would change from positive to negative, similar to the case of spinodal decomposition in a binary system. When the system is highly non-ideal, the individual diffusivity values would be highly variable with composition even if the mixture is stable. For ideal and nearly ideal

Theoretical Background of Diffusion in Minerals and Melts

41

systems, the diffusion equation for an iV-component system (n = N-1) is of the following form: ^

= IV(D#VC,)

(92)

For one-dimensional diffusion and constant Du values, the diffusion equation becomes:

Therefore, the concentration evolution of component i depends on the concentration gradients of other components. Equation (93) contains coupled equations. Hence, it is necessary to solve the concentration evolution of all n components simultaneously in the ^-component system. In the matrix form, the concentrations can be solved as follows. The one-dimensional diffusion equation with constant diffusivity matrix can be written as:

dt

dx

where C is the concentration vector, or the transpose of (C 1 ; C2,..., Cn), and D is the diffusivity matrix. If the W-component mixture is stable, it can be shown that all n eigenvalues of the D matrix are real and positive (e.g., De Groot and Mazur 1962), and D can be diagonalized: D = TXT- 1

(95)

where X is a diagonal matrix made of the eigenvalues of D, and T is a matrix made of the eigenvectors of D. Replacing Equation (95) into Equation (94) leads to a7

, d2u a 7

= x

(96)

where u = T C is the transformed composition vector. Because X is a diagonal matrix, the above equation is equivalent to:

du. „ 3'm. 1>7 = ^

,„„ (97)

where i can be 1, 2, ..., n. Hence, in the transformed compositional space, the diffusion equation for each w, depends only on its own concentration gradient with a real and positive diffusivity of If the initial and boundary conditions can also be transformed, Equation (97) is in the same form as binary diffusion and w, can be solved. After solving for every w„ the final solution is C = Tu (98) When the diffusivity matrix is not constant (e.g., some of the elements in the matrix depend on concentration, which is common), the above analytical solution is not possible, and the multicomponent diffusion equation must be solved numerically (this complexity also exists for the effective binary treatment). In principle, the diffusivity matrix (i.e., diffusion of all elements in a diffusivity matrix) can be obtained from experimental diffusion studies, similar to binary or effective binary diffusivities, by fitting experimental concentration profiles using the diffusivities of elements as fitting parameters. This has been done for some simple melts (e.g., Vignes and Sabotier 1969; Sugawara et al. 1977; Liang et al. 1996; Liang and Davis 2002). However, for natural silicate melts, due to the large number of fitting parameters involved (e.g., the diffusivity matrix of a 10-component system is made of 81 individual values), this is a daunting task. Strategies

42

Zhang

have been proposed (Trial and Spera 1994) and bold attempts have been made (Kress and Ghiorso 1995; Mungall et al. 1998), but the diffusivity matrices have not been verified and more follow-up work is necessary. More on multicomponent diffusion as well as empirical models for multicomponent diffusion matrices can be found in Liang (2010, this volume). Activity-based diffusivity matrix approach Even the complicated treatment of multicomponent diffusion using diffusivity matrices is not enough to treat phase separation in multicomponent systems. A more fundamental approach is to use the activity-based diffusivity approach as in the binary system (Eqn. 65). The diffusive flux is expressed as (Zhang 2008) " , meaning that logD (or InD) versus 1/7 (or 1000/7) is a straight line with a negative slope, as shown in Figure 2 for Ar diffusivities in water, basalt melt, rhyolite melt and the mineral hornblende. (Fig. 17a in a later section also shows some diffusion data in an Arrhenius plot.) The Arrhenius equation can be derived from either the collision theory or the transition state theory (e.g., Lasaga 1998), in which the activation energy is identified to be the necessary enthalpy for forming the activated complex. The preexponential factor D 0 is proportional to 7I/2 in the collision theory and to 7 in the transition state theory. There were efforts in the early years to test how the preexponential factor D 0 depends on temperature (e.g., Perkins and Begeal 1971; Shelby and Keeton 1974), but it requires high-quality data (e.g., with < 10% relative error in D) in a large temperature range (e.g., 400-1200 K), and the results are inconclusive: high-quality data over a large temperature range may show a small curvature in InD versus 1/7, indicating that either D 0 or E depends on temperature, but the exact relation is not well constrained. On the other hand, for most diffusion data in melts and minerals in the geological literature, the uncertainty in D is often of the order of 30% and the temperature range is not large enough, so that the Arrhenius relation works well within uncertainty. In limited cases when the temperature range is large from glass to melt, the Arrhenius equation fits the data well and it is rarely necessary to invoke dependence of D 0 on 7, or E on 7, or a discontinuity due to glass transition (e.g., Zhang et al. 1991a; Zhang and Behrens 2000; Behrens and Zhang 2009). In minerals, although some authors proposed discontinuities in InD versus 1/7 (e.g., Buening and Buseck 1973 forFe-Mg interdiffusion in olivine), later studies show no such discontinuity is present (e.g., Chakraborty 2010, this volume). In summary, the temperature dependence of all experimental diffusivity data in the geological literature can be summarized well by the Arrhenius relation in the form of D = D0e-E/(RT> with positive D 0 and E. Pressure dependence of diffusivities The dependence of diffusivities on dry pressure (P) is more complicated than the temperature dependence. (Hydrous pressure, on the other hand, has two effects: one is pressure, and the other is the presence of water, which may accelerate diffusion.) In a small pressure range, InD is often linear in P. Such a relation is consistent with the transition state theory

44

Zhang

because the enthalpy is linear to pressure in liquid and solid phases, leading to Equation (11), D = D0e~.

• D,

where d = 2-h and Cr° and Cs° represent the initial concentrations at time t = 0 in the thin film and the substrate, respectively. If Dt and D s are equal the solution is greatly simplified (see, e.g., Crank 1975, p. 31) and only one function C(x,t) is required to describe the whole profile where the distance x is now measured from the surface: C(x,Q-C° 0

cr°-c. '

erf

^ h—x 2-JdÎ

^

-erf

( n( , -+L . x 2 -JETt

(6)

If the film thickness is very small compared to the profile length and if diffusion is sufficiently fast within this layer, the solution for an infinitely thin instantaneous source at the surface can be used (see Crank 1975): C(x,t) =

M

^•exp

( -x 2 A +C 4DJ

(7)

Experimental Methods for Characterizing Diffusion

67

where M is the total amount of diffusant contributed by the source over and above that already present in the layer. Fitting uncertainties The analytical solutions given in Equations (5)-(7) can be used directly to fit a measured depth profile, either by a simple "visual" fit or by performing a regression minimizing the sum of the squares of the residuals. The error in the diffusion coefficient obtained from fitting a single profile is usually smaller than 0.2 log unit, but this can depend on the statistical error in the individual analyses and the complexities of the specific diffusion couple (Ganguly et al. 2007; Zhang et al. 2010). However, a serious systematic error can be introduced if the effective spatial resolution of the profile analysis is not considered. The effective spatial resolution may be considerably lower than the nominal resolution of the analytical method if the geometry of the film is imperfect owing to, for example, non-uniform deposition, grain growth, reactions, etc. Such analytical artifacts may lead to a smearing of the measured concentration profile. This can be explicitly modeled following the approach of Ganguly et al. (1988), which was originally developed to consider the convolution effect arising from the limited spatial resolution of the electron microprobe. For SIMS depth profiles, Hofmann (1994) presented an analogous approach. The main principle is that the broadening of the measured profile C'(x) can be predicted by convolving the real concentration profile, C(x), with a resolution function g(x - x'): (8) The resolution function can be often approximated as a Gaussian in which the spatial resolution is characterized by the value of o, which represents the standard deviation of the Gaussian: (9) The integration in Equation (8) can be performed numerically to either determine the value for o on reference samples with a steep concentration gradient (e.g., as illustrated in Fig. lb) or when this o is known to simulate the measured profile with a forward modeling. A simple estimate of how much the diffusion coefficient retrieved from directly fitting the profile, denoted as Dc, deviates from the true diffusion coefficient, D, was provided by Ganguly et al. (1988): (10) where L is the length of the diffusion profile measured from the interface. This equation is strictly valid for cases in which L = 4(Dt)l/2, such as a diffusion couple or diffusion within a substrate from a constant course (Fig. 1), but it can be used for approximate calculation of the convolution effect for diffusion from a thin film. It is clear from Equation (10) that for a given value of a the effect is strongest for very short times and/or very small diffusion coefficients. A simple calculation using Equation (10) demonstrates that to keep the error induced by the spatial averaging below ~10%, the measured profile length L should be at least ten times the value of a. Otherwise, the convolution should be explicitly considered in the numerical model according to Equations (8) and (9). If the profile length is very close to o, one should simply run longer experiments. A simplified way to consider convolution is to replace in the analytical solution 2Dt by 2Dt + a2. The procedures described above can be also applied to estimate the errors induced by the roughness of a sample because the morphology of a rough surface can be approximately represented by a Gaussian function in a depth profile analysis. Convolution effects have two general implications for diffusion studies. First, the error from inadequate spatial resolution always leads to an overestimate of the true value of the diffusion coefficient, as one can easily see from Equation (10). Second, the activation energy

68

Watson & Böhmen

resulting from multiple experiments over a range in temperature tends to be lower than the true one. The latter point is related to the Arrhenian behavior of diffusion. In most diffusion studies the measured profile lengths are smaller at lower temperatures even if the run duration is much longer, which implies that the resulting diffusion coefficients deviate further (in a positive direction) from the true diffusion coefficients. Pulsed laser ablation: a versatile method for thin film deposition From the above discussion it is clear that a prerequisite for the measurement of small diffusion coefficients by the thin-film method is a film that is geometrically and chemically uniform. The ease of creating such a film depends on the deposition method. Many techniques have been used; these can be broadly subdivided as either chemical or physical in nature. A complete overview is beyond the scope of this chapter; the reader is referred to published monographs for details: e.g., Smith (1995), Soriaga et al. (2002), and Kern and Schuehgraf (2002). For diffusion experiments specifically on minerals, radio frequency sputtering has been used to study diffusion in olivine (Jaoul et al. 1981, 1995), diopside (Dimanov and Jaoul 1998), and monazite (Gardes et al. 2006). Thermal vapor deposition has been used to deposit a variety of diffusants on minerals, including MgO (Schwandt et al 1993; Zhang et al. 2010), KOH (Ito and Ganguly 2004), and Cr 2 0 3 (Ganguly et al. 2007). Evaporation of an aqueous solution containing the tracer has also been successfully employed to deposit layers of simple oxides, phosphates and silicates (e.g., Sneeringer et al. 1984; Cygan and Lasaga 1985; Chakraborty and Ganguly 1992; Van Orman et al. 2001, 2002; Ganguly et al. 1998; Tirone et al. 2005). There are pros and cons to the various deposition methods (for a short summary see Dohmen et al. 2002a), but the common disadvantage is that a new protocol must be developed for a given desired film composition, and complex compositions may be unattainable. These methods do not lend themselves to the study of a diverse and complex set of minerals and diffusants. The difficulty of controlling the composition of a precipitate from aqueous solution means that the resulting film and the substrate may be thermodynamically unstable together. This can lead to the formation of a reaction layer that interferes with the diffusion process of interest. As discussed in the "Fitting Uncertainties" and "Analytical Considerations, Advantages and Drawbacks" sections in this chapter, the presence of a reaction product can degrade the information obtained by depth profiling because mixture and convolution effects must be addressed. Many of the problems of thin-film production can be circumvented by pulsed laser deposition (PLD), which is being used with increasing frequency in materials science to deposit layers of insulators, metals, superconducting materials, semiconductors, polymers and even biological materials (e.g., see review of Norton 2007). The principle of PLD was discovered in the late 1960s (see, for example, the review of Sankur and Hall 1985), but it was not until in the late 1980s that the potential of PLD to produce superconducting thin films was fully recognized (Dijkkamp et al. 1987). The method is very flexible and has the particular advantage that thermal fractionation effects are minimized. This allows production of thin films that contain elements with strongly divergent chemical properties—different volatility, in particular. Sumit Chakraborty was the first geoscientist to recognize the potential applications of PLD to studies in mineral kinetics, and in early 2001 a PLD facility was established at the Institut für Geologie, Mineralogie und Geophysik, Ruhr-Universität Bochum. This setup has been used successfully to deposit thin films of numerous mineralogical compositions, ranging from simple oxides such as Ti0 2 and Zr0 2 (Dohmen 2008; Dohmen et al. 2009) to various complex silicates, including olivine, pyroxenes, and garnet (Dohmen et al. 2002a; Milke et al. 2007) as well as aluminous spinel (Vogt 2008). The principle of PLD is straightforward, but the physical processes involved are complex and not well understood theoretically (see, e.g., Schou 2010): it is, in principle, a physical vapor deposition process that is carried out in a vacuum chamber (Fig. 3a). A short (0.1 - 20 ns) high-energy laser pulse impinges on a target that has the desirable composition for the

Experimental

pulsed beam of excimer laser ___ wavelengths: 193 nm, 248 nm

Methods for Characterizing

Diffusion

69

rotating j j target holder

batch of 4-8 AljOj

\ . \

/ / /'„ / ™

Target: pressed, sintered pellet of material to be deposited; e.g., Isotopscally enriched olivine, garnet, etc.

Figure 3. Illustration of a PLD thin film setup: (a) basic principle; (b) photo through the vacuum window of the PLD vacuum chamber in Bochum during a deposition process. Note the strongly directed bright plasma (green in color in visible light), which is formed here from an olivine target.

thin film. Both single crystals and polycrystalline pressed pellets have been used as targets. The ablation process induces a plasma plume of the target composition perpendicular to the target surface. The plume propagates into the vacuum chamber (Fig. 3b) and impinges on the chosen substrate a few centimeters away (in mineralogical applications this is typically a polished and oriented single crystal). A series of laser pulses transfers the desired amount of nominally stoichiometric material from the target to the substrate. For a given material a number of process parameters still must be optimized, but the most critical parameters are the wavelength and the energy flux of the laser pulse into the target, because this determines the amount of energy absorbed per volume of target material. The combination of the very short, high-energy pulses and strongly efficient absorption of this energy by the target leads to a non-equilibrium process that avoids thermal fractionation effects. Typical energy fluences for PLD are 0.1 - 10 J/crrr, which is the optimum range for most materials. A specific threshold fluence exists for each material: above that threshold a value that is too low or too high can lead to non-stoichiometry of the film (e.g.. Schou 2010). From experience with laser-ablation ICP-MS. many geoscientists know that wavelengths in the UV range are most appropriate to ablate silicates, carbonates and also most oxides. The setup in Bochum uses an excimer laser (Lambda Physik LPX305i) that can be operated at three different wavelengths (193. 248. and 351 nm) with pulse energies usually of hundreds of mj. Experience at Bochum over the last decade has demonstrated that for deposition of most silicates a wavelength of 193 nm is optimal. A wavelength of 248 nm often yields thin films characterized by micrometer-sized droplets of quenched molten material from the target (Dohmen et al. 2002a) called splashing (e.g., Chen 1994). Because this problem can now be avoided, PLD provides a way to produce thin films of silicates and oxides that have minimal surface roughness (< 1 nm; Dohmen et al. 2002a). However, the directed plume produces a non-uniform deposition rate with an angular distribution, which can be described by a cos"9- relationship where the angle 9- is measured normal to the target surface (e.g., Saenger 1994; Dohmen et al. 2002a). This distribution leads to a central deposition area of at least -4x6 mm. within which the film thickness varies by no more than a few %. This allows simultaneous deposition on several mm-sized crystals that will have almost the same film thickness; one of these can be kept as a reference sample to measure the initial thickness and composition. The deposition rate is roughly constant, and typical deposition rates are slightly less than 0.1 nm/pulse. For pulse frequencies of 5 - 20 Hz and laser fluencies of a few J/cm2, this means that a 100-nm layer of olivine (for example) is produced within a minute or two. The calibration of the deposition rate is relatively simple. If a substrate is used that has a higher reflectivity than the thin film (nmm < nsub), then destructive and con-

70

Watson

&

structive interference between the light reflected at the substrate film interface and at the thin film surface imparts color to the film. A silicon wafer or even A1 foil are suitable substrates for gauging the thickness of a thin silicate or oxide film from its color. The specific color depends on the difference in the optical path length (for a normal incidence of light given by 2« ri]n // (i]m ). On larger substrates interference color fringes appear due to the continuous thickness variation of the thin film (Fig. 4). By comparison with the Michel Levy color chart used for polarization microscopy, the retardation, g, can be inferred from the color. The thickness of the layer is then given simply by the ratio g/2nmm. In this way a very good estimate of thickness and thickness distribution can be made. This technique is also useful for identifying the optimal location in the vacuum chamber to obtain the highest and most uniform deposition rate.

Dohmen

20 mm Figure 4. Si wafer of one inch diameter with interference color fringes (see color image online) from a thin film made from an anorthite pellet. The gradual change in the grey scale indicates the gradual change in the film thickness here from about 150 nm in the central area down to about 50 nm.

For depositions at room temperatures the films are expected to be amorphous, which has been documented, for example, by Dohmen et al. (2002a) for olivine and by Marquardt et al. (2010) for YAG. In most PLD setups the crystal substrates can be heated in situ before, during, and/or after the deposition. An effective way to achieve this heating is by placing the substrates on thin (~1 mm) electrical insulators (e.g.. S i 0 2 glass) positioned above heating elements in the vacuum chamber. Heating during or after deposition may produce crystalline or even epitaxial thin films. Heating before deposition can be used to "prepare" the substrate by evaporating adsorbed components, particularly water. The latter treatment has been shown to be essential for producing geometrically well defined diffusion couples because heating a couple with trapped volatiles leads to degassing and bubble formation within the layer. Additional technical refinements used in materials science are summarized by Chrisey and Hubler (1994) and Norton (2007); these include: (i) background gas (e.g., 0 2 ) to form specific molecules or to lower the kinetic energy of the plasma species (this may be required to ensure stoichiometry and epitaxial growth when the film is deposited on a heated substrate); (ii) rotation of the substrate to produce a more homogeneous layer thickness: (iii) real-time characterization of the thin film using reflection high-energy electron diffraction (RHEED) to control the crystallinity and the smoothness of the surface; (iv) real-time monitoring of film thickness using oscillating quartz crystals; and (v) in situ production of multi-layers using an automatic stage to change ablations targets. Only the first of these techniques has been implemented in the setup at Bochum, but the possibilities for future adoption in mineralogical studies are clear.

Application of PLD to diffusion studies - examples Pulsed laser deposition provides a convenient way to produce well-defined contacts between two or more different solids. The possibilities are numerous, and a number of different thin-film geometries have already been produced—including ones with double layers. Below we present a few examples of diffusion studies that make use of various thin film configurations. We focus here on the basic design principles of the experiments as well as the potential pitfalls in the determination of diffusion coefficients. With respect to the latter, it is important to characterize the thin film diffusion couples both before and after the diffusion anneal. Information on the

Experimental Methods for Characterizing

Diffusion

71

surface morphology can be directly obtained by using reflected light microscopy, white-light interference microscopy, scanning electron microscopy ( S E M ) , and. in particular, atomic force microscopy ( A F M ) . In addition, the novel technique focused ion beam thinning ( F I B ; Wirth 2 0 0 4 ) enables preparation o f thin slices ( ~ 100 nm) o f the thin film samples at a desired location. These cross sections can be further investigated with a common S E M or preferably with a T E M that provides information on the geometry, crystallinity, and microstructure o f the film. Diffusion profiles could also be analyzed on the cross section using analytical transmission electron microscopy [ A T E M ] (Meissner et al. 1998; Dohmen et al. 2 0 0 2 a ; Marquardt et al. 2 0 1 0 ) with a spatial resolution o f about 2 0 - 5 0 nm. or the new generation o f electron microprobes with a field emission gun ( F E G probe). More commonly, however, R B S and S I M S have been used to measure depth profiles o f the elements or isotopes o f interest. Single l a y e r c o n f i g u r a t i o n s To date. P L D thin films have been used mainly to measure crystal lattice and grain boundary diffusion coefficients in the classical configuration (Fig. 2a) where a film is deposited on an oriented single crystal (e.g., Dohmen et al. 2 0 0 2 a . b . 2 0 0 7 ; t e r H e e g e et al. 2 0 0 6 ; Chakraborty et al. 2 0 0 8 ; Dohmen et al. 2 0 0 9 ) , a bi-crystal (see Fig. 5). or a polycrystalline aggregate (Dobson et al. 2 0 0 8 ; Shimojuku et al. 2 0 0 9 ) . I f a polycrystalline aggregate or a bi-crystal is used this enables simultaneous investigation o f diffusion in the crystal lattice and in grain boundaries (Dohmen and Milke 2 0 1 0 , this volume). T h e strategy in this type o f experiment is to create a single-phase diffusion couple in which the film serves as a source o f components for either chemical diffusion or isotopic exchange. Examples o f the former include a fayalite source to measure F e - M g interdiffusion in olivine (Dohmen et al. 2 0 0 7 ) , and T i 0 2 enriched in S m 2 0 3 , N b 2 0 5 and T a 2 O s to measure effective tracer diffusion o f Nb, Ta, and S m in rutile (Dohmen et al. 2 0 0 9 ) . Isotopic exchange examples include use o f films containing 2 9 S i and l s O to measure Si and O self diffusion in olivine (Dohmen et al. 2 0 0 2 b ) , ringwoodite. wadsleyite (Shimojuku et al. 2 0 0 9 ) , and M g perovskite (Dobson et al. 2 0 0 8 ) . A particular strength o f P L D is that it readily accommodates simultaneous doping with several isotopes or trace elements, which enables a direct comparison o f the diffusivities for various elements and enhances the accuracy o f the relative diffusion coefficients. This made it possible, for example, for Dohmen et al. ( 2 0 0 9 ) to reproduce a relative difference o f only 5 0 % in the diffusion coefficients o f the geochemical

V

\

I

b

pWinu"»

tbiSpm:

Yb-XAG

- interfacegrain b o u n d a r y

200 nm

200 nm

V

Figure 5. Bright-fiefd TEM images of YAG bi-crystals deposited with (Yb, J 7 Y| S3)Al50|i before and after annealing at 1450 °C for 2 hours (a and b, respectively ). The film is perfectly epitaxial and the grain boundary continues through the initially amorphous layer. Unpublished images kindly provided by K. Marquardt.

72

Watson & Dohnen

twins Nb and Ta. This difference is small by some standards—and would be difficult to quantify in independent experiments—but the Dohmen et al. (2009) results demonstrate unequivocally that Nb and Ta can be fractionated significantly by diffusion. In these experiments the thin film was also enriched in Sm. which was shown to be relatively immobile and therefore was used as a tracer to identify analytical artifacts due to convolution effects. Use of a single-phase diffusion couple within its stability field ensures that the film portion of the couple acts purely as a diffusion reservoir. Undesired chemical gradients in tracer diffusion studies are avoided using this approach. A potential downside is that, because the film is initially amorphous, it will tend to re-crystallize during the diffusion anneal. This may lead to problems in the profile measurement and fitting in two ways. First, the chemical potential (|a,) of a diffusing component i within an amorphous layer is different from that in a crystalline medium of the same composition. If the crystallization rate is slow compared to the total run duration, the gradient in (.i, may change or even be inverted during the diffusion anneal, leading to complex evolution of the diffusion profile. In such a case, determination of the diffusion coefficient would not be straightforward (but complications would be recognizable, at least, from a time series). The second complication that may arise due to crystallization of an amorphous film is that different materials recrystallize in different ways (see examples in Fig. 6). This leads to a unique microstructure and geometry of a given film, which, in the case of polycrystalline films, also varies with time during a diffusion anneal due to grain growth. The microstructure of the thin film may complicate depth profiling by RBS or SIMS, because the effective depth resolution depends on the roughness and thin film/substrate interface irregularity. Consequently, the

Figure 6. SEM images of various annealed thin films (top view) on single crystals of clinopyroxene: (a) Sample Cpxlc32, - 1 0 0 nm cpx film, 1306 °C, 10 min; (b) Sample Cpxlc56,'~120 nm ol film, 1105 °C, 4 days; (c) Sample Cpxlc35, - 1 0 0 nm cpx film, 1200 °C, 18 hours; (d) Cpxlc51, - 2 0 0 nm ol film, 1200 °C, 1.5 hours. [Note: The film thicknesses indicated above are those of the initially amorphous layer as measured from the reference sample, which was deposited together with the individual sample.]

Experimental Methods for Characterizing Diffusion

73

resulting diffusion profile may be strongly affected by convolution effects as illustrated in the "Fitting Uncertainties" section, and this could lead to large uncertainties or even erroneous diffusion data. [Note: The powder-source method combined with profiling by RBS or NRA (see "The Powder-Source Technique" section) is less susceptible to this effect, because, within limits, roughness on the surface of an experimental run product—as opposed roughness at the interface of a diffusion couple—has little effect on the RBS or NRA spectrum]. With respect to the two points discussed above, olivine thin films are almost ideal. They recrystallize to a polycrystalline layer of uniform microstructure and thickness, thereby providing a geometrically well-defined diffusion couple (Dohmen et al. 2002a). The time required for complete recrystallization depends upon temperature, of course, but observations also indicate that the crystallization rate increases with the fayalite content. For example, a Fo 80 layer is fully crystalline after only a few minutes at 1000 °C. At lower temperatures, however, the crystallization rate for such an olivine becomes more sluggish relative to the time scale of FeMg diffusion, which has the effect of inverting the chemical potential gradient of the diffusant between the thin film and the crystalline substrate during the initial stage of the experiment [this inversion has been detected by RBS (unpublished observation by RD)]. This problem was solved by Dohmen et al. (2007) for experiments between 700 and 900 °C by making a film with a much higher fayalite content (Fa70) than that of the substrate (Fa10). This has two advantages: (i) recrystallization of the film is accelerated; and (ii) the chemical potential of the fayalite component in the film is always higher than in the substrate, even when the film is amorphous. Sluggish recrystallization appears to be even more problematic for experiments with pyroxene thin films at temperatures below 1000 °C. This may be due to the higher silica content of pyroxenes relative to olivines, which results in a lower diffusion rate within the amorphous phase. For this reason, ter Heege et al. (2006) and Chakraborty et al. (2008) used Fo 30 thin films to measure FeMg interdiffusion in orthopyroxene (opx) and clinopyroxene (cpx). Olivine coexists stably with opx or cpx at the experimental conditions, and Fe-Mg partitioning between olivine and either pyroxene is nearly ideal—i.e., Kn is fairly close to one (e.g., Perkins and Vielzeuf 1992; Von Seckendorff and O'Neill 1993). The latter point ensures that the chemical potential gradient always has the same sign during the diffusion anneal. The olivine layer is easily recognized by RBS and, as shown by Dohmen et al. (2007) for Fe-Mg diffusion in olivine, an Fe depth profile can be extracted from the RBS spectrum (Fig. 7). Because of these advantages, a consistent data set of data for Fe-Mg diffusion in both opx and cpx were obtained using olivine thin films (ter Heege et al. 2006; Chakraborty et al. 2008). Vogt (2008) demonstrated that thin films having a spinel composition [(Mg,Fe)Al 2 0 4 ], are even better suited than olivine, which was confirmed by Marquardt et al. (2010) for Yb or Nd-doped YAG. Simple oxides also perform very well as thin films: e.g., Ti0 2 (Dohmen et al. 2009) and Zr0 2 (Dohmen 2008). Epitaxial growth was actually observed in the case of thin films on a YAG bicrystal where the grain boundary in the substrate was continuous through the film at the end of the experiment (Fig. 5). Sluggish recrystallization rates appear to be more problematic for experiments with pyroxene at lower temperatures. An additional problem related to higher chemical potential of the diffusant within the substrate than in the thin film [item (ii) in the preceding paragraph] arises in experiments designed to measure the small diffusivities governing Si in cpx, which requires temperatures approaching the melting point of cpx. The silicate chains in cpx may be the reason for pronounced anisotropic growth leading to isolated micrometer-sized idiomorphic crystals on the surface (Figs. 6a,c). In thinfilm terminology, this type of growth mechanism is called island growth (Venables 2000). Fast surface diffusion may ensure a homogeneous surface concentration on the large single crystal, but the islands (up to a few |im high) produce strong convolution effects in the depth profile. Again, olivine thin films were used as the diffusant source, but even the relatively smooth olivine films show considerable roughness (Figs. 6b,d), which generally increases with increasing temperature due to enhanced grain growth. Quickly annealed samples were

Watson &

74

Dohnen

distance from thin-film/single-crystal interface (nm) Figure 7. Fe concentration depth profiles from olivine thin film/orthopyroxene diffusion couples obtained from the RBS spectra using the software RBX (Kotai 1994). Shown are profiles from an un-annealed reference sample and a sample annealed at 900 °C for 73 hours. For opx with 12 mol% Fs a Fe-Mg diffusion coefficient of 5.8xl0~ 21 n r / s is obtained from the best fit.

used to calibrate the effective depth resolution, which was considered in the profile fitting following the procedure illustrated in the "Fitting Uncertainties" section (see Fig. 8). During the short anneals diffusion is insignificant, so any smearing of a sharp concentration step is related to the roughness of the sample and other processes such as atomic mixing effects arising f r o m the ion bombardment during sputtering. Thin films deposited by P L D have also been used successfully in diffusion experiments run in high- and ultra-high pressure experiments such as the piston-cylinder apparatus (Costa and Chakraborty 2008) and a multi anvil setup (Dobson et al. 2008: Shimojuku et al. 2009). An inert material on top of the thin film can ensure the stability of the film source and a low roughness of the surface even at extreme conditions. Metal capsules (Jaoul et al. 1995: Bejina et al. 1997), Au foil (Shimojuku et al. 2009). or an additional thin film of inert material like ZrC>2 (Costa and Chakraborty 2008) have been used for this purpose. The effectiveness of an inert layer was demonstrated by Shimojuku et al. (2009). who used this technique to limit the roughness of a wadsleyite sample surface to - 1 0 nm during an experiment in a multi-anvil device at 16 GPa and 1400 °C for 50 hours.

Double layer configurations For high-pressure experiments on olivine, some workers have deposited an additional layer of Z r 0 2 on top of the diffusant source to minimize any interaction with the chemical environment and thus limit dissolution/precipitation processes. Double-layer ("sandwich") setups have also been implemented to study processes other than diffusion in the substrate, including reaction rim growth (Milke et al. 2007) and element exchange mediated through an inert polycrystalline matrix (e.g., Dohmen 2008). In their study of the growth kinetics of enstatite reaction rims, Milke et al. (2007) deposited a 300-500 nm thick olivine layer on top of a 20-100 nm enstatite layer, which itself had been deposited on quartz single crystal (see Fig. 9). The experimental protocol involved two different deposition and annealing stages to prepare uniform polycrystalline thin films of olivine and enstatite as the starting material for the diffusion anneal (Fig. 9a). During diffusion anneals at temperatures between 1000 °C and 1300 °C the enstatite layer grows continuously at the expense of the olivine layer as shown in Figures 9b-d. Because the enstatite layer is present from the beginning, possible nucleation problems with this phase are avoided. Two additional advantages of this thin-film configuration

Experimental Methods for Characterizing Diffusion

1.4

1

1

75

1—

• Cpxlc51-5: 1200°C, 1,5 hours A Cpxlc51-3: 1250°C, 4 days

1.2 1.0

fsj 28

08

-«

Si 0,6

clinopyroxene single crystal

•

0.4

0.2

_ olivine thin film

distance from thin-film/single-crystal interface (nm) Figure 8. Depth profiles of the 29 Si/Si fraction measured with SIMS on two clinopyroxene single crystals deposited simultaneously with a 29 Si-enriched olivine film: Sample Cpxlc51-5, 200 nm, 1200 °C, 1.5 hours; Cpxlc51-3, 200 nm, 1250 °C, 4 days. The brief anneal of sample Cpxlc51-5 served only to recrystallize the olivine layer. Diffusion of Si was insignificant during this anneal, so the profile was used to calibrate the depth resolution of SIMS, also taking into account the effects of sample roughness and thin-film/substrate interface irregularity. The corresponding fit using Equations (8) and (9) assuming a sharp compositional jump (dotted line) at the interface is indicated as the dashed line. The solid line is the fit to the depth profile of Cpxlc51-3 using the calibrated depth resolution of a = 50 nm and the analytical solution given in Equation (5). The best fit gives a diffusion coefficient for Si tracer diffusion in cpx of 1.7x10~2° m 2 /s.

initial state (reference sample) QzOI25: 1200°C, 30 m in

QzOI24: 1200°C, 160 min

QzOI27: 1200°C, 360 min

Figure 9. Bright-field TEM images of the cross sections of four thin-film samples prepared to investigate the kinetics of enstatite rim formation: a reference sample showing the initial stage before the diffusion anneal and three samples annealed at 1200 °C and an/02 = 10 -1 P a f ° r different durations. For more details see Milke et al. (2007). [Used with kind permission from Springer-Science+Business Media: Fig. 4 from Milke et al. 2007.]

76

Watson & Dohmen

over classic rim growth experiments are worth noting (see also Dohmen and Milke 2010, this volume). First, as in the case of studies addressing purely diffusion phenomena, the investigation of the reaction/diffusion processes on the nm scale makes possible the acquisition of data at lower temperatures. In the case involving the growth of enstatite rims (Milke et al. 2007), it was possible to compare rates obtained at a given temperature from experiments performed "wet" at 0.1 -0.2 GPa (and yielding micrometer-scale growth) with those obtained under completely "dry" conditions at 0.1 MPa (yielding only nm-scale growth). This comparison documented unequivocally the dramatic effect of even traces of water on the diffusivity of elements at grain boundaries (about four orders of magnitude) and the corresponding acceleration of reaction rates. Note that increasing the pressure often has the opposite effect (compare with Zhang 2010, this volume) and thus cannot account for this observation. The second advantage of the thin-film technique for measuring reaction rates arises from the fact that growth of a reaction rim or corona is a coupled diffusion process controlled by the mobile chemical components of the system (see references in Dohmen and Milke 2010, this volume). In principle, the rate-controlling component(s) may be unique to the specific system under study. By simultaneous doping of the reacting top layer with stable isotopes, the most mobile components can be identified (Abart et al. 2004; Milke et al. 2007). Isotopically enriched materials can be very expensive, so an added practical advantage of combining isotopic doping with the thin-film technique is that thin films use very little material. The double-layer or sandwich configuration can also be used in instances where an intervening polycrystalline layer acts only as a passive medium for the exchange of elements between the top layer and the substrate (Fig. 10a). This setup was used by Dohmen (2008) to study Fe-Mg exchange by grain-boundary diffusion through inert, polycrystalline Z r 0 2 layers (note that these experiments have conceptually similarities to the detector-particle approach described in "The Detector-Particle Method for Studies of Grain-Boundary Diffusion" section). In this case the top layer was a fayalite-rich olivine (a source for Fe, a sink for Mg), intended to exchange Fe and Mg with a single crystal of San Carlos olivine (a sink for Fe, a source for Mg) mediated through the inert Zr0 2 layer. This model system served as a prototype for studying the kinetics of exchange reactions between two minerals sitting in a polycrystalline matrix—a typical situation for many rock types. The depth profiles of the sample were collected using both RBS and SIMS, which gave consistent results. Diffusive transport through the polycrystalline Z r 0 2 layer could be identified, but grain-boundary transport was not efficient because local equilibrium between the olivine thin film and the surface of the olivine single crystal was achieved during the course of the exchange reaction, thus eliminating the chemical potential gradient across the Z r 0 2 layer. This kinetic behavior can be classified as rate-controlled by combined grain-boundary and lattice diffusion (Dohmen and Chakraborty 2003). Because of this, Dohmen (2008) was able to obtain the time-integrated diffusive properties of the inert layer from fitting of the concentration depth profiles for Fe and Mg (Fig. 10b). In general, the configuration shown in Figure 10a allows characterization of the diffusion properties of the inert "sandwiched" film as long as the transport through the layer governs the bulk rate of exchange. Such a setup would be particularly useful to measure the diffusivity of incompatible elements that strongly segregate into the grain boundaries (e.g., Watson 2002; Hiraga et al. 2003; Hayden and Watson 2007, 2008). An appropriate source and sink for the element of interest must be chosen as the top layer and substrate; the thickness of the middle layer can be varied to confirm that transport is controlled by diffusion through the layer. Many of the considerations of this double-layer strategy are similar to those pertaining to the detector-particle method (see "The Detector-Particle Method for Studies of Grain-Boundary Diffusion" section). Similarities to the both the single- and double-layer techniques can also be found in the approach used by Hwang et al. (1979) to characterize grain-boundary diffusion of Ag in Au.

Experimental Methods for Characterizing

Diffusion

77

Figure 10. (a) Basic principle of the setup to examine element exchange along the grain boundaries of an inert layer. Depth profiles of the Fe content (b) and the mole fraction of Fe (c) for three olivine-ZrOi-olivine diffusion couples in direct comparison. The solid lines are best fits using a diffusion model as described in Dohmen (2008). The fitting also includes the modeling of the convolution effect as calibrated by the reference sample.

These workers deposited an epitaxial Ag film on a polycrystalline Au layer (substrate) and annealed the setup at temperatures where grain-boundary diffusion was effective but lattice diffusion was not. Silver atoms diffused through the Au layer along grain boundaries and accumulated on the opposite side, where they were detected by Auger spectroscopy. This. then, was a source-sink experiment in which an epitaxial layer served as the source of diffusant (Ag) and the free surface on the opposite side of the diffusion medium of interest (polycrystalline Au) served as the sink. The mathematical framework for interpreting the results of these experiments was provided in a companion paper by Hwang and Balluffi (1979). In summary, the examples of double-layer setups described above effectively illustrate the power of the PLD method. Researchers now have the opportunity to study not only the kinetics but also, in principle, the equilibrium properties of multi-phase systems at a scale that was not possible previously. Clearly, we are still at the beginning of this endeavor and have not yet realized the full potential of this versatile method.

78

Watson & Dohmen THE POWDER-SOURCE TECHNIQUE

Overview and history The powder-source technique does not rely upon new or sophisticated technologies at the execution stage of an experiment, so its application to the study of diffusion in crystals pre-dates that of the thin-film method implemented using PLD. As its name suggests, the powder-source technique involves the use of finely-ground crystalline material as the source of diffusant in contact with the surface of a large single crystal of interest. The strategy differs from the thinfilm approach described in "The Thin-Film Method and Pulsed Laser Deposition" section in that the source is effectively infinite rather than deliberately finite. Early applications of the powder-source approach in geochemistry involved studies of FeMg interdiffusion in olivine (Buening and Buseck 1973; Misener 1974), and were conducted by packing millimeter-sized natural olivine crystals in powder sources consisting of either MgO (Misener) or synthetic Fe olivine. The samples were held at high temperatures (up to 1400 °C in the Misener study), which resulted in FeMg exchange between the olivine crystals and the powder source surrounding them. In this case, because Fe secondary collisions O

primary knock-on atom (PKA)

Figure 16. Schematic illustration of the interactions between ions and target material. The sequence of events following an elastic (nuclear) collision is depicted in (a) at an atomic scale (after Dearnaley et al. 1973). These collisions create Frenkel defects, as discussed in the text. A more general view is shown in (b), where the distinction is made between low doses creating non-interconnected damage regions and high doses leading to overall amorphization of the target (after Nastasi et al. 1996).

Watson & Dohmen

90

resulting from implantation of the diffusant might affect the measured diffusion coefficient. This depends on the extent and nature of the damage and its thermal stability relative to the effectiveness of the diffusion process of interest. It is clear that introduction of heavy-atom diffusants (e.g., Xe, Pb) by implantation can affect the resulting diffusion coefficient in at least some minerals. In pioneering studies of noble-gas diffusion in minerals, for example, Melcher et al. (1981, 1983) implanted Xe in feldspar and olivine and profiled the annealed samples by RBS. These studies demonstrated that retention of implanted Xe depends on the implantation dose for both minerals. The same is true of Pb in zircon. Cherniak and Watson (2001) used the powder-source technique to revisit the topic of Pb diffusion in zircon that had been explored previously by Cherniak et al. (1991) using ion-implanted Pb. Comparison of the two data sets reveals that the Arrhenius relation from the implantation study lies well above the powder-source line, with the two converging only at T > 1500 °C. In their powder-source study, however, Cherniak and Watson (2001) also reported the results of experiments done by hot implantation of Pb at 800 °C (the 1991 implantation results were obtained from samples implanted at room temperature). Maintaining the sample at elevated temperature during implantation enables constant healing of damage as mobile displaced atoms are able to return to normal lattice sites. For the case of zircon specifically, 800 °C is hot enough for the lattice to resist amorphization even at quite high doses of heavy ions (Wang and Ewing 1992; Weber et al. 1994); other materials will have different critical amorphization temperatures depending on bonding and other crystal characteristics. Cherniak and Watson (2001) were able to show that diffusion results in the 1200-1350 °C range obtained from hot-implanted samples are consistent with those obtained by the powder-source method. In any given diffusion study conducted using an ion-implanted source, care must be taken to ensure that the results reflect diffusion in the material of real interest, which is usually the undamaged crystal lattice. If lattice damage is introduced at the implantation stage, then subsequent diffusion anneals may involve the simultaneous occurrence of two separate kinetic processes: healing of implantation-induced lattice damage and diffusion of the implanted species. In some instances, diffusion in radiation-damaged material may be the process of actual interest. In this context, we note that ion implantation can be used to simulate the damage caused by natural radioactive decay: for example, heavy Xe ions of appropriate energy can cause lattice damage (atom displacements) similar to those resulting from a-recoil in Uand Th-bearing minerals. In principle, ion implantation thus provides the means to introduce a controlled amount of lattice damage—albeit over a relatively localized region within the crystal—and evaluate the consequences for diffusion.

THE DETECTOR-PARTICLE METHOD FOR STUDIES OF GRAIN-BOUNDARY DIFFUSION Context and history Compared with the other techniques described in this chapter, the detector-particle method for grain-boundary diffusion is relatively new: in fact, it is still undergoing development to enable extraction of more quantitative information. Some of the ongoing refinements are described here for the first time. Because of its wide-ranging technological and scientific applications, the study of grainboundary diffusion has a long history that has led to both a robust mathematical basis (e.g., Fisher 1951; Whipple 1954; Harrison 1961; Leclaire 1963) and numerous experimental measurements (see Dohmen and Milke 2010, this volume). A conceptually valuable classification of grainboundary diffusion into discrete "types" (kinetic regimes) was provided by Harrison (1961) and augmented by Mishin and Herzig (1995), as discussed by Dohmen and Milke 2010 (this volume; see their Fig. 5). The case of type C grain-boundary diffusion involves essentially

Experimental Methods for Characterizing

Diffusion

91

no diffusion in the lattice (i.e., transport occurs in grain boundaries only)—a situation that could arise either (transiently) because the grain-boundary diffusivity (Dgh) is overwhelmingly larger than the lattice diffusivity (Aar) o r because the diffusant of interest is highly incompatible in the grains (for more quantitative constraints, see Dohmen and Milke 2010, this volume). This situation may be commonplace in geological settings, including instances such as grainboundary diffusion of siderophile or large-ion lithophile elements in the mantle. Diffusion of rare-earth and high field-strength elements along grain boundaries of major crustal phases such as quartz and feldspars is a plausible crustal scenario. Experimental characterization of type C kinetics presents a particularly difficult problem because the existing 2-D solution to the non-steady state diffusion equation developed for grain-boundary diffusion (e.g., that of Leclaire 1963) assumes uptake of diffusant in the crystal lattice. In the case of type C kinetics, in contrast, the diffusant remains exclusively in the grain boundaries essentially by definition. Accordingly, for a grain boundary resembling a thin, infinite slab and having a source of diffusant uniformly distributed along one edge (see Fisher 1951; Leclaire 1963; Dohmen and Milke 2010, this volume), transport along the boundary conforms to 1-D diffusion into a semi-infinite medium. However, the lack of diffusive "leakage" into the contacting grains means that traditional grain-boundary diffusion experiments involving type C kinetics require an analytical technique capable of quantitative determination of diffusant at challenging length scales and/or very low concentrations (see Dohmen and Milke 2010, this volume; Farver and Yund 2000). The detector-particle approach: general considerations and examples Watson (1986) put forth an alternative approach for characterizing type C grain-boundary diffusion—logically called the "detector-particle" method—which is contrasted with the type B case in Figure 17. The approach was originally conceived for the purpose of quantifying oxygen transport along grain boundaries in fluid-absent rocks. Strictly speaking, this is not the best geochemical example of type C kinetics (because oxygen is compatible in silicate minerals), but grain boundary diffusion of oxygen was expected to be many orders of magnitude faster than lattice diffusion under dry conditions. The detector particles in Watson's (1986) experiments were CuO or Fe 2 0 3 grains dispersed in synthetic dunite or quartzite. The synthetic rocks were encapsulated in graphite or Fe metal, and grain-boundary diffusion of oxygen was assessed from the width of the zone in contact with the container where the detector particles were reduced to Cu or FeO. Watson (2002) later used the detector method to quantify transport of incompatible siderophile elements (Pt, Pd, Au, W) along grain boundaries of mantle rock analogs (MgO and forsterite). The extreme incompatibility of these elements in oxide minerals precludes the use of conventional grain-boundary diffusion experiments. The proposed alternative approach was to synthesize polycrystalline MgO samples containing dispersed small particles in which the diffusants of interest were compatible, even though they were excluded to the point of being immeasurable in the major-phase (MgO) grains. If elements of interest—say Pt and Pd—are mobile along MgO grain boundaries, then small Pd particles dispersed in polycrystalline MgO should "communicate" (i.e., form alloys) with distant Pt particles by diffusion along MgO grain boundaries, provided diffusion in the particles themselves is sufficiently fast (see the "Numerical Simulation: Constant-Surface Model" section). A sectioned sample and analytical results from the original experiments of Watson (2002) are shown in Figure 18. To date, the main application of the detector-particle method has been to estimate grainboundary diffusivities of 10 siderophile elements in polycrystalline MgO (Hayden and Watson 2007) and also of carbon in MgO and dunite (Hayden and Watson 2008). In these studies, chemical potential gradients were set up in experimental samples by placement of mutually soluble "sources" and "sinks" separated by dense polycrystalline aggregate of the bulk material of interest. In most instances the sources and sinks were small ( - 1 0 jam) metal particles, but

Watson & Dohnen

92 conventional: Type B kinetics crystal

» t

» t+

4

crystal

detector particle approach crystal

Figure 17. Characterization of grain-boundary diffusion in the "type B" regime (top) relies upon uptake of the diffusant of interest in the crystal lattice (small flux arrows). The detectorparticle strategy (bottom) addresses situations where the diffusant is incompatible in the lattice and therefore immeasurable in the bulk sample. A particle of another phase in which the grainboundary diffusant is compatible is placed on the grain boundary; diffusion in the boundary is estimated from the diffusant uptake in this particle. See text for further discussion.

crystal

Figure 18. Backscattered-electron images of polycrystalline MgO from a 5-hour piston-cylinder experiment at 1600 °C and 2.3 GPa. The sample contains two horizons of metal particles, initially pure Pt and Pd. that have alloyed at run conditions by grain-boundary diffusion in the MgO. Images are from Watson (2002); see Hayden and Watson (2007) for details of similar detector-particle experiments.

metal foils and wires were also used as sinks for carbon, and massive graphite was used as a carbon source. Dohmen (2008) designed a conceptually similar approach to examine Fe-o-Mg interdiffusion in olivine and through the grain boundaries of (inert) polycrystalline Zr0 2 (see Fig. 10). In that case, the PLD technique was to deposit thin layers of Zr0 2 and Fo30 olivine on the polished surface of a single crystal of San Carlos olivine (Fosl)). Interdiffusion of the two olivine compositions occurred by transport through the Zr0 2 grain boundaries, and the grainboundary diffusivity was obtained by mathematical analysis of the Fe depth profile—obtained

Experimental Methods for Characterizing

Diffusion

93

by RBS and SIMS—in the single-crystal olivine. Dohmen was able to extract information not only on diffusion but also on the "storage capacity" of Z r 0 2 grain boundaries for the impurity diffusants Fe and Mg. In the Hayden and Watson studies, approximate grain-boundary diffusion coefficients were calculated in two ways. The simple approach was to estimate the length scale (x) over which distant particles communicated with one another by diffusion. The grain-boundary diffusivity was then computed as Dgb ~ x2/t [Note: the tortuosity was not considered in this approximate calculation; in principle, this should result in underestimation of the diffusivity by a factor of ~3 (Watson's 1991 study suggested a tortuosity of -1.7)]. Electron microprobe analysis of sink particles (e.g., Pt) at various distances from the source of a diffusant of interest (e.g., Pd metal) provided an estimate of the diffusive length scale and hence an order-of-magnitude diffusivity. The second approach used by Hayden and Watson involved conversion of diffusant concentrations in the detector particles to time-integrated grain-boundary fluxes. This strategy can be appreciated with reference to Figure 17, which depicts a single detector (sink) particle intersecting a grain boundary with a diffusant source at one end. The diffusant atoms enter the grain boundary at the source (left) and diffuse along the boundary until they encounter the sink phase at the right. The time-integrated grain-boundary flux (Jgl>) can be calculated by measuring the total number of diffusant atoms (n) accumulated in the sink over a given time; then (21)

where A is the effective cross-sectional area of the grain boundary over which the diffusant is intercepted by the sink phase (for this calculation the grain-boundary width is taken as 1 nm; Hiraga et al. 2002). The flux alone is valuable as a qualitative indicator of the effectiveness of grain-boundary diffusant transport, but additional information or assumptions are needed to calculate an actual diffusivity. Assuming the system quickly approaches a steady-state condition (see the "Numerical Simulation: Constant-Surface Model" section), the diffusivity in the grain boundary is given simply by

where (dc / dx)|

is the concentration gradient of the diffusant in the grain boundary.

Complications arise in this simplistic approach because neither the absolute concentration in the grain boundary nor (dc I dx)| is likely to be directly measurable or strictly constant. Another approach is to assume that local partitioning equilibrium exists between sink grains and grain boundaries (see, e.g., Hiraga et al. 2004; Hiraga and Kohlstedt 2007), so the diffusant concentration in sinks at various distances from the diffusant source will reflect (i.e., be proportional to) the concentration profile of the diffusant in the grain boundaries. Such a profile can be fit to a simple diffusion model as was done by Hayden and Watson (2008) to obtain an effective bulk diffusivity that has practical value in addressing geochemical problems. The effective length scale of grain-boundary diffusion is well constrained even if the absolute grainboundary diffusivities are not. The detector particle method as described above could be implemented to yield qualitative results for a wide variety of systems, the main requirements being: 1) diffusion in the detector particles themselves must be reasonably fast; 2) the diffusant of interest must partition into them sufficiently to enable measurement; and 3) the detector particles must be in chemical equilibrium with the major phases of the rock of interest. In view of the fact that many problems involving chemical transport in the Earth are completely unconstrained at present—is it plausible, for example, that the mantle and core communicate by diffusion?— even qualitative results have significant value.

94

Watson & Dohmen

Numerical simulation: constant-surface model Since the initial efforts of Watson (2002) and Hayden and Watson (2007, 2008), additional insight into the behavior of detector particles during grain-boundary diffusion has been obtained at RPI through numerical simulations. The geometry and boundary conditions are set up in a manner much like that used by Fisher (1951) in his pioneering analysis. Figure 19a is a realistic, 2-D representation of a polycrystalline aggregate containing dispersed detectors particles and a source of diffusant at the left (x = 0). The model system (Fig. 19b) contains regularly-spaced detector particles of uniform size, and the grain boundaries are in three orthogonal sets: one parallel to the diffusion direction (solid lines) and another perpendicular to these (dashed lines). The third (implied) set is parallel to the plane of the figure (i.e., in the x-y plane). Given a uniform planar source, diffusion in the x direction advances in an identical manner along parallel grain boundaries, so there are no sustained grain-boundary fluxes in the y or z directions once the y- and z-parallel grain boundaries are locally "filled" with diffusant. For this reason, the behavior of an entire aggregate can be captured with reasonable accuracy by modeling a single, tabular grain boundary as was done in the case of type B kinetics (Fisher 1951; Leclaire 1963). Such a grain boundary, depicted in Figure 20, has a finite width (5) in the }• direction, and detector particles of radius r are spaced along it at interval Ax (the distance between the first particle and the source, Axj = Ax — r —i.e., the same as the distance from the surface of one particle to the center of the next). For the numerical simulations, the grain boundary of the model system shown in Figure 20 is divided into discrete volume elements (5x = 5z; by = 5) that are allowed to interact diffusively. Each volume element is treated individually by calculation of fluxes to or from the four neighboring elements with which it shares a side. Relative to the center of the volume element under consideration at any moment, the coordinates of the volume elements with which it interacts are -5x, +5x, -8z, +5z; the node-to-node fluxes in dt are given by the product of the governing diffusivity, Dgh, and the concentration gradients, AC/Ax and AC/Az [Note that this approach amounts to an explicit finite difference method in the terminology of Crank (1975)]. The detector particles are imbedded in the grain boundary as shown in the figure, and local equilibrium between the grain boundary and the particle surface is assumed to be governed by a partition coefficient: c*f

(23)

where is the concentration of the diffusant of interest at the surface of the particle and Cgl, is the local concentration in the grain boundary near the particle surface. The latter quantity is taken as the average concentration in the grain-boundary volume elements contacting the particle (see Fig. 20). This concentration is assumed to be in equilibrium with the entire (spherical) surface area of the particle, which amounts to assuming infinitely fast diffusion around the particle surface. Diffusion within the particle itself is assumed to be governed by the lattice diffusivity, Dprh and is treated, independently for each particle, as diffusion in a sphere with a locally time-dependent surface concentration. The reader is referred to Watson et al. (2010) for more detailed discussion of the finite-difference approach using discrete volume elements. The intent here is to illuminate the essential features of type C grain-boundary diffusion as captured by concentration changes in detector particles. In order to assess consistency with actual measurements, simulations were run using durations, particle spacings and particle sizes similar to those used in ongoing laboratory experiments at RPI. Figure 21a shows the results of a 7-day numerical simulation made using plausible input parameters as indicated on the figure. The bulk concentration of diffusant (Cprt) in the first of 5 particles spaced at ~200-mm intervals along the grain boundary is shown as a function of time. For the input parameters used, Cpn depends almost linearly upon time for all values of Dgb (Ixl0~ l u to 3xl0~ 9 cm 2 /s). The linearity indicates an essentially steady-state condition, as was

Experimental Methods for Characterizing Diffusion "real" sample

A —

95

model system

'detector' particles grain boundaries

+ + + ++ t

c ro i/i

y

I

^I

t j-

I

I

^ I ! lì

i I

j

I

I

^

^

I j -

Figure 19. 2-D illustration of a "real" polycrystalline sample (a) containing dispersed detector particles, compared with a model system (b ) having a uniform grid of grain boundaries and regularly-spaced detector particles. Because all grain boundaries oriented parallel to the x direction are the same, the diffusion behavior of the bulk sample is captured by treating just one of them (e.g., the region enclosed in the dotted rectangle). See text for discussion.

single grain boundary y

.i

5oy g.b. thickness S t

computational grid

Figure 20. A single grain boundary (upper panel) isolated from the model system in Figure 19 and assigned properties and boundary conditions. The concentration in the grain boundary at the source (left) is held constant; the detector particles are assigned a specific radius and spaced at regular intervals along the boundary. The lower panel illustrates the 2-D array of volume elements in the grain boundary and imbedded particles used for numerical computation (see text and Fig. 25).

Watson & Dohnen

96

a

Dft (crc2^) particle radius: 6 um 3E-9 1E-9 6E-10 3E-10 IE-10

D prt /D gb :

0.0001

g.b, width (8):

ix;

1 nm

200 firn

Kdprt"b= 10 /

X

y ?

3 4 time (days)

days» 0.7 2.1 3.5 4.9 6.3

O O O • —•—• ÜJ •

• 10 1 vKdprt/gb . -0.1

Dgb = 1E-9 cm 2 /s

rv 1 200

2

3

4

400 600 800 distance from source (microns)

Figure 21. (a) Results of numerical simulations showing the time-evolution of the bulk diffusant concentration (Cpn) in the particle nearest the source (no. 1 in graph b) for several values of the grainboundary diffusivity, Dsll. Note that the plots are almost linear, indicating a near steady-state condition was reached early in the simulations. Graph (b) shows the time-evolution of diffusant concentration in five particles as a function of their distance f r o m the source. Three values of the particle/grain-boundary partition coefficient K/'r/sl' are considered, but the influence of this parameter is minor. Note that the main effect of increased time is to raise Cpn in the first and second particles: penetration further along the boundary is limited, as evidenced by the minimal changes in particles 3-5 with increasing time. See text for details and discussion, and compare (b) with actual data shown in Figure 22.

5 1000

assumed by Hayden and Watson (2007) in interpreting their experimental results for siderophile element diffusion in polycrystalline MgO. Note, however, that all Cprt vs. t lines intersect the time axis at positive values, especially at Dgb < 10~9 cm 2 /s. The non-zero intercept represents an initial transient during which diffusion in the grain boundary reaches the first particle. Figure 21b shows the time evolution of diffusant concentration in five particles spaced at 200-(.tm intervals for three values of KJ'r,/gh (10, 1, and 0.1). This figure demonstrates that, within the limited range of parameters examined, uptake of diffusant in the particles is insensitive to KJ"',/gb. There is essentially no difference in behavior for values of KJ'n/gb > 1, and even modest »/compatibility (KJ"',/gb = 0.1) in the particles relative to the grain boundary affects diffusant uptake in the particles by only -30%. The reason for this insensitivity to KJ'r,/gh is that the capacity of 12-(im particles to incorporate diffusant overwhelms the capacity of 1-nm grain boundaries to supply that diffusant: the overall process is regulated by the diffusive flux in the grain boundary. Despite a high diffusivity, this flux is small because the cross-sectional area of the grain boundary is minute. Figure 21b also reveals that concentration changes are most apparent in the particles nearest the diffusant source: these intercept most of the diffusant, so particles further from the source show only minor changes in concentration with time. This means that the time evolution of the overall system is characterized not by progressive penetration deeper into the bulk sample

Experimental Methods for Characterizing

Diffusion

97

(as is the case with type B grain-boundary diffusion), but by progressive steepening of particle concentration profiles like those shown in the figure. Use of the bulk-sample penetration depth (i.e., diffusive length scale) to estimate grain boundary diffusivities would yield a deceptively low apparent grain-boundary diffusivity. This principle is illustrated in Figure 22. which shows unpublished data from the RPI lab pertaining to M g - o F e interdiffusion along grain boundaries in synthetic quartzite. In this study, the surface source of Mg 2 + at one edge of the quartzite was MgF 2 and the detector particles dispersed in the quartzite were fayalite (Fe 2 Si0 4 ). The summary panel of Figure 22 clearly shows only minor increases in Mg penetration depth with increasing time, but the profile of M g O in the particles vs. depth in the quartzite steepens markedly as the experiment duration progresses from 1 to 8 days. In future experiments using the detector particle approach, the optimal strategy might be to choose particles in which the diffusant of interest is only slightly compatible—enough to result in measurable diffusant concentrations in the particles, but not so much that the particles near the source act as "infinite sinks" for the diffusant. This goal might be achieved by using particles having Kfn/sh « 1: however, a priori knowledge of K f r , / g b is not likely to be available in most cases. Detector particles in which the diffusant is sparingly (but measurably) soluble would be ideal, as suggested by our preliminary results for grain boundary diffusion of Ca in quartzite using fayalite detector particles. Figure 23 is a composite diagram from the 7-day numerical simulation shown in Figure 21b (Dgb = lxl0~ 9 cm 2 /s). The top and middle panels of this figure portray diffusant distribution 0

100

200

300

400

500 0

100

200

300

400

500

Figure 22. Resufts of a time series of experiments on FeMg exchange along grain boundaries of quartzite at 1125°C and 1.5 GPa. In a-c the time progression of the experiments is 1—>2-»8 days; the plotted data are MgO contents of fayalite detector particles (initially FeiSiOiO in the quartzite as a function of distance from the Mg source to the left (in this case a layer of MgFi). The graph in d is a summary of the three curves drawn through the data in panels a-c. Note that with increased time the slopes of these curves become progressively steeper near the source, but the depth of penetration into the quartzite changes relatively little. This evolution is exactly as predicted by the numerical simulations shown in Figure 21. The data shown are unpublished results from the RPI lab of a study described by Thomas et al. (2009). [Note that fayalite detector particles react with their quartz host when the MgO content reaches about 3 wt%, so particles with higher MgO contents are no longer fayalite but orthopyroxene.]

Watson & Dohnen

98 1.0-

\ Cgb srf c gb

0.6-

0.2-

|7days|

\

\

grain boundary

V Dprt/Dgb: g.b. width (8): f-srf . gb • Kd:

1 1E-9 cm2/s 0.001 1 nm 10000 10

Figure 23. Summary diagram from a numerical simulation showing the distribution of diffusant in both the grain boundary itself and the 6-|im radius detector particles placed along it. The top panel is an axial concentration profile (dotted line in middle panel) along the grain boundary. The middle panel is a contoured concentration surface of the entire grain boundary in which the dramatic effect of the detector particles is very clear: they cause "cones of depression" in the diffusant concentration in the grain boundary. The bottom panel show the diffusant concentrations in the particles aligned with the top panels. The input parameters for the model are shown on the figure.

within the grain boundary itself, both as an axial concentration profile and a contoured concentration surface. The reason for the effectiveness of the particles in scavenging diffusant is quite clear from these diagrams: much of the diffusant that gets past the first particle diffusing away from the source is subject to "back-diffusion" on the far side of the particle. A final numerical test was to explore the effect of the diffusivity in the detector particle itself (D prt ) on overall behavior. This was done by re-running the 7-day simulation shown in Figure 23 with the same grain-boundary diffusivity (10~9 cm 2 /s) but a much smaller diffusivity in the particles. The results are shown in Figure 24a as a comparison of bulk uptake in the first particle as a function of time for i)pt/i),,i, = 0.001 and 0.000001. Compared in this way using bulk particle concentrations, the results are nearly identical, indicating essentially no role for the diffusivity in the particle in controlling overall behavior. When compared in terms of the internal distribution of diffusant in the particles, however, the results are dramatically different (Fig. 24b). The higher value of Dpn (10~12 cm 2 /s) results in a nearly homogeneous radial distribution of diffusant. In contrast, the lower Dpn value (10~15 cnr/s) results in diffusant uptake only in the outer micrometer or so of the particle. As a practical matter of analyzing particles after an experiment (e.g., with an electron microprobe), the high-diffusivity particles are vastly superior.

Experimental

Methods for Characterizing

99

Diffusion

D gb = 1E-9 cm 2 /s

1.0-

K„ = 10

0.8 Ugb

0.6

= 0.000001 s

0.4

^=0.001

0.2

.11.

0.0-1 0

1

2

3

4

time (days)

5

6

7

0

1

2

3

4

5

6

distance from particle center ( u r n )

Figure 24. Results of numerical simulations to evaluate the effect the diffusivity in the detector particles (Dpn) on overall behavior. In (a) two cases are compared in which Dpn / Dgi, differs by a factor of 1000. The plotted data are the bulk concentration in the particle nearest the source, which is essentially unaffected by changes in Dpn. The reason for this is clear in (b), which shows radial concentration profiles within the particle after 7 days. The integrated amount of diffusant under the two curves is nearly the same, but the distribution within the high-diffusivity particle is almost uniform and that within the low-diffusivity particle is concentrated near the rim. Simulation parameters are the same as those used to generate Figure 23. See text for discussion.

A simple analysis of the detector-particle method As noted previously, Hayden and Watson (2007) assumed that a steady-state condition is established between a diffusant source and sink in order to extract grain-boundary diffusion information from their experiments. The numerical simulations described here confirm that this assumption is largely valid for a substantial range of grain-boundary diffusivities (~10~'° cm2/s or higher; see Fig. 21) and dimensions typical of experimental systems. This warrants taking the detector-particle method one step further in order to facilitate design of future experiments and extraction of diffusion data. We begin with the initial assumption that the number of diffusant atoms captured by a detector particle intersecting a grain boundary in time t is

C(*=0fz) Figure 25. Reference illustration of the 1-particle system used to arrive at Equations (29) and (30). showing the spatial coordinates, concentrations, and key dimensions A, d. Ax and A.ï|. The arrows are approximate flux vectors within the grain boundary, whose width d is greatly exaggerated. See text for discussion.

where A = Ax-5 (see Fig. 25). An underlying assumption here is that diffusion in the grain boundary—not diffusive uptake in the particle—is the rate-limiting process in the overall behavior. Numerical simulations using varying ratios of Z>s;/D;„t indicate that this will be the case for all plausible ratios (see the "Numerical Simulation: Constant-Surface Model" section and Fig. 24). This conclusion is somewhat counterintuitive, but it is important to remember that it is not the diffusion coefficient alone

100

Watson & Dohmen

that determines the flux through a given medium or path: the volume of the medium and the diffusant concentration in it are equally important in determining the total flux. The capacity of ~10-|^m detector particles to take in even modestly compatible elements is very large relative to the "carrying capacity" of grain boundaries only ~1 nm wide. The steady-state assumption allows us to describe the source-to-particle flux along the grain boundary as the product of the grain-boundary diffusivity, Dgh, and the concentration gradient in the boundary: J g =-D "

C - CsSTC —^ — Ajq

(25) ( }

Combining Equations (24) and (25), we obtain D

C - Csr s !

,

1 A

^-tAxj

A

= - C ,rV p r t

(26)

If the particles are small relative to the distance between them, then Axj » Ax, so ^ ( C , i > - c ; t > f - 8 = -C;„TV;„t

(27)

Assuming the concentration in the grain boundary is low where it contacts the particle, then

DghC^-t-d^-Cprypr,

(28)

so 4

C r Cprl •-V V pri P«'^ o prl _ D _ p" p" 5J = s " C -6-, C cw

(29)

and

DSh • C"l S •8•?

\ \

— nr

(30)

3

At first glance, this simple result is encouraging: detector particle properties can be chosen to optimize the chances that Cpn will be measurable in an experimental run product. There is, however, a serious obstacle to accurate calculation of Dgb: C'^—the concentration of diffusant in the grain boundary at the source—is truly knowable only by direct measurement, and concentrations within grain boundaries can be extremely difficult to measure (see, e.g., Hiraga et al. 2003). Indeed, the main reason for implementing the detector-particle method is to avoid the challenge of quantifying concentrations in grain boundaries. It is nevertheless encouraging that the grain-boundary concentration must be measured only near the source where it is highest. It should be noted, in addition, that C'l^ can be expressed as Csrc / KsJc'sh, where Csrc is the diffusant concentration in the source material itself. Under some circumstances the source will be a phase that forms a solid solution with the detector particles, as in the experiments of Hayden and Watson (2007), in which case KsJclsb ~ K^T"sh. An approach similar to that developed by Hiraga and Kohlstedt (2007) might be used to estimate Kd, thereby constraining Csrc. Concluding remarks on detector particles Throughout this section we have assumed that the interaction between grain boundaries and detector particles occurs only by diffusive exchange—that is, the detector particle remains in equilibrium with the dominant phase in the system despite diffusive fluxes through the grain boundaries. This assumption is probably realistic for the experiments described by Hayden

Experimental Methods for Characterizing Diffusion

101

and Watson (2007, 2008) in which the diffusants of interest were metal or C atoms capable of forming alloys with one another but essentially inert to the oxide or silicate matrix phase (MgO; olivine). In other instances, grain-boundary diffusive fluxes may lead to chemical reactions. As discussed in "The Detector-Particle Approach: General Considerations and Examples" section (Fig. 22), Thomas et al. (2009) examined Fe-o-Mg exchange along quartzite grain boundaries using fayalite (Fe 2 Si0 4 ) detector particles dispersed in the quartzite and an MgF 2 source. At the P-T conditions of the experiments (1125 °C, 1.5 GPa), pure fayalite is stable with quartz, but when the MgO content of the fayalite locally exceeds ~2-3% it reacts with the quartz to form enstatite. Additional supply of Mg 2+ (exchanging for Fe 2+ ) along the quartzite grain boundaries thus leads to growth of enstatite rims on the fayalite combined with continuing FeMg exchange. The influx of Mg 2+ along the grain boundaries thus leads not only to diffusive exchange in the fayalite detector particles but conversion of those particles to a new phase. Apart from minor complications arising from AVrxn, this difference should not dramatically affect the systematics of bulk-sample behavior. The numerical simulations described in the "Numerical Simulation: Constant-Surface Model" section reveal that the grain-boundary flux limits the rate of diffusant uptake in detector particles: in general, it should make little difference whether the element of interest is captured by diffusive uptake or by reaction. This conclusion is strongly suggested by Figure 23, which demonstrates insensitivity of diffusant uptake to the diffusivity in the detector particle, Dprt. The case of low Dprt is not very different from overgrowth of a thin rim of diffusant-rich material on the surface of the detector particle. A final observation regarding the detector-particle method is that, in principle, it can be implemented using a single grain boundary and a limited number of particles, essentially as shown in the bottom panel of Figure 17. This would provide the opportunity to characterize Dgb as a function of grain boundary misorientation.

ACKNOWLEDGMENTS The work on grain-boundary diffusion at RPI using the detector-particle method was supported by the U.S. Department of Energy, Office of Basic Energy Sciences grant no. DE-FG02-94ER14432 to E.B. Watson. The work with PLD thin films was funded by the Deutsche Forschungsgemeinschaft, mainly within the SFB526. We thank Daniele Cherniak, Jay Thomas and Thomas Miiller for helpful discussions and Jiba Ganguly, Youxue Zhang and an anonymous reviewer for insightful official reviews. We are grateful to Katharina Marquardt for allowing us to use the TEM images shown in Figure 5. Above all, we thank Daniele and Youxue for their heroic efforts in bringing this RiMG volume to completion.

REFERENCES Abart R, Kunze K, Milke R, Sperb R, Heinrich W (2004) Silicon and oxygen self diffusion in enstatite pofycrystafs: the Mifke et af. (200f) rim growth experiments revisited. Contrib Mineraf Petrof f47:633-646 Baxter E F (2010) Diffusion of noble gases in minerals. Rev Mineral Geochem 72:509-557 Bejina F, Raterron P, Zhang J, Jaoul O, Liebermann RC (1997) Activation volume of silicon diffusion in San Carlos olivine. Geophys Res Lett 24:2597-2600 Bohr N (1913) II. On the theory of the decrease of velocity of moving electrified particles on passing through matter. Philos Mag 25:10-31 Bowen NL (1921) Diffusion in silicate melts. J Geol 29:295-317 Brady IB (1995) Diffusion data for silicate minerals, glasses, and liquids. In: Mineral Physics and Crystallography: A Handbook of Physical Constants. American Geophys Union, Washington, D.C., p. 269-290 Buening DK, Buseck PR (1973) Fe-Mg lattice diffusion in olivine. J Geophys Res 78:6852-6862 Chakraborty S, Dohmen R, Miiller T, Becker HW, ter Heege J (2008) Fe-Mg interdiffusion coefficients in clinopyroxene: experimental determinations using nanoscale thin films. EOS Trans AGU 89(53) Abstract #MR21C-04

102

Watson & Dohmen

Chakraborty S, Ganguly J (1992) Cation diffusion in aluminosilicate garnets — experimental determination in spessartine-almandine diffusion couples, evaluation of effective binary diffusion coefficients, and applications. Contrib Mineral Petrol 111:74-86 Chen L-C (1994) Particulates generated by pulsed laser ablation. In: Pulsed Laser Deposition of Thin Films. Chrisey DB, Hubler GK (eds) John Wiley & Sons, New York, p. 167-198 Cherniak DJ and Watson EB, Wark DA (2007) Ti diffusion in quartz. Chem Geol 236:65-74 Cherniak DJ, Hanchar JM, Watson EB (1997) Diffusion of tetravalent cations in zircon. Contrib Mineral Petrology 127:383-390 Cherniak DJ, Hervig R, Koepke J, Zhang Y, Zhao D (2010) Analytical methods in diffusion studies. Rev Mineral Geochem 72:107-169 Cherniak DJ, Lanford WA, Ryerson FJ (1991) Lead diffusion in apatite and zircon using ion implantation and Rutherford backscattering techniques. Geochim Cosmochim Acta 55:1663-1673 Cherniak DJ, Ryerson FJ (1993) A study of strontium diffusion in apatite using Rutherford backscattering spectroscopy and ion implantation. Geochim Cosmochim Acta 57:4653-4662 Cherniak DJ, Watson EB (2001) Pb diffusion in zircon. Chem Geol 172:5-24 Cherniak DJ, Watson EB (2007) Ti diffusion in zircon. Chem Geol 242:473-483 Cherniak DJ, Watson EB (2010) Li diffusion in zircon. Contrib Mineral Petrol 159, doi: 10.1007/s00410-0090483-5 Cherniak DJ, Watson EB, Thomas JB (2009) Diffusion of helium in zircon and apatite. Chem Geol 268:155-166 Chrisey DB, Hubler GK (1994) Pulsed Laser Deposition of Thin Films, John Wiley & Sons Costa F, Chakraborty S (2008) The effect of water on Si and O diffusion rates in olivine and implications for transport properties and processes in the upper mantle. Phys Earth Planet Inter 166:11 -29 Costa F, Chakraborty S, Dohmen R (2003) Diffusion coupling between trace and major elements and a model for calculation of magma residence times using plagioclase. Geochim Cosmochim Acta 67:2189-2200 Costa F, Dohmen R, Chakraborty S (2008) Time scales of magmatic processes from modeling the zoning patterns of crystals. Rev Mineral Geochem 69:545-594 Crank J (1975) The Mathematics of Diffusion, 2 Edition. Clarendon Press, Oxford Cygan RT, Lasaga AC (1985) Self-diffusion of magnesium in garnet at 750 °C to 900 °C. A m J Sci 285:328-350 Dearnaley G, Freeman JH, Nelson RS, Stephen J (1973) Ion Implantation. North Holland, Amsterdam Dijkkamp D, Venkatesan T, Wu XD, Shaheen SA, Jisrawi N, Minlee YH, Mclean WL, Croft M (1987) Preparation of Y-Ba-Cu oxide superconductor thin films using pulsed laser evaporation from high-T c bulk material. Appl Phys Lett 51:619-621 Dimanov A, Jaoul O (1998) Calcium self-diffusion in diopside at high temperature: Implications for transport properties. Phys Chem Miner 26:116-127 Dobson DP, Dohmen R, Wiedenbeck M (2008) Self-diffusion of oxygen and silicon in M g S i 0 3 perovskite. Earth Planet Sci Lett 270:125-129 Dohmen R (2008) A new experimental thin film approach to study mobility and partitioning of elements in grain boundaries: Fe-Mg exchange between olivines mediated by transport through an inert grain boundary. Am Mineral 93:863-874 Dohmen R, Becker HW, Chakraborty S (2007) Fe-Mg diffusion in olivine I: experimental determination between 700 and 1,200 °C as a function of composition, crystal orientation and oxygen fugacity. Phys Chem Miner 34:389-407 Dohmen R, Becker HW, Meissner E, Etzel T, Chakraborty S (2002a) Production of silicate thin films using pulsed laser deposition (PLD) and applications to studies in mineral kinetics. Eur J Mineral 14:1155-1168 Dohmen R, Chakraborty S (2003) Mechanism and kinetics of element and isotopic exchange mediated by a fluid phase. Am Mineral 88:1251-1270 Dohmen R, Chakraborty S, Becker HW (2002b) Si and O diffusion in olivine and implications for characterizing plastic flow in the mantle. Geophys Res Lett 29: art. no. 2030 Dohmen R, Marschall H, Ludwig T (2009) Diffusive fractionation of Nb and Ta in rutile. Geochim Cosmochim Acta 73:A297 Dohmen R, Milke R (2010) Diffusion in polycrystalline materials: grain boundaries, mathematical models, and experimental data. Rev Mineral Geochem 72:921-970 Farley KA (2000) Helium diffusion from apatite: General behavior as illustrated by Durango fluorapatite. J Geophys Res Solid Earth 105:2903-2914 Farver JR, Yund RA (2000) Silicon diffusion in forsterite aggregates: Implications for diffusion accommodated creep. Geophys Res Lett 27:2337-2340 Fisher JC (1951) Calculation of diffusion penetration curves for surface and grain boundary diffusion. J Appl Phys 22:74-77 Futagami T, Ozima M, Nagai S, Aoki Y (1993) Experiments on thermal release of implanted noble gases from minerals and their implications for noble gases in lunar soil grains. Geochim Cosmochim Acta 57:31773194

Experimental Methods for Characterizing Diffusion

103

Ganguly J (2010) Cation diffusion kinetics in aluminosilicate garnets and geological applications. Rev Mineral Geochem 72:559-601 Ganguly J, Bhattacharya RN, Chakraborty S (1988) Convolution effect in the determination of compositional profiles and diffusion coefficients by microprobe step scans. Am Mineral 73:901-909 Ganguly J, Cheng WJ, Chakraborty S (1998) Cation diffusion in aluminosilicate garnets: experimental determination in pyrope-almandine diffusion couples. Contrib Mineral Petrol 131:171-180 Ganguly J, Ito M, Zhang X Y (2007) Cr diffusion in orthopyroxene: experimental determination, 53 Mn- l3 Cr thermochronology, and planetary applications. GeochimCosmochimActa71:3915-3925 Gardés E, Jaoul O, Montel JM, Seydoux-Guillaume AM, Wirth R (2006) Pb diffusion in monazite: An experimental study of Pb 2+ + Th 4+ = 2Nd3+ interdiffusion. Geochim Cosmochim Acta 70:2325-2336 Harrison LG (1961) Influence of dislocations on diffusion kinetics in solids with particular reference to alkali halides. Trans Farad Soc 57:1191-1199 Hayden LA, Watson EB (2007) A diffusion mechanism for core-mantle interaction. Nature 450:709-711 Hayden LA, Watson EB (2008) Grain boundary mobility of carbon in Earth's mantle: A possible carbon flux from the core. Proc Nat Acad Sci USA 105:8537-8541 Hiraga T, Anderson IM, Kohlstedt DL (2003) Chemistry of grain boundaries in mantle rocks. Am Mineral 88:1015-1019 Hiraga T, Anderson IM, Kohlstedt DL (2004) Grain boundaries as reservoirs of incompatible elements in the Earth's mantle. Nature 427:699-703 Hiraga T, Anderson IM, Zimmerman ME, Mei S, Kohstedt DL (2002) Structure and chemistry of grain boundaries in deformed, olivine plus basalt and partially molten lherzolite aggregates: evidence of meltfree grain boundaries. Contrib Mineral Petrol 144:163-175 Hiraga T, Kohlstedt DL (2007) Equilibrium interface segregation in the diopside-forsterite system I: Analytical techniques, thermodynamics, and segregation characteristics. Geochim Cosmochim Acta 71:1266-1280 Hofmann AW, Magaritz M (1977) Diffusion of Ca, Sr, Ba, and Co in a basalt melt — implications for geochemistry of the mantle. J Geophys Res 82:5432-5440 Hofmann S (1994) Atomic mixing, surface roughness and information depth in high-resolution AES depth profiling of a GaAs/AlAs superlattice structure. Surf Interface Anal 21:673-678 Houlier B, Cheraghmakani M, Jaoul O (1990) Silicon diffusion in San Carlos olivine. Phys Earth Planet Inter 62:329-340 Hwang JCM, Balluffi RW (1979) Measurement of grain-boundary diffusion at low temperatures by the surface accumulation method. 1. Method and analysis. J Appl Phys 50:1339-1348 Hwang JCM, Pan JD, Balluffi RW (1979) Measurement of grain-boundary diffusion at low temperature by the surface-accumulation method. 2. Results for gold-silver system. J Appl Phys 50:1349-1359 Ito M, Ganguly J (2004) Potassium diffusion in melilite: Experimental studies and constraints on the thermal history and size of planetesimals hosting CAIs. Meteor Planet Sci 39:1911-1919 Jambon A (1982) Tracer diffusion in granitic melts — experimental results for Na, K, Rb, Cs, Ca, Sr, Ba, Ce, Eu to 1300 °C and a model of calculation. J Geophys Res 87:797-810 Jaoul O, Bertran-Alvarez Y, Liebermann RC, Price GD (1995) Fe-Mg interdiffusion in olivine up to 9 GPa at T = 600-900 °C — experimental data and comparison with defect calculations. Phys Earth Planet Inter 89:199-218 Jaoul O, Poumellec M, Froidevaux C, Havette A (1981) Silicon diffusion in forsterite: A new constraint for understanding mantle deformation. In: Anelasticity in the Earth. Stacey FD, Paterson MS, Nicolas A (eds) American Geophysical Union, p 95-100 Kern W, Schuehgraf KK (2002) Deposition technologies and applications: Introduction and overview. In: Handbook of Thin Film Deposition Processes and Techniques. Seshan K (ed) No Yes Publications/William Andrew Publishing, p. 11-44 Koga KT, Walter MJ, Nakamura E, Kobayashi K (2005) Carbon self-diffusion in a natural diamond. Phys Rev B 72: art. no. 024108 Leclaire AD (1963) Analysis of grain boundary diffusion measurements. Brit J Appl Phys 14:351-356 Liermann HP, Ganguly J (2002) Diffusion kinetics of Fe 2 + and Mg in aluminous spinel: Experimental determination and applications. Geochim Cosmochim Acta 66:2903-2913 Lindhard J, Scharff M (1961) Energy dissipation by ions in keV region. Phys Rev 124:128 Lindhard J, Scharff M, Schiott ME (1963) Range concepts and heavy ion ranges (Notes on Atomic Collisions II). Mat Fys Medd Dan Vid Selsk 33:3 Lovering TS (1936) Heat conduction in dissimilar rocks and the use of thermal models. Bull Geol Soc Am 47:87-100 Lowry RK, Henderson P, Nolan J ( 1982) Tracer diffusion of some alkali, alkaline-earth and transition element ions in a basaltic and an andesitic melt, and the implications concerning melt structure. Contrib Mineral Petrol 80:254-261 Magaritz M, Hofmann AW (1978) Diffusion of Sr, Ba and Na in obsidian. Geochim Cosmochim Acta 42:595605

104

Watson & Dohmen

Marquardt K, Petrishcheva E, Abart E, Gardés E, Wirth R, Dohmen R, Becker H-W, Heinrich W (2010) Volume diffusion of ytterbium in YAG: thin-film experiments and combined TEM - RBS analysis. Phys Chem Miner DOT: 10.1007/s00269-010-0373-4 Matano C (1933) On the relation between the diffusion-coefficients and concentrations of solid metals (the nickel-copper system). Japan J Phys 8:109-113 Meissner E, Sharp TG, Chakraborty S (1998) Quantitative measurement of short compositional profiles using analytical transmission electron microscopy. Am Mineral 83:546-552 Melcher CL, Burnett DS, Tombrello TA (1981) Xenon diffusion following ion implantation into feldspar: dependence on implantation dose. Proc Lunar Planet Sci 12B: 1725-1736 Melcher CL, Tombrello TA, Burnett DS (1983) Retention of ion-implanted xenon in olivine — dependence on implantation dose. Geochim Cosmochim Acta 47:1707-1712 Milke R, Dohmen R, Becker HW, Wirth R (2007) Growth kinetics of enstatite reaction rims studied on nanoscale, Part I: Methodology, microscopic observations and the role of water. Contrib Mineral Petrol 154:519-533 Miro S, Studer F, Costantini JM, Berger P, Haussy J, Trouslard P, Grob JJ (2007) Effect of gold ion irradiation on helium migration in fluoroapatites investigated with nuclear reaction analysis. J Nucl Mater 362:445-450 Miro S, Studer F, Costantini JM, Haussy J, Trouslard P, Grob JJ (2006) Effect of composition on helium diffusion in fluoroapatites investigated with nuclear reaction analysis. J Nucl Mater 355:1-9 Misener DJ ( 1974) Cationic diffusion in olivine to 1400 °C and 35 kbar. In: Geochemical Transport and Kinetics. Giletti BJ, Yoder HS Jr, Yund RA (eds) Carnegie Institution of Washington, Washington, D.C., p 117-129 Mishin Y, Herzig C (1995) Diffusion in fine-grained materials: Theoretical aspects and experimental possibilities. Nanostruct Mater 6:859-862 MoriokaM, Nagasawa H (1991) Diffusion in single crystals of melilite. 2. Cations. Geochim Cosmochim Acta 55:751-759 Nastasi MA, Mayer JW, Hirvonen JK (1996) Ion-solid Interactions: Fundamentals and Applications. Cambridge University Press Norton DP (2007) Pulsed laser deposition of complex materials: progress towards applications. In: Pulsed Laser Deposition of Thin Films: Application-Led Growth of Functional Materials. Eason R (ed) John Wiley & Sons, p. 3-32 Ouchani S, Dran JC, Chaumont J (1998) Exfoliation and diffusion following helium ion implantation in fluorapatite: implications for radiochronology and radioactive waste disposal. Appl Geochem 13:707-714 Perkins D, Vielzeuf D (1992) Experimental investigation of Fe-Mg distribution between olivine and clinopyroxene — implications for mixing properties of Fe-Mg in clinopyroxene and garnet-clinopyroxene thermometry. Am Mineral 77:774-783 Reiners PW, Spell TL, Nicolescu S, Zanetti KA (2004) Zircon (U-Th)/He thermochronometry: He diffusion and comparisons with 40 Ar/ 39 Ar dating. Geochim Cosmochim Acta 68:1857-1887 Ryerson FJ (1987) Diffusion measurements: Experimental methods. Methods Exp Phys 24A:89-130 Ryssel H, Ruge I (1986) Ion Implantation. John Wiley & Sons Saenger KL (1994) Angular distribution of ablated material. In: Pulsed Laser Deposition of Thin Films. Chrisey DB, Hubler GK (eds) John Wiley & Sons, New York, p. 199-228 Sankur H, Hall R (1985) Thin-film deposition by laser-assisted evaporation. Appl Optics 24:3343-3347 Sautter V, Jaoul O, Abel F (1988) Aluminum diffusion in diopside using the 27 Al(p,y) 28 Si nuclear reaction preliminary results. Earth Planet Sci Lett 89:109-114 Schou J (2010) Physical aspects of the pulsed laser deposition technique: the stoichiometric transfer of material from target to film. Appl Surf Sci 255:5191 -5198 Schwandt CS, Cygan RT, Westrich HR (1993) A thin-film approach for producing mineral diffusion couples. Pure Appl Geophys 141:631-642 Shewmon PG (1963) Diffusion in Solids. McGraw-Hill, New York Shimojuku A, Kubo T, Ohtani E, Nakamura T, Okazaki R, Dohmen R, Chakraborty S (2009) Si and O diffusion in (Mg,Fe) 2 Si0 4 wadsleyite and ringwoodite and its implications for the rheology of the mantle transition zone. Earth Planet Sci Lett 284:103-112 Smith DB (1995) Thin Film Deposition: Principles and Practice. McGraw-Hill Sneeringer M, Hart SR, Shimizu N (1984) Strontium and samarium diffusion in diopside. Geochim Cosmochim Acta 48:1589-1608 Soriaga MP, Stickney J, Bottomley LA, Kim Y-G (2002) Thin Films : Preparation, Characterization, Applications. Springer Tannhauser DS (1956) Concerning a systematic error in measuring diffusion coefficients. J Appl Phys 27:662 ter Heege JH, Dohmen R, Becker H, Chakraborty S (2006) Experimental determination of Fe-Mg interdiffusion coefficients in orthopyroxene using pulsed laser ablation and nanoscale thin films. EOS Trans AGU 87(52) Abstract # M R 2 1 A - 0 0 0 4

Experimental Methods for Characterizing Diffusion

105

Thomas JB, Gannon R, Watson EB (2009) Grain boundary diffusion in dry quartzites using the dispersed "sink" method: sequestration of diffusant in near-source sink minerals. EOS Trans AGU 90(52) Thomas JB, Watson EB, Spear FS, Shemella PT, Nayak SK, Lanzirotti A (2010) TitaniQ under pressure: the effect of pressure and temperature on Ti-in-quartz solubility. Contrib Mineral Petrol, doi: 10.1007/s00410010-0505-3 Tirone M, Ganguly J, Dohmen R, Langenhorst F, Hervig R, Becker HW (2005) Rare earth diffusion kinetics in garnet: experimental studies and applications. Geochim Cosmochim Acta 69:2385-2398 Van Orman JA, Grove TL, Shimizu N (2001) Rare earth element diffusion in diopside: influence of temperature, pressure, and ionic radius, and an elastic model for diffusion in silicates. Contrib Mineral Petrol 141:687703 Van Orman JA, Grove TL, Shimizu N, Layne GD (2002) Rare earth element diffusion in a natural pyrope single crystal at 2.8 GPa. Contrib Mineral Petrol 142:416-424 Venables JA (2000) Introduction to Surface and Thin Film Processes, Cambridge University Press Vogt K (2008) Mg-Fe Diffusion in Spinellen. MSc thesis, Ruhr-Universität Bochum, Germany Von Seckendorff V, O'Neill HSC (1993) An experimental study of Fe-Mg partitioning between olivine and orthopyroxene at 1173 K, 1273 K and 1423 K and 1.6 GPa. Contrib Mineral Petrol 113:196-207 Wagner C (1969) Evaluation of data obtained with diffusion couples of binary single-phase and multiphase systems. Acta Metall 17:99-107 Wang LM, Ewing RC (1992) Detailed in situ study of ion beam-induced amorphization of zircon. Nucl Instrum Methods Phys Res Sect B 65:324-329 Wark DA, Watson EB (2006) TitaniQ: a titanium-in-quartz geothermometer. Contrib Mineral Petrol 152:743754 Watson EB (1986) An experimental study of oxygen transport in dry rocks and related kinetic phenomena. J Geophys Res 91:14117-14131 Watson EB (1991) Diffusion in fluid-bearing and slightly-melted rocks - experimental and numerical approaches illustrated by iron transport in dunite. Contrib Mineral Petrol 107:417-434 Watson E B (2002) Mobility of siderophile elements in grain boundaries of periclase and periclase/olivine aggregates. EOS Trans AGU, Spring Meeting Suppl, Abstract # V52B-05 Watson EB, Wanser KH, Farley KA (2010) Anisotropic diffusion in a finite cylinder, with geochemical applications. Geochim Cosmochim Acta 74:614-633 Weber WJ, Ewing RC, Wang LM (1994) The radiation-induced crystalline-to-amorphous transition in zircon. J Mater Res 9:688-698 Whipple RTP (1954) Concentration contours in grain boundary diffusion. Philos Mag 45:1225-1236 Wirth R (2004) Focused Ion Beam (FIB): A novel technology for advance application of micro- and nanoanalysis in geosciences and applied mineralogy. Eur J Mineral 16:863-876 Zhang XY, Ganguly J, Ito M (2010) Ca-Mg diffusion in diopside: tracer and chemical inter-diffusion coefficients. Contrib Mineral Petrol 159:175-186 Zhang Y (2010) Diffusion in minerals and melts: theoretical background. Rev Mineral Geochem 72:5-59 Ziegler J F (1992) Handbook of Ion Implantation Technology. North Holland, New York Ziegler JF, Biersack JP (2006) SRIM: The Stopping and Range of Ions in Matter. Computer Program (http:// www.srim.org/SRIM/SRfMINTRO.htm)

4

Reviews in Mineralogy & Geochemistry Vol. 72 pp. 107-170, 2010 Copyright © Mineralogical Society of America

Analytical Methods in Diffusion Studies Daniele J. Cherniak Department of Earth and Environmental Sciences Rensselaer Polytechnic Institute chernd @ rpi.edu

Richard Hervig School of Earth and Space Exploration Arizona State University

Jiirgen Koepke Institutfiir Mineralogie Eeibniz Universitaet Hannover

Youxue Zhang Department of Geological Sciences University of Michigan

Donggao Zhao Department of Geological Sciences University of Texas at Austin

INTRODUCTION In this chapter, we provide an overview of analytical methods used in diffusion studies. These methods fall into two broad categories - direct profiling and bulk release/exchange. Direct profiling methods, where the concentration of diffusant vs. depth is determined, can employ a variety of techniques, and can access diffusivities ranging from ~10 -9 to 10~24 m2/s. Among these methods are the "classical" techniques of serial sectioning and autoradiography, which, despite a long history of application in diffusion studies, are now rarely used, having been superseded by other analytical methods with better depth resolution and greater flexibility. Direct profiling can also be subdivided into methods involving step-scanning, with measurements made by stepping across a sample cut normal to the diffusion interface, and depth profiling, where analyses are performed parallel to the diffusion direction. Step-scanning may be used in cases where diffusion profiles are at least many tens of micrometers long, as it is limited by the beam spot size (typically > 2 (im; although transmission electron microscopes have much better spatial resolution, they are not used extensively in diffusion studies dues to the difficulties in sample preparation) of the analytical tool used and the minimum number of points necessary to define the diffusion profile. The electron microprobe is used for diffusion measurements by step-scanning, as is the ion microprobe (Secondary Ion Mass Spectrometry, SIMS), and IR spectroscopy. SIMS instruments can also be used in depth profiling mode when smaller diffusion distances are measured. The accelerator-based ion beam techniques Rutherford Backscattering Spectrometry (RBS), Nuclear Reaction Analysis (NRA) and Elastic Recoil Detection (ERD) are also applied in measuring short diffusion distances. This group of techniques is capable of high depth resolution (~ 10 to a several tens of nm) and thus can be used 1529-6466/10/0072-0004$ 10.00

DOI: 10.2138/rmg.2010.72.4

108

Cherniak, Hervig, Koepke, Zhang, Zhao

to measure the slow diffusivities characteristic of many species in minerals. In addition, other promising techniques have been developed, including synchrotron x-ray fluorescence (SRXRF), primarily used in measuring diffusion in glasses, and laser-ablation inductively coupled mass spectrometry (LA-ICPMS), which is being applied in diffusion studies of minerals and metals. In the other broad group of methods, involving bulk release or exchange, mineral grains or crushed fragments of material are exposed to a reservoir containing a radiotracer or isotopically tagged tracer (typically an aqueous solution or gas source, depending on the diffusing species), with the degree of exchange determined using mass spectrometry (in the case of tracer isotopes) or counting of activity (in the case of radiotracers). Variants of these methods have been used, for example, in early studies of C and O diffusion in carbonates (e.g., Anderson 1969, 1972) and alkali diffusion in feldspars (e.g., Foland 1974; Lin and Yund 1972), more recently in sulfur diffusion in sulfides (Hoeppener et al. 1990), and for oxygen diffusion in a variety of mineral phases (e.g., Hayashi and Muehlenbachs 1986; Fortier and Giletti 1991). However, such "bulk exchange" methods have several limitations, including the difficulty in evaluating diffusional anisotropy, and of sorting out potential contributions from transport along extended defects or other diffusional "fast paths" from those of lattice diffusion when profiles are not directly measured. In addition, assumptions are often made regarding diffusion models used to evaluate diffusivities, the effective surface areas of crushed grains or powders, and compositions of reservoirs acting as sources or sinks for the diffusing species. A similar approach, involving bulk loss, is taken in measuring noble gas diffusion in minerals. The gas released from sized mineral grains or grain fragments over a series of heating steps is measured by mass spectrometry, with the amount of gas released under these controlled time-temperature conditions used to calculate diffusivities. These methods, with numerous refinements, have been used over many decades to characterize diffusion of Ar (e.g., Fechtig and Kalbitzer 1966; Giletti 1974; Harrison 1981; Foland and Xu 1990; Baldwin et al. 1990; Lovera et al. 1991; Grove and Harrison 1996; McDougall and Harrison 1999) and He (e.g., Wolf et al. 1996; Reiners and Farley 1999; Farley 2000; Shuster et al. 2003; Reiners et al. 2004; Boyce et al. 2005; Blackburn et al. 2007; Farley 2007) in a range of mineral phases. Along with these methods, a variety of "indirect" techniques have been used to evaluate diffusion coefficients in minerals, including observation of homogenization of exsolution lamellae in feldspars and pyroxene, which have provided information about K-Na and NaSi-CaAl interdiffusion in feldspars (e.g., Brady and Yund 1983; Grove et al. 1984; Yund 1986; Liu and Yund 1992; Baschek and lohannes 1995), and Ca-Mg diffusion in pyroxene (Brady and McCallister 1983), electron paramagnetic resonance to constrain Al diffusion in quartz (Pankrath and Florke 1994), electrical conductivity measurements to determine diffusivities of alkalis in quartz (e.g., Verhoogen 1952), and study of Fe-Mg reaction kinetics between M l and M2 sites in orthopyroxene to infer Fe-Mg interdiffusion coefficients (Ganguly and Tazzoli 1994). Analytical methods used in diffusion studies of geological materials have previously been reviewed by Ryerson (1987). Since that time, SIMS instruments have made considerable technological advances with the development of Focused Ion Beam and nanoSIMS capabilities, which further improve spatial resolution. Accelerator-based ion beam techniques have found increased application in diffusion studies of geological materials, with RBS used to measure diffusion of many medium to heavy elements in a variety of mineral phases, and NRA employed in the investigation of diffusion of light elements and isotopic tracers; additional refinements of extant methods have enhanced their utility in studying geological samples. Improvements in instrumentation and analytical protocols have also benefited the application of IR spectroscopy, synchrotron XRF, and electron microprobe analysis in measurement of diffusion profiles. In the sections that follow, we will present abrief overview of several analytical techniques, beginning with classical approaches of serial sectioning and autoradiography, followed by the

Analytical Methods in Diffusion

Studies

109

electron microprobe, secondary ion mass spectrometry, laser ablation ICPMS, acceleratorbased ion beam methods (RBS, NRA and ERD), IR spectroscopy, and synchrotron XRF. In each section, the basic principles of the techniques will be outlined, along with practical considerations for their use in diffusion studies and examples of applications. A brief summary of the quantitative analytical capabilities of each instrument or method as applied in geological studies is given below. •

Serial Sectioning and Autoradiography: For measurement of radioactive tracer isotopes with resolution to about a micrometer, for diffusivities down to -lxlO" 18 m2/s.

•

Electron Microprobe (EMPA): For analyses of major and minor elements from Be to U with spatial resolution of a few micrometers, for diffusivities down to ~lxl0~ 17 m2/s.

•

Ion Microprobe (SIMS): For analyses of essentially all elements and isotope ratios at major or trace level with spatial resolution of about 10 ^m in step-scanning mode and about 20 nm in depth profiling mode. Newly developed nanoSIMS has the potential to improve spatial resolution.

•

Laser Ablation Inductively Coupled Mass Spectrometry (LA-ICP-MS). For analyses of essentially all elements and isotope ratios at major or trace level with spatial resolution of a few tens of micrometers in step-scanning mode and about 0.5 urn in depth profiling mode.

•

Rutherford Backscattering Spectroscopy (RBS): For analyses of heavy to mediummass elements in a matrix of predominantly lighter elements, in depth profiling mode with depth resolution to -10 nm.

•

Nuclear Reaction Analysis (NRA): For analyses of light to medium-mass elements (H to ~Ti) and isotopes in depth profiling mode, with depth resolution -10-50 nm.

•

Elastic Recoil Detection (ERD): For analyses of light elements (H to Ne) in depth profiling mode, with depth resolution ~20-50 nm.

•

Infrared Spectroscopy and Raman Spectroscopy: For analyses of H- and C-bearing species such as OH, H 2 0, NH3, NH4+, C0 2 , C0 3 2 - in glasses and minerals in stepscanning mode, with spatial resolution of a few tens of micrometers.

•

Synchrotron X-ray Fluorescence Microanalysis (n-SRXRF): For analyses of heavy elements (from Mn to Pb) in step-scanning mode, with spatial resolution of ~2 l^m depending on capillary used and sample geometry. "CLASSICAL" METHODS FOR MEASURING DIFFUSION PROFILES USING RADIOACTIVE TRACERS

Serial sectioning The technique of serial sectioning has long been used in diffusion studies (e.g., McKay 1938; Miller and Banks 1942), coming into use in the mid-1930s with the availability of artificial radioisotopes produced in accelerators, and later in nuclear reactors. The first applications were in the area of materials science, studying self-diffusion in metals. Among geologically significant materials, the method has been used primarily in investigations of simple oxides (e.g., Lindner and Parfitt 1957; Chen and Peterson 1975, 1980, Peterson et al. 1980), fluorides (Baker and Taylor 1969; Berard 1971), phosphates (e.g., den Hartog et al. 1972), sulfide minerals (e.g., Gobrecht et al. 1967; Chen and Harvey 1975), glasses (e.g., Jambon 1983; Magaritz and Hofmann 1978a;b), as well as a variety of silicates (e.g., Sippel 1963; Sneeringer et al. 1984; Behrens et al. 1990; Morioka and Nagasawa 1991; Morioka et al. 1997).

110

Cherniak, Hervig, Koepke, Zhang, Zhao

The method is employed by depositing a radioactive tracer on a sample surface or otherwise exposing the surface to the radiotracer, performing the diffusion anneal, then sequentially removing sample material in the direction normal to the diffusion direction, for a "one-dimensional" configuration to measure diffusion. The sequential removal of material is typically performed by grinding or dissolution. The thickness or weight of material removed is measured at each step, and the concentration of diffusant is measured either in the material removed or the remaining portion of the sample (with the latter called the residual method). Since a radiotracer is generally used as the diffusing species, the activity of the tracer measured is used to determine its concentration, for example with a scintillation detector for y radiation and a surface-barrier detector for (3 particles. Serial sectioning tends to have relatively limited depth resolution, with resolution constrained by the thickness of the material that can be reproducibly removed and adequately quantified. If only the activity of the removed portion is measured, this results in a stepped, rather than smooth, profile, with step width dependent on the thickness of the removed layers. Better depth resolution can be obtained with the residual method, where both the thickness removed and the activity in the remaining material are measured, but the limitations in depth resolution are ultimately based on the amount of material that can be sequentially removed and evaluated, so depth resolutions are typically larger than 0.5 with the finest scale range achieved by measurement with micrometers or careful weighing of materials having well-known density and fixed geometries. Diffusivities down to ~lxl0~ 1 9 m 2 /sec have been achieved for Sr in diopside (Sneeringer et al. 1984) and to ~3xlO~20 m2/sec for Ce diffusion in obsidian (Jambón 1983); but the latter approach (involving HF etching of the glass), is not applicable for most silicate minerals. Along with limitations in depth resolution, other potential problems with the serial sectioning method include the possible presence of diffusional fast paths, which cannot always be readily distinguished from volume diffusion when relatively coarse-scale depth profiling methods are used. Surface diffusion of the tracer to the back and sides of samples can also occur, as can transport of the tracer in the volatile components in the salts or solution in which they were deposited on the sample surface. Some of these can be circumvented by lowtemperature anneals to stabilize the diffusant in a near-surface layer, and by grinding away sample sides and edges to remove surface-diffused tracer. Another practical problem is the availability and expense of suitable radiotracers, as well as radiation safety considerations. Autoradiography Autoradiography is another radiotracer method, based on the detection of distributions of radiotracer species with photographic emulsions or other spatially sensitive detectors. The species detected may be from decay of radiotracers, or from fission of high-Z (where Z is the atomic number) elements during neutron irradiation. Typically, diffusion samples are cut parallel to the diffusion direction, polished and placed in contact with a track detector. Most studies employing this method have used beta-emitting radiotracers, such as 14C, 61Ni, 151Sm, 45 Ca (e.g., Hofmann and Magaritz 1977; Medford 1973; Watson 1979a,b, Watson et al. 1982); Seitz (1973) used mapping of fission tracks from neutron irradiated samples to measure U and Th diffusion in apatite and pyroxene. Track densities from track detectors have been determined by a variety of methods over time, including microdensitometers, optical examination, SEM, or microprobe traverses of silver halide emulsions. Limitations of the technique include broadening of beta distributions due to non-zero beta ranges, multiple decays, where a beta emitter decays to another species that is also a beta-emitter (e.g., 21l)Pb decay to 210Bi) which may lead to miscounting, and non-linear detector response, where saturation above certain concentrations may lead to undercounting of the decayed species, and, as in the case of serial sectioning, the availability of radiotracers. Because beta-particle ranges are typically on the

Analytical Methods in Diffusion

Studies

111

order of 100 |im (International Commission on Radiation Units and Measurements 1984), the length of diffusion profiles must be on the order of 1 mm or more for diffusivities to be reliably determined. Some early data in the literature using the autoradiography method do not meet this criterion, because beta-particle ranges were thought to be much shorter. Profiling using autoradiography does not have the spatial resolution of the serial sectioning method; practical lower limits for diffusivities measured by autoradiography is ~ lxl0~ 1 4 m 2 /sec.

ELECTRON MICROPROBE ANALYSIS Electron microprobe analysis (EMPA), also known as electron probe microanalysis, is an electron microbeam technique used primarily to quantitatively determine the chemical composition of solid materials on the micrometer scale and to qualitatively map elemental distributions in materials. It is an in situ, almost non-destructive method, and can be used to determine concentrations of elements from Be to U. Electron microbeam techniques, which also include scanning electron microscopy (SEM), transmission electron microscopy (TEM) and scanning transmission electron microscopy (STEM), use a focused (SEM, EMPA and STEM) or parallel (TEM) electron beam to bombard a sample, and the different types of signals generated by the beam-sample interactions are detected using appropriate detectors. The term "electron microscope" was first used by Knoll and Ruska (1932), who made the concept of an electron lens a reality. Ruska was awarded the Nobel Prize in 1986 for this technical innovation and its development. For further details on SEM, TEM or STEM, readers may refer to Goldstein et al. (2003) and Williams and Carter (1996), two widely-read and recent books on electron microscopy. References for EMPA include Smith (1976), Potts (1987), Reed (1993, 1995, 2005), and the EMPA special issue of Microscopy and Microanalysis (2001). In this section we follow the paths or trajectories of signals, first the electron and then X-ray, to discuss the basic principles, instrumentation and applications of EMPA. We will start from the top of an electron microprobe, at the electron gun, then move down along the electron column to electromagnetic lens and beam current and regulation, and then move down further to the sample chamber, vacuum system, and electron beam-sample interaction. Once the electroninduced X-ray is generated, we will shift our discussion to X-ray detection in EMPA, i.e., the wavelength dispersive spectrometer (WDS), and matrix effects. Finally we will present a few examples to illustrate the applications of EMPA in diffusion studies of geological materials.

Principles of EMPA The development of the first electron microprobe was reported by Castaing (1951) in his Ph.D. thesis at the University of Paris. An electron microprobe allows elemental concentrations on the micrometer-scale in materials to be measured routinely at levels as low as 100 ppm or 0.01 wt%. Using long count times at high beam currents, a detection limit of 10 ppm or 0.001 wt% is achievable for some specific applications, for example the measurement of Ti in quartz for the Ti-in-quartz thermometer (Wark and Watson 2006). The basis for qualitative analysis, 1.e., the identification of individual elements, is the so-called Moseley law. The X-rays emitted during inner-shell electron ionization and transition are called characteristic X-rays because their energies and wavelengths are unique to the excited element. Moseley (1913) discovered that the wavelength of the characteristic X-ray emitted from an element is inversely related to its atomic number Z:

y=

B

(z-cf

(1)

where B and C are constants that differ for each X-ray family (from different groups of electron shells), and y is the wavelength of the characteristic X-ray. By comparing the intensity of a

112

Chemiak, Hervig, Koepke, Zhang, Zhao

characteristic X-ray from an unknown sample with that from a known standard, quantification of elemental compositions can be achieved. In EMPA, characteristic X-rays are generated by a focused electron beam that bombards and interacts with solid materials. During inelastic collisions of the incident electrons with atoms in a sample, the incident electrons lose part of their energy. If the lost energy is high enough to overcome the critical ionization energy of an element, it will be able to remove an inner-shell electron from the atom, which leaves an inner-shell vacancy. The removed electron is called a secondary electron. The excited atom is not stable and a higher-shell electron will fall into the vacancy, resulting in release of the extra energy, either as an X-ray photon or as an Auger electron. The generation of the characteristic X-ray from an element is illustrated schematically in Figure 1. EPMA analysis is generally considered to be nondestructive. However, the electron beam can damage soft materials, such as glasses, and cause migration and loss of components in alkali- or volatile-bearing phases, which could result in underestimation of the concentration of such components. The emission of characteristic X-rays is a relatively inefficient process. Most interactions occur between the primary electron beam and electrons in outer shells of atoms within the sample. The primary beam electrons are decelerated and lose energy under the influence of the electric fields of the outer orbital electrons, resulting in the production of continuous or bremsstrahlung X-rays. This band of continuous X-rays extends in energy from zero up to the incident energy of the electron beam. In this regard, the electron microprobe could be considered as a giant X-ray tube with the sample as the target.

Scattered Primary Electron

Figure 1. Inner shell ionization in an atom and subsequent de-excitation during electron beam-material interaction. The difference in energy from an electron transition is released either as a characteristic X-ray photon or as an Auger electron (from Goldstein et al. 2003).

Analytical Methods in Diffusion

Studies

113

Instrumentation for EMPA The first electron microprobe built by Castaing (1951) used a Geiger counter to detect the X-rays generated from the sample. Since the Geiger counter could not distinguish among different characteristic X-rays, Castaing added a quartz crystal between the sample and the detector to differentiate different characteristic X-rays by using Bragg diffraction. He also added an optical microscope to view the beam impact. The first commercial microprobe was manufactured by Cameca in 1956, and they remain, along with JEOL, as the major electron microprobe manufacturers today. An electron microprobe consists of the following major components: 1) electron gun, such as a tungsten filament, LaB 6 or field emission gun, used to generate electrons, 2) electron column, composed of a series of electromagnetic lenses, used to manipulate the electron beam in a way similar to light optics, 3) sample chamber with a sample stage adjustable in x, y, and z directions, with a sample exchange door and a variety of detectors around the sample stage, 4) vacuum system for the column and chamber, 5) wavelength dispersive spectrometers (WDS), installed outside of the sample chamber around the electron column, which are used for detecting characteristic X-rays, 6) light microscope for optical observation of the sample, and 7) a control system composed of control interfaces and panel, and computers for data acquisition and processing. In addition to the computer operating system provided by the vendors, some electron microprobes may also use software for data acquisition and processing, for example, Probe for EPMA, previously called Probe for Windows. The components 1 to 4 above are also common constituents of a typical SEM. A modern electron microprobe now includes almost all functions of a SEM, equipped with secondary electron (SE), backscattered electron (BSE) and energy dispersive spectrometry (EDS) detectors, and sometimes with a cathodoluminescence (CL) and/or electron backscattered diffraction (EBSD) system. Hence, an electron microprobe is often considered as a specialized or modified SEM with the addition of wavelength dispersive spectrometers (WDS). Although some SEMs may also be equipped with a WDS detector, the primary applications of a SEM are to acquire images of a sample, and in some cases to qualitatively or semi-quantitatively obtain elemental compositions using an EDS detector, while an electron microprobe is designed primarily for quantitative chemical analysis with spatial resolution on the micrometer scale and detection limits down to 0.001 wt%. Major components of an electron microprobe are shown in Figure 2. Electron gun. There are two types of electron guns - thermionic and field emission guns. Most electron microprobes use thermionic tungsten or LaB 6 filaments, but field emission gun electron microprobes are now also available (MacRae et al. 2006). Thermionic electron emission occurs when the filament material is heated to a temperature of 2600-2700 K so that electrons have a sufficiently high thermal energy to overcome the work function energy of the filament material. A thermionic electron gun is composed of three parts - the filament or cathode at the top, Wehnelt cap in the middle and anode at the bottom. A negative electrical potential (-500 V) is applied to the Wehnelt cap to create so-called space charge, a collection of electrons in the space between the filament tip and Wehnelt cap, and to ensure that electrons only emit from a small area at the tip of the filament. The electrons at the bottom of the space charge emit from a hole in the center of the Wehnelt cap, and then are attracted and accelerated rapidly by high voltage of the anode through the electron column. Between the Wehnelt cap and the anode, the divergent electron beam emitted from the filament is focused into a crossover image of diameter 10 to 100 (im, which is subsequently demagnified by the rest of the electron column. The accelerating voltage of the anode is from 10 to 30 kV for SEM and EMPA and up to 300 kV for TEM and STEM. The electron gun has to be saturated at the beginning of the probe current plateau in order to preserve long and stable filament life (Fig. 3). A thermionic tungsten filament for an EMPA usually lasts for a few months; high quality of vacuum within the gun area provided

Cherniak, Hervig, Koepke, Zhang, Zhao

114

—I Wehnelt | J Ion pump I Alignment

coils

-| Liner tube | CCD camera |

Z autofocus I [Optical -| Beam regulator |

ieflectory objective~j Diffracting crystal

coils Probe Forming lens

I

Gas Flow Proportionnal counter

|

| Sample stage |

[BSE Detector I

Figure 2. (top) Schematic diagram of a Cameca S X 5 0 electron microprobe, showing the electron column with gun and lenses, sample chamber with sample exchange door, control console, and control computer monitors (image from a manual published by Cameca). ( bottom ) A J E O L JXA-8200 electron microprobe (Photo courtesy of the Electron Microbeam Laboratories at the University of Texas at Austin).

