Mechanisms of Diffusional Phase Transfmns. in Metals, Alloys
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Mechanisms of Diffusional Phase Transformations in Metals and Alloys
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Mechanisms of Diffusional Phase Transformations in Metals and Alloys
0]JMZ\1)IZWV[WV Hubert I. Aaronson 5I[I\W-VWUW\W Masato Enomoto 2WVO34MM Jong K. Lee
CRC Press Taylor & Francis Group Boca Raton London York London New NewYork
CRC Press is an imprint of the the business Taylor & Francis Group, an informa business
CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-6300-4 (Ebook-PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
To the late Professor Hubert Irwin Aaronson, and his sister, Barbara McMurray
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Contents Preface............................................................................................................................................. xv Authors.......................................................................................................................................... xvii
Chapter 1
Applied Thermodynamics ........................................................................................... 1 1.1 Free Energy–Composition Relationships for Binary Substitutional Solid Solutions.................................................................................................. 1 1.1.1 Basic Free Energy–Composition Relationship.................................... 1 1.1.2 Gibbs Free Energy of the Standard States of Pure Elements ............. 1 1.1.3 Mixing Free Energy, DGM .................................................................. 4 1.1.3.1 Fundamentals ....................................................................... 4 1.1.3.2 Ideal Solution Approximation ............................................. 7 1.1.3.3 Regular Solution Approximation......................................... 7 1.1.3.4 Subregular and Other-Type Solutions ............................... 12 1.1.3.5 Relationships for Partial Molar Free Energies, Phase Equilibria, and Critical Temperature ....................... 12 1.1.3.6 Comparison between Regular Solutions and Nonregular Solutions .................................................. 16 1.2 Free Energy–Composition Diagram and Applications to Driving Force Calculations ........................................................................ 17 1.2.1 Some Considerations on the Free Energy vs. Composition Curve..... 17 1.2.2 Total Free Energy Change Attending Precipitation .......................... 21 1.2.3 Free Energy Change Attending the Precipitation of a Small Amount of a .................................................................... 24 1.2.4 Division of the Total Free Energy Change between Capillarity and Diffusion..................................................................................... 25 1.2.5 Influence of Capillarity upon Solubility............................................ 27 1.2.6 Division of DG between Diffusion and Uniform Interfacial Reaction ............................................................................................. 29 1.2.7 Permissible Range of Nonequilibrium Precipitate Compositions ..... 30 1.3 Thermodynamics of Interstitial Solid Solutions through Application to the Proeutectoid Ferrite Reaction in Fe–C Alloys ..................................... 31 1.3.1 Introduction ....................................................................................... 31 1.3.2 Free Energy and Positional Entropy of Ideal Interstitial Solid Solutions .................................................................................. 32 1.3.3 Free Energy and Positional Entropy of Nonideal Interstitial Solid Solution................................................................... 33 1.3.4 Evaluation of Constants in a Partial Molar Free Energy Equation ..... 35 1.3.5 Application of Partial Molar Free Energy Equations........................ 36 1.3.5.1 Calculation of the g=(a þ g) or Ae3 Phase Boundary....... 36 1.3.5.2 Driving Force for the Massive Transformation in Fe–C Alloys................................................................... 37 1.3.5.3 Driving Force for the Precipitation of Proeutectoid Ferrite ................................................................................. 37 1.3.5.4 Graphical Presentations of the Results of Sections 1.3.5.1 through 1.3.5.3.................................... 38 vii
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1.3.6
Interpretation of z in Terms of Carbon–Carbon Interaction Energy................................................................................................ 39 1.3.7 More Sophisticated Treatments of Interstitial Statistical Thermodynamics ............................................................................... 41 References ................................................................................................................. 46 Chapter 2
Diffusional Nucleation in Solid–Solid Transformations........................................... 49 2.1 Introduction through Qualitative General Statements .................................... 49 2.2 Brief Comparative Survey of Nucleation in the Four Basic Types of Phase Transformation................................................................................. 52 2.2.1 Vapor-to-Liquid Transformation ....................................................... 52 2.2.2 Vapor-to-Solid Transformation ......................................................... 52 2.2.3 Liquid-to-Solid Transformation (Solidification) ............................... 52 2.2.4 Solid-to-Solid Transformation........................................................... 53 2.2.5 General Remarks ............................................................................... 53 2.3 Outline of Approach for Development of Nucleation Theory ....................... 53 2.4 Proof That the Equilibrium Concentration of Critical Nuclei Is Proportional to exp(DG*=kT)....................................................... 53 2.5 Fictitious Equilibrium Nucleation Rate .......................................................... 55 2.6 Derivation of Steady-State Nucleation Rate................................................... 55 2.7 Estimation of b*............................................................................................. 59 2.8 Time-Dependent Nucleation Rate .................................................................. 59 2.9 Feder et al.’s Treatment of t........................................................................... 63 2.9.1 Relationships for d and for td ........................................................... 64 2.9.2 Relationship for t0 .............................................................................. 65 2.9.3 Total Value of t................................................................................. 66 2.10 Time-Dependent Nucleation Rate for Homogeneous Nucleation with Isotropic g .............................................................................................. 66 2.10.1 Introduction ....................................................................................... 66 2.10.2 Activation Energy of Nucleation DG*.............................................. 67 2.10.2.1 Introduction to the Critical Nucleus Shape Problem......... 67 2.10.2.2 G–x Diagram Approach ..................................................... 67 2.10.2.3 Introduction to the Volume Strain Energy Incorporation Problem ....................................................... 68 2.10.2.4 Conventional Gibbsian Approach...................................... 69 2.10.2.5 Wulff Volume Approach for DG*..................................... 69 2.10.2.6 Nucleus Volume Approach for DG* ................................. 71 2.10.3 Frequency Factor b* ......................................................................... 72 2.10.4 Zeldovich Factor, Incubation Time and the Re-Derivation of Frequency Factor .......................................................................... 72 2.10.5 Nucleation Site Density, N ................................................................ 75 2.10.6 Time-Dependent Nucleation Rate ..................................................... 75 2.11 Ancillary Parameters....................................................................................... 75 2.11.1 Volume Diffusivity............................................................................ 75 2.11.2 Volume Free Energy Change ............................................................ 76 2.11.3 Volume Strain Energy ....................................................................... 79 2.11.3.1 Elementary Calculation of Dilatational Strain Energy for a Plate-Shaped Nucleus................................................ 80 2.11.3.2 Volume Strain Energy of Fully Coherent Precipitates ...... 82
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2.12 2.13
2.14
2.15
2.11.3.3 Volume Strain Energy of Incoherent Precipitates ............. 89 2.11.3.4 Volume Shear Strain Energy ............................................. 90 2.11.3.5 Unsolved Major Problems in Volume Strain Energy........ 91 2.11.4 Interfacial Energy .............................................................................. 91 2.11.4.1 Scope.................................................................................. 91 2.11.4.2 Energy of Coherent Interphase Boundaries....................... 92 2.11.4.3 Energy of Partially Coherent Interphase Boundaries ...... 125 2.11.4.4 Energy of Disordered Interphase Boundaries.................. 126 Preliminary Consideration of the Approximation for f ¼ DGv þ W ........... 132 Nonclassical Nucleation Theory................................................................... 134 2.13.1 Continuum Theory .......................................................................... 134 2.13.1.1 General Introduction ........................................................ 134 2.13.1.2 Calculation of the Nonclassical DG*............................... 135 2.13.1.3 General Properties of the Critical Nucleus ...................... 140 2.13.1.4 Nucleation in a Regular Solution .................................... 149 2.13.1.5 Applicability Region of Classical Theory ....................... 155 2.13.2 Discrete Lattice Point Theory.......................................................... 156 Modifications of Homogeneous Nucleation Kinetics by Anisotropic Interfacial Energy ................................................................ 159 2.14.1 Equilibrium Shape Problem ............................................................ 159 2.14.1.1 Rudimentary Solution of the Equilibrium Shape Problem ............................................................................ 160 2.14.1.2 g-Plot and Some Properties ............................................. 161 2.14.1.3 Wulff Construction .......................................................... 163 2.14.1.4 Simple Approach to Calculation of g-Plots..................... 164 2.14.1.5 Wulff Construction of Equilibrium Shape vs. Temperature in a Regular Solution, fcc Miscibility Gap .......................................................... 165 2.14.2 Sphere Faceted at One Boundary Orientation................................. 169 2.14.2.1 g-Plot and Force Balance ................................................ 170 2.14.2.2 Calculation of r* and DG* .............................................. 171 2.14.2.3 Calculation of b* ............................................................. 173 2.14.2.4 Calculation of Zeldovich Factor, Z, and Incubation Time, t ........................................................... 174 2.14.3 Nonspherical Critical Nucleus Shapes with a Finite Number of Interfacial Energies and Analytically Describable Interfaces ......................................................................................... 179 Nucleation Kinetics at the Faces of Disordered Grain Boundaries ............. 180 2.15.1 Introductory Comments on Grain Boundary Geometry and Structure.................................................................................... 180 2.15.2 Equilibrium Shape Problem at Grain Boundaries........................... 180 2.15.3 Shape-Dependent Nucleation Kinetics Factors for the Two Spherical Cap Nucleus with Isotropic gab ............................. 182 2.15.3.1 DG* and R* ..................................................................... 182 2.15.3.2 b*, Z, t, and Transport by Interfacial Diffusion ............. 183 2.15.4 Nucleation Kinetics of the Double Spherical Cap Faceted at One Boundary Orientation .......................................................... 184 2.15.4.1 When f fc1 and gcab > gbb =2 .................................... 184 2.15.4.2 Two-Dimensional Nuclei................................................. 185 2.15.4.3 Three-Dimensional Nuclei When f > fc1 ...................... 191
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2.16 Comparative Nucleation Kinetics at Grain Faces, Edges, and Corners Relative to Homogeneous Nucleation: Trade-Offs between N and DG* When gab Is Isotropic.............................. 196 2.17 Nucleation at Dislocations............................................................................ 201 2.17.1 Incoherent Nucleation...................................................................... 202 2.17.1.1 The Cahn Treatment ........................................................ 202 2.17.1.2 The Gomez-Ramirez and Pound Treatment .................... 205 2.17.2 Coherent Nucleation........................................................................ 210 2.18 Comparisons of Theory and Experiment ..................................................... 211 2.18.1 Homogeneous Nucleation ............................................................... 211 2.18.1.1 Homogeneous Nucleation of Co-Rich Precipitates in Cu-Rich Cu–Co Alloys ............................................... 212 2.18.1.2 Homogeneous Nucleation of Ni3Al Precipitates in Ni-Rich Ni–Al Alloys ................................................. 226 2.18.1.3 Homogeneous Nucleation in Liquids .............................. 228 2.18.2 Nucleation at Grain Boundaries ...................................................... 230 2.18.2.1 Nucleation of Proeutectoid Ferrite at Austenite Grain Boundaries in Fe–C Alloys ............................................. 230 2.18.3 Nucleation at Grain Faces vs. Grain Edges .................................... 238 2.18.4 Nucleation at Dislocations............................................................... 239 2.18.5 Secondary Sideplate Selectivity ...................................................... 244 References ............................................................................................................... 245 Chapter 3
Diffusional Growth.................................................................................................. 249 3.1 Basic Differences between Diffusional Nucleation and Diffusional Growth ................................................................................ 249 3.2 A General Theory of Precipitate Morphology ............................................. 249 3.3 Disordered Interphase Boundaries................................................................ 252 3.3.1 Introduction ..................................................................................... 252 3.3.2 Volume Diffusion–Controlled Growth Kinetics ............................. 253 3.3.2.1 Mathematics for Diffusion and Flux Equations .............. 253 3.3.2.2 Comparisons with Experiment......................................... 303 3.3.3 Growth Faster than Volume Diffusion Control Allows.................. 332 3.3.3.1 Grain Boundary Allotriomorphs ...................................... 332 3.3.3.2 Dissolution of Grain Boundary Allotriomorphs .............. 340 3.3.3.3 Plate Lengthening ............................................................ 342 3.3.4 Growth Slower than Volume Diffusion Control Allows ................ 343 3.4 Partially and Fully Coherent Interphase Boundaries.................................... 347 3.4.1 Introduction ..................................................................................... 347 3.4.2 Misfit Dislocations at Partially Coherent Interphase Boundaries ... 347 3.4.2.1 Theory .............................................................................. 347 3.4.2.2 Comparisons of Theory and Experiment......................... 354 3.4.3 Acquisition of the Misfit Dislocation Structure of Partially Coherent Interphase Boundaries...................................................... 368 3.4.3.1 Theory .............................................................................. 368 3.4.3.2 Comparisons of Theory with Experiment ....................... 378 3.4.4 Growth Ledges at Partially and Fully Coherent Interphase Boundaries ....................................................................................... 386 3.4.4.1 Prevalence and Role in Interface Crystallography .......... 386 3.4.4.2 Visibility Conditions for Ledges ..................................... 387
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xi
3.4.4.3 Sources of Ledges............................................................ 389 3.4.4.4 Ledge Heights .................................................................. 394 3.4.4.5 Inter-Ledge Spacings ....................................................... 395 3.4.5 Structural Ledges at Partially Coherent Interphase Boundaries ....................................................................................... 397 3.4.6 Migration of Partially and Fully Coherent Interphase Boundaries by Growth Ledges........................................................ 409 3.4.6.1 Theory .............................................................................. 409 3.4.6.2 Comparison of Theory and Experiment for Growth of Ledged Interphase Boundaries .................................... 419 3.5 Relative Growth Kinetics of Disordered and Partially Coherent Interphase Boundaries .................................................................................. 425 References ............................................................................................................... 427 Chapter 4
Precipitation............................................................................................................. 433 4.1 Introduction................................................................................................... 433 4.2 Metastable Equilibrium Phase Boundaries................................................... 433 4.2.1 Types of Metastable Equilibrium Phases ........................................ 433 4.2.2 Calculation of Metastable Equilibrium Phase Boundaries.............. 435 4.3 GP Zones ...................................................................................................... 438 4.3.1 Definition ......................................................................................... 438 4.3.2 Early History and Methods of Experimental Detection.................. 438 4.3.3 GP Zone Solvus Curves .................................................................. 442 4.3.4 Description of GP Zones: Morphology, Size, Number Density, and Composition.............................................................................. 444 4.3.5 Kinetics of GP Zone Formation ...................................................... 448 4.3.6 Origins of GP Zone Formation ....................................................... 451 4.4 Transition Phases .......................................................................................... 458 4.4.1 Definition and Basic Characteristic ................................................. 458 4.4.2 Occurrence and Thermodynamics ................................................... 459 4.4.3 Crystallography ............................................................................... 459 4.4.4 Nucleation Sequence of Transition Phases ..................................... 462 4.4.4.1 From the Viewpoint of DGv ............................................ 463 4.4.4.2 From the Viewpoint of Interfacial Energy ...................... 463 4.4.4.3 Nucleation Sites of Successive Transition Phases........... 464 4.5 Nucleation Sites ............................................................................................ 465 4.5.1 Homogeneous Nucleation ............................................................... 465 4.5.2 Nucleation at Large-Angle Grain Boundaries................................. 466 4.5.3 Nucleation Kinetics at Small-Angle Boundaries............................. 467 4.5.4 Nucleation at Dislocations............................................................... 469 4.5.4.1 General Remarks.............................................................. 469 4.5.4.2 Stacking Fault Nucleation and Edgewise Growth........... 469 4.5.4.3 Nucleation on Displaced and Freshly Generated Dislocations...................................................................... 470 4.5.5 Nucleation at Point Defect Clusters ................................................ 472 4.5.6 Nucleation on Precipitates............................................................... 475 4.5.6.1 Nucleation on Precipitates of a Different Phase.............. 475 4.5.6.2 Sympathetic Nucleation ................................................... 478 4.6 Successive Reactions Involving Different Phases........................................ 478 4.7 Precipitate Free Zones .................................................................................. 479
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4.8 4.9
Coarsening (Ostwald Ripening) ................................................................... 482 Overall Evolution of the Microstructure ...................................................... 488 4.9.1 Effects of the Ratio of Growth Rates of Disordered-to-Partially Coherent Boundaries ............................. 488 4.9.2 Effects of Diffusion Distance-to-Matrix Grain Radius Ratio ......... 488 4.9.3 Effects of the Lever Rule and of Nucleation-to-Growth Rate Ratio ........................................................................................ 490 References ............................................................................................................... 491 Chapter 5
Massive Transformation .......................................................................................... 495 5.1 Definition and History .................................................................................. 495 5.2 Phase Diagrams ............................................................................................ 495 5.3 Thermodynamics .......................................................................................... 497 5.3.1 Free Energy Composition Diagram................................................. 497 5.3.2 Experimental Evaluation of Enthalpy Change Associated with Massive Reaction .................................................................... 497 5.4 Overall Reaction Kinetics and the Existence Range.................................... 499 5.5 Nucleation of Massive Transformation ........................................................ 508 5.5.1 Nucleation during Continuous Cooling .......................................... 508 5.5.2 Nucleation during Isothermal Transformation ................................ 514 5.5.3 Nucleation Sites and Massive Crystal Morphology........................ 516 5.6 Growth Kinetics............................................................................................ 523 5.6.1 Theory.............................................................................................. 523 5.6.1.1 Disordered Interphase Boundaries ................................... 523 5.6.1.2 Partially Coherent Interphase Boundaries ....................... 524 5.6.2 Comparison with Experiment.......................................................... 525 5.7 Interfacial Structure, Habit Planes, Orientation Relationships, and Growth Mechanisms.............................................................................. 532 5.8 Note on the Driving Force for Trans-Interphase Boundary Diffusion during Massive Transformation in a Two-Phase Field ................................ 540 References ............................................................................................................... 541
Chapter 6
Cellular Reaction ..................................................................................................... 543 6.1 Definition and Introduction .......................................................................... 543 6.2 Systematics of Cellular Reactions ................................................................ 545 6.3 Nucleation of Cellular Reactions.................................................................. 552 6.3.1 Crystallography-Based Mechanisms ............................................... 552 6.3.1.1 Tu–Turnbull Replacive Mechanism ................................ 552 6.3.1.2 Proposals of Aaronson and Aaron................................... 554 6.3.2 Noncrystallographic Mechanism of Fournelle and Clark ............... 555 6.3.3 Other Mechanisms for Inducing Grain Boundary Motion before and during Cellular Reaction, Including DIGM .................. 556 6.4 Growth Kinetics of Cells.............................................................................. 561 6.4.1 Introductory Comments................................................................... 561 6.4.2 Turnbull Theory of Cell Growth Kinetics....................................... 562 6.4.3 Cahn Theory of Cell Growth Kinetics ............................................ 564 6.4.4 Comparisons with the Experiment of the Turnbull and Cahn Theories........................................................................... 567
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6.4.5 6.4.6
Hillert Theory of Growth Kinetics .................................................. 570 Volume versus Boundary Diffusion Control and Interactions with Continuous Precipitation ......................................................... 572 References ............................................................................................................... 573 Chapter 7
Pearlite Reaction...................................................................................................... 575 7.1 Systematics ................................................................................................... 575 7.1.1 Definition ......................................................................................... 575 7.1.2 Pearlite in the Context of Eutectoid Decomposition Products........ 575 7.1.3 Occurrence of Pearlite ..................................................................... 575 7.1.3.1 In Steel ............................................................................. 575 7.1.3.2 In Other Alloy Systems ................................................... 575 7.1.4 Preliminary Discussion of Pearlite Nucleation ............................... 576 7.1.5 Morphology of Pearlite ................................................................... 576 7.2 Crystallography, Nucleation, and Growth Mechanisms............................... 582 7.2.1 Hull and Mehl Concept ................................................................... 582 7.2.2 Hillert Concept ................................................................................ 584 7.2.3 Reprise ............................................................................................. 586 7.3 Edgewise Growth Kinetics of Pearlite ......................................................... 588 7.3.1 Comparative Thermodynamics with Cellular Reaction .................. 588 7.3.2 Role of Growth in Pearlite Reaction ............................................... 589 7.3.3 Approximate Treatments of Edgewise Growth............................... 590 7.3.3.1 Volume Diffusion–Controlled Growth ............................ 590 7.3.3.2 Growth Controlled by Diffusion along Colony Boundary.......................................................................... 592 7.3.4 Improved Treatments....................................................................... 593 7.3.4.1 Volume Diffusion–Controlled Growth ............................ 593 7.3.4.2 Boundary Diffusion–Controlled Growth ......................... 594 7.3.5 Experimental Measurements for Growth Kinetics Studies ............. 595 7.3.5.1 Interlamellar Spacing ....................................................... 595 7.3.5.2 Rate of Growth ................................................................ 595 7.3.6 Comparisons of Theory and Experiment and the Problem of S ..... 596 References ............................................................................................................... 599
Chapter 8
Martensitic Transformations.................................................................................... 601 8.1 Definition ...................................................................................................... 601 8.2 Salient Characteristics (Described Briefly)................................................... 601 8.2.1 Crystallography ............................................................................... 601 8.2.2 Morphology ..................................................................................... 601 8.2.3 Surface Relief Effects...................................................................... 602 8.2.4 Time-Dependence of Martensite Formation ................................... 602 8.2.5 Temperature Dependence of Martensite Formation........................ 604 8.2.6 Reversibility..................................................................................... 606 8.2.7 Influence of Applied Stress ............................................................. 606 8.2.8 Thermal Stabilization ...................................................................... 608 8.3 Thermodynamics of Martensite Transformation .......................................... 608 8.3.1 T0 Temperature ................................................................................ 608 8.3.2 Difference between T0 and Ms ........................................................ 612
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8.4
Overall Kinetics of Martensite Transformation............................................ 615 8.4.1 Qualitative Kinetics ......................................................................... 615 8.4.2 Quantitative Kinetics ....................................................................... 615 8.4.2.1 Anisothermal Martensite Formation ................................ 615 8.4.2.2 Isothermal Martensite Formation..................................... 617 8.5 Nucleation of Martensite .............................................................................. 620 8.6 Crystallography and Growth (or Propagation) of Martensite ...................... 622 8.6.1 Physics of Phenomenological Theory of Martensite Crystallography ............................................................................... 622 8.6.2 Comparisons with Experiment ........................................................ 625 References ............................................................................................................... 632 Chapter 9
Bainite Reaction and Role of Shear in Diffusional Phase Transformations........... 635 9.1 Introduction................................................................................................... 635 9.2 Three Definitions of Bainite ......................................................................... 635 9.2.1 Generalized Microstructural Definition ........................................... 635 9.2.2 Kinetic Definition of Bainite ........................................................... 643 9.2.3 Surface Relief Definition of Bainite................................................ 649 9.3 Upper Bainite versus Lower Bainite, and Inverse Bainite........................... 651 9.4 Sources of Carbide Precipitation .................................................................. 651 References ............................................................................................................... 654 Further Readings ..................................................................................................... 655
Index............................................................................................................................................. 657
Preface This book is a collection of the lecture notes of the late Hubert I. Aaronson, professor emeritus at Carnegie Mellon University, Pittsburgh, Pennsylvania, who taught graduate courses on phase transformations at Michigan Technological University, Houghton, from 1972 to 1979 and at Carnegie Mellon University from 1979 to 1992. These lecture notes are devoted to solid–solid phase transformations in which elementary atomic processes are diffusional jumps, and these processes occur in a series of so-called nucleation and growth through interface migration. The main sources are individual papers and review articles. Prof. Aaronson did not merely introduce well-established theories or knowledge in a systematic way, but also described, in great detail, how a new idea or interpretation of a phenomenon has emerged, evolved, and gained its current status among the theories. The original descriptions were retained as much as possible unless they were too colloquial. Prof. Aaronson, simply known as ‘‘Hub’’ to his friends and colleagues, was a man of passion— passion for science, and passion to teach his students and research colleagues. He influenced the field of physical metallurgy through his teaching, research, and organizing a number of excellent conferences. He was particularly well known for his pioneering contributions to the mechanisms of phase transformations in crystalline solids. He was a superb experimentalist, and at the same time he never ceased to question the ‘‘whys and hows.’’ This drove him to passionately follow theories or mathematical models and, if not available, to pursue the truth by himself. Checking experimental findings to theories for explanation became his habit. Such a passion for scientific truth is well reflected in this book. If the published version of a theory or a model was too condensed, Hub worked things out in painstaking detail so that graduate students could follow the derivations. We hope that the readers do not miss the unique ‘‘Aaronsonian idiosyncrasies’’ shining throughout the book. We also believe that the book appropriately mirrors his distinguished professional life of over five decades. Hub passed away at the age of 81 on December 13, 2005. Dr. George Spanos at the U.S. Naval Research Laboratory and Prof. William T. Reynolds, Jr. at Virginia Polytechnic Institute and State University provided valuable contributions to the idea of publication, without which this book would not have been born. We would like to express our special thanks to Prof. Kenneth C. Russell at Massachusetts Institute of Technology and Prof. Mark R. Plichta of Michigan Technological University for kindly reviewing some chapters and providing useful suggestions. Jong K. Lee also wishes to thank Prof. Barry C. Muddle for the warm support he received at Monash University, Melbourne, Victoria, Australia, during the preparation of the manuscript. All the royalties of this book will be donated to the Hub Aaronson Memorial Fund set up at both Carnegie Mellon University and Michigan Technological University. We also wish to express our sincere thanks to Taylor & Francis for the opportunity to publish these lecture notes as a textbook. Masato Enomoto Department of Materials Science and Engineering, Ibaraki University, Hitachi City, Japan Jong K. Lee Department of Materials Science and Engineering, Michigan Technological University, Houghton, Michigan
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Authors Hubert I. Aaronson (Hub) (deceased) received his BS, MS, and PhD in metallurgical engineering at Carnegie Institute of Technology, Pittsburgh, Pennsylvania (now Carnegie Mellon University). He was a worldwide leader in the field of phase transformations of metals and alloys for more than half a century. He published more than 300 technical papers, organized numerous symposia and conferences, served on a number of technical committees, and was recognized with many awards. He was a member of the U.S. National Academy of Engineering, Washington, D.C.; a fellow of both The Minerals, Metals & Materials Society and ASM International; and an honorary member of the Japan Institute of Metals, Sendai, Japan. As R.F. Mehl Professor Emeritus at Carnegie Mellon University, Hub continued his professional activities to the very end until his passing in December 2005. Masato Enomoto received his BS and MS in physics from Tokyo University, and his PhD from Carnegie Mellon University, Pittsburgh, Pennsylvania. He has received many honors and awards in both the United States and Japan for his research on phase transformations in metallic materials. He authored a book, Phase Transformations in Metals, in Japanese, and has served on the editorial boards of several technical journals, including ISIJ International as editor in chief. He was elected a fellow of ASM International. Dr. Enomoto is currently a professor of materials science and engineering, Ibaraki University, Hitachi City, Japan. Jong K. Lee received his BS from Seoul National University, Seoul, South Korea; his MS from the University of Washington, Seattle; and his PhD from Stanford University, California. He taught at Michigan Technological University, Houghton, for over three decades. He is a fellow of ASM International, and a foreign member of both the Korean Academy of Science and Technology and the National Academy of Engineering of Korea. Dr. Lee continues his research activities as a professor emeritus and research professor at the Department of Materials Science and Engineering, Michigan Technological University.
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1 Applied Thermodynamics 1.1 FREE ENERGY–COMPOSITION RELATIONSHIPS FOR BINARY SUBSTITUTIONAL SOLID SOLUTIONS 1.1.1 BASIC FREE ENERGY–COMPOSITION RELATIONSHIP A useful manner of writing the general equation connecting the molar Gibbs free energy, Ga, of an a solid solution with the atom fraction of solute, x, in the A–B system, where A is defined (at least on the A-rich side of the phase diagram) as the solvent and B is taken to be the solute is Ga ¼ (1 x)GaA þ xGaB þ DGM, a
(1:1)
where G is equal to H TS, H is molar enthalpy S is molar entropy T is the absolute temperature GA and GB are the Gibbs free energies (hereafter simply referred to as ‘‘free energies’’) of pure A and pure B in the crystal structure of a, and DGM is the free energy of mixing A and B to form the a solid solution.
1.1.2 GIBBS FREE ENERGY
OF THE
STANDARD STATES OF PURE ELEMENTS
The two types of Gi in Equation 1.1, GA and GB, represent standard states. According to Lupis [1], there is no point to the assignment of absolute values of Gi since they are about 10–12 orders of magnitude greater than the free energy changes attending phase transformations usually encountered. In dealing with phase transformations in terms of free energy–composition curves, the free energy change of interest is that involved in passing from one curve to another—to keep matters initially as simple as possible by restricting consideration to a transformation taking place without a change in composition. Confining our interest here to the situations at pure A (x ¼ 0) and pure B (x ¼ 1) (see Figure 1.1), it is apparent that the free energy changes attending the spontaneous transformation of b to a at pure A and and DGa!b , respectively. When both a and b actually exist as also a to b at pure B are DGb!a B A b!a may be obtained, for instance, directly from Hultgren equilibrium phases, values of DHA and DSb!a A et al. [2]. When such data are either unavailable or suspect, or a given pure element never appears (say, at the usual standard one atmosphere pressure) in a particular crystal structure involved in a phase transformation taking place in one of its alloys, some maneuvering is required to extract the necessary information. An example of the latter situation is the transformation of body-centered cubic (bcc) bCu– Zn to face-centered cubic (fcc) a of the same composition; pure Cu does not appear in bcc form and pure Zn does not occur as either a bcc or an fcc structured phase. This type of manipulation has been done by Kaufman* for decades. He has shown that when the transformation temperature in a pure metal lies above the Debye temperature and there are either no magnetic changes in either phase or the same changes occur in both,y the DH and the DS attending a transformation between any two of the crystal structures fcc, bcc, and hexagonal close-packed (hcp), tends to be about the same for all elements having the same group number in the periodic table of elements. In the plots shown in Figure 1.2, this finding is * This is known as the CALPHAD approach. y Here ‘‘magnetic’’ is rather broadly interpreted to include any non-transformation effect that perturbs the usual form of specific heat vs. temperature curves.
1
2
Mechanisms of Diffusional Phase Transformations in Metals and Alloys G
G αβ
α
ΔGB
β
GA β
β
α
ΔGA
β
GB α
β
GA
α
A (x = 0)
B (x = 1)
x
FIGURE 1.1 Free energy curves of a and b phases.
5
6
7
8
9
10 Group number
8000
4.0
6000
3.0 ΔS β ε (Right scale)
4000
2.0
2000
1.0 0
0 ΔH β ε (Left scale)
–2000
–4000
–1.0
–2.0
Zr
–3.0
–6000 –8000 Hf Zr Hf
= (S ε–S β), entropy diference (J/g–atom.K)
= (H ε–H β), enthalpy diference (J/g–atom)
4
No Ta
Mo W
Tc Re
Ru Os
Rh Ir
Pd Pt
Ag Au
–4.0 Cd
Element
FIGURE 1.2 Enthalpy and entropy differences between the hcp (e) and bcc (b) phases of transition metals of the fifth and sixth periods. (This article was published in Kaufman, L. and Bernstein, H., Computer Calculation of Phase Diagrams, Academic Press, New York, 1970. With permission. Copyright Elsevier.)
3
Applied Thermodynamics 5
6
7
8
10
Group number
ΔS α ε (Right scale)
3000
= (H ε–H α), enthalpy diference (J/g–atom)
9
3.0
2000
2.0
1000
1.0
0
0
–1000
–1.0 ΔH α ε (Left scale)
–2000
–2.0
–3000 Zr Hf
= (S ε–S α), entropy diference (J/g–atom.K)
4
–3.0 No Ta
Mo W
Tc Re
Ru Os
Rh Ir
Pd Pt
Ag Au
Cd
Element
FIGURE 1.3 Enthalpy and entropy differences between the hcp (e) and fcc (a) phases of transition metals of the fifth and sixth periods. (This article was published in Kaufman, L. and Bernstein, H., Computer Calculation of Phase Diagrams, Academic Press, New York, 1970. With permission. Copyright Elsevier.)
demonstrated for refractory transition metals of group numbers 4–10 and also for certain elements in group number 1 (Ag, Au) and group number 2 (Cd) all for the hcp (e) to bcc (b) transition. An equivalent plot for the hcp (e) to fcc (a) transition in the same pure elements is presented in Figure 1.3. The counterpart plots for the bcc to fcc transition are readily deduced from these plots, which taken from Kaufman and Bernstein [3]. Michaels et al. [4] have extended the Kaufman–Bernstein (K–B) compilations, emphasizing the group number 1–5 region. Their results are shown in Figures 1.4 through 1.6 for all three pairs of phase transformation. No explanation has been offered yet for the particular forms exhibited by the original K–B plots or for the additional complexities of shape Michaels et al. [4] found in the group number 1–5 region. Among the latter three plots, some deviations are observed from the K–B rule of same ‘‘stability parameters’’ (as DH and DS are sometimes termed) for a given group number. Those for Li and Na are readily explained, since the bcc-to-hcp transformation (Figure 1.4) takes place below the Debye temperatures for these elements. Kaufman suggested that the large DHfcc!hcp for Al (see Figure 1.6) may be related to the relatively high stacking fault energy of this element, hence the somewhat different type of interatomic bonding exhibited by this element relative to such elements as Cu and Ag. However, a number of other examples of disagreement are found, e.g., for some transformations in Tl, Be, and Mg. Hence, the K–B correlations must be used with some caution.
4
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 4.0
8000
Tb, Nb
3.0
hcp hcp
Kaufman and Bernstein Hultgren et al. Kaufman Wood
4000
2.0
1.0
2000
0
Li, Na; Na
0
TI TI
–2000 Zn Cd
Cu, Ag, Au –4000
Cu, Ag, Au
La Sc
Ba Be, Mg Zn, Cd
–6000
AI Sc; Y Y AI
Be, Mg Ba
La
1
2
3 Group number
–2.0
Pr, Nd Gd Dy Zr, Ti Ho Gd, Dy; Tb Nd, Ho Pr Tb
–3.0 Ta, Nb
Hf, Zr, Ti Hf
–8000
–1.0
hcp = entropy
Li
4
difference, hcp–bcc (J/deg.mol)
ΔH bcc ΔS bcc
ΔS bcc
ΔH bcc
hcp = enthalpy
difference, hcp–bcc (J/mol)
6000
–4.0 5
FIGURE 1.4 Enthalpy and entropy differences of elements between the hcp (e) and bcc (b) phases plotted against the group number of the periodic table. (With kind permission from Springer Science þ Business Media: From Michaels, K.F. et al., Metall. Trans., 6A, 1843, 1975.)
In summary, the available data on the free energy change attending the transformation of pure elements from one crystal structure to another may be utilized to establish the vertical (DG) differences between the endpoints of the G–x curves of Figure 1.1. Let us turn our attention to the curves themselves.
1.1.3 MIXING FREE ENERGY, DGM 1.1.3.1 Fundamentals Formally, the resolution of this problem is perfectly straightforward. A more usual form of Equation 1.1 is Ga ¼
n X
xai G ai
(1:2)
i¼1
where n denotes the number of atomic species present xai is the atom fraction of the ith species G ai represents the partial molar free energy, i.e., the chemical potential of the ith species all in the a phase
5
Applied Thermodynamics Be. Mg
16,000
5.0
AI
4.0 Zr, Hf
ΔH fcc
Pu
2.0
Th, Yb Ca
Ag, Au, Cu
4,000
Ag, Au, Cu 0
TI La Zn, Cd, Be; Mg
Cd Zn Ca Sr; Sr
Hf
Ce h Pu, Yb
1.0
Zr
TI
0
–1.0
–4,000
–2.0
ΔH fcc bcc ΔS fcc bcc Kaufman and Bernstein [3] Hultgren et al. [2]
–8,000
difference, bcc–fcc (J/deg·mol)
8,000
3.0
Ce
bcc = entropy
bcc = enthalpy
difference, bcc–fcc (J/mol)
AI La
Nb. Ta
ΔS fcc
12,000
Nb, Ta –3.0
–12,000 –4.0
–16,000
–5.0 1
2
4
3 Group number
5
FIGURE 1.5 Enthalpy and entropy differences of elements between the fcc (a) and bcc (b) phases plotted against the group number of the periodic table. (With kind permission from Springer Science þ Business Media: From Michaels, K.F. et al., Metall. Trans., 6A, 1843, 1975.)
Replacing G ai by the definitional expression Ga ¼
n X i¼1
xai Gai þ RT ln aai
(1:3)
where Gai is the free energy of pure i in the crystal structure of a R is the gas constant aai is the thermodynamic activity of the ith species in the a crystal structure If aai has been experimentally established as a function of xai at a given temperature, the full curve in the G–x space for the a phase in Figure 1.1 can now be drawn. Furthermore, if the relative partial molar heat of solution of i in a, DH ai , is known as a function of composition through the standard relationship
6
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
3000
3.0
AL (5476)
2.0
2000
TI
Nb, Ta 0 Zr, Hf
TI
–1000
Cu, Ag, Au
–1.0
Zn, Cd Cd Be, Mg Zn
–2000
ΔH fcc hcp ΔS fcc hcp Kaufman and Bernstein Hultgren et al. Jayaraman et al.
–3000 1
2
3 Group number
hcp = entropy
0
Ga Ga La La
ΔS fcc
difference, hcp–fcc (J/mol)
Cu, Ag, Au
ΔH fcc
hcp = enthalpy
1.0
1000
difference, hcp–fcc (J/deg·mol)
AL
Be, Mg
–2.0
Nb, Ta –3.0
Zr, Hf 4
5
FIGURE 1.6 Enthalpy and entropy differences of elements between the fcc (a) and hcp (e) phases plotted against the group number of the periodic table. (With kind permission from Springer Science þ Business Media: From Michaels, K.F. et al., Metall. Trans., 6A, 1843, 1975.)
q ln aai DH ai ¼ 1 R q T
(1:4)
the Ga vs. xi curve (in a binary system, surface in a ternary system) can be computed at any other temperature in the context of Figure 1.1. The same considerations apply to the Gb vs. xi curve. However, at the present time, the activity information for many alloy systems is not good enough. Unless a new, accurate, and very high-speed method of wide generality is developed for the measurement of ai and adequate funding and personnel become available to utilize actively such instruments for many years, it is likely that the ai data available in the literature will be deficient in completeness. Accordingly, a straightforward problem in classical thermodynamics is converted to an exceedingly difficult problem in the solid-state physics of phase stability. Because small differences between large numbers are involved (see Figures 1.2 through 1.6), it seems equally unlikely that the physics approach will yield the requisite information in the foreseeable future. Therefore, it is left for the applied thermodynamicist, be he a physical chemist, metallurgist,
7
Applied Thermodynamics
ceramist, or whatever, to struggle virtually unendingly to develop approximations with which to fill the gap. Insufficient though the results of this struggle may often be, in the context set forth here, it is appropriate to view the results of such efforts with sympathy and appreciation as much as with caution and criticism! 1.1.3.2 Ideal Solution Approximation Noting that DGM ¼ DHM TDSM, where DHM is the enthalpy of mixing and DSM is the entropy of mixing, the ideal solution behavior requires that DH M ¼ 0
(1:5)
DSM ¼ R{x ln x þ (1 x) ln (1 x)}
(1:6)
and
Here DSM is the standard formulation for the ideal entropy of mixing in a binary substitutional solid solution. Not surprisingly, this approximation is really good only for such highly specialized cases as the isomorphous phase diagrams of rare earth elements, which are nearest neighbors in the same period of the periodic table. Shiflet et al. [5] have reported phase diagram calculations for intra-rare earth systems using the Kaufman regular solution approach. In such cases, the regular solution constant becomes small and thus approaches the ideal solution case. The regular or ideal solution calculations of these phase diagrams can be more reliable than many of those experimentally reported because of the difficulties encountered in sufficiently purifying rare earths of their neighbors. Even in the less simple, less ideal cases of intra-refractory transition metal binary alloys, Kaufman and Bernstein found many cases of surprisingly good qualitative or even semiquantitative agreement between phase diagrams calculated on the ideal solution approximation and those experimentally determined. But of course, in general, the ideal solution approximation is of quite limited utility except in some very dilute alloys. 1.1.3.3 Regular Solution Approximation This approximation, originally due to Hildebrand [6,7], assumes the following: DH M ¼ Cx(1 x)
(1:7)
where C is the regular solution constant and DSM ( ¼ Sideal ) ¼ R{x ln x þ (1 x) ln (1 x)}
(1:6)
If the solid solution is indeed regular, a plot of experimentally measured DHM values vs. x(1 x) should be linear with a slope of C. Often, however DHM data are also unavailable. On the other hand, terminal solid solubility data of reasonable accuracy can usually be obtained. Servi and Turnbull [8] have proposed that when a binary phase diagram exhibits only two phases, both dilute terminal solid solution and both having the same crystal structure, DH aB , the relative partial molar heat of solution of B in a can be extracted from an activation plot of the terminal solid solubility of B in a through the relationship q ln xab DH aB a ¼ 1 R q T
(1:8)
8
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
where xab a is the atom fraction of B in a at the a=(a þ b) phase boundary. Lee et al. [9,10] have presented a detailed derivation of Equation 1.8 and made clear the limitation of its applicability to solid solutions in which the activity coefficient is independent of composition. A further note is that the application of the equilibrium conditions to two phases equilibrated with respect to each other results in the disappearance of the standard state terms from both equations, when the crystal structures of these phases, denoted as a and b, are the same. Thus, instead of equilibrating the partial molar free energy of B (say) in the a phase with that in the b phase at the boundaries of the (a þ b) region, we may go directly to the equilibration of activities ba aab B,a ¼ aB,b
(1:9)
Equivalently, ba ba ab gab B,a xa ¼ gB,b xb
(1:10)
where the a’s represent activities the g’s represent activity coefficients at the designated compositions Taking natural logarithms of Equation 1.10, differentiating with respect to 1=T and rearranging, we obtain q ln
xab a
xba DH bB DH aB b ¼ 1 R q T
(1:11)
where the counterpart of Equation 1.4 for the activity coefficient q ln gi DH i ¼ 1 R q T
(1:12)
has been used to replace the differentials of the activity coefficient with respect to 1=T. Following Wagner [11], when the b precipitate is very solute rich, 2 B DH bB ¼ 1 xba b
(1:13)
b as xba b ! 1, where B is the regular solution constant for the b phase, it is apparent that DH B ! 0. Thus, Equation 1.8 is recovered from Equation 1.11. Here, derivations are presented for Equation 1.13 and its counterpart for H A , following again Wagner [11] and filling in some details. We first note that the basic thermodynamic relationships for G A and G B , Equations 1.32 and 1.33, are equally applicable to the relative partial molar free energies, DG i ¼ G i Gi . Likewise, both the partial and the relative partial enthalpies, H i and DH i , are described by equations of exactly the same form. Here we write in terms of DH i :
DH A ¼ DH M x
qDH M qx
DH B ¼ DH M þ (1 x)
qDH M qx
(1:14) (1:15)
9
Applied Thermodynamics
Also, recognize that the relative integral molar enthalpy, or the enthalpy of mixing DH M is given by DH M ¼ Cx(1 x)
(1:16)
H ¼ (1 x)HA þ xHB þ Cx(1 x)
(1:17)
and that the integral molar enthalpy is
carrying out the operations first of Equation 1.14 and then of Equation 1.15 DH A ¼ Cx(1 x) xC(1 2x) ¼ Cx2
(1:18)
DH B ¼ Cx(1 x) þ (1 x)C(1 2x) ¼ C(1 x)2
(1:19)
Equation 1.19 reproduces Equation 1.13. Also, Equation 1.18 is its counterpart for H A . Hence, these relationships are rigorous in the context of the regular solution model. When xab a 1, Equation 1.8 becomes q ln xab C a ¼ 1 R q T
(1:20)
Note that Equations 1.7 and 1.20 both incorporate a mixing enthalpy. Now violating the definitional strictures of regular solution theory, Lee et al. proposed to replace DHM with GXS where the latter quantity is equal to DHM TSXS, where SXS is the excess entropy of mixing, i.e., the extra entropy beyond that given by Equation 1.6, and DHM can vary in any legitimate manner with x(1 x). Hence, they proposed that C¼
GXS x(1 x)
(1:21)
Alternatively, they suggested, again following Wagner, that activity coefficient data being related to the regular solution constant as ln gaB
2 C 1 xab a ¼ RT
(1:22)
where gaB is the activity coefficient of B in a, might be similarly employed. Henry’s law states that gaB approaches a constant as xa approaches zero. Hence, C is best approximated as C ¼ RT ln ga,0 B
(1:23)
XS a,0 a a where ga,0 present, B is the limiting value of gB when x ! 0. Since gB incorporates any G Equation 1.23 should be equivalent to Equation 1.21. Locating a suitable binary system upon which to test these considerations proved difficult. Finally, the Cu–Co system was selected. This choice is a fortunate one for it will provide, in Chapter 2, our only thorough-going test of nucleation theory via homogeneous nucleation. Only the two terminal solid solutions are present and both are fcc. The values of C in Table 1.1 were extracted from the literature with the various equations for determining this constant. Even though the DHM data are not available, it is clear that the equally strict regular solution approach of Equation 1.8, via H B ,
10
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
TABLE 1.1 Regular Solution Constant in the Cu–Co System Determined from Different Methods C, J=mol End of Phase Diagram
Solubility Method (Equation 1.8)
Standard DHM Method (from Equation 1.7)
GXS Method (Equation 1.21)
gB Method (Equation 1.23)
55,500 [8] 54,700 [12]
NA NA
32,900 [2] 33,700 [2]
31,200 [2] 38,100 [2]
Cu-rich Co-rich
Source: Reprinted from Lee, Y.W. et al., Scr. Metall., 15, 723, 1981. With permission from Elsevier.
yields values markedly higher than the excess free energy and the activity coefficient methods. The latter, on the other hand, produces rather similar results, and expect for the Co-rich terminal solid solution region utilized through the activity coefficient method yield adequately similar values of the regular solution constant. We must now decide whether the ‘‘strict construction’’ partial molar heat of solution method, which incorporates only the enthalpic contribution to C, or the Lee approaches, which include both nonparabolic contributions to the mixing enthalpy as well as excess entropy, yields the better result. To do this, we first calculate the terminal solid solubility curves with the two different overall levels of regular solution constant value provided by the foregoing table. To accomplish this, we invoke the result with which we are already familiar that the miscibility gap is symmetrical about x ¼ 0.5 on the regular solution approximation. Even though it may be interrupted by a eutectoid or a peritectoid at a higher temperature, this gap may still be viewed as providing the terminal solvi. To secure an equation for this gap, a result shortly to be derived will be utilized, namely, the partial molar free energy for a species, say A, on the regular solution approximation: 2 G aA ¼ GaA þ RT ln 1 xab þ C xab a a
(1:24)
From the symmetry property of the gap,
ba xab a ¼ 1 xb
(1:25)
ba G ab A,a ¼ G A,b
(1:26)
On the Gibbs condition for equilibrium,
Since the same standard state is used on both sides of the gap, inasmuch as both have the same crystal structure, Equations 1.24 and 1.25 are substituted together into Equation 1.26:
Rearranging, we get
2 2 RT ln 1 xab ¼ RT ln xab þ C xab þ C 1 xab a a a a RT 1 xab a þ 2xab ln a ¼ 1 C xab a
(1:27)
(1:28)
As values of C, we select from Table 1.1 the average value obtained from Equation 1.8, 55,100 J=mol, and also use the average value from Equation 1.21, 33,300 J=mol. The dashed lines in Figure 1.7 are
11
Applied Thermodynamics 1800
Temperature (K)
1600 L 1400
1200
1000 0 Cu
0.2
0.6
0.4 XCo
0.8
1.0 Co
FIGURE 1.7 Terminal solid solubilities in Cu–Co system. Cu-rich solvus. Co-rich solvus. (Reprinted from Lee, Y.W. et al., Scr. Metall., 15, 723, 1981. With permission from Elsevier.) ---- solvi calculated from Equation 1.28 and 33,000 J=mol; ——— solvi calculated from Equation 1.28 and 55,100 J=mol.
clearly in reasonable agreement with the experimental values of the terminal solvi (the Cu-rich solvus was taken from the literature consolidation of Servi and Turnbull, while the recently redetermined Co-rich solvus is due to Old and Haworth [12]). These lines were calculated from Equation 1.21 or the GXS approach, where C ¼ 33,300 J=mol. On the other hand, the dash–dot curves, obtained from Equation 1.8 or the DH B approach, are clearly grossly in error. Hence, taking account of a substantial contribution of SXS and likely also nonparabolic DHM yielded a much improved calculation of the terminal solid solubilities, though by no means exact results. A second approach to discriminating between the two different average values of the regular solution constant is through the calculation of the chemical interfacial free energy of fully coherent interphase boundaries between the Cu-rich and the Co-rich phases, both of which are fcc and differ little in lattice parameter. Loss of full coherency between these phases is likely to be accomplished only with great difficulty. This calculation is made through the regular solution approach to the discrete lattice method employed by Lee and Aaronson [13] and will be discussed in detail in Chapter 2. For present purposes, it is thus practicable only to quote the results: when C ¼ 33,300 J=mol, the boundary energy is ca. 185 mJ=m2 (¼185 ergs=cm2). When C ¼ 55,100 J=mol, the energy becomes 415 mJ=m2. In both cases, the boundary energy is nearly independent of boundary orientation. Now the energy of an average, high-angle, likely much disordered boundary in pure Cu is ca. 600 mJ=m2 [14]. The energy of a disordered grain boundary in even a dilute Cu–Co alloy is lower because of the anticipated segregation of Co to these boundaries. Furthermore, Smith [15] has shown that normally the energy of a disordered interphase boundary in a given alloy is somewhat lower (average 0.7–0.9) than that of a disordered grain boundary. Hence, 415 mJ=m2 is essentially within the energy range of disordered interphase boundaries. On the other hand, 185 mJ=m2 will be seen in Chapter 2 to be comparable to the energy of nucleus–matrix boundaries in this system as determined by (indirect) measurements of homogeneous nucleation kinetics of Co-rich particles in Cu-rich Cu–Co alloys. Hence, the Lee ‘‘deviationism’’ from the classical frame of regular solution theory to incorporate excess entropy and nonparabolic entropy has proved useful at least in this alloy system. It remains to be noted that Kaufman and Bernstein have evolved from the literature and their own considerations an elaborate, essentially nonadjustable parameter system for calculating regular solution constants
12
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
without any data whatsoever on the properties of the solid solution. Only the coordination number of the solid solution, molar volumes, and vaporization enthalpies of the pure elements (plus the application of a plot of vaporization enthalpy vs. group number) are required. Through numerous phase diagram calculations, they have shown that this system works well, particularly for the refractory transition metals in the group number range 4–10. However, Michaels et al. [4] found that this method can yield disastrously erroneous results in the group number 1–5 region as a result of the wrong prediction for the vaporization enthalpy vs. the group number plot. Unpublished efforts to use this system of regular solution constant calculation to predict phase diagrams between elements in group number 9 or 10 and group number 1 or 2 columns have also not been successful. Hence, while the Kaufman–Bernstein system is of substantial utility in situations where no solution thermodynamics data are available, it does have considerable limitations and should be used with appropriate caution. 1.1.3.4 Subregular and Other-Type Solutions These versions of the excess free energy of mixing take the general form ðE0 þ E01 T þ E02 T 2 þ Þ þ xðE1 þ E11 T þ E12 T 2 þ Þ M GXS ¼ x(1 x) þ x2 ðE2 þ E21 T þ E22 T 2 þ Þ
(1:29)
where the various Es are constants, particularly in value to a given alloy system and M ideal GM ). Hardy [16] has defined the subregular solution model by truncating XS ¼ DH T(S S this series as GM XS ¼ x(1 x)[E0 þ E01 T þ x(E1 þ E11 T)]
(1:30)
Lumsden [17] has gone further to produce the following version of the subregular solution model
GM XS
2
3 2 E þ E T E þ E T x(1 x)(E þ E T) 2 21 5 51 21 2 5 þ ¼ x(1 x)4 8 (1 x) þ xr 2 (1 x) þ xr 5 ZRT 1 x þ xr 3
(1:31)
where r is the ratio of the atomic radius of B atoms to the atomic radius of A atoms Z is the coordination number E2, E2l, E5, and E5l are constants to be determined from either the phase diagram or activity data Various investigators have obtained a better agreement with experimentally determined phase diagrams by means of these more elaborate models. For example, the regular solution model can only predict a symmetrical miscibility gap, whereas the subregular models can predict the assymetrical ones normally observed. But in the regular=subregular context, such models begin to become increasingly like exercises in curve fitting. More and detailed experimental data are needed to use the more elaborate models of this type to full advantage. It is, however, the sometime unavailability of such data that is the major incentive for such approximations in the first place. 1.1.3.5
Relationships for Partial Molar Free Energies, Phase Equilibria, and Critical Temperature The standard thermodynamic relationships for partial molar free energies are G aA ¼ Ga x
qGa qx
(1:32)
13
Applied Thermodynamics
G aB ¼ Ga þ (1 x)
a qG qx
(1:33)
In the ideal solution case, partial molar free energies can be derived from Equations 1.1, 1.5, and 1.6, or simply written by inspection as G aA ¼ GaA þ RT ln aaA ¼ GaA þ RT ln (1 xa )
(1:34)
G aB ¼ GaB þ RT ln aaB ¼ GaB þ RT ln xa
(1:35)
In the regular solution case, for the partial molar free energy of A, Equations 1.6 and 1.7 are first substituted into Equation 1.1 Ga ¼ (1 x)GaA þ xGaB þ Cx(1 x) þ RT[x ln x þ (1 x) ln (1 x)]
(1:36)
Carrying out the operations dictated by Equation 1.32 G aA ¼ GaA þ RT ln (1 x) þ Cx2
(1:37)
G aB ¼ GaB þ RT ln x þ C(1 x)2
(1:38)
Similarly,
To compute the equations for binary phase equilibria, let us begin with the regular solution approximation. The equilibrium relationships are ba G ab A,a ¼ G A,b
(1:39)
ba G ab B,a ¼ G B,b
(1:40)
Equations 1.37 and 1.38 are now substituted into these relationships to secure the phase boundaries ba of, say, the a þ b region, whose compositions are xab a at the a=(a þ b) phase boundary and xb at the b=(a þ b) phase boundary. Beginning by substituting Equation 1.37 into Equation 1.39, 2 b ba ab 2 ¼ G þ RT ln 1 x þ A x þ B xba GaA þ RT ln 1 xab a a A b b
(1:41)
where A is the regular solution constant for a B is the regular solution constant for b, instead of using C for any phase Rearranging, we obtain DGa!b A
þ RT ln
1 xba b 1
xab a
2 2 ¼ A xab B xba a b
(1:42)
Substituting Equation 1.38 into Equation 1.40 2 ba ab 2 GaA þ RT ln xab ¼ GbB þ RT ln xba a þ A 1 xa b þ B 1 xb
(1:43)
14
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Rearranging, we obtain þ RT ln DGa!b B
xab a xba b
2 2 ¼ A 1 xab B 1 xba a b
(1:44)
A simultaneous analytic solution for Equations 1.42 and 1.44 is not possible. Trial and error methods of solution often do not work well because of the often remarkable sensitivity of these relationships to guesses as to the correct phase boundary compositions. For more sophisticated approaches to solve them, see Lupis [1]. Under ideal solution conditions, A ¼ B ¼ 0 and Equations 1.42 and 1.44 reduce to RT ln
1 xab a
1 xba b
RT ln
xab a xba b
a!b ¼ DGA
(1:45)
¼ DGa!b B
(1:46)
Simultaneous solution of these equations yields
xba b
! DGa!b A 1 exp RT ! ! ¼ a!b DGa!b DG B A exp exp RT RT
(1:47)
and
xab a
¼
DGa!b A exp RT
!
1
DGa!b DGa!b B A exp RT
!
(1:48) 1
T0 is the temperature at which two phases have the same molar free energy and the same composition, but they are in an unstable equilibrium with each other under stress-free conditions. This concept, due at least in a metallurgical context to Zener [18], is very important in the massive and martensitic transformations and is also useful in the construction of phase diagrams. As a rough estimate, the T0–x curve bisects a two-phase field. Clearly, this estimate is more accurate, the narrower the two-phase region. But more accurate estimates of this curve are readily made. Beginning with the T0 condition, one may write simply, Ga ¼ Gb
(1:49)
Substituting Equation 1.36 with appropriate superscripts, (1 x)GaA þ xGaB þ RT[x ln x þ (1 x) ln (1 x)] þ Ax(1 x)
¼ (1 x)GbA þ xGbB þ RT[x ln x þ (1 x) ln (1 x)] þ Bx(1 x)
(1:50)
15
Applied Thermodynamics
Consolidating terms, we obtain þ xDGa!b þ (B A)x(1 x) ¼ 0 (1 x)DGa!b B A
(1:51)
This equation can be solved for x by means of the quadratic formula. The composition thus obtained is and DGa!b from the that of the preselected T0 temperature employed in the calculation of DGa!b B A stability parameters. In the ideal solution case, again A ¼ B ¼ 0 and þ xDGa!b ¼0 (1 x)DGa!b B A
(1:52)
where x¼
DGa!b A DGa!b DGBa!b A
(1:53)
In respect of miscibility gaps on the regular solution approximation, it suffices for present purposes to note that they will not occur when C 0 as seen in Figure 1.8. The ideal solid solution G–x curve must be concave upward because the {x ln x þ (1 x) ln(1 x)} term is always negative. If the Cx(1 x) term is also negative, the concavity is merely deeper, or more pronounced. However, if C is positive since the TSideal component of DGM must decline in importance with decreasing temperature, at a sufficiently low temperature, an upward bulge in the G–x curve is inevitable, see Figure 1.9. An experimental manifestation of the resulting division of the solid solution into two phases of compositions, xl and x2 (with the same crystal structure), requires, of course, that (Dt)1=2 (where D is the appropriate diffusivity and t is the diffusion time) be sufficiently large. Repeating the above construction at different temperatures yields the variation of xl and x2 with temperature, thereby permitting the construction of the miscibility gap portion of the diagram, as shown in Figure 1.10. The highest temperature of the gap is termed the critical temperature, TC. At this temperature, it may be shown that the second and third derivatives of G with respect to composition are zero. Differentiating Equation 1.36 three times and making use of these relationships demonstrates that the critical temperature occurs at x ¼ 1=2 and that TC ¼
C 2R
(1:54)
Hence, TC > 0 only when C > 0.
G Ideal solution
C≤0
x
FIGURE 1.8 Free-energy curve of an ideal solution with C < 0.
16
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
G
A
B
x2
x1
FIGURE 1.9 Free-energy curve of an ideal solution with C > 0.
TC T
A
FIGURE 1.10
x
B
Miscibility gap constructed from the free energy curve in Figure 1.9.
1.1.3.6 Comparison between Regular Solutions and Nonregular Solutions These brief, qualitative considerations are based upon reviews by Kapoor [19] and Ansara [20] as well as upon Chapter 15 of Lupis [1]. The regular solution model assumes that every atom has the same percentage of nearest neighbors of each species. Hence, a random distribution of atoms on the available substitutional lattice sites, with neither clustering nor ordering, is postulated. The total enthalpy of a solid solution is assumed to be the sum of all bond energies. All bonds are taken to have a fixed energy, independent of the chemical identity and distribution of the nearest neighboring atoms. The internal partition function of atoms is also taken to remain unchanged upon mixing; more specifically, the rotation and vibrational characteristics of atoms about their mean positions remain unchanged when they enter a solid solution. In the quasichemical model, although every atom again has Z nearest neighbors, the number of nearest neighbors of a given species is not necessarily the average of the solid solution. A little more specifically, the quasichemical model assumes that, aside from the bulk proportion of each species
17
Applied Thermodynamics
present, the probability of site occupation is proportional to exp(Ui=kT) where Ui, the energy of an atom at this site, is taken to be equal to the sum of the energies, or better the bond enthalpies with respect to the atoms nearest neighboring to this site. Pairwise interaction is assumed which implies that the bond enthalpy of two atoms is independent of the atoms surrounding them. Also, other types of energy contribution, e.g., vibrational contribution, are neglected. The deviation from random distribution or the maximum disorder is the source of excess entropy, which is always negative. The main constant in quasichemical theory is the quantity K ¼ exp{(2HAB HAA HBB)=kT}. K must be evaluated on the basis of comparisons of experimental thermodynamic data with curves calculated from various K values, hence restricting the utility of the quasichemical theory. Nonetheless, the mathematics are not terribly complex and this approach has been shown to be consistent with the experimentally measured excess free energies of mixing in the binary systems among the elements Sn, Bi, Sb, Ag, Cu, Au, and Pb and also among binaries from elements including Cr, Mn, Co, Fe, and Ni. The central atoms or surrounded atom model, independently developed by Lupis and Elliot [21] and Mathieu et al. [22], improves considerably upon the quasichemical model albeit at considerable cost in complexity. Here, the basic unit is the atom rather than the bond between two atoms. This approach permits the use of any suitable function for calculating the energy of a nearest neighboring atom and also permits taking into account the changes in the vibrational degrees of freedom upon mixing. If the calculation of the energy of a nearest neighboring atom is made on the assumption of a linear relationship between the potential energy of an atom and the composition of the nearest neighbors, the regular solution model is recovered if the Bragg– Williams statistical thermodynamic treatment is employed and the quasichemical one is obtained when Guggenheim statistics are used. The use of a parabolic relationship between potential energy and composition of nearest neighbor may be more realistic but can lead to computational difficulties. The central atoms model is useful for explaining the thermodynamic behavior of asymmetric systems, as well as that of systems containing both an interstitial and a substitutional solute when the interstitial is present in small concentrations, and of the different interaction energy between adjacent interstitial carbon atoms in different hosts, i.e., substitutional lattices. However, the mathematics can be very complex and some fundamental issues—which are getting close to the ‘‘core’’ solid-state physics problems—remain to be resolved. The central atom model is receiving increasing attention, but neither easily nor rapidly because of inherent difficulties involved in this more sophisticated and realistic approach to the intrinsically very difficult problem of the free energy of the mixing of solid solutions.
1.2 FREE ENERGY–COMPOSITION DIAGRAM AND APPLICATIONS TO DRIVING FORCE CALCULATIONS The principal references to this subject are the articles by Hillert [23,24], Cahn [25], and Baker and Cahn [26].
1.2.1 SOME CONSIDERATIONS
ON THE
FREE ENERGY VS. COMPOSITION CURVE
These considerations will be restricted to situations in which the G–x curves are concave upward. In the context of a regular solution model, C is thus negative, or if positive then the temperature at which the G–x curve is constructed must be greater than TC. In more general terms, G–x curves will be employed for phases that are ‘‘internally stable,’’ i.e., for which separation into two phases would increase the total free energy of the system. In Figure 1.11, the b phase of composition xb is supposed to have split into two phases of composition xl and x2 (but with the same crystal structure—otherwise they would not lie on this G–x curve). Connecting their free energies with a straight line then permits an increase in free energy, DG, of the alloy as a whole to be obtained as the
18
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
d2G > 0 dx2
β
x
G
ΔG
A
x2
xβ
x1
B
Initial composition
FIGURE 1.11
Change in free energy when the free energy curve of the b phase is concave upward.
difference between G on the Gb vs. x curve and that on the straight line at composition xb. A more compact way of writing the same generalization is simply that d2Gb=dx2 > 0. Note that when this inequality is reversed, decomposition into xl and x2 will occur spontaneously, see Figure 1.12. In the absence of a composition dependence of lattice parameter, which permits the influence of elastic strain energy to be ignored, the foregoing situation is that of spinodal decomposition (Figure 1.13). This takes place when the compositions lie between the two at which d2G=dx2 ¼ 0. The G–x curves of interest here, i.e., those of the type shown in Figure 1.11, are sometimes approximated as parabolas in their lower G region. Unless a specific model for the solid solution is employed, no such specification other than the restrictions noted above, is employed here. The regular solution model is now used to demonstrate that G–x curves must be tangent to the G-axis at x ¼ 0 and x ¼ 1. Taking the first derivative of Equation 1.36 with respect to x, qGa x ¼ G aA þ G aB þ RT ln þ C(1 2x) qx 1x
(1:55)
qGa ¼ 1 qx
(1:56)
At x ¼ 0
d 2G < 0 dx2
G
ΔG
β
x1
FIGURE 1.12
xβ
x2
Change in free energy when the free energy curve of the b phase is convex upward.
19
Applied Thermodynamics
d2G = 0 dx2
G
A
xcs
B
xcs
(cs = chemical spinodal)
FIGURE 1.13
Spinodal composition of a binary alloy.
At x ¼ 1 qGa ¼ þ1 qx
(1:57)
Finally, it is demonstrated through a variational method that the lowest common tangent between the G–x curves for two phases, say b and a, represents an equilibrium between these phases in a two-phase region at a given temperature and thereby defines the compositions of these phases, as shown in Figure 1.14. Here we have two G–x curves; they represent phases differing discretely in crystal structure and=or in degree of long-range order. First, let the alloy be divided into b phase of composition xb1 and a phase of composition xa1 . The free energy of this mixture of phases is given by the ‘‘height’’ at which the vertical line erected at the average composition intercepts the straight
β
α G av 1
xβ1 xβ
xβ
G
xα
av
G2
2
Gav 3
3
1
xα
2
βα xβ
xα3
av Gequil
αβ
xα
A
xave
B
FIGURE 1.14 Common tangent construction for determination of equilibrium composition of a and b phases.
20
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
line connecting the G–x points denoting xb1 and xa1 . This average free energy, Gav 1 , is thus the ruleof-mixtures average between G at xb1 , say Gb1 , and G at xa1 , say Ga1 . av av Now obtain similarly Gav 2 for b of composition xb2 and a of composition xa2 . Since G2 < G1 , the compositions of the two phases comprising the mixture are clearly being displaced in directions likely to lead toward an equilibrium. Continuing the process, it is observed that the lowest value of ab av Gav is reached when the compositions of the two phases become xba b and xa , yielding Gequil . By inspection, the straight line connecting the b and a phases of these compositions is the lowest common tangent to the Gb–x and Ga–x curves. The same result can be achieved through the alternate route provided by partial molar free energies. To introduce this route, the point is first made (as a matter of review from undergraduate thermodynamics courses) that the partial molar free energies of A and B, say of the b phase, are given by the intercepts of the tangent to the Gb–x curve with the G-axis at x ¼ 0 (pure A) and x ¼ 1 (pure B), see Figure 1.15, where G bA and G bB represent the partial molar free energies of A and B, respectively, at composition xb. To prove that partial molar free energies can indeed be secured in this manner, one may proceed as follows. At the tangent point a in the G–x curve of the diagram below, the value of G is equal to the length of the line bc. The slopeof the tangent line is equal to qG cd=ac ¼ cd=(1 x). From Equation 1.33, i.e., G B ¼ G þ (1 xb ) , making these substituqx tions yields G bB ¼ bc þ (1 xb )
cd ¼ bd 1x
(1:58)
Now consider again G–x curves for the b and a phases (Figure 1.16). To avoid overcomplicating the diagram, introduce only a pair of nonequilibrium compositions, xb1 and xa1 . Construct tangents to the G–x curves at these compositions to obtain their partial molar free energies and do the same at ab the equilibrium compositions, xba b and xa . First consider only the partial molar free energies of A and B at the nonequilibrium compositions. Note that the partial molar free energy of A at xb1 is greater than that of A in xa1 . Hence, A will diffuse from nonequilibrium b to nonequilibrium a. Conversely, the partial molar free energy of B in xa1 is greater than that in xb1, resulting in B diffusion from a to b. Thus, the nonequilibrium b is enriched in solute while the nonequilibrium a is simultaneously depleted in solute (taking B as solute across the phase diagram). When this
β –β d GB G
a
c
–β GA
A
FIGURE 1.15
xβ
B
b
Construction of partial molar free energies for a component species in a binary alloy.
21
Applied Thermodynamics – GA(xβ1)
β
– αβ – βα GAα = GAβ
α
– GB (xα1) – ΔGAβ1
– ΔGBα1
α1
β1
– αβ – βα GBα = GBβ – GA (xα1)
– GB (xβ1) A
FIGURE 1.16
xβ1
βα
xβ
αβ
xα
xα1
B
Driving force for diffusion of component species in a binary alloy.
ab process has driven xb1 to xba b and xa1 to xa , the partial molar free energy of A will be the same in both phases and so will the partial molar free energy of B as shown in the above diagram, thereby fulfilling the equilibrium conditions ba G ab A,a ¼ G A,b
(1:26)
ba G ab B,a ¼ G B,b
(1:40)
and
and again establishing the lowest common tangent as the determinant of the compositions of twophase equilibrium in a binary alloy.
1.2.2 TOTAL FREE ENERGY CHANGE ATTENDING PRECIPITATION In this and the following sections through 1.2.7, b will be designated as the matrix phase and a will be designated as the precipitate phase. Following solution annealing in the b phase region, an alloy of bulk composition xb is quenched into the a þ b region. The reaction b ! a þ b1, where b1 is now definitely, albeit informally designated as b of equilibrium composition xba b at temperature T. The relevant phase diagram configuration is shown in Figure 1.17. Drawn not to scale, compositionwise, the corresponding G–x diagram is illustrated in Figure 1.18. As implied in previous discussions, the total free energy change attending the completed b ! a þ b1 reaction is the vertical distance between the point of tangency to the Gbx curve at xb and the common tangent drawn to both Gb and Ga curves. A method of calculating this DG is now presented in detail. On the standard thermodynamic P relationship G ¼ i xi G i , the free energy of the b phase (prior to transformation) at xb is Gb ¼ (1 xb )G bA þ xb G bB
(1:59)
22
Temperature
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
α
β
T xβ A xβα β
αβ xα B
B mole fraction
FIGURE 1.17
Phase diagram of a eutectic binary alloy. –β GB G β
– αβ – βα GAα = GA
α
β
β
α + β1
ΔGTotal
–β GA
– βα – αβ GB = GBα β
A
FIGURE 1.18
βα
xβ
αβ
xβ
xα
B
Free energy diagram at temperature T of Figure 1.17.
Equation 1.59 can also be obtained graphically. Figure 1.18 has been redrawn and the similar triangles cde and hje are demarcated in Figure 1.19. From the law of similar triangles, cd ed
¼
hj ej
(1:60)
Substituting the quantities equal to the length of the various sides of these triangles utilized, Gb G bA G bB G bA ¼ xb 1
(1:61)
23
Applied Thermodynamics β
h
– αβ – βα GAα = GA
–β GB
β
c –β GA e
j
d α
G
– βα – αβ GB = GBα β
A
FIGURE 1.19
xβ
B
Total free energy change attending precipitation.
Rearranging, Equation 1.59 is readily obtained: Gb ¼ G bA þ xb G bB G bA ¼ (1 xb )G bA þ xb G bB
(1:62)
The free energy of the equilibrium mixture of a and b1 can be annoying to calculate if the path is followed utilizing the proportion of each phase formed. By ignoring such minor matters as differences in molar volume of the two phases and any dependence thereof upon composition, the relationships of equilibrium, Equations 1.39 and 1.40, are invoked to write Gaþb1 in terms of one phase, b1. ba Gaþb1 ¼ (1 xb )G ba A,b þ xb G B,b
(1:63)
The reason for this choice will soon become apparent. A construction similar to that shown in Figure 1.19 can be used to derive this relationship graphically. Subtracting Equation 1.59 from Equation 1.60 and recalling that G i ¼ Gi þ RT ln ai , where Gi is the free energy of pure i in the given crystal structure and ai is the thermodynamic activity of the ith species in an A–B solid solution with the same crystal structure, b ba b G G G DGb!aþb1 ¼ (1 xb ) G ba þ x b B A,b A B,b " # aba aba A,b B,b ¼ RT (1 xb ) ln b þ xb ln b aA aB where ba aba ib is the activity of the ith species in b at the b=(a þ b) phase boundary (i.e., at xb ) abi is the activity of the ith species at xb
(1:64)
24
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Note that as a direct consequence of writing partial molar free energies in terms of the b phase for the product phases as reactant phase, the standard state terms cancelled out during the sequence of operations described in Equation 1.64. Had Equation 1.63 been written in terms of the a phase as ab Gaþb1 ¼ (1 xb )G ab A,a þ xb G B,a
(1:65)
then Equation 1.64 would have developed instead as b ab b DGb!aþb1 ¼ (1 xb ) G ab A,a G A þ xb G B,a G B " # " # aab aab A,a B,a b!a b!a ¼ (1 xb ) DGA þ RT ln b þ xb DGB þ RT ln b aA aB
(1:66)
Thus, knowledge of both DGb!a and DGb!a would be required in addition to activity data, and the B A latter would be required with reference to two different standard states. Hence, the possibilities for error are appreciably increased while the amount of effort needed to complete the calculation is also much greater when different reference phases are used for the matrix and the products. Returning now to Equation 1.64, this free energy change is accurately evaluated when activity data are available. When they are not, as is all too often the case, we must resort to the various solution models considered for DGM. For example, on the assumption that the b phase is an ideal solution, Equation 1.64 becomes
DGb!aþb1
2
3 xba b 5 þ xb ln ¼ RT 4(1 xb ) ln (1 xb ) xb 1 xba b
(1:67)
If regular solution behavior of the b phase is assumed, Equations 1.37 and 1.38 for the regular solution partial molar free energies are substituted into the first line of Equation 1.64 to yield
DGb!aþb1
2
3 2 xba b 5 þ xb ln þ B xba ¼ RT 4(1 xb ) ln b xb xb (1 xb ) 1 xba b
(1:68)
Note that when B ¼ 0, Equation 1.67 is recovered.
1.2.3 FREE ENERGY CHANGE ATTENDING
THE
PRECIPITATION
OF A
SMALL AMOUNT
OF
a
This situation obtains during nucleation (though some additional complexities will be described later) and can also be regarded as the free energy change driving growth, since in effect only a small amount of a is in the process of undergoing diffusional growth at a given position along a moving interphase boundary. An appropriate G–x diagram for this case is illustrated in Figure 1.20. Whereas free energy change is calculated at the average or bulk composition of the alloy, xb, when the precipitation process goes to completion, as in Section 1.2.2, in the present instance concern is with the earliest stage of precipitation; hence, the calculation is performed at (the equivalent of the immediate vicinity of) xab a . Since the calculation does not differ otherwise from that performed in
25
Applied Thermodynamics β α
G
–β GB β
αβ
G x α –βα – αβ GA,β = GA,α
ΔG
–β GA
– αβ – βα GB,α = GB,β
A
FIGURE 1.20
βα
xβ
xβ
αβ
xα
B
Free energy change attending precipitation of a very small amount.
connection with the G–x diagram of Figure 1.18, we can move immediately to the stage of Equation 1.64, but now writing it as ba b G A,b G bA þ xab G ba DG ¼ 1 xab a a B,b G B " # ba ba a a A,b B,b (1:69) ¼ RT 1 xab ln b þ xab a ln b a aA aB The replacement of the activities with expressions derived from a particular solution model, if required, proceeds as in the preceding example.
1.2.4 DIVISION
OF THE
TOTAL FREE ENERGY CHANGE BETWEEN CAPILLARITY
AND
DIFFUSION
In the example in Section 1.2.2, the entire free energy change was available to drive the interdiffusion of B and A needed to produce solute-rich a and solute-impoverished b1. However, when the a precipitates are very small (say, a micron or less) in one, two, or three dimensions, a proportion of the free energy change is stored or withheld as interfacial energy, thereby reducing the driving force available for diffusion. To begin the calculation of the division of the free energy change thereby engendered on a G–x diagram, the convention is adopted that all of the alteration in free energy change resulting from the interfacial energy effect, known as capillarity, is effected in the precipitate phase. Since concern here is only with differences in free energy, and the ‘‘bill’’ for the capillarity effect must be ‘‘paid’’ in terms of free energy in any circumstances, it is clearly pointless to worry about partitioning of the capillarity effect between the matrix and precipitate phases. Thus, a diagram of the type shown in Figure 1.18 is modified by the incorporation of the dashed G–x curve for the a phase; the height (in terms of G) of this curve relative to the solid (capillarity-free) Ga–x curve represents the increase in free energy, as a function of composition, due to capillarity. The appropriately modified diagram is shown in Figure 1.21. Once again making calculations at xb, the difference between the height at which the vertical at this composition is intercepted by the line tangent to the Gb and the capillarity-corrected curve for the a phase, Gar , and the height at which the tangent to the Gb and the Ga curve cuts this vertical is clearly the free energy change (total) stored as capillarity, DGCapillarity. Hence, the remainder of the total free energy change is that
26
Mechanisms of Diffusional Phase Transformations in Metals and Alloys –β GB
β
– αβ – βα GAα = GAβ ΔGDifusion
– αβ – βα GAα,r = GAβ,r
αr ΔGCapillarity
–β GA
α
A
βα xβ
– βα – αβ GAβ,r = GB α,r
αβ xα
xβ βα
B
αβ x α,r
– βα – αβ GBβ = GBα
xβ,r
FIGURE 1.21
Division of total free energy change between capillarity and diffusion.
available to drive diffusion, DGDiffusion. The r-subscripts on the partial molar free energies denote the effect of capillarity (through radius r) upon them. Note that, when the a phase is richer in solute ba than the b phase, the above diagram shows it to be geometrically necessary that xba b,r > xb . The reverse is of course true when the b phase is solute-rich. Beginning with the diffusion component of the free energy change, b!ar þbr b ba b ¼ (1 xb ) G ba DGDiffusion (1:70) Ab,r G A þ xb G Bb,r G B
where both ar and br represent the metastable equilibrium phases whose compositions are altered by capillarity. Now writing the capillarity component, ar þbr !aþb1 ba ba ba ¼ (1 xb ) G ba DGCapillarity (1:71) A,b G Ab,r þ xb G B,b G Bb,r
As it should, the sum of Equations 1.70 and 1.71 yields the first line of Equation 1.64. To make clear the influence of capillarity upon xba b , and to do so with the minimum level of complexity, the partial molar free energies in Equation 1.71 are now replaced with their ideal solution equivalents in the manner of Equations 1.34 and 1.35 ar þbr !aþb1 DGCapillarity
"
¼ RT (1 xb ) ln
1 xba b
1 xba b,r
þ xb ln
xba b xba b,r
#
(1:72)
ba where xba b,r is the capillarity-altered version of xb . As shown in the G–x diagram of Figure 1.21, ba graphically determining xb,r as a function of temperature will yield the capillarity altered b=(a þ b) solvus, termed br=(ar þ br) displaced to higher solute contents, see Figure 1.22. The calculation of xba b,r is investigated in Section 1.2.5.
27
Applied Thermodynamics β/(α + β)
βr /(αr + βr)
T
A
FIGURE 1.22
βα
xβ
βα
x
xβ,r
Shift of b=(a þ b) solvus due to capillarity.
1.2.5 INFLUENCE OF CAPILLARITY
UPON
SOLUBILITY
This treatment is due to Hillert [23]. The special case of G–x curves that are so narrow that capillarity displaces them vertically upward without a change in shape is utilized here. Nonetheless, ba the result obtained is adequately accurate when xba b,r xb is small. The increase in free energy due to capillarity is equivalent to an increase in pressure. The pressure rise, DP, is related to the interfacial energy of the matrix:precipitate or interphase boundary by 1 1 (1:73) þ DP ¼ Pa Pb ¼ g r1 r2 where Pa and Pb are the pressures associated with the individual phases g is the specific interfacial free energy (customarily abbreviated as just interfacial energy) rl and r2 are the principal radii of curvature These radii are those formed at any selected point on even a complexly curved surface by the intersection of two orthogonal planes passed through a line erected normally to the point with the surface. At the edge of a plate, r2 ¼ 1; in the case of a sphere, r1 ¼ r2 ¼ r. In the present derivation, it will be convenient to use the sphere as an example of a finely curved surface. In the presence of pressure, the free energy of a phase is increased by the amount PV, where V is the molar volume of the phase. The upper sketch in Figure 1.23 shows that even when the G–x curves are displaced vertically without a change in shape by capillarity pressure, the curves are displaced, respectively, by PbVb and PaVa, which are normally unequal. These unequal displacements are seen to change the relationships among the partial molar free energies. The bottom sketch of Figure 1.23 shows selected features copied from the upper one, redrawn for the sake of clarity, with composition differences now specifically indicated and the tangent line between the G–x curves drawn in the absence of pressure displaced upward in parallel to its original location. It is now used to define the ratio of the base to the altitude of the triangle whose apexes are denoted by hollow circles and to do the same for the base and altitude of the triangle defined by the filled circles. These triangles are similar, hence their ratios are equal: DG B,r Pa V a 1 xab a
¼
DG B,r Pb V b 1 xba b
(1:74)
28
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
β,r
– αβ – βα GAα,r{P α, P β} = G A β,r{P α,P β} – ΔGA,r
– βα – αβ GB {P α,P β} = GB {P α,P β}
Gα{P α}
G β{P β}
G
α,r
P αV α P βV β
Gβ{P β = 0}
Gα{P α = 0}
– αβ – βα GAα{ P α = P β = 0 } = GAβ{ P α = P β= 0 }
– ΔGB,r
– βα – αβ G Bβ{P α = P β= 0} = G Bα{P α = P β= 0 } – ΔGB,r – P αV α – ΔGB,r – P βV β
αβ
1 – xα 1–x
βα β
G
βα β
A
FIGURE 1.23
βα
αβ
x ≈ xβ,r
αβ
xα ≈ xα ,r
B
Influence of capillarity upon solubility.
Rearranging to solve for DG B,r ,
DG B,r ¼
b b 1 xba Pa V a 1 xab a P V b ba xab a xb
(1:75)
Since interest here is only in DP, as already noted, all pressure will be assigned to the precipitate. Therefore, take Pb ¼ 0 and DP ¼ Pa ¼ 2g=r, for the sphere being used as an example morphology. Making these substitutions in Equation 1.75 and replacing DG B,r with a difference indicated in the right-hand side of the upper sketch,
DG B,r
2gV a 1 xba b ¼ ba r xab a xb
(1:76)
Replacing the partial molar free energies with the relationship connecting them with activity, noting that the standard state terms thus introduced cancel, and then replacing activities with atom fractions because ideal solution behavior can be assumed with some degree of safety under the restriction that the composition change due to pressure is small
RT ln
xba b,r xba b
2gV a 1 xba b ¼ ba r xab a xb
(1:77)
29
Applied Thermodynamics
Rearranging, we get
xba b,r Since ey ¼ 1 þ
3 2gV a 1 xba b 4 5 ¼ xba b exp ab RTr xa xba b 2
y y2 þ þ 1 þ y when y 1, 1! 2! 3 2 2gV a 1 xba b ba 4 5 xba 1þ b,r ¼ xb ab RTr xa xba b
(1:78)
(1:79)
This result is appropriate when a has a limited existence range and when both b and a can be considered dilute solutions. It should be noted that the form in which Equation 1.79 is often used, b
xba b,r
¼
xba b
2gV B 1þ RTr
!
(1:80)
b
where V B is the partial molar volume of solute in the b matrix, is not correct for solid–solid transformations.
1.2.6 DIVISION
OF
DG BETWEEN DIFFUSION
AND
UNIFORM INTERFACIAL REACTION
Many investigators are of the opinion that precipitate growth with kinetics less rapid than those allowed by volume diffusion-control can be explained by a slow reaction, e.g., difficult diffusional jumps, occurring continuously and uniformly over the area of the interphase boundary. It was felt, however, by Aaronson that this type of growth barrier is rare, at least in phase transformations investigated in detail so far, but that more proved examples of it could be discovered in the future. Let a uniform interfacial reaction barrier yield an interface composition in the b matrix of xIb instead of xba b . Thus, the free energy change is partitioned between that for interfacial reaction and that for diffusion. In Figure 1.24, this is shown on a G–x diagram for both the total driving force and for the driving force for a very small amount of a precipitation, i.e., the driving force for growth (or for nucleation). Writing the free energy change only for the total driving force case, the two components of this change are Total,b!aI þbI
DGDiffusion
¼ (1 xb ) G IA,b G bA þ xb G IB,b G bB
(1:81)
and Total,aI þbI !aþb1 ba ba I I DGInterfacial reaction ¼ (1 xb ) G A,b G A,b þ xb G B,b G B,b
(1:82)
where aI and bI represent the (uniform) compositions of the a and b phases after transformation has progressed as far as it can under the stricture of uniform interfacial reaction G Ii,b is the partial molar free energy of A or B in the b phase at the aI:bI boundary during growth or after growth has ceased and uniform composition has been achieved throughout this nonstable phase
30
Mechanisms of Diffusional Phase Transformations in Metals and Alloys β
α
– βα – αβ GA,β = GA,α
–I GA,β
–β GB
Growth
ΔGDiffusion Total
ΔGDiffusion G Total
ΔG Interfacial
–I GB,β
reaction
Growth
ΔG Interfacial reaction
– βα – αβ GB,β= GB,α
–β GA βα
A
FIGURE 1.24
xβ
I
xβ
αβ
xβ
xα
B
Division of the total free energy change between diffusion and interfacial reaction.
αβ
xαI = xα
α
β
xβ I
xβ βα
xβ
FIGURE 1.25
Solute concentration profile during the growth of a.
A concentration–penetration curve through an aI: bI boundary is shown schematically by the solid curve in Figure 1.25; the dashed curve is its counterpart for a and b, where a distinction between a and aI has not been made. Equations 1.81 and 1.82 can be converted to their counterparts for the precipitation of a small amount of a by simply replacing xb with xab a .
1.2.7 PERMISSIBLE RANGE OF NONEQUILIBRIUM PRECIPITATE COMPOSITIONS To form a small amount of a precipitate it is only necessary that, in the style shown in Figure 1.20, the Ga–x curve lie below the tangent drawn to the Gb–x curve at xb. As illustrated in Figure 1.26, this requirement is fulfilled by a phase compositions lying between xa,min and xa,max.
31
Applied Thermodynamics
β α
G
A
FIGURE 1.26
xβ
xα,min
xα,max B
Possible range of precipitate composition.
1.3 THERMODYNAMICS OF INTERSTITIAL SOLID SOLUTIONS THROUGH APPLICATION TO THE PROEUTECTOID FERRITE REACTION IN FE–C ALLOYS 1.3.1 INTRODUCTION The proeutectoid ferrite reaction involves the transformation of austenite, an fcc solid solution in which up to ca. 9 at.% of carbon can be interstitially dissolved, to a mixture of ferrite, a bcc solid solution that can dissolve no more than ca. 0.1 at.% C, and carbon-enriched austenite. This reaction is the initial one occurring during the decomposition of austenite in hypoeutectoid steel during continuous cooling (unless the austenite contains appreciable amounts of one or more carbide-forming alloying elements) and is thus, by far, the world’s most important industrial phase transformation. Many measurements have been reported on the activity of carbon in austenite as a function of carbon content and of temperature, and far more theoretical and computational treatments have been made of these data, usually couched in terms of statistical thermodynamics. The minute solubility of carbon in ferrite has restricted experimental attention to carbon activity in this phase to a lower but by no means negligible level. While the treatment to be developed is generally applicable to cubic-type interstitial solid solutions in which consideration is restricted to interactions between nearest neighboring interstitial atoms and in which these interactions are repulsive, once the necessary expressions have been developed for the activity of the substitutional solvent and the interstitial solute, they will be particularized and applied to Fe–C alloys. This will give you a sense of the numerical values involved. Until a change of source is indicated, the treatment presented is that of Kaufman et al. [27]. This paper, nicknamed KRC, builds in an exceptionally lucid fashion upon many earlier attempts to solve the central statistical thermodynamic problem, of which that by Speiser and Spretnak [28] provided the first correct solution. The approach utilized restricts exact applicability to dilute solutions, defined as those in which there is negligible overlap between the coordination shells containing the nearest neighboring interstitial sites of adjacent interstitial atoms. The result of treatments that do take account of such overlap will be quoted later; the escalation of complexity that they require, however, effectively prohibits their utilization for the purpose of pedagogy. Another deficiency of this and virtually all other treatments of interstitial solid solutions is that it focused detailed attention solely upon the positional entropic contribution to the free energy of these
32
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
solutions. Vibrational, electronic, magnetic, and other contributions to excess entropy, which are far more difficult to treat and upon which research is consequently much less advanced, are handled with utmost crudity, as are all aspects of the excess enthalpy of solution. The reason why so much attention has been lavished upon the positional entropy of interstitial solid solutions, other than the conceptual definiteness of this contribution, is that it is obviously of great importance. Unlike binary substitutional solid solutions, (a) the positions of substitutional and interstitial atoms are either largely or entirely noninterchangeable and (b) whereas 99.99% or more substitutional sites are occupied in disordered substitutional solid solutions, only a small proportion of the available interstitial sites are normally occupied. Even in aTi–N, which can dissolve ca. 20 at.% N, only ca. 40% of the interstitial sites are filled.
1.3.2 FREE ENERGY
AND
POSITIONAL ENTROPY OF IDEAL INTERSTITIAL SOLID SOLUTIONS
For initial purposes, the substitutional solvent atoms will be designated as A, the interstitial solute atoms will be designated as C, and the solid solution will be termed b, without specific reference to detailed crystal structure. The free energy of this solid solution may be written as nþ1 nþ1 b b x GA þ xGbACn þ GbE TSbp (1:83) G ¼ 1 n n where GbA is the free energy of pure A in the crystal structure of b n is the number of interstitial sites per substitutional site (for bcc, n ¼ 3; for fcc, n ¼ 1) x is the atom fraction of C atoms When all interstitial sites are occupied, x ¼ n=(n þ 1) and this corresponds to ACbn . Whether or not ACbn actually exists is not important in the present context. When x ¼ n=(n þ 1), the mole fraction of ACbn ¼ 1. When x is less than n=(n þ 1), the mole fraction of ACbn is (n þ 1)x=n, and the mole fraction of A is thus 1 (n þ 1) x=n. GbACn is the free energy of pure ACbn in the crystal structure of b, GbE is all contributions to the free energy of mixing other than the positional entropy, and Sbp is the positional entropy of interstitial atoms. When only one species of substitutional atom is present, as in the present situation, and the only vacancies present are due to thermal fluctuations, on Equation 1.6, the positional entropy of these atoms is essentially zero—assuming, of course, that substitutional and interstitial atoms cannot occupy each other’s sites. Qualitatively, positional entropy refers to uncertainty in the location of a certain number of interstitial atoms among a larger number of interstitial sites (in the present context). Quantitatively, the general relationship for Sp is provided by the Boltzmann equation Sp ¼ k ln W
(1:84)
where k is Boltzmann’s constant W is the total number of distinguishable ways in which the system may be arranged, while maintaining the same macroscopic properties To use this equation, let N0 ¼ the total number of atoms (A þ C)=mol xN0 ¼ the number of interstitial (C) atoms=mol (1 x)N0 ¼ the number of substitutional (A) atoms=mol n(1 x)N0 ¼ the number of interstitial sites=mol
33
Applied Thermodynamics
Equation 1.84 may be descriptively written as Sp ¼ k[ln (total number of ways) ln (number of indistiguishable ways)] In the context of the present b solid solution,
Sp ¼ k ln ¼ k ln
(Total number of interstitial sites)! (Number of filled sites)!(Number of empty sites)! [n(1 x)N0 ]! (xN0 )![n(1 x)N0 xN0 ]!
(1:85)
In writing this relationship, the theorem is used so that the number of ways in which m objects can be arranged is m! The natural logarithms of the factorial terms are individually evaluated through the application of Stirling’s approximation ln y! ffi y ln y y when y 1
(1:86)
Performing these operations yields the relationship for the ideal positional entropy of the b interstitial solid solution x x(n þ 1) x(n þ 1) ¼ R x ln ln 1 Sb,ideal n(1 x) ln (1 x) þ n 1 p n n n
(1:87)
Note how markedly this equation differs from that of the ideal positional entropy of a disordered binary substitutional solid solution Sb,ideal ¼ R{x ln x þ (1 x) ln (1 x)} p
(1:88)
Although Equation 1.87 establishes a base equation for the positional entropy of interstitial solid solutions, it will not be utilized further since there is strong evidence that neither carbon nor nitrogen in austenite, nor carbon in ferrite behaves in an ideal manner.