Analytical

Methods

in Diffusion

Studies

115

Figure 3. Effect of the filament temperature or current (x axis) on electron probe current from the gun (a) and current reaching the sample (b). The gun becomes saturated at the beginning of the current plateau (Reed 2005).

t

Temperature

by an ion pump could make a tungsten filament last up to one year. Such a long life time is usually achieved by a LaB 6 filament. Stable emission and long lifetime may also be obtained by positioning the filament further away from the Wehnelt cap. Electromagnetic lens. Electromagnetic lenses are used to focus and demagnify the electron beam emitted from the gun to an electron probe of approximately 1 to 10 nm on the specimen. An electromagnetic lens is composed of a hollow cylinder made from magnetic material, and copper wire coil through which the current is running. The lens is excited by passing a current through the copper coil, creating an intense electromagnetic field across the pole pieces to focus the electron beam. When a negatively charged electron passes through an excited lens, it is deflected to the direction perpendicular to the axis of the lens. This deflection, combined with electron velocity, results in the electron beam being focused at a point on the axis of the lens. Similar to an optical lens, the strength of an electromagnetic lens is measured by its focal length. The stronger the lens, the shorter the distance between focal point and pole pieces. All lenses suffer from manufacturing defects or imperfections, so an electron beam may never be brought into perfect focus by an electromagnetic lens. Electrons in trajectories further away from the lens axis are bent more strongly by the lens magnetic field than those near the axis. For an imperfect lens, electrons are focused at different points in a horizontal disk rather than at a point where all electron rays converge. This phenomenon is called spherical aberration. Another type of aberration is chromatic aberration, which is caused by variations in the energy of electrons emitted by the gun. Effects of aberrations cannot be completely canceled by using combinations of lenses as in light optics. However, in EMPA the aberration effects can be minimized by using concentric apertures to intercept the unwanted part of the electron beam. Beam current and its regulation. The accuracy of quantitative analysis depends partly on the stability of the electron probe current. Probe current must be regularly monitored and corrected during an analytical session. One way to achieve high current stability over long analytical sessions is to use a beam regulating system. A beam regulator is composed of two apertures. The fraction of the electron beam passing through the first aperture falls on the inner aperture, which is used to continuously monitor the probe current. If this current changes, a signal generated in a feedback circuit will cause the current passing through the condenser lens be adjusted for the current drift. Several definitions related to current are used in discussions of electron microprobe analyses, and these are often not clearly explained. For example, in analytical methods sections of published papers, both beam current and probe current are used, and they can refer either to a Faraday cup current or the current on sample or on the brass of a sample holder.

116

Chemiak, Hervig, Koepke, Zhang, Zhao

To avoid confusion. Potts (1987) distinguished uses of these definitions of current as follows: 1) Filament current is the current used to heat the filament for electron emission. 2) Lens current is the current passing through the coil of an electromagnetic lens. An increase in the lens current causes an increase in the strength of the electromagnetic field, which reduces the focal length. 3) Beam current is the total current emitted by the filament. 4) Probe current is the total current delivered to the specimen, which can be measured by using a Faraday cup, i.e.. probe current detector (PCD). The probe current represents only a portion of the original beam current from the filament. The probe current must be maintained at a constant value throughout an analytical session as emphasized above, and is a value that should be reported in published papers. In the literature, probe current is often referred as beam current. 5) Specimen current is the residual fraction of the probe current that stays within the sample. Some of the probe current bombarding the sample is backscattered out of the sample. For a constant probe current, the specimen current may vary from sample to sample depending on the mean atomic number and conductivity of the sample. Electron beam-sample interaction. The capabilities and limitations of electron microprobe analysis are related to interactions that occur within the sample in a volume of approximately 300 |.im\ in which a variety of collisions occur. The region typically has a truncated pear shape (Fig. 4) and is called the interaction or excitation volume. The shape and size of the interaction volume are of interest since they represent the source from which analytical signals originate. The range of electrons and X-rays within a sample depends on the energy of the electron beam and average atomic number of the sample. In addition to inelastic collisions of the incident electrons with atoms in a sample, which result in generation of secondary electrons. X-ray photons and Auger electrons (Fig. 1), a small fraction of the collisions are elastic, in which the primary electrons are scattered back from the sample without losing any appreciable energy. However, the majority of the primary electrons stay in the sample. Because of this, non-conducting samples must be coated with a thin (~25 nm) layer of carbon to prevent charging of the specimen surface. The transfer of energy during collisions between primary beam electrons and sample atoms may also cause phenomena such as sample heating and emission of photons in cathodoluminescence. Wavelength dispersive spectrometers (WDS). X-ray wavelength dispersive spectrometers (WDS). the key component of an electron microprobe. utilize Bragg diffraction through a diffracting crystal to disperse wavelengths of characteristic X-rays. Once a specific X-ray wavelength is selected, it is focused onto gas-flow or sealed proportional counters for

Figure 4. Monte Carlo simulations of interaction between a 20 keV electron beam and silicon. The beam electron trajectories that emerge as backscattered electrons are shown as thick traces (Goldstein et al. 2003).

Analytical

Methods

in Diffusion

Studies

117

measurement. Only a small portion of the X-ray photons from the sample reach a diffracting crystal: and the intensity of the X-ray in a WDS is generally lower than that in an energy dispersive spectrometer (EDS) detector for a given beam current. For a specific interplanar spacing d of the diffracting crystal and an X-ray wavelength y. there is an angle 0 at which the diffracted characteristic X-rays are in phase and their intensity is constructively enhanced. As illustrated in Figure 5, the wavefront of an X-ray beam approaches the diffracting crystal and is then reflected from the parallel crystal planes. Of the two X-ray paths shown in Figure 5, the difference in path length between these two paths is ABC = 2d sin 9 where d is distance between adjacent lattice planes of the diffracting crystal. If this distance ABC equals to an integral multiple n of wavelength X, then the reflected X-ray beams will be in phase and an intensity maximum will be detected by the proportional counter. This relationship is known as Bragg's law: n"k = 2d sin 9 (2) where 0 is the angle of incidence of the X-ray beam on the diffracting crystal and n is the order of reflection. A reflection is called the first order when n = 1, which is usually the strongest reflection and normally used in EMPA. Higher orders of reflection create unwanted peaks in the WDS spectrum, which can be suppressed by pulse-height analysis (PHA) through a singlechannel analyzer (SCA).

d B

Figure 5. Diffraction according to Bragg's law. Strong scattering of X-rays of wavelength nX occurs only at angle 9. At all other angles scattering is weak (Goldstein et al. 2003).

Diffracting crystals. Most electron microprobes are equipped with multiple WDS spectrometers with more than one crystal, which are necessary not only for analyzing multiple elements simultaneously, but also for optimizing performance in different wavelength ranges. Figure 6 lists some of the most commonly used crystals from major electron microprobe manufacturers. Mathematics tells us that sin 9 cannot exceed unity; therefore. Bragg's law, nX = 2d sin 0. establishes an upper limit of 2d for the maximum wavelength diffracted by a given crystal. More practical limits are imposed by the spectrometer design itself. A lower wavelength limit is imposed by Bragg's equation because it becomes impractical to physically move the diffracting crystal too close to the specimen, which limits the value of theta. Most naturally occurring crystals like LiF are limited to shorter wavelengths because their interplanar spacings are small. Synthetic crystals, like TAP, provide a very useful extension of the measurable wavelength range, but still do not have sufficiently large d spacings to

118

Chemiak, Crystal d spacing

2d (nm)

LIF

.4027

PET

.875

TAP

2.576

PCO

4.45

PC1

6.22

PC2

9.80

PC3

19.95

Hervig,

Koepke, Zhang,

Zhao

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Ne P Ca Mn Zn Br Zr Rh Sn Cs Nd Tb Yb Re Hg At Th

La UF PET TAP

Lithium fluoride 200 Pentaerythritol Thallium acid phthalate

Crystal d spacing

2d (nm)

LIFH

0.4027

LIF

0.4027

PETH

0.8742

PET

0.8742

TAPH

2.576

TAP

2.576

6 C I

14 Si I

PCO PC1 PC2 PC3

22 Ti I

30 Zn I

38 Sr I

2oCa H H I 3 1 Ga

uSi^RzT»

9F m,M

Ku. ß

46 Pd I

W/Si Multilayer crystal W/C Multilayer crystal V/C Multilayer crystal Mo/B4C Multilayer crystal

54 Xe I

62 Sm I

78 Pt I

86 Rn I „

soSn H M H H M M M M 7 9 A U

37Rb;!s»äSäSäi/ä//5SBa

24CrHHHl3SBr I1A9

La, ß

70 Yb I

"Hf

70Yb

M a , ß, K

Figure 6. Analysis ranges of crystals for a Cameca electron microprobe (top), and for a JEOL electron microprobe (bottom) (adapted from figures in manuals published by Cameca and JEOL).

cover the longest wavelengths encountered, for example, the Ka for Be, B, C. and N. The measurement of these long wavelengths requires the use of "pseudocrystals." i.e.. layered synthetic microstructures with large d spacings: lighter elements (H, He and Li) with even longer wavelengths cannot be analyzed by EMPA. X-ray source and matrix correction. For quantitative EMPA, the concentration of a given element in an unknown sample is obtained by comparing the intensity of a characteristic X-ray of the element with that from a standard with known chemical composition. The intensities of an X-ray line of both sample and standard must be corrected for dead time, background and instrumental drift, overlapping and matrix effects in order to obtain accurate results. According to the Castaing approximation, the intensity I of a characteristic X-ray is proportional to the mass concentration C of the element measured. To first approximation, the concentration for a given element^ in a sample can be expressed as: r>, i ^ ^ (/¿(unknown)") A C A ( u n k n o w n ) = CA ^ I A (standard)

(3)

where (^(unknown) is the uncorrected concentration in the unknown sample. The ratio /i(unknown)//^(standard) is called the k ratio. The measured X-ray intensity of an element in

Analytical Methods in Diffusion

Studies

119

EMPA is affected by the concentrations of all the other elements present in the sample. These so called matrix effects include the influence of atomic number (Z), X-ray absorption (A) and secondary fluorescence (F) and must be corrected for in order to obtain the actual concentration of an element. There are several matrix correction methods, which include calibration curves, these are Bence-Albee, ZAF and PAP. In the ZAF correction, the measured X-ray intensity is multiplied by a number of factors that model the influences of the effects listed above. The atomic number correction (Z) deals with differences in the behavior of incident electrons between an unknown sample and a standard, and includes two factors: the stopping power of the sample and backscattering of electrons from the sample. The stopping power is the ability of a material to reduce the energy of an incident electron by inelastic scattering. The transfer of energy from electrons to the sample is related to the generation of X-rays. As a result, the X-ray intensity increases with atomic number. Backscattering of electrons, an elastic process, is strongly affected by atomic number. The effect of backscattering is to decrease the X-ray intensity with increasing atomic number, which is opposite to the effect of stopping power. However, since the backscattering effect is greater; the overall atomic number correction follows the same trend as the backscattering effect. Since these two factors tend to cancel each other, the atomic number correction factors are not very large in most cases. The absorption correction (A) considers effects of the mass absorption coefficient, incident electron energy, and X-ray take-off angle. X-rays are generated throughout the analytical volume during an analysis. X-rays produced at depth must pass through a certain distance within the sample and may be absorbed during that time. Hence, the intensity of X-rays will be reduced by absorption from the elements in the material. In the ZAF correction, absorption is also one of the terms in the correction, and is often the largest correction made to the X-ray intensities in quantitative microanalysis. Therefore, its accuracy most directly influences the accuracy of the quantitative results. In the last few decades much work has been done to describe the absorption correction as a function of the depth distribution of the generated X-rays within the sample. Finally, the fluorescence correction (F) needs to be considered if the characteristic X-rays from an element B have high enough energy to excite another element A. Fluorescence excitation may also be caused by continuous X-rays, but this effect usually is negligible. The ZAF correction does not account for depth distribution of X-rays and is not very reliable for light elements, i.e., elements with X-ray energies less than 1 keV. A modified ZAF correction procedure called the PAP method (Pouchou and Pichoir 1986a,b, 1988, 1991) can better account for the depth distribution of energies of X-rays produced from a sample. The PAP method uses a modified version of the phi(rho-Z) polynomial used in standard ZAF correction procedures. To further optimize quantitative analyses for elements with low X-ray energies, it is best to use a standard of similar composition to minimize matrix effects, for example, a fluorapatite standard rather than a fluorphlogopite or fluorite standard to analyze F in apatite. X-ray proportional counter. The last major component is the gas proportional counter detector, shown in Figure 7. This highly efficient detector has an excellent dynamic range (050,000 counts per second or more) and covers a wide range of energies. It consists of a gasfilled tube with a coaxial thin wire, usually tungsten, held at a 1 to 3 kV potential. When an x-ray photon enters the tube through a thin window, it ionizes atoms of the gas and produces photoelectrons. The photoelectrons are then accelerated towards the anode wire and further ionize other gas atoms producing more electron-ion pairs. This "avalanche" effect produces an amplification of the initial signal, resulting in a charge pulse appearing on the anode. The amplitude or height of the pulse depends on the number of ionizations, which is related to the energy of X-ray photons. If the gas used by the counter is P10 (90% argon-10% methane), approximately 28 eV is needed to create one electron-ion pair. For the Mn Ka, which has an energy of 5.895 keV, about 210 electron-ion pairs will be directly created by the absorption

120

Chemiak, Hervig, Koepke, Zhang, Zhao

X-Ray

Figure 7. Schematic drawing of a gas flow proportional counter as used in WDS (Goldstein et al. 2003).

of a single Mn Ka photon. Electron multiplication in the counter depends on the anode voltage. At certain ranges of anode voltages in the counter, the pulse height is directly proportional to the energy of the incident X-ray. The voltage distribution of pulses for a pre-selected time period can be obtained using a single-channel analyzer (SCA), which is used to select and transmit pulses within a predetermined voltage range for further processing. The SCA also serves as an output driver, causing the selected pulses to become rectangular pulses of a fixed voltage and time duration. The pulse height analysis process with a SCA is illustrated schematically in Figure 8. Applications and limitations of EMPA

Figure 8. Schematic representation of a pulse-height analyzer, a) Main amplifier output; b) single-channel analyzer output. EL = 5 V; A£ = 2 V; E„ = 7 V (Goldstein et al. 2003).

Quantitative determination of the composition of an unknown phase by EMPA requires several steps: 1) a setup procedure which includes selection of elements to be analyzed. X-ray lines to be measured, standards and their compositions, accelerating voltage(s). probe current, analytical mode, beam size, diffracting crystals, background, count times, and other parameters, 2) standardization or calibration based on standards of known composition, 3) secondary standard check and evaluation of quantitative results, and 4) analysis of samples and assessment of errors. The EMPA technique is probably mostly used by materials (including earth materials) scientists for accurate chemical analysis of both natural and synthetic materials on small scales (Smith 1976; Potts 1987; Reed 1993, 1995, 2005; Goldstein et al. 2003; Mehrer 2007). For example, variations in elemental concentrations on the micrometer-scale from the center to the edge of a crystal can provide information on the history of crystal growth and elemental diffusion.

Analytical

Methods

in Diffusion

Studies

121

In experimental diffusion studies, elemental concentration profiles are measured using EMPA by stepping across a sample along the diffusion direction (i.e.. perpendicular to the diffusion surface or interface), also referred to as line scans. Such scans can be set up automatically by specifying the beginning and ending points and the number of points in between, with equal-distance spacing. If the profile is not perpendicular to the diffusion surface or interface, the distance from each point to the diffusion surface or interface can be calculated or measured optically after the microprobe analyses. Since the spatial resolution of the electron microprobe is a few micrometers, the half-length of the diffusion profile, roughly (Dt)m (e.g.. Zhang 2008), must be > 15 (im for the diffusivity to be accurately extracted from the profile (Ganguly et al. 1988). More generally, the half-length of the diffusion profile must be > 10 times the spatial resolution (Ganguly et al. 1988). If the length of the diffusion profile is shorter, the averaged concentration that is measured would be significantly different from the actual concentration at the intended point, and the extracted diffusivity may differ significantly from the actual value because of this "convolution" effect. Due to the constraints on the minimum length of diffusion profiles and practical limits on experimental durations (e.g., experiments lasting no more than 6 months), only relatively large diffusion coefficients (D > 10~17 m2s_1. e.g., Carroll 1991). such as those for diffusion in melts and inter-diffusion of rapidly diffusing elements in minerals, or diffusion at relatively high temperatures, can be measured using EMPA. Smaller diffusivities require other methods, such as Rutherford Backscattering or SIMS in depth-profiling mode. Figure 9a shows an Ar concentration profile measured by EMPA (Behrens and Zhang 2001). The profile is a result of Ar diffusion into a dry Ar-free rhyolite melt at high Ar pressure and high temperature (Behrens and Zhang 2001). The profile is long and well-resolved. The data can be fit well by an error function (solid curve), indicating that the Ar diffusivity is independent of Ar concentration. Figure 9b shows an interdiffusion profile of FeO measured by EMPA between a high-silica rhyolite (77 wt% Si0 2 ) and a lower-silica rhyolite (73 wt% Si0 2 ) melts (Van Der Laan et al. 1994). These data show con-

A r d i f f u s i o n in d r y r h y o l i t e m e l t

! 375 K: 0.5 GP;u 1800 s

Behrens :ind Zhang 2001

.1,5 Rhyolite diffusion coupic 1523 K: 1 GPa; 1800 s

Figure 9. Two concentration profiles measured by EMPA. (a) Ar diffusion profile in dry rhyolitic melt (Behrens and Zhang 2001); (b) FeO interdiffusion profile for a rhyolite diffusion couple (Van der Laan et al. 1994). Experimental temperatures, pressures and durations are shown in the figures.

Cherniak, Hervig, Koepke, Zhang, Zhao

122

siderably more scatter than the example of Figure 9a. Within uncertainties, the data can also be fit well by an error function (solid curve), allowing the extraction of an effective binary diffusivity. Points analyzed by EMPA can be seen optically after the analyses, and can be used to confirm the spots analyzed (e.g.. distance from the surface or an interface) and the distance between points. The application of EMPA in diffusion studies may be affected by factors such as the phase boundary fluorescence effect (Zhao 1998; Reed 2005). As illustrated in Figure 4, the electron beam-sample interaction volume is limited to a few micrometers in depth. However, the fluorescence volume, in which secondary X-ray fluorescence occurs, is much larger than the interaction volume in which the primary X-rays are generated. When the interaction volume or fluorescence volume crosses a phase boundary, fluorescence corrections of the matrix effects for quantitative analysis may not be correct. For example, in a common mafic and ultramafic assemblage with chromite-olivine pairs (olivine is a common inclusion in chromite hosts), analyses of Cr 2 0 3 concentrations in tiny olivine inclusions in the chromite host could be affected by secondary fluorescence (Zhao 1998). When the electron beam bombards the olivine, the primary Fe K a X-ray generated from the olivine will penetrate into a much larger fluorescence volume and may cross the olivine-chromite grain boundary. Since the chromite host has a high Cr 2 0 3 concentration and the chromium's critical ionization energy, 5.989 keV, is slightly lower than the energy of Fe Ka X-rays (6.403 keV) the Cr in the chromite will be excited by the primary Fe Ka X-rays, and secondary Cr Ka X-rays will be generated and detected. Therefore, the secondary fluorescence of Cr Ka in chromite by the Fe K a from olivine will dramatically increase the apparent C ^ O , concentration of the olivine inclusion, making a correction of Cr^O, concentration in the olivine necessary to account for this effect. Figure 10 shows that the apparent increase in C ^ O , concentration from the center of the olivine inclusion to the olivine-chromite boundary, illustrating that the apparent C ^ O , concentration could be artificially increased even at the center of an olivine inclusion of 50 urn in size. The secondary fluorescence effect was confirmed when higher apparent Cr 2 O t

0.6

0.4

Cr:03 wt % 0.2

0.0

0

50

100 Distance (nm)

150

Figure 10. Variations of Cr content across an olivine inclusion in a chromite host. The samples were analyzed at two different accelerating voltages — 10 and 30 kV. The apparent increase in Cr concentration toward the chroniite-olivine boundary is due to the effect of secondary fluorescence, where X-rays generated from Fe in the olivine excite Cr X-rays in the adjacent chromite. The effect becomes more pronounced with increasing accelerating voltage and when nearer the olivine-chromite boundary (Zhao 1998).

Analytical Methods in Diffusion

Studies

123

concentrations were obtained for same sample analyzed at 30 kV, a higher energy where the effects of secondary fluorescence are more pronounced. The effects of secondary fluorescence became more evident when the olivine was analyzed after removal of the chromite host, which resulted in Cr 2 0 3 concentrations for the olivine inclusion of ~0.10±0.02 wt%. Therefore, the phase boundary fluorescence effect must be considered when small mineral inclusions are analyzed by electron microprobe, as well as when performing analyses near interfaces where there are compositional differences and the potential for secondary excitation.

Summary Electron microprobes can analyze almost all elements (from Be to U) with a spatial resolution of a few micrometers and a typical detection limit of 100 ppm (with detection limits of 10 ppm achievable for some elements under certain circumstances). They are the most widely used instrument for major and minor element analyses in the geological sciences, but are unable to analyze most elements at trace levels. Furthermore, they are unable to detect the lightest elements (H, He and Li). For example, hydrous components in minerals cannot be analyzed using EMPA. There are also overlaps of X-ray lines for some elements in WDS detection, making quantitative analysis difficult due to these interfering signals. In addition, EMPA cannot distinguish between the different valence states of an element, as in the case of Fe 3+ /Fe 2 .

SECONDARY ION MASS SPECTROMETRY (SIMS) Secondary ion mass spectrometry (SIMS), also known as the ion microprobe, is a powerful microanalytical tool. It has found considerable application in characterizing the concentration and distribution of dopants in semiconductors, and is a crucial instrument in quality control (and research and product development) in this industry. The capabilities that make it essential in this context (high sensitivity and depth resolution) also make it well-suited to the characterization of diffusion profiles in geological materials.

Basic principles of SIMS In SIMS, atoms from a sample are ejected, or "sputtered" by the impact of energetic (several keV) primary ions on the sample surface. A schematic of this process (based on the Cameca SIMS design; http://www.cameca.fr/) is shown in Figure 11. The analyst has a choice of primary ion species and the polarity of the secondary ion beam; different primary beams enhance the ion signal for different elements. For example, elements with high electron affinities (e.g., F, S, As) show the best sensitivity when negative secondary ions are detected and an electropositive primary beam (such as Cs + ) is used, while elements with low ionization potentials (e.g., Li, Be, Y) yield low levels of detection when positive secondary ions are sputtered by an electronegative primary species such as oxygen. The schematic (Fig. 11) shows the effect of changing the primary ions from l6 0~ (most commonly used in geological studies) to l 6 0 2 + (most commonly used in the semiconductor industry). Most sputtered atoms are ejected as neutral species, some as molecular ions, some secondary electrons are generated, but some atoms in the sample form elemental ions in the sputtering process. Placing a high voltage on the sample accelerates sputtered ions away from the sample, and into a mass spectrometer where the ions are separated by their massxharge ratio and then counted. Changing the primary beam polarity has several effects on the analysis, but the two most obvious from Figure 11 are the impact angle and the impact energy. Tutorials on SIMS are available on the web, with one from Evans Analytical Group (http://www.eaglabs. com/training/tutorials/sims_theory_tutorial/) being commonly used. Sputtered neutral mass spectrometry (SNMS) is a related technique that ionizes the neutral species after ejection from the sample using electron impacts, ion impacts, or lasers. While not discussed in this chapter, the interested reader is referred to Williams and Streit (1986) and Vad et al. (2009).

124

Cherniak, Hervig, Koepke, Zhang, Zhao

Sample (+4500 V)

Sample (-¡-45QGV) b) Figure 11. Schematic diagram of the sputtering process in SIMS, (a) Sample held at + 4 5 0 0 V and sputtered with a primary beam of l 6 0 ~ ions accelerated from a duoplasmatron held at - 1 2 . 5 kV. The primary ions strike the sample at an angle of - 2 5 ° from the sample normal, (b) Same as (a) except that the primary beam is l 6 O i + with an impact angle - 3 9 ° from the sample normal. Sputtered particles are indicated in light shades. The sample is ~5 mm from the grounded plate in front of it. This creates a strong extraction field (~1 V/pm) that results in efficient collection of secondary ions but also deflects the primary beam as it encounters this voltage gradient, requiring substantial changes in primary beam steering (not shown) as the primary polarity is changed.

Analytical

Methods

in Diffusion

Studies

125

Using SIMS to measure diffusion profiles There are two approaches to using SIMS for diffusion studies: 1) line scans and 2) depth profiling. Line scans (or "step" scans) are used in those cases where the diffusion length is many times larger than the diameter of the primary ion beam. In such cases, chemical, isotopic, and tracer diffusion can be determined. Depth profiles are used most often when the diffusion length is less than a few micrometers. Line scans. One example of a diffusion experiment characterized by a line scan is shown in Figure 12. For this study, a platinum capsule was filled with rhyolite powder and melted at 1 atmosphere to produce a nominally dry cylinder of glass. The cylinder was then sealed in a larger capsule filled with D 2 0 and taken to 750 bars pressure, 850 °C in a cold seal pressure vessel (where the pressurizing medium was also D 2 0). Deuterium diffused into the molten rhyolite cylinder for 12 hours, after which the experiment was quenched. The quenched glass cylinder was embedded in epoxy, cut longitudinally, and polished (see Fig. 12). Step scans were conducted using SIMS to determine how far the deuterium had diffused into the silicate melt, and the resulting analytical craters are shown on Figure 12 while Figure 13 displays the analytical results). The different spacings between craters in earlier line scans represent attempts to examine the zoning of different elements and/or to test different analytical parameters. For example, while the experiment was designed to measure the diffusion of D 2 0 (and compare to related experiments on H 2 0 diffusion) it was realized in the mid-1990s that noble metal capsules are often contaminated with Li and B, so that these samples might show diffusion profiles for other light elements. Subsequent analyses revealed high boron only at the epoxy/glass interface of this 12-hour duration experiment, but the diffusion of Li is pronounced, and reaches the baseline Li concentration in the starting rhyolite at about the same distance as D 2 0 (Fig. 13; Hervig, unpublished data).

Figure 12. Polished sample of rhyolite glass showing several step scans across glass (reflected light image). Note the epoxy mounting medium on the left. The liquid meniscus at 750 bars was convex against the D i O fluid. The most recent step scans show up as dark craters. The scan at the bottom represents 25 |im steps - 1 4 0 0 |im long ( - 2 0 |am beam diameter). Most of the older step scans (illustrating a wide range of step sizes and primary beam diameters as specified by the operator) show brighter craters because the sample was re-coated with gold prior to the most recent work scans (sample from Stanton 1989). All craters were made using a primary beam of 16 0~.

Cherniak, Hervig, Koepke, Zhang, Zhao

126

Chemical Diffusion in Rhyolite 12 hours at 750 bars. 800 °C

D 2 0 (wt.%)

500

1000

Distance (|jm) Figure 13. Variation in D 2 0 and Li contents as determined by the step scan shown in Figure 2 above. The 2 + H signal was converted to D 2 0 (wt%) by comparison with the 2 H + signals obtained on bulk-analyzed rhyolitic glasses synthesized in the presence of D 2 0 fluid at different pressures (Stanton 1989). The Li concentration in the starting glass was - 3 0 ppm (Hervig, unpublished data).

The results in Figure 13 illustrate the capability of SIMS in that both volatile and trace light elements can be determined with a lateral resolution typically around 5-30 |am. The lateral resolution limitations are dominated by the performance of the primary ion source (how small one can focus the beam) and the amount of sample one needs to consume to obtain statistically meaningful analyses on a reasonable time scale (i.e., the useful ion yield; Hervig et al. 2006). The results are also interesting from a diffusion standpoint in that D 2 0 shows the well known concentration-dependent, concave-down diffusion profile (Behrens et al. 2007; Ni and Zhang 2008) while the lithium profile is not concentration-dependent, and concave up. SIMS line scans can also be used for isotopic diffusion (Lesher et al. 1996; Richter et al. 2003). An example from Lesher et al. (1996) is displayed in Figure 14. These experiments required the synthesis of basaltic glasses doped with 30 Si and 1 8 0 and creation of a diffusion couple by placing these nominally anhydrous samples against basaltic glass with normal isotopic abundances. After treating the couple at high pressure and temperature, the experiment was quenched, and the resulting isotopic gradient was characterized using a primary beam of Cs + and detection of negative secondary ions. As the results show, there are gaps in the line scan. These are caused by the development of cracks in the glass quenched from high pressure, and point out the need for flat surfaces when making SIMS analyses. If the sample surface is variably tilted, measured ion intensities of different secondary species can change significantly (these outliers were removed from Fig. 14). Careful optical study of run products prior to analysis can reveal areas likely to suffer from such problems. In the case of the data in Figure 14, the results were used to document similar diffusion of silicon and oxygen in highpressure silicate melts (Lesher et al. 1996). SIMS line scans of the run products from Richter et al. (2003) were used to document gradients in Li content and the Li isotope ratio in a chemical diffusion experiment.

Analytical Methods in Diffusion

Studies

127

Self-Diffusion in Basalt

Figure 14. Isotopic diffusion of silicon and oxygen in basaltic liquids at 1450 °C, 1 GPa (35 minute duration). The line scans used 50 |am steps, but overlap of the beam on numerous cracks that developed in the run product during quench led to some unusable analyses. The figure was redrawn from published data (Lesheretal. 1996).

Limitations of SIMS line scans. Despite the advantages of using SIMS in line-scan mode to measure diffusion profiles, there are prices to pay: 1) analyses take a long time, 2) calibrations may not be linear, 3) molecular ion interferences may be difficult to resolve and may change along the profile, and 4) the diffusion length scale should be much longer than the lateral resolution as defined by the diameter of the primary ion beam. Analysis duration. The time required to collect a measurement of sufficient precision at each analysis point depends on several parameters. One important value is the sputter yield, S, defined as the atoms ejected from the sample surface/primary ion impact. The sputter yield changes as a function of the target density, primary beam species, and primary beam impact energy. When an analysis is initiated, the addition of the primary beam to the sample changes the chemistry of the target. This leads to significant changes in the observed secondary ion intensities for a few minutes. For each analysis in a line scan, waiting until the ion signals of interest reach steady-state conditions (known as the "pre-sputter" time) can take as few as 1 or as many as 10 minutes (depending on primary species and intensity). For a ~60-step profile such as shown in Figure 13, this would translate into a total analysis time of ~ 10 hours or more, depending on how long the analyst sets the actual data collection portion within each crater. Linearity of secondary ion signal with concentration. Yields of ions are generally linear (i.e., ion intensity increases with element abundance) in constant matrices (where the species of interest represents a dilute component) making measurement of concentration profiles relatively straightforward in many applications of SIMS in diffusion studies. However, there are examples where this is not true. For example, major elements (Steele et al. 1981), hydrogen in high-H glasses (Hauri et al. 2002; Tenner et al. 2009), and Li isotope ratios in olivine (Bell et al. 2009) can show non-linear effects. These are typically related to matrix effects (i.e., major element variations influencing the ionization probability of other elements, which

128

Cherniak, Hervig, Koepke, Zhang, Zhao

can also be related to changes in the way the primary beam interacts with the sample). This problem can be avoided if experiments are designed to examine diffusion of multiple elements at trace concentrations. Interfering molecular ions. As indicated in Figure 11, the sputtered flux includes molecular as well as elemental ions, and some molecular ions may have the same mass/charge ratio as the diffusing species. For many potential diffusion experiments, this might not pose a problem. For example, if one wanted to measure the isotope diffusion of 40 Ca vs. 44 Ca across two otherwise identical basaltic glass samples, the most important interferences on the calcium isotopes would be 2 4 Mg 1 6 0 and 28 Si 16 0, at the same nominal mass/charge ratios, respectively. If the Si and Mg contents were constant across the diffusion couple, these molecular species would add a constant amount to the ion signals for both calcium isotopes throughout the profile, allowing accurate diffusion coefficients to be derived in spite of the background signal. In contrast, if the Mg or Si concentration were also changing across the sample, it would be critical to eliminate these molecular ions from the mass spectrum before interpreting the SIMS analyses in the context of calcium diffusion. The analyst has two choices for separating molecular ions from the elemental ion signal: 1) high mass resolution or 2) energy filtering. The former solution takes advantage of the fact that there is a difference in mass between an isotope of interest and a molecular ion at the same nominal mass/charge ratio. In the case of 40 Ca + (mass 39.962 u) vs. 2 4 Mg l 6 0 + (mass 39.979 u), the mass difference is 0.017 u (the exact masses of the isotopes, based on defining 12C =12.000, can be found in charts of the nuclides, textbooks, and web sites). Magnetic-sector secondary ion mass spectrometers allow the operator to close the entrance slits to the mass spectrometer to resolve these two ion signals at a nominal mass/charge ratio of 40. The mass resolution required is calculated as M/AM; in the case of the present example, 40/0.017 yields a mass resolving power requirement of -2300. Operating the SIMS at high mass resolving powers unambiguously eliminates the molecular ion, but operation at these conditions also decreases signal intensity and decreases the width of the peak of interest, placing high demands on the stability of the secondary magnet. Planning any diffusion experiment that will utilize SIMS for characterization requires the experimentalist to carefully consider the potential molecular ion interferences and how to separate these ions from the species of interest. In some cases, energy filtering of the secondary ion beam (Schauer and Williams 1990; Shimizu et al. 1978) will reduce molecular ion intensities to negligible levels while still leaving sufficient signal from the elemental ion to allow high-quality analyses to be obtained. This approach selects only those secondary ions that are sputtered from the sample with energies higher than can be accounted for by the potential placed on the sample. That is, some ions are ejected via the energetic collisions (initiated by primary ion impact) and have energies tens to hundreds of eV higher than the ions that are accelerated only by the potential placed on the sample. An example of an energy spectrum is shown in Figure 15. Energy filtering is generally effective at removing molecular ions composed of 3 or more atoms (e.g., 2 8 Si 2 l 6 0 2 + on 88Sr+) but does not eliminate all molecular ions from the mass spectrum. Most molecules composed of two atoms are not removed by energy filtering, such as the 28 Si 2 molecule in Figure 15 (which would overlap with 56 Fe), 2 4 Mg 1 6 0 and 2 8 Si 1 6 0 interfering with Ca isotopes, and the light rare earth element oxide ions which interfere with mid- to heavy rare earth elemental ions (Zinner and Crozaz 1986). However, if very high-energy secondary ions are examined (Eiler et al. 1997; Hervig et al. 1989, 1992; Lesher et al. 1996; Schauer and Williams 1990) even these molecular ions can be removed, but at great expense to the signal of the elemental ion. Regardless, care must be taken to ensure that there is an appropriate approach for elimination of these molecular ions when designing diffusion experiments meant for SIMS characterization. The examples shown for characterizing diffusion using SIMS line scans are only for silicate melts. As a general rule, diffusivities of elements in crystalline materials are orders of magnitude

Analytical Methods in Diffusion Studies i

1 1 1

i

1 1 1

i

1

i

129

T—I 1—I 1 1—|—I 1—I—f

1 1

10 7

10 5

Intensity (counts/s) 10 3

10

101 -40

-20

0

20

40

60

80

100

120

Initial Kinetic Energy (eV) Figure 15. Secondary ion energy spectra showing the variation in intensity of elemental , n Si + ions compared to 2 8 Si 2 + and 2 8 Si 3 + ions as a function of their initial kinetic energy. In this case, + 1 0 0 0 0 V was applied to a sample of quartz, and ions with a range of - 4 0 eV were allowed into the mass spectrometer (the energy range is a variable selected by the operator). The sample voltage was varied from 1050 V (far left side of the figure) to - 9 9 0 0 V (right side). Ions with positive initial kinetic energy represent those ions ejected with high relative energies from energetic collisions in the near-surface of the sample. The signal for the elemental ion decreases with initial kinetic energy, but remains relatively intense. The complex molecular ion, 2 S Si 3 + shows a decrease in intensity by >100x when high-energy ions are selected. In this case, these collisions also cause the molecular ion to break apart. Note that molecular ions composed of only two atoms, such as the dimer 2 8 Sii + , are not efficiently removed by this approach. Energy filtering cannot completely remove multiply-charged species, but becomes more effective as the number of atoms in the molecular ion increases. The box on the figure represents the range of energies selected for typical analyses using energy filtering. The zero point on the figure was defined by the center of mass of the "'Si + energy spectrum.

smaller than in melts, and so the scale of diffusion is not likely to be significantly larger than the diameter of the primary beam. In this case, it may be necessary to abandon line scans in favor of depth profiling analyses. Note, however, that one SIMS (NanoSIMS manufactured by Cameca) provides sub-micrometer primary beam diameters, and this instrument may be suitable for the characterization of short diffusion profiles in line scan mode. Depth profile analyses Depth profile analyses take advantage of the fact that S I M S samples atoms derived from the surface of the target. As the analysis progresses, atoms from deeper and deeper in the sample are collected, providing a measure of changing chemistry with depth. The typical analysis involves focusing a primary beam of ions to a point, where those primary beams may be Cs + , 0 + , O . 0 2 + , or 0 2 . Alternative beams are more rarely employed (Hervig et al. 1989, 2004; Hervig and Williams 1986). The primary beam is swept, or rastered, over the sample surface, generally in a square pattern from - 1 0 0 to 2 5 0 (.im on a side, although the primary beam can also be shaped by an aperture to make a nearly cylindrical crater without rastering, as shown in Figure 16 (Clement and Compston 1990; Genareau et al. 2007). The process of

130

Cherniak, Hervig, Koepke, Zhang, Zhao

Figure 16. Sputtered crater in surface of volcanic plagioclase 4 0 |im deep by 70 |im in diameter obtained on the 3f S I M S in ~ 6 hours using "aperture illumination," the 0 2 + primary beam, and the normal-incidence electron gun for charge neutralization (analysis and image by Dr. Kimberly Genareau, ASU). Note volcanic glass adhering to crystal surface, and the steps in the bottom of the crater. The overlap between these steps and the region of the crater floor allowed into the mass spectrometer will define the depth resolution of this profile.

sputtering the sample erodes the surface at a rate depending on the sputter yield S (defined above) together with the area of the crater and the primary current used. To make certain that the data from these depth profiles can be used for the accurate characterization of diffusion, several requirements must be fulfilled: the floor of the crater must be flat, the crater depth must be measured to calibrate the rate of erosion, any ions originating from the walls of the crater must be eliminated, the properties of the secondary ion beam should be tailored to the problem, and the effects of sputtering on the diffusion profile itself must be considered. Crater depth measurement. After the analysis, the total crater depth can be measured using stylus profilometry or interferometry. An example of stylus profilometry on a sputtered crater produced by a rastered primary ion beam is shown in Figure 17. The profile shows that the crater floor is flat (unlike the SEM image of the crater in Fig. 16) with a depth of 2 |am. It is difficult to

Depth

(Mm)

-1 o

Distance (pm)

Figure 17. Reflected light image of a sputtered crater in Lake County plagioclase ( - 1 0 0 x 1 0 0 |im2) and profilometry scan showing the ~2 |im depth of the crater after the depth profile (redrawn from Genareau et al. 2007).

Analytical Methods in Diffusion

Studies

131

measure the crater depth during the analysis (although the Cameca Wf SIMS instrument offers an option for in situ crater depth measurements using laser interferometry). The analyst must assume that the erosion rate is constant over the duration of the analysis, which is not a bad assumption when the crater is several hundred nm deep, the sample is uniform in major element chemistry, and the primary beam is stable. In fact, since many geological samples are insulators, it is necessary to coat the sample surface with a thin film of carbon or gold, and the sputtering rate of these films is much different than typical silicates (Au shows a high sputter yield, while carbon has a low sputter yield). Because the Au or C coats are -20-40 nm thick and the total crater depth is several hundred nm, the error in the sputtering rate (e.g., nm/second) is small (in addition, a gold coat can be easily removed prior to measurement of the crater depth; e.g., Van Orman et al. 2001). However, if the major element chemistry of the sample varies (for example, if diffusion of major elements occurs in the profile), the assumption of linear erosion rates must be tested (at the minimum, by determining the sputter yield on a range of homogeneous targets with appropriately varying major element chemistry). Contribution from crater walls. During a depth profile, the sides (as well as the bottom) of a crater are sputtered by primary ions and so secondary ions also originate from the walls (which represent all depths in the crater). It is important to allow only those ions derived from the flat-bottomed floor of the crater into the mass spectrometer over the analysis time (= depth). The Cameca design uses stigmatic ion optics to form an image of the sample surface beyond the entrance slits of the mass spectrometer. By placing a variably-sized "field" aperture at this plane, ions from the edges of the crater floor and the crater walls can be eliminated. For those SIMS instruments that are not ion microscopes, electronic gating of the secondary ion detector can be used to eliminate ions sputtered when the primary beam is near the crater walls (note that electronic gating is not appropriate for craters formed by aperture illumination; Fig. 16). A test of whether the selected analysis condition permits crater walls to contribute a signal can be performed by sputtering an ion-implanted sample (Williams 1983). Such samples have approximately Gaussian distributions of the implanted species with depth. The baseline signal should be approximately a factor of 104 below the peak intensity (this ratio is known as the "dynamic range"). Higher background intensity for the Cameca instruments most likely suggests either an instrumental "memory" of the isotope of interest, but could also indicate that there is a contribution from the crater walls (see Fig. 18 for an example of a good depth profile through an ion-implanted silicon wafer). The above "memory effect" is a consequence of the design of the high-transmission Cameca SIMS. Depth (A) If a sample rich in a particular element is analyzed, some sputtered Figure 18. Depth profile into silicon wafer implanted with atoms of the element may be de1014 atoms B/cm 2 ( 0 2 + primary beam, positive secondary ions detected). The background signal at - 6 0 0 0 A depth posited on the electrically groundcorresponds to - 0 . 1 ppm B. This low signal indicates that the ed plate in front of the sample (see depth profile was conducted in a manner limiting or excluding Fig. 11). Later analyses of other the contribution of boron from memory effects or the walls of samples exploring the diffusion of the sputtered crater (Hervig 1996).

132

Cherniak, Hervig, Koepke, Zhang, Zhao

this same element may result in these deposits being "sputter-cleaned" from the plate and potentially landing on the floor of the crater in the new sample. Subsequent primary impacts may sputter these atoms as ions, showing up as a background signal derived from a previous sample. In addition, background signal may also be encountered if molecular ions at the same mass/ charge are present. The latter topic was covered above under "Interfering molecular ions". Selection of secondary species. The yields of positive or negative ions during sputtering scales exponentially with the ionization potential or electron affinity of the element (Williams 1983). As such, the sensitivity of SIMS varies over a wide range (Hervig et al. 2006) and the analyst can select the polarity to maximize the signal for the study of trace element diffusion. Yields of secondary ions can be strongly influenced by variations in the bulk composition of the solid; such "matrix effects" were briefly described above. A typical diffusion experiment should not be strongly influenced by matrix effects, as the diffusing species is ideally contained within an unchanging matrix. However, it is common to apply a thin layer very rich in this species to the surface of a phase, and this represents a different composition than the phase of interest. Thus the secondary ion signals may fluctuate dramatically during the beginning of sputtering. These transient variations can make locating the position of the original sample surface in SIMS depth profiles difficult. This is discussed in more detail later in this volume by Ganguly (2010). For some studies, it may not be necessary to maximize the sensitivity for a particular diffusing species if interfering molecular ions are minimal and the gradient is large. In such cases, the analyst may consider conducting the profile using simple, reproducible conditions instead of maximizing sensitivity (for example, operating at the conditions for highest sensitivity may lead to difficulties in controlling sample charging during the depth profile, as discussed below). Effect of sputtering. When considering the effect of the sputtering process on a diffusion profile, there are two important items to consider. One is that the ions sputtered from the sample are dominantly derived from the top two monolayers of the sample surface (Dumke et al. 1983). The initial impression one gets from this observation is that SIMS profiles should have resolution on a similar scale. However, the other item of note is that energetic primary ions will penetrate beneath the surface and thus cause a collision cascade (multiple collisions of the atoms in the sample). The primary beam is added to the initial chemistry of the surface with the result being a more-or-less uniformly mixed chemistry over a depth corresponding to the projected range of the primary ions (Nastasi and Mayer 1994), where the projected range represents the most common depth achieved by primary ions in a sputtered sample (highest concentration of the implanted primary species). The range is a function of the impact energy and mass of the primary beam and the mass of the target. The range shows large variations; consider the two situations shown in Figure 11. In the first case, a primary beam of 16 0", accelerated to -12.5kV in the duoplasmatron primary ion source, is attracted toward the sample at +4.5 kV and has a total impact energy of 17 keV. In contrast, Figure 1 lb portrays a beam of 0 2 + , initially accelerated to +12.5 keV in the duoplasmatron but which is decelerated as it approaches the sample, striking with a total energy of 8 keV. In addition, the molecular beam will break into two separate oxygen projectiles upon impact, each having 4 keV energy. The decrease in impact energy translates into a much lower projected range (average depth of penetration) for 0 2 + compared to O . Note that the angle of impact is different (Fig. 11), so the normal component of the impact energy is smaller for the positive primary species. The result is two-fold: the range for an 0~ impact is deeper, so that the sample atoms get mixed to a greater depth than when using 0 2 + , and, because the energy of the molecular 0 2 + beam is restricted to near-surface interactions (and only atoms from the top two atomic layers are sputtered), more atoms are ejected per primary ion impact (higher sputter yield). The conditions presented in Figure 1 lb result in a sputter yield for Si metal ~4x greater than in Figure 1 la (Sobers et al. 2004). The implications of these changes in the instrument parameters on ion beam mixing and depth resolution are shown in Figure 19. Other modifications to the parameters are possible:

Analytical Methods in Diffusion

Studies

133

Depth (nm) Figure 19. Depth profile into San Carlos olivine comparing the intensity of iron ions using O" and 0 2 + primary beams. Only the first 200 nm of the profiles are shown. The initial 20-40 nm represent sputtering through the overlying gold coat. The secondary ion signal for iron appears earlier in the profile when O" is used because this primary species penetrates deeper into the sample and, through ion beam mixing, churns iron (and other elements in the olivine) to the surface where they can be sputtered and detected. In comparison, the smaller projected range of 0 2 + mixes atoms at a shallower level, improving the depth resolution of the profile as revealed by the 54 Fe signal appearing at a greater depth. The spacing between data points is ~3x greater when 0 2 + is used because the sputter yield of the molecular primary beam is ~3x larger than for O" on this phase. Note the larger transient peak for iron between 30-40 nm when O" is used. Varying initial signals for matrix (and diffusing) elements is a common observation in depth profiles of silicates using an O" primary species (Ganguly et al. 1998; Van Orman et al. 2001; Ito and Ganguly 2006). Using the 0 2 + beam requires an auxiliary electron gun for charge neutralization (Genareau et al. 2007) and adds significant complexity to the depth profiling analysis. It also delivers the best depth resolution. However, most diffusion experiments can be designed so that the most important part of the profile is observed below the transient sputtering region, allowing straightforward use of the O" species.

the primary beam potential can be decreased from 12.5 kV, and in some SIMS instruments, the sample potential can also be decreased to help control the impact energy (and hence the penetration depth of the primary beam) and maximize the depth resolution. Regardless of the variations in instrumental set-up, most diffusion experiments can be engineered to make the diffusion profile long compared to the short-range artifacts of sputtering. Sample charging. A significant number of geologic phases are bulk insulators, and so addressing the problem of sample charging is very important. If the degree of charging varies throughout the depth profile, the secondary ion signals will vary, and different elements may be affected differently. Charge compensation is most simply addressed when a primary beam of , 6 0~ is used with the detection of positive secondary ions. In this case, accumulation of negative charge in the crater is accommodated because of the abundant secondary electrons produced during sputtering. These electrons can "hop" to the conducting gold or carbon coat surrounding the crater to drain excess negative charge. If it is necessary to use a positive primary beam, the analyst will need to use an auxiliary electron gun to alleviate positive charge build-up in the crater. This increases the complexity of the analysis, but useful depth profiles can be obtained (Genareau et al. 2009).

134

Cherniak, Hervig, Koepke, Zhang, Zhao

Ion implantation and SIMS As discussed in this volume's chapter on experimental methods by Watson and Dohmen (2010), ion implantation represents one possible experimental approach to introducing diffusants into materials for the determination of diffusion coefficients. Figure 18 shows an example of how SIMS can be used to characterize the distribution of boron in a silicon wafer after implantation. If this sample is heated, the boron will diffuse, and subsequent SIMS depth profiles in aliquots of the implanted wafer treated for different times and temperatures could be used to quantify the diffusion. While ion implantation has been coupled with analyses of profiles by RBS, NRA and ERD for diffusion studies in minerals (e.g., Cherniak et al. 1991, 2009; Ouchani et al. 1998), this type of characterization (i.e., ion implantation with SIMS analyses) has mostly been limited to the semiconductor industry, where it is important to activate the electrical properties of dopants by annealing and to know how these trace elements (e.g., Be, B, and S) have migrated during the annealing step (e.g., Tsai et al. 1979; Oberstar et al. 1982; Wilson 1984). These studies demonstrate that in general, the damage to the crystal lattice from the implantation step results in more rapid diffusion of the implanted species to the surface than into the bulk. In the case of geological samples, if the depth of the implant can be made deep enough to avoid transient effects observed during the first few nm of sputtering, and damage to crystal lattices (in the case of diffusion in minerals) can be annealed faster than the implanted species can diffuse, SIMS could be used to characterize subsequent diffusion.

Summary comments Secondary ion mass spectrometry has been an effective analytical tool for characterizing diffusion profiles. Active dialog between experimenter and SIMS laboratory is essential for designing experiments to make best use of the unique capabilities of this tool.

LASER ABLATION ICP-MS (LA ICP-MS) Laser ablation ICP-MS (LA ICP-MS) is a microanalytical technique for the determination of trace elements in solid materials. The sample to be analyzed is placed in a sample chamber with a lid transparent to UV light, and a pulsed laser beam is used to ablate a small quantity of sample material. The fine particles produced in the ablation are transported into the Ar plasma of an inductively coupled plasma mass spectrometer (ICP-MS) instrument by a stream of carrier gas (typically Ar and/or He), where they are ionized and then mass-analyzed. In the ablation process, the laser beam leaves behind an ablation crater (typically on the order of a few to tens of |im in diameter) where the analyzed material has been removed. Additional information on ICP-MS instrumentation and application can be found in numerous references (e.g, Montasser 1998; Taylor 2000; Sylvester 2001, 2008; Nelms 2005; Thomas 2008). Lasers were first used with ICP-MS instruments in the 1980s. The initial ablation systems used solid-state ruby lasers (operating at 694 nm, in the infrared region), but these were found to be unsuccessful for applications in trace-element analysis due to poor laser stability, large beam diameters, and low power density, among other factors. Over the next decade, commercial laser ablation systems were developed employing Nd-YAG lasers, which produce IR laser light with a fundamental wavelength of 1064 nm. However, these faced continued limitations because IR laser light does not interact very efficiently with most solids, so further developments in the field explored the use of UV laser light (which couples more efficiently with most materials) for ablation systems given its greater potential for effective use in trace-element analysis. The interaction between UV laser light and most solids tends to involve mostly mechanical breakup of the ablated area, whereas IR laser light may produce a greater degree of sample heating and melting, which can contribute significantly to elemental fractionation and limit capabilities for analysis. More recently, some laboratories have begun to use gas-filled (excimer) lasers operating in the UV region of the electromagnetic spectrum. These lasers, including XeCl (308

Analytical Methods in Diffusion

Studies

135

nm), KrF (238 nm), and ArF (193 nm) exhibit better absorption capabilities for UV-transparent materials (e.g, silicates, calcite, fluorite), and possibly less elemental fractionation in ICP-MS than longer wavelength lasers because particles produced in ablation are smaller and easier to volatilize. Since the UV range is the fundamental wavelength for the excimer lasers, there is higher energy transfer than for Nd-YAG lasers, where UV wavelengths are produced by directing laser beams through crystals to quadruple or quintuple their frequency. The less coherent nature of the excimer beam produces better optical homogenization and therefore cleaner, flatter ablation craters, a critical factor in depth profiling. Improvements in laser optics have also provided more controlled, smaller beam spots, important in step scans and depth profiling of materials, the two methods used to obtain diffusion profiles. Laser ablation instruments require an accurate optical system of lenses, prisms and mirrors to direct and focus the laser beam onto the sample. Several parameters are typically adjusted in analyses to optimize for a particular material and the species of interest. These parameters include laser power, laser pulse repetition rate and the number of laser pulses fired in succession. In addition, there is generally a system of apertures of different diameters that can be used to vary the beam diameter and hence the diameter of the ablation crater generated. Although the size of an ablation crater can be made very small, and small size would be advantageous in step scans of diffusion profiles, the usefulness of small volumes of material for quantitative analysis is strongly dependent on the sensitivity of the ICP-MS instrument to the elements of interest, since signal intensities for small amounts of ablated material may be low. In addition, the walls of ablation pits may yield ionic concentrations that differ from the center region because energy interactions are slightly different at the edge of the ablation spot. Further, the physical barrier of the pit edge becomes increasingly significant as ablation proceeds deeper into a sample. Because smaller spot sizes have a larger wall-to-center ratio, a greater fraction of the signal in small spot sizes will be due to contributions from the walls than would be the case for larger spots. In depth profiling of diffusion samples, apertures and rastering may be used to vary the size of the ablated area, with a larger areas removed near the sample surface (typically containing the highest concentration of diffusant) and smaller ablated areas at depth. This helps to avoid contamination at depth in the sample from the ablated material in the upper layers of the sample and from the sides of the ablation crater. Sample chambers for laser ICPMS are typically mounted on a stage that allows the sample to be positioned relative to the laser beam in x-y-z coordinates and move to a particular region of interest or to perform step or area scans. In diffusion measurements, concentration profiles can be measured either through depth profiling or step scans in the direction normal to the interface between the sample and diffusant source. It should be noted that step scans are only practical for relatively fast diffusivities (down to ~lxl0~ 1 6 m 2 /sec) given typical sizes of the ablated area, so they are most useful in studies of diffusion in glasses or melts, or for some fast-diffusing species in crystalline materials. In depth profiling, since depth resolutions may range from several tenths to several |am, diffusion profiles will generally need to be on order of several to tens of micrometers in length, thus limiting accessible diffusivities to ~lxlO~ 18 m 2 /sec. Hence LA-ICPMS measurements using current instrumentation cannot access the very slow diffusivities that may be measurable by depth profiling with RBS or SIMS, and because of larger spot sizes cannot measure step scans on as fine a scale as EMPA or nanoSIMS. In most instances, sample preparation for LA-ICPMS analysis is very simple. Samples used are typically in the form of epoxy mounts or petrographic thin sections similar to those used in electron microprobe analysis. For diffusion studies, samples would generally need to be flat and well-polished to avoid any effects on depth resolution due to surface roughness when depth profiling, and to avoid sampling problems in step scans. While sample ablation cell (the region above the sample where the carrier gas removes the ablated material) design continues to evolve and improve, it is critical that samples be configured in such a manner that they do not create eddies or pockets that could trap or fractionate the analyte, which may

136

Cherniak, Hervig, Koepke, Zhang, Zhao

compromise the ablation cell's ability to efficiently transport ablated material to the mass analyzer so that an optimal signal can be produced. Standards are required for analytical calibration of LA-ICPMS, with standardization in most cases using various synthetic or natural reference materials of known composition. Materials commonly employed as standards are the set of silicate glasses produced by the US National Institute of Standards and Technology (NIST) containing various trace elements in a range of concentrations, standards from the US Geological Survey, or various other natural or synthetic minerals or glasses of well-determined composition. Appropriate standards will depend on the type of materials and elements to be analyzed, with ideal calibration standards matching the composition of the bulk chemical matrix of the sample (e.g., Koenig 2008). It has been found that for many crystalline silicates and glasses that 193 nm (ArF excimer) laser systems are less matrix-dependent than 213 nm (Nd:YAG) systems, and these are less matrix-dependent than longer-wavelength laser systems. Nonetheless, all LA-ICPMS analyses require calibration, and optimal reference materials that are reasonably close in bulk chemical composition to the sample, well characterized for all species of interest, homogeneous at the required scale of analysis, and available in sufficient quantities to last more than a few analyses. Along with this external standardization, internal standardization (using elements in the sample analyzed) is typically also required in LA-ICPMS to correct raw data for differences in the ablation characteristics among standards and samples and between different elements, as well as to correct for general instrumental drift. Different groups of elements may need to be normalized against different internal standard elements to achieve high quality analyses; this will depend largely on the relative volatility of the elements. It should be noted that the elements used as internal standards must be quantified by an alternate analytical technique (e.g., electron microprobe or SXRF) in both the sample material and the external reference materials. This adds another level of complexity to the process of standardization. Like SIMS and solution ICPMS, LA-ICPMS is subject to various mass spectrometric interferences, including isobaric, molecular and doubly-charged ion interferences. Mass spectrometric interferences are more of a concern for light elements and most of the interferences in ICP-MS systems arise from compounds made from the torch, carrier gas, and gases in the chamber atmosphere. Of secondary concern are interfering compounds generated from the major components of the matrix. Interferences may be minimized or avoided through judicious selection of isotopic species to analyze, and optimization of instrumental operating conditions. It should also be noted that when considering mass interferences in analysis that even state-of-the-art quadrupoles used in ICPMS instruments still cannot match the mass resolution of magnetic sector machines. LA-ICPMS detection limits vary with laser power and the volume of material analyzed. Theoretical detection limits for most elements are typically in the ppb to low ppm range. LA-ICPMS provides better trace element sensitivity than the EPMA or XRF for much of the periodic table, and analyses are generally fast, with detection of the entire periodic table (except for the noble gases, H, N, O and F) at low concentrations possible in less than a minute. Despite the generally excellent trace element sensitivity of LA-ICPMS there are still cases where interferences, poor ionization in the ICP and other problems lead to compromised sensitivity, especially with small ablation spot sizes. While instrumentation has improved significantly and rapidly, questions still remain regarding control of ablation of certain types of materials, mobilization of some species in material beyond the ablated area, and, in the case of depth profiling, the effects on measured concentrations of contamination from pre-ablated material and mixing of components from several depths. Although LA-ICPMS is finding wide application for trace-element analysis in the geosciences (e.g., Sylvester 2001, 2008), it is has not yet been used extensively in diffusion studies of geological materials. Some examples include studies of Ar diffusion in K-feldspar

Analytical Methods in Diffusion

Studies

137

(Wartho et al. 1999) using depth profiling for profiles less than 50 |im and step scans for longer profiles, Ar diffusion in quartz (Baxter 2010, this volume) with depth profiling using Nd-YAG (213 nm) and ArF excimer (193 nm) lasers, diffusion of a suite of siderophile elements (Cu, Co, Ni, Ge, Ga, As, Ru, Pd, Pt, Ir, Au) in FeNi metal (Righter et al. 2005) measured with step scans, and diffusion of rare earth elements in olivine and chromite (Spandler et al. 2007) also measured using step-scanning. With advances in instrumentation and refinements of analytical protocols, the application of this technique to investigations of diffusion in geological materials is likely to become increasingly common.

RUTHERFORD BACKSCATTERING SPECTROMETRY (RBS) Rutherford Backscattering Spectrometry (RBS) is a method for determining chemical composition and elemental distributions in the outer few micrometers of a material. It is based on elastic collisions from interactions between light energetic ions (typically helium, produced in a small particle accelerator) and nuclei in a sample material. 4 He in the energy range of 1 to 4 MeV is the most common beam used, but other ions, including protons, deuterons, 3 He, as well as those heavier than He, are also used for backscattering in certain applications. For example, protons can achieve larger depth ranges but at the expense of mass and depth resolution, and heavy ions can achieve high mass resolution, but can probe only shallow depths in a material. General overviews of RBS can be found in Chu et al. (1978) and Leavitt and Mclntyre (1995). RBS has been used to investigate diffusion in minerals for over a quarter of a century. The earliest-reported applications include the work by Melcher et al. (1983) in measuring diffusion rates of Xe in forsteritic olivine, and the study of Sneeringer et al. (1984), in which Sr diffusion in diopside was measured and results from RBS, SIMS, and radiotracer methods compared. RBS and Nuclear Reaction Analysis (NRA, to be discussed in the next section), have depth resolutions typically ranging from a few to several tens of nm, thus permitting the measurement of diffusion coefficients down to relatively low temperatures (e.g., < 700 °C) applicable in a wide range of geologic settings, and diffusivities down to ~10~23 m 2 sec _1 , avoiding in many cases the uncertainties of large down-temperature extrapolations. In addition, these methods are essentially "non-destructive" since there is no physical removal of material during the analysis.

Basic principles of RBS In RBS analysis, the energies of ions from scattering events are measured with a detector which is usually positioned at an angle of nearly 180° with respect to the incident beam, so it is referred to as "backscattering." These collisions can be described by the equations of classical kinematics. The energies of the detected backscattered particles (En) will depend on the masses of the target atoms, related to the incident energy (E0) by K„E0 = E„, where Kn is the kinematic factor: (m] - Mf sin2 e)" 2 + Mi cos9

M.+M i

(4)

n

Here, 0 is the laboratory angle through which the ion is scattered with respect to its incident direction, and M, and Mn are the masses of the incident particle and the target atom from which it is scattered, respectively. It can be clearly seen that backscattered particle energies will be higher for heavier target atoms by simplifying the expression in taking the limiting case of 9 = 180°:

M n +M.i

138

Cherniak, Hervig, Koepke, Zhang, Zhao

The probabilities of occurrence of scattering events, or Rutherford scattering cross-sections, can also be determined from first principles. The scattering cross-sections are functions of target and incident particle mass and charge, as well as the scattering angle and the incident particle energy at the time of scattering. The differential scattering cross-section, or crosssection per unit solid angle Q, is described by the following:

(6) Rutherford scattering cross-sections are larger for higher-Z elements (the cross-section varies as Z2), so the technique is more sensitive for high-Z (and larger mass) species. Detection limits can be down to a few tens of (atomic) ppm, but are considerably poorer for light elements. Since scattering cross-sections can be readily quantified and directly relate atomic concentrations to scattered particle yields, it is possible with RBS to perform quantitative elemental analysis without reference to standards. Only a fraction of the ions comprising the ion beam incident on the sample will be backscattered from target atoms, and detected in the solid angle subtended by the detector. The detected spectrum for a particular element will include signal from scattering at the sample surface (which will have energy K„E0, where Kn is the K-factor for the element n, and E0 is the energy of the incident ion [i.e., initial energy of the ion beam]), as well as signal produced by scattering events with this element from greater depth in the material. Those ions not scattered from the surface will lose energy traveling deeper into the sample, primarily through inelastic collisions with the electrons of the target material, at a rate dependent on the material's density and composition. These lower-energy ions can then scatter from target atoms at depth, losing additional energy as they make their way out of the sample to the detector. If the backscattered particle energies are measured, and the energy loss rates are known or can be determined, information on the depth distribution of this element in the sample can be evaluated as follows: (7) where (dE/dx)in and (dE/dx)au, represent the energy loss per unit distance in the material for the ions going into the sample (before scattering) and out of the sample (after scattering), also referred to as the stopping power, of the material. Stopping powers have been evaluated for ions in all elemental targets based on semi-empirical fitting of experimental data (Ziegler et al. 1985), with stopping powers for compound targets obtained through application of the Bragg Rule, which weights the contribution to the stopping power of each element in the target according to its mole fraction in the material. These values can be calculated for compounds for a range of ions and energies using, for example, the software SRIM-2006 (Ziegler and Biersack 2006; www.srim.org). An example RBS spectrum, from an experiment on Dy diffusion in fluorite, is shown in Figure 20. The RBS spectrum is a superposition of signals from all of the elements present in the outer several micrometers of the sample. Those elements with uniform distribution in the sample will each appear as a "step" (with a parabolic curve upward at lower energies because of the E~2 dependence of scattering cross-section) with their edge (at highest energy) at K„Ea. In this spectrum, examples of these are the Ca and F steps for the fluorite matrix. The edge of the step for Ca, the higher mass species, is at higher energy (greater channel number) than the step edge for F. The step height is also smaller for F despite its higher molar abundance than Ca, due to its smaller scattering cross-section. Dy, of considerably greater mass than either of the major matrix elements, is at an even higher energy position, and the signal from this distribution

Analytical

Methods

in Diffusion

139

Studies

Energy (MeV) 0.5

1.0

1.5

2.0

Channel Figure 20. An example of an RBS spectrum taken of a sample from a diffusion experiment, in this case measuring Dy diffusion in fluorite. The "steps" toward the lower channel numbers (lower energies) represent contributions from the major elements (Ca and F). The energies of the step rises are proportional to the elemental masses, and step heights depend on Rutherford scattering cross sections, proportional to the square of the atomic number. The contribution from Dy (also shown in the inset figure) indicates a nonuniform distribution of this element, a result of it diffusing a few hundred nm into the fluorite. See text for additional details.

(which only extends a few hundred nm into the material) is well-separated from the RBS spectra constituents due to the major elements. The Dy peak maximum is about one-fifth of the nearsurface height of the Ca step, but its actual concentration is much lower, as the scattering crosssection for Dy is nearly 11 times that of Ca. Elements with non-uniform distribution, like Dy in this case, will have their leading edge at K„E0 provided there is significant concentration of the species near the sample surface, but yields will not be step-shaped. For example, a profile for a species diffusing into a material will appear as a peak with count yield decreasing with depth, extending only to the depth of penetration of the diffusing species. In order to obtain depth distributions and elemental concentrations from RBS spectra, channel number (for the detection of particles recorded in a multichannel analyzer as in this example) is directly related to detected particle energy through calibration with major element edge positions from the sample itself or from standards analyzed in the same session. The energies in turn can be related to depth in the material (with lower detected energies of particles scattering from a given element corresponding to greater depth in the material) by using the energy loss rates in the material, as outlined above (e.g.. Eqn. 7). The count yields for the element of interest can be directly related to concentrations through the scattering cross-sections for that element. As this example illustrates, RBS is well-suited for analysis where there is a heavy impurity or species in a relatively light matrix. In contrast to some other depth profiling methods, the entire profile, including all elements present above detection limits, will be obtained in a single

140

Cherniak, Hervig, Koepke, Zhang, Zhao

analysis, so potential interferences must be considered. In diffusion studies, an ideal case would be a material composed of relatively light elements with a heavy diffusant. Diffusion profiles under best circumstances would be sufficiently short such that they would be separated from the contributions to the RBS spectrum from the major elements comprising the sample material, leaving the diffusion profile with little or no background, as is the case in the example shown in Figure 20. Although the simultaneous collection of signals from all elements may be a drawback under some circumstances, it can also be a benefit, as sample stoichiometry can be monitored, as well as changes in concentrations when multiple species are involved in chemical diffusion.

Depth and mass resolution Depth resolution of standard RBS (a few MeV energy incident helium beam, standard solid-state surface barrier detector to detect backscattered ions) in the near-surface region is about 10-20 nm. Solid-state surface barrier detectors are most common for RBS analysis given their low cost and reliability, but other types of detectors may also be used. Time-of-flight detectors, for example, may be used for backscattering when heavy ions are the incident beam, as ions heavier than Li seriously degrade solid-state surface barrier detector performance, and electrostatic analyzers have found application in various areas, including lower-energy ion scattering studies (e.g., Leavitt and Mclntyre 1995). With magnetic spectrometers, electrostatic analyzers and other high-resolution detection systems (e.g., Lanford et al. 2000), near-surface resolution can be improved by about an order of magnitude over that provided by typical surface barrier detectors. At greater depth in materials, depth resolution will degrade, primarily due to energy spread of the incident beam as it travels through the sample, referred to as "straggle." The potential for degradation of depth resolution in this manner also has to be considered in designing experiments to be analyzed by RBS. For example, diffusion couples or "thick film" (greater than several tens of nm) sources are generally precluded because the beam either may not "see through" a thick later to the profile beneath, or depth resolution in measurements may be seriously compromised if the beam must go through a very thick surface layer to reach the profile. In addition, samples should be well-polished (or good natural mineral growth or cleavage faces used) to avoid additional loss of resolution due to surface roughness. However, when films on sample surfaces are thin, these sources can be employed quite successfully in diffusion studies using RBS (and NRA) (e.g., Dimanov et al. 1996; Bejina and laoul 1996). Other studies have used powder sources (e.g., Cherniak and Watson 1992, 1994), diffusion couples where materials can be readily separated (e.g., Cherniak et al. 2007), ion implantation of diffusants (e.g., Cherniak et al. 1991; Martin et al. 1999), gas sources (e.g., Watson and Cherniak 2003), and fluid media (provided care is taken to avoid dissolution of the sample and precipitation on sample surfaces; e.g., Watson and Cherniak 1997); in short, a range of source types can be used, provided a clear, smooth sample surface or surface area can be retained for analysis following diffusion experiments. Beam spots for standard RBS are typically in the range of 0.5-2 mm in diameter. Smaller beam spots (down to a few micrometers) can be obtained on specially designed microbeam lines with additional focusing, but usually with poorer detection limits because of much lower beam currents typical of these configurations. Detection limits for heavy elements are good with RBS (typically to a few tens of ppm), but mass resolution becomes poorer in higher mass ranges because of the nature of the mass dependence for the kinematic factor. Therefore, there can be difficulties in separating out signals in RBS spectra from elements whose masses are in close proximity (e.g., the rare earths), and isotopes cannot be readily distinguished under most circumstances; some exceptions are described below. However, high resolution detection systems can improve mass, as well as depth, resolution. As noted above, detection limits for light elements are relatively poor in RBS,

Analytical Methods in Diffusion

Studies

141

but in some cases non-Rutherford scattering (to be discussed below) can be used to improve these detection limits, and another ion beam technique, nuclear reaction analysis (NRA), to be discussed in the next section of the chapter, may be used in some circumstances for profiling of low to medium Z atomic number elements. Above certain energies, depending on the projectile, scattering cross-sections will depart from classical Rutherford values as a result of short-range nuclear forces. For a standard backscattering energy for 4 He of 2 MeV, cross-sections for elements with Z greater than ~6 will be Rutherford, while for 2 MeV protons this value will be Z ~20; for 4 He of 4 MeV energy non-Rutherford behavior will be observed for elements with Z less than ~15. Departure from Rutherford scattering behavior often results in large increases in scattering cross-sections; a good example is the non-Rutherford scattering cross section of 4 He from 1 6 0 for a 4 He energy just above 3 MeV (e.g., Cheng et al. 1993; Leavitt and Mclntyre 1991; Demarche and Terwagne 2006) where scattering cross-sections are more than an order of magnitude greater than Rutherford values. While these enhanced cross-sections have not yet found significant direct use in diffusion studies in minerals, they have been exploited to supplement other measurements, as in the study of Cherniak (2000), where non-Rutherford a scattering from 28Si at 6.6 MeV was used to explore substitutional processes involved in REE diffusion in fluorapatite. Example applications of RBS in diffusion studies Diffusion of elements of geochronologic interest, such as Pb, can present optimal circumstances for RBS analysis, since these elements are generally of high atomic mass. Much of the focus on measuring diffusivities of atomic species of importance in geochronology has been directed toward diffusion in accessory minerals, including zircon, apatite, and monazite. Given the very slow diffusion rates of most atomic species in many accessory phases, RBS, with its superior depth resolution, has found effective application in Pb diffusion measurements (e.g., Cherniak et al. 1991, 2004a; Cherniak and Watson 2001; Gardes et al. 2006, 2007). Since the work of Sneeringer et al. (1984) on diopside, Sr diffusion in other minerals has been investigated with RBS, including feldspars (Cherniak and Watson 1992, 1994; Cherniak 1996), apatite (Cherniak and Ryerson 1993), fluorite (Cherniak et al. 2001), and calcite (Cherniak 1997). In some of these studies, insight into substitutional processes has been obtained by monitoring changes in major constituents in RBS spectra; for example, changes in near-surface Ca in K in calcic plagioclase and K-feldspar, respectively, following Sr diffusion experiments (Cherniak and Watson 1992, 1994) suggest exchange of Sr for these species in feldspars (Fig. 21). Diffusion of a range of other minor and trace elements, including rare-earth elements (e.g., Cherniak et al. 1997a; Martin et al. 1999; Cherniak 2000), high field strength elements (e.g., Cherniak et al. 2007) and actinides (Cherniak et al. 1997b; Cherniak and Pyle 2008) have been characterized with RBS. Measurements of diffusion of noble gases were among the earliest applications of RBS in diffusion studies in minerals, where Melcher et al. (1983) measured Xe diffusion in minerals present in meteorites (olivine, feldspar and ilmenite) in order to better understand the early chronology of the solar system. More recently, the diffusion of Ar has been measured in quartz (Watson and Cherniak 2003) as well as olivine, enstatite and corundum (Thomas et al. 2008; Watson et al. 2007) with RBS, to evaluate the potential of these major rock-forming minerals as reservoirs for noble gases within the earth. RBS measurements of Ar diffusion in quartz have also recently been coupled with measurements of Ar by laser ablation ICPMS (Baxter et al. 2006). Since these analytical methods address different length scales, their application to measurements of diffusion in the same samples can increase understanding of transport processes for noble gases in minerals. Unfortunately, because of the close energetic proximity of signals from Ar to those from K and Ca in RBS spectra, measuring Ar diffusion in minerals of interest in 40 Ar/ 39 Ar dating (e.g., K-feldspar) with this method is problematic.

Chemiak, Hervig, Koepke, Zhang, Zhao

142

Energy (MeV) 0.5

1.0

Channel

1.5

Figure 21. RBS spectrum of Sr diffusion in anorthite. In (a), the full spectra from a Sr diffusion experiment (grey line) and an untreated specimen of anorthite (black line) are plotted. The figure in (b) illustrates complementary Ca-Sr exchange in the process of Sr chemical diffusion, where the spectrum from the diffusion experiment shows Sr uptake accompanied by a decrease in near-surface Ca.

Channel

In certain cases, RBS can be used for measurements of major element diffusion and interdiffusion in minerals. For example, isotopic tracers can be used when there is sufficient mass separation between them and the dominant natural isotope of the element of interest (as in the case of -"'Si or 44Ca), and interdiffusion may be studied when one of the species significantly differs in mass from the other (and the heavier species is introduced through a thin film or removable source), as in the case of Fe-Mg interdiffusion. Ca self-diffusion has been measured by RBS in natural and synthetic diopside (Dimanov and Ingrin 1995; Dimanov et al. 1996; Dimanov and Jaoul 1998) using a 44Ca tracer deposited in a RF-sputtered isotopically enriched thin film of diopside composition. Also in diopside, RBS has been used to measure (Fe.Mn)-Mg interdiffusion (Dimanov and Sautter 2000), while Fe-Mg interdiffusion has been measured by RBS in both orthopyroxene (ter Heege et al. 2006) and olivine (Bertran-Alvarez et al. 1992; Jaoul et al. 1995a; Dohmen et al. 2007). Silicon diffusion has been measured by RBS. using 10Si tracers, in a range of mineral phases, including olivine (Houlier et al. 1988. 1990), pyroxene (Bejina and Jaoul 1996). zircon (Cherniak 2008). quartz and feldspars (Bejina and Jaoul 1996; Cherniak 2003). Recently. S diffusion in pyrite and sphalerite (Watson et al. 2009) has been measured by RBS using a 14S tracer.

Analytical Methods in Diffusion

Studies

143

In addition to single crystal studies, RBS has also been used in investigations of diffusion in polycrystalline materials, for example in characterizing diffusion of heavy elements such as cadmium, the rare earths, and actinides with applications to environmental problems and radioactive waste storage. RBS has been used by Toulhoat et al. (1996) to measure Cd diffusion in hydroxyapatite; diffusion of colloids in granitic rocks has been studied by RBS (Alonso et al. 2007a,b) to better understand transport from waste repositories and migration of other contaminants. Although geologic applications of RBS in diffusion studies have largely focused on diffusion in crystalline materials, RBS has also been used to investigate diffusion controlledprocesses in glasses, including dynamic oxidation in Fe-bearing aluminosilicate glasses and basaltic melts (Cooper et al. 1996; Cook and Cooper 2000).

NUCLEAR REACTION ANALYSIS (NRA) In certain respects, NRA is similar to RBS in that a beam of ions produced in an accelerator is used to probe the chemical composition of the outer few micrometers of a material. However, for NRA, the ions are sufficiently energetic to overcome the Coulomb barrier and interact with specific target nuclei, resulting in a nuclear reaction. Products of these reactions (which may be gamma rays or light charged particles) are then detected, providing information about the depth distribution and concentrations of specific species in the material. Because this method relies on inducing nuclear reactions, it is isotope-specific. NRA has been applied in studies of diffusion in geologically significant materials over several decades. Among the earliest studies was an investigation of oxygen diffusion in quartz using the nuclear reaction 1 8 0(p,a) 1 5 N (Choudhury et al. 1965). By the 1980s, applications of Nuclear Reaction Analysis to diffusion problems became more frequent. Given the beam energies accessible to accelerators commonly used for these techniques (typically up to several MeV), NRA is most often exploited for the detection of light elements. Many useful nuclear reactions, especially those induced by protons, require beam energies of only a few hundred keV, so relatively low-energy accelerators can be used for some types of measurements. Instrumentation required is generally similar to that for RBS, although nuclear reaction analysis methods that rely on the measurement of gamma rays induced in the reaction require scintillation detectors rather than the charged particle detectors used for RBS, and for NRA when charged particles are the product from the nuclear reaction detected. The isotopespecificity of NRA makes it suitable for a range of applications involving a tracer species, including diffusion studies. A good example is the use of an , 8 0 tracer and the nuclear reaction 18 0(p,a) 1 5 N to measure oxygen diffusion in minerals. The expression in the previous sentence is the abbreviated way of writing the reaction 15 p + 180 N+a (8) where a proton beam is used to induce a nuclear reaction with an 1 8 0 atom. The products of the reaction are a 15N atom and an alpha particle, with the latter species being detected.

NRA can be done in either resonant or non-resonant modes, depending on the reaction used and its cross-section. The reaction cross-section describes the probability of occurrence of a specific reaction as a function of incident particle energy and the angle between the incident particle and measured reaction product. As a function of energy, reaction cross-sections can have narrow energy-width regions of large cross-section ("resonances"), with regions of relatively small cross section in energy regions above and below these resonances; for some reactions there may also be broader-width energy regions with enhanced cross-sections (e.g., Fig. 22). For application of NRA in the resonant mode, the ideal reaction would have a large cross-section at the resonance energy Er and negligible or comparatively small cross-section

Cherniak, Hervig, Koepke, Zhang, Zhao

144 70

18

0(p,a) N

60 E c 50

165°

o o

0 = 500

15

600

700

800

Proton energy (keV)

900

1000

Figure 22. An example of a reaction cross section as a function of incident ion energy, in this case the reaction l8 IS, 0 ( p , a ) b N at a laboratory angle (angle between incident beam and detector) of 165°. The reaction cross-section (which describes the probability of the reaction occurring) shows a broad region of enhanced cross-section in the range of 800 keV, and a narrow energy region of enhanced cross-section (a resonance) at -629 keV. The cross section is plotted as millibarns (where a barn is 10~24 cm 2 ) per unit of solid angle (in steradians). Cross section data are from Amsel and Samuel (1967).

at other nearby energies. Perhaps the best example is the 'H + 15N —> 12C + 4 He + y resonant reaction (resonance energy 6.385 MeV), which is used to profile 'H (with a 15N beam), one of the few means to obtain quantitative depth profiles of H in materials. This method was used by Laursen and Lanford (1978) in their pioneering study on the hydration of natural obsidian, in which they explored the mechanism of hydration and developed an interdiffusion model. With an incident proton beam, this same reaction can also be employed to profile l5 N. The application of resonant profiling is illustrated schematically in Figure 23. When the incident beam is at the resonance energy En the concentration of the species of interest at the sample surface is detected. Depth profiling is done by increasing the incident beam energy, thus increasing the depth in the material at which the resonance (and enhanced yield of the products of the nuclear reaction) occurs. Depth scales are determined by the difference between the beam energy and the resonance energy, and the rate of energy loss for the incident ions in the material. Concentrations of the element or isotopic species of interest at a particular depth are determined from the detected yields of the product of the nuclear reaction for the number of particles of the incident beam delivered to the sample, with detected yields of products of the nuclear reaction dependent on the reaction cross-section at the resonance energy, beamdetector geometry, and efficiency of the detector. Resonant profiling is also used with a range of other nuclear reactions that have found application in diffusion studies, such as 27Al(p,y)28Si (using the 992 keV resonance, e.g., Sautter et al. 1988) to measure A1 diffusion in diopside, 30 Si(p,y) 31 P (using the 620 keV resonance, e.g., Jaoul et al. 1995b; Bejina and Jaoul 1996; Cherniak 2003, 2008) to measure Si diffusion in quartz, diopside and zircon, and 48Ti(p,y)49V (Cherniak and Watson 2007) to measure Ti diffusion in zircon. When reaction cross-sections are sufficiently large and vary smoothly over an extended energy range, the entire depth profile may be obtained using a single incident beam energy. This is referred to as non-resonant profiling. In non-resonant mode, the depth scales are determined by energy loss rates of incident and outgoing product particles from the nuclear reaction. Non-resonant profiling can be used with many reactions in which particles (e.g., protons, deuterons, 3 He, a particles), rather than y rays, are the detected reaction product, since y rays do not experience the energy losses particles do in passing through the sample material and thus cannot provide depth information. The measurement of 1 8 0 using the nuclear reaction 1 8 0(p,a) 1 5 N is a typical application of non-resonant profiling. 1 8 0 profiling is most often done in non-resonant mode at incident

Analytical

in Diffusion

Studies

145

Sample Surface

a

15

Methods

N Beam depth

b X

15

N Beam=s>

At

E 0 >E,

Resonance Energy Figure 23. Schematic drawings illustrating the method of resonant NRA. This example shows the use of the reaction 'H( b N,ay) l 2 C to measure 'H. An incident b N beam is used, and y rays of characteristic energy produced in the reaction are detected. When the 15N beam is at the resonant energy (£,.), hydrogen concentrations at the sample surface are measured. At higher incident b N energies, H concentrations at depth in the material are sampled, with the depth (x) determined by the difference in energy between the incident beam and the resonant energy, divided by the energy loss rate for LYN in the sample material.

energies of around 800 keV (e.g.. Reddy et al. 1980; Ryerson et al. 1989). taking advantage of a region of fairly large and smoothly varying reaction cross-section (Amsel and Samuel 1967) (Fig. 22). A spectrum of a particles over a range of energies is collected, representing contributions from 1 8 0 at various depths in the material (Fig. 24). This reaction has been used for oxygen diffusion studies in many mineral phases, including olivine (Reddy et al. 1980: Gerard and Jaoul 1989: Ryerson et al. 1989; Jaoul et al. 1980), zircon (Watson and Cherniak 1997). rutile (Derry et al. 1981; Moore et al. 1998). monazite (Cherniak et al. 2004b) and titanite (Zhang et al. 2006). It should be noted that 1 8 0 also can be profiled using the resonant technique, with, for example, the sharp resonance at 629 keV (Fig. 22). In addition to resonant and non-resonant approaches described above, "hybrid" methods are sometimes used in which energies are varied, but the cross-section is sufficiently broad that the signal is comprised of integrated contributions from a significant range of depths in the material at each energy step. This approach is usually employed in cases of faster diffusing species where diffusion distances are comparatively long. Some examples are measurement of He diffusion using the 3 He(d,p) 4 He reaction (e.g., Cherniak et al. 2009, Miro et al. 2006). Interpretation of NRA spectra and conversion to concentration profiles is slightly more complicated than for RBS for a few reasons. In RBS, Rutherford scattering cross-sections can be described analytically; in contrast, cross-sections for nuclear reactions depend on nuclear structure and can vary quite dramatically with incident particle energy and with elemental and isotopic species. Cross-sections must be empirically determined through careful measurement as functions of incident energy and beam-detector angle. Fortunately, cross-sections for many useful reactions have been measured and tabulated for analytical purposes. Tables of reactions

Cherniak, Hervig, Koepke, Zhang, Zhao

146

12000 10000 W

8000

o

6000

c

-

4 backscattered ; protons

4000

a particles (x10)

2000

0

.

0

500

1000

.

,

1500

.

2000

Channel Figure 24. An example spectrum for non-resonant nuclear reaction analysis, in this case measuring oxygen diffusion in olivine using the 1 8 0(p,a) 1 5 N reaction. The count yields in the alpha peak (toward the right of the figure) are proportional to the l s O concentration in the material, with yields at lower channel numbers (and lower energies) corresponding to l s O at greater depths in the material. Also detected in the spectrum are protons backscattered from the constituents of the olivine; these are toward the lower channel number (lower energy) end of the spectrum. Because the energy released in the nuclear reaction is relatively large (3.98 MeV) the alpha particles produced in the reaction will be fairly energetic, with much larger energies than the backscattered protons. However, reaction cross-sections are generally much smaller than Rutherford scattering cross-sections, so the count yields of alpha particles produced in the reaction are considerably less than those of the backscattered protons. Samples from Ryerson et al. (1989).

used in NRA can be found in Cherniak and Lanford (2001) and Tesmer and Nastasi (1995); the online database IBANDL (the Ion Beam Analysis Nuclear Data Library, at http://www-nds. iaea.org/ibandl/) has a frequently updated compilation of reaction cross-sections as a function of energy for a range of nuclei and projectiles. An additional complication in interpreting spectra is that the incident particles and product species from the nuclear reaction will not be the same, as they are in elastic scattering, and thus can have quite different energy loss rates in the sample material. Using the example of the ls O(p,a) 15 N reaction, the incident protons will lose energy at a much smaller rate while traversing a given distance in the material than the alpha particles that are the product of the reaction. When gamma rays are the species measured, for example when profiling H and Al with the reactions ' H ^ N , ay) 12 C and 27Al(p, y)28Si, respectively, depth information can only be obtained through varying the energy of the incident beam since the gammas will travel through the material without losing energy as will product particles such as protons, alphas, or deuterons. In some cases, a multiple ion beam techniques can be employed in analysis of a single sample or set of samples to provide added information about the diffusional process. For example, RBS measurements of Pb and Sr diffusion in feldspars (Cherniak and Watson 1992, 1994; Cherniak 1995a) were supplemented by NRA measurements of Al and Na in order to provide insight into substitutional mechanisms involved in Pb and Sr exchange in alkali feldspars and sodic plagioclase. Similarly, measurements of phosphorus using the reaction 31 P(a,p) 34 S were made to accompany RBS measurements of REE diffusion profiles in zircon to investigate the role of the substitution REE +3 + P+5 —> Zr+4 + Si+4 in rare-earth element diffusion in zircon

Analytical Methods in Diffusion

Studies

147

(Cherniak et al. 1997b). In addition, in some studies of Si diffusion, both RBS and NRA (using the 30 Si(p,y) 3l P reaction) have been employed to measure Si diffusion on the same samples, with good agreement among results (e.g., Bejina and Jaoul 1996; Cherniak 2003, 2008).

ELASTIC RECOIL DETECTION (ERD) ERD (Elastic Recoil Detection) is another ion-beam method for measuring depth profiles of light elements (Z = 1 to -10). It is like RBS in that it relies upon elastic scattering; however, the incident ions must be heavier than the element to be profiled. Typical incident beams employed (e.g., Petit et al. 1990; Cookson 1991) are 4 He at a few MeV (for profiling hydrogen) or heavier ions such as C, Si or CI at higher energies (~ 1 MeV/amu). The light target atoms will recoil in a forward direction upon collision with the heavier incident ions; if the sample is tilted these recoiled atoms will escape the sample and can be detected. ERD can also be performed in transmission mode, where the detector is placed behind the sample and recoiled atoms exiting the back of the sample are detected, but this can only be done for thin (typically a few micrometers) self-supporting films (such as polymers), so this approach is of limited use in geological studies. The energy of the detected recoils Er will provide information about the depth distribution of the recoiling species, as described in the expression:

(MI+M2) where Er is the energy of the recoiled atom, E0 is the energy of the incident ion, Aij and M2 are the masses of the incident and recoiled species, respectively, and 180° - t(> is the angle between the incident beam and the detector used to detect the recoiled atoms. In general, c|) will be set at angles between 10 and 30°. To prevent scattered incident ions from interfering with the signal from the light recoils, an absorber foil is typically placed in front of the detector to stop these heavier ions. When the absorber foil is used, the detected particle energy is not Er, but a lower value Edet, with

Eilet=Er-5fSf

(10)

where 5^-and 5^-are, respectively, the thickness of the foil, and "stopping power" or energy loss for the ions in the foil per unit thickness. As with RBS and NRA, depth scales for the detected particles are constructed from information on energy loss in the sample material. If an incident ion penetrates to a specific depth (measured normal to the sample surface) before a recoil event occurs, the incident ion will have lost energy in inelastic collisions while traveling through the material. Similarly, the recoil atom will lose energy traveling out of the sample to its surface. If the energy-averaged stopping powers for the incident and recoil species are 51,- and Sr, respectively, the initial depth of the recoil species can be determined from the expression depth = (Edet + 5,S, - w

i

— + I sin a smp I

(1')

where a and (3 are the acute angles between the sample surface and the incident beam, and the sample surface and the detector. Depth resolution will depend on the energy loss variations of both the incident and recoiling atoms in the material and the recoiling atoms in the foil, and will typically be in the range of tens of nm. Better resolution (by up to an order of magnitude) can be obtained by eliminating the stopper foil and using other detection systems, such as a magnetic spectrometer, rather than standard surface barrier detectors. Although it holds considerable promise for analysis of light elements, thus far ERD has been used to only a limited extent in diffusion studies of geological materials. An example is

148

Cherniak, Hervig, Koepke, Zhang, Zhao

measurement of He diffusion in fluorapatite by Ouchani et al. (1998). Additional information on the technique (along with schematics of experimental configurations for ERD analysis) can be found in Barbour and Doyle (1995).

FOURIER TRANSFORM INFRARED SPECTROSCOPY Infrared (IR) spectroscopy uses infrared radiation to probe the chemical species and molecular clusters in solids, liquids or gases by exciting the vibrational modes (which absorb in the infrared region) in molecules or clusters. Infrared radiation is electromagnetic radiation with frequencies lower than visible light but higher than radio wave frequencies. The infrared spectrum covers the wavelength range from 0.75 to 100 (im, longer than visible light (400-750 nm) but shorter than the radio wave. The infrared spectrum is often subdivided into the nearinfrared (NIR, wavelength 0.75-3 urn), mid-infrared (MIR, 3-30 urn), and far-infrared (FIR, 30-100 (im) regions. Both the frequency and wavelength are related to the infrared radiation energy. The most common way to express infrared energy in plots of infrared spectra is by wavenumber, which in spectroscopy equals the number of waves per unit length (most often expressed in units of crrr 1 , but mm will be used here). Infrared spectroscopy is widely applied in identification of specific chemical bonds, species and clusters. Its application in quantitative analyses (the focus of this chapter) is mostly in the measurement of H 2 0 and C 0 2 concentrations, as well as concentrations of the individual species (molecular H 2 0 and the hydroxyl ion for total H 2 0 , and molecular C 0 2 and the carbonate ion for total C0 2 ). In the geological literature, these analytical capabilities were pioneered by Stolper (1982a,b), Aines and Rossman (1984), Fine and Stolper (1985, 1986), Newman et al. (1986), Paterson (1986), Rossman (1988), and Bell and Rossman (1992). In addition to the aforementioned species, NH 4 + concentrations in mica (Busigny et al. 2003, 2004) have also been measured by IR spectroscopy. Although the number of components that can be analyzed by infrared spectroscopy is limited, H 2 0 and C 0 2 are major volatile components in the Earth and in melts and minerals, and they cannot be analyzed easily by other methods. Furthermore, infrared spectroscopy is a non-destructive method (but preparation of doubly-polished sections for analysis will cause sample loss) with high sensitivity, high precision, and high spatial resolution. Recent development has resulted in nanoSIMS having even better sensitivity and spatial resolution than infrared spectroscopy (e.g., Saal et al. 2008), but infrared spectroscopy is still the only method available to quantitatively determine the concentrations of species, which may provide critical structural information and further insight into diffusional processes. Below we introduce the basic principles describing the molecular vibrational modes that often cause infrared absorption, the instrumentation used in infrared spectroscopy, and applications to measurement of H 2 0 and C 0 2 concentrations and their use in diffusion studies.

Vibrational modes and infrared absorption Basic principles of vibrational motion in multi-atom "molecules" (including ionic clusters) have been discussed in numerous texts and papers (e.g., Colthup et al. 1990; Ihinger et al. 1994; Beran and Libowitzky 2004). The vibrational motion absorbs in the infrared region. For an OH group in a structure, there is a stretching mode (Fig. 25), absorbing near 355 m m 4 . For an H 2 0 molecule (H 2 O m ), there are 3 vibrational modes (symmetric stretching at 365 m n r 1 , asymmetric stretching at 375 mm -1 , and bending at 163 mm" 1 ; Fig. 26). For a C 0 2 molecule, because it is a linear molecule, there are 4 vibrational modes (symmetric stretching at 139 m m 4 that is IR inactive, asymmetric stretching at 235 mm" 1 , and two identical bending modes at 67 mm" 1 ; Fig. 27). For a carbonate ion (C0 3 2 ", a triangular ion), there are 6 independent vibrational modes (symmetric stretching at 106 mm" 1 that is IR inactive, two asymmetric stretching at about 142 mm" 1 that may split into a pair of peaks, two in-plane bending modes at about 68 mm" 1 , and one out-of-plane bending mode at 88 mm" 1 ).

Analytical

Methods

in Diffusion

Studies

149

Figure 25. Stretching mode of an OH cluster, where the small solid circle represents H, and the large open circle represents O. The diagram only shows one direction of motion during vibration. By reversing the direction of all arrows, one gets the other type of motion in this and other examples below. Symmetric stretch

Asymmetric stretch

Bending

Figure 26. Vibrational modes of the HiO molecule, where the small solid circles represent H, and large open circles represent O. Symmetric stretch

Asymmetric stretch

Bending

t

J

Figure 27. Vibrational modes of the COi molecule, where small solid circles represent C, and large open circles represent O.

Based on quantum mechanics, the quantized energy level of a harmonic oscillator (an approximation for a vibrational mode) can be expressed as follows (Fig. 28): E = (?i+l/2)lm, where n = 0, 1, 2,...

(12)

where E is the quantized energy level, n is the quantum number, h is the Planck constant, and v is the characteristic vibrational frequency, which is related to the force constant k as:

where m" is the reduced mass, for a diatomic cluster defined as m'=

1

'

t

(14)

—+— »i,

in2

where m^ and m2 are the masses of atom 1 and atom 2 in the diatomic cluster. Based on this definition, the reduced mass m is smaller than either m^ or m2. For example, for O-H stretching. mi= 1.0079 u, and m2 = 15.9994 u, leading to a reduced mass m" = 0.9482 u. On the basis of Equation (12), the lowest energy (ground state) of a harmonic oscillator is (l/2)/iV when n = 0, which is called the zero-point energy. By absorbing an energy of hv (corresponding to the fundamental vibrational band in an IR spectrum), the harmonic oscillator can be excited to the energy of 1.5hv (Fig. 28). By absorbing an energy of 2hv (corresponding to the first overtone in an IR spectrum), the harmonic oscillator can be excited to the energy of 2.5hv, and so on. If the oscillator is perfectly harmonic, the selection rule states that overtones should not occur. Because the energy versus distance relation for a chemical bond is slightly different from that of a harmonic oscillator (Fig. 29). (1) the overtones do occur with an

Cherniak, Hervig, Koepke, Zhang, Zhao

150

Harmonic approximation

Energy level SJD U CJ

s a

Interatomic distance Figure 28. Quantized energy levels of a harmonic oscillator.

Figure 29. Potential energy of a diatomic molecule as a function of distance between the two atoms. The solid curve is the actual energy and the dashed curve is the harmonic approximation. The r that corresponds to the energy minimum is the bond length (or equilibrium distance r c ).

intensity about two orders of magnitude lower than the fundamental modes, and (2) the first overtone may occur at a slightly different wavenumber from exactly 2 times the fundamental band. A photon may also simultaneously excite two modes (combination modes). The intensity of the combination modes is between those of the fundamental modes and overtones. Not all vibrational modes can be detected by infrared spectroscopy. If a molecule has a zero dipole moment (or a center of symmetry), and if the vibrational mode is symmetric so that it does not generate a dipole moment, it would be IR-inactive. meaning that in theory it cannot be excited by infrared radiation. In practice, however. IR-inactive modes still absorb in the IR region but the absorption is very weak. For example, H 2 , N 2 and Oi are all IR-inactive. On the other hand, if a molecule does not have a center of symmetry (such as the case of a CO molecule), or if the vibration is asymmetric for a molecule with center of symmetry

Analytical

Methods

in Diffusion

Studies

151

(such as asymmetric stretching of the 0 = C = 0 molecule), the vibrational mode is IR-active. When a vibrational mode is IR-inactive. it is Raman-active. When a vibrational mode is IRactive, its absorption in Raman is weak. Hence. IR spectroscopy and Raman spectroscopy are complementary. For example, H 2 is IR inactive but can be easily detected by Raman for diffusion studies (e.g., Shang et al. 2009). Furthermore, the spatial resolution of Raman spectroscopy is much better than IR spectroscopy, especially because the Raman beam can be focused to a specific spot (including a specific depth in a sample) with 1-2 |am resolution (e.g.. Di Muro et al. 2006). However, the reliability of using Raman spectroscopy for quantitative analyses has been debated due to complications in its calibration when internal calibration has been used (e.g.. Thomas 2000; Arredondo and Rossman 2002). Recent attempts using external calibration (or the "comparator o n fundamental technique") show promise for (HXL and OH) determining total H 2 0 contents (Di Muro et al. 2006: Thomas et al. 2008). Even though B determining total H 2 0 contents may be possible, extracting concentrations of hydrous •fi < species (H 2 O m and OH) does not seem to be possible (Behrens et Non-H,0,„ al. 2006). Due to its high spatial OH resolution and its ability to be focused at specific depths in a 550 600 300 350 400 450 500 250 material, future development Wavenumber (mm"1) may potentially make Raman 0.14 spectroscopy a much more powerful tool in quantitative analyses of volatile species and in diffusion studies. Two infrared spectra are shown in Figure 30, one for a low-H 2 0 glass in which the fundamental OH stretching band (at 355 mm -1 ) can be seen clearly, and one for a highH 2 0 glass, in which the first overtone (about 710 mm -1 ) of the fundamental OH stretching band, and the combination bands at 452 mm"1 for X-OH (this band is characteristic of OH not associated with H 2 0 molecules) and at 523 m n r 1 for H-O-H (this peak is characteristic of molecular H 2 0) can be observed. The fundamental stretch peak is too high (over-scale) to be shown in Fisure 30b.

0.04 400

450

500

550

600

650

Wavenumber (mm' 1 )

700

750

Figure 30. Two IR spectra, one for low H 2 0 content and one for high H 2 0 content. The sharp peak at 235 mm - 1 is the asymmetric stretching of molecular C 0 2 ; the peak at 355 m n r 1 is the fundamental stretching of OH (which indicates the sum of H 2 O m and OH because H 2 O m also contains OH bonds); the peak at 450 m n r 1 is a combination mode indicating OH, the peak at 523 mm - 1 is a combination mode indicating H 2 O m , and the peak at 710 m n r 1 is the first overtone of OH fundamental stretching. From Zhang and Behrens (2000) and Ni et al. (2009a).

152

Cherniak, Hervig, Koepke, Zhang, Zhao

Instrumentation for Infrared Spectroscopy The classical infrared spectrometer uses the wavelength dispersive method to measure infrared spectra. Even though it is no longer used, we introduce the principles of the wavelength dispersive infrared spectrometer because their explanation is straightforward. The dispersive infrared spectrometer consists of four parts: energy source, prism, sample chamber, and detector. An energy source (which depends on whether one is interested in MIR or NIR), emits energy at all wavenumbers in the MIR or NIR region. The energy passes through a prism (or diffraction grating or monochromator) so that only the energy in a narrow wavenumber range is allowed through. The energy then passes through an aperture to limit the beam diameter, and then into the sample chamber with or without a sample. Finally, the remaining intensity is determined by a detector. To measure a full spectrum, the monochromator must be able to adjust so that radiation of different wavelengths can be let through. By adjusting the monochromator, the entire wavenumber range can be scanned. If no sample is present, the intensity is denoted as 70. If a sample is present, the detected intensity is denoted as 7. Transmission is defined as /// 0 . Absorbance is defined as -log(/// 0 ) = log(/(/7). Hence, if the sample does not absorb the energy at a given wavenumber, the transmission is 100%, and the absorbance is 0. If the samples absorbs significantly, the transmission becomes much less than 1 and absorbance would have a positive value. When the absorbance is greater than 2 or 3, there is very little energy detectable after passing through the sample; the error then becomes very large and absorbance loses its quantitative meaning. Since the 1990s, Fourier transform infrared spectrometers have become more widely available and classical infrared spectrometers are no longer commonly used. Nowadays, when IR instrumentation is discussed, it is implicitly FTIR, not the classical wavelength-dispersive IR. Classical IR spectrometers have distinct disadvantages compared with FTIR: (i) they only use a small fraction of the total energy from the source at every instant of time, wasting most of the energy and leading to large noise/signal ratios, (ii) one scan takes a long time (several minutes to an hour, depending on the precision needed), (iii) they require intensity calibrations, and (iv) their signals are affected by stray light. Fourier transform infrared (FTIR) technology eliminates all of the above disadvantages. In FTIR, the monochromator is replaced by a unit (Michelson interferometer) that uses a beamsplitter to split the input beam into two equal parts, which are reflected back by mirrors and recombined. The recombined light then goes to the sample and then to the detector. The recombination leads to interference, so that the recombined signal is equivalent to the Fourier transform of the input signal. That is, in the input beam the intensity depends on the wavenumber, which cannot be detected by the detector, whereas in the recombined beam the intensity depends on time, which can be detected by the detector. After the detector records the intensity versus time signal, the digital data are then Fourier transformed back through software into a function of intensity versus wavenumber. Without the sample, the same signal as the input is obtained. With absorption by the sample, the spectrum would display the absorption bands. Different types of IR spectra IR spectra can be taken either on powders mixed and pressed with KBr into a disc, or on glass or single crystal wafers. Powder spectra are for qualitative identification, but for quantitative analyses of geological materials, spectra of glass plates or single-crystal wafers are most often obtained. If the sample is isotropic (including glass and isometric minerals), then a typical FTIR configuration with a nonpolarized (or partially polarized because the mirrors and beamsplitter can cause polarization) beam works well for quantitative analyses. If the sample is anisotropic (all minerals with symmetry lower than isometric minerals), the absorption band intensity will depend on the orientation of the crystal as well as how the E (electric field) vector is oriented.

Analytical Methods in Diffusion

Studies

153

Libowitzky and Rossman (1996) discussed the principles of quantitative IR measurements of anisotropic minerals. In addition to transmission spectra, in which light passes through the sample and goes to the detector, reflectance spectra can also be taken (e.g., Moore et al. 2000), in which light strikes the surface of a sample, is reflected, and then reaches the detector. Peaks in reflectance spectra are usually weak and reflectance spectra are more difficult to quantify. For quantitative analyses, transmission spectra have been the method of choice except for remote sensing. Depending on the infrared wavenumber region one is interested in, it is necessary to decide whether to obtain MIR spectra (typically 40-400 m m - 1 ) , using the energy source, beamsplitter and detector for the MIR region, or to obtain NIR spectra (typically 200-900 m m 4 ) . A typical IR instrument is set up for MIR analyses, with a Globar (a silicon carbide rod heated electrically to high temperature so that it glows) energy source, a KBr beamsplitter, and a D T G S (Deuterated Triglycine Sulfate) detector. If one needs to obtain NIR spectra, a special energy source (tungsten light source), beamsplitter, and detector (LN 2 -cooled InSb) must be added. For C 0 2 analyses, MIR bands are used. For H 2 0 analyses, NIR bands are often used (except for the bending mode) for analyzing water-rich glasses (e.g., > 0.3 wt% H 2 O t ), whereas either NIR or MIR can be used for water-poor glasses (e.g., < 0.5 wt% H 2 O t ) and most minerals. To measure small samples, an FTIR microscope system is necessary, typically equipped with a LN 2 -cooled M C T (mercury-cadmium-telluride) detector that can be used with either MIR or NIR source and beamsplitter. If large samples are available and there is no need to measure the spatial variation of species concentration, then the main chamber of the spectrometer is often used. An advantage of the main chamber is that the rays in the incidence beam are roughly parallel and the beam is perpendicular to the sample surface. Hence, the light will not bend much and the light path length is similar to the sample thickness so results are more reproducible f r o m one lab to another. On the other hand, if the sample is small (e.g., < 0 . 1 m m in diameter) or one needs high spatial resolution (e.g., measurement of a diffusion profile), then a FTIR microscope must be used, in which the incidence beam is focused (converged) to a thin beam. However, the rays in the beam are not parallel but are first converging, reaching a minimum beam diameter, and then diverging. Therefore, the beam will be refracted in the sample, and the beam path is longer than the sample thickness, so reproducibility between different laboratories is not as good as in the case when main chambers are used. Furthermore, because light rays in the beam focused by a microscope are not parallel, they diverge in the sample, leading to a spatial resolution inferior to that indicated by the aperture size. For example, Ni and Zhang (2008) showed that with a 20 |im wide aperture and a sample thickness of 200 (im, the actual spatial resolution (FWHM) is about 30 ^m.

Calibration In order to convert an IR absorption peak intensity f r o m an infrared spectrum to a concentration, it is necessary to carry out a calibration using samples with known concentrations of the species of interest. The calibration for total H 2 0 content (H 2 O t ) and species concentrations is used as example here. No method is available to directly determine the species concentrations in Fe-bearing glasses (i.e., natural glasses), which is a difficulty in the calibration. (For Fe-free glasses, N M R is able to determine the concentration ratio of the two species based on peak area ratio without calibration so that once H 2 O t content is known, concentrations of both H 2 O m and OH can be determined, see Schmidt et al. 2001; Yamashita et al. 2008.) Concentrations of H 2 O t may be determined by manometry (e.g., Epstein and Taylor 1970; Newman etal. 1986), or by Karl-Fischer titration (e.g., Turek et al. 1976; Westrich 1987; Behrens et al. 1996). According to Beer's law, for dilute solutions, the concentration of a species i (C,) and the absorbance at a peak for species i (A,) are related as follows:

154

Cherniak, Hervig, Koepke, Zhang, Zhao 18.015A

(15)

pdtt

where p is the density, d is the thickness, and s, is the molar absorptivity at this peak. The thickness of the sample is measured using a micrometer. Glass density is measured as a function of H 2 O t : for glasses with the same anhydrous composition but different H 2 O t , the density can often be expressed as p = p 0 (l - aC) where C is the mass fraction of H 2 O t and a is a constant (often about 0.6 , e.g., see the summary in Table 3 of Zhang 1999). A more accurate expression uses the partial molar volume of H 2 0 of 12.0(±0.5)xl0~ 6 m 3 (Richet et al. 2000). Either the peak height (linear absorbance) or the peak area (integrated absorbance) may be used for A In the literature, peak height is more often used because it is simpler and the precision is about the same (at least for H 2 0 species). If C is the H 2 O t concentration (mass fraction or weight percent), C, is the H 2 O m concentration, and C 2 is the concentration of H 2 0 present in the form of OH (i.e., it is the mass of two OH groups minus one oxygen), H 2 O t concentration as the sum of H 2 O m and OH can be expressed as follows, since the H 2 O m peak occurs at - 5 2 3 mm - 1 , and the OH peak occurs at - 4 5 2 mm" 1 : 18.015A

(16)

pde452 where A523 and A 452 are the absorbance for the 523 mm 1 and 452 mm the molar absorptivities for the 523 mirr 1 and 452 irmr 1 peaks.