1.3.3 FREE ENERGY
AND
POSITIONAL ENTROPY OF NONIDEAL INTERSTITIAL SOLID SOLUTION
As noted before, only the case in which the nearest neighboring interstitial atoms repel each other will be considered. Under this circumstance, an average of se sites about each interstitial atom can be considered unoccupiable. Each interstitial atom accounts for z ¼ 1 þ se sites. Thus, the compound forms at x ¼ n=(n þ z), rather than at x ¼ n=(n þ 1), and the mole fraction of compound is x(n þ z)=n. Hence, the free energy equation becomes x(n þ z) b n þ z GA þ x GbACn=z þ GbE TSbp G ¼ 1 n n b
(1:89)
34
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Phase ACn=z has all of the nonexcluded interstitial sites filled, in contrast to phase ACn, wherein all interstitial sites are filled. z is taken to be independent of temperature. The positional entropy now becomes
Sbp
(Total no: of nonexcluded interstitial sites)! (No: of ways C’s are arrangeable, each among z sites) ¼ k ln (No: of filled interstitial sites)!(No: of nonexcluded empty sites)! n(1 x)N0 xN0 !z z ¼ k ln n(1 x)N0 xN0 ! (xN0 )! z
(1:90)
The term zxN0 is obtained as follows. Each C atom can be at any one of z sites, i.e., the one it actually occupies plus se others. Two C atoms isolated from each other can be arranged in z2 different ways. Applying Stirling’s approximation, 8 9 n(1 x)N0 n(1 x)N0 n(1 x)N0 > > > > ln > > xN0 ln z þ > > > > z z z > > > > > > = < n(1 x)N 0 b Sp ¼ k xN0 ln (xN0 ) þ xN0 þ xN0 > > z > > > > > > > > > > n(1 x)N n(1 x)N > > 0 0 > > xN0 ln xN0 ; : z z R x(n þ z) x(n þ z) x ¼ ln 1 n(1 x) ln (1 x) þ xz ln (1:91) n 1 z n n n Substituting Equation 1.91 into Equation 1.89 and applying Equation 1.32 yields G bA ¼ GbA þ GbE x
qGbE RTn n x(n þ z) þ ln qx z n(1 x)
(1:92)
Similarly, G bC ¼
n þ z z x qGb þ GbE þ (1 x) E GbACn=z GbA þ RT qx n n n (n þ z)x
(1:93)
Since GbACn=z and GbA are functions of temperature but not of x, rewrite Equation 1.93 as G bC ¼ f{T} þ RT
x qGb þ GbE þ (1 x) E qx n (n þ z)x
(1:94)
For the terms involving GbE , the following approximations are employed: HEb ¼ xH b
(1:95)
SbE ¼ xSb
(1:96)
where Hb and Sb are constants independent of both T and x. Utilizing these relationships and that between partial molar free energy and activity, Equation 1.92 thus becomes
35
Applied Thermodynamics
RTn n (n þ z)x ln z n(1 x)
G bA GbA ¼ RT ln abA ¼
(1:97)
Similarly, when f{T} in Equation 1.94 is set equal to GbC , where GC is the free energy of pure C in the crystal structure of b, this relationship becomes G bC
GbC
¼ RT
ln abC
x ¼ RT ln þ H b TSb n (n þ z)x
(1:98)
Hence, in this approach, x affects the activities only through its influence upon positional entropy.
1.3.4 EVALUATION
OF
CONSTANTS
IN A
PARTIAL MOLAR FREE ENERGY EQUATION
At this point in the development, Equations 1.97 and 1.98 are specialized to the case of austenite in Fe–C alloys; hence, n ¼ 1. From Equation 1.98, experimental data on the activity of carbon in austenite, agC , as a function of x at a given T are used to construct a plot of ln agC vs. ln xg={1 (1 þ z) xg}; z is methodically varied on a computer until a least squares slope of unity is obtained. Using the accurate data on agC reported by Smith [29], Scheil [30] found that z ¼ 5, i.e., se ¼ 4, gives the best fit. Thus, on average, 4=12 of the nearest neighboring interstitial sites to a given carbon atom are excluded from occupancy. Repeating such matching as a function of temperature yields, in cal=mol G gC GgC ¼ RT ln agC ¼ RT ln
xg þ 10, 580 4:01T 1 6xg
(1:99)
To interpret the other two constants, the standard relationship q ln agC DH gC H gC HCg ¼ ¼ q(1=T) R R
(1:100)
is first employed, where DH gC is the relative partial molar heat of solution of carbon in austenite and HCg is the enthalpy of C in the crystal structure of austenite in its standard state. Dividing Equation 1.99 by RT and applying Equation 1.100 is seen to yield DH gC ¼ 10,580 cal=mol (¼ 44,220 J=mol). Noting next that qG gC ¼ S gC qT
(1:101)
Application to Equation 1.99 produces S gC ¼ 4:01 R ln
xg , 1 6xg
cal=degmol
(1:102)
where S gC is the partial molar entropy of carbon in austenite. The excess partial molar entropy is S gC,XS ¼ S gC S gC,ideal
(1:103)
Applying Equation 1.101 to Equation 1.93 under the assumption of ideal solution behavior, wherein GbE ¼ 0 and z ¼ 1, S gC,ideal ¼ R ln
xg 1 2xg
(1:104)
36
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Substituting into Equation 1.103, S gC,XS ¼ 4:01 þ R ln
1 6xg 1 2xg
(1:105)
At the eutectoid composition (0.80 wt.% C or x ¼ 0.0361), it is interesting to note that the amount of the second term is only 0.34 cal=degmol (¼1.42 J=degmol). In the case of ferrite, KRC deemed the activity data insufficient to warrant the use of this approach. As will be seen later, additional, more fundamental cause is provided to support this viewpoint. Accordingly, an expression for G C will be obtained through an indirect route. Noting that na ¼ 3 and that in the argument of the logarithm of Equation 1.98, na (na þ za) xa in the ferrite region since x < 0.001, Equation 1.98 can be simplified to G aC GG C ¼ RT ln
xa þ A þ BT 3
(1:106)
where GG C ¼ f(T) is the free energy of pure carbon as graphite A ¼ Ha B ¼ Sa Since the partial molar free energy of C in ferrite at the a=(a þ g) phase boundary must be equal to that in austenite at the g=(a þ g) phase boundary, see Equation 1.40, equating Equations 1.106 and 1.99 RT ln
xga xag g a þ A þ BT ¼ RT ln þ 10,580 4:01T 3 1 6xg
(1:107)
ga where xag a and xg are the atom fractions of C in ferrite and in austenite at the aforementioned phase boundaries. Substituting phase diagram values of these atom fractions at two temperatures enabled KRC to evaluate A and B:
G aC GG C ¼ RT ln
xag a þ 26,160 9:75T 3
(1:108)
In the case of G aFe , the presence of z in the prelogarithmic term of Equation 1.97 prevents the evaluation of this equation even in approximate form. Fortunately, this relationship will not be required in the following applications. On the other hand, the relationship for G gFe may be written directly from Equation 1.97 G gFe GgFe ¼
1.3.5 APPLICATION
OF
RT 1 6xg ln 1 xg 5
(1:109)
PARTIAL MOLAR FREE ENERGY EQUATIONS
1.3.5.1 Calculation of the g=(a þ g) or Ae3 Phase Boundary Since the partial molar free energy of Fe in austenite at the g=(a þ g) boundary must equal the partial molar free energy of iron in ferrite at this phase boundary, the following equality can be written on the basis of Equation 1.109: g GaFe þ RT ln aag Fe,a ¼ GFe þ
RT 1 6xga g ln 1 xga 5 g
(1:110)
37
Applied Thermodynamics
where aag Fe,a is the activity of Fe in ferrite at the a=(a þ g) phase boundary xga g is the atom fraction of carbon in austenite at the g=(a þ g) phase boundary Since aag Fe,a 1, DGg!a ¼ Fe
RT 1 6xga g ln 1 xga 5 g
(1:111)
Rearranging, we obtain 5DGg!a Fe 1 RT ¼ 5DGg!a Fe 6 exp RT exp
xga g
(1:112)
g!a DGFe is a complicated function. An empirical expression for this function, determined on the basis of Fisher’s tabulation [31] is given by Aaronson et al. [32]. More recently, though not very different, information on this free energy change has been tabulated by Kaufman et al. [33]. See also the thermodynamic information on Fe tabulated by Hultgren et al. [2] and the analytical expressions on the Gibbs energy of pure iron by Ågren [34].
1.3.5.2 Driving Force for the Massive Transformation in Fe–C Alloys The transformation at issue is the transformation of austenite to ferrite without a change in composition by diffusional jumps rather than by shear. This driving force is zero at the T0 temperature, where the free energies of the two phases are the same. From Equation 1.49, DGg!am ¼ (1 x) G aFe G gFe þ x G aC G gC
(1:113)
where am denotes the product of this massive transformation. Substituting for the partial molar free energies involved in Equations 1.97, 1.99, 1.108, and 1.109, DGg!am ¼ (1 x)DGg!a þ x(15,580 5:7T) Fe 3 x(3 þ za ) 1 x 1 6x 3(1 x) 3 x(3 þ za ) ln þ ln RT x ln 1 6x 5 1x za 1 6x
(1:114)
where za is the unknown value of z for ferrite. KRC evaluated za values from 1 to 10. 1.3.5.3 Driving Force for the Precipitation of Proeutectoid Ferrite Except that the precipitate is solute-poor and the matrix is relatively solute-rich, this calculation is the numerical counterpart to the graphical calculation shown earlier. This calculation is thus made for the total driving force for the transformation g ga g G G G DGg!aþg1 ¼ (1 xg ) G ga þ x g Fe,g Fe C,g C
(1:115)
where G ga i, g refers to the partial molar free energy of either C or Fe in austenite at the g=(a þ g) boundary G gi refers to the partial molar free energies of the two species at the bulk composition of the alloy or the composition of the matrix prior to transformation
38
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Substituting Equations 1.99 and 1.109, DG
g!aþg1
xga g ¼ (1 1 6xga g 1 xg 1 6xg xg ln þ xg ln RT 5 1 xg 1 6xg g!a xg )DGFe
þ xg RT ln
(1:116)
1.3.5.4 Graphical Presentations of the Results of Sections 1.3.5.1 through 1.3.5.3 The Ae3 curve (denoted as xga ) and the T0 curve are among the results incorporated by KRC, as shown in Figure 1.27. Note that T0 lies above Ms at all compositions, as it should, and roughly midway between the xag and xga curves. The xag ( ¼ xag a ) curve is essentially the ordinate axis. The xga (¼ xga ) curve is seen to curve downward at an accelerating rate, approaching its limiting g value of n=(n þ z) ¼ 1=6. This corresponds to Fe5C, which does not exist. Both the curvature and the prediction of a compound not found experimentally probably arise from the impingement of exclusion shells. As already noted, the present statistics are not equipped to deal with this complication. Figure 1.28 presents the free energy change vs. temperature for the massive transformation (DGg!a ) and the proeutectoid ferrite reaction (DGg!a=g ), as well as transformations not calculated 0 here, including the martensite reaction (DGg!a ) and the decomposition of austenite of equilibrium g!a=cem ferrite and cementite (DG ) for an Fe–C alloy containing 4 at.% (0.89 wt.%) carbon.
Weight percent carbon 0
0.43
0.89
1.35
1.83
2.34
2.87
3.41
3.95 1000
γ
1200
γ + Cm 900
1100
800 996 K (723°C) 700
900
x γα
600
800
500
700
400 ρ
600
x γα(ρ ~ 15Å)
x αγ 500
To(ΔGγ
α΄
Temperature (°C)
Temperature (K)
1000
300 200
= 0)
400
100 Ms 0 Fe
2
4
6
8
10
12
14
16
18
Atomic percent carbon
FIGURE 1.27 Metastable equilibrium in the Fe–C system. (From Kaufman, L. et al., Decomposition of Austenite by Diffusional Processes, Interscience, New York, 1962, pp. 318–331. With permission.)
39
Applied Thermodynamics
–3000
–2500
γ
Free energy change (J/mol)
γ
α/cem
α/γ
–2000 γ
α΄
–1500
–1000
γ
α
–500
0
+500 300
400
500
600 700 Temperature (K)
800
900
1000
FIGURE 1.28 Free energy change attending the decomposition of austenite in an Fe-4 at.% (0.89 wt.%) C alloy. (From Kaufman, L. et al., Decomposition of Austenite by Diffusional Processes, Interscience, New York, 1962, pp. 318–331. With permission.)
As expected, the free energy change accompanying the massive transformation is less negative at a given temperature than is that for the proeutectoid ferrite reaction. The free energy changes accompanying the precipitation of ferrite and cementite below the eutectoid temperature are the most negative of all. Because of an ordering reaction accompanying martensite formation, the chemical free energy change attending this transformation is less negative than that for the massive transformation. However, Ms temperature lies below the upper limit for the massive transformation (T0) because of the substantial elastic shear strain energy attending the formation of martensite.
1.3.6 INTERPRETATION
OF Z IN
TERMS OF CARBON–CARBON INTERACTION ENERGY
Earlier, z was defined as 1 þ se, where se is the average number of nearest neighboring interstitial sites about a particular interstitial atom whose occupancy by another interstitial is excluded. Here, a more realistic interpretation of z is developed through a statistical thermodynamic analysis due to Darken [35], and to some extent recast by KRC. This analysis is based upon partition functions. The partition function of the ith (C) atom can be defined as fi ¼
i X j¼1
wj
gj e kT
(1:117)
40
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
where j represents a particular energy state gj is the degeneracy factor for the jth state (i.e., the number of distinguishable arrangements that have the same energy) wj is the interaction energy ( ¼ an internal energy, E) characterizing the jth state Two energy states are recognized: j ¼ 1, where w1 is the energy required to bring a carbon atom to an interstitial site in austenite with no carbon nearest neighbors and j ¼ 2, where w2 is the energy needed to bring a carbon atom to a site nearest neighboring to another carbon atom. To describe the statistics of this situation, consider the solid solution to be formed by the addition, one at a time, of m carbon atoms (m ¼ xN0) to interstitial sites in an iron lattice containing N1(¼(1 x)N0) iron atoms. For the first carbon added, g1 ¼ N1, i.e., since n ¼ 1 in austenite there is an energetically equivalent interstitial position available for a carbon atom corresponding to each of the Nl iron atoms present. For compactness, set exp(w1=kT) ¼ u1. Thus, the partition function for the first carbon atom is f1 ¼ N1 u1
(1:118)
With respect to the second carbon atom, there are N1 13 interstitial sites available that are neither occupied by a carbon atom nor have a carbon atom as a nearest interstitial neighbor. There are 12 interstitial sites, i.e., 12 ‘‘ways of addition,’’ that have a carbon atom as a nearest neighbor, with interaction energy w2. Let exp(w2=kT) ¼ u2. Hence, f2 ¼ (N1 13)u1 þ 12u2
(1:119)
Accordingly, for the third carbon atom: f3 ¼ (N1 2 13)u1 þ 2 12u2
(1:120)
fm ¼ (N1 13(m 1))u1 þ 12(m 1)u2
(1:121)
And for the mth atom:
Assuming that each of these operations can be considered independent of the others, i.e., a sufficiently dilute solution, the ‘‘complete partition function,’’ f, is f ¼
f1 f2 f3 fm m!
Qm
i¼1 fi
m!
(1:122)
To set a standard interaction energy level, assume w1 ¼ 0, i.e., u1 ¼ 1. Substituting Equations 1.118 through 1.120 into Equation 1.122, [N1 (N1 13 þ 12u2 )(N1 26 þ 24u2 ) ] m! (13 12u2 )m N1 N1 N1 1 2 ¼ 13 12u2 13 12u2 13 12u2 m! N1 ! (13 12u2 )m 13 12u2 ¼ N1 m! m ! 13 12u2
f ¼
(1:123)
41
Applied Thermodynamics
G gC and f are connected through the thermodynamic relationship G gC ¼ RT
q ln f qm T,V,N1
(1:124)
Taking natural logarithms of both sides of Equation 1.123 and truncating Stirling’s approximation (see Equation 1.86),
N1 N1 ln ln f ¼ m ln (13 12u2 ) m ln m þ 13 12u2 13 12u2 N1 N1 m ln m 13 12u2 13 12u2
(1:125)
Differentiating Equation 1.125 with respect to m, substituting into Equation 1.124, and rearranging with m ¼ xgN0 and N1 ¼ (1 xg)N0, G gC
¼ RT ln
xg 1 (14 12u2 )xg
(1:126)
For austenite, Equation 1.98 is rewritten as G gC
¼
GgC
þ RT
ln agC
xg þ A þ BT ¼ RT ln 1 (1 þ zg )xg
(1:127)
Noting that the standard state terms, A þ BT, are readily incorporated in Equation 1.126 through rewriting in terms of agC . A comparison of the arguments of the logarithms in Equations 1.126 and 1.127 shows that w 2 zg ¼ 13 12u2 ¼ 13 12 exp kT
(1:128)
Rearranging, we get
12 wg ¼ w2 ¼ kT ln 13 zg
(1:129)
Thus, the carbon–carbon interaction energy in austenite wg, must vary with temperature in an appropriate manner, if zg is to be independent of temperature. This question will be investigated in Section 1.3.7, but on the basis of more advanced statistical thermodynamic treatments of the activity of carbon in austenite.
1.3.7 MORE SOPHISTICATED TREATMENTS
OF INTERSTITIAL
STATISTICAL THERMODYNAMICS
Many such treatments have been made, and numerous ones have been applied to analyze experimental data on the activity of carbon in austenite. Here only two excellent treatments are briefly noted and utilized. Lacher [36] and Fowler and Guggenheim [37] developed an elaborate statistical
42
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
thermodynamic treatment of the thermodynamics of interstitial solid solutions in which overlapping exclusion shells were taken into account. Their relationship for agC is ln agC
wg
1 2xg 6wg DH gC DS gC T þ þ ¼ 5 ln xg RT RT h i12 1 2(1 þ 2Jg )xg þ (1 þ 8Jg )x2g 1 þ 3xg þ 6 ln h i12 1 2(1 þ 2Jg )xg þ (1 þ 8Jg )x2g þ 1 3xg
(1:130)
where Jg ¼ 1 eRT . McLellan and Dunn (MD) [38] improved upon the Lacher–Fowler–Guggenheim (LFG) treatment using the quasichemical approach. Their expression for the activity of carbon in austenite corresponds correctly to that for the complete blocking or exclusion of nearest neighboring interstitial sites around individual carbon atoms, whereas that of LFG does not. The relationship obtained appears to differ greatly from Equation 1.130. However, when particularized to the fcc crystal structure and rearranged, the MD equation is seen to be quite similar to, though still significantly different from that of LFG: ln agC
1 2xg 6wg DH gC DS gC T þ þ ¼ 11 ln xg RT RT h i12 1 2(1 þ 2Jg )xg þ (1 þ 8Jg )x2g 1 þ (1 þ 2Jg )xg þ 6 ln h i12 1 2(1 þ 2Jg )xg þ (1 þ 8Jg )x2g 1 þ 2Jg þ (1 4Jg )
(1:131)
Shiflet et al. [38], who did the above rearrangement, compared the activity of carbon in austenite calculated from Equations 1.130 and 1.131 as a function of wg at different levels of xg. This is shown in Figure 1.29. Even at the lowest carbon content, the activities calculated from the two relationships do not diverge until wg becomes too large to be operative in austenite. Aaronson, Domian, and Pound (ADP) [40] performed the Gibbs–Duhem integration upon Equation 1.130 to secure an expression for the activity of Fe in austenite, agFe . Shiflet et al. [39] did the same for Equation 1.131, and it was disconcerting to obtain exactly the same result:
Activity of carbon in γ
αγ × 102
αγ × 103
αγ × 104
αγ × 105
84
LFG/MD
LFG MD
xγ = 0.05
83 88
LFG/MD
87
LFG
xγ = 0.01
MD LFG/MD
77
LFG MD
xγ = 0.001
76
LFG/MD
76
LFG MD
xγ = 0.0001
75 0
10
20
30
40
50
60
70
80
ωγ × 10–3 (cal per mol)
FIGURE 1.29 Comparison of Equations 1.130 and 1.131 for various values of xg. (With kind permission from Springer Science þ Business Media: From Shiflet, G.J. et al., Metall. Trans. A, 9A, 999, 1978.)
43
Applied Thermodynamics
1 xg ln agFe ¼ 5 ln 1 2xg þ 6 ln
h i12 1 2Jg þ (4Jg 1)xg 1 2(1 þ 2Jg )xg þ (1 þ 8Jg )x2g 2Jg (2xg 1)
(1:132)
Hence, the calculation of the Ae3, e.g., yields identical results for the LFG and the MD treatments. Shiflet et al. [39], performing their study a decade after ADP, had considerably more data on both the activities of carbon both in austenite and in ferrite [41,42] and were able to investigate certain questions, such as the following one. The original evaluation of wg, by Darken from Equation 1.126 on the basis of Smith’s agC data, yielded an average value of ca. l500 cal=mol (¼ 6.27 103 J=mol). Evidence for a temperature dependence of this quantity was found, but seeing no fundamental reason for such behavior Darken chose to ignore it. ADP, using much the same database, obtained similar results and urged further consideration of the problem. For example, the calculated wg’s might be apparent interaction energies with two (or more) temperature independent interaction energies being actually operative as a consequence, perhaps, of occupation of the somewhat smaller tetrahedral interstitial sites, in addition to the usual octahedral sites by carbon atoms with increasing temperature.* Shiflet et al. [39] found that wg was not independent of temperature on the basis of the CO=CO2 data from Ban-ya et al. but a temperature dependence could not be discerned. Shortly after publication of their work, however, Leitnaker [43] privately pointed out that the equilibrium constant for the C þ CO2 $ 2CO reaction used in their analysis was outdated. A reanalysis of aC was done by Shiflet et al. [44]; the results are shown in Figure 1.30. Note that there is now a perceptible increase in wg with decreasing temperature. From these 11,000 wγ = –4.4 (T–273) + 11,550 (J/mol)
wγ (J/mol)
10,000 9,000 8,000 7,000
-ADP calculations Ref. 8 (R.P. Smith Data) -SBA calculations Ref. 1 (BEC Data) -Corrected values present investigation
6,000 5,000
300
400
500
600
700
800
900
1,000
1,100
1,200
Temperature (°C)
FIGURE 1.30 Temperature dependence of the interaction energy between nearest neighboring carbon atoms in austenite, wg. (With kind permission from Springer Science þ Business Media: From Shiflet, G.J. et al., Metall. Trans. A, 15A, 1287, 1984.)
* ADP, as did KRC earlier, and Darken and Smith originally, had trouble deciding whether to use activity data based upon equilibration with CO=CO2 mixtures or CH4=H2 mixtures. KRC pointed out that the latter data are likely to be less accurate because of the formation of other hydrocarbons. These differences persist among even the most recent data; e.g., Shiflet, et al. find an average wg from the LFG=MD models of 1925 ca1=mol (¼ 8.05 103 J=mol) from CO=CO2 data and only 415 cal=mol (¼ 1.73 103 J=mol) from CH4=H2 data. The latter activity data will be ignored in further considerations.
44
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
30,000
1
2
3 4 5 1
2
bcc nearest-neighbor interstitial position 3 fcc nearest-neighbor interstitial position
Pair interaction energy (J/mol)
20,000 342,000 cos (kFr – 1.05)/(kFr)3
10,000 0
1
2
3
4
5
6
7
kFr
–10,000 –20,000 –30,000 –40,000
FIGURE 1.31 Carbon–carbon interaction potential as a function of interatomic distance. (From Keefer, D. and Wert, C.A., J. Phys. Soc. Jpn., 18, 110 1963. With permission.)
results, it was found that wg=T, and hence both the zg of the KRC treatment and the Jg of the LFG and MD treatments, is also temperature-dependent rising from 1.33 at 10008C to 1.79 at 8008C. A fundamental reexamination of this unexpected temperature dependence now seems to be in order. Whereas all values of wg reported are positive, indicating repulsion between nearest neighboring carbon atoms in austenite, ADP found that wa was negative, indicating an attraction between such atoms in ferrite. However, their values based upon the limited data on carbon activity in ferrite obtained by Smith [29] varied erratically with temperature. It was suspected at the time that the small carbon content involved and the apparent insufficiency of data were responsible. However, Shiflet et al. [38] redid the analysis upon the basis of the many large volumes of data provided by Lobo and Geiger [42] and also by Dunn and McLellan [45] and secured essentially the same results. It now appears likely that the source of the difficulty lies in the use of statistical thermodynamic analyses suited only for repulsive interaction. Further research based upon a more appropriate analysis is now required. It should, however, be noted that there is support for the conclusion of ADP and Shiflet et al. that wa < 0. Carbon association in pairs along octahedral directions to reduce strain energy in ferrite has been suggested by Eshelby [46]. Keefer and Wert [47] used internal friction to show that pairs and even triplets of carbon atoms are present in significant proportions in ferrite (5%–10% of each configuration), with a binding energy of ca. 2000 cal=mol (¼8.36 103 J=mol) of C–C bonds. Finally, Machlin [48] calculated the electrostatic interaction potential between a pair of carbon atoms. As shown in Figure 1.31, at the interatomic distance corresponding to nearest neighboring carbon atoms in ferrite, the potential indicates a strong attractive interaction, greater than 10,000 cal=mol (4.18 104 J=mol), whereas in austenite a repulsion energy of ca. 2,000 cal=mol (¼8.36 103 J=mol) is seen to obtain at the somewhat greater distance between the nearest neighboring interstitial sites in austenite. The latter value is in excellent agreement with the wg data discussed before and suggests that Machlin’s explanation, i.e., the differences between wg and wa are an essentially coincidental result of the differences in the geometries (and to some extent the lattice parameters) of the fcc and bcc lattices, is correct. Shiflet et al. reevaluated DH gC and DS gC in both of their studies [39,44]. On the CO=CO2
45
Applied Thermodynamics 1000
900 γα
Hansen, xγ
Temperaturre (°C)
800
700 KRC, ωγ = 1405 cal/mol LFG/MD, ωγ = 415 cal/mol
600
LFG/MD, ωγ = 1925 cal/mol KRC, ωγ = 375 cal/mol
500
ADP, Ref. [1] ωγ = 1500 cal/mol
400
300
200
0
0.04
0.08
0.12
0.16
0.20
0.24
0.28
γα xγ
FIGURE 1.32 xga g vs. temperature curves calculated from KRC (or Darken–Smith statistics), LFG=MD, and the optimum LFG curve of ADP analysis. (With kind permission from Springer Science þ Business Media: From Shiflet, G.J. et al., Metall. Trans. A, 9A, 999, 1978.)
basis, their best average values differ little from those obtained earlier by KRC: 10,150 cal=mol (¼ 4.24 104 J=mol) and 3.73 cal=degmol (¼1.56 10 J=degmol), respectively. Figure 1.32 shows Ae3 curves calculated through the method of KRC from various statistical thermodynamic models, from Shiflet et al. [39]. Note how similar the curves obtained from all models are with a wide range of wg’s at high temperatures and how much they can diverge at lower temperatures, where they presently cannot be evaluated experimentally. For example, the KRC treatment yields xga g ffi 0:09 at 2008C, whereas the LFG=MD Ae3 with wg ¼ 1925 cal=mol (¼8.05 103 J=mol) falls at xga g ffi 0:20. The more recent Shiflet et al. work [44], incorporating a temperature-dependent wg, reduces xga g to ca. 0.19 with the LFG=MD statistics. Note also that the LFG=MD approach yields an Ae3 that increases about linearly at low temperatures, whereas that of KRC actually bends backward toward lower carbon contents. These differences in xga g at low temperatures are most important when analyzing measurements of ferrite plate lengthening rates since the diffusivity of carbon in austenite rises exponentially with carbon content. Calculations of the a=(a þ g) equilibrium curve and its metastable equilibrium extrapolation below the eutectoid temperature have not yielded good results, probably in part because of the modeling problem already noted. Lobo and Geiger obtained DH aC ¼ 26,800 cal=mole (¼1.12 105 J=mol) and DS aC ¼ 12:29 cal=degmol (¼ 5.14 10 J=degmol), the former in particularly good agreement with the KRC value. McLellan, on the other hand, argues from his thermodynamic analyses that DH aC ¼ 20,000 cal=mole (¼8.36 104 J=mol). Although the Lobo–Geiger works strongly supports the KRC result, this issue too remains in need of further clarification. Sample calculations of the total free energy change associated with the proeutectoid ferrite reaction in a Fe-2 at.% C alloy as a function of temperature as calculated from the various models are shown in Figure 1.33. The differences between models are seen to be small, in contrast to those
46
Mechanisms of Diffusional Phase Transformations in Metals and Alloys –800 –3200 Xγ = 0.02
–700
–2800 KRC, ωγ = 375 cal/mol LFG/MD, ωγ = 415 cal/mol
ΔG (cal/mol)
–500
–2400
ADP, Jγ = 0.474; (Ref. [1])
–2000
LFG/MD, ωγ = 1925 cal/mol
–400
–300
–200
–1600
ΔG (J/mol)
–600
–1200 Equation 1.19, ωγ = 1405 cal/mol
–800
–100
0 200
–400
300
400
500
600
700
0 800
Temperature (°C)
FIGURE 1.33 Free energy change attending ferrite transformation, DGg!aþg1 , calculated from KRC, LFG= MD, and ADP analysis (based on LFG). (With kind permission from Springer Science þ Business Media: From Shiflet, G.J. et al., Metall. Trans. A, 9A, 999, 1978.)
displayed in the calculations of the Ae3 curve. The circumstance that natural logarithms of ratios govern the DG calculation, whereas exponentials dominate the Ae3 calculation is responsible for damping difference in the former and accentuating them in the latter. Shiflet et al., and ADP previously, also calculated T0x curves on the various models. Differences are not large, and their results will be presented in Chapter 5, where they will prove useful in discriminating between shear and nucleation-and-diffusional growth mechanisms of this transformation.
REFERENCES 1. C. H. P. Lupis, Chemical Thermodynamics of Materials, North-Holland, New York, 1983. 2. R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelly, and D. D. Wagman, Selected Values of the Thermodynamic Properties of the Elements, ASM, Metals Park, OH, 1973. 3. L. Kaufman and H. Bernstein, Computer Calculation of Phase Diagrams, Academic Press, New York, 1970, pp. 47–48. 4. K. F. Michaels, W. F. Lange III, J. R. Bradley, and H. I. Aaronson, Metall. Trans., 6A, 1843 (1975). 5. G. J. Shiflet, J. K. Lee, and H. I. Aaronson, CALPHAD, 3, 129 (1979). 6. J. H. Hildebrand, J. Am. Chem. Soc., 51, 66 (1925); 57, 866 (1935). 7. J. H. Hildebrand and S. E. Wood, J. Chem. Phys., 1, 816 (1933). 8. I. S. Servi and D. Turnbull, Acta Metall., 14, 161 (1966). 9. Y. W. Lee, H. I. Afaronson, and K. C. Russell, Scr. Metall., 15, 723 (1981). 10. Y. W. Lee, Master thesis at Michigan Technological University, Houghton, MI, 1979. 11. C. Wagner, Thermodynamics of Alloys, Addison-Wesley Press, Reading, MA, 1952, pp. 17 and 35. 12. C. F. Old and C. W. Haworth, J. Inst. Met., 94, 303 (1966). 13. Y. W. Lee and H. I. Aaronson, Acta Metall., 28, 539 (1980). 14. M. C. Inman and H. R Tipler, Metall. Rev., 8, 105 (1963).
Applied Thermodynamics 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.
47
C. S. Smith, Trans. AIME, 175, 15 (1948). H. K. Hardy, Acta Metall., 1, 202(1953). J. Lumsden, Thermodynamics of Alloys, Institute of Metals, London, U.K., 1953. C. Zener, Trans. AIME, 167, 550 (1946). M. L. Kapoor, Int. Metall. Rev., 20, 150 (1975). I. Ansara, Int. Metall. Rev., 24, 20 (1979). C. H. P. Lupis and J. F. Elliot, Acta Metall., 15, 265 (1967). J.-C. Mathieu, F. Durand, and E. Bonnier, J. Chim. Phys., 11–12, 1289, 1297 (1965). M. Hillert, Lectures on the Theory of Phase Transformations, TMS-AIME, Warrendale, PA, 1975, p. 1. M. Hillert, The Mechanism of Phase Transformations in Crystalline Solids, Institute of Metals, London, U. K., 1969, pp. 231. J. W. Cahn, Acta Metall., 7, 25 (col. 1) (1959). J. C. Baker and J. W. Cahn, Solidification, ASM, Metals Park, OH, 1971, pp. 30–35. L. Kaufman, S. V. Radcliffe, and M. Cohen, Decomposition of Austenite by Diffusional Processes, Interscience, New York, 1962, pp. 318–331. R. Speiser and J. W. Spretnak, Trans. ASM, 47, 493 (1955). R. P. Smith, J. Am. Chem. Soc., 68, 1163 (1946). E. Scheil, Arc. Eisenhütt., 22, 37 (1951). J. C. Fisher, Trans. AIME, 185, 688 (1949). H. I. Aaronson, H. A. Domian, and G. M. Pound, Trans. TMS-AIME, 236, 772 (1966). L. Kaufman, E. V. Cloughty, and R. J. Weiss, Acta Metall., 11, 323 (1963), Table 3. J. Ågren, Metall. Trans. A, 10A, 1847 (1979). L. S. Darken, in an Appendix to the paper by R. P. Smith, J. Am. Chem. Soc., 68, 1172 (1946). J. A. Lacher, Proc. Camb. Philos. Soc., 33, 518 (1937). R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics, Cambridge University Press, New York, 1939. R. B. McLellan and W. W. Dunn, J. Phys. Chem. Solids, 30, 2631 (1969). G. J. Shiflet, J. R. Bradley, and H. I. Aaronson, Metall. Trans. A, 9A, 999 (1978). H. I. Aaronson, H. A. Domian, and G. M. Pound, Trans. TMS-AIME, 236, 753 (1966). S. Ban-ya, J. F. Elliott, and J. Chipman, Trans. TMS-AIME, 245, 1199 (1969); Metall. Trans., 1, 1313 (1970). J. A. Lobo and G. H. Geiger, Metall. Trans., 7A, 1347, 1359 (1976). J. M. Leitnaker, ORNL, private communication. G. J. Shiflet, J. R. Bradley, and H. I. Aaronson, Metall. Trans., 15A, 1287 (1984). W. W. Dunn and R. B. McLellan, Metall. Trans., 2, 1079 (1971). J. D. Eshelby, Acta Metall., 3, 487 (1955). D. Keefer and C. A. Wert, J. Phys. Soc. Jpn., 18(Suppl III), 110 (1963). E. S. Machlin, Trans. TMS-AIME, 242, 1845 (1968).
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Nucleation in 2 Diffusional Solid–Solid Transformations 2.1 INTRODUCTION THROUGH QUALITATIVE GENERAL STATEMENTS* (1) Nucleation is the process by which the smallest survivable aggregate of a more stable phase arises from the parent phase. (2) Nucleation is the only process involving more than one atom, which proceeds up a gradient of free energy. Such is feasible only because nucleation takes place on a very small scale (involving from a few to a few hundred atoms) entirely by fluctuations. By way of contrast, diffusional growth proceeds down a gradient of free energy. Thus, there is no driving force urging atoms to join a cluster of atoms developing into a nucleus, whereas there is such a driving force, expressed as gradients of partial molar free energy activity or concentration, governing diffusional growth. (3) The unit atomic process of the nucleation phenomenon to be considered in this section is the individual, thermally activated diffusional jump. (4) A cluster of atoms organized as a new phase is termed an ‘‘embryo’’ until it has reached a critical size, defined in (6), whereupon the cluster is termed a ‘‘nucleus.’’ (5) Irrespective of whether the matrix or the product phase is a solid, a liquid, or a gas, nucleation can be viewed as a series of biatomic or bimolecular reactions. An embryo ‘‘grows,’’ by statistical fluctuations, toward the critical nucleus size or dissolves back to individual atoms (which chemists term ‘‘monomers’’ in the present context) by the addition or subtraction of single atoms or molecules. On a probability basis, triatomic and higher-order reactions, involving the addition of two or more atoms or molecules at the same time, are sufficiently improbable so they can be disregarded. In substitutional solid solutions, for example, the simultaneous addition of two atoms would require the collaboration of two vacancies. This could occur even during the absence of a structural defect in the matrix, particularly if there was a binding energy between vacancies, but the probability of such an event would usually be appreciably less than that of single atom additions executed through the agency of single- or mono-vacancies. (6) The driving force for nucleation is the negative free energy change attending the formation of a more stable phase. However, as an embryo develops, at least one and usually two opposing forces simultaneously appear. The most important opposition comes from the nucleus:matrix interfacial energy, which arises because the embryo differs from the surrounding matrix in the crystal structure, composition, or both. The difference in the number and type of chemical bonds between an atom at the ‘‘surface’’ of an embryo and one in its interior may be viewed, in the quasichemical theory, as the source of the interfacial energy. The second opposing force develops when the average volume per atom in the embryo differs from that in the matrix, resulting in elastic strain energy. Both the free energy change per unit volume of the embryo (termed the ‘‘volume free energy change,’’ DGV) and the elastic strain energy per unit volume of the embryo, W, vary as the third power of the embryo radius, r3 (to use a sphere-based embryo as a convenient example). The interfacial energy, g, however, varies as r2. For small embryo sizes, the free energy added to the system by the interfacial energy is therefore larger than the reduction in free energy provided by DGV. (Like g, W is always positive. As will be demonstrated later, W can be viewed as an algebraic additive under usual circumstances to DGV: therefore, W reduces the driving force for nucleation.) * The references that are useful for this chapter are several review papers and portions of a textbook from Russell [1,4], Aaronson and Lee [2], Aaronson et al. [3], and Christian [5].
49
50
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
ΔGInterfacial = γA r 2
ΔG °
energy
+
ΔG* kT δ n* r*
n r
– ΔG°
ΔGvolumetric = (ΔGV +W ) . V r 3
FIGURE 2.1 Variation of DG* for embryo formation with radius, r, and the number of atoms in the embryo, n. The symbol a means ‘‘proportional.’’
The free energy of the embryo increases until the r3-dependent DGV can finally halt the rise in DG8, the total standard free energy change associated with the formation of an embryo, at a sufficiently large embryo size. This rise is halted when q(DG8)=qr ¼ 0; the corresponding radius is termed, r*, the critical radius, or the radius of the critical nucleus size. At larger values of r, DG8 diminishes continuously and eventually becomes negative. Such behavior is shown graphically in Figure 2.1. In this figure, the total increase in interfacial energy due to the embryo or nucleus formation is taken to be the product of g, the interfacial free energy per unit nucleus:matrix interfacial area, and A, the area of this interface. The volumetric contribution to DG8 is taken as the product of DGV þ W and V, the volume of the critical nucleus. The abscissa may be calibrated in terms of either the radius, r, or the number of atoms in an embryo, n. Thus n* is the number of atoms in the critical nucleus. Even when an embryo reaches r*, a thermal fluctuation of strength, kT, can remove an atom and start the embryo on its way back to the monomer. Not until the size of the nucleus has reached a value of r such that DG8 ¼ DG* kT can the nucleus be considered essentially proof against fluctuational disruption. The width of the DG8 vs. r curve at the level of DG* kT is denoted as d. (7) When an embryo reaches size n, which can be anywhere between 2 and n* 1 atoms, the probability that it will retreat to size n 1 is always greater than the probability that it will advance to size n þ 1. Hence, the probability that an embryo ‘‘struggling’’ up the DG8 vs. n hill will eventually dissolve back into its component monomers is enormously greater than the probability that it will reach the critical nucleus size. In a crude form, the reason is that the addition of an atom to an embryo occurs at a rate proportional to exp [(DGn!nþ1 þ DGD )=kT], while the subtraction of an atom from an embryo occurs at a rate proportional to exp(DGD=kT). Here, DGn!nþ1 is the increase in DG8 attending passage from an n-mer to an (n þ 1)-mer and DGD is the free energy of activation for the diffusion of an individual atom. Since DGn!nþ1 can readily be comparable to DGD, the rate of departure of atoms is far greater than the accretion rate. This situation is shown graphically for the transition from n to n þ 1 in Figure 2.2. Accordingly, the DG8 curve shown in Figure 2.1 is not smooth, but contains ‘‘jerks’’ in the form of activation humps for diffusion,
51
Diffusional Nucleation in Solid–Solid Transformations
ΔGD ΔG °
ΔG °n
(n+1)
+
n+1
n
n
–
FIGURE 2.2 DG8 up-hill for embryo growth, incorporated with DGD for the addition of a single atom.
ΔG°
+ I
n* r*
n r
–
FIGURE 2.3
A schematic ‘‘fine structure’’ of the DG8 vs. r curve for the evolution of an embryo.
sketched in Figure 2.3. This schematic curve shows that once n* þ 1 has been passed, the probability of shrinkage diminishes rapidly. (8) Homogeneous nucleation occurs in a structurally perfect region of the matrix phase. Heterogeneous nucleation takes place with the assistance of a structural imperfection such as a grain boundary, a sub-grain boundary, a twin boundary, dislocation, vacancy aggregate, etc. The imperfection(s) reduce DG*, the value of DG8 at r ¼ r*, and the free energy of activation for critical nucleus formation by providing a portion of the interfacial energy and=or strain energy needed to accomplish the nucleation process. When vacancies present at the thermal equilibrium at the transformation temperature are consumed by a nucleus, no net reduction in DG* is accomplished, since such vacancies must be regenerated elsewhere in the matrix. Excess vacancies that have been quenched-in from a higher temperature or otherwise formed, on the other hand, do accomplish a net diminution of DG*. In solids, all nucleation should be considered to be heterogeneous unless it can be proved to be homogeneous.
52
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
(9) Nucleation is the greatest known example of the free enterprise system! So many atoms are present per unit volume, ca. 1023=cm3, so few atoms are usually present in a critical nucleus (of order 102 or less), and so few nuclei are present per unit volume, unless nucleation is homogeneous (about 1018=cm3 for homogeneous nucleation, but easily 104–6=cm3 or less for heterogeneous nucleation), that the system can attempt at least once to begin forming any physically possible design of the nucleus. Variations in the crystal structure, composition, orientation relationship, and interfacial structure can be attempted. The nuclei that actually form are not necessarily those of either the most stable phase or the lowest free energy composition (and other specifications), but simply those that form most rapidly. Normally, though not always, these nuclei have the lowest DG*, since this is by far the most powerful variable term in the overall equation for the nucleation rate. (10) Essentially following Gibbs, there are two basically different modes through which a solid– solid transformation can be initiated: spinodal decomposition and nucleation and growth. Spinodal decomposition requires no nucleation; it begins everywhere at once when the proper thermodynamic and kinetics are provided and proceeds spontaneously through the evolution of sinusoidal composition waves. Both modes of transformation can produce a product phase with a composition different from that of the matrix. Only transformations initiated by nucleation, however, can also directly accomplish a change in the crystal structure and=or in the orientation relationship.
2.2 BRIEF COMPARATIVE SURVEY OF NUCLEATION IN THE FOUR BASIC TYPES OF PHASE TRANSFORMATION 2.2.1 VAPOR-TO-LIQUID TRANSFORMATION The kinetic theory of gases is an important component in the structure of nucleation theory in this case. Nucleation can be homogeneous but often takes place heterogeneously on ions (formed by cosmic rays, for example) or on bits of foreign solid or liquid.
2.2.2 VAPOR-TO-SOLID TRANSFORMATION The kinetic theory of gases again plays an important role, but the structure and energy of the nucleus:substrate interface also enter when nucleation of this transformation is heterogeneous. Vacua of 1010 torr are needed, together with degassing of the substrate, to ensure that heterogeneous nucleation occurs on the substrate itself. The direct impingement of atoms from the vapor onto an embryo growing on a substrate is found to play an unimportant role in nucleus formation. The actual sequence of atomic transport processes is the adsorption of atoms onto the substrate, surface diffusion on the substrate, and then two-dimensional nucleation by statistical fluctuations among the adsorbed atoms.
2.2.3 LIQUID-TO-SOLID TRANSFORMATION (SOLIDIFICATION) Fluctuations of composition, structure, and energy in the liquid are now involved. The basic theory of such fluctuations in a liquid is due to Einstein. Nucleation is often heterogeneous, on particles of foreign solids, or on the container walls. Once nucleation begins, growth is so fast (being controlled by diffusion in the liquid phase, it readily reaches values of 1 cm=s and more) that evolution of the heat of fusion can make solidification highly anisothermal. There seems as yet to be no way in which to directly measure nucleation kinetics in a bulk opaque liquid. To solve this problem and also to limit the effects of the heat of fusion release, a technique was developed for breaking up the liquid into fine droplets, e.g., by passage through a nozzle and then retaining the droplets as such through the addition of a surface-active additive. Because growth kinetics is so rapid, a single nucleus can be held responsible for the solidification of a droplet. Because the droplets are typically
Diffusional Nucleation in Solid–Solid Transformations
53
a micron in diameter, however, anisothermality will be limited. If the nucleation rate is proportional to the surface area of the droplets, it has occurred on the droplet surfaces, i.e., heterogeneously; if the nucleation rate is proportional to the droplet volume, it may be considered to be homogeneous, though some recent results have made the latter conclusion somewhat questionable.
2.2.4 SOLID-TO-SOLID TRANSFORMATION Fluctuations of structure, composition, and energy occur by solid-state diffusion. Many sources of heterogeneous nucleation are available. Homogeneous nucleation usually occurs only when the driving force for nucleation is high and=or the crystal structures of the matrix and the nucleus are sufficiently similar to make the nucleus:matrix interfacial energy low. The anisotropy of interfacial energy and volume strain energy are the major differences between this and the three preceding types of nucleation. Heat effects attending solid–solid nucleation are usually small, though not always. Kinetics of solid–solid nucleation can be experimentally measured. Careful collection of such data and comparison with nucleation theory, however, is still in a relatively early stage of development.
2.2.5 GENERAL REMARKS The same basic theoretical framework is applicable to all of the foregoing types of nucleation. Adaptation to solid–solid transformations, however, affects important details in this framework and can enormously affect the calculated rates of nucleation.
2.3 OUTLINE OF APPROACH FOR DEVELOPMENT OF NUCLEATION THEORY The following 11 points will be addressed during the development of the nucleation theory: (1) demonstration that the equilibrium number of critical nuclei is proportional to exp(DG*=kT), (2) calculation of the fictitious equilibrium nucleation rate, (3) derivation of the (real) steady-state nucleation rate, (4) estimation of the frequency factor, b*, (5) the problem of the time-dependent nucleation rate, (6) formation of a complete nucleation rate equation for homogeneous nucleation with isotropic interfacial energy, (7) the ancillary parameters: diffusivity, volume free energy change, volume strain energy, and interfacial energies, (8) calculation of DG*, b*, and Z (the Zeldovich nonequilibrium factor) are t (the incubation time) under various crystallographic conditions and for both homogeneous nucleation and nucleation at grain boundaries, (9) the combined effects of nucleation at grain boundaries upon nucleation site density and DG*, (10) the nucleation kinetics at dislocations, and (11) the comparisons of nucleation theory and experiment: homogeneous nucleation and nucleation at grain boundaries.
2.4 PROOF THAT THE EQUILIBRIUM CONCENTRATION OF CRITICAL NUCLEI IS PROPORTIONAL TO exp(DG*=kT) This proof is included primarily to emphasize that nucleus formation is preceded by a series of biatomic reactions. Let A be a single atom and An be an embryo consisting of n atoms. The critical nucleus, An*, of n* atoms is formed by the following series of biatomic reactions: A þ A > A2
A2 þ A > A3 A3 þ A > A4 ... ...
An* 1 þ A > An*
54
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Adding n*A > An*
(2:1)
From standard chemical thermodynamics, the equilibrium constant, K, for this reaction is K¼
[x(n* )]
(2:2)
*
[x(1)]n
where x(n*) is the mole fraction of the critical nuclei, x(1) is the mole fraction of the single atoms (monomers). By replacing the mole fractions, we get K¼
Cn* =(C1 þ Cn* )
*
[C1 =(C1 þ Cn* )]n
(2:3)
where Cn* is the number of critical nuclei per unit of volume, C1 is the number of monomers per unit of volume. Since C1 Cn*, C1 þ Cn* C1. Thus Kffi
Cn* C1
(2:4)
From the van’t Hoff isotherm, DGm* ¼ RT ln K
(2:5)
DG* ¼ kT ln K
(2:6)
for 1 mole of An*’s. For a single An*,
where DG* is the standard free energy change per individual critical nucleus when both reactants and products are in their standard state of unit activity. Combining Equations 2.6 and 2.4, we get Cn* ¼ C1 exp
DG* kT
(2:7)
Since C1 is so much larger than Cn*, C1 may be safely replaced by N, defined as the number of atomic nucleation sites per unit of volume. Hence, the equilibrium number of critical nuclei per unit of volume is DG* (2:8) Cn* ¼ N exp kT Although the equilibrium number of critical nuclei per unit volume is a useful concept, it will become apparent in a later section that this number is unlikely to exist experimentally.
Diffusional Nucleation in Solid–Solid Transformations
55
2.5 FICTITIOUS EQUILIBRIUM NUCLEATION RATE The nucleation rate defined as the number of nuclei formed per unit of volume of unreacted and compositionally unaffected matrix per unit time is customarily termed J*, since in reality this rate is given by the flux of critical nuclei through the phase space, DG8 vs. n. As a transition to the next section, however, here an equilibrium nucleation rate is considered to exist. This rate is equal to the product of the equilibrium number of nuclei per unit of volume and the frequency with which single atoms join these nuclei, b*: DG* * ¼ b*Cn* ¼ b*N exp (2:9) Jequil kT
2.6 DERIVATION OF STEADY-STATE NUCLEATION RATE This derivation follows that of Becker and Doering [6], as modified by Zeldovich [7]. An equivalent result was achieved earlier by Farkas [8]. The original concept for the theoretical progress was given by Gibbs [9]. Again consider that embryos form and dissolve by biatomic reactions. The number of embryos of size n actually presents per unit of volume of the matrix is designated as Cn; the equilibrium number of embryos of size n is denoted by Cn and is obtained by simply rewriting Equation 2.7 as DGn (2:10) Cn ¼ C1 exp kT where C1 is the number of monomers per unit volume under the equilibrium condition. The net rate at which the embryos of size n become the embryos of size n þ 1 is J ¼ bn Cn anþ1 Cnþ1
(2:11)
where J represents the net flux of embryos from size n to size n þ 1, bn is the rate at which single atoms are absorbed by an embryo containing n atoms, i.e., an n-mer, anþ1 is the rate at which single atoms leave an embryo of size n þ 1. The dependence of a upon n exists because the driving force for departure from an embryo is capillarity, whose effectiveness is dependent upon size. Since it is more difficult to evaluate than b, a will be replaced as follows. This process begins by rewriting Equation 2.11 for equilibrium conditions, i.e., when J ¼ 0 0 ¼ bn Cn anþ1 Cnþ1
(2:12)
By rearranging, we get anþ1 ¼
bn Cn Cnþ1
(2:13)
This seems to be a reasonable approach, inasmuch as the relationship between a and b ought to be the same irrespective of whether or not equilibrium conditions have been attained. By substituting Equation 2.13 into Equation 2.11, we get bn Cn Cnþ1 Cnþ1 Cn ¼ bn C n J ¼ bn Cn Cnþ1 Cn Cnþ1
(2:14)
56
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
(y2)
Cn+1 C °n+1
C C°
(y1)
Cn C°n
n (x1)
n
n+1 (x2)
FIGURE 2.4 Linear approximation for y ¼ Cn =Cn .
When n is large, the expression in brackets can be rewritten as a differential. Recall from analytic geometry that y1 y2 ¼ m(x1 x2) is the expression for a straight line with a slope m. The situation is graphically represented in Figure 2.4. Hence, the equation of the straight line becomes q Cn =Cn Cn Cnþ1 q Cn =Cn ¼ [n (n þ 1)] ¼ Cn Cnþ1 qn qn
(2:15)
By substituting into Equation 2.14 and restoring the n subscripts in a general sense, we get
J¼
bn Cn
q Cn =Cn qn
(2:16)
Alternatively, Equation 2.16 can be obtained from a first-order Taylor expansion. As long as the supersaturation of the system is reasonably low, the supply of monomers can be considered inexhaustible. A quasi-steady state is thus set up in which all values of Cn are independent of time. In order to maintain this condition, J must be independent of both n and time. Therefore, Equation 2.16 can be written as
J* ¼
bn Cno
q Cn =Cno qn
(2:17)
where J* is the flux or rate of formation of the critical nuclei. Before integrating this equation, the difference between the steady-state concentration of embryos, Cn, and the equilibrium concentration of embryos, Cn , should be schematically noted as a function of n in Figure 2.5. The Cn vs. n curve passes through a minimum at n* because DGn is a maximum at DGn* . Cn falls below Cn beyond n* – d=2 because the probability now becomes appreciable that the embryos will reach n* þ d=2 and disappear into growth.
57
Diffusional Nucleation in Solid–Solid Transformations
–
C °n = C°1e
Cn or C n°
ΔG °n kT
C°n* Cn 1 C° 2 n*
n
0
n* – δ 2
n* + δ 2
n*
FIGURE 2.5 Dependence of the steady-state concentration, Cn, and the equilibrium concentration, Cn , as a function of n.
To integrate Equation 2.17, first consider the boundary conditions. Since the transition to growth has not yet appreciably affected Cn, at small values of n, Cn =Cn approaches 1 as n goes to l. Also, since no embryos of infinite size are present, Cn =Cn approaches 0 as n goes to 1. By rearranging Equation 2.17 and utilizing these boundary conditions, we get
J*
1 ð 1
0 ð0 dn Cn Cn ¼ bn d ¼ bn ¼ bn Cn Cn 1 Cn
(2:18)
1
Rearranging again, we obtain J* ¼
bn dn C 1 n
1 Ð
(2:19)
By substituting Equation 2.10 for Cn , we get J* ¼ 1 Ð 1
bn
(2:20)
dn exp (DG =kT)=C1
Recalling that DG is a function of n, this quantity is now expanded in a Taylor’s series about n ¼ n*, retaining terms to the second power in the exponent, i.e., using the ‘‘quadratic approximation.’’ An appropriate form of Taylor’s series is f (z) ¼ f (zo ) þ (z zo )f 0 (zo ) þ
(z zo )z 00 f (zo ) þ 2!
58
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
where f(z) is a function of z and f 0 (z) ¼ df(z)=dz. Let n* ¼ zo and n ¼ z. Then D G ¼ DG* þ (n n*) At n ¼ n*,
q(DG ) (n n* )2 q2 (DG ) þ jn¼n* þ 2! qn qn2 n¼n*
q(DG ) ¼ 0, qn n¼n*
and
q2 (DG ) 6¼ 0 qn2 n¼n*
(2:21)
(2:22)
By substituting the remaining two terms of the series in Equation 2.21 into Equation 2.20, we get J* ¼
bn* C1 exp (DG*=kT)
1 Ð (n n* )2 q2 (DG ) dn exp 2kT qn2 n¼n* 1
(2:23)
The integral in the denominator is of the form 1 ð
exp (aU 2 )dU
1
1 q2 DG where U ¼ n n*, dU ¼ dn, and a ¼ . By replacing the lower limit by 1, 2kT qn2 n¼n* which is really not a poor approximation, since even half of infinity is very large, the resulting integral has a known exact solution 1 ð
2
exp (aU )dU
1
1 ð
1
rffiffiffiffi p exp (aU )dU ¼ a 2
(2:24)
Substituting into Equation 2.23 yields the final result known as the Becker–Döring equation: Jss* ¼ bn* C1
1=2 1 q2 DG DG* exp 2pkT qn2 n¼n* kT
(2:25)
The evaluation of the second derivative in this equation requires knowledge of the critical nucleus shape and will accordingly be delayed. Customarily, the term in (square brackets) is known as the ‘‘Zeldovich nonequilibrium factor’’: Z¼
1=2 1 q2 (DG ) 2pkT qn2 n¼n*
(2:26)
This term converts the fictitious ‘‘equilibrium’’ nucleation rate to the real steady-state nucleation rate. Z accounts for the steady-state concentration of the critical nuclei being about half the equilibrium concentration and for the decomposition of the supra-critical nuclei whose DG8 > DG* kT when a chance thermal fluctuation starts them back toward the monomers. Equation 2.25 can now be consolidated as
59
Diffusional Nucleation in Solid–Solid Transformations
* ¼ bn* ZC * ¼ b* ZC * ¼ b*ZN exp Jss n n
DG* kT
(2:27)
where bn* ¼ b* is the rate at which single atoms are added to the critical nucleus, Cn* is the equilibrium concentration of the critical nuclei, and Z converts the equilibrium concentration of the critical nuclei to the steady-state concentration. For clarity, compare Equation 2.27 to Equation 2.9.
2.7 ESTIMATION OF b* b* is simply the product of the number of solute atoms in the matrix at the nucleus:matrix boundary in contact with that portion of this boundary across which atomic transfer is permitted multiplied by the jump frequency of such solute atoms across the boundary. The first factor is written as follows: The effective number of solutes at the nucleus: matrix boundary ¼ S*xb =a2
(2:28)
where S* is the ‘‘effective’’ area of the nucleus:matrix boundary (across which atomic transfer is feasible), a is one lattice parameter, S*=a2 is approximately the ‘‘effective’’ number of atoms of both species at the nucleus:matrix boundary, xb is the atom fraction of the solute in the matrix. The second factor is obtained from the atomic theory of diffusion, see, for example, Shewmon [10], G¼
6D 6D 2 a2 a
(2:29)
where G is the atomic jump frequency, D is an appropriate volume diffusivity for the present, a is the jump distance equal to ca. a. By multiplying Equations 2.28 and 2.29 and ignoring the numerical factor in Equation 2.29 as a reasonable approximation, we get b* ¼
S*Dxb a4
(2:30)
Modification of this result to accommodate the situation in which mass transport takes place primarily by diffusion along grain and interphase boundaries is facilitated by an explicit expression for S*. This requires knowledge of the critical nucleus shape and hence will be accomplished during the discussion of nucleation at grain boundaries.
2.8 TIME-DEPENDENT NUCLEATION RATE Unlike the steady-state nucleation rate, Jss* , the time-dependent nucleation rate, J*, cannot be directly derived. The appropriate differential equation is readily obtained, but an analytic solution to it has yet to be achieved, and at present does not appear possible. Hence, a less direct approach to
60
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
the presentation of this material becomes desirable. We begin by writing the most useful general form of the steady-state nucleation rate equation: DG* * Jss ¼ b*ZN exp kT
(2:31)
The generally accepted equation for the time-dependent rate is t DG* exp J* ¼ b*ZN exp kT t
(2:32)
where t is the incubation time, t is the isothermal reaction time. Let us now consider the physical origin of t. Because a critical nucleus is assembled by a series of biatomic reactions, time is required for the necessary diffusional processes to take place. In the gas phase, this time, known as the incubation time, can be of the order of microseconds; in the solid state, seconds may be needed even when the kinetics are those of interstitial diffusion. Accordingly, the concentration of the various n-mers can be visualized as varying with time as shown in Figure 2.6. Turnbull originally drew diagrams of this type, though without maxima [11]. Subsequent studies by Abraham on nucleation in the vapor phase [12] and by Russell and Hall on void nucleation [13] showed that a maximum obtains in each plot when n < n* because of losses to the larger embryos. The only exception is that for C1, since the concentration of monomers is the base from which all others develop, C1 diminishes slightly with time. The differential equation that completely describes the variation of Cn, the concentration of n-mers, with time will now be derived. This derivation is originally due to Zeldovich [7] and the treatment here follows Zeldovich’s derivation given by Christian [5]. This derivation begins with Equation 2.16: q Cn =Cn (2:16) J ¼ bn Cn qn C1
C2
In Cn
C3
C4 Cn*
Reaction time
FIGURE 2.6 A schematic for the evolution of various n-mers against reaction time.
61
Diffusional Nucleation in Solid–Solid Transformations
Carrying through the indicated partial differentiations, J ¼ bn Cn Now recall Equation 2.10 Cn
Cn (qCn =qn) Cn qCn =qn 2 Cn
¼
C1
DG exp kT
(2:33)
(2:10)
Since qCn C1 DG q(DG ) Cn q(DG ) ¼ exp ¼ qn kT kT kT qn qn Equation 2.33 becomes J ¼ bn
qCn bn Cn q(DG ) qn kT qn
(2:34)
The rate of change in the concentration of embryos of a given size is qCn ¼ Jn1 Jn qt
(2:35)
By substituting Equation 2.34 into Equation 2.35, we get qCn qCn1 qCn bn1 Cn1 q DGn1 bn Cn q DGn ¼ bn1 þ bn þ qt qn qn kT kT qn qn
(2:36)
As before, differences are converted to derivatives, see Figure 2.7. By applying this conversion and dropping the subscripts, we get
βn
∂cn ∂n βc
β
βn –1
(a)
∂c ∂n
∂(ΔG°) ∂n
∂cn–1 ∂n
n–1
n
n
(b)
n–1
n
FIGURE 2.7 A linear approximation for the conversion from differences to derivatives.
n
62
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
qC q(DG ) q b q bC qCn 1 qn qn [n 1 n] [n 1 n] ¼ qn qn qt kT qC q(DG ) q b q bC 1 qn qn ¼ þ qn qn kT
(2:37)
This equation is known as the Fokker–Planck or ‘‘master’’ equation. It is formally equivalent to the diffusion equation for particles moving in a field of force under the potential DG8 along the n-axis. Persistent attempts to develop approximate solutions to Equation 2.37 in lieu of the apparently unattainable exact one, have proved to be persistently troublesome. For example, Wakeshima [14] obtained a quite accurate expression for t, but it was not in the form of Equation 2.32. Here, the approximation developed by Kantrowitz [15] is utilized; this result fulfills our immediate need but has other obvious deficiencies. The first term on the right-hand side (r.h.s) of Equation 2.37 represents the overcoming of kinetic obstacles to embryo growth whereas the second term is the thermodynamic barrier. Since the first term is a function of exp(DG8=kT), it is considerably larger than the second, which is a function of the roughly parabolic variation of DG8 with n. Hence, one hazards dropping the second term on the r.h.s. of Equation 2.37, leaving qC q2 C ffib 2 qt qn
(2:38)
A further approximation was made that b is independent of n, since it varies with n only as the 2=3 power (i.e., the area of the nucleus:matrix boundary), whereas C varies with n according to a usually much higher exponential. Formally, Equation 2.38 is Fick’s second law. The applicable boundary conditions are as follows (see Figure 2.8): Cn,t¼0 ¼ 0
when n > 0;
Cn¼0,t¼t ¼ C1 ; Cn¼s,t¼t ¼ 0
(2:39)
Under these conditions, the solution to Equation 2.38, from Carslaw and Jaeger [16] is 1 mpn n 2C1 X 1 m2 p2 bt sin Cn ¼ C1 1 exp p m¼1 m s s s2
(2:40)
Cn
Dropping the second term on the r.h.s. of Equation 2.34 in accordance with the discussion of Equation 2.37, qC J* ffi b* (2:41) qn s
n
FIGURE 2.8 Appropriate boundary conditions for Cn.
s
Diffusional Nucleation in Solid–Solid Transformations
63
By performing a differentiation of Equation 2.40 with respect to n, we get 1 mpn qCn C1 2C1 X m2 p2 bt ¼ cos exp qn s s m¼1 s s2
(2:42)
Setting n ¼ s, and substituting into Equation 2.41, we get " # 1 X b*C1 m2 p2 b*t m 1þ2 J* ¼ (1) exp s s2 m¼1
(2:43)
This series, however, converges too slowly. It is therefore transformed into one converging more rapidly by means of Poisson’s summation formula (Courant and Hilbert [17]), yielding J* ¼
2C1
1=2 X 1 b* (2m 1)2 s2 exp 4b*t pt m¼1
(2:44)
For short times, only m ¼ 1 is sufficient: J* ffi 2C1
1=2 t b* exp pt t
(2:45)
where t ¼ s2=4b*. To within an order of magnitude tffi
(n*)2 : b*
for n* < s;
(2:46)
Equation 2.45 is clearly defective, as it does not even include the most important term in a J* equation, exp(DG*=kT). However, Equation 2.45 does incorporate t in the correct manner, and the expression for t does not differ much from the relationships that are more accurate. The results of more sophisticated treatments yield expressions of the form t¼
kTk q2 (DG ) b* qn2 n*
(2:47)
where the dimensionless constant, k, has been found by various investigators to lie between 0.5 and 5. In the next section, such a relationship will be obtained through a more physical approach to the problem.
2.9 FEDER ET AL.’S TREATMENT OF t Following Feder et al. [18], the incubation time, t, can be divided into two components, t0 and td, as shown in Figure 2.9. A distinction is made between these components because passage of an embryo through these regions of DG8 vs. n space can be considered to take place under the impetus of different driving forces. Initially, q(DG )=qn is large and passage can be regarded as drifting under a definite driving force, and t0 is the time for this duration td is the time for the second stage, that is, the time to grow from n* d=2 to n* þ d=2. At this second stage, q(DG )=qn is sufficiently small so that random walk (Brownian motion) is the primary means for growth. Consequently, as will be shown shortly, td is much greater than t0 .
64
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
ΔG ° Time to traverse = t΄
Time to traverse = τδ
ΔG* kT δ
+
n* – δ 2
n* + δ 2
n*
n
– FIGURE 2.9 A schematic for the time to the initial evolution, t0 , and the time to traverse, td.
2.9.1 RELATIONSHIPS
FOR
d
AND FOR
td
From Figure 2.9, kT ¼ DGn* DGn* þd=2
(2:48)
The second term in the r.h.s of this equation is expanded to the second order around n* in a Taylor’s series, which is written as f (z þ h) ¼ f (z) þ hf 0 (z) þ
h2 00 f (z) þ 2!
and recalling that q(DG8)=qn ¼ 0 at n ¼ n*, this Setting n* ¼ z, d=2 ¼ h, and f (z þ h) ¼ DGn* þd=2 expansion modifies Equation 2.48 to kT ¼
DGn*
By rearranging, we get
DGn*
d2 q2 (DG ) 8 qn2 n*
1=2 1 q2 (DG ) d¼ 8kT qn2 n*
(2:49)
(2:50)
Considering a passage across d as a random walk process, we recall the relationship among the self-diffusivity, D, diffusion distance, s, and diffusion time, t, s ¼ (2Dt)1=2
(2:51)
65
Diffusional Nucleation in Solid–Solid Transformations
By analogy, d ¼ (2b*td )1=2
(2:52)
where b* is again being used as a diffusivity, d is the equivalent of distance in DG8 vs. n space, td is the diffusion time. By rearranging, we get td ¼
d2 2b*
(2:53)
Substituting Equation 2.50 for d, we get td ¼
2.9.2 RELATIONSHIP
FOR
t0
4kT q2 (DG ) b* qn2 n*
(2:54)
On the principle of microscopic reversibility, also termed the principle of time reversal, fluctuant processes evolve and decay over the same path and with the same kinetics [18]. (This principle is well grounded in the fundamentals of statistical mechanics.) Therefore, the following treatment of the decay kinetics of an embryo will be taken as applicable to the growth of an embryo through the t0 region. The rate at which an embryo drifts from n* – d=2 down to the monomer in DG8 vs. n space, dn=dt, is the negative of the product of the ‘‘mobility’’ of the embryo and the driving force for the drift: dn q(DG ) ¼ M dt qn
(2:55)
where M is mobility. From the Einstein relationship between diffusivity and mobility, D ¼ kTM ¼ b
(2:56)
where b is again the equivalent of a diffusivity. By substituting Equation 2.56 into Equation 2.55, we get dn b q(DG ) ¼ dt kT qn
(2:57)
By integrating, we get
0
t ¼
ð1
n* d=2
b q(DG ) kT qn
1
dn
(2:58)
66
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
2.9.3 TOTAL VALUE OF t By definition, t ¼ t 0 þ td
(2:59)
For a homogeneously formed spherical nuclei, Feder et al. calculated that td > 2t 0
(2:60)
Russell [19] concluded that for the three nuclei models applicable to nucleation at a grain boundary—two spherical caps abutting the boundary, one spherical cap whose base is coplanar with the grain boundary plane, and a disc with one broad face coplanar with the grain boundary— td 4t 0
(2:61)
Intuitively, these results are as expected: td represents random walk, whereas t0 is applicable to growth under a free energy gradient and hence should be significantly shorter. It thus appears reasonable to conclude that t td
(2:62)
2.10 TIME-DEPENDENT NUCLEATION RATE FOR HOMOGENEOUS NUCLEATION WITH ISOTROPIC g 2.10.1 INTRODUCTION The considerations of previous sections are incorporated into this section to produce a complete equation for J* in terms of independently measurable quantities. This relationship will not include any arbitrary or disposable terms. Nucleation will be assumed to occur homogeneously and isotropic nucleus:matrix interfacial energy, g, will be taken to be independent of the boundary orientation. The critical nucleus will be assumed to be spherical. Nucleation in such a condition is experimentally rare, although the only example of a comparison between the homogeneous nucleation theory and the experiment presently available, which will later be discussed in much detail, is taken to be of exactly this description. However, it will subsequently be seen that a very wide variety of nucleation situations can be quantitatively described in terms of the parameters of homogeneous nucleation multiplied by some constants, most of which reflect the other interfacial energies involved. Hence, the mathematics developed in this section will provide a base for much of what is to follow. This point in the development is also a convenient one to indicate that not entirely obvious assumptions are currently made in the use of the Gibbsian nucleation theory. In seeking an expression for DG*, only the total interfacial energy of the nucleus:matrix interface will be minimized. The roles of DGV and W in determining the smallest possible value of DG*, and the interactions among these three factors, which could be directed toward this end, will be ignored. Such justifications for this simplification, as can presently be offered, will be presented in the next Section 2.11 dealing with ancillary parameters on an individual basis. The second assumption made is that macroscopic values of these parameters, and also the diffusivity term in b*, are exactly applicable on the near atomistic scale of solid–solid nucleation. What little has so far been accomplished in the way of assessing this assumption will also be considered in the ancillary parameters section. For present purposes, it will be sufficient to remark that: (a) none of these assumptions appear, at this time, to be the source of a serious error and (b) both of these assumptions, though already investigated in part, are worthwhile topics for future research in the context of the solid–solid nucleation theory.
67
Diffusional Nucleation in Solid–Solid Transformations
2.10.2 ACTIVATION ENERGY
OF
NUCLEATION DG*
This is normally the most potent single term in any relationship for J*. Four different methods for deriving DG* will be developed. The first is undertaken in terms of a G–x diagram and places DG* in a thermodynamic context. The second is the most familiar method from Gibbs. The third is a newly developed, less difficult means for obtaining the same result, and will thus be useful in subsequent derivations. The fourth, also from Gibbs, provides a relationship that is particularly useful in grasping the relationship between DG* and the size of the critical nucleus; this one will be helpful to keep in mind as the critical nucleus shape is repeatedly modified in succeeding sections in a search for lower DG* morphologies. 2.10.2.1 Introduction to the Critical Nucleus Shape Problem As shown by the plot of DG8 vs. r or n in Figure 2.1, the principal barrier to nucleation is the interfacial energy of the nucleus:matrix interface. The critical nucleus shape, which minimizes this barrier, will have the lowest DG*. Almost invariably, the shape with the lowest DG* is the one most likely to form. Describing the energy of the nucleus:matrix interface as an isotropic means that the energy of the interface is independent of its orientation. Hence, the critical nucleus morphology, which minimizes the interfacial area for a fixed nucleus volume, will also require the lowest interfacial energy. In the absence of a heterogeneity that catalyzes nucleation, this morphology is a sphere. The spherical morphology will be utilized in each of the succeeding sections. 2.10.2.2 G–x Diagram Approach This method is from Hillert [20]. In a situation wherein there is an appreciable DG* for the nucleation of a in a b matrix, the radius of a spherical a precipitate can be reduced sufficiently so that it is in a metastable equilibrium with a b matrix of the bulk composition, as illustrated in the G–x diagram of Figure 2.10. Such a sphere may be considered to have the so-called ‘‘critical radius,’’ r*; a sphere of a still smaller radius would dissolve (unless, of course, it were ‘‘struggling’’ up the free energy hill toward r* via statistical fluctuations).
αnucleus αr * G β
1 2
P*αV α
αr = ∞ P*αV α
A
xβ
xα
B
FIGURE 2.10 Activation energy for nucleation. (From Hillert, M., in Lectures on the Theory of Phase Transformations, TMS-AIME, Warrendale, PA, p. 1, 1975. With permission.)
68
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
From the Gibbs–Thomson equation for a sphere, with Pb ¼ 0 as per the convention P*a Pb ¼ P*a ¼
2g r*
(2:63)
Noting in Figure 2.10 that P*a V a is equal to DGVVa for nucleation, and rearranging, r* ¼
2g DGv
(2:64)
Hence, spheres of r* are in a temporary form of equilibrium with the matrix phase of the bulk composition. These spheres are the nuclei of the critical radius. Their G–x curve is denoted as ar* . The total free energy, G, needed to form a critical nucleus of a spherical shape containing the m* fraction of a mole is
G¼
m ð* 0
fGa (Pa ¼ 0) þ Pa V a gdm
(2:65)
where Ga(Pa ¼ 0) is the free energy of a in the absence of capillarity. Substituting Equation 2.63 for P*a and noting that only m ¼ 4pr3=3Va of a mole of a is to be formed, G ¼ Ga (Pa ¼ 0)m* þ
m ð* 0
2g a V dm r
(2:66)
Since dm ¼ 4pr2 dr=Va, a
a
G ¼ G (P ¼ 0)m* þ
rð* 0
2g a 4pr 2 dr ¼ Ga (Pa ¼ 0)m* þ 4pr*2 g V Va r
¼ Ga {Pa ¼ 0}m* þ
4pr*3 g 3 ¼ Ga {Pa ¼ 0}m* þ P*a V a m* r* 2
(2:67)
From Figure 2.10, P*a V a will raise m* moles of a in the form of a sphere of radius r* to the thermodynamic equilibrium with the b of composition xb. Hence, P*a V a =2 remains from the free energy derived beginning with Equation 2.65 as the activation free energy, DG*. Recasting into a form that is more conventional, we get 1 a a 4pr*2 g 4pg 2g 2 16pg3 ¼ ¼ DG* ¼ P* V m* ¼ 2 3 3 DGV 3DG2V
(2:68)
Therefore, one may incorporate in the G–x diagram another curve, located P*a V a =2 above the critical nucleus G–x curve, denoting the free energy level, which must be attained in order to form a critical nucleus. Note that this curve lies above the line tangent to Gb–x at xb and thus that it cannot represent any form of equilibrium with the b matrix, as already described. 2.10.2.3 Introduction to the Volume Strain Energy Incorporation Problem Here this problem is solved in a preliminary fashion with an argument that is no better than semiquantitative. In Section 2.11.3, a more rigorous approach will be developed to achieve the same result. Again, considering g to be isotropic and nucleation to be homogeneous, the spherical
Diffusional Nucleation in Solid–Solid Transformations
69
shape, which minimizes the total interfacial free energy, will not necessarily minimize the volume strain energy associated with the transformation. It will be noted, in fact, that for a coherent nucleus whose shear modulus is less than that of the matrix in which it is embedded, the minimum volume strain energy shape is a flat disc. However, if the volume strain energy is incorporated by algebraically adding it to DGv, the sum of these two terms is raised only to the second power in the expression for DG* (see Equation 2.68), whereas the interfacial energy is taken to the third power. Hence, it appears reasonable to posit that, unless strain energy denoted W, is a large fraction of DGV, it will have a minute effect upon the critical nucleus shape, and will serve only to make DG* somewhat larger. Accordingly, W will be included in the relationship for DG* through the following relationship: f ¼ DGV þ W
(2:69)
2.10.2.4 Conventional Gibbsian Approach DG* is calculated through the approach described graphically in Figure 2.1: the contributions of interfacial energy and of volume free energy change plus volume strain energy are separately calculated and summed; the total free energy change, DG8, is then minimized with respect to r in order to calculate DG*. Thus, the total standard free energy change associated with the formation of an embryo, DG8, is given by DG ¼ Ag þ Vf
(2:70)
where A is the interfacial area, V is the volume of the embryo. Substituting expressions for A and V appropriate to a sphere, we get DG ¼ 4pr 2 g þ
4pr 3 f 3
(2:71)
Differentiating with respect to r, q(DG ) ¼ 0 ¼ 8prg þ 4pr 2 f qr
(2:72)
Solving for r, which is thus the critical radius, r* r* ¼ 2g=f
(2:73)
16pg3 3f2
(2:74)
Substituting into Equation 2.71, we get DG* ¼
Note that Equation 2.74 is the same as Equation 2.68, obtained via the G–x diagram approach, and similarly Equation 2.73 is identical to Equation 2.64, in both cases replacing DGV by f. 2.10.2.5 Wulff Volume Approach for DG* In order to carry out the conventional Gibbsian approach for the calculation of DG*, it is necessary to know the area as well as the volume of the critical nucleus. Although this is a trivial problem for a
70
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
sphere, it can become quite tedious when the critical nucleus shape is more complex and the area of a number of different portions of the nucleus:matrix interface, each with its own interfacial energy, must be calculated. The following approach, developed by Lee and Aaronson [21], allows the omission of this portion of the calculation. Derivation of this approach begins by rewriting Equation 2.70 or 2.71 in a generalized form: DG ¼ KA r m1 g þ KV r m f
(2:75)
where KA is a multiplying factor expressing the area of the nucleus:matrix interface when multiplied by rm1, KV is also a multiplying factor similarly expressing the volume of the embryo, m is dimensionality (3 for three dimensions). As in the usual Gibbs method, Equation 2.75 is first differentiated with respect to r, and the resulting expression is set equal to zero: q(DG ) ¼ 0 ¼ (m 1)KA r m2 g þ mKV r m1 f qr
(2:76)
Solving for r* r* ¼
(1 m)gKA mfKV
(2:77)
By substituting Equation 2.77 into Equation 2.75 and rearranging, we get (1 m)KA g m1 (1 m)KA g m g þ KV f mKV f mKV f (1 m)KA g m1 (1 m)KA g (1 m)KA g m1 þ ¼ KA g mKV f m mKV f m1 (1 m)KA g (1 m)KA g KA g þ ¼ mKV f m m1 (1 m)KA g gKA (1 m) m1 gKA m ¼ ¼ m m mKV f fKV
DG* ¼ KA
(2:78)
By multiplying both the numerator and the denominator by r(m1)m, we get DG* ¼
(1 m)m1 (KA r m1 g)m fm1 mm (KV r m )m1
(2:79)
Noting that E, the total interfacial energy of the nucleus, and V, the volume of the nucleus, are given by E ¼ KA r m1 g
(2:80)
V ¼ KV r m
(2:81)
Diffusional Nucleation in Solid–Solid Transformations
71
DG* can be rewritten as DG* ¼
(1 m)m1 Em fm1 mm V m1
(2:82)
While developing a more general proof of the Wulff theorem, Herring derived the following relationship [21]: ð 1 m1 m E ¼ gdA ¼ mV m VW
(2:83)
where VW is the volume of the Wulff construction. This construction, presented later in detail, is geometrically similar to the equilibrium shape, but plotted in g-space rather than in r-space with g being substituted for r on a one-for-one basis. Hence, VW is the volume of the nucleus in g-space, whereas V is the volume of the nucleus in r-space. By substituting Equation 2.83 into Equation 2.82, we get DG* ¼
(1 m)m1 mm V m1 VW (1 m)m1 ¼ VW V m1 fm1 mm fm1
(2:84)
4 VW f2
(2:85)
In three dimensions with m ¼ 3, DG* ¼
To apply this equation to a spherical nucleus, it is only necessary to write VW ¼
4pg3 3
(2:86)
16pg3 3f2
(2:87)
and substitute this relationship into Equation 2.85 DG* ¼
thus recovering Equations 2.74 and 2.68. Hence, the calculation of the interfacial area has been eliminated; computation of only VW is now required. Recently, Offerman et al. [23] introduced as a nucleation barrier factor, which is equal to the product of DG* and f2. Comparing to VW in Equation 2.85, it is clear that ¼ 4VW, and the use of bears no new merits. Therefore, in honor of Wulff’s seminal contribution to surface science, it is appropriate to keep using the Wulff volume, VW, as the nucleation barrier factor, as pointed out by Enomoto and Yang [24]. 2.10.2.6 Nucleus Volume Approach for DG* This relationship was developed by Gibbs. Rewriting Equation 2.70 or 2.71 in another generalized, but strictly three-dimensional form, we get DG ¼ Cr2 þ Dr3
(2:88)
72
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Differentiation with respect to r and setting the result equal to zero 2C 3D
(2:89)
4C3 27D2
(2:90)
D 8C 3 D ¼ r*3 2 27D3 2
(2:91)
Dr*3 ¼ fV*
(2:92)
r* ¼ By substituting into Equation 2.88, we get DG* ¼ This result may be rewritten as DG* ¼ But
where V* is now designated as the volume of the critical nucleus in r-space. By substituting into Equation 2.91, we get DG* ¼
V* f 2
(2:93)
The simple proportionality between DG* and V* should prove repeatedly useful in appreciating the critical nucleus shape problem. In quantitative terms, however, this relationship is no more useful than the original approach of Gibbs, since the elimination of r* from Equation 2.93 requires an evaluation of the interfacial area as well as the volume of the nucleus (note Equation 2.89).
2.10.3 FREQUENCY FACTOR b* Recalling Equation 2.30, b* ¼ S*Dxb=a4, and replacing S* with the area of a spherical critical nucleus, we get b* ¼ 4pr*2
Dxb a4
(2:94)
Dxb f 2 a4
(2:95)
Then by replacing r* from Equation 2.64, we get b* ¼ 16pg2
2.10.4 ZELDOVICH FACTOR, INCUBATION TIME AND THE RE-DERIVATION OF FREQUENCY FACTOR Z is given by Equation 2.26 and t is given by Equation 2.54 via Equation 2.62. Both contain [q2(DG8)=qn2]n*. It was previously noted that the evaluation of this second derivative requires knowledge of the critical nucleus shape. Hence, this procedure can now be undertaken. To begin, it is necessary to reformulate DG8 in terms of the number of atoms in a spherical embryo, n, rather than the radius of the embryo, r. By rewriting the volume of a spherical embryo in terms of n and of va, the average volume of an atom in the nucleus, we get 4pr 3 ¼ nva 3
(2:96)
73
Diffusional Nucleation in Solid–Solid Transformations
By rearranging, we get r¼
3nva 1=3 4p
(2:97)
Hence, Equation 2.71 becomes 3nva 2=3 4pf 3nva ¼ (4p)1=3 (3nva )2=3 g þ nva f þ DG ¼ 4pg 4p 4p 3
¼ Bn2=3 þ Hn
(2:98)
where B ¼ (4p)1=3 g(3va )2=3 , H ¼ va f. To find n*, the number of atoms in the critical nucleus, and DG*, the conventional Gibbsian procedure was followed: q(DG ) 2 ¼ 0 ¼ Bn1=3 þ H qn 3
(2:99)
Hence, n* ¼
2B 3H
3
(2:100)
By substituting Equation 2.100 into Equation 2.98, we get 2B 2 2B 3 4B3 þH ¼ DG* ¼ B 27H 2 3H 3H
(2:101)
The differentiation of Equation 2.99 at n ¼ n* yields the desired second derivative: q2 (DG ) 2 ¼ Bn*4=3 2 qn 9 n*
(2:102)
This may be cast into a more convenient form. By rewriting Equation 2.101 for n*, we get DG* ¼ Bn*2=3 þ Hn*
(2:103)
DG* ¼ Bn*4=3 þ Hn*1 n*2
(2:104)
Dividing through by n*2, we get
Noting from Equation 2.99 that H ¼ 23 Bn*1=3 DG* 2 1 ¼ Bn*4=3 Bn*4=3 ¼ Bn*4=3 2 3 3 n*
(2:105)
74
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Multiplying through by 2=3, we get 2 DG* 2 q2 (DG ) 4=3 ¼ Bn* ¼ 3 n*2 9 qn2 n*
(2:106)
To evaluate the Zeldovich nonequilibrium factor, Z¼
1=2 1 q2 (DG ) 2pkT qn2 n¼n*
By substituting Equation 2.106, we get
Z¼
DG* 3pkTn*2
1=2
(2:26)
(2:107)
By substituting Equations 2.100 and 2.101, we get Z¼
3H 2 4(pkTB3 )1=2
(2:108)
By substituting the relationship for B and D from Equation 2.108, we get Z¼
va f2 8p(kTg3 )1=2
(2:109)
To obtain t, the incubation time, from t¼
4kT q2 (DG ) b* qn2 n*
(2:54)
By substituting Equations 2.106 and 2.95, we get t¼
4kT 2DG* 16pg2 Dxb 3n*2 f 2 a4
(2:110)
By substituting Equations 2.100 and 2.101, and then substituting the relationships for B and D from Equation 2.98, we get t¼
8kTga4 n2a f2 Dxb
(2:111)
To evaluate b* again, this time in terms of n rather than r, write S* via Equation 2.98: b* ¼
Dxb Bn*2=3 Dxb B(2B=3H)2 4B3 Dxb 16pg2 Dxb ¼ 4 ¼ ¼ a4 a 9H 2 ga4 g g f 2 a4
(2:112)
Diffusional Nucleation in Solid–Solid Transformations
75
2.10.5 NUCLEATION SITE DENSITY, N For a substitutional solid solution, N¼
No Vb
(2:113)
where No is Avogadro’s number and Vb is the molar volume of the b phase.
2.10.6 TIME-DEPENDENT NUCLEATION RATE Substituting into Equation 2.32 the relationships for the various components, including Equations 2.68 and 2.112, we get J* ¼ b*ZN exp (DG*=kT) exp (t=t)
! na f 2 16pg2 Dxb 16pg3 8kTga4 exp 2 2 N exp 2 ¼ f 2 a4 3f kT na f Dxb t 8p(kTg3 )1=2 ! 2Dxb g1=2 16pg3 8kTga4 exp 2 2 exp 2 ¼ 3f kT na f Dxb t a4 (kT)1=2
(2:114)
where vaN ¼ 1. Note that the pre-exponential diffusivity, D ¼ Do exp(Q=RT), where Q ¼ the activation energy for diffusion, introduces a third exponential. The balance of the pre-exponential factor is clearly not very sensitive to temperature. Attention is again called to the greater potent of g3 than of f2 in the first exponential, as well as to the more important relative role occupied by f2 in the second exponential; the major role of D in the latter exponential is also important. Finally, Equation 2.114 is seen to be free of either arbitrary constants or unevaluable quantities. Each term in this equation can be independently measured or calculated.
2.11 ANCILLARY PARAMETERS Equation 2.114 shows that knowledge of the appropriate diffusivity in the matrix phase, the volume free energy change, the volume strain energy, and the nucleus:matrix interfacial energy is essential to the calculation of J*. Data on the lattice parameter and on the average volume per atom in the nucleus are also required, but are usually readily available and are not subject to major errors. The first four quantities, on the other hand, are less easily obtained, can involve substantial calculation, and provide opportunities for introducing serious errors. These quantities will be discussed individually.
2.11.1 VOLUME DIFFUSIVITY Consideration in this section will be restricted to the question of which volume diffusivity is appropriate in various situations. The discussion of grain boundary and interphase boundary diffusivities will be delayed until calculation of b* for nucleation at grain boundaries is considered; similarly, diffusivity in dislocation pipes will be discussed in conjunction with the section on nucleation at dislocations. When the solute in a binary alloy is interstitial, the diffusivity at composition xb, determined from the Boltzman–Matano analysis, is appropriate. In the case of substitutional diffusion, two different situations must be distinguished. In the first, a disordered or incoherent boundary separates the nucleus and the matrix. A ready source and sink for vacancies is thus available, and the conventional interdiffusion coefficient, again obtained from the Boltzmann–Matano analysis, is applicable: e ¼ xb DA þ (1 xb )DB D¼D
(2:115)
76
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
where e is the interdiffusivity, D DA and DB are the intrinsic diffusivities of A and B.
and
d ln gi * Di ¼ Di 1 þ d ln xi
(2:116)
where Di* is the self-diffusivity of either A or B (as determined, for example, by the radiotracer technique), gi is the activity coefficient of A or B. Equations 2.115 and 2.116 result from the Darken analysis. When the interphase boundary of an embryo is coherent but mobile—a situation usually obtained only when the crystal structure and lattice orientation of the nucleus and matrix phases are the same—the absence of a vacancy source and sink at or within the embryo prevents the maintenance of the equilibrium vacancy concentration in the vicinity. (An embryo or even nucleus is usually far too small to contain a vacancy source or sink in its interior.) For this situation, Russell [19] has derived a ‘‘constrained’’ or ‘‘coherent’’ diffusivity ec ¼ D
DA DB xb DA þ (1 xb )DB
(2:117)
e and D e c differ appreciably only when DA and DB do so. This is unlikely unless A and B Note that D differ sharply in melting point and=or vapor pressure.
2.11.2 VOLUME FREE ENERGY CHANGE This is the free energy change per unit volume of precipitate attending the formation of a minute volume of the precipitate. As described before, DGV is calculated at the nucleus composition. On Equation 2.68, DG* is smallest when, other factors being held constant, DGV takes the most negative value possible. The composition of the nucleus yielding this result is determined on a G–x diagram from the ‘‘parallel tangent’’ construction. A line is first drawn tangent to the Gb–x curve at composition xb, the bulk composition of the alloy. A second line is then constructed parallel to the first and tangent to the Ga–x curve. The composition at the latter point of tangency, xna , is that of the critical nucleus. As shown in the upper figure of Figure 2.11, when Ga–x is narrow, xna xab a . However, the bottom figure shows that a wide Ga–x curve can produce a significant difference between xna and xab a , and also between the DGV’s associated with them. Mathematically, the parallel tangent construction is described by the following relationship: qGa qGb ¼ (2:118) qx xna qx xb
Evaluations of both xna and DGV are readily accomplished if b and a phases are assumed to be regular solutions. Even with this simplification, however, it will be seen that analytic expressions are not obtained. Expressions for the free energy of b at xb and for the free energy of a at xna are differentiated and are equated in accordance with Equation 2.118. By rearranging the yields, we get b!a DGb!a þ B(1 2xb ) þ RT ln DGA B
xb xn ¼ A 1 2xna þ RT ln a n 1 xb 1 xa
(2:119)
77
Diffusional Nucleation in Solid–Solid Transformations
β α
G
ΔG = ΔGVV α Parallel tangent
A
βα
xβ
αβ
xβ
xα
B
xαn
β
n
ΔG at xα = ΔGVV α
α
G
ΔG at xααβ = ΔGVV α
Parallel tangent
A
βα
xβ
xβ
αβ
xα
n
xα
B
FIGURE 2.11 DGv, volume free energy change for nucleation: the top sketch is for the case with a narrow a curve, and the bottom one is for the case with a wide a curve.
When DHib!a and DSb!a are known, this relationship can be solved for xna by trial and error. The i volume free energy change is then obtained by substituting this value of xna in the regular solution version of the volume free energy change equation:
i n 1 h n n n n n 1 x G G G G þ x a Aa Ab Ba Bb a Va h n o 2 1 ¼ a 1 xna GaA þ RT ln 1 xna þ A xna GbA RT ln (1 xb ) B(xb )2 V n o 2 þ xna GaA þ RT ln xna þ A 1 xna GbB RT ln xb B(1 xb )2
DGv ¼
¼
n 2 1 1 xna b!a 2 n DG þ RT ln B(x ) 1 x þ A x b a a A 1 xb Va xna b!a n 2 n 2 þ xa DGB þ RT ln þ A 1 xa B(1 xb ) xb
(2:120)
78
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
The maximum jDGVj criterion for the volume free energy change attending nucleation is appropriate only when the partial molar volumes of A and B in the precipitate phase are equal, i.e., a a V A ¼ V B . Capillarity can change the (unstable) equilibrium composition of the critical nucleus. In such a case, Equation 2.119 and the method of the parallel tangent line construction for jDGVj in Figure 2.11 are not valid. In the G–x diagram of Figure 2.12, the free energy change due to capillarity is again assumed (for convenience) to be concentrated wholly within the precipitate phase, a, and to be P*aVa. To determine xna , the tangent to the Ga curve (at Pa ¼ 0) must be positioned a a so that the intercepts with the x ¼ 0 and x ¼ 1 axes permit DGAr =DGBr to be equal to V A =V B , where two definitions of DGir are given on the diagram [20]. Note that, as in the case of free energies, a tangent line to the molar volume of the a phase, Va, yields to the x ¼ 0 and 1 axes partial molar volumes of A and B, respectively. To find the nucleus composition, xna , from this construction, a
DGAr P*a V A ¼ a DGBr P*a V B
(2:121)
From Figure 2.12, n
b
GA GAa n
b
GB GBa
a
¼
VA a VB
(2:122)
By multiplying both sides by the denominator of the left-hand side, and making the customary substitutions for the partial molar free energies, we get ab þ RT ln nA ¼ DGa!b A aAa
a
VA a VB
ab DGa!b þ RT ln nB B aBa
!
αr*
β G
–β GB α
–β GA – P*αVAα
(2:123)
– – P *αVBα = ΔGB
P *αV α
– = ΔGAr
r
–n GB
α
–n GA
α
A
βα
xβ
xβ
xααβ xαn
B
FIGURE 2.12 Construction for finding the nucleus composition, xna , when the two partial molar volumes are Aa 6¼ V Ba . (From Hillert, M., in Lectures on the Theory of Phase Transformations, TMS-AIME, different, i.e., V Warrendale, PA, p. 1, 1975. With permission.)
79
Diffusional Nucleation in Solid–Solid Transformations
On the ideal solution approximation (1 xb ) a!b ¼ þ RT ln DGA 1 xna
a VA xb a!b DG þ RT ln a B xna VB a
(2:124)
a
This relationship is solved for xna . Note that only when V A ¼ V B , Equation 2.124 becomes Equation 2.119. The precipitate phase that has the most negative total molar free energy change when formed from a given matrix phase is the most stable precipitate, and thus is the one found on equilibrium phase diagrams. As indicated in the foregoing, however, the free energy change associated with nucleation is calculated at the composition of the nucleus rather than at the composition of the bulk alloy prior to transformation as in the case of the total free energy change. As demonstrated in Figure 2.13, this difference can make the free energy change associated with the nucleation of a transition phase, a0 , more negative than that associated with the nucleation of the equilibrium phase, a. Since the total free energy change resulting from the formation of the metastable equilibrium proportion of the transition phase is less negative than is the total free energy change attending precipitation of the equilibrium proportion of the equilibrium phase, the transition a0 phase must eventually disappear. Since transition phases usually exhibit conspicuously better lattice matching to the matrix phase across their interphase boundaries (discussed in more detail in Chapter 4), they have a lower total interfacial energy. The addition of an advantage in DGV would further assist in the nucleation of such a phase in preference, initially to the equilibrium phase.