1

peaks, s 5 2 3 and £452 are

Based on Equation (16), from the measurement of H 2 O t concentration by an absolute method, and measurement of peak heights A523 and A452 in an infrared spectrum, one equation relating C, A523 and A452 can be obtained. Because there are two unknowns (s 523 and s 452 ) in Equation (16), it is necessary to conduct the analyses over a large range of values of C to yield numerous linearly independent equations (meaning that Cj and C 2 concentrations must not be proportional as C increases) so that both s 5 2 3 and s 452 can be determined. A feature of the species equilibrium in the melt (quenched to glass) between H 2 O m and OH is that the concentrations of the two species are not proportional to each other, allowing the determination of both s 523 and e 452 . One method to obtain s 5 2 3 and s from data is through direct multi-linear regression of C versus A523 and A452 (Newman et al. 1986) by making the intercept be zero. Another method is to rewrite Equation (16) in the following form (Behrens et al. 1996): 4 5 2

18.015AS2J p dC

e523 18.015A'452 523

e452

pdC

(17)

and plot 1 8 . 0 1 5 A w / ( p d C ) on the v-axis versus 18.015A 4 5 2 /(pdC) on the x-axis. The >'-intercept is s 523 , the x-intercept is s 452 , and the slope is -£¡23^452The above treatments assume that s 5 2 3 and s 4 5 2 are constant (independent of H 2 O t ). Because H 2 O t in glasses is relatively high (typically a couple of wt%), the assumption may or may not be accurate. For example, Zhang et al. (1997) showed that the s523/s452 ratio in rhyolite glasses derived from a calibration based on different samples and the ratio from heating the same sample to different temperatures are different, indicating either the calibrations are not accurate, or s 523 and s 4 5 2 depend on H 2 O t . The uncertainty on s 5 2 3 and s 452 affect the species concentrations more than H 2 O t content. By combining manometry data and heating data, Zhang et al. (1997) carried out a calibration by treating s 523 and s 452 to be a function of A523 and A452. The calibration by Zhang et al. (1997) works well at H 2 O t < 3.0 wt%, but the error increases as H 2 O t increases above 3.0 wt% (Zhang and Behrens 2000). Schmidt et al. (2001) found that the molar absorptivities in alkali aluminosilicate glasses are independent of H 2 0 concentrations, whereas Yamashita et al. (2008) showed that the molar absorptivities in sodium silicate glasses depend on concentrations. The issue of whether and how molar absorptivities

Analytical Methods in Diffusion

Studies

155

in various hydrous glasses vary with H 2 O t at greater H 2 O t remains to be resolved. At low H 2 O t concentrations, the 355 mm"1 peak (fundamental stretching of OH) can be used, which is about a factor of 50 larger than the NIR peaks (452 mm"1 and 523 mm -1 ). At low H 2 O t , OH is the dominant species. Hence, the calibration involves only one molar absorptivity (e.g.,Dobsonetal. 1989; Dixon etal. 1995). For a sample of about 1 mm thickness, the detection limit of H 2 O t using the 355 mm 1 peak is about 1 ppm. Examples for the use of the 355 mm 1 peak include investigation of water diffusion in basalt melt (Zhang and Stolper 1991). For the IR measurement of molecular C 0 2 and C 0 3 2 - concentrations, the calibration procedures are similar when only one of the two species is present in the given glass. Sample preparation for infrared analyses is in general straightforward. The sample must be doubly polished and the thickness well-determined. For transmission infrared spectra, the surface of the sample must be cleaned without leaving a residue that may contribute to the species to be measured. The surface along the IR path must not be covered by plastic tape or double-sided tape. Cracks, especially cracks filled by epoxy, must be avoided because epoxy has a large IR signal that would superimpose on the bands of interest. Applications to geology The most important geological applications of FTIR are to analyze total H 2 0 concentrations, concentrations of H 2 0 species, total C 0 2 concentrations, and concentrations of C 0 2 species in natural silicate glasses and minerals. Absorption bands involving H and C are often at much higher wavenumbers and far separated from structural IR bands due to silicate network vibrations and are hence easily quantified. This is because the masses of H and C are small (leading to small values of reduced mass, Eqn. 14), so that at the same bond strength (force constant k), the vibrational frequencies (or wavenumbers) are much higher than those for other species (Eqn. 13). Stolper (1982a,b) pioneered infrared studies of dissolved water in natural silicate melts and glasses, and was first in the earth sciences to discover the presence of both molecular H 2 0 and hydroxyl groups in silicate melts and glasses and to study the equilibrium between H 2 0 molecules and OH groups. At high total H 2 0 contents (such as > 4 wt%), the concentration of H 2 O m becomes higher than that of OH. With the advancement of IR microbeam techniques, H 2 0 diffusion profiles can be measured, along with species concentrations and total concentration using step scans (Zhang et al. 1991a; Zhang and Stolper 1991) or automatic line scans (e.g., Zhang and Behrens 2000). The analyses of data on species concentration demonstrated that H 2 O m is the diffusing species and that OH is largely immobile compared to H 2 O m (Zhang et al. 1991a;, Doremus 1995; Zhang and Behrens 2000; Behrens et al. 2004; Liu et al. 2004b; Ni and Zhang 2008; Ni et al. 2009a,b; Wang et al. 2009), an inference reached earlier by some glass scientists based on the shape of diffusion profiles representing total H 2 0 at low H 2 O t contents (Doremus 1969, 1973; Ernsberger 1980; Smets and Lommen 1983; Nogami and Tomozawa 1984). The comparison between H 2 0 diffusion and , 8 0 "self' diffusion also led to the realization that oxygen diffusion in melts and minerals in the presence of H 2 0 often occurs through H 2 0 diffusion (Zhang et al. 1991b; Behrens et al. 2007; see also ab initio calculation results by McConnell 1995). Infrared spectroscopy has also been applied to investigate diffusion of hydrous components in olivine (Mackwell and Kohlstedt 1990), pyroxene (Skogby and Rossman 1989; Ingrin et al. 1995), and garnet (Wang et al. 1996). Other applications include speciation studies (e.g., Ihinger et al. 1999; Liu et al. 2004a; Hui et al. 2009), solubility studies (e.g., Blank et al. 1993; Tamic et al. 2001; Liu et al. 2005), and kinetic and geospeedometry studies (Zhang et al. 1997b; Zhang et al. 2000; Wallace et al. 2003). IR measurements of H 2 0 species concentrations and profiles are much more difficult in glasses with high FeO concentration because (i) the degree of transparency of the glass is

156

Cherniak, Hervig, Koepke, Zhang, Zhao

reduced and (ii) the broad Fe absorption band in the NIR overlaps with the 523 mm"1 peak (e.g., Ohlhorst et al. 2001; Liu et al. 2004a,b; Ni et al. 2009b). One way to get around this difficulty is to use Fe-free melts using equimolar amounts of MgO and CaO to replace FeO (e.g., Ni et al. 2009a). Stolper and coworkers also pioneered the infrared study of C 0 2 in silicate melts and glasses (Fine and Stolper 1985, 1986). It was found that dissolved C 0 2 in polymerized (or silicic) melts is present as molecular C0 2 . As the melt composition changes from rhyolite to andesite to dacite to basalt, increasingly more of the dissolved C 0 2 is present in the form of carbonate ions (CO; 2 ). Infrared measurements of C 0 2 and CO,1 concentrations have been applied to investigate total C 0 2 diffusion (Blank 1993; Sierralta et al. 2002; Nowak et al. 2004) and solubility (e.g., Blank et al. 1993; Tamic et al. 2001; Behrens 2010).

SYNCHROTRON X-RAY FLUORESCENCE MICROANALYSIS (jj-SRXRF) The n-SRXRF ("microscopic"-synchrotron radiation X-ray fluorescence) technique is a well-established microanalytical technique for trace element analysis, which in the last decades has been continuously improved in its instrumentation and in the quantification of X-ray spectra (see reviews by Smith and Rivers 1995, Haller and Knöchel 1996, Janssens et al. 2000, Hansteen et al. 2000, Sutton et al. 2002). Due to the outstanding properties of synchrotron radiation, such as high brightness, high degree of polarization and extremely low divergence, ^-SRXRF offers a number of interesting advantages in comparison with other trace element analytical tools: (1) non-destructive analysis which enables long-time acquisitions under steady-state conditions, even for fragile biological materials, (2) easy quantification by using a standard-free fundamental-parameter approach, (3) the possibility of calculating matrix effects by using fundamental-parameter approaches, (4) the collection of multi-element spectra with one acquisition; (5) low detection limits (ppm level) at a high spatial resolution, and (6) sensitivity ranging over six orders of magnitude. In the last decades, the (i-SRXRF technique has been applied to many disciplines of the earth sciences (see the overviews by Smith and Rivers 1995 and Sutton et al. 2002). Baker and Watson (1988) and Baker (1989, 1990) were the first to apply (i-SRXRF to study trace element diffusion in silicate glasses, measuring diffusion profiles of a few selected elements. Koepke and Behrens (2001) improved the technique by using a modern energy dispersive detector to simultaneously analyze 18 trace element diffusivities in a single experiment. With this multielement approach, the relative errors of the diffusion coefficients are minimized, resulting in "internally consistent" data sets (Mungall et al. 1999; Koepke and Behrens 2001).

Instrumental setup, spectra acquisition and data processing Most studies on diffusion in silicate glasses using (i-SRXRF as an analytical tool have been performed at the HASYLAB synchrotron source of the DESY in Hamburg, Germany (Koepke and Behrens 2001; Koepke et al. 2003; Hahn et al. 2005; Behrens and Haack 2007; Behrens and Hahn 2009). Therefore, the experimental u-SRXRF setup installed at beamline L of the HASYLAB is described here in detail (Fig. 31). Similar setups can be found in other synchrotron radiation facilities around the world (Brown et al. 2006) For the measurements, the "white" X-ray continuum of a bending magnet is used, which shows maximum brilliance at 16.0 keV (critical energy). During routine analysis, the resulting X-ray spectral distribution enables simultaneous K-shell excitation of elements with atomic numbers from 25 (Mn) to 82 (Pb). The excitation conditions are normally optimized for single elements or element groups by using suitable absorbers. For minimizing the background radiation caused by Rayleigh and Compton scattering, the "horizontal" geometry is used, (for details see Haller and Knöchel 1996; Janssens et al. 2000), for which the radiation is detected

Analytical

Methods

in Diffusion

Studies

157

HPGe detector Microscope + Camera Collimator

Cross slits

i

Beam monitor

45°

Capillary Synchrotron source

Absorber

Pinhole

\

\ 45° \ . S-'

Beam monitor

Sample

Beam stop

Figure 31. Schematic drawing of the |j,-SRXRF analytical setup installed at beaniline L of D O R I S III synchrotron radiation source at H A S Y L A B (view from the top).

in the plane of maximum polarization at an angle of 90° (Fig. 31). For this geometry, the angle of incidence of the synchrotron beam on the sample is 45°. For a sufficiently high spatial resolution, a reduction of the diameter of the incoming synchrotron beam is necessary, which can be accomplished by using different types of glass capillaries as collimators (e.g., Haller et al. 1995; Janssens et al. 2000). For most diffusion studies, the beam size of the incoming synchrotron beam on the sample varies between 2 and 20 (.im. Samples analyzed are typically thin sections of diffusion couples, usually oriented so that the incoming synchrotron beam is perpendicular to the direction of diffusion (Fig. 32). Because the high-energy synchrotron beam penetrates through the whole thickness of the

Diffusion Couple

i

Figure 32. Sample geometry for the analysis of trace element diffusion profiles in the | i - S R X R F spectrometer.

158

Cherniak, Hervig, Koepke, Zhang, Zhao

sample without significant loss of energy, the variation with depth cannot be resolved. This characteristic is similar to transmission infrared spectroscopy, but differs from EMPA, LA-ICPMS or SIMS, where volumes from near-surface spots are analyzed. Therefore, for diffusion studies using ^-SRXRF analyses, the contact plane of the diffusion couple is aligned parallel to the plane of the incoming beam and the direction of detection. Fluorescence-derived X-ray photons, the K^ lines of elements, are detected with an energydispersive high-purity germanium detector. Due to the high brilliance of the synchrotron source, acquisition times are generally low, e.g., 240 seconds in the studies of Koepke and Behrens (2001) and Hahn et al. (2005) resulting in detection limits in the range of 1 to 15 ppm for the measured elements. Peak fitting and determination of the net peak areas are routinely performed using commercial software (e.g., Van Espen et al. 1977), with the option of processing multi-element spectra (Fig. 33a). For example, in the study of Hahn et al. (2005) 24 trace elements were measured simultaneously. Net intensities are routinely normalized to an internal standard (typically Ca or Fe) to correct for variations in synchrotron beam intensity, dead time and thickness of the sample. For most trace element diffusion studies using diffusion couples, the major element matrix is identical throughout the sample. Therefore, the normalized intensities are directly proportional to the concentration of trace elements in the glass, and the determination of absolute concentrations is not necessary. Typical profiles of trace elements with different geochemical behaviors are presented in Figure 33b. Normalized peak areas may be used directly for the calculation of diffusion coefficients. Sample preparation Samples for typical trace element diffusion studies using |a-SRXRF are couples of trace element-doped and undoped silicate melts (for experimental details see Koepke and Behrens 2001; Koepke et al. 2003; Hahn et al. 2005). After experimental runs, capsules are cut perpendicular to the contact plane of the couple for preparation of doubly polished thin sections. Due to the complete penetration of the incoming beam through the sample, the spatial resolution of this method is strongly dependent on the sample thickness. Moreover, since self-absorption is relatively high for the elements used for internal standardization, and practically negligible for elements of high atomic numbers, it must be ensured that the samples are of homogeneous thickness. Therefore, samples for (i-SRXRF studies have to be prepared as reasonably thin sections; for example, 50 ± 1 (im thick sections were used in the studies of Koepke and Behrens (2001) and Hahn et al. (2005). It should be noted that a decrease of sample thickness will also reduce the intensity of the fluorescence lines, resulting in lower precision and poorer detection limits of the analyzed elements. Applications of u-SRXRT for measuring trace element diffusivities in silicate melts Pioneering studies using (i-SRXRF to investigate diffusion in silicate glasses were conducted by Baker and Watson (1988), who measured diffusion profiles in complex CI- and F-bearing silicate melts, Baker (1989) who investigated tracer versus trace element diffusion using Sr isotopes, and Baker (1990) who analyzed Rb, Zr, Sr and Nb concentration profiles generated during chemical interdiffusion between a dacite and a rhyolite melt. Carroll et al. (1993) measured Kr diffusion in different silicic melts including pure Si0 2 and rhyolitic compositions. All of these studies considered only a few selected elements, which were analyzed by (i-SRXRF. Ten years later, Koepke and Behrens (2001) benefited from advancements in li-SRXRF permitting the collection of multi-element spectra with one acquisition, and started to obtain "internally consistent" data sets for trace element diffusivities in a synthetic haploandesitic melt with and without added water. This approach minimizes the relative errors of the diffusion coefficients, and such datasets provide the opportunity to explore systematic dependencies of diffusivities on ionic charge and radius. Doped elements can be grouped geochemically as LFSE (Rb, Sr, Ba), REE (La, Nd, Sm, Eu, Gd, Er, Yb, Y), HFSE (Ti, Zr, Nb, Hf, Ta), and transition elements

Analytical Methods in Diffusion Studies

159

10000

1000

100 12

17

22

27

32

37

42

47

52

57

Energy [keV]

1.0 "O 03

J

ro E

•

X

(b)

0.8

0.6

J c

0.4

» 8 R * * g 8 B B I I) 8 • AA °

8

x Zr o La o Zn

0.2 0.0! M

I

• ov

„ „

°

x

«Sr

0000^_nau B n jasasrinftssgs 1000 2000

3000

4000

5000

Distance [|jm] Figure 33. (a) n - S R X R F spectrum of a haploandesitic glass doped with various trace elements at the ~ 3 0 0 ppm level, which was used as starting material in the trace element diffusion study of Koepke and Behrens (2001). Only the Ka-lines are indicated. Above - 3 2 keV both K a i and K a 2 lines are visible. Peaks not labeled correspond to Kp-lines. Conditions for the |j,-SRXRF measurements: 20-|j,m capillary; acquisition time of 4 0 0 seconds (real time); Al absorber of l mm thickness. The presence of Ag is a due to a contamination resulting from sample polishing, (b) Diffusion profiles for selected trace elements measured with n - S R X R F obtained from a diffusion couple of haploandesitic melt containing 5 wt% H 2 0 . Experimental conditions: Temperature = 1583 K; pressure = 5 0 0 MPa; run time = l hour. Conditions for the | i - S R X R F measurements: Acquisition time = 2 4 0 seconds (real time) for each spectrum, resulting in ~ 4 h for the whole profile; absorber = l mm thick Al; capillary = diameter of 2 0 urn, non-focusing. The profiles are normalized to the peak net area of Ca. Shown are examples for trace elements of different chemical behavior with fast (Sr), medium (Zn, La), and slow (Zr) diffusivity. For details see Koepke and Behrens (2001).

160

Cherniak, Hervig, Koepke, Zhang, Zhao

(Sc, Cr, Fe, Ni, Zn). Most of the investigated elements showed a linear relation between log diffusivity and log viscosity, permitting the prediction of diffusivities in hydrous andesite systems at various conditions. This relation could potentially be used to estimate trace element diffusivities for silicate melts with different compositions provided viscosity data are available. Due to the simultaneous measurement of all trace elements, ratios between diffusion coefficients can be obtained with high precision. The precision of the analytical method is determined mainly by the scatter of data points along a profile, which decreases with increasing intensity of the fluorescence signal of the element (e.g., high fluorescence signal for elements like Y and Zr; and low signals for elements like Ta of Hf; see Fig. 33). This results in a slightly different fit parameter when evaluating duplicate analyses of concentration profiles from the same sample. Based on duplicate profiles in different samples, the elements can be divided into three groups: those with high reproducibility of diffusion coefficients (within 3-5 %): Rb, Sr, Y, Zr, Nb; those with intermediate reproducibility (within 10-15 %): Ba, Cr, Fe, Ni, Zn, light REE; and those with low reproducibility (within 30-50 %): Yb, Er, Hf (for details and individual errors of estimated diffusion coefficients see Koepke and Behrens 2001). Difficulties arise in systems with many trace elements in complex matrices due to spectral interferences between elements of interest. Behrens and Hahn (2009) overcame this problem by separating the conflicting trace elements into two sets. Hahn et al. (2005) used the same analytical equipment in measuring 24 trace elements, representing different geochemical groups, in hydrous rhyolitic glasses. These authors applied both (J.-SRXRF and SIMS in analyzing the concentration profiles, and found that multiple diffusivities derived from both techniques are in very good agreement for most elements. They showed that some trace elements could not be reliably quantified with (i-SRXRF. These include Ta and Pb (these elements are used for detector collimator material), Ti, V (low energy of K„ lines), Co (K a lines for Co have overlaps with Fe K„ lines) and Cr, Ni, Cu, Zn (overlaps with L-lines of REEs). In contrast to (i-SRXRF, in SIMS analyses elements are measured sequentially, which is much more time-consuming, so fewer trace elements could be analyzed in a single session. In addition, some elements are difficult to analyze with SIMS due to isobaric interferences (for example, interference of NaSi with V, and interference of CaO with Ni) or very low yields (e.g., Sn). Recently Behrens and Haack (2007) and Behrens and Hahn (2009) obtained "internally consistent" data sets for trace element diffusivities using (i-SRXRF in soda-lime-silicate glass melts and in potassium-rich trachytic/phonolitic melts, respectively.

ACKNOWLEDGMENTS We thank Harald Behrens for his careful review of the chapter and insightful comments. RLH acknowledges NSF (EAR-0622775) for support of the ASU SIMS facilities and helpful advice from J. Ganguly (University of Arizona), J. Boyce (UCLA) and Y. Guan (Caltech). YZ acknowledges support by NSF (EAR-0838127). DJC thanks Jon Price for helpful advice and comments on various sections of the chapter.

REFERENCES Aines RD, Rossman GR (1984) Water content of mantle garnets. Geology 12:720-723 Alonso U, Missana T, Patelli A, Rigato V (2007a) Bentonite colloid diffusion through the host rock of a deep geological repository. Phys Chem Earth 32:469-476 Alonso U, Missana T, Patelli A, Rigato V, Ravagnan J (2007b) Colloid diffusion in crystalline rock; an experimental methodology to measure diffusion coefficients and evaluate colloid size dependence. Earth Planet Sci Lett 259:372-383 Amsel G, Samuel D (1967) Microanalysis of the stable isotopes of oxygen by means of nuclear reactions. Anal Chem 39:1689-1698

Analytical Methods in Diffusion

Studies

161

Anderson TF (1969) Self-diffusion of carbon and oxygen in calcite by isotope exchange with carbon dioxide J Geophys Res 74:3918-3932 Anderson TF (1972) Self-diffusion of carbon and oxygen in dolomite. J Geophys Res 77:857-861 Arredondo EH, Rossman GR (2002) Feasibility of determining the quantitative OH content of garnet with Raman spectroscopy. Am Mineral 87:307-311 Baker DR (1989) Tracer versus trace element diffusion: Diffusional decoupling of Sr concentrations from Sr isotope composition. Geochim Cosmochim Acta 53:3015-3023 Baker DR (1990) Chemical interdiffusion of dacite and rhyolite: Anhydrous measurements at 1 atm and 10 kbar, application of transition state theory and diffusion in zoned magma chambers. Contrib Mineral Petrol 104:407-423 Baker DR, Watson EB (1988) Diffusion of major and trace elements in compositionally complex CI- and F-bearing silicate melts. J Non-Cryst Solids 102:62-70 Baker M, Taylor A (1969) Cation diffusion in fluorite single crystals. J Phys Chem Solids 30:1003-1007 Baldwin SL, Harrison TM, Fitz Gerald JD (1990) Diffusion of 40 Ar in metamorphic hornblende. Contrib Mineral Petrol 105:691-703 Barbour JC, Doyle BL (1995) Elastic recoil detection: ERD. In: Handbook of Modern Ion Beam Materials Analysis. Tesmer JR, Nastasi MA (eds) Materials Research Society, p 83-138 Baschek G, Johannes W (1995) The estimation of NaSi-CaAl interdiffusion rates in peristerite by homogenization experiments. Eur J Mineral 7:295-307 Baxter EF, Clay PL, Kelley SP, Watson EB, Cherniak DJ, Thomas JB (2006) Two diffusive pathways for argon in quartz and feldspar. Abstracts of the 16th annual V. M. Goldschmidt conference. Geochim Cosmochim Acta 70:A41 Baxter E F (2010) Diffusion of noble gases in minerals. Rev Mineral Geochem 72:509-557 Behrens H (2010) Ar, C 0 2 and H 2 0 diffusion in silica glass at 2 kbar pressure. Chem Geol 272: 40-48 Behrens H, Haack M (2007) Cation diffusion in soda-lime-silicate glass melts. J Non-Cryst Solids 353:47434752 Behrens H, Hahn M (2009) Trace element diffusion and viscous flow in potassium-rich trachytic and phonolitic melts. Chem Geol 259:63-77 Behrens H, Johannes W, Schmalzried H (1990) On the mechanisms of cation diffusion processes in ternary feldspars. Phys Chem Miner 17:62-78 Behrens H, Romano C, Nowak M, Holtz F, Dingwell DB (1996) Near-infrared spectroscopic determination of water species in glasses of the system MAlSi 3 0 8 (M=Li, Na, K): an interlaboratory study. Chem Geol 128:41-63 Behrens H, Roux J, Neuville DR, Siemann M (2006) Quantification of dissolved H2O in silicate glasses using confocal microRaman spectroscopy. Chem Geol 229:96-112 Behrens H, Zhang Y (2001) Ar diffusion in hydrous silicic melts: implications for volatile diffusion mechanisms and fractionation. Earth Planet Sci Lett 192:363-376 Behrens H, Zhang Y, Leschik M, Miedenbeck M, Heide G, Frischat GH (2007) Molecular H2O as carrier for oxygen diffusion in hydrous silicate melts. Earth Planet Sci Lett 254:69-76 Behrens H, Zhang Y, Xu Z (2004) H2O diffusion in dacitic and andesitic melts. Geochim Cosmochim Acta 68:5139-5150 Béjina F, Jaoul O (1996) Silicon self-diffusion in quartz and diopside measured by nuclear micro-analysis methods. Phys Earth Planet Inter 97:145-162 Bell DR, Hervig RL, Buseck PR, Aulbach S (2009) Lithium isotope analysis of olivine by SIMS: Calibration of a matrix effect and application to magmatic phenocrysts. Chem Geol 258:5-16 Bell DR, Rossman GR (1992) Water in Earth's mantle: The role of nominally anhydrous minerals. Science 255:1391-1397 Beran A, Libowitzky E (2004) Spectroscopic Methods in Mineralogy: University Textbook. Eotvos University Press, Budapest Berard M F (1971) Self-diffusion of C a i n single-crystal CaF 2 . J Am Ceram Soc 54.144-146 Bertran-Alvarez Y, Jaoul O, Liebermann RC (1992) Fe-Mg interdiffusion in single crystal olivine at very high pressure and controlled oxygen fugacity; technological advances and initial data at 7 GPa. Phys Earth Planet Inter 70:102-118 Blackburn TJ, Stockli DF, Walker JD (2007) Magnetite (U-Th)/He dating and its application to the geochronology of intermediate to mafic volcanic rocks. Earth Planet Sci Lett 259:360-371 Blank JG (1993) An Experimental Investigation of the Behavior of Carbon Dioxide in Rhyolitic Melt, Ph.D. dissertation, California Institute of Technology, Pasadena Blank JG, StolperEM, Carroll MR (1993) Solubilities of carbon dioxide and water in rhyolitic melt at 850 °C and 750 bars. Earth Planet Sci Lett 119:27-36 Boyce JW, Hodges KV, Olszewski WJ, Jercinovic MJ (2005) He diffusion in monazite: implications for (U-Th)/ He thermochronometry. Geochem Geophys Geosyst 6.Q12004

162

Cherniak, Hervig, Koepke, Zhang, Zhao

Brady JB, McCallister RH (1983) Diffusion data for clinopyroxenes from homogenization and self-diffusion experiments, A m Mineral 68:95-105 Brady JB, Yund RA (1983) Interdiffusion of K and Na in alkali feldspars; homogenization experiments. Am Mineral 68:106-111 Brown Jr GE, Sutton SR, Calas G (2006) User facilities around the world. Elements 2:9-14 Busigny V, Cartigny P, Philippot P, Javoy M (2003) Ammonium quantification in muscovite by infrared spectroscopy. Chem Geol 198:21-31 Busigny V, Cartigny P, Philippot P, Javoy M (2004) Quantitative analysis of ammonium in biotite using infrared spectroscopy. Am Mineral 89:1625-1630 Carroll MR (1991) Diffusion of Ar in rhyolite, orthoclase and albite composition glasses. Earth Planet Sci Lett 103:156-168 Carroll MR, Sutton SR, Rivers ML, Woolum DS (1993) An experimental study of krypton diffusion and solubility in silicic glass. Chem Geol 109:9-28 Castaing R (1951) Application des Sondes Électroniques à une Méthode D'analyse Ponctuelle Chimique et Crystallographique. Ph.D. Thesis. Univ. of Paris Chen JH, Harvey WW (1975) Cation self-diffusion in chalcopyrite and pyrite. Metallurg Trans B (Process Metallurgy) 6B:331-339 Chen WK, Peterson NL (1980) Iron diffusion and electrical conductivity in magnesio-wustite solid solutions (Mg, F e ) 0 . J Phys Chem Solids 41. 335-339 Chen WK, Peterson NL. (1975) Effect on the deviation from stoichiometry on cation self-diffusion and isotope effect in wustite, Fe,.„O. J Phys Chem Solids 36:1097-1103 Cheng H, Shen H, Yang F, Tang J (1993) Cross sections for non Rutherford backscattering of 4 He from some light elements. Chinese J Nucl Phys 15:333-336 Cherniak DJ (1995a) Diffusion of Pb in plagioclase and K-feldspar measured by Rutherford Backscattering spectroscopy and resonant nuclear reaction analysis. Contrib Mineral Petrol 120:358-371 Cherniak DJ (1996) Strontium diffusion in sanidine and albite, and general comments on Sr diffusion in alkali feldspars. Geochim Cosmochim Acta 60:5037-5043 Cherniak DJ (1997) An experimental study of Sr and Pb diffusion in calcite, and implications for carbonate diagenesis and metamorphism. Geochim Cosmochim Acta 61:4173-4179 Cherniak DJ (2000) Rare earth element diffusion in apatite. Geochim Cosmochim Acta 64:3871-3885 Cherniak DJ (2003) Silicon self-diffusion in single-crystal natural quartz and feldspar. Earth Planet Sci Lett 214:655-668 Cherniak DJ (2008) Si diffusion in zircon. Phys Chem Miner 35:179-187 Cherniak DJ, Lanford WA (2001) Nuclear Reaction Analysis. In: Non-Destructive Elemental Analysis. Alfassi Z, (ed.) Blackwell Science , p 308-338 Cherniak DJ, Pyle JM (2008) Thorium diffusion in monazite. Chem Geol 256:52-61 Cherniak DJ, Ryerson FJ ( 1993) A study of strontium diffusion in apatite using Rutherford Backscattering and ion implantation. Geochim Cosmochim Acta 57:4653-4662 Cherniak DJ, Watson EB ( 1992) A study of strontium diffusion in K-feldspar, Na-K feldspar and anorthite using Rutherford Backscattering Spectroscopy. Earth Planet Sci Lett 113:411-425 Cherniak DJ, Watson EB (1994) A study of strontium diffusion in plagioclase using Rutherford Backscattering Spectroscopy, Geochim Cosmochim Acta 58:5179-5190 Cherniak DJ, Watson EB (2001) Pb diffusion in zircon. Chem Geol 172:5-24 Cherniak DJ, Watson EB (2007) Ti diffusion in zircon. Chem Geol 242:473-486 Cherniak DJ, Hanchar JM, Watson EB (1997a) Diffusion of tetravalent cations in zircon. Contrib Mineral Petrol 127:383-390 Cherniak DJ, Hanchar JM, Watson EB (1997b) Rare earth diffusion in zircon. Chem Geol 134:289-301 Cherniak DJ, Lanford WA, Ryerson FJ (1991) Lead diffusion in apatite and zircon using ion implantation and Rutherford Backscattering techniques. Geochim Cosmochim Acta 55:1663-1673 Cherniak DJ, Watson EB, Grove M, Harrison TM (2004a) Pb diffusion in monazite: a combined RBS/SIMS study. Geochim Cosmochim Acta 68:829-840 Cherniak DJ, Watson EB, Thomas JB (2009) He diffusion in zircon and apatite. Chem Geol 268:155-166 Cherniak DJ, Watson EB, Wark DA (2007) Ti diffusion in quartz. Chem Geol 236:65-74 Cherniak DJ, Zhang XY, Nakamura M, Watson EB (2004b) Oxygen diffusion in monazite. Earth Planet Sci Lett 226:161-174 Cherniak DJ, Zhang XY, Wayne NK, Watson EB (2001 ) Sr, Y and REE diffusion in fluorite. Chem Geol 181:99111 Choudhury A, Palmer DW, Amsel G, Curlen H, Baruch P ( 1965) Study of oxygen diffusion in quartz by using the nuclear reaction l s O(p,a) l 5 N. Solid State Commun 3:119-122 Chu WK, Mayer JW, Nicolet MA (1978) Backscattering Spectrometry, Academic Press

Analytical Methods in Diffusion

Studies

163

Clement SWJ, Compston W(1990) SIMS at high sensitivity and high mass resolution. In: Secondary Ion Mass Spectrometry, SIMS VII. Benninghoven A (ed). J. Wiley & Sons, Chichester, p 815-819 Colthup NB, Daly LH, Wiberly SE (1990) Introduction to Infrared and Raman Spectroscopy. 547 p. Academic Press, Inc., Boston Cook GB, Cooper RF (2000) Iron concentration and the physical processes of dynamic oxidation in an alkaline earth aluminosilicate glass. A m Mineral 85:397-406 Cookson JA (1991) Nuclear microprobe and competing techniques. Nucl Instr Meth 54:433-439 Cooper RF, Fanselow JB, Weber JKR, Merkley DR, Poker DB (1996) Dynamics of oxidation of a Fe 2+ -bearing aluminosilicate (basaltic) melt. Science 274:1173-1176 Demarche J, Terwagne G (2006) Precise measurement of the differential cross section from the 1 6 0 ( a , a ) 1 6 0 elastic reaction at 165° and 170° between 2.4 and 6.0 MeV. J Appl Phys 100:124909-1 -7 den Hartog, H, Welch DO, Royce BSH (1972) The diffusion of calcium, phosphorus, and OD-ions in fluorapatite. Phys Status Solidi 53:201-212 Derry DJ, Lees DG, Calvert I M (1981) A study of oxygen self-diffusion in the c-direction of rutile using a nuclear technique. J Phys Chem Solids 42:57-64 Di Muro A, Villemant B, Montagnac G, Scaillet B, Reynard B (2006) Quantification of water content and speciation in natural silicic glasses (phonolite, dacite, rhyolite) by confocal microRaman spectroscopy. Geochim Cosmochim Acta 70:2868-2884 Dimanov A, Ingrin J (1995) Premelting and high-temperature diffusion of calcium in synthetic diopside: an increase of the cation mobility. Phys Chem Miner 22:437-442 Dimanov A, laoul O (1998) Calcium self-diffusion in diopside at high temperature: implications for transport properties. Phys Chem Miner 26:116-127 Dimanov A, Jaoul O, Sautter V (1996) Calcium self-diffusion in natural diopside single crystals. Geochim Cosmochim Acta 60:4095-4106 Dimanov A, Sautter V (2000) "Average" interdiffusion of (Fe,Mn)-Mg in natural diopside. Eur J Mineral 12:749-760 Dixon JE, Pan V (1995) Determination of the molar absorptivity of dissolved carbonate in basanitic glass. Am Mineral 80:1339-1342 Dobson PF, Epstein S, Stolper EM (1989) Hydrogen isotope fractionation between coexisting vapor and silicate glasses and melts at low pressure. Geochim Cosmochim Acta 53:2723-2730 Dohmen R, Becker HW, Chakraborty S (2007) Fe-Mg diffusion in olivine; I, Experimental determination between 700 and 1,200 degrees C as a function of composition, crystal orientation and oxygen fugacity. Phys Chem Miner, 34:389-407 Doremus RH (1969) The diffusion of water in fused silica. In: Reactivity of Solids. Mitchell JW, Devries RC, Roberts RW, Cannon P (eds) Wiley, New York, p. 667-673 Doremus RH (1973) Glass Science. Wiley, New York Doremus RH (1995) Diffusion of water in silica glass. J Mater Res 10:2379-2389 Dumke MF, Tombrello TA, Weller RA, Housley RM, Cirlin EH (1983) Sputtering of the gallium-indium eutectic alloy in the liquid phase. Surf Science 124 407-412 Eiler IM, Graham C, Valley IW (1997) SIMS analysis of oxygen isotopes: Matrix effects in complex minerals and glasses. Chem Geol, 138:221-244 Epstein S, Taylor Jr. HP (1970) The concentration and isotopic composition of hydrogen, carbon and silicon in Apollo 11 lunar rocks and minerals. Apollo 11 Lunar Science Conference Proc 2:1085-1096 Ernsberger F M (1980) The role of molecular water in the diffusive transport of protons in glasses. Phys Chem Glasses 21:146-149 Farley KA (2000) Helium diffusion from apatite: General behavior as illustrated by Durango fluorapatite. J Geophys Res 105:2903-2914 Farley KA (2007) He diffusion systematics in minerals; evidence from synthetic monazite and zircon structure phosphates. Geochim Cosmochim Acta 71:4015-4024 Fechtig H, Kalbitzer S (1966) The diffusion of argon in potassium-bearing solids. In: Potassium Argon Dating. Schaeffer OA, Zahringer J (eds) Springer-Verlag, New York, p 68-107 Fine G, Stolper EM (1985) The speciation of carbon dioxide in sodium aluminosilicate glasses. Contrib Mineral Petrol 91:105-121 Fine G, Stolper EM (1986) Dissolved carbon dioxide in basaltic glasses: concentrations and speciation. Earth Planet Sci Lett 76:263-278 Foland KA (1974) Alkali diffusion in orthoclase. In: Geochemical Transport and Kinetics. Hofmann AW, Giletti BI, Yoder, Ir HS, Yund RA (eds) Carnegie Institution of Washington Publication 634, Carnegie Institution of Washington, Washington DC, p 77-98 Foland KA, Xu Y(1990) Diffusion of 40 Ar and 39 Ar in irradiated orthoclase. Geochim Cosmochim Acta 54:31473158 Fortier SM, Giletti BJ (1991) Volume self-diffusion of oxygen in biotite, muscovite, and phlogopite micas. Geochim Cosmochim Acta 55:1319-1330

164

Cherniak, Hervig, Koepke, Zhang, Zhao

Ganguly J (2010) Cation diffusion kinetics in aluminosilicate garnets and geological applications. Rev Mineral Geochem 72:559-601 Ganguly J, Bhattacharya RN, Chakraborty S (1988) Convolution effect in the determination of compositional profiles and diffusion coefficients by microprobe step scans. Am Mineral 73:901-909 Ganguly J, Tazzoli V (1994) Fe 2+ -Mg interdiffusion in orthopyroxene: retrieval from data on intracrystalline exchange reaction. Am Mineral 79. 930-937 Ganguly J, Tirone M, Hervig RL ( 1998) Diffusion kinetics of samarium and neodymium in garnet, and a method for determining cooling rates in rocks. Science 281:805-807 Gardés E, Jaoul O, Montel J, Seydoux-Guillaume A, Wirth R (2006) Pb diffusion in monazite; an experimental study of Pb 2+ Th 4+ 2 Nd , + interdiffusion. Geochim Cosmochim Acta 70:2325-2333 Gardés E, Montel JM, Seydoux-Guillaume AM, Wirth R (2007) Pb diffusion in monazite: New constraints from the experimental study of Pb 2+ H2O1 I S proportional to H 2 O t under otherwise identical conditions (see also Wang et al. 1996). Zhang and Stolper (1991) investigated H 2 0 diffusion in basalt melt at 1573-1773 K, 1.0 GPa, and H2O,Nowak and Behrens (1997) carried out diffusion couple experiments and investigated H 2 0 diffusion in a synthetic rhyolite melt (AOQ) at 1073-1473 K, 0.05-0.5 GPa, and < 9.0 wt% H2Ot. They demonstrated that the H 2 O t diffusivity at H 2 O t > 3 wt% increases exponentially with H 2 O t , implying that the H 2 O m diffusivity cannot be constant but must increase with H 2 O t content at H 2 O t > 3 wt%. They fit logZ)n.0| as a polynomial function of H 2 O t . Zhang and Behrens (2000) studied H 2 0 diffusion in rhyolite melts at 673-1473 K, 0.0001-0.81 GPa, and < 7.7 wt% H2Ot. They confirmed the observations of Nowak and Behrens (1997) and found that the diffusion profiles can be accurately modeled by assuming H 2 O m is the diffusing species and _DH2om increases exponentially with H 2 O t , Ai 2 o m = A)eH2om value as X approaches zero, and a is a parameter characterizing how rapidly Ai 2 o m increases with H2Ot. The exponential increase means that at low H 2 O t , Ai 2 o m increases only slowly with H 2 O t and can be treated roughly as a constant (e.g., for a = 25 and from 0 to 1 wt% H 2 O t , e"x varies from 1 to 1.56, only slightly outside the experimental uncertainty of diffusivity determinations at high temperatures), consistent with the observations of Zhang et al. (1991a). The exponential increase is also consistent with the diffusivities of neutral molecular species such as Ar (Behrens and Zhang 2001) and C0 2 (Watson et al. 1982; Watson 1991). Later experimental studies in general followed the framework established in Zhang et al. (1991a), Zhang and Stolper (1991), and Zhang and Behrens (2000). Freda et al. (2003) investigated H z O diffusion in a trachyte melt at 1334-1601 K, 1 GPa, and < 2.0 wt% H2Ot. Okumura and Nakashima (2004, 2006) developed the in situ FTIR measurements of H 2 0 loss due to dehydration to obtain diffusivities, and reported results on rhyolite melts at < 4.1 wt% H 2 O t and on dacite, andesite and basalt melts at < 1.1 wt% H2Ot. Their method works well for rapidly obtaining diffusion data but cannot resolve how DH20i depends on H2Ot. Liu et al. (2004b) explored H 2 0 diffusion in dacite melts by dehydration experiments at 824-910 K, < 0.15 GPa, and < 2.5 wt% H 2 O t . Behrens et al. (2004) studied H z O diffusion in dacite and andesite melts at 1458-1858 K, 0.5-1.5 GPa, and < 6.3 wt% H 2 O t ; some of their data are consistent with concentration-independent D H20l . Behrens et al. (2007) obtained some H 2 0 diffusion data in rhyolite melts in a study comparing oxygen and H 2 0 diffusion. Ni and Zhang (2008) further resolved the pressure effect on H 2 0 diffusion in rhyolite melts and constructed a general H 2 0 diffusivity model over a large range of T, P and H 2 O t . Behrens and Zhang (2009) and Wang et al. (2009) quantified H 2 0 diffusion in peralkaline rhyolite melts. Ni et al. (2009a,b) examined H 2 0 diffusion in dacite and haploandesite melts. All the melt compositions that have been studied for H 2 0 diffusion are listed in Table 1. H 2 0 diffusion, theory and data summary Strictly speaking, silicate melts are multicomponent systems, and H 2 0 diffusion should fall into the category of multicomponent diffusion. However, the anhydrous melt composition generally does not vary along a diffusion profile in the typical design of the experiments, and addition of H 2 0 mainly causes a dilution effect. Therefore, H 2 0 diffusion in the literature is treated as effective binary diffusion, or the first kind of effective binary diffusion defined by Zhang (2010, this volume) and Zhang et al. (2010, this volume). Because £>H2O, depends on H 2 O t concentration, the general equation for one-dimensional diffusion is as follows: (9)

Diffusion

of H, C, O Components

in Silicate

Melts

181

where t is time, x is distance, Dh2o, is the diffusivity (effective binary diffusivity) of H 2 O t , and X = [H2OJ (see Eqn. 3a). The above diffusion equation can be solved given how £>H2O, varies with H 2 O t , and boundary and initial conditions. For hydration and dehydration experiments, the equilibrium surface concentration and the initial concentration (usually uniform) are the boundary and initial conditions, respectively. For diffusion couple experiments, no boundary condition is needed and the initial concentrations in the two starting halves are the initial condition. If D H20t were independent of H 2 O t (or X). then the diffusion profile could be fit by an error function solution. However, this is rarely the case: the variation of D H20t with H 2 O t turns out to be complicated. Experimental results in the glass and geological literature show that when H 2 O t is sufficiently low (< 2 wt%, but depending on temperature), ¿?H2O, is proportional to H 2 O t (or proportional to X in Eqn. 9). Such an equation can be solved numerically (e.g.. Crank 1975) and the solution has been applied to fit diffusion profiles. If a profile can be fit within experimental uncertainty (e.g.. Fig. 4), then the proportionality relation is assumed to describe how ¿?H2O, varies with H2Ot.

Figure 4. Experimental H 2 0, diffusion profile (points) in a peraluminous rhyolite melt and two fits. The dashed curve is an error function fit (assuming D E , ( ) | is independent of H 2 O t ). The solid curve is a fit assuming D h ,o t is proportional to HiO,. The solid curve agrees well with experimental data. From Behrens and Zhang (2009).

jr (mm)

At high H 2 O t (such as >3 wt%, especially at relatively low temperatures such as 800 K). the proportionality relation often does not work well (Nowak and Behrens 1997: Zhang and Behrens 2000: Liu et al. 2004b: Ni and Zhang 2008; Ni et al. 2009a,b; Wang et al. 2009)? One example is given in Figure 5. Hence, more complicated relations are proposed to describe how D|| lflt varies with H2Ot. Based on our knowledge of H 2 0 speciation. the general and mechanistic approach is to consider the diffusion of both species:

®L = JL\D ot

cit(

^ ¡ L

-

civ

+IÖ 2

civ

(10)

where Xm is the mole fraction of H 2 O m , Xqh is the mole fraction of OH, and ¿?H2om and D(m are the diffusivities of H2Om and OH. The factor 1/2 is due to the fact that one mole of H 2 O m reacts to form two moles of OH (Reaction 1). Experimental data show D,,u « i?H2om in rhyolite melt so that diffusive flux due to OH diffusion can be ignored (Zhang et al. 1991a). (The small diffusivity of OH does not mean that OH profile is flat, because interconversion between OH and H 2 O m can change OH concentration.) Therefore, Equation (10) can be simplified as:

182

Zhang & Ni 5

Error function lit Figure 5. Experimental H 2 O t diffusion profile (points) in a peralkaline rhyolite melt and three fits. The long-dashed (blue in online version) curve is an error function fit (assuming D E , ( ) | is independent of H 2 O t ). The short-dashed (red in online version) curve is a fit assuming DH , 0 is proportional to HiO,. The solid curve is a fit assuming £>H2om = D(Se"x (Eqn. 12). The solid curve agrees well with experimental data. From Wang et al. (2009).

CBS-DC2 0

-0.6

-0.4

-0.2

0

x

0.2

0.4

0.6

0.8

(mm)

(11) The above equation means that Dh 2 o, = D a , 0 m d X J d X if equilibrium for Reaction (1) is reached at every point along a profile (dXJdX for K= 0.5 is shown in Fig. 3). (Equilibrium for Reaction (1) is necessary because without equilibrium the partial differential dXm/dX would depend on time and cannot be simplified to dX^/dX.) At low H 2 O t (e.g.. below 2 w t % but depending on the temperature). A t 2 o m is roughly independent of H 2 O t . and dXJdX is proportional to H 2 O t (Fig. 3), leading to D H l ( ) t proportional to H 2 O t . However, at high H 2 O t , must increase with H 2 O t (Nowak and Behrens 1997) and has been shown to increase exponentially with H 2 O t (Zhang and Behrens 2000: Liu et al. 2004b; Ni and Z h a n g 2008: Ni et al. 2009a.b: Wang et al. 2009):" D

= D eaX

(12)

where a is a fitting parameter that may vary with T and P. N u m e r o u s studies have f o u n d that a increases with decreasing temperature, indicating a stronger dependence of D H , 0 m on H 2 O t at lower temperatures. This finding is not surprising since at low 7 the presence of H 2 0 has a larger impact on melt properties such as viscosity (e.g.. Z h a n g et al. 2003). which is not simply related to the formation of the n u m b e r of O H but is also related to the larger effect of each O H and H 2 O m on the melt viscosity at low temperatures. T h e above approach can fit all H 2 O t profiles within experimental uncertainty (as well as species concentration profiles if the species concentrations can b e preserved during quench), including those profiles that cannot b e fit by assuming £ > H 2 O , I S proportional to H 2 O t (Fig. 5). Furthermore, experiments using samples with different H 2 O t generally yield the same set of parameters Du and a, which further supports the validity of Equation (12). T h e exponential dependence of the diffusivity of H 2 O m as well as other neutral molecules (Watson 1991: Behrens and Z h a n g 2001) on H 2 O t concentration might b e related to the dramatic change of melt structure, as evidenced by the significantly reduced melt viscosity caused by the addition of H 2 0 (e.g., Shaw 1974; Z h a n g et al. 2003). If equilibrium for Reaction (1) is reached at every point along a profile, the general relation between D H , 0 t and A t 2 o m is (Wang et al. 2009): (0.5 - X ) il/2

[X(l - X)((4 I K ) - 1) + 0.25]

(13)

Diffusion

ofH, C, O Components

where K is the equilibrium constant of Reaction (1) and depends on temperature (to a lesser extent, on pressure or H2Ot). and X and Xm are mole fractions of 112(), and H 2 O m on a single oxygen basis. Hence, if K is known or fixed, ¿?H2O, CAN be calculated from £)H2om using the above equation. Figure 6 illustrates how ¿?H2om a n d ^h 2 o, v a r Y with H2Ot concentration at given K and a. At low II 2 0,. Z)||.()| is approximately proportional to fTCV while at higher H 2 O t . the relation transitions to exponential. For rhyolite, dacite and haploandesite melts. K is given by Equations (7a) to (7c). Diffusion data in various melts are summarized below.

in Silicate

—i

20

1

Melts

183

r-

K = 0.5

a = 30 10

H2Om/

h 2 O, _l

0

1

2

3

I

4

H 2 O t (wt%)

L-

5

6

7

Figure 6. D H l 0 m and DhiO, as function of H 2 0 , for a given set of AT and a, calculated f r o m Equations (12) and (13). The relation between D H l o, and H 2 0 , shifts from quasi-proportional to quasi-exponential as H 2 O t increases. £>Hoot is lower than DH2()m-

If equilibrium for Reaction (1) is not reached (at relatively low temperatures and low H2Ot), one would have to consider the kinetics of Reaction (1) and couple the kinetic equation with the diffusion equation 11 to solve the concentration profiles. Such full treatment has not been carried out yet, but experimental data have been noted where these conditions exist (Jambon et al. 1992; Zhang et al. 2007).

Metaluminous and peraluminous rhyolite melts. "Normal" metaluminous rhyolite melts including haplorhyolite melts (compositions 1 -3 in Table 1) have been studied most extensively (Shaw 1974; Delaney and Karsten 1981; Karsten et al. 1982; Zhang et al. 1991a; Nowak and Behrens 1997; Zhang and Behrens 2000; Okumura and Nakashima 2004; Behrens et al. 2007; Ni and Zhang 2008). Figure 7 compares Ar 2 o, i n rhyolite melts at 1 wt% H 2 O t from various laboratories. It can be seen that the original diffusivity data from different laboratories are largely consistent at this H 2 O t when cast in the same way. Furthermore, there is a significant pressure effect on _DH2O,. especially at low temperature. Ni and Zhang (2008) combined their new experimental data and literature data at 676-1900 K, 0-1.9 GPa, 0-8 wt% H 2 O l (reflecting the coverage of conditions by the experimental data), using A'from Equation (7a), and obtained ¿?H2om (m2/s) as follows: rhyolite

= exp - 1 4 . 2 6 + 1 . 8 8 8 P - 3 7 . 2 6 X -

12939 + 3626P - 75884X

(14)

where T is in K, P is in GPa, and X is mole fraction of H 2 O t on a single oxygen basis. The H2Ot diffusivity in peraluminous rhyolite melts (composition 4 in Table 1) at < 0.5 GPa is indistinguishable from that in "normal" rhyolite melts and can be described by the above expression as well (Behrens and Zhang 2009). Hence, £>H2o, for metaluminous and peraluminous rhyolite melts can be calculated from the above D H ,()m using Equation (13) with if from Equation (7a). The 2o uncertainty in lnD Hl0t is about 0.5 for this and other melts discussed below. The above equation implies that DH,()m increases rapidly with H 2 O t at low temperatures but slowly at higher temperatures. Therefore, at higher temperatures, £)h2o, is proportional to H2Ot from zero to higher concentrations of H2Ot. Because the proportionality equation is easy to apply and works well at low H 2 O t (which means < 1 wt% at 773 K and < 3 wt% at 1473 K). it is given below (Ni and Zhang 2008):

184

Zhang & Ni

——-0.0001 GPa * 0.05-0.25 GPa - B - - 0 . 5 GPa ° ; .81-0.95 GPa - - • - * 1.9 GPa a 0.S GPa; Nowak » 0.07 GPa; Karsten * 0.2 GPa; Shaw

1 0 0 0 / 7 ( 7 in K) F i g u r e 7. Comparison of experimental data on HiO, diffusivities in "normal" rhyolite melts at 1 wt% H 2 0,. Data sources: Data are from Zhang et al. (1991a), Zhang and Behrens (2000), and Ni and Zhang (2008), unless otherwise indicated as: Nowak and Behrens (1997); Karsten et al. (1982); Shaw (1974).

rhyolite

,

(

,_ _

9699 + 3626P

"H,0,

(15)

where Tis in K. P is in GPa. and Cw is wt% of H2Ot (Cw = 1 for 1 wt% H 2 O t ). Peralkaline rhyolite melts. Behrens and Zhang (2009) and Wang et al. (2009) investigated H 2 0 diffusion in peralkaline (PA) rhyolite melts (compositions 5 and 6 in Table 1) covering 789-1516 K, 0-1.4 GPa, 0-4.6 wt% H2Ot. Behrens and Zhang (2009) investigated diffusion at low H 2 O b and Wang et al. (2009) expanded the studied range of H 2 O t and pressure. Even though these studies are not numerous, the effort was coordinated and systematic so that the data are sufficient to allow the construction of a general model for ¿ ? H O (and hence ¿3H2O,)- Figure 8 compares the H 2 O t diffusivities in peralkaline rhyolite melts to metaluminous rhyolite melts. On average, ¿3h2o, a l 0.5 GPa and 1 wt% H 2 O t in peralkaline rhyolite melt is - 2 times that in metaluminous rhyolite melt. The effect of pressure on ¿?H2O, in peralkaline rhyolite melts is smaller than that in metaluminous rhyolite melts. For example, lnD^o, decreases by -1.6 as pressure increases from 0.0001 to 0.5 GPa for metaluminous rhyolite melts; but by only -0.3 for peralkaline rhyolite melts. At various P-T conditions, the difference between ¿?H2O, in peralkaline rhyolite melts and that in metaluminous rhyolite melts is often a factor of 2 or smaller. 2

M

Using results from both Behrens and Zhang (2009) and Wang et al. (2009), and K from Equation (7a). Wang et al. (2009) constructed the following expression for D H , 0m (m2/s): n

Dh o

'Ad^

f

„

nQnv =exp -12.79-27.87X

13939+1230P-60559X^1

(16)

where T is in K, P is in GPa, and X is mole fraction of H 2 O t on a single oxygen basis. Z) n . 0| for peralkaline rhyolite melts can be calculated from the above ¿?H2om using Equation (13) with K from Equation (7a). The resulting ¿?H2om a n d ¿?h2o, are slightly larger than that in "normal" rhyolite melts, but no more than a factor of 2. By combining the data in both Behrens and Zhang (2009) and Wang et al. (2009), a simple equation to calculate directly ¿?H2O, l ° w H20 t (which means < 1 wt% at 773 K and < 3 wt% at 1473 K) is as follows:

Diffusion

of H, C, O Components

0.8

1

in Silicate

Melts

185

12

1000/r (T in K) Figure 8. Comparison between HiO, diffusivities in peralkaline rhyolite melt at 0.5 GPa (filled circles) and those in metaliiminous (or "normal" ) rhyolite melts at 0.0001 and 0.5 GPa (open circles and squares). Data sources for peralkaline rhyolite: Behrens and Zhang (2009) and Wang et al. (2009 ). See Figure 7 for data sources for metaluminous rhyolite.

Du

= CW exp - 1 6 . 5 5 -

10870+ 1101P

(17)

where Tis in K, P is in GPa, and Cw is wt% of H2Ot. Dacite melts. Liu et al. (2004b), Behrens et al. (2004), Okumura and Nakashima (2006), and Ni et al. (2009b) quantified H 2 0 diffusion in dacite melts (compositions 7 and 8 in Table 1), covering 786-1800 K, 0-1 GPa, 0-8 wt% ll2(>,. Liu et al. (2004b) and Okumura and Nakashima (2006) conducted dehydration experiments at intermediate temperature and low pressure; Behrens et al. (2004) carried out diffusion couple experiments at high temperature and high pressure; and Ni et al. (2009b) investigated H 2 0 diffusion at intermediate temperature and high pressure. All data have been combined to evaluate the effect of temperature, pressure and H 2 O t on H 2 0 diffusion. Figure 9 shows some experimental diffusion data and compares these with data in rhyolite melts. At 1 wt% 11 ( ) . /^- n. in dacite melts is lower than that in rhyolite melts at < 1470 K. Furthermore, for both rhyolite and dacite melts, -Dh2o, decreases as pressure increases. As pressure increases from 0.0001 to 0.5 GPa, lnDH2ot decreases by ~ 1.6 for rhyolite melt; but only by -1.0 for dacite melts. That is, the pressure effect becomes smaller from rhyolite melt to dacite melt. Using results from Liu et al. (2004b), Behrens et al. (2004) and Ni et al. (2009b), and K from Equation (7b). D H , 0m at 786-1800 K, 0-1 GPa, and 0-8 wt% H 2 O t (reflecting conditions covered by the experimental data) can be expressed as (Ni et al. 2009b): Dh

o

dacilc

= exp I - 9 . 4 2 - 6 2 . 3 8 X -

19064 + 1477P -108882X

(18)

where T is in K, P is in GPa, and X is the mole fraction of 112(), on a single oxygen basis. ¿3h2o, for dacite melts can be calculated from the above ¿?H2om using Equation (13) with K from Equation (7b). At H2[H2])/(DH2om[H2Om]) as a function of temperature and oxygen fugacity buffer is plotted in Figure 13c. It can be seen that when oxygen fugacity is at NNO or higher, diffusion of the total hydrogen component is dominantly due to H 2 O m diffusion except for temperatures below 600 K. Under more reducing conditions (that can be encountered in terrestrial melts), at typical magmatic temperatures (> 1100 K), H 2 O m diffusion still dominates the total hydrogen component flux, but molecular H 2 diffusion may dominate the diffusive flux of the total hydrogen component at temperatures below typical magmatic temperatures. It is emphasized that this conclusion is tentative because of the many assumptions involved. Previous experimental diffusion studies (e.g., Shaw 1974; Delaney and Karsten 1981; Karsten et al. 1982; Zhang et al. 1991a; Zhang and Stolper 1991; Nowak and Behrens 1997; Zhang and Behrens 2000; Behrens et al. 2004; Liu et al. 2004b; Okumura and Nakashima 2004, 2006; Ni and Zhang 2008; Behrens and Zhang 2009; Ni et al. 2009a,b; Wang et al. 2009) on the H 2 0 component were conducted without detailed characterization of the oxygen fugacity. Nonetheless, the consistency of these experimental data (e.g., Ni and Zhang 2008) show that H 2 diffusion probably did not play a major role in any of these studies. Furthermore, simultaneous H z O and 1 8 0 diffusion studies in the same experiments (Behrens et al. 2007) show that H 2 O m that carries O, not H 2 that does not carry O, is the diffusing species for the hydrous component. In the future, it will be important to investigate diffusion of the total hydrous component under very reducing conditions with well-characterized oxygen fugacity (e.g., oxygen fugacity that is uniform in the sample and independent of time). In short, it is possible that molecular H 2 diffusion could play a major role in transporting the hydrous component under reducing conditions that may be encountered in terrestrial melts. However, this inference is uncertain at present because of the many assumptions involved. New experimental studies are necessary to resolve this issue.

Diffusion

ofH, C, O Components

in Silicate

Melts

197

DIFFUSION OF THE C0 2 COMPONENT Zhang et al. (2007) reviewed C 0 2 diffusion in silicate melts. Not much new work has been published on natural melts since then. For completeness of this chapter and this review volume, a brief summary is provided here. Dissolved C 0 2 is present in silicate melts in either C 0 2 molecules or carbonate groups (Fine and Stolper 1985; Blank and Brooker 1994). Hence, C 0 2 diffusion is also a multi-species diffusion. However, the presence of multiple species seems to contrive to simplify the diffusion properties of C0 2 , contrary to the case of H 2 0 diffusion (Zhang et al. 2007). Watson et al. (1982) were the first to investigate tracer diffusion of carbonate (using ,4 C tracer diffusion) in silicate melts, and made the surprise discovery that C 0 2 diffusivity does not depend on the anhydrous melt composition (from a haplobasalt to melt containing 30 wt% Na 2 0). More extensive 14C tracer diffusion studies by Watson (1991) confirmed this conclusion but also showed that the diffusivity depends strongly on H 2 O t . Watson et al. (1982) and Watson (1991) determined concentration profiles using P-track maps made by exposing nuclear emulsion plates to the quenched and sectioned diffusion capsules. As discussed in Mungall (2002), the (3-particle range is of the order of a hundred |am (International Commission on Radiation Units and Measurements 1984), much more than initially thought. Hence, tracer diffusion data using P-track maps may be compromised by this effect (those diffusivities extracted from concentration profiles longer than 1 mm are less affected). Furthermore, Zhang et al. (2007) showed that the data are inconsistent with effective binary diffusivities of C0 2 . Hence, l4 C tracer diffusion data of Watson et al. (1982) and Watson (1991) are not discussed further in quantitative treatment. Blank (1993), Sierralta et al. (2002) and Nowak et al. (2004) (as well as Fogel and Rutherford 1990, Zhang and Stolper 1991, and Liu 2003) investigated effective binary diffusion of C 0 2 at C 0 2 concentration levels of hundreds to thousands of ppm. For the purpose of modeling natural magmatic processes (e.g., C 0 2 bubble growth in a basalt melt), the effective binary diffusivities of total C 0 2 are the necessary diffusivities. These studies further confirmed the rough independence of C 0 2 diffusivity on anhydrous melt composition. This independence is even more surprising considering that C 0 2 is present as C 0 2 molecules in rhyolite melt but as C 0 3 2 - ion in basalt melt. Nowak et al. (2004) explained the dependence as follows: Assume molecular C 0 2 is the diffusing species whereas CO ;2 is roughly immobile. From rhyolite melt to basalt melts, molecular C 0 2 diffusivity increases (similar to Ar diffusivity), which increases total C 0 2 diffusivity, but the fraction of dissolved C 0 2 present as molecular C 0 2 decreases (the rest is C0 3 2 - ), which decreases the total C 0 2 diffusivity. The two factors roughly cancel each other, leading to a total C 0 2 diffusivity roughly independent of the anhydrous melt composition. Baker et al. (2005) carried out three carbonate dissolution experiments and measured total C 0 2 concentration profiles by difference-from-100% method, and two diffusion couple experiments and measured C 0 2 concentration profiles by FTIR. The compositions for which C 0 2 diffusion has been investigated are listed in Table 2. Because of its rough independence of the anhydrous melt composition, C 0 2 is likely to be the first component for which the diffusivity in various melts can be predicted based on a limited number of studies, which can be compared to the large number of experimental studies on H 2 0 diffusion reviewed in the previous section. Existing experimental data on the effective binary diffusion of C 0 2 cover 723-1623 K and < 1 GPa. However, they are mostly on dry melts, with only 4 data points on hydrous melts containing > 1 wt% H 2 0„ of which two points with 5 wt% H 2 O t by the reconnaissance experiments of Baker et al. (2005) apparently showed that adding 5 wt% H 2 O t does not affect C 0 2 diffusivity in a trachyte melt, contrary to results by Sierralta et al. (2002) on albite melt and Watson (1991) on other melts. A possible explanation of the discrepancy is that determination of C0 2 concentration profiles by the difference-from-100% method used by Baker et al. (2005) does not work well for hydrous

Zhang & Ni

198

Table 2. Chemical composition (wt% on dry basis) for C0 2 diffusion studies. ID

Comp.

Si0 2

Ti0 2 a i 2 o 3 FeO MnO MgO

CaO

Na 2 0

k2o

P20 5

Ref

1

Rhyolite

76.45

0.08

12.56

1.02

.08

.06

.25

4.21

4.78

a

2

Rhyolite

77.5

0.07

13.0

0.56

0.04

0.05

0.52

4.10

4.18

b

3

Basalt

50.6

1.88

13.9

12.5

0.23

6.56

11.4

2.64

0.17

4

Ab

69.03

19.33

0.21

c

11.41

d

5

AbNal

68.87

19.24

12.45

d

6

AbNa2

68.56

18.58

13.38

d

7

AbNa3

68.43

18.62

13.92

d

8

AbNa4

66.66

18.63

15.24

d

9

AbNa6

67.38

18.18

15.95

d

10

AbNa7

65.73

17.60

17.57

11

Rh ( C 0 2 )

75.09

0.22

13.83

1.47

1.40

3.86

4.10

e

12

Da ( C 0 2 )

71.78

0.40

15.31

2.26

3.09

4.52

2.97

e

13

DaAn (C0 2 )

66.13

0.68

16.18

4.59

4.79

4.26

1.90

e

14

An (C0 2 )

63.37

1.11

16.90

8.08

6.81

4.04

0.88

e

15

AnTh (C0 2 )

61.92

1.55

15.22

8.79

6.74

3.73

1.33

e

16

Th ( C 0 2 )

60.48

2.01

14.54

9.95

8.79

3.32

1.61

e

17

Ha ( C 0 2 )

53.55

3.74

16.93

11.32

9.19

4.71

1.44

e

18

Trachyte

59.9

0.39

18.0

3.86

2.92

4.05

8.35

0.89

0.12

d

0.21

f

The compositions are listed on the anhydrous basis. References: a. Fogel and Rutherford (1990); b. Blank (1993); c. Zhang and Stolper (1991); d. Sierralta et al. (2002); e. Nowak et al. (2004); f. Baker et al. (2005).

melts and hence C 0 2 diffusivities in hydrous trachyte melt are incorrect. Because total C 0 2 diffusivity in all melts are similar to Ar diffusivity in silicic melts (Zhang et al. 2007), and extensive Ar diffusion data in silicic melts are available, covering a wide range of temperature, pressure and H 2 0 contents (673-1773 K, < 1.5 GPa, and < 5 wt% H 2 O t ), Zhang et al. (2007) used such data to derive the following equation, which is proposed to be applicable for total effective binary diffusivity of C 0 2 in basalt to rhyolite melts: lngailmcl,_ I f ) , ,

{

1399 "

17367 + 1944.8P rp

|

(855.2+ 2 7 1 . 2 P ) C J W

where D is in m2/s, Tis in K, P is in GPa, and C w is wt% of H 2 O t . Figure 14 compares experimental data and the above equation, especially the data by Baker et al. (2005) because the comparison with Baker et al. (2005) was not made in Zhang et al. (2007). It can be seen that the calculated line and the experimental data of the same color (the online version) are in rough agreement, except for the two data points at 5 wt% H 2 O t by Baker et al. (2005) (purple short-dashed line and purple solid squares in Fig. 14; the color can be seen in the online version). Excluding these two data points, the maximum uncertainty of the above equation in predicting lnDC()2total (Fogel and Rutherford 1990; Zhang and Stolper 1991; Blank 1993; Sierralta et al. 2002; Nowak et al. 2004; Baker et al. 2005) is 1.13. This 2o uncertainty is larger than typical experimental data uncertainty (with a 2a of about 0.6 in InD), because Dc„2Rital depends weakly on melt compositions, which is ignored in the above treatment. New experimental data on C 0 2 diffusion under conditions of high H 2 O t content are necessary to further constrain C 0 2 diffusivities.

Diffusion

of H, C, O Components

in Silicate

Melts

199

Figure 14. Comparison of experimental data of effective binary diffusivity of bulk C 0 2 with Equation (41) under some conditions. The lines in matching color with data points (the colors can be seen in the online version) are calculated from Equation (41); solid line corresponding to solid circles; long dashes to open circles; medium dashes to squares with plus inside; short dashes (the uppermost line) to solid squares; sparse dot line to solid diamonds; and dense dot line to open diamonds. For clarity, low-temperature data are not shown. Data sources; -'0.1-0.2 wt% H 2 0 ; 0.05-0.25 GPa": Blank (1993) and Fogel and Rutherford (1990); "< 0.14 wt% H.O; 0.5 GPa": Sierralta et al. (2002) and Nowak et al. (2004); "2 wt% H.O and 0.5 GPa": Sierralta et al. (2002); "5 wt% H 2 0 ; 1 GPa": Baker et al. (2005); "< 0.2 wt% H , 0 ; 1 GPa": Zhang and Stolper (1991 ) and Baker et al. (2005); "1 wt% H 2 0 ; 1.2 GPa": Baker et al. (2005).

Comparison of C02 and H20 diffusivities. Comparison between DC(),tl,tll, O H ^ . and D H l 0 t using data and equations summarized above reveal that DHh0m is greater than DC(),tl,nl, and D H l 0 t is usually greater than D C()2tonl (Fig. 15). This is not unexpected since the effective radius of H 2 0 molecules (1.37 A, Shannon 1976; Zhang and Xu 1995) is smaller than the effective radius of C0 2 molecules (1.7 A. Behrens and Zhang 2001; Zhang et al. 2007) that dominates C 0 2 diffusion. However, because DC(H [ouJ approaches a non-zero constant but DH2O, approaches essentially zero as H2Ot approaches zero, DCQ2 ^ can become larger than ¿?H2O, AT magmatic temperatures at H 2 O t < 0.2 wt% with this content depending temperature, pressure and anhydrous melt composition. In natural silicate melts, H 2 O t is usually greater than 0.2 wt% and hence £>H2o, is greater than DC02ML. The diffusion of the two most important volatile components, H 2 0 and C0 2 , shows similarities and distinctions (Zhang et al. 2007). In both cases, speciation plays a critical role, the neutral species (molecular H 2 0 or C0 2 ) is the dominant diffusing species, and the diffusivity of the neutral species increases with H2Ot. However, there are also major differences. Most importantly, H 2 0 speciation results in ¿3H2O, that depends strongly on H 2 O t (as well as melt composition): while C0 2 speciation results in C0 2 diffusivity that is independent of C0 2 concentration, and more surprisingly, even independent of the anhydrous melt composition, which is a blessing for quantifying C 0 2 diffusion.

OXYGEN DIFFUSION Oxygen is the major constituent and a framework element in silicate melts. Therefore, oxygen diffusivities are essential for characterizing reactions and transport in silicate melts.

200

Zhang & Ni

H,O t (wt%) Figure 15. Comparison of H 2 O m , H , 0 , diffusivity and total C 0 2 diffusivity at 1400 K and 0.5 GPa. CO2.101al diffusivity is from Equation (41), H 2 O m diffusivity in rhyolitic melt is from Equation (14), and H 2 O t diffusivity in rhyolitic melt is calculated from Equations (13), (14), and (7a).

Because oxygen can be present in many different components and species, including molecular H 2 0, 0 2 , C0 2 , ionic CO, 2 - and OH, as well as oxygen bonded to Si, Ti, Al, Fe, M g ^ a , Na, K, and P (often simplified as bridging oxygen, non-bridging oxygen and free O 2- ). oxygen diffusion kinetics can be complicated but may also yield structural information for silicate melts. Hence, oxygen diffusion has been investigated extensively in the geological, materials science, and glass science literature. However, the various reports can be confusing. Some authors investigated chemical (or effective binary) diffusion of various oxygen species and components, and others investigated self-diffusion of oxygen, which is sometimes due to chemical diffusion of H 2 0. In earlier studies, bulk exchange and analytical methods often using mass spectrometry are applied; and in more recent studies, profiling methods typically using the ion microprobe are applied. As a general rule, the studies employing profiling techniques provide more reliable data (e.g.. Zhang 2008). In addition to studies by geologists, there is also a significant body of glass and materials science literature on oxygen diffusion. Here, we focus on the geological literature. Geochemists have investigated oxygen diffusion in melts with natural rock compositions as well as mineral or mineral mixture compositions. Dacite, Di5SAn42, \a 2 Si |0,,. jadeitc and basalt melts have been investigated more systematically. Limited data are available for diopside, CaOA l 2 0 r S i 0 2 . nephelinite. andesite, and rhyolite melts. Because diffusion under wet conditions and that under dry conditions have different mechanisms, they are discussed separately. l s O self-diffusion and oxygen chemical diffusion are also discussed separately. Self-diffusion of oxygen in silicate melts under dry conditions There is a large literature on oxygen self-diffusion in dry silicate glasses by glass scientists, which is not covered here. Muehlenbachs and Kushiro (1974) were the first in the geological literature to explore self-diffusion of 1 8 0 (or 1 8 0- 1 6 0 exchange diffusion) in various melts at high temperature and room pressure (about 0.1 MPa). Even though the diffusion data turned out to have large errors (see discussion below), the method is introduced below because many other authors used the same or a similar experimental technique. A melt blob suspended from a Pt loop is first equilibrated with a gas (either C0 2 or 0 2 ) of some normal 5 , 8 0 in a one-atmosphere furnace. This equilibrated melt is the initial state with known initial l 8 0 / l 6 0 ratio. Then a gas with a new and constant 8 l s O is continuously led into the furnace so that l s O and 1 6 0 in the gas exchange with those in the melt blob through diffusion. That is. the experiments investigate

Diffusion

ofH, C, O Components

in Silicate

Melts

201

the interdiffusivities between , 8 0 and 16 0, which are also oxygen self-diffusivities. After some duration, the sample is quenched and average 5 l s O in the whole melt blob is measured by mass spectrometry. The result is compared with the analytical solution for diffusion in a sphere: M , , 6 ^ 1 TT=1"—Z-e*p( Mr 7T rr

n-tcDt —) r

(42)

where M, and M„ are the amount of , 8 0 entering the sphere of radius r at time t and time specific equation is often written in the following form:

6 On - 5

Ogas

7i /(=i ii

\

r

The

)

where the subscripts 0 and t mean experimental time 0 (i.e., the initial state of equilibration) and time /. and the subscript "gas" means the gas phase. The final 5 1 8 0 in the melt is assumed to be the same as that of the gas by treating gas as an infinite reservoir and ignoring isotopic fractionation between the gas and the melt at high experimental temperatures. The above equation is used to fit experimental data of 8 l s O, versus f, from which the diffusivity is extracted. Muehlenbachs and Kushiro (1974) reported diffusion data in basalt melt (Fig. 16) and the mineral anorthite at room pressure and 1553-1803 K. as well as andesite. rhyolite, and Di-ioAihoAbjo melts (and plagioclase, anorthite, diopside, enstatite, and forsterite minerals) at 1553 K. The diffusivity does not depend on whether the gas phase is 0 2 or C0 2 . For basalt melt, the activation energy was found to be 377 kJ/mol, which is very high for a melt. Comparison of different melts at 1553 K indicates that 1 8 0 diffusivity in basalt melt is highest, followed by andesite melt and Di-KAnjoAbw melt (about an order of magnitude lower than in basalt), then rhyolite melt (about three orders of magnitude lower than in basalt) (Fig. 16). The mineral data are not of concern here; they will be discussed in another chapter (Farver 2010, this volume).

l O O O / n r i n K) Figure 16. Oxygen self-diffusivity in dry basalt, andesite. dacite and rhyolite melts. Data sources: Basalti M), Andesite(M) and Rhyolite(M): room pressure data by Muehlenbachs and Kushiro (1974); Basalt(C): room pressure data by Canil and Muehlenbachs (1990); Basalt(L): 1 G P a d a t a b y Lesheret al. (1996); Dacite(T): 1 GPa data by Tinker and Lesher (2001 ). Calculated line is for dacite at room pressure using Equation (44) to compare with the data by Muehlenbachs and Kushiro ( 1974) and Canil and Muehlenbachs (1990).

202

Zhang & Ni

Dunn (1982) used the experimental and analytical approach of Muehlenbachs and Kushiro (1974) and investigated 1 8 0 diffusion in Di58An42, Di40An60, and diopside melts at room pressure. Canil and Muehlenbachs (1990) followed with an Fe-rich basalt melt. The activation energy for l8 0 diffusion in dry basalt was estimated to be about 251 kJ/mol by Canil and Muehlenbachs (1990), much lower than that of Muehlenbachs and Kushiro (1974), but the two data sets are in good agreement (Fig. 16; compare Basalt(M) and Basalt(C)). This comparison shows that activation energy based on limited data is not reliable especially when there is large data scatter. Compared to more recent diffusion data based on the profiling method, these early data based on bulk gain or loss methods show large scatter (2a error is a factor of 3 in D, or about 1.1 in InD, Fig. 16) and often do not provide much quantitative constraint in our discussion below on how oxygen diffusivities depend on various parameters. Major advancements in l s O self-diffusion studies came with the development of the ion microprobe for microscopic measurement of isotopic ratios, although with the requirements of using materials highly enriched in 18 0. Coles and Long (1974) and Hofmann et al. (1974) were the first to apply the ion microprobe to study self-diffusion in minerals. Shimizu and Kushiro (1984) were the first to determine 1 8 0 self-diffusivities in silicate melts by carrying out diffusion couple experiments and by measuring 1 8 0/ 1 6 0 ratios using an ion microprobe. They examined oxygen self-diffusion ( 1 8 0- 1 6 0 exchange diffusion) in jadeite and diopside melts. Rubie et al. (1993), Lesher et al. (1996), Liang et al. (1996), Poe et al. (1997), Reid et al. (2001), Tinker and Lesher (2001), and Tinker et al. (2003) followed the approach of Shimizu and Kushiro (1984). Rubie et al. (1993) and Poe et al. (1997) measured oxygen self-diffusivities in dry Na 2 Si 4 0 9 melt at 1898-2800 K and 2.5-15 GPa, as well as in Na 3 AlSi 7 0| 7 and albite melts. Lesher et al. (1996) determined oxygen (and silicon) self-diffusion in dry basalt melt at 1593-1873 K and 1-2 GPa. Liang et al. (1996) examined oxygen (as well as Ca, Al and Si) self-diffusivities in various Ca0-Al 2 0 3 -Si0 2 melts at 1773 K and 1 GPa. Reid et al. (2001) studied oxygen (and silicon) self-diffusion in dry diopside melt at 2073-2573 K and 3-15 GPa. Tinker and Lesher (2001) investigated oxygen (and silicon) self-diffusion in dry dacite melt at 1628-1935 K and 1-5.7 GPa. Tinker et al. (2003) reported oxygen (and silicon) diffusivities in dry Di58An42 melt at 1783-2037 K and 1-4 GPa. These new data generally have higher precision, with the possible exception of Reid et al. (2001) (see below). Studies on wet melts are not summarized here and will be discussed in a later section. The melt that has been most systematically investigated for oxygen self-diffusion is dry dacite even though the number of data points is not extensive (Tinker and Lesher 2001). Figure 17 shows all available experimental data. The data indicate that oxygen self-diffusivity exhibits Arrhenian behavior within the T-P range investigated, and increases with increasing pressure from 1 to 4 GPa and then decreases at pressures above 5 GPa. The decrease of the diffusivity with pressure at > 5 GPa may have structural implications, such as possible end of the formation of highly coordinated Si or Al species (Tinker and Lesher 2001). For the purpose of predicting oxygen self-diffusion in dacite melt, there are not enough data at > 5 GPa to constrain the relation. Hence, the data at 1628 to 1935 K and 1 to 4 GPa are fit to obtain the following expression for the self-diffusivity of oxygen in dry dacite melt: In

me

" = -(6.57 + 2.148P) -

40830

"5223P

(44)

where T is in K and P is in GPa. The activation energy decreases with pressure as (339 - 43P) kJ/mol. The activation volume is -RTd(\nD)/dP = -(43.4 - 0.0187)x 10"6 m3/mol, a fairly large negative activation volume, in contrast with the positive activation volumes for H 2 0 and C0 2 diffusion. The maximum error by Equation (44) to reproduce the experimental data at 15001950 K and < 4 GPa is 0.32 in terms of InD. Equation (44) cannot be extrapolated at all to P > 4 GPa because the linearity between InD and P does not hold at higher pressures.

Diffusion

ofH, C, O Components

in Silicate

203

Melts

Oxygen self-diffusion in dry Di 58 An 42 melt has also been examined systematically (Dunn 1982: Tinker et al. 2003). Figure 18 shows all the available data. It can be seen that the data by Dunn (1982) at room pressure are scattered and do not provide much constraint. The Di 58 An 42 melt is often treated as a basalt-like melt or an FeO-free basalt. However, when oxygen selfdiffusivities in dry basalt (solid diamonds and a single plus in Fig. 18) are compared with those in dry Di 5 8 An 4 2 , those in basalt are greater by about 1.2 InD units. The diffusivity data on Di 58 An 42 melt by Tinker et al. (2003) at 1738 to 2037 K and 1 to 4 GPa can be expressed as:

-25

1

—I

•--.

•

-26

^

X

1

1

1

Oxygen self-diffusion in dacite melt

EH

-S5 -27 ri -28

=

-29 -30

-

\

— • - 1.0 GPa - S - - 2 . 0 GPa - - • - - 4 . 0 GPa a 5.2 GPa • 5.7 GPa

•

^ ^

**

-31 0.5

0.52

0.54

0.56

0.58

0.62

0.6

iooo/r(rinK) F i g u r e 17. Oxygen self-diffusion data in dacite melt (Tinker and Lesher 2001).

-23 -24 o VI

-25

ri

E

-26

-27

e

-28

-29

-•— 1.0 GPa -S--2.0 GPa -••-3.5 GPa a 4.0 GPa O 0.1 MPa - • - Basalt (1 GPa) + Basalt (2 GPa)

o o

Oxygen self-diffusion in Di58An42 melt

oo

-30 0.5

0.55

0.6

0.65

iooo/r(rinK) F i g u r e 18. Oxygen self-diffusion data in DijgAoii melt compared with those in basalt. Data for Di 5 sAn 4 i melt are from Dunn (1982) (0.1 MPa; open diamonds) and Tinker et al. (2003) (1.0, 2.0, 3.5 and 4.0 GPa). Data for basalt are from Lesher et al. (1996).

204

Zhang & Ni l n D

^ , , o

l l =

_

| 0

52_26331-240P

j

•

where Tis in K and P is in GPa (adding an additional P term such as that in Equation (44) does not significantly improve the fit). The activation energy is (219 - 2.0P) kj/mol. The activation volume is -RTd{\nD)/dP = -2.0x10~ 6 mVmol, an order of magnitude smaller than that for oxygen diffusion in dacite melt. The maximum error by the above equation to reproduce the data of Tinker et al. (2003) is 0.18 in terms of InD. Extrapolation of the above equation to lower pressures of 0-1 GPa is likely acceptable, but probably not to higher pressures of > 4 GPa. Rubie et al. (1993) and Poe et al. (1997) studied oxygen self-diffusion in dry Na 2 Si 4 09 (or Na 2 0-4Si0 2 , NS4) melt. The data (Fig. 19) covering 1893-2800 K and 2.5-15 GPa can be fit by In / ) 1v

N:

"*" = - 1 7 . 1 9 -

o

12693

" Y

3 6 0 P

(46)

where D is in m2/s, T is in K. and P is in GPa. The activation energy is surprisingly low. only about 100 kJ/mol (depending on pressure). The activation volume is negative and small, about -3.Ox 10~6 mVmol. The maximum error of the above equation in reproducing the experimental data is 0.22 in terms of InD. Three papers explored oxygen self-diffusion in dry basalt melt (Muehlenbachs and Kushiro 1974; Canil and Muehlenbachs 1990: Lesher et al. 1996). However, the earlier data by Muehlenbachs and Kushiro (1974) and Canil and Muehlenbachs (1990) are scattered (Fig. 16). Lesher et al. (1996) reported three data points at 1 GPa and one datum at 2 GPa. Oxygen self-diffusivities at 1593-1873 K and 1 GPa (solid squares in Fig. 16) can be represented by the following equation (Lesher et al. 1996): ]nD

wu1,n=_12 ls

5

o

_ 20447 Y

where 71s in K and D is in m2/s. The activation energy is 170 kJ/mol, smaller than those obtained from other l s O diffusion data in basalt melt at room pressure (377 kJ/mol by Muehlenbachs

Oxygen self-diffusion in Na 2 Si 4 0 9 melt

• 2.5 GPa - e - - 4.0 GPa • 6.0 GPa a 8.0 GPa —o— 10 GPa X 12.5 GPa • 15 GPa -23.5 035

0.4

• ^o o 0.45

0.5

0.55

i o o o / r ( r i n K ) Figure 19. Oxygen self-diffusivity in NaiSijOq melt with data from Rubie et al. (1993) (with 1000/7" > 0.45) and Poe et al. (1997) (with 1 0 0 0 / r < 0.45).

Diffusion

of H, C, O Components

in Silicate

Melts

205

and Kushiro 1974, and 251 kJ/mol by Canil and Muehlenbachs 1990). The large difference in activation energy is likely due to data uncertainty rather than being real. Because the activation volume is highly variable from dacite to Di58An42 melts (-1 lxlO - 6 to -2.0x10~ 6 m'/mol; see also Tinker et al. 2003), it is not possible to use information from other melts to constrain the pressure effect on oxygen diffusivity in basalt. Three papers investigated oxygen self-diffusion in diopside melt (Dunn 1982; Shimizu and Kushiro 1984; Reid et al. 2001). However, the temperature dependence was not constrained: only at two pressures (0.1 MPa by Dunn 1982 and 3 GPa by Reid et al. 2001) were there diffusion data at different temperatures. The data at 0.1 MPa are scattered and those at 3 GPa are strange in that the diffusivity does not change much from 2073 to 2273 K, with an implied activation energy of only 4 kJ/mol (Fig. 20). An error of a factor of 2.5 on individual D values is needed to allow a more reasonable activation energy of about 250 kJ/mol. Hence, the precision of the oxygen self-diffusion data by Reid et al. (2001) is not high. The reason is not clear. Shimizu and Kushiro (1984) reported oxygen self-diffusion data in jadeite melt at 1673 to 1883 K at 1.5 GPa, and 0.5 to 2.0 GPa at 1673 K (Fig. 21). The diffusivity at 1.5 GPa can be expressed as: In Dffalc o

mc

" = -10.84 -

31815

(48)

J

where D is in m2/s, and T is in K. The activation energy is 265 kJ/mol. At 1673 K, the activation volume is -6.4xl0~ 6 mVmol. Assuming the activation volume is independent of temperature, the P i d e p e n d e n c e of l s O diffusivity in jadeite melt may be written as: l n £ ) jadei,e

md,

37Q70 - 770P

_ 1 Q 34 _ 3-V/U

=

O

J

'

where D is in rrr/s. Tis in K, and P is in GPa. Liang et al. (1996) investigated the compositional effect of l s O self-diffusion in various C a 0 - A l 2 6 r S i 0 2 melts at 1773 K and 1 GPa. They found that l s O self-diffusivity increases -20

o- • O

-21 • " !»

»

o

-22

E -23 s

'â

-24 -25

— • -

0 n

—

o O

0.1 MPa 1 .OGPa 1,7GPa 3GPa 6GPa 7GPa

o • • » • a

9GPa 1 1GPa 12GPa 13GPa 14GPa 15GPa

-26

-27 0.38

O self-diffusion in dry diopside melt 0.42

0.46

0.5

0.54

0.58

iooo/r(rinK) F i g u r e 20. Oxygen self-diffusivities in dry diopside melt. Data sources are: 0.1 MPa from Dunn (1982); 1.0 GPa and 1.7 GPa from Shimizu and Kushiro ( 1984); and the rest from Reid et al. (2001).

Zhang & Ni

206 -27.5

-28

^

rl

-28.5

E

.9

Q, 9

s

-29

"29.5 -30

-30.5 0.53

0.54

0.55

0.56

0.57

0.58

0.59

0.6

i o o o / r ( r i n K ) Figure 21. Oxygen self-diffusivities in jadeite melt (Shimizu and Kushiro 1984).

with decreasing silica and alumina content, as expected. As NBO/T increases from 0.3 to about 0.9 in CaO-AUOrSiOi system, l s O self-diffusivity increases by about an order of magnitude. Figure 22 compares oxygen self-diffusivity in all melts for which the temperature dependence of diffusivity has been determined well at 1 GPa. Oxygen self-diffusivities increase with decreasing Si0 2 and AljO, contents, or from polymerized to depolymerized melts. From dacite melt (NBO/T « 0.1) to basalt melt (NBO/T « 1), oxygen self-diffusivity increases by two orders of magnitude. It may be inferred that oxygen self-diffusivities depend on oxygen speciation, increasing from bridging oxygen (BO) to non-bridging oxygen (NBO) and then to free oxygen (O 2- ). The diffusivity of each oxygen species may depend on the overall melt composition. In terms of oxygen self-diffusion, basalt melt (NBO/T » 1) is similar to diopside melt (NBO/T = 2) and Na 2 Si 4 0 9 melt (NS4 melt. NBO/T = 0.5), whereas dacite melt (NBO/T = 0.1) is similar to jadeite melt (NBO/T = 0) (Fig. 22). But the similarity in each group is not close enough for diffusivities to be merged for a combined fitting. For different melts, there does not seem to be a single compensation temperature where l s O diffusivities in all melts are the same. There are not enough data yet to contemplate a general relation between oxygen selfdiffusivity and natural melt composition under dry conditions. To achieve such a goal, it is necessary to investigate , 8 0 diffusion in dry basalt, dry andesite, and dry rhyolite as a function of temperature and pressure systematically (on par with the investigation of dacite melt), and to examine the compositional dependence. In highly polymerized melts (rhyolite and pure silica), it will be important to make the dry system very dry, e.g., less than 50 ppm H 2 O t (depending on temperature), so that the diffusive flux is due to true l s O self-diffusion, not H 2 0 chemical diffusion (see later sections). One may try to use self-diffusivities of other elements such as Si to constrain those of oxygen. However, even though self-diffusivities of oxygen and silicon (both are structural elements) in dry melts are often similar (e.g.. Lesher et al. 1996: Poe et al. 1997; Tinker et al. 2003), they may also be significantly different (e.g.. Tinker and Lesher 2001; see review by Lesher 2010 and Zhang et al. 2010). It has been shown that oxygen self-diffusivity in dry melts is similar to the Eyring diffusivity defined as D = kTI("kx\) where X is the jump distance and r| is the melt viscosity (e.g., Shimizu

Diffusion

of H, C, O Components

-23

1 A

—

1

1

in Silicate —i

r

Melts

207

1

-24 -25

fH -26 c Q -27 Ci

—•— Basalt (L) A Diopside (S) • -OI58An42 (T) -B- - Dacite (T) • Jadeite (S) — -Na2Si40,

•N.

-28

"V -29

Oxygen

X.

v

««. ^

self-diffusion

-30

at 1 G P a i i

-31 0,5

0.52

0.54

® " -a. i

i

•


n0X[ 1

H

m ^

(53) V '

where X is the mole fraction of H 2 O t on a single oxygen basis. The expression Xm&XIAXm is shown in Figure 3 For K = 0.5. At low 112(), (e.g., when X < 0.01), the above can be simplified as (Eqn. 16 in Zhang et al. 1991b):

D,, O ~Dn ' " O . a n h y d r o u s +—D„„ 9 A 1

H

(54)'

v

Zhang et al. (199 lb) summarized literature data to evaluate the role of H 2 0 diffusion in l s O "self' diffusion. Behrens et al. (2007) experimentally investigated H 2 l s O sorption into a rhyolite melt, from which both H 2 O t and l s O diffusion profiles were measured. Their results show that both 112( ), and l s O profiles indicate the same H 2 O m diffusivity (Fig. 24) under the assumption that H 2 O m is the diffusing species and there is chemical and isotopic equilibrium. Thus, their experimental data confirm that H 2 O m is the diffusing species in both 112(), diffusion and l s O "self' diffusion, as well as the quantitative theory of Zhang et al. (1991b) presented above. In summary. H 2 0 chemical diffusion (or effective binary diffusion) carries an l 8 0 flux, which contributes to an apparent l 8 0 diffusivity. If the H 2 0 content is uniform in the sample and there is no chemical diffusion of H 2 0 (e.g., diffusion couple with similar starting H 2 O t but different l 8 0 / l 6 0 in the two halves, or sorption of 18 0-enriched H 2 0 into a sample already containing the equilibrium concentration of H 2 0). l 8 0 - , 6 0 exchange can still be due to the

Figure 24. Experimental data on H 2 0 , and Ri = ls O/( l 6 0 + l 7 0 + l 8 0 ) profiles (data points) during hydration using l s O-enriched H 2 0 . The solid lines are fit by Equation (11) (for the H 2 O t profile) and Equation (51) x (for the Ri profile) assuming H 2 O m is the diffusing species and DH^O,,, = Dite" with the same a and Dn values for both profiles. From Behrens et al. (2007).

Diffusion

ofH, C, O Components

in Silicate

211

Melts

mobility (self-diffusion) of H 2 0. and the diffusion may be said to be self-diffusion. In both cases, the diffusing species is H 2 O m . The resulting l 8 0 diffusivities from H 2 0 chemical diffusion and from H 2 0 self-diffusion are not much different. In the next section, we examine the conditions for , s O diffusion in natural silicate melts to be true oxygen "self" diffusion or through H 2 0 diffusion, and quantify apparent l 8 0 diffusivity in hydrous rhyolite and dacite melts. "SelF' diffusion of oxygen in natural silicate melts in natural environments Using the theory presented above, if we know both the true l 8 0 self-diffusivity under dry conditions and the H 2 0 diffusivity, we can compare £>i«o,anhydrous with DH2()mXm. or £>i«o,anhydrous with DH20J(mdXldXm, to determine whether the true , 8 0 self-diffusion or the apparent l 8 0 flux due to H 2 0 diffusion dominates the l 8 0 flux. Below, we employ data on H 2 0 diffusion and dry l8 0 self-diffusion to evaluate quantitatively the conditions when H 2 0 diffusion dominates l 8 0 "self" diffusion. (1) Dacite melt. This melt is considered first because extensive data are available. Figure 25 compares dry l s O self-diffusivity (Tinker and Lesher 2001) and apparent 1 8 0 diffusivity contributed by H 2 0 diffusion. First consider a numerical case at 1600 K and 1 GPa (which are close to the covered experimental conditions of both l s O self-diffusion and H 2 0 diffusion). From Equation (44), dry l s O self-diffusivity is 3.5xl0~ 14 m2/s. Based on Equations 18 and 53, about 0.32 wt% H 2 O t is required to contribute 3.5xl0~ 14 m2/s to the oxygen diffusivity. Because synthesized experimental dry melts typically contains only < 0.1 wt% H 2 O t , the experimental dry l 8 0 self-diffusivities at such high temperatures by Tinker and Lesher (2001) are true ^o.anhydrous- Now consider 1200 K (a more reasonable magmatic temperature for natural dacite) and 1 GPa. Data by Tinker and Lesher (2001) do not cover such a low temperature (metastable melt). Dry l 8 0 self-diffusivity extrapolated using Equation (44) is 2.1xl0~ 17 m2/s. Based on Equation (18), about 0.05 wt% (500 ppm) H2Ot is required to contribute 2.1xl0~ 17

05

0.6

0.7

0.8

0.9

I

1.1

1.2

1.3

i o o o / r ( r i n K) Figure 25. Oxygen and H 2 O t diffusion at i GPa in dacite melt. The open circles and the solid line are H 2 0, diffusivity at 0.95 to 1 GPa from Behrens et al. (2004) and Ni et al. (2009b). The two dashed lines are calculated oxygen diffusivities due to H 2 0 diffusion based on the HiO diffusivity of Ni et al. (2009b). The filled diamonds (in red in the online version) with a best-fit line (in red in the online version) are experimental l s O self-diffusion data at 1 GPa from Tinker and Lesher (2001). The dotted line is Eyring diffusivity calculated using the viscosity model of Hui and Zhang (2007) at 0.1 MPa to 1 GPa assuming a jumping distance of 2.8xl0~ l n m.

212

Zhang & Ni

m2/s to the oxygen diffusivity. The difference at the two temperatures is due to lower activation energy for H 2 0 diffusion and higher activation energy for dry , 8 0 diffusion. As the temperature is lowered further, H 2 0 diffusion would dominate the apparent , 8 0 "self' diffusion (Fig. 25) at even lower H 2 O t . Because most natural dacite melts are expected to contain more than 0.2 wt% H 2 O t , we conclude that apparent , 8 0 diffusion in natural dacite melts is almost always dominated by H 2 0 diffusion. (2) Basalt melt. Both 1 8 0 and H 2 0 diffusion data are limited on basalt melt. Consider 1600 K and 1 GPa. From Equation (47), dry l s O self-diffusivity is about l.lxlO - 1 1 m2/s. Based on Equation (22), about 1 wt% H 2 O t is necessary to contribute l . l x l 0 ~ u m 2 /s to apparent ls O diffusivity. That is, for basalt melt, due to high dry l s O self-diffusivity and magmatic temperatures, about 1 wt% H 2 O t is needed for H 2 0 diffusion to dominate l s O "self' diffusion. Therefore, in almost all mid-ocean ridge basalt melts (typically containing 0.2 to 0.7 wt% H 2 O t , but can be up to 1.2 wt%; Dixon et al. 1988; Michael 1988; Workman et al. 2006) and most ocean island basalt melts (often more volatiles than MORB; Dixon et al. 1997; Hauri 2002; Workman et al. 2006), l s O diffusivity is most likely dominated by true 1 8 0 self-diffusion. However, in preemptive island arc basalt (IAB) melts, and even in melt inclusions in mantle megacrysts likely influenced by subduction, H 2 O t concentration are 2-6 wt% (Stolper and Newman 1994; Wang et al. 1999; Newman et al. 2000; Gurenko et al. 2005; Wallace 2005), meaning apparent l s O self-diffusion in these melts is dominated by H 2 0 diffusion. (3) Rhyolite melt. There are only limited 1 8 0 diffusivity data in dry rhyolite melt at 1553 K and 0.1 MPa (Muehlenbachs and Kushiro 1974) and the quality of the data is not high (see discussion in an earlier section). On the other hand, H 2 0 diffusion in rhyolite melt has been investigated extensively (e.g., comprehensive model by Ni and Zhang 2008). Figure 26 compares the contribution to 1 8 0 diffusion by the H 2 0 flux with that by true network l s O self-diffusion. Oxygen self-diffusivities by Muehlenbachs and Kushiro (1974) are higher by a factor of about 24 than the Eyring diffusivity for dry melt calculated from the viscosity model of Zhang et al. (2003). If the data by Muehlenbachs and Kushiro (1974) are accurate and the melt was indeed dry (much less than 0.1 wt% H 2 O t ), oxygen flux due to chemical diffusion of H 2 0 when H 2 O t content is about 0.1 wt% would be roughly the same as the oxygen flux due to true oxygen self-diffusion. On one hand, oxygen self-diffusivities obtained by Muehlenbachs and Kushiro (1974) are close to effective binary diffusivity of P, which seems to imply the Eyring diffusivity limit. On the other hand, typical "dry" rhyolite glasses (either natural or synthesized) contain > 0.1 wt% H 2 O t . If the rhyolite used by Muehlenbachs and Kushiro (1974) also contained about 0.1 wt% H 2 0„ their measurements would actually mean apparent l s O "self" diffusivity due to H 2 0 diffusion, implying true dry l s O self-diffusivities are yet to be determined. Regardless how this issue is resolved, at typical rhyolite melt temperatures (such as 1200 K), H 2 0 diffusion would dominate oxygen transport at much lower H 2 O t , such as 10 to 100 ppm level. That means, in natural rhyolite melt in which H 2 0 content if often 4-6 wt% (e.g, Wallace et al. 2003), diffusive transport of oxygen isotopes is through H 2 0 diffusion. There are no , 8 0 diffusivity data in dry andesite melts, and hence similar quantitative comparisons cannot be carried out. Nonetheless, it is expected that H 2 0 diffusion plays a more important role in transporting l s O flux as temperature is lowered and as the melt becomes more silicic. Hence, the behavior of andesite melt is expected to be between that of basalt melt and dacite melt. When H 2 O t is high enough (e.g., > 0.5 wt%), the apparent l s O diffusivity in andesite melt is likely dominated by H 2 0 diffusion. Because pre-eruptive natural andesite melt often contains > 0.5 wt% H 2 O t , we expect the , 8 0 flux in natural andesite melt to be often dominated by H z O chemical diffusion. In summary, in nature, true l s O self-diffusion is the diffusion mechanism only for relatively dry basalt melt (such as MORB and OIB). The realization that diffusive transport of l s O in natural rhyolite and dacite melts is almost always due to H 2 0 diffusion means it is possible to predict 1 8 0 diffusive transport using Equation

Diffusion

ofH, C, O Components

in Silicate

Melts

213

-24

H 2 O t dif -26

_]wt% H 2 O t

-28

-i r i- -30

c

-32

4

Oxygen dif wt% H2Ot

P dif; 0.1 wt% \ t ^ O self dif (dry)

C5

\

s

Oxygen dif wt%H,O t

Vs

-36 . Eyring dif^v ". .dry to 0.1 w l ^ ^ -38 -40

0.6

Rhyolite melt I I I

0.7 l O O O / n r0.8 i n K)

I I I

0.9

1

Figure 26. Oxygen and HiO, diffusion at 0.1-100 MPa in rhyolite melt. The pressure effect in this small pressure range is smaller than 0.2 InD units. The solid line is the calculated HiO, diffusivity. The two dashed lines (roughly parallel to the solid line) are oxygen diffusivities due to HiO diffusion calculated at 0.1 GPa based on Ni and Zhang (2008). The Eyring diffusivity at 0-0.1 wt% H 2 0 , is calculated from viscosity data (solid circles; Neuville et al. 1993; Schulze et al. 1996) and the viscosity model (lines; Zhang et al. 2003) assuming a jumping distance of 2.8x10~ ln m. The two filled diamonds are experimental l s O self-diffusion data at 0.1 MPa from Muehlenbachs and Kushiro (1974). The open squares and short dashed line are effective binary diffusion data of P and fit at 0.8 GPa and 0.1 wt% HiO, (Harrison and Watson 1984), which are somewhat higher than the Eyring diffusivity.

(52) or (53) because H 2 0 diffusion in these melts has already been investigated well (e.g., Behrens et al. 2004: Liu et al. 2004b: Ni and Zhang 2008: Ni et al. 2009b). For the convenience of the readers, calculated apparent , 8 0 diffusivities in hydrous rhyolite and dacite melts as a function of T, P and H 2 O t are listed in Tables 3 and 4. In the calculation, the diffusivity of oxygen not associated with H is approximated by Eyring diffusivity. The calculated diffusivities have a 2 a uncertainty of about 0.6 in InD and may be applied in rough estimation of diffusion rates. For calculation of the , 8 0 diffusion profile, it is more accurate to use Equation (51). Prediction of l 8 0 diffusion in basalt and andesite melts require more experimental work in terms of both H 2 0 diffusion and dry , 8 0 diffusion. Contribution of C 0 2 diffusion to l s O transport in C0 2 -bearing melts Because carbon species ( C 0 2 molecule and C 0 3 2 - ) also carry oxygen, oxygen transfer in natural systems may also be realized through the diffusion of these species, especially molecular C 0 2 . One-atmosphere experiments on , 8 0 diffusion often use 18 0-enriched C 0 2 gas as the source for , 8 0 (Muehlenbachs and Kushiro 1974: Canil and Muehlenbachs 1990). Not withstanding the quality of such data, one may wonder whether the extracted diffusivity is true , 8 0 self-diffusivity. or just a reflection of C 0 2 diffusion carrying l 8 0 into the sample. No simultaneous investigation of l 8 0 and C 0 2 diffusion has been carried out yet. Although experimental data are lacking, the role of C 0 2 diffusion in transporting l 8 0 can be treated similarly as the role of H 2 0 diffusion in transporting l 8 0 . However, it is more convenient to consider total C 0 2 diffusion rather than the diffusion of C 0 2 molecules because D c „ 2 t o t , is roughly independent of melt composition. The diffusion equation for the isotopic fraction of , 8 0 can be written as follows (comparing with Eqn. 51):

Zhang & Ni

214

Table 3. Calculated loDis 0 (D in m2/s) in hydrous rhyolite melt. H2Ot (wt%) 0.1 0.3 0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0

0.1 GPa

0.5 GPa

1 GPa

1000 K

1300 K

1600 K

1000 K

1300 K

1600 K

1000 K

1300 K

1600 K

-37.36 -35.11 -34.02 -32.45 -31.43 -30.63 -29.34 -28.26

-35.02 -32.78 -31.73 -30.24 -29.32 -28.63 -27.58 -26.76

-33.43 -31.31 -30.28 -28.86 -28.02 -27.40 -26.51 -25.85

-38.06 -35.80 -34.72 -33.15 -32.13 -31.33 -30.03 -28.95 -27.99 -27.1 -26.26 -25.47

-35.38 -33.14 -32.08 -30.60 -29.68 -28.99 -27.94 -27.12 -26.42 -25.79 -25.22 -24.69

-33.56 -31.46 -30.43 -29.01 -28.16 -27.55 -26.66 -25.99 -25.46 -25.00 -24.59 -24.22

-38.93 -36.67 -35.59 -34.02 -33.00 -32.19 -30.90 -29.82 -28.86 -27.97 -27.13 -26.34

-35.83 -33.59 -32.53 -31.05 -30.13 -29.44 -28.39 -27.56 -26.86 -26.24 -25.67 -25.14

-33.72 -31.64 -30.61 -29.19 -28.34 -27.73 -26.84 -26.17 -25.64 -25.18 -24.77 -24.41

Each cell lists In/) ( values where /) ,, is calculated total : 0 diffusivity in m 2 /s using Equation (52), with i>u ,, from Equation (14), K from Equation (7a), and /) ( , , ,, , approximated by / ) | . calculated using the viscosity model of Zhang et al. (2003). Values of InD at other temperatures can be obtained using the Arrhenius relation.

Table 4. Calculated lnDis 0 (D in m2/s) in hydrous dacite melt. H2Ot (wt%) 0.1 0.3 0.5 1.0 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0

0.1 GPa 1000 K -38.62 -36.33 -35.20 -33.54 -32.43 -31.54 -30.08 -28.83

1300 K -34.60 -32.47 -31.42 -29.94 -29.02 -28.34 -27.30 -26.48

0.5 GPa

1 GPa

1600 K

1000 K

1300 K

1600 K

1000 K

1300 K

1600 K

-29.51 -28.54 -27.91 -26.93 -26.32 -25.87 -25.21 -24.72

-39.21 -36.92 -35.79 -34.13 -33.02 -32.13 -30.67 -29.42 -28.30 -27.25 -26.27 -25.32

-34.95 -32.86 -31.82 -30.34 -29.43 -28.75 -27.72 -26.91 -26.21 -25.58 -25.01 -24.47

-29.53 -28.60 -27.99 -27.06 -26.47 -26.05 -25.42 -24.96 -24.59 -24.27 -23.99 -23.74

-39.95 -37.65 -36.53 -34.87 -33.76 -32.87 -31.40 -30.16 -29.04 -27.99 -27.00 -26.06

-35.33 -33.31 -32.28 -30.80 -29.90 -29.23 -28.22 -27.42 -26.74 -26.13 -25.56 -25.03

-29.54 -28.65 -28.07 -27.18 -26.63 -26.22 -25.64 -25.21 -24.86 -24.57 -24.31 -24.07

from Each cell lists lnZ> 0 values where ! 1 ,, is calculated total h O diffusivity in m 2 /s using Equation (52), with ! , Equation (18), K from Equation (7b), and !) ,, h i i ; . approximated by ., calculated using the viscosity model of Whittington et al. (2009).

dR d2R d2(R-Xc — —— + Dco h 8t = D 18nO.lloncarhoimlcd C02„, — -^2

0

)

C02.„„ R

d2Xc

0

(55)

where ^c 05 o lola i is mole fraction of total C 0 5Otot£1] (each C 0 5Otot£1| contains a single oxygen and two C 0 5 O total makes one C 0 2 , t o t a i ) on a single oxygen basis, and Di8 0 mhydl.ous is diffusivity of ls O unassociated with carbon. X C ( 1 5 o t 0 t a l , is calculated as (C cd /22.005)/[C cd /22.005+(100Cci)!W] where C cd is wt% of C 0 2 .total a r i d W is the molar mass of the dry melt on a single oxygen basis (Table 1). X C02t)tal may be used instead of XC()50total, but a factor of 2 would have to be incorporated because each C 0 2 molecule carries two oxygen atoms. DC02ma] does not depend on C 0 2 concentration or anhydrous melt composition and hence can be taken out of

Diffusion

o f H , C, O Components

the differentials. Based on the above equation, the (comparing with Eqn. 52): Dn, s O *Dn

O.noncartionated

,8

in Silicate

Melts

215

0 diffusivity can often be estimated as

+Drn \

'ri a 9 -S co —i — ^o t - 0 s H ¡ î 8 T3 en -d M

CM On

ö ö

oí

• ïï CT

co ™ O.

- i z : © (N - h © —

i n m ~1 m, ooc Ö ©

~~1 co co ^

0< -

o" ^ y -g M I >> . 53

o m — j -a i—l 0) ¿j- S

1 1 1 I .s

Ms l a

g m 3

a o

. 55 s S s S

s Q-

^§

es ©

S

g

s i s

'S

'vO i n © Tt m, © © — ' © © °

S

< SS ^ co tì

r-; S —. t^oo^os^goq^prvoincnvovop^vnc^oocsivo co © ^o 0 r-: ^ 2 c¿ ^ ^ co - J 'fj. «n «n i n c m co 2

•n

oo co ^ r- o oo oo ^

« I

: o ; F3

© ©

o©©

1

-

çm

co ^ m, co • co o m ^ , 3

m -S" 5 , q §

S o C3 —i . . • M n o 5

• - ON ^ V ^ CT S

I Í

TT ^

a

es

o

"O Ü a n « n §

vi n o . s Ooo O _ z: © c ri • -o ^ • .2 o

on

ü

S

g

S SQ 'S

a oo

T. (U -o •a s s S S K > Rb > Cs) in silicate melts. This is similar to noble gas diffusion: the diffusivity of noble gases increases as the size decreases, from Xe to Kr to Ar to Ne to He (e.g., Perkins and Begeal 1971; Roselieb et al. 1995; Behrens 2010, this volume). This trend may sound intuitive, but cannot be explained by the StokesEinstein equation (see Zhang 2010, this volume) because the diffusivity variation is orders of magnitude, much larger than the relative change in the ionic radius. Furthermore, this trend cannot be generalized to other elemental series: e.g., in alkali earth elements, REE, and most other series, the trend can be opposite. The alkali earths (Be, Mg, Ca, Sr, Ba, Ra) Be diffusion. Only one study (Mungall et al. 1999) reported Be tracer diffusion data on three melt compositions: HR7, wet HR7 with 3.6 wt% H z O, and HR7+Na (Table 1). The data are shown in Figure 9, together with the line for Cs diffusivities in dry HR7 for comparison. Even though Be is a small cation, Be diffusivities are very low, much lower than the lowest diffusivities of alkali elements (Cs). As will be seen later, Be diffusivities are also the smallest among the alkali earth elements. Be tracer diffusivity in dry HR7 at 1410-1873 K and 0.1 MPa can be expressed as follows: -sHR7 Be TD

(-8.38 ± 2.62) _(38690 ±4236)

(18)

The maximum error of Equation (18) in reproducing the three experimental diffusivities is 0.21 in InD. The activation energy is high, 322±35 kj/mol. Mg diffusion. Mg is a major element in most natural silicate melts. Extensive Mg diffusion data in silicate melts are available, with 13 papers and 166 points (of which 15 are for hydrous melts). Zhang et al. (1989) determined Mg SEBD in dry andesitel melt during crystal dissolution experiments. Kubicki et al. (1990) studied SEBD of Mg in dry diopsideanorthite melts. Sheng et al. (1992) explored Mg self diffusion in dry CMAS melts at 15341826 K and in air. Baker and Bossanyi (1994) examined the effect of H 2 0 and F on Mg SEBD in rhyolite7-dacitel diffusion couple at 1373-1673 K and 1 GPa. van der Laan et al. (1994) obtained Mg SEBD in dry rhyolite3-rhyolitel6 and andesite3-rhyolitel6 diffusion couples at 1523 K and 1 GPa. LaTourrette et al. (1996) and LaTourrette and Wasserburg (1997) studied Mg self diffusion in dry HB1 melt at 1623-1773 K and in air. Mungall et al. (1999) examined Mg tracer diffusion in dry HR7 and dry HR7+Na at 0.1 MPa and wet HR7 (3.6 wt% H z O) melts at 1 GPa. Van Orman and Grove (2000) reported a datum of Mg SEBD in lunar basalt (LB1 in Table 1) during clinopyroxene dissolution. Roselieb and Jambon (2002) investigated Mg tracer diffusion in dry albite and jadeite melts at 1073-1293 K and 0.1 MPa. Lundstrom (2003) acquired SEBD of Mg in dry basalt7-basanite couple at 1723 K and 0.9 GPa. Chen and Zhang (2008, 2009) determined MgO SEBD in basaltl 1 (MORB) melt during olivine and clinopyroxene dissolution. Mg diffusion data are shown in Figure 10.

Diffusion

Data in Siliate

Melts

331

1 0 0 0 / 7 ( 7 in K) Figure 8. Li, Na, K, Rb and Cs tracer diffusivities in rhyolite and albite melts. See Figures 1-7 for data sources. In (b), Rb and Cs diffusion in albite melts were studied in two papers (Jambon and Carron 1976; Roselieb and Jambon 1997), with small differences.

Examination of the data shows the following. First, the pressure effect at < 2 GPa is negligible (e.g.. open circles, triangles, squares and diamonds in Fig. 10a). Second, differences between self-diffusivities and SEBD are less than 1 InD unit (e.g., self diffusivities in HB1 and SEBD in basaltl 1 in Fig. 10a) when Si0 2 concentration difference across the profile is < 6 wt%, although there are no data at exactly the same bulk composition for a direct comparison. (LB1, even though also a basalt, is compositionally very different from terrestrial basalts.) Thirdly, Figure 10b shows that Mg diffusivities vary significantly with melt composition (and temperature), especially from andesitel to HR7. Mg SEBD in dry basaltl 1 (MORB) melt (containing 0.2 to 0.4 wt% H 2 0) during olivine

Zhang, Ni, Chen

332

ì o o o / n r in K ) Figure 9. Be tracer diffusivities in dry HR7 at 0.1 MPa, wet HR7 (containing 3.6 wt% H , 0 ) at 1 GPa, and dry H R 7 + N a at 0.1 MPa. All data are from Mungali et al. ( 1999). The short dashed (and purple) line with no corresponding points are for Cs diffusion in dry HR7, shown for comparison.

dissolution at 1543-1753 K and 0.5-1.4 GPa can be expressed as (Chen and Zhang 2008): 7 - j b a s a l l l l loliv diss) _ Ma S O J D

(

~~

,.. e

n

x

P

(26222 + 2470) (-7.92+1.50)--

(19)

The maximum error of Equation (19) in reproducing the diffusion data is 0.27 in InD. Mg SEBD values in the same basaltll melt during clinopyroxene dissolution at 15441790 K and 0.5-1.9 GPa are lower than that during olivine dissolution by about 0.37 in InD and can be expressed as follows (Chen and Zhang 2009) r-\ b a s a t i l i l e p \ diss) Mg SOJD

_ ~~

C A

P

(-6.65 + 1 . 4 6 ) -

(28922 + 2420)

(20)

The maximum error of Equation (20) in reproducing the diffusion data is 0.16 in InD. Although the same initial melt composition (basaltl 1) is used in the olivine and clinopyroxene dissolution studies by Chen and Zhang (2008, 2009). the interface melt during clinopyroxene dissolution is more Si0 2 -rich than that during olivine dissolution. Therefore, it is expected that Mg SEBD values during clinopyroxene dissolution are lower than those during olivine dissolution. In andesitel melt, Mg SEBD during olivine dissolution at 1488-1673 K and 0.5-1.5 GPa can be expressed as: r-\ a n d e s i t e l (oliv diss) _ M » SF.BT)

~~

„

Y n

C A

P

(-3.53 + 3 . 9 6 ) -

(35049 + 6283)

(21)

The maximum error of Equation (21) in reproducing the experimental diffusivities is 0.34 in InD. Mg SEBD values in the same andesitel melt during clinopyroxene dissolution are not significantly different from those during olivine dissolution. In highly silicic melt HR7, Mg tracer diffusivities can be expressed as follows:

Diffusion

Data in Siliate

-22

333

• • • O v • O

SEBO; basalt 11 (ol); 0.5 GPa SEBD; basalti 1(o!); 0.9 GPa SEBO; basalti 1 (ol); 1.4 GPa SEBD; basalt 11 (cpx); 0.5 GPa SEBD; basalt 11 (cpx); 0.9 GPa SEBO; basalti 1 (cpx); 1.4 GPa SEBB; basalti 1 (cpx); 1.9 GPa —a— SD; HB1; 0.1 MPa SEDB; LBI(cpx); 1.3 GPa X SEBD; b-b; 0.9 GPa

-23

e

Melts

-24

-25

Mg diffusion in basalt

a

-26

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

1 0 0 0 / 7 ( T in K )

-22 -24

'to

N s=

-26 -28

\

• - • - - S D in HB1 -SEBD; basalt! 1(ol) SEBD; basalt 11 (cpx) - • - SEBD; andesitel (ol)

—* -SEBD; andesitel (cpx) - O • - TD in HR7 •-©-•TD in albite — & - TD in jadeite

ov

-30 -32

•o

cf

Ì.

Sxg

-34 b -38 0.5

M g d i f f u s i o n in dry m e l t s

0.6

0.7

0.9

0.8

i o o o / r ( r i n

K)

Figure 10. M g diffusivities in silicate melts, basalt 11 (ol) means D in basalt! 1 melt during olivine dissolution; L B l ( c p x ) means D in lunar basalt during clinopyroxene dissolution; b-b means basalt7basanite couple. Data sources can be found in the text and Table 1. Pressure is not shown in (b) because (a) shows that the pressure effect is negligible.

(29341 ±7457) (—11.77 ± 4.51) —

(22)

The maximum error of Equation (22) in reproducing the experimental diffusivities at 14101873 K and 0.1 MPa is 0.49 in InD. We made an effort to model the compositional effect on Mg diffusivities (including self and tracer diffusivities, and SEBD) in natural and nearly natural dry basalt to rhyolite melts (rather than melts with mineral compositions), including SEBD in andesitel melt (Zhang et

334

Zhang, Ni,

Chen

al. 1989). self diffusivities in HB1 melt (LaTourrette et al. 1996; LaTourrette and Wasserburg 1997). tracer diffusivities in HR7 (Mungall et al. 1999), SEBD in lunar basalt (Van Orman and Grove 2000). SEBD in basalt7-basanite melts (Lundstrom 2003). and SEBD in basalt 11 (Chen and Zhang 2008. 2009). We also included the SEBD data by Kubicki et al. (1990) on Di-An melts and self diffusion data by Sheng et al. (1992) on "basic" melts because the compositions of these melts are not far from the HB1 melt of LaTourrette et al. (1996) and LaTourrette and Wasserburg (1997). Data by van der Laan et al. (1994) are not included because the 1120 content is fairly high (0.3-1.2 wt%). For SEBD values, there is a range of compositions along a profile. For the modeling purpose, the melt composition is taking to be the average of the two diffusion ends if a diffusion couple profile is fit by assuming a constant diffusivity. or the average of the far-field and interface melts if the experiment method is crystal dissolution. The diffusion data can be well fit by the following: = -5.17-11.37XSi-2A6Xm -

10993+

^7839*sa

(23)

where X is cation mole fraction, A"FM = Fe + Mn + Mg, and XSA = Si + Al. Fits of the data by Equation (23) are shown in Figure 11, with a reproducibility within 0.76 laD units (or 0.33 logD units) except for the four outlier points (one point on basalt7-basanite couple by Lundstrom 2003 is off by 0.9 laD units, one point for lunar basalt melt by Van Orman and Grove 2000 is off by 1.9 InD units, one single point for andesite melt during quartz dissolution by Zhang et al. 1989 is off by 1.2 InD units, and one point in Kubicki et al. 1990 is off by 1.2 InD units). The maximum error in using Equation (23) to predict Mg tracer diffusivities is 0.7 InD units for HR7 + Na melt, 1.1 InD units for jadeite melt, 1.6 InD units for albite melt. Mg diffusion data in hydrous melts are limited, with only 15 points. Adding 1 wt% H 2 0 increases the diffusivity by about 2 InD units (van der Laan et al. 1994); adding 3.6 wt% H 2 0 increases the diffusivity by about 4 InD units (Mungall et al. 1999); and adding 3.2

20

• HB1 B HR7 X basalti 1 (ol) A basalti 1 (cpx) s andesitel (ol) • Di50An50

1

22 -i r-4

E •E

24 -26

i. 1 -28

Q s

En

— -30 -32 0.5

Dry melts 0.55

0.6

0.65

0.7

0.75

1000/7" (7" in K ) Figure 11. Fitting Mg diffusion data in natural and nearly natural basalt to rhyolite melts by Equation (23). basaltll(ol) means D in basaltll melt during olivine dissolution. Data sources: LB1 (LaTourrette et al. 1996; LaTourrette and Wasserburg 1997); HR7 (Mungall et al. 1999); basalti 1 (of) (Chen and Zhang 2008); basalti l(cpx) (Chen and Zhang 2009); andesitel(ol) (Zhang et al. 1989); Di 50 An 50 (Kubicki et al. 1990).

Diffusion

Data in Siliate

Melts

335

wt% and 6.0 wt% H 2 0 increases the diffusivity by 3.1 and 3.5 InD units, respectively (Baker and Bossanyi 1994). The limited data indicate that adding H 2 0 increases MgO diffusivity significantly, but the effect cannot be quantified yet. Ca diffusion. Sixteen papers reported Ca diffusion data in silicate melts (129 points, of which 20 are for hydrous silicate melts). Medford (1973) explored Ca SEBD in mugearitel and mugearite2 melts (Table 1) using the diffusion couple method at 1503-1696 K and 0.1 MPa. Hofmann and Magaritz (1977) studied Ca tracer diffusion in dry basalt3 melt at 15231723 K and 0.1 MPa. Jambon (1982) acquired Ca tracer diffusivities in rhyolite5 melt at 905-1204 K and 0.1 GPa. Harrison and Watson (1984) reported Ca SEBD in rhyolitel2 melt during apatite dissolution at 1473-1673 K, 0.8 GPa, and 0.1 and 1.0 wt% H 2 0. Zhang et al. (1989) extracted Ca SEBD in andesitel melt during diffusive clinopyroxene dissolution experiments. Kubicki et al. (1990) studied SEBD of Ca in binary systems of dry diopsideanorthite melts at 1633-1963 K and 0.0001 to 2 GPa, with most data at 0.2 GPa. Behrens (1992) reported Ca tracer diffusivities in dry Ab 40 An 60 melt at 993-1123 K and 0.1 MPa. Baker and Bossanyi (1994) examined the effect of H 2 0 and F on Ca SEBD in the same diffusion couple at 1373-1673 K and 1 GPa. van der Laan et al. (1994) obtained Ca isotopic effective binary diffusivities in rhyolite3-rhyolitel6 and andesite3-rhyolitel6 diffusion couples at 1523 K and 1 GPa. LaTourrette et al. (1996) characterized Ca self diffusion in dry HB1 melt (Table 1) at 1623-1773 K and in air. Liang et al. (1996a) explored Ca self diffusion in various dry Ca0-Al 2 0 3 -Si0 2 melts at 1773 K and 1 GPa. Mungall et al. (1999) examined Ca tracer diffusion in dry HR7 and dry HR7+Na at 0.1 MPa and wet HR7 (3.6 wt% H 2 0) melts at 1 GPa. Roselieb and Jambon (2002) studied tracer diffusion of Ca in dry jadeite and albite melts at 923-1293 K and 0.1 MPa. Lundstrom (2003) obtained SEBD of Ca in dry basalt7-basanite couple at 1723 K and 0.9 GPa. Gabitov et al. (2005) examined Ca SEBD in two haplorhyolite (HR6 and HR9 in Table 1) melts during fluorite dissolution at 1173-1273 K, 0.075-0.1 GPa, and 1.2-4.8 wt% H 2 0. Morgan et al. (2006) measured Ca SEBD in low-Ti and high-Ti lunar picrite melts (labeled LP1 and LP2 in Table 1) during anorthite dissolution at 1673 K and 0.6 GPa. Chen and Zhang (2008, 2009) determined Ca SEBD in basaltl 1 melts during olivine and clinopyroxene dissolution. During olivine dissolution, the Ca concentration gradient is small, and Ca diffusion is dominated by cross-diffusivities, and sometimes even show uphill diffusion (e.g., Zhang et al. 1989). Hence, Ca effective binary diffusivities extracted by Chen and Zhang (2008) may be useful in some aspects but are not comparable with Ca diffusion controlled by its own concentration gradient. Ca SEBD values during olivine dissolution are hence not modeled quantitatively below; only Ca SEBD values during clinopyroxene dissolution are considered. Melt compositions are listed in Table 1 except for diopside-anorthite and CaOAl 2 0 3 -Si0 2 melts. Ca diffusion data in dry natural and nearly natural melts are shown in Figure 12. SEBD data of Ca (close to FEBD) in mugearite melt at 1503-1696 K and 0.1 MPa by Medford (1973) are scattered and do not define a good Arrhenius trend (Fig. 12a), reflecting the experimental and analytical difficulties in the early years of diffusion studies. Ca self diffusivities in HB1 melts at 1623-1773 K and 0.1 MPa (LaTourrette et al. 1996), SEBD in lunar picrite melts during anorthite dissolution at 1673 K and 0.6 GPa (Morgan et al. 2006), and SEBD in basaltl 1 during clinopyroxene dissolution at 1509-1790 K and 0.5-1.9 GPa (Chen and Zhang 2009) are similar and independent of pressure, and can be fit by the following equation: InDvbasalt C a ST) & S E B D

~~

(11.79+1.43)

(19138 + 2349) T

(24)

The maximum error of Equation (24) in reproducing diffusivities in the three experimental studies (LaTourrette et al. 1996; Morgan et al. 2006; Chen and Zhang 2009) is 0.30 in InD.

Zhang, Ni, Chen

336 -20

SB SEBD; Mu; 0.1 MPa; M73

• •

-21 09 S

—•— SD; HB1; 0.1 MPa; L96 • SEBD; b-b; 0.9 GPa; L03

O SEBD; LB1; 0.6 GPa; M06

-SEBD; basil;0.5-1.9 GPa; C09

-••-SEBD; andl; 1.0-2.2 GPa; Z89

-22 -23

-25 -26 0.55

0.6

0.65

1000/7 ( T in K )

-22

- - • " T D ; rhy5; 0.1 GPa; J82 —•—SEBD; rhyl 2(ap); 0.1 GPa; H84 a IEBD; r-r; 1 GPa; V94 O IEBD; a-r; 1 GPa; V94 - « - - T O ; HR7; 0.1 MPa; M99

-24 -26 ri

£

-28 -30

9

5

-32 -34 -36 -38 -40 0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1000/7 ( T in K ) Figure 12. Ca diffusivity in dry (a) mafic melts (Si0 2 < 60 wt%) and (b) silicic melts (SiOi > 60 wt%). The legend contains information about the type of diffusion, the melt composition (Mu means mugearite; b-b means basalt7-basanite couple; andl=andesitel; rhy5=rhyolite5; rhyl2(ap) means D in rhyolite 12 during apatite dissolution; r-r means rhyolite3-rhyolitel6 couple; a-r means andesite3-rhyolitel6 couple), pressure, and authors (M73: Medford 1973; L96: LaTourrette et al. 1996; L03; Lundstrom 2003; M06: Morgan et al. 2006; C09: Chen and Zhang 2009; Z89; Zhang et al. 1989; J82: Jambon 1982; H84; Harrison and Watson 1984; V94: van der Laan et al. 1994; M99; Mungall et al. 1999).

However, two Ca SEBD values in basalt7-basanite diffusion couple (Lundstrom 2003) are greater than those predicted by Equation (24) by 1.5 to 2.0 InD units (Fig. 12a). and those in mugearite melt (Medford 1973) are lower by up to 1.2 InD units. Ca diffusivities in rhyolite melts are more variable than in basalt melts. In dry HR7 melt (among the most silicic rhyolite), Ca tracer diffusivity at 1410-1873 K and 0.1 MPa (Mungall et al. 1999) can be expressed as:

Diffusion Data in Siliate Melts In D^,™ 7 = - ( 1 0 . 5 8 + 2.63)

337

(29830 + 4356) T

(25)

The maximum error Equation (25) in reproducing the four experimental diffusivities is 0.28 in InD. Adding H 2 0 enhances Ca diffusivity significantly. Sr diffusion. Ten papers reported Sr diffusion data in silicate melts (110 points, all except for one are tracer or self diffusivities). Hofmann and Magaritz (1977) studied Sr tracer diffusion in dry basalt 10 melt at 1532-1719 K and 0.1 MPa. Magaritz and Hofmann (1978a) determined Sr tracer diffusivities in dry rhyolitel5 melt at 948-1226 K and 0.1 MPa. Lowry et al. (1982) explored tracer diffusion of 85 Sr in dry basaltl and andesite2 melts at 1566-1676 K and 0.1 MPa. Lesher (1994) investigated Sr self diffusion in dry rhyolite2 melt at 1528-1738 K and 1 GPa, and in dry basalt3 melt at 1528 K and 1 GPa. Perez and Dunn (1996) examined Sr tracer diffusion in rhyolite8 melt (containing < 0.8 wt% H 2 0 , but H 2 0 was not reported for every sample) at 1273-1723 K and 1 GPa. Nakamura and Kushiro (1998) reported tracer diffusivities of Sr in jadeite and diopside melts at a single temperature (1673 K for jadeite and 1863 K for diopside) and some pressures. Mungall et al. (1999) investigated Sr tracer diffusion in dry HR7 at 1410-1873 K and 0.1 MPa, wet HR7 (3.6 wt% H 2 0 ) at 1 GPa and two temperatures (1573 and 1873 K), and dry H R 7 + N a at 1083-1773 K and 0.1 MPa. Roselieb and Jambon (2002) examined Sr tracer diffusivities in albite and jadeite melts at 918-1293 K and 0.1 MPa. Lundstrom (2003) obtained a single S E B D of Sr between basalt7-basanite melt at 1723 K and 0.9 GPa. Behrens and Hahn (2009) characterized Sr tracer diffusion in dry and wet trachyte and phonolite melts at 1323-1573 K and 0.5 GPa. Sr diffusivity data are shown in Figure 13. Sr diffusivities are the lowest in dry HR7 melt among the melts investigated. Sr self diffusivities in dry rhyolite2 are similar to Sr tracer diffusivities in rhyolite8 containing < 0 . 8 wt% H 2 0 and in dry rhyolitel5. Self and tracer diffusivities of Sr increase from dry HR7, to dry natural metaluminous rhyolite and dry trachyte, then dry andesite, then dry basalt and dry phonolite, then wet trachyte, and then wet phonolite. Some of the equations are given below. Self and tracer diffusivity of Sr in rhyolite2 at 1 GPa (Lesher 1994), rhyolite8 at 1 GPa (Perez and Dunn 1996) and rhyolite 15 at 0.1 MPa (Magaritz and Hofmann 1978a) can be fit as follows: srYD&™

D

51 GPa

= exp ( - 1 4 . 8 0 + 0.65)

(18745 + 824) T

(26)

The maximum error of Equation (26) in reproducing the diffusion data is 0.89 in InD (0.39 in logD). The activation energy is 156±7 kj/mol. Sr tracer diffusivities in dry basaltl and basalt 10 melts at 1567-1675 K and 0.1 MPa (Hofmann and Magaritz 1977; Lowry et al. 1982) can be fit as follows: = exp ( - 1 1 . 4 0 + 2.40)

(20361 + 3881) T

(27)

The maximum error of Equation (27) in reproducing the diffusion data is 0.30 in InD. In dry and wet trachyte melts, Sr tracer diffusivities at 1323-1527 K, 0.5 GPa and < 1.7 wt% H 2 0 (Behrens and Hahn 2009) are roughly linear to H 2 0 content, and can be fit as follows: D

Jry tmdlytcl = ^

n

. l \ ± 5.23)

(22435 ± 7 5 7 1 ) - ( 3 4 3 0 + 545) w T

The maximum error of Equation (28) in reproducing the diffusion data is 0.75 in InD.

(28)

Zhang, Ni, Chen

338

-34 0.5

0.6

0.7

0.8

0.9

1.1

1 » « 0 / 7 " ( 7 " in K ) Figure 13. Sr diffusivities in silicate melts. The default pressure is 0.1 MPa. Data sources can be found in the text.

There are not enough data to resolve how Sr tracer diffusivities in phonolite melts depend on H 2 0 . Assuming the dependence is similar to trachyte melts, then Sr tracer diffusivities in dry and wet phonolite at 1373-1528 K, 0.5 GPa and < 1.9 wt% H 2 0 (Behrens and Hahn 2009) can be fit as follows: nphonolilcl

Sr ÏD

_ - CAp

(-8.98 ± 1.56)

(?4708±6624)-(1644±299)vv

(29)

The maximum error of Equation (29) in reproducing the diffusion data is 0.44 in InD. Ba diffusion. Twelve papers reported Ba diffusion data in silicate melts (117 points), most of which are tracer or self diffusivities. Hofmann and Magaritz (1977) explored Ba tracer diffusion in dry basaltlO melt at 1523-1723 K and 0.1 MPa. Magaritz and Hofmann (1978a) determined Ba tracer diffusivities in dry rhyolitel5 melt at 973-1226 K and 0.1 MPa. Jambon (1982) reported Ba tracer diffusivities in rhyolite5 melt at 905 and 1083 K and 0.1 GPa. Lowry et al. (1982) characterized Ba tracer diffusion in basalti and andesite2 melts at 1572-1673 K and 0.1 MPa. Henderson et al. (1985) studied Ba tracer diffusion in dacite2 (1573-1672 K) and pantellerite 1 (1470-1575 K) melts at 0.1 MPa. LaTourrette et al. (1996) examined Ba self diffusion in dry HB1 melt at 1623-1773 K and in air. Nakamura and Kushiro (1998) obtained Ba tracer diffusivities in dry jadeite and diopside melts at 1573-1723 K and 0.75-2.0 GPa. Mungali et al. (1999) characterized Ba tracer diffusion in dry HR7 and dry HR7 + Na at 0.1 MPa and wet HR7 (3.6 wt% H 2 0 ) melts at 1 GPa. Koepke and Behrens (2001) measured Ba tracer diffusivities in wet HAI melt (4.5-5.2 wt% H 2 0 ) and one datum for dry HAI. Roselieb and Jambon (2002) investigated Ba tracer diffusion in dry albite and jadeite melts at 10731293 K and 0.1 MPa. Lundstrom (2003) acquired SEBD of Ba in dry basalt7-basanite couple at 1723 K and 0.9 GPa. Behrens and Hahn (2009) examined Ba tracer diffusion in dry and wet trachyte and phonolite melts. Ba diffusion data are shown in Figure 14. Ba diffusivities increase with decreasing viscosity (from polymerized silicic melt to depolymerized melt, and from dry to wet melt). Ba tracer and self diffusivities in dry basalti (Lowry et al. 1982), basaltlO (Hofmann and Magaritz 1977) and HB1 (LaTourrette et al. 1996)

Diffusion

Data in Siliate

Melts

339

melts are consistent and can be fit by the following equation: d:

dry basalt; 0.1 MPa _ ( - 1 4 . 6 8 ± 2 . 3 1 ) - (15691±3800)1 Ba ST) and TD = exp

(3Q)

Equation (30) can reproduce the data to within 0.56 InD units. Experimental data in trachyte and phonolite melts are not enough to resolve how D depends on H 2 0 . We assume that InD increases linearly with H 2 0 to fit the data. For dry and

- a - basalti —»—basalti 0 —m- -HB1 - andesite2 - • » • • dacite4 • pantelleritel -HR7 -m- -HR7+Na •jadeite -O-albite -rhyolitel 5 --X- -pho; 0.5 GPa •-o •tra; 0.5 GPa

>

tfP -26 S

c e

-28

-30 -32 -34

Nominally dry melts default pressure is 0.1 MPa -j i i i

-36 0.5

0.6

0.7

0.8

1«00/7-(7in

-20

-2">

• o

dry andesite2; 0.1 MPa dry HA1; 0.5 GPa -wet HA1; 0.5 GPa -•--dry tra; 0.5 GPa • wet tra; 0.5 GPa

i.l

0.9

K)

• dry pho; 0.5 GPa - IB—wet pho; 0.5 GPa —0—dry HR7; 0.1 MPa w e t HR7; 1 GPa

Dry & wei melts -24

"'•Si- ^ -26

a.

Q c

"a..

-28

\

-30 0.5

0.55

0.6

0.65

_LZ 0.7

L_ 0.75

0.8

1 0 0 0 / 7 * (7" i n K ) Figure 14. Ba diffusivities in selected melts. Melt compositions can be found in Table 1. (a) Dry melts, (b) Comparison of dry and wet melts. The H 2 0 contents in wet melts are as follows: wet H A i : 4.5-5.2 wt%; wet tra (trachyte!): 1.1-1.7 wt%; wet pho (phonolitel): 1.6-1.9 wt%; wet HR7: 3.6 wt%.

340

Zhang, Ni, Chen

hydrous trachyte 1 and phonolite 1 melts, B a tracer diffusivities at 1323-1528 K, 0.5 GPa, 0 - 1 . 9 wt% H 2 0 can be fit as: ( - 1 4 . 3 4 ± 4.67) - (20885 ± 6 7 9 7 ) - ( 3 0 6 5 ± 4 8 9 ) w r-\trachytel ; 0.5GPa _ Ba TD -CAp

(31)

rvphonolilcl: 0.5GPa Ba TD ~

(32)

_

eXP

( - 8 63 + 3 35)

( 2 5

"

6

*

3412>

" 0

805

*

1

Equations (31) and (32) can reproduce the data to within 0 . 6 0 and 0.23 InD units, respectively. Ra diffusion.

No R a diffusion data in silicate melts are known.

Summary of alkali earth diffusion. For the diffusion o f alkali earth elements in dry H R 7 + N a melt (Mungali et al. 1999), the diffusivity decreases from B a to Sr, to Ca, to Mg, and to B e . That is, the diffusivity sequence for the alkali earth elements is: B a > Sr > Ca > Mg > B e . In dry H R 7 melt (Mungali et al. 1999), the diffusivity sequence is: B a « S r > Ca > Mg > B e (Fig. 15a). In wet HR7 melt containing 3.6 wt% H 2 0 (Mungali et al. 1999), the diffusivity sequence is: Sr > B a « C a > Mg > B e (Fig. 15b). In basaltlO melt (Hofmann and Magaritz 1977), the diffusivity sequence is: C a « C o > Sr > Ba, opposite to the trend in HR7 + Na. In HB1 melt (LaTourrette and Wasserburg 1996), the diffusivity sequence is: Mg > Ca > Ba, similar to the trend in basaltlO. In albite and jadeite melts (Roselieb and Jambon 1995), the diffusivity sequence is: Sr = Ca > B a > Mg. Hence, for the alkali earth elements, there is no simple relation between diffusivity and cation size: the diffusivity may increase or decrease with increasing size, depending on the melt composition and other factors. That larger cations with the same valence may diffuse more rapidly in some melts may be counterintuitive to some. Nonetheless, this trend will be encountered in diffusion of other isovalent series ( R E E , B , Al, Ga, Si-Ge, etc.). This and other trends will be discussed in a later summary section.

B, Al, Ga, In, and T1 B diffusion. Three papers reported B diffusion data in silicate melts (70 points). Baker (1992a) investigated B tracer diffusion in dacitel and rhyolitel4 melts at 1573-1873 K and 1 GPa. Chakraborty et al. ( 1 9 9 3 ) studied F E B D o f B in diffusion couples with one half made of H R 7 and the other half made o f HR7 plus 5 wt% or 10 wt% B 2 0 3 at 1473-1873 K and 0.1 MPa. Mungali et al. ( 1 9 9 9 ) examined B tracer diffusion in dry HR7 melt at 0.1 M P a and 1410 and 1673 K, dry H R 7 + Na melt at 1083-1473 K and 0.1 MPa, and wet HR7 (3.6 wt% H 2 0 ) at 1573-1873 K and 1 GPa. The data are shown in Figure 16. As shown by Chakraborty et al. (1993), F E B D values o f B decrease as S i 0 2 concentration increases along a diffusion couple. When F E B D values of B at the HR7 end with 7 5 - 7 9 wt% S i 0 2 (Chakraborty et al. 1993) are compared with tracer diffusivities of B in H R 7 (Mungali et al. 1999), they are consistent (Fig. 16). B tracer diffusivities in rhyolitel4 (76 wt% S i 0 2 ) are similar to those in H R 7 + B melts containing 7 0 - 7 5 wt% S i 0 2 . B diffusivities ( F E B D by Chakraborty et al. 1993 and tracer diffusivities by Mungali et al. 1999) in dry HR7 melt at 1410-1873 K, 0.1 MPa, and B 2 0 3 concentration < 10 wt% can be fit as: n(iiyHR7,0.1MPa

B TD&T'liBD

CAF

;.56-0.486C-(39664-"62C)

(33)

where C is wt% o f B 2 0 3 . All B diffusion data in Chakraborty et al. ( 1 9 9 3 ) and Mungali et al. ( 1 9 9 9 ) can be reproduced by Equation (33) to within 0 . 6 2 laD units. B diffusivity in rhyolitel4 melt at 1573-1873 K and 1 GPa (Baker 1992) is about 6 times the diffusivity calculated using Equation (33).

Diffusion

Data in Siliate

Melts

341

—•—Be -Mg -»-Ca --•--Sr • ••• • Ba

Qc

Diffusion in HR7 I L 0.6 0.65 1 0 0 0 / 7 ( 7 i n K)

1000/7" (T in K ) Figure 15. Comparison of Be, Mg, Ca. Sr and Ba diffusivities in dry and wet HR7 melt (Mungall et al. 1999).

When compared with other cations, B diffusivities are similar to diffusivities of trivalent cations Lu and Ga in HR7 melts. Al diffusion. Twelve papers reported Al diffusion datain silicate melts (127 points). Cooper and Kingery (1964) explored Al diffusion (SEBD) in CaO-AliOrSiOi melt during sapphire dissolution at 1618-1823 K and 0.1 MPa. Cooper and coworkers also investigated diffusion in other synthetic systems of interests to ceramic and glass scientists, such as K 2 0-Sr0-Si0 2 , etc. These works are not covered here because we focus on geologically relevant silicate melts. Baker and Watson (1988) investigated Al diffusion (SEBD) in rhyolitel-rhyolite8 and HD2-rhyolite8 couples (Table 1) at 1171-1273 K and 0.01 GPa and 1373-1473 K and 0.2-1 GPa. Zhang et al. (1989) determined Al SEBD in andesitel melt during dissolution of olivine, clinopyroxene, spinel and quartz at 1488-1673 K and 0.55-2.15 GPa. Because the interface

342

Zhang, Ni, Chen -24 -FEBD; 71-75% -FEBD; 75-79% - X -TD; HR7 - o - TD; rhyolitel 4 — • -TD;HR7+Na - o - -TD; wet HR7 -TD; dacitel ~ffi-

-26

-36 0.5

0.6

0.7

0.8

1000/7(7in

0.9

1

K)

Figure 16. B diffusivities in silicate melts. Data sources: FEBDat 71-79 wt% SiOi at 0.1 MPa ( Chakraborty et al. 1993); tracer diffusivities in dry HR7 at 0.1 MPa, in HR7 + Na at 0.1 MPa, and in wet HR7 (3.6 wt% HiO) at 1 GPa (Mungall et al. 1999); and in rhyolitel4 and dacitel at 1 GPa (Baker 1992a).

melt composition changes from basalt (during olivine dissolution) to rhyolite (during quartz dissolution), it is important to include the interface melt composition variation to understand the data. Kubicki et al. (1990) obtained A1 SEBDin diopside-anorthite (Di-An) melts at 16331923 K and 0.1-2 GPa. Most of their data are at 0.2 GPa, and compositional change from Di l l K r Di 8 0 An 2 0 couple to Di 60 An 4ir Di4 0 An 60 couple does not affect Al diffusivities in a major way. Baker and Bossanyi (1994) examined the effect of H 2 0 and F on Al SEBD in rhyolite8dacitel diffusion couples at 1373-1673 K and 1 GPa. van der Laan et al. (1994) reported Al SEBD in rhyolite3-rhyolitel6 and andesite3-rhyolitel6 diffusion couples. Liang et al. (1996a) investigated Al self diffusion in CaO-AI 2 O r Si() 2 systems at 1773 K and 1 GPa. Van Orman and Grove (2000) obtained a single datum for Al SEBD in a lunar basalt melt (LB 1 in Table 1) during clinopyroxene dissolution at 1623 K and 1.3 GPa. Lundstrom (2003) obtained two data points for Al SEBD in basalt7-basanite couple. Morgan et al. (2006) determined Al SEBD in LP1 and LP2 melts during anorthite dissolution at 1673 K and 0.6 GPa. Chen and Zhang (2008. 2009) obtained Al SEBD data in basalt 11 melt (with 0.3-0.4 wt% H 2 0 ) during olivine and diopside dissolution at 1543-1790 K and 0.5-1.9 GPa. Despite the numerous papers on Al diffusivities, most Al diffusivity data were obtained as side-products. Except for Liang et al. (1996a) who investigated Al self diffusion in a ternary system at a single temperature (1773 K). other studies are all on SEBD of Al in various melts. No tracer diffusivities or FEBD on Al are available. Furthermore, Al diffusion data in some papers are scattered. Hence. Al diffusion in natural silicate melts is not very well constrained. Experimental Al SEBD data in dry melts are shown in Figure 17. Al diffusivities decrease from lunar picrites and basalt-basanite couple, to basalt 11, and to andesitel. For basalt to andesite melts, the data are less scattered and the trends with Si0 2 content is consistent. Al SEBD data for Di-An melts (ranging from Di ]0 ( r Di 8n An 2 0 couple to Di 6 oAn 4 ( r Di 4n An 6 o couple) vary more widely, but are roughly the same as those in basalt 11 melt. Al diffusivities (SEBD) in basalt 11 melt (containing about 0.3-0.4 wt% H 2 0 ) during both olivine and clinopyroxene dissolution at 1509-1790 K and 0.5-1.9 GPa (Chen and Zhang 2008. 2009) are similar and roughly independent of pressure, and can be expressed as:

Diffusion Data in Siliate Melts -21

343

• -ffl- • basalt 11 ; 0.5-1.4 GPa; C08 — " b a s a l t ! 1 ; 0.5-1.9 GPa; C09 » b-b; 0.9 GPa; L03 •S" LB1; 1.3 GPa; V00 • LP1-LP2; 0.6 GPa; M06

-22

-27

1000/7(Fin

K)

Figure 17. S E B D of A1 in dry melts. Indicated in the legends are the melt composition listed in Table 1 (b-b means basalt7-basanite couple with 4 6 wt% average SiCK), pressure, and references (C08: Chen and Zhang 2008; C09: Chen and Zhang 2009; L 0 3 : Lundstrom 2003; M06; Morgan et al. 2006; V 0 0 : Van Orman and Grove 2000; K 9 0 : Kubicki et al. 1990; Z89; Zhang et al. 1989).

D:

, =exp (-5.75 + 1.71)-

(31293 + 2599)

(34)

Equation (34) reproduces the experimental data to within 0.44 InD units. Furthermore. Equation (34) can also roughly predict Al S E B D in Di-An melts (Fig. 17) with a maximum error of 1.1 InD units. Al diffusivities ( S E B D ) in andesitel melt (containing about 0.04 wt% H 2 0 ) during olivine, clinopyroxene and spinel dissolution at 1488-1673 K and 0.5-2.1 GPa (Zhang et al. 1989) are similar and roughly independent of pressure, and can be expressed as:

rr:

= exp ( - 2 . 5 2 + 4 . 9 5 ) -

(37649 + 5848)

(35)

Equation (35) reproduces the experimental data to within 0.79 InD units. Al diffusivity during quartz dissolution (Zhang et al. 1989) is lower than that predicted by Equation (35) by 2.05 InD units, attributed to a large difference in the interface melt composition during quartz dissolution compared to dissolution of the other minerals. Data in Figure 17 show systematic dependence of Al S E B D on S i 0 2 , although the data are scattered. We fit all S E B D data on natural silicate melts as follows: D

(23111 + 5 9 1 8 X , ) basalt to andesite = exp ( - 0 . 8 8 - 1 8 . 0 2 X j , ) -

(36)

Equation (36) reproduces the experimental data of Zhang et al. (1989). Van Orman and Grove (2000), Lundstrom (2003), Morgan et al. (2006). and Chen and Zhang (2008, 2009) to within 1 InD unit (or 0.43 logD units). Al is a network-forming element. When compared to the diffusion of the most common major network-forming element Si, Al diffusivity is often slightly higher than Si diffusivity.

344

Zhang, Ni, Chen

about 1.5 to 2.5 times the Si diffusivity (Zhang et al. 1989; Liang et al. 1996a; Chen and Zhang 2008, 2009). Hence, when Al diffusivity in a melt is not known but Si diffusivity in the melt is known, Al diffusivity can be roughly estimated to be two times the Si diffusivity. Ga diffusion. Two papers reported Ga tracer diffusion data in silicate melts (18 points). Baker (1992a) investigated Ga diffusion in dacitel and rhyolitel4 melts (Table 1) at 1573-1873 K and 1 GPa. Baker (1995) examined Ga diffusion in albite melt at 1427-1775 K and 0.1 MPa. The data are summarized in Figure 18. Mungall (2002) dismissed Ga diffusion data by Baker (1992a).

1000/r (T in K) Figure 18. Ga tracer diffusivities in silicate melts. Data sources: albite (Baker 1995); rhyolite!4 and dacitel (Baker 1992a).

In diffusion. No In diffusion data in silicate melts are known. Tl diffusion. Only one paper reported T1 diffusion data in silicate melts. MacKenzie and Canil (2008) studied T1 tracer diffusion in dry CMAS1 and NMAS1 (Table 1) melts using devolatization (desorption) experiments at 1473-1623 K and 0.1 MPa. The diffusion data are summarized in Figure 19. There is considerable scatter in the data. Comparison of diffusivities ofB, Al, Ga, In, and Tl. There are not enough data to compare diffusivities of B, Al, Ga, In and Tl because (i) there are no data on In diffusion; (ii) the only Tl diffusion data are for CMAS1 and NMAS1 systems, different from melt compositions studied for the other elements; (iii) the only common melts that have been investigated for B and Ga diffusion are dacite l and rhyolitel4 melts; and (iv) Al diffusion data in dacite and rhyolite melts are SEBD data and are highly scattered. Hence, only B and Ga diffusivities are compared in Figure 20. In dacitel melt, diffusivities of the two elements are similar. However, in rhyolite 15 melt, B diffusivity is lower than Ga diffusivity by a factor of 2 to 4. The latter is consistent with the trend shown by the alkali earth elements with smaller cations having smaller diffusivity. Nonetheless, the difference in diffusivity is small. It is likely that Al diffusivities lie between those of B and Ga, meaning that B, Al and Ga all have similar diffusivities. Because Tl is volatile, its diffusivities are expected to be higher than those of B. Al and Ga.

Diffusion

Data in Siliate

Melts

345

-24 Figure 19. T1 diffusivities in silicate melts. Data source: MacKenzie and Canil (2008).

25 -

0.61

0.62

0.63

0.64

0.65

0.66

0.67

0.68

1 0 0 0 / 7 ( 7 " in K )

Figure 20. Comparison of B and Ga diffusivities in dacitel and rhyolitel4 melts.

—1•—B in dacitel - o -Ga in dacitel - » - B in rhyolite14 --ffl-Ga in rhyolite14 0.52

0.54

0.56

0.58

0.6

0.62

0.64

i o o o / r ( r i n K) C, Si, Ge, Sn and Pb C diffusion. C diffusion in silicate melts in the form of dissolved molecular C 0 2 and carbonate ion C 0 3 2 - is reviewed in another chapter (Zhang and Ni 2010). Data for carbon diffusion in other forms (such as dissolved molecular CO) are not available. Si diffusion. Si is the second most major element in silicate melts (next to oxygen), and it controls the melt structure. Hence, its diffusion has been investigated extensively, with about 23 papers and more than 262 diffusivity values (some of the diffusivities are reported as smooth trends rather than data points). Lesher and Walker (1986. 1991) obtained Si SEBD from Soret diffusion experiments on nephelinite. basalt8, limburgite, basalt 11, leuconorite. nordmarkite, trachyte, pantellerite2. FaLcQ (fayalite-leucite-quartz), rhyolite4. and rhyolitelO melts (Table 1). Baker and Watson (1988) investigated Si SEBD in rhyolite 1-rhyolite8 and HD2-rhyolite8 couples at 1171-1273 K and 0.01 GPa and 1273-1473 K and 0.2-1 GPa. Koyaguchi (1989) reported Si SEBD in basalt-dacite and basalt-rhyolite couples at 1473-1773 K, 1 GPa, and 0.39-

346

Zhang, Ni, Chen

13.77 wt% H 2 0 . The compositions used by Koyaguchi (1989) include basalt2, basalt9, dacite3, and rhyolitel3. Zhang et al. (1989) obtained Si SEBD in andesitel melt during dissolution of olivine, spinel, rutile and quartz at 1488-1673 K and 0.55-1.5 GPa. Because the interface melt composition changes from basalt (during olivine dissolution) to rhyolite (during quartz dissolution), it is important to include the interface melt composition variation to understand the data. Baker (1991, 1993) and Baker and Bossanyi (1994) examined the effect of H z O, F and CI on Si SEBD in rhyolite8-dacite3 diffusion couples at 1373-1673 K and 1 GPa, and found that the effect of F and CI is negligible (D increases by less than a factor of 2 when F and CI concentrations are 1 wt% or less), and that of H 2 0 is significant. Kubicki et al. (1990) reported Si SEBD in diopside-anorthite melts at 1873 K and 0.2 GPa. Baker (1992a) studied Si tracer diffusion in rhyolite8 and dacitel melts at 1573-1773 K and 1 GPa. In these experiments, a layer of sodium silicate glass powder (with 73 mol% Si0 2 and 27 mol% Na 2 0, or close to N a ^ i s C ^ ) with thickness < 0.1 mm enriched in 30 Si (a stable silicon isotope, meaning that the experiments are more like self diffusion experiments) is loaded on a dacite or rhyolite glass cylinder. The sample is loaded into a piston-cylinder assemblage and heated up. Because the typical Si diffusion length in the experiments of Baker (1992a) is about 0.05 mm (Fig. 1 in Baker 1992a), which might be thinner than the thickness of the loaded film (< 0.1 mm), the experiments may not be true tracer diffusion experiments, but are close to diffusion couple experiments between a synthetic sodium silicate on one half and either dacite or rhyolite on the other half, with an Si isotopic gradient between the two halves. The extracted diffusivities are likely isotopic effective binary diffusivities (IEBD), although such diffusivities based on isotopic fraction profiles are often not too far off self diffusivities (Zhang 1993; Lesher 1994; van der Laan et al. 1994). van der Laan et al. (1994) reported Si SEBD in rhyolite3-rhyolitel6 and rhyolite 16dacite3 diffusion couples. Baker (1995) investigated Si SD in albite melt at 1438-1831 K and 0.1 MPa. Lesher et al. (1996) investigated Si self diffusion in basalt6 melt (Table 1) at 15931873 K and 1 GPa, and 1673 K and 2 GPa. Liang et al. (1996a) investigated Si self diffusion in the C a 0 - A l 2 0 r S i 0 2 system. Liang et al. (1996b) extracted multicomponent diffusivity matrices in the Ca0-Al 2 0 3 -Si0 2 system (see review in Liang 2010, this volume), and also SEBD of Si, but the SEBD data were only shown in a figure without the melt composition. Poe et al. (1997) measured Si self diffusivity in NS4 (Na 2 Si 4 0 9 ) melt at 2100 to 2800 K and 10-15 GPa. Van Orman and Grove (2000) estimated Si SEBD in a lunar basalt melt (LB 1 in Table 1) during clinopyroxene dissolution at 1623 K and 1.3 GPa. Reid et al. (2001) studied Si self diffusion in diopside melt at 2073-2573 K and 3-15 GPa. Tinker and Lesher (2001) examined Si self diffusion in dry HD2 melt at 1628-1935 K and 1.0-5.7 GPa. Lundstrom (2003) obtained one datum of Si SEBD in basalt7-basanite couple. Tinker et al. (2003) investigated Si SD in diopside-anorthite (Di52An48) melt at 1783-2037 K and 1-4 GPa. Morgan et al. (2006) determined Si SEBD in two lunar picrite melts (LP1 and LP2 in Table 1) during anorthite dissolution at 1673 K and 0.6 GPa. Chen and Zhang (2008, 2009) obtained Si SEBD in nominally dry basaltll melt (about 0.3-0.4 wt% H 2 0 ) during olivine and clinopyroxene dissolution at 1543-1790 K and 0.5-1.9 GPa. Si self diffusivities. In dry silicate melts, silicon self diffusivity is similar to and often slightly lower than oxygen self diffusivity (Fig. 21). The maximum difference is about 1.5 InD units and this difference occurs in silicic melts with 69 wt% Si0 2 . The limited data apparently indicate that when diffusivities are low ( 4 GPa to constrain how Si self diffusivity varies with pressure. Hence. restricting our attention to 1-4 GPa, Si self diffusivity in dry HD2 melt at 1628-1935 K can be fit as follows: 7-)®y HLU _ ^Si SD ~~ u x r ( - 3 . 9 1 3 ± l ,603)P -

(54245 ± 690)

-(8523

± 2984)P

(37)

Equation (37) can reproduce the experimental data at 1-4 GPa to within 0.66 InD units. It can be extrapolated to 0 GPa. but cannot be extrapolated to >4 GPa. Effective binary diffusivities of Si. Most Si diffusivities are SEBD values from diffusion couples made of rhyolite-basalt, rhyolite-dacite, or basalt7-basanite, or mineral dissolution experiments in basalt and andesite melts. In these experiments, as well as in nature, the Si0 2 gradient is often the largest gradient, or one of the largest. Hence, even though it is expected that the gradients of other concentration gradients would affect Si0 2 diffusion due to cross diffusion effects, such effects are not expected to completely overwhelm Si0 2 diffusion. Figure 22a compares different kinds of Si diffusivities and the maximum difference is 1.6 InD units (0.7 logD units). Therefore, Si0 2 SEBD in most geological cases is expected to be roughly the same as the self diffusivity. However, for synthetic melts (such as the CaO-Al 2 O r Si0 2 system. Liang et al. 1996a,b) where Si0 2 gradient may be small compared to gradients of other components, the difference between self diffusivities and SEBD can be very large. Experimental SEBD values of Si in diffusion couple or Soret diffusion experiments usually decrease from basalt to rhyolite, and InD versus Si0 2 for typical natural melts is roughly linear (Fig. 22b) (Lesher and Walker 1986. 1991; Koyaguchi 1989). Lesher and Walker (1986, 1991) showed based on Soret diffusion experiments that lnD si decreases linearly with increasing

Zhang, Ni, Chen

348

SD in basalt6 SEBD in basalti 2 SEBD in b-d • SEBD in b-b X SEBD in basalti 1

• s

dry basalt (48.5±2.6 w t % _27

052

.

I

0.54

.

I

.

I

I

Si0 2 ) I

.

I

0.56 0.58 0.6 0.62 1 0 0 0 / 7 ( T in K )

.

I

0.64

.

0.66

SiO, (wt%) Figure 22. (a) Comparison between SD (self diffusivities) and SEBD of Si. Data sources: SD in basaltó (Lesher et al. 1996); SEBD in basalt 12 (Lesher and Walker 1986); SEBD in b-d (basalt2-dacite3; Koyaguchi 1989); SEBD in b-b (basalt7-basanite; Lundstrom 2003); SEBD in basalti 1 (Chen and Zhang 2008', 2009). (b) SEBD of Si versus SiOi content. Data sources; LW86 (Soret diffusion results of Lesher and Walker 1986); K89 (diffusion couple results of Koyaguchi 1989).

Si0 2 : in dry melts at 1748 K, lnD si (D in m 2 /s) decreases from -24.2 in a basalt with 51 wt% Si0 2 to -26.8 in a rhyolite with 75 wt% Si0 2 (a factor of 13. or 1.1 orders of magnitude). Koyaguchi (1989) showed from diffusion couple experiments that lnD si decreases linearly with increasing Si0 2 : in dry melts at 1773 K. lnD si (D in m 2 /s) decreases from -24.2 in a basalt with 50 wt% Si0 2 to -27.6 in a rhyolite with 74 wt% Si0 2 (a factor of 30. or 1.5 orders of magnitude).

Diffusion

Data in Siliate

Melts

349

Baker (1993) and Baker and Bossanyi (1994) examined the effect of H 2 0. F and CI on Si SEBD. Adding 3 wt% H 2 0 increases Si SEBD by a factor of about 10. In dry and wet melts, adding 1 wt% of F or CI increases Si SEBD by less than a factor of 2, within experimental error of the diffusion data by Baker and coworkers. Even though the data seem to indicate a significant effect on the activation energy of Si SEBD by F and CI, the activation energy based on a small temperature range and scattered data is not very reliable (e.g., see discussion in Zhang and Ni 2010). Overall, the effect of F and CI can be ignored unless Si diffusion data accuracy can be significantly improved or F and CI concentrations are much higher than 1 wt%. Ge diffusion. Only one paper (Mungall et al. 1999) reported Ge tracer diffusion data in silicate melts (9 points), in dry HR7 and HR7 + Na, and wet HR7 (Table 1). The data are summarized in Figure 23. Ge diffusivity in HR7 can be expressed as: DdryHR7

Gc TD - e X P

(-11.27 + 5.16)- ( ^ 4 1 4 + 8558)

(38)

In dry HR7 melt, Ge diffusivities are similar to those of Hf, and they are similar to Eyring diffusivities (Fig. 23).

1 0 0 0 / 7 ( F i n K) Figure 23. Ge diffusion data in silicate melts. Data source: Mungall et al. (1999). Eyring diffusivity line is calculated from the viscosity data of Hess et al. (1995 ) on dry HR7 melt.

Sn diffusion. Only one paper reported good-quality Sn tracer diffusion data in silicate melts (Behrens and Hahn 2009). Two other papers reported highly scattered SEBD of Sn (Linnen et al. 1995, 1996) obtained from cassiterite dissolution in haplorhyolite melts, but it was likely that the data were compromised by convection. Sn in silicate melts may be in the form of Sn 2+ or Sn 4+ . Hence, Sn diffusivity may depend on oxygen fugacity. Because the data of Linnen et al. (1995, 1996) are the only data examining the effect of / 0 „ they are shown in Figure 24a. Their data indicate a decrease of Sn diffusivity with increasing/ 0 „ meaning that Sn 2+ diffuses more rapidly than Sn 4+ , as expected. Behrens and Hahn (2009) characterized Sn tracer diffusivities in dry and wet trachyte 1 and phonolitel melts (Fig. 24b). The oxygen fugacity in these experiments was near the MnOMn ? 0 4 buffer but somewhat variable. Assuming D Sn increases linearly with H 2 0, Sn tracer

Zhang, Ni, Chen

350

diffusivities in dry and wet trachyte 1 and phonolite 1 melts at 1323-1528 K, 0.5 G P a and < 1.9 wt% H 2 0 (Behrens and Hahn 2009) can b e fit as follows: rjtrachytel Sn TD

eA

j-jpllOUOlilcl _ Sil TD ~

P (—11.08 ± 7.10) —

eX

(27290 ± 1 0 2 7 5 ) - ( 2 9 9 4 ± 740)w

(30799 + 8555) - (1492 + 478) vv

P (-6.34 + 5.98)-

(39)

(40)

Equations (39) and (40) reproduce experimental data to within 0.88 and 0.57 InD units, respectively.

-25 -26 -27

c e

-28 -29

X -30 Qs -31 -32

,

— • •

•

-I

1

|

-i

a •

•

•• • •

•

. \ ? • •

• • • •

• •

• Apparent Sn "diffusivities" in haplorhyolite melts

8/o

2 if Ol

i n

•

•

" 1125 K; 0.2 G P a ; ~ 6 w t % H 2 O t -33 -10 -9 -8 -7 -6 , 0

• •

P a

• •

-5

)

-24

—1»—dry trachyte 1 ; 0.5 GPa — e - wet trachyte 1; 0.5 GPa —•—dry phonolite; 0.5 GPa " H " w e t phonolite; 0.5 GPa 0.7 1000/7(Tin

K)

Figure 24. Sn diffusivities in silicate melts. Data in (a) are fromLinnen et al. (1995, 1996) and might be incorrect due to convection in the mineral dissolution experiments. Data in (b) are from Behrens and Halin (2009).

Diffusion

Data in Siliate

Melts

351

Pb diffusion. Four papers reported Pb self and tracer diffusion data in silicate melts (40 points). Jambon (1982) obtained one datum of Pb tracer diffusivity in dry rhyolitel5 melt (Table 1). LaTourrette et al. (1996) reported one datum of Pb self diffusion in HB1 melt. Perez and Dunn (1996) investigated Pb tracer diffusivities in rhyolite8 melt (containing < 0.8 wt% H 2 0, but H 2 0 is not always measured) at 1273-1723 K and 1 GPa. MacKenzie and Canil (2008) characterized tracer diffusion of Pb on dry CMAS1 and NMAS1 at 1473-1623 K and 0.1 MPa. The diffusion data are shown in Figure 25. For dry melts, the diffusivity increases from rhyolite to basalt and CMAS1 to NMAS1 melts. Pb tracer diffusivities in dry and wet rhyolite8 melts at 1273-1723 K. 1 GPa. and < 2.6 wt% 1120 (Perez and Dunn 1996) can be expressed as: nrhyolilc8.

Ph TD

1 GPa _ C X ,-v, ~~ P -9.08 - 4.32 h'

28512-7900vv

(41)

The fits by Equation (41) to Pb diffusivities in dry and wet rhyolite are shown in Figure 25. Except for two outliers. Equation (41) reproduces experimental data to within 0.72 InD units. Pb diffusion data of Perez and Dunn (1996) on rhyolite8 at 1 GPa and those of Jambon (1982) on rhyolite5 at 0.1 MPa lie roughly on the same trend. Comparison of diffusivities of C, Si, Ge, Sn and Pb. Diffusion data of C. Si, Ge, Sn, and Pb are not directly comparable. C in silicate melt is tetravalent and diffuses as a linear molecule (C0 2 ) in most situations even when there is significant CO? 2- (Nowak et al. 2004; Zhang et al. 2007). whereas Si and Ge are tetravalent and often in tetrahedral sites, but Sn and Pb are often divalent. Hence, one may compare Si and Ge diffusivities as a group, and Sn and Pb diffusivities as another group. For Si and Ge, the only Ge diffusion data are tracer diffusivities in dry HR7, wet HR7. and dry HR7+Na melts, for which Si diffusion data are not available. Hence, no direct comparison can be made. However, because both Ge and Si diffusivities are similar to the Eyring diffusivity. Si and Ge diffusivities are similar. For Sn and Pb diffusion, no direct comparison can be made. r -24

-5"

-26

r+>yolite8; 1 GPa rhyolite! 5

-30

-32 0.55

SD in H 6 1

X

CMAS1 NMAS1

wet rhyolite8(1%); 1 GPa wet rhyolite8 (2.5wt%); 1 GPa

0.6

0.65

0.7

0.75

0.8

0.85

1 0 0 0 / 7 ( T in K ) F i g u r e 25. Pb diffusion data in silicate melts. If not indicated, the melt is dry. the pressure is 0.1 M P a and the diffusivities are tracer diffusivities. Data sources: dry and wet rhyolite8 at 1 G P a (Perez and Dunn 1996): rhyolite 15 (Jambon 1982); self diffusivity in HB1 melt (LaTourrette et al. 1996); C M A S 1 and N M A S 1 (MacKenzie and Canil 2008). The lines are calculated f r o m Equation (41) for dry and wet rhyolite8.

Zhang, Ni, Chen

352 N, P, As, Sb, Bi

A' diffusion. No N diffusion data in natural silicate melts are known. P diffusion. Four papers reported P diffusion data (SEBD) in silicate melts (35points). Harrison and Watson (1984) investigated SEBD of P in dry and wet (up to 10 wt% H 2 0) rhyolitel2 melt (Table 1) during apatite dissolution at 1373-1773 K and 0.8 GPa. Rapp and Watson (1986) examined SEBD of P in wet rhyolitel2 melt (containing 1-6 wt% H 2 0) during monazite dissolution at 1273-1673 K and 0.8 GPa. Pichavant et al. (1992) obtained two SEBD values of P in wet synthetic haplorhyolite melt during apatite dissolution at 1173 K and 0.2 GPa; the data are likely compromised by convection. Lundstrom (2003) reported two SEBD values in basalt7-basanite diffusion couple at 0.9 GPa. The data are summarized in Figure 26. Interestingly, as noted by Rapp and Watson (1986). P diffusivity during apatite dissolution in rhyolite melt is significantly faster than that during monazite dissolution in the same melt (comparing solid and open symbols in Figure 26; e.g., at 1273 K, 0.8 GPa, and 6 wt% H 2 0, P diffusivity in rhyolite melt is ~4.3xl0~ 14 m 2 /s during apatite dissolution, but only 1.5xl0~15 m 2 /s during monazite dissolution, with a difference of a factor of 28), even though one might expect that they would be similar. The data are from the same laboratory, and are hence free of inter-laboratory inconsistencies. One possibility is that the difference in the diffusing species of P during monazite versus apatite dissolution may lead to the difference in the diffusivity. The dependence of P diffusivities on H 2 0 content is shown in Figure 27a. It is not clear whether the relation between lnD P and H 2 0 is linear or curved; some of the scatter is likely due to uncertainty in determining H 2 0 content in the early years. Because P is a network former and diffuses slowly, one might expect that DP is the same as Aiyrina calculated from viscosity r|. Diffusivity of P is indeed among the lowest of all

-24

0

dry rhyl 2(Ap) wet rhy12(Ap) 1% wet rhyl 2(Ap); 2% wet rhyl 2(Ap); 6% wet rhy12(Mon); 1% wet rhyl 2(Mon); 2% wet rhyl 2(Mon); 4% wet rhyl 2Eyring can be almost two orders of magnitude. Hence, it is impossible to use viscosity or Eyring diffusivity to predict P diffusivity. Although Dp * Airing, it is possible to relate P diffusivity to viscosity or Eyring diffusivity under specific conditions. For example, for P diffusivity in wet rhyolite containing 6 wt% H 2 0 during monazite dissolution, I n i J f ^ ^ ^ ^ = (20.64±2.26) + (1.509±0.064)ln£)1;ylina

H,Ot(wt%)

" •

P d i f f u s i o n in rhyolite melt 1023-1773 K 0.2-0.8 G P a

• • + •

-38

A 0.1 wt% 8 A 1 wt% ffl A 2 wt% X A 6 wt% O 1:1 line

Ml wt% M 2 wt% M 4 wt% M 6 wt%

-36 -34 -32 -30 logDEyring(£> i n m V s )

Figure 27. P diffusivities in rhyolite 12 melts. In the legend of (b), A means apatite and M means monazite; solid symbols represent data during apatite dissolution; open symbols represent data during monazite dissolution. D EyIius is calculated using the viscosity model of Zhang et al. (2003) and a jump distance of 2.8xl0- 1 "m.

Zhang, Ni, Chen

354

where viscosity needed to estimate r>Eyring is from Zhang et al. (2003). However, no universal relation exists between P diffusivity and Eyring diffusivity. Hence, such relations would have to be established at each specific H 2 0 contents, meaning that there is not much advantage for doing so. As diffusion. No As diffusion data in silicate melts are known. Sb diffusion. Two papers reported Sb tracer diffusion data in silicate melts (18 points). Koepke and Behrens (2001) investigated Sb diffusion in HA1 (Table 1) with one datum in dry HA 1 and 4 points in wet HA 1 (4.5-5.2 wt% H 2 0 ) . MacKenzie and Canil (2008) studied Sb diffusion in dry CMAS1 and NMAS1 melts. The data are summarized in Figure 28. Sb diffusivity in wet HA1 melt at 1373-1673 K. 0.5 GPa, and 4.5-5.2 wt% H 2 Q can be described by: wet HAI Sb TD

CA

P

(-12.56 ± 1.28) _ (1^423 ±1934)

(42)

Sb diffusivity in andesite melt is slightly greater than (about 1.5 to 2 times) Ba diffusivity in the same melt.

[ T T 1 1 1 J 1 I T T 7 1 • I |'"l 1 • | •

11

| >

• CMAS1 O NMAS1 X dry HAI L - • — w e t HAI

-23

C

O

.24

_a -25

•

X

^

• •

-26

K

,

O • •

•

o

• -27 0.58

« • 0.6

0.62

0.64

0.66

î o o o / r c r i n

0.68

0.7

0.72

0.74

K)

Figure 28. Sb tracer diffusivities in silicate melts.

Bi diffusion. No Bi diffusion data in silicate melts are known.

O, S, Se, Te, Po O diffusion. Oxygen diffusion in silicate melts is reviewed in another chapter (Zhang and Ni 2010). S diffusion. Sulfur in silicate melts can be present in various species controlled mainly by oxygen fugacity (the valence of sulfur can be - 2 . + 4 . and +6. with corresponding species of S2~. S 0 2 and S 0 4 2 + ) . Sulfur diffusivity is expected to depend on the sulfur species and hence on oxygen fugacity. which is a main complication. Three papers reported S diffusion data in silicate melts (41 points). In a review paper. Watson (1994) previewed sulfur diffusivity in a variety of melts, including rhyolite. dacite, haplodacite, haploandesite, and a lunar ultramafic melt using (i) the devolatization technique (obtaining tracer diffusivity or FEBD), (ii) the

Diffusion

Data in Siliate Melts

355

diffusion couple technique (obtaining tracer diffusivity or FEBD). and (iii) the thin source technique (obtaining tracer diffusivity). Increasing oxygen fugacity leads to decreasing sulfur SEBD. Winther et al. (1998) characterized S tracer diffusion in dry albite melt at 1573-1773 K. 1 GPa, and oxidized conditions. Freda et al. (2005) investigated S tracer diffusion in Etna and Stromboli basalt melts. Some diffusion data are summarized in Figure 29.

1000/nrin

K)

Figure 29. S tracer diffusion in silicate melts. Data sources and conditions: dry and wet basalt melts (from Etna and Stromboli) at reduced conditions of QFM-3 (Freda et al. 2005); dry albite under "oxidized" condition (Winther et al. 1998).

Due to the complexity of S speciation as well as the limited number of studies, sulfur diffusion in silicate melts is not well understood. Furthermore, errors for S diffusion data are larger than those of other elements, likely also due to the various sulfur species. Zhang et al. (2007) obtained the following equation to describe sulfur tracer diffusivity in Etna and Stromboli basalt melts (ignoring the small compositional difference between the two) at 14981723 K. 0.5-1 GPa (with negligible pressure effect in this range), oxygen fugacity of 3 log units below QFM (hence sulfur is in the form of S2~), and 0-4 wt% H 2 0: _ u^basalts s TD - exp - 8 . 2 1 001

27692-651.6w

(43)

The maximum error of Equation (43) in reproducing the experimental diffusivities is 0.77 in laD. The activation energy is about 230 kJ/mol in dry basalt melt, decreasing to 209 kJ/mol in wet basalt melt containing 4 wt% H 2 0. Se diffusion. No Se diffusion data in silicate melts are known. Te diffusion. Only one experimental study reported Te diffusion data (SEBD). Te can exist in different valences, similar to S. Hence, its diffusivity may depend on the oxygen fugacity. MacKenzie and Canil (2008) carried out devolatization (desorption) experiments of CMAS1 melt (Table 1) at 1523-1623 K and in air. and extracted Te diffusivities. The data are shown in Figure 30. As can be seen, there is considerable scatter in the data (about a factor of 5 difference in diffusivity values at 1573 K), similar to the large scatter in S diffusivity. Because CMAS1 composition is close to a haplobasalt, the Te diffusion data in CMAS1 melt

Zhang, Ni, Chen

356 -23,5

—

-24 -24.5 -

-26.5 -

-27.5

1

0.58

'

1

0.6

1

1

0.62

l 0.64

0.66

l O O O / n r i n K) Figure 30. Te tracer diffusivities in CMAS1 melt (Table 1) under oxidized conditions (MacKenzie and Canil 2008) compared with S tracer diffusivities in dry Etna and Stromboli basalt melts under reduced conditions (Freda et al. 2005).

are compared in Figure 30 with S diffusion data in basalt melt, though the condition was oxidized for Te diffusion and reduced for S diffusion. The two sets of data are comparable. Po diffusion. No Po diffusion data in silicate melts are known. Summary of O, S, Se and Te diffusivities. The diffusivities of O. S, Se and Te are not readily comparable because of limited and scattered data on S and Te and no data on Se. as well as different speciation and oxidation states of O, S and Te. F, CI, Br, I, At F diffusion. Five papers reported F diffusion data in silicate melts (84 points). Dingwell and Scarfe (1984. 1985) explored FEBD of F in jadeite and albite melts (containing up to 6.3 wt% F) using the diffusion couple method at 1473-1673 K and 1.0-1.5 GPa by packing F-bearing and F-free jadeite powders into a Pt capsule. Dingwell and Scarfe (1985) reported more FEBD values of F in jadeite, albite and a peraluminous silicate melt (45.5 wt% O; 5.4 wt% F; 31.58 wt% Si; 10.87 wt% Al; and 6.14 wt% Na) using devolatization experiments at 1473-1673 K and 0.1 MPa of pure oxygen gas. Gabitov et a l (2005) studied SEBD of F during fluorite dissolution into a haplorhyolite (similar to HR7 in Table 1) melt at 1173-1273 K, 0.1 GPa. and 1.2 to 4.8 wt% H 2 0, and showed that dissolved H 2 0 enhances F diffusivity significantly. Alletti et al. (2007) characterized F tracer diffusion in dry and wet basalt4 melts at 1523-1723 K and 0.5-1.0 GPa. Balcone-Boissard et al. (2009) investigated F tracer diffusion in Na-phonolite and K-phonolite melts at 1473-1723 K and 0.5-1 GPa and examined the effect of Na/K ratio and H 2 0 content. In addition. Baker and Balcone-Boissard (2009) reviewed halogen diffusion in silicate melts (also Baker et al. 2005). Figure 31 shows F diffusion data in dry jadeite, albite and basalt melts. F diffusivities in jadeite melt at 0.1 MPa are greater than those in albite melt by 3.4 InD units. In the pressure range of 1.0 to 1.5 GPa, F diffusivities in jadeite melt are almost independent of pressure within 0.27 InD units, but are significantly (1.8 InD units) higher than those at 0.1 MPa. The independence of F diffusivity on pressure at high pressure but the dependence on pressure from 0.1 MPa to 1 GPa may imply either a structural change in jadeite melt with pressure, or a

Diffusion

Data in Siliate

Melts

357

F diffusion in dry melts

Q

—•— jadeite; 1.5 GPa —© -jadeite; l .25 GPa -•«--jadeite; 1.0 GPa jadeite; 0.1 MPa - » - a t b i t e ; 0.1 MPa • ••• • basalt4; 1 GPa

cT

0.58

0.62

0.64

0.66

0.68

1 0 0 0 / 7 (7" in K) Figure 31. F diffusivities in dry albite, jadeite. and basalt4 melts. Data sources: jadeite 1.0-1.5 GPa (Dingwell and Scarfe 1984); jadeite and albite at 0.1 MPa (Dingwell and Scarfe 1985); basalt4 at 1 GPa (Alletti et al. 2007).

change in the diffusion species from the devolatization method in 0.1 MPa pure 0 2 gas to the diffusion couple method in Pt capsule (Baker and Balcone-Boissard 2009). or experimental error. Another surprise is that F diffusivities in polymerized jadeite melt are similar to those in less polymerized basalt melt (for comparison, the O self diffusivity in dry basalt melt is greater than that in jadeite melt by about 2 orders of magnitude, Shimizu and Kushiro 1984; Lesher et al. 1996). Taking together, it is possible that F diffusivities in jadeite melt at 1.0-1.5 GPa (Dingwell and Scarfe 1984) are compromised somehow, e.g.. because powders (rather than glass cylinders) were packed to make the diffusion couples or because of the presence of unknown amounts of H 2 0 in high-pressure melts (Baker and Balcone-Boissard 2009). F diffusivities in dry basalt4 and jadeite melts can be expressed as (Alletti et al. 2007); (22967 + 5384) dry basall-l: 1 GPa _ i id - exp (-9.46 + 3.32)-

n

(44)

Equation (44) can reproduce the experimental data to within 0.5 InD units. The activation energy is 191 ±45 kj/mol. A single datum at 0.5 GPa shows that the pressure effect within this range is negligible. Adding 3 wt% H 2 0 in basalt melt increases InD by 0.54 units. Figure 32 summarizes F diffusion data in natural silicate melts. Experimental F diffusion data at 0.5 and 1 GPa (Balcone-Boissard et al. 2009) show that the pressure effect is negligible. Diffusion data in dry and wet melts suggest that InD is linear to H 2 0 . The diffusion data in dry and wet Na-phonolite and K-phonolite melts at 1473-1723 K. 1 GPa (likely applicable to 0.5-1.5 GPa). and 0-5 wt% H 2 Q can be fit as follows: ,J

(17624+ 2 3 4 7 ) - ( 3 1 2 + 68) H7-)Na-phonoli[c: 1 GPa _ \ SLUD ~~ CA1J ( - 1 3 . 0 4 + 1 . 4 7 ) -

(45)

(21110+ 4 3 5 0 ) - ( 4 8 1 + 123)vv ,K-phonolite: 1 GPa D. = exp (-11.43 + 2 . 7 1 ) -

(46)

Zhang, Ni, Chen

358 -22

—•—dry basalt4 —© -wet basalt4; 3% — *• - dry Na-phonolite -•S--wet Na-phonolite; 2% • -ffl- - wet Na-phonolite; 5% —*- - dry K-phonolite —O-wet K-phonolite; 2% - X - wet K-phonolite; 5% * wet HR7; 1.3%

-23 ^

-24

•= Q

-25

CS* -26 -27 -28

G Pa 0.6

0.64

0.68 0.72 0.76 1 0 0 0 / 7 (7" in K )

0.8

0.84

Figure 32. F effective binary diffusivities in basalt and phonolite melts. Data sources: dry and wet basalt4 at 0.5 to 1 GPa: Alletti et al. (2007); dry and wet Na and K phonolites at 0.5 to 1 GPa: Balcone-Boissard et al. (2009); and wet HR7 at 0.1 GPa (Gabitov et al. 2005).

Equations (45) and (46) can reproduce the experimental data to within 0.34 and 0.47 InD units, respectively. CI diffusion. Four papers reported CI diffusion data in silicate melts (98 points). In addition, there was an early abstract by Watson and Bender (1980) reporting some results, which are not included in this review because no details are available. Bai and Köster van Groos (1994) investigated SEBD (close to tracer diffusion or FEBD) of CI in rhyolitel7 and HR8 melts (Table 1) in contact with NaCl melt or NaCl solution at 0.0001 to 0.46 GPa. Lundstrom (2003) reported a SEBD datum of CI from basalt7-basanite diffusion couple experiment at 1723 K and 0.9 GPa. Alletti et al. (2007) examined CI tracer diffusion in dry and wet basalt4 melts at 1523-1723 K and 0.5-1.0 GPa. Balcone-Boissard et al. (2009) determined CI tracer diffusivities in dry and wet Na-phonolite and K-phonolite melts at 1473-1723 K and 0.5-1 GPa. Furthermore, Baker and Balcone-Boissard (2009) reviewed halogen diffusion in silicate melts (also Baker et al. 2005). Bai and Köster van Groos (1994) inferred that lnD cl in rhyolitel7 increases rapidly with H 2 0 from 0 to 2 wt%. and then does not increase much as H 2 0 increases from 2 to 6 wt% (Fig. 33a). However. Balcone-Boissard et al. (2009) deduced that lnD cl in Na- and K-phonolite melts is almost linear to H 2 0 from 0 to 5 wt% H 2 0 (Fig. 33a). Both studies used the method of difference-from-100% in electron microprobe analyses to estimate H 2 0. Hence, there is some uncertainty in H 2 0 contents in both studies. On the other hand, in some experiments by Bai and Köster van Groos (1994), the pressure is too low to keep H 2 0 content in the rhyolitel7 melt (e.g.. 6.9 wt% H 2 0 at a pressure of 0.1 GPa). indicating possible problems in the experiments. Tentatively, we suggest that Da depends linearly on H 2 0 based on BalconeBoissard et al. (2009) and that the inference by Bai and Köster van Groos (1994) is incorrect. For dry and wet Na-phonolite and K-phonolite melts at 1523-1723 K, 1 GPa and 0-5 wt% H 2 0. CI diffusivity can be expressed as follows: u cVl iTDpin • r 11 •• I i I • : : 1 G P-a exp _( 11 57 + 3 2°)

('7963±5I44)-,-

n

-

aryHK/ _ Ti T D ~~

C A

cx P (-10.08 + 2 . 2 3 ) - (2460CH:3800)

(74)

(39776 ±4960) (-9.1612.99)--

(75)

F

Ti SEBD values in basaltl 1 melt during clinopyroxene dissolution is about 1.6 times, and those during olivine dissolution is about 4 times the self diffusivities in dry HB1 (Eq. 52a). Ti SEBD in dry andesitel melt during rutile dissolution at 1.5 GPa is 0.6 times the diffusivity from Equation (74). Adding 3.6 wt% H 2 0 or 20 wt% N a 2 0 to HR7 melt increases Ti diffusivity by roughly the same amount. Ti diffusivity increases from HR7 (a haplorhyolite) melt to basalt melt by about 4 orders of magnitude. The effect of 5-12 wt% H 2 0 on Ti duffusivity in silicic melts is greater than that in mafic melts. Zr diffusion. Nine papers reported Zr diffusion data in silicate melts (100 points). Harrison and Watson (1983) extracted SEBD (close to FEBD and tracer diffusivities) of Zr in rhyolite 12 melt (Table 1) during zircon dissolution at 1473-1673 K, 0.8 GPa and 0.1-6.3 wt% H 2 0 . Baker and Watson (1988) determined SEBD (close to FEBD) of Zr using the rhyolitelrhyolite7 diffusion couple at 1171-1673 K and 0.01-1 GPa. Baker et al. (2002) studied SEBD (close to FEBD) of Zr in rhyolite 14 melt and two other rhyolite melts with slightly different concentrations of FeO (difference of 1.2 wt%), F (difference of 1.2 wt%) and CI (difference of 0.35 wt%) during zircon dissolution at 1323-1673 K, 1 GPa, and < 5.1 wt% H z O. No difference was found in Zr diffusivity among the three rhyolite melts used by Baker et al. (2002). LaTourrette et al. (1996) investigated Zr self diffusion in dry HB1 melt at 1623-1773 K and in air. Nakamura and Kushiro (1998) characterized Zr tracer diffusion in jadeite melt at 1673 K and 1.25-2 GPa. Mungall et al. (1999) examined Zr tracer diffusion in dry HR7, wet (3.6 wt% H 2 0 ) HR7, and dry H R 7 + N a melt (Table 1) at 1083-1873 K, 0.0001 and 1 GPa. Koepke and Behrens (2001) measured Zr tracer diffusivities in wet HA1 melt with about 4.55.2 wt% H 2 0 at 1373-1673 K and 1 GPa and one datum in dry HA1 melt. Lundstrom (2003) reported a single SEBD value of Zr from a dry basalt7-basanite diffusion couple experiment at 1723 K and 0.9 GPa. Behrens and Hahn (2009) investigated Zr tracer diffusion in dry and wet trachyte 1 and phonolitel melts at 1323-1528 K and 1 GPa. Zr diffusion in rhyolite melts has been investigated extensively (Harrison and Watson 1983; Baker and Watson 1988; Baker et al. 2002) and hence inter-laboratory comparisons can be made. The data for rhyolite melts are shown in Figure 53. Some data are difficult to reconcile. For example, the huge pressure effect from 0.01 to 1 GPa in the Zr diffusion data of Baker and Watson (1988) (solid squares and open triangles) are difficult to explain. In dry rhyolite melts of similar compositions, Zr diffusivities in samples containing < 0.7 wt% halogens (Baker and Watson 1988) are about 2 orders of magnitude higher than those obtained by Harrison and Watson (1983). Baker and Watson (1988) attributed the difference to the effect of halogens, but a later study (Baker et al. 2002) indicated that adding even 1.2 wt% F has only a minor effect on Zr diffusivities. Assuming that the more recent paper by the same lead author is more likely correct, it appears that the data in Baker and Watson (1988) are erroneous, and the effect of F and CI at the level of < 1 wt% is minor on Zr diffusion. For zircon dissolution in wet rhyolite melts (solid triangles and circles with center dot), there is no consistency either: Zr diffusivities at 6.0 wt% H 2 0 (Harrison and Watson 1983) are smaller than those at 4.2-4.8 wt% H 2 0 (Baker et al. 2002). Two possible explanations can be advanced although it is difficult to judge which data are more reliable. First, there might be convection in the zircon dissolution experiments (Harrison and Watson 1983; Baker et al. 2002), which could have compromised the diffusion data (see Zhang et al. 1989), although the solubility of zircon

Diffusion

Data in Siliate

Melts

379

--»-dry rhyl2; 0.8 GPa; HW83 -Hi -dry r-r; 0, and X, = 0 if Na+K-Al < 0. this is because peralkalinity seems to affect diffusivity more than peraluminity). The available data cannot resolve the compositional dependence of the activation energy. The maximum error in reproducing the experimental data by Equation (84) is 0.37 lnD units (Fig. 61). The Mn SEBD data in rhyolite at 1 GPa (Baker and Watson 1988) deviate from Equation (84) (e.g., the data would indicate a negative activation energy), probably due to experimental or analytical problems. Mn and Co diffusivities are similar in all the melts that have been investigated (basalt, andesite, dacite and pantelleritel). Fe diffusion. Eleven papers reported Fe diffusion data in silicate melts (165 points). In silicate melts. Fe can be present as ferrous (Fe 2+ ) or ferric (Fe 3+ ). Diffusivities of the two different states are likely significantly different. Hence, it is important to characterize Fe oxidation state before the experiment (e.g., whether the two halves of a diffusion couple have the same ferric/ferrous ratio), control oxygen fugacity during the experiments, and/or measure the oxidation state of Fe after the experiment. Lowry et al. (1982) examined Fe radioactive tracer diffusion in basaltl and andesite2 melts under oxidized condition at 1570-1675 K and in air. Henderson et al. (1985) studied Fe diffusivities in dacite4 and pantelleritel melts under oxidized condition at 1475-1672 K and in air. Baker and Watson (1988) investigated Fe SEBD in rhyolite l-rhyolite7 and HD2-rhyolite7 couples (Table 1) at 1171-1273 K and 0.01 GPa and 1373-1473 K and 1 GPa (/"0, not characterized). The pressure effect is insignificant compared to data scatter. Dunn and Ratliffe (1990) investigated tracer diffusion of ferrous iron in a peraluminous sodium aluminosilicate (HR1 in Table 1) at 1516-1716 K and 0.1 MPa using a C 0 - C 0 2 gas mixture to control oxygen fugacity to be near the FMQ buffer, and at 1523-1723 K and 0.6-2 GPa in graphite capsules. Watson (1991b) examined Fe transport in slightly melted dunite. Because the extracted diffusivity is not for pure liquid, the data will not be used further due to limitations of the scope of this chapter. Baker and Bossanyi (1994) examined the effect of H 2 0 and fluorine on Fe SEBD in dacite 1 -rhyolite7 diffusion couples at 1373-1673 K and 1 GPa. van der Laan et al. (1994) reported a single Fe diffusivity

Zhang, Ni, Chen

386

(SEBD) in a rhyolite3-rhyolitel6 couple at 1523 K and 1 GPa. Koepke and Behrens (2001) obtained Fe tracer diffusivities in hydrous HA 1 melt at 1373-1583 K and 0.5 GPa with oxygen fugacity roughly at NNO + 3. Lundstrom (2003) obtained two Fe S E B D values in basalt7basanite couple. Ruessel and Wiedenroth (2004) characterized the compositional effect on Fe tracer diffusivities in N a 2 0 - C a 0 - M g 0 - A l 2 0 3 - S i 0 2 melts (Table 1) in air and 1300-1873 K, but diffusivity values were only reported at 1573 K. Morgan et al. (2006) determined Fe S E B D in lunar picrites (LP1 and LP2) during anorthite dissolution at 1673 K and 0.6 GPa. Due to the presence of both F e 2 + and F e 3 + in typical geological conditions, with F e 3 + likely having lower diffusivities (due to higher bond strength than F e 2 + ) , Fe diffusivity is expected to lie between that of F e 2 + and F e 3 + . If oxygen fugacity is uniform in the melt, equilibrium between F e 2 + and F e 3 + would lead to uniform ferric to ferrous ratio. Following Zhang et al. (1991b), Fe diffusivities can be expressed as follows: I)

= ,Y, . /)

+ X^uD^-

(85)

where X1Ie2+ = F e 2 + / ( F e 2 + + F e 3 + ) . Based on valence and size consideration, it can be argued that F e 2 + diffusivities lie between that of Mn 2 + and Co 2 + (ionic radii of Mn 2 + , F e 2 + and Co 2 + in octahedral sites at high spin are 0.083, 0.078 and 0.0745 nm, Shannon 1976). If F e 3 + is in the tetrahedral site, F e 3 + diffusivity is likely similar to that of Ga 3 + (ionic radii of F e 3 + and Ga 3 + in tetrahedral sites are 0.049 and 0.047 nm). If F e 3 + is in the octahedral sites, F e 3 + diffusivity is likely similar to that of Cr 3 + , or V 3 + , or Ga 3 + (ionic radii of F e 3 + , Cr 3 + , V 3 + , and Ga 3 + in octahedral sites are 0.0645, 0.0615, 0.062, and 0.64 nm). In experiments and nature, Mn and Co are divalent, and Ga and Cr are trivalent. Hence for Mn, Co, Cr and Ga, we do not need to worry about the oxidation state. Furthermore, Mn and Co diffusivities are similar (Fig. 62). Hence, Fe diffusivities in melts are expected to lie between those of Mn and Ga, or of Co and Ga; Figure 62 is consistent with this expectation. Furthermore, Figure 62 shows that Fe 2 + diffusivity is about 6 times Fe 3 + diffusivity in dacite melts. Oxygen fugacity was not always controlled in experimental studies. For example, no care was taken to control oxygen fugacity in the experiments of Baker and Watson (1988), Baker and Bossanyi (1994) and van der Laan et al. (1994) because the main goal of their studies was not on Fe diffusion. Some Fe diffusion data are shown in Figure 63. Lowry et al. (1982) and Henderson et al. (1985) conducted the tracer diffusion experiments in air. There might still be some uncertainty about the oxidation state of the diffusion experiments because the initial glass was not equilibrated in air before diffusion experiments. Nonetheless, Fe diffusivities are significantly lower (by a factor of about 3) than both Mn and Co diffusivities, and only slightly higher than Ga diffusivities (Fig. 62), consistent with the expectation that Fe is mostly trivalent in these experiments. Fe diffusivities (TD) in metaluminous basalt 1-andesite2-dacite4 melts by Lowry et al. (1982) and Henderson et al. (1985) in air can be expressed as: exp 5 . 3 0 - 1 7 . 6 4 X j

29879 T

(86)

where X, is the sum of cation mole fractions of Si +Ti + Al + P. The available data cannot resolve the compositional dependence of the activation energy (about 248 kJ/mol). The maximum error in reproducing the experimental data by Equation (86) is 0.42 InD units (Fig. 63). Dunn and Ratliffe (1990) studied F e 2 + diffusion in HR1 melt (a haplorhyolite melt) by using reducing conditions at 0.0001 and 0.6-2 GPa so that F e 2 + is the dominant iron species. They found that the pressure effect is insignificant from 0.6 to 2 GPa (diffusivity variation is within 0.3 InD units). However, Fe tracer diffusivity values at 0.6-2 GPa using graphite capsules is 5 times those at 0.1 MPa at FMQ buffer. Dunn and Ratliffe (1990) attributed the

Diffusion

Data in Siliate

38 7

Melts

-25.5 -26

-26.5 -27 Q Q

-27.5 -28

-28.5

»... -Mn in dacite4 --ffl"Co in dacite4 - • - F e in d a c i t e 4 - • • - G a in d a c i t e l

-29 0.59

0.6

0.61

0.62

0.63

0.64

1 0 0 0 / 7 ( 7 " in K ) Figure 62. Tracer diffusivities of M n and Co in dacite4 melt (Henderson et al. 1985). those of Ga in dacitel melt (Baker 1992a) compared with those of Fe in dacite4 melt in air (Henderson et al. 1985). -23 -24

M E

-25 -26

—

-27

Q iSm

basalti ; air —a— andesite2; air • dacite4; air pantelleritel ; air — * HR1; 0.1 MPa; FMQ X HR-r; 0.6-2 GPa - - B - - wet HAI; 0.5 GPa; 5%

X

- •

-28

X —

-29 -30 -31 0.58

— m -

0.6

0.62

0.64

0.66

0.68

1 0 0 0 / 7 ( 7 in

0.7

0.72

0.74

K )

Figure 63. Some experimental data on Fe diffusion in silicate melts. Data for dry basalti and dry andesite2 melts are f r o m Lowry et al. (1982); dry dacite4 and pantelleritel melts are f r o m Henderson et al. (1985); dry H R 1 at 0.1 M P a and 0.6-2 GPa are f r o m Dunn and Ratliffe ( 1990); and wet H A I are f r o m Koepke and Behrens (2001 ).

difference to a sudden jump from 0.0001 to 0.6 GPa and then no additional pressure effect above 0.6 GPa. a somewhat ad hoc explanation. More likely, the sudden jump is due to experimental difficulties. The oxygen fugacity in the experiments of Koepke and Behrens (2001) using hydrous haploandesite (4.9-5.2 wt% H 2 0) was estimated to be about NNO + 3, at which the ferric to ferrous ratio is roughly 1. Hence, these data are roughly the average Fe 2+ and Fe 3+ diffusivities

Zhang, Ni, Chen

388 and can be expressed as:

r - j w e t H A I ; 0.5 M P a ; N N 0 + 3 ; 4 . 9 - 5 . 2 % Fe T D

_ ~~

e

X

P

(24421 ±10053) (-7.83 ± 6 . 8 4 ) - -

(87)

where % in the superscript indicates wt% of H 2 0 . The maximum error in reproducing the three experimental data by Equation (87) is 0.28 InD units. Ruessel and Wiedenroth (2004) produced extensive Fe tracer diffusion data in Na 2 0C a 0 - M g 0 - A l 2 0 r S i 0 2 melts at 1300-1873 K. The experiments were implicitly conducted in air, though it was not clearly specified. Hence, the diffusivities are close to Fe 3 + diffusivities. Ruessel and Wiedenroth (2004) reported activation energy and pre-exponential factors for melts with various compositions, but only reported diffusivity values at 1573 K. The activation energy varies from 209 to 284 kj/mol as the melt composition varies. Due to errors in extracted activation energies and pre-exponential factors, it is easier to treat original diffusion data than to model how activation energies and pre-exponential factors depend on melt composition. The available data of Fe diffusion, though numerous, are not systematic enough to understand both Fe 2 + and Fe 3 + diffusion. In order to better understand Fe diffusion, it is necessary to conduct Fe diffusion studies under a range of oxygen fugacities so that diffusivities of both Fe 2 + and Fe 3 + and their dependence on temperature, pressure and composition can be extracted. Then Fe diffusivity at any given f0l (or given ferric/ferrous ratio) can be inferred using Equation (85). Co diffusion. Three papers reported Co diffusion data in silicate melts (29 points). Hofmann and Magaritz (1977) studied Co tracer diffusion in dry basaltlO melt (Table 1) at 1553-1713 K and 0.1 MPa. Lowry et al. (1982) investigated Co tracer diffusion in dry basaltl and andesite2 melts at 1567-1672 K and 0.1 MPa. Henderson et al. (1985) studied Co tracer diffusion in dry dacite4 and pantelleritel melts at 1473-1672 K and 0.1 MPa. No EBD data of Co are available. Co tracer diffusion data in two different dry basalts (basaltl by Lowry et al. 1981 and basaltlO by Hofmann and Magaritz 1977) are in good agreement (Fig. 64) and can be fit as follows: n

d r y basalts: 0.1 M P a Co I D

_

~

eX

(202"±3706) P -(11.05±2.29)-

(88)

The maximum error in reproducing the experimental data by Equation (88) is 0.16 InD units. The activation energy is 168±31 kj/mol. Co tracer diffusivity increases from pantelleritel to dacite4 to andesite2 to basaltl melts; the increase from pantellerite to dacite is a little strange because increasing alkalinity is usually thought to cause an increase in diffusivity. All diffusion data are shown in Figure 64 and can be roughly fit by the following equation:

DCo ID

d r v melts; 0.1 M P a

, (4352 =eXP -l1.02-42.07X,-

+

2 5 l l 6 X

')

(89)

where is the sum of cation mole fractions of S i + T i + A l + P , and X2 = max(Na+K-Al,0) (i.e., X2 = N a + K - A l i f Na + K - A l > 0 , and A" = Oif N a + K - A l 0, respectively; [E] is a diagonal matrix with its diagonal terms given by

Multicomponent

Diffusion

in Molten

Silicates

415

i E„ = erf and the off-diagonal terms Ey = 0 for i * j;

(24b)

2 ^kf

> 0 are the eigenvalues of the diffusion matrix.

Case 2. Diffusion between two rods initially having uniform but different compositions. This is a case of a finite diffusion couple with impermeable walls at the two ends. Let x = 0 be the original interface between the two rods, solutions to Equation (3), in this case, can be written as w„L„ «

(25a) V

where w„ is the initial composition of rod a (~La < x < 0); w6 is the initial composition of rod b (0 < x < Lhy, [F] is a diagonal matrix with its diagonal elements given by „ /

1 . f mnL n ,„=, m

Ax+L„)

exp

L + Lh

m 7t~kkt

'K^hf

(25b)

Xk > 0 are again the eigenvalues of the diffusion matrix. Equations (25a) and (25b) are identical to the component form given by Trial and Spera (1994). Case 3. Diffusion of a melt pocket (composition wb) initially confined in the region -h f> see Figs. 1 a and Id), we have an approximate relation between the concentration gradients of components 2 and 3, D

ox + D ^civ*

^

0

(29b)

Hence concentration profiles of the fast component 2 are locked into the concentration profiles of the slow component 3. This is shown in Figures 3a and 4a where concentration profiles of component 2 are linearly correlated with the concentration profiles of the slow component 3 for the two diffusion couples considered (solid and dashed lines, respectively). The slopes of the nearly parallel straight lines, dw2/dwi„ are 0.264 (0.260) a t x = 0 and t = 1 (t = 5) for the finite diffusion couple (Fig. lb) and 0.271 (0.259) at xitm = ±1 and t = 1 (t = 5) for the diffusion couple shown in Figure le. This is in excellent agreement with the prediction from Equation (29b).

2 dw} dw

=

[X Aw, ^ h (d Aw, 1 Aw, wJ l 22 A

°1> | Q — [ d D„

1

L

2 2

21

O

J~

4

D

(30)

Multicomponent

Diffusion

in Molten

419

Silicates

tl = 0.001 = 0.05 — 1 3 =1 t4 = 5 — 1

2

Figure 3. Variations of component 2 (a) and component 1 (b) as a function of the slow diffusing component 3 at the four selected times. The solid lines are diffusion profiles from the finite diffusion couple shown in Figures l a - l c . The dashed lines are from diffusion profiles shown in Figures l d - l f . 0.28

0.27 •

0.26 •

0.25 •

0.24

0.25

>CO

0.15

J4

^

V ~ 0.1

\

' (c)

t3

J \

^

1 v

ti

\

y

0 x/sqrt(t)

_ _ 1

U

x

\

t2~

/ / !

1 \ 1

J

_ S

\ \

t2\

s A ]

( 1 1 \

\

0.2

""cm

;

t4

\

V I

J i -

(d)

v \

J /

t4 - -

2

N^

1

A

s/.

11 -

1

0

1

2

x/sqrt(t)

Figure 4. Plots of concentration derivatives and concentration ratios calculated from the diffusion profiles 12 shown in Figures l a - l c (solid lines) and l d - l f (dashed lines) as a function of x/t ' (in 105 ms~ 1 / 2 ) for the two diffusion couples (see text for details).

Liang

420 where O stands for order of magnitude.

A similar exercise can be carried out for quasi steady-state diffusion of the intermediate component 1 at long times by setting

J ^ - D J ^ - D J - ^ - D J ^ 0 ox

ox

(31a)

ox

In this case both concentration gradients of components 2 and 3 contribute to the diffusive flux of component 1. Since the concentration gradients of component 2 is locked into the concentration gradient of component 3 via Equation (29a), we have J ^ - D ^ +D ox

^ ^ D22 OX

D

^

0

(31b)

OX

which can be further simplified to DW

I

dw3

D D

=

N 23~DND22

(32)

Z), XD^2

Substituting the elements of the diffusion matrix from Equation (27) into Equation (32), we find a slope dwy! dw3 = -0.188, which is in reasonable agreement with the numerical values calculated directly from the diffusion profiles shown in Figure l a - l f (-0.350 and -0.485, for the two diffusion couples respectively, at t = 1; and - 0 . 2 4 7 and -0.211 at t = 5, all at x = 0). Figures 3b and 4b show that concentration profiles of component 1 are approximately linearly correlated with the concentration profiles of component 3 for the two diffusion couples at long time (solid and dashed lines, respectively). The slow convergence rate in this case is due to the moderate diffusion rate of component 1 relative to component 3 such concentration profiles of component 1 were still not relaxed at t = 1 (i.e., l/jl > 0). Diffusive behaviors of the fast, intermediate, and slow diffusing components illustrated in Figures 1-3 are broadly similar to those for diffusion between basalt and molten feldspar or granite (Watson 1982; Watson and Jurewicz 1984; Watson and Baker 1991, see their Fig. 1), if we take the fast component 2 as the sum of Na z O + K 2 0 , the intermediate component 1 as CaO + MgO + FeO, and the slow component 3 as Si0 2 . Watson and coworkers noted that alkalies are preferentially enriched or partitioned into the felsic melt relative to those in the basalt, by a factor of two to three, after an initial short time, irrespective of their initial abundances in the two starting melts. This "transient equilibrium partitioning" of the fast diffusing alkalies between the high- and low-Si melts in a finite diffusion couple (Watson and Baker 1991) can be understood in terms of quasi steady-state diffusion in a multicomponent melt in which the diffusive fluxes of the alkalies are strongly coupled to the concentration gradient of Si0 2 . Since the system reaches a quasi steady-state, not a true thermodynamic equilibrium state, concentration variation ratios or slopes in the concentration correlation diagram (e.g., dw2/dw3 and dwjdw3) approach constants (e.g., Eqns. 30 and 32). The concentration ratios (e.g., w2/w3 and w^wj), on the other hand, depend on the boundary conditions (e.g., type of diffusion couple and choice of starting compositions). This can be further demonstrated by the examples below. Figures 4c and 4d display the variations of the concentration ratio w 2 /w 3 and w]/w 3 as a function of position in the two diffusion couples at the four selected times. In contrast to the nearly constant slopes ( d w 2 / d w 3 and dwl/dw3 in Figs. 4a and 4b) discussed in the proceeding examples (see also Figs. 3a and 3b), the concentration ratios w 2 /w 3 and w^/wj at the quasi steady-state (t3 or t4) not only vary within a given diffusion couple, but more importantly also depends strongly on the setup or choice of the diffusion couple (cf. the solid and dashed lined in each panel). For example, at t4 = 5, w 2 /w 3 ~ 0.158 and wYlw3 a 0.43 for the finite diffusion couple (Figs, l a - l c and 4c), whereas w 2 /w 3 ~ 0.238 and wylw 3 ~ 0.33 for the finite

Multicomponent

Diffusion

in Molten

Silicates

421

melt sandwiched between two semi-infinite melt reservoirs (Figs. Id-If and 4d), all at x = 0. Another advantage of using the concentration variation ratios rather than concentration ratios in the interpretation of "transient equilibrium partitioning" is that the former also applies to diffusion in an effectively infinite melt reservoir (see Figs, lc-d, 2, also solid lines in Figs. 3, 4a-b). Hence it may be more appropriate to use the phrase "transient gradient partitioning" to describe element distributions during quasi steady-state diffusion in a multicomponent melt. Transient gradient partitioning during quasi steady-state diffusion (e.g., Eqns. 30 and 32) may be helpful in designing special chemical diffusion experiments that can be used to better constrain selected elements of the diffusion matrix. Multiple time-scales of diffusion and effective binary diffusion. The contrasting diffusive behaviors between the fast and slow diffusing components in the proceeding examples have already demonstrated the multiple time-scale nature of diffusion in multicomponent melts. The three unique eigenvalues of the hypothetical diffusion matrix Equation (27) imply three independent diffusion time scales for a given length scale: t^, t2°, 1In theory, one can conduct a more rigorous analysis of quasi steady-state diffusion by setting the diffusive flux associated with the fast diffusing eigenvector to zero at long time (e.g., J2 = -A, 2 3t/ 2 /dx = 0 at t > f2°). In practice, since the elements of diffusion matrix are usually not known a priori, it is often difficult to define the fast diffusing eigenvector. To further demonstrate the concept of multiple time-scale of diffusion, we take the effective binary approach. Chemical diffusion in multicomponent molten silicates has commonly been modeled in terms of effective binary diffusion (Cooper 1968; see also reviews by Watson and Baker 1991; Chakroborty 1998; Zhang et al. 2010). The multicomponent melt is treated as a pseudo-binary system in which the component of interest is taken as the independent component or variable and all other components are taken together as the second component or dependent variable. The diffusive flux of the component of interest (/) is then assumed to obey a relation of the form, J,=-Df^dx

(33)

where Df is the effective binary diffusion coefficient (EBDC) for component i. Since Df must be positive for diffusion equation to be stable, the effective binary simplification is not capable of modeling uphill diffusion. Cooper (1968) showed how the EBDC defined in Equation (33) could be related to the elements of the diffusion matrix. For example, the EBDCs for component 1 and 3 in the quaternary melt can be calculated using information from diffusion matrix and slopes in concentration correlation diagram, D f = D

u +

D

ow3

+

n ^ owl

+

D

ow3

^ ' owl +

(34a) (34b)

Since the concentration derivatives, dwjdwj, depend on geometry and starting compositions of the diffusion couple, as well as time and spatial coordinates in the diffusion couple (see Figs. 4a and 4b), the EBDCs defined in Equation (34a) and (34b) are in general not constant and uniform in a given diffusion couple even when the elements of the diffusion matrix are constant and uniform. Thus, EBDCs often depend on the direction of diffusion in composition space (Cooper 1968; see also Zhang et al. 2010). For example, the experimentally measured EBDC of Si0 2 in molten Ca0-Al 2 0 3 -Si0 2 (1500 °C and 1 GPa) differs by almost an order of magnitude for diffusion along the directions of constant CaO and constant A1 2 0 3 (Liang et al. 1996a). An equally important property of effective binary diffusion is that EBDCs vary as a function of time even at a fixed position in a given diffusion couple. This is illustrated in Figures 5a and 5b where the EBDCs of component 1 and 3 for the quaternary system calculated using

422

Liang

Figure 5. Plots of calculated efficient binary diffusion coefficients (EBDCs) of component 1 (a) and component 3 (b) as a function of diffusion time for three different diffusion couples (see text for details). The EBDCs were calculated using Equations (34a) and (34b) and concentration derivatives from solutions to the coupled diffusion equations using the hypothetical diffusion matrix Equation (27).

the hypothetical diffusion matrix Equation (27) and Equations (34a) and (34b) are shown as a function of time. Three diffusion couples are considered: a finite melt pocket of length h sandwiched between two semi-infinite melt reservoirs (solid blue lines labeled as Case 3), a finite diffusion couple in which LJh = L2/h = 2 (dashed blue lines labeled as Case 2a), and a short finite diffusion couple in which LJh = L2lh = 1 (solid red lines labeled as Case 2b). Selected concentration profiles of Case 2a and Case 3 are shown in Figures la-lc and Id-If, respectively. For comparison, starting melt compositions for the three diffusion couples are the same and the EBDCs are calculated at the original interface of the diffusion couples using Equations (34a) and (34b) (x/t 1 / 2 = ±1 for Case 3, x = 0 for Case 2a and Case 2b). Three interesting observations can be readily made from Figures 5a and 5b. First, there are two time scales of diffusion: the EBDCs of components 1 and 3 at short time when the melt reservoirs are effectively semi-infinite are 2-6 times larger than those at long time. This again can be understood in terms of coupled diffusion. At short time, the concentration gradients of component 1, dw]/8x, are significant. The diffusive flux of component 1 is dominated by the diagonal term, as the magnitudes of the off-diagonal terms are relatively small. Hence, £)f ~ Du = 2x 10~u m 2 /s, which is in good agreement with the three cases shown in Figure 5a. Since the sign of dwi /8x is opposite to that of dw}/dx (Figs, la-lf) and D}] < 0, the presence of a significant concentration gradient dw\ /8x elevates the apparent diffusion rate of component 3 (hence an increase in EBDC of component 3). As the magnitude of dwi /dx becomes smaller with increasing time, EBDC of component 3 decreases (Fig. 5b). Contributions of the (relatively) large concentration gradient dw}/dx to the diffusive flux of component 1 then become significant, reducing the apparent diffusion rate of component 1 at long time, as D ] 3 > 0 (Fig. 5a). At long time, EBDC of component 1 (also component 2) is approximately the same as EBDC of component 3 for the two finite diffusion couples (Case 2a and Case 2b in Figs. 5a and 5b). This is consistent with the observation of Watson and Baker (1991) for diffusion between basalt and molten feldspar or granite: "If the diffusion reservoirs are of limited dimension, the initial rapid transfer of alkalies ceases, and the concentration gradients assumed by all elements become equal in length to (though possibly in the opposite direction of) that ofSi02."

Multicomponent

Diffusion

in Molten

Silicates

423

Although the relative diffusion information among the independent components appears lost in terms of effective binary diffusion at long time, the nearly constant slopes in the concentration correlation diagrams still offer valuable information regarding the elements of the diffusion matrix (e.g., Eqns. 30 and 32). Also at long time, EBDCs derived from the finite diffusion couples (Case 2a and Case 2b in Figs. 5a and 5b) are different from (lower than) those derived from Case 3 that has two semiinfinite melt reservoirs. This is because the slope, Sw,-/8wj, for Case 3 is larger than for Cases 2a and 2b, even though in both cases d w j d w j approach their asymptotic values (Figs. 4a and 4b). Finally, Figures 5a and 5b also show that transition from short time to long time diffusive behaviors occurs at different times depending on the size of the diffusion couple (cf. Case 2a and Case 2b). The time- and geometry-dependent natures of EBDC further complicate direct applications of laboratory derived EBDCs for molten silicates to geological problems: one has to match not only the direction of diffusion in composition space, but also the time scale and geometry of diffusion between the laboratory and nature in order to use the EBDCs. That is a lot of requirements. After all, effective binary diffusion is not as simple as it appears.

EXPERIMENTAL STUDIES OF MULTICOMPONENT DIFFUSION Experimental design and strategy The experimental study of chemical diffusion in molten silicates involves juxtaposing two melts of different compositions in an inert container placed in a high temperature (and very often high pressure) furnace. In general, an accurate estimate of the diffusion matrix in an n component system requires at least n-1 different diffusion couples because there are (n-1)2 unknowns in the diffusion matrix and each diffusion couple has only n-1 independent concentration profiles (Gupta and Cooper 1971). Ideally, one should select a set of diffusion experiments such that the n-1 diffusion couples not only cross at the composition point of interest but also are orthogonal to each other in the eigen-space of the diffusion matrix (Liang 1994; Trial and Spera 1994). But, since the diffusion matrix, and thus the eigen-space, is not known a priori, the n-1 directions are often chosen so that one of the components is constant along each of the directions (e.g., Kirkaldy and Young 1987, page 179). Figure 6 shows an example of the starting melt compositions used to form diffusion couples (pairs of open circles joined by thin solid lines) in the ternary CaO-A^Oj-SiC^ (Liang et al. 1996a; Liang and Davis 2002). Three diffusion couples formed by juxtaposing compositions 3 against 5 (designated as 3/5), 4 against 2 (4/2), and 6 against 1 (6/1) were used to determine the diffusion matrix at composition 7, whereas two diffusion couples (16/14 + 17/10) were used to determined the diffusion matrix at composition 12. The advantage of using multiple diffusion couples to determine the elements of a diffusion matrix can be demonstrated by Monte Carlo simulations in which computer generated diffusion profiles, to which Gaussian noise representing analytical errors has been added, are inverted simultaneously for the elements of the diffusion matrix (Liang 1994; Trial and Spera 1994). Figure 7 compares the relative errors derived from simultaneous inversion of synthetic concentration profiles from single-direction (diffusion couple 6/ 1, open circles), twodirection (3/5 + 6 / 1 open triangles, 4 / 2 + 6 / 1 open diamonds, 4 / 2 + 3/5 crossed squares), and three-direction diffusion experiments (4/2 + 3/5 + 6/1, open squares) for a range of prescribed analytical uncertainties (Liang 1994). The diffusion couples intercept at composition 7 in composition space (Fig. 6). As shown in Figure 7, for a given analytical uncertainty, uncertainties of the estimated diffusion matrix from simultaneous inversion of concentration profiles from two-direction diffusion experiments are reduced by 220-370% compared to those derived from the single diffusion couple 6 / 1 using the same inversion procedure. Uncertainties in the inverted diffusion matrix are further reduced when concentration profiles from all three diffusion couples are used together in a joint inversion. Overall, uncertainties in the measured diffusion matrix from single-direction experiments are 2-6 times greater than those obtained

Liang

424

sio2

SiO->

Ca'

CaO

40

30

20

A1 2 0 3

Figure 6. Composition space (in wt%) showing the starting compositions (open symbols) used to form diffusion couples for studying multicomponent diffusion in the ternary CaO-AliCh-SiOi. Tie-lines join starting compositions from which diffusion couples were formed. These diffusion couples intercept at compositions 7 and 12, respectively in composition. Adapted from Liang and Davis (2002) and with permission of Elsevier, http://www.sciencedirect.com/science/journal/00167037.

from two-direction or three-direction diffusion experiments. Another important result of the Monte Carlo simulations shown in Figure 7 is the sensitivity of the inverted diffusion matrix to the quality of the measured concentration profiles. To accurately determine the elements of a diffusion matrix, one needs not only multiple diffusion couple experiments but also high quality concentration profiles. The latter can be achieved by long counting times and high spatial resolution in electron microprobe analysis of the diffusion charge. Several other factors are important in determining the best compositions to juxtapose in a given diffusion experiment. From the point of view of accurately measuring concentration changes along a diffusion profile, one should choose diffusion couples that have large concentration differences, and thus large signal to analytical noise ratios. This, however, is limited by the compositional dependence of the diffusion matrix, which poses real difficulties in terms of determining the diffusion matrix corresponding to a single composition. Hence, the compositions used for diffusion couples are usually a compromise between concentration differences large enough to allow for accurate measurement but small enough to minimize the compositional dependence of the diffusion matrix. Stability with respect to convection is another crucial factor, given the possibility of isothermal double-diffusive convection that can occur even when the lower density melt is placed above the more dense one (McDougall 1983; Turner 1985; Liang et al. 1994; Liang 1995; Richter et al. 1998). The difficulty here is that predicting convectively stable directions requires knowledge of the diffusion matrix, which is unknown at the outset. As a result, some of the initial experiments designed for the purpose of determining the diffusion matrix will prove unsuitable because of convection. Twodimensional X-ray concentration maps of the diffusion couples (e.g.. Liang et al. 1994; Liang 1995; Richter et al. 1998) allowed easy identification of convective flow. It is often possible to obtain stable chemical diffusion runs in convectively unstable directions by using short run

Multicomponent 0.06

1/ QJ

•

1

Q


5) for the biotite to act as a homogeneous infinite reservoir. From Hauzenberger et al. (2005), reprinted with permission from European Journal of Mineralogy (website: http://www.schweizerbart.de).

120C

(1.004

0.12 0.11 -

0.10L 0.09 "

Figure 18. Simulation of the evolution of compositional profiles of divalent cations in a garnet crystal from the Micaschist-Marble complex in the Wolz Tauern (part of the middle Austroalpine unit) using T-t paths that are produced by different average values of subduction plus exhumation rates and a subduction angle of 60°. The measured data are shown by symbols. The dotted lines define the assumed initial profiles. The numerical simulations were carried out within the framework of multicomponent diffusion theory. From Faryad and Chakraborty (2005), reprinted with permission from Springer-Verlag.