2.11.3 VOLUME STRAIN ENERGY This energy, W, is also computed on the basis of per unit volume of precipitate. The source of volume strain energy usually considered is that arising from a difference in the average volume of an atom in the precipitate and matrix phases in the absence of elastic constraints. This is known as the dilatational strain energy. A second source that can be even more important but which will be considered only briefly, results from a difference in the structure of the crystals of the two phases (again in the absence of external constraints) and is termed the shear strain energy. Both dilatational and shear strain energies are taken to be solely elastic since the necessary α΄ β α
ΔGn n
ΔG G
A
xββα xβ
αβ
n
xα ≈ xα
xαα΄΄β ≈ xαn΄
B
FIGURE 2.13 Free energy basis for possible nucleation of a transition phase, a0 , before nucleation of the equilibrium phase, a.
80
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
dislocation source(s) for plastic deformation are unlikely to be available because of the very small size of the critical nuclei. 2.11.3.1
Elementary Calculation of Dilatational Strain Energy for a Plate-Shaped Nucleus The mathematics of volume strain energy calculations are very complex. Here only a simple case is treated, using only simple tools. Hereafter, attention will be largely focused on the statements of results obtained from much more elaborate calculations. Consider a precipitate plate wholly imbedded within the matrix phase. Assume that Young’s modulus, E, and Poisson’s ratio, n, are the same in both phases. Furthermore, assume that the misfit is confined to the plane of the plate. (This is physically reasonable, since the misfit is usually much worse around the edges of a plate, hence, the principal volumetric distortions are radial.) An expression for the volume strain energy of a plate under elastic stresses in the x- and y-directions (taken as orthogonal directions in the plane of the plate) is developed as follows. The area under a load (P)-deformation (d) curve is defined as the strain energy of the stressed specimen, see Figure 2.14. The total elastic strain energy, U, is equal to the shaded area: Pd 2
(2:125)
P A
(2:126)
DL d ¼ Lo Lo
(2:127)
U¼ From the definitions of stress, s, and strain, e, s¼ e¼
where A is the area over which the load is applied, Lo is the original length of the specimen, DL ¼ d is the change in length as a result of the application of P. Note that all the deformations are taken to be purely elastic. By substituting into Equation 2.125, we get
P
1 1 U ¼ se(ALo ) ¼ seVo 2 2
δ
FIGURE 2.14
Load, P, against displacement, d.
(2:128)
81
Diffusional Nucleation in Solid–Solid Transformations
where Vo is the original volume of the stressed body. Hence, W, the volume strain energy, is W¼
U 1 ¼ se Vo 2
(2:129)
Specialized to the x-direction 1 W ¼ sx e x 2
(2:130)
If the body is now stressed in the y-direction as well, by the superposition principle 1 W ¼ (sx ex þ sy ey ) 2
(2:131)
where sx is the stress in the x-direction, ex is the strain in the x-direction, sy and ey are analogous quantities for the y-direction. Noting that a strain in the x-direction requires a proportional strain in the y-direction, and vice versa, in order to maintain constancy of the volume, with the constant of proportionality being Poisson’s ratio ex ¼
1 (sx vsy ) E
(2:132)
ey ¼
1 (sy vsx ) E
(2:133)
Now further assume that the stresses, and thus the strains, are the same in the x- and y-directions. Equation 2.131, thus, is reduced to W ¼ se
(2:134)
and Equations 2.132 and 2.133 become e¼
s (1 n) E
(2:135)
Ee 1n
(2:136)
By rearranging Equation 2.135, we get s¼
By substituting Equation 2.136 into Equation 2.134, we find W¼
Ee2 1n
(2:137)
Following the notation of Laszlo [25], whose results will be summarized next, l=n is set equal to m. Also, take e ¼ d, the linear misfit between precipitate and matrix. Thus W¼
mEd 2 m1
(2:138)
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Mechanisms of Diffusional Phase Transformations in Metals and Alloys
2.11.3.2 Volume Strain Energy of Fully Coherent Precipitates When full coherency obtains, the interface plane serves as an integral part of the normal stacking order in the direction orthogonal to the boundary plane for both lattices, and accomplishes this without the need to introduce misfit dislocations in the boundary plane. (An elastic misfit between the two lattices is permitted by this definition; this merely distorts, without fundamental alteration, the interface plane.) The usual definition of full coherency as one-to-one matching of atoms across an interphase boundary, if taken literally, means an A–B–C stacking sequence, for example, in a faced-centered cubic lattice. Eshelby [26] has shown that Equation 2.138 is applicable independently of precipitate morphology if E and n are the same in both phases. When this situation does not obtain, as should normally be the case, W is then a function of morphology. Table 2.1, reproduced from Laszlo [25], gives W for the matrix, the precipitate, and both together for the three basic morphologies—the plate, the cylinder, and the sphere. In this table, i designates the precipitate and o designates the matrix. Laszlo used the shear modulus m¼
mE E ¼ 2(m þ 1) 2(1 þ n)
(2:139)
as a tool for comparing the strain energies of various precipitate shapes. It was concluded that Wplate < Wcylinder < Wsphere when mppt < mmatrix. The strain energies fall in the reverse order when mppt > mmatrix. Barnett et al. [27] reported a complete expression for W of coherent ellipsoids of revolution as a function of their c=a (major=minor) axis ratio. This relationship permitted, as shown in Figure 2.15, plots for various relationships between the shear modulus for the precipitate (m*) and that for the matrix (m). These results are in good agreement with those of Laszlo under circumstances where they can be compared: when the precipitate has the higher shear modulus, the sphere is the minimum strain energy shape, whereas when the precipitate has the lower shear modulus, the disc (or plate) has the minimum W. As already noted, W is seen to be independent of c=a when the elastic constants are the same in both phases. All of the foregoing considerations assume that both the matrix and precipitate phases are elastically isotropic. Using Eshelby’s inclusion model [28], Lee et al. [29] investigated the elastic strain energy of coherent ellipsoidal precipitates in the presence of anisotropic elasticity. Misfit was again considered to be purely dilatational. The elastic moduli used were those for cubic-type * , C12 * , and C44 * in the systems; hence, only the constants C11, C12, and C44 in the matrix and C11 precipitate were required. (Note that the shear modulus, again designated by m and m*, equals C44, Poisson’s ratio, n ¼ C12=2(C12 þ C44), and the Zener anisotropy ratio, A ¼ 2C44=(C11 – C12)). The results are presented by normalizing the W calculated from anisotropic elasticity considerations at a given aspect ratio b ( ¼ c=a) by Wo, the strain energy calculated for assuming both isotropic elasticity and the same elastic constants in both phases, i.e., Equation 2.137. Figure 2.16 presents W=Wo vs. b under conditions of isotropic elasticity, whereas Figure 2.17 shows W=Wo vs. b for assorted mixtures of isotropic and anisotropic elasticity. Figure 2.16 is that of Figure 2.15 recalculated for slightly different values of the elastic constants and rescaled congruently with Figures 2.17 and 2.18. In Figure 2.17, the anisotropy ratio for Cu, A ¼ 3.209, is sometimes employed; this ratio is considered to be relatively high. Both phases are first taken to have this A, then precipitate, and finally the matrix is assumed to have the anisotropic ratio of 1.0 while the other phase retains the A of Cu. A unitary dilatational misfit strain is assumed to be present (eij ¼ dij). Now even when m* ¼ m, the ratio W=Wo is no longer independent of b. In this situation, the disc (b ¼ 0) is the minimum W morphology for two of the conditions studied. In the third case, that of an isotropic precipitate and an anisotropic matrix, the disc becomes the high W morphology: this is because in the case of b ¼ 0, all strain energy is concentrated in the precipitate phase and the original result of isotropic elasticity with A ¼ 1 (see Table 2.1) is recovered. While the behavior of
Row 1.
2.
Strain Energy
Shape of Precipitate Lamellar
Total
mi Ei 2 l mi 1
Precipitate
mi Ei 2 l mi 1
Cylindrical 3 mi þ 1 1 mo þ 1 þ 2 mi Ei 2 mo Eo l2 mi þ 1 mi 2 1 mo þ 1 þ mi E i mi E i Ei mo Eo " # 1 mi 2 mi þ 1 2 mi þ 1 mo þ 1 1 mo þ 1 2 þ 3 þ2 2 mi E i mi E i mi E i m o E o 2Ei mo Eo l2 mi þ 1 mi 2 1 mo þ 1 2 þ mi Ei mi Ei Ei mo Eo 2 mi þ 1 mo þ 1 mi E i mo E o 2 l mi þ 1 mi 2 1 mo þ 1 2 þ mi E i mi E i E i mo Eo Ei 9 mo Eo þ l2 2 4 mo þ 1
Spherical 3 l2 mi 2 mo þ 1 2 þ mi Ei mo Eo mi 2 mi Ei 2 l mi 2 mo þ 1 2 2 þ mi E i mo Eo 6
mi þ 1 mi Ei 2 l mi 2 mo þ 1 2 þ 2 mi E i mo Eo 3
3.
Matrix
Nought
4.
Total for mi ¼ 2
2Eil2
5.
Precipitate for mi ¼ 2
2Eil2
Ei 2 l 2
Nought
6.
Matrix for mi ¼ 2
Nought
9 mo Eo 2 l 4 mo þ 1
3
Source: Laszlo, F., J. Iron Steel Inst., 164, 5, 1950. With permission. o ¼ matrix phase i ¼ precipitate phase
3
Diffusional Nucleation in Solid–Solid Transformations
TABLE 2.1 Strain Energy for Lamellar, Cylindrical, and Spherical Precipitates
mo E o 2 l mo þ 1
mo E o 2 l mo þ 1
83
84
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
ν = ν* = 0.291 μ = 8.6 × 1011 dynes/cm2
W/W0
3.0
2.0 μ* = 3μ
μ* = μ
1.0
μ* = μ/3
0.0 0.0 1.0 2.0 (Disc) (Sphere)
3.0
5.0
4.0
6.0
∞ (Needle)
7.0
β (= c/a)
FIGURE 2.15 A variation of the volume strain energy of an ellipsoid of revolution as a function of the aspect ratio, b ¼ c=a. m and n are the shear modulus and Poisson’s ratio for the matrix phase, while m* and n* are those of the precipitate phase. (From Barnett, D.M. et al., Scr. Metall., 8, 1447, 1974. With permission.)
A* = 1.00 = A ν* = 0.308 = ν μ = 7.54 × 1011 dynes/cm2
2.0
1.5 W/W0
μ* = 3μ
μ* = μ
1.0
μ* = μ/3 0.5
0
0
0.5
β
1.0
∞
FIGURE 2.16 Normalized strain energy of an isotropic coherent ellipsoidal precipitate in an isotropic matrix vs. aspect ratio. A, n, and m are anisotropy ratio, Poisson’s ratio, and shear modulus, respectively. (With kind permission from Springer Science þ Business Media: From Lee, J.K. et al., Metall. Trans., 8A, 963, 1977.)
85
Diffusional Nucleation in Solid–Solid Transformations
ν* = 0.308 = ν μ = 7.54 × 1011 dynes/cm2
2.0
A* A 3.209 3.209 1.000 3.209 3.209 1.000
W/W0
1.5
μ* = 3μ 1.0
μ* = μ 0.5
μ* = μ/3 0
0
1.0
0.5
∞
β
FIGURE 2.17 Normalized strain energy of a coherent precipitate vs. aspect ratio. The solid lines indicate an anisotropic precipitate in an anisotropic matrix and the dotted lines show the results for an isotropic precipitate in an anisotropic matrix. The broken lines indicate an anisotropic precipitate in an isotropic matrix (the anisotropy ratio is either 3.209 or 1.0). (With kind permission from Springer Science þ Business Media: From Lee, J.K. et al., Metall. Trans., 8A, 963, 1977.)
m* ¼ m=3 curves is qualitatively the same as in the isotropic elasticity case in that all three fall off as b ! 0, when the trend of m* ¼ 3m, the isotropic results are reversed when the precipitate is anisotropic and the matrix is not. In Figure 2.18, two sets of calculations were repeated for A ¼ 1.215, the anisotropy ratio for aluminum; this ratio is considered to represent mild elastic anisotropy. Under the conditions investigated, the results were similar to those for complete elastic isotropy. These calculations were next extended to anisotropic precipitates with cubic-type crystal structures in Al and Cu matrices with various orientation relationships. The elastic constants of the precipitates are given in Table 2.2, while the orientation relationships assumed are given in Table 2.3. Note that the orientation relationships are designated by numbers, to be used in the graphs that follow. Figure 2.19 shows the results of calculations for a pure Cu precipitate in a pure Al matrix, using the correct A for each, under the various orientation relationships tabulated. A remarkable range of behaviors is displayed. For orientation relationships (ORs) 4–7, the sphere is the minimum W morphology. The ORs 1–3, however, make the disc the minimum W morphology, with OR #1 having a slight advantage over the other two. This relationship corresponds to the parallel cube planes in the two lattices. The calculations for this figure are a good simulation of the precipitation of pure Cu GP zones in dilute Al-rich Al–Cu matrices where the zones do form as discs and have the predicted OR [29]. As in the case of isotropic elasticity, the volume strain energy associated with the formation of an elastically anisotropic precipitate resides wholly within the precipitate when b ! 0, i.e., when the precipitate is a vanishingly thin disc. Thus, W for this case depends upon the elastic constants only of the precipitate. However, when the precipitate is elastically anisotropic, the applicable elastic
86
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
ν* = 0.341 = ν μ = 2.85 × 1011 dynes/cm2
2.0
A* A 1.215 1.215 1.000 3.209 1.5
W/W0
μ* = 3μ
1.0
μ* = μ
μ* = μ/3 0.5
0
0
1.0
0.5
∞
β
FIGURE 2.18 Normalized strain energy of a coherent precipitate vs. aspect ratio. The solid lines indicate an anisotropic precipitate in an anisotropic matrix, while the broken lines are for an isotropic precipitate in an anisotropic matrix (the anisotropy ratio is either 1.215 or 1.0). (With kind permission from Springer Science þ Business Media: From Lee, J.K. et al., Metall. Trans., 8A, 963, 1977.)
constants depend upon the orientation relationships between the ellipsoidal principal axes and those of the precipitate lattice. The strain energy of the precipitate under this circumstance is independent of the matrix elastic constants. This is demonstrated in Figure 2.20, where W for Ag precipitates imbedded in Cu and in Al under various OR conditions is calculated vs. b. Note that for TABLE 2.2 Elastic Constants of Some Cubic Elements Crystal Al Cu Au Nb Cr KCl Ag Fe
C11
C12
C44
10.82 16.84 18.60 24.60 35.00 3.98 12.40 24.20
6.13 12.14 15.70 13.40 5.78 0.62 9.34 14.65
2.85 7.54 4.20 2.87 10.10 0.625 4.61 11.20
A¼
2C44 C11 C12 1.215 3.209 2.897 0.513 0.691 0.372 3.013 2.346
Source: With kind permission from Springer Science þ Business Media: Lee, J.K. et al., Metall. Trans., 8A, 963, 1977. Unit: 1011 dyn=cm2ð¼ 10 GPa Þ.
87
Diffusional Nucleation in Solid–Solid Transformations
TABLE 2.3 Orientation Relationships Examined Ellipsoid Axes Number 1. 2. 3. 4. 5. 6. 7.
Remarks
a
b
c
[100]m[100]i [101]m[100]i [121]m[010]i [101]m[111]i [121]m[101]i [101]m[101]i [101]m[332]i
[010]m[010]i [121]m[010]i [111]m[001]i [121]m[101]i [111]m[121]i [121]m[121]i [121]m[113]i
[001]m[001]i [111]m[001]i [101]m[100]i [111]m[121]i [101]m[111]i [111]m[111]i [111]m[110]i
(111)==(0001)
Nishiyama OR
Source: With kind permission from Springer Science þ Business Media: From Lee, J.K. et al., Metall. Trans., 8A, 963, 1977. i ¼ ellipsoidal precipitate phase. m ¼ matrix phase.
1.100 4,7
1.095
5,6
5,6
4,7
1.090 S
W/W0
1.085
2, 3
1.080
1 2, 3
1.075 1 Cu in Al
1.070 1.065 1.060 0.2
0.4
0.6
0.8
1.0 β
1.2
1.4
1.6
1.8
FIGURE 2.19 Normalized strain energy of a Cu coherent precipitate in an Al matrix vs. aspect ratio under various orientation relationships. (With kind permission from Springer Science þ Business Media: From Lee, J. K. et al., Metall. Trans., 8A, 963, 1977.)
each of the ORs examined, the variation of W with b is very different for the two matrices. However, at each OR the same value of W is finally reached when b ! 0. In general, the minimum W morphology under conditions of anisotropic elasticity (for both matrix and precipitate) was found to be controlled in a manner similar to those found earlier by Laszlo [25] for isotropic elasticity. Thus, discs are favored when the precipitate has the lower bulk modulus, (C11 þ 2C12)=3, (this situation is termed one of being elastically ‘‘softer’’), and the sphere
88
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
5
1
S
5 4, 7
16 7
W/ε 2 (1011 ergs/cm3)
Ag in Cu Ag in Al 14
4
S 12 1
5 4, 7
1 10
8 0
0.4
0.8
1.2
1.6
β
FIGURE 2.20 Normalized strain energy of an Ag precipitate in a Cu and in an Al matrix vs. b for different orientations. Each number indicates an orientation relationship listed in Table 2.3. The S denotes a spherical precipitate. (With kind permission from Springer Science þ Business Media: From Lee, J.K. et al., Metall. Trans., 8A, 963, 1977.)
is favored when the precipitate is elastically ‘‘harder,’’ i.e., has a higher bulk modulus. When these differences in bulk moduli are pronounced, the indicated morphologies are favored regardless of orientation relationships. Only when the difference is relatively small is W crucially affected by other factors such as the OR. Two hcp elements, two tetragonal elements, and an hcp compound, Ag2Al, were also considered in Al and in Cu matrices. The non-cubic crystal structures and highly anisotropic elastic constants (summarized in Table 2.4), led to W values even more strongly dependent upon b and OR. This is TABLE 2.4 Elastic Constants of Some Non-Cubic Elements investigated Crystal Mg (hcp) Zn (hcp) Ag2Al (hcp) Sn (tetrag) In (tetrag)
C11
C12
C13
C33
C44
C66
5.649 16.368 14.15 8.60 4.45
2.316 3.64 8.47 3.50 3.95
1.810 5.30 7.46 3.00 4.05
5.873 6.347 16.85 13.30 4.44
1.681 3.879 3.408 4.90 0.655
(C11-C12)=2 (C11-C12)=2 (C11-C12)=2 5.30 1.22
Source: With kind permission from Springer Science þ Business Media: Lee, J.K. et al., Metall. Trans., 8A, 963, 1977. Unit: 1011 dyn=cm2ð¼ 10 GPa Þ.
89
Diffusional Nucleation in Solid–Solid Transformations 1.030 Ag2 Al in Al
1.025 1, 2
1, 2
W/W0
5, 6 3, 7 1.020 4 5, 6
1.015
4
S
3, 7 1.010 0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
β
FIGURE 2.21 Normalized strain energy of an Ag2Al precipitate in an Al matrix vs. b. (With kind permission from Springer Science þ Business Media: From Lee, J.K. et al., Metall. Trans., 8A, 963, 1977.)
illustrated for the Ag2Al-in-Al case in Figure 2.21. (Precipitation of this compound from Al-rich Al–Ag alloys is an interesting research subject because of its clear and simple behavior.) While for cubic-type precipitates a spherical shape is always either a maximum or a minimum W morphology, this generalization does not hold true for non-cubic precipitates. For example, notice that the precipitate morphology with an extremum W occurs either at 0 < b < 1 or 1 < b < 1.6. 2.11.3.3 Volume Strain Energy of Incoherent Precipitates In this situation, the precipitate–matrix boundary is considered to be disordered or incoherent, with an atomic structure depending upon local, compromised atomic arrangements and not giving rise to long-range elastic strains. Such a boundary has a relatively ‘‘open’’ structure. This case was first analyzed by Nabarro [30], and was later further investigated by Kröner [31] and by Lee and Johnson [32], using anisotropic elasticity. Figure 2.22, from Lee and Johnson, shows that variations of strain energy with aspect ratio are quite different from those characteristic of coherent precipitates. Cu, Al, and KCl precipitates were taken to be imbedded in a Cu matrix. W is seen always to become zero at b ¼ c=a ¼ 0 (thin disc) and to pass through a maximum at c=a ¼ 1 (sphere). The height of the maximum is found to increase with increasing values of the elastic constants. It was reasoned [30] that qualitatively the vanishing of W at c=a ¼ 0 is the yielding of the faces of a very thin, incoherent disc to internal pressure by elastically ‘‘popping out,’’ as sketched in Figure 2.23. Equivalently, the excess material at the broad faces can be thought of as diffusing along the incoherent broad faces to the disc edges, which now become still thinner and make a contribution to W, which tends to zero as b approaches zero. An alternative view is that an oversized liquid droplet spreads along its perimeter under a hydrostatic pressure. Russell [33] has questioned the relevance of these calculations to nucleation. DG* of the successful nuclei will always be the lowest possible value (as long as DG* is less than ca. 60 kT).
90
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 0.30 Cu 0.25 AI Cu
W/(C44 ε2)
0.20 AI 0.15
0.10
KCI
0.05
0
KCI
Cu matrix
0
0.2
0.4
0.6
0.8 β
1.0
1.2
1.4
∞
FIGURE 2.22 Normalized elastic strain energy, W=(C44e2), per unit volume of the incoherent precipitate vs. the aspect ratio, b ¼ c=a, for the Cu precipitate, Al precipitate and KCl precipitate in a Cu matrix. (Reprinted from Lee, J.K. and Johnson, W.C., Acta Metall., 26, 541, 1978. With permission from Elsevier.)
FIGURE 2.23 A schematic diagram for a disklike (b ! 0) incoherent precipitate embedded in a matrix phase.
Thus, when the average volume of an atom within the nucleus is greater than that within the matrix, W can be eliminated because of the migration of a sufficient number of vacancies to the nucleus, where they are destroyed at the incoherent boundaries while relieving strain energy. In the inverse situation, migration of atoms toward the nucleus will accomplish the same purpose. 2.11.3.4 Volume Shear Strain Energy Eshelby [26] found that even when the matrix and precipitate have the same elastic constants the volume shear strain energy does depend upon the morphology. For a coherent, oblate ellipsoid, this energy is given by 2 E eT13 p(2 v) c W¼ (1 þ v) 4(1 v) a
(2:140)
where c and a are plate thickness and diameter, eT13 is the stress-free tensor shear strain, which is half the engineering shear strain. By making c=a sufficiently small, W can be greatly reduced, but at the
Diffusional Nucleation in Solid–Solid Transformations
91
cost of a marked increase in interfacial energy, often a more important factor in determining the nucleation rate. Shear strain energy is probably a dominant factor in the energetics of the martensitic type of phase transformation. 2.11.3.5 Unsolved Major Problems in Volume Strain Energy Despite much effort and progress in this field, much remains to be done. The strain energy associated with a faceted sphere, a likely precipitate morphology during homogeneous nucleation, has yet to be analyzed for the case of anisotropic elasticity. The influence of even a disordered grain boundary upon volume strain energy associated with nucleation, in any morphology, at that boundary also remains to be studied. Effects upon volume strain energy when a portion of the nucleus or precipitate has a coherent interfacial structure and the remainder has a disordered structure also require investigation. Finally, the effects upon W of replacing a fully coherent with a partially coherent (i.e., periodically spaced misfit dislocations along an otherwise fully coherent boundary) need to be understood. Much research of great importance thus remains in this area for the metallurgically attuned and mathematically able!
2.11.4 INTERFACIAL ENERGY 2.11.4.1 Scope For homogeneous nucleation, information is needed only on the energies of interphase boundaries. When nucleation at grain boundaries, a major interest in this course, is considered, data on grain boundary energies is also required. The primary focus of attention here, however, will be on the interphase boundaries. The following is a brief summary of the various major types of interphase boundaries, which include: (1) Coherent—a plane at the interphase that is part of the stacking order of both crystals forming the boundary, with only elastic distortions in the boundary region needed to accommodate the mismatch between the two lattices. (2) Partially coherent—same as coherent, except that the mismatch between the two lattices is sufficient so that it must be periodically ‘‘absorbed’’ by one or more arrays of periodically spaced misfit dislocations in the plane of the boundary. The resultant consolidation of the misfit permits the remainder of the boundary area to be described as fully coherent without severe elastic distortions except in the vicinity of the misfit dislocations. (3) Disordered or incoherent—matching is sufficiently poor between the two lattices at the particular lattice and boundary orientations describing this interface so that atomic positions are compromises between the demands of the two bulk lattices. There should be no long-range elastic strains associated with this type of boundary (unlike the coherent boundary, where they can be quite long range, or the partially coherent boundary, where the elastic strains extend to the order of the inter-dislocation spacing). However, by analogy to high-angle grain boundaries, it is quite possible that the boundary structure may be resolved into a succession of polyhedra of a limited number of types, each containing but a few atoms. Data on coherent interphase boundary energies are still largely, if not entirely, lacking. However, considerable work has been done on the calculation of these energies; the essentials of this are presented in some detail in the next section. The position with respect to the energies of partially coherent interphase boundaries is similar. However, the structure and energetics of these boundaries when they are large enough in extent to ensure that an equilibrium structure can at least be approached is a topic that belongs properly in the province of Chapter 3. Here, only the energy of an interphase boundary that has been made partially coherent by the addition of a single dislocation loop will be briefly noted. In the case of disordered interphase boundaries, theory is largely lacking but there is extensive experimental data on the energies of these boundaries relative to those of disordered grain boundaries in the same system, as well as some data on the absolute energies of disordered interphase boundaries. This literature data will be summarized, and, in conjunction with it, descriptions will be given of the basic experimental technique for the
92
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
measurement of the energies of interphase boundaries. In the case of grain boundaries, the theoretical and experimental literature on their structure and energies are much larger. It will be necessary to restrict attention here to the experimental data per se and especially to correlations that have been developed among these data. 2.11.4.2
Energy of Coherent Interphase Boundaries
2.11.4.2.1 Two-Plane Discrete Lattice Model The relationship used almost universally among metallurgists to describe the energy of a coherent interphase boundary is from Becker [34]. Although the derivation will be shown in the next section the results are found to yield interfacial energies that are significantly too high except for the lowest temperatures. Nonetheless, both its continued use and the applicability of methodology to a more advanced treatment make presentation of this treatment desirable in some detail. This treatment assumes that both phases, a and b, are compositionally homogeneous throughout and are ‘‘welded’’ together without alteration in the composition of the two planes forming the boundary. The usual assumption is made that interaction is restricted to the nearest neighbors. The lattice parameter is tacitly assumed to be constant throughout both phases, thereby avoiding the need to incorporate a strain energy contribution. In this situation, Aaronson et al. [35] have shown that there is no positional entropic component of the interfacial energy; hence, consideration can be restricted to contributions to the interfacial enthalpy. Because full coherency is assumed, there is no ‘‘structural’’ contribution to the interfacial energy (or more accurately to the interfacial free energy). Only a ‘‘chemical’’ contribution is present. This contribution results from the difference in the bond enthalpy an atom at the ‘‘surface’’ of a given phase experiences relative to the bond enthalpy the atom encounters in the interior of the phase. The proportion of the nearest neighbors of a given species that a ‘‘surface’’ atom sees is different from that seen by an ‘‘interior’’ atom. Finally, the boundary is assumed for convenience to be planar and of infinite extent. Let HAA be the enthalpy of an A–A bond, HBB be the enthalpy of a B–B bond, and HAB be the enthalpy of an A–B bond. Now consider first the interfacial enthalpy (¼ interfacial free energy) of a boundary between two a crystals of identical composition and orientation, i.e., of a boundary within the a phase of an a:b bicrystal. The atom fraction of B in a is xa and the atom fraction of A is thus (1 – xa). The number of A–A pairs that are the nearest neighboring ones across this boundary are the product of two terms: (1) ns(1 – xa)—the number of A atoms per unit area of boundary in one of the two a crystals; and (2) zs(1 – xa)—the product of zs, the interfacial coordination number or the number of atoms (of both species) in the boundary plane of the second a crystal, which is the nearest neighbor to a given atom in the interface plane of the first a crystal, and (1 – xa), the proportion of these nearest neighboring atoms, which is of species A. Hence, the enthalpy contributed by the A–A pairs per unit area of boundary is equal to nszs(1 – xa)2HAA. Similarly, the enthalpy arising from the B---B pairs is equal to ns zs x2a HBB . The enthalpy contributed by the A–B pairs is the sum of two pairs (one for B–A pairs and the other for A–B Pairs) and has two terms: nsxazs(1 – xa)HAB and ns(1 – xa)zsxaHAB. The total enthalpy per unit area of the a:a boundary is given by the sum of all the foregoing contributions: Eaa ¼ ns zs (1 xa )2 HAA þ ns zs x2a HBB þ 2ns zs xa (1 xa )HAB ¼ ns zs (1 xa )2 HAA þ x2a HBB þ 2xa (1 xa )HAB
(2:141)
h i Ebb ¼ ns zs (1 xb )2 HAA þ x2b HBB þ 2xb (1 xb )HAB
(2:142)
Equivalently, for a boundary between two b crystals of identical composition of xb,
93
Diffusional Nucleation in Solid–Solid Transformations
The enthalpy of an a:b boundary has just the same form except that in every term one xa is replaced by an xb (or vice versa): Eab ¼ ns zs (1 xa )(1 xb )HAA þ xa xb HBB þ {xa (1 xb ) þ xb (1 xa )}HAB
(2:143)
The interfacial energy of a coherent a:b boundary is then given by Eaa þ Ebb gc ¼ Eab 2
(2:144)
By substituting Equations 2.141 through 2.143, we get
1 1 gc ¼ ns zs (xa xb ) HAB (xa xb )2 HAA (xa xb )2 HBB 2 2 1 ¼ ns zs (xa xb )2 HAB (HAA þ HBB ) 2 2
(2:145)
To evaluate HAB (HAA þ HBB)=2, we assume a random solid solution (as is assumed in the case of a regular solution) and express the enthalpy, H, in terms of the number of pairs, PAA, PBB, and PAB of each type and the corresponding bond enthalpies: H ¼ PAA HAA þ PBB HBB þ PAB HAB
(2:146)
If Z is the coordination number in the bulk solid solution, NA is the number of A atoms present and NB is the number of B atoms present: 2PAA þ PAB ¼ ZNA
(2:147)
2PBB þ PAB ¼ ZNB
(2:148)
By substituting Equations 2.147 and 2.148 into Equation 2.146 in terms of PAA and PBB, we get 1 1 H ¼ (ZNA PAB )HAA þ (ZNB PAB )HBB þ PAB HAB 2 2 1 1 1 ¼ ZNA HAA þ ZNB HBB þ PAB HAB (HAA þ HBB ) 2 2 2
(2:149)
The molar enthalpy of mixing is equal to the enthalpy of one mole of solid solution minus the enthalpies of the component species in the pure state. Hence, 1 DH M ¼ PAB HAB (HAA þ HBB ) 2
(2:150)
Also, on a molar basis, PAB
2No ¼ x(1 x)ZNo ¼ 2x(1 x) 2
(2:151)
94
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
where No is Avogadro’s number. The combined probability of an A atom having a B atom as a nearest neighbor and a B atom having an A atom as its nearest neighbor is 2x(1 – x) and the number of bonds (avoiding double counting) is ZNo=2 per mole. Substituting into Equation 2.150, 1 DH M ¼ x(1 x)ZNo HAB (HAA þ HBB ) 2
(2:152)
ZNo HAB 12 (HAA þ HBB ) is the molar regular solution constant, V. Use of V in Equation 2.145 yields for the expression of coherent interfacial free energy: 2
gc ¼
ns zs (xa xb ) V ¼ ZNo
2 ba V ns zs xab x a b ZNo
(2:153)
2.11.4.2.2 Multi-Plane Discrete Lattice Model On this approach, the composition change encountered in passing from one phase to the other does not take place abruptly but is instead spread out over several planes when T > 0 K. However, a specific composition is found for each plane parallel to the interphase boundary. Hence, the gradual transition in composition takes place stepwise rather than in a continuous fashion. This approach was first developed by Ono [36]. It was later used by Hillert [37] to make a discrete lattice analysis of a spinodal decomposition. Wynblatt and Ku [38] employed a version of this approach to analyze the analogous problem of solute segregation to a free surface. Lee and Aaronson [39] further developed the Wynblatt–Ku work and applied it to calculate the concentration profile normal to coherent a:b boundaries and the energy of these boundaries. Consider that a block of random solid solution a phase that is homogeneous through the surface layer is joined to a similarly described block of b phase. The two phases are taken to have the same crystal structure and lattice orientation, and the interface between them is assumed to be flat and fully coherent. The effects of strain energy are eliminated by setting the lattice parameters of the two phases independent of composition and equal. Both a and b are assumed to be regular solid solutions whose compositions correspond to the phase boundaries of a symmetrical coherent miscibility gap. Since the abrupt change in composition across the interface will be shown not to be an equilibrium configuration, atomic rearrangements are required to reach the equilibrium state. Solute atoms must segregate toward or away from the interface region in order to reach this state. Since full coherency is assumed, the atoms displaced from the interface region must be removed to the interior of the a or b crystal, and counterpart movements must occur, on a one-for-one basis, from the interior to the interface region. When these atomic interchanges have reached the stage at which the rate of free energy change with respect to solute concentration at each lattice plane in the segregated region about the interface has become zero, the equilibrium state has been achieved. Hence, the problem is reduced to calculating DH and DS, the enthalpy and entropy changes accompanying the net atomic movements required to attain equilibrium. As the free energy change per atom accompanying the atomic movements from homogeneous b to the equilibrium state can be shown to be equal to that from homogeneous a to the equilibrium configuration on the regular solution model, homogeneous a is taken, for convenience, to be the initial or reference state. The a:b couple is assumed to consist of Q solute (B) atoms in N atomic sites; all solute atoms are taken to be substitutionally dissolved. The boundary region is described as those atomic planes whose solute concentration differs from that of either homogeneous phase. When an equilibrium is achieved throughout the system, the ith layer of the boundary region contains qi of B atoms per unit area. Thus, xi ¼
qi ns
(2:154)
95
Diffusional Nucleation in Solid–Solid Transformations
where xi is the atom fraction of solute in the ith boundary layer, ns is the total number of atoms per unit area in this layer. The movement of solute atoms from the bulk to the boundary region requires that an equal number of solvent atoms move in the opposite direction because the system, being coherent, is closed. The excess number of B atoms per unit area of the ith atomic plane relative to homogeneous a is simply ns xi xab a
(2:155)
Figure 2.24 shows schematically how the nearest neighboring atoms are distributed around an atom, p, in the ith boundary plane. Zl is the horizontal or lateral coordination number within the ith plane and Zj is the coordination number to the nearest neighboring atoms in the jth plane. Thus, the total coordination number is Z ¼ Zl þ 2
X
Zj
(2:156)
j
To compute the bond enthalpy difference between a B atom in the ith boundary plane and a B atom in homogeneous a, it is first noted that the number of bonds around a B atom in the ith plane is Number of B---B bonds ¼ xi Zl þ (xiþ1 þ xi1 )Zl þ (xiþ2 þ xi2 )Z2 þ X ¼ xi Zl þ (xiþj þ xij )Zj
(2:157)
Number of A---B bonds ¼ (1 xi )Zl þ
(2:158)
j
X j
[(1 xiþj ) þ (1 xij )]Zj
(h k l) Xi+j
Zj
Xi+2
Z2 (i)
Zl/2
Z1 P
Z1
Zl/2
Xi+1 = qi+1/ns Xi = qi/ns Xi–1 = qi–1/ns
Z2
Xi–2
Zj
Xi–j
FIGURE 2.24 A schematic diagram showing the distribution of Z nearest neighbors about P in adjacent planes parallel to (hkl). Zj is the unidirectional vertical coordination number to the jth plane as determined from plane i. (Reprinted from Lee, Y.W. and Aaronson, H.I., Acta Metall., 28, 539, 1980. With permission from Elsevier.)
96
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
The number of bonds around a B atom in bulk a is Number of B---B bonds ¼ xab a Z Number of A---B bonds ¼ 1 xab a Z
(2:160)
" # X 1 ab (xiþj þ xij )Zj xa Z (HBB HAB ) xi Zl þ 2 j
(2:161)
(2:159)
The required bond enthalpy difference is thus the sum of Equations 2.157 and 2.158 minus the sum of Equations 2.159 and 2.160, with a multiplying factor of 1=2 applied to cancel double counting of the number of bonds:
Equation 2.156 is used to obtain Equation 2.161. Similarly, the bond enthalpy change undergone by an A atom displaced from the ith boundary plane to the bulk is " # X 1 ab (xiþj þ xij )Zj xa Z (HAA HAB ) xi Zl þ 2 j
(2:162)
The sum of Equations 2.161 and 2.162 is the enthalpy change, Dhi, associated with the segregation of a B atom to the ith boundary plane: " # X 1 ab (xiþj þ xij )Zj xa Z (HAA þ HBB 2HAB ) xi Zl þ Dhi ¼ 2 j " # X HAA þ HBB ab (xiþj þ xij )Zj HAB ¼ xa Z x i Z l 2 j " # X (xiþj þ xij )Zj DE ¼ xab a Z xi Z l
(2:163)
j
where DE ¼ HAB (HAA þ HBB)=2. The influence of the boundary orientation upon the enthalpy change is exerted through Zj and Zl; the evaluation of these quantities will be considered shortly. The total enthalpy change required to reach the equilibrium state from the a reference state is thus on Equation 2.155: DH ¼ ns
X i
xi xab a Dhi
(2:164)
where S denotes the summation over the entire boundary region. The entropy change, DS, attending segregation is the configurational entropy difference between the equilibrium state and the reference state:
DS ¼ k
( X i
) ns ! (N Lns )! N! P P þ ln ln ln qi !(ns qi )! (Q i qi )!(N Lns Q þ i qi )! Q!(N Q)!
(2:165)
97
Diffusional Nucleation in Solid–Solid Transformations
where L is the number of boundary atomic layers. Applying Stirling’s approximation, ln y! y ln y – y, DS X ¼ [ns ln ns qi ln qi (ns qi ) ln (ns qi )] k i X X X X Q qi qi ln N Lns Q þ qi N Lns Q þ qi ln Q i
i
i
i
N ln N þ Q ln Q þ (N Q) ln (N Q) þ (N Lns ) ln (N Lns )
(2:166)
Since the boundary region is narrow relative to the bulk of the a and b crystals, the following approximations are reasonable: ln (N Lns ) ln N X qi ) ln Q ln (Q i
ln (N Lns Q þ
X i
qi ) ln (N Q)
By applying these approximations and then consolidating terms, X ns qi qi N Q qi Q þ ln DS kns ln ln þ ln N ns qi ns ns qi ns N Q i
(2:167)
Since ns 1 1 ¼ ¼ ns qi 1 qi =ns 1 xi
qi qi =ns xi ¼ ¼ ns qi 1 qi =ns 1 xi NQ Q ¼ 1 ¼ 1 xab a N N
Q Q=N xab a ¼ ¼ N Q 1 Q=N 1 xab a Rearranging and consolidating terms, DS ¼ kns
X
xi ln
i
xi xab a
þ (1 xi ) ln
1 xi
1 xab a
(2:168)
The equilibrium solute distribution in the boundary region is achieved when qDG qDH qDS ¼ T ¼0 qxi qxi qxi
(2:169)
Recall that DH ¼ ns Dhi ¼
"
xab a Z
X i
xi Zl
xi xab a Dhi
X j
(2:164) #
(xiþj þ xij )Zj DE
(2:163)
98
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Since Dhi is a function of xij, . . . , xi1, xi, xiþ1, . . . , xiþj, the terms in DH must be considered from the layer i j to the layer i þ j when taking the partial differential of DH with respect to xi, q xijþ1 xab 1 qDH q xij xab a Dhij a Dhijþ1 ¼ þ þ ns qxi qxi qxi q xi xab q xiþ1 xab a Dhi a Dhiþ1 þ þ þ qxi qxi q xiþj1 xab q xiþj xab a Dhiþj1 a Dhiþj þ þ qxi qxi ab ¼ xij xab a Zj DE xijþ1 xa Zj1 DE ab þ Dhi xi xab a Zl DE xiþ1 xa Zi DE ab xiþj1 xab a Zj1 DE xiþ1 xa Zj DE " # X ab (xiþj þ xij )Zj DE ¼ 2Dhi ¼ Dhi þ xa Z xi Zl
(2:170)
j
Differentiating the TDS term with respect to xi, T qDS 1 1 ¼ kT ln 1 þ ln ab 1 ns ns qxi xi xa
(2:171)
From Equations 2.169 through 2.171, we get " # X 1 qDG 1 1 ab xiþj þ xij Zj DE kT ln ¼0 ¼ 2 xa Z xi Zl 1 ln ab 1 ns qxi xi xa j
(2:172)
This is a set of difference equations that must be solved simultaneously for xi. Interfacial energy is by definition the free energy difference between an equilibrated mixture of a þ b and the homogeneous, continuous a or b phase with the equilibrium composition. Hence, DG ¼ DH TDS
(2:173)
represents the coherent boundary energy when the equilibrium concentration profile is invoked from the solutions to Equation 2.172. To place Equation 2.173 into a usable Pform, the following steps are taken. By rewriting Equation 2.163 by subtracting and adding 2xi j Zj to the term in the square brackets, we get Dhi ¼ ¼
"
"
xab a Z
xi Zl 2xi
xab a xi Z þ
X j
X j
Zj þ 2xi
X j
Zj
X
#
j
2xi xiþj xij Zj DE
#
xiþj þ xij Zj DE (2:174)
99
Diffusional Nucleation in Solid–Solid Transformations
where Equation 2.156 is employed. By substituting into Equation 2.164, we get X xi xab DH ¼ ns a Dhi i " # X X ab ab xi xa 2xi xiþj xij Zj DE ¼ ns x a xi Z þ i
¼ ns
X i
"
j # ab 2 X ab 2xi xiþj xij Zj DE xa x i Z þ x i x a
(2:175)
j
In the computer calculations needed to evaluate the xi from Equation 2.172, it is necessary to assume ba that xi becomes identical to xab a or xb beyond a boundary region of a predetermined number of planes thick. To take into account the effects of this assumption, the summation of i in Equation 2.175 is extended from the present range of i ¼ 1 to i ¼ i out to the range i ¼ i j to i ¼ i þ j. This does not change DH since there is no solute segregation in the extended regions. Some simplification of Equation 2.175 does result, however, through the second term in square brackets. Consider first the second component of this term: XX xab a (2xi xiþj xij )Zj j
i
¼
xab a
X i
[(2xi xiþ1 xi1 )Z1 þ (2xi xiþ2 xi2 )Z2 þ þ (2xi xiþj xij )Zj ]
As a sample, perform a representative portion of the summation over i for j ¼ l, X X X X xi1 ¼ 0 xiþ1 xi (2xi xiþ1 xi1 )Z1 ¼ Z i
i
i
(2:176)
(2:177)
i
Hence, all terms cancel as long as the i ¼ 1 and I planes are taken far from the boundary region. The same result is obtained for all j. Now examine the first component of the foregoing term: X XX [xi (2xi xiþ1 xi1 )Z1 þ xi (2xi xiþ2 xi2 )Z2 þ þ xi (2xi xiþj xij )Zj ¼ i
i
j
xi (2xi xiþj xij )Zj ]
(2:178)
As before, perform a representative portion of the summation over i for j ¼ l, when i ¼ i 1, i ¼ i, i ¼ i þ 1, and i ¼ i þ 2. Hence, X xi (2xi xiþ1 xi1 )Z1 ¼ Z1 ½ þ x2i1 þ x2i1 2xi xi1 þ x2i þ x2i i
2xi xiþ1 þ x2iþ1 þ x2iþ1 þ X (xi xiþ1 )2 Z1 ¼
(2:179)
i
For the first component as a whole, XX xi (2xi xiþj xij )Zj i
j
¼
¼
X i
(xi xiþ1 )2 Z1 þ (xi xiþ2 )2 Z2 þ þ (xi xiþj )2 Zj
XX i
j
(xi xiþj )2 Zj
(2:180)
100
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
By substituting Equations 2.177 and 2.180 into Equation 2.175, we get
DH ¼ ns
X i
"
xab a
2
xi Z þ
X j
2
#
(xi xiþj ) Zj DE
(2:181)
By substituting Equation 2.181 for DH and Equation 2.168 for DS into Equation 2.173 for DG, we get
DG ¼ gc ffi ns
X i
"
DE
xab a
2
xi Z þ DE
X j
2
(xi xiþj ) Zj þ kT xi ln
xi xab a
þ (1 xi ) ln
1 xi
1 xab a
#
(2:182)
Evaluation of the quantities Zl, Zj, and ns is required in order to determine the orientation dependence of the concentration profile and the interfacial energy. Any atom in a crystal has nearest neighbors that, in most orientations, are not confined to the adjacent parallel atomic planes. The method developed here describes the distribution of the nearest neighbors. Although it can be applied to other cubic-type systems, it is presented here only for the fcc case. Figure 2.25 shows an fcc unit cell. Consider the (hkl) plane i parallel to the interface. It is assumed that plane i contains an atom P that has 12 nearest neighbors (only one of which is shown pv ¼ 12 < 110 > is the position vector and af is the fcc lattice pv where ~ for clarity) in positions af~ ~ is a normal vector to (hkl) projected from P. Since [hkl] is perpendicular to (hkl) in a parameter. m pv , is cubic type system, components of ~ m are the same as those of (hkl). Since j~ pv j, the magnitude of ~ ~ is a lattice vector, the angle uv between ~ the same for all v (v ¼ 112, for an fcc lattice), and m pv and ~ or the scalar product m ~ ~ m pv can be used to determine the perpendicular distance between each of the nearest neighbor and plane i. This distance, in turn, determines the order of atomic plane j denoted in Figure 2.24.
(hkl), Plane i
P af Pv
θv
m
~ is the FIGURE 2.25 A unit cell of fcc in which ~ pv is the position vector to the nearest atoms from P and m normal vector to (hkl) projected from P. uv is the angle between the two vectors. (Reprinted From Lee, Y.W. and Aaronson, H.I., Acta Metall., 28, 539, 1980. With permission from Elsevier.)
Diffusional Nucleation in Solid–Solid Transformations
101
Calculations can be confined to planes within or on the boundaries of the unit stereographic triangle with corners at (100), (110), and (111) because of cubic symmetry. Thus, only those planes with (hkl) such that h k l 0
(2:183)
~ ~ need to be considered. Because of symmetry, those producing a negative m pk have the same meaning as those yielding a positive value except that their direction is opposite. Therefore it is sufficient to consider only those ~ pv for which ~ ~ m pv 0
(2:184)
~ ~ A set of atoms having the same m pv lies on the same plane parallel to plane i with the perpendicular distance, dv, between the two planes given by dv ¼ af j~ pv j cos uv
(2:185)
If d, the spacing between two adjacent, parallel, crystallographically equivalent planes, which is in general not equal to the conventional inter-planar spacing, is known for a given (hkl), dv=d determines the order of plane j in Figure 2.24. Further development of Equation 2.185 is necessary in order to determine j unambiguously. As shown in Figure 2.25 cos uv ¼
~ ~ m pv j~ mjj~ pv j
(2:186)
Equation 2.185 then becomes dv ¼
~ ~ af m pv j~ mj
(2:187)
Since d0 , the conventional interplanar spacing is d0 ¼ af =j~ mj, Equation 2.187 becomes ~ ~ dv ¼ d0 m pv
(2:188)
~ ~ If h, k, and l are mixed, the scalar products of m pv are also mixed. When dv =d (not d0 ) is not an integer, there must exist a plane between two adjacent (hkl) planes, say (2h, 2k, 2l). Thus, 1 d ¼ d0 2
for h, k, and l mixed
(2:189)
for h, k, and l odd
(2:190)
From similar reasoning, d ¼ d0
With the aid of Equations 2.189 and 2.190, Equation 2.188 becomes dv =d ¼ 2~ m ~ pv ¼ j for h, k, and l mixed ~ ~ pk ¼ j for h, k, and l odd ¼m
(2:191)
In the former case, a (2h, 2k, 2l) plane is the j ¼ 1 plane. ~ ~ Thus, the scalar product m pk directly determines the order of the atomic planes on which the ~ ~ atoms nearest neighboring to atom P lie. Hence, a set of atoms having m pk ¼ 0 determines Zl regardless of whether h, k, and l are all odd or mixed, and Zj is simply the number of atoms having ~ ~ the same and nonzero m pk where j is determined by Equation 2.191.
102
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
After defining d from Equation 2.189 or 2.190, ns is obtained from the following relationship: n s ¼ nV d
(2:192)
where nV ¼ 4=a3f for the fcc lattice. In order to apply the foregoing considerations to calculate gc as a function of boundary orientation, it is necessary to solve Equation 2.172, or more specifically the set of difference equations for all the boundary layer compositions, xi. These xi are then substituted into Equation 2.182 to obtain gc. The number of lattice planes differing in composition from xab a and increases with temperature, finally becoming infinite at T , the critical temperature of the xba c b coherent miscibility gap. This temperature is given by Tc ¼ V=2R
(2:193)
on the regular solution model, where V is the regular solution constant and R is the gas constant. When the solvi of a eutectic system are modeled as ‘‘arms’’ of a miscibility gap, Tc can lie high above the melting temperature of the alloy, or even of the pure metals. In the regular solution model, Tc always occurs at x ¼ 0.5. Hence, in the interest of holding the computation time to a reasonable level, it is necessary to limit the number of planes for which xi is determined. Experience has shown that L, the thickness of the boundary region, does not significantly exceed the following number of (111) planes as a function of T=Tc, 8 at 0.25, 10 at 0.5 and 12 at 0.75. Since regular solution modeling makes the concentration profile normal and the interface symmetrical, only half of these numbers of simultaneous difference equations need to be solved. However, the number of planes representing the effective L is determined by L=d(hkl); hence, for high index boundary orientations, the number of equations needed to compute half the concentration profile is 14, 20, and 25 at 0.25, 0.50, and 0.75 Tc, respectively. Figure 2.26 shows half the concentration profile normal to the a:b boundary, i.e., the profile in only one phase for (100), (110), and (111) interfaces. Ignoring the results from the continuum 0.5
0.4
0.3 xi 0.2 T = 0.75Tc x αβ α = 0.1122
3
0.1
T = 0.25Tc
T = 0.5Tc x αβ α = 0.0212
x αβ α
2
= 0.337 ×
af
10–3
1
0
0
FIGURE 2.26 Half of the concentration profile normal to the a:b boundary. . —(100); ~—(110); —(111); the continuum method. (Reprinted from Lee, Y.W. and Aaronson, H.I., Acta Metall., 28, 539, 1980. With permission from Elsevier.) ~—from
103
Diffusional Nucleation in Solid–Solid Transformations
method of Cahn and Hilliad, which will be discussed later, two points are immediately obvious. The thickness of the boundary region increases with temperature, and at a given T=Tc, the effect of the boundary crystallography is exerted solely through the effect of interplanar spacing, thus, the concentration profile is almost independent of the boundary orientation. Also, xi converges to the bulk concentration at much smaller distances than the assumed L; hence, the equilibrium values of the profiles and of gc are assured. Because L increases with T=Tc, and the number of planes comprising L rises with (hkl) at a given T=Tc, the number of boundary orientations that it is practicable to examine diminishes with rising temperatures. The calculation of contour plots of gc on a stereographic projection thus proved feasible only at T=Tc ¼ 0.25 and 0.50; at 0.75 too few orientations could be calculated in the computer time available. Figure 2.27 presents the results obtained; a unit stereographic triangle suffices to display the results. All interfacial energy values were normalized by division by the value at (100). The level of anisotropy, defined here as the ratio of the maximum to the minimum gc, falls from 1.30 at 0 K through 1.14 at 0.25 T=Tc, 1.03 at 0.50 T=Tc, and less than 1.006 at 0.75 T=Tc. Although there are no experimental data against which to test this prediction, the thermodynamic nature of the surface energy is the same as that of the coherent interphase boundary energy with identical lattices and orientations on both sides of the boundary. Hence, it is appropriate to note that McLean found just this behavior for the surface energy of Cu [40]. 0.866
0.895 0.940
0.925 0.950
0.960
0.975
0.980
1.000
0.990 1.000
1.050
1.005 (210) 1.000 1.075 1.050 (a)
1.118 1.100 1.115
(510) 1.061
1.000 1.017 1.015 1.015 1.0173 (b)
1.010
1.008
0.970 0.982 0.984 0.986 0.988 0.990
1.000 (c)
0.998
0.996 0.994
0.992
FIGURE 2.27 Coherent interface free energy, gc, plot: (a) for 0 K, (b) for 0.25 Tc, and (c) for 0.5 Tc. (Reprinted from Lee, Y.W. and Aaronson, H.I., Acta Metall., 28, 539, 1980. With permission from Elsevier.)
104
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
The gc plots show that a point energy cusp obtains at (111) for all temperatures studied. The maximum gc occurs at (100) at and above 0.50 Tc. At lower temperatures, the maximum moves along the (100)–(110) boundary, reaching (210) at 0 K. These results are not in agreement with some experimental data on the anisotropy of the surface energy of pure metals. But they do give a good accounting for the excellent work of this type on Au by Winterbottom and Gjostein [41], who found the high temperature maximum surface energy to lie roughly halfway between (510) and (100) on the (100)–(110) boundary. 2.11.4.2.3 Continuum Model This is the work of Cahn and Hilliard [42]. This paper, with strain energy incorporated in a subsequent paper, provided the basis for the celebrated Cahn theory of spinodal decomposition. Unfortunately, this chapter, though providing the most accurate closed-form analytic relationship for gc available even today, has received only ‘‘lip service’’ so far in the literature of physical metallurgy. Here, the complex manipulations used to obtain this important result will be studied at some length. Then, the integral form of the Cahn and Hilliard (C–H) relationship will be developed in detail in the manner of the previous Lee and Aaronson approach from the foregoing multi-plane discrete lattice treatment. Cahn and Hilliard began by criticizing the two-plane discrete lattice treatment. They noted that the assumption of a specified thickness of the interface is ‘‘incorrect in principle since once the temperature and pressure of the system are specified, the interfacial thickness is no longer an independent variable.’’ The local free energy per atom, g, in a region of nonuniform composition was proposed to depend upon both the local composition and the composition of the immediate environment. Accordingly, g was expressed as the sum of two contributions that are functions of the local composition and the local composition derivatives, respectively. The composition gradient was assumed small compared with the reciprocal of the interatomic spacing. Composition, x, and its derivatives were taken as independent variables. Providing that g is a continuous function of these variables, it can be expanded in a Taylor series about go, the free energy per atom of a solution with uniform composition x. Following the useful review by Hilliard [43], the general form of a multivariable Taylor series is
f (y, z, . . . ) ¼ fo þ y
qf qf 1 q2 f q2 f þ z þ y2 2 þ z2 2 þ qy qz 2 qy qz
(2:194)
In one dimension, this series for g becomes
g ¼ go (x) þ
2 dx qg dx qg 1 dx 2 q2 g þ 2 þ dx ds ds2 2 ds d2 x dx q q q 2 ds ds ds
(2:195)
where s is the distance along the diffusion direction (the letter x is reserved for atom fraction of solute) and the third and higher order terms have been neglected. Rewritten in a more compact form through the introduction of symbols for the square-bracketed terms, we get 2 2 dx d x dx þ K2 g ¼ go (x) þ E þ K1 2 ds ds ds
(2:196)
Diffusional Nucleation in Solid–Solid Transformations
105
where qg E¼ dx q ds qg K1 ¼ 2 dx q ds2 K2 ¼
1 q2 g 2 2 dx q ds
For crystals with a center of symmetry, the free energy must be invariant with respect to a change in the sign of the axis, i.e., with a reversal of the slope of the profile. Hence, the coefficient E must be zero. Although this statement was originally made with cubic-type crystals in mind, Cahn [44] pointed out that all but 10 or 11 of the 32 point groups have a center of symmetry. Even those that would give rise through retention of E(dx=ds) are small in magnitude and important only near crystal surfaces. In the interior of the crystals, the interphase boundary energy would be essentially unaffected by this contribution. Integrating Equation 2.196, an expression is obtained for the total free energy of the systems of cross-sectional area A: ð G ¼ Anv g ds ð"
2 2 # d x dx ds þ K2 go (x) þ K1 ds2 ds
¼ Anv
(2:197)
where nv is the number of atoms per unit volume. parts, in the usual formula for such, Ð Integrating Ð the second term in the integrand by u dv ¼ uv v du, let u ¼ K1, v ¼ dx=ds, and dv ¼ d2x=ds. Hence, ð
K1
ð 2 dx dx dx dK1 dx ds ds ¼ K 1 2 ds ds ds dx ds ð 2 dx dx dK1 ¼ K1 ds dx ds ds
(2:198)
Assuming that the system is homogeneous at the surface, the first term on the r.h.s. must vanish, and in a macroscopic system, it is negligible elsewhere even if it is not zero at the surface. Therefore, Equation 2.197 becomes 2 # ð" dx dK1 dx 2 G ¼ Anv go (x) þ K2 ds dx ds ds þ1 2 # ð " dx go (x) þ K ds (2:199) ¼ Anv ds 1
106
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
where K ¼ K2
dK1 1 ¼ dx 2
q2 g q2 g 2 2 dx dx qxq 2q ds2 ds
This equation, which is the central C–H result, although rewritten into one dimension rather than in the three dimensions that they employed, shows that, to a first approximation, the free energy of a small length of the nonuniform solid solution can be expressed as the sum of two contributions, one being the free energy that this length would have in a homogeneous solution and the other being a ‘‘gradient energy,’’ which is a function of the local composition. Using the regular solution model, C–H have shown that K, the gradient energy coefficient, is 2 M 2 ro K ¼ DH0:5 3
(2:200)
where M DH0:5 is the enthalpy of mixing per unit volume at x ¼ 0.5, ro is the nearest neighbor distance. Now assume that the free energy of the solution of the nonequilibrium composition lying between ba the extremes xab a and xb can be represented by a continuous function go(x) of the form shown in Figure 2.28. By definition, interfacial free energy is the difference per unit area of interface between the actual G of the system and that which it would have if the properties of the phases were
T < TC
μB(xi)
ΔGM
Δgi
αβ
μA(xα ) =
αβ
a
βα μA(xβ )
μB(xα ) =
b
βα
μB(xβ )
μA(xi)
A
αβ
xα
xi
0.5
βα
xβ
B
xB
FIGURE 2.28 Free energy of mixing vs concentration diagram (schematic). Common tangent line, a–b determines the equilibrium bulk concentration.
107
Diffusional Nucleation in Solid–Solid Transformations
ba uniform throughout, i.e., if the system were all a single phase of composition xab a or xb . Hence, Equation 2.199 must be divided by the area, A, and the free energy of homogeneous a (say) of composition x, and the equilibrium partial molar free energies (obtained from the common tangent line) must be subtracted from the integrand
G gc ¼ ¼ nv A
¼ nv
þ1 ð "
# 2 dx ab ab xGBa (1 x)GAa ds go (x) þ K ds
þ1 ð "
2 # dx Dg(x) þ K ds ds
1
1
(2:201)
where ab
ab
Dg(x) ¼ go (x) xGBa (1 x)GAa a
ab
ab
¼ xGB þ (1 x)GAa xGBa (1 x)GAa
a ab a ab ¼ x GB GBa þ (1 x) GA GAa a
Gi , the partial molar free energy of component i at composition x, may be regarded as the free energy referred to the reference state of an equilibrium mixture of a and b, or as the free energy change per atom involved in transferring a minute amount of material from an infinite reservoir of equilibrium a to composition x. According to Equation 2.201, the more diffuse the interface is, the smaller the contribution of the gradient energy term, K(dx=ds)2, to gc. However, this decrease in free energy can only be achieved by introducing more material in the boundary region that is of nonequilibrium composition, thereby increasing Dg(x). At equilibrium, the composition variation will be such that the integral of Equation 2.201 is a minimum. (This is equivalent to the requirement that the partial molar free energies be constant throughout the system.) This minimum must be achieved subject to the condition that the average composition of the system remains constant. The minimum is found by substituting the integrand, I, of Equation 2.201 into an appropriate form of the Euler equation dx qI ¼ constant ¼ z I ds qðdx=dsÞ
(2:202)
Let y ¼ dx=ds. Hence,
q Dg(x) þ Ky2 Dg(x) þ Ky y qy 2
¼ Dg(x) þ Ky2 2Ky2 ¼ z
Or Dg(x) Ky2 ¼ z
(2:203)
108
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
However, z must be zero since both Dg(x) and dx=ds tend to zero as s goes to 1. The minimum value of gc is then achieved when Dg(x) ¼ K
2 dx ds
(2:204)
By substituting this relationship into Equation 2.201, we get
gc ¼ 2nv
þ1 ð
Dg(x)ds
(2:205)
1
To change the integration limits from s to x, multiply both sides of Equation 2.204 by (ds)2. Dg(x)(ds)2 ¼ K(dx)2
(2:206)
Taking the square root of both sides, we get [Dg(x)]1=2 ds ¼ K 1=2 dx
(2:207)
By multiplying both sides by [Dg(x)]1=2, we get Dg(x)ds ¼ [KDg(x)]1=2 dx
(2:208)
By substituting into the integrand of Equation 2.205 and replacing the integration limits accordingly, we get xba b
gc ¼ 2nv
ð
[KDg(x)]1=2 dx
(2:209)
xab a
This is the general form of the Cahn–Hilliard equation for the coherent interphase boundary energy between two crystals with the same cubic-type structure. To evaluate the dependence of interfacial energy upon temperature over a wide range of temperature, it will be necessary to employ a specific solution model (once again that of the regular solution) as well as some numerical analysis. However, it is possible to secure, without much approximation, an analytic expression for the variation of energy with temperature in the vicinity of the critical temperature, Tc, at which the two phases attain the same composition, xc. This analysis begins by expanding the two components of the Dg(x) term in Equation 2.209 by means of the multivariable Taylor series expansion of Equation 2.194. From Equation 2.201, when go(x) is symmetric about xc in an abbreviated form Dg(x) ¼ go (x) go xab a
(2:210)
where the zero subscript continues to denote reference to a homogeneous solution, x is the composition of the alloy, xab a indicates an equilibrium composition associated with one arm of the miscibility gap.
109
Diffusional Nucleation in Solid–Solid Transformations
The expansions are performed about Tc and xc. Addressing first go(x) to its 4th order qgo qgo go (T, x) ¼ go (Tc , xc ) þ (T Tc ) þ (x xc ) qT Tc qx xc " 2 2 1 q2 go 2 q go 2 q go (T Tc ) þ þ (x xc ) þ 2(T Tc )(x xc ) 2! qT 2 qx2 qTqx Tc
xc
Tc ,xc
#
"
1 q3 go q3 go q3 go (T Tc )3 3 þ (x xc )3 3 þ 3(T Tc )2 (x xc ) 2 qT Tc qx xc qT qx Tc ,xc 3! # q3 go 2 þ 3(x xc ) (T Tc ) 2 qx qT Tc ,xc " 4 4 1 q4 go 4 q go 3 4 q go (T Tc ) þ þ (x xc ) þ 4(T Tc ) (x xc ) 3 qT 4 Tc qx4 xc qT qx Tc ,xc 4! # 4 q4 go 2 2 q go 3 þ 4(x xc ) (T Tc ) 3 þ 6(T Tc ) (x xc ) 2 2 qx qT Tc ,xc qT qx Tc ,xc þ
(2:211)
qn go An equivalent expression can be found for ¼ 0 for all n, . We now note that qxn xc qn go are cancelled by each as shown in Figure 2.29. Furthermore, terms involving go(Tc, xc) and qT n Tc kþj go k jq other, and those associated with (T Tc ) (x xc ) k j with an odd j should vanish because qT qx Tc , xc go(T, x) is symmetric about xc. The latter may be understood through an examination of the situation for two compositions symmetrically disposed about xc. Let one be xc þ a and the other be xc a. For the first (x xc) ¼ xc þ a xc ¼ a and the for second, (x xc) ¼ xc a xc ¼ a. Thus, the difference is 2a, which is impossible if go(T, x) is symmetric about xc. In practice, this means qkþj go ¼ 0 when j is odd, which can be expected in cubic and in isotropic crystals. qT k qxj
go T, xab a
Tc , x c
Neglecting the terms involving (T Tc)2 (x xc)2 as T approaches Tc and higher order terms, we obtain Dg(x) ¼ go (x) go xab a
h ab 2 i q3 go ab 4 iq4 go 1 1h 2 4 (x xc ) xa xc þ ¼ (T Tc ) (x xc ) xa xc qx2 qT Tc ,xc 4! qx4 xc 2! 2 2 4 1 1 4 2 q go 4 q go ¼ (T Tc ) Dx Dxe 2 þ Dx Dxe qx qT Tc ,xc 4! qx4 xc 2! (2:212) ¼ b(Tc T) Dx2 Dx2e þ d Dx4 Dx4e
where Dx ¼ x xc, Dxe ¼ xab a xc ,
110
Mechanisms of Diffusional Phase Transformations in Metals and Alloys T = TC
T > TC
ΔGM
T < TC
0
∂ΔGM 0 ∂xB
0
0
∂2ΔGM ∂x2B
0 0
0
0
0
0
B A
B A
∂3ΔGM ∂x3B A
B
FIGURE 2.29 Effect of temperature, T, on the derivative terms of the free energy of mixing, DGM, with respect to composition. (From Gaskell, D.R., Introduction to the Thermodynamics of Materials, 2nd ed., McGraw-Hill, New York, p. 392, 1981. With permission.)
1 q2 go , 2! qx2 qT Tc ,xc 1 q4 go , d¼ 4! qx4 xc ba Dg(x) is a minimum (i.e., zero) at xab a and xb . b¼
Hence, qDg qx ¼ 0 at these compositions. Applying this consideration to Equation 2.212, and noting that Dxe is a constant, qDg ¼ 0 ¼ b(Tc T)[2Dx] þ 4dDx3 qx x¼xab a
Diffusional Nucleation in Solid–Solid Transformations
111
Or b(Tc T) Dx2e TTc ¼ 2d
Through the use of this equation, Equation 2.212 yields Dg(x)jTTc ¼ 2dDx2e Dx2 Dx2e þ d Dx4 Dx4e 2 ¼ d Dx2e Dx2
(2:213)
(2:214)
By substituting Equation 2.214 into Equation 2.209, we get Dx¼þDxe ¼xc xba b
ð
gc ¼ 2nv
Dx¼Dxe ¼xab a xc
(Kd)1=2 Dx2e Dx2 d(Dx)
(2:215)
Taking K, d, and Dxe as constants and substituting Equation 2.213 for Dxe, we get þDxe Dx3 4Dx3e gc ¼ 2nv (Kd) ¼ 2nv (Kd)1=2 3 3 Dxe pffiffiffi 3=2 1=2 3=2 8 b(Tc T) 2 2 nv K b ¼ nv [Kd]1=2 (Tc T)3=2 ¼ 3d 3 2d 1=2
Dx2e Dx
(2:216)
Thus, it is predicted that near the critical temperature the interfacial energy is proportional to (Tc T)3=2. It can be shown, as noted by C–H in their seminal work [42], that if a model of an interface is used that confines the thickness of the interface to a fixed number of planes, say L, the interfacial energy will be proportional to (Tc T)=(L þ 1). The Becker equation of Equation 2.153 takes L ¼ 2. The composition profile and thickness of the interphase boundary are now evaluated. The variation of the composition with distance, s, at the boundary is such that dx Dg(x) 1=2 ¼ ds K
(2:217)
In Figure 2.30, an inspection of the variation of Dg with the composition makes it clear that the composition profile must be sigmoidal in shape in order to satisfy Equation 2.217, as sketched in Figure 2.31. In the vicinity of the critical point, Equation 2.214 can be substituted for Equation 2.217: dx ¼ ds Since d(Dx) ¼ d(x xc) ¼ dx,
1=2 2 d Dxe Dx2 K
d(Dx) ¼ ds Or d(Dx=Dxe ) ¼ ds
1=2 2 d Dxe Dx2 K
2 # 1=2 " d Dx 1 Dxe K Dxe
(2:218)
112
g0(x)
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Δg
0
xα
xβ
1
FIGURE 2.30 g0 (x) vs. x for T < Tc. (From Cahn, J.W. and Hilliard, J.E., J. Chem. Phys., 28, 258, 1958. American Institute of Physics. With permission.)
xβ
x
xα
L S
FIGURE 2.31 Interface composition profile. (From Cahn, J.W. and Hilliard, J.E., J. Chem. Phys., 28, 258, 1958. American Institute of Physics. With permission.)
Integrating over s from s ¼ 0 to s ¼ s, " # " #
Dx d 1=2 b(Tc T) 1=2 ¼ tanh Dxe s ¼ tanh s Dxe K 2K
(2:219)
where Equation 2.213 is utilized and the distance, s, is measured from an origin at Dx ¼ 0, i.e., x ¼ xc. The interface thickness, L, is expressed (for convenience) in terms of the gradient at xc as pffiffiffiffi 1=2 ba xab K 2Dxe K a xb ¼ pffiffiffi ¼ 2Dxe L¼ (dx=ds)xc Dg(max) d Dx2e Dx2
where Dg(max) is the maximum of Dg.
(2:220)
113
Diffusional Nucleation in Solid–Solid Transformations
Since Dx ¼ 0 when Dg ¼ Dgmax ,
pffiffiffiffi 2 K L ¼ pffiffiffi dDxe
Substituting Equation 2.213, we get
LTTc ¼ 2
2K b(Tc T)
1=2
(2:221)
Thus, the thickness of the interface is seen to increase with temperature and to become infinite at the critical temperature. To effect the numerical evaluation of gc, its temperature dependence, and L outside the range of T Tc requires the use of a particular solution model in order to secure expressions for K and Dg. The regular solution model is used for this purpose. Calculation of the free energy of a nonuniform regular solution begins by determining its enthalpy in the context of a two-component cubic lattice. This calculation is performed assuming that the lattice parameter is independent of the composition (to avoid strain energy problems) and the distribution of the atoms on the lattice sites is random. Let C(R) ¼ the probability of finding a B atom on site R, let C(S) be the probability of finding a B atom on site S. The probability that an A–B bond will be formed between an A atom on site S and a B atom on site R is PAB ¼ C(R)[1 C(S)]
(2:222)
If~ r is the radius vector of site S relative to site R, then C(S) can be obtained as a function of C(R) by expanding about R: C(S) ¼ C(R) þ~ r
dC(R) ~ r 2 d2 C(R) ~ r 3 d3 C(R) þ þ þ dR 2! dR2 3! dR3
(2:223)
k, where ~i, ~j, and ~ k are unit vectors in the x, y, and z directions, and Since ~ r ¼ rx~i þ ry~j þ rz~ q q q ~ r r ¼ rx þ ry þ rz , Equation 2.224 can be written as qx qy qz C(S) ¼ C(R) þ~ r rC(R) þ
(~ r r)2 (~ r r)3 C(R) þ C(R) þ 2! 3!
(2:224)
Consider now the Zn atoms in the nth coordination shell at a radius rn from R. The probable number of A–B bonds, Zn(PAB)n, between a B atom at R and the A atoms in the nth shell is sought. By substituting Equation 2.224 and multiplying by Zn, we get Zn (PAB )n ¼ Zn C(R)[1 C(S)] X 1 X ~ (~ r r)2 C(R) ¼ Zn C(R) 1 C(R) r rC(R) 2!
(2:225)
where third and higher order terms are omitted. Equation 2.225 can be expressed in terms of its vector components before the indicated summation can be made. From the above relationship for ~ r r, ~ r rC(R) ¼ rx (~ r r)2 C(R) ¼ rx2
qC(R) qC(R) qC(R) þ ry þ rz qx qy qz
(2:226)
2 2 q2 C(R) q2 C(R) q2 C(R) q2 C(R) 2 q C(R) 2 q C(R) þ r þ r þ 2r r r r þ 2r þ 2r x y y z z x y z qx2 qy2 qz2 qxqy qyqz qzqx (2:227)
114
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
By factoring C(R) from Equation 2.225 and then substituting Equations 2.226 and 2.227, we get
X qC(R) qC(R) qC(R) þ ry þ rz rx Zn (PAB )n ¼ Zn C(R) 1 C(R) qx qy qz
2 2 1 X 2 q2 C(R) 2 q C(R) 2 q C(R) þ r þ r rx y z 2! qx2 qy2 qz2
þ 2rx ry
q2 C(R) q2 C(R) q2 C(R) þ 2ry rz þ 2rz rx qxqy qyqz qzqx
(2:228)
As discussed before, all terms with odd derivatives must vanish because of the center of inversion in a cubic lattice. Also, all mixed derivatives vanish for the same reason. By replacing the probability that a B atom is present at site R, C(R) with the atom fraction of B, c(R), we get
2 2 Zn c(R) X 2 q2 c(R) 2 q c(R) 2 q c(R) rx þ r þ r y z 2 qx2 qy2 qz2 Zn c(R) X 2 2 ¼ Zn c(R){1 c(R)} rn r c(R) 2
Zn (PAB )n ¼ Zn c(R){1 c(R)}
(2:229)
In a cubic lattice, the average contribution per atom pair is rn2 =3, instead of rn2 . Hence, Equation 2.229 becomes Zn (PAB ) ¼ Zn c(R){1 c(R)}
Zn c(R) 2 2 rn r c(R) 6
(2:230)
n n n When en ¼ EAB (EAA þ EBB )=2, where Eijn is the bond energy between i and j in the nth shell, the total energy per atom at u(R), relative to the pure components, i.e., (EAA þ EBB )=2, is given by
u(R) ¼ c(R){1 c(R)}
X n
en Zn
c(R) 2 X c(R)Zn rn2 en r 6 n
(2:231)
Defining v¼
X
en Zn
(2:232)
n
P 2 n Zn rn en l ¼ P 3 n en Zn 2
(2:233)
By substituting these definitions into Equation 2.231, we get
u(R) ¼ vc(R){1 c(R)}
c(R)vl2 2 r c(R) 2
(2:234)
115
Diffusional Nucleation in Solid–Solid Transformations
For a liquid solution, Zn is replaced by 4pr2 r(r) dr=v for the probable number of atoms between r and r þ dr, where r(r) is the reduced radial distribution function, assumed independent of composition and species involved, and v is the volume per atom. A substitution for Zn yields the following new definitions for v and l2: 4p v¼ v
1 ð
er r(r)r 2 dr
(2:235)
0
Ð1 er r(r)r 4 dr l ¼ Ð0 1 3 0 er r(r)r 2 dr 2
(2:236)
The parameter, l, has a dimension of length and represents the root mean square effective interaction distance for the energy in a concentration gradient. If consideration is restricted to the nearest neighbor interactions, the integrals become unnecessary and l2 ¼
ero r(ro )ro4 ro2 ¼ 3ero r(ro )ro2 3
(2:237)
If the assumption is made that en is proportional to rn and the radial distribution function is approximated to be zero for r < ro and one for r > ro, then Ð 1 4 n r r dr n3 2 l ¼ Ð0 1 2 n ¼ r 3 0 r r dr 3(n 5) o 2
(2:238)
p When n ¼ 6, Equation 2.238 yields l ¼ ro, as compared with l ¼ ro= 3 from Equation 2.237. If a repulsive term proportional to r12 is added to en, l becomes l¼
11 7
1=2
ro
(2:239)
Hence, l is sensitive to the exact nature of atomic interactions. So far, only the enthalpy of a nonuniform regular solution has been considered. Attention is now turned to the entropy. Let the lattice be composed of p layers, each of which has its own composition, cp. Let such a layer contain Np atoms. The number of ways, Wp, in which the atoms can be arranged within the layer is Vp ¼
Np ! (cp Np )![(1 cp )Np ]!
(2:240)
Since the composition of each layer is fixed, the total number of ways in which all of the atoms can be arranged is thus, Y W¼ Vp (2:241) p
Substituting for W in the Boltzmann entropy equation, S ¼ k ln V ! Y X S ¼ k ln Vp ¼ k ln Vp p
p
(2:242)
116
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
By applying the Stirling approximation, we get S¼k
X
ln
p
¼ k
X p
Np ! (cp Np )![(1 cp )Np ]!
[cp ln cp þ (1 cp ) ln (1 cp )]Np
(2:243)
On a per atom basis at a particular lattice point, say R, S ¼ k[c ln c þ (1 c) ln (1 c)]
(2:244)
However, this equation is identical to that for the ideal positional entropy of a uniform solid solution of composition c. Hence, there is no contribution to the entropy from a composition gradient. Now comparing coefficients in Equation 2.234 and the three-dimensional version of Equation 2.196, and using the subscript ‘‘R’’ to denote the regular solution version of a parameter, K2R ¼ 0 and K1R ¼ cvl2=2. Hence, performing the differentiation indicated in the portion of Equation 2.199, reading K ¼ K2 dK1=dx, but replacing dx by dc, KR ¼ vl2 =2
(2:245)
Substituting this equation into a three-dimensional version of Equation 2.199 (with AnV now replaced by nV) yields the total G of a nonuniform solid solution on the regular solution formulation: G ¼ nV
ð
gR þ (vl2 =2)(rc)2 dV
(2:246)
To obtain the interfacial energy of a regular solution, the following properties of a regular solution are employed: 1. xc ¼ cc ¼ 1=2 V v 2. Tc ¼ ¼ 2R 2k 3. GA ¼ GA þ kT ln(1 x) þ vx2 4. GB ¼ GB þ kT ln x þ v(1 x)2 xab v (2xab a 1) 5. ln a ab ffi kT 1 xa Three more parameters must now be reevaluated on a regular solution basis. The first of these is secured by substituting (2) and (3) into the relationship for Dg(x) given in association with Equation 2.201: 9 8 < h i
= 2 2 (1 x)
þ v x2 xab þ (1 x) kT ln DgR ¼ x kT ln ab þ v (1 x)2 1 xab a a ; : xa 1 xab a 2 3 2 x (1 x)
5 v x xab ¼ gR (x) gR xab (2:247) ¼ kT 4 x ln ab þ (1 x) ln a a ab xa 1 xa
x
Diffusional Nucleation in Solid–Solid Transformations
117
A regular solution evaluation of b, which was defined in association with Equation 2.212, is made by performing the indicated differentiations on Equation 2.247: 1 q3 gR 2! qTqx2 2 3 qg x (1 x) 5 ¼ k4 x ln ab þ (1 x) ln qT xa 1 xab a
bR ¼
1 xab q2 g x a ¼ k ln qTqx (1 x) xab a q3 g k ¼ 2 qTqx x(1 x)
Recalling that Equation 2.212 is applicable near Tc and xc and that xc ¼ 1=2 in a regular solution, bR ¼
1 q3 gR ¼ 2k 2 qTqx2
(2:248)
Turning attention now to a counterpart regular solution evaluation of d, which was also defined in connection with Equation 2.212
d¼
1 q4 g 4! qx4 2
3 qg x (1 x)
5 2v x xab ¼ kT 4ln ab ln a ab qx xa 1 xa
q2 g 1 1 þ 2vxab þ ¼ kT a qx2 x 1x q3 g 1 1 ¼ kT 2 þ qx3 x (1 x)2
q4 g 1 1 ¼ 2kT þ qx4 x3 (1 x)3 At Tc and xc,
dR ¼
1 q4 gR 4kTc ¼ 3 4! qx4
(2:249)
A regular solution expression for the coherent interphase energy can now be written by substituting Equation 2.245 and others into Equation 2.209:
118
Mechanisms of Diffusional Phase Transformations in Metals and Alloys xba b
gR ¼ 2nv
ð
vl2 DgR 2
xab a
1=2
dx
xba b
¼ 2nv lkTc
ð DgR 1=2 dx kTc
xab a
¼ 2nv lkTc sR
(2:250)
where xba b
ð DgR 1=2 sR ¼ dx kTc
(2:251)
xab a
This integral was evaluated numerically and the results are plotted as sR vs. T=Tc in Figure 2.32. A version for gR appropriate to the region T ! Tc can be secured by substituting Equations 2.245, 2.248, and 2.249 into Equation 2.216: gR(T!Tc ) ¼
1=2 p 2 2nV vl2 Tc T 3=2 (Tc T)3=2 (2k)3=2 ¼ 2nV lkTc Tc 4kTc 2
(2:252)
An interfacial energy equation appropriate to T ! 0 can be found where it is recognized that the ba entropy term in Equation 2.247 becomes negligible and that xab a ! 0 and xb ! 1 on the regular solution approximation. This derivation begins by repeating the last line in the derivation of
0.6
σR = (γR/2nvλ kTc )
0.5
0.4
0.3
0.2
0.1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 T/Tc
FIGURE 2.32 Reduced interfacial free energy, sR, vs. T=Tc for a regular solution. (From Cahn, J.W. and Hilliard, J.E., J. Chem. Phys., 28, 258, 1958. American Institute of Physics. With permission.)
119
Diffusional Nucleation in Solid–Solid Transformations
Equation 2.247 and substituting for each term on the r.h.s. the usual form of the equation for the free energy of a regular solution, but specialized to either x or xab a : DgR(T!0) ¼ gR (x) gR xab a
ab ¼ vx(1 x) þ kT[x ln x þ (1 x) ln (1 x)] vxab a 1 xa ab ab kT xab ln 1 xab a a ln xa þ 1 xa ffi vx(1 x) þ kT[x ln x þ (1 x) ln (1 x)]
(2:253)
By substituting this result into Equation 2.250, we get xba b
gR(T!0) ¼ 2nV lkTc
ð
xab a
vx(1 x) þ kT[x ln x þ (1 x) ln (1 x)] kTc
1=2
dx
ffi 2lnV kTc
ð1
vx(1 x) kTc
1=2
kT[x ln x þ (1 x) ln (1 x)] 1þ vx(1 x)
ffi 2lnV kTc
ð1
vx(1 x) kTc
1=2
kT[x ln x þ (1 x) ln (1 x)] 1þ dx 2vx(1 x)
0
0
1=2
dx
) ð1 ( 2kTc x(1 x) 1=2 kT[x ln x þ (1 x) ln (1 x)] þ dx ¼ 2lnV kTc kTc 2[2kTc x(1 x)]1=2 0
p
¼ 2 2lnV kTc
ð1 0
[x(1 x)]1=2 þ
T[x ln x þ (1 x) ln (1 x)] dx Tc [16x(1 x)]1=2
(2:254)
The first term in this integral is equal to ð1 0
[x(1 x)]1=2 dx ¼
p 8
(2:255)
Using C–H’s numerical solution for the second term, gR(T!0) ffi 2lnV kTc
p T p 0:426 4 2 Tc
(2:256)
Cahn and Hilliard asserted that an approximate expression for gR, which is valid over the whole temperature range, can be obtained by noting that sR, as given by Equation 2.251, can be written as DgR (max) 1=2 Dxe sR ¼ f kTc
(2:257)
120
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
where f is 4=3 at T ¼ Tc and p=2 at T ¼ 0. Considering first the Tc result, we see from Equation 2.252 that near Tc sRTc
Tc T 3=2 ¼ Tc
(2:258)
Hence, to check the C–H statement we write Tc T 3=2 Tc
4 DgR (max) 1=2 Dxe 3 kTc
?
(2:259)
An equation for Dg near Tc given by Equation 2.214. On the regular solution approximation, DgR is a maximum at x ¼ 0.5, where Dx in Equation 2.214 is equal to 0. Hence, substituting for DgR(max) yields Tc T 3=2 Tc
?
1=2 4 dDx4e Dxe 3 kTc
By substituting the first Equation 2.249 for dR, then Equation 2.214 for Dx2e , and finally using again the relationship for dR and Equation 2.214 for bR, we get
Tc T Tc
3=2
?
0
11=2 4 4 3=2 kT Dx 4 B3 c e C 4 4 b(Tc T) 3=2 3 Dxe ¼ @ A Dxe ¼ kTc 3 3 3 2d
0
13=2
B4 2k(Tc T)C ¼@ A ¼ 4 3 2 kTc 3
Tc T Tc
3=2
Thus, the C–H statement that f ¼ 4=3 near Tc is proved. To check their statement that f ¼ p=2 as T ! 0, the lack of a pre-derived relationship for DgR in this regime requires a direct derivation of DgR(max); however, this has already been obtained as Equation 2.253. Conversion to the maximum value of DgR requires only the substitution of x ¼ 0.5, yielding v 1 1 1 1 1 DgR (max) ¼ þ kT ln þ ln ¼ kTc kT ln 2 4 2 2 2 2 2 As T ! 0, Dxe ! 0.5. Hence, from Equation 2.256, we get p sR(T!0) ¼ p 4 2
(2:260)
Thus, the counterpart to Equation 2.259 near T ¼ 0 is p p
4 2
?
p DgR (max) 1=2 p kTc =2 1=2 1 p Dxe ¼ ¼ p 2 kTc 2 kTc 2 4 2
(2:261)
121
Diffusional Nucleation in Solid–Solid Transformations
thereby validating the C–H statement that f ¼ p=2 as T ! 0. To secure an equation for the interfacial energy applicable from Tc to 0 K, C–H used a linear interpolation for f: f¼
4 p T p þ 3 2 Tc 2
(2:262)
By substituting into Equation 2.257 and then into Equation 2.250, we get dR(T!0) ¼ plnV [kTc DgR (max)]
1=2
8 T Dxe 1 1 3p Tc
(2:263)
Cahn and Hilliard noted that this equation provides a convenient means of calculating the regular solution interfacial energy with an error not exceeding 1%. We conclude with the calculation of the interface profile for a regular solution. As noted in connection with Lee and Aaronson’s discrete lattice plane calculation of interfacial energy, the interface profile is symmetrical at x ¼ 0.5. This manipulation begins by rewriting the expression for the interface thickness, Equation 2.220, into which Equation 2.217 has been incorporated pffiffiffiffi 1=2 ab xba K 2Dxe K b xa ¼ pffiffiffi ¼ 2Dxe L¼ Dg(max) (dx=ds)xc d Dx2e Dx2
(2:220)
The regular solution value of LR is developed by first replacing K with its regular solution equivalent, given by Equation 2.245: LR ¼ 2 Dxe
v 2 Dg(max)
1=2
l
(2:264)
By rearranging, we get 1=2 1=2 pffiffiffi LR pffiffiffi v 2kTc 1=2 kTc ¼ 2Dxe ¼ 2Dxe ¼ 2Dxe l Dg(max) Dg(max) Dg(max)
(2:265)
From Equation 2.247 2 3 2 1 1 1 1 1
5 þ kT 4 ln ab þ ln Dg(max) ¼ v xab a 2 2 2xa 2 2 1 xab a
(2:266)
Hence, 2 Dg(max) 1 T ab ab ¼ 2 xa ln 4xab a 1 xa kTc 2 2Tc 3 2 ab ab 2 ab 1 2xa 6 T ln 4xa 1 xa 7 ¼ 41
2 5 2 Tc ab 1 2xa
(2:267)
122
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 16 14 12
LR/λ
10 8 6 4 2
0
0.2
0.4 0.6 T/Tc
0.8
1.0
FIGURE 2.33 LR=l vs. T=Tc. (From Cahn, J.W. and Hilliard, J.E., J. Chem. Phys., 28, 258, 1958. American Institute of Physics. With permission.)