Cation Diffusion Kinetics in Aluminosilicate

Garnets

589

It was found that very good matches for all profiles could be obtained simultaneously using the T-t path that was calculated for an average subduction plus exhumation velocity of ~ 4-5 cm/yr, which yields a total time span of 0.8-0.9 m.y. for the development of the diffusion profiles. This result is compatible with the average vertical exhumation rate of -4.6-7.4 cm/ yr that was deduced by Dachs and Proyer (2002) from modeling compositional zoning across core-overgrowth segments of garnets from the Penninic Tauern Window, as discussed above. Modeling partially modified growth zoning of garnets in metamorphic rocks Garnets that cooled from granulite fades conditions typically show homogeneous cores and retrograde zoning profiles near the rim. All prograde thermal histories are erased from the compositional properties of these garnets as these were completely homogenized at the peak metamorphic conditions. In contrast, garnets in medium grade metapelites show growth zonings that were modified by intracrystalline diffusion (e.g., Chakraborty and Ganguly 1991). These partially modified growth zoning profiles preserve the P-T-t records for both prograde and retrograde cycles, which may be retrieved from numerical modeling that involves garnet nucleation and growth in metapelites, development of growth zoning and its modification due to intracrystalline diffusion along a P-T-t path. Florence and Spear (1993) were the first, I believe, to develop a forward modeling computer program (DiffGibbs) to carry out the numerical simulation of partially relaxed growth zoning in garnet using multicomponent diffusion theory and apply it to retrieve both prograde and retrograde P-T-t histories of staurolite schists from the Littleton Formation, northwestern New Hampshire. An example of their simulation of garnet zoning profiles and the associated T-t path is shown in Figure 19. In these simulations, Florence and Spear (1993) used the self-diffusion data for the divalent cations in garnet from Loomis et al. (1985), and assumed, following their suggestion, that DCi = 0.5 DFc. They mentioned that using the Arrhenius relations of Chakraborty and Ganguly (1992), which incorporates the experimental data of Loomis et al. (1985), necessitates raising the peak temperature by 25 °C and slowing down the second stage of cooling by a factor of 2 to produce acceptable match to the observed profiles. The slower cooling rate is permissible by the geochronological data. In the program developed by Florence and Spear (1993), the equilibrium phase assemblage has to be inferred from observational data at a reference point in the P-T space, and subsequent changes of the assemblage along a P-T path need to be deduced from petrographic observation. Thus, their approach relies heavily on the user's petrographic ability instead of the quality of the thermodynamic data. Additionally, they assumed that all garnet crystals nucleate at the same time. Recently Gaidies et al. (2008) developed an alternative software package (THERIA_G) that serves the same purpose but uses an internally consistent thermodynamic data base, bulk composition and diffusion kinetic properties of garnet (Chakraborty and Ganguly 1992) and also incorporates the effect of nucleation history that may be constrained from the observed crystal size frequency distribution or CSD analysis (Cashman and Ferry 1988). Using the diffusion data of Chakraborty and Ganguly (1992), Okudaira (1996) modeled Mn-zoning profiles of garnet crystals of different sizes from the low-pressure/high temperature Ryoke metamorphic rocks in south western Japan. Because the Fe and Mn components of garnet constitute -88% of garnet composition, Okudaira normalized the garnet composition to the Fe-Mn binary, and treated the diffusion problem as a quasibinary process between Fe and Mn. On the basis of observations of spatial distribution of garnet grains and CSD analysis, Okudaira (1996) concluded that garnet grains of 0.1-0.5 mm radius were formed by continuous nucleation and diffusion-controlled growth. The activation energy for intergranular or grain boundary (gb) diffusion was taken to be 83.7 kj/mol, and a value of growth constant of 6.29 x 10~8 m/(s) ,/2 was inferred from observational data. (The growth constant, k2, is inversely proportional to the pre-exponential factor for grain boundary diffusion, D() gb, so that error in

Ganguly

590

I.DC

8864: st + grt + bt + chl 0.50

200

400

600

800

1000

Radius (jim)

Model T-t Paths 620

—1——1

1

1—

1

b

580

P

-

^8866 jT * * 8835b

540

-

500 460

-

10

j— 20

' 30

1 40

1 50

>

60 70

duration (Ma) Figure 19. (a) Simulation of diffusion modified growth zoning profiles in a garnet crystal (#8864) f r o m the Littleton area, N e w H a m p s h i r e , and (b) the assumed T-t path, along with those for t w o other crystals that p r o d u c e d successful simulations of the m e a s u r e d profiles. F r o m F l o r e n c e and Spear (1993).

the value of > . O O !l M £g 1 rS— T3 ir> 'S60 & ja a n i c -S

n O o oo

o o o

» > "p u uSes 3 ' X S. — -SS es ss .ta wo oa y : : S o —• P bjj 01S ) D ° P SS C . u cd cg „^u bij c-3 o .¡S! S c^Oi 5 jg ra > ÇJ í c5 o o ^ o ^ a5- a o « t; a O 3o^ -S 2 M 8 O O J SPS bJJ Já U á - g -O w es ÛJj .. I O a -e a p : m• o a 'O Pt»a c a & o es S O ü 0íT «J B J Ü ö -S 1 8 a " Cp "O .ti bjj O J< a sí ; a cl H _ y3 O h >•> ••31. S o 5 >< X

608

Chakraborty

The transition from intrinsic to TaMED diffusion mechanisms is likely to be marked by a large change in Arrhenius slope (change in activation energies of several 100 kJ/mol), but is unlikely to ever be realized in nature in olivines and its polymorphs because of the high temperature and high degree of purity required for truly intrinsic diffusion. The transition from TaMED to pure extrinsic is a smeared transition with a small change in activation energy (several 10 kJ/mol). The exact temperature range over which this transition occurs depends on the relevant trace element concentrations (Dohmen and Chakraborty 2007). The main conclusion from these analyses is that extrapolation of diffusion data to lower temperatures is unlikely to be affected by large errors. This is seen by the fact that a single equation can be used (see below for Fe-Mg diffusion) to describe diffusion rates over the temperature ranges where pure extrinsic or TaMED mechanisms prevail. It is worth noting two aspects of diffusion by TaMED mechanism in olivine. Generally, vacancies can diffuse rapidly through a crystal and therefore defect concentrations are expected to equilibrate rapidly to ambient P-T-f(h conditions. However, when diffusion occurs by the TaMED mechanism, equilibration of vacancies at low temperatures may be hindered by the overall kinetics of a net transfer reaction such as the one noted above for the creation of a vacancy (involving incorporation of Si0 2 and 0 2 to maintain charge and site balance). Dohmen and Chakraborty (2007) discuss these aspects for diffusion of Fe-Mg in olivine, Costa and Chakraborty (2008) discuss defects in the context of Si and O diffusion. Secondly, diffusion of divalent cations in olivine in the TaMED domain occurs by a cation randomly exchanging positions with an adjacent vacancy; abundance of vacancies is constrained by the thermodynamic criteria described above. Therefore, to calculate diffusion coefficients from point defects, four sets of information are required (a) knowledge of crystal structures (jump lengths), (b) concentration of vacancies, (c) jump frequencies and (d) correlation factors. Of these, the structure of olivine is well known and it is now possible to calculate quantitatively the vacancy concentrations for Fe-Mg olivines from macroscopic models. The last two sets of parameters may be obtained in principle from in situ spectroscopic observations or computer calculations. Although neither tool has delivered quantitative data appropriate for calculation of diffusion coefficients in Fe-bearing olivines yet, both are on the verge of major developments at this point of time and therefore significant advances can be expected in the near future. Given the expected composition dependence of jump frequencies (e.g., Hermeling and Schmalzried 1984, see below the discussion on Fe tracer diffusion), a lot of work remains to be done in this direction. However, at the present time it has been possible to determine empirically a composite parameter that is a product of the jump frequency and correlation factors from one set of experiments, and use it to calculate diffusion coefficients at a number of other conditions (Dohmen and Chakraborty 2007). Costa et al. (2008) provide a step by step tutorial of this approach in their Appendix II. This ability to quantitatively calculate diffusion coefficients from point defect based mechanistic models is an important tool for evaluating the quality and consistency of defect and diffusion data in olivines. Consequently, olivine has become somewhat of a Drosophila of diffusion and defect studies in silicates (Chakraborty 2008). Diffusion of divalent cations Over the long history of study of diffusion in olivine many sets of data have accumulated and not all of these are consistent with each other. Several data compendiums and reviews report all data available until the time of publication of each review (Freer 1981; Brady 1995; Bejina et al. 2003). Availability of improved technology has now allowed the use of multiple approaches (different crystals, different source materials, different analytical techniques for measuring concentration gradients and different schemes of data analysis and modeling of profiles) to measure the same quantity. Moreover, theoretical constraints (relationships between tracer and chemical diffusion coefficients, quantitative point defect models) can now be used to evaluate data. In combination, these have helped to establish a consistent set of diffusion

Diffusion in Olivine, Wadsleytie,

Ringwoodite

609

coefficients for olivine. These aspects are illustrated below in the context of individual data sets, and it is shown that in spite of these advances much work remains to be done in certain areas. Mg and Fe. By far the biggest effort has been spent to measure diffusion of Fe and Mg in olivine crystals. Experiments have been carried out to measure DUg, Z)pe and Are-Mg F e ~Mg bearing olivine. Pure forsterite (Fo100) or mantle olivine (~ Fo 90 ) was used in most studies, while a few experiments were carried out with fayalite or synthetic crystals of other compositions. In addition, grain boundary diffusion of Mg and Si in forsterite has been measured. This is discussed in the chapter by Dohmen and Milke (2010, chapter 21 this volume) and will not be considered here. D*c was measured by Hermeling and Schmalzried (1984) using polycrystalline pellets (Fig. 3). Their measurements were at 1130 °C, at different f0l and for samples across the entire Fe-Mg solid solution series. They found that the diffusion coefficients from these polycrystalline samples corresponded approximately to diffusion rates measured along the è-axis of a single crystal. The diffusion rates were given by l o g o ; = -10.143 + 2.705X + 0.21og/ O2 logD*:* = -11.815 + 2.255X

for 0 . 2 < X < 0 . 9

for 0 . 2 < X < 0 . 8

at

= 10~994

Here X is mole fraction of the fayalite component a n d / 0 , is fugacity of oxygen in bars. represents the self-diffusion coefficient, which could be distinguished from the tracer diffusion coefficient, DTc, in this work. These rates were found to be consistent with Equation (1) when Z>Fc_Mg from Nakamura and Schmalzried (1984) and DUo (reported in the paper as unpublished results of Schnehage from the same laboratory) were used. The data were consistent with

1400 —I—

1200

T[Fc_Mi! is also shown for comparison.

610

Chakraborty

diffusion by a vacancy mechanism where the vacancy concentrations obeyed the point defect model of Nakamura and Schmalzried (1983). DT/DMG was found to be constant across the solid solution series at 3.6. Two other significant results from this study are that (i)

Close to the end member composition of pure fayalite, DTC (and also the vacancy concentrations) increases more strongly with X than the exponential behavior indicated by the equations above.

(ii) Subject to certain assumptions, the correlation factor,/, as well as the jump frequency of the Fe and Mg atoms could be calculated. These are found to decrease rapidly as the Mg content of the olivine increases so that transport of any atom is less efficient in Mg rich olivines. The higher jump frequency of Fe atoms at any given composition of olivine cause them to be transported faster than Mg atoms, although there is hardly any ionic size difference between the two (note, for example, that these two ions mix close to thermodynamically ideally in many silicates). The difference in the jump frequency of the two kinds of atoms leads to strongly correlated motion when Fe and Mg are exchanged. This is one instance where the availability of the correlation factor,/, allows the tracer diffusion coefficients, D . to be distinguished from the self-diffusion coefficients, D \ D*MS in forsterite (Sockel and Hallwig 1977; Sockel et al. 1980; Morioka 1981; Andersson 1987; Andersson et al. 1989; Chakraborty et al. 1994) and in San Carlos olivine (Chakraborty et al. 1994) has been measured in a number of studies (Fig. 3). The study of Sockel and Hallwig (1977) was intended to be a preliminary study to demonstrate the feasibility of measurement of small diffusion coefficients using the ion microprobe. Sockel et al. (1980) provide plots of concentration profiles and summarized Arrhenius relations so that individual diffusion coefficients from their study cannot be evaluated. Morioka (1981) determined D"m„ in forsterite at three temperatures between 1400 and 1300 °C. Andersson etal. (1989) report the results from the thesis of Andersson (1987). The data shown in the thesis indicate that diffusion coefficients could be reproduced within a couple of orders of magnitude in some of the experiments carried out at same conditions. Chakraborty et al. (1994) carried out experiments not only to determine diffusion coefficients at different conditions, but also to evaluate the effects of various possible experimental parameters (e.g., polishing technique, pre-annealing, different heating/ cooling protocols, reproducibility of data from anneals done in different labs) in diffusion measurements. In comparing their results with those of the earlier studies, Chakraborty et al. (1994) found that the diffusion coefficients measured by Morioka (1981) and Sockel and Hallwig (1977) were too high and the results of Andersson (1987) and Anderson et al. (1989) were comparable to those of Chakraborty et al. (1994) although the data of Andersson (1987) and Anderson et al. (1989) scattered over a much larger range and showed less reproducibility. For olivine of composition ~Fo90, the only available data for DMo is that of Chakraborty et al. (1994). Given the preliminary nature of the study of Sockel and Hallwig (1977) and Sockel et al. (1980), the limited temperature range of the study of Morioka (1981), and the general agreement of diffusion coefficients obtained by Andersson (1987) with the results of Chakraborty et al. (1994), it is considered that the dataset of Chakraborty et al. (1994) is currently the most reliable set of values for D"m„ in forsterite as well as Fe-bearing olivine (Fogy). Diffusion rates in forsterite were much slower and activation energies of diffusion were much higher compared to rates in Fe-bearing olivine. The Arrhenius parameters in D = DA e x p ( - Q / R T ) that describe the two diffusion processes along [001] are: Forsterite (with 10-180 ppm Fe): D() = 9.61Q-4 m2/s; Q = 400±60 kJ/mol San Carlos olivine (~ Fo90): A> = 5.6x 10"8 m2/s; Q = 275±25 kJ/mol The anisotropy of diffusion was found to be Z>r0011 > Dnooi > D [m01 . There is no break in slope on an Arrhenius plot between 1000-1300 °C, indicating there is no change of diffusion

Diffusion in Olivine, Wadsleytie,

Ringwoodite

611

mechanism over this temperature range for Mg tracer diffusion. Pressure dependence of this diffusion coefficient in forsterite is weak, with an activation volume of 1-3.5 cm 3 /mol. The mechanism of diffusion in forsterite and Fe-bearing olivine is different. Diffusion in both kinds of materials could be by a vacancy mechanism, but the reaction involved in vacancy formation, its energetics, and the energy of migration could all be different. However, even in the nominally pure forsterite there is some Fe (on the order of 60 ppm) and this is found to be enough to influence its point defect structure. Therefore, diffusion of Mg in "pure forsterite" was found to be dependent on f0l but the dependence was different (weaker) from that of Febearing olivine. As a result, by controlling the oxygen fugacity it is possible to control the defect structure of Fe-bearing olivine much better than the defect structure of "pure forsterite," where the small amount of Fe may also be heterogeneously distributed. In addition, other impurities may also control the diffusion behavior (see discussion above on point defects). This is evident in the larger scatter of the measured diffusion coefficients in "pure forsterite" compared to those in San Carlos olivine even when the experiments were carried out simultaneously using the same protocol. The result is the larger uncertainty in the activation energy that is determined: ± 60 kj/mol for forsterite vs. ± 25 kj/mol for San Carlos olivine. The observation o f / 0 , dependence of diffusivity in nominally pure forsterite has major implications for comparing experimental diffusion data to results of computer simulations. Calculations carried out on pure Mg bearing olivines cannot be compared to any set of measured data, because truly Fe-free olivines have not been studied in diffusion experiments. The small amounts of Fe present in nominally "Fe-free" olivines appear to control the diffusion behavior. This applies to all other results on "pure forsterite" crystals described later. Analysis of the data led Chakraborty et al. (1994) to conclude that the observed diffusion process was by an extrinsic rather than an intrinsic diffusion mechanism. In particular, the observations with forsterite containing only very minor amounts of Fe pointed to the fact that truly intrinsic diffusion would be observed only at very high temperatures and in crystals of an extreme degree of purity—situations that are unlikely to be commonly encountered in natural minerals. A e-Mg has been measured at atmospheric pressure by Buening and Buseck (1973), Misener (1974), Nakamura and Schmalzried (1984), Jurewicz and Watson (1988), Chakraborty (1997) and Dohmen et al. (2007). Dohmen and Chakraborty (2007) considered the data in terms of a comprehensive point defect model and discussed many of the experimental aspects of these studies. Figure 4 summarizes results from all of these studies. Two very detailed studies concerned with measurement of D l e . M g in olivine appeared in the literature (Buening and Buseck 1973; Misener 1974) practically simultaneously. Both evaluated diffusion coefficients as a function of composition and documented anisotropy of diffusion. Buening and Buseck (1973) discovered that the diffusion process depended on oxygen fugacity. They analyzed the results in terms of diffusion mechanisms to reach the important conclusion that diffusion in olivine probably occurred by a vacancy mechanism. Misener (1974) carried out experiments at high pressures as well and obtained the first estimates of activation volume of diffusion in a silicate mineral. The experimental setups were different in the two studies. I will discuss the essential features of their experiments in view of the long lasting discussion that the two studies initiated. Buening and Buseck (1973) embedded oriented, polished single crystals of olivine from San Carlos, Arizona in a compacted powdered matrix of pure fayalite composition and annealed these under controlled oxygen fugacity at atmospheric pressure. After the experiments, they measured concentration profiles in the single crystal part only and analyzed the data in terms of compositionally dependent diffusion. Misener (1974) used olivines of similar composition from the St. John Island in Red Sea (an area historically well known for producing many Peridot gems) and had several different experimental setups. In the first of these, he used two

612

Chakraborty

1400

ire] 1000

1200

800 1: 2: 3: 4: 5: 6: 7: 8:

Dohmen & Chakraborty 2007 (Fo90) Chakraborty 1997 Nakamura & Schmalzried 1984 Buening & Buseck 1973 Misener 1974 (Fo93) Misener 1974 (Fo80) Jurewicz & Watson 1989 (Fe) Spandler & O'Neill 2009

1 -20

6

7

8 10 4 /T[1/K]

9

10

Figure 4. Arrhenius plot showing best fits to F e - M g diffusion coefficients [Dfe-Mg] in olivine from different studies carried out at 1 atmosphere total pressure. Data sources are indicated in the legend, along with compositions of olivine. Please note that comparison of results f r o m different studies may additionally require taking into account effects of different compositions and oxygen fugacities at which the data were obtained.

single crystals of different compositions (the second one being one of nearly pure fayalite from Rockport, Mass., U.S.A.) in contact with each other, wrapped in a metal foil (Ag-Pd alloy or Pt) tied with Pt wires to ensure contact between the two. In the second setup, the more Mg rich crystals were placed in contact with MgO disks (pellets of pressed MgO powder) in Pt capsules and annealed at higher temperatures. All of these diffusion couples were placed in evacuated silica glass tubes to prevent oxidation during diffusion anneals. High pressure experiments were carried out in solid media piston cylinder apparatus using olivine couples encased in graphite capsules (expected to control the ambient/o 2 ). Misener (1974) carried out experiments to test whether there was any difference in diffusion rates obtained from crystalcrystal couples vs. crystal-powder experiments. The results indicated that the results from the two kinds of experiments were comparable. The compositional dependence (diffusion rates increase with Fe content) and anisotropy (diffusion fastest along the [001] direction) found in the two studies were similar. But beyond that, there were important differences in the absolute values of diffusion coefficients measured in the two studies. More significantly, Buening and Buseck (1973) found a break in the Arrhenius slope (i.e., slope of the best fit line to data when the logarithm of diffusion coefficient is plotted against inverse temperature, 1/7) at about 1125 °C—diffusion coefficients along all three crystallographic directions plotted with a high slope (= activation energy, ca. 240 kj/mol) at T > 1125 °C and a lower slope (activation energy of ca. 125 kJ/mol) at temperatures below. Buening and Buseck (1973) themselves provided two alternative explanations for this observation—either a change of diffusion mechanism from intrinsic to extrinsic, or a change of diffusion from volume

Diffusion in Olivine, Wadsleytie,

Ringwoodite

613

diffusion control to grain boundary diffusion control—and left the issue open. But because they had measured concentration profiles only in the single crystal part of their diffusion couples, and perhaps because Misener (1974) had shown that there was little difference in diffusion rates obtained from crystal-crystal vs. crystal-powder diffusion couples, in the subsequent literature this break in slope was widely interpreted as an example of change of diffusion mechanism in a mineralogical system. Some early measurements of D"Mg (Sockel and Hallwig 1977; Morioka 1981 - see above for discussion) and Co-Mg diffusion (Morioka 1980) were combined to further bolster this interpretation. It ultimately gained the status of demonstrated fact and found its place in influential review articles on diffusion in minerals (e.g., Lasaga 1981) and mineralogy textbooks (e.g., Putnis 1992). This profoundly influenced the application of diffusion data in the Earth sciences (Can diffusion coefficients measured at high temperatures be extrapolated to model geological processes that occur at lower temperatures, or does a change of Arrhenius slope due to change of mechanism intervene?). The situation was actually even more complicated when the two data sets are considered together. First, data obtained by Misener (1974) at temperatures below 1125 °C indicated the activation energy of diffusion to be higher, comparable to the ~ 240 kj/mol found by Buening and Buseck (1973) at higher temperatures. Indeed, the higher temperature data of Buening and Buseck (1973) could be almost smoothly extrapolated to the lower temperature data of Misener (1974) without any break in slope. In numerous subsequent studies that made use of diffusion data in olivine, there were discussions of substantial length as to which of these two data sets were more reliable. Secondly, Misener (1974) had found two sets of diffusion coefficients as a function of composition—one for the compositional range of 0.1 < X Mg < 0.8, and another, slower, set of diffusion coefficients for more magnesian olivines (e.g., X Mg = 0.9, corresponding to typical mantle rocks). An equation for calculation of activation energy of the first set of diffusion coefficients was provided in the abstract of the paper, and this was used indiscriminately in the subsequent literature for calculation of diffusion coefficients of olivines of all compositions, including mantle olivines. This was the diffusion coefficient that was used largely in the discussion comparing the studies of Buening and Buseck (1973) and Misener (1974) as well. The discussion on the quality of data of the two studies was not straightforward because each study had its strengths and weaknesses. Buening and Buseck (1973) had controlled oxygen fugacity in their experiments, but they had used diffusion couples of single crystals with powders, and only measured concentration gradients in the single crystal part of the couple. They assumed the powder matrix to have behaved as an infinite medium. Misener (1974) had used single crystals and measured concentration profiles on both sides of the diffusion couple, but the oxygen fugacity in his experiments were not explicitly controlled. There was also widespread use of Pt in his experiments which could have led to Fe loss or at least, modification of the point defect structure of the olivines. Because his experiments were carried out in silica capsules with fayalite as part of the diffusion couple, some later workers interpreted his diffusion data to be valid for oxygen fugacities corresponding to the quartz-fayalite-magnetite buffer. There were even attempts to reconcile the two datasets and come up with new expressions describing diffusion rates in olivine where the f0l dependence was taken f r o m the study of Buening and Buseck (1973) and the pressure dependence from the study of Misener (1974), assuming his data were measured at the Q F M buffer (e.g., see Weinbruch et al. 1993). In spite of all these attempts, the fact remained that below 1125 °C there was an unresolved discrepancy between the data sets of Buening and Buseck (1973) and Misener (1974). Even at temperatures above this, there was considerable difference between the results from the two studies, the exact difference depending on exactly which data sets were compared at which composition. A third study on diffusion of Fe-Mg in olivines appeared in the material science literature (Nakamura and Schmalzried 1984). These authors measured diffusion coefficients between

614

Chakraborty

1050-1280 °C at oxygen fugacities between 10~8 and 10~"-2 bars using olivines with XFc between 0.9 and 0.1. They used synthetic polycrystalline pellets as diffusion couples, but they argued that, because of the coarse grain size (30-50 ^m) of the pellets, the contribution from grain boundary transport was minimal, and the diffusion they observed was essentially volume diffusion. Obviously, anisotropy of diffusion could not be resolved and the diffusion coefficient that was obtained was an average of diffusion rates along the different crystallographic directions. Not surprisingly then, these rates were somewhat slower than those obtained by Buening and Buseck (1973) and Misener (1974) in some of their experiments along the [001] direction. However, the activation energy (200 kj/mol) did not correspond to that obtained from either of the previous studies. It was intermediate between the high and the low temperature values of Buening and Buseck (1973). Thus, this study did not resolve the issue about which diffusion coefficients to use for practical modeling purposes. However, these data did provide the basis for a very successful point defect thermodynamic model for olivines. Indeed, that work (Nakamura and Schmalzried 1983) laid out the formalism of how point defects in silicates can be modeled. But in spite of these advances and the wide use of Fe-Mg diffusion coefficients in olivine for various applications, there remained considerable uncertainty as to the value of diffusion coefficients at different temperatures and compositions. The next set of measurements came from Jurewicz and Watson (1988). They placed olivine crystals in contact (by wrapping with Fe-doped Pt wire) with melt that was in equilibrium with the olivine to study the diffusion of a number of major and trace elements over a restricted range of temperature. They fitted the concentration profiles of Fe and Mg separately to obtain different diffusion coefficients for the two elements. The difference in profile lengths of Fe and Mg may have resulted from significant concentrations of the minor elements accounting for charge balance, in which case the problem was one of multicomponent diffusion. Because of the nature of the experiments involving melts, it was not possible to carry out the experiments over a large temperature range and the question of change of diffusion mechanism and Buening and Buseck (1973) vs. Misener (1974) remained open after this study. Recently, Spandler and O'Neill (2009) have determined Fe-Mg diffusion rates in olivine coexisting with melt and found rates identical to those of lurewicz and Watson (1988). Chakraborty (1997) set up a set of experiments to specifically test the controversy related to the Buening and Buseck (1973) and Misener (1974) data sets. He used single crystal diffusion couples held together in a container-less set up by spring-loaded alumina rods. The crystals were annealed under controlled oxygen fugacity for long enough durations that diffusion profiles obtained could be analyzed using an electron microprobe without further corrections for convolution or other effects. The diffusion profiles were analyzed in terms of a compositionally dependent diffusion process. Notably, two different kinds of diffusion couples were used. These required slightly different methods for data analysis, but both spanned the important compositional range of mantle olivines—Fo88-Fo92. The rationale was that if all relevant parameters were properly controlled, then both of these couples should yield the same diffusion coefficients for the same compositions. This was found to be the case and therefore it could be further demonstrated that once the relevant set of parameters were controlled (P, T,f0l, X12 etc.), then the data are likely to be more robust. The surprising result from this study was that the diffusion coefficients obtained here were slower than the data of both Buening and Buseck (1973) and Misener (1974) for the Ferich olivines (XMg üO) o > o ¿i' °< O O)o g o 5 CN f ^ o ó E cu o o

o _

í; cj> T- n

£ 0 1- §

a 1 S •a '> oí a u S^ 8ë OÏ » SS c «•oí S 2 ü 5 SB < U£ "Os S s l C 0)1 i-S •i n s s s S o f ! -a tS0 Ä o rt m)1 — I < r in a O C3 a a) •o • =s c SÜ > o E60 C 5 — O ci aj " c^ C D ^ oÖ O ° us S c "O co co o ¡2 u S o o. -o pI S S J O 60 O C a o se

CN^

Fe-Mg ••--.(10%Fe,-12)

.m=0.15

E, Q-20

m- 0.17

O)

'""••-.

o -21

-

-22

-

Cr 2.6%Fe

Orthopyroxenes C-ax¡s

Mg (11%Fe,IW)

pO = 10"12 MPa -23 7

8

10

10 000/7(K) Figures l a and lb. See caption on facing

page.

648

Cherniak & Dimanov

1300

1250

1200

i

-19-

Ca 1 -lntr \(2%Fe, EPM) (c,b) ,

1150

1100

1050

1000

950^

r

v /

oT-20CN £

Ca'-Trans :%Fe,EPM) (c,b) Fe. (l,8%Fe) (b,c)

/

Ca-lntr (0,4%Fe,Syn)>. (b)

Q D) O

Fe

(0%Fe,Syn)

(a)

-21 •

Clinopyroxenes

(2%Fe) (c,b)

a-axis -22-

b-axis

Ca-Extr (0,4%Fe,Syn) (b)

c-axis pO - 10 1 2 MPa —I— 6.5

—I 7.0

1

1— 7.5

3.0

Ca (2.6%Fe,IW

B.w5

10 000/T(K) 1100

1000

900

800^

10 000/7~(K) F i g u r e s l c a n d I d . See caption

on previous

page.

75CC

Diffusion in Pyroxene, Mica, Amphibole

649

log p02 (MPa) Figure 2. Oxygen fugacity dependence of diffusion coefficients for major cations in iron-bearing pyroxenes. Data are reported for T = 1100 °C. On the top axis are the p02 values imposed by the major solid buffers QFI, IW, WM, QFM and M H at 1100 °C. Between p02 = 1 0 " l f l M P a a n d p 0 2 = 10"8 MPa ironbearing clinopyroxene (2 at% Fe) is in its stability field—i.e., in equilibrium with oxygen—which fixes the Fe 2+ /Fe 3+ ratio and point defect populations (see text for additional discussion). Beyond the stability field limits, the clinopyroxene presents EPM (early partial melting, which is the exsolution of silica microdroplets, see text for further information), which prevents the oversaturation of point defects. Within the stability field, diffusion in diopside (Cpx) of Fe (Azough and Freer 2000) and (Fe,Mn)-Mg interdiffusion (Dimanov and Wiedenbeck 2006) exhibit a positive dependence on p 0 2 , indicating vacancy mechanisms. In contrast, diffusion of Ca in diopside (Dimanov et al. 1996) and of Cr in enstatite (Ganguly et al. 2007) exhibit a negative dependence on p02 in the stability field, indicating interstitial mechanisms. Beyond the stability field limits p02 dependencies vanish. Mg diffusion in diopside (Gasc et al. 2006) is also reported, and the data correspond to Fe diffusion after extrapolation to higher oxygen fugacity. In enstatite (Opx), interdiffusion of Fe-Mg (ter Heege et al. 2006) shows a positive p0 2 -dependence and the data correspond closely to the data of Ganguly and Tazzoli (1994) when extrapolated to higher temperatures.

defects. This interpretation was supported by the so called premelting phenomenon, which was observed during calorimetric investigations of diopside and other silicates (Richet and Fiquet 1991; Richet et al. 1994), and which manifests itself by an anomalous increase of heat capacity when approaching the melting point. Premelting is usually attributed to intrinsic (thermal) point defects, cation disordering, phase transitions, or sublattice melting (Ubbelohde 1978). Dimanov and Ingrin (1995) showed that the enthalpy excess associated with the anomalous increase of heat capacity may be compared to the formation enthalpy of Ca Frenkel defects and related to the observed enhancement of Ca self-diffusion above ~ 1240 °C. Hence, the authors attributed diopside premelting to a sort of Ca disorder among M2 and interstitial sites. The authors calculated that close to the melting point, the Ca-Frenkel defect concentrations may be as high as a few mole percent, which in turn is the basis for sub-lattice melting. Their suggestion that dominant Ca Frenkel type defects are responsible for the intrinsic diffusion

650

Cherniak & Dimanov

regime (or premelting regime) was later corroborated by Pacaud et al. (1999) and Ingrin et al. (2001), who measured oxygen self-diffusion in the same synthetic Fe-free diopside and did not detect any diffusional enhancement in the premelting regime, thus precluding Schottky type defects as dominant. Dimanov et al. (1996) used natural iron-bearing diopside single crystals with different iron contents and measured 44Ca diffusion along the c- and b-axes. Specimens were annealed between 1000 °C and 1250 °C, under controlled oxygen fugacity, between 0.021 MPa (air) and 10~18 MPa. Diffusion profiles were analyzed by RBS. As observed earlier for Sr diffusion by Sneeringer et al. (1984), Ca diffusion was found to be isotropic over the investigated temperature range and to be faster in natural than in synthetic crystals (Fig. lc). For iron contents Xi;e between 0.004 and 0.025, Ca diffusion rates were found to depend on Fe content, described by the relationship (A-^)0-24. The authors reported two distinct diffusion regimes, with a transition temperature (Tc) ranging between 1100 °C and 1150 °C, depending o n p 0 2 . (Figs, la, lc, 2) The authors noticed that above Tc, diopside exhibits the early partial melting (EPM, see below) phenomenon, the precipitation of small (< ~ 100 nm) silica droplets within the diopside crystal, which occurs in response to over-saturation of extrinsic cation vacancies related to the Fe oxidation state, and hence to p02 (Raterron and Jaoul 1991; Jaoul and Raterron 1994). In the EPM regime, Ca selfdiffusion is characterized by an activation energy of E.d = 396±38 kJ/mol, and is independent of p02. Below Tc, in the EPM-free regime, Ca self-diffusion has a lower activation energy, £ a = 264 ±33 kJ/mol, and varies as (p02)AU4 between p02 = 10~17-10~7 MPa. Based on their observations and the point defect model (see below) refined by Jaoul and Raterron (1994), Dimanov et al. (1996) proposed an interstitial mechanism for Ca self-diffusion. As noted later by several authors (Cherniak 1998a; Zhang et al. 2010), the data of Dimanov et al. (1996) do not strongly appear to support a change of diffusion regime; given the experimental scatter the entire dataset may also be fitted by a single linear relation rather than two distinct Arrhenius lines. However, Dimanov et al. (1996) could discriminate between the two distinct regimes on the basis of a statistically sound data set (77 independent experimental data points), analyzed with two different data inversion procedures. One of the inversions was based on the Bayesian probabilistic approach (Weber and Taupin 1995; Taupin 1998), while the other was a classical least-squares non-linear inversion (Sotin and Poirrier 1984). Both procedures could provide the complete fitting parameter set (Da, m, E¡), the corresponding errors, and statistical analysis of the residuals, in order to evaluate the adequacy of the fitting model. In particular, the presence of two diffusion regimes was indicated by a bi-modal distribution of residuals when performing a least-squares fit based on a single diffusion regime. In addition to the purely statistical aspect, the authors also considered that the observed change of p02 dependence justifies data analysis in terms of two different regimes, because this implicitly indicates either a change of diffusion mechanism, or at least a different dominant reaction for formation of point defects. Preliminary work by Stimpfl et al. (2003) and the more recent and extensive study of Zhang et al. (2010) reported results for Ca self-diffusion in diopside. Both studies were performed in the same laboratory and are based on similar experimental procedures and analytical techniques, but given the greater scope of the investigation of Zhang et al. (2010), this study supersedes the findings of Stimpfl et al. (2003) (J. Ganguly, personal communication). 44Ca enriched CaO thin films were deposited by thermal evaporation onto oriented single crystals containing ~ 2.5 at% Fe. The diffusion experiments were performed between 950-1150 °C, in a dry environment, with C 0 / C 0 2 gas mixtures adjusted to buffer the system at an oxygen fugacity equivalent to the IW buffer. Zhang et al. (2010) reported a lower activation energy (£ a = 264 ±23 kJ/mol) and faster diffusion rates for diffusion along the c-axis, while diffusivities along the a*- and b-axes were comparable (Fig. lc). Diffusion along the c-axis was about half an order of magnitude faster at 950 °C, but due to the difference in activation energies for the different orientations the diffusion anisotropy vanishes above 1050 °C, which is in good agreement with the observations of Dimanov et al. (1996) and Dimanov and Jaoul (1998), who reported isotropic diffusion along

Diffusion in Pyroxene, Mica, Amphibole

651

the c- and ¿-axes above 1000 °C. However, the diffusion rates reported by Zhang et al. (2010) are about 1.5 orders of magnitude faster than those reported by Dimanov et al. (1996) and Dimanov and Jaoul (1998). Zhang et al. (2010) highlighted the discrepancies between their results and those of Dimanov et al. (1996), but noted the relatively good agreement between their results and the high pressure Ca-(Mg,Fe) interdiffusion data of Brady and McCallister (1983), after providing corrections to compare with findings of the latter study to account for the effects of pressure and non-ideal mixing. However, the corrections may be called into question because they were performed with different activation volumes (varying between 4-7 cm 3 /mol, Zhang et al. 2010), and because they do not account for the differences in oxygen fugacity among the studies. Indeed, the high temperature (1100-1200 °C) and pressure (2.5 GPa) data of Brady and McCallister (1983), might correspond to lower oxygen fugacity than the IW conditions of Zhang et al. (2010); experiments run under graphite-buffered conditions in piston-cylinder apparatus have been recognized to differ substantially from what would be expected for a pure C - C O buffer (Holloway et al. 1992; Hirschmann et al. 2008; Medard et al. 2008). For example, Medard et al. (2008) report that the potential range of conditions varies from C - C O to four orders of magnitude lower. Similarly, Zhang et al. (2010) do not adequately compare their results with those from Dimanov et al. (1996) and those for Fe diffusion from Azough and Freer (2000), because the data of Zhang etal. (2010) correspond to an oxygen fugacity that varies with temperature (between ~ 10~ l7 -10~ 13 MPa, i.e., along the IW buffer), while the data of Dimanov et al. (1996) and of Azough and Freer (2000) correspond to fixed oxygen fugacities (of 10~12 MPa and 10 11 MPa, respectively). Both Dimanov et al. (1996) and Azough and Freer (2000) demonstrated marked effects of oxygen fugacity, so corrections to account for differences in p02 are critical for direct comparison. In general, diffusion data obtained at oxygen fugacity conditions which vary with temperature (such as with a solid buffer) present only an apparent activation energy, because the variation of point defect populations, and hence of diffusion rates, are not solely determined by temperature. However in natural systems, for example the cooling of a self-buffered natural assemblage, oxygen fugacity will vary, and correspondingly the populations of extrinsic point defects will vary so diffusion rates are affected as well. Hence for geological applications it may be convenient and useful to present diffusion data under conditions that correspond to solid buffers. But it is important to recognize that an apparent activation energy determined under such conditions corresponds to behavior under the given buffer and does not represent the thermal activation of the diffusion process itself. The latter can only be determined at fixed p02, when populations of extrinsic point defects are independent of temperature. Therefore, comparison of different data sets might only be fairly performed at comparable thermodynamic conditions, either along the same oxygen fugacity buffer, or at a fixed oxygen fugacity. This type of comparison is illustrated in Figure la, where all diffusion data for major cations in pyroxenes were recalculated to p02 = 10~12 MPa, for cases where dependence on oxygen fugacity was investigated and quantified. For instance, since Zhang et al. (2010) did not report EPM, we corrected their data according to the p02 exponent m = - 3 / 1 6 , which is the theoretical value calculated by Jaoul and Raterron (1994) in the EPM-free regime, and is also close to the experimental values (m = - 0 . 1 8 , Dimanov et al. 1996; m = - 0 . 1 9 , Dimanov and Jaoul 1998). The data of Zhang et al. (2010) fall in much better agreement with the previously published data of Dimanov et al. (1996) after this correction (Fig. 1). However, because the samples and the experimental techniques employed in these studies are different, it is not surprising that some differences may remain. Dimanov and Jaoul (1998) further investigated Ca self-diffusion in natural iron-bearing diopside under controlled oxygen fugacity, and at temperatures up to 1320 °C (i.e., 30° below the nominal melting point). As in the work of Dimanov and Ingrin (1995), they observed two distinct diffusion regimes, characterized by activation energies of 284 ± lOkJ/mol and 1006 ± 7 5 kJ/mol below and above T = 1230± 15 °C, respectively (Fig. la, lc). In the lower temperature regime, Ca self-diffusion was proportional to (/>O 2 )~ a l 9 ± a 0 3 . Following their previous studies (Dimanov

652

Cherniak & Dimanov

and Ingrin 1995; Dimanov et al. 1996), the lower temperature, p02 dependent diffusion regime was interpreted as an extrinsic one, while the higher temperature regime was interpreted in terms of intrinsic diffusion and premelting conditions, as was previously proposed for synthetic Fe-poor diopside (Dimanov and Ingrin 1995). On the basis of their work, Dimanov and Jaoul (1998) alternatively suggested that the p02 independent diffusion regime, characterized by an intermediate activation energy = 396 ±38 kJ/mol between ~ 1150 °C and 1250 °C (Dimanov et al. 1996) may actually represent a transitional mixed mode between extrinsic and intrinsic regimes (Fig. lc). Based on the crystal structure and diffusional isotropy the authors also suggested that in both regimes Ca self-diffusion might involve the well-known M3 (distorted octahedron) interstitial sites, located in the octahedral layers, but also new octahedral interstitial sites that the authors identified within the tetrahedral layers and termed M4. In summary: i) when point defect concentrations are determined by the Fe oxidation state, Ca extrinsic self-diffusion operates by an interstitial mechanism, in which the activation energy does not depend on low Fe concentrations (< 2.5 at%); ii) when point defect concentrations are essentially thermally activated, Ca-Frenkel type defects are dominant, and hence Ca intrinsic self-diffusion still operates by an interstitial mechanism involving M3 and newly recognized M4 interstitial sites; iii) owing to the interstitial mechanism, Ca diffusion is isotropic at temperatures above 1000 °C, but could become anisotropic at lower temperatures, due to a somewhat lower activation energy for diffusion along the c-axis; v) Ca is the slowest-diffusing octahedral cation and is therefore a kinetically limiting species. Iron. Azough and Freer (2000) measured 54 Fe tracer diffusion along the b and c-axis directions in synthetic Fe-free single crystals of diopside and natural diopside containing 1.8 at% Fe. Diffusion couples were produced by deposition of thick films (0.1-0.2 |im) of Fe citrate gel, obtained by mixing and heating 54 Fe enriched iron chloride and citric acid, on oriented, polished and chemically cleaned diopside samples. Subsequent annealing for 2 hours at 1050 °C in a reducing atmosphere resulted in crystallization of the gel, ensuring good interfacial contact, evidenced by the significant enrichment in Ca, Mg and Si. Diffusion couples were placed in Pt crucibles, with diffusion experiments performed at room pressure at temperatures between 950-1100 °C, with oxygen fugacity controlled by flowing H 2 /C0 2 gas mixtures. An activation energy for diffusion was determined at a fixed p02 = 10~14 MPa, and the dependence of Fe diffusion on oxygen fugacity was determined at 1050 °C between p02 = 10_11 MPa and 10~17 MPa. After the longer experimental anneals the composition of the deposited surface layer approached that of diopside. Diffusion profiles, extending over ~ 0.1-0.6 |im, were acquired by SIMS. The data were analyzed with different analytical solutions to the diffusion equation, with thin, thick or infinite surface source boundary conditions, depending on the comparison between diffusion lengths and thicknesses of surface films; film thickness was determined with SIMS prior to diffusion anneals. The different solutions resulted in diffusion coefficients spanning an order of magnitude, yielding pre-exponential factors within a factor of ~ 5 and similar activation energies. Their recommended Fe diffusion coefficient was taken as the mean of the two most reliable analytical treatments. As in the case of Ca (Dimanov et al. 1996), Fe diffusion rates in the natural diopside were found to be effectively the same parallel to the c and b axes (Fig. lc), and to depend on p02 (Fig. 2). Between p02 = 10~u-10~17 MPa diffusion of Fe in the natural (iron-bearing) diopside crystals at 1050 °C was determined to be proportional to (pO2) a229±a036 . Based on this observation and on the previous work of Jaoul and Raterron (1994) and Dimanov et al. (1996) the authors proposed a vacancy mechanism for Fe diffusion in diopside. From results of atomistic simulations of vacancy formation in diopside (Azough et al. 1998) the authors inferred that the low activation energy (Ea = 162 ± 35 kJ/mol obtained at p02 = 10~14 MPa) was representative of the migration energy for vacancy related extrinsic diffusion. Fe diffusion at 1050 °C in the iron-free synthetic samples was approximately one order of magnitude slower than in the natural crystals (Fig. lc).

Diffusion in Pyroxene, Mica, Amphibole

653

Magnesium self-diffusion. Mg self-diffusion in diopside at ambient pressure was measured with NRA by the group of Olivier Jaoul (Pacaud 1999; Gasc et al. 2006). In these studies 26 Mg-enriched synthetic diopside was deposited as a thin film (~ 50 nm) by RF sputtering onto polished and chemically cleaned iron-bearing (~ 3.6 at% Fe) natural diopside and the iron-poor synthetic diopside single crystals used by Dimanov and Ingrin (1995). Natural samples oriented for diffusion along the a*- and c-axes were annealed at 7 = 1100 °C. Synthetic samples, oriented for diffusion along the a-, b- and c-axes, were annealed at T = 1200 - 1300 °C. An oxygen fugacity p02 = 10~6 MPa was controlled by flowing Ar. Diffusion profiles were analyzed by the nuclear reaction 26Mg(p,y)27Al; diffusion profiles extended up to 150 nm. Consistent with the findings for diffusion of Ca (Dimanov et al. 1996, Dimanov and Jaoul 1998) and Fe (Azough and Freer 2000), no obvious anisotropy was observed for the synthetic samples at the investigated temperatures. The limited data for natural diopside also suggest little anisotropy (Fig. lc). The Mg diffusion coefficients in the synthetic diopside samples (£>Me = 7.8 xlO- 1 9 - 4.2xlO- 1 8 m 2 /s at T = 1300 °C and DM& = 3.6xlO" 1 9 - 4.2x KM 9 m 2 /s at T= 1200 °C) and in natural samples (Du& = 6.03 x lO"20 - 1.4 x 10"19 m 2 /s at T = 1100 °C) proved to be substantially higher (about two orders of magnitude) than those for calcium in the same materials (Dimanov and Ingrin 1995; Dimanov et al. 1996; Dimanov and Jaoul 1998). As previously observed for calcium (Dimanov et al. 1996) and for iron (Azough and Freer 2000), Mg is significantly faster in natural iron-bearing diopside than in synthetic iron-free or iron-poor material. Combining the limited data sets of Pacaud (1999) and Gasc et al. (2006) one can obtain an activation energy of E.d = 748 ±87 kj/mol for Mg self-diffusion in synthetic diopside between T = 1200-1300 °C. This value is much higher than proposed by Azough and Freer (2000) for extrinsic vacancy-mediated diffusion, but it is consistent with the findings of Dimanov and Ingrin (1996) for Ca self-diffusion in the intrinsic regime (E.d = 950 kj/mol, above T ~ 1230 °C), where Ca-Frenkel defects dominate. The combined results of Pacaud (1999) and Gasc et al. (2006) suggest that the Mg sublattice may also be affected by the premelting regime. In an abstract in 2003, Stimpfl et al. reported preliminary results of Mg self-diffusion in diopside, measuring isotopic exchange between diopside and 26 Mg-enriched MgO thin films, with the films deposited by thermal evaporation onto oriented diopside single crystals. Diffusion experiments were performed at 0.1 MPa under conditions near the IW buffer, and diffusion profiles were measured by SIMS. This preliminary work has been superseded by a more comprehensive study of Mg diffusion (Zhang et al. 2010) from the same laboratory. Zhang et al. (2010) measured diffusion along the three crystallographic axes, between 9501150 °C at 0.1 MPa, and with p02 near the IW buffer, controlled using C 0 - C 0 2 gas mixtures. As in the case of their Ca diffusion results, Zhang et al. (2010) observed that Mg diffusion along c-axis was somewhat faster than along a*- and b-axes (Fig. lc). The activation energy for Mg self-diffusion along the three crystallographic axes was found to vary between 150 ±22 kJ/mol and 231 ±23 kJ/mol. These values correspond reasonably well to those measured by Azough and Freer (2000) for Fe self-diffusion (£ a = 162 ±35 kJ/mol) and their estimates for Mg self-diffusion (£ a = 214 kJ/mol) based on their previously published atomistic simulations of vacancy mediated mechanisms (Azough et al. 1998). In Figures 1 a and 1 c, it can be seen that the Mg diffusion data of Gasc et al. (2006) obtained at 1100 °C are within half an order of magnitude of the recent Mg diffusion data of Zhang et al. (2010) when extrapolated down to their temperature range (900-1000 °C) with the activation energy (214 kJ/mol) proposed by Azough and Freer (2000). However, this agreement may be fortuitous since the studies were not performed under the same conditions of oxygen fugacity. The numerical and experimental findings of Azough et al. (1998) and Azough and Freer (2000) indicate that vacancy related mechanisms might be favorable for both Fe and Mg diffusion. For these mechanisms the hypothetical oxygen fugacity exponent of the Mg diffusion coefficient should be positive, as it is in the case of Fe diffusion. If the differences in oxygen fugacities

654

Cherniak & Dimanov

among data sets are considered in order to directly compare the data at an intermediate p02, the diffusivities of Gasc et al. (2006) obtained at high oxygen fugacity (p02 = 10~6 MPa) should be lowered, while those of Zhang et al. (2010) obtained at low oxygen fugacity (IW conditions) should be raised. Unfortunately, since the dependence on oxygen fugacity of Mg diffusion in diopside was not investigated in any of the extant studies, it is difficult to quantify the appropriate corrections. (Fe,Mn)-Mg and Fe-Mg interdiffusion. Dimanov and Sautter (2000) and Dimanov and Wiedenbeck (2006) investigated (Fe,Mn)-Mg interdiffusion along the c axis in natural diopside single crystals (A"Fc ~ 0.03). In both studies the diffusion couples were prepared by RF-sputtering of ferro-johannsenite (Mn-rich hedenbergite) thin (< 50 nm) films onto polished and chemically cleaned oriented iron-bearing diopside samples. Iron and manganese surface films contained Fe/(Fe + Mn + Mg) ~ 0.48. Diffusion couples were placed in sintered diopside crucibles and annealing was performed at ambient pressure, temperatures between 900-1240 °C, and at low oxygen fugacities controlled between p02 = 10~7-10~19 MPa by either C 0 / C 0 2 (Dimanov and Wiedenbeck 2006), or Ar-H 2 /Ar-H 2 0 (Dimanov and Sautter 2000) gas mixtures. Dimanov and Sautter (2000) determined the diffusion profiles of Fe and Mn (ranging in length from ~ 30 to 350 nm) by both RBS and SIMS, while Dimanov and Wiedenbeck (2006) used only SIMS. Fe and Mn from the surface layer exchanged with Mg from the diopside substrate. As M2 sites were saturated with calcium in both the substrate and the surface layer, the exchange involved essentially only the Ml sublattice. During this pseudo-binary (Fe,Mn)-Mg interdiffusion process, Fe and Mn diffused at similar rates. Diffusion profiles were successfully evaluated with concentration-independent diffusion coefficients and a thin source solution to the diffusion equation. However, in all experiments Dimanov and Sautter (2000) used the same gas mixture to set oxygen fugacity, resulting in temperature-dependent oxygen fugacity (see previous sections). They noted that if the diffusion mechanism is vacancy-related the reported activation energy (406 ±64 kl/mol) might be overestimated. Dimanov and Wiedenbeck (2006) continued the study of Dimanov and Sautter (2000), focusing on the effects of oxygen fugacity. They investigated both EPM-present and EPM-free regimes, at temperatures between 1000-1200 °C and/?0 2 = 10~7-10~19 MPa. As shown in Figure 2, at both high and extremely low oxygen fugacities EPM was observed and the interdiffusion process was p02 independent. In an intermediate regime, where diopside is in equilibrium with respect to point defect concentrations and is EPM-free, the interdiffusion coefficient depends on p02 with an exponent m = 0.22 ±0.02, which is very similar to that reported by Azough and Freer (2000) for Fe self-diffusion. Dimanov and Wiedenbeck (2006) found that in the EPMfree regime, at fixed p02 = 10~14 MPa, the activation energy is E.d = 297±31 kJ/mol, which is substantially lower than the previously reported value (Dimanov and Sautter 2000). However, in Figure la it can be seen that the data of Dimanov and Sautter (2000) and Dimanov and Wiedenbeck (2006) are in reasonably good agreement both in terms of magnitude of diffusivities and activation energy after p02 corrections are made to the former dataset. Both datasets still differ by about half an order of magnitude, but this moderate discrepancy may result from the different analytical techniques (e.g., SIMS versus RBS). Some difference may also result from the fact that Dimanov and Wiedenbeck (2006) controlled oxygen fugacity with dry C 0 / C 0 2 gas mixtures, while Dimanov and Sautter (2000) employed Ar-H 2 /Ar-H 2 0 mixtures, in which hydrogen was present. It has been shown (Hier-Majumder et al. 2004; Demouchy et al. 2007) that hydrogen incorporation within the crystal structure is accommodated by iron reduction and concomitant point defect formation, resulting in the enhancement of cation diffusion in ironbearing silicates and oxides. Figure la also shows that in the range 900-1100 °C the (Fe,Mn)-Mg interdiffusion rates (Dimanov and Sautter 2001; Dimanov and Wiedenbeck 2006) and self-diffusion rates of Fe (Azough and Freer 2000) and Mg (Gasc et al. 2006; Zhang et al. 2010) are all within less than an order of magnitude. In contrast, Ca diffusion rates (Dimanov et al. 1996; Dimanov and

Diffusion in Pyroxene, Mica, Amphibole

655

Jaoul 1998; Zhang et al. 2010) are substantially lower, which confirms that Cais the kinetically limiting species among octahedral cations. In comparison with other silicates, the diffusion rates in diopside are unusually slow for interdiffusion processes involving Fe and Mg. Dimanov and Sautter (2000) observed that as well as being comparatively slow, all diffusion rates for major cations in diopside reported thus far are also closely grouped (within less than 3 orders of magnitude, see Fig. la), while cation diffusion rates at similar conditions in olivines and garnets and much more widely spread. The authors argued that this behavior reflects a strong crystal-electrochemical coupling between the octahedral and tetrahedral sublattices, with many divalent and trivalent cations able to occupy all of the different crystal sites present in clinopyroxenes. Conversely, in olivine and garnets, octahedral and tetrahedral sublattices are virtually independent in terms of cation occupancy; hence the species occupying octahedral sites diffuse much faster than those occupying tetrahedral sites. Dimanov and Sautter (2000) further argued that due to this crystal-chemical coupling, clinopyroxenes should be diffusionally unreactive, and hence their compositional re-equilibration should also involve net transfer reactions (e.g., exsolution, dissolution-precipitation) rather than purely diffusional exchange with surrounding phases. This is potentially of great importance when modeling compositional re-equilibration in natural assemblages assuming only diffusion controlled kinetics. Dimanov and Wiedenbeck (2006) also compared the data from previous studies on transport properties (electrical conductivity, single crystal creep and cation diffusion) in diopside versus oxygen fugacity. Based on their data and previous work, the established point defect model for iron-bearing diopside of Jaoul and Raterron (1994) was strengthened and further extended to the case of Fe- and Mn-bearing diopside. The authors also showed that at 1000-1150 °C and very low oxygen fugacity (below QFI), irrespective of the diffusion mechanism considered, the diffusion rates of the major octahedral cations (Ca; Fe; Fe,Mn-Mg) tend to be grouped within less than an order of magnitude (Fig. 2). Conversely, the diffusion rates of the same species strongly diverge at oxidizing conditions, which is due to the opposite dependences of interstitial and vacancy mediated mechanisms on oxygen fugacity. Hence, for cation exchange involving both interstitial and vacancy mechanisms, for example Ca-(Fe,Mg) exchange, important compositional effects are expected to appear primarily at high oxygen fugacities. Recent preliminary investigations of Fe-Mg interdiffusion in iron-rich diopside (Di 95 Hd 5 ) were reported by Chakraborty et al. (2008). Thin films (20-50 nm) of iron-rich olivine (Fo3oFa70) were deposited by laser ablation onto oriented diopside samples. Samples were annealed at 8001000 °C under flowing C 0 / C 0 2 controlling p02 = 10~17 MPa. Fe-Mg interdiffusion between olivine films and diopside substrates along c-axis was measured by RBS. A single datum was reported at 800 °C and yielded Ae-Mg = 2.2 x 10~22 m2/s, which is at least one and half orders of magnitude higher than data from Dimanov and Sautter (2000) and Dimanov and Wiedenbeck (2006) extrapolated to lower temperatures (Fig. la). It is unfortunately impossible to further comparatively discuss these data due to different experimental conditions and lack of details provided in the abstract of Chakraborty et al. (2008). Ca-Mg interdiffusion. Except for the early study of Brady and McCallister (1983), where the authors evaluated the pseudo-binary Ca-(Fe,Mg) interdiffusion coefficient (see the "Pioneering Approaches" section) from the kinetics of lamellar homogenization, there has not been to date any direct measurement of Ca-Mg interdiffusion rates in pyroxenes. However, Zhang et al. (2010) measured both Ca and Mg self-diffusion coefficients under the same experimental conditions (see previous sections), and hence they could model the CaMg exchange process. The authors used the formalism of Brady and McCallister (1983), where Ca-Mg interdiffusion is described as a serial process, depending on Ca- and Mg selfdiffusion coefficients, the corresponding element fractions, and a thermodynamic factor (TF), which accounts for the non-ideality of mixing of Ca and Mg, in relation to the large pyroxene

656

Cherniak & Dimanov

solvus. Brady and McCallister (1983) and Zhang et al. (2010) demonstrated that T F has a quite substantial impact on the Ca-Mg interdiffusion rates. For instance, Zhang et al. (2010) show that without accounting for the T F the calculated Ca-Mg interdiffusion coefficients are very similar for moderate variations in diopside content. When the T F was considered, the interdiffusion coefficients differed substantially both in magnitude and activation energy for the same compositions. For example, the authors calculated that at 1100 °C the activation energy was increased from 252 kj/mol to 427 kj/mol, while the magnitude of the interdiffusion coefficient decreased by an order of magnitude when the diopside component fraction decreased from 0.8 to 0.7. These considerations, together with the fact that Ca- and Mg self-diffusion show temperature dependent anisotropy (Zhang et al. 2010), may be of fundamental importance when modeling Ca-Mg and Ca-(Fe,Mg) interdiffusion processes in natural assemblages.

Diffusion in synthetic versus natural crystals It is worthwhile to recall some systematic observations from the studies reported in the previous sections. As general rule it seems that cation diffusion is always substantially faster in natural, iron-bearing crystals than in pure synthetic ones. In the studies of Azough and Freer (2000), Dimanov et al. (1996), Pacaud (1999) and Gasc et al. (2006) it appears that Fe, Ca and Mg self-diffusion coefficients at temperatures above 1000 °C were found to be half an order to an order of magnitude faster in natural iron-bearing diopside (~ 2-3 at% Fe) than in pure synthetic iron-free (or iron-poor) diopside. Similar observations are noted for trace elements, as Sr diffusion in diopside (Sneeringer et al. 1984) and for other silicates such as olivine, where it was observed that Mg diffusion in San Carlos olivine was substantially faster than in pure synthetic forsterite (Chakraborty et al. 1994). It may be generally stated that, except if they are specifically doped, synthetic materials always contain much lower concentrations of any kind of impurities, and hence of any related point defects. Similarly, synthetic crystals are also virtually free of higher order crystalline defects (free dislocations, dislocation arrays and sub-grains) related to the sometimes complex thermo-mechanical histories of the natural crystals; these extended defects may act as shortcut diffusion pathways. For these reasons it appears reasonable to observe lower diffusion rates in synthetic, pure crystals than in natural minerals which inherently contain numerous types and higher concentrations of crystalline defects.

Major element cation diffusion in orthopyroxenes From the beginning of the 1990s to date a few studies have investigated major cation diffusion in enstatite. These were also motivated by the interest in interpreting petrogenetic and cooling histories of pyroxene bearing assemblages, as well by the need to constrain the closure temperatures of the well-known orthopyroxene-garnet geothermometer, based on FeM g exchange (Ganguly et al. 1994; Pattison and Bégin 1994). Fe-Mg interdiffusion. Ganguly and Tazzoli (1994) made the first indirect estimates of FeMg interdiffusion rates in orthopyroxenes on the basis of experimentally determined kinetics of Fe and Mg fractionation between non-equivalent M l and M2 sites (Besancon 1981; Saxena et al. 1987; Anovitz et al. 1988). Ganguly and Tazzoli (1994) interpreted the rate constants of Fe-Mg order-disorder, or crystal site exchange reactions, in terms of intracrystalline Fe-Mg interdiffusion. Considering enstatite crystallography and nearly ideal mixing properties of Fe and Mg, the authors assumed an equivalence of interdiffusion rate constants and order-disorder rate constants at the microscopic level. They further argued that the overall Fe-Mg exchange rate constant might represent essentially an average of microscopic rate constants along the c- and fc-crystallographic directions. The authors also assumed that the Fe-Mg interdiffusion coefficients along the c- and ¿-directions might be comparable, and directly related to the macroscopic order-disorder rate constants. Experiments were carried out at temperatures between 500-800 °C; in some of the experiments oxygen fugacity was controlled by a gas mixture of H 2 and C 0 2

Diffusion in Pyroxene, Mica, Amphibole

657

within ~ 0.5-0.8 log units above that of the IW buffer, while in other experiments it was defined by vacuum conditions in sealed silica capsules. The iron contents of the enstatite crystals ranged between 0.13 and 0.49. Rate constants, and hence diffusivities, increased with iron content, but the activation energy for diffusion was constant (Eñ = 241 ± 8 kj/mol). More recently, Stimpfl et al. (2005) addressed the effect of oxygen fugacity on Fe-Mg order-disorder kinetics and reported an oxygen fugacity exponent m ~ 1/6 (m = 0.17). Figure lb shows the interdiffusion data (for 10% Fe and IW conditions) originally obtained by Ganguly and Tazzoli (1994) and the corresponding data recalculated for a fixed p02 = 10~'2 MPa with m = 0.17. Klügel (2001) estimated the Fe-Mg interdiffusion rates in iron-rich orthopyroxene from comparative analysis of chemical zoning in olivine and orthopyroxene in natural harzburgite xenoliths. The author observed Fe-Mg zoning in both olivine and orthopyroxene that he interpreted as diffusional exchange between the xenoliths and surrounding mafic, silicaundersaturated melt. He estimated temperatures between 1050-1200 °C and QFM buffered oxygen fugacity. From the olivine zoning and the known Fe-Mg interdiffusion coefficients he retrieved the diffusion times, which were used to calculate Fe-Mg interdiffusion rates from the orthopyroxene zoning. He reported D l t M g = 3 x 10~19 m 2 /s at T = 1130 °C and QFM-buffered conditions and found good agreement with the (Fe,Mn)-Mg interdiffusion data in diopside (Dimanov and Sautter 2000) and the Fe-Mg interdiffusion data for orthopyroxene from Ganguly and Tazzoli (1994), when the latter are extrapolated to higher temperatures (Fig. lb). More recently, preliminary investigations of Fe-Mg interdiffusion in iron-rich orthopyroxene (En90Fs10) were reported by ter Heege et al. (2006). Thin films (20-50 nm) of iron-rich olivine (Fo3oFa7ü) were deposited by laser ablation onto oriented orthopyroxene samples. Samples were annealed at 800-1000 °C under flowing C 0 / C 0 2 controlling p02 to 10" 17. ]q-i3 MPa. Fe-Mg interdiffusion between olivine films and orthopyroxene substrates along the c-axis was measured by RBS. Reported interdiffusion coefficients were A"e-Mg = 4 x 10~22 m2/s to 2 x 10-2() m2/s at temperatures of 800 °C to 1000 °C a n d p 0 2 = 10"17 MPa. At 800 °C, the data are in relatively good agreement (within half an order of magnitude) with the previously estimated Fe-Mg interdiffusion rates from Ganguly and Tazzoli (1994). ter Heege et al. (2006) also reported that the interdiffusion rates at 1000 °C decreased by a factor of 4 when oxygen fugacity decreased from 10~13 to 10~17 MPa, thus suggesting a vacancy related mechanism. The very limited data provide only a rough estimate of the oxygen fugacity exponent, m = 0.15, which is consistent with the findings of Stimpfl et al. (2005), who studied the effect of oxygen fugacity on Fe-Mg order-disorder rates in orthopyroxene and found m between 1/5.5 and 1/6.5. These values agree with the general framework of the point defect model of Jaoul and Raterron (1994), which can be similarly applied to clinopyroxene or to orthopyroxene with low Fe content. The data of ter Heege et al. (2006) and of Ganguly and Tazzoli (1994) are still in relatively good agreement when both data sets are recalculated to the same oxygen fugacity (Fig. lb). Unfortunately additional comparison and discussion remains speculative due to the lack of details and the very limited data provided in the abstract of ter Heege et al. (2006). Magnesium self-diffusion. Schwandt et al. (1998) experimentally determined magnesium self-diffusion coefficients in iron-rich orthopyroxene (XFc ~ 0.11), with centimeter-sized crystals oriented to study diffusion along the a-, b- and c-axes. The polished samples were pre-annealed for 24 hours at temperature and oxygen fugacity conditions corresponding to the experimental conditions for diffusion, and thin layers (100-200 nm) of 25 Mg-enriched MgO were deposited onto the pre-annealed substrates by thermal evaporation in high vacuum. Diffusion experiments were performed at temperatures between 750-900 °C, at oxygen fugacity equivalent to the IW buffer, controlled by C 0 - C 0 2 gas mixtures, and diffusion profiles measured by SIMS. The effects of composition and oxygen fugacity were not investigated. Although the surface films were relatively thin, the diffusion profiles could be modeled by thick source solutions. Mg diffusion coefficients were found to be comparable, irrespective of the crystallographic

658

Cherniak & Dimanov

direction of diffusion. However, the authors speculated that the observed isotropy of diffusion may only correspond to the experimental conditions, and, in spite of the large uncertainties due to limited data and a restricted temperature range for experiments, the authors argued that the activation energy for diffusion along the c-axis is consistently lower (Ea = 265 ±66 kj/mol) than for diffusion along the ¿-axis (Ea = 339±77 kj/mol) and the c-axis (Ea = 360±52 kj/mol). The data were found to be in relatively good agreement with the previous estimates of Fe-Mg interdiffusion rates by Ganguly and Tazzoli (1994). In extrapolation to higher temperatures, both data sets are also in relatively good agreement with the Fe-Mg interdiffusion rates estimated from naturally zoned assemblages by Kliigel (2001). Chromium. Recently, Ganguly et al. (2007) determined Cr diffusion coefficients in iron bearing natural orthopyroxene (X^ ~ 0.026) in order to constrain closure temperatures for application of the 53Mn-53Cr chronometer in understanding early solar-system processes. The authors investigated diffusion parallel to the a-, b-, and c-axes. Temperatures ranged between 900 °C and 1100 °C, and p02 , corresponding to that of the IW buffer and up to 4.5 orders of magnitude above, was controlled by C0-C0 2 gas mixtures. The authors indicated that in these conditions Cr is likely in the +3 valence state. Oriented samples were polished and pre-annealed at conditions similar to those of the diffusion experiments in order to heal polishing damage and to set point defect concentrations. Thin Cr films (~ 50 nm) were deposited onto samples by thermal evaporation under high vacuum. Diffusion profiles, typically several tens of nm, were obtained by SIMS depth profiling. Two analytical solutions were applied to retrieve diffusion coefficients. The authors considered either infinite or finite sources of diffusant, with the former model providing the best fits. Cr was assumed to diffuse within the octahedral sub-lattice alone. Diffusion was found to be slightly anisotropic, with the fastest rates along the c-axis and the slowest rates along the a-axis. Fits to the data for experiments buffered at IW yield an activation energy of 237± 19 kJ/mol and pre-exponential factor 2.62x 10"" m2/s (log Dn = -10.58±0.67) for diffusion parallel to b, and an activation energy of 277 ± 14 kl/mol and pre-exponential factor 3.87 x 10~H) m2/s (log Da = -9.41 ±0.35) for diffusion parallel to the c-axis. The activation energy for diffusion along the a-axis could not be adequately constrained, but Ganguly et al. (2007) suggested that the activation energies are likely to be similar for all the crystallographic directions. They proposed the mean value of 257 ±12 kl/mol, which was used to reprocess the data to obtain the pre exponential factors of 7.41 x 10~u m2/s (log Da = -10.13±0.39) for diffusion parallel to the a-axis, 1.78 x 10~H) m2/s (log Da = -9.75 ±0.78) for diffusion parallel to b, and 3.10x 10~'° m2/s (log Da = -9.51 ±0.56) for diffusion parallel to the c-axis. Ganguly et al. (2007) investigated the effects of p02 for diffusion parallel to c at 950 °C and 1050 °C. They observed a decrease of diffusion rates by a factor of 2 to 3 for an increase in log p02 from IW to IW + 4.5, and hence an oxygen fugacity exponent m —0.07 to -0.1. A decrease in diffusion coefficients with increasing oxygen fugacity indicates an interstitial mechanism as for Ca in diopside (Dimanov et al. 1996), but the low value of the oxygen fugacity exponent appears inconsistent with the previously reported point defect model for clinopyroxene (Jaoul and Raterron 1994; Dimanov and Wiedenbeck 2006), where m = -3/16 for an interstitial mechanism. The authors proposed an alternative point defect model, which also predicted a much higher p02 dependence (m —2/3) than observed (m —0.07 to -0.1). Given this discrepancy they suggested a mixed mode for Cr diffusion involving both vacancy and interstitial mechanisms. Pyroxene point defect chemistry As reported in the previous sections, diffusion data for many major cations in both clinoand ortho-pyroxenes exhibit pronounced dependences on oxygen fugacity, with m ~ 0.10.22. Similar observations are reported for diffusion of rare earths and Pb in clinopyroxenes (Van Orman et al. 2001; Cherniak 1998a, 2001), as well as for Pb diffusion in orthopyroxene (Cherniak 2001). The influence of oxygen fugacity is also evidenced in creep (Jaoul and

Diffusion in Pyroxene, Mica, Amphibole

659

Raterron 1994) and electrical conductivity (Duba et al. 1973, 1976; Huebner and Voigt 1988), and is a fundamental control of transport properties o f pyroxenes. In comparison with olivine (Smyth and Stacker 1975; Nakamura and Schmalzried 1983, Hirsh and Shankland 1991; Tsai and Dieckmann 2002), however, fewer models of point defect chemistry have been developed for iron-bearing pyroxenes (Huebner and Voigt 1988). The present section summarizes the best developed point defect model for iron bearing clinopyroxene (Jaoul and Raterron 1994). As Nakamura and Schmalzried (1983) proposed for fayalite, Jaoul and Raterron (1994) proposed that for Fe-bearing diopside (described as M e S i 0 3 , where M e stands for Ca, Mg or Fe) in equilibrium with gaseous oxygen, the dominant extrinsic point defects are cation vacancies and electron holes in the valence band, while the oxygen sub-lattice is considered a perfect framework. In the standard point defect notation of Kroger and Vink (1956) the vacancies are V " M c and V " " i , and the electron holes are h*. The subscript indicates the vacant crystallographic site and the superscript represents the charge deficiency o f the point defect. A prime (') denotes a negative charge excess, a dot (•) represents a positive one, while x stands for normal charge. S

When a system is open to oxygen alone (i.e., other component activities are not buffered), increasing oxygen fugacity results in adsorption of oxygen atoms at the sample surface. These may be incorporated in the crystal structure by the formation of new, but incomplete diopside molecules. In these molecules all cation sites are vacant. Their charge deficiency is compensated by electron holes localized on substitutional Fe, written as F e ' M e (i.e., F e 3 + located on the octahedral M1 and/or M 2 site). The latter are also called small polarons. Conversely, lowering oxygen fugacity leads to oxygen desorption from the sample surface, resulting in the reduction of Fe 3 + , with a concomitant annihilation of cation vacancies. In this model, cation vacancies and small polarons are the majority point defects. However, Raterron and Jaoul (1991), and Doukhan et al. (1993) report that, depending on oxygen fugacity and temperature, but considerably below the nominal melting point, small amounts ( « 1 vol%) of partially molten phase is exsolved as tiny droplets within the host diopside. At the beginning of this phenomenon, called E P M (early partial melting), the droplets are only a few tens of nm large and composed of nearly pure silica, although with time and increasing temperature these grow and their composition progressively evolves, draining impurities (Fe, Al) and Ca from the surrounding matrix. Jaoul and Raterron (1994) proposed that the formation o f major point defects has certain limits, beyond which the vacancy-supersaturated diopside tends to stabilize its crystalline structure by silica exsolution, and hence the onset o f E P M . The model has two concurrent relations for point defect formation equilibria: 6 F e ^ c + M e „ c + Sf S l + | o

2Fe*Me + Me*Me + S i 0 2 +

2

4 » M e S i O , + V ^ + V™'+ 6 F e ; c

* * MeSi03 +

+ 2Fe' Me

(1)

(2)

The first reaction represents either the formation of new diopside molecules, plus cation vacancies which are charge balanced by ferric iron (to the right), or oxygen desorption accompanied by vacancy annihilation and reduction o f ferric to ferrous iron (to the left). This operates alone, within the limits o f the stability field of diopside plus gaseous oxygen. But if the equilibrium is displaced enough to the right so that the concentrations o f cation vacancies become higher than allowed by the second equilibrium condition, then reaction (2) starts operating to the left. This represents consumption o f octahedral vacancies and reduction of ferric to ferrous iron through desorption o f oxygen and formation of silica. Reactions (1) and (2) can be written more concisely as:

660

Cherniak & Dimanov 6Fe^c + | o 2 2Fe^ c + Si0 2 +

30Q + V I + V™+ 6Fe'Mc

(1')

o 30Q + SfSl + V ^ + 2Fe^ c

(2')

Any such chemical solid state reaction must obey three conservation rules: (i) the conservation of matter, (ii) the respect for electroneutrality, and (iii) the conservation of the ratio of anion to cation sites. For the MeSiOj diopside one has Me:Si:0 = 1:1:3. Expressed differently, the reaction must create or annihilate entire molecules in the crystallographic sense, though these can be incomplete in the chemical sense. Jaoul and Raterron (1994) showed that the concentrations of major point defects (cation vacancies and small polarons) related to reaction (1) vary as (p0 2 ) 3/16 . Conversely, the minority point defects (interstitial cations) show a p02 dependence with exponent m = -3/16. In the following we will derive the p02 dependence of major point defects associated with reaction (1) alone, i.e., in EPM absent conditions. The reader is referred to Jaoul and Raterron (1994) for further details of conditions involving reaction (2), i.e., in cases where EPM is present. The majority point defects involved in the chemical reaction (1) satisfy the charge neutrality condition. 4[vr] + 2 K ] = [Fe^]

(3)

where the brackets represent site fractions (which are actually identical to mole fractions for cations in diopside written as MeSi0 3 ). The mass action law associated with the chemical reaction (1) is: [^"]-[v:](Fe;£]6=^4cPOf

(4)

where e (atomic fraction Fe/(Ca + Mg + Fe)) is nearly the Fe mole fraction [Fe x M J (because the number of Fe3+ is negligible as compared with Fe2+). A' = exp(-Q]/RT) is the equilibrium constant for the reaction (1) and QL is the (standard state) Gibbs free energy of the reaction. We note that in reaction (1), considering an ideal solid solution, the activity of the components FeSi0 3 and MeSi0 3 are assumed to be respectively xFc and 1, which is reasonable since the overall Fe concentration in diopside is low. The system is incompletely defined if only two reactions govern the concentrations of three major point defects. Non-stoichiometry of pseudo-ternary systems is completely determined if one introduces two parameters describing departures from stoichiometry between anions and cations and between octahedral cations and tetrahedral cations (Nakamura and Schmalzried 1983). These two parameters are approximated by Jaoul and Raterron (1994) as functions of cation vacancy concentration, assuming that the oxygen lattice forms a perfect framework, as follows:

"Si T "Mc ^

H

where n, represents the number of moles of a species i in the pyroxene system. It is assumed that diopside has some initial departure from stoichiometry; however this parameter is not directly measurable. Reaction (1) shows that, with respect to oxygen exchange between diopside and its surrounding environment, octahedral and tetrahedral cation vacancies

Diffusion in Pyroxene, Mica,

Amphibole

661

are both created or annihilated at the same rate. Thus, the initial state of octahedral to tetrahedral cation non-stoichiometry remains preserved. Hence, for convenience, it is possible to consider the simplest case and take the initial cationic ratio Si:Me as equal to 1 (i.e., the initial octahedralto tetrahedral cation departure from stoichiometry is zero). It follows that all cationic vacancies are related to reaction (1) and obey the relationship: [vr]=K]

(7)

Combining Equations (3) and (7) allows us to rewrite Equation (4) and to obtain the expression of the concentration of cation vacancies and oxidized iron as a function of p0 2 [VMe]-[vr]-[FeL]x^>or6

(8)

The model of Jaoul and Raterron (1994) provides a valuable description of the influence of point defects on transport properties of Fe-bearing diopside. For instance, Equation (8) shows that the concentrations of cation vacancies vary as (p02)m with m = 3/16 = 0.188, which is in very good agreement with the experimental observations for electrical conductivity (Huebner and Voigt 1988) and cation diffusion (Dimanov et al. 1996; Azough and Freer 2000; Cherniak 2001; Dimanov and Wiedenbeck 2006). Diffusion of minor and trace elements in pyroxene Along with major element constituents, diffusion of a range of minor and trace elements has been measured in pyroxene. These include Li, useful in geospeedomtery and tracing weathering and aqueous alteration, and elements important in geochronology (Pb, Sr, U and Th) and as geochemical tracers, such as the rare earth elements. As in the case of major elements, dependence of diffusion on oxygen fugacity and other parameters can shed light on diffusion mechanisms, and has been explored in some of these studies. The findings of these investigations are discussed below and diffusion parameters for minor and trace elements are presented in Table A2. Lithium. Li diffusion was measured in natural diopside (with -0.7 at% Fe) by Coogan et al. (2005) as part of a study of Li partitioning between clinopyroxene and plagioclase undertaken in order to develop a lithium geospeedometer. Diffusion experiments were conducted in 1-atm furnaces with flowing C0-C0 2 gas to fix oxygen fugacity within the stability field of diopside and prevent oxidation of Fe+2. p02 was lO"11 Pa at 800-900 °C and 10"7 Pa at 1000-1100 °C. The source of Li for diffusion experiments was a mixture of crushed San Carlos olivine, Li 2 C0 3 (enriched in 6Li) and Li 2 Si0 3 powders. The diopside crystals, cut normal to the [010] axis and polished, were annealed in alumina crucibles with the powder mixtures. 6Li profiles were measured with SIMS. From four experiments performed (one each at 800, 900, 1000, and 1100 °C), Coogan et al. (2005) obtained an activation energy of 258 kJ/mol and pre-exponential factor of 2.9 x 10~2 m2/s. No uncertainties are reported, but these may be significant given the limited size of the dataset. Li diffusion in clinopyroxene is about 3 orders of magnitude slower than Li diffusion in plagioclase (Giletti and Shanahan 1997), but considerably faster than diffusion of most other elements in clinopyroxene. Strontium. Sneeringer et al. (1984) measured Sr chemical diffusion in synthetic and natural diopside, using a variety of analytical techniques, including a 85Sr radiotracer source and serial sectioning, and stable Sr tracers with RBS or SIMS depth profiling. This study was one of the first applications of the accelerator-based ion beam technique RBS to studies of diffusion in geological materials. The radiotracer experiments were conducted with a natural diopside (~ 1 at% Fe) and a synthetic diopside containing less than 0.1 at% Fe. The radiotracer was in the form of 85SrCl2 in a dilute HC1 solution and was deposited on sample surfaces (oriented to measure diffusion parallel to c); the tracer was dried and heated to 800 °C (a temperature considerably lower than

662

Cherniak & Dimanov

the diffusion anneals) in flowing N 2 gas to induce bonding of the tracer to the sample. Excess tracer was removed following this step, and diffusion anneals were conducted in 1-atm furnaces in flowing N 2 gas. Samples were sequentially polished and diffusivities determined by evaluating residual activity as a function of depth from the initial sample surface. For the natural diopside over the temperature range 1200-1300 °C, an activation energy of 406 ± 7 1 kJ/mol and preexponential factor of 5.4 x 10~3 m 2 /s (log D 0 = - 2 . 2 7 ) are reported. The details of the procedures used in fitting are not fully elaborated by Sneeringer et al. (1984), but it should be noted that in some cases reported Arrhenius parameters appear inconsistent with those that would be obtained given the tabulated diffusivities and uncertainties. For the synthetic diopside over the same temperature range, Sneeringer et al. (1984) report an activation energy of 4 5 6 ± 7 5 kJ/mol and pre-exponential factor of 2.5 x 10~3 m 2 /s (log D0= -2.61). Diffusion in synthetic diopside is about 2 orders of magnitude slower than diffusion in natural diopside, and the activation energy for diffusion in the synthetic material is higher. The natural diopside used by Sneeringer et al. (1984) was found (through examination of etch pits) to have a dislocation density two orders of magnitude higher than the synthetic diopside. This higher vacancy concentration is likely associated with electron holes needed to charge balance Fe 3+ and other impurities. Experiments were also run with a stable Sr tracer at 1-atmosphere, conducted by applying the tracer solution to polished synthetic diopside, permitting it to dry, heating at low temperature on a hotplate, and placing the tracer-deposited sample face to face with another polished diopside section inside a Pt envelope. The experimental assembly was then annealed in air in 1 -atm furnaces. Samples were analyzed with an ion microprobe, and diffusion was investigated in three crystallographic orientations. For transport along the a-axis, Sneeringer et al. (1984) report an activation energy of 4 5 2 ± 4 2 kJ/mol and pre-exponential factor of 6 . 4 x 10 -4 m 2 /s (log D()= - 3 . 1 9 ) . For the ¿-axis, values reported are an activation energy of 565 ± 3 8 kJ/mol and pre-exponential factor of 1.2x 10' m 2 /s (log Dn = 1.08), and for the c-axis an activation energy of 511 ± 2 9 kJ/mol and pre-exponential factor of 1.2x 10~' m 2 /s (log D0 = - 0 . 9 2 ) are determined. These findings suggest that diffusion along the a-axis may be somewhat slower than diffusion along the b and c axes, but there is considerable overlap and scatter in the data and little evidence of strong anisotropy. Some of the samples analyzed by SIMS were also analyzed with Rutherford Backscattering Spectrometry (RBS). For these data, which measured diffusion parallel to c, an activation energy of 5 4 4 ± 2 5 kJ/mol and pre-exponential factor of 3.1 m 2 /s (log D() = 0.49) were reported. Sneeringer et al. (1984) also conducted experiments at high pressure (0.8-2 GPa) using solid-media piston-cylinder apparatus. Diffusion couples of polished synthetic diopside were used in experiments, with one half of the couple an undoped diopside and the other half doped with Sr and Sm in concentrations of 450 ppm and 250 ppm respectively. Sr and Sm profiles were obtained with an ion microprobe. The authors reported some difficulty in separating the halves of the diffusion couple and locating the interface between doped and undoped material, which may contribute to experimental uncertainty. The results for high-pressure experiments at 2 GPa suggest pronounced anisotropy for Sr diffusion, with diffusion along the c-axis slower than diffusion along the b- and a-axes. Diffusion along the c-axis was also found to have a higher activation energy for diffusion. For these data at 2 GPa, the Arrhenius parameters are 2 6 0 ± 5 0 kJ/mol and 1.7x 10"9 m 2 /s (log D0 = - 8 . 7 7 ) for diffusion along the a-axis, 3 8 1 ± 8 4 kJ/mol and 5.2 x 10"5 m 2 /s (log D0 = - 4 . 2 8 ) for diffusion along the ¿-axis, and 607 ± 3 3 k j / mol and 8.7 x 102 m 2 /s (log Da = 2.94) for diffusion along the c-axis. For diffusion at 1.4 GPa parallel to the c-axis, an activation energy of 728 ± 1 3 4 kJ/mol and pre-exponential factor 1.2 x 107 m 2 /s (log D0 = 7.08) are obtained. These data suggest anisotropy in activation volumes (with diffusion along the c-axis having the smallest values), and negative activation volumes (apparent faster diffusion at higher pressure), with values ranging f r o m - 2 0 . 6 cm 3 /mol to - 2 . 3 cm 3 /mol. However, Sneeringer et al. (1984) note that there was cracking of the samples (most likely during pressurization or depressurization of experiments) which could increase

Diffusion in Pyroxene, Mica, Amphibole

663

measured diffusivities. Diffusivities for Sr and Sm (discussed below) from the same specimens in these high-pressure experiments were for the most part similar in value, as were Arrhenius parameters, which suggests the possibility of influence of experimental artifacts on measured profile lengths and thus determinations of diffusivities. Lead. Pb diffusion in a range of pyroxene compositions and under various p02 conditions was measured by Cherniak (1998a, 2001). The sources of diffusant in these experiments were fused and finely ground mixtures of PbS powder and ground pyroxene of the same composition as the pyroxene samples. Experiments were run in 1 -atm furnaces in sealed silica glass capsules with solid buffers (to buffer at QFM, IW or MH); a few experiments were "self-buffered", with only the sample and PbS-pyroxene source sealed in a silica glass capsule with no solid buffer. Rutherford Backscattering Spectrometry (RBS) was used to measure Pb diffusion profiles. Pb diffusion in a near-end member diopside (~ 0.3 at% Fe) under QFM-buffered conditions over the temperature range 800-1100 °C was described by an activation energy of 544+40 kJ/ mol and pre-exponential factor of 4.03 x 102 m 2 /s (log D() = 2.61 + 1.65) for diffusion normal to the (110) cleavage plane. For transport normal to (001), an activation energy of 512 ±23 kJ/mol and pre-exponential factor 2.17x10' m2/s (log D0 = 1.34+0.52) were determined; little anisotropy was evident comparing diffusion in these two orientations. A fit to both data sets yields the values 519+19 kJ/mol for the activation energy and 4.40x 10' m2/s (log D 0 = 1.64 ±0.49) for the pre-exponential factor. The "self-buffered" data, from experiments with the PbS-diopside source and no solid buffer, yield a somewhat higher activation energy of 609+67 kJ/mol and larger pre-exponential factor of 5.38 x 104 m 2 /s (log Da = 4.73±2.74) for diffusion normal to (110). The single datum for the (001) orientation under self-buffered conditions does not differ significantly from diffusivities found for (110). Cherniak (1998a) also investigated Pb diffusion in a more Fe-rich clinopyroxene composition (-40% diopside component; ~5 at% Fe). Over the temperature range 800-1050 °C, for QFM-buffered experiments for diffusion normal to (110), an activation energy of 387 + 31 kJ/mol and pre-exponential factor of 2.20xl0~ 4 m 2 /s (log Da = -3.66±1.38) were obtained. Diffusivities from the "self-buffered" experiments for this pyroxene composition did not differ substantially from those for the QFM-buffered experiments; for these data the activation energy is 410+36 kJ/mol and the pre-exponential factor is 1,90x 10~3 m2/s (log D 0 = - 2 . 7 2 + 1.46). Diffusion in the more iron-rich clinopyroxene is faster (by about half a log unit) and activation energy for diffusion lower than that for the near-end-member diopside. In the nearend-member diopside, Pb diffusivities under QFM-buffered and "self-buffered" conditions differ, but this is not the case for the Fe-rich clinopyroxene, for which they are quite similar. The latter observation suggests that the system containing the Fe-rich clinopyroxene may have a greater ability to buffer itself because of the higher Fe content of the source (since it contains ground clinopyroxene of the same composition as the sample). In contrast, the source used for the near-end-member diopside contains ground diopside of low Fe content, a material which likely has much poorer buffering capacity. Additional work on Pb diffusion in pyroxene was conducted by Cherniak (2001), with investigation of diffusion in a chromian diopside, an augitic pyroxene, and a near-end member enstatite. The experimental and analytical approaches were similar to those of Cherniak's 1998 study, with sources of diffusant consisting of PbS powder and ground pyroxene (of the same type as the sample), source and sample contained in an open silica capsule which was sealed under vacuum with solid buffers (to buffer at QFM, IW, or MH) in an outer silica glass capsule. Prepared sample capsules were annealed in 1 -atm furnaces, and Pb distributions in the pyroxene specimens were profiled by RBS. For the QFM buffered experiments on the chromian diopside (transport normal to the (110) cleavage plane) over the temperature range 850-1050 °C an activation energy 351 ±36 kJ/mol

664

Cherniak & Dimanov

and pre-exponential factor 8.66 x 10~7 m 2 /s (log D() = - 6 . 0 6 ± 1.49) were obtained. For transport parallel to c, an activation energy of 309+40 kJ/mol and pre-exponential factor 2.00x 10~8 m2/s (log D0 = - 7 . 7 0 + 1.67) were determined. There appears little anisotropy when comparing diffusion in two orientations. Cherniak's (2001) fit to data for both orientations yields the values 336+27 kJ/mol for the activation energy and 2.36x 10~7 m2/s (log D() = -6.63+1.10) for the pre-exponential factor. For the augitic pyroxene, an activation energy of 372 ±15 kJ/mol and pre-exponential factor of 3.78x 10~5 m2/s (log D 0 = -4.42+0.64) were found for transport normal to (110) under QFM-buffered conditions over the temperature range 850-1050 °C. Diffusion measurements on enstatite over the temperature range 850-1100 °C yielded an activation energy of 366±29 kJ/mol, and pre-exponential factor of 6.63 x 10~7 m 2 /s (log Da = -6.18 + 1.18) for transport normal to (210) under QFM-buffered conditions. As in the 1998 study, Cherniak (2001) found that Pb diffusion appears faster in more Fe-rich pyroxenes. Diffusivities in the augitic pyroxene are higher (by nearly an order of magnitude) than Pb diffusion in the Cr diopside. Dimanov et al. (1996) found a positive correlation between Ca diffusion rates and Fe content in diopside, with diffusion differing by nearly a log unit between diopsides having 0.4 and 2.4 at% Fe, when the data are normalized to constant p02 and T. Activation energies for Ca diffusion in these diopsides were, however, found to be quite similar. In the case of Pb diffusion, comparable trends may broadly apply. Activation energies for Pb diffusion in these pyroxenes under QFM buffered conditions are also similar (-390-350 kJ/mol), with the exception of the near end-member diopside, which has a higher value. (Fig. 3) Since diffusion is controlled by defect chemistry, as discussed above, it is certainly likely that the differences in major and minor element concentrations and variations in defect populations between the pyroxenes will have significant effects on Pb transport rates. Investigation of the dependence of oxygen fugacity on diffusion can help to shed light on the involved diffusion mechanisms, thus Cherniak (2001) explored the relationship between Pb diffusion and oxygen fugacity, measuring diffusion under more oxidizing and more reducing (MH and IW buffered) conditions than QFM for Cr diopside, augitic pyroxene and enstatite, and conducted comparable experiments on the diopside and Fe-rich clinopyroxene studied in the 1998 work. Cherniak (2001) found a positive dependence of Pb diffusion rates on p02 for all of the pyroxene compositions investigated. This is clearly shown in Figure 4, where values for log _D1>b at constant temperature (1000 °C for the enstatite, 950 °C for other pyroxenes) for these pyroxene compositions are plotted as a function of log p02. The slopes of these lines are similar, ranging from 0.141 to 0.203, and these values of the exponent m where D cc (p02)m, are presented in Table A2. Comparison of the Arrhenius relation for Pb diffusion in near end-member diopside under "selfbuffered" conditions (Cherniak 1998a) with diffusivities found under controlled buffered conditions suggests that the "self-buffered" system (with its relatively iron-poor natural diopside) is more reducing than QFM, in contrast to the conclusion drawn in Cherniak (1998a). Interestingly, the exponential term for diopside is 0.19, which is in good agreement with the value +3/16 determined from point defect models for diffusion mechanisms controlled by cation vacancies (Jaoul and Raterron 1994). Despite differences in major and minor element concentrations and variations in defect populations among these pyroxenes, the similarities in p02 dependence on Pb diffusion rates suggest a common controlling defect mechanism, likely vacancy-controlled (octahedrally coordinated vacancies) mechanism for Pb diffusion in pyroxene. This is in contrast to the finding of Dimanov et al. (1996) for Ca diffusion, in which a negative dependence on p02 was determined for natural Fe bearing (~2 at% Fe) diopside, suggesting diffusion controlled by an interstitial mechanism. These dependences of D on p02 can also be used to obtain activation energies for diffusion at constant p02 (rather than under the conditions imposed by the solid buffers, for which p02

Diffusion in Pyroxene, Mica, Amphibole

665

T(°C) -18

-19

JT -20

1100

1000

900

800

Pb diffusion pyroxene

v. Di 40 Augite ^ ^ "NV Cr DioRside^^s^

CM

E,

Q

-21

O) o

N

X

-22

\ Enstatite

-23

-24

^ Di

Figure 3. Summary of Pb diffusion data for pyroxene for experiments buffered at QFM. Pb diffusion is generally faster in more Fe-rich pyroxene, while diffusion in enstatite is slower than in the clinopyroxene compositions investigated. Data from Cherniak (1998a, 2001 ).

95

Q F M Buffer

i

i

i

i

7.5

8.0

8.5

9.0

9.5

4

1/T (X10 /K)

Figure 4. Dependence of Pb diffusion on p 0 2 for various pyroxene compositions, measured at 1000 °C for enstatite and 950 °C for other compositions. In all cases, there is a positive dependence of diffusion coefficients on p02 , with exponents m ranging from 0.141 to 0.203 (Table A2), where D oc (p02 )'". This dependence is consistent with a vacancy controlled mechanism for Pb diffusion. Data from Cherniak (2001).

Enstatite

-22

Cr diopside -23 -15

-10

log f0 2 varies with temperature) as was done for major elements in Figure 1. If the diffusivities are normalized to a constant p02 of 1 x 10~12 MPa, we obtain activation energies of 417 kJ/mol for diffusion parallel to c in the near end-member diopside. 294 kJ/mol for diffusion in the Fe-rich clinopyroxene (for diffusion normal to (110)), 275 kJ/mol for the augitic pyroxene, and 259 kJ/mol for the Cr diopside (for diffusion normal to ( 110). These values are lower than those determined for diffusion under QFM-buffered conditions. However, it should be emphasized that in self-buffered natural assemblages, p02 will generally vary with temperature rather than being of fixed value, so Arrhenius relations determined for conditions corresponding to solid buffers may better describe the temperature dependence of diffusion in geologic systems.

666

Cherniak & Dimanov Rare earth elements.

Diopside. Sneeringer et al. (1984) measured Sm diffusion in synthetic diopside ( CM

E O)

o

7

8 1/T (x104/K)

T(°C) 1400

-16

-

1200

\ Sr(s)

1000

900 Diopside

Sr(n)

w -18 fe,Mn)-Mg

U) 2

-20

-22

DeKalb

U, Th

-

6

7

8 1/T

9

(x104/K)

Figure 10. Diffusion data for major, trace and minor elements in diopside. Sources for data: Ca-Mg - Brady and McCallister (1983); (Fe.Mn)-Mg - Dimanov and Wiedenbeck (2006); Fe - Azough and Freer (2000); Ca ( p 0 2 = 10- 12 MPa) - Dimanov et al. (1996); Al - Sautter et al. (1989); Si - Bejina and Jaoul (1996); Sr (n - natural, s - synthetic) - Sneeringer et al. (1984); Ce. Yb, Dy (QFM) - Van Orman et al. (2001); Pb (DeKalb and chromian diopsides, QFM) - Cherniak (1998a, 2001); U, Th (QFM) - Van Orman et al. (1998).

Diffusion

in Pyroxene,

Mica,

675

Amphibole

T(°C) 1400 -16

1200

\ \

1100

1000

900

Sr - natural diopside \

\

-22

Pb-CrDi(QFM)

7

8

N ^

9

1/T (x10 4 /K) Figure 11. Potential effects of oxygen fugacity on Sr diffusion. The Arrhenius relation for the natural diopside in the Sneeringeret al. (1984) study is "normalized" to lower/>0 2 conditions using the dependence ofDonpO? derived for Pb diffusion in diopside by Cherniak(2001) (i.e., m = 0.19), producing the family of curves shown for MH, QFM, and IW buffered conditions (dashed lines). An up-temperature extrapolation from the diffusion data for Pb in Cr diopside (experiments buffered at QFM) intersects the calculated QFM line for Sr diffusion. The calculated line for the most reducing conditions (i.e., IW), is nearly coincident with the line for Sr diffusion in the synthetic diopside. This agreement may be merely fortuitous, but perhaps points to the importance of changes in defect chemistry in influencing diffusion in pyroxenes. See text for additional discussion.

Diffusion rates for the QFM buffered and the "self-buffered" Pb diffusion experiments on near end-member (DeKalb) diopside suggest that oxidation or reduction of multivalent minor elements may in some ways affect Pb transport (although Pb itself should remain in divalent state [e.g.. Otto 1966]). Impurity levels tend to be much higher in natural samples (the diopside used by Cherniak (2001) has appreciable amounts of both Na and Mn) and could influence diffusion. Effects would largely manifest themselves in elevated pre-exponential factors, as was found in the work of Dimanov et al. (1996) on Ca diffusion. Activation energies would likely be unaffected by the aforementioned factors in the extrinsic diffusion regime. The activation energy for Sr (607 + 33 kj/mol) diffusion in synthetic diopside agrees within uncertainty with both buffered and self-buffered Pb diffusion results for diopside in the (110) orientation. CaMg interdiffusion rates, obtained through homogenization of pigeonite lamellae in diopside (Brady and McCallister 1983), are about an order of magnitude slower than Pb diffusion data for diopside. but interdiffusion coefficients could vary by as much as an order of magnitude due to the large thermodynamic effect of non-ideal mixing on diffusion near a solvus (Brady and McCallister 1983; Zhang et al. 2010). The most striking differences among diffusion rates of the divalent cations, however, may be noted when comparing results for Ca self-diffusion (Dimanov et al. 1996; Zhang et al. 2010) in natural Fe-bearing diopside with other findings. Ca diffusion is about one and half (Zhang et al. 2010) to more than two (Dimanov et al. 1996) orders of magnitude slower than Pb transport in the Cr diopside and DeKalb diopside. Based on point defect models and the observed p02

676

Cherniak & Dimanov

dependence of Ca diffusion, Dimanov et al. 1996 advocate an interstitial mechanism for Ca transport. As it seems less likely that the much larger cations Pb and Sr (0.17 and 0.14 A larger than Ca in 8-fold coordination, respectively; Shannon 1976) would be able to move through the lattice in this manner, diffusion rates and activation energies for Ca and other divalent cations might not be expected to be comparable. Dimanov et al. (1996) report an activation energy of 396+38 kJ/mol for temperatures above —1150 °C, and 264+33 kJ/mol for temperatures below 1150 °C. These activation energies are lower than that reported for Pb diffusion in diopside, but this is not inconsistent, as noted above, with the possibility of a different transport mechanism for Pb (and possibly Sr as well). Perhaps movement via the interstitial mechanism also accounts for the slower diffusion rate for Ca with respect to Pb and Sr; if these species traveled via a similar diffusion mechanism it might be expected that Ca would diffuse more rapidly than both given its smaller size. Diffusion of cations having higher valence states is slower than that of Pb. The tetravalent cations U and Th (Van Orman et al. 1998), which likely substitute for Ca on M2 sites, diffuse several orders of magnitude more slowly than Pb. The trivalent REE (Van Orman et al. 2001), and especially the LREE with large ionic radii, also diffuse more slowly than Pb. Pronounced decreases in diffusivities with higher cation charge have been noted in the feldspars (e.g., Giletti and Shanahan 1997; Giletti and Casserly 1994; Cherniak and Watson 1992,1994; Foland 1974), zircon (Cherniak et al. 1997), and calcite (Cherniak 1997, 1998b). Silicon diffusion is also quite sluggish in diopside (Bejina and Jaoul 1996), consistent with observations made in other minerals, including olivine (Houlier et al. 1990; Dohmen et al. 2002), quartz (Cherniak 2003; Bejina and Jaoul 1996), and anorthite (Cherniak 2003). However, the activation energy for Si diffusion in diopside is relatively low (211 kJ/mol) when compared with activation energies for diffusion of other cations, but, as noted by Bejina and Jaoul (1996), and earlier in this chapter, this value may represent an apparent activation energy, as experiments were not all performed at constant oxygen fugacity. Al diffusion in diopside at 1 atmosphere (Sautter et al. 1989; Jaoul et al. 1991) is also comparatively slow. In sum, several factors may influence cation diffusion in clinopyroxenes. Variations in diffusion can be a consequence of the sites the ions occupy (tetrahedral vs. lower-energy M l or M2 sites, or interstitial positions). Pyroxene composition has an influence, with higher diffusivities for more Fe-rich clinopyroxenes found for Ca (Dimanov et al. 1996), Sr (Sneeringer et al. 1984), and Pb (Cherniak 1998a, 2001). However, above about 1% Fe, relatively large changes in Fe content appear to produce comparatively small changes in diffusivities (e.g., Pb diffusion in cpx with 17 wt% FeO is up to half an order of magnitude faster than Pb diffusion in diopside with ~ 1 wt% FeO), suggesting that cation diffusion rates are only reduced significantly when Fe content is quite low (< ~ 1 wt%). It may be the case that cation vacancies associated with Fe 3+ become saturated at relatively low Fe contents. Further, it appears that the energy required to jump to an M3 or M4 (e.g., Dimanov and Jaoul 1998) interstitial site is not that much different from the energy required for an M2 cation to move to an adjacent vacancy, and as a result the diffusion mechanism an ion prefers may be very sensitive to size and charge.

DIFFUSION IN AMPHIBOLES AND MICAS The structures of amphiboles and micas are closely related to pyroxenes. Amphibole group minerals are common in metamorphic and igneous rocks, and in a broad range of pressuretemperature conditions. Micas fall into several distinct groups, and studies of diffusion have primarily focused on muscovite, biotite, and phlogopite. While micas and amphiboles are common minerals, and quantifying the diffusion behavior of cations and anions is important in applications of geochronology and geothermometry and in understanding metasomatic processes, there exist relatively few diffusion data for these minerals, which might be attributed to the difficulty in maintaining stability of hydroxyl-bearing phases under a broad range of

Diffusion in Pyroxene, Mica, Amphibole

677

experimental conditions, and, in the case of micas, of assessing potential anisotropy of diffusion in materials with platy geometries. Among early measurements were those of K. Sr and Rb diffusion in biotite by Hofmann and Giletti (1970) and Hofmann et al. (1974). initially using bulk-exchange methods and later ion microprobe analysis to characterize diffusion. Two decades later. Brabander et al. (1995) and Brabander and Giletti (1995) measured F-OH interdiffusion in tremolite and Sr diffusion in tremolite and hornblende, respectively, developing methods to minimize the effects of dissolution-reprecipitation on sample surfaces during diffusion anneals. More recently, Hammouda and Cherniak (2000) circumvented the difficulty of dealing with hydroxyl-bearing phases by using synthetic fluorphlogopite, stable at elevated temperatures at 1-atm, for their Sr diffusion experiments. However, given the ubiquity of hydroxyl in most phlogopites found in nature, it should be stressed that results for diffusion in fluorphlogopite may have limited application to most geologic systems, although these data can provide useful constraints. In addition to the findings on cation and OH-F diffusion reported here, plotted in Figure 12, and summarized in Table A4, there have also been studies of H diffusion in micas (Graham 1981: Graham et al. 1984), oxygen diffusion in micas and hornblende (Fortier and Giletti 199 UFarver and Giletti 1985; Connolly and Muehlenbachs 1988: Giletti and Anderson 1975), and Ar diffusion in micas and amphibole (Giletti 1974; Giletti and Tullis 1977; Harrison 1981: Harrison et al. 1985, 2009; Baldwin et al. 1990; Grove and Harrison 1996). F-OH interdiffusion in tremolite Brabander et al (1995) measured F-OH interdiffusion in tremolite over the temperature range 500-800 °C and 200 MPa pressure, with experiments conducted in cold-seal pressure vessels. Tremolite was exposed to F-poor (distilled water) or F-rich (created through reaction of PdF2 in the sample capsule) fluids in Au or Pt capsules, with F profiles measured by electron microprobe. A second set of experiments, which were analyzed with SIMS, used HF or CsF in aqueous solution to create the F-rich solution. The stability of crystal surfaces was maintained through the use of a silica-buffering assemblage (either quartz or forsterite + diopside +

T(°C) -18

1200

800

1000

700

500

600

OH-F - tremolite

-19

-20 CM £

Sr - tremolite

o

Q

K - biotite parallel to c

O) - 2 1 o Srhornblende -22

Srfluorophlogopite

-23 10

11

12

13

1/T(x107K) Figure 12. Diffusion in amphiboles and micas. Sources for data: Sr - fluorphlogopite: H a m m o u d a and Cherniak (2000): R b - biotite: H o f m a n n and Giletti (1970): K - biotite: H o f m a n n et al. (1974); Sr tremolite, hornblende: Brabander and Giletti (1995); O H - F - tremolite: Brabander et al. ( 1995).

678

Cherniak & Dimanov

tremolite) in the fluid, with a double Pt capsule assembly used to contain sample, solid buffer and fluid. For diffusion parallel to c, an activation energy of 41 ±5 kj/mol and pre-exponential factor of 3.4xl0~ 17 m2/s (log D() = -16.46±0.36) were obtained. Diffusivities normal and parallel to c agreed within experimental uncertainties, as did diffusivities for two tremolites of differing composition. This isotropy of diffusion is in contrast to results for O diffusion (Farver and Giletti 1985) where diffusion parallel to c is 20 times faster than diffusion normal to c. Oxygen likely diffuses as molecular water, while fluorine and hydroxyl migrate as charged species. However, it should be noted that in the study of Brabander et al. (1995) there was a difference of a factor of 5 between diffusivities from experiments conducted on cleaved surfaces and those on polished surfaces, with diffusion in experiments with cleaved surfaces the slower. This suggests that surface damage through polishing may contribute to apparent enhanced diffusivities, or reflects effects of the comparative instability of polished surfaces under the hydrothermal conditions of the experiments. Sr diffusion in tremolite and hornblende Brabander and Giletti (1995) measured Sr diffusion in hornblende and tremolite over the temperature range 700-960 °C at 200 MPa pressure. Experiments were conducted with a quasihydrothermal technique designed to avoid the effects of solution-precipitation and consequent degradation of sample surfaces. A SrCl2 solution was evaporated on polished or cleaved sample surfaces and dried, and samples were sealed inside welded Au capsules; no free water was added to the capsules to avoid dissolution of the amphiboles, but a water fugacity high enough to maintain amphibole stability was created by diffusion of hydrogen through the walls of the noble metal capsule and water released through dehydration reactions on unpolished surfaces of the amphibole. Experiments at temperatures up to 800 °C were conducted in cold-seal pressure vessels, while those at higher temperatures were run in TZM pressure vessels with Ar-CH4 gas as the pressure medium. Cold-seal experiments were buffered at ~Ni-NiO by the cold-seal vessel walls, and the TZM experiments were buffered at Ni-NiO, Mn0-Mn 3 0 4 or magnetitewiistite using a double-capsule technique. Sr depth profiles in the amphiboles were measured with SIMS. For diffusion in hornblende parallel to c, Brabander and Giletti (1995) obtain an activation energy of 260± 12 kj/mol and pre-exponential factor of 4.9x 10 ^ m2/s (log DQ= -7.31 ±0.67). Experiments at 800 °C on two different natural tremolites (the former with a higher Fe content; 2.17 vs. 0.66 wt% FeO) yielded diffusivities (for transport parallel to c) of 3.2 x 10~21 and 2.0xl0~ 21 m2/s. Diffusion in tremolite at this temperature is a factor of 3-5 slower than in hornblende. There is little evidence of diffusional anisotropy for any of the amphiboles, as diffusivities parallel and normal to c agree within experimental uncertainty. Sr diffusion appears relatively insensitive to oxygen fugacity in hornblende, as experiments buffered at Mn0-Mn 3 0 4 and at magnetite-wiistite (with oxygen fugacities differing by 6 x 107) yield diffusivities agreeing with those for Ni-NiO buffered experiments within a factor of 2. This lack of dependence on p02 suggests that Sr diffuses via a vacancy mechanism, and the differences in Sr diffusion among the amphiboles are attributed to the higher Mg to Fe and Mn ratio of the tremolite. The smaller ionic radius of Mg (in the M2 site) compared with these other divalent cations decreases the length of the M(4)-0 bond, making these bonds more difficult to break and reducing the frequency of successful cation jumps to the M4 sites, which Sris likely to occupy in substitution for Ca. Sr diffusion in tremolite is about 2 orders of magnitude slower than OH-F diffusion at 800 °C, a variance that would likely increase with decreasing temperature because of the large differences in activation energies for diffusion between Sr and OH-F diffusion.

Diffusion in Pyroxene, Mica, Amphibole

679

Sr diffusion in fluorphlogopite Hammouda and Cherniak (2000) measured Sr diffusion in synthetic fluorphlogopite over the temperature range 550-1200 °C. Sources for one set of experiments consisted of powder mixtures of oxides and fluorides in proportions of stoichiometric fluorphlogopite, with Sr added as a carbonate (with about 6000 wt ppm Sr in the source mixture). A second set of experiments was conducted by depositing an aqueous SrCl2 solution on the sample surfaces (cleavage planes or growth surfaces) and drying the samples on a hotplate. 1-atm experiments with both sources were annealed in vertical tube furnaces in welded Pt capsules. Sr profiles were measured with RBS. Similar results were obtained for experiments run with the two sources, with an infinite source model used in determining diffusivities for both configurations. For the experiments with the surface-applied SrCl2 source, this solution was used because diffusion profiles were comparatively short, and high concentrations of the tracer remained on the sample surface throughout experiments. A few experiments at modest pressure (28 MPa) were run using fastquench cold seal pressure vessels; these experiments used the powder source, with Pt capsules pressurized with Ar in the cold end of the cold-seal bomb prior to heating. For diffusion parallel to c, Hammouda and Cherniak (2000) obtained an activation energy of 136 ±3 kJ/mol and preexponential factor of 2.7 x 10~14 m2/s (log Da = -13.57). The 1-atm and 28 MPa experiments yielded similar results, but the pressure range was too narrow to establish whether there are any significant pressure effects on Sr diffusion. Sr diffusivities for amphiboles and mica are within an order of magnitude over the temperature range in which experimental measurements overlap (Fig. 12), but the activation energy for Sr diffusion in fluorphlogopite is considerably smaller than that for hornblende (Brabander and Giletti 1995), which is in turn smaller than that for Sr diffusion in diopside (Sneeringer et al. 1984); these variations may be reflective of the influence on Sr transport of structural differences among these minerals. K and Rb diffusion in biotite Hofmann and Giletti (1970) conducted experiments to measure isotopie exchange of Rb, Sr and K between natural biotite and hydrothermal solutions doped with alkali chloride solutions enriched in 41 K and 87Rb, 84Sr, or 85Sr and 86Rb. Experiments were run in cold-seal pressure vessels at 200 MPa and temperatures ranging from 550-700 °C. The relative weights and isotopie ratios in fluids and bulk samples were measured and diffusivities determined by using planar or cylindrical models and assuming that all exchange was due to diffusion. The experiments yielded a diffusivity for Rb in the range of 2 x 10~19 to 7 x 10~2() m2/s at 650 °C, with a similar value for K. There is considerable uncertainty in these determinations due to some evidence of dissolution. In addition, large uncertainties are inherent in bulk exchange experiments such as these, where profiles are not directly measured and assumptions are necessarily made in the modeling. Hofmann et al. (1974) continued these studies, using SIMS instead of bulk exchange methods in analyses. In this work they measured self-diffusion of potassium in biotite by inducing isotopie 41K exchange between the biotite and a 41K-enriched hydrothermal alkali chloride solution at 650 °C and 200 MPa pressure. K isotope profiles in the mica were measured following diffusion anneals. In the c direction, a diffusion coefficient of D = 1 x 10 2 l m7s was determined, with diffusivities in the a and b directions greater than in the c direction by 2 to 4 orders of magnitude. This large difference in K diffusion for different crystallographic orientations is consistent with the findings by Hofmann and Giletti (1970) for Rb diffusion, which are 2-3 orders of magnitude faster than K diffusion parallel to c. It might be assumed that the large alkali ions K and Rb would diffuse at similar rates, and with similar anisotropy. The bulk exchange method used by Hofmann and Giletti (1970) does not permit direct evaluation of such anisotropy, and diffusivities determined via this method would be biased toward the fastest diffusion directions.

680

Cherniak & Dimanov ACKNOWLEDGMENTS

We thank Jibamitra Ganguly for his thorough review of the manuscript, and Youxue Zhang for his editorial efforts for this chapter. We also gratefully acknowledge the legacy and career of Olivier Jaoul, who was one of the pioneering leaders in the study of diffusion in minerals, an exceptionally talented teacher, and a very motivated and motivating person.