Substituting into Equation 2.265 3 ab 1=2 ab 6 T ln 4xa 1 xa 7 LR pffiffiffi ¼ 2 Dxe 1 2xab
2 5 a 41 l Tc 1 2xab a 2
(2:268)
ab ab Since Dxe ¼ xba b xa ¼ 1 2xa , see Equation 2.212. Reproducing Equation 2.221 to obtain LR near TC
LTTc
2K ¼2 b(Tc T)
1=2
By substituting the regular solution relationships for K and b and rearranging, we get 1=2 1=2 LR v 2kTc Tc 1=2 ¼2 ¼2 ¼2 2k(Tc T) l 2k(Tc T) Tc T
(2:269)
The variation of LR=l with T=Tc is shown in Figure 2.33. As T ! Tc, the interface thickness is clearly shown to increase to infinity. 2.11.4.2.4 Comparison of Discrete Lattice and Continuum Model Lee and Aaronson [39] have employed the following development to show that the continuum and the multi-plane discrete lattice models become identical at high temperatures when the regular solution approximation obtains. Referring to Equation 2.183 for gc, a comparison of the first and
123
Diffusional Nucleation in Solid–Solid Transformations
third terms in square brackets in this equation with DGb!aþb and the relationship in Equation 2.152 indicates that these terms are equal to Dgi, on a per atom basis. Hence, ( ) X X 2 gc ¼ ns (Dxij ) Zj Dgi þ DE (2:270) i
i
where Dxij ¼ xi xiþj. Dgi represents the free energy difference between the homogeneous, metastable solution with solute concentrations xi and the equilibrium homogeneous solution with concentration xab a . The following rearrangement of the second term in the above equation is now developed as X j
(Dxij )2 Zj ¼ d 2
X Dxij 2 d
j
Zj
x x 2 xi xiþ1 2 i iþ2 Z1 þ Z2 þ ¼d d d 2
(2:271)
Since (xi xiþ1)=d dx=ds, (xi xiþ2)= d 2dx=ds, and (xi xi þ j)=d jdx=ds, where d is the inter planar spacing we have X j
" # dx 2 dx 2 dx 2 (Dxij ) Zj d Z1 þ 2 Z2 þ þ j Zj ds ds ds X dx2 ¼ d2 Zj j ds j 2
2
(2:272)
when the concentration gradient is small, i.e., when the boundary width is large as T ! Tc. By substituting Equation 2.272 and Equation 2.192 into Equation 2.270 and approximating the resulting equation in integral form, we get
gc ¼ nV
þ1 ð (
1
2
Dgi þ d DE
X j
2 ) dx j Zj ds ds 2
(2:273)
where ds is the differential of d in Equation 2.191. As in the case of Equation 2.201, from the continuum model, the interfacial energy is that which obtains when the sum of the two terms in the integrand is a minimum. Applying the Euler equation as before, this yields 2
Dgi ¼ d DE
2 dx j Zj ds
X
2
j
(2:274)
By substituting into Equation 2.273, we get gc ¼ 2nv ¼
þ1 ð
d 2 DE
X
j2 Zj
j
1
4nv d 2 kTc Z
P 1
2
j
j2 Zj
2 dx ds ds þ1 ð
dx ds
1
2
ds
(2:275)
124
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
where DE ¼ 2kTc=Z. To change the integration limits, multiply both sides of Equation 2.274 by (ds)2 Dgi (ds)2 ¼ d2 DE
X
j2 Zj (dx)2
(2:276)
j
By taking the square root of both sides, we get
(Dgi )
1=2
ds ¼
2d2 kTc X 2 j Zj z j
!1=2
dx
(2:277)
Dividing through by (Dgi)1=2 and substituting the expression thus obtained for ds in Equation 2.275, and replacing (dx=ds)2 with the relationship secured by rearranging Equation 2.274 yields
gc ¼ 2nV kTc
2d kTc X 2 j Zj z j 2
!1=2 xðba b Dgi 1=2 dx kTc
(2:278)
xab a
where "
2d2 X 2 j Zj l¼ z j
#1=2
This equation now reproduces Equation 2.252, the regular solution C–H continuum expression for gc. In brief, the two-layer model of Becker, Equation 2.153, fails to predict that near Tc, the interfacial energy is proportional to (Tc – T)3=2. On the other hand, the multi-plane discrete lattice model correctly follows the prediction of the C–H continuum result. For fcc systems, l is independent of both the crystal structure and the boundary orientation and is p equal to afcc= 6 as shown in the following two examples. For (100): d ¼ d0=2 ¼ af=2, j ¼ 1, Z1 ¼ 4, and 1=2 p 2(afcc =2)2 12 4 ¼ afcc = 6 l¼ 12 p For (111): d ¼ d0 ¼ af = 3, j ¼ 1, Z1 ¼ 3, and l¼
1=2 p p 2(afcc = 3)2 12 3 ¼ afcc = 6 12
The integral in Equation 2.275 and hence in Equation 2.278 is also independent of both crystal structure and orientation; the latter property is demonstrated in Figure 2.34. Hence, the continuum model, and its high-temperature discrete lattice equivalent predict an isotropic gc at a high T=Tc. Figure 2.34 compares the two-plane discrete lattice model (d) from Equation 2.153, the C–H continuum model evaluated via the regular solution approximation (c) from Equation 2.252, and the multi-plane discrete lattice model (a) for (111) and (b) for (100) from Equation 2.182, as functions of T=Tc. The two-plane model, (d), is seen to be too high above 0.2 T=Tc and to remain seriously in error except for a certain orientation, e.g. (100) in the figure and at low temperature interfaces limited orientation. Often diffusional processes will be unable to occur within the temperature range in which
125
Diffusional Nucleation in Solid–Solid Transformations 1.4 1.333 1.2
(d)
(b) 1.155
γc (kTc /a2f )
(c)
(a)
1.0
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1.0
T/Tc
FIGURE 2.34 Coherent interfacial free energy, gc, in an fcc lattice. Curves (a) and (b) are for (111) and (100) from the multi-plane discrete lattice model, (c) is from the Cahn–Hilliard continuum model, and (d) is from the two-plane discrete lattice model. (Reprinted from Lee, Y.W. and Aaronson, H.I., Acta Metall., 28, 539, 1980. With permission from Elsevier.)
the two-plane model, Equation 2.153, is useful. Above ca. 0.7 T=Tc the continuum and the multi-plane discrete lattice models are seen to yield essentially identical results. A more detailed exploration of the two components of the continuum model shows that they do become equal, as required by the theory, only above ca. 0.85 Tc; the larger range of usefulness of this theory is due to the divergence of the two components in opposite directions to an approximately equal extent down to 0.7 Tc. 2.11.4.3 Energy of Partially Coherent Interphase Boundaries Turnbull [45] has suggested that the energy of a partially coherent boundary can be reasonably approximated as the sum of the chemical and the structural components of this energy. The chemical component is derived from the bond enthalpy and entropy differences that an atom at or near a coherent interphase boundary experiences relative to an atom well within the interior of either crystal forming the boundary; these were discussed in detail in the preceding section. The structural component consists of the misfit dislocations and any other imperfection systematically and repeatedly introduced along the interface, which concentrates the misfit between the two crystals in the immediate vicinity of such imperfections and permits the remainder of the partially coherent boundary to be fully coherent. Later, in dealing with diffusional growth, the structure and energy of partially coherent interphase boundaries that have a large enough area to support many misfit dislocations will be discussed in detail. A partially coherent boundary enclosing part of a critical nucleus, on the other hand, being at the most the order of 10 10 nm2 and usually presumably far smaller, would not readily support more than one or a very few misfit dislocations. Therefore, it is appropriate to confine consideration of partially coherent interphase boundaries in the context of nucleation to the treatment of Brown et al. [46] who analyzed the energetics of a single prismatic dislocation loop about a spherical precipitate. They took the energy of this loop to have two components: the line tension or self-energy
126
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
gself ¼
mb2 rl 8rl 3 2v ln 1þ 2(1 v) ro 4(1 v)
(2:279)
and the interaction energy with the stress field of the precipitate gint ¼ 4pmbr 2 e
(2:280)
In these equations, m is the shear modulus, assumed to be the same in both phases, b is the Burgers vector, rl is the loop radius, ro is the dislocation core cut-off radius, r is the radius of the nucleus, and e is the misfit parameter defined as 2(a1 – a2)=3a1 where a1 is the precipitate lattice parameter and a2 is the matrix lattice parameter. The structural component of the boundary energy is the sum of these two energies. This result, however, may be less accurate when the loop size is very small, as should usually be the case. The interface containing the loop would be dominated by the dislocation core, and additional interactions among different portions of the loop may be expected. 2.11.4.4 Energy of Disordered Interphase Boundaries Calculation of the energy of such boundaries has yet to be addressed in detail. Even qualitative considerations of their structure are still in an elementary state. However, current developments in the structure of high-angle grain boundaries likely provide a useful forecast of coming developments in this closely related area. The term ‘‘disordered’’ and its synonym ‘‘incoherent’’ have largely fallen out of fashion in the grain boundary structure area. One reason for this is experimental: TEM studies persistently reveal some form of interfacial structure in nearly all high-angle boundaries, and are suspected of being able to do so eventually in the remainder as the resolution of these instruments improves further. Part of these observations, though, may be due to the virtually unavoidable texturing that repeatedly occurs, beginning with the solidification and concluding with the deformation texture developing during the final rolling of the material destined to become a thin foil and the recrystallization texture that succeeds it when the specimen is solution annealed subsequently. However, analytic studies are repeatedly showing that, particularly in grain boundaries between cubic-type materials, the symmetry of the lattices permits nearly any grain boundary to be not many degrees away from a high-angle but good matching lattice misorientation such as the coincident site lattice misorientations provide. The introduction of ledges and dislocations can therefore complete the task of achieving partial coherency. More recently, the coincident site lattice (and the related coincident axial disorientation) concept have been increasingly replaced by the idea that the structure of high-angle boundaries can be described in terms of successions of small polyhedra, each containing but a handful of atoms; even at especially simple boundaries, more than one type of polyhedron may be required to describe the boundary structure. Deviations from such lattice and=or boundary orientations can be compensated for by the introduction of a few more types of polyhedra, appearing at appropriate intervals. But the total number of polyhedron types involved is limited, and the sequence of polyhedra developed occurs repeatedly as long as the boundary orientation remains unchanged from a work by Pond et al. [47]. Figure 2.35 shows the five basic so-called ‘‘random close packed’’ polyhedra originally discovered by Bernal from hard sphere models of liquid structures [48], as well as some related polyhedral structures. Figure 2.36 shows two-dimensional representations of (110) tilt boundaries formed by symmetrical rotations of the bounding grains. These boundary structures were deduced by computer simulation. Different research groups are coming up with quite similar results. It should also be noted that an fcc metal can be described as an array of octahedra and tetrahedra; hence, many of these polyhedra represent structures different from those of the bulk material. Some elaboration of the foregoing theme may well prove appropriate for interfacial structures presently described as ‘‘disordered interphase boundaries.’’ At the present time, the energies of the boundaries that we shall continue to term ‘‘disordered interphase boundaries’’ come almost entirely from experiments. Virtually the only reliable data of this type are not the absolute energies needed
127
Diffusional Nucleation in Solid–Solid Transformations
Tetrahedron
Octahedron
Trigonal prism and related form
Archimedean square antiprism and related form
Tetragonal dodecahedron
FIGURE 2.35 Five ideal polyhedrals and two related forms that are the basis for random closed packed structures according to Bernal. (From Pond, R.C. et al., Scripta. Metall., 12, 699, 1978. With permission from Elsevier.)
k
(a)
l
(b)
m n
(c)
(d)
FIGURE 2.36 Two-dimensional representations of (110) tilt boundaries formed by symmetrical rotations of the bounding grains. (From Pond, R.C. et al., Scripta. Metall., 12, 699, 1978. With permission from Elsevier.)
for use in calculations of nucleation kinetics but are instead relative energies. Smith [49] devised a simple but clever method through which the influence of nucleation upon orientation relationships and thus upon the orientation-dependency of interphase boundary energy could be overthrown and the energies of all boundaries of a given type made sufficiently similar (with some obvious and easily discarded exceptions) so that measurements could be readily made. The method is to transform an alloy to the full extent allowed by the lever rule in the a þ b region and then quench it to halt further transformation at a lower temperature. The alloy is then heavily cold worked
128
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
(say, cold rolled 50%), then reheated at the same temperature originally employed until one or both phases has recrystallized and the boundaries have been able to develop the geometry characteristic of an equilibrium with an isotropic, i.e., boundary orientation-independent interfacial energy. When the proportion of the precipitate phase is kept sufficiently low relative to the matrix grain size achieved so that excessive impingement of the grain boundary precipitates does not occur, the precipitate develops an equilibrium shape consisting of two abutting spherical caps, whose dihedral angle, c, satisfies the force or energy balance equation gbb ¼ 2gab cos c
(2:281)
where gbb is the grain boundary energy, gab is the interphase boundary energy. If the interphase boundary energy is much larger than the grain boundary energy, the equilibrium shape will approximate that of a sphere. If, on the other hand, the interphase boundary energy is onehalf or less than that of the grain boundary energy, c ¼ 0 and the a particles will spread continuously along or ‘‘wet’’ the grain boundary. However, the considerable amount of data generated through this technique on a wide variety of matrix=precipitate pairs (see Inman and Tipler [50] for a reasonably complete review) indicate that 0:7
gab 0:9 gbb
(2:282)
is the range normally encountered experimentally. Although, sometimes this ratio can reach unity, it is a good generalization that in a given alloy the energy of a disordered interphase boundary is less than that of a disordered grain boundary. Inasmuch as the grain boundary energy has only a structural component whereas that of the interphase boundary has both a structural and a chemical component, this finding suggests that solute adsorption preferentially at the interphase boundary and=or some special type of crystallographic rearrangement (say, in a form of polyhedron) is made possible at the boundaries formed by two different types of crystal structure that does not occur at the grain boundaries [51]. To date, absolute energies of disordered (and other types of) interphase boundaries have been determined by such techniques as coarsening kinetics (which is an interfacial energy driven process), growth kinetics (which can be influenced by interfacial energy if one or two dimensions of the precipitate continue to have radii on the order of a micron or less during growth such as the lengthening of needles and plates), homogeneous nucleation kinetics (which should yield the precipitate:matrix interfacial energy if this energy is boundary-orientation independent), calorimetry (the enthalpy of transformation is increased by the additional interfacial area when a product such as a pearlite is formed to a sufficient extent to be measurable), and various polyphase equilibria (see the review by Hondros [52]). However, none of these have proved to be as accurate or as reliable as the zero-creep method now to be described. This method is from the developments of a sequence of investigators [53–58]. The basic concept is as follows. Thin wires or foils are hung in a vacuum or in a protective atmosphere at a temperature usually approaching its melting point, Tm; a tiny weight is often attached. Because the specimens are so thin (less than 0.01 cm), surface energy exerts an appreciable contractile force on them. Their own and the added weight, on the other hand, tend to extend them under the influence of gravity. Both contraction and expansion should proceed entirely by the Herring–Nabarro creep, i.e., vacancies migrate from the extended regions to the compressed regions of the specimen and atoms flow in the reverse direction. (Plastic deformation by dislocations would damage the accuracy of the experiment.) When neither expansion nor contraction is occurring, the zero creep
129
Diffusional Nucleation in Solid–Solid Transformations
condition has been achieved and the surface energy can be computed from the resulting balance of forces. Grain boundaries serve as both sources and sinks of vacancies; in the absence of grain boundaries, Pranatis and Pound [57] have shown that these specimens do not undergo Herring–Nabarro creep. To analyze the zero creep method quantitatively, consider a thin wire annealed to the point where the grain boundaries form a ‘‘bamboo structure.’’ Then, the stress parallel to the wire axis, sv, is sv ¼
2prgs w pr 2
(2:283)
where r is wire radius, gs is surface energy, w is weight of the wire plus the added weight. And the stress perpendicular to the wire axis, sh, is sh ¼
gs ggb ngb þ r l
(2:284)
where ggb is average grain boundary energy, ngb is number of grain boundaries, l is average grain length. Under the zero creep condition, the two stresses are equal, i.e., sh ¼ sv. Solving for w rggb ngb w ¼ pr gs l
(2:285)
Hence, this experiment yields two unknowns: gs and ggb. Another relationship between gs and ggb is accordingly required. This is obtained through the thermal grooving phenomenon. As described by Chalmers et al. [59], a groove is formed at the junction line between a grain boundary and a free surface due to the equilibration of the two surface energies with the grain boundary energy as shown in Figure 2.37. Since the surface energy is typically threefold larger than the grain boundary energy, the half-groove angle, u=2, is large. When gs is independent of surface orientation, the relationship between the two energies is given by ggb ¼ 2gs cos
u 2
(2:286)
Equations 2.285 and 2.286 can then be solved simultaneously to obtain the absolute values of both gs and ggb.
Vapor Solid
θ 2 γs
γs
γgb
FIGURE 2.37 A schematic for the force balance at a triple junction between a surface and a grain boundary.
130
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
The following two improvements of this method should be noted. One is purely experimental: Inman and Tipler [50] used electron shadowgraphs to measure more accurately the rather large (ca. 1608) thermal groove angles. This is far more readily and conveniently accomplished on wire specimens than on foil specimens; also the force balance problem can be solved more accurately on wires because the grain geometry is so much simpler. The other arises from a simple calculation due to McLean [60]. Based upon Herring’s equilibrium equation, McLean showed that a surface energy anisotropy of only l% can cause a 30% error in the measurement of a typical grain boundary energy in copper. This error will be larger for the partially and fully coherent interphase boundaries that are of such importance in the nucleation process (the latter especially, it is presently believed). Herring’s equation also applies to the situation in which three interfaces meet at a line but the interfacial energy of two or three of them is boundary-orientation dependent: gab þ
qgag qgbg sin f þ sin u ¼ gag cos f þ gbg cos u qf qu
(2:287)
qgag qgbg and are known as ‘‘torque terms’’ and reflect the resistance of an interface to qf qu rotation. For grain boundaries, these terms are found, in Al, to range from 0 to 0.3 of the average grain boundary energy by Miller and Williams [61]; similar values were found by Bishop et al. for grain boundaries in Zn [62]. With none but the crudest qualitative evidence, it is rather strongly suspected that partially and fully coherent interphase boundaries between crystals with different structures have far higher torque terms (Figure 2.38). As to the measurement of the anisotropy of surface free energy (which should be essentially that given as a function of T=Tc for coherent interphase boundaries, and thus modest, though in the present context still significant at the near-Tm temperatures at which the zero creep method is practicable), the first rigorous analysis of the problem and application of this analysis was reported by Winterbottom and Gjostein [41,63]. Subsequently, the amount of experimental data that had to be acquired in order to measure the anisotropy of gs was reduced through the measurement method developed by McLean and Gale [64]. Zero creep wires were mounted in a specially designed electron microscope stage. The wires were rotated in this stage, and the groove angle was measured every 58 of rotation. Laue backreflection measurements, using a micro-focus camera, were employed to determine crystal orientations. gs was written as a double Fourier series of finite order. By substituting this equation into the where
γαγ
α
γαβ
γαγ φ
γ φ
γβγ θ θ β
γβγ
FIGURE 2.38 boundaries.
A schematic for the force balance at a triple junction among the a:b, b:g, and g:a interphase
131
Diffusional Nucleation in Solid–Solid Transformations
Herring relationship, Equation 2.287, a relationship relating grain boundary energy to the parameters of the Fourier surface energy series is obtained. Only ratios of the surface energy of a given plane to that of a ‘‘reference plane,’’ say (100), can be obtained. However, once the ratio of the grain boundary energy to that of the (100) surface plane is accurately evaluated through the inclusion of the torque terms whose determination the McLean–Gale analysis allows to be accomplished with reasonable economy of effort, a quite accurate absolute value of the grain boundary energy can be determined by simply using an average value of the surface energy, because the differences among the various surface energy values may be only a few percent. The temperature dependence of the surface free energy of metals and alloys is sufficiently weak so that it is not easy to establish through the scatter in the surface energies obtained from the zero creep method. For Al, Cu, Ag, and Au, dgs=dT ranges from 0.4 to 0.5 mJ=m2K; in Fe-3 wt.% Si, the value is 0.36; a value of 1.76 is reported for Fe-18% Cr-8% Ni [65]. Now suppose that the temperature-dependence of gs has been established, and a wire is transformed to the equilibrium proportion of a þ b. Further assume that coarsening has effectively ceased. Now let the various a:b boundaries be exposed to the vapor phase under conditions wherein the surface layers of the wire will not undergo a change in composition due to the evaporization of one component preferentially or some other reaction within the vacuum or vapor phase. The configuration thus developed will have the appearance shown in Figure 2.39. This sketch is drawn so as to exaggerate considerably the plausible difference in the surface energies of the two phases, ga and gb. Clearly, then, evaluation of gab requires that both ga and gb, and temperature dependencies thereof, be evaluated: gab ¼ ga cos w1 þ gb cos w2
(2:288)
Further, in order to avoid significant errors in gab, it is necessary to evaluate the orientation dependence of both surface energies in order to determine the torque terms and thus permit the use of Equation 2.287, Herring’s equilibrium equation. Referring to Figure 2.38, we note that replacement of gb by gag, gb by gbg, w1 by u, and w2 by f converts the diagram of Figure 2.39 to the diagram of Figure 2.38. Thus, it is not necessary to evaluate the torque term of gab in order to make an accurate determination of gab. However, this torque term could be very large, if gab represents a low energy (coherent or partially coherent) interphase boundary. The foregoing procedure will be employed to secure the absolute values of the grain and interphase boundaries, as well as the surface energies, in Al-rich Al–Ag alloys wherein a is fcc and g is hcp, thereby ensuring that partially coherent interphase boundaries with particularly low interfacial energies can develop.
γβ
ψ2
ψ1
γα
γαβ β
α
FIGURE 2.39 A schematic for the force balance at a triple junction among an a free surface, a b free surface, and a a:b interphase boundary.
132
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
2.12 PRELIMINARY CONSIDERATION OF THE APPROXIMATION FOR f ¼ DGV þ W
The results on volume strain energy vs. morphology have tempted direct application to critical nucleus shape evaluation. However, Lee et al. [29] have investigated the role of W in this regard through examination of its interaction with g as a function of DGV in determining the critical nucleus shape and thus DG*. In effect, they tested the validity of Equation 2.69, f ¼ DGV þ W. The mathematical complexity restricted this treatment to a particularly simple yet fundamental case: homogeneous, fully coherent nucleation with isotropic interfacial energy, both isotropic and anisotropic elasticity, purely dilatational transformation strain, but different elastic constants allowed in both phases in order to make W dependent upon shape. In the absence of strain energy, this nucleus would be a sphere. As discussed in the section on coherent dilatational strain energy, when the bulk modulus of the nucleus is less than that of the matrix, the minimum strain energy shape is a disc; in the inverse case it is a sphere. Hence, when the nucleus has a smaller bulk modulus than the matrix, a conflict situation between g and W is established. After developing a general mathematical framework for considering the combined roles of these factors in determining the minimum DG* morphology, the m* < m case will be numerically examined. The standard free energy change associated with the formation of an ellipsoid of revolution embryo is 4 DG ¼ pa3 b(DGv þ W) þ pa2 g[2 þ g(b)] 3
(2:289)
where a is the semimajor axis, b is the aspect ratio, is equal to the semiminor=semimajor axis, and g (b) is a function given by 2b2 g(b) ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanh1 (1 b2 ) when 1 b2
b1
(2:292)
DG* is a function of two nucleus shape parameters, a and b. Hence, the minimum DG8 shape is that which satisfies both of the following conditions: q(DG ) ¼0 qG b q(DG ) ¼0 qb a
(2:293)
(2:294)
The differentiation specified by Equation 2.293 yields a* ¼
g[2 þ g(b)] 2b(DGV þ W)
(2:295)
However, the differentiation required by Equation 2.294 cannot be undertaken analytically because of the complicated function dependence of W upon b previously noted. Accordingly, the critical aspect ratio, b*, was determined numerically and so was DG*, which is given by
133
Diffusional Nucleation in Solid–Solid Transformations
DG* ¼
pg3 [2 þ g(b)]3 12b2 (DGV þ W)2
(2:296)
In order to avoid the use of absolute values of DGV and g, the normalized quantity, DG*=DG*h , is used: DG* [2 þ g(b)]3 ¼ DGhu * [8b(1 þ W=DGV )]2
(2:297)
* ¼ 16pg3 =3DG2V is the free energy of activation for the critical nucleus formation in the where DGhu absence of W. The application of this analysis was undertaken for the situation in which the shear modulus of the nucleus is l=3 that of the matrix and Poisson’s ratio for both phases is 0.291, the value for a Fe. * with the aspect ratio, b, at various ratios of Figure 2.40 shows the variation of DG*=DGhu Ws=DGV. Ws indicates the strain energy per unit volume for a spherical nucleus. Close inspection of this plot shows that until this ratio reaches ca. 0.82, the minimum DG* shape is that of a sphere. Only at higher ratios—where nucleation may be very difficult because so little driving force is available—will the nucleus become an oblate ellipsoid. Figure 2.41 recalculates this calculation for anisotropic elasticity wherein an Ag nucleus forms within an Al matrix in orientation No. 1 of Figure 2.20. Again, deviation from the spherical shape begins when Ws=DGV exceeds ca. 0.81. Finally, undertaking this calculation once more, now for an incoherent ellipsoid of revolution (see Figure 2.22), and assuming that vacancies will not reduce or eliminate W, and using the same values
100
S
90
–Ws/ΔGv 0.90
80 70
0.88
ΔG */ΔG *hu
60 Isotropic coherent precipitate
50 40 0.82
30 20
0.77
10 0.50 0 0.0
0.2
0.4
0.6
0.8 β
1.0
1.2
1.4
1.6
FIGURE 2.40 Variation of DG*=DG*hu with b, where DGh* is the critical free energy of formation for a spherical unstrained nucleus, for an isotropic coherent nucleus in an anisotropic matrix. (With kind permission from Springer Science þ Business Media: From Lee, J.K. et al., Metall. Trans., 8A, 963, 1977.)
134
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 100 Ag coherent precipitate in Al matrix
90
S 80 –Ws/ΔGV
70
ΔG*/ΔGhu*
0.88 60 0.86
50 40 30
0.81
20 10 0 0.0
0.61 0.2
0.4
0.6
0.8 β
1.0
1.2
1.4
1.6
* with b for an Ag coherent nucleus in an Al matrix. (With kind FIGURE 2.41 Variation of DG*=DGhu permission from Springer Science þ Business Media: From Lee, J.K. et al., Metall. Trans., 8A, 963, 1977.)
of the shear modulus and Poisson’s ratio as was used in the previous two examples, the results show that the deviation from sphericity now begins at an only slightly different value of Ws=DGV, namely, 0.75. These considerations strongly support the validity of Equation 2.69, besides its simplicity, in which it was assumed that DGV and W are additive under conditions usually encountered. Hence, in this situation the DG* calculated without regard to the interplay between two of the ancillary factors, g and W, does normally turn out to have the smallest value. Before closing, we note the further detailed studies reveal shape bifurcation phenomena where Ws =DGV is large [2,139].
2.13 NONCLASSICAL NUCLEATION THEORY 2.13.1 CONTINUUM THEORY This section is based almost entirely upon Cahn and Hilliard’s seminal work on nucleation in a twocomponent incompressible fluid [66]. Although strictly applicable only to nucleation in fluids, it should be useful for fcc ! fcc or bcc ! bcc homogeneous nucleation when T=Tc is sufficiently high and the properties of the solid solution are such that the polar g-plot is effectively spherical and when the lattice parameter is the same and independent of composition in both phases, i.e., W ¼ 0. 2.13.1.1 General Introduction Gibbs [9] identified two different approaches to the decomposition of a supersaturated matrix phase. In the current terminology, these are (1) spinodal decomposition, wherein changes take place that are small in degree but large in extent to, in particular, relatively small composition fluctuations spread out over a large volume and (2) diffusional nucleation, wherein changes take place that are large in degree but small in extent, e.g., local composition fluctuations occurring relatively rarely but yielding nuclei whose composition and other properties approach those of the bulk equilibrium precipitate phase. Whereas a fluctuation attending spinodal decomposition can immediately grow, one produced during a nucleation process can do so only after it exceeds critical nucleus size. As described before,
Diffusional Nucleation in Solid–Solid Transformations
135
spinodal decomposition occurs within the T–x region in which d2G=dx2 < 0, whereas diffusional nucleation takes place outside this region, though of course, still within the miscibility gap. Gibbs recognized that a critical nucleus might be so small that no part of it could be homogeneous. Nevertheless, because of his definition of surface tension (g in our case), he was able to develop a self-consistent formulation for the properties of a critical nucleus as if it were homogeneous up to a sharp boundary with the exterior (i.e., matrix) phase. The reason why he resorted to this artificial model was to allow application of thermodynamic principles that had been developed for homogeneous phases. In particular, the requirement that the chemical potential of each component be constant throughout a system in a stable or an unstable equilibrium uniquely determines the composition and pressure of a homogeneous critical nucleus. To avoid diversion from the direction in which we are moving namely that of distinguishing between classical and nonclassical nucleation theory, but to take advantage of this very important statement, the following remarks are here noted. This requirement is readily met despite the kinetic nature of nucleation because deviation from it sets up a ‘‘downhill’’ driving force for atomic diffusion, which should promptly reverse the deviation. Inasmuch as such deviations are to be anticipated as an embryo climbs the free energy ‘‘hill,’’ the activation energy for a deviational diffusional jump will be much greater than that for the jump that reverses the deviation. Another consequence of this statement is that the ‘‘trade-offs’’ among volume free energy change, interfacial energy, and=or volume strain energy often proposed [137,138] with the intention of further decreasing DG* are, in general, inadmissible. For example, if the nucleus were to assume a composition less than that calculated assuming equality of partial molar free energies between the nucleus and the matrix with which it is in contact, this would make the volume free energy change less negative, but at the same time open the possibility of reducing the kinetically much more important nucleus:matrix interfacial energy and perhaps also the volume free energy change as well. However, as the foregoing kinetic argument should make clear, the deviation from the nucleus composition corresponding to the equality of partial molar free energies required to accomplish this trade-off will be prevented from occurring beyond the level of isolated deviations which, sooner or later, will be either directly reversed or otherwise neutralized. In classical nucleation, g is taken as appropriate to a flat interface separating the two coexisting stable phases. DP ¼ DGV is also assumed to be appropriate for bulk nucleus material in an equilibrium with the matrix. However, this assumption is strictly correct only when DGV ! 0, at which the composition condition applies and the critical nucleus becomes so large that it can be safely regarded as exhibiting bulk properties. At appreciable departures from the equilibrium, i.e., from zero undercooling below the solvus, relationships such as r* ¼ 2g=DGV can become nothing more than a definition of r* and g in terms of DGV. Gibbs was of course aware of this limitation but did not pursue the point further since he believed that as long as DG* is finite and positive a phase could remain indefinitely in the metastable condition. The C–H nonclassical approach is based upon the three-dimensional version of Equation 2.199, their relationship for the free energy of a compositionally nonuniform solution. This expression will permit the properties of a critical nucleus to be calculated without any assumption about its homogeneity and without dividing the energy of a nucleus into surface and volume terms. 2.13.1.2 Calculation of the Nonclassical DG* Before following the C–H treatment of this central problem, it is necessary to digress briefly to consider the meaning of what Gibbs and then Cahn and Hilliard describe as W (which we rewrite as W*—not the volume strain energy), the work associated with the formation of a critical nucleus, W. Several different terms can be used to describe W*, but all yield Gibbs’ result, W* ¼ gA*=3, where A* is the surface area of the nucleus, provided that the nucleation process is taking place under the appropriate constant conditions. These conditions, different for each, are the following: (1) DF* is correct when T and volume, V, are constant—these are the conditions assumed by Cahn and
136
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Hilliard; (2) DG* is correct when T and P are constant; (3) DE* is correct when entropy, S, and V are constant; and (4) DS* is correct when E and V are constant, as in an isolated system. We shall continue to utilize DG*, just as we used G in the derivation of the central C–H relationship in Equation 2.199. By rewriting Equation 2.199 in three dimensions, replacing x with c as the symbol for solute concentration and, with C–H, expressing the free energy of a nonuniform solution on a per unit volume instead of a per atom basis, we get G¼
ð
go (c) þ K(rc)2 dV
(2:298)
where K¼
1 q2 g q2 g 2 qjrcj2 qcq(r2 c)
Recall that K is the gradient energy coefficient. Equation 2.298 has many of the properties of a nucleation relationship. Consider that G is integrated over the volume of a fluctuation. When the concentration fluctuations, Dc, are small in extent (the average composition of the system being kept constant), the integrated value of go(c) is insufficiently negative to offset the positive contribution from the gradient energy term, K(rc)2. Thus, G initially increases regardless of the size of the fluctuations. However, when fluctuations become sufficiently large in extent, the situation is reversed when the spatial extent of the individual fluctuations is great enough to diminish the gradient energy term sufficiently to permit the integrated go(c) to make the total value of G negative. To find the lowest value of DG*, a functional dependence of composition upon spatial position is sought, which makes Equation 2.298 stationary under the constraint that the average composition of the system remains constant: ð
(c co ) dV ¼ 0
(2:299)
where co is the initial composition, which is the average composition. This constraint is introduced into the integrand, Ig, of Equation 2.298 by adding that of Equation 2.299, Ic, as multiplied by a Lagrangian multiplier, l, i.e., lIc: I ¼ Ig þ lIc
(2:300)
To find the extremal of I, Euler’s equation for multivariables is applied. In the present context, the variables are c, x, y, z, qc=qx, qc=qy, and qc=qz. This equation, which is a three-dimensional version of Equation 2.202, is written as follows:
qI d qI d qI d qI þ þ þ ¼0 qc dx q(qc=dx) dy q(qc=dy) dz q(qc=dz)
(2:301)
Summarizing, where appropriate, the various differentiations and others, and noting that rc and C are independent variables, we get I ¼ go (c) þ K(rc)2 þ l(c co )
(2:302)
qI qgo (c) qK ¼ þ (rc)2 þ l qc qc qc
(2:303)
Diffusional Nucleation in Solid–Solid Transformations
qI q qc ¼ go (c) þ K(rc)2 þ l(c co ) ¼ 2K q(qc=qx) q(qc=qx) qx
137
(2:304)
qI qc ¼ 2K q(qc=qy) qy
(2:305)
qI qc ¼ 2K q(qc=qz) qz
(2:306)
q qI qK qc q2 c qK qc 2 q2 c þ 2K 2 ¼2 þ 2K 2 ¼ 2 qx q(qc=qx) qx qx qx qc qx qx
(2:307)
q qI qK qc 2 q2 c ¼2 þ2K 2 qy q(qc=qy) qc qy qy d qI qK qc 2 q2 c ¼2 þ2K 2 dz q(qc=qz) qc qz qz
(2:308)
(2:309)
By substituting Equations 2.303 and 2.307 through 2.309 into Equation 2.301, we get qgo (c) qK qK 0¼ (rc)2 l þ 2 qc qc qc 2 2 2 qc q c qc þ 2K þ þ qx2 qy2 qz2
" # qc 2 qc 2 qc 2 þ þ qx qy qz (2:310)
By consolidating and rearranging terms in two steps, we obtain
qgo (c) qK qK (rc)2 l þ 2 (rc)2 þ 2Kr2 c ¼ 0 qc qc qc qK qgo (c) (rc)2 þ 2Kr2 c ¼ þl qc qc
(2:311) (2:312)
To evaluate the Lagrangian multiplier, l, the standard assumption is made that the system is large enough so that the change in the matrix composition during nucleation is negligible. Thus, in a three-dimensional counterpart to the reasoning employed to evaluate the constant z in Equation 2.203, at sufficiently large distances from any fluctuation both rc and r2c equal zero in Equation 2.312 and c ¼ co even during embryo growth. Hence, qgo (c) l¼ qc c¼co
(2:313)
By substituting into Equation 2.312, we have
qK qgo (c) qgo (c) qgo (c) qgo (co ) (rc)2 þ 2Kr2 c ¼ ¼ qc qc qc c¼co qc qc
(2:314)
The solution to this equation, subject to the appropriate boundary conditions, describes the spatial variation of composition in the critical nucleus. To evaluate DG*, the free energy of a uniform
138
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
solution with composition co is subtracted from that of a nonuniform solution as described by Equation 2.298: ð DG* ¼ [go (c) go (co ) þ K(rc)2 ]dV (2:315) where c must satisfy the Euler equation, Equation 2.314. A quantity Dg is now defined as ( ) qgo (c) Dg ¼ go (c) go (c) þ (c co ) (2:316) qc cco
In terms of the g–c (¼ g–x) diagram shown in Figure 2.42, Dg is the difference between go(c) at composition c (here ca) and the tangent to the g–x diagram at co. By substituting this equation into Equation 2.315, we get ) ð( qgo (c) 2 þK(rc) dV DG* ¼ Dg þ (c co ) (2:317) qc cco qgo (c) is a constant, it can be rewritten as Since qc c¼co ð ð qgo (c) DG* ¼ [Dg þ K(rc)2 ]dV þ (c co )dV (2:318) qc c¼co From Equation 2.299, the second integral is zero, and thus we obtain ð DG* ¼ [Dg þ K(rc)2 ]dV
(2:319)
Δgn g
Δg
cα co
cs
ca
cb
xααβ
cβ
=
=
A
c
cn = xnβ
B
xββα
FIGURE 2.42 A schematic for a g–c diagram showing Dg at ca. Note that both x and c are interchangeably used for concentration.
139
Diffusional Nucleation in Solid–Solid Transformations
By differentiating Equation 2.316 with respect to c, we get qDg qgo (c) qgo (co ) ¼ qc qc qc
(2:320)
By substituting this result into Equation 2.314, we have qK qDg (rc)2 þ 2Kr2 c ¼ qc qc
(2:321)
Equations 2.319 and 2.321 are the equations of the continuum nonclassical nucleation theory, stated in the most general form. The Euler equation, Equation 2.321, is solved for c as a function of position. Substitution of the (numerical) results thus obtained in Equation 2.319 then permits the calculation of DG*. The essential difference between these equations and those of classical nucleation theory is that no assumption is made in the derivation of the nonclassical equations about the homogeneity of the nucleus with respect to the composition. Note also that the nonclassical equations do not explicitly include an interfacial energy term. For an isotropic system, the critical nucleus will be a sphere. Hence, Equations 2.319 and 2.321 can be usefully rewritten in terms of the embryo radius, r. This process begins by utilizing that (rc)2 ¼ rc rc ¼
2 qc qr
(2:322)
and r2 c ¼
q2 c 2 qc þ qr 2 r qr
(2:323)
Using the above results, DG* ¼
ð"
2 # qc Dg þ K dV qr
(2:324)
Since V ¼ 4pr3=3 and dV ¼ 4pr2dr, DG* ¼ 4p
ð"
2 # qc Dg þ K r 2 dr qr
(2:325)
By substituting Equations 2.322 and 2.323 into Equation 2.321, we get qK qc 2 q2 c 4K qc qDg ¼ þ 2K 2 þ qc qr qr r qr qc
(2:326)
The boundary conditions are qc ¼0 qr
at r ¼ 1
(2:327)
c ¼ co
at r ¼ 1
(2:328)
140
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Returning once more to DG*, a more convenient form of this relationship is now derived. A physically reasonable assumption is first made that qK=qc ¼ 0. With the assistance of Equation 2.322, this simplifies Equation 2.301 to 2K
q2 c 4K qc qDg ¼ 2Kr2 c ¼ þ 2 qr r qr qc
(2:329)
Now using a vector identity, r w~ u ¼ wr ~ u þ~ urw
(2:330)
r [(c co )rc] ¼ (c co )r rc þ rc rc
(2:331)
(rc)2 ¼ (c co )r2 c þ r [(c co )rc]
(2:332)
and letting w ¼ c co and ~ u ¼ rc, By rearranging, we get
By substituting into Equation 2.319, we obtain ð ð DG* ¼ [Dg K(c co )r2 c]dV þ K r [(c co )rc]dV In the general form, the divergence theorem is ð ð ~ F ~ ndS r FdV ¼ ~
(2:333)
(2:334)
where ~ F is an arbitary vector and ~ n is the unit normal vector of a surface element, dS. Use of the divergence theorem for the 2nd term of Equation 2.333 yields ð ð r [(c co )rc]dV ¼ [(c co )rc] ~ ndS (2:335)
Choosing a surface far from the nucleus and recalling that in the boundary condition, Equation 2.327, rc ¼ 0, Equation 2.333 becomes ð (2:336) DG* ¼ [Dg K(c co )r2 c]dV By substituting Equation 2.329, and again replacing dV as in Equation 2.325, we find DG* ¼ 4p
1 ð 0
(c co ) qDg 2 r dr Dg 2 qc
(2:337)
2.13.1.3 General Properties of the Critical Nucleus 2.13.1.3.1 Properties Determined by Equation 2.326 (a) Dg at the center of the nucleus is negative In the context of the g–c diagram in Figure 2.42, the statement is made that the solute concentration of the nucleus must exceed cb at which it can form from a matrix of composition co with a decrease in free energy. This statement can be proved through integrating Equation 2.326, r ¼ 0 to r ¼ 1: # cðo " cðo 2 qK qc q2 c 4K qc qDg dc (2:338) þ 2K 2 þ dc ¼ qc qr qr r qr qc cn
cn
141
Diffusional Nucleation in Solid–Solid Transformations
By replacing dc by (dc=dr)dr and dividing both sides by 2, we get 1 2 # ð " 3 1 qK qc q c qc 2K qc 2 dr þK þ 2 qc qr qr 2 qr r qr 0
1 ¼ [Dg(r ¼ 1) Dg(r ¼ 0)] 2
(2:339)
But 1 ð" 0
" #1 2 # 3 1 qK qc q c qc K qc 2 þK dr ¼ 2 qc qr qr 2 qr 2 qr
(2:340)
0
However, from Equation 2.327, dc=dr ¼ 0 at r ¼ 0 and r ¼ 1. Hence, Equation 2.340 is equal to zero. Note also that Dg(r ¼ 1) ¼ 0, since c ¼ co at r ¼ 1. Equation 2.339 reduces to 1 ð 0
2K qc 2 1 dr ¼ Dg(r ¼ 0) r qr 2
(2:341)
Since K and r are always greater than zero and so is (qc=qr)2 irrespective of sign, Dg at r ¼ 0 must be negative in order to satisfy Equation 2.341. (b) At low supersaturation, the critical nucleus approaches the classical nucleus in the following respects: b-1. Composition, cN, at the center of the nucleus approaches the equilibrium composition, cb ¼ xba b . From the g–c diagram of Figure 2.42, as co ! ca, cb ! cb. Since cn > cb, cn also approaches cb and does so more closely. (Note: since the partial molar volumes of A and B in the nucleus are assumed to be the same, the parallel tangent construction applies in the determination of cn; hence, this result can be obtained with equal clarity simply by noting that as co ! ca, cn > cb.) b-2. The nucleus:matrix interfacial energy approaches that of a flat interface. This proof begins by replacing dr by (dr=dc)dc in Equation 2.341, yielding cðo
cN
2K qc 1 dc ¼ Dgn (cn ) ! 0 r qr 2
as co ! ca
(2:342)
Since 2qc=rqr is always negative, this term must also approach zero. Hence, the 4Kqc=rqr term in the Euler equation, Equation 2.326, can be omitted. Further, since r ! 1 as cn ! cb, r can be replaced with s, the distance 2 qK qc q2 c qDg þ 2K 2 ¼ (2:343) qc qs qs qc By multiplying both sides by qc=qs, we get 2 3 qK qc q c qc qDg ¼ þ 2K qc qs qs2 qs qs
(2:344)
But " # 2 q qc 2 qK qc 3 q c qc K þ 2K ¼ qs qs qc qs qs2 qs
(2:345)
142
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
By substituting into Equation 2.344, we get " # q qc 2 qDg K ¼ qs qs qs
(2:346)
Integrating 2 qc K ¼ Dg qs
(2:347)
This relationship is equivalent to Equation 2.204, the Euler equation for a flat interphase boundary. b-3. The critical nucleus radius approaches 1. The mean value theorem states that if some function, g(x), does not change signs between a and b and if c lies within this interval then, ðb a
ðb
f (x)g(x) dx ¼ f (c) g(x) dx
(2:348)
a
By applying this theorem to Equation 2.341, we get 1 ð 0
1 ð 2 2K qc 2 2 qc 1 K dr ¼ dr ¼ Dg(r ¼ 0) r r qr qr 2
(2:349)
0
where r is the mean radius of the nucleus. From Equations 2.204 and 2.205, on a per unit volume than a per atom basis, we have gc ¼ 2
þ1 ð
1
2 qx K ds qs
(2:350)
Converting from ds to dr under the condition that r is very large, we obtain gc ¼ 2
1 ð 0
2 qc K dr qr
(2:351)
By substituting this equation into Equation 2.349, we get gc 1 ¼ Dg(r ¼ 0) r 2
(2:352)
By rearranging, we find r ¼
2gc Dg(r ¼ 0)
(2:353)
As co ! ca, Dg ! 0 hence, r ffi R* ! 1. 2.13.1.3.2 Properties Near the Critical Temperature In the Cahn–Hilliard I on the interfacial energy of an interphase boundary [42], the free energy of the homogeneous solid solution was expanded about the critical composition, cc, of the miscibility gap near the critical temperature, Tc. The expansion was expressed in terms of the reduced composition:
Diffusional Nucleation in Solid–Solid Transformations
1 c co 2 ca c c
143
(2:354)
Thus, the expansion provides a basis for reformulating the Euler equation in spherical coordinates, Equation 2.326, in a dimensionless form. First, the foregoing expansion is differentiated with respect qK qc 2 to the composition to provide the right-hand side of the reduced equation. Next, the term qc qr on the left-hand side of this equation can be omitted, when qK=qc remains finite at the critical point or its square is less than ca. (cc – ca)2. The radial distance, r, is then replaced with the parameter: 1=2 2h(ca cc )2 1 q4 g t¼r , h¼ (2:355) Kc 4! qc4 Tc , cc where Kc is the value of K at T ¼ Tc and c ¼ cc , denoted KTc , cc . On this basis, the Euler equation becomes d2 X 2 dX 3 (4X X) 4Xo3 Xo ¼ 0 þ (2:356) 2 dt t dt Similarly, in the vicinity of the critical point Equation 2.337 is transformed to 3=2
32pKc (ca cc )Ic DG*c ¼ (2h)1=2
(2:357)
where Ic ¼
1 ð 0
(Xo þ X)(Xo X)3 t 2 dt
(2:358)
An analogue computer was used to solve Equation 2.356 for several values of Xo (the reduced composition of the alloy) between the equilibrium composition, Xa ¼ 1=2, and the spinodal composition, Xs ¼ (1=12)1=2 ¼ 0.29 (obtained by taking the second derivative of g with respect to c and equating to zero). As shown in Figure 2.43, as Xo at a given temperature approaches the spinodal, the compositional thickness of the interface increases, the region in the center of the nucleus where the solute concentration is essentially constant and also distinctly different from that of the matrix vanishes, and the excess solute concentration at the center of the nucleus approaches zero. Figure 2.44 shows the corresponding variation in Ic via curve (a). As shown by Equation 2.357, DG* is directly proportional to Ic in the vicinity of the critical point of the miscibility gap. Hence, DG* is infinite at the equilibrium composition (where undercooling is zero) but zero at the spinodal composition. Note that curve (b), which represents the behavior of DG* on the classical model, behaves similarly as the equilibrium composition is approached. However, the classical DG* remains finite at the spinodal composition. As will become clearer later, this result must be incorrect. In the Cahn–Hilliard I [42], it was shown that 3 qg (2:359) 4h(ca cc )2 TTc ¼ (Tc T) qTqc2 Tc , cc Rearranging to obtain an expression for ca cc and then substituting into Equation 2.357 yields 3 3=2 16pKc Ic qg 1=2 * (T Tc ) DGc ¼ h qTqc2 Tc , cc
Hence, DG*c is seen to be proportional to (T Tc)1=2 at a given Xo.
(2:360)
144
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
X0 = –0.45
0.8 0.4 0 0.8 X
X0 = –0.40 0.4 0 X0 = –0.35
0.4 0 0.4 0
X0 = –0.30 2
4 t
6
8
FIGURE 2.43 Reduced composition profiles of the critical nucleus in the region T Tc. (From Cahn, J.W. and Hilliard, J.E., J. Chem. Phys., 31, 688, 1959. American Institute of Physics. With permission.)
2 (b)
Ic
(a)
1
0 –5
–4
–3 Xs X0
FIGURE 2.44 Reduced free energy of formation for a critical nucleus with Xo: (a) Ic from Equation 2.358 and (b) I for a classical nucleation nucleus. (From Cahn, J.W. and Hilliard, J.E., J. Chem. Phys., 31, 688, 1959. American Institute of Physics. With permission.)
2.13.1.3.3 Properties Near the Spinodal Composition Since both the Euler equation, Equation 2.326, and that for DG*, Equation 2.337, in the general relationships for spherical nuclei contain the term qDg=qc, specialization of these relationships to the present situation (co ! cs) is begun with a Taylor series expansion of Dg about co, with an advantage later being taken of the circumstance that q2Dg=qc2 ¼ 0 at c ¼ cs:
Diffusional Nucleation in Solid–Solid Transformations
Dgco cs
qDg 1 q2 Dg 1 q3 Dg 2 ¼ Dg(co ) þ (c co ) þ (c co ) þ (c co )3 þ qc co 2! qc2 co 3! qc3 co
145
(2:361)
From Figure 2.42, Dg(co) ¼ 0 and (qDg=q c)co ¼ 0. Hence, Dgco cs ¼
1 q2 Dg 1 q3 Dg 2 (c c ) þ (c co )3 þ o 2 qc2 co 6 qc3 co
(2:362)
Taking the first derivative of Equation 2.362 while bearing in mind that the various derivatives taken at c ¼ co are constants, 3 qDg q2 Dg 1 2 q Dg ¼ (c co ) 2 þ (c co ) 3 qc qc co 2 qc co
(2:363)
where terms of an order higher than the third order are omitted. Taking the second derivative of Equation 2.362 with respect to c, we get q2 Dg q2 Dg q3 Dg ¼ þ (c c ) o qc2 qc2 co qc3 co
At c ¼ cs,
q2 Dg q3 Dg 0¼ þ(cs co ) 3 qc2 co qc co
(2:364)
(2:365)
where q2Dg=qc2 as well as q2g=qc2 is equal to zero at the spinodal composition because Dg is computed by subtracting the free energy along a straight line tangent to the g–c curve at c ¼ co from the ‘‘shoe string’’-shaped g–c curve for the solid solution. By substituting the relationship for (q3 Dg=qc2 ) co obtained by rearranging this equation into Equation 2.362, we get Dgco cs Let
Hence,
1 q3 Dg 1 q3 Dg 2 (c co )3 ¼ (c co ) (cs co ) 3 þ 2 qc co 6 qc3 co 1 q3 Dg j¼ 6 qc3 co Dgco cs ¼ j 3(cs co )(c co )2 (c co )3
(2:366)
(2:367)
(2:368)
To obtain DGs*, the activation free energy for the nucleus formation in the vicinity of the spinodal, begin by substituting Equation 2.368 into Equation 2.363 in order to eliminate the second derivative at c ¼ co thence simplifying Equation 2.363 with Equation 2.367, and substituting the relationship thus obtained for qDg=q c together with Equation 2.368 for Dgco cs into Equation 2.337: DGs* ¼ 2pj
1 ð 0
(c co )3 r 2 dr
(2:369)
146
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Now let c co cs co
(2:370)
j(cs co ) 1=2 r Ks
(2:371)
Y¼ and let
fcn ), instead of by the two-facet morphology. The results demonstrate that when gcab =gab is small, puckering permits a marked reduction in DG*. This occurs because the facet on the puckered nucleus occupies a much larger proportion of the interphase boundary area, particularly at small values of f. The introduction of the second facet, however, soon diminishes this advantage. Recently, using the electron backscattering diffraction technique, Adachi et al. [86] studied the variant selection of intergranular bcc-Cr precipitates in a Ni-43 wt.% Cr alloy and described its underlying mechanism in terms of the nucleation barrier as pictured in Figure 2.81.