REFERENCES Anovitz LM, Essene EJ, Dunham WR (1988) Order-disorder experiments in orthopyroxenes: Implications for the orthopyroxene geospeedometer. Am Mineral 73:1060-1073 Azough F, Freer R (2000) Iron diffusion in single-crystal diopside. Phys Chem Miner 27:732-740 Azough F, Freer R, Wright K, Jackson R (1998) A computer simulation study of point defects in diopside and the self-diffusion of Mg and Ca by a vacancy mechanism. Mineral Mag 62:59-606 Baldwin SL, Harrison TM, Fitz Gerald JD (1990) Diffusion of 40 Ar in metamorphic hornblende. Contrib Mineral Petrol 105:691-703 Baxter E F (2010) Diffusion of noble gases in minerals. Rev Mineral Geochem 72:509-557 Bejina F, Jaoul O (1996) Silicon self-diffusion in quartz and diopside measured by nuclear micro-analysis methods. Phys Earth Planet Inter 97:145-162 Besancon JR (1981) Rate of cation ordering in orthopyroxenes. Am Mineral 66:965-973 Boyd FR, Schairer JF (1964) The system MgSi0,-CaMgSi 2 0 6 . J Petrol 5:275-309 Brabander DJ, Giletti BJ (1995) Strontium diffusion kinetics in amphiboles and significance to thermal history determination. Geochim Cosmochim Acta 59:2223-2238 Brabander DJ, Hervig RL, Jenkins DM (1995) Experimental determination of F-OH interdiffusion in tremolite and significance to fluorine-zoned amphiboles. Geochim Cosmochim Acta 59:3549-3560 Brady JB (1983) Intergranular diffusion in metamorphic rocks. Am J Sci 283 A:181-200 Brady JB, McCallister RH (1983) Diffusion data for clinopyroxenes from homogenization and self-diffusion experiments. Am Mineral 68:95-105 Buening DK, Buseck PR (1973) Fe-Mg lattice diffusion in olivine. J Geophys Res 78:6852-6862 Cameron M, Papike JJ (1981) Structural and chemical variations in pyroxenes. Am Mineral 66:1-50 Chakraborty S, Dohmen R, Miiller T, Becker HW, ter Heege J (2008) Fe-Mg interdiffusion coefficients in clinopyroxene: Experimental determinations using nanoscale thin films, (abstr.) AGU Fall Meeting, San Francisco, CA, abstract #MR21C-04 Chakraborty S, Farver JR, Yund RA, Rubie DC (1994) Mg tracer diffusion in synthetic forsterite and San Carlos Olivine as a function of P, r a n d / 0 2 . Phys Chem Miner 21:489-500 Chakraborty S, Rubie DC (1996) Mg tracer diffusion in aluminosilicate garnets at 750-850 degrees C, 1 atm. and 1300 degrees C, 8.5 GPa. Contrib Mineral Petrol 122:406-414 Cherniak DJ (1995) Diffusion of lead in plagioclase and K-feldspar: an investigation using Rutherford Backscattering and resonant nuclear reaction analysis. Contrib Mineral Petrol 120:358-371 Cherniak DJ (1997) An experimental study of Sr and Pb diffusion in calcite, and implications for carbonate diagenesis and metamorphism. Geochim Cosmochim Acta 61: 4173-4179 Cherniak DJ (1998a) Pb diffusion in clinopyroxene. Chem Geol 150:105-117 Cherniak DJ (1998b) REE diffusion in calcite. Earth Planet Sci. Lett 160: 273-287 Cherniak DJ (2001) Pb diffusion in Cr diopside, augite, and enstatite, and consideration of the dependence of cation diffusion in pyroxene on oxygen fugacity. Chem Geol 177:381-397 Cherniak DJ (2003) Silicon self-diffusion in single-crystal natural quartz and feldspar. Earth Planet Sci Lett 214:655-668 Cherniak DJ, Hanchar JM, Watson EB (1997) Rare earth diffusion in zircon. Chem Geol 134:289-301 Cherniak DJ, Liang, Y (2007) Rare earth element diffusion in natural enstatite. Geochim Cosmochim Acta 71:1324-1340 Cherniak DJ, Watson EB (1992) A study of strontium diffusion in K-feldspar, Na-K feldspar and anorthite using Rutherford Backscattering Spectroscopy. Earth Planet Sci Lett 113:411-425 Cherniak DJ, Watson EB (1994) A study of strontium diffusion in plagioclase using Rutherford Backscattering Spectroscopy. Geochim Cosmochim Acta 58:5179-5190 Choudhury N, Ghose S, Chowdhury CP, Loong CK, Chaplot SL (1998) Lattice dynamics, Raman spectroscopy, and inelastic neutron scattering of orthoenstatite Mg 2 Si 2 0 6 . Phys Rev B 58:756-765 Connolly C, Muehlenbachs K (1988) Contrasting oxygen diffusion innepheline, diopside and other silicates and their relevance to isotopic systematics in meteorites. Geochim Cosmochim Acta, 52:1585-1591 Coogan LA, Kasemann SA, Chakraborty S (2005) Rates of hydrothermal cooling of new oceanic upper crust derived from lithium-geospeedometry. Earth Planet Sci Lett 240:415-424

Diffusion in Pyroxene, Mica, Amphibole

681

Demouchy S, Mackwell SJ, Kohlstedt DL (2007) Influence of hydrogen on Fe-Mg interdiffusion in (Mg,Fe)0 and implications for Earth's lower mantle. Contrib Mineral Petrol 154:279-289 Dimanov A, Dresen G (2005) Rheology of synthetic anorthite-diopside aggregates: implications for ductile shear zones. J Geophys Res 110:B07203, D01:10.1029/2004JB003431 Dimanov A, Ingrin J (1995) Premelting and high-temperature diffusion of calcium in synthetic diopside: An increase of the cation mobility. Phys Chem Miner 22:437-442 Dimanov A, Jaoul O (1998) Calcium self-diffusion in diopside at high temperature: implications for transport properties. Phys Chem Miner 26:116-127 Dimanov A, Jaoul O, Sautter V (1996) Calcium self-diffusion in natural diopside single crystals. Geochim Cosmochim Acta 60:4095-4106 Dimanov A, Lavie MP, Dresen G, Ingrin J, Jaoul O (2003) Creep of polycrystalline anorthite and diopside. J Geophy. Res 108 (Bl):2061, D01:10.1029/2002JB001815 Dimanov A, Rybacki E, Wirth R, Dresen G (2007) Creep and strain-dependent microstructures of synthetic anorthite-diopside aggregates, J Struct Geol 29:1049-1069 Dimanov A, Sautter V (2000) "Average" interdiffusion of (Fe,Mn)-Mg in natural diopside. Eur J Mineral 12:749-760 Dimanov A, Wiedenbeck M (2006) (Fe,Mn)-Mg interdiffusion in natural diopside: effect of p02. Eur J Mineral 18:705-718 Dodson M H (1973) Closure temperature in cooling geochronological and petrological systems. Contrib Mineral Petrol 40:259-274 Dodson MH (1986) Closure profiles in cooling systems. Mater Sci Forum 7:145-154 Dohmen R, Chakraborty S, Becker H W (2002) Si and O diffusion in olivine and implications for characterizing plastic flow in the mantle. Geophys Res Lett 29(21), doi 10.1029/2002GL015480 Doukhan N, Doukhan JC, Ingrin J, Jaoul O, Raterron P (1993) Early partial melting in pyroxenes. Am Mineral 78:1247-1257 Duba A, Boland J, Ringwood AE (1973) The electrical conductivity of pyroxene. J Geol 81:727-735 Duba A, Piwinskii AJ, Heard HC, Schock RN (1976) The electrical conductivity of forsterite, enstatite and albite. In: The Physics and Chemistry of Rocks and Minerals. Strens RGJ (ed) John Wiley & Sons, London p 249-260 Farver JR (2010) Oxygen and hydrogen diffusion in minerals. Rev Mineral Geochem 72:447-507 Farver JR, Giletti BJ (1985) Oxygen diffusion in amphiboles. Geochim Cosmochim.Acta 49:1403-1411 Foland KA (1974) Alkali diffusion in orthoclase. In: Geochemical Transport and Kinetics. Hofmann AW, Giletti BJ, Yoder Jr HS, Yund RA (eds) Carnegie Institution of Washington Publication, 634, Carnegie Institution of Washington, Washington, DC, p 77-98 Fortier SM, Giletti BJ (1991) Volume self-diffusion of oxygen in biotite, muscovite, and phlogopite micas. Geochim Cosmochim Acta 55:1319-1330 Fraser DG, Lawless PJ (1978) Paleogeotherms: implication of disequilibrium in garnet lherzolite xenoliths. Nature 273:220-222 Freer R (1979) An experimental measurement of cation diffusion in almandine garnet. Nature 280:220-222 Freer R, Carpenter MA, Long JVP, Reed SJB (1982) "Null result" diffusion experiments with diopside: implications for pyroxene equilibria. Earth Planet Sci Lett 58:285-292 Ganguly J, Cheng W, Chakraborty S (1998) Cation diffusion in aluminosilicate garnets; experimental determination in pyrope-almandine diffusion couples. Contrib Mineral Petrol 131:171 -180 Ganguly J, Dasgupta S, Cheng W, Neogi S (2000) Exhumation history of a section of the Sikkim Himalayas, India: records in the metamorphic mineral equilibria and compositional zoning of garnet. Earth Planet Sci Lett 183:471-486 Ganguly J, Ito M, Zhang X (2007) Cr diffusion in orthopyroxene; experimental determination, 5 , Mn- 53 Cr thermochronology, and planetary applications. Geochim Cosmochim Acta71:3915-3925 Ganguly J, Tazzoli V (1994) Fe 2+ -Mg interdiffusion in orthopyroxene: Retrieval from the data on intracrystalline exchange reaction. A m Mineral 79:930-937 Ganguly J, Tirone M (2001) Relationship between cooling rate and cooling age of a mineral; theory and applications to meteorites. Meteoritics Planet Sci 36:167-175 Ganguly J, Yang, H, Ghose S (1994) Thermal history of mesosiderites: constraints from compositional zoning and Fe-Mg ordering in orthopyroxenes. Geochim Cosmochim Acta 48:2711-2723 Gasc J, Béjina F, Ingrin J (2006) Magnesium diffusion in diopside. 11th International Conference on Experimental Mineralogy, Petrology and Geochemistry Gasparik T (1984) Experimentally determined stability of clinopyroxene + garnet + corundum in the system CaO-MgO A 1 0 r S i 0 2 . Am Mineral 69:1025-1035 Gasparik T, Lindsley DH (1980) Phase equilibria at high pressure of pyroxenes containing monovalent and trivalent ions. Rev Mineral 7:309-339

682

Cherniak & Dimanov

Giletti BJ (1974) Studies in diffusion; 1, Argon in phlogopite mica. In: Geochemical Transport and Kinetics. Carnegie Institution of Washington Publication 634. HofmannAW, Giletti BJ, Yoder HS Jr., Yund RA (eds) Carnegie Institution of Washington, Washington, DC, p 107-115 Giletti BJ, Anderson TF (1975) Studies in diffusion; II, Oxygen in phlogopite mica. Earth Planet Sei Lett 28:225-233 Giletti BJ, Casserly JED (1994) Strontium diffusion kinetics in plagioclase feldspars. Geochim Cosmochim Acta 58:3785-3793 Giletti BJ, Shanahan TM (1997) Alkali diffusion in plagioclase feldspar. Chem Geol 139:3-20 Giletti BJ, Tullis J (1977) Studies in diffusion; IV. Pressure dependence of Ar diffusion in phlogopite mica. Earth Planet Sei Lett 35:180-183 Graham CM (1981) Experimental hydrogen isotope studies III: Diffusion of hydrogen in hydrous minerals, and stable isotope exchange in metamorphic rocks. Contrib Mineral Petrol 76:216-228 Graham CM, Harmon RS, Sheppard SMF (1984) Experimental hydrogen isotope studies: Hydrogen isotope exchange between amphibole and water. A m Mineral 69:128-138 Grove M, Harrison TM (1996) 40 Ar* diffusion in Fe-rich biotite. Am Mineral 81:940-951 Hammouda T, Cherniak DJ (2000) Diffusion of Sr in fluorphlogopite determined by Rutherford backscattering spectrometry. Earth Planet Sei Lett 178:339-349 Harrison TM (1981) Diffusion of 40 Ar in hornblende. Contrib Mineral Petrol 178:324-331 Harrison TM, Celerier J, Aikman AB, Hermann J, Heizier MT (2009) Diffusion of 40 Ar in muscovite. Geochim Cosmochim Acta 73:1039-1051 Harrison TM, Duncan I, McDougall I (1985) Diffusion of 40 Ar in biotite; temperature, pressure and compositional effects. Geochim Cosmochim Acta 49:2461-2468 Hier-Majumder S, Anderson IM, Kohlstedt DL (2004) Influence of protons on Fe-Mg interdiffusion in olivine. J Geophys Res 110, DOI: 10.1029/2004JB003292 Hirsch LM, Shankland TJ (1991) Equilibrium point-defect concentrations in MgO: understanding the mechanisms of conduction and diffusion and the role of Fe impurities. J Geophys Res 96:385-403 Hirschmann MM, Ghiorso MS, Davis FA, Gordon SM, Mukherjee S, Grove TL., Krawczynski, M, Medard E, Till CB (2008) Library of experimental phase relations (LEPR): a database and web portal for experimental magmatic phase equilibria data. Geochem Geophys Geosyst 9 (3), DOI: 10.1029/2007GC001894 Hofmann AW, Giletti BJ (1970) Diffusion of geochronologically important nuclides in minerals under hydrothermal conditions. Eclogae Geologicae Helvetiae 63:141-150 Hofmann AW, Giletti BJ, Hinthorne JR, Andersen CA, Comaford D (1974) Ion microprobe analysis of a potassium self-diffusion experiment in biotite. Earth Planet Sei Lett 24:48-52 Hofmann AW, Hart SR (1978) An assessment of local and regional isotopic equilibrium in the mantle. Earth Planet Sei Lett 38:44-62 Holloway JR, Pan V, Gudmundson G (1992) High pressure fluid absent melting experiments in the presence of graphite: oxygen fugacity, ferric/ferrous ratio and dissolved C 0 2 . Eur J Mineral 4:105-114 Houlier B, Cheraghmakani M, Jaoul O (1990) Silicon diffusion in San Carlos Olivine. Phys Earth Planet Inter 62:329-340 Huebner JS (1976) Diffusively rimmed xenocrysts in 77115. Lunar Planet Sei Conf 7:396-398 Huebner JS, Nord GL (1981) Assessment of diffusion in pyroxenes: what we do and do not know. Lunar Planet Sei XII, p 479-481 Huebner JS, Ross M, Hikling N (1975) Cooling history and significance of exsolved pyroxene in lunar noritic breccia 77215. Lunar Planet Sei Conf 6:408-410 Huebner JS, Voigt DE (1988) Electrical conductivity of diopside: Evidence for oxygen vacancies. Am Mineral 73:1235-1254 Ingrin J, Pacaud L, Jaoul O (2001) Anisotropy of oxygen diffusion in diopside. Earth Planet Sei Lett 192:347361 Jantzen CM (1984) On spinodal decomposition in Fe-free pyroxenes. Am Mineral 69:277-282 Jaoul O, Raterron P (1994) High-temperature deformation of diopside crystal. 3. Influences of p02 and S i 0 2 precipitation. J Geophys Res 99:9423-9439 Jaoul O, Sautter V (1999) A new approach to geospeedometry based on the "compensation law". Phys Earth Planet Inter 110:95-114 Jaoul O, Sautter V, Abel F (1991) Nuclear microanalysis: a powerful tool for measuring low atomic diffusivity with mineralogical applications. In: Diffusion, Atomic Ordering and Mass Transport—Selected Topics in Geochemistry. Advances in Physical Geochemistry, Vol 8. Ganguly J (ed) Springer, New York, p 198-220 Joesten R (1991) Grain-boundary diffusion kinetics in silicate and oxide minerals. In: Diffusion, Atomic Ordering and Mass Transport—Selected Topics in Geochemistry. Advances in Physical Geochemistry, Vol 8. Ganguly J (ed) Springer, New York, p 345-395 Klügel A (2001) Prolonged reactions between harzburgite xenoliths and silica-undersaturated melt: implications for dissolution and Fe-Mg interdiffusion rates of orthopyroxene. Contrib Mineral Petrol 141:1-14

Diffusion in Pyroxene, Mica, Amphibole

683

Kröger FA (1964) The Chemistry of Imperfect Crystals. North Holland, Amsterdam Kröger FA, Vink HJ (1956) Relations between the concentrations of imperfections in crystalline solids. In: Solid State Physics Advances in Research and Applications, Vol. 3. Seitz F, Turnbull D (eds) Academic Press, New York, p 307-435 Lasaga AC (1983) Geospeedometry: An Extension of Geothermometry. In: Kinetics and Equilibrium in Mineral Reactions, Vol 3. Saxena SK (ed) Springer, New York, p 81-114 Lasaga AC, Richardson SM, Holland HD (1977), The mathematics of cation diffusion and exchange between silicate minerals during retrogrde metamorphism. In: Saxena SK, Bhattachanji S (eds.), Energetics of Geochemical Processes, 353-388 LaTourrette T, Wasserburg GJ (1998) Mg diffusion in anorthite; implications for the formation of early solar system planetesimals. Earth Planet Sei Lett 158:91-108 Liang Y, Cherniak DJ, Morgan ZT, Hess PC (2004) Eu 2+ and R E E , + diffusion in enstatite, diopside, anorthite, and a silicate melt: A database for understanding kinetic fractionation of REE in the lunar mantle and crust. In: Lunar Planet Sei XXXV. Lunar Planet. Inst, Houston. CD-ROM #1894 (abstract) Lindner R (1955) Studies on solid state reactions with radiotracers. J Chem Phys 23:410-411 McCallister RH (1978) The coarsening kinetics associated with exsolution in an iron-free clinopyroxene. Contrib Mineral Petrol 65:327-331 McCallister RH, Brady JB, Mysen BO (1979) Self-diffusion of calcium in diopside. Carnegie Institution of Washington Yearbook 78:574-577 McCallister RH, Nord G (1981) Subcalcic diopsides from kimberlites: Chemistry, exsolution microstruetures, and thermal history. Contrib Mineral Petrol 78:118-125 McCallum IS, O'Brien HE (1996) Stratigraphy of the lunar highland crust: Depths of burial of lunar samples from cooling-rate studies. Am Mineral 81:1166-1175 Médard E, McCammon CA, Grove TL (2008) Oxygen fugacity, temperature reproducibility and H 2 0 content for nominally dry piston-cylinder experiments using graphite capsules. Am Mineral 93:1838-1844, DOI: 10.2138/am.2008.2842 Miyamoto M, Takeda H ( 1977) Evaluation of a crust model of eucrites from the width of exsolved pyroxene. Geochem J 11:161-169 Miyamoto M, Takeda H (1994) Evidence for excavation of deep crustal material of a Vesta-like body from Ca compositional gradients in pyroxene. Earth Planet Sei Lett 122:343-349 Nakamura A, Schmalzried H (1983) On the nonstoichiometry and point defects of olivine. Phys Chem Miner 10:27-37 Nakamura E, Kushiro I (1998) Trace element diffusion in jadeite and diopside melts at high pressures and its geochemical implication. Geochim Cosmochim Acta 62:3151-3160 Otto EM (1966) Equilibrium pressures of oxygen over oxides of lead at various temperatures. J Electrochem Soc 113:525-527 Pacaud L (1999) Mesure de l'autodiffusion de l'oxygène et du magnésium dans le diopside et applications. PhD Thesis, Université de Paris 11, Orsay, France Pacaud L, Ingrin J, Jaoul O (1999) High-temperature diffusion of oxygen in synthetic diopside measured by nuclear reaction analysis. Mineral Mag 63:673-686 Pattison DRM, Begin NJ (1994) Zoning patterns in orthopyroxene and garnet in granulites: implications for geothermometry. J Metamorph Geol 12:387-410 Raterron P, Jaoul O (1991) .High-temperature deformation of diopside single crystal. 1. Mechanical data. J Geophys Res 96:14,277-14,286 Richet P, Fiquet G (1991) High-temperature heat capacity and premelting of minerals in the system MgO-CaOA l 2 0 r S i 0 2 . J Geophys Res 96:445-456 Richet P, Ingrin J, Mysen BO, Courtial P, Gillet P (1994) Premelting effects in minerals: an experimental study. Earth Planet Sei Lett 121:589-600 Rietmeijer FJM (1983) Inter-diffusion coefficients parallel to the oaxis in iron-rich clinopyroxenes calculated from microstructures. Contrib Mineral Petrol 83:169-176 Sautter V, Harte B (1988) Diffusion gradients in an eclogite xenolith from the Roberts Victor kimberlite pipe: (1) mechanism and evolution of garnet exsolution in A^Oj-rich clinopyroxene. J Petrol 29:1325-1358 Sautter V, Harte B (1990) Diffusion gradients in an eclogite xenolith from the Roberts Victor kimberlite pipe: (2) kinetics and implications for petrogenesis. Contrib Mineral Petrol 105:637-649 Sautter V, Jaoul O, Abel F (1989) Aluminium diffusion in diopside using the 27 Al(p,y) 28 Si nuclear reaction: preliminary results. Earth Planet Sei Lett 89:109-114 Saxena SK, Tazzoli V, Domeneghetti MC (1987) Kinetics of Fe 2+ -Mg distribution in aluminous orthopyroxenes. Phys Chem Miner 15:140-147 Schreiber HD (1977) Redox states of Ti, Zr, Hf, Cr, and Eu in basaltic magmas; an experimental study. In: Proceedings of the Eighth Lunar Science Conference. Pergamon Press, New York, NY, p 1785-1807 Schwandt CS, Cygan RT, Westrich HR (1998) Magnesium self-diffusion in orthoenstatite. Contrib Mineral Petrol 130:390-396

684

Cherniak & Dimanov

Schwartz JM, McCallum IS (2005) Comparative study of equilibrated and unequilibrated eucrites: Subsolidus thermal histories of Haraiya and Pasamonte. Am Mineral 90:1871 -1866 Seitz MG (1973) Uranium and thorium diffusion in diopside and fluorapatite. Year Book - Carnegie Institution of Washington 72:586-588 Shannon RD (1976) Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. ActaCryst A32: 751-767 Smyth DM, Stacker RL (1975) Point defects and non-stoichiometry in forsterite. Phys Earth Planet Inter 10: 183-192 Smyth JR (1974) Low orthopyroxene from a lunar deep crustal rock: a new pyroxene polymorph of space group P2!ca. Geophys Res Lett 1: 27-29 Sneeringer M, Hart SR (1978) Sr diffusion in diopside. EOS Trans Am Geophys Union 59:402 Sneeringer M, Hart SR, Shimizu N (1984) Strontium and samarium diffusion in diopside. Geochim Cosmochim Acta 48: 1589-1608 Sotin C, Poirrier JP (1984) Analysis of high temperature creep experiments by generalized nonlinear inversion. Mech Mater 3 : 311-317 Stanford RF, Huebner JS (1979) Reexamination of diffusion processes in 77115 and 77215. Lunar Planet Sci X, p 1052-1054 Stimpfl M, Ganguly J, Hervig R (2003) Ca and Mg tracer diffusion in diopside: experimental determination and application to cooling history of planetary samples. XXXIV Lunar Planet Sci Conf, League City, Texas, abstract no. 1497 Stimpfl M, Ganguly J, Molin G (2005) Kinetics of Fe 2+ -Mg order-disorder in orthopyroxene; experimental studies and applications to cooling rates of rocks. Contrib Mineral Petrol 150:319-334 Stacker RL (1978) Variation of electrical conductivity in enstatite with oxygen partial pressure: Comparison of observed and predicted behaviour. Phys Earth Planet Inter 17:34-40 Taupin D (1998) Probability estimations for fitted parameters - the EXPHER package. J Microscopy 190:19-27 ter Heege JH, Dohmen R, Becker H, Chakraborty S (2006) Experimental determination of Fe-Mg interdiffusion coefficients in orthopyroxene using pulsed laser ablation and nanoscale thin films. AGU Fall Meeting, San Francisco, CA. abstract #MR21A-0004 Tsai TL, Dieckmann R (2002) Variation of the oxygen content and point defects in olivines, (Fe x Mg 1 _ x ) 2 Si0 4 , 0.2 < x < 1.0. Phys Chem Miner 29:680-694 Ubbelohde AR (1978) The Molten State of Matter. John Wiley and Sons, New York Van Orman JA, Grove TL, Shimizu N (1998) Uranium and thorium diffusion in diopside. Earth Planet Sci Lett 160:505-519 Van Orman JA, Grove TL, Shimizu N (2001) Rare earth element diffusion in diopside; influence of temperature, pressure, and ionic radius, and an elastic model for diffusion in silicates. Contrib Mineral Petrol 141:687703 Watson EB, Liang Y (1995) A simple model for sector zoning in slowly grown crystals; implications for growth rate and lattice diffusion, with emphasis on accessory minerals in crustal rocks. Am Mineral 80:1179-1187 Weber P, Taupin D (1995) EXPHER (Experimental PHysics ERror analysis): a declaration language and a program generator for the treatment of experimental data. J Phys III 5(5):605-622 Weinbruch S, Miiller W F (1995) Constraints on the cooling rates of chondrules from the microstructure of pyroxene and plagioclase. Geochim Cosmochim Acta 59:3221-3230 Weinbruch S, Miiller WF, Hewins R H (2001) A transmission electron microscope study of exsolution and coarsening in iron-bearing clinopyroxene from synthetic analogues of chondrules. Meteoritics Planet Sci 36:1237-1248 Weinbruch S, Styrsa V, Miiller W F (2003) Exsolution and coarsening in iron-free clinopyroxene during isothermal annealing. Geochim Cosmochim Acta 67:5071-5082 Yang H, Ghose S (1995) High temperature single crystal X-ray diffraction studies of the ortho-proto phase transition in enstatite, Mg 2 Si 2 0 6 at 1360 K. Phys Chem Miner 22:300-310 Yund RA, Tullis J (1991) Compositional changes of minerals associated with dynamic recrystallization. Contrib Mineral Petrol 108:346-355 Zhang X, Ganguly J, Ito M (2010) Ca-Mg diffusion in diopside: tracer and chemical inter-diffusion coefficients. Contrib Mineral Petrol 159:175-186, DOI: 10.10007/s00410-009-0422-5

Diffusion in Pyroxene, Mica,

Amphibole

APPENDIX On the following pages, Tables A1-A4 contain the diffusion data and selected Arrhenius relations for the pyroxene, mica, and amphibole mineral phases discussed in this chapter.

685

Cherniak

686

&

Dimanov

Table A l . Diffusion data for major cations in pyroxenes where D = D0 (p02)m e x p O X F c ) Q\\)(-Q/RT) = D 0 'exp(-g//?7). T

P

(°C)

(MPa)

(MPa)

Ca-(Mg,Fe)

1150-1250

2500

c-o 2

Al-(Si,Mg?)

1000-1180

0.1

10-14

Si

1040-1250

0.1

10-14-10-17

44

Ca

1000-1242 1242-1380

0.1 0.1

10-7 10-7

|| to b,c

44

Ca

1000-1130 1130-1250

0.1 0.1

10-7-10-18 0.02-10-18

Diopside 2% Fe

|| to b,c

44

Ca

1000-1230 1230-1320

0.1 0.1

10-"-10- |fl 10- |3 -10" 18

Diopside 1.8% Fe Synthetic

11to b,c || to a

54

950-1100 1050

0.1 0.1

10-"-10- 16 10-14

Diopside 2.9% Fe

II to c

(Fe,Mn)-Mg

900-1240

0.1

lo-^-io- 19

Diopside 2.9% Fe

II to c

(Fe,Mn)-Mg

1000-1190

0.1

10-7-10-19

Opx 10-50% Fe

|| to b/c

Fe-Mg

500-800

0.1

IW

Enstatite 11% Fe

|| to a || to b II to c

750-900

0.1

IW

Enstatite 2.6% Fe

|| to a || to AII to c

1000-1100 900-1100 900-1100

0.1

IW

Diopside Synthetic

11 to a,b,c

26

Mg

1200-1300

0.1

0.02

Diopside Natural 3.6% Fe

|| to a*,b

26

Mg

1100 1000-1100

0.1 0.1

10-5 10-5

Diopside 1.8% Fe

|| to a* || to b II to c

44

Ca

950-1150

0.1

IW

Diopside 1.8% Fe

|| to a* || to 6 II to c

26

Mg

950-1150

0.1

IW

Mineral

Orientation

Species

Cpx 2.8% Fe

1 to (001)

Diopside 2.4% Fe

II to c

Diopside 1.8% Fe

II to c

30

Diopside Synthetic

|| to b

Diopsides 0.4-2.4% Fe

25

Fe

Mg

Cr

pOi

t when possible, recalculated to p02=10~ 12 M Pa ** Mg diffusion coefficient calculated from the data of Pacaud (1999) and Case et al. (2006). *** Mg diffusion coefficient calculated from the data of Gasc et al. (2006) at 1100 °C and the activation energy £.,= 214 kj/ mol from Azough and Freer (2000).

Diffusion in Pyroxene, Mica, Amphibole

Do

logD'o

(m 2 /s)

(m2/s)t

(kj/mol)

m

n

3.9 xlO" 7

361 ± 1 9 0

—

—

2.11x10-"

273

—

—

687

References Brady and McCallister (1983) Sautter etal. (1989), Jaoul et al. (1991)

2.3 xlO- 1 4

-13.6±4 (variable p02)

211 + 110

—

—

Bejinaand Jaoul (1996)

1.58x10"" 2.5x10'2

-6.8 ±1 16.4±2.9

280±26 951 ± 8 7

—

—

Dimanov and Ingrin (1995)

7.04 xlO- 1 4 2.38 xlO- 7

-10.96+1.3 -6.14+1.4

264±33 396±38

-0.14±0.01 0

55.3 55.3

2.63 xlO"' 3 5.62x 1014

-10.3±0.4 18.75 ±2.46

284± 10 1006 ± 7 5

- 0 . 1 9 ±0.03 0

1.03x10"" 5.7 xlO- 2 2

— 13.75 + 1.13 - 2 1 . 2 4 ±0.18

162 ± 3 5

0.23 ±0.04

9.55 x 10-5

- 4 . 0 2 ±0.32 (variable p02)

406 ± 6 4

1.62x lO"6

- 8 . 4 3 ±0.32

297 ± 3 1

2.88x10"'°

- 9 . 2 8 (10% Fe) (variable p 0 2 )

240 ± 8

1.1 xlO- 4 6.93 x 10-6 4.34 xlO- 9

-3.96±2.48 5.16 + 3.69 -8.36±4.34

360 ± 5 2 339±77 265 ± 6 6

7.41x10"" 1.78x10-'° 3.10x10-'°

—10.13 + 0.39 - 9 . 7 5 ±0.78 - 9 . 5 1 ±0.56

257 ± 12 257 ± 12 257 ± 12

-0.09

4.07 xlO 7

7.61 ±2.97

748 ± 8 7

—

58x10" 2 0 1.2x10""

-19.07±0.18 -10.92

214

1.99 xlO" 8 3.2 xlO" 7 2.4x10-'°

- 7 . 0 7 ±0.70 - 6 . 5 2 ±1.29 - 9 . 6 2 ±0.91

319+18 350 ± 3 2 265 ± 2 3

Zhang etal. (2010)

6.76 xlO"' 5 1.23x10"" 1.44x10-"

-14.17±0.49 -10.91 ±0.93 -12.84±0.68

150±22 231 ± 2 3 176+ 18

Zhang et al. (2010)

—

0.22 ±0.02

—

—

Dimanov et al. (1996) Dimanov and Jaoul (1998) Azough and Freer (2000)

—

— 5.99

Dimanov and Sautter (2000) Dimanov and Wiedenbeck (2006) Ganguly and Tazzoli (1994) Schwandt et al. (1998)

Ganguly et al. (2007)

—

**Pacaud (1999), Gasc et al. (2006) Gasc et al. (2006) ***£ a from Azough and Freer (2000)

Cherniak & Dimanov

688

Table A2. Arrhenius relations for trace and minor elements in pyroxenes,

of the form D = D0exp(-EJRT). T

P

(°C)

(MPa)

Li

800-1100

0.1

[110]

Pb

800-1100

0.1

Diopside, 0.3 at% Fe

[001]

Pb

800-1100

0.1

Diopside, 0.3 at% Fe

[110], [001]

Pb

800-1100

0.1

Clinopyroxene 5.4 at% Fe

[110]

Pb

800-1050

0.1

Clinopyroxene 5.4 at% Fe

[110]

Pb

800-1050

0.1

Cr diopside, 0.4 at% Fe

[110]

Pb

850-1050

0.1

Cr diopside, 0.4 at% Fe

[001]

Pb

850-1050

0.1

Augitic pyroxene, 1.2 at% Fe

[110]

Pb

850-1050

0.1

Enstatite, 0.2 at% Fe

[210]

Pb

850-1100

0.1

Diopside, 0.2 at% Fe

[001]

La

1200-1300

0.1

Diopside, 0.2 at% Fe

[001]

Ce

1150-1450

0.1

Diopside, 0.2 at% Fe

[001]

Nd

1200-1300

0.1

Diopside, 0.2 at% Fe

[001]

Dy

1100-1300

0.1

Diopside, 0.2 at% Fe

[001]

Yb

1050-1300

0.1

Diopside, 1 at% Fe

Mineral

Orientation

Species

Diopside, 0.7 at% Fe

[010]

Diopside, 0.3 at% Fe

[001]

Sr

1200-1300

0.1

Diopside,

A k

9

5 // M g

- Ak lie (c)

O A k / -17

Mg - Ak 7 0 Gh 3 0 / l l a

-18

AlAI^MgSi(gh)] -19 -20

¡ ¡ 0 % M g - Ak lla

Co-gh / (coupled e x c h a n g e )

Mg - Gh lla Mg - Ak lie (s)

-21

Mg

Gh lie

1/T(x107K) Figure 6. Data for A1 + A1 Mg + Si interdiffusion in melilite from Nagasawa et al. (2001) for various compositions along the gehlenite-akermanite solid solution, and for Mg diffusion from the studies of LaTourette and Hutcheon (1999), Ito et al. (2001) and Ito and Ganguly (2009). Al + Al Mg + Si diffusivities for near-end member gehlenite are the slowest, with intermediate compositions having - 8 0 mol% akermanite component the fastest. Activation energies for Al + Al Sr = 4.1x10~ exp(-151 ±21 kJ/mol/^7) m /s D Ua = 1.5xl0~ exp(-133±25 kJ/mol//?7) m /s

Diffusion

in Quartz, Melilite,

Perovskite,

Midlife

749

T ( C) 1300

1200

1100

1000

900

w

CN

E, o ¡3) O

6.0

6.5

7.0

7.5

8.0

8.5

9.0

4

1/T(X10 /K) Figure 8. Diffusion of Ca, Sr. Ba and K in akermanite (dashed lines) and gehlenite (solid lines). Diffusion of all of these species is faster in akermanite than in gehlenite. Among the large divalent cations Ca. Sr and Ba, there is no systematic trend of decreasing diffusivities with increasing ionic radii. For K, there is slight anisotropy of diffusion in gehlenite. with diffusion parallel to the «-axis slightly faster than diffusion parallel to the c -axis. Sources for data: Ca, Sr, Ba in akermanite - Morioka and Nagasawa (1991); Ca, Sr, Ba in gehlenite - Morioka et al. (1997); K - Ito and Ganguly (2004).

Diffusivities of these elements are about an order of magnitude slower in gehlenite than in akermanite over the investigated temperature range, but activation energies for diffusion in akermanite are considerably higher than in gehlenite. Activation energies for diffusion also decrease with increasing ionic radius from Ca to Ba for both end-member melilite compositions. Morioka and Nagasawa (1991) argue that the loose structure of melilites may permit easier migration of large cations on M-sites and thus there is little difference in diffusivities among likecharged cations, but it remains unclear why the largest divalent cation (Ba) would have the lowest activation energies for diffusion in both akermanite and gehlenite, when this is the opposite of trends observed for other mineral systems, where larger cations frequently have higher activation energies for diffusion. Further, there is little evidence of systematic behavior when comparing Ca, Sr and Ba with Mn and Fe, as it has been suggested that the latter two elements also diffuse on the M site. However, it should be noted that there is some scatter in the diffusion data, and relatively few data points to constrain each Arrhenius relation. Potassium Ito and Ganguly (2004) measured potassium diffusion in melilite, motivated by the detection in melilite from CAIs of excess 41K produced by decay of short-lived 41Ca. Results from this study are plotted in Figure 8. Polished synthetic akermanite (cut normal to the c-axis) and gehlenite (cut normal to the a- or c-axes) crystals were coated with a thin film of potassium by thermal evaporation of KOH under vacuum. Coated samples were sealed in silica glass tubes and annealed in 1 -atm furnaces. 39K profiles in the melilites were measured by ion microprobe. For K diffusion parallel to the c-axis in akermanite, over the temperature range 900-1077 °C. an activation energy for diffusion of 272±27 kj/mol was obtained, with a pre-exponential factor of 2.51x10~8 m2/s (log A) = -7.6± 1.1). For gehlenite (parallel to the c-axis), Arrhenius parameters

750

Cherniak

obtained were 284±49 kJ/mol for the activation energy and 7.94xl0~ 9 m2/s (log D0 = -8.1±2.0) for the pre-exponential factor; parameters for diffusion parallel to the a-axis are 292±17 kJ/mol and 3.98xl0~ 8 m2/s (log Du = -7.4±0.7) for the activation energy and pre-exponential factor, respectively. Diffusion of K in gehlenite is anisotropic, with diffusion parallel to the a-axis about a factor of 5 faster than diffusion parallel to the c-axis; similar anisotropy has been observed for oxygen diffusion in ákermanite (Yurimoto et al. 1989). For transport parallel to the c-axis, K diffusion in gehlenite is about an order of magnitude slower than diffusion in ákermanite, again similar to trends for other cations, and for oxygen diffusion in melilite (Yurimoto et al. 1989). Diffusion in melilite - a summary Diffusion data for various cations in gehlenite and ákermanite are summarized in Figures 9 and 10. Among all of the extant data, the slowest diffusion is found for K in gehlenite, Co diffusion in gehlenite involving coupled Co + Si N O C Mr-C M H Ha r3 "O O Q_ Q_ Ph p N N D. S 3C i •uC UE P5 C O —O — •s s "H « Ä ' r-

M O. B a S o P6 H OX) „ C $ o

I

o

CN ON in -H 00 (N O On ON

o

-1 C/5 O P-

I I

O O MD O) © m m m n 00 00 in in ru-j u-j

m c-» in m m

xt- ir, ir, xt-

eo m en m m m c-» 00 00 c-» rr- r- r- r- so CM \o •r•>o >o TÍ- Ti- in in Ci OS ers m m n m en en CM c-» rrrr- m en O O en

m c^ in m m

Sí x„

e

a

"Ò "« O O c

C c

U U
0(m2/s) 1

log£>, -0.028±0.014

Reference Cherniak et al. (2004a)

Pb

1100-1350

592±39

9.38X10-

Pb

1200-1500

509±24

3.89X10- 4

- 3 . 4 1 ±0.77

Gardés et al. (2006)

Pb-Ca

1300-1500

447±33

5.37x10" 6

-5.27±0.99

Gardés et al. (2007)

Pb

1200-1500

494±17

1.38X10- 4

-3.86±0.53

combination of datasets of

Th

1350-1550

740±63

6.76x10'

1.83±1.88

Gardés et al. (2006, 2007) Cherniak and Pyle (2008)

metallic materials is determined by the competing effects of short-range covalent and longrange ionic forces, with the latter contributing to greater resistance to irradiation. However, despite this robustness, the U-Th-Pb system in monazite is complex (e.g.. Zhu et al. 1997; Bingen and Van Breemen 1998; Zhu and O'Nions 1999a). and can be disturbed by other means. For example, distinct age and/or compositional domains in monazites are frequently observed, and it has been established that recrystallization, generally mediated by fluids, can affect the U-Th-Pb system in monazite (e.g., Teufel and Heinrich 1997; Townsend et al. 2001). rather than volume diffusion. Calcium and Lead Attempts to measure Pb diffusion in monazite by bulk release methods were made in the 1960s and early 1970s (Voronovskiy and Magomedov 1969; Shestakov 1969,1972;Magomedov 1972), in which grains were heated and released Pb measured by mass spectrometry. The first direct depth profiling study was by Smith and Giletti (1997), who measured Pb diffusion in natural monazite using a 2l)4Pb tracer deposited on sample surfaces. Samples were annealed in Pt capsules in 1-atm furnaces, and the 204Pb concentration was profiled with SIMS. They obtained an Arrhenius relation of D = 6.6/10 15 exp(-180±48 k.l/mol/W/) m2/s over the temperature range 1000-1200 °C.

846

Cherniak

In 2004, Cherniak et al. reported measurements of Pb diffusion in both synthetic ( C e P 0 4 ) and natural monazites in experiments run under dry, 1-atm conditions. Powdered mixtures of pre-reacted C e P 0 4 and PbZrO, were used as the source of Pb diffusant for "in-diffusion" experiments, with monazites with large flat natural growth faces sealed with the source material in Pt capsules and annealed in 1 -atm furnaces. Following the diffusion anneals, Pb concentration profiles were measured with Rutherford Backscattering Spectroscopy (RBS), supplemented by measurements with secondary ion mass spectrometry (SIMS). Over the temperature range 1100 to 1350 °C, the Arrhenius relation determined for in-diffusion experiments on synthetic monazite for diffusion normal to the ( 100) face was D = 0.94 e x p ( - 5 9 2 ± 3 9 kJ/mol//?7) m 2 /s. In order to evaluate potential compositional effects upon Pb diffusivity and simulate diffusional Pb loss that might occur in natural systems, Cherniak et al. (2004a) also conducted "out-diffusion" experiments on Pb-bearing natural monazites. In these experiments, monazite grains were surrounded by a synthetic zircon powder to act as a "sink". Monazites from these experiments were analyzed with SIMS. Diffusivities for synthetic and natural monazites were found to be similar, as were results of measurements made on the_same samples using both R B S and SIMS. Diffusion anneals at 1250 °C for ( 110), ( 101 ) and (111) faces (with diffusivities measured normal to these faces) of a natural monazite yielded diffusivities indistinguishable from those measured normal to the (100) face, indicating little anisotropy for Pb diffusion in monazite. The activation energy for Pb diffusion determined in this study was found to be more than three times (180 vs. 592 kJ/mol) that reported by Smith and Giletti (1997). It is not clear why this discrepancy between these datasets exists, but analytical artifacts may be responsible, and the activation energy determined by Smith and Giletti (1997) is not well-constrained because there is considerable scatter (about an order of magnitude in each case) among diffusivities measured at both the highest (1200 °C) and lowest (1000 °C) temperatures they investigated. In addition, Smith and Giletti (1997) attributed some of the rapid diffusivities they measured to the presence of microcracks in their samples. Gardés et al. ( 2 0 0 6 , 2 0 0 7 ) also conducted studies of Pb diffusion in monazite, and obtained similar results to those of Cherniak et al. (2004a). In their 2006 study, Gardés and co-workers measured Pb diffusion in synthetic N d P 0 4 crystals, using an epitaxial thin film of composition Nd a 6 6 Pb 0 1 7 T h 0 1 7 P O 4 deposited on the monazite (101) surface as the source of diffusant. Experiments were run in Pt capsules in 1-atm furnaces, with sintered N d 0 6 6 P b 0 1 7 T h 0 1 7 P 0 4 pellets enclosed in the capsule with the samples to inhibit Pb evaporation in the thin film during the diffusion anneals. Experiments were analyzed with Rutherford Backscattering Spectrometry ( R B S ) and Transmission Electron Microscopy coupled with Energy Dispersive X-ray Spectrometry (TEM-EDS). From the data the following Arrhenius relation was obtained over the temperature range 1200-1500 °C: D = 3 . 9 x l 0 " 4 e x p ( - 5 0 9 ± 2 4 kJ/moy^7) m 2 /s The authors conclude, based on their T E M - E D S measurements, that Pb diffusion proceeds via the substitutional mechanism Pb + 2 + Th + 4 —> 2REE+ 3 However, this interpretation is inconsistent with the findings of Cherniak and Pyle (2008), who determined that Th diffuses much more slowly in monazite than does Pb, regardless of the substitutional mechanism forTh. These findings will be discussed in greater detail in the next section. Gardés et al. (2007) investigated Pb + 2 Ca + 2 interdiffusion in monazite, performing experiments on sintered polycrystals of a Ca-brabantite-Nd monazite solid solution, which were coated with thin films of Pb-brabantite-Nd monazite solid solution. Experiments were run in Pt capsules, with sintered Pb-brabantite-Nd monazite pellets enclosed in the capsule, as in their 2006 study, to inhibit Pb evaporation in the thin films. Pb and Ca profiles were measured on individual grains in the polycrystalline material (to avoid contributions from grain boundary

Diffusion in Accessory Minerals

847

diffusion) by performing Scanning Electron Microscopy coupled with Energy Dispersive X-ray Spectrometry (STEM-EDS) line scans. The Ca and Pb profiles appear complementary, indicating Ca+2 RFF 15 exchange in REE-bearing apatite (with no net change in total REE content), which requires no charge compensation, is considerably faster than REE diffusion in coupled exchange. Diffusion of F-C1-OH, as well as oxygen (Farver and Giletti 1989), is faster than diffusion of cations, with diffusivities 1-3 orders of magnitude faster than the fastest-diffusing cations (Pb and Sr) in apatite over temperature ranges of geologic interest.

COMPARISON OF DIFFUSIVITIES AMONG ACCESSORY MINERALS Since several accessory minerals from a rock may be studied and analyzed for trace elements or isotopes in the same investigation (for example. U-Pb isotopes may be determined for multiple phases), it is important to consider the relative diffusivities of various elements in accessory minerals. We compare here extant diffusion data for the geochemically and geochronologically important elements Pb, U. Th and the rare earth elements in a range of accessory phases.

Lead Data for Pb diffusion in accessory minerals is presented in Figure 17. The data fall into two broad groups - minerals in which Pb diffuses relatively rapidly (apatite, titanite, rutile), and those in which it diffuses more slowly (zircon, monazite, xenotime), with most in the latter group having higher activation energies for diffusion. The activation energy for Pb diffusion in titanite is greater than those for either rutile or apatite, so Pb diffusivities in titanite will approach those of rutile at lower temperatures but will be about 2 orders of magnitude faster than rutile at 900 °C. With respect to apatite, an opposite trend will apply, with titanite and

862

Cherniak T(°C) 1400

1000

600

800

500

~Pb -18

apatite

\ xenotime

\

^ rutile

monazite

Figure 17. Summary of Pb diffusion data for accessory minerals. Diffusion in zircon, monazite and xenotime are considerably slower than diffusion in apatite, rutile and titanite, which is also reflected in the closure temperatures of Pb in these minerals. Sources for data: apatite - Cherniak et al. (1991); titanite - Cherniak (1993); rutile - Cherniak (2000b); zircon - Cherniak and Watson (2001); monazite - Cherniak et al. (2004a); xenotime - Cherniak (2006b).

titanite zircon

-24 6

8

10

1/T (x104/K) apatite diffusivities approaching one another at higher temperature but with a difference of about 2 orders of magnitude in diffusion coefficients at 500 °C. Over all temperatures Pb diffusion coefficients for apatite and rutile differ by about 2 orders of magnitude. Pb diffusion in xenotime is slowest of all minerals studied over the temperature range investigated, but the activation energy for diffusion is relatively low. so Pb will diffuse faster in xenotime than in monazite or zircon below temperatures of ~ 1000 and 800 °C, respectively. Zircon and monazite have activation energies that are similar, so Pb diffusivities in these minerals will differ by roughly an order of magnitude over the temperature range of geologic interest, with diffusion in monazite the slower. The differences in Pb diffusivities among the "fast" and "slow" groups of mineral phases can be quite pronounced: for example, at 800 °C. Pb will diffuse about 8 orders of magnitude faster in apatite than in zircon. The results from these experimental studies of Pb diffusion display trends that broadly correspond to field-based determinations of Pb closure temperatures, which are largely diffusion controlled. Rare Earth Elements (REE) A summary of diffusion data for the rare earth elements is presented in Figure 18. Like Pb. diffusion of the rare earths differs with mineral type, with diffusion in apatite fastest, followed by sphene. with zircon the slowest. In the case of zircon, diffusion appears quite sensitive to REE ionic radius, which is not the case for either xenotime or apatite. Although zircon and xenotime are isostructural, the differences in size and charge of the cations for which the REEs substitute in the respective structures appear to influence REE diffusivities and may account for this contrasting behavior. The ionic radius of Zr+4 (0.84 A) in zircon is considerably smaller than that ofY + 3 (1.019 A) in xenotime (both 8-fold coordination; Shannon 1976). The rare earth elements likely substitute on these sites in zircon and xenotime, but given the differences in ionic radii of these species and the ionic radii of the rare earth elements (Yb+3 = 0.985 A; Dy +? = 1.027 A; Sm +3 = 1.079 A; Shannon 1976), the rare earth elements are a closer match in size with Y in xenotime than Zr in zircon. Calculated differences in ionic radii between Zr and the substituent REE range from 0.145 A (for AYb-Zr) to 0.239 A (for ASm-Zr) in zircon, contrasting

Diffusion in Accessory

-18 apatite - LREE exchange

-20

O) o

-22 Nd - titanite Sm - xenotime Yb - xenotime

apatite REE coupled exchange

Minerals

863

Figure 18. Summary of REE diffusion in accessory minerals. As with Pb, the REE show the general trend of diffusion being fastest in apatite, slowest in zircon and xenotime, and intermediate in titanite. In apatite, diffusion appears strongly dependent on a substitutional mechanism, which may also be the case for titanite given the dependence of REE diffusion on oxygen fugacity. In zircon, REE diffusion exhibits a strong dependence on REE ionic radius, but in xenotime, despite its similar structure, the REE diffuse at similar rates. Sources for data: apatite - Cherniak (2000a); zircon - Cherniak et al. (1997a); titanite - Cherniak (1995); xenotime -Cherniak (2006b).

1/T (x10 4 /K) with values ranging from -0.034 A (for AYb-Y) to 0.060 A (for ASm-Y) for xenotime. These smaller differences between the ionic radii of the lattice cation and the substituent cation may lead both to lower activation energies for diffusion, as well as to a less pronounced dependence of activation energy on ionic radius due to lower lattice strain (e.g.. Van Orman et al. 2001). In addition, the trivalent REE can exchange directly with Y + \ with no charge compensating species necessary as would be required when the REE substitute for Zr +4 . Rare earth elements also are easily accommodated in Ca sites in apatite, but, as noted in previous sections, REE diffusion in apatite appears sensitive to a substitutional mechanism, with diffusion slower when coupled substitutions are required in REE chemical diffusion. Thorium and Uranium A comparative plot of U and Th diffusion in monazite, zircon and apatite is shown in Figure 19. As in the case of Pb, U and Th diffusivities in monazite and zircon are much slower than U diffusion in apatite, and have higher activation energies for diffusion. In the experiments run to measure U diffusion in zircon, temperatures were sufficiently high that U remained in the tetravalent state, and diffused at rates similar to Th. Because of the very slow diffusivities of U in zircon, experiments could not be run at sufficiently low temperature to explore the the diffusional behavior of U in the hexavalent state and evaluate whether differences in U diffusivities between hexavalent and tetravalent states might be observed as in apatite. Diffusion of Th in zircon is somewhat slower than diffusion of Th in monazite. This may be a consequence of the larger relative difference in size between Th and Zr, for which Th substitutes in the zircon lattice, compared with the smaller size differential between Th and the REE in monazite. Although charge compensating species are required to facilitate the exchange of Th for the trivalent REE in monazite, in contrast to the direct Zr 3900 Ma detrital grains. Geochim Cosmochim Acta 70:5601-5616 Cherniak DJ (1993) Lead diffusion in titanite and preliminary results of the effects of radiation damage on Pb transport. Chem Geol 110:177-194 Cherniak DJ (1995) Diffusion of Pb in plagioclase and K-feldspar measured by Rutherford backscattering spectroscopy and resonant nuclear reaction analysis. Contrib Mineral Petrol 120:358-371 Cherniak DJ (1996) Strontium diffusion in sanidine and albite, and general comments on Sr diffusion in alkali feldspars. Geochim Cosmochim Acta 60:5037-5043

Diffusion in Accessory

Minerals

865

Cherniak D J (1998a) Rare earth element and gallium diffusion in yttrium aluminum garnet. Phys Chem Miner 26:156-163 Cherniak D J (1998b) R E E diffusion in calcite. Earth Planet Sci Lett 160:273-287 Cherniak DJ (2000a) Rare earth element diffusion in apatite. Geochim Cosmochim Acta 64:3871-3885 Cherniak D J (2000b) Pb diffusion in rutile. Contrib Mineral Petrol 139:198-207 Cherniak DJ (2002) B a diffusion in feldspar. Geochim Cosmochim Acta 6 6 : 1 6 4 1 - 1 6 5 0 Cherniak D J (2003) R E E diffusion in feldspar. Chem Geol 193:25-41 Cherniak DJ (2005) Uranium and manganese diffusion in apatite. Chem Geol 219:297-308 Cherniak D J (2006a) Zirconium diffusion in titanite. Contrib Mineral Petrol 152:639-647 Cherniak DJ (2006b) Pb and rare earth element diffusion in xenotime. Lithos 88:1-14 Cherniak D J (2008) Si diffusion in zircon. Phys Chem Miner 35:179-187 Cherniak DJ, Hanchar JM, Watson E B (1997a) Rare-earth diffusion in zircon. Chem Geol 134:289-301 Cherniak DJ, Hanchar JM, Watson E B (1997b) Diffusion of tetravalent cations in zircon. Contrib Mineral Petrol 127:383-390 Cherniak DJ, Lanford WA, Ryerson F J ( 1991 ) Lead diffusion in apatite and zircon using ion implantation and Rutherford Backscattering techniques. Geochim Cosmochim Acta 55:1663-1673 Cherniak DJ, Manchester J, Watson E B (2007) Zr and Hf diffusion in rutile: Earth Planet Sci Lett 2 6 1 : 2 6 7 - 2 7 5 Cherniak DJ, Pyle J, Rakovan J (2004c). Synthesis of R E E and Y phosphates by Pb-free flux methods, with utilization as standards for electron microprobe analysis, and in design of monazite chemical U-Th-Pb dating protocol. Am Mineral 89:1533-1539 Cherniak DJ, Pyle J M (2008) Thorium diffusion in monazite. Chem Geol 256:52-61 Cherniak DJ, Ryerson F J (1993) A study of strontium diffusion in apatite using Rutherford Backscattering and ion implantation. Geochim Cosmochim Acta 57:4653-4662 Cherniak DJ, Watson E B ( 1992) A study of strontium diffusion in K-feldspar, Na-K feldspar and anorthite using Rutherford Backscattering Spectroscopy. Earth Planet Sci Lett 113:411-425 Cherniak DJ, Watson E B (1994) A study of strontium diffusion in plagioclase using Rutherford Backscattering Spectroscopy. Geochim Cosmochim Acta 58:5179-5190 Cherniak DJ, Watson E B (2001) Pb diffusion in zircon. Chem Geol 172:5-24 Cherniak DJ, Watson E B (2003) Diffusion in zircon. Rev Mineral Geochem 53:113-143 Cherniak DJ, Watson E B (2007) Ti diffusion in zircon. Chem Geol 242:473-481 Cherniak DJ, Watson E B (2009) The fast and the slow, the high and the low - investigations of Pb and He diffusion in accessory minerals and implications for geochronology. American Geophysical Union, Fall Meeting 2009, abstract #V52C-01 Cherniak DJ, Watson E B (2010) Li diffusion in zircon. Contrib Mineral Petrol 160(3):383-390 Cherniak DJ, Watson E B , Grove M, Harrison T M (2004a) Pb diffusion in monazite: a combined R B S / S I M S study. Geochim Cosmochim Acta 6 8 : 8 2 9 - 8 4 0 Cherniak DJ, Watson E B , Thomas J B (2009) Diffusion of helium in zircon and apatite. Chem Geol 2 6 8 : 1 5 5 - 1 6 6 Cherniak DJ, Zhang XY, Nakamura M, Watson E B (2004b) Oxygen diffusion in monazite. Earth Planet Sci Lett 226:161-174 Cherniak DJ, Zhang X Y , Wayne NK, Watson E B (2001) Sr, Y and R E E diffusion in fluorite. Chem Geol 181:99111 Coogan LA, Hinton RW (2006) Do the trace element compositions of detrital zircons require Hadean continental crust? Geology 34:633-636 Crocombette J P (1999) Theoretical study of point defects in crystalline zircon. Phys Chem Miner 27:138-143 Davis D, Williams IS, Krogh T E (2003) Historical development of zircon geochronology. Rev Mineral Geochem 53:145-181 den Hartog H, Welch DO, Royce B S H ( 1972) The diffusion of calcium, phosphorus, and OD-ions in fluorapatite. Phys Status Solidi 53:201-212 Dennis P F (1984) Oxygen self-diffusion in quartz under hydrothermal conditions. J Geophys Res B 89:40474057 Donovan JJ, Hanchar JM, Picolli PM, Schrier MD, Boatner LA, Jarosewich E (2003) A re-examination of the rare-earth-element orthophosphate standards in use for electron-microprobe analysis. Can Mineral 41:221232 Elliott J C (2002) Calcium phosphate biominerals. Rev Mineral Geochem 48:427-453 Ewing RC, Meldrum A, Wang L M , Weber W J , Corrales L R (2003) Radiation effects in zircon. Rev Mineral Geochem 53:387-425 Ewing RC, Wang L M (2002) Phosphates as nuclear waste forms. Rev Mineral Geochem 48:673-699 Farley K A (2000) Helium diffusion from apatite: General behavior as illustrated by Durango fluorapatite. J Geophys Res 105:2903-2914 Farley KA (2007) He diffusion systematics in minerals; evidence from synthetic monazite and zircon structure phosphates. Geochim Cosmochim Acta 7 1 : 4 0 1 5 - 4 0 2 4

866

Cherniak

Farver JR (1989) Oxygen self-diffusion in diopside with application to cooling rate determinations. Earth Planet Sci Lett 92:386-396 Farver JR (2010) Oxygen and hydrogen diffusion in minerals. Rev Mineral Geochem 72:447-507 Farver JR, Giletti BJ (1989) Oxygen and strontium diffusion kinetics in apatite and potential applications to thermal history determinations. Geochim Cosmochim Acta 53:1621-1631 Farver JR, Giletti BJ (2001) Calcium and strontium diffusion in apatite. 11th annual V. M. Goldschmidt conference. LPI Contribution, abstract no. 3734 Ferry JM, Watson E B (2007) New thermodynamic models and revised calibrations for the Ti-in-zircon and Zrin-rutile thermometers. Contrib Mineral Petrol 154:429-437 Fletcher IR, McNaughton NJ, Aleinikoff JN, Rasmussen B, Kamo SL (2004) Improved calibration procedures and new standards for U/Pb and Th/Pb dating of Phanerozoic xenotime by ion microprobe. Chem Geol 209:295-314 Foland KA (1974) Alkali diffusion in orthoclase. In: Geochemical Transport and Kinetics. Hofmann AW, Giletti, BJ, Yoder HS Jr, Yund RA (eds) Carnegie Institute of Washington, p 77-98 Fritsch S, Navrotsky A (1996) Thermodynamic properties of manganese oxides. J AmCeram Soc 79:1761-1768 Gardés E, Jaoul O, Montel J, Seydoux-Guillaume A, Wirth R (2006) Pb diffusion in monazite; an experimental study of Pb 2+ -o- Th 4+ -o- 2 Nd3+ interdiffusion. Geochim Cosmochim Acta 70:2325-2333 Gardés E, Montel JM, Seydoux-Guillaume AM, Wirth R (2007) Pb diffusion in monazite: New constraints from the experimental study of Pb 2+ o - Ca 2+ interdiffusion. Geochim Cosmochim Acta 71:4036-4043 Giletti BJ, Casserly JED (1994) Strontium diffusion kinetics in plagioclase feldspars. Geochim Cosmochim Acta 58:3785-3793 Giletti BJ, Semet MP, Yund RA (1978) Studies in diffusion. III. Oxygen and feldspars, an ion microprobe determination. Geochim Cosmochim Acta 42:45-57 Giletti BJ, Shanahan TM (1997) Alkali diffusion in plagioclase feldspar. Chem Geol 139:3-20 Giletti BJ, Yund RA (1984) Oxygen diffusion in quartz. Chemical effects of water on the strength and deformation of crustal rocks. J Geophys Res B 89:4039-4046 Gross KA, Berndt CC (2002) Biomedical application of apatites. Rev Mineral Geochem 48:631-672 Hanchar JM, Finch RJ, Hoskin PWO, Watson EB, Cherniak DJ, Mariano AN (2001) Rare earth elements in synthetic zircon. Part 1. Synthesis, and rare earth element and phosphorus doping. Am Mineral 86:667-680 Harrison TM, Aikman A, Holden P, Walker AM, McFarlane C, Rubatto D, Watson EB (2005) Testing the Tiin-zircon thermometer. Trans. Am. Geophys. Union 86 (Fall Meeting Supplement, abstract V4IF-1540) Harrison TM, Catlos EJ, Montel JM (2002) U-Th-Pb dating of phosphates. Rev Mineral Geochem 48:523-558 Hauri EH, Shimizu N, Dieu JJ, Hart SR (1993) Evidence for hotspot-related carbonatite metasomatism in the oceanic upper mantle. Nature 6443:221-227 Hawkins DP, Bowring SA (1997) U-Pb systematics of monazite and xenotime: case studies from the Paleoproterozoic of the Grand Canyon, Arizona. Contrib Mineral Petrol 127:87-103 Hawkins DP, Bowring SA (1999) U-Pb monazite, xenotime, and titanite geochronological constraints on the prograde to post-peak metamorphic thermal history of Paleoproterozoic migmatites from the Grand Canyon, Arizona. Contrib Mineral Petrol 134:150-169 Hayden LA, Watson EB, Wark DA (2008). A thermobarometer for sphene (titanite). Contrib Mineral Petrol 155:529-540 He Y, Cormack AN (1999) Atomistic simulation study of the defect structure of zircon as a high-level nuclear waste host form. J China Univ Geosci 10:309-313 Headley TJ, Arnold GW, Northrup CJM (1982) Dose-dependence of Pb-ion implantation damage in zirconolite, hollandite and zircon. In: Scientific Basis for Radioactive Waste Management V. Lutze W (ed) Elsevier, p 379-388 Heinrich W, Andrehs G, Franz G (1997) Monazite-xenotime immiscibility gap thermometry. I. An empirical calibration. J Metamorph Geol 15:3-16 Higgins JB, Ribbe PH (1976) The crystal chemistry and space groups of natural and synthetic titanites. Am Mineral 61:878-888 Hinton RW, Upton B G J (1991) The chemistry of zircon: Variations within and between large crystals from syenite and alkali basalt xenoliths. Geochim Cosmochim Acta 55:3287-3302 Hopkins M, Harrison TM, Manning CE (2008) Low heat flow inferred from >4 Gyr zircons suggests Hadean plate boundary interactions. Nature 456:493-496 Hughes JM, Bloodaxe ES, Hanchar JM, Foord EE (1997) Incorporation of rare earth elements in titanite: Stabilization of the A2/a dimorph by creation of antiphase boundaries. Am Mineral 82:512-516 Hughes JM, Cameron M, Crowley KD (1991) Ordering of divalent cations in the apatite structure: crystal structure refinements of natural Mn- and Sr-bearing apatite. Am Mineral 76:1857-1862 Hughes JM, Cameron M, Mariano AN (1991) Rare-earth element ordering and structural variations in natural rare-earth bearing apatites. Am Mineral 76:1165-1173 IqdariA, VeldeB, Benalioulhaj N, Dujon SC, El YamineN (2003) Exchange of light rare earths for Ca in apatite. Comptes Rendus - Acad Sci Geosci 335:381-390

Diffusion in Accessory

Minerals

867

Jercinovic M J , Williams M L (2005) Analytical perils (and progress) in electron microprobe trace element analysis applied to geochronology: back-ground acquisition, interferences, and beam irradiation effects. Am Mineral 90:526-546 Kamber B S , Frei R, Gibb AJ (1998) Pitfalls and new approaches in granulite chronometry: an example from the Limpopo Belt, Zimbabwe. Precamb Res 91:269-285 Koppel V, Grünenfelder M (1975) Concordant U-Pb ages of monazite and xenotime from the central Alps and the timing of the high-temperature Alpine metamorphism, a preliminary report. Schweiz Mineral Petrogr Mitt 55:129-132 Kramers J, Frei R, Newville M, Kober B , Villa I (2009) On the valency state of radiogenic lead in zircon and its consequences. Chem Geol 2 6 1 : 4 - l 1 Lang M, Zhang F, Lian J, Trautmann C, Neumann R, Ewing R C (2008) Irradiation-induced stabilization of zircon ( Z r S i 0 4 ) at high pressure. Earth Planet Sei Lett 269:291-295 Lasaga AC (1979) Multicomponent exchange and diffusion in silicates. Geochim Cosmochim Acta 43:455-469 Lee JKW, Williams IS, Ellis D J (1997) Pb, U and Th diffusion in natural zircon. Nature 390:159-162 Lumpkin GL, Ewing R C (1988) Alpha-decay damage in minerals of the pyrochlore group. Phys Chem Miner 16:2-20 Mackie PE, Young R A (1973) Location of Nd dopant in Fluorapatite, Ca 5 (P0 4 ) 3 F:Nd. J Appl Cryst 6:26-31 Magomedov SA (1972) Aspects in determination of diffusion parameters of lead atoms in natural minerals. Int Geol Rev 14:109-111 Manning J R (1968) Diffusion Kinetics for Atoms in Crystals. Van Nostrand, Princeton Martin P, Carlot G, Chevalier A, Den-Auwer C, Panczer G (1999) Mechanisms involved in thermal diffusion of rare earth elements in apatite. J Nucl Mater 275:268-276 Medenbach O (1976) Geochemic der elemente in Zirkon und ihre Rung-Eine Untersuchung mit der elektronenstrahlmikrosonde. Ph.D. thesis, Ruprecht Karl Universität, 58 pp Meis C, Gale J D (1998) Computational study of tetravalent uranium and plutonium lattice diffusion in zircon. Mater Sei Eng B 5 7 : 5 2 - 6 1 Meldrum A, Boatner LA, Ewing R C (1997a) Electron-irradiation-induced nucleation and growth in amorphous L a P 0 4 , S c P 0 4 , and zircon. J Mater Sei 12:1816-1827 Meldrum A, Boatner LA, Ewing RC (1997b). Displacive radiation effects in the monazite- and zircon-structure orthophosphates. Phys Rev B 5 6 : 1 3 8 0 5 - 1 3 8 1 4 Meldrum A, Boatner LA, Weber W J , Ewing R C (1998) Radiation damage in zircon and monazite. Geochim Cosmochim Acta 62:2509-2520 Meldrum A, Wang L M , Ewing RC (1996) Ion beam induced amorphization of monazite. Nucl Instr Methods Phys Res B 116:220-224 Mojzsis SJ, Harrison TM, Pidgeon KT (2001) Oxygen isotope evidence from ancient zircons for liquid water at the Earth's surface 4 3 0 0 Myr ago. Nature 409:178-181 Moncoffre N, Barbier G, Leblond E, Martin P, Jaffrezic H (1998) Diffusion studies using ion beam analysis. Nucl Instr Methods Phys Res B 140:402-408 Montel JM, Foret S, Veschambre M, Nicollet C, Provost A (1996) Electron microprobe dating of monazite. Chem Geol 1 3 1 : 3 7 - 5 3 Morishita Y, Giletti B J , Farver JR ( 1996) Volume self-diffusion of oxygen in titanite. Geochem J 30:71 -79 Mullen J G (1966) Theory of diffusion in ionic crystals. Phys Rev 143:658-662 Murakami T, Chakoumakos B C , Ewing RC, Lumpkin GR, Weber W J (1991) Alpha-decay event damage in zircon. Am Mineral 76:1510-1532 O'Reilly SY, Griffin W L (2000) Apatite in the mantle: implications for metasomatic processes and high heat production in Phanerozoic mantle. Lithos 53:217-232 Otto E M (1966) Equilibrium pressures of oxygen over oxides of lead at various temperatures. J Electrochem Soc 113:525-527 Ouchani S, Dran JC, Chaumont J (1998) Exfoliation and diffusion following helium ion implantation in fluorapatite: implications for radiochronology and radioactive waste disposal. Appl Geochem 13:707-714 Parrish R (1990) U-Pb dating of monazite and its application to geological problems. Can J E a r t h Sei 17:14311450 Parrish R R , Noble SR (2003) Zircon U-Th-Pb geochronology by isotope dilution; thermal ionization mass spectrometry (ID-TIMS). Rev Mineral Geochem 53:183-213 Petit JC, Dra JC, Delia Mea G (1987) Effects of ion implantation on the dissolution of minerals. Part II: Selective dissolution. Bull Fr Minéral Cristallogr 110:25-42 Pyle JM, Spear FS (1999) Yttrium zoning in garnet: coupling of major and accessory phases during metamorphic reactions. Geol Mater Res 1:1 -49 Pyle JM, Spear F S (2000) An empirical garnet (YAG)-xenotime thermometer. Contrib Mineral Petrol 138:51-58 Pyle JM, Spear F S , Wark DA, Daniel CG, Storm L C (2005) Contributions to precision and accuracy of monazite microprobe ages, Am Mineral 90:547-577

868

Cherniak

Redely SM, Timms NE, Pantleon W, Trimby P (2007) Quantitative characterization of plastic deformation of zircon and geological implications. Contrib Mineral Petrol 153:625-645 Reddy SM, Timms NE, Trimby P, Kinny PD, Buchan C, Blake K (2006) Crystal-plastic deformation of zircon; a defect in the assumption of chemical robustness. Geology 3 4 : 2 5 7 - 2 6 0 Reich M., Ewing RC, Ehlers TA, Becker U (2007) Low-temperature anisotropic diffusion of helium in zircon: implications for zircon (U-Th)/He thermochronometry. Geochim Cosmochim Acta 7 1 : 3 1 1 9 - 3 1 3 0 Reiners PW, Spell TL, Nicolescu S, Zanetti K A (2004) Zircon (U-Th)/He thermochronometry; He diffusion and comparisons with 4 0 Ar/ 3 9 Ar dating. Geochim Cosmochim Acta 68:1857-1878 Ribbe PH (1980) Titanite. Rev Mineral: 137-154 Robinson G W (1979) The occurrence of rare earth elements in zircon. Ph.D. dissertation, Queen's University, Kingston, ON Ronsb0 J G (1989) Coupled substitutions involving R E E s and Na and Si in apatites in alkaline rocks form the Ilimaussaq intrusion, South Greenland, and the petrological implications. Am Mineral 74:896-901 Ryerson F J , McKeegan K D (1994) Determination of oxygen self-diffusion in akermanite, anorthite, diopside, and spinel; implications for oxygen isotopic anomalies and the thermal histories of Ca-Al-rich inclusions. Geochim Cosmochim Acta 58:3713-3734 Saadoune I, de Leeuw NH (2009) A computer simulation study of the accommodation and diffusion of He in uranium- and plutonium-doped zircon ( Z r S i 0 4 ) Geochim Cosmochim Acta 73:3880-3893 Saadoune I, Purton JA, de Leeuw NH (2009) He incorporation and diffusion pathways in pure and defective zircon Z r S i 0 4 : a density functional theory study. Chem Geol 2 5 8 : 1 8 2 - 1 9 6 Seifert W, Kramer W (2003) Accessory titanite: an important carrier of zirconium in lamprophyres. Lithos 71:81-98 Seitz M G (1973) Uranium and thorium diffusion in diopside and fluorapatite. Year Book - Carnegie Institution of Washington 72, p 586-588 Seydoux-Guillaume AM, Goncalves P, Wirth R, Deutsch A (2003) Transmission electron microscopic study of polyphasic and discordant monazites: site specific specimen preparation using the Focused Ion Beam technique. Geology 31:973-976 Seydoux-Guillaume AM, Wirth R, Deutsch A, Schärer U (2004) Microstructure of 24-1928 M a concordant monazites: implications for geochronology and nuclear waste deposits. Geochim Cosmochim Acta 68:2517-2527 Seydoux-Guillaume AM, Wirth R, Heinrich W, Montel J M (2002a) Experimental determination of thorium partitioning between monazite and xenotime using analytical electron microscopy and X-ray diffraction Rietveld analysis. Eur J Mineral 14:869-878 Seydoux-Guillaume AM, Wirth R, Ingrin J (2007) Contrasting response of T h S i 0 4 and monazite to natural irradiation. Eur J Mineral 19:7-14 Seydoux-Guillaume AM, Wirth R, Nasdala L, Gottschalk M, Montel JM, Heinrich W (2002b) An X R D , T E M and Raman study of experimentally annealed natural monazite. Phys Chem Miner 29:240-253 Sha L K , Chappell B W (1999) Apatite chemical composition, determined by electron microprobe and laserablation inductively coupled plasma mass spectrometry, as a probe into granite petrogenesis. Geochim Cosmochim Acta 63:3861-3881 Shannon R D (1976) Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Crystallogr A32:751-767 Shestakov GI (1969) On diffusional loss of lead from a radioactive mineral. Geochem Int 6:888-896 Shestakov GI (1972) Diffusion of lead in monazite, zircon, sphene, and apatite. Geochem Int 9:801-807 Shuster DL, Farley KA (2009) The influence of artificial radiation damage and thermal annealing on helium diffusion kinetics in apatite. Geochim Cosmochim Acta 7 3 : 1 8 3 - 1 9 6 Shuster DL, Farley KA, Sisterson JM, Burnett DS (2003) Quantifying the diffusion kinetics and spatial distribution of radiogenic 4 He in minerals containing proton-induced 3 He. Earth Planet Sei Lett 217:19-32 Smith HA, Giletti B J (1997) Lead diffusion in monazite. Geochim Cosmochim Acta 61:1047-1058 Spear F S , Pyle J M (2002) Apatite, Monazite, and Xenotine in Metamorphic Rocks. Rev Mineral Geochem 48:523-558 Speer JA (1982) Zircon. Rev Mineral 5:67-112 Suzuki K, Adachi M (1991) Precambrian provenance and Silurian metamorphism of the Tsubonosawa paragneiss in the south Kitakami terrane, Northeast Japan, revealed by the chemical Th-U-total Pb isochron ages of monazite, zircon and xenotime. Geochem J 2 5 : 3 5 7 - 3 7 6 Suzuki K, Kouta H, Nagasawa H (1992) Hf-Zr interdiffusion in single crystal zircon. Geochem J 26:99-104 Tailby N, Mavrogenes J, Hermann J, Evans K, O'Neill H (2007) Site occupancy and distribution of Ti in zircon. Eos Trans AGU, 88(52), Fall Meet. Suppl: Abstract V 3 1 D - 0 6 8 9 Tepper JH, Kuehner S M (1999) Complex zoning in apatite from the Idaho batholith: A record of magma mixing and intracrystalline trace element diffusion. Am Mineral 84:581-595 Teufel S, Heinrich W (1997) Partial resetting of the U-Pb isotope system in monazite through hydrothermal experiments: An S E M and U-Pb isotope study. Chem Geol 137:273-281

Diffusion in Accessory

Minerals

869

Townsend KJ, Miller CF, D'Andrea JL, Ayers JC, Harrison TM, Coath CD (2001) Low temperature replacement of monazite in the Ireteba granite, Southern Nevada: geochronological implications. Chem Geol 172:95112 Trachenko K (2004) Understanding resistance to amorphization by radiation damage. J Phys Condens Matter 16:R1491-R1515 Trail D, Mojzsis SJ, Harrison TM (2007) Thermal events documented in Hadean zircons by ion microprobe depth profiles. Geochim Cosmochim Acta 71:4044-4065 Trail D, Thomas JB, Watson EB (2009) OHin zircon. Geochim Cosmochim Acta 73 Supplement 1:A1343 Ushikubo T, Kita NT, Cavosie AJ, Wilde SA, Rudnick RL, Valley JW (2008) Lithium in Jack Hills zircons; evidence for extensive weathering of Earth's earliest crust. Earth Planet Sci Lett 272:666-676 Van Orman JA, Crispin KL (2010) Diffusion in oxides. Rev Mineral Geochem 72:757-825 Van Orman JA, Fei Y, Hauri EK, Wang J (2003) Diffusion in MgO at high pressures: Constraints on deformation mechanisms and chemical transport at the core-mantle boundary. Geophys Res Lett 30:28-1-28-4 Van Orman JA, Grove TL, Shimizu N (2001) Rare earth element diffusion in diopside: Influence of temperature, pressure and ionic radius, and an elastic model for diffusion in silicates. Contrib Mineral Petrol 141:687703 Van Orman JA, Grove TL, Shimizu N, Layne GD (2002) Rare-earth element diffusion in a natural pyrope single crystal at 2.8 GPa. Contrib Mineral Petrol 142:416-424 Voronovskiy SN, Magomedov SA (1969) Diffusion of the radioactive decay products in monazite. Geochem Int 6:134-139 Wang LM, Cameron M, Weber WJ, Crowley KD, Ewing RC (1994) In situ TEM observation of radiation induced amorphization of crystals with the apatite structure. In Hydroxyapatite and Related Materials. Brown PW, Constantz B (eds), CRC Press, p 243-249 Wang LM, Ewing RC (1992) Detailed in-situ study of ion-beam induced amorphization of zircon. Nucl Instrum Methods Phys Res B 65:324-329 Watson EB, Cherniak DJ (1997) Oxygen diffusion in zircon. Earth Planet Sci Lett 148:527-544 Watson EB, Cherniak DJ, Hanchar JM, Harrison TM, Wark DA (1997) The incorporation of Pb into zircon. Chem Geol 141:19-31 Watson EB, Green TH (1981) Apatite/liquid partition coefficients for the rare-earth elements and strontium. Earth Planet Sci Lett 56:405-421 Watson EB, Harrison TM (1984) Accessory minerals and the geochemical evolution of crustal magmatic systems: A summary and prospectus of experimental approaches. Phys Earth Planet Inter 35:19-30 Watson EB, Harrison TM (2005) Zircon thermometer reveals minimum melting conditions on earliest Earth. Science 308:841-844 Watson EB, Harrison TM, Ryerson FJ (1985) Diffusion of Sm, Sr, and Pb in fluorapatite. Geochim Cosmochim Acta 49:1813-1823 Watson EB, Wark DA, Thomas J B (2006) Crystallization thermometers for zircon and rutile. Contrib Mineral Petrol 151:413-433 Weber WJ (1990) Radiation-induced defects and amorphization in zircon. J Mater Res 5:2687-2697 Weber WJ, Ewing RC, Meldrum A (1997) The kinetics of alpha-decay-induced amorphization in zircon and apatite containing weapons grade plutonium and other actinides. J Nucl Mater 250:147-155 Weber WJ, Ewing RC, Wang LM (1994) The radiation-induced crystalline-to-amorphous transition in zircon. J Mater Res 9:688-698 Williford RE, Weber WJ, Devanthan R, Cormak AN (1999) Native vacancy migrations in zircon. J Nucl Mater 273:164-170 Young EJ, Myers AT, Munson EL, Conklin NM (1969) Mineralogy and geochemistry of fluorapatite from Cerro de Mercado, Durango, Mexico. USGS Prof. Paper 650-D:D84-D93 Zener C (1952) Theory of diffusion. In: Imperfections in Nearly Perfect Crystals. Shockley W, Hollomon JH, MaureR, Seitz F (eds) Wiley, New York, pp.289-314 Zhang M, Boatner LA, Salje EKH, Honda S, Ewing RC (2008) Pb + irradiation of synthetic zircon (ZrSi0 4 ): infrared spectroscopic investigation. Am Mineral 93:1418-1423 Zhang XY, Cherniak DJ, Watson E B (2006) Oxygen diffusion in titanite: lattice diffusion and fast-path diffusion in single crystals. Chem Geol 235:105-123 Zhu XK, O'Nions RK (1999). Monazite chemical composition: Some implications for monazite geochronology. Contrib Mineral Petrol 137:351-363 Zhu XK, O'Nions RK, Belshaw NS, Gibb AJ (1997) Significance of in situ SIMS chronometry of zoned monazite from the Lewisian Granites, northwest Scotland. Chem Geol 135:35-53

19

Reviews in Mineralogy & Geochemistry Vol. 72 pp. 871-897, 2010 Copyright © Mineralogical Society of America

Diffusion in Carbonates, Fluorite, Sulfide Minerals, and Diamond D.J. Cherniak Department of Earth & Environmental Sciences Rensselaer Polytechnic Institute Troy, New York 12180, U.S.A. chernd @ rpi.edu

INTRODUCTION This chapter focuses on diffusion in non-silicate minerals that do not fall under any of the other chapter categories. This includes minerals that are not oxides (fluorite, sulfide minerals, diamond) and those, like the carbonates, that are not among the oxide minerals discussed in Chapter 17 of this volume (Van Orman and Crispin 2010). Although there are numerous minerals in these categories, limited diffusion data exist for many of them, so the primary phases that will be considered are carbonates, fluorite, diamond, and sulfide minerals.

CARBONATES Carbonate minerals are major constituents of sediments and many sedimentary rocks. Fractionation of O and C isotopes between carbonate rocks and fluids can provide insight into paleoclimates and sedimentary histories; the behavior of the isotopes of major elements in calcite and aragonite has become of increasing interest due to the important role of C 0 2 and carbonate dissolution/precipitation in the oceans and the connections to global climate (e.g., Arita and Wada 1990; Chacko et al. 1991; Dickson 1991; Beck et al. 1992; Cole 1992; Graham et al 1998; Cole and Chakraborty 2001). Ca-Mg diffusion rates affect the process of dolomitization (Hardie 1987; Fisler and Cygan 1999), and characterization of diffusion of these species is important for refinements in application of the calcite-dolomite geothermometer (e.g., Essene 1982; Farver and Yund 1996) and better understanding of deformation mechanisms. For example, calcium isotopic fractionation between seawater and calcite and aragonite has been observed in marine samples (Zhu and MacDougall 1998; de la Rocha and DePaolo 2000; Schmitt et al. 2003) with possible applications as a thermometer. Ca isotopic fractionation also occurs in biologic systems, as in the growth of foraminifera (Skulan et al. 1997; Zhu and Macdougall 1998). Diffusion of major elemental species plays an important rate-limiting role in deformational mechanisms and can influence the strength of carbonate rocks at crustal conditions (e.g., Schmid et al. 1977). Sr isotopes in carbonates can provide information about weathering and fluid interactions, and can be used to explore climate change, biomineralization, lithification of carbonate sedimentary grains into limestone, and subsequent processes of diagenesis and alteration. Sr, as an indicator of ocean chemistry, has found extensive application in paleothermometry (e.g., Elderfield et al. 1996, 2000, 2002; Hampt Andreasen and Delaney 2000; Stoll et al. 2002), and Ca isotopes, when coupled with Sr concentrations in calcite, can provide an environmental multi-proxy in inorganic systems (e.g., Tang et al. 2008) Carbonates incorporate many other minor and trace elements, including the REEs, Y, Pb, Mn, and Fe, which provide invaluable information about the environment and processes 1529-6466/10/0072-0019$05.00

DOI: 10.2138/rmg.2010.72.19

Cherniak

872

that influence mineral formation and growth, as well as the circumstances characterizing subsequent alteration. For example. Pb isotopes have been used to determine rates of carbonate diagenesis (e.g.. Smith et al. 1991) and in other geochronological applications (e.g.. Jahn and Cuvellier 1994). Given the utility and broad applications of stable isotope and trace element analyses of calcite, diffusion of many major and minor elements has been investigated, among these C. O. Ca. Mg. Sr. Pb and the rare earth elements. These findings are reviewed below, and selected data are summarized in Table 1.

Carbon Diffusion data for C in calcite and dolomite are summarized in Figure 1. Early studies of carbon and oxygen diffusion in carbonates were conducted by Anderson (1969, 1972). who determined carbon diffusion over the temperature range 250-750 °C by measuring isotope exchange between a 1 4 C 0 2 reservoir and crushed calcite. Surface activity of the crushed calcite (and surface area for exchange) was evaluated by back-exchange following diffusion anneals, which consisted of brief, low-temperature (30 min at 250 °C) heating in a 14 C free- C 0 2 gas environment. The amount of carbon diffused into the sample was determined by dissolution of the sample in nitric acid and measurement of the activity of 14 C. Diffusivities were also evaluated using stable isotope exchange, where crushed, sized calcite was exposed to B C and l s O enriched C 0 2 and analyzed with mass spectrometry. The low-temperature (less than 550 °C) , 4 C experiments appeared to exhibit a very low activation energy for diffusion, but this was likely a consequence of limitations on measuring very small diffusivities (and shallow penetrations of l 4 C into the sample) with this method. Observed uptakes of isotopically distinct carbon in these experiments may also have been complicated by surface effects and the presence of cracks, dislocations or other diffusional fast-paths, which cannot be easily distinguished from the exchange due to volume diffusion in studies using bulk-exchange methods. The higher temperature data appear consistent with the work of Haul and Stein (1955). who took an experimental approach similar to Anderson's 'C isotope exchange method. From these two datasets. Anderson (1969) calculated an Arrhenius relation of 1.3xl0 _ 1 exp(-368 kJ/mol/i?7) nr/s. Oxygen diffusion appears similar to carbon over the range of investigated conditions.

dolomite - Anderson (1972)

U)

atm - Labotka et al. (2004)

E,

Q

O) o

F i g u r e 1. Summary o f carbon diffusion data for calcite and dolomite. The difference in activation energy between C diffusion at 0.1 and 100 M P a observed by Labotka et al. ( 2 0 0 0 . 2 0 0 4 ) may be attributable to a change in diffusion mechanism. See text for further details.

-20

CM

-22

100 MPa et al. (2000)

-24

-26

- C

d i f f u s i o n

Kronenberg et al. (1984)

C a r b o n a t e s -28 '

!

'

1

10

11

1

12

1/T (X104/K)

1

13

14

Diffusion in Carbonates, Fluorite, Sulfides, Diamond

873

o o\ c\ oo\ 7S

Q U T3

0J -D

(J

rON On

©

__

co

o X

co

O in

-o "o

rYb = 3.1x10"' exp(-395±14 kJ/mol/i?7) m /s

As in the case of Sr diffusion, diffusivities of Nd for three natural fluorites of differing minor and trace-element compositions were quite similar. Diffusion of Y and the three rare earth elements in fluorite were also found to be quite similar. This is similar to findings reported for REE diffusion in calcite (Cherniak 1998b, and above), fluorapatite (Cherniak 2000), and yttrium aluminum garnet (Cherniak 1998a), but differs from observations for REEs in zircon (Cherniak et al 1997a). In all of the former instances, the REE (and Y) substitute for species (Ca or Y) that they closely match in size. In fluorite, the REE (and Y) range in size from 0.985 to 1.109 A (for 8-fold coordination; Shannon 1976), and substitute for Ca (1.12 A ) in the lattice. In contrast, REE substitute in the zircon lattice for Zr, which has a considerably smaller ionic radius (0.84 A), and a pronounced dependence of REE diffusion rates in zircon on ionic radius is observed (Cherniak et al. 1997a). Diffusion of Sr is slightly slower than diffusion of trivalent REE and Y. This observation is in contrast with those made for other mineral systems, where marked decreases in diffusion rates with increasing cationic charge have been noted, for example, among feldspars (Foland 1974; Giletti 1991; Giletti and Casserly 1994; Giletti and Shanahan 1997; Cherniak and Watson 1992; Cherniak 2003), and zircon (Cherniak et al 1997a,b). However, the ionic radius of Sr+2 (1.26 A) is larger than that of Y +3 (1.019 A) and the investigated REE (0.985-1.109 A), suggesting that differences in cation size may exert greater influence on diffusion rates in fluorite than do differences in cation charge. This size dependence is weak, however, perhaps due to similarities in size between these cations and Ca, for which they likely substitute in the fluorite lattice, or as a consequence of the relative flexibility of the fluorite lattice in accommodating cations of varying sizes. The zircon lattice is considerably stiffer than the fluorite lattice (the adiabatic bulk modulus for zircon is ~3 times that of fluorite, e.g., Bass

Diffusion

in Carbonates,

Fluorite,

Sulfides,

Diamond

885

1995), which will increase the difficulty of migration of larger ions and likely lead to a greater dependence of diffusion rates on cation size. Diffusivities from the earlier diffusion studies of Sr from Baker and Taylor (1969) and Ca (Berard 1971) do not differ significantly from the diffusion rates measured by Cherniak et al. (2001) for REE and Y. Activation energies for diffusion determined in these studies—401 kJ/mol for Ca self-diffusion (Berard 1971), and 422 kJ/molfor Sr diffusion (Baker and Taylor 1969)—fall within the range of activation energies for diffusion determined by Cherniak et al. (2001) (385-454 kJ/mol) for the REEs and Y, suggesting some similarity of diffusion mechanism. Trivalent REE have been found to substitute for Ca in the fluorite lattice (e.g., Meary et al. 1984; Vinokurov et al. 1963). The RBS spectra from Cherniak et al. (2001) for Sr, Y, and the REE diffusion experiments provide additional clues to diffusion mechanisms, showing decreases in the Ca concentration complementary to Sr and REE uptake, which would be expected if the diffusing cations were substituting for Ca. The trivalent REE and Y require charge compensation when substituting for Ca in the fluorite lattice; compensation could be achieved through anion interstitials, or possibly by substitution of altervalent anions (i.e., oxygen). ions diffuse quite rapidly in fluorite (e.g., Matzke 1970; Jacucci and Rahman 1978), so diffusion would likely be rate-limited by cation rather than anion exchange. Matzke (1970) has noted that diffusion in fluorite may be influenced by the interrelation between cation and anion defects, observing that both cation and anion diffusivities increase in fluorite doped with NdF 3 .

DIAMOND Given its refractory nature, limited conditions of stability, as well as the very slow diffusivities characteristic of diamond (the diffusivity of C is among the slowest ever measured for a solid material), few measurements have been made of diffusion in this material. Koga et al. (2003) report measurements of C self-diffusion in diamond at 10 GPa over the temperature range 1800-2100 °C. Polished diamond was coated with a film of °C, and heated in a multianvil apparatus, with l3 C profiles measured by ion microprobe. They report an Arrhenius relation of 4.1xl0~ 5 exp(-656 kJ/mol/RI) m 2 /s. They estimate N diffusion in diamond from kinetic studies of nitrogen aggregation using the results of Evans and Qi (1982) and Taylor et al. (1996), and obtain the Arrhenius relation D = 9.7xl0- 8 exp(-579 ki/moVRT) m 2 /s. These data suggest that the diffusivities of N and C are comparable at temperatures typical of the subcontinental mantle, but diffusion of N is slower than C at higher temperatures.

SULFIDE MINERALS Diffusion of sulfur, major element cations and trace elements has been measured in a range of sulfide minerals, among these pyrite, pyrrhotite, sphalerite, galena and chalcopyrite. Pyrite is the most abundant of the sulfide minerals, ubiquitous in earth's crust and found in igneous, metamorphic and sedimentary rocks. It can be formed both by inorganic crystallization and as by-product of bacterial sulfate reduction. The other common iron sulfide mineral, pyrrhotite, occurs most frequently in basic igneous rocks but is also found in pegmatites, contact metamorphic zones and high-temperature hydrothermal veins. Its stoichiometric endmember, troilite (FeS), is found in many meteorites and lunar rocks; most terrestrial pyrrhotite is slightly Fe deficient, with a Fe:S ratio less than one. Chalcopyrite, the most widely occurring Cu-bearing mineral and an important copper ore, occurs in veins in mafic and ultramafic rocks, hydrothermal vein deposits, metamorphosed massive sulfide deposits, and in carbonate sedimentary rocks. Sphalerite, the dominant ore for zinc, most often occurs in hydrothermal deposits and is commonly associated with galena, among the most abundant and widely distributed sulfide minerals. Various coexisting sulfide phases in ores, including sphalerite,

886

Cherniak

pyrite and pyrrhotite, have long been used as a geobarometers and geothermometers (e.g., Barton and Toulmin 1966; Scott and Barnes 1967; Craig and Scott 1974; Toulmin et al. 1991). Sulfur isotope fractionation between pyrite and other phases is large and strongly temperature dependent, and sulfur isotope ratios in sulfide minerals can provide information about temperatures of igneous and metamorphic systems, precipitation temperatures and other conditions of formation (e.g., Coleman 1977; Hoeppener et al. 1990; Imai 2000; Jamieson et al. 2006). The behavior of sulfur isotopes in troilite and other sulfides in extraterrestrial materials can provide insight into processes in the early solar system, the moon and on Mars (Farquhar et al. 2000; Hochleitner et al. 2004; Zolotov and Shock 2005). Sulfur isotope ratios have also been used as biosignatures to identify microbial activity both in recent times and in the geologic past (e.g., Bottcher and Lepland 2000; Canfield 2001; Hurtgen et al. 2005; Canfield et al. 2006; Fenton et al. 2007; Johnston et al. 2007; Philippot et al. 2007). Diffusion of Os has also been investigated in sulfide minerals, because of the importance of the Re-Os geochronometer in determining the timing of melting and solidification events in magmatic systems containing sulfide phases, the timing of diagenetic events and depositional ages, and in enhancing understanding of ore-forming processes (e.g., Watanabe and Stein 2000; Stein et al. 1998, 2003; Brenan et al. 2000; Selby and Creaser 2001; Zhang et al. 2005; Morelli et al. 2007; Yang et al. 2008). Summaries of diffusion data for pyrite and pyrrhotite are plotted in Figure 7, and for sphalerite and chalcopyrite in Figure 8. Cation and S diffusion data for these and other common sulfide minerals are shown in Figures 9 and 10 and selected data summarized in Table 3. Pyrite Iron. Chen and Harvey (1975) measured Fe self-diffusion in natural pyrite over the temperature range 100-300 °C using a 59Fe radiotracer. The 59Fe was produced by neutron irradiation of pyrite specimens, and a piece of the irradiated pyrite with the 59Fe tracer was set in contact with an unirradiated pyrite specimen in a diffusion couple. The diffusion couples were annealed in 1 -atm furnaces with He gas flowing through the furnace to prevent oxidation of the pyrite. 59Fe is a gamma emitter, and the 59Fe uptake in the initially undoped pyrite sample was determined by measuring gamma activity with a Nal detector. From these experiments, Chen and Harvey (1975) obtained an activation energy of 42 kJ/mol and pre-exponential factor of 2.5x 10"16 m 2 /s (log D0 = -15.60). Sulfur. Watson et al. (2009) measured self-diffusion of sulfur in pyrite over the temperature range -500-725 °C (-0.1 MPa pressure) by immersing natural pyrite specimens in a bath of molten elemental 34S in sealed silica capsules and characterizing the resulting diffusive-exchange profiles by RBS. They obtained an activation energy of 132.1 ± 12.5 kJ/mol and pre-exponential factor of 1.74xl0~14 m2/s (log D0 = -13.76±0.71). Diffusion of sulfur is significantly slower than diffusion of Fe in pyrite (Chen and Harvey 1975), with an activation energy for diffusion about a factor of 3 higher. Osmium. Brenan et al. (2000) made attempts to measure Os diffusion in pyrite, with experiments sealed in evacuated silica glass tubes and annealed in 1-atm furnaces. Sources consisted of pyrite and OsS2 powders, and experiments were analyzed by RBS. Brenan et al. (2000) noted increases in Os surface concentration with increasing experimental annealing times, and this, along with observations from AFM imaging of pyrite surfaces, led the authors to conclude that the Os detected in the pyrites was primarily Os incorporated into the near-surface region of the material as a consequence of additional pyrite growth during diffusion anneals. The authors then attempted "relaxation" experiments on the samples with near-surface Os by heating them in evacuated silica capsules. RBS analyses of the samples were performed before and after diffusion anneals to evaluate thermal broadening of the Os distributions. Little change in measured Os profiles was observed in spectra taken before and

Diffusion in Carbonates, Fluorite, Sulfides, Diamond

o oo I

2

; I < ^
B 2 —> B 4 —> A. However, if d « L" (classified as fine-grained crystals) a different sequence occurs and B 2 is followed by the new B 2 '-regime until the A'-

Diffusion

(a)

The model Source Î. V ,6/5 or y (as in Fig. 9). This can be only observed when the system is in the B-regimes. The appearance of two different profile regions has thus been used as a criterion for assessing whether the total flux is affected by fast diffusion paths, e.g., grain or twin boundaries (e.g., Zhang et al. 2007). Although the relationship in Equation (19) of Le Claire has been derived from the isolated grain boundary model (the Fisher model) and Whipple's exact solution for a fixed surface concentration, it has been analytically or numerically shown that a similar relation also applies to (i) other boundary conditions (IFS-condition or finite source and infinitely fast surface diffusion, FSD; Suzuoka 1964; Kaur et al. 1995, p. 51 ff), (ii) the parallel slab model (Gupta et al. 1978), (iii) 2D isotropic coarse-grained polycrystals with random grain boundary orientation relative to the surface (Mishin 1992) (iv) a 2D fine-grained isotropic spherical grain model (Bokshtein et al. 1958) (v) a 3D isotropic fine-grained cubic grain model (Belova and

Table 2. List of the parameters q and p for the Le Claire equation for different boundary conditions and model geometries. For constraints to apply the Le Claire equation see text and Equations (22)-(23).

Model Isolated grain boundary Le Claire (1963) Suzuoka(1964) Mishin (1992)

2D Coarse-grained poly crystals Mishin (1992)

Fine-grained poly crystals 2D spheres; Bokshtein et al. (1958) 3D; Levine and MacCallum (1960)

Boundary condition

s8Dsh = q • J5Jt • (8 ln(C,„„ )/dx" y2"' P

1

FSC IFS FSC IFS FSD

1.2 1.2 1.2 1.2 1.2

1.332 1.308 1.298 1.224 1.142

FSC IFS FSD

1.2 1.2 1.2

1.517 1.436 1.346

FSC FSC

1.000 1.200

1.332 1.946

940

Dohmen

& Milke

Murch 2009), and (vi) 3D isotropic fine-grained polycrystals (Levine and MacCallum 1960). The main difference is the constant factor q in Equation (21) (shown in Table 2 for various models) primarily affecting the slope in the tail region obtained for a given set of diffusion coefficients. It is obvious that for the different models the values do not change dramatically. The corresponding difference for the derived grain boundary diffusion coefficient using Equation (21) is rather small and usually below the reproducibility of diffusion data. Therefore the detailed boundary condition and geometry of a polycrystalline aggregate is only of second order relevance for the determination of the product s8Dgb with the method of Le Claire. In the work of Belova and Murch (2009) a three-dimensional cubic grain model is studied, and the authors point out that when the solutions of two-dimensional models are applied to threedimensional systems the factor q should be multiplied by an empirical factor 1.5 or 2 if p is set equal to 6/5 and 1, respectively. Further potential uncertainties in the Le Claire approach due to grain boundary migration or the presence of dislocations will be discussed in the next sections. Although the general shape is the same, there are still significant differences in the concentration vs. depth profiles obtained from a polycrystalline material as compared with probing an isolated grain boundary. Depending on the lateral resolution of the analytical method relative to the average grain size, several grain boundaries are probed simultaneously and the total contribution of the grain boundaries to the measured signal is therefore larger than for an isolated grain boundary, which basically shifts the latter flat part of the profile to higher values. This can lead to apparently larger volume diffusion coefficients if the first part of the profile is used to derive them (Gupta et al. 1978). For coarse-grained crystals (d > L") as long as d > 60 (AO" 2 the error is less than 5% (Suzuoka 1964) and hence insignificant. If the diffusion penetration distance exceeds half the grain size, the system is in the transition to the A-regimes. There was some debate about the exact upper time limit on the use of the Le Claire approach for polycrystals, but a conservative value is (dl20) 2 ID, (Belova and Murch 2009). By recalling the definition of t" and L", it can be shown that this value is only larger than f"/25 (compare with Eqn. 23a), if L" > dl2, which also defines the transition from coarse-grained to fine-grained material (Table 2). In other words, for coarse-grained polycrystals the constraint on the run duration and profile segment to apply the Le Claire approach is also defined by Equation (23a). For other cases the upper time limit has to be replaced by (d/2Q)2/Dh To summarize, from the mathematical viewpoint the Le Claire approach is a very robust determination of the product s8Dgh, if D ; is known and the system is within the main part of the B2- or B 2 '-regime. The lattice diffusion coefficient could be simultaneously determined from the first part of the profile if accurately measured and not affected by inadequate spatial resolution (Watson and Dohmen 2010, this volume). An alternative way to determine D ; for a FSC boundary condition was suggested by Fielitz et al. (2003). The procedure is to fit the lower part of the profile with Le Claire's equation and then use the intercept at x = 0 of the line, cgb(0) (see also Fig. 9), to calculate Dh as follows: (25) where c° is the fixed surface concentration and A. is the length of the grain boundary trace per unit area, which has to be determined from SEM image processing (see Fielitz et al. 2003). Complexities of real and polyphase systems Obvious simplifications of the models presented so far compared to a rock are the simplified geometry, a mono-phase system, a stationary microstructure (ignoring grain growth, etc.), and the absence of other short cut diffusion paths like dislocations. Various new analytical models have been developed to consider these effects, a selection of which is illustrated in Figure 10, where the effect of a single additional complexity has been studied separately. Within this

Diffusion la\ \"/

in Polycrysìalline

Asymmetrical isolated grain boundary qrain

/K\ \ /

Materials Moving isolated grain boundary

Source

Source *

>

1

v




II Mishin and Razumovskii (1992b)

Mishin and Razumovskii (1992a) (c) * '

941

Structural inhomogeneity (e.g., dislocations)

(d)

Element exchange in a closed system (FGB model)

Source y

y v

** V

I « t') equivalent to the B 4 regime. Within the applicable time domain, new kinetic regimes, MC and MB 2 , were discovered and the constraints for these depend on the size of vt relative to L{ and 8 (Table 1, Fig. 11). Two characteristic velocities can be defined, = 2Z);/5 and v2 = AD2IDgh?>, that separate different regime sequences. First of all, it should be emphasized that as long as v < v2 the system behaves as if the grain boundary was stationary and the time sequence of the kinetic regimes was exactly the same as for the isolated grain boundary model above. If v > the C-regime is followed by the MC regime where similar to the C regime, the diffusive flux into the lattice can be ignored but the element/isotope of interest will be distributed within the lattice due to the combination of diffusion through the grain boundary and its migration. For t > ViS/v the system proceeds into the MB 2 -regime and the diffusive flux perpendicular to the grain boundary becomes significant as in the B2-regime, but in addition the migration of the boundary distributes new atoms into the lattice that once were transported along the grain boundary. For values of v between and v2 a third sequence occurs, where following the standard sequence C —> B 2 the system develops into MB2. In any case, as for a stationary grain boundary, the system finally develops into the B4-regime where lattice diffusion dominates the overall process. The main difference here is that depending on the value of v, the regime starts before f (Table 1, Fig. 11). An approximate solution was obtained for constraints equivalent to the MB 2 -regime. From the integration of this solution (see Eqn. 18) a formula for the sectioning profile can be obtained:

Diffusion

in Polycrystalline

Materials

943

1.E-02 1.E-04 1.E-06 1.E-08

Figure 11. Diagram illustrating the range of the kinetic regimes as a function of v and r for the model considering a migrating grain boundary as shown in Figure 10c. Two different parameter sets were chosen: (a) D, = 10~2°

1.E-10 1.E-12 1.E-14 1.E-16

nr/s and Dg„ID, = 104 (b) D, = 10"25 nr/s and DglJD,= 106.

1.E-03

The grain boundary width, 8, is set to 1 nm. The arrows in (a) indicate the different sequences of kinetic regimes for various different constant migration velocities.

1.E-06 1.E-09 1.E-12 1.E-15 1.E-18 1.E-21 1.E-24

C

(x,t) = vtexp

SDs„

(28)

This equation predicts that within the MB 2 regime an exponential decay of the depth profiles is observed. Interestingly this equation is exactly the same as the one for the MC regime derived by Glaeser and Evans (1986). also for the FSC boundary condition. There is only the difference of the factor d since Glaeser and Evans (1986) considered a parallel slab model. This implies that for the integrated diffusive flux even within the MB 2 -regime the contribution of lattice diffusion is negligible and transport is dominated by the combination of grain boundary diffusion and migration as in the MC-regime. The only difference is that within the MB 2 -regime the depth profile consists, as in all B-regimes, of two different regions where the first region is dominated by the lattice diffusion flux (Fig. 9). With Equation (27) this first region of the profile cannot be described since for the derivation of the equation the parallel diffusion flux from the source has been ignored. Therefore Equation (28) describes only the second region of the profile (the tail, compare with Fig. 9) and this would marginally imply a curvature in a plot of InC vs. xm. Consequently, if for an experimental sample v> v2. depending on the scatter of the experimental data points the tail region could be also well fit linearly if erroneously analyzed with the Le Claire approach. This has three implications for the analysis of experimental profiles with the Le Claire approach: (i) In the MB 2 -regime the standard Le Claire equation would yield an apparent value of 5D, which is a factor of about v?/(2(Zty)1/2) smaller than the "real" one. (ii) In the MC-regime the exponential decay could be misinterpreted as a B-regime profile, although the steep first region of the profile should not appear. However, for example, the first parts of SIMS depth profiles are subject to various analytical artifacts, which also depend critically on

944

Dohmen

&

Milke

the surface quality/roughness of the sample such that a steep profile region typically occurs in the concentration profile (more details on this topic are given in the chapter of Watson and Dohmen 2010, this volume), (iii) When several grain boundaries are probed simultaneously, only a fraction of them might be mobile and the resulting depth profile within the MB 2 -regime would consist of three regions, where the second is an intermediate region dominated by the contribution of grain boundary migration (Giithoff et al. 1993). This intermediate region has a steeper slope compared with the flat part, region 3, which could still be used to derive reliable products for 5D, whereas the migration rate could be extracted from region 2 (Mishin and Razumovskii 1992b). We can now further speculate how element or isotope redistribution in a polycrystalline rock would be affected by the migrating grain boundaries. Clearly during the MC- and the MB-regime, grain boundary migration would enhance the total flux of material. For a polycrystalline system an interesting case occurs when vt > d, Lgh* > d and t < t". In this case the bulk concentration profile should be roughly given by Equation (28). This would imply that the total flux into a rock with a changing microstructure would be much faster compared to a stationary system, which was already experimentally shown for calcite, quartzite, and dunite rocks by various groups (Nakamura et al. 2005, McCaig et al. 2007, and Ohuchi et al. 2010, respectively). This could be classified as a MB 2 '-regime (as extension of the scheme of Mishin and Herzig 1995). It would be worth developing a model for a polycrystalline system that considers migrating boundaries and in addition grain boundary segregation. It is obvious that mobile grain boundaries may provide a more efficient mechanism than volume diffusion alone to equilibrate rocks, as long as the time scale is shorter than the time scale for equilibrating each grain by diffusion (t < 150 (PlDi). It should be emphasized here that the model assumes a constant migration velocity, which is not realistic since the classic formula treating steady state grain growth is a parabolic law for the average grain size (Burke and Turnbull 1952). This implies that the migration velocity decreases with time and hence the sequence of the kinetic regimes would be different, and in many cases grain boundary migration would be only relevant in the early stage of a system. On the other hand, if external stresses are applied, re-crystallization and deformation of the grains could enhance the role of grain boundaries for diffusive mass transfer and hence also accelerate chemical equilibration rates.

Presence of dislocations/sub-grain boundaries No real crystal is perfect—all of them contain one-dimensional defects in the form of dislocations, which could also provide short circuit diffusion paths. The density of these dislocations strongly varies depending on, for example, the phase, the growth history, stresses to which the material was exposed, and thermal annealing. Depending on these circumstances pipe diffusion effects play either a large or a negligible role. A mathematical model which attempts to consider these effects for diffusion in polycrystalline materials was developed by Klinger and Rabkin (1999) (Fig. 10c). In their model, diffusion along the dislocations was explicitly treated but their detailed geometrical distribution was not. They introduced two parameters, m and fd, where the latter stands for the volume fraction of the material located within the pipe diffusion paths and the former is a phenomenological parameter describing the exchange of material between the network of dislocations and the crystal lattice. The governing equations consider, in addition to the Fisher model (Eqn. 4, 5), the direct diffusive flux from the grain boundary into the dislocations/sub grain boundaries (characterized by a diffusion coefficient Dd) and the diffusive exchange of material between the dislocation/sub-grain boundary network and the crystal lattice:

Diffusion

in Poly cry stalline

945

Materials

(30)

dt

>— 2

(31)

where Cd and Dd denote the concentration and diffusion coefficient for the dislocations. For further analysis again approximations were made that correspond to the B 2 -regime (second partial derivatives of C, and Cd after x vanishes and a steady state approximation for Cgb) and the FSC boundary condition was assumed. The exchange parameter m was evaluated in case of dislocations using the model of Mimkes and Wuttig (1970), as follows: (32) where rd and rm are the radius of the dislocation core and the half-spacing between the dislocations, respectively. For sub-grain boundaries m can be estimated assuming a system of parallel sub-grain boundaries with half distance, ds, as (Klinger and Rabkin (1999):

If the volume fraction of dislocations fd > DJDd, the overall behavior changes compared to the Fisher model. To convert this critical volume fraction to the more familiar 2D dislocation density in numbers of dislocations per area, an average cross section of the dislocation has to be assumed; for example, for a ratio of D¡ and Dd of 10 1 and 10 \ and a core radius of 1 nm, dislocation densities of 1015/m2 and 10 u /m 2 are required to yield effects for the overall diffusion behavior. Three new kinetic regimes, DB¡, DB2, and DB3, were discovered at time scales comparable to those for the B-regimes (in the original paper of Klinger and Rabkin, D ]? D2, and D3, were used to denote the regimes, but we added a B to distinguish it clearly from the symbol for diffusion coefficients and to indicate that the assumptions were equivalent to those valid for the B-regimes). The sequence of kinetic regimes depends on the parameter fd (Table 1). In the original paper of Klinger and Rabkin (1999) two additional sequences were claimed based on a further subdivision if fd is greater or smaller than a certain value fd*, which is time dependent, and in the former case the C-regime would be absent. But it can be shown from the definition of the parameters that when D¡ < (fd)2Dd it follows fd (fd)2Dd accordingly fd >fd*, hence we have ignored these other cases in the following. In all DB-regimes the distribution of material within the bulk is more efficient than lattice diffusion alone. The difference between the regimes is that the penetration distance within the grain boundary and the efficiency of material redistribution within the bulk changes. For the D B r andtheDB 3 -regime, Equation (12) can be used to calculate the grain boundary penetration distance, but with D¡ replaced by fd2Dd and fdDd, respectively (Table 1). For the DB 2 -regime there is no relation given for Lgh* but it was claimed that it is a very weak function of time. The simple interpretation of this different Lgh* in the presence of dislocations with fd > D¡/Dd is that the leakage from the grain boundary into the bulk is dominated by the diffusive flux into the dislocations. During the DB 2 -regime essentially a quasi-steady state exists where the total flux into the grain boundary is balanced by the leakage flux into the dislocations, leading to a roughly constant concentration profile within the grain boundary. During this stage the element redistribution into the bulk is most efficient. As in all B-regimes, within the DB-regimes an approximately linear relation of InC vs. x6/5

946

Dohmen

& Milke

is obtained for depth profiles. But as in the case of migrating grain boundaries, the exact slope is different (Table 1). In a manner similar to the penetration distance, for the D B r and the DB3regimes in the Whipple-Le Claire equation D, has to be replaced by fd2Dd and fdDd, respectively. Compared to these, the slope within the DB 2 -regime would only change very weakly, but again there is no relation given by Klinger and Rabkin (1999). In any case the standard Le Claire approach would yield quite incorrect results for the values of 5Dgb. It should be noted that all the above considerations are only strictly applicable for non- segregating elements (s = 1). The presence of dislocations could potentially also have an influence on the slope for the first (steep) region of the profile, but this was not evaluated by Klinger and Rabkin (1999). Element/isotope exchange mediated by grain boundary diffusion All models discussed so far have treated a situation where an external reservoir is the diffusion source. This could be either an infinite reservoir, which keeps the contact/interface at a constant composition (FSC boundary condition) or it is a finite reservoir (e.g., IFS or FSD boundary conditions) where the flux into the system changes with time. Such a situation would be applicable to model, for example, the diffusive influx of elements during contact metamorphism or the chemical flux during a metasomatic process. These are essentially open system processes. In contrast, for a closed system we are interested in the internal redistribution of elements/isotopes (Fig. lOd) due to the change of an intensive thermodynamic parameter (essentially P or 7) or due to the decay of a radiogenic isotope. Here we could be interested in the closure temperature of a geothermometer or a radiogenic system used for dating (e.g., Dodson 1973; Ganguly and Tirone 1999). Grain boundaries have been considered in all the models for closure temperatures in a very simplified way, where the surrounding polycrystalline matrix is usually assumed to be a homogeneous infinite reservoir. In the more sophisticated fast grain boundary (FGB) models of Eiler et al. (1992) and Jenkin et al. (1994) it is assumed that grain boundary diffusion is infinitely fast compared to volume diffusion. A very significant outcome of these simulations was that the closure temperature of a multi-phase system depends also on the modal abundance of the various minerals and is not only determined by the mineral with the lowest diffusion coefficient as one would expect. Diffusion within each of the minerals was considered explicitly in these "fast grain boundary" diffusion models, but it was assumed that between the various mineral surfaces local equilibrium is always attained instantaneously, which is equivalent to an effectively homogeneous chemical potential within the grain boundary. The boundary condition for each mineral then follows from the mass balance of the diffusive fluxes out of/into the various grains. From the insights we obtained from the Fisher model it is clear that a much faster diffusion along the grain boundary (e.g., a condition like DglJDi » 1) is not sufficient to maintain the basic assumption of these models. The leakage from the grain boundary into the adjacent mineral grains cannot be ignored. To fulfill the main assumption of the FGB models the grain boundary penetration distance, Lgb*, has to be several orders of magnitude larger than the average distance between the mineral grains. The latter means, in terms of the model of Mishin and Herzig (1995), that the FGB model is only applicable to fine, ultra-fine and special ultrafine-grained material for time scales within the B 2 ' regime. It is however not straightforward to evaluate Lgh* by Equation (12) because we have to consider different minerals with different Dh In another model (Dohmen and Chakraborty 2003) for element exchange between two minerals, the grain boundary diffusion and the interface kinetics were explicitly considered. In this model, the two minerals were placed in fluid filled baths connected by a transport channel. The aim of this model was to address a situation as illustrated in Figure lOd where two solids separated from each other by an inert polycrystalline matrix exchange elements or isotopes (e.g., Fe-Mg exchange between garnet and biotite embedded in a matrix of quartz and feldspar). The fluid represented the intergranular fluid, the geometry of the channel (length and diameter) represented the distance between the minerals, and the diameter relative to the mineral surface

Diffusion

in Poly cry stalline

Materials

947

area represented the integrated grain boundary width (equivalent to the porosity for wetted grain boundaries or to 8/d for "dry" grain boundaries). The main assumption here was that the diffusive flux from the grain boundaries into the surrounding matrix minerals (e.g., quartz) is ignored. The latter condition would be easily justified if the relevant elements within the surrounding matrix strongly segregate into the grain boundary (strongly incompatible elements with respect to the matrix minerals, e.g., Fe and Mg in quartz). The matrix grains could then be classified according to Mishin and Herzig (1995) as special ultra-fine-grained crystals and hence the bulk diffusion coefficient of this matrix would be always given by Dgb. For this specific situation the simulations of Dohmen and Chakraborty (2003) show that a steady state concentration gradient within the grain boundary quickly develops but that this gradient could be either very small, implying effectively local equilibrium between the two separated mineral surfaces (if the exchange between the surface and the grain boundary is not kinetically inhibited) or large, implying non-equilibrium for the surfaces. The former case was classified as the solid-state diffusion controlled case and this would be equivalent to the assumptions of the FGB models. The latter case was classified as fluid diffusion controlled. These are only two of three endmember and three intermediate kinetic regimes identified by Dohmen and Chakraborty (2003); for more details regarding these regimes we refer interested readers to the original paper. The FGB model of Eiler et al. (1992) and the exchange model of Dohmen and Chakraborty (2003) were briefly introduced here to illustrate how diffusion in multi-phase polycrystalline systems can be treated in a simplified way and how the underlying assumptions of these models can be tested for a given system, applying the characteristic parameters and the classifications from the model of Mishin and Herzig (1995).

EXPERIMENTAL METHODS Various possible experimental setups for diffusion studies are presented and discussed in detail in Watson and Dohmen (2010, this volume). Therefore here only a very brief overview is given, with the focus on the data processing of experimental results using the mathematical models presented above. The fundamental difference between grain boundary diffusion studies and those investigating diffusion within crystal lattices or melts is that since the grain boundary width is about a nanometer, it is usually not possible to resolve the concentration within the grain boundary. Therefore grain boundary diffusion is commonly investigated indirectly by measuring the integrated flux along the grain/interphase boundaries, which is visible in the volume of crystals or in the form of an integrated mass transport (e.g., a deformed sample in case of Coble creep). The various methods can be broadly subdivided into two categories depending on the detection of the integrated flux after the experiment: (1) direct measurements of concentration gradients within the sample and (2) measurement of the total amount transported along grain boundaries. Experiments belonging to the second category are normal grain growth experiments, creep experiments in the grain boundary diffusion regime (Coble creep), and rim growth experiments, all performed with polycrystalline material. In these studies the kinetic process is controlled by diffusion as identified through the evolution of the system with time (e.g., the reaction rate follows a parabolic rate law), but the concentration contours or profiles of elements are not directly measured. The principle of these experiments and how to derive grain boundary diffusion coefficients from their outcome is, for example, presented in Joesten (1991) or for the case of rim growth in Fisler and Mackwell (1994) and Abart et al (2009). The problem with these methods is in general that certain assumptions and indicators from experimental observations are required to infer the element/chemical component that dominates the overall process (rate-determining step) and to calculate a diffusion coefficient from the observed rate (reaction rate, deformation rate, etc.) (e.g., Yund 1997). Consequently, there is usually a large uncertainty involved in the derived diffusion data. In general it cannot be excluded that various components simultaneously contribute to the overall process. While

948

Dohmen

& Milke

these kinds of experiments are very valuable to understand and quantify the studied kinetic process, the derived grain boundary diffusion data are frequently ambiguous. In addition, they rely on constitutive laws, which are well established for metals and simple oxides but less so for silicates (e.g., Kohlstedt 2006). The various assumptions can only be tested if grain boundary diffusion data for the various elements are independently known from other studies. Since it is beyond the scope of the present review to discuss constitutive laws for, e.g., Coble creep of silicates and carbonates, or discuss various possible mechanisms for rim growth, the data summarized are only those obtained from direct measurements of concentration contours and profiles where the mathematical models presented above were applied. These experiments can be further subdivided into experiments where the concentration distribution is mapped across a section of the sample perpendicular to the grain boundaries and the diffusion source, and those where only sections perpendicular to the diffusion source are analyzed by using depth profiling methods like SIMS. The latter approach is more common and gives integrated information as a function of depth (Fig. 9, Eqn. 18). Almost all data presented here for minerals were obtained by this way. These studies differ in the choice of the diffusion source, which was either an aqueous solution enriched with a stable tracer isotope (e.g., I 8 0 in pure H 2 0) or an isotopically enriched thin film deposited on the sample surface (for various deposition methods and related experimental procedures see Watson and Dohmen 2010, this volume). In the former case a fixed surface concentration is usually assumed for data processing, whereas in the latter case the instantaneous source would most often be a more appropriate boundary condition. Most of the early grain boundary diffusion data for metals were obtained using radiotracers as diffusants, which is still a common approach in metallurgy to study solid-state diffusion. Finally, it should be mentioned that as for lattice diffusion or diffusion in melts, computational investigation of grain and interphase diffusion processes is fundamental for understanding the detailed diffusion mechanisms and the grain boundary structure, and to identify parameters that affect them. Ideally, using molecular dynamics, diffusion coefficients can be calculated. The various methods, limitations and applications are summarized in de Koker and Stixrude (2010, this volume). Here we will refer to computational studies when the parameters affecting grain boundary diffusion are discussed. Setup with bi-crystals The best defined geometry for a grain boundary diffusion experiment is a bi-crystal with a diffusion source on top, since the Fisher model can be directly applied, and depending on the boundary conditions for the bi-crystal's surface, the analytical solutions of Suzuoka (1961) or Whipple (1954) can be used to derive diffusion coefficients. However, the preparation of bicrystals with a defined grain boundary orientation is a challenging and time consuming task (e.g., Heinemann et al. 2005), but was, for example, used to measure the dependence of grain boundary diffusion coefficients on the grain boundary orientation (tilt or twist angle, etc.) for metals (e.g., Budke et al. 1989). The time scale of the experiment has to be within the B2regime, since in the C-regime the concentration of the diffusing species will be too small to be detected (if it is not a highly segregating impurity), and finally within the B 4 -regime there is no effect on the grain boundary diffusion visible at all. An example of the latter phenomenon is shown in Chakraborty (2008) where Fe was mapped in a forsterite bi-crystal (Heinemann et al. 2005) after being embedded in fayalite-rich olivine powder for 72 hours at 1300 °C; no increased diffusive flux for Fe close to the boundary was visible. This observation is consistent with published Fe-Mg diffusion rates in olivine (Dohmen et al 2007; Dohmen and Chakraborty 2007a,b) and the classification of Mishin and Razumovskii (1992a). According to the latter authors, if t approaches f (Eqn. 7) the penetration distance (see also Fig. 3d and 4) along the grain boundary becomes similar to that parallel to the grain boundary within the lattice. For Fe we can assume 5 = 1 and for 5 = 1 nm, then the time constraint is equivalent to t ~ 2.5xl0~ 19 m2 Dgh2/D^. At 1300 °C, D, ~ 10"16 m2/s, which implies that even for a ratio of Dg,JD, ~ 104 we

Diffusion

in Poly cry stalline

Materials

949

approach the B 4 regime after 70 hours and accordingly the transport within the grain boundary ceases to be more efficient than lattice diffusion. This indirectly implies another constraint for diffusion studies where the grain boundary diffusion coefficient may be obtained from mapping concentration contours: the time constraint for the experiment is also limited by the spatial scales that can be resolved by the analytical methods. For techniques like electron microprobe or SIMS that are often used to map concentrations across such samples on the micron scale, the diffusion penetration distance L, has to be at least 10 times the spatial resolution, o, to resolve the concentration gradients without any convolution effects (Ganguly et al. 1988; see also Watson and Dohmen 2010, this volume). This defines a lower time constraint for the experiment duration of t > a2/Db which can be combined with the upper time constraint for the B2 regime to derive the minimum spatial resolution as a function of the ratio DglJD,: o < (s8)/2 DglJDh For the example discussed previously, we obtain a minimum spatial resolution of 5 |am, if the ratio of DglJDl = 104 as obtained for Mg tracer diffusion in olivine (see below). As discussed above in the section on kinetic regimes, to shift the system more towards the B2 regime at similar run duration experiments could be done at lower temperatures where the ratio of Dgbl Dj is expected to increase due to the potentially higher activation energy for lattice diffusion, but this also significantly reduces D, and hence the minimum run duration can become too long to achieve any measureable diffusion profile within the crystal. Therefore grain boundary diffusion studies that attempt to map the concentration distribution might only be useful in a very limited temperature-time window dictated also by the spatial resolution of the method. However, application of new analytical techniques like Nano-SIMS, FEG electron microprobe or synchrotron XRF can significantly improve the spatial resolution down to the nanoscale and allow experimental studies under conditions previously not feasible. Setup with a polycrystalline aggregate Nearly all of the data presented and discussed below were obtained using a monophase polycrystalline aggregate, either synthetic or separated from a mono-mineralic rock, and depth profile measurements from SIMS analysis. Most of these experiments were performed in the B2- or B2'-regimes, where two different regions can be observed in the depth profile (Fig. 9). The product s8Dgh is then determined using Equation (21) in accordance with the models of Levine and MacCallum (1960) and Le Claire (1963). It is obvious that the accuracy of the derived value for s8Dgh depends not only on the accuracy in determining the slope but also critically on the accuracy of Dh Typical uncertainties for the lattice diffusion coefficient are ±0.2 log units, which translates using error propagation into a relative uncertainty for the product s8Dgh of only ±0.1 log units. In some cases D ; has been determined from the first part of the profile as illustrated in Figure 9 (e.g., Yamazaki et al. 2000), which is also the common method in materials science. However, this part of the profile could be strongly affected by convolution effects, depending on the ratio of the diffusion penetration distance within the lattice, Lh and the effective spatial resolution of the depth profile measurements, which also depends on the surface roughness (Watson and Dohmen 2010, this volume). Therefore in various studies values for Dt were taken from experiments that were specifically designed to measure this diffusion coefficient (e.g., Farver and Yund 1991, 1995a). This could be another pitfall, as in some cases lattice diffusion data from different diffusion studies can differ by orders of magnitude, e.g., due to the dependence on thermodynamic parameters like/o 2 or/ H2() or sometimes due to analytical artifacts. A few studies were also performed in the Type C or C' regime (e.g., Farver and Yund 2000a,b) where Dgb is directly obtained from fitting the diffusion profile. A critical element of the diffusion experiments is the microstructure of the polycrystalline aggregate. We have seen that migrating boundaries produce unusual concentration contours leading to artifacts in the data obtained from depth profiles. Therefore, the polycrystalline sample is normally hot-pressed at relatively high temperature before the diffusion anneal to evolve a "stable" microstructure. Usually the average grain sizes of the starting material and

950

Dohmen & Milke

those following the diffusion anneal are compared. In a particular study by Farver and Yund (1992), the microstructure was even varied using different pressure mediums for the synthesis to investigate the effects of the detailed grain boundary geometry on the diffusion findings. Source-sink studies An alternative way of measuring grain boundary diffusion of strongly segregating elements is to detect the total amount of the element transported through a polycrystalline matrix. The methods are implicitly based on the assumption that the system is in the C' or A0-regime (Table 1), where the diffusive leakage flux into the "inert" polycrystalline matrix can be completely ignored, which is commonly justified for incompatible elements. The requirement for these studies is a source containing the segregating/incompatible element placed in contact with the polycrystalline aggregate and a method to detect the transported material. Detection can be done at various positions within the matrix or only at one position, but in the latter case a steady state flux must be achieved to obtain a diffusion coefficient. There are various configurations possible. One was suggested by Hwang and Baluffi (1979) where the material accumulated at the surface of a polycrystalline thin film is detected by XPS (X-ray photo electron spectroscopy). A different technique, called the detector-particle method, was developed to measure diffusion of C and highly siderophile elements through MgO and olivine aggregates (Hayden and Watson 2007, 2008). In a third technique a double layer was used to investigate Fe-Mg interdiffusion through a Zr0 2 aggregate (Dohmen 2008). The latter two methods, in particular the detectorparticle method, are described in detail in Watson and Dohmen (2010, this volume). EXPERIMENTAL DATA Parameters affecting grain boundary diffusion coefficients Before we present an overview of experimental diffusion data for various minerals we will discuss the parameters that potentially affect grain boundary and heterophase diffusion, based on various experimental and computational studies. This is also essential for appropriate comparison of different data sets obtained under different experimental conditions. For this purpose we also consider experimental studies from materials science research on metals and oxides since some observations can be generalized, although caution should be applied in transferring observations from metals to ionic compounds where the chemical bonding is completely different. The grain boundary diffusion coefficient is in principle sensitive to the detailed grain boundary structure. The diffusivity will be enhanced if the chemical bonding is weaker and/or open sites to jump into are available. Parameters affecting the grain boundary structure are the tilt or twist angle (low or high-angle grain boundary) and the presence of impurities, e.g., water related species. Hence the detailed grain boundary chemistry might also be relevant. Pressure and temperature. Like volume diffusion in a crystalline solid, grain boundary diffusion is a thermally activated process, so temperature is one of the most significant parameters. This dependence is described, along with that of pressure, by an Arrhenius relation (see also Zhang 2010, this volume): (34) where AH and AV are the activation enthalpy at 10s Pa and the activation volume, respectively. Recall that with common experimental techniques only the product s8Dgb is determined, and the data are presented in the form:

Diffusion

in Poly cry stalline

sòD„h = (s8D)0 • exp

Materials

951 (35)

with Q - activation energy, which at atmospheric conditions is effectively equal to the activation enthalpy. While the effective grain boundary width is likely to be less affected by the experimental parameters, the segregation coefficient depends in principle on P and Tas well, and furthermore may depend on the grain boundary chemistry due to interactions of the diffusing element with other chemical species. Most of the data presented here are determined for major elements, where we can realistically assume that the segregation coefficient is effectively equal to one. In most experimental studies activation energies are determined, and typical values for activation energies for grain boundary diffusion in silicates and oxides are 100-200 kJ/mol (see below). The vast amount of experimental data for pure metals show that the activation energy for grain boundary diffusion of an element is between 40-60% of the activation energy of lattice diffusion, and within the investigated temperature regions the ratio of DglJD, is about 104-106. This range can be roughly generalized for other materials, but for silicates and carbonates the activation energies of grain boundary and lattice diffusion for O, Ca, Mg, and K are generally very similar (see below) whereas activation energies for Si diffusion are quite different. For metals empirical correlations between grain boundary tracer diffusion coefficients and the melting temperature and the crystallographic structure have been established. For example, the activation energy is roughly between 75-95 J/(mol K) times the melting point in K (compiled in Mishin and Herzig 1999). Much less is known about the activation volume. There are so far only a very few systematic experimental studies that have tried to measure this quantity (e.g., Erdelyi et al. 1987; Farver et al. 1994; Klugkist et al. 2001). Erdelyi et al. (1987) and Klugkist et al. (2001) measured the grain boundary diffusion of Zn in polycrystalline Al and Al bi-crystals, respectively, at pressures up to 1.2 GPa and found that the activation volume is on the order of the atomic volume of Al (AV = 0.4-0.8 i l « ~ 4-8 cm3/mol). It was argued that such an activation volume indicates a vacancy mechanism. The only study of geologically relevant material, i.e., forsterite (Farver et al. 1994), found a very small activation volume of about 1 cm3/mol for Mg grain boundary diffusion in forsterite polycrystals. If the activation volume of grain boundary diffusion is smaller than that for lattice diffusion, as is the case for the activation energy at atmospheric pressures, this can provide us at least an upper limit. Even for lattice diffusion in silicates, only few measurements of activation volumes exist, but those determined are also typically less than 10 cm3/mol (e.g., Bejina et al. 2003; Holzapfel et al. 2007). The effect of a positive activation volume is twofold: at a constant Tit lowers Dgh with increasing P, but for experiments at constant P the apparent activation energy becomes larger. For reasonable activation volumes, a significant effect occurs only at pressures in the GPa range. For example, for an activation volume of 5 cm3/mol, Dgh at 1200 °C and 1, 10, and 100 GPa decreases by a factor of 0.66, 0.017, 1.8xl0~18, respectively, compared with 0.1 MPa. The activation energy at these pressures would be 5, 50, and 500 kJ/mol larger, respectively, than those at 0.1 MPa. Therefore, at extreme pressures like 100 GPa, other diffusion mechanisms with a smaller positive activation volume or even a negative activation volume might dominate; hence extrapolations from lower pressures can be misleading. Effect of water. Increasing pressure may have another indirect but very strong influence on the diffusivity. The net effect of increasing pressure at water-saturated conditions is that the solubility of water in nominally anhydrous minerals increases significantly (as does the solubility of water in melt) because of the strong dependence of solubility on water fugacity (e.g., Kohlstedt et al. 1996; Keppler and Bolfan-Cassanova 2006). Since hydrogen is an incompatible element for most minerals, it is very likely that it strongly segregates into grain boundaries, profoundly affecting the grain boundary structure. At this point we have to distinguish between different

952

Dohmen

& Milke

situations when water is present within the system: at water-saturated conditions, depending on the solid/fluid interface energies, the fluid is present either in isolated pockets within the three dimensional arrangement of the grains or in triple junctions forming an interconnected three dimensional network (e.g., Watson and Brenan 1987). In the latter case the fluid network with physical properties close to a bulk fluid may act as a very efficient diffusivity path and dominate the bulk diffusivity of the polycrystalline system (e.g., Watson 1991). However, even if the system is undersaturated, meaning fHl0 < /H2() sat, fHl0 can still be large and hence significant concentrations of water may be present in grain boundaries and affect the diffusion mechanisms and rates of other elements in grain boundaries. It is well known from various experimental studies that even small amounts of water present within the structure of silicates can have a drastic influence on the mobility of Si (e.g., Kohlstedt 2006; Costa and Chakraborty 2008), and this effect increases with increasing concentrations of water-related defects in minerals. There have already been a number of attempts to explore this effect for grain boundary diffusion, especially in the context of reaction rim growth. Figure 12a shows the growth rates of polycrystalline orthopyroxene rims when they evolve by a diffusion-controlled reaction between olivine and quartz. By displaying the growth rate data without calculating a diffusion coefficient for the respective rate-controlling components, a model assumption regarding the rate-limiting species is not necessary, but we assume that the bulk diffusivity of the rate limiting species is dominated by grain boundary diffusion. The results from six studies fall in two groups; within experimental scatter they define two linear bands in the Arrhenius diagram. The steeper slope (indicating an apparent activation energy of the rate-limiting process of around 400 kj/mol) refers to dry experiments at 0.1 MPa (Fisler et al. 1997; Milke et al. 2007) and a high-pressure study (Gardes et al. 2010) where water-adsorbing crushed alumina was used as the pressure medium surrounding the sample capsules. The less steep slope (apparent activation energy around 200 kJ/mol) refers to high-pressure studies in solid media presses using various media (graphite, pyrophyllite, fluorite, boron nitride) and different initial water contents of the material (defined addition of water to solid reactants - Yund 1997; only absorbed water in the starting powder - Milke et al. 2001; or dried material - Yund 1997, Milke et al. 2009). In the initially dry experimental assemblies small amounts of water entered the capsules during the experiments, evidenced by locally present open micro- and nanopores after the experiment, or by increase in intracrystalline aqueous defects within the reactant olivine as indicated by FTTR measurements (Milke et al. 2009). Remarkably, there are no experimental data available that fall between these two lines in the diagram, thus indicating that a sharp divide between "dry" and "wet" grain boundary diffusion regimes. Moreover, these data show that the effect of pressure on diffusivities in polycrystals is almost negligible in dry systems, but if only traces of water are present there is a large implicit effect of pressure. However, a thermodynamically rigorous definition of "wet" and "dry" grain boundaries in terms of its diffusion properties is not straightforward and the transition from "dry" to "wet" may depend also on the element of interest (Costa and Chakraborty 2008). The divide between "wet" and "dry" can be made in terms of the water fugacity, which in (local) thermodynamic equilibrium controls the concentration of H 2 0 within the grain boundary and the minerals. The concept is illustrated in Figure 12b,c, where starting from a critical water fugacity,/H2O", the diffusion coefficient increases with increasing water fugacity. The exact dependence and point of transition will depend in general on the specific grain boundary and the diffusing element. Here we show two possible examples where in the first case (Fig. 12b) the diffusion coefficient (or any other kinetic property) changes smoothly with a constant slope right after the transition and in the second case (Fig. 12c) at the transition a sharp increase of the diffusion coefficient occurs following a minor dependence on water fugacity. The data shown in Figure 12a indicate a transition of the latter type for the growth of opx rims, which suggests that the grain boundary structure strongly changes at the "dry" to "wet" transition similar to a phase transition. Studies of the rheological behavior of geomaterials have long emphasized hydrolytic weakening of polycrystalline silicates (e.g., Kronenberg and Tullis 1984). In the grain boundary

Diffusion

in Polycrystalline

953

Materials

•

Yund 1997 0.7 GPa, add. 1-5wt%H20

A

Yund 1997 0.7 GPa, add. 0.1wt%H20

O

Yund 1997 0.7 GPa, vac. dried

• ™

Milke etal. 2001 1 GPa, undrled Milke et al 2009 1.9 GPa, 600 °C dried

Gardes et al 2010 1.5 GPa, 700 °C dried alumina assembly A v

4

Milke et al. 2007 0.1 MPa

| ~ | Fisleretal. 1997 0.1 MPa

10 /7[K]

Figure 12. Illustration of the concept of "dry" and "wet" grain boundary diffusion regimes: (a) experimental data, (b) and (c) schematic view of the separation between "dry" and "wet" for two possible cases. In (a) the rate constants are compared for the growth of enstatite rims between olivine and quartz applying a square root rate dependence. Two groups emerge, where the upper group is considered "wet" based on observations of the materials after reaction irrespective of previous drying, and the lower group is indeed dry. The dotted lines are not fits, but drawn lines to illustrate the two different trends of the data. The results of Gardes et al. (2010) are better fitted to a rate law that also includes an increase in grain size during the experiments. This explains the slight scatter of their data in this figure, where the coarsening effect is ignored because compared to the dominant effect of water its influence is small. In (b) and (c) the kinetic property (e.g., diffusion coefficient or rate constant) denoted by D becomes dependent from the water fugacity at a critical water fugacity,/ H ,o*. The two cases illustrated are mainly different during the transition from "dry" to "wet" (see text). The case illustrated in (b) is similar to what was proposed for lattice diffusion in olivine (Costa and Chakraborty 2008). In case (c) the behavior is similar to a phase transition and the data shown in (a) are more consistent with this case compared with the case shown in (b).

diffusion controlled regime, deformation of fine-grained anorthite aggregates (around 5 |am) evolves with high activation energy (almost 600 kj/mol) at "dry" conditions (40 wt ppm H 2 0). but with significantly lower activation energy if the system is considered "wet" (700 wt ppm H 2 0) (Rybacki and Dresen 2000). The lower and upper boundaries in these studies were drawn in hindsight from FTIR measurements of prepared starting samples with specific compositions and might not reflect the entire range of the action of water on grain boundary diffusion. These results clearly indicate that the experimental investigation and understanding of the effects of very low water concentrations in the range below 10 ppm and the 10-100 ppm range will be

954

Dohmen & Milke

mandatory for indirect measurements of grain boundary diffusion by reaction rim growth, or for direct measurements in order to establish a robust definition of "dry" and "wet" in the sense of grain boundary diffusion. Grain boundary orientation and geometry. Experimental studies that investigate in detail the effect of the grain boundary orientation on the diffusivity (e.g., using bi-crystals with different tilt angles) are rare even for metals (e.g., Hofmann 1956; Budke et al. 1996; Klugkist et al. 2001) and completely absent for geological materials. Synthetic bi-crystals of relevant materials have now become available (Heinemann et al. 2001, 2005; Hartmann et al. 2010) and experimental studies are carried out but are still in the early stages (Chakraborty 2008; Hartmann et al. 2009). For the investigated systems (Cu in Ag) the effect is within a factor of 2-3 (see also the study of Nakagawa et al. 2006 for O diffusion in a-alumina), but for certain orientations a very sharp minimum occurs, changing s8Dgh by almost an order of magnitude (Budke et al. 1996). From the standard experiments using polycrystalline aggregates only average information is obtained. Since strong effects occur only for a very small range of grain boundary orientations, these minima probably do not significantly affect the grain boundary diffusion coefficient obtained from a polycrystalline aggregate. In triple junctions the mobility of elements could potentially be significantly higher than in interfaces between two grains, but for example the molecular dynamics study of Frolov and Mishin (2009) found for copper that diffusion within the triple junction is only enhanced by a factor of two. Such an affect would be not enough to enhance the bulk diffusivity, considering the small volume fraction of the triple junctions. However, a completely different situation occurs if the triple junction network is filled with a fluid as mentioned above. In this case the diffusivity can be approximated by the diffusivity of a free aqueous fluid (e.g., Watson 1991; Brady 1983) and will be many orders of magnitude higher than a completely "dry" grain boundary. In addition the volume fraction of the triple junction increases. However, it should be also considered that elements have strongly different solubilities within the fluid. Some elements might be much more compatible with respect to the fluid (s « 1) and therefore transport may become inefficient although the diffusivity is high (e.g., see the numerical example for Fe and Mg in granulite facies rock of Dohmen and Chakraborty 2003). Effect of other impurities. Since a rock usually contains a large number of different elements in concentrations above the ppm level, and a number of the minor and trace elements may be strongly incompatible, leading to significant concentrations within the grain boundary, the situation appears to be rather complicated. We have already speculated about the dramatic effect of water for the mobility of Si (and possibly other elements) within the grain boundary. It is clear that for these "dirty" systems the grain boundary chemistry can be drastically different compared with the adjacent mineral volumes. The net effect for the diffusivity of a particular element can be twofold: For example, the presence of Y in a-alumina retards the mobility of oxygen in grain boundaries by a factor of about 10, as shown by experiments of Nakagawa et al. (2007). In another study, it is indicated that the segregation of Al in yttria-stabilized zirconia (YSZ) enhances the diffusivity of oxygen (Chokshi et al. 2003). The detailed mechanism of how grain boundary diffusion is affected by impurities is unclear. It should be noted that in these studies the impurity concentrations were quite large (in the wt% level) and studies using natural minerals from different origins with different minor and trace element contents do not indicate a strong effect of impurities for natural concentration ranges (e.g., Farver and Yund 1996). However, Freund et al. (2004) found a positive effect of Mn, even for hundreds of ppm, on the diffusion creep of calcite. Clearly, more systematic studies for minerals have to be performed to examine in detail the potential effect of impurities. To summarize, the net effect of pressure on grain boundary diffusion at crustal conditions is of similar magnitude as the range of diffusivities for different grain boundary orientations, but the effect of pressure becomes much more significant in the upper mantle and the transition zone. In addition to temperature, we argued that the water fugacity (saturated or undersaturated)

Diffusion

in Poly cry stalline

Materials

955

of the system is the dominant variable influencing grain boundary diffusion in silicates. Direct measurement of tracer diffusion in poly crystals of geological relevance The body of data from direct measurements of tracer diffusion in natural rocks or synthetic polycrystals of geological relevance is limited. Over the decade from 1991 to 2000 J. Farver and R. A. Yund authored a series of nine papers on this topic. Their general technique was to apply an isotopically doped diffusant to a polished sample surface and measure concentration profiles after diffusion anneals by ion microprobe (SIMS). In recent years, a few similar investigations on polycrystals increased this pioneering effort. The focus of these works was on the most obvious silicate minerals: quartz, feldspars, olivine, and on the non-silicate calcite. Recent studies have focused on major phases of the upper and lower mantle as well as the transition zone, such as wadsleyite and ringwoodite (Shimonjuku et al. 2009) and MgSi0 3 perovskite (Dobson et al. 2008). Diffusion was studied for oxygen, silicon and other major cations. Experiments were performed under dry and hydrothermal conditions, and from atmospheric to lower mantle pressures. In addition to the measurement of tracer diffusion coefficients, the investigation of diffusion of impurities at mineral grain boundaries has just started, but the experimental methods and models to quantify these rates are still developing and emerging (Hayden and Watson 2007; Dohmen 2008; Tominaga et al. 2009; Watson and Dohmen 2010, this volume). The Earth's crust and mantle can in a most simplified way be considered a close packing of oxygen atoms held together by Si (or Al) with many other cations distributed within this structure. Thus, studies of O or Si grain boundary diffusion have a prominent status in the existing body of data. Among the other cations, Mg has especially come into focus due to the dominance of Mg-silicates in the Earth's mantle. Notably, the grain boundary diffusion of Al in aluminosilicates or other mineral aggregates relevant to rocks is still unstudied. In the following we reference direct measurements of grain boundary diffusion for quartz, feldspars, calcite, and Mg-silicates, and compare these results to diffusion within the crystal lattices. Most of the experiments were performed with a polycrystalline aggregate in one of the Type B regimes and values of 5D gh were reported, although actually only the product s8Dgh can be determined. The authors implicitly assumed that the segregation coefficient is one or at least close to one, which is a reasonable assumption for the major elements studied (Hiraga and Kohlstedt 2007). Therefore we also assumed here that s = 1, but this assumption might be inadequate when hydrous species are dominant within the grain boundary or when the intergranular fluid behaves in a manner similar to a bulk aqueous fluid (Brady 1983). Furthermore, to show Dgh together with D ; in one diagram we make the assumption that the effective grain boundary width, 8, is 1 nm. The error bar from this arbitrary number is much smaller than the typical difference between Dgh and D; and in this respect appears negligible. Grain boundary diffusion in quartz. There are studies of self-diffusion of oxygen (Farver and Yund 1991, 1992) and silicon (Farver and Yund 2000a) in polycrystalline quartz (Fig. 13). Fine-grained (1.2 and 4.9 |im) natural quartz aggregates (novaculite) served as the matrix for the oxygen diffusion measurements under hydrothermal conditions (100 MPa) with 18 0-enriched water as the source of the diffusant. The measurements were performed in the type B regime. The starting material was in its natural state (Farver and Yund 1991) or pre-annealed with C0 2 , water, or water + 6M NaCl (Farver and Yund 1992) in order to modify the microstructure to a non-wetted, partially wetted, and wetted state, respectively. In general a ratio DgbIDi = 10410 5 is valid for O diffusion in quartz under hydrothermal conditions (Fig. 13). In the untreated novaculite a higher 5Dgh by a factor of 3 was obtained for the coarser rock type and explained by thermal microcracking (Farver and Yund 1991). However, in the light of the expanded scheme of diffusion regimes one must note that according to the classification of Mishin and Herzig (1995), the latter rock type is a coarse-grained polycrystal, whereas the 1.2 |am rock is a finegrained polycrystal where the diffusion penetration distance during these experiments is larger

956

Dohmen

13001100

-9

•

&

Milke

Temperature [°C] 900

700

500 1D..O

1

A

-11

•

a

i=;-i3 -52 . CN

8

E

°

100 MPa: sat.;

—I

A

Farver and Yund {1991)

a

lie

A g

A v

A A

150 MPa; sat.; 1.2 Farver and Yund (2000a)

•

^ ~ * ~ « 1 0 0 MPa; sat.; //c

He

\

D„Si

0.1 MPa; 1.2 urn Farver and Yund (2000a)

Giletti & Yund (1984)

D,Si\ 0.1 MPa;

100 MPa; sat.; 5 ^im Farver and Yund (1991)

A

D,0

\

100 MPa;

-21

100 MPa; sat.; 1.2 nm

1

Dennis (1984a)

Q -17

g). 19

A

§

100 MPa; sat.; 5 MNaCI Farver and Yund (1992)

A A

lie

Dennis (1984b)

•Cherniak (2003)

-23 -25

9

11

13

15

10 4 /T[K]

Figure 13. Comparison of grain boundary and volume diffusion coefficients of Si and O in quaitz (all tracer diffusivities). Experimental conditions are indicated either in the legend on the right hand side of the plot or next to the data where sat. indicates experiments at water saturated conditions. To convert the reported values of 5D,,h to D,,b, 8 was set to 1 n m in all cases. For Dsh the raw data are plotted along with the linear fit. shown as a thin solid line. The data for D, of Si and O (at water saturated conditions and at nominally dry conditions) are shown as thick lines. The dotted line is the extrapolation of the Si data. For more information on the grain boundary diffusion data see the legend next to the diagram and Table 3.

than the grain size (Type B2" regime). This must lead to an additional leakage flux from the grain boundaries in the direction parallel to the source, which essentially produces a smaller slope in a Le Claire plot compared to a 5 |_im sample. The most obvious effect on O grain boundary diffusion in quartz was initiated by changing the microstructure. Whereas pre-annealing with C 0 2 did not affect the diffusion properties of the starting material, pre-annealing with pure H 2 0 slightly increased 5 w h i l e pre-annealing with 6 M NaCl increased 5D 1012, Fig. 13), the experiments took place in the type C regime. The measurements resulted in similar activation energies near 150 kJ/mol (see Table 3 for a summary of activation energies of various studies) while 8Dgb in the hydrothermal anneal was about lOx larger than in the dry experiment, leading to the conclusion that the effect of water on Si grain boundary diffusion is only about an order of magnitude. However, the actual water content of the novaculite was not determined before or after the diffusion anneals and the virtual absence of porosity does not mean that the material behaves "dry" in the sense of grain boundary diffusion. It cannot be excluded that hydrogen defects segregated to the grain boundaries and molecular water clusters invisible to the TEM investigations were present that may contain enough water to make the grain boundaries "wet". The overall Si and O diffusion in quartz grain boundaries is very similar, in strong contrast to Si and O lattice diffusion in quartz. This suggests that if the grain boundary is not completely

Diffusion

13001100

-9

in Polycrystalline

Temperature [°C] 900

700

Materials

500

957

¡K DgbK 100 MPa; sat.; Kfs F a r v e r a n d Y u n d (1995)

-11 o

-13 cv

U) -19 o

i

*

0

..

s r • — • ».

-25

A

K

0.1 M P a ;

Kfs

F a r v e r a n d Y u n d (1995)

^ DghO

"'V. o

100 M P a ; sat.; A b 9 7 F a r v e r a n d Y u n d (1995)

— . _ D,O

O

D

B l J

Ca

G i l e t t i et al. (1978

0.1 M P a ; A n 9 3 F a r v e r a n d Y u n d (1995)

100 M P a ; sat.; A t

D , C a 0.1 M P a ;

An60

B e h r e n s et al. (1990)

r.1 M P a ; A n 9 3 * ' - % : h e r n i a k (2003)

l b

^ DrlhO 100 MPa; sat.; Kfs

&

X

-21 • D , S i \ -23

;

X

8

\

D

F a r v e r a n d Y u n d (1995)

6

'**'*•.

iEi Q -17

I

a

*.

-15

X

X

D , C a 0.1 M P a ; \

An95

LaTourette and W a s s e r b u r g (1998)

9

11 4

10 /7[K]

13

15

D,K

100 M P a ; sat.; K f s F o l a n d (1974)

Figure 14. Comparison of grain boundary and lattice diffusion coefficients of K, Ca and O in various feldspars (feldspar, albite. labradorite, and anorthite, all tracer diffusivities). In addition lattice diffusion data for Si are shown. The same conventions were used as in Figure 13 to illustrate and indicate the various data. Dotted lines are extrapolated data for lattice diffusion coefficients.

wetted, but there is still some water present in the grain boundaries, Si0 2 might be the dominant diffusing species, and in these cases the oxygen grain boundary diffusivity might not be controlled by diffusion of H 2 0, but of Si0 2 . The former species was suggested by Farver and Yund (1991, 1992) to be the dominant carrier for oxygen. Grain boundary diffusion in feldspar. Data are available for oxygen, potassium and calcium in both natural and synthetic feldspar aggregates, with some experiments performed under hydrothermal conditions and others under dry conditions (Farver and Yund 1995a) (Fig. 14). Most measurements were made in the type B regime using Dt from previous studies. Oxygen grain boundary diffusion was studied hydrothermally with an H 2 l s O source as in the quartz experiments described above, and yielded similar results in the temperature range of the diffusion anneals but somewhat lower activation energies (110 kJ/mol for quartz vs. 80 kJ/mol for orthoclase and albite polycrystals), and again with a general ratio D^JDj = 104-105 (Fig. 14). O diffusion occurs at rates similar to K diffusion in the investigated temperature range, but with a different activation energy (around 210 kJ/mol for K in K-spar-aggregates), signaling that the diffusing species are not coupled. Nearly identical O diffusion coefficients were found for K-feldspar and albite, with the latter being hot-pressed material of natural origin, or polycrystals synthesized from a glass. This was interpreted to mean that the grain boundaries in these feldspar aggregates always assume a similar overall structure (Farver and Yund 1995a). The direct comparison of K grain boundary diffusivities determined at 100 MPa water pressure and at 0.1 MPa in air reveals only an increase by a factor of 5 in the hydrothermal experiments (Farver and Yund 1995a). Compared to other effects of trace amounts of water, for example on the results of rim growth studies or Mg diffusion in forsterite polycrystals (see below), this is a surprisingly small difference. In hindsight this again points to the necessity of determining the actual water contents of the materials used for grain boundary diffusion measurements by spectroscopic methods. K and O diffusion in K-feldspar-quartz aggregates yielded similar results to those for pure feldspar polycrystals, but shifted by about a factor of five towards faster diffusion, which was interpreted to be a result of decreasing crystallographic

958

Dohmen

&

Milke

Table 3. List of experimental conditions, Arrhenius parameters (D 0 and Q), and references for tracer diffusion coefficients of various elements in minerals. Phase

i

Diff. source

Exp. Setup

Trange [K]

P H2O [MPa]

Qtz (Novaculite)

O

H 2 I S O fluid

Cold-seal

723-1073

100

sat.

B2'

Qtz (Novaculite)

O

H 2 1 8 0 fluid

Cold-seal

823-1073

100

sat.

B2

Qtz (Novaculite)

Si

30

SiO2 thin film

Cold-seal

873-1073

150

sat.

C"

Qtz (Novaculite)

Si

30

SiO2 thin film

1 atm furnace

1073-1373

0.1

Tanco Ab97

O

H 2 I S O fluid

Cold-seal

873-1073

100

sat.

B2-B2'

Reg*

C"

Amelia Ab (hot pressed)

o

H2

0 fluid

Cold-seal

873-1073

100

sat.

B2-B2'

Ab (syn.)

o

H 2 1 8 0 fluid

Cold-seal

873-1073

100

sat.

B2-B2'

Cold-seal

723-1073

100

sat.

B2-B2'

1 atm furnace

723-973

0.1

823-973

100

Or (syn.) Or (syn.) Or (syn.)

O

H2

K

41

18

1S

O fluid

KC1 thin film 41

K

fluid with K

Cold-seal

An (syn.)

Ca

42

latm furnace

873-1373

0.1

Cc Solnhofen

Ca

42

CaC0 3 thin film

latm furnace

923-1123

0.1

Cc Solnhofen

Ca

42

CaC0 3 thin film

Cold-seal

923-1073

100

Cc (syn.)+5wt% A1 2 0,

Ca

42

CaC0 3 thin film

1 atm furnace

923-1073

0.1

Cc Solnhofen

O

Cc (syn.)+5wt% A1203

CaO thin film

B2-B2'

sat.

B2-B2' B2-B2' B2

sat.

B2 B2-B2'

H2

IS

O fluid

Cold-seal

573-773

100

sat.

B2

O

H2

18

0 fluid

Cold-seal

623-773

100

sat.

B2-B2'

Mg

26

MgO thin film

1 atm furnace

1273-1473

0.1

B2

FolOO (syn.)

Mg

26

MgO thin film

FolOO (syn.)

Si

30

FolOO (syn.)

l8

Si O 2 thin film 29

18

1 atm furnace

1273-1573

0.1

B2

1 atm furnace

1173-1473

0.1

C'

Wd (XMg=0.9)

Si

(Fe,Mg)2 Si 04 thin film

Multi-Anvil

1673-1823

16000

B2

Wd (X„g=0.9)

O

(Fe,Mg)229Si1804 thin film

Multi-Anvil

1673-1823

16000

B2

Rw (XMg=0.9)

Si

(Fe,Mg) 2 29 Si l8 0 4 thin film

Multi-Anvil

1673-1823

22000

B2

Rw (XMg=0.9)

o

(Fe,Mg)229Si1804 thin film

Multi-Anvil

1673-1823

22000

B2

MgSiOj (Pv, syn.)

Si

29

Multi-Anvil

1673-2073

25000

B2

Si0 2 thin film

* Reg. - Kinetic regime according to the classification of Mishin and Herzig (1995). see Table 1. $

Reference is given here for the lattice diffusion data used to calculate from the slope in the depth profile the h i according to Equation (21). This is only required for experiments within the B 2 or B 2 '- regime.

959

Diffusion in Poly crystalline Materials

^ S o u r c e f o r />,

dD0 [m 3 /s]

»0 [m 2 /s]

fi [kj/mol]

Reference

Farver and Yund (1991b) Giletti and Yund (1984)

2.60E-17

113±4

Farver and Yund (1991a)

Farver and Yund (1991b) Giletti and Yund (1984)

3.40E-17

109+12.5

Farver and Yund (1991a)

6.20E-09

178+38

Farver and Yund (2000a)

3.70E-10

137+18

Farver and Yund (2000a)

Giletti et al. 1978

3.20E-18

83±20

Farver and Yund (1995a)

Giletti et al. 1978

1.10E-18

73±14

Farver and Yund (1995a)

Giletti et al. 1978

1.10E-18

68+12

Farver and Yund (1995a)

Giletti et al. 1978

2.60E-18

78±8

Farver and Yund (1995a)

Foland (1974)

4.00E-11

205+15

Farver and Yund (1995a)

Foland (1974)

5.10E-10

216+9

Farver and Yund (1995a)

Behrens et al. (1990)

6.80E-11

291+13

Farver and Yund (1995a)

Farver and Yund (1996)

1.50E-09

267±47

Farver and Yund (1996)

Farver and Yund (1996)

1.50E-09

267±47

Farver and Yund (1996)

Farver and Yund (1996)

1.50E-09

267±47

Farver and Yund (1996)

Farver (1994)

3.80E-14

127+17

Farver and Yund (1998)

Farver (1994)

3.80E-14

127+17

Farver and Yund (1998)

Chakraborty et al. (1994)

7.70E-10

376±47

Farver et al. (1994)

Chakraborty et al. (1994)

2.10E-10

343±27

Farver et al. (1994)

5.40E-09

203±36

Farver and Yund 2000b

Shimojuku et al. (2009)

1.26E-15

327±101

Shimojuku et al. (2009)

Shimojuku et al. (2009)

1.58E-17

244±86

Shimojuku et al. (2009)

Shimojuku et al. (2009)

6.31E-14

402±88

Shimojuku et al. (2009)

Shimojuku et al. (2009)

7.94E-18

246±70

Shimojuku et al. (2009)

311±48

Yamazaki et al. (2000)

Yamazaki et al. (2000)

7.12E-17

960

Dohmen & Milke

order in heterophase boundaries vs. homophase boundaries (Farver and Yund 1995b). The activation energy for Ca diffusion (around 290 kJ/mol) is even higher than that for K, and at identical run conditions within the studied range of temperatures, 5D gh for Ca is 4-5 orders of magnitude smaller than for K, indicating strong effects of ion size and charge, where ion charge appears to be more important than ion size. In comparison with lattice diffusion coefficients for Ca in plagioclase measured in different crystallographic orientations and for different plagioclase compositions (Behrens et al. 1990; LaTourette and Wasserburg 1998) it appears that DglJDt « 104 for Ca in anorthitic plagioclase, with similar activation energy for grain boundary and lattice diffusion (Fig. 14). Dt for Si in dry anorthite is smaller than in dry quartz by about two orders of magnitude (Cherniak 2003); it would be interesting to compare this finding with grain boundary diffusion data. Grain boundary diffusion in calcite. Diffusion in polycrystalline calcite was measured for calcium (Farver and Yund 1996) and oxygen (Farver and Yund 1998). In the two studies, both hot-pressed calcite aggregates and natural limestone were used and effectively no difference between these materials was observed (Fig. 15). Ca grain boundary diffusion was only addressed in dry diffusion experiments (Farver and Yund 1996) within the type B regime. In the same study, Ca volume diffusion was quantified from single crystals using identical techniques. Comparing grain boundary and lattice diffusion, D gl JDi « 106-107, leads to the conclusion that in calcite rocks, grain boundary diffusion is the dominant diffusion mechanism for Ca up to grain sizes of about 1 cm (Farver and Yund 1996). In contrast, oxygen diffusion in calcite polycrystals was exclusively studied under hydrothermal conditions (100 MPa) (Farver and Yund 1998). In calcite single crystals the diffusivity of oxygen is proportional to water fugacity, and water is thus considered the oxygenbearing transport species (Farver 1994) which would also be expected for grain boundaries. The presence of water has a strong effect in decreasing activation energy for oxygen lattice diffusion in calcite (Anderson 1969; Farver 1994) (Fig. 15) thus enhancing O diffusion in calcite at relatively low temperatures. The activation energy for O diffusion in calcite aggregates is only slightly smaller than in single crystals at 100 MPa water pressure, which might indicate that there is a similar effect of water and dissociated protons at work both within the calcite lattice and grain boundaries. However, at identical water fugacity DgbIDl « 10 6 -10 7 for O, indicating that as for Ca the grain boundaries are the dominant diffusion pathways up to a grain size of 1 cm in calcite marbles. The available data leave open the question whether there is a comparable effect of "dry" vs. "wet" for calcite polycrystals as in calcite lattice diffusion, and how much water might be needed to change the diffusion properties. Grain boundary diffusion in olivine and its high-P polymorphs. The Mg0(Fe0)-Si0 2 system, and especially compounds of stoichiometry Mg 2 Si0 4 , is presently the most studied chemical system with respect to grain boundary diffusion at various P and T due to its key insight into mechanisms highly relevant to properties of the upper and lower mantle and the transition zone. Diffusion in polycrystals within this system is still far from being completely understood. Due to the connection with mantle petrology and petrophysics, experimental data exist over a very wide range of pressures that shed light on the pressure effect on grain boundary diffusion, i.e., the activation volume. Grain boundary diffusion of Mg has so far only been studied in olivine (Farver et al. 1994). The starting materials for 26 Mg diffusion experiments were hot-pressed synthetic forsterite aggregates. Experiments were performed at 0.1 MPa and 10 GPa, and those at ambient pressure both in C 0 + C 0 2 and H 2 +C0 2 atmosphere, where the former mixture is essentially dry and the latter provides water fugacities in the 0.2 bar range. The diffusion anneals were evaluated considering type B behavior, with Dt data taken from earlier studies. In experiments run in both gas mixtures, relatively high activation energies (around 360 kJ/mol) were found. However, the

Diffusion

in Polycrysìalline

Materials

961

Temperature [°C] 900

700

300

500

A

DgbO

100 MPa; sat.; limestone Farver and Yund (1998)

A

s

§

A

DgbO

100 MPa; sat.; calcite Farver and Yund (1998)

A k

h

Q

D g l J C a 0.1 MPa; limestone

O

D

Farver and Yund (1996)

A

C a 100 MPa; sat.; limestone Farver and Yund (1996)

•

\

^

^

O

100 MPa; sat.

y

^

D g l J C a 0.1 MPa; calcite Farver and Yund (1995)

^ F a r v e r (1994)

\ s

D,Ca \

N

0.1 MPa Farver and Yund (1996)

12

10

18

16

14

10 4 /T[K]

Figure 15. Comparison of grain boundary and lattice diffusion coefficients of Ca and O in calcite (all tracer diffusivities). The sanie conventions were used as in Figure 13 to illustrate and indicate the various data.

Temperature [°C] 1500

1100

1300

900

A

A

DghO

^

DgbMg

0.1 MPa; Fo100 C o n d i t e t al. {1985)

A

0 A

'

B

^

O •

8

°

t

g

B

Fo100"-

•

. •

^

• 0.1 MPa; Fo95

^

***

QQ

0.1 MPa Chakraborty et al. (1994)

0.1 MPa; Fo95

Dohmen et al. (2002)

Dohmen e t a l . (2002)

4

10 /7TK1

D g ö M g 0.1 MPa; C 0 / C 0 2 ; Fo100 Farver et al. (1994) D J V I g 1 . 2 - 1 0 GPa; Fo100

o 0g

B

D ,S i

0.1 MPa; H 2 /C0 2 ; Fo100 F a r v e r e t al. (1994)

A

Farver et al. (1994)

DgbSi

0.1 MPa; Fo100 Farver and Yund (2000b)

A D,, O •

16 GPa; wadsleyite

A •

22 GPa; ringwoodite

Shimojuku et al. (2009) 16 GPa; wadsleyite

D,„ S i S h i m o j u k u et al. (2009)

D„„ O

Shimojuku et al. (2009)

Q

gj

22 GPa; ringwoodite S h i m o j u k u et al. (2009)

Figure 16. Comparison of grain boundary and lattice diffusion coefficients of Mg. Si. and O in the polymorphs of (Fe.MgtiSiC^ (olivine, wadsleyite. and ringwoodite, all tracer diffusivities). The same conventions were used as in Figure 13 to illustrate and indicate the various data.

small water fugacity in the H 2 +C0 2 experiments led to a reproducible increase of Dgb by a factor of 5, which is a remarkably strong effect when compared with the effect of the larger water fugacity in hydrothermal experiments. Compared to lattice diffusion of Mg in forsterite, DsiJDi ~ 104, but again similar activation energies were obtained (Chakraborty et al. 1994). Farver et al. (1994) related their Mg diffusivity results at 0.1 MPa from experiments run in

962

Dohmen & Milke

a gas-mixing furnace to diffusivities determined at 10 MPa in a multi anvil press and from this derived an activation volume of ~ 1cm3, indicating a very small pressure dependence for Mg diffusion in forsterite. It must be kept in mind, however, that the decrease of D with increasing P might be compensated by increasing water solubility into the grain boundaries with increasing water fugacity. At present this problem cannot be completely solved and depends on the experimental setup. For future studies it is necessary that FTIR measurements be performed on the respective polycrystals before and after experiments. The water concentrations can be converted to a water fugacity using the experimental calibrations for the water solubility in olivine and polymorphs (e.g., Kohlstedt et al. 1996). Silicon diffusion in forsterite polycrystals has been measured in synthetic aggregates, and no detectable difference was found in diffusion rates between samples buffered by excess MgO or Si0 2 , or samples run in dry N 2 or H 2 /C0 2 atmosphere (Farver and Yund 2000b). The activation energy of about 200 kj/mol is much less than that for Si lattice diffusion in forsterite (Dohmen et al. 2002), and in the investigated temperature range of 900-1200 °C Dg,JDt« 1081010 (Type C regime). Although Si lattice diffusion is many orders of magnitude smaller than Mg lattice diffusion (at 1200° C about 6 orders of magnitude), the grain boundary diffusion coefficient of Si in forsterite almost approaches that of Mg at temperatures below 1000 °C. Silicon and oxygen diffusivities in the high-P polymorphs of forsterite, i.e., wadsleyite and ringwoodite, were determined in a recent study by Shimojuku et al. (2009), at pressures of 16 and 22 GPa, respectively. Their Si and O grain boundary diffusion coefficients are much smaller than the Si and Mg data for olivine measured at 1 bar, with Dgh0 > DghSi but with similar activation energy. These high-P data contain the pressure dependence of Si and O grain boundary diffusion, which leads to a lower diffusivity depending on the respective activation volumes. The apparent activation energy is also higher because of the PAV term (Eqn. 34). If we assume that the structural transformations at increasing pressure affect mainly the lattice, and less so the structure of the grain boundaries, and considering that the grain boundary chemistry for the pure phases is the same, we can use these data sets to derive the pressure dependence for Si grain boundary diffusion in the Mg 2 Si0 4 polymorphs. As discussed above, a large dependence on the presence of water should be expected especially in the case of Si. The multi-anvil experiments with ringwoodite and wadsleyite were nominally dry, but still small amounts of water within mineral lattices were detected with FTIR before and after experiments (Shimojuku et al. 2009). Since the Si diffusion data for olivine were measured at nominally dry conditions in a gas-mixing furnace at 0.1 MPa, large effects of the presence of water are not expected. Hence, if we assume that the difference in the diffusivities of Si between the polymorphs is solely related to the P dependence, we can predict an activation volume for Si diffusion. A simultaneous fitting of the data to Equation (34) results in an activation volume of 3.17 cm 3 /mol, at AH of 224 kJ/mol and 5D a of 3.59 10"17 m 3 /s (see Fig. 17 for the quality of the fit for the three pressures 0.1 MPa, 16 GPa, and 22 GPa). Synopsis of silicon and oxygen grain boundary diffusion in some geologically relevant phases. Si and O grain boundary diffusion data are summarized in Figure 17 and 18 as 8Dgb (taking 5 = 1). In addition to quartz and Mg 2 Si0 4 polymorphs, Figure 17 also contains silicon diffusion data determined in dry multi-anvil experiments at 25 GPa in MgSi0 3 perovskite (Yamazaki et al. 2000). These data are in striking accordance with what would be predicted based on the above pressure dependence on grain boundary diffusion in Mg 2 Si0 4 . This coincidence is probably not accidental, but might indicate that the diffusion properties of grain boundaries both in chemically pure Mg 2 Si0 4 and MgSi0 3 high-P phases are largely identical. Despite the possible importance of orthopyroxenites in mantle processes, no experiments on grain boundary diffusion in orthopyroxene aggregates are available so far. While the data for the mafic minerals are effectively similar when the pressure dependence is considered, the data for quartz are strikingly different. Whether this is related to a different grain boundary structure

Diffusion

in Polycrysìalline

Temperature [°C]

-20

170015001300

1100

900

963

Materials

700 ©

a

w CO

E

8

-22 \ 0 . 1

Qtz; 150 MPa; sat. Farver and Yund (2000a)

o

O

1

•

o

M P a

Qtz; 0.1 MPa Farver and Yund (2000a)

•

Fo; 0.1 MPa Farver and Yund {2000b)

Q

-24

x '\ x

O)

o

%

-26

I H Wd; 16 GPa Shimojuku et al. (2009)

\

X X

K 2 2

G P a

Rw; 22 GPa Shimojuku et al. (2009)

\ \

'••.. 1 6

Pv; 25 GPa Yamazaki et al. (2000)

G P a '

-28

10

12

4

10 /T[K] Figure 17. Comparison of grain boundary diffusion coefficients (as 8D,,h) for Si in various silicates (Qtz quartz, Fo - forsterite, Wd - wadsleyite, Rw - ringwoodite, and Pv - MgSiO, perovskite). The dotted lines are calculated for the pressures indicated at the line from a global fit of the data for Fo, Wd, and Rw with Equation (34).

Temperature [°C]

180015001200 900

600

300

Q

MgO; 1 6 + 25 GPa Van Orman et al. (2003)

•

Fo; 0.1 MPa Condit et al. (1985) Pv; 25 GPa Dobson et al. (2008)

À

Ce: 100 MPa; sat. Farver and Yund (1998)

•

Kfs; 100 MPa; sat. Farver and Yund (1996)

^

Ab; 100 MPa; sat. Farver and Yund (1996)

Q

Qtz; 100 MPa; sat.

+

Wd; 16 GPa

Farver and Yund {1991 )

Shimojuku et al. (2009)

1077"[K]

X

Rw; 22 GPa Shimojuku et al. (2009)

Figure 18. Comparison of grain boundary diffusion coefficients (as SDsl, ) for O in various silicates ( Qtz quartz, Kfs - Orthoclase, Ab - albite, Fo - forsterite, Wd - wadsleyite, Rw - ringwoodite, and Pv - M g S i 0 3 perovskite ), Ce - Calcite and MgO.

964

Dohmen

& Milke

and the diffusion mechanism of Si (due to the presence of Mg), or due to effectively wet grain boundaries even in the nominally dry study, could be the focus of future studies. In Figure 18 5Dgh for oxygen are collected from the studies discussed and a few other investigations. Van Orman et al. (2003) measured Mg and O diffusivities in single and polycrystals of periclase using synthetic single/polycrystalline diffusion couples. The O data shown, determined at pressures of 16 and 25 MPa, yielded an activation volume for grain boundary diffusion of 3.3+2.4 cm3/mol, similar to the activation volume derived here for Si grain boundary diffusion. The data sets for O diffusion in forsterite polycrystals (Condit et al. 1985) and for O diffusion in MgSi0 3 perovskite polycrystals (Dobson et al. 2008), both for nominally dry conditions, cannot be fitted to an Arrhenius relation. In the latter case the analyzed area for SIMS depth scans was approximately the same as the grain size, thus the contribution of grain boundaries to the measured volume is uncertain and probably responsible for the scatter (Dobson et al. 2008). Concluding remarks In conclusion, we point to some observations from the existing data of direct grain boundary diffusion measurements that allow on the one hand some generalization, but on the other hand lead to questions that may define new research areas. In addition, by combining the general trends of diffusion data and the mathematical models discussed, we briefly address two processes in polycrystalline systems of great geological interest: (i) rates and spatial scales of equilibration (chemical or isotopic) and (ii) chemical as well as isotopic fractionation controlled by diffusion. This short discussion is not comprehensive, but from the current state of knowledge we seek a future outlook. •

In the body of data discussed it appears to be a general observation that grain boundary diffusion coefficients of oxygen and the various cations are closer to each other than for lattice diffusion of these elements within the same compounds. The same observation holds for diffusion in melts, and reflects the decreasing influence of atomic order on mobility. Lattice diffusion can be highly anisotropic, while it is isotropic in melts, and grain boundaries fall in-between. The effect of grain boundary structure in phases relevant to the Earth's mantle and crust is a future topic coming into reach, with the improving spatial resolution of analytical methods that will allow measurements of concentration contours on the scale of a single grain boundary.

•

For oxygen and diverse cations it appears that Dg}JDt is generally constant over the investigated temperature intervals in the phases discussed here. Typical ratios of DgtJ Di are between 104 and 107. Consequently, independent of temperature, the bulk diffusion coefficient of these elements (with a segregation coefficient close to one) for monophase polycrystalline aggregates (quartzite, dunite, etc.) with grain sizes larger than about 100 (im should be dominated by the lattice diffusion coefficient (Fig. 7, 8), and the contribution of grain boundary diffusion to the bulk diffusion flux is not significant. A remarkable exception is Si diffusion in quartz and olivine, and probably also in other silicates. It might be possible that there is indeed an extreme difference between the covalently bonded lattice and the weaker bonds in disordered grain boundaries. For "dry" conditions, activation energies of Si lattice diffusion in silicates are typically larger than 500 kJ/mol (e.g., Dohmen et al. 2002; Cherniak 2003), which makes Si lattice diffusion extremely inefficient at lower temperatures; hence the bulk diffusion flux would be dominated by grain boundary diffusion. It is, however, also possible that bonds within the grain boundaries are preferentially weakened by segregated hydrogen, and that thus small traces of water present in the experiments are responsible for the observed effect. Given the importance of minerals like quartz or olivine in the Earth, additional research is necessary.

•

The effect of traces of water on grain boundary diffusion makes quantification of

Diffusion

in Poly cry stalline

Materials

965

the water contents within the polycrystals used in diffusion experiments a necessary task before and after experiments (e.g., by FTIR or SIMS). Assuming equilibrium between the water in the mineral and the fluid, a water fugacity for the experiment can be calculated from experimental calibrations of the water solubility in the respective mineral (e.g., Kohlstedt et al. 1996). By this means a thermodynamically more rigorous definition of the "dry" to "wet" transition for the grain boundary diffusion of a particular element could be found. The low end of water's influence on diffusivities (what is the critical water fugacity,/H2O* separating "dry" from "wet") is a goal for future investigations. For the application of grain boundary diffusion data in geologically relevant cases it could be that for some elements (e.g., Si) only "wet" data should be considered. •

Grain boundary diffusion experiments for the most important rock-forming minerals have mostly addressed the major elements so far (and we have only focused on these in this review). The increasing sensitivity of analytical tools on a very small scale will make measurement of grain boundary diffusion of trace elements more accessible.

•

Grain boundary diffusion of trace elements is of interest due to the expected large grain boundary segregation effects, which are relevant for the mobility of these elements within the bulk rock. The model and classification scheme of Mishin and Herzig (1995) for polycrystals, in alliance with the thermodynamic description of the segregation of incompatible elements (and experimental data, Hiraga et al. 2004) provide us with a new tool to describe bulk diffusion in rocks, to estimate spatial scales of diffusive equilibration as well as diffusive fractionation. From the model it is expected that these scales will be strongly different for incompatible elements when compared to major elements. Other than in the case of major elements in rocks (Joesten 1991, see above), grain boundary diffusion will dominate the diffusive flux for segregating species, including incompatible trace elements, over a wide range of geological time-scales. Segregation in olivine aggregates is prominent for many elements (Hiraga and Kohlstedt 2007; Hiraga et al. 2007) and not yet investigated for other rock-forming mineral assemblages.

•

Differences in diffusion lengths during trace element transport within the grain boundary network have the potential to induce kinetically controlled changes in trace element ratios in polycrystalline materials (for example, rocks) on geologically relevant scales far from equilibrium fractionation. This research is just beginning.

•

In addition to diffusion-controlled fractionation between diverse elements, stable isotopes might be subject to isotopic fractionation by grain boundary diffusion (Richter et al. 2009; Mueller et al. 2010, this volume). For example, it has been shown that Li isotopes are affected by diffusive fractionation (Richter et al. 2003; Teng et al. 2006). Consequently, ''LiA'Li ratios in subduction-related mantle melts might contain information on the diffusive interaction between the Li-hosts via the grain boundaries, depending on the grain boundary diffusivity and the respective segregation coefficients.

•

All of the considerations above regarding diffusive equilibration ignore the effect of dislocations and migrating grain boundaries. Here we discussed mathematical models (Mishin and Razumovskii 1992b; Klinger and Rabkin 1999) that predict these effects. New kinetic regimes have been discovered in which the diffusive flux through dislocations or due to grain boundary migration becomes significant. However, the appearance of these regimes depends on critical threshold values for the dislocation density (here expressed as the volume fraction of dislocations) and the migration velocity (Table 1). The potential effect of these complexities on real polycrystalline systems can now be explored for the system of interest using the classification scheme of these models.

966

Dohmen & Milke ACKNOWLEDGMENTS

The authors like to thank both editors and Dave Kohlstedt for detailed and very helpful comments that significantly improved the clarity of the manuscript. The TEM image in Figure la was kindly provided by Katharina Marquardt. Both authors also thank the DFG for funding in various ways (in particular within the Research Collaborative Centre 526, D0777/1, and FOR741).

REFERENCES Abart R, Petrishcheva E, Fischer FD, Svoboda J (2009) Thermodynamic model for diffusion controlled reaction rim growth in a binary system: Application to the forsterite-enstatite-quartz system. Am J Sci 309:114-131 Anderson TF (1969) Self-diffusion of carbon and oxygen in calcite by isotope exchange with carbon dioxide. J Geophys Res 74:3918-3932 Behrens H, Johannes W, Schmalzried H (1990) On the mechanisms of cation diffusion processes in ternary feldspars. Phys Chem Miner 17:62-78 Bejina F, Jaoul O, Liebermann RC (2003) Diffusion in minerals at high pressure: a review. Phys Earth Planet Inter 139:3-20 Belova IV, Murch GE (2009) Analysis of kinetics regimes in grain boundary self-diffusion. Philos Mag 89:665675 Benoist P, Martin G (1975) Atomic model for grain-boundary and surface diffusion. Thin Solid Films 25:181197 Blundy J, Wood B (1994) Prediction of crystal-melt partition-coefficients from elastic-moduli. Nature 372:452454 Bokshtein BS, Magidson IA, Svetlov IL (1958) On the diffusion in grains and grain boundaries. Fiz Met Metalloved 6(6): 1040-1052 (in Russian) Bollmann W (1970) Crystal Defects and Crystalline Interfaces. Springer, New York Brady JB (1983) Intergranular diffusion in metamorphic rocks. Am J Sci 283:181-200 Budke E, Herzig C, Prokofjev S, Shvindlerman LS (1996) Orientation dependence of 191 Au and M Cu diffusion along symmetric [001] tilt grain boundaries in Cu. Mater Sci Forum 207:465-468 Burke JE, Turnbull D (1952) Recrystallization and grain growth. Prog Metal Phys 3:220-292 Cahn JW, Balluffi RW (1979) Diffusional mass-transport in polycrystals containing stationary or migrating grain-boundaries. ScrMetall 13:499-502 Chakraborty S (2008) Diffusion in solid silicates: a tool to track timescales of processes comes of age. Annu Rev Earth Planet Sci 36:153-190 Chakraborty S, Farver JR, Yund RA, Rubie DC (1994) Mg tracer diffusion in synthetic forsterite and San Carlos olivine as a function of P, T, and f 0 2 . Phys Chem Miner 21:489-500 Chen Y, Schuh CA (2007) Geometric considerations for diffusion in polycrystalline solids. J Appl Phys 101:063524 Cherniak (2003) Silicon self-diffusion in single-crystal natural quartz and feldspar. Earth Planet Sci Lett 214:655-668 Chokshi AH, Yoshida H, Ikuhara Y, Sakuma T (2003) The influence of trace elements on grain boundary processes in yttria-stabilized tetragonal zirconia. Mater Lett 57:4196-4201 Chung YC, Wuensch BJ (1996) Assessment of the accuracy of Le Claire's equation for determination of grain boundary diffusion coefficients from solute concentration gradients. Mater Lett 28:47-54 Condit RH, Weed HC, Piwinskii AJ (1985) A technique for observing oxygen diffusion along grain boundary regions in synthetic forsterite. In: Point Defects in Minerals. Schock RN (ed) Am Geophys Union, Geophysical Monographs 31:97-105 Costa F, Chakraborty S(2008) The effect of water on Si and O diffusion rates in olivine and implications for transport properties and processes in the upper mantle. Phys Earth Planet Inter 166:11-29 de Koker N, Stixrude L (2010) Theoretical computation of diffusion in minerals and melts. Rev Mineral Geochem 72:971-996 Dennis PF (1984a) Oxygen self-diffusion in quartz under hydrothermal conditions. J Geophys Res B 89:40474057 Dennis PF (1984b) Oxygen self diffusion in quartz. 6th progress report of research supported by N.E.R.C., 1981-1984. Prog Exp Petrol 6:260-265 Dobson DP, Dohmen R, Wiedenbeck M (2008) Self-diffusion of oxygen and silicon in M g S i 0 3 perovskite. Earth Planet Sci Lett 270:125-129 Dodson MH (1973) Closure temperature in cooling geochronological and petrological systems. Contrib Mineral Petrol 40:259-274

Diffusion

in Poly cry stalline

Materials

96 7

Dohmen R (2008) A new experimental thin film approach to study mobility and partitioning of elements in grain boundaries: Fe-Mg exchange between olivines mediated by transport through an inert grain boundary. A m Mineral 93:863-874 Dohmen R, Becker H-W, Chakraborty S (2007) Fe-Mg diffusion in olivine. I. experimental determination between 700 and 1200 °C as a function of composition, crystal orientation and oxygen fugacity. Phys Chem Miner 34:389-407 Dohmen R, Chakraborty S (2003) Mechanism and kinetics of element and isotopic exchange mediated by a fluid phase. A m Mineral 88:1251-1270 Dohmen R, Chakraborty S, Becker H W (2002) Si and O diffusion in olivine and implications for characterizing plastic flow in the mantle. Geophys Res Lett 29, doi:10.1029/2002GL015480 Dohmen R, Chakraborty S (2007a) Fe-Mg diffusion in olivine II: point defect chemistry, change of diffusion mechanisms and a model for calculation of diffusion coefficients in natural olivine. Phys Chem Miner 34:409-430 Dohmen R, Chakraborty S (2007b) Fe-Mg diffusion in olivine II: point defect chemistry, change of diffusion mechanisms and a model for calculation of diffusion coefficients in natural olivine (vol. 34, pg 409, 2007). Phys Chem Miner 34:597-598 Eiler JM, Baumgartner LP, Valley JW (1992) Intercrystalline stable isotope diffusion - a fast grain-boundary model. Contrib Mineral Petrol 112:543-557 Erdelyi G, Lojkowski W, Beke DL, Godeny I, Kedves FJ (1987) The pressure-dependence of grain-boundary diffusion of Zn-65 in polycrystalline aluminum. Philos Mag A 56:673-680 Evans B, Hay RS, Shimizu N (1986) Diffusion-induced grain-boundary migration in calcite. Geology 14:60-63 Farver JR (1994) Oxygen self-diffusion in calcite: Dependence on temperature and water fugacity. Earth Planet SciLett 121:575-587 Farver JR, Yund RA (1991) Measurement of oxygen grain-boundary diffusion in natural, fine-grained, quartz aggregates. Geochim Cosmochim Acta 55:1597-1607 Farver JR, Yund RA (1992) Oxygen diffusion in a fine-grained quartz aggregate with wetted and nonwetted microstructures. J Geophys Res [Solid Earth] 97:14017-14029 Farver JR, Yund RA (1995) Grain-boundary diffusion of oxygen, potassium and calcium in natural and hotpressed feldspar aggregates. Contrib Mineral Petrol 118:340-355 Farver JR, Yund RA (1995b) Interphase boundary diffusion of oxygen and potassium in K-feldspar/quartz aggregates. Geochim Cosmochim Acta 59:3697-3705 Farver JR, Yund RA (1996) Volume and grain boundary diffusion of calcium in natural and hot-pressed calcite aggregates. Contrib Mineral Petrol 123:77-91 Farver JR, Yund RA (1998) Oxygen grain boundary diffusion in natural and hot-pressed calcite aggregates. Earth Planet Sci Lett 161:189-200 Farver JR, Yund RA (2000a) Silicon diffusion in a natural quartz aggregate: constraints on solution-transfer diffusion creep. Tectonophysics 325:193-205 Farver JR, Yund RA (2000b) Silicon diffusion in forsterite aggregates: Implications for diffusion accommodated creep. Geophys Res Lett 27:2337-2340 Farver JR, Yund RA, Rubie DC (1994) Magnesium grain-boundary diffusion in forsterite aggregates at 1000 °C - 1300 °C and 0.1 MPa to 10 GPa. J Geophys Res [Solid Earth] 99:19809-19819 Fielitz P, Borchardt G, Schmucker M, Schneider H (2003) How to measure volume diffusivities and grain boundary diffusivities of oxygen in polycrystalline oxides. Solid State Ionics 160:75-83 Fisher JC (1951) Calculation of diffusion penetration curves for surface and grain boundary diffusion. J Appl Phys 22:74-77 Fisler DK, Mackwell SJ (1994) Kinetics of diffusion-controlled growth of fayalite. Phys Chem Mineral 21:156165 Fisler DK, Mackwell SJ, Petsch S (1997) Grain boundary diffusion in enstatite. Phys Chem Mineral 24:264-273 Foland KA (1974) Alkali diffusion in orthoclase. In: Geochemical Transport and Kinetics. Hofmann AW, Giletti BJ, Yoder HS Jr, Yund RA (eds), Carnegie Inst. Washington 634:77-98 Frank FC (1951) The resultant content of dislocations in an arbitrary intercrystalline boundary. In: A Symposium on Plastic Deformation of Crystalline Solids. Carnegie Institute of Technology and Office of Naval Research, p 150-154 Freund D, Wang ZC, Rybacki E, Dresen G (2004) High-temperature creep of synthetic calcite aggregates: influence of Mn-content. Earth Planet Sci Lett 226:433-448 Frolov T, Mishin Y (2009) Molecular dynamics modeling of self-diffusion along a triple junction. Phys Rev B 79, doi: 10.1103/PhysRevB.79.174110 Ganguly J, Bhattacharya RN, Chakraborty S (1988) Convolution effect in the determination of compositional profiles and diffusion-coefficients by microprobe step scans. Am Mineral 73:901-909 Ganguly J, Tirone M (1999) Diffusion closure temperature and age of a mineral with arbitrary extent of diffusion: theoretical formulation and applications. Earth Planet Sci Lett 170:131-140

968

Dohmen & Milke

Gardés E, Wunder B, Wirth R, Heinrich W (2010) Growth of multilayered polycrystalline reaction rims in the M g 0 - S i 0 2 system. Part I: Experiments. Contrib Mineral Petrol doi.10.1007/s00410-010-0517-z Gibbs JW (1928) Collected Works. Longman, New York Gilleti BJ, Semet MP, Yund RA (1978) Studies in diffusion, III, Oxygen in feldspars, an ion microprobe determination. Geochim Cosmochim Acta 42:45-57 Gilletti BJ, Yund RA (1984) Oxygen diffusion in quartz. J Geophys Res 89:4039-4046 Glaeser AM, Evans JW (1986) Effect of grain-boundary migration on apparent boundary diffusion-coefficients. Acta Metall 34:1545-1552 Gupta D, Campbell DR, Ho PS (1978) Grain boundary diffusion. In: Thin Films: Interdiffusion and Reactions. Poate JM, Tu KN, Mayer JW (eds) Wiley, New York, p 161-208 Güthoff F, Mishin Y, Herzig C ( 1993) Self-diffusion along stationary and moving grain-boundaries in a-HF. Z Metallkd 84:584-591 Harrison LG (1961) Influence of dislocations on diffusion kinetics in solids with particular reference to alkali halides. Trans Faraday Soc 57:1191-1199 Hart E W ( 1957) On the role of dislocations in bulk diffusion. Acta Metall 5:597-597 Hartmann K, Ramasse Q, Kilielowski C, Wirth R, Heinrich W (2009) Yttrium-ytterbium interdiffusion along a near S5 grain boundary at the nm-scale. Eos Trans AGU, 90 (52), Fall Meet. Suppl., Abstract V43C-2273 Hartmann K, Wirth R, Heinrich W (2010) Synthetic near 55 (210)/[100] grain boundary in YAG fabricated by direct bonding: structure and stability. Phys Chem Miner 37:291-300 Hashin Z, Shtrikman S ( 1962) A variational approach to theory of effective magnetic permeability of multiphase materials. J Appl Phys 33:3125-3131 Hashin Z, Shtrikman S (1963) Conductivity of polycrystals. Phys Rev 130:129-133 Hayden LA, Watson EB (2007) A diffusion mechanism for core-mantle interaction. Nature 450:709-711 Hayden LA, Watson EB (2008) Grain boundary mobility of carbon in Earth's mantle: A possible carbon flux from the core. Proc Nat Acad Sei USAa 105:8537-8541 Heinemann S, Wirth R, Dresen G (2001) Synthesis of feldspar bicrystals by direct bonding. Phys Chem Miner 28:685-692 Heinemann S, Wirth R, Gottschalk M, Dresen G (2005) Synthetic [100] tilt grain boundaries in forsterite: 9.9 to 21.5 degrees. Phys Chem Miner 32:229-240 Hiraga T, Anderson IM, Kohlstedt DL (2003) Chemistry of grain boundaries in mantle rocks. Am Mineral 88:1015-1019 Hiraga T, Anderson IM, Kohlstedt DL (2004) Grain boundaries as reservoirs of incompatible elements in the Earth's mantle. Nature 427:699-703 Hiraga T, Anderson IM, Zimmerman ME, Mei S, Kohlstedt DL (2002) Structure and chemistry of grain boundaries in deformed, olivine + basalt and partially molten lherzolite aggregates: evidence of melt- free grain boundaries. Contrib Mineral Petrol 144:163-175 Hiraga T, Hirschmann MM, Kohlstedt DL (2007) Equilibrium interface segregation in the diopside-forsterite system II: Applications of interface enrichment to mantle geochemistry. Geochim Cosmochim Acta 71:1281-1289 Hiraga T, Kohlstedt DL (2007) Equilibrium interface segregation in the diopside-forsterite system I: Analytical techniques, thermodynamics, and segregation characteristics. Geochim Cosmochim Acta 71:1266-1280 Hoffman RE (1956) Anisotropy of grain boundary self-diffusion. Acta Metall 4:97-98 Holzapfel C, Chakraborty S, Rubie DC, Frost DJ (2007) Effect of pressure on Fe-Mg, Ni and Mn diffusion in (FexMgi_ x ) 2 Si0 4 olivine. Phys Earth Planet Inter 162:186-98 Houlier B, Cheraghmakani M, Jaoul O (1990) Silicon diffusion in San Carlos olivine. Phys Earth Planet Inter 62:329-340 Hwang JCM, Balluffi RW ( 1979) Measurement of Grain-Boundary Diffusion at Low-Temperatures by the Surface Accumulation Method. 1. Method and Analysis. J Appl Phys 50:1339-1348 Jenkin GRT, Farrow CM, Fallick AE, Higgins D (1994) Oxygen-isotope exchange and closure temperatures in cooling rocks. J Metamorph Geol 12:221-235 Joesten R (1991) Grain-boundary diffusion kinetics in silicate and oxide minerals. In: Diffusion, atomic ordering, and mass transport. Ganguly J (ed) Springer, New York, p 345-395 Kaur I, Mishin Y, Gust W (1995) Fundamentals of Grain and Interphase Boundary Diffusion. John Wiley and Sons Ltd. Keppler H, Bolfan-Casanova N (2006) Thermodynamics of water solubility and partitioning. Rev Mineral Geochem 62:193-230 Klinger L, Rabkin E (1999) Beyond the Fisher model of grain boundary diffusion: Effect of structural inhomogeneity in the bulk. Acta Mater 47:725-734 Klugkist P, Aleshin AN, Lojkowski W, Shvindlerman LS, Gust W, Mittemeijer EJ (2001) Diffusion of Zn along tilt grain boundaries in Al: Pressure and orientation dependence. Acta Mater 49:2941-2949

Diffusion

in Poly cry stalline

Materials

969

Kohlstedt DL (2006) The role of water in high-temperature rock deformation. Rev Mineral Geochem 62:377396. Kohlstedt DL, Keppler H, Rubie DC (1996) Solubility of water in the a , p and y phases of (Mg,Fe) 2 Si0 4 . Contrib Mineral Petrol 123:345-357 Kronenberg AK, Tullis I (1984) Flow strengths of quartz aggregates - grain-size and pressure effects due to hydrolytic weakening. J Geophys Res 89:4281-4297 Lalena JN, Cleary DA (2005) Principles of Inorganic Materials Design, Wiley & Sons, Inc. LasagaAC (1979) Multicomponent exchange and diffusion in silicates. Geochim Cosmochim Acta 43:455-469 LaTourette T, Wasserburg GJ (1998) Mg diffusion in anorthite: implications for the formation of early solar system planetesimals. Earth Planet Sci Lett 158:91-108 Le Claire AD (1963) Analysis of grain boundary diffusion measurements. British J Appl Phys 14:351-356 Levine HS, MacCallum CI (1960) Grain boundary and lattice diffusion in polycrystalline bodies. I Appl Phys 31:595-599 Lojkowski W, Fecht HI (2000) The structure of intercrystalline interfaces. Prog Mater Sci 45:339-568 McCaig A, Covey-Crump SI, Ben Ismail W, Lloyd GE (2007) Fast diffusion along mobile grain boundaries in calcite. Contrib Mineral Petrol 153:159-175 McLean D (1957) Grain Boundaries in Metals. Clarendon Press, Oxford Milke R, Dohmen R, Becker HW, Wirth R (2007) Growth kinetics of enstatite reaction rims studied on nanoscale, Part I: Methodology, microscopic observations and the role of water. Contrib Mineral Petrol 154:519-533 Milke R, Kolzer K, Koch-Miiller M, Wunder B (2009) Orthopyroxene rim growth between olivine and quartz at low temperatures (750-950 °C) and low water concentration. Mineral Petrol 97:223-332 Milke R, Wiedenbeck M, Heinrich W (2001) Grain boundary diffusion of Si, Mg, and O in enstatite reaction rims: a SIMS study using isotopically doped reactants. Contrib Mineral Petrol 142:15-26 Miller DP, Marschall HR, Schumacher IC (2009) Metasomatic formation and petrology of blueschist-facies hybrid rocks from Syros (Greece): Implications for reactions at the slab-mantle interface. Lithos 107:53-67 Mimkes I, Wuttig M (1970) Exact solution for a model of dislocation pipe diffusion. Phys Rev B 2:1619-1623 Mishin Y (1992) A model of grain-boundary diffusion in coarse-grained polycrystals. Phys Stat Solidi A-Appl Res 133:259-267 Mishin Y, Herzig C (1995) Diffusion in fine-grained materials: Theoretical aspects and experimental possibilities. Nanostruct Mater 6:859-862 Mishin Y, Herzig C (1999) Grain boundary diffusion: recent progress and future research. Mater Sci Eng A 260:55-71 Mishin YM, Razumovskii IM (1992a) Analysis of an asymmetrical model for boundary diffusion. Acta Metall Mater 40:597-606 Mishin YM, Razumovskii IM (1992b) A model for diffusion along a moving grain-boundary. Acta Metall Mater 40:839-845 Mortlock AI (1960) The effect of segregation on the solute diffusion enhancement due to the presence of dislocations. Acta Metall 8:132-134 Mueller T, Watson EB, Harrison TM (2010) Applications of diffusion data to high-temperature earth systems. Rev Mineral Geochem 72:997-1038 Murch GE, Belova IV (2003) The lattice model for addressing phenomenological diffusion problems associated with grain boundaries. Interface Sci 11:91 -97 Nakagawa T, Sakaguchi I, Matsunaga K, Yamamoto T, Haneda H, Ikuhara Y (2006) Oxygen diffusion along symmetric [0001] tilt grain boundaries in alpha-alumina. Key Eng Mater 317-318:415-418 Nakagawa T, Sakaguchi I, Shibata N, Matsunaga K, Mizoguchi T, Yamamoto T, Haneda H, Ikuhara Y (2007) Yttrium doping effect on oxygen grain boundary diffusion in alpha-Al 2 0 3 . Acta Mater 55:6627-6633 Nakamura M, Yurimoto H, Watson EB (2005) Grain growth control of isotope exchange between rocks and fluids. Geology 33:829-832 Ohushi T, Nakamura M, Michibayashi K (2010) Effect of grain growth on cation exchange between dunite and fluid: implications for chemical homogenization in the upper mantle. Contrib Mineral Petrol doi: 10.1007/ s00410-010-0517-z Philibert 1(1991) Atom Movements: Diffusion and Mass Transport in Solids. Les Editions de Physique, Les Ulis Poirier I P (1985) Creep of Crystals, Cambridge Univ Press. Porter DA, Easterling KE (1981) Phase Transformations in Metals and Alloys, Van Nostrand Reinhold, Wokingham UK Richter FM, Dauphas N, Teng F-Z (2009) Non-traditional fractionation of non-traditional isotopes: Evaporation, chemical diffusion and Soret diffusion. Chem Geol 258:92-103 Richter FM, Davis AM, DePaolo DI, Watson EB (2003) Isotope fractionation by chemical diffusion between molten basalt and rhyolite. Geochim Cosmochim Acta 67:3905-3923

970

Dohmen & Milke

Rybacki E, Dresen G (2000) Dislocation and diffusion creep of synthetic anorthite aggregates. J Geophys Res 105:26017-26036 Shimojuku A, Kubo T, Ohtani E, Nakamura T, Okazaki R, Dohmen R, Chakraborty S (2009) Si and O diffusion in (Mg,Fe) 2 Si0 4 wadsleyite and ringwoodite and its implications for the rheology of the mantle transition zone. Earth Planet Sci Lett 284:103-112 Stauffer D, Aharony A (1994) Introduction to Percolation Theory. Taylor & Francis Ltd, London Sutton AP, Baluffi RW (1996) Interfaces in Crystalline Materials. Clarendon Press, Oxford Suzuoka T (1961) Lattice and grain boundary diffusion in polycrystals. Trans Japan Institute Metals 2:25-33 Suzuoka T (1964) Exact solutions of 2 ideal cases in grain boundary diffusion problem + application to the sectioning method. J Phys Soc Japan 19:839-851 Swiler TP, Tikare V, Holm EA (1997) Heterogeneous diffusion effects in polycrystalline microstructures. Mater Sci Eng A 238:85-93 ten Grotenhuis SM, Drury MR, Peach CJ, Spiers CJ (2004) Electrical properties of fine-grained olivine: Evidence for grain boundary transport. J Geophys Res - Solid Earth 109: doil0.1029/2003jb002799 Teng FZ, McDonough WF, Rudnick RL, Walker RJ (2006) Diffusion-driven extreme lithium isotopic fractionation in country rocks of the Tin Mountain pegmatite. Earth Planet Sci Lett 243:701-710 Thorvaldsen A (1997) The intercept method. 1. Evaluation of grain shape. Acta Mater 45:587-594 Tominaga A, Kato T, Kubo T, Kurosawa M (2009) Preliminary analysis on the mobility of trace incompatible elements during the basalt and peridotite reaction under uppermost mantle conditions. Phys Earth Planet Inter 174:50-59 Van Orman JA, Fei YW, Hauri EH, and Wang JH (2003) Diffusion in MgO at high pressures: Constraints on deformation mechanisms and chemical transport at the core-mantle boundary. Geophys Res Lett 30, doil0.1029/2002gl016343 Watson EB (1991) Diffusion in fluid-bearing and slightly melted rocks - experimental and numerical approaches illustrated by iron transport in dunite. Contrib Mineral Petrol 107:417-434 Watson EB, Brenan JM (1987) Fluids in the lithosphere. 1. Experimentally-determined wetting characteristics of C 0 2 - H 2 0 fluids and their implications for fluid transport, host-rock physical properties, and fluid inclusion formation. Earth Planet Sci Lett 85:497-515 Watson EB, Dohmen R (2010) Non-traditional and emerging methods for characterizing diffusion in minerals and mineral aggregates. Rev Mineral Geochem 72:61-105 Whipple RTP (1954) Concentration contours in grain boundary diffusion. Philos Mag 45:1225-1236 Wolski K, Laporte V (2008) Grain boundary diffusion and wetting in the analysis of intergranular penetration. Mater Sci Eng A 495:138-146 Yamazaki D, Kato T, Yurimoto H, Ohtani E, Toriumi M (2000) Silicon self-diffusion in MgSiOj perovskite at 25 GPa. Phys Earth Planet Inter 119:299-309 Yan MF, Cannon RF, Bowen HK (1977) Grain boundary migration in ceramics. In: Ceramic Microstructures. Fulrath RM, Pask JA (eds) Westview Press, Boulder CO, p 276-307 Yund RA (1997) Rates of grain boundary diffusion through enstatite and forsterite reaction rims. Contrib Mineral Petrol 126:224-236 Zhang XY, Watson EB, Cherniak DJ (2007) Oxygen self-diffusion "fast-paths" in titanite single crystals and a general method for deconvolving self-diffusion profiles with "tails". Geochim Cosmochim Acta 71:15631573 Zhang Y (2010) Diffusion in minerals and melts: theoretical background. Rev Mineral Geochem 72:5-59 Zhu JH, Chen LQ, Shen J, Tikare V (2001) Microstructure dependence of diffusional transport. Comput Mater Sci 20:37-47

22

Reviews in Mineralogy & Geochemistry Vol. 72 pp. 971-996, 2010 Copyright © Mineralogical Society of America

Theoretical Computation of Diffusion in Minerals and Melts Nico de Koker Bayerisches Geoinstitut Universität Bayreuth, Germany nico. de koke r@ uni-bayreuth. de

Lars Stixrude Department of Earth Sciences University College London, United Kingdom l. stixrude @ ucl. ac. uk

INTRODUCTION Chemical diffusion is a fundamental process in the evolution of planets. Equilibration within and among phases in response to changes in physical conditions requires the comprising chemical species to be spatially rearranged, over distances comparable to the grain size. Quantitative description of such processes demands diffusivities of these chemical species to be accurately known, while detailed insight into the mechanisms of diffusion at the atomic scale elucidate their dependence on pressure, temperature and composition. By applying our understanding of chemical bonding in condensed systems to numerically simulate diffusivity over a range of pressures and temperatures of planetary interest, we can obtain direct constraints on diffusivities at these extreme conditions, and self consistently assess the models used to extrapolate experimental data. Such computations further serve as a proving ground for testing the robustness of the various levels of theory applied in the characterization of bonding and dynamics of a material. The mathematical description of diffusion was first developed in the context of thermal transport by Fourier (1822). Its applicability to chemical transport was recognized by Fick (1855), who cast Fourier's law of thermal conduction in terms of chemical transport and applied it to experiments on the diffusion of salt in a column of water. In an anisotropic system o f « components, Fick's description is most generally given by (1)

where cA is the concentration of species A, and D\b is the relevant component of the diffusivity matrix (e.g., Crank 1975; Lasaga 1998; Zhang 2010). However, this description of diffusion is entirely in the continuum limit, and does not give any insight into the mechanisms by which chemical transport occurs at the atomic level. The description of chemical diffusivity was cast in an atomistic context by studies of Brownian motion (Einstein 1905; Smoluchowski 1906; Langevin 1908; Chandrasekhar 1943), which was described as colloidal particles undergoing stochastic migrations, punctuated by instantaneous elastic collisions. The Stokes-Einstein relation between particle diffusivity and fluid viscosity derived from this works well for simple liquids such as metals (Poirier 1988; 1529-6466/10/0072-0022$05.00

DOI: 10.2138/rmg.2010.72.22

972

de Koker &

Stixrude

Dobson et al. 2000), but discrepancies for fluids of more complex composition suggests that diffusion involves mechanisms not captured by such a simple model (Zhang 2008). Computer-aided atomistic simulation has made an enormous contribution to statistical mechanics and condensed matter physics. Early work coincided with the development of computers as tools for numerical analysis, with Metropolis et al. (1953) using the MANIAC at Los Alamos National Laboratory to calculate the equation of state of a periodic system of hard disks (two dimensional hard spheres), using the Monte Carlo method that now bears his name. This work was soon followed by a Monte Carlo calculation of the equation of state of a Lennard-Jones fluid (Wood and Parker 1957), molecular dynamics simulations of the fluid-solid transition in a system of hard spheres (Alder and Wainwright 1957), and molecular dynamics simulation of a Lennard-Jones system to compute the chemical diffusivity of liquid Argon (Rahman 1964). The groundwork for first-principles methods was set in place with the development of density functional theory (DFT; Hohenberg and Kohn 1964; Kohn and Sham 1965), which steadily increased in popularity throughout the 1970s as an effective tool for characterizing electronic structure and bonding in simple solids. Car and Parrinello (1985) combined DFT with the formalism of molecular dynamics in a computation of the physical properties of crystalline silicon, representing the first dynamical application of electronic structure based computations and pointing the way to direct first-principles treatment of fluids. Today, computational resources are sufficient for more accurate Born-Oppenheimer molecular dynamics (Payne et al. 1992) to be readily performed for systems of relatively complex chemical composition. The purpose of this review is to introduce the condensed matter physics used in performing atomistic simulations, together with the statistical mechanics applied in its analysis to determine diffusivities. To illustrate these methods we will discuss a selection of applications of atomistic modeling to the characterization of diffusivity in both liquids as well as in solids. THEORETICAL FOUNDATIONS Thermodynamic description As an irreversible process, diffusion is readily described in the framework of nonequilibrium thermodynamics. Consider a system of N particles, each one of n compounds, at pressure P and temperature T. The system has volume V, entropy S and chemical potentials ll ,. |_i/( li„. By the Euler equation the internal energy for the system is E = TS-PV

+ fjnANA

(2)

A

Diffusion will result in entropy production of

( 9f jPJ.

N

T ^

Where J^ is the mass flux and V\x/T its conjugate force. J^ is a vector giving the number of particles of component A passing through a unit surface in a unit time JA(t) = mAfjRaA(t)

(4)

where mA and Ra (f) is respectively the mass and velocity of particle A, with the over-score dot denoting a time derivative.

Theoretical Computation of Diffusion in Minerals & Melts

973

Conservation o f mass requires I J , = 0

(5)

A

so that only n-1 flux vectors are independent. A similar argument applies to the chemical potentials due to the Gibbs-Duhem relation. Forming a Taylor expansion of the mass flux in terms of its conjugate force, and assuming the chemical potential does not vary strongly with position, we have (6) 1

B

where LAB are the n-1 x n-1 independent components o f the symmetric phenomenological matrix (Onsager 1931 a,b). In an anisotropic material, each o f these components is itself a second rank tensor in spatial components. Laboratory measurements as well as computational simulations o f diffusion are more readily characterized in terms of concentration gradients than chemical potentials. Applying the chain rule to Equation (6), we may also write J, = " E ^

b

V c

(7)

b

B

where cB is the concentration o f component B, and DAB are the n-1 x n-1 independent components of the diffusivity matrix given by S

T

c

f ^ dcB

(8)

Note that unlike the phenomenological matrix, the diffusivity matrix is not symmetrical. In describing J it is important to define a spatial frame of reference, relative to which the flux occurs. I f R a is the velocity of species A with concentration cA, the corresponding mass flux is J,=c4R,-R„)

(9)

where R 0 is the reference velocity. For a constant number of particles with velocities described relative to an absolute reference frame independent of particle distribution (the system described in Eqns. 2-8), the reference frame is referred to as mass-fixed or barycentric (de Groot and Mazur 1969; Lasaga 1998). This choice is most amenable to theory, and relevant to molecular dynamics simulations. In contrast, across coexisting phase boundaries or compositional couples, diffusion is described relative to the phase/couple interface, which moves in response to the flux (Kirkendall effect; Smigelskas and Kirkendall 1947), with regions further from the interface serving as chemical reservoirs. In this volume-fixed reference frame the flux is weighted by the partial molar volume VA of each component A in the mass conservation condition t Z

JA

A

=o

(io)

In order to transform the diffusivity matrix between these reference frames one applies the coordinate transformation (de Groot andMazur 1969)

D\0 = D„

T7

f s

V

A

VN

^

(11)

C

A

Cn

J

where DVM is the diffusivity in the volume fixed reference frame.

974

de Koker &

Stixrude

DAB is generally referred to as the chemical diffusivity matrix, the diagonal components are the main-, diagonal- or self-diffusivities1, with off-diagonal components known as o f f diagonal- or cross-diffusivities. When the off-diagonal diffusivities are comparable in magnitude to the diagonal values, changes in concentration of one component will strongly affect the distribution of the other components. In stable phases, the eigenvalues of the diffusivity matrix are all positive, but in unstable systems some eigenvalues of the matrix are negative, resulting in diffusion against the concentration gradient, also referred to as uphill diffusion. Unmixing of fluids and exsolution in minerals are example scenarios which involve uphill diffusion (e.g., Lasaga 1998; Zhang 2008). In the foregoing discussion we assumed that the only force experienced by diffusing particles is due to the chemical potential. Should other forces be present (electromagnetic, gravity, etc.) these can be incorporated by additional terms in the Euler equation describing the force-potential conjugate pair (Lasaga 1979; Callen 1985). Statistical mechanical description Diffusion in liquids. The statistical mechanical description of diffusion was placed on a sound theoretical platform during the 1950s with the development of linear response theory (Callen and Welton 1951; Green 1952; Kubo 1957; Zwanzig 1965). The response function formalism allows the diffusivity, inherently a non-equilibrium property, to be expressed in terms of the time evolution of an equilibrium system

Jly

02)

A a

where (0 • R„A ) = lim - J 0 X A (0R Ba (t + t')dt'

(13)

is the velocity autocorrelation function. It can be readily shown that

^ r S i

< R ^ - R

a

> = iim\

L

(14)

which then yields the result obtained by Einstein (1905), Smoluchowski (1906) and Langevin (1908) in their studies of Brownian motion (|RA(0)-RA(O|2)

DA= lim-J

61

L

(15)

with (|R,t(0) - R^it)!2) the mean square displacement of the particle; Equation (15) is known as the Einstein relation (Hansen and McDonald 2006; Allen and Tildesley 1987). A more general description is given by the mass flux correlation function, in terms of which the full phenomenological coefficient matrix is (Zhou and Miller 1996; Wheeler and Newton 2004) L

AB=^^]{iÀt)-iB)dt

(16)

with the diffusivity matrix following via Equation (8). 1 Although the self-diffusivity is often defined in terms of gradients in isotopie concentrations, it is standard in computational work to use the term for the diagonal values of the diffusivity matrix, denoting these simply as DA. We will follow this convention in this chapter as well.

Theoretical

Computation

of Diffusion

in Minerals

& Melts

975

Shear viscosity is similarly given by the average of the stress autocorrelation function (17)

-"V >*i

where a,y are the off-diagonal components of the stress tensor. This relation allows one to directly compute the liquid viscosity, rather than relying on simple scaling relations between D and r|, such as the Stokes-Einstein (Einstein 1905; Langevin 1908) or Eyring relations (Eyring 1936). Diffusion in solids. In contrast to liquids, solid-state diffusion is generally much slower and anisotropic. Migration occurs by individual jumps of atoms into either vacancies or interstitial sites, which are associated with activation energies of several eV. For typical vibrational frequencies, the Boltzmann distribution predicts jumps to occur with characteristic times of 10~4-10~9 seconds. We will consider vacancy self-diffusion here; the equations describing interstitial diffusion are very similar. The rate and geometry of jump events depends on the crystal lattice, the nature (size, charge, etc.) of the migrating atom, and on the concentration of vacancies in the lattice. In addition to simple jumps of atoms into vacancies on their own sublattice, diffusion in solids can also involve cyclic mechanisms of multiple steps and combinations of jump events onto different sublattices. A common example, the six-jump cycle, is well known in alloys (Elcock 1959), and involves a set of six jump events by which an atom and a vacancy end up swapping positions (Fig. 1). • X X

_

»X X

^

X «X

_

X 9X

^

XX •

^

XX

^ «

X

X *

Figure 1. Schematic of the individual steps of the six jump cycle of solid-state diffusion. • and x denote two different chemical species, with • ultimately exchanging positions with a vacancy. This sequence of jumps is energetically more favorable than a single diagonal jump of • into the vacancy.

The self-diffusivity of component A migrating though vacancy diffusion is given by (e.g., Philibert 1991; Glicksman 2000) DA=X™D™

(18) 1

where X™ is the molar fraction of vacancies on the vA-site, and D™ is the vacancy diffusivity. To evaluate D™1 we view the diffusion process as a reaction in which the system is activated from equilibrium to an activated state. In the activated state, the migrating atom will be at a saddle point on the potential energy surface corresponding to the highest potential energy it takes on during a migration event. Let vatt be the attempt frequency, which is simply the vibrational frequency of the atom in the direction of the candidate jump, and V j the frequency with which these attempts result in successful jumps. By the Boltzmann distribution, the fraction of successful attempts is u m p

= expf — | = e x p f ^ — l e x p f - - ^ — | I k j ) UbJ I k j )

(19)

where G™, H™ and S1™ are respectively the Gibbs free energy, enthalpy and entropy of vacancy migration. The vacancy diffusivity can now be related to the attempt frequency via

^-^fM-v) with Z is the coordination (i.e., the number of candidate jump directions) and X is the jump

de Koker &

976

Stixrude

distance. fc is a correlation factor accounting for the possibility that, following a successful jump, the atom can jump back into its previous position. The value o f / c depends on the crystal structure, varying between 0.5 and 1.0 (Borg and Dienes 1988; Shewmon 1989). H™ is the height difference on the potential energy surface between the equilibrium and saddle points, while S™ can be shown from transition state theory (Vinyard 1957) to be

nv (21)

- i f

where N is the number of atoms in the system, v, and v,' are phonon frequencies at the equilibrium position and at the saddle point respectively, and the product in the denominator excludes the unstable mode frequency associated with the migration. The intrinsic vacancy fraction can be obtained as H vf \

rrvf XT =

ex

P

k j

= exp V

fB J ft

e x p

V

r Bi t

(22)

with CA', S*, and H * respectively the Gibbs free energy, entropy and enthalpy of vacancy formation. H f is the energy difference between a perfect, infinite lattice and an infinite lattice containing a single vacancy. The difference in configurational entropy between the perfect and imperfect lattices is negligible, so that S* is given by the differences in vibrational entropy between the two geometries as (Burton 1972; Gillan and Jacobs 1983) 3 iV

nv,

•

exp V ^B

37V

(23)

r>;

Note that the assumption in these formulas is that the crystal contains no trace element impurities, i.e., all vacancies are intrinsic. If these are of different charge than the ion they substitute, additional vacancies are required in order for charge balance to be maintained. Indeed, the concentration of such extrinsic impurities are often much higher than intrinsic vacancies, so that can be estimated from the trace element concentration instead.

COMPUTATIONAL APPROACHES Atomistic simulation involves the numerical investigation of materials by applying the principles of condensed matter theory to characterize bonding and dynamics. Used wisely, these computational techniques afford understanding of macroscopic material properties in the context of behavior at the atomic level. A variety of methods exist, each suited to investigating a specific subset of physical properties. Important considerations in choosing a suitable technique of simulation include the nature of the property of interest, whether thermal and/or electronic effects need to be characterized, the desired accuracy of the result, the quality and availability of experimental measurements, and the available computational resources. Commonly applied simulation techniques include static energy calculation, structural optimization, lattice dynamics, Monte Carlo computation, molecular dynamics, and electronic band structure determination. All these methods require as input the chemical composition, a reasonable initial geometry, and information on bonding via parameterized potentials or electronic structure.

Theoretical Computation of Diffusion in Minerals & Melts

977

We will begin by discussing various methods by which bonding can be characterized, and then consider methods through which thermal effects in materials can be simulated in more detail. This will lead us into a discussion of the various methods by which diffusivity may be computed and characterized. Characterization of bonding The physical conditions on and within Earth are such that bonding must be accurately characterized in order to capture the essential physics of condensed terrestrial phases (Birch 1952; Knopoff and Uffen 1954). Simple kinetic models that neglect bonding are not sufficient, and specialized descriptions either via parameterized potentials or directly by means of the electronic structure must be used. Parameterized potential approach. Perhaps the simplest characterization of bonding involves describing interatomic interaction in terms of a set of analytical equations with parameters constrained from the known physics of the system of interest. A canonical example of such a potential is that due to Lennard-Jones (1924) (24) where s represents the ground state potential energy and o is a radial scaling parameter. By minimizing the misfit between properties computed using Equation (24) and experimental data for the system of interest (e.g., liquid Ar), optimal values for s and o can been determined. Numerous similar expressions exist by which the potential energy may be parameterized in terms of two-, three-, and four-body interactions (e.g., Gale 1998, 2001), common examples being the Born-Mayer, Buckingham, Morse and Stillinger-Weber functional forms. Such descriptions can consist of many free parameters, the accurate constraint of which requires large and preferably varied data sets for the materials in question. Standard potential sets exist for some common materials such as water (Berendsen et al. 1987), silica (van Beest et al. 1990), and also silicate melts (Matsui 1994). While these potential sets may be used to compute physical properties in close agreement with experimental data, they give only atomistic insight, not electronic, and are severely limited in their predictive power when applied to conditions outside the range of chemical and/or physical conditions at which they were constrained. First-principles approach. First-principles or ab initio calculations apply electronic structure theory to characterize bonding. The potential energy of the material associated with bonding is obtained in situ by computing the electronic structure of the material. Of the electronic structure-based methods that have been developed DFT remains the state of the art, and has been widely applied in the physical sciences. In the standard quantum mechanical formalism embodied by the time independent Schrodinger equation (25) representation of bonding requires characterization of the total electronic energy T and wave function >F for the Hamiltonian of the system of interest

2m

(26)

with me the electronic mass, V includes Coulomb interactions among the nuclei and electrons, and r, and R,- are the coordinates of the Ne electrons and N„ nuclei, respectively.

978

de Koker &

Stixrude

Constructing the wave function for a system of fully interacting electrons is prohibitively difficult for more than a few tens of electrons (Kohn 1999). This presents a fundamental obstacle in solving the Schrodinger equation even for relatively simple materials. DFT proposes a solution to this dilemma by recasting the problem with the charge density p f (r) as independent variable. The foundation of the theory is the Hohenberg-Kohn theorem (Hohenberg and Kohn 1964), which states that the ground-state charge density of a system of interacting electrons in some external potential determines this potential uniquely, i.e., 'V(r 1 ,r 2 ,...,r J V t ,R 1 ,R 2 ,...,R J V n )->'V[p c (r)]

(27)

This allows the charge density to be used as independent variable, instead of the wave function itself, and guarantees an injective mapping between the ground state charge density and observable properties. The theorem further holds that the ground state total energy of a system is variational with respect to the electron density, i.e., the exact charge density provides the minimum possible energy for the ground state. The theorem provides the theoretical underpinning for the Kohn-Sham approach (Kohn and Sham 1965), by which the Hamiltonian may in principle be solved exactly. The approach involves replacing the difficult interacting many body system with a non-interacting system which can be more readily solved, and then relying on the variational principle to iteratively converge on the exact solution of the interacting system. The Kohn-Sham electronic potential is then ^KS[Pe(r)] = X , ( R p R . - ^ , J + \ a r t r c e [ P e ( r ) ] + X [ P e ( r ) ]

(28)

"Xxi being the external potential due to nuclear changes, VH 0. By considering the diffusivity of a single sphere in a fluid continuum as a function of cell size, Yeh and Hummer (2004) and Zhang et al. (2004) showed that such an extrapolation should have the form D 2 = D

n

/

+

^

oixri e

(35)

Theoretical Computation of Diffusion in Minerals & Melts where ^ is a constant using Equation (17).

981

2.837297), and r| is the viscosity which is best determined directly

Solids. In order to describe the statistical mechanics by which diffusion in solids occurs, the enthalpies, entropies and attempt frequencies for defect/vacancy formation and migration must be computed (Eqns. 19, 22). For a chemically pure system, the enthalpy and entropy of vacancy formation can be obtained as differences between an infinite perfect lattice and an infinite lattice containing a single vacancy (Glicksman 2000). The enthalpy of formation is then given by H ^ H ^ - H ?

(36)

In practice the boundary condition introduces periodic images of the vacancy, so that //™c contains a small contribution from interaction of the vacancy with its periodic images. This interaction can be corrected for; in a cubic unit cell the correction is gZ/vi^o^Z™ 2s(/

(37)

where f? is the size of the periodic cell, zvac is the charge of the vacancy, a M is the Madelung constant for the particular lattice site, and s 0 is the dielectric constant of the perfect crystal (Leslie and Gillan 1985; Brodholt 1997). The migration enthalpy is the minimum barrier height of the migration event, and is associated with a saddle point in the potential energy surface. Although the location of the saddle point may be symmetrically evident in simple cases, this is generally not the case in more complex structured materials, where the minimum energy path often involves movement of the atom along a curved, non-symmetric path. An efficient solution may be obtained by the so-called "nudged elastic band" method (Jonsson et al. 1998), by which a trial migration path is adapted such that its barrier height is minimized. When combined with the "climbing image" scheme (Henkelman et al. 2000) the location and energy for the saddle point can be precisely found. Once the position of the saddle point is known, H™ follows as the energy difference between the saddle point and the equilibrium position M vm H

A

F7Saddle =

t

A

/-¡.'qui ~

h

A

/ o o \ (

3

8

)

The jump frequency is similarly obtained by computing the vibrational frequencies of the equilibrium and saddle point geometries. This approach neglects anharmonicity, although anharmonic corrections have been developed (Sangster and Stoneham 1984). It is often assumed that anharmonic effects are sufficiently similar between the equilibrium position and the saddle point to effectively cancel.

SELECTED APPLICATIONS Liquids and melts Viscosity of the liquid outer core. The Earth's magnetic field is induced and maintained by convection in the liquid outer core, and the viscosity of liquid iron at core conditions is of key importance to understanding this process (e.g., Merrill et al. 1998). Outer core viscosities determined from geophysical observations are disparate, with values ranging between 10~210in P a s . In contrast, values determined using condensed matter theory suggest values to be between 10~4-10° Pa-s (Gans 1972; Poirier 1988; Vocadlo 2007, and references therein). These theoretical estimates are in close agreement with low pressure experimental measurements, which are very challenging above pressures of about 5 GPa (e.g., Dobson et al. 2000; Rutter et al. 2002; Terasaki et al. 2002).

982

de Koker &

Stixrude

Vocadlo et al. (1997), de Wijs et al. (1998) and Alfè et al. (2000) performed FPMD simulations of liquid Fe at pressures characteristic of the Earth's liquid outer core. They computed the Fe self-diffusivity using the mean square displacement slope (Eqn. 15; Fig. 2), and from this obtained the viscosity via the Stokes-Einstein relation, which has been shown to hold well for liquid metals (Dobson et al. 2000). Diffusivities thus calculated are around 5xl0~ 9 m2/s, corresponding to viscosities of 1.3xl0~2 Pas, with a very similar value obtained by computing viscosity directly from the stress autocorrelation function (Eqn. 17). This value is only about ten times higher than the typical viscosities of liquid metals at ambient pressure, consistent with the suggestion of Poirier (1988) that transport properties in metals remain relatively constant along the melting curve. It is interesting to note that these first-principles estimates are not very different from the values estimated using approximate condensed matter physics based arguments (e.g., Gans 1972). What's more. Alfe et al. (2000) compared their results for liquid structure and dynamics to values computed using a simple inverse power potential and found that such a potential gives a good representation of these physical properties in the liquid, indicating that even at the extreme pressures present in the core. Fe behaves as a simple metallic liquid. However, this result can only be derived after the fact, since there is no guarantee that the bonding behavior of Fe will remain similar at extreme pressure conditions. This is an important point to note about atomistic simulations: much insight can be gained about the atomistic behavior of systems using relatively simple stripped-down potentials, but only methods which represent the physics of bonding directly from first principles can be truly predictive.

Figure 2. Diffusion coefficient for liquid Fe at 4300 K and 132 GPa, expressed as the mean square displacement (r(t)) divided by 6 x the elapsed time interval, as describe by the Einstein relation (Eqn. 15). The firstprinciples result (solid line) be compared to that computed using a simple inverse power potential (dotted line). The diffusivity converges to a plateaux value, the uncertainty of which is theoretically expected to decrease as r " 2 for longer simulation times. The initial peak is related to the movement of the atoms about their equilibrium positions, and gives an indication of its amplitude. Adapted from Alfe et al. (2000).

i(ps) Melts on the Mg0-Si02 join. The early thermal and chemical history of a terrestrial planet such as Earth is intimately related to processes associated with planetary-scale magma oceans. Indeed, sufficient energy sources were present for the Earth to have been entirely molten during the late stages of accretion (Urey 1955: Hanks and Anderson 1969: Ruff and Anderson 1980: Tonks and Melosh 1993). Magma ocean dynamics depend strongly on viscosity, yet values for silicate melts have only been measured up to about 15 GPa (Liebske et al. 2005). Extrapolation of these values to higher pressures is very uncertain, and depend crucially on the activation volume used in the Arrhenius relation. MgO and Si0 2 are the two most abundant oxide components in the mantle, and account for a dominant compositional fraction in a terrestrial magma ocean.

Theoretical Computation

of Diffusion in Minerals & Melts

983

Due to the tremendous technical challenges associated with the laboratory investigation of silicate melts at elevated pressures (e.g., Stebbins et al. 1984; Lange and Carmichael 1987; Rivers and Carmichael 1987; Rigden et al. 1989; Reid et al. 2001; Liebske et al. 2005; Ai and Lange 2008), theoretical approaches to the study of silicate melts using empirical and semiempirical parameterized potential sets were extensively performed during the 1980s and early 1990s (e.g., Angelletal. 1987; Kieffer and Angell 1989;StixrudeandBukowinski 1989;Kubicki and Lasaga 1991; Cohen and Gong 1994; Matsui and Anderson 1996; Nevins and Spera 1998); these methods are now applied to very large systems for very long simulation times (e.g., Lacks et al. 2007; Adjaoud et al. 2008; Martin et al. 2009; Nevins et al. 2009). However, the speed and power of computational resources have now developed to a point where Born-Oppenheimer first-principles molecular dynamics (FPMD) can be readily performed on silicate liquids, albeit for small systems sizes, simple compositions and relatively short run times. In a series of studies (Stixrude and Karki 2005; Karki et al. 2006, 2007, 2009; de Koker et al. 2008, 2010; Mookheijee et al. 2008; de Koker and Stixrude 2009; Stixrude et al. 2009), FPMD simulations for liquids along the MgO and S i 0 2 binary were performed over the range of pressures relevant to the entire mantle. Compositions considered are MgO, Mg 5 Si0 7 , Mg 2 Si0 4 , Mg 3 Si 2 0 7 , MgSiO ;. MgSi 2 O s , MgSi 3 0 7 , M g S i , O n and Si0 2 , as well as 12MgSiO, + 8 H 2 0 (10 wt% water). These simulations were performed in the WVr-ensemble for silicate liquid systems consisting of between 72 and 112 atoms, and MgO liquid systems of 64 atoms (Stixrude and Karki 2005; Karki et al. 2006, 2007; de Koker et al. 2008, 2010). Simulations were run for at least 3000 fs, with time steps of 1 fs (0.5 fs for hydrous systems), and the first 600 fs used for equilibration. Si0 2 , MgSi0 3 , and hydrous systems were run for longer times in order to test convergence of computed diffusivities with respect to runtime (Karki et al. 2007, 2009), while system sizes of up to 400 atoms have been considered in a number of compositions to assess finite size effects. Self-diffusivities for these systems were determined from the mean squared displacement slope (Eqn. 15), and show a consistent trend of Dsi < D0 < DMg along the binary (Fig. 3). Self-diffusivities for all species are enhanced by about a factor of four in the hydrous melt, with the H value higher than those of the other species by almost an order of magnitude. Self-diffusivities for all species are lower at elevated pressures, although for H the decrease is significantly smaller. At low temperatures D s i , and to a lesser extent D () , initially increases upon compression for Xsi()2>o.5. Analysis of finite size effects suggests that computed diffusivities underestimate values extrapolated to infinite system sizes by about 10-20%. With the exception of the low P-T behavior, self-diffusivities are well represented by an Arrhenian relation

DA(P,T) = D(>exp

K+PK k„T

(39)

E\ and VI being the activation energy and volume of species A, with D° the limiting diffusivity as T —> oo; the low P-T increases in self-diffusivity imply activation volumes that are locally negative. In the dry melt compositions, activation energies for Mg is consistently the lowest and for Si the highest. Values decrease notably with S i 0 2 content, flattening out for X si o 2 < 0.4 (Fig. 4). Activation energy values for Si and O are very similar at high S i 0 2 concentrations, but diverge as concentration decreases. Relationships in Arrhenian parameters between Mg, Si and O exhibited in the dry melt are also seen in the hydrous melt. Activation volumes and limiting diffusivities are very similar to the anhydrous melts, but activation energy of the hydrous melt is notably depressed.

984

de Koker &

Stixrude

P (GPa)

P (GPa)

Figure 3. Self-diffusivities in hydrated MgSiO, melt with 10 wt% H 2 0 , with lines indicating the Arrhenius fit. Data from Karki et al. (2009) and Mookherjee et al. (2008).

The relative mobilities of various species can be understood in the context of differences in the character of bonding and changes in liquid structure with Si0 2 content. The most notable structural trend is the change in oxygen speciation, as expressed by the oxygen coordination with respect to silicon ( Z _ i ) . At all degrees of compression the Z()_si distribution shifts to lower values as Si0 2 content decreases, with polyhedral species (Z0_si > 2) especially rare for X si() , < 0.4 (de Koker et al. 2010). Therefore, at temperatures close to melting, self-diffusion occurs more readily in the absence of a highly polymerized framework, while the presence of free oxygen (Z0_si = 0) in addition to non-bridging oxygen (Z 0 . si = 1 ) does not have a notable effect on diffusivity. Interestingly, analysis of bond lifetimes reveal that free oxygen is notably longer lived than more highly coordinated oxygen species (de Koker et al. 2008). 0

S

Because diffusion occurs through continuous breaking and forming of old and new bonds, one cannot describe self-diffusion of a given species without also invoking mechanisms by which other species migrate. Visualization and analysis of liquid dynamics indicate that diffusion mechanisms can be well described in terms of increases and decreases in coordination. One pair of events would be2 O,

+ |Z|Si

0

^ON+|z+1|Si°

(40)

followed by 2 We describe the local structure about Si atoms as i/jSi Q , with [Z] the coordination number and Q the number of bridging oxygens; O speciation is denoted by O h for free oxygen, O n for non-bridging oxygen and O b for bridging oxygen.

Theoretical

Computation

of Diffusion

in Minerals

& Melts

985

Figure 4. Activation energies ( £ / ) , volumes ( V / ) and limiting diffusivities (DA") for fits of the Arrhenius equation to nine compositions along the M g 0 - S i 0 2 binary. Also shown are the corresponding values for the hydrous melt (offset by X = 0.02 for clarity), together with results from previous empirical potential work, colored by chemical element as for the FPMD results: KL91 - Kubicki and Lasaga (1991), L07 - Lacks et al. (2007), A08 - Adjaoud et al. (2008), M09 - Martin et al. (2009), N09 - Nevins et al. (2009). The V / values for Si and O at MgSi 9 O w and SiO, compositions are negative.

o

ft —

0.0

0.2

0.4

0.6

0.8

0.0

1.0

X(Si0 2 )

'I

I

I

I1

0.2

0.4

0.6

0.8

1.0

X(Si02)

oB, +

lz+llsi

y

^oN*

+ IZI si

y 1

-

(41)

with * denoting a distinct oxygen atom. Reversed event pairs are also observed. O n

+

. Z . s ì ^ O . +

^ S Ì

0

(42)

followed by oN, +

lz_llsi

y

^o;*

+

l z l si

y+i

(43)

Over time, a single atom will traverse large distances by repeated working of a combination of these types of events. However, free oxygen is unique in this scheme: it can only undergo increases in coordination, and thus have only half the general diffusion pathways available to it. This may explain its prolonged mean lifetime, as well as the observation that activation energies level off at low Si0 2 concentrations. Higher diffusivities in the hydrated melt can also be well understood in terms of liquid structure. With the presence of additional oxygen, the average oxygen-silicon coordination is decreased, resulting in a less polymerized liquid in which hydrogen binds especially to the free- and non-bridging oxygen species (Karki et al. 2009). As seen in the dry melt, a decrease in bridging oxygen results in decreased activation energy for all species present in the melt, as non-bridging and free oxygens more readily serve as hosts for transition states associated with the various diffusion mechanisms. Indeed Karki et al. (2009) identified two main hydrogen diffusion mechanisms. The first involves hydrogen atoms forming and breaking bonds with polyhedral oxygen (Z 0 . si > 1), thus always staying directly associated with the silica framework via H-O-Si bonds (Fig. 5). The second involves the special case when a hydrogen atom binds

986

de Koker &

O: Large spheres H: Small spheres Si-O coordination: Polyhedra Ms; Points

Stixrude

ZAB

—

•

D

1 1

H

Figure 5. Snapshots o f F P M D simulations of hydrous MgSiO, melts, illustrating the transfer of a hydrogen atom between two silica polyhedra, as indicated by the circles. From Karki et al. (2009). [Used by permission of Springer, from Karki et al. (2009), Physics and Chemistry of Minerals, Vol. 37, p. 103-117.]

to a free oxygen (Z0_Si = 0). to form hydroxyl species that are unusually long lived for similar reasons to those described for free oxygen. The high self-diffusivity of H suggests that hydrated melts could have high conductivity, which would make them geophysically detectable in the upper mantle by magnetotelluric sounding. Using the Nemst-Einstein relation, Mookherjee et al. (2008) computed that hydrated melt should have a conductivity of 59 c/10 S/m, so that a 20 km thick layer of 5 vol% partial mantle melt containing 3 wt% water will have a conductivity of 18 S/m, and would be geophysically observable. Hydrothermal fluids and aqueous solutions. Hydrothermal fluids are responsible for many rare and unusual mineralogical deposits. They accumulate, transport, fractionate and concentrate incompatible trace elements from the crust and upper mantle, and are the source for many ore deposits of noble metals and pegmatitic minerals. Ascending hydrothermal fluids associated with subduction processes are very reducing, commonly at temperatures of around 400-1100 K (e.g.. Manning 2004), and dissolve large concentrations of silica while also scavenging trace incompatible elements from the country rock. This dissolved load is deposited at shallow depths as the fluid cools, decompresses, and becomes oxidized. Almost all theoretical studies of water have focused on ambient conditions. This is in part because water at these conditions is of central importance to many fields of chemistry, but also because water is in fact a rather unusual substance, with bonding characteristics not easily captured with existing first-principles techniques (Silvestrelli and Parrinello 1999; Grossman et al. 2004). It is well known that the structure of pure water is disrupted by a solute; dissolved ions are surrounded by water molecules arranged to form multiple solvation shells. The structure of these solvation shells has been studied in detail both experimentally as well theoretically (e.g., Ohtaki and Radnai 1993; Bakker 2008). The characteristic residence time of a water molecule in the first solvation shell is a key parameter in models describing the structural dynamics by which solute diffusion in aqueous solutions takes place. However, this parameter is difficult to constrain uniquely from experiment, and molecular dynamics simulations have been indispensable in elucidating general trends between diffusivity and parameters describing the first solvation shell. Although a number of theoretical studies have considered aqueous silica solutions (Cheng

Theoretical Computation of Diffusion in Minerals & Melts

987

et al. 2002; Tossel 2005), the diffusive properties of dissolved silica have only been partly addressed. Doltsinis et al. (2007) considered the diffusivities of various candidate molecular species that silica may form in aqueous solution at ambient pressure using Car-Parrinello molecular dynamics (CPMD). Their results indicate that the diffusivity of silica complexes (e.g., Si0 4 H 4 ), which diffuse as single structural units at these temperatures, decreases if the size of the silica complex is increased (e.g., to Si 2 0 7 H 6 or Si 3 O ]0 H 8 ). This result can be understood in terms of the Stokes-Einstein relation: increasing the size of the silica complex results in a large effective diameter of the solute, and therefore a decrease in diffusivity. Although our understanding of chemical diffusion has progressed greatly since the experiments of Graham (1833) and Fick (1855), saline aqueous solutions continue to form a cornerstone in investigating the mechanisms and interesting effects associated with the phenomenon. We will highlight two examples, the first relating to mass-dependent diffusion, and the second to the characterization of cross-diffusivity and the full diffusion matrix. While differences in nuclear mass among isotopes have a negligible effect on bonding, differences in inertia can be sufficient to bring about isotopic fractionation by diffusion. Examples include the correlated enrichment in heavy isotopes of Mg and Si within Ca-Al rich inclusions (CAI's) found in meteorites (Richter et al. 2007), major element isotope fractionation in melts (Richter et al. 2008, 2009), and fractionation of ionic species in water (Richter et al. 2006). For a dilute gas, kinetic theory of diffusion (e.g., McQuarrie 1984) predicts that the ratio of diffusion coefficients DA and DB of two species of respective molar mass values mA and mB is given by (44) with (3 = 0.5. From experimental measurements for aqueous solutions (e.g., Richter et al. 2006), the value of (3 for diffusion in liquid water is known to be notably less than 0.5, yet still large enough for mass dependent isotope fractionation to occur. Molecular dynamics simulations of solute diffusion in liquid water have helped elucidate the mechanisms by which such fractionation takes place (Nuevo et al. 1995; Willeke 2003; Bourg and Sposito 2007, 2008). Indeed, numerical simulation is especially powerful for investigating mass dependent diffusion because one can access a large range of hypothetical nuclear mass values that do not occur in nature, greatly improving the accuracy with which mass dependent fractionation is constrained. Using a standard description of the water molecule (Berendsen et al. 1987), together with very simple Lennard-Jones type potential representations of the interactions among molecules and solute ions, Bourg and Sposito (2007, 2008) considered the mass dependent fractionation in water of Li + , Mg +2 , Cl~, He, Ne, Ar and also Xe. The simplicity of the potential allows for simulation over very long runtimes (8 ns) with relatively large system sizes (550-1000 atoms). The results of Bourg and Sposito (2007, 2008) show excellent agreement with the available experimental data for Li, Mg, and CI (Fig. 6) and further predict strong mass dependent fractionation of noble gasses in water, for which no experimental data is available yet. Cross-diffusivities are rather challenging to determine from experiment, so that various approximate treatments are used to describe the effect (e.g., Zhang 2008, 2010). A statistical mechanical description of multicomponent diffusion useful for computational studies was only recently derived (Zhou and Miller 1996; Wheeler and Newton 2004). To implement and test this set of equations, Wheeler and Newton (2004) used molecular dynamics simulations very similar to those of Bourg and Sposito (2007) to compute the full diffusion matrix for aqueous solutions of NaCl and KC1. Results show very good agreement with experimental measurements at room temperature (Fig. 7), especially in the light of the relative simplicity of the potentials used.

988

de Koker &

Stixrude

0.60 r

u "Ho o

Figure 6. Determination of the mass dependence of diffusivities using M D can be done readily by accessing very large and very small hypothetical molar masses not available to experimental measurement. The solid line represents the expected trend from experimental data of Richter et al. (2006). Diffusivities are in 10~9 m 2 /s. Adapted from Bourg and Sposito (2007).

0.55 -

0.50

0.45

log(mci)

Figure 7. Diagonal and off-diagonal components for the diffusion matrix of KC1 aqueous solutions computed using classical molecular dynamics simulations, compared to a compilation of experimental data. Adapted from Wheeler and Newton (2004).

Concentration (molai)

Solids Deep mantle phases. An issue of great importance to deep Earth science concerns the rheology of the mantle. Modeling of geoid, post-glacial rebound, and tectonic plate velocity data, suggests the viscosity of the lower mantle to vary between 1021 -1023 P a s (Richards and Hager 1984; Hager et al. 1985: Mitrovica and Forte 2004). and also reveal the presence of lateral variations in viscosity at the core-mantle boundary (Cadek and Fleitout 2006: Tosi et al. 2009). However, the mineralogical underpinnings of these observations are not well understood. Based on a variety of geodynamic and mineralogical arguments, the dominant mechanism of solid-state flow in the lower mantle is believed to be diffusion creep (e.g., Karato et al. 1995). A thorough understanding of solid-state diffusion in the lower mantle is therefore the key to understanding the rheology of the lower mantle in an atomistic context. As is often the case in theoretical work, the early studies have focussed mostly on MgO periclase because of its simple structure and computational economy. Initial studies applied relatively simple parameterized potentials at ambient pressures (Sangster and Stoneham 1984; Vocadlo et al. 1995), with more recent work using ab initio and first-principles methods to consider diffusion at lower mantle pressures (Ita and Cohen 1998; Karki and Kanduja

Theoretical Computation of Diffusion in Minerals & Melts

989

2006; Ammann et al. 2009b, 2010). These studies found good agreement with the available experimental data, and highlighted the fact that in mantle minerals the role of minor and trace elements in vacancy formation (extrinsic) dominates over formation of simple Schottky defects in a pure system (intrinsic). Therefore vacancy concentrations can be obtained from measured impurity concentrations, rather than by calculations of the free energy of vacancy formation. Early studies of perovskite notably overestimated the experimental Si activation energy as a result of incorrect identification of saddle point locations (Wright and Price 1993; Karki and Kanduja 2007). Ammann et al. (2009b) identified the possible diffusion paths for selfdiffusion of Mg, Si and O, and located the relevant saddle points by applying the climbingimage nudged elastic band method. After characterizing the migration enthalpy, entropy and vibrational frequencies, they obtained vacancy self-diffusivities for each of these species in the various crystallographic directions. They found that Mg and O self-diffusion occur respectively as simple jumps into vacancies on Mg and O lattice sites, while Si diffuses most readily via a six jump cycle (Fig. 1). After further analysis they found these cycles to break 1-20% of the time due to electrostatic repulsion, and speculate that the effect can be reduced by the presence of proton defects (i.e., H + ions). Building on their earlier results, Ammann et al. (2010) characterized diffusivity in MgSi0 3 post-perovskite. Diffusivities for Si and Mg were found to be extremely anisotropic. Interestingly, this anisotropy is not directly related to the layered structure but rather to the fact that, because Si-0 octahedra share edges in the [100] direction and corners in the [001] direction, a series of channels exist in the [100] direction along which migration enthalpies are low. Diffusion of both Si and Mg in the [010] direction (across layers) was found to occur by six jump cycles. Self-diffusivities of Mg, Si and O in periclase, perovskite and post-perovskite are shown in Figure 8. When combined with experimental estimates of extrinsic vacancy concentration values, the low pressure periclase and perovskite diffusivities of Ammann et al. (2009a,b) show excellent agreement with measurements. Mantle viscosity values computed from their results along the mantle geotherm further show close agreement with the profile derived from post-glacial rebound and the geoid. He in zircon: thermochronometry. The ability to model the denudation history of tectonically active regions using thermochronometry has revolutionized the fields of geomorphology and tectonics, providing a deeper understanding of the the coupling between the evolution of the Earth's surface, interior and atmosphere (e.g., Parrish 1985; Farley et al. 2001; Hodges 2003; Ehlers 2005; Ehlers and Poulsen 2009). The premise for thermochronometry is the insight that at temperatures below some characteristic value (closure temperature) diffusion in minerals is so slow that that noble gas daughter isotopes of radioactive decay processes are retained in crystal grains and fission tracks do not anneal (Dodson 1973). Measurements of daughter isotope- or fission track retention in a mineral grain can therefore provide a thermal age for its host rock, essentially dating the time that has passed since the rock was last at the closure temperature. Experimental measurements of He diffusivity in zircon (ZrSi0 4 ) indicate a closure temperature of 450 K for the zircon (U-Th)/He thermochronometer (Reiners et al. 2004). However, these measurements also suggest that He diffusion in zircon is not strictly Arrhenian below about 650 K, and assume diffusivity in zircon to be isotropic. Zircon is tetragonal with an open structure forming a series of channels in the [100], [101], and [001] directions, along which He could possibly migrate with relative ease. To better characterize He diffusion in zircon, Reich et al. (2007) determined the migration enthalpy of He diffusion in zircon as a function of crystallographic direction, and performed molecular dynamics simulations to characterize the temperature dependent behavior of

990

de Koker & Stixrude

Figure 8. a) Vacancy diffusion coefficients in MgO periclase, M g S i 0 3 perovskite and MgSiO, postperovskite computed along a geotherm. Upper bounds - LDA; Lower bounds - GGA. b) Associated viscosity profile for the mantle, computed assuming diffusion creep, compared to estimates of Mitrovica and Forte (2004) from inverse modeling of geophysical data. Post-perovskite values are end-member viscosities along crystallographic axes. [Reprinted from Macmillan Publishers Ltd: Nature, Ammann et al. (2010). doi: 10.1038/nature09052]

anisotropy. Their calculations employed a standard combination of the Born-Mayer and Coulomb potential, for which they constrained the parameters using experimental data for zircon, and quantum mechanical calculations of the energies of He-Zr, He-Si and He-O interaction pairs. They found the migration enthalpy for He diffusion to be smallest in the [001] direction (13.4 kJ/mol), followed by the [100] and [101] directions (44.8 kJ/mol and 101.7 kJ/ mol, respectively). Their molecular dynamics simulations revealed that at room temperature He migrates only along the channels in the [001] direction, while hops in the [100] direction to neighboring [001] channels become increasingly common at temperatures of around 600

Theoretical Computation of Diffusion in Minerals & Melts

991

K; at temperatures above 850 K He moved freely in both the [100] and [001] directions. This change in the extent of anisotropic diffusion with temperature may explain the apparent nonArrhenian behavior seen in the experimental measurements, and highlights the need to also consider anisotropy when interpreting zircon (U-Th)/He thermochronometry data.

A VIEW TO THE FUTURE Looking back 50 years to the first atomistic simulation studies, the degree to which the state of the art has advanced is staggering, and one is left somewhat daunted by the prospect of what will be possible 50 years into the future. More perspective can perhaps be gained by looking back only 15 years to when Born-Oppenheimer FPMD started to be applied to a broad variety of problems. These early calculations generally involved system sizes of around 60 atoms and run times of about 2000 time steps, yet required very large computational resources to perform. Since then, accessible system sizes and run times have increased by almost a factor of 20, but more importantly, the computational resources required to perform FPMD for a modest system of around 100 atoms and run times on the order of 5000 time steps have become small enough for simulations to be readily performed in parallel on 16 processors in 24 hours. For system sizes of around 100 atoms, the uncertainty in diffusivities determined with FPMD is on the order of 5-10%. Theoretically, the uncertainty should decrease with simulation run time as tm„ 1/2. Run durations of around 4 times longer are therefore required to reduce uncertainties by a factor of two. In the context of system size, a factor two reduction in the magnitude due to the finite size of the simulation system would require system sizes 8 times larger (Eqn. 35). Density functional theory based codes generally scale as nc3 of the number of electrons treated in the system. Projections of system sizes accessible to these methods place the largest systems in the order of 10,000 atoms by the year 2020, while the "average" FPMD run projects to around 1000 atoms (Head-Gordon and Artacho 2008). However, over the last 10 years codes that scale more efficiently with nc have been under development, and are already reaching sizes on the order of 10,000 atoms. As these methods improve, accurate and robust treatment of very large systems with complex chemistry at colder temperatures (close to or below melting) would become viable. An exciting prospect of such developments for geological systems would be the ability to directly model processes in natural melts. New developments in the theory of electronic structure calculations can also be expected, with the treatment of highly localized states and strongly correlated systems coming to mind as two promising improvements. This would make the treatment of transition metals more reliable, and open numerous new avenues to explore the mineral physics and geochemistry of systems where these elements play a key role.

ACKNOWLEDGMENTS This work was made possible by support from the European Commission under contract MRTN-CT-2006-035957 to NDK. We are grateful to Michael Ammann for sharing his draft manuscripts, and to Youxue Zang, James van Orman and an anonymous reviewer for insightful comments.

REFERENCES Adjaoud O, Steinle-Neumann G, Jahn S (2008) Mg 2 Si0 4 liquid under high pressure from molecular dynamics. Chem Geof 2 5 6 : f 8 5 - f 9 2

992

de Koker &

Stixrude

Ai Y, Lange RA (2008) The Compressibility of C a 0 - M g 0 - A l 2 0 3 - S i 0 2 liquids from new acoustic velocity measurements: réévaluation of the equation of state of CaMgSi 2 0 6 -CaAl 2 Si0 8 liquids to 25 GPa. J Geoophys Res 113:B04203 Alder BJ, Wainwright TE (1957) Phase transition for a hard sphere system. J Chem Phys 27:1208-1209 Alfè D, Kresse G, Gillan (2000) Structure and dynamics of liquid iron under Earth's core conditions. Phys Rev B 61:132-142 Allen MP, Tildesley DJ (1987) Computer Simulation of Liquids. 1st Edition. Oxford University Press, Oxford, Ammann MW, Brodholt JP, Dobson DP (2009a) DFT study of migration enthalpies in M g S i 0 3 perovskite. Phys Chem Miner 36:151 -158 Ammann MW, Brodholt JP, Dobson DP (2009b) Simulating diffusion. Rev Mineral Geochem 71:201-224 Ammann MW, Brodholt JP, Wookey J, Dobson DP (2010) First-principles constraints on diffusion in lowermantle minerals and a weak D" layer. Nature 465:462-465, doi: 10.1038/nature09052 Angell CA, Cheeseman PA, Kadiyala RR (1987) Diffusivity and thermodynamic properties of diopside and jadeite melts by computer simulations. Chem Geol 62:83-92 Bakker HJ (2008) Structural dynamics of aqueous salt solutions. Chem Rev 108:1456-1473 Berendsen HJC, Grigera JR, Straatsma TP (1987) The missing term in effective pair potentials. J Phys Chem 91:6269-6271 Birch F (1952) Elasticity and constitution of the Earth's interior. J Geoophys Res 57:227-286 Borg RJ, Dienes GJ (1988) Solid State Diffusion. Academic Press, Boston Bourg IC, Sposito G (2007) Molecular dynamics simulations of kinetic isotope fractionation during the diffusion of ionic species in liquid water. Geochim Cosmochim Acta 71:5583-5589 Bourg IC, Sposito G (2008) Isotopic fractionation of noble gases by diffusion in liquid water: Molecular dynamics simulations and hydrologie applications. Geochim Cosmochim Acta 72:2237-2247 Brodholt J (1997) Ab initio calculations on point defects in forsterite (Mg 2 Si0 4 ) and implications for diffusion and creep. Am Mineral 82:1049-1053 Burton JJ (1972) Vacancy-formation entropy in crystals. Phys Rev B 5:2948-2957 Cadek O, Fleitout L (2006) Effect of lateral viscosity variations in the core-mantle boundary region on predictions of the long-wavelength geoid. Studia Geophys Geod 50:217-232 Callen HB (1985) Thermodynamics and an Introduction to Thermostatistics. 2nd Edition. John Wiley & Sons, New York Callen HB, Welton TA (1951) Irreversibility and generalized noise. Phys Rev 83:34-40 Car R, Parrinello M (1985) Unified approach to molecular dynamics and density-functional theory. Phys Rev Lett 55:2471-2474 Ceperley DM, Alder BJ (1980) Ground-state of the electron-gas by a stochastic method. Phys Rev Lett 45:566569 Chandrasekhar S (1943) Stochastic problems in physics and astronomy. Rev Mod Phys 15:1-89 Cheng HP, Barnett RN, Landman U (2002) Structure, collective hydrogen transfer, and formation of Si(OH) 4 in Si0 2 -(H 2 0)„ clusters. J Chem Phys 116:9300-9304 Cohen RE, Gong Z (1994) Melting and melt structure of MgO at high pressures. Phys Rev B 50:12301-12311 Crank J (1975) The Mathematics of Diffusion. 2nd Edition. Oxford University Press, Oxford de Groot SR, Mazur P (1969) Non-Equilibrium Thermodynamics. North-Holland, Amsterdam de Koker N, Karki BB, Stixrude L (2010) Mixing, structure and freezing of M g 0 - S i 0 2 liquids in Earth's lower mantle, (in prep) de Koker N, Stixrude L (2009) Self-consistent thermodynamic description of silicate liquids, with application to shock melting of MgO periclase and M g S i 0 3 perovskite. Geophys J Int 178:162-179 de Koker N, Stixrude L, Karki BB (2008) Thermodynamics, structure, dynamics, and freezing of Mg 2 Si0 4 liquid at high pressure. Geochim Cosmochim Acta 72:1427-1441, doi : 10.1016/j .gca.2007.12.019 de Wijs GN, Kresse G, Vocadlo L, Dobson D, Alfè D, Gillan MJ, Price GD (1998) The viscosity of liquid iron at the physical conditions of the Earth's core. Nature 392:805-807 Dobson DP, Crichton WA, Vocadlo L, Jones AP, Wang Y, Uchida T, Rivers M, Sutton S, Brodholt J (2000) In situ measurement of viscosity of liquids in the Fe-FeS system at high pressures and temperatures. Am Mineral 85:1838-1842 Dodson MH (1973) Closure temperature in cooling geochronological and petrological systems. Contrib Mineral Petrol 40:259-274 Doltsinis NL, Burchard M, Maresch WV, Boese AD, Fockenberg T (2007) Ab initio molecular dynamics study of dissolved S i 0 2 in supercritical water. J Theor Comput Chem 6:49-62 Ehlers TA (2005) Crustal thermal processes and the interpretation of thermochronometer data. Rev Mineral Geochem 58:315-350 Ehlers TA, Poulsen CJ (2009) Influence of Andean uplift on climate and paleoaltimetry estimates. Earth Planet Sei Lett 281:238-248 Einstein A (1905) Über die von der Molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annal Phys 322:549-560

Theoretical Computation of Diffusion in Minerals & Melts

993

Elcock E W (1959) Vacancy diffusion in ordered alloys. Proc Physical Soc 7 3 : 2 5 0 - 2 6 4 Eyring H (1936) Viscosity, plasticity and diffusion as examples of absolute reaction rates. J Chem Phys 4:283291 Farley KA, Rusmore M E , Bogue S W (2001 ) Post-10 M a uplift and exhumation of the northern coast mountains, British Columbia. Geology 2 9 : 9 9 - 1 0 2 Fick A (1855) Ûber Diffusion. Poggendorff's Annalen der Physik und Chemie 94:59-86 Fourier J (1822) Théorie Analytique de la Chaleur. Firmin Didot, Paris Frenkel D, Smit B (1996) Understanding Molecular Simulation: From Algorithms to Applications. I s1 Edition. Academic Press, San Diego Gale J D (1998) G U L P - a computer program for the symmetry adapted simulation of solids. J Chem Soc Faraday Trans 93:629-637 Gale J D (2001) Simulating the Crystal Structures and Properties of Ionic Materials from Interatomic Potentials. Rev Mineral Geochem 4 2 : 3 7 - 6 2 Gans R F (1972) Viscosity of the Earth's core. J Geoophys Res 7 7 : 3 6 0 - 3 6 6 Gillan M J , Jacobs P W M (1983) Entropy of a point defect in an ionic crystal. Phys Rev B 28:759-777 Glicksman M E (2000) Diffusion in Solids: Field Theory, Solid-State Principles, and Applications. l s l Edition. John Wiley and Sons, New York Graham T (1833) Researches on the arseniates, phosphates, and modifications of phosphoric acid. Philos Trans Royal Soc 123:253-284 Green M S (1952) Markoff random processes and the statistical mechanics of time-dependent phenomena. J Chem Phys 2 0 : 1 2 8 1 - 1 2 9 5 Grossman JC, Schwelger E, Draeger EW, Gygi F, Galli G (2004) Towards an assessment of the accuracy of density functional theory for the first principles simulations of water. J Chem Phys 120:300-311 Hager B , Clayton R, Richards M, Comer R, Dziewonski A (1985) Lower mantle heterogeneity, dynamic topography and the geoid. Nature 3 1 3 : 5 4 1 - 5 4 5 Hanks TC, Anderson D L (1969) The early thermal history of the Earth. Phys Earth Planet Inter 2:19-29 Hansen JP, McDonald IR (2006) Theory of Simple Liquids. 3 rd Edition. Academic Press, London Head-Gordon M, Artacho E (2008) Chemistry on the computer. Physics Today 61:58-63 Henkelman G, Uberuaga BP, Jônsson H (2000) A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J Chem Phys 113:9901-9904 Hodges K V (2003) Geochronology and thermochronology in orogenic systems. In: Treatise of Geochemistry, Volume 3: The Crust. Rudnick R L (ed) Elsevier, p 2 6 3 - 2 9 2 Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136:864-871 Hoover W G (1985) Canonical dynamics: equilibrium phase-space distributions. Phys Rev A 3 1 : 1 6 9 5 - 1 6 9 7 Ita J, Cohen R E (1998) Diffusion in MgO at high pressure: implications for lower mantle rheology. Geophys Res Lett 2 5 : 1 0 9 5 - 1 0 9 8 Jônsson H, Mills G, Jacobsen K W (1998) Nudged elastic band method for finding minimum energy paths of transitions. In: Classical and Quantum Dynamics in Condensed Phase Simulations. B J Berne, G Ciccotti, D F Coker (eds), World Scientific, Singapore, p 385-404 Karato SI, Zhang S, Wenk H-R (1995) Superplasticity in Earth's lower mantle: evidence from seismic anisotropy and rock physics. Science 270:458-461 Karki B B , Bhattarai D, Mookherjee M, Stixrude L (2009) Visualization-based analysis of structural and dynamical properties of simulated hydrous silicate melt. Phys Chem Miner 37:103-117 Karki B B , Bhattarai D, Stixrude L (2006) First principles calculations of the structural, dynamical and electronic properties of liquid MgO. Phys Rev B 7 3 : 1 7 4 2 0 8 Karki B B , Bhattarai D, Stixrude L (2007) First-principles simulations of liquid silica: Structural and dynamical behavior at high pressure. Phys Rev B 7 6 : 1 0 4 2 0 5 Karki B B , Kanduja G (2006) Vacancy defects in MgO at high pressure. Am Mineral 9 1 : 5 1 1 - 5 1 6 Karki B B , Kanduja G (2007) A computational study of ionic vacancies and diffusion in M g S i 0 3 perovskite and post-perovskite. Earth Planet Sci Lett 2 6 0 : 2 0 1 - 2 1 1 Kieffer J, Angell CA (1989) Structural incompatibilities and liquid-liquid phase separation in molten binary silicates: A computer simulation. J Chem Phys 9 0 : 4 9 8 2 - 4 9 9 1 Knopoff L, Uffen R J (1954) The densities of compounds at high pressures and the state of the Earth's interior. J Geoophys Res 5 9 : 4 7 1 - 4 8 4 Kohn W (1999) Nobel lecture: electronic structure of matter - wave functions and density functionals. Rev Mod Phys 7 1 : 1 2 5 3 - 1 2 6 6 Kohn W, Sham L J (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140:1133 Kubicki JD, Lasaga AC (1991) Molecular-dynamics simulations of pressure and temperature effects on M g S i 0 3 and M g 2 S i 0 4 melts and glasses. Phys Chem Miner 17:661-673 Kubo R (1957) Statistical-mechanical theory of irreversible processes, i. general theory and simple applications to magnetic and conduction problems. J Phys Soc Japan 12:570-586

994

de Koker &

Stixrude

Lacks DJ, Rear DB, Van Orman JA (2007) Molecular dynamics investigation of viscosity, chemical diffusivities and partial molar volumes of liquids along the M g 0 - S i 0 2 join as functions of pressure. Geochim Cosmochim Acta 71:1312-1323 Lange RA, Carmichael ISE (1987) Densities of N a 2 0 - K 2 0 - C a 0 - M g 0 - F e 0 - F e 2 0 3 - A l 2 0 3 - T i 0 2 - S i 0 2 liquids new measurements and derived partial molar properties. Geochim Cosmochim Acta 51:2931 -2946 Langevin P (1908) Sur la théorie du mouvement brownien. CR Acad Sci 146:530-533 LasagaAC (1979) Multicomponent exchange and diffusion in silicates. Geochim Cosmochim Acta 43:455-469 Lasaga AC (1998) Kinetic Theory in the Earth Sciences, l s l Edition. Princeton University Press, Princeton, NJ Lennard-Jones JE ( 1924) On the determination of molecular fields. II. From the equation of state of a gas. Proc R S o c A 106:463-477 Leslie M, Gillan MJ (1985) The energy and elastic dipole tensor of defects in ionic crystals calculated by the supercell method. J Phys C Solid State Phy 18:973-982 Liebske C, Schmickler B, Terasaki H, Poe P, Suzuki A, Funakoshi Kl, Ando R, Rubie DC (2005) Viscosity of peridotite liquid up to 13 GPa: Implications for magma ocean viscosities. Earth Planet Sci Lett 240:589604 Manning CE (2004) The chemistry of subduction-zone fluids. Earth Planet Sci Lett 223:1-16 Martin B, Spera FJ, Ghiorso MS, Nevins D (2009) Structure, thermodynamic and transport properties of liquid Mg 2 Si0 4 : Molecular dynamics simulations and model EOS. A m Mineral 94:693-703 Matsui M (1994) A transferable interatomic potential model for crystals and melts in the system CaO-MgOA l 2 0 r S i 0 2 . Mineral Mag 58A:571-572 Matsui M, Anderson OL (1996) Molecular dynamics simulation of structures, bulk moduli, and volume thermal expansivities of silicate liquids in the system C a 0 - M g 0 - A l 2 0 3 - S i 0 2 . G e 0 p h y s Res Lett 23:395-398 McQuarrie DA (1984) Statistical Mechanics. University Science Books, Sausalito, CA Mermin N D (1965) Thermal properties of inhomogeneous electron gas. Phys Rev 137:1441 Merrill RT, McElhinny MW, McFadden PL (1998) The Magnetic Field of the Earth, l s l edition. Academic Press, London Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21:1087-1092 Mitrovica JX, Forte AM (2004) A new inference of mantle viscosity based upon joint inversion of convection and glacial isostatic adjustment data. Earth Planet Sci Lett 225:177-189 Mookherjee M, Stixrude L, Karki BB (2008) Hydrous silicate melt at high pressure. Nature 452:983-986 Nevins D, Spera FJ (1998) Molecular dynamics simulations of molten CaAl 2 Si 2 0 8 : dependence of structure and properties on pressure. Am Mineral 83:1220-1230 Nevins D, Spera FJ, Ghiorso MS (2009) Shear viscosity and diffusion in liquid MgSiOj: transport properties and implications for terrestrial planet magma oceans. Am Mineral 94:975-980 Nosé S (1984) A unified formulation of the constant temperature molecular dynamics methods. J Chem Phys 81:511-519 Nuevo MJ, Morales JJ, Heyes DM (1995) Mass dependence of isotope self-diffusion by molecular dynamics. Phys Rev E 51:2026-2032 Ohtaki H, Radnai T (1993) Structure and dynamics of hydrated ions. Chem Rev 93:1157-1204 O n s a g e r L (1931a) Reciprocal relations in irreversible processes. I. Phys Rev 37:405-426 Onsager L (1931b) Reciprocal relations in irreversible processes. II. Phys Rev 38:2265-2279 Parrish RR (1985) Some cautions which should be exercised when interpreting fission-track and other dates with regard to uplift rate calculations. Nucl Tracks Radiation Measure 10:425 Payne MC, Teter MP, Allan DC, Arias TA, Joannopoulos JD (1992) Iterative minimization techniques for ab initio total-energy calculations - molecular-dynamics and conjugate gradients. Rev Mod Phys 64:10451097 Philibert J (1991) Atom Movements, Diffusion and Mass Transport in Solids, 1st Edition. Les Editions de Physique, Paris Poirier JP (1988) Transport properties of liquid metals and viscosity of the Earth's core. Geophys J 92:99-105 Rahman A (1964) Correlation in the motion of atoms in liquid argon. Phys Rev 136:60-66 Reich M, Ewing RC, Ehlers TA, Becker U (2007) Low-temperature anisotropic diffusion of helium in zircon: Implications for zircon (U-Th)/He thermochronometry. Geochim Cosmochim Acta 71:3119-3130 Reid JE, Poe BT, Rubie DC, Zotov N, Wiedenbeck M (2001) The self-diffusion of silicon and oxygen in diopside (CaMgSi 2 0 6 ) liquid up to 15 GPa. Chem Geol 174:77-86 Reiners PW, Spell TL, Nicolescu S, Zanetti KA (2004) Zircon (U-Th)/He thermochronometry: He diffusion and comparisons with 40 Ar/ 39 Ar dating. Geochim Cosmochim Acta 68:1857-1887 Richards M, Hager B (1984) Geoid anomalies in a dynamic earth. J Geoophys Res 89:5987-6002 Richter FM, Janney PE, Mendybaev RA, Davis AM, Wadhwa M (2007) Elemental and isotopic fractionation of type B CAI-like liquids by evaporation. Geochim Cosmochim Acta 71:5544-5564 Richter FM, Watson EB, Christensen JN, Hutcheon ID, Williams RW, Sturchio NC, Beloso AD Jr (2006) Kinetic isotopic fractionation during diffusion of ionic species in water. Geochim Cosmochim Acta 70:277-289

Theoretical Computation of Diffusion in Minerals & Melts

995

Richter FM, Watson E B , Mendybaev R, Dauphas N, Georg B , Watkins J, Valley J (2009) Isotopic fractionation of the major elements of molten basalt by chemical and thermal diffusion. Geochim Cosmochim Acta 73:4250-4263 Richter F M , Watson E B , Mendybaev R, Teng FZ, Janney P E (2008) Magnesium isotope fractionation in silicate melts by chemical and thermal diffusion. Geochim Cosmochim Acta 7 2 : 2 0 6 - 2 2 0 Rigden SM, Ahrens TJ, Stolper E M (1989) High-pressure equation of state of molten anorthite and diopside. J Geoophys Res 9 4 : 9 5 0 8 - 9 5 2 2 Rivers M L , Carmichael I S E (1987) Ultrasonic studies of silicate melts. J Geoophys Res 9 2 : 9 2 4 7 - 9 2 7 0 Ruff L, Anderson D L (1980) Core formation, evolution and convection: A geophysical model. Phys Earth Planet Inter 21:181-201 Rutter MD, Secco RA, Liu H, Uchida T, Rivers M, Sutton S, Wang Y (2002) Viscosity of liquid F e at high pressure. Phys Rev B 6 6 : 0 6 0 1 0 2 Sangster M J L , Stoneham A M (1984) calculation of absolute diffusion rates in oxides. J Phys C 17:6093-6104 Shewmon P (1989) Diffusion in Solids. Minerals, Metals & Materials Society, Warrendale, PA Silvestrelli PL, Parrinello M (1999) Structural, electronic, and bonding properties of liquid water from first principles. J C h e m P h y s 111:3572-3580 Smigelskas AD, Kirkendall E D (1947) Zinc diffusion in alpha brass. Trans Am Inst Min Metall Eng 171:130142 Smoluchowski M (1906) Essai d'une théorie cinétique du mouvement Brownien et des milieux troubles. Bull Int Acad Sci Cracovie p 577-602 Stebbins IF, Carmichael ISE, Moret L K (1984) Heat-capacities and entropies of silicate liquids and glasses. Contrib Mineral Petrol 86:131-148 Stixrude L, Bukowinski M S T (1989) Compression of tetrahedrally bonded S i 0 2 liquid and silicate liquidcrystal density inversion. Geophys Res Lett 16:1403-1406 Stixrude L, de Koker N, Sun N, Mookherjee M, Karki B (2009) Thermodynamics of silicate liquids in the deep Earth. Earth Planet Sci Lett 2 7 8 : 2 2 6 - 2 3 2 Stixrude L, Karki B B (2005) Structure and Freezing of MgSiOn liquid in the Earth's lower mantle. Science 310:297-299 Terasaki H, Kato T, Urakawa S, Funakoshi K, Sato K, Suzuki A, Okada T (2002) Viscosity change and structural transition of molten Fe at 5 GPa. Geophys Res Lett 29:1227 Tonks W B , Melosh H J (1993) Magma ocean formation due to giant impacts. J Geoophys Res 9 8 : 5 3 1 9 - 5 3 3 3 Tosi N, Cadek O, Martinec Z (2009) Subducted slabs and lateral viscosity variations: effects on the longwavelength geoid. Geophys J Int 179:813-826 Tossel IA (2005) Theoretical study on the dimerization of Si(OH) 4 in aqueous solution and its dependence on temperature and dielectric constant. Geochim Cosmochim Acta 69:283-291 Urey HC (1955) The cosmic abundances of potassium, uranium and thorium and the heat balances of the Earth, the Moon, and Mars. Proc Nat Acad Sci U S A 4 1 : 1 2 7 - 1 4 4 van Beest B W H , Kramer GI, van Santen R A (1990) Force fields for silicas and aluminophosphates based on ab initio calculations. Phys Res Lett 6 4 : 1 9 5 5 - 1 9 5 8 Verlet L (1967) Computer "experiments" on classical fluids. I. thermodynamical properties of Lennard-Jones molecules. Phys Rev 159:98-103 Vinyard GH (1957) Frequency factors and isotope effects in solid state rate processes. J Phys Chem Solids 3:121-127 Vocadlo L (2007) Mineralogy of the Earth - The Earth's core: iron and iron alloys. In: Treatise of Geophysics, Volume 2: Mineral Physics. Price G (ed) Elsevier, p 9 1 - 1 2 0 Vocadlo L, de Wijs GA, Kresse G, Gillan M, Price G D (1997) First principles calculations on crystalline and liquid iron at Earth's core conditions. Faraday Discuss 106:205-217 Vocadlo L, Wall A, Parker SC, Price G D (1995) Absolute ionic diffusion in MgO - computer calculations via lattice dynamics. Phys Earth Planet Inter 8 8 : 1 9 3 - 2 1 0 Wheeler D R , Newton I (2004) Molecular dynamics simulations of multicomponent diffusion. 1. equilibrium method. J Phys Chem 108:18353-18361 Willeke M (2003) Limits of the validity of the mass ratio independence of the Stokes-Einstein relation: molecular dynamics calculations and comparison with the Enskog theory. Molec Phys 101:1123-1130 Wood WW, Parker F R (1957) Monte Carlo equation of state of molecules interacting with the Lennard-Iones potential. I. A supercritical isotherm at about twice the critical temperature. J Chem Phys 2 7 : 7 2 0 - 7 3 3 Wright K, Price G (1993) Computer simulations of defects and diffusion in perovskites. J Geoophys Res 98:22245-22253 Yeh IC, Hummer G (2004) System-size dependence of diffusion coefficients and viscosities from molecular dynamics simulations with periodic boundary conditions. I Phys Chem B 1 0 8 : 1 5 8 7 3 - 1 5 8 7 9 Zhang Y (2008) Geochemical Kinetics. Princeton University Press, Princeton, NJ Zhang Y (2010) Diffusion in minerals and melts: theoretical background. Rev Mineral Geochem 7 2 : 5 - 5 9

996

de Koker &

Stixrude

Zhang Y, Guo G, Refson K, Zhao Y (2004) Finite-size effect at both high and low temperatures in molecular dynamics calculations of the self-diffusion coefficient and viscosity of liquid silica. J Phys Condens Matter 16:9127-9135 Zhou Y, Miller GH (1996) Green-Kubo formulas for mutual diffusion coefficients in multicomponent systems. J Phys Chem 100:5516-5524 Zwanzig R (1965) Time-correlation functions and transport coefficients in statistical mechanics. Annu Rev Phys Chem 16:67-102

23

Reviews in Mineralogy & Geochemistry Vol. 72 pp. 997-1038, 2010 Copyright © Mineralogical Society of America

Applications of Diffusion Data to High-Temperature Earth Systems Thomas Mueller New York Center for Astrobiology Rensselaer Polytechnic Institute Troy, New York 12180, U.S.A. Present address: Institute for Geology, Mineralogy & Geophysics Ruhr-University Bochum Universitaetsstr. 150; NA 03/586 D-44780 Bochum, Germany [email protected]

E. Bruce Watson Department of Earth and Environmental Sciences Rensselaer Polytechnic Institute Troy, New York 12180, U.S.A.

T. Mark Harrison Department of Earth and Space Sciences University of California, Los Angeles Los Angeles, California 90095, U.S.A.

INTRODUCTION Diffusion in general is such a widespread phenomenon in nature that a basic knowledge can serve very effectively to address well known applications not only to thermochronology and petrology, but also to environmental geology (e.g., paleoclimate), sedimentology, and applied geosciences (e.g., assessing contamination of gas stations). Indeed, the list of potential applications of diffusion phenomena is so long that an exhaustive treatment would quickly exceed the scope of this review. The reader is therefore referred to reviews covering various aspects of diffusion in the Earth sciences, including diffusion in melts (Watson 1994; Chakraborty 1995), diffusion processes during diagenesis (Berner 1980; Krom and Berner 1980; Boudreau 2000) and diffusion phenomena in general (e.g., Lasaga 1998; Ganguly 2002; Watson and Baxter 2007; Zhang 2008). All of these treatments are based on the same general concepts of diffusion, which do not need restating here in great detail. Applications of diffusion profiles in igneous systems have been thoroughly reviewed a recent Reviews in Mineralogy and Geochemistry volume (RiMG 69 - Putirka and Tepley 2008) and there is also an excellent volume on low temperature thermochronology (RiMG 58 - Reiners and Ehlers 2005) describing the impact and application of diffusion theory for age and thermal history determination. With these precedents in mind, we focus the present review on the application of diffusion in high temperature regimes (i.e., systems at T> 500 °C). Both igneous and metamorphic systems are addressed, but because of the recent precedents noted above, we emphasize recent developments and challenges in metamorphic studies. In a broad sense, diffusion describes the statistical, non-directional movement of molecules (or heat) through a medium of interest. As independent theories of diffusion were developed in 1529-6466/10/0072-0023$05.00

DOI: 10.2138/rmg.2010.72.23

998

Mueller,

Watson,

Harrison

the early 19th century and compared with the principles of statistical variance, it was recognized that the partial differential equations describing the two are identical (Narasimhan 2009). This equivalence is due to the fact that both phenomena are rooted in randomness. In the physical case, the predictability resulting from large numbers of random motions endows the diffusion model with a predictive capability unlike any other transport mechanism. While it is impossible to accurately predict the outcome of a single, random event, the motion of large numbers of particles or phonons results in a functional form that is diagnostic of conductive transport and the integral of that function is unique to every thermal history. Advection—which refers to the motion of the medium itself—and diffusion represent the full description of any mass transport process. A few examples highlight the large variety of length and time scales relevant to mass transport in the Earth: mass transfer in the mantle related to convection, magma mixing in the crust, infiltration of fluids into rocks, and the movement of single atoms within the crystal lattice of a single mineral grain. The role of diffusion in transport processes and the effective length scales at which it operates are thus of widespread interest in the study of natural systems; this is one of the subjects explored in the section "Mass transport in geological systems." A large number of studies—most of them experimental in nature—have determined diffusion coefficients for elements and isotopes and their dependence upon intensive variables such as pressure (P), oxygen fugacity (/b2), and, most notably, temperature (7). Characterization of these dependencies provides the key link from the laboratory to natural systems: in principle, experimentally-derived data can be applied to natural rocks whose /'-'/-lime path is known to evaluate the extent of element or isotope mobility at the scale of interest and vice versa. A parallel example of the value of laboratory data in interpreting natural systems is the use of thermodynamic calculations to determine the equilibrium state of a mineral or rock as a function of pressure, temperature and composition. Most thermodynamic data are based on experiments and, together with the assumption of equilibrium; these data provide the basis for geothermobarometry. Today, there are several public software packages [e.g., Gibbs (Spear and Menard 1989), Melts (Ghiorso and Sack 1995), PERPLEX (Connolly 1990), Theriak-Domino (de Capitani and Brown 1987), or Thermocalc (Powell et al. 1998)] available which makes it easy to calculate phase relations based on those internally consistent data bases. However, the use of thermodynamics to deduce P-T conditions assumes that the system of interest is able to adjust its composition infinitely fast at any given pressure or temperature up to a certain point in time, and remains perfectly closed thereafter (i.e., without any subsequent chemical exchange). In fact, virtually every dynamic system evolves along a (/'-)r-time path and is continuously subject to the tendency for reequilibration by diffusive adjustment. In other words, due to the vagaries of diffusion and atom mobility, a given mineral may be either unable to adjust its composition fast enough, or continue exchange with its surroundings on the retrograde path. Either way, the mineral is unlikely to perfectly preserve the equilibrium composition defined at any single pressure and temperature. Rather, it will exhibit a range of compositions (i.e., characteristic zoning patterns) indicating changes in pressure and temperature during crystallization. The tendency for dynamic but incomplete diffusive adjustment could be regarded purely as a limitation, but at the same time it provides a large number of opportunities. Compositional profiles within crystals contain a record of non-equilibrium states. While it is impossible to decipher equilibrium/'-Tconditions from these profiles, they provide information on mechanisms and rates of processes, such as element or isotope exchange or even mineral reactions. In cases where either the transport of material to the mineral or homogenization within the mineral is limited by diffusion, one can use the resulting compositional profiles to extract information on the crystallization history, as has been successfully done in thermochronology (RiMG 58 - Reiners and Ehlers 2005), geospeedometry (RiMG 69 - Putirka and Tepley 2008) and metamorphic petrology (e.g., Skora et al. 2006; Miiller et al. 2008). Deductions are based on the application of kinetic theory to generate models of preserved compositional heterogeneities

Applications of Diffusion Data to High-T Systems

999

and textures in the rock under study. The petrological community has been aware of the principles and potential of this approach for some time, but it is only recently that practical tools have been developed for a wide range of applications. An increasing number of studies are now showing that knowledge of trace element diffusion data can be used to extract growth histories and/or dissolution rates. This approach will be discussed in more detail in the section " D i f f u s i o n in minerals." We will also discuss the equally promising strategy of exploiting the variation in diffusivities among different elements or isotopes within a single system, such as a magma, a rock matrix or even a single crystal lattice. Differences in diffusivity can produce elemental and/or isotopic fractionations that are telltale indicators of specific geologic processes. All applications that involve diffusion phenomena are inherently linked to temperature and time. Thermochronology and thermobarometry are common approaches for constraining rock histories. Modified mineral compositions that do not represent equilibrium conditions— and thus introduce bias into any age or P-T determination—are generally a consequence of diffusion-controlled processes. To assure a correct interpretation of measured elemental or isotopic data, it is crucial to know the extent of diffusive gain or loss experienced by a mineral or rock since the event of interest. This subject will be covered in detail in the section "Thermochronology." Closely related is the exploitation of the strong temperature dependence of diffusion processes, leading to isotopic closure of a mineral system over a narrow temperature range (Dodson 1973). Whereas a closed system isotopic chronometer can tell us how much time has passed since a given event, the non-equilibrium record preserved in compositional or isotopic diffusion profiles enables us to determine rates and timescales of geologic processes. Because the extent of diffusion is linked to time and temperature, brief or low temperature events may result in very limited diffusion profiles that are difficult to characterize. Geological processes that occur at high temperatures or over large time spans, on the other hand, can leave a diffusive 'imprint' on a scale that is easier to measure. These processes can also lead to complete erasure of the non-equilibrium signal. The extent of diffusion that can and cannot be evaluated depends upon the analytical capabilities available, to which significant improvements have been made in recent decades. Because of past analytical limitations, geospeedometry focused mainly on processes in the deep Earth or those related to environments subject to protracted heating. More recently, sophisticated micro-analytical tools allow detection of compositional variations and textural features on increasingly smaller scales and in three dimensions. The combination of new analytical and theoretical tools, in addition to the growing database of experimentally derived diffusion coefficients, allow quantification of the temporal evolution of igneous and metamorphic systems with unprecedented resolution. For example, compositional diffusion profiles can now be resolved down to the scale of a few nanometers, which makes it possible to resolve residence times of minerals in magma chambers down to periods as short as a few days (Costa et al. 2008). We do not attempt an exhaustive review of diffusion applications in this chapter. Instead, the objective is to address particularly exciting areas that have seen rapid expansion in the past 10-15 years, and that we believe will continue to generate interest in the decade ahead.

DECIPHERING KINETICALLY CONTROLLED PROCESSES USING DIFFUSION Mass transport in geological systems Mass transport due to diffusion affects almost every geological system, including the atmosphere, sediments and lithosphere as well as the Earth's deep interior. In these Earth systems, diffusion contributes to transport on an enormous range of scales. At the lower end,

1000

Mueller,

Watson,

Harrison

for example, diffusion on the scale of nanometers may be important within the lattices of single crystals or along grain boundaries in rocks. At the other extreme, diffusion may play a key role in more large-scale processes such as fluid infiltration on the outcrop scale, or in magma mixing in batholiths on the scale of hundreds of meters or more (when combined with advection). In the deep Earth where temperatures are very high, diffusion may even operate on the scale of many kilometers given time frames of hundreds of millions of years. Hofmann and Hart (1978), for example, first applied diffusion data to the then controversial question of preservation of chemical heterogeneities in the mantle in the face of diffusion and mantle convection. Diffusion is a manifestation of the random movement of atoms or molecules within the reference frame of their host material. It is commonly stated that diffusion is "driven" by a gradient (e.g., in concentration, chemical potential, temperature, or even pressure). The danger in this description is that it can be interpreted to mean that gradients are required for diffusion to occur, which is a conceptual trap (e.g., Carmichael et al. 1977). Unfortunately, many geochemists and petrologists misleadingly think of diffusion as the net transport of elements in the direction of a compositional gradient. A more accurate statement is that gradients are required to produce a net diffusive flux, which follows simply from considering diffusion as a randomwalk phenomenon. Treatment of diffusion as a random-walk process leads to the well-known Gaussian distribution for spreading of an initial, spatially restricted concentration distribution: x2

c(x,f) = A-exp

aD

>

(1)

where D is the diffusion coefficient, x is distance, t is time, A is a constant, and a is a parameter related to the dimensionality of the space in which diffusion occurs. The local concentration of a diffusing species is a function of both position and time, and also depends on D, which is characteristic of a given diffusant, diffusion medium, temperature and pressure. Any concentration profile that develops as a consequence of diffusion can be described by Equation (1), regardless of the initial or boundary conditions. It is worth noting here that Equation (1) can be normalized so that the integral from minus infinity to plus infinity equals one, which provides the basis for using the well-known (Gaussian) error-function to be used for analytical solutions of simple diffusion problems. Eventually the random movements of a diffusant will lead to its homogenization within the system: that is, pre-existing gradients in an initially heterogeneous system will be eliminated. This means that the process just described involves a net flux of diffusant. In 1855, Fick first formulated a description of one-dimensional diffusion in terms of a flux, which is now wellknown as Fick's first law: J

,

ox

=

-

D

t

(

2

)

Here, the flux /, of species i per unit area through its host medium is taken as proportional to the concentration gradient dcjdx, with the constant of proportionality being the diffusion coefficient D, (the units of which are distance2-time_1—usually m2-s~'). This equation is written for the case of a concentration gradient in one dimension only (which corresponds to a = 2 in Eqn. 1), and indicates no explicit dependence on time or temperature. Equation (2) implies that there will be no flux of material if no concentration gradient is present. However, given the random-walk nature of diffusion, it is clear that this actually refers only to the net flux of diffusant in a given direction. On the other hand, Equation (2) does confirm that whenever there is a gradient present in a system, that gradient will create a flux. The latter statement has important consequences for geothermobarometry and geochronology and will be discussed in more detail later on ("Bulk closure" section). Introducing time dependence leads to the nonsteady state diffusion equation (Fick's second law), which, in one dimension, is:

Applications

of Diffusion

dt

Data to High-T

Systems

1001

(3)

' dx'

The local concentration changes with time in proportion to rate of change of the local concentration gradient. A large number of solutions to Equation (3) have been obtained for a variety of geometries (such as for a sphere, cylinder, etc.), as well as a variety of initial- and boundary conditions (see Crank 1975 for more details). Most transport processes in earth sciences involve contributions to dcj/dt from other sources besides diffusion, and the 1-D transport equation (Eqn. 3) can be extended to dt

dx

' dx

dx

(4)

Equation (4) includes an advective contribution, v-(dcjdx), resulting from bulk flow of the medium of interest at a constant velocity v in the x direction. Note that driving potential, the concentration gradient dcjdx, is the same as that causing the diffusive flux. Equation (4) also includes a term, S, referring to the rate of reaction, which can be imagined as a source/sink term. In the analogous case of diffusion of heat, this source/sink term would be latent heat of crystallization, heat produced by radioactive decay, shear heating, etc. In considering mass transport through a polycrystalline medium, S could represent, for example, local dissolution or precipitation affecting c,. Within a single crystal, S might be local in-growth of diffusant due to the decay of a radioactive parent (in this case the advective term would be zero). The length scale of processes to which Equation (4) is applied can vary widely. The same basic equation has been used to describe relatively large-scale phenomena such as magma mixing in igneous systems or fluid infiltration along veins during metamorphism, as well as micro- or even nanoscale processes occurring at the moving interface of a growing crystal. The magnitude of the source/sink term can be small, and neglecting it for the moment emphasizes that mass transport can be characterized essentially by the ratio of advection to diffusion. This ratio is formalized in the dimensionless chemical Peclet number

where I is the length scale of the system or phenomenon under consideration. Like Equation (4), the Peclet number is general in terms of both the length scale at which it can be applied and the range of applications for which it has been used. The introduction of the length scale, which can be either the size of the system or simply the appropriate unit of length, makes the resulting number dimensionless and general when the units of the velocity and diffusivity are chosen accordingly. Systems or phenomena for which Pec > 1 are dominated by advective transport and those in which Pec < 1 are dominated by diffusion. The advantage of the dimensionless character of Pec is that it can be invoked for processes taking place at the scale of several km (as in many hydrogeologic applications) while at the same time it describes adequately the movement of reaction fronts on a few nanometers within a crystal lattice. The effect of the chemical Peclet number on the characteristics of transport processes is well illustrated by the example of fluid infiltration into a rock column causing either an isotope exchange front (e.g., Baumgartner and Rumble 1988; Ferry and Dipple 1992; Baumgartner and Valley 2001) or a reaction front driving a fluid-controlled mineral reaction (Ferry and Dipple 1992; Ferry and Rumble 1997; Ferry and Gerdes 1998; Ferry et al. 2005; Müller et al. 2009). In both cases, the rate of advancement and, more importantly, the shape of the advancing front will largely depend on Pec. Fluid infiltration is dominated by advection (i.e., large Pec), the resulting front will have a sharp, step-like shape. Increasingly significant contributions from diffusion (reflected by small Pec) will cause the initially sharp front to broaden into a profile resembling half of a Gaussian distribution. It is important to bear in mind that, in the one-dimensional form of

1002

Mueller,

Watson,

Harrison

Equation (4), advection is efficient only in the x direction. In practice, this assumption might be a valid approximation for some cases, but diffusion is likely to operate with equal effectiveness both parallel and perpendicular to the migration direction of the infiltration front (Gerdes et al. 1995; Yardley and Lloyd 1995; Ferry and Gerdes 1998). Thus, to adequately address transport processes in nature it is crucial to define the direction of fluid flow. In this connection, systematic isotope data analyses have been useful to identify the nature of fluid flow in contact aureoles, for example (e.g., Ferry and Dipple 1992; Roselle et al. 1999; Cook and Bowman 2000; Ferry et al. 2002). Texture-forming processes involving mineral reactions driven by fluid infiltration can potentially be deciphered by examining characteristics of transport processes in metamorphic rocks, as shown by Miiller et al. (2009). Diffusion in minerals The picture of a fluid infiltrating a rock column in a single front is descriptive and useful, but it is a much more complex process in terms of the possible pathways (intra- and intergrain) and interactions of a fluid 'diffusing' through a rock (Fig. 1). Progress along various pathways occurs at different rates and will be recorded on different scales, which potentially makes the rock a versatile recorder of its own temporal evolution. The diffusivities of elements and isotopes in (mostly silicate) minerals depend not only on atomic (ionic) size, mass and charge but also on the actual diffusion mechanism in the mineral lattice. In other words, the same element or isotope can travel through the crystal lattice at a different rate depending upon the atomistic mechanism or sometimes, also, the crystallographic direction. Three different mechanisms can be distinguished (Fig. 2). Ions can exchange positions, either directly or in coupled (ringlike) configuration, but they can also travel as interstitial atoms or simply jump into adjacent vacant sites in the lattice. The last mechanism is generally the fastest way for an ion to move through a crystal lattice because it requires less energy to break existing bonds. It is important to note that every crystal lattice contains a significant concentration of point defects—resulting, for example, from aliovalent impurities in the crystal lattice or simply as thermal vacancies. According to Boltzmann, in fact, a perfect crystal lattice free of point defects cannot exist at temperatures above absolute zero (see Kittel 1966, chapter 18). The reason for this is the higher entropy of a crystal containing point defects, which lowers the free energy and thus tends to stabilize the defect-bearing crystal relative to the ideal 'perfect' crystal lattice. The point defect chemistry of a crystal is also affected by the oxygen fugacity (fo2). Some elements, such as iron or manganese, will change their oxidation state, depending on whether oxidizing or reducing conditions, i.e., high or low partial oxygen pressures, are present. This in turn can lead to the creation of vacancies to maintain neutral charge balance in the crystal. As a result, diffusivities are highly dependent on the oxygen fugacity in cases where transport through vacancies is the preferred diffusion mechanism. The random-walk concept discussed in the prior section has an important consequence for diffusion: there is always a finite probability that an ion vibrating in a crystal lattice will either jump into a vacant site or exchange positions in the crystal lattice even in the absence of any chemical gradient present in the crystal. This jump constitutes one step in a random walk through a lattice, and the phenomenon is referred to as self-diffusion of an element or isotope. This is contrasted with what is called chemical diffusion, which involves net transport in response to a 'driving force' such as a gradient in concentration or chemical potential. The conceptualization of diffusive transport in crystals accomplished by site-to-site jumping of atoms underscores a characteristic feature of diffusion in solids: it is generally inefficient at large spatial scales. At short length scales, in contrast, site-to-site jumping can be an efficient transport mechanism, as the net motion of an atom is essentially instantaneous. However, longrange progress through a random walk is limited because the eventual net displacement is much less than the sum of all jumps of the atom (Fig. 3). In addition to the situation where diffusion takes place in the absence of a driving

Applications

of Diffusion

Bulk-Rock Diffusion Pathways

Data to High-T Intragrain

Systems

1003

Pathways

lattice Figure 1. Schematic illustration of possible diffusion paths in a rock (modified after Watson and Baxter 2007). Diffusion operates both within single grains and along grain boundaries. Grain boundaries are commonly more efficient pathways for diffusive transport (e.g., Eiler et al. 1994). Note that a mineral grain can be regarded as scaled-down version of a rock with the same pathway properties.

Figure 2. Schematic illustration modified after Watson and Baxter (2007) of three possible types of diffusion mechanism. Diffusion rates will depend on the size, charge and mass of the element/ion moving through the crystal. Diffusion through vacancies or along interstitial sites are commonly faster because less energy is needed for an atom to jump.

force, there also exists the possible scenario in which one or more gradients is present in the system and Fick's first law states that there will be a flux due to a gradient in concentration. In thermodynamic terms, however, gradients are normally represented in terms of chemical potentials or activities rather than simple concentrations. At relatively low concentrations (e.g., trace elements) and within a single phase, this difference is generally unimportant, because the activity of a diffusant is proportional to its concentration. In contrast, as soon as diffusion occurs between crystals—i.e.. across an interphase boundary—or major elements are involved, the difference between concentration and chemical potential often becomes crucial. The difference

1004

Mueller,

Watson,

Harrison

velocity v

Figure 3. Schematic illustration of the efficiency of diffusion modified after Chakraborty (2008). On a short timescale, the jump of an atom from one site to another can be regarded as almost instantaneous. With respect to longer times, the random walk process makes the net displacement of the atom much smaller than the sum of all single jumps.

can even lead to a phenomenon called 'uphill diffusion,' in which diffusion occurs down a gradient in chemical potential, but in opposition to a concentration gradient. Examples for this kind of diffusion are quite common, as indicated by studies of diffusion in melts (Kress and Ghiorso 1993; Chakraborty et al. 1995; Liang et al. 1996) or between phases of different composition (Fisher 1973). Experimental results typically reveal a log-linear dependence of D upon for best-fit solutions. This relationship is commonly expressed in an Arrhenius type equation of the form (6)

E is the activation where DT is the diffusivity at temperature T, D 0 is the diffusivity at T = energy, and R is the universal gas constant. The values of Dn and E depend on the diffusion mechanism relevant to the species of interest in the crystal under consideration, so these vary widely even for a single mineral. The diffusivity governing Mn diffusion into a dolomite crystal, for example, will depend on whether it is exchanging with Mg only or with Mg and Ca. as would be the case for diffusive equilibration between rhodochrosite and dolomite. The diffusivity will be effectively lower for Mn exchanging between dolomite and ankerite. which also has a significant amount of Ca (so Ca in the dolomite need not exchange with Mn as in the previous case). Thus, the diffusivity of Mn in dolomite may depend on the specific mechanism of chemical exchange. Given that the activation energy is related to the amount of energy needed to break bonds and to overcome the energy barrier for the diffusive jump to occur it is clear that the temperature dependence of the diffusivity will also depend on the diffusion mechanism. Typical values for activation energies in melts are around 200 k j moH (Liermann and Ganguly 2002; Dohmen and Chakraborty 2007) and increase significantly to values up to several hundred kJ mol 1 for diffusion of tetravalent ions in silicate minerals (Chemiak et al. 1997). Even though both Du and E are directly related to the diffusion mechanism, it is difficult to invert this observation for structurally and chemically complex crystals like minerals (Costa and Chakraborty 2008).

Applications

of Diffusion

Data to High-T

Systems

1005

From Equation (6) it is clear that the diffusivity of a given species in a given medium increases with increasing temperature. For this reason diffusion is expected to become more efficient with increasing depth in the Earth because of the geothermal gradient. Although generally correct, this generalization neglects the pressure effect on the diffusion coefficient. To account for both variables. Equation (6) is expanded to Dtp

= On

exp

-(E

+

AVJP-IQ-) RT

(7)

where P is pressure (in Pa) and AVa is the activation volume, which, like the activation energy, is related to the diffusion mechanism. In the case of a vacancy diffusion mechanism, A Va is the sum of the partial molar volume of the vacancies and that of the 'activated' diffusant atom—that is, the dilation that occurs as ions passes through the space between lattice atoms during a diffusive jump. The activation volume may in fact vary withP and '/'under certain circumstances. Typical values for the activation volume have a positive sign and are on the order of the molar volume of cations (~1-I0xl0~ 6 m3-mol_1) [see Bejina et al. (2003) for a review of pressure on diffusion in solids]. Taken together, the exponential term in Equation (7) suggests that the pressure effect can be interpreted as an added energy barrier which counteracts the thermal energy available for a diffusive jump. The effect of pressure on diffusion is generally small over the range of most geological environments in the lithosphere. and becomes significant mainly under conditions in the Earth's deep interior (as shown in Fig. 4; Watson and Baxter 2007). Control of solid-state reaction rates and compositions of reaction products by diffusion Limiting factors controlling the reaction rate. The crystallization history of a rock is typically recorded in several ways and on different scales as described in the previous section. On the scale of a thin section, interpretation of mineral textures of igneous and metamorphic rocks is commonly based on equilibrium thermodynamic calculations that lead to information on P-T-X conditions (where X equals composition) during crystal growth. Compositional changes within single metamorphic mineral grains are taken as a record of evolving P-T conditions that are constrained by appropriate mineral reactions and phase equilibria (see Spear 1993 and references therein). For igneous systems, elemental and isotopic profiles in

Figure 4. Dependence of diffusion rates as a function of pressure and along the geothermal gradient from the surface to the core (modified after Watson and Baxter 2007). Parameters used for Equation (7) are as follows: D0 = l x l O - 6 n r s - ' ; E = 300 k J m o H ; Va = 3x10-" m ' m o H . The geotherm in the insert is calculated after Duffy and Hemley (1995).

mainly T-effect

0

20

40 60 80 100 120 140 pressure (GPa)

1006

Mueller,

Watson,

Harrison

magmatic minerals have been used to constrain parental magma compositions (e.g., Davidson et al. 2007 and references therein) and temperatures of crystallization can be inferred using equilibrium thermodynamics. Again, these are powerful methodologies, but an important assumption made in implementing them is that equilibrium was attained at the scale of interest. An increasing number of recent studies reveal non-equilibrium textures, which highlights the need to invoke kinetic effects on texture formation and compositional evolution (e.g., Roselle et al. 1997: Müller et al. 2004. 2008; Watson and Müller 2009). Exploring the kinetic effects on compositional profiles and textures can lead to variety of information on timescales and rates of crystallization. Reactions among minerals in a rock or growth of a single mineral from magma can be thought of as a series of steps that includes volume diffusion, surface reaction, and intergranular transport (Fig. 5). As one mineral is consumed at the expense of another, components of the reactant phase must be available on the grain surface—or supplied to the surface by volume diffusion—if they are to participate in the reaction. Components of the reactant phase must then be 'liberated' by reaction at the surface and transported across the reaction boundary to the growing mineral site, where they are incorporated into the surface—again by surface reaction. Finally, the growing phase will attempt to homogenize its composition by volume diffusion. Most igneous processes do not involve mineral dissolution at the expense of a newly-nucleated phase: apart from this difference, however, the sequence of steps described above applies to both mineral growths from magmas and metamorphic mineral reactions. Once a mineral has nucleated, all the steps take place as a serial sequence, but they cannot all be expected to occur at the same rate. The overall reaction will be rate-limited by a specific step or mechanism in the process. In crustal metamorphic rocks, two steps have been identified as possibly rate-limiting: 1) the reaction at the surface: or 2) the diffusive flux of reactants. In the former case the kinetics of dissolution or precipitation of material at the mineral surface may be the slowest process.

volume diffusion & dissolution

precipitation & volume diffusion Figure 5. Schematic illustration of processes taking place during a possible mineral reaction. Processes include nucleation of new minerals, transport of elements to the surface of reactants, dissolution of material on the interface of reactants, transport of components through the melt or along grain boundaries, precipitation on the growing crystal and homogenization by volume diffusion. Note that, despite nucleation, all processes are serial, i.e., the slowest of each of the processes involved determines the overall reaction rate.

Applications of Diffusion Data to High-T Systems

1007

leading to what is called an interface- (or surface-) controlled reaction. In the latter case, the overall reaction rate might be limited by sluggish diffusion of components either within the grains or in the intergranular space and the overall reaction progress is said to be diffusion (or transport) controlled. The type of rate-limiting step can affect development of texture in both metamorphic and igneous rocks. The best example may be crystal size distributions (CSDs). If minerals form in a regime that is controlled by the surface-reaction rates, they should be randomly distributed and exhibit similar radial compositional profiles from one crystal to the next. The other extreme is the case where growing crystals compete with one another for material because diffusion is not fast enough to homogenize the matrix from which they grew. In this instance, the vicinity of each crystal becomes depleted in those elements the crystal needs for growth. This in turn has implications for the subsequent textural and chemical evolution of the system: 1)

The developing compositional depletion halo around the growing crystal has a significantly lower probability for nucleation of a new mineral, since there is not enough material available. Consequently, the minerals in the rock are no longer randomly distributed and exhibit some sort of spatial ordering (e.g., Kretz 1969, 1993; Hirschetal. 2003).

2)

The compositional profile of the diffusion-limited component(s) within the growing grains will decrease monotonically from the core to the rim of the crystal, potentially exhibiting characteristics that are fundamentally different from profiles generated by simple Rayleigh fractionation processes in an equilibrium environment. In addition, small crystals are expected to exhibit the same element zonation as the outer portion of larger crystals.

3)

The bulk composition of the depletion halo will differ from the surrounding, unaffected matrix. In theory, this has important implications for the final texture due to the local reduction in reaction affinity (i.e., unavailability of reactant components). In principle, this slows down the growth rate, which in turn allows the depletion halo to relax by diffusion. The relaxation then re-initiates growth in what amounts to a feedback loop, which should be recorded in the growing crystal with an oscillatory zoning pattern.

The last scenario above is an extreme case that is not frequently realized in nature. Nevertheless, scenarios 2 and 3 both underscore the fact that the compositional profiles of certain elements record key information on the crystallization history—that is, mechanisms and temporal evolution of mineral growth. A more detailed example of this effect is discussed below for the case of REE uptake during metamorphic garnet growth (Skora et al. 2006). Geothermometry. Thermodynamic calculations can be used to define equilibrium mineral compositions in a system where minerals are growing from a magma or which is described by solid-state reaction among minerals. For the latter, the simplest type of reaction is that of two minerals that share a common crystal chemical exchange (e.g., Fe-Mg, K-Na, CaMg). However, in both cases the final composition of minerals will depend on the partition coefficient (K D ) and thus on intrinsic variables such as pressure and temperature that affect its value. This phenomenon provides the basis for many geothermobarometers. For example, FeMg exchange between garnet and biotite or the Mg-content of calcite coexisting with dolomite are highly temperature dependent and widely used to estimate metamorphic peak temperatures (Ferry and Spear 1978; Anovitz et al. 1987). In general, exchange reactions of this type that are exploited for thermobarometry are well calibrated by experiments, and the studies establishing the calibrations present evidence that equilibrium was in fact reached in the experiments. Sometimes it can be demonstrated that the exchange of elements between minerals was fast enough to fully adjust the mineral compositions over the duration of the experimental run. It is crucial to keep in mind, however, that scaling issues may arise in applying the laboratory

1008

Mueller,

Watson,

Harrison

experiments to the natural environment. The grain sizes used in the experiments are typically much smaller (1000 |am), facilitating attainment of equilibrium, but there is a tradeoff because the time available in nature is always much greater. The outcome of this tradeoff is not always obvious. Rim compositions of crystals may indeed reflect local equilibrium expressed by the predicted partition coefficient, but if bulk exchange did not go to completion, erroneous temperatures may result if careful attention is not paid to the spatial resolution of the acquired data, as, for example, when bulk analyses are used. While many applications indeed consider that only the rims of minerals are in equilibrium, the general application of geothermobarometry is based on the assumption that a mineral is able to adjust its composition infinitely fast until peak (or closure) temperature is reached and does not exchange after that. In natural systems, several indicators, such as the presence of element zoning, suggest that the minerals do not always fully adjust their compositions to achieve equilibrium. Various strategies have been developed (e.g., use of rim compositions only or contacting mineral grains) to estimate temperatures based on the observed incomplete equilibration. Consequently, accurate and meaningful estimation of temperatures on the basis of mineral compositions requires identification of the exact mineral couples undergoing compositional adjustment, as well as the corresponding equilibrium domains with respect to the element of interest. Textural features in thin sections are helpful in this regard, but the application of any geothermometer or barometer can nevertheless yield substantial variations in estimated temperatures or pressures within a single rock sample or even a single mineral. The failure to reach complete equilibrium, however, can in fact provide enormous opportunities. Several studies have shown, for example, that calcite-dolomite thermometry returns temperature estimates varying by up to 200 °C for a single rock sample in calc-silicates surrounding contact aureoles (e.g., Cook and Bowman 1994; Miiller et al. 2008). Taking kinetics into account allows one to deduce peak metamorphic temperatures from the same compositional data. Miiller et al. (2008) suggested that the temperatures obtained on single calcite crystals can be successfully related to different mineral-forming reactions even on the scale of a thin section. Calcite crystals formed at lower temperatures are shown to maintain their composition during subsequent high-temperature contact metamorphism, and can readily be distinguished from calcite formed at higher temperatures within the same sample. In addition, Miiller et al. (2008) modeled the effect of retrograde cooling and exsolution on the mineral composition and resulting analytical data. The results emphasize the potential use of diffusion profiles not only to determine a single P,T condition of a rock or mineral but also to decipher entire crystallization histories recorded in the non-equilibrium element profiles. Mechanisms of element exchange between minerals. The sometimes sluggish response of minerals in adjusting their composition raises the additional question as to exactly how element exchange is accomplished in a given instance. For example, mineral replacement reactions, which can greatly enhance element exchange, take place by dissolution-reprecipitation rather than by volume diffusion, when a fluid phase is present. Experimental studies investigating mineral replacement processes by dissolution-precipitation initially focused on salt crystals in which the reaction could be monitored in-situ (Putnis 2002). These experiments, performed at room conditions, showed well-defined reaction fronts moving toward the center of the crystals surrounded by a fluid phase that exchanged cations with the crystal. The progress of reaction is here a function of the volume change during the exchange of cations. The work of Putnis and co-workers summarized in his review (Putnis 2009) was at least partially inspired by observations made by O'Neil and Taylor (1967) who measured oxygen isotope exchange profiles in feldspar which could not be explained by volume diffusion alone. On the other hand, the measured isotope ratios did not define abrupt step profiles indicating that dissolutionreprecipitation was not the sole replacement mechanism, either (O'Neil and Taylor 1967). Recently, experiments at elevated temperatures and pressures on cation exchange mechanisms in feldspar suggest that surface reactions are still the dominant process for element exchange (Labotka et al. 2008; Niedermeier et al. 2009). Nevertheless, results at elevated temperatures

Applications of Diffusion Data to High-T Systems

1009

yielded an increasingly broadened element exchange front, indicating that a significant (albeit variable) amount of volume diffusion contributes to the exchange mechanism. This observation is consistent with the increasing effectiveness of diffusion at progressively higher temperatures. Diffusion is expected to play a more significant role in any exchange process taking place at elevated P-T conditions in geological settings. It is important to note, however, that diffusion and reaction are inherently linked (Eqn. 4). In the present context the velocity term would dominate migration of the reaction front from the crystal surface to its center. Note, however, that migration of the reaction front can be attributed to fast diffusion in a fluidfilled porosity network that develops in the replaced portions of the crystal. Because Equation (4) does not specify any specific properties of the diffusion medium (single- or polycrystal; fast-path present or absent, etc.), the element-exchange front moving through a mineral will follow the same rules as the infiltration of fluid into a rock column (e.g., Korzhinskii 1965; Frantz and Weisbrod 1974; Baumgartner and Valley 2001). The kinetics controlling both element exchange and reaction between minerals are governed by the interplay of intracrystalline transport, transport to or from the margin of a mineral, and surface reaction, as discussed above. Dohmen and Chakraborty (2003) presented a model describing the mechanism and kinetics of element (or isotope) exchange between minerals mediated by a fluid phase. Their model consists of two physically separated grains communicating with each other by a fluid channel of given length and diameter (Fig. 6), much like the actual system modeled by Watson and Wark (1997) in their experiments on diffusion of aqueous Si0 2 solute. Dohmen and Chakraborty (2003) included the contributions of the various potentially rate-limiting mechanisms, and they found that there exist at least six reaction-limiting mechanisms which they could describe with a reaction mechanism map (Fig. 6) using two parameters 5 and y. These parameters can be calculated independently from thermodynamic, kinetic, and transport properties. In simple terms, they represent the effects of surface reaction vs. diffusion. The Dohmen and Chakraborty study was thus the first to provide a basis for predicting the rate-limiting mechanism in mineral exchange reactions under various P-T regimes. Interpretation of element and isotope zonation in minerals. Element profiles recorded in mineral grains might represent a sequence of non-equilibrium stages during progressive crystallization as discussed above, but there are other possible causes. The process of crystal growth itself and the movement of a crystal interface with respect to its growth medium can result in characteristic compositional profiles containing substantial information on the crystallization history. Element uptake during crystal growth is controlled primarily by the partition coefficient (KD), which can be computed from thermodynamics for a given pressure and temperature. The KD describes the equilibrium concentration ratio of an element in the crystal relative to its matrix or growth medium, and it is a constant for a given temperature, pressure, and phase compositions. However, a "depletion halo" of highly compatible elements (KD » 1) around a growing crystal can develop if diffusion of the elements of interest toward the advancing interface is not fast enough to meet the demand of the growing mineral. Similarly, highly incompatible elements (KD « 1) can accumulate in advance of the moving crystal interface because slow diffusion prevents dispersal into the growth medium of atoms not incorporated into the crystal. This particular type of disequilibrium, initially discussed in the materials sciences (Tiller et al. 1953; Smith et al. 1955), was first introduced to the geosciences by Albarede and Bottinga (1972). These authors called attention to the possibility of disequilibrium partitioning of trace elements between phenocrysts and surrounding melt and documented a correlation between grain size and trace-element zoning patterns in igneous rocks. While this work opened up new ways of thinking and interpreting element uptake during crystal growth, the initial models were limited to one dimension, and extrapolation to natural systems was difficult because unrealistic growth and/or diffusion rates were necessary to match observed element profiles. Recently, the increasing recognition of disequilibrium profiles within single crystals has brought renewed attention to disequilibrium element uptake during crystal growth (Singer et al. 1995; Skora et

1010

Mueller,

Watson,

Harrison

Figure 6. Conceptual model for element exchange between two minerals mediated by a fluid phase and resulting reaction mechanism map, after Dohmen and Chakraborty (2003). The rate limiting reaction mechanism can be classified using two parameters 8 and y (defined in the figure). The a parameter represents the surface reaction rate constant, K is the partition coefficient, r is the characteristic diffusion radius and D equals the diffusion coefficient of the species of interest within the mineral. The surface area of the mineral and fluid are denoted by s s „ M and s j , respectively. L is the length of the connecting fluid channel and D lluid represents the diffusion coefficient of the species in the fluid phase. flll (i

al. 2006; Müller et al. 2008). In addition, new sources of information—including theoretical calculations on rate data (Lasaga and Rye 1993) and the increased ease and potential of numerical calculations—have provided an improved basis for extracting information. For example. Baker (2008) was able to reproduce systematic changes in the composition of basaltic melt inclusions captured during laboratory growth of plagioclase and pyroxene crystals at different rates (in this case, the melt inclusions sampled the diffusive boundary layer against the growing crystals). Very recent models have been extended to two and three dimensions in simple geometries (e.g., Skora et al. 2006. 2008: Müller et al. 2008), and these have the potential to explain diffusive fractionation during crystal growth (Watson and Müller 2009). as will be discussed later in the section on "Chemical diffusive fractionation!' Modeling studies by Watson and Liang (1995) and Watson (2004) called attention to a different sort of kinetic disequilibrium arising from near-surface enrichment in the crystal lattice even if the growth medium is uniform in both time and space. In the so-called "growth entrapment" model, it is considered that the near-surface of a crystal is likely—even at equilibrium—to be characterized by a trace-element (and isotopic) composition that differs significantly from the bulk lattice. If the interface is moving, as during crystal growth, the anomalous near-surface composition may simply be "captured" by the crystal unless diffusion in the lattice is fast enough to continuously restore equilibrium. The phenomenon of growth entrapment thus depends critically on the trade-off between the rate of crystal growth and the rate of diffusion in the newly-formed crystal. In an initial paper addressing sector zoning specifically, Watson and Liang (1995) formalized the process in terms of a spatially-variable

Applications of Diffusion Data to High-T Systems

1011

equilibrium concentration, C„ which was assumed to change in close proximity to the surface according to the relation (8) where I is the half-thickness of the chemically anomalous near-surface layer, C,0 is the equilibrium concentration in the lattice well away from the surface, x is the distance from the surface (increasing negative numbers moving away from the surface at x = 0 into the crystal), and F is the surface enrichment factor. This last quantity is analogous to an equilibrium partition coefficient and is defined as C,(I = 0)/C,0. During growth of the crystal (i.e., advancement of the surface into the growth medium), the governing diffusion/mass-conservation equation is

(9) where Vg is the linear growth rate of the crystal. Watson and Liang (1995) showed that the extent of growth entrapment (that is, the deviation from the bulk equilibrium partition coefficient in the final crystal) is critically dependent on the dimensionless ratio VgllDh which they called the growth Peclet number (Peg). Depending somewhat on the value of F, growth entrapment becomes potentially important at Peg > 1 and inevitable at Peg > 1 0 . Watson (1996) later pointed out that sector zoning is just a visible manifestation of growth entrapment that develops specifically when adjacent crystal surfaces (facets) have differing values of F. The phenomenon could be more widespread, but if F is similar for all crystal surfaces, growth entrapment may occur with no telltale evidence in the form of observable sector zoning. Watson (1996) also suggested that the most likely natural situations for growth entrapment to occur would be lowtemperature aqueous environments, where Vg can be high but lattice diffusion is likely to be slow, leading to large Peg and Stoll et al. (2002), used the growth entrapment model to explain growth-rate dependent uptake of Sr in calcite. Watson (2004) refined the growth entrapment model with specific reference to calcite as an important biogenic mineral, and three new aspects were introduced. First, Z), was allowed to vary with depth in the crystal because it was realized that diffusion in the very near-surface of a crystal may be faster than that in the bulk lattice (much as grain-boundary diffusion is faster than lattice diffusion). Second, F was not restricted to values > 1, so "growth depletion" might occur for some elements. Finally, because nearsurface equilibrium isotopic composition of calcite (and other minerals) may be distinct from the crystal interior, growth entrapment may apply to isotope ratios as well as elements. Several studies successfully applied the growth entrapment model to experimental data on the effect of growth rate on trace-element and isotope uptake during carbonate crystallization (Gabitov and Watson 2006; Gaetani and Cohen 2006; Gabitov et al. 2008; Tang et al. 2008). For the purposes of this review, the important take-away points concerning growth entrapment are that: 1) information on diffusion in crystals can be used to model certain types of compositional disequilibrium records related to transport entirely within (but close to the surface of) the crystal of interest; and 2) the concept of growth entrapment applies generally to mineral growth at all conditions, but may be particularly important under circumstances of relatively rapid growth at modest temperatures where near-surface lattice diffusion is slow. Metamorphic example of diffusion-limited uptake: REE behavior during garnet growth Garnets are commonly used to estimate peak pressure-temperature conditions of metamorphic rocks in various geological settings (see Spear 1993 and references therein). Furthermore, compositional zoning in these common minerals can be used to decipher P-T histories of metamorphic rocks, assuming that the garnets grew in equilibrium with their surrounding matrix

1012

Mueller,

Watson,

Harrison

at any given time (e.g., Spear and Selverstone 1983). Together with geochronologic methods (e.g., Lu/Hf or Sm/Nd) this information can relate the conditions of metamorphic crystallization to the timing of a geological event (Vance and O'Nions 1990). As discussed in the section "Control of solid-state reaction rates and compositions of reaction products by diffusion," garnet growth may be regulated to some extent by the availability of components and/or the surface kinetics of either the growing crystal or those being consumed (e.g., Kretz 1969, 1993; Fisher 1978; Carlson and Denison 1*992). Skora et al. (2006) investigated mechanisms of metamorphic garnet growth in rocks from the Alps using compositional profiles of trace elements. Their approach was based on the assumption that trace elements should be particularly good indicators of garnet growth because they are not essential structural components of the mineral, and thus low concentrations are not likely to influence the rate-limiting mechanism. While a number of earlier studies—mainly using CSD's as record of kinetically controlled settings—suggested that metamorphic garnet growth is most likely kinetically controlled by limited diffusion (see extensive work of Carlson and co-workers, e.g., Carlson 1989; Carlson and Denison 1992; Chernoff and Carlson 1999; Carlson and Gordon 2004; Meth and Carlson 2005), this study warrants special attention here because it is the first to use compositional profiles to show that garnet growth was indeed ratelimited by diffusive supply from the matrix, which underscores the value of diffusion data and modeling. The samples investigated by Skora et al. (2006) came from the Zermatt-Saas Fee ophiolite in the European Alps. Peak metamorphic conditions for these rocks are estimated at ~550600 °C and -15-20 kbar on the basis of phase relations and chemical and textural evidence (e.g., Reinecke 1998). Skora et al. (2006) measured element zoning profiles on central cuts of garnet crystals, which revealed systematic zoning in both major and trace elements. Matching of the features of chemical profiles from different garnets was used by Skora et al. (2006) to conclude that the same radial fraction of each garnet grew in a given time interval (dr/dt = constant). This observation of linear garnet growth is expected for cases where the growth rate is controlled by the interface reaction (Kretz 1969). However, Skora et al. (2006) showed using numerical models that diffusion-controlled growth at high temperatures (where matrixtransport is effective) can also result in a nearly linear growth rate. The garnets examined by Skora et al (2006) exhibit characteristic zoning patterns in the rare-earth elements (REE). Compositional profiles for the more compatible heavy REE are characterized by sharp peaks in the core of the crystal and an additional maximum towards the rim (Fig. 7). In contrast, light REE are typically low in the center and show a sharp increase towards the rim of centrally-cut crystals. The observed profiles cannot be explained by simple Rayleigh fractionation. Therefore, the authors modeled the effects of REE uptake in metamorphic garnets controlled by the growth medium by varying growth rates of the crystal and REE diffusivities in the matrix. Their approach followed that of previous one-dimensional models of diffusion-limited trace-element uptake (Tiller et al. 1953; Smith et al. 1955; Albarede and Bottinga 1972) but was extended to a spherical geometry for both the crystal and the growth medium. For simplicity, volume diffusion within the crystal was ignored and the equilibrium partition coefficient was assumed to be constant despite a significant change in temperature (450-600 °C). The effective diffusivity was calculated using a simple Arrhenius equation to express the temperature dependence. The modeling results matched the observed compositional profiles for the HREE in showing a sharp peak in the center and a second maximum towards the rim (Fig. 7). The authors explained the development of these characteristic profiles in terms an initial period in which the vicinity of the crystal is depleted in heavy REE (e.g., Yb or Lu), producing a pronounced peak in the center of the crystal and an exponential decrease toward the rim (Fig. 8). Increase in temperature accelerates diffusion and allows relaxation of the depletion halo in the matrix, resulting

Applications

of Diffusion

Data to High-T

Systems

1013

in the observed second concentration maximum near the rim of the crystal. Further growth then continues the depletion of the REE in the matrix and lowers its concentration at the rim of the garnet. The compositional profiles for the light REE (e.g.. Sm) lack the central peak, but they show even more pronounced maxima toward the shoulders. These findings are broadly consistent with those derived for the heavy REE. The authors suggested several reasons for the absence of a visible central peak. First, the expected peak would be very narrow (< 1 |.im) and such a peak in the core of the grain would likely be missed if the crystal was cut slightly off center. Secondly, even small amounts of volume diffusion might erase the original central peak during the subsequent high-temperature metamorphism. The presence of other refractory minerals—epidote. for example—might have limited the availability of Sm during initial garnet growth. The details of the REE profile shape (i.e.. width of the central peak and the position and height of the second maximum) depend on a number of factors. For example, the authors used different growth rate equations (r cc t and r oc t1'1) to illustrate the sensitivity of the resulting profile to the growth mechanism (Fig. 9). Other factors affecting the appearance of the second maximum are the diffusion coefficient and its temperature dependence. Higher activation energies

Radius [cm] Figure 7. Lu and Y profiles on central cuts of garnets of the Zermatt-Saas Fee ophiolithe (from Skora et al. 2006). Circles and squares represent the two values of each measured profile. Note the characteristic shape of the profile exhibiting a sharp peak in the core of the crystal and an additional maximum towards the rim. Solid lines show best fits of modeling diffusion-limited uptake of these elements during metamorphic garnet growth. The model included prograde increasing temperature change from 450 to 600 °C and a linear growth rate (constant radius per time increment ). Effective diffusion coefficients are calculated using an simple Arrhenius relationship. See Skora et al. (2006) for details.

Figure 8. Schematic sketch illustrating the model proposed by Skora et al. (2006) of matrix diffusion-limited uptake of REEs during metamorphic garnet growth. Trace elements become depleted in the vicinity of the crystal during an initial growth period. Increasing temperatures during prograde metamorphism cause enhanced diffusion and result in a relaxation of the diffusion halo in the matrix. Further growth eventually depletes the matrix further and generates a further decrease in the REE content of the crystal. r

1014

Mueller,

Watson,

Harrison

CL CL

0.00

0.05

0.10

0.15

0.20

0.25

0.00

0.05

0.10

0.15

0.20

0.25

Radius r [cm] Figure 9. Trace element profiles for Lu and Y using different growth rates for metamorphic garnet growth modeled by Skora et al. (2006). Linear growth rates (proxy for surface controlled growth) result in wider central peaks. Also note that growth rates using the square root of time (proxy for diffusion controlled growth) produce a shift of the secondary maximum towards the rim, i.e., to later stages in the crystallization history of the garnet.

shift the secondary concentration peak towards the center (i.e., to earlier stages in the garnet growth history). Given the same 3+ charge on different REE ions, their diffusion coefficients are likely to depend mainly on ionic radius and perhaps slightly on mass. Cherniak (2010) suggested only little difference among the diffusion rates of REE's in olivine and Van Orman et al. (2002) also find little evidence for variations of diffusivities for REE's in garnet. However, other studies of REE lattice diffusion in silicate minerals indicate significant variations across the series (Cherniak et al. 1997; Van Orman et al. 2001), and the same could occur for REE diffusion in the matrix. At present there are few constraints on REE grain-boundary diffusion, but differences across the REE series could explain the observed shapes in the compositional profiles. In summary, the high sensitivity of REE profiles to both growth rates and diffusivities underscores their potential as recorders of crystallization histories in metamorphic rocks. Chemical diffusive fractionation Diffusion of an atom or ion depends on a number of factors related to the size and charge of the diffusant and the nature of the bonds it forms with the atoms of the host medium. It is well known, that isotopes of a given element have slightly different diffusivities because of small differences in bond energy and vibrational frequency. A substantial literature on the isotopic mass effect exists in other disciplines—most notably metallurgy (see Mullen 1961; Peterson and Rothman 1967; Richter et al. 2009a)—but studies of 3 He and 4 He diffusion in silicate minerals were among the first to reveal the effect in a geological material (Trull and Kurz 1993). However, because the large relative mass difference between He isotopes, this study did not dispel the general belief that in geological settings mass-dependent isotope fractionation by diffusion is rare. Richter et al. (1999) first presented convincing evidence for mass-dependent fractionation of relatively heavy Ca-isotopes in Ca0-Al 2 0 3 -Si0 2 melts. Recent experimental studies have revealed that significant diffusive isotope fractionation is also observed in melts of basaltic and rhyolitic composition (Richter et al. 2003, 2009a). Richter et al. (1999) developed a mathematical description for the relative diffusivities of isotopes of a given element that is based (for lack of better analogs) on the kinetic theory of gases: D, =

M2

m;

(10)

The relative diffusivity DreJ relates the ratio of the diffusion coefficients (D\, D2) of the two

Applications of Diffusion Data to High-T Systems

1015

isotopes to the inverse of their mass ratio to the power of (3. The value of (5 for gases is 0.5, but because there is not yet any physical basis or interpretation of this parameter for condensed materials, it must be determined empirically. To date, only isotopes of Li, Ca, Mg, Fe and Ge have been characterized, specifically for diffusion in silicate melts (aqueous solutions have also been studied, but the mass effect on diffusion is small for most ions; Richter et al. 2006). In silicate melts, the range of measured (3 values to date is from -0.025 for Ge to 0.215 for Li (isotopes of the former may diffuse as large germanate tetrahedra, which would markedly 'dilute' the mass difference between the two isotopes themselves, 76 Ge and 70 Ge). The (5 values for divalent cations fall in the 0.05-0.08 range. These values imply small differences in the diffusivities of geochemically-important isotopes (e.g., of Sr. Ca, Mg and Fe), but under some circumstances these differences may lead to significant isotopic fractionation, as discussed below. Profiles of isotopic composition in igneous minerals have seen increasing use in constraining parental magma compositions. In most cases, isotopic zoning in igneous crystals is attributed to processes such as magma mixing or fractional crystallization (e.g.. Gagnevin et al. 2005: Teng et al. 2008). Core-to-rim isotopic profiles are interpreted as time markers in the crystallization history—a concept known as 'Crystal Isotope Stratigraphy' (CIS; Davidson et al. 1998). The possibility that core-to-rim-varying isotope ratios within single crystals could be due instead to fractionation during diffusion-controlled growth has been recognized for some time, but the effect has generally been thought to be negligibly small (Faure 1986: Dickin 2005: Davidson et al. 2007). Building on measurements of mass-dependent diffusive fractionation in melts (Richter et al. 2003; Richter et al. 2008b), Watson and Muller (2009) developed a model to explain disequilibrium uptake of isotopes in terms of kinetically controlled mineral growth. Their model uses a spherical grain growing from a reservoir of specified size and concentration of the element and isotopes of interest (Fig. 10). The model system was similar to that used by Skora et al. (2006) (see also the "A metamorphic example of diffusion-limited uptake: REE behavior during garnet growth" section), but was extended to include multiple elements and volume diffusion within the growing crystal. As the spherical crystal grows, incompatible elements accumulate in advance of the moving interface and the concentrations of compatible elements are depleted. Depending on their diffusivities in the reservoir the effect is smaller for the faster-diffusing (lighter) isotope, so isotopic fractionations can occur within

interface s y s t e m boundary (zero-flux)

/

/

t=0

c

ra crystal c CD o c o o

/

growth medium

interface

I 1

\

V

\

diffusive boundary layer \

\

growth medium

\

-r=rSi

distance

Figure 10. Model system for growth of a spherical crystal in a generic growth medium. Schematic graphs at the right show evolution of concentration profiles in the crystal and growth medium for an incompatible element.

1016

Mueller,

Watson,

Harrison

the compositional boundary layer against the growing crystal. Assignment of appropriate (3 values of 0.05 to 0.08 reveals small differences in the diffusivities of isotopes of key interest For example, Dre, for 40 Ca/ 44 Ca = 1.007; Drd for 86Sr/87Sr = 1.0009 (Eqn. 10). These values are close to 1, but potentially measurable differences in the isotope ratios can nevertheless develop in the compositional boundary layer against the growing crystal if the element of interest is either highly compatible or highly incompatible. Consequently, significant isotopic fractionation is predicted to occur within the boundary layer during progressive crystal growth (Fig. 11; 5 values range up to 0.3 for 86Sr/87Sr, and up to 3.5 for 40Ca/44Ca). The assumption of local equilibrium at the interface between the crystal and its growth medium implies that a fastgrowing crystal will record isotopic fractionation in its resulting zoning profile. The amount of fractionation is determined not only by Drd but also by the growth rate (R), so the extent of isotopic fractionation can be related to the Peclet number given by Equation (5) in which the velocity of the moving interface (R) is related to the diffusivity of one of the isotope D, (Fig. 11). It is important to note that the formation of a significant diffusive boundary layer requires relatively slow diffusion of the element of interest in the growth medium and fairly large growth rates, but this combination is not unrealistic in fast-cooling viscous magmas, such as lava lakes or small igneous bodies. Watson and Miiller (2009) extended their modeling for a static system to cases in which the compositional boundary layer was limited in width due to 'erosion' resulting from convection of the growth medium. Based on their numerical modeling results, Watson and Miiller (2009) determined an analytical expression for isotopic fractionation in boundary layers of fixed width (BL), which is a plausible condition in convecting fluid systems. In conventional delta notation for isotopic systems: (11) In summary, core-to-rim variation in isotope composition could be the result of diffusive fractionation during growth, and does not necessarily require changes in the overall reservoir (growth medium) composition. However, whether this effect can explain natural variations awaits detailed modeling. Moreover, this finding does not exclude the possibility of isotopic zoning caused by magmatic events (e.g., mixing) that do cause changes in the bulk reservoir composition. The Watson-Miiller analysis may nevertheless prove useful for deciphering dynamic crystallization processes, and it is also as a potential window into growth rates, mechanisms and more exact constraints on T-f-path in a variety of geological settings, including not only igneous processes but also metamorphism and growth from aqueous solutions. There is growing evidence that solid-state diffusion—both along grain boundaries and within individual crystals—can also fractionate isotopes. Teng et al. (2006) suggested that diffusive isotope fractionation of Li isotopes occurred in the country rock of a pegmatite body, where 8 7 Li drops by ~20%o from the pegmatite/host contact to - 1 0 meters away. Dohmen et al. (2010) presented convincing experimental evidence that Li isotopes are fractionated by volume (lattice) diffusion in olivine. These authors proposed that the difference in diffusivity between 7 Li and 6Li is due to Li diffusing through the crystal lattice using different sites, namely an octahedral (LiMc) and an interstitial one (Lii). The octahedral site can also be a vacant site (Li v ) and Li will partition into these different sites, maintaining an equilibrium distribution according to Li v = LiMe + Lij. The latter reaction is assumed to equilibrate almost instantaneously relative to the timescales of all other diffusion process occurring in the crystal. As a result, the partitioning—and thus the actual diffusion path—is controlled by external parameters such as fo2 or aLi4sio4- Modeling revealed that 6Li diffuses about 5% faster on the interstitial site than 7Li, creating a significant amount of diffusive isotope fractionation within the crystal. This model suggests, further, that the timescales needed to equilibrate isotope ratios in olivine

Applications of Diffusion Data to High-T Systems 2.0

4

_!Ca Ca

ÜSr Sr

40

86

^ > > i ;

1017

*

0.5 -

-4.0 0.990 0.992 0.994 0.996 0.998

1.000 1.002

1.004

1.006

DJD, Figure 11. The maximum deviation from isotopic equilibrium, S max , for crystals of 0.3 and 1.0 cm radius grown at R/D = 0.5, 1.0 and 2. The deviation is plotted against D'el (= DAIDB), the ratio of the diffusivities of the two isotopes being compared. The diffusivity ratio for 44 Ca/ 40 Ca measured by Richter et al. (2003) and the estimated value for 87 Sr/ 8fl Sr are indicated as dotted vertical lines.

may be different from equilibration of the overall Li concentration. The combination of Li fast diffusion in minerals such as olivine with diffusive isotope fractionation might then be used to constrain rapid late-stage magmatic processes (Costa et al. 2008) or the length scale of mantle heterogeneities (Halama et al. 2008). In recent years, geochemists have resumed the effort initiated by metallurgists in the 1960s to consider the isotope mass effect on diffusion in metals, focusing on Fe-Ni alloys as models for meteorites and planetary materials. Dauphas (2007), for example, has shown that the small fractionations of Fe and Ni isotopes between taenite and kamacite in the Toluca (IAB) iron meteorite (Poitrasson et al. 2005; Horn et al. 2006; Cook et al. 2007) can be explained by differences in the diffusivities of the relevant isotopes. Diffusive fractionation in a thermal gradient Intermittently over much of the last century, petrologists have speculated that magmatic components can be fractionated from one another in response to an imposed thermal gradient. This phenomenon—commonly referred to as "thermal" or "Soret" diffusion—is well documented in laboratory experiments [Note: Taken literally, the term "thermal diffusion" implies transport of heat itself, and is therefore confusing in the context of a discussion of chemical transport. Unfortunately, however, this term is well entrenched in the literature in reference to the chemical response of a system to a thermal gradient; we include it for completeness but shift to more explicit terms hereafter]. Soret diffusion was dismissed by Bowen (1921) as a major factor in magmatic differentiation. He argued that because conduction of heat in rocks is so much faster than diffusion of magmatic components (which he was the first to constrain by experiment), the temperature of a closed system would rarely, if ever, be maintained long enough to allow appreciable chemical separation by diffusion. Nevertheless, for a many years the Soret effect was considered by some to be a plausible explanation for chemical differentiation in subvolcanic magma chambers of large ash-flow sheets like the Bishop tuff (Hildreth 1979), and it was only recently abandoned (Hildreth and Wilson 2007).

1018

Mueller,

Watson,

Harrison

Chemical fractionation by diffusion in a thermal gradient in magmatic liquids is now well understood, largely through the efforts of D. Walker and colleagues in the 1980s and 90s addressing thermal diffusion in basaltic systems (e.g., Walker and Delong 1982; Lesher and Walker 1986; Lesher et al. 1991). Bowen would not be disappointed by their general conclusion that fractionation of magmatic components by thermally induced diffusion is not a pervasive process, but instead one that is restricted mainly to diffusive boundary layers in periodically replenished and/or double diffusive systems. More recently, experimentalists have turned their attention to isotopic fractionation driven by diffusion in thermal gradients (Richter et al. 2008b; Watson et al. 2008; Richter et al. 2009b), and significant effects have been documented for some isotopes of O, Mg, Si, S, Ca and Fe, mainly in basaltic systems. Kyser et al. (1998) investigated simple compositions in the leucitefayalite-silica system. Observed fractionations range from 0.6 to 3.6%c per 100 °C per a.m.u. (a larger effect is suggested for C isotopes, but the measurement needs confirmation; Watson et al., 2008). For ~8 mm long samples with end-to-end temperature differences of 100-160 °C (mean T ~ 1450 °C), steady-state thermal fractionation is reached in a few tens of hours and persists as long as the temperature gradient is maintained. For all elements except C, the heavier isotopes are enriched at the cold end of the experimental system, with Mg showing the largest effect. Interestingly, the fractionation of Mg isotopes by thermal diffusion is substantially more effective than fractionation resulting from the mass effect on chemical diffusion (see the "Chemical diffusive fractionation" section). Compared using Equation (10), the exponent (3 describing the mass effect is 0.05 and that describing the thermal effect is 0.65. The overall conclusions of Richter et al. (2008a, 2009b) regarding isotope fractionation by diffusion in a thermal gradient in basaltic melts can be summarized as follows: 1) measurable fractionations can result from modest temperature differences (a few tens of degrees) if the T gradient is maintained for sufficient time; 2) as in the case of major-element Soret fractionation, effects in natural systems are probably limited to diffusive boundary layers and double-diffusive systems; 3) thermal diffusion effects on isotopes are sufficiently large and distinctive to serve as indicators of probable Soret effects on the major elements.

THERMOCHRONOLOGY Background Just as the outcome of a single, random diffusion jump cannot be predicted, individual radioactive decay events cannot be predicted either. Neverthetheless, the statistically well behaved outcome of large numbers of events leads to our routine ability to date geologic events via the ratio of abundances of radiogenic daughter to parent isotope. A mineral that remains stable and quantitatively retains both parent and daughter isotopes constitutes an ideal geochronometer. A few mineral systems, such as zircon U-Th-Pb dating, approach this ideal under crustal conditions, but most minerals do not completely retain their radiogenic daughter products. The reason for this relates to the nature of the transmutations caused by radioactive decay. For example, beta decay results in the conversion of the parent element into an adjacent isobar, with either one proton added or lost. The shift to a different column of the periodic table generally results in the daughter product becoming geochemically incompatible in the host mineral and thus conducive to exchange. It is this incompatibility and intrinsic 'leakiness' of daughter species that endows some radiogenic systems with the capacity to act as a temperature history monitor. The parent-daughter system can be disturbed by many mechanisms involving mineral instability (e.g., dissolution-reprecipitation, deformation) but a stable phase can only exchange isotopes via diffusion. This latter mechanism is the most common source of discrepancy between a radiometric mineral date and the age of the rock in which it formed.

Applications of Diffusion Data to High-T Systems

1019

Bulk closure Both diffusion and radioactive decay are highly predictable processes given appropriate time and length scales. Coupled with the strong temperature dependence of diffusion (Eqn. 6), these qualities lead to a distinctive spatial daughter/parent distribution being produced within a solid following a given thermal history. Consider the case in which a mineral containing a radioactive parent experiences a complex thermal evolution, possibly involving heating as well as cooling. Within the sample, the daughter product is continually produced by radioactive decay and is lost or exchanged via diffusion. The outcome of the competition between radiogenic production and diffusion loss is given by solutions of the production-diffusion equation which have the form: (12) where Tc is a characteristic apparent closure temperature of the system (Kelvin), A is a geometric factor, and t relates to the form of the thermal history. In the case of monotonic cooling of the form T