191
Diffusional Nucleation in Solid–Solid Transformations c γαβ φc1 γαβ 0.900
1.0
φcn φc2
0.9 φc1
0.8
φc2 0.700
ΔG*/ΔG*s
0.7
0.6
φc1
0.526
0.5
φc2
0.4 0.300 0.3 φ c2
0.2
0.100 γββ = 1.05 γαβ
0.1 0.0
0
10
20
30
40 50 φ (°)
60
70
80
90
FIGURE 2.81 The normalized nucleation barrier, DG*=DGs*, as a function of f for a two-dimensional grain boundary allotriomorph. (Reprinted from Lee, J.K. and Aaronson, H.I., Acta Metall., 23, 799, 1975. With permission from Elsevier.)
2.15.4.3
Three-Dimensional Nuclei When f > fc1
2.15.4.3.1 Planar Grain Boundary To solve this problem, the simplifying assumption is made that the faceted portion of the nucleus is a truncated spherical cap with a fixed shape. Further, gab is taken to apply to the curved areas of both the faceted and the unfaceted portions of the nucleus. Hence, the problem is reduced to finding an equation for the shape of the unfaceted portion of the nucleus that yields the minimum interfacial area to volume ratio. This equation can be obtained through the variational calculus. The free energy of the lower portion of the nucleus can be expressed as G¼
ðð
[gab (1 þ p2 þ q2 )1=2 þ lu]dxdy
(2:499)
where x and y are orthogonal coordinates in the grain boundary plane, u is the coordinate normal to the grain boundary plane, p ¼ qu=qx, q ¼ qu=qy, l is a Lagrangian multiplier which is shown to be equal to DGV. A surface element of the interface is (1 þ p2 þ q2)1=2 dxdy and a volume element is udxdy.
192
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Minimization of the integrand is accomplished by applying the appropriate form of the Euler– Lagrange equation [87]: qf q qf q qf ¼0 qu qx qp qy qq
(2:500)
where f ¼ gab (1 þ p2 þ q2 )1=2 þ lu. Hence, l ¼ gab
q qf q qf þ qx qp qy qq
(2:501)
By carrying out the indicated differentiations q=qx(qf =qp) ¼ q=qx{P(1 þ p2 þ q2 )}1=2 ¼ r(1 þ p2 þ q2 )1=2 P(Pr þ qs)=(1 þ p2 þ q2 )3=2 etc. and then consolidating terms, we get r(1 þ q2 ) þ t(1 þ p2 ) 2pqs ¼
l (1 þ p2 þ q2 )3=2 gab
(2:502)
where r ¼ q2 u=qx2 , t ¼ q2 u=qy2 , and s ¼ q2u=qxqy. This is a quasi-linear, second-order elliptic partial differential equation for which there is no analytic solution under the condition of a given ‘‘domain’’ boundary, i.e., the trace of the faceted portion of the nucleus on the grain boundary plane. The equation was accordingly solved numerically, by means of the finite difference method, for chosen values of the relative interfacial energies involved. Before presenting representative results, it will be shown that this equation fulfills the basic condition of equilibrium shape, namely, that its mean radius of curvature is the same as that of the spherical part of the faceted portion of the nucleus. From the theory of differential geometry, the principal radii of curvature, R1 and R2, are the roots of the following quadratic equation [88]: R2 (rt s2 ) R[r(1 þ q2 ) þ t(1 þ p2 ) 2pqs](1 þ p2 þ q2 )1=2 þ (1 þ p2 þ q2 )2 ¼ 0
(2:503)
Denoting the mean curvature as H, and by means of the well known relationships of the sum and the product of the two roots of the equation, extracting the roots of Equation 2.503, we obtain 1 1 1 r(1 þ q2 ) þ t(1 þ p2 ) 2pqs ¼ þ H¼ 2 R1 R2 2(1 þ p2 þ q2 )3=2
(2:504)
Comparing Equations 2.502 and 2.504, we get 2H ¼
l DGV ¼ gab gab
(2:505)
2gab 2gab ¼ DGV f
(2:506)
Since H ¼ l=Rmean, Rmean ¼
Thus, the curvature of the lower interface is equal to the curvature of the spherical area of the faceted portion of the nucleus at R ¼ R*, as required by the condition of the equilibrium.
Diffusional Nucleation in Solid–Solid Transformations
(a)
193
(b)
(c)
FIGURE 2.82 A three-dimensional perspective of a faceted allotriomorph at a planar grain boundary: (a) the upper portion, (b) the lower portion, and (c) the contour map of the lower portion for a case with f ¼ 158, gbb ¼ 1.07 gab, and gcab ¼ 0:3gab . (From Lee, J.K. and Aaronson, H.I., Acta Metall., 23, 809, 1975. With permission.)
From the numerical solution to Equation 2.502, for the lower interface, and the analytic solution to the faceted spherical cap shape of the upper interface, computer-drawn perspective plots were obtained. Figure 2.82 shows the upper and lower portions in (a) and (b), and a contour map of the lower portion in (c), for a morphology in which one facet intercepts the grain boundary. The line along which the facet meets the grain boundary is indicated by an arrowhead. Note that the shape of the lower portion departs markedly from sphericity only near the center of the facet. Far from the facet, the lower portion is more nearly spherical. In the intermediate regions, the interface merges smoothly with those of the extremum regions. The contour map of the lower portion shows the smooth merging of the various ‘‘regions’’ with clarity. Figure 2.83 is a contour map of a lower portion when two facets are present. The interior or central region is now ellipsoidal-like rather than spherical-like in shape. In making these calculations, and those of DG* to be presented, the value of c1 for the faceted portion of the nucleus was taken as the one computed from the two-dimensional nucleus shape study under the same conditions of relative interfacial energy and f. Empirically varying c1 on either side of this value yielded only small reductions in DG*, indicating that the choice that was initially made was reasonable, and that the inability to integrate over the entire interface of the nucleus had not resulted in a serious error. Figure 2.84 shows the effect of DG*=DG*s (where DG*s is the activation free energy for the formation of an unfaceted, double spherical cap) for the planar boundary case. Comparison with the counterpart plot for the two-dimensional nuclei on Figure 2.81 indicates that
FIGURE 2.83 A contour map for the lower portion of an allotriomorph with two facets at a planar grain boundary: f ¼ 608, gbb ¼ 1.07 gab, and gcab ¼ 0:3gab . (From Lee, J.K. and Aaronson, H.I., Acta Metall., 23, 809, 1975. With permission.)
194
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
1.0
c γαβ γαβ
0.9
φeff C
2
0.7 0.8
φC1
ΔG*/ΔGs*
0.7
0.5
0.6
φeff C
2
0.5
0.3
0.4 0.3 0.1
0.2
γββ γαβ = 1.07
φeff C
2
0.1 0.0
0
10
20
30
40 50 φ (°)
60
70
80
90
FIGURE 2.84 The normalized nucleation barrier, DG*=DGs*, as a function of f for a three-dimensional allotriomorph at a planar grain boundary. (From Lee, J.K. and Aaronson, H.I., Acta Metall., 23, 809, 1975. With permission.)
the differences between them are largely quantitative. The principal qualitative difference is that the two-faceted morphology develops less readily at a given gcab =gab , fails to do so at all when this ratio reaches 0.6, and fcn appears when this ratio reaches 0.7 rather than 0.9 as in the two-dimensional case. However, the basic conclusions drawn from the two-dimensional plot are seen to be equally applicable to the three-dimensional nuclei. 2.15.4.3.2 Puckered Grain Boundary The situation here is basically different in one vital respect from that which obtains at grain boundaries that are puckered in two dimensions. Because the grain boundary must now meet the nucleus along a line (rather than at point junctions), must be everywhere continuous, is required to meet the nucleus at a range of levels at and ‘‘below’’ the grain boundary, and must make different angles with different combinations of upper and lower portion interphase boundaries, the puckered grain boundary in three dimensions really is puckered and thereby adds a significant amount of grain boundary area to the system. Two problems must thus be considered: (1) how to join the grain boundary to the nucleus so as to do the best practicable job of force=energy balancing and (2) how to minimize the area of additional grain boundary while simultaneously ensuring that the grain boundary does not attempt to migrate, i.e., in an equilibrium configuration. As shown in Figure 2.85, if the grain boundary meets the facets with the normal to its surface 0 parallel to OO , it will not have a component of force parallel to the facets. Immobilizing the grain boundary and minimizing the added grain boundary area is accomplished by giving the boundary zero a mean curvature. However, there is no such surface that has the same normal along the entire
195
Diffusional Nucleation in Solid–Solid Transformations
γαβ 0΄ B΄
B P
P h C΄
C 0 γαβ
FIGURE 2.85 A cross-section of a grain boundary allotriomorph with two facets at a puckered grain boundary. (From Lee, J.K. and Aaronson, H.I., Acta Metall., 23, 809, 1975. With permission.)
junction between the upper and lower portions of the nucleus. Hence, a zero mean curvature boundary was sought that does meet the junctions and adds minimally to the grain boundary area. Such a boundary, it was shown, can be adequately approximated by a catenoid: z2 ¼ c2 cosh2
y x2 c
(2:507)
where c is an adjustable constant. Use of this relationship allows for analytic treatment of the added grain boundary area. An exact relationship for the zero mean curvature minimal interface is given by Equation 2.502 with l ¼ 0 to denote the absence of a driving force. But an analytic solution of even this simplified version of Equation 2.502, though apparently possible, is very difficult. Numerical treatment indicates that Equation 2.507 represents a very good approximation of the exact solution. The added grain boundary area due to puckering, DAbb, multiplied by gbb, must thus be added to DG8.
DAbb ¼
ðð
(1 þ p2 þ q2 )1=2 1 dxdy
(2:508)
where p ¼ qz=qx, q ¼ qz=qy and the relationship for z is given by Equation 2.507. DG*=DG*s for puckered three-dimensional nuclei as a function of f at various ratios of gcab =gab is shown in Figure 2.86. A comparison with the counterpart plot for planar grain boundaries on Figure 2.84 shows that nucleation at a planar grain boundary is favored for most of the ranges in which f and gcab =gab were examined. The pucker mechanism has a lower DG* only when 08 f 188 with gcab < gab =2 and 08 f fc1 < 188 with gcab > gab =2. Hence, a situation is finally at hand in which the critical nucleus shape (and the associated puckering of the grain boundary), which more closely approximates equilibrium shape, has a higher DG*.
196
Mechanisms of Diffusional Phase Transformations in Metals and Alloys φCeffn 1.0 c γαβ
0.9
γαβ 0.8
0.7 φC1
ΔG*/ΔGs*
0.7 0.6
0.5
0.5 0.4 0.3
γββ
0.3
γαβ
= 1.07
0.2 0.1 0.0
0.1 0
10
20
30
40 50 φ (°)
60
70
80
90
FIGURE 2.86 The normalized nucleation barrier, DG*=DG*s , as a function of f for a three-dimensional allotriomorph at a puckered grain boundary. (From Lee, J.K. and Aaronson, H.I., Acta Metall., 23, 809, 1975. With permission.)
2.16 COMPARATIVE NUCLEATION KINETICS AT GRAIN FACES, EDGES, AND CORNERS RELATIVE TO HOMOGENEOUS NUCLEATION: TRADE-OFFS BETWEEN N AND DG* WHEN gab IS ISOTROPIC The basic reference is the work of Cahn [89]. See also Christian [5]. In this section, the term N, the number of atomic nucleation sites per unit volume in the nucleation rate equation, is examined in detail for the first time. However, N is necessarily considered in conjunction with DG*, particularly on the present approach of comparing nucleation kinetics at different types of grain boundary sites. It will be seen that sites offering low N values also offer significantly reduced values of DG*. This trade-off leads to a marked dependence upon supersaturation (DGV) and upon the relative values of the interfacial energies involved of the type of site at which nucleation is most rapid. All models of critical nuclei employed are based upon spherical caps, i.e., upon the assumption that gab is isotropic. This approach both permits the essentials of the problem to be grasped uncluttered by specialized crystallographic details and avoids the question, not yet adequately investigated, of the influence of orientation-dependent interphase boundary energy upon critical nucleus shapes at grain edges and grain corners. The critical nucleus shapes and their DG* equations employed by Cahn [89] are those of Clemm and Fisher [90]. The faces of a matrix grain are formed by the mating of pairs of planar or curved surfaces; again we assume that these surfaces are planar. The abutting pair of spherical caps used as a model in the previous section applies at grain faces when gab is isotropic. At the equilibrium between grain boundaries whose interfacial energies are equal and isotropic, a grain edge is formed by three grains, and thus by three grain boundaries meeting along a common line. The grain boundaries form 1208 angles with respect to each other. Here the grain boundaries are
197
Diffusional Nucleation in Solid–Solid Transformations
β
β α
β (a)
(b)
FIGURE 2.87 The shape of an a embryo formed at a three-grain junction in b: (a) a general view and (b) a cross-section view of the normal to grain edge. (From Clemm, P.J. and Fisher, J.C., Acta Metall., 3, 70, 1955. With permission from Elsevier.)
assumed to be planar and thus the grain edges are taken to be linear. The critical nucleus is formed by portions of three spheres, illustrated in an overview (a) and in a cross-section (b) in Figure 2.87. Grain corners are formed by the junction of four grains meeting at a point. The four grain edges, each being the junction of three of the grains, are assumed to radiate symmetrically from the corner. Hence, an embryo bounded by spherical surfaces will be a spherical tetrahedron as shown in Figure 2.88. Homogeneous nucleation is termed three-dimensional (3-D); grain face nucleation is 2-D; grain edge nucleation is 1-D, and grain corner FIGURE 2.88 The shape nucleation is 0-D. As the dimensionality of the nucleation decreases, of an a embryo formed at a four-grain junction in b. the assistance provided by the destruction of grain boundaries (From Clemm, P.J. and increases from nothing to areas of one, three, and six grain boundaries, Fisher, J.C., Acta Metall., 3, respectively. However, N simultaneously decreases. The following 70, 1955. With permission calculation assesses the competitive effects of these two factors as a from Elsevier.) function of cos c ¼ gbb =2gab . Begin by estimating approximately the ratio of the density of nucleation sites at the various grain boundary locations to that for homogeneous nucleation. For simplicity, and with insignificant loss of accuracy, assume that the matrix grains are cubes rather than tetrakaidecahedra, the equilibrium shape for grains in a three-dimensional structure with isotropic grain boundary energy. Assume a cubic grain to have an edge length of d and thus a volume of d3. Taking the grain boundary to have a width d, the volume of the grain faces is 6d2d=2, where the 1=2 enters because each face is shared by two grains. Similarly, the total volume of the grain boundary edges is 12dd2=4 and the total volume of the grain boundary corners associated with a single grain is 8d3=8. The ratio of the volume of a given type of grain boundary nucleation site to the volume for homogeneous nucleation is thus roughly (d=d)3–j, where j is the dimensionality of the nucleation site. Hence, the number of atomic (substitutional) nucleation sites per unit volume is expressed as No(d=d)3–j, where No is the number of atomic sites per unit volume for homogeneous nucleation (as before). The ratio of the number of atomic sites=unit volume of dimensionality i to that of dimensionality j is (d=d)3–i=(d=d)3–j ¼ (d=d)i–j. Clemm and Fisher [90] wrote the free energy of nucleus (a) at grain boundaries of the matrix phase (b) as, DG ¼ Cg3 f þ Bg2 gab Ag2 gbb obtained the following expression for DG* for the nucleus morphologies under consideration: 4(gab B gbb A) DG* ¼ 27C2 f2
(2:509)
where the expressions for A, B, C, and subsidiary constants for the three types of grain boundary nucleation site are given in Table 2.6. Figure 2.89 plots the ratio of DG* for nucleation at the various
198
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
TABLE 2.6 Expressions for A, B, and C Face
Edge 2
A ¼ p(1 k )
B ¼ 4p(1 k) C¼
2p (2 3k þ k3 ) 3
p
B ¼ 12 a kb 2 k2 C ¼ 2 p 2a þ (3 4k 2 )1=2 3 2 bk(3 k )
Corner 2 A ¼ 3 2f(1 k ) " #) 1=2 R R pffiffiffi R 1 k 2 4 8 p
B ¼ 24 kf e 3 " # ( 2 1=2 p
R R pffiffiffi C ¼ 2 4 e þ kR 1 k2 4 3 8 )
a ¼ sin1
R¼
2 1=2
2
A ¼ 3b(1 k ) k(3 4k )
2kf(3 k 2 )
1=2 4 3 2 k 2k 2 3 2 3 R f ¼ sin1 2(1 k2 )1=2 pffiffiffi 2 k(3 R2 )1=2 e ¼ cos1 R(1 k2 )1=2
1 2(1 k2 )1=2 k b ¼ cos1 {3(1 k 2 )}1=2
Source: Clemm, P.J. and Fisher, J.C., Acta Metall., 3, 70, 1955. With permission from Elsevier. gbb Note: k ¼ cos c ¼ 2gab
1.0
* ΔG*/ΔGhu
0.8
0.6 Face 0.4 Edge 0.2 Corner
0
0.5
1.0
1.5
2.0
γββ/2γαβ
* , at the various types of grain boundary FIGURE 2.89 The normalized nucleation barrier, DG*=DGhu nucleation sites as a function of cos c ¼ gbb=2gab. (From Cahn, J.W., Acta Metall., 4, 449, 1956. With permission from Elsevier.)
types of nucleation sites to DG* hu (the activation free energy for homogeneous nucleation of spheres) as a function of cos c at the three types of grain boundary nucleation sites. As expected, this ratio decreases with decreasing dimensionality of the nucleation site and increasing gbb=2 gab. If the principal effects of the nucleation site dimensionality upon Jss* can be considered to be encompassed by influences upon N and DG* (since the role of Z should be minor and that of b*,
Diffusional Nucleation in Solid–Solid Transformations
199
limited to less than an order of magnitude, changes in S*), then the ratio of the steady-state nucleation rate at i-type sites to that at j-type sites can be written as J*ss,i ¼ J*ss,j
ij * d (Ki Kj )DGhu exp kT d
(2:510)
Taking the natural logarithm of both sides of this equation, we get ln
* J*ss,i d (Ki Kj )DGhu ¼ (i j) ln kT d J*ss,j
(2:511)
By rearranging, we obtain kT J*ss,i kT d ln ¼ (i j) ln (Ki Kj ) * * * DGhu Jss,j DGhu d
(2:512)
Defining R as R¼
kT d ln * d DGhu
(2:513)
Using R *i kT Jss, ln ¼ (i j)R (Ki Kj ) * Jss, *j DGhu
(2:514)
(i j)R ¼ (Ki Kj )
(2:515)
For J*ss,i ¼ J*ss,j ,
Using this equality, Figure 2.90 is constructed on the basis now illustrated. Let i ¼ 2 and j ¼ 1, with k ¼ gbb=gab (¼ 2 cos c) ¼ 0.5. From Figure 2.89, Ki ¼ 0:63 and Kj ¼ 0:50. Substituting into Equation 2.515 yields R ¼ 0.13, the location in Figure 2.90 at k ¼ 0.5 of the curve separating the * i > Jss, * j , Equation 2.515 is ‘‘boundary’’ (i ¼ 2) from the ‘‘edge’’ ( j ¼ l) regions. To make Jss, rewritten as (i j)R < (Ki Kj ) when i > j
(2:516)
The effects of the foregoing conditions on J* can be conveniently illustrated in terms of a minimum detectable nucleation rate, 1=cm3s. (Even with optical microscopy as applied to a relatively rapid reaction such as the formation of proeutectoid ferrites in Fe–C alloys, this is likely an overoptimistic estimate of the lowest detectable rate.) During homogeneous solid–solid nucleation, the pre-exponential factor is ca. 1030=cm3s. For grain boundary nucleation, i3 * d Ki DGhu 30 *i ¼ Jss, 10 exp d kT
(2:517)
When d ¼ 108 cm and d ¼ 102 cm, the pre-exponential factor as a whole is successively reduced to 1024, 1018, and 1012 as the dimensionality is decreased to 2, 1, and 0.
200
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
0.25
Homogeneous
0.30
Boundary
0.20
log (d/δ)
R=
ΔG*nu/kT
Face
0.15
0.10 Edge
0.05 Corner
0
0.5
1.0
1.5
2.0
γββ/γαβ = k
FIGURE 2.90 R vs. k ¼ gbb=gab ¼ 2cos c. (From Cahn, J.W., Acta Metall., 4, 449, 1956. With permission from Elsevier.)
* ¼ 1=cm3 s decreases Figure 2.91 shows that as gbb is increased relative to gab, the R at which Jss rapidly once the grain boundary nucleation range is entered. Thus, the type of site at which nucleation first appears when the driving force for nucleation is just large enough to make it detectable depends upon gbb=gab. When this ratio exceeds 0.9, the grain boundary energy is high enough relative to the interphase boundary energy to make corner nucleation preferable. In the range of 0.6–0.9, the more numerous edge sites are favored; the corner sites are now too few to compensate for the higher—but not high enough—nucleation kinetics at corners. Similarly, in the range of 0.3–0.6, nucleation at grain faces (i.e., grain boundaries) is favored. And at lower values of the ratio, homogeneous nucleation predominates. Finally, keeping the interfacial energy ratio, d and d constant, an inquiry is made as to the effects of undercooling below the equilibrium temperature upon the preferred type of nucleation site. These effects * . The assumption is made that nucleation takes place are exerted through the DGV component of DGhu in a temperature range and in an alloy that permits one to write DGV ¼ DHV TDSV. Since at the equilibrium temperature, DGV ¼ DHV (1 T=Teq ) ¼ DHV (1 u). Hence, R may be manipulated as follows: kT d kTlnðd=dÞ ln ¼ 3DHV2 (1 u)2 3 * 16pgdb DGhu d kuTeq lnðd=dÞ ¼ 3DHV2 (1 u)2 ¼ Constant u(1 u)2 3 16pgdb
R¼
(2:518)
201
Diffusional Nucleation in Solid–Solid Transformations
0.25
a 0.20
Homogeneous
0.30
b
log (d/δ)
0.15
c
R=
ΔG*hu/kT
Face
0.10
Edge d
0.05 Corner
0
0.5
1.0
1.5 e
2.0
γββ/γαβ = k
FIGURE 2.91 An illustration for the conditions under which various types of sites make the greatest contribution to the nucleation rate. (From Cahn, J.W., Acta Metall., 4, 449, 1956. With permission from Elsevier.)
R is seen to become zero at u ¼ 1 (i.e., at Teq) and at u ¼ 0. By differentiating Equation 2.518 with respect to u and setting the resulting relationship equal to zero, it is found that qR=qu ¼ 0 at u ¼ 1 and 1=3. According to Gjostein [85], the lattice diffusion at T=Tm ¼ 0.5, the commonly accepted (approximate) temperature at which diffusivity becomes negligible, corresponds to 1015 cm2=s. This diffusivity is reached at ca. 0.33 Tm for grain boundary diffusion in bcc metals and at ca. 0.27 Tm for grain boundary diffusion in fcc metals. Since Teq < Tm, it seems fair to conclude that R increases with increasing undercooling throughout the temperature region in which nucleation occurs with detectable kinetics. Hence, referring to Figures 2.90 and 2.91, on the spherical cap based models of the critical nucleus used when gbb=gab < 1.5 2.0 with increasing undercooling, nucleation is initially most rapid at corners, then at edges and finally at grain faces. Only when gbb=gab < 0.4 does homogeneous nucleation enter this picture, succeeding grain face nucleation at the largest undercoolings. In view of the extent of the foregoing discussions, it is important to note explicitly that replacement of the spherical cap models with appropriately faceted ones is required before the results of this treatment can be quantitatively compared with experimental observations.
2.17 NUCLEATION AT DISLOCATIONS This topic is important not only in connection with nucleation at isolated dislocations in the interiors of matrix grains but also at sub-grain boundaries, small-angle grain boundaries, and, as more recent research on grain boundary structure indicates, at many high-angle grain boundaries as well. The catalytic effect of dislocations on nucleation is evidently fundamentally different from that exerted by disordered-type grain boundaries. Whereas destruction of the grain boundary area by the critical nucleus reduces the interfacial energy attending the nucleation process, elimination of or interaction with a dislocation by a critical nucleus makes available some of the strain energy associated with the dislocation to reduce DG8 and also diminishes the transformation strain energy, W, attending
202
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
nucleation. Even on the scale of critical nucleus dimensions, the area of a dislocation in contact with the nucleus appears to be too small to permit a significant contribution of interfacial energy to the nucleation process by the dislocation. Hence, the primary addition to the nucleation mathematics in the case of dislocations is that of describing the strain energy of dislocations and the strain energy interactions between the dislocations and the critical nucleus. The complexity of the latter mathematics, particularly, is such that the critical nucleus shapes employed tend not to be crystallographically realistic. And theoretical studies to date have been largely confined to isolated, straight dislocations serving as nucleation sites. Nonetheless, important contributions have been made and a considerable predictive capability will be seen to have developed in a later section.
2.17.1 INCOHERENT NUCLEATION Incoherent nucleation is shorthand for nuclei whose interphase boundary structure is of the isotropic, disordered type, and, as noted before, is more relevant to those of a liquid or gas phase. For a crystalline case, it appears physically unrealistic and not useful for a direct comparison with the experiment, but this assumption nonetheless permits focusing of attention upon interactions between the critical nucleus and the dislocation rather than upon the complexities of the critical nucleus shape arising from the anisotropy of interphase boundary energy. 2.17.1.1 The Cahn Treatment This is the first quantitative treatment of nucleation at dislocations and continues to be very important [91]. The dislocations at which nucleation is considered are straight, isolated edge, or screw dislocations. Both the nucleus and the matrix are taken to be elastically isotropic with the same elastic constants. Volume strain energy associated with critical nucleus formation is implicitly assumed to be relieved by the dislocation. Unlike the nucleation processes so far considered, three terms now comprise the free energy change, DG8, associated with the formation of an embryo: (1) the usual nucleus:matrix interfacial energy term (which is again positive), (2) the usual volume free energy term (which is again negative), and (3) the dislocation strain energy term (which is negative since the dislocation strain energy eliminated by the nucleus is contributed to the free energy required by the nucleation process). As a consequence of the presence of these three terms, the free energy change vs. radius behavior for embryo formation is altered. Dislocation strain energy is most important at small values of r, the radius of a cylindrical embryo. The volume free energy change exerts its major influence at large r values and the interfacial energy of nucleus:matrix boundaries is most important at intermediate values of r. If the dislocation and volume free energy change terms, both negative, do not overlap sufficiently in their major regimes, a plot of the type indicated by Curve A in Figure 2.92 results. The radius corresponding to the minimum in this curve is interpreted as that of a subcritical metastable cylinder of the nucleus phase surrounding the dislocation line. The radius, ro, is taken as the energetic starting point for nucleation. When the dislocation strain energy and the volume free energy change do overlap sufficiently, Curve B behavior results and there is no activation barrier to nucleation. Formation of a precipitate can then be regarded as a growth process ab initio. A preliminary calculation is now undertaken to ascertain the conditions under which the two types of behavior develop. DG8 for the circular cross-section of the cylindrical embryo of length l is written as r (2:519) DG ¼ 2prlg pr 2 lg Al ln 0 1 r where r is the radial distance from the dislocation line (r l), g is equal to DGV, r 0 is the dislocation core radius, A is equal to mb2=4p (1 v) for edge dislocation,
203
Diffusional Nucleation in Solid–Solid Transformations
ΔG °
A
B
r
FIGURE 2.92 DG8 vs. r for a cylindrical embryo surrounding a dislocation line: (A) with an activation barrier and (B) with no activation barrier for nucleation. (From Cahn, J.W., Acta Metall., 5, 169, 1957. With permission from Elsevier.)
A is equal to mb2=4p for screw dislocation, m is elastic shear modulus, b is Burgers vector length, n is Poisson’s ratio. Taking qDG8=qr ¼ 0, Al r
(2:520)
2pgr 2 2pgr þ A ¼ 0
(2:521)
" # g 2Ag 1=2 g ro ¼ 1 1 2 ¼ [1 (1 a)1=2 ] 2g pg 2g
(2:522)
0 ¼ 2plg 2prlg By rearranging, we get
Solving
where a ¼ 2 Ag=pg2. A minimum in DG8 vs. r occurs when a < 1 at ro ¼
g [1 (1 a)1=2 ] 2g
(2:523)
The extraneous, nonphysical solution with 1 þ (1 a)1=2 was dropped. No minimum appears when a > 1. Under this circumstance, DG8 decreases continuously with r and there is no barrier to nucleation (Curve B behavior). When a < 1, a nucleation barrier does obtain. Nucleation may take place by local thickening of the cylinder in the customary fluctuational manner. When this ‘‘bulge’’ develops beyond a certain size, continued growth with a decrease in free energy can again take place. DG* will depend upon the shape of the bulge and its length along the dislocation; the problem is thus to determine the shape and size of the fluctuation that requires the small increment of free energy. Let the shape of the nucleus be given by the radius, r, as a function of the distance z along the dislocation line. The additional free energy that must be acquired to form the bulge is
204
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
DG ¼
þ1 ð
1
pffiffiffiffiffiffiffiffiffiffiffiffiffi
2 r 2 2 2pg r 1 þ r_ ro pg r ro A ln dz ro
(2:524)
where r_ ¼ dr=dz. The integrand is the free energy of the bulged cylinder minus that of the unbulged cylinder. To minimize DG8, an appropriate form of the Euler–Lagrange equation is taken to be r_
qf f ¼ C1 q_r
(2:525)
where f is the integrand, C1 is a constant and is applied subject to the relevant boundary conditions. Substituting the result into Equation 2.524 yields pg3 DG* ¼ 3 (1 b2 ) g where y ¼
1 ð 0
h
(1 þ y)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 q2 dy
(2:526)
r 1 and b2 ¼ 1 a. ro (1 þ b) ln (1 þ y) y(1 b)(2 þ y) q ¼ (1 þ y) 1þ þ C1 2 4
Equation 2.526 was numerically integrated and the resulting values of DG* normalized through * are plotted against a in Figure 2.93. Recall that increasing values of a correspond division by DGhu to large values of A and g and lower values of r. Figure 2.94 shows the variation of ro with a and two shape parameters, r1 and l, defined in the inset sketch of the critical nucleus morphology,
1.0 0.9 0.8
ΔG*/ΔG*hu
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 α
* vs. a for nucleation at a dislocation line. (From Cahn, J.W., Acta Metall., 5, 169, FIGURE 2.93 DG*=DGhu 1957. With permission from Elsevier.)
205
Diffusional Nucleation in Solid–Solid Transformations 22 20 18 16
Nucleus size
14 12 rh 10 8 r0 6
r1 r1
4 2 0
r0 0.1
0.2
0.3
0.4
0.5 α
0.6
0.7
0.8
0.9
1.0
FIGURE 2.94 The dimensions of a critical nucleus vs. a at a dislocation line: ro is the radius of the original precipitate pipe, r1 is the maximum radius of the nucleus, l is the length of the nucleus, and rh is the radius of a homogeneous nucleus. (From Cahn, J.W., Acta Metall., 5, 169, 1957. With permission from Elsevier.)
together with rh, the critical radius of an unfaceted sphere. The upturn in l as a approaches unity is due to the merger of the critical nucleus pictured with a cylinder of infinite length, which does not require a nucleation step to form. Substituting ‘‘reasonable’’ numbers in an approximate nucleation rate equation, Cahn finds a homogeneous nucleation rate of 1070=cm3s when the rate of nucleation on dislocation is 1 per cm of dislocation line per second. 2.17.1.2 The Gomez-Ramirez and Pound Treatment Gomez-Ramirez and Pound [92] investigated in more detail the strain energy contribution that a dislocation can make to incoherent nucleation. Although they assumed the nucleus:matrix boundary energy to be independent of boundary orientation, neither the Wulff construction nor the Euler– Lagrange equation was employed to deduce the critical nucleus shape, thereby limiting the utility of the results achieved. The standard free energy change associated with the formation of an embryo at a dislocation was written as ð 2m(1 þ v) T 2 Ec mb2 l r mb(1 þ v) sin u e V 2 Vc ln dV DG ¼ gab Aab þ DGV V þ 9(1 v) 4p rc 3p(1 v) r prc (2:527) The first and second terms on the right-hand side of this equation are of standard form for the interfacial energy and the volume free energy change. The two terms are left in schematic form since they used a variety of critical nucleus shapes. The third term is the volume strain energy associated with nucleation: this is readily converted to Equation 2.137 with eT ¼ 3e and E ¼ 2m(1 þ n):
206
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
W¼
2m(1 þ v) T 2 E Ee2 V e V¼ (3e)2 V ¼ 9(1 v) 9(1 v) 1v
(2:528)
The 4th term is the strain energy contributed by the destruction of the dislocation core. Let Ec ¼ the core energy per unit length of dislocation. Hence, the core energy per unit volume of dislocation core is Ec =prc2 , where rc is the core radius. Taking Vc as the volume of core eliminated by the formation of an embryo, the total core energy made available is equal to the 4th term. (The factor of 2 in the original work appears to be a typographical error.) The authors estimate rc 2b and Ec mb2=4p. The 5th term is the strain energy contributed to the nucleation process by the elimination of the dislocation strain field in the region now occupied by the embryo. The elastic strain energy of the strain field associated with a screw dislocation per unit length of the dislocation line is W1 ¼
mb2 R0 ln 4p rc
(2:529)
where R0 represents the effective radius of the strain field. The assumption is made that the dislocation segment present prior to the formation of an embryo is ‘‘smeared out’’ over the surface of the embryo, as a continuous distribution of infinitesimal dislocations. The core energy of the smeared-out dislocation is reasonably considered to have been destroyed, since the smeared-out core would not add further to the energy of an already disordered interphase boundary. The strain field in the matrix associated with the smeared-out dislocation is W2 ¼
mb2 R0 ln 4p r
(2:530)
where r is the radius of the embryo. The reduction in strain energy is therefore, DW ¼ W2 W1 ¼
mb2 r ln 4p rc
(2:531)
The 5th term is then completed by multiplying this relationship by l, the length along the dislocation that the embryo extends, in order to obtain the total strain energy contribution from this cause. Finally, the 6th term is the interaction energy between the dislocation strain field in the matrix and the strain field arising from the volume change associated with the transformation. The authors realized that DG* can be obtained from Equation 2.527 through the application of the Euler–Lagrange equation. However, reaching the usual conclusion that the resulting equation cannot be solved analytically, they chose another route. They hypothesized that DG* is not a strong function of the critical nucleus shape (contrary to the conclusion repeatedly reached) and that it would suffice simply to choose some plausible smooth shapes, such as would preserve cylindrical symmetry for nucleation on screw dislocations and allow asymmetry in the case of nucleation on edge dislocations. DG* was then calculated numerically for these shapes at different levels of interfacial energy. Figure 2.95 shows the critical nucleus shapes employed. In Figure 2.95, the drawings of (a), (b), and (c) were used for screw dislocations and the rest are for edge dislocations. Figure 2.96 shows for edge in (a) and for screw dislocations in (b) that DG* is ca. independent of the critical nucleus shape for the shape families employed. Curiously, though, the example employed in the published calculations was the a to g transformation in pure Fe. The most negative jDGVj for this transformation is only ca. 3 cal=cm3 (¼12.6 mJ=m3). Further, the energy of a disordered a:g boundary is ca. 750 ergs=cm2 (¼750 mJ=m2), appreciably above the highest interfacial energy used in these calculations, 560 ergs=cm2 (¼560 ergs=cm2). Note that the 60 kT observability limit for DG*, which is ca. 5eV, falls far below even the (reasonably extrapolated) lowest set of curves at the 3 cal=cm3 level of ‘‘Specific Volume Free Energy,’’ the maximum
207
Diffusional Nucleation in Solid–Solid Transformations h
2t
ℓ
h
(a)
(c) ℓ θ
h
h+x=r h–x (d)
(b)
h m n
h
h (e)
ℓ
ℓ
FIGURE 2.95 Critical nucleus shapes: (a) prorate spheroidal shape, (b) circular shape, (c) trapezoidal shape, (d) heart shape, and (e) ellipsoidal shape. (With kind permission from Springer Science þ Business Media: From Gomez-Ramirez, R. and Pound, G.M., Metall. Trans., 4, 1563, 1973.)
1
3
2
10
3 ΔGV (eV/nm ) 5 4
7
8
γ = 560 ergs/cm2
Ellipsoidal shape Heart shape
9
6
1 10 9
8
8
6 γ = 400 ergs/cm
5
2
4 3
γ = 240 ergs/cm2
5
6
7
8 γ = 560 ergs/cm2
γ = 400 ergs/cm2
Fe screw dislocation
γ = 240 ergs/cm2
4 3
1 150
4
Trapezoidal shape Circular shape Prolate spheroidal shape
5
1 100
ΔGV (eV/nm3)
6
2
50
3
7
2
0
(a)
Nucleation barrier (eV)
Nucleation barrier (eV)
Fe edge dislocation 7
2
200
250
0
300
Speciic volume free energy (cal/cm3)
(b)
50
100
150
200
250
300
Speciic volume free energy (cal/cm3)
FIGURE 2.96 Nucleation barriers for various critical nucleus shapes at an edge dislocation in (a) and at a screw dislocation in (b). (With kind permission from Springer Science þ Business Media: From Gomez-Ramirez, R. and Pound, G.M., Metall. Trans., 4, 1563, 1973.)
208
Mechanisms of Diffusional Phase Transformations in Metals and Alloys ΔGV (eV/nm3) 10 9
1
2
3
4
5
6
7
8
γ = 560 ergs/cm2
Fe Edge dislocation Screw dislocation Homogeneous
8
Nucleation barrier (eV)
7 6 5 γ = 400 ergs/cm2 4 3 2 γ = 240 ergs/cm2 1 0
50
100 150 200 250 Speciic volume free energy (cal/cm3)
FIGURE 2.97 The nucleation barrier along an edge and a screw dislocation as compared with the barrier for homogeneous nucleation. (With kind permission from Springer Science þ Business Media: From Gomez-Ramirez, R. and Pound, G.M., Metall. Trans., 4, 1563, 1973.)
available for this transformation. This finding is a particularly striking example of the unviability of incoherent nuclei, admittedly under unusually stringent conditions. Figure 2.97 compared DG* for nucleation at edge and screw dislocations with that for homogeneous nucleation. Because of the smaller strain field associated with a screw than with an edge dislocation (i.e., by a factor of 1=(1 v)), DG* is seen always to be smaller for edge dislocations. The advantage accruing to dislocation nucleation relative to homogeneous nucleation is seen to be small for these unfaceted nuclei. Note how rapidly DG* increases at a given DGV with increasing gab, and also how DG* at a given gab rises swiftly as DGV approaches zero, thereby causing DG* ! 1. The authors assessed nucleation rates at dislocations by taking the ratio of the nucleation rate at dislocations to that of homogeneously formed, unfaceted spheres under the same thermodynamic conditions. All pre-exponential terms in this ratio were taken to cancel other than N, where nucleation at dislocations also suffers a severe ‘‘dimensionality’’ disadvantage. N at dislocations was computed as the product of the volume of the cylindrical core of a dislocation of unit length, pr2 1, the number of atoms per unit volume, No=Vb (Equation 2.113), and the dislocation density, r. If the dislocations are assumed to be straight and parallel, the number of dislocations per unit area is also the number of dislocations per unit volume. * Jscrew * Jhu* . The effectiveness of increasing gab upon Figure 2.98 demonstrates that Jedge diminishing J* is clearly shown. However, the most striking result is the enormous diminution in J* with jDGVj. (It is clear that at 3 cal=cm3, J* would be truly negligible under any of the circumstances considered.) Figure 2.99 shows that J* rises rapidly with an increasing shear
209
Diffusional Nucleation in Solid–Solid Transformations Fe ρ = 1010disln’s/m2 T = 1250 K
20 16
Logarithm of the rate nucleation
12 8 4 –ΔGV 2
4
6
(eV/nm3)
8
–4 –8 –12 –16 0.560 J/m2 0.400 J/m22 0.240 J/m
Edge Screw Homogeneous
–20
FIGURE 2.98 Nucleation rates for a homogeneous case, for nucleation at an edge dislocation, and for nucleation at a screw dislocation. (With kind permission from Springer Science þ Business Media: From Gomez-Ramirez, R. and Pound, G.M., Metall. Trans., 4, 1563, 1973.)
G (1010 N/m2) 5
1 Δ GV = –5
20 Logarithm of the rate of nucleation
2
3
4
5
6 7 8 9 10
3
eV/nm
3
m
Δ GV
10
Δ
Sr
TI 0
20
n eV/ = –3
–6 G V=
Co Fe
Ti
Ca 40
3
m eV/n
60
80 100
200
400
U 600
3
nm
–10
V/ 8e
ΔG
=–
V
Screw Edge
0.560 J/m2 0.400 J/m2 0.240 J/m2
ρ = 1010 disln’s/m2 T = 1.1Tequil
–20 Shear modulus (eV/nm3)
FIGURE 2.99 The nucleation rate as a function of the shear modulus, m. (With kind permission from Springer Science þ Business Media: From Gomez-Ramirez, R. and Pound, G.M., Metall. Trans., 4, 1563, 1973.)
210
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
modulus, even though DG8 is but linearly proportional to m in various terms of Equation 2.527. Physically this occurs because the larger the shear modulus is, the longer is the strain energy that the destruction of a portion of a dislocation can contribute to the reduction of DG*. Unlike the Cahn treatment, the present approach usually did not produce a metastable minimum in the DG8 vs. R curve. Typical curves that did obtain are shown in Figure 2.100. The ‘‘normal’’ curve, (a), appeared most frequently. However, in the presence of a high shear modulus, metastable embryos were found, even in undersaturated systems, (b).
2.17.2 COHERENT NUCLEATION When all nucleus:matrix interfaces are fully coherent, the nucleation process is termed ‘‘coherent.’’ Since the lattice planes are continuous through the two phases, a coherent nucleus cannot relieve the strain energy associated with a dislocation. Hence, both the strain energy and the core energy of the dislocation remain during nucleation. The assistance that a dislocation affords coherent nucleation has been investigated several times [93–96]. Barnett [95] and later Larché [96] wrote the schematic equation for the free energy change associated with embryo formation as DG ¼ (DGV þ W)V þ Wint þ gab A
(2:532)
where W is the usual elastic strain energy associated with a dilatational transformation in the absence of a dislocation, Wint is the interaction strain energy when a nucleus forms in the presence of a dislocation.
Free energy of formation
Free energy of formation
ΔG o
ΔG*
ΔG o
R
R Radius
(a)
(b)
R*
R (c)
Radius
Free energy of formation
ΔG o
Free energy of formation
ΔG o
Radius
(d)
R Radius
FIGURE 2.100 DG8 vs. R for nucleation at a dislocation line: (a) a common case, (b) an undersaturated case with a high shear modulus, (c) and (d) supersaturated cases with a high shear modulus. (With kind permission from Springer Science þ Business Media: From Gomez-Ramirez, R. and Pound, G.M., Metall. Trans., 4, 1563, 1973.)
Diffusional Nucleation in Solid–Solid Transformations
211
Larché [96] notes that once again the Euler–Lagrange equation is needed to determine exactly the shape of the critical nucleus (gab is taken to be independent of boundary orientation, just as in the incoherent case). Lyubov and Solv’Yev [93] solved the variational calculus problem with an axially symmetric approximation, yielding a second order ordinary differential equation that could be solved. The basic minimal shape is cylinder-based. The cylinder is tangent to the dislocation line, making the interaction strain energy negative if it appears on the compressive side of the dislocation when the volume change is negative and on the expansion side of the dislocation when the volume change is positive. Forming the nucleus in an appropriately strained region removes the need to produce the strain as part of the nucleation process. In effect, Wint relieves W through being of the opposite sign. Thus, the dislocation does not ‘‘back add’’ any strain energy to the nucleation process in the sense of providing extra energy that can be used to create interfacial energy. The unusual shape and severe strains of the dislocation core explain why the nucleus will not normally incorporate the core when the nucleus forms coherently. The position that the nucleus takes in the dislocation strain field will thus be just that which relieves the largest proportion of the dilatational transformation strain. Larché [96] expressed the results obtained in terms of an effective interfacial energy of an a:b boundary, geff. For a spherical nucleus, geff ¼ gab A=3
(2:533)
geff ¼ gab A
(2:534)
and for a cylindrical nucleus
where A¼
mb (1 þ v) T e 3p (1 v)
(2:535)
A is in the range of ca. 103eT mJ=m2. Hence, if eT ¼ 0.05, a 50 mJ=m2 reduction in gab will be achieved. Since the coherent interphase boundary energies usually encountered may be of about this magnitude, the effect upon DG*, and thus upon J*, can be very large indeed. This finding also makes clear that edge dislocations, particularly, can be effective catalysts only for those nuclei that form with an appreciable stress-free transformation strain.
2.18 COMPARISONS OF THEORY AND EXPERIMENT Since such comparisons are still sparse, those that are available will be dissected in some detail. Emphasis will be incorporated on the principles of the experimental techniques utilized.
2.18.1 HOMOGENEOUS NUCLEATION Because interfacial energy is almost invariably not only the most important ancillary parameter but also the one about which the least data are available, homogeneous nucleation represents the most attractive vehicle for comparing measured and calculated nucleation kinetics, since the comparison in this case requires knowledge of only nucleus:matrix interfacial energy. If the critical nucleus shape can be safely approximated as a sphere, a single such energy is needed. When gab is significantly anisotropic, several energies may be needed. In either case, however, the nucleus: matrix boundary during homogeneous nucleation is almost certainly fully coherent. Hence, the consideration of the ‘‘multi-plane discrete lattice model,’’ and particularly the plots of the anisotropy of gab vs. T=Tc presented on Figure 2.27, are applicable. The principal problem remaining in the evaluation of gab is the selection of the most useful regular solution constant or, of course, the employment of a more accurate model of the solid solution, if it is available.
212
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
2.18.1.1 Homogeneous Nucleation of Co-Rich Precipitates in Cu-Rich Cu–Co Alloys 2.18.1.1.1 Servi–Turnbull Investigation The references are the studies of Servi and Turnbull [97] and Tanner and Servi [98]. Their work remains the best available study of its type; despite excellence, however, it cannot be regarded as fully definitive. In part this is because a multilevel indirect approach was used to secure the nucleation data. However, the skill with which this approach was developed encourages its detailed examination, particularly since this permits inclusion of topics that would not otherwise be covered because of our philosophic objections to such an approach. Since the following recounting is rather lengthy and involved, it will be helpful to provide a ‘‘road map’’ of what is to be discussed before actually undertaking this enterprise: (1) The source of the basic data on f, the fraction of the matrix transformed to precipitate, namely, measurements of electrical resistivity, will be noted and the relationship through which resistivity data are converted to f will be given. (2) An equation expressing f in terms of reaction time, t, and a rate constant, G, will be written and it will be shown how it is extracted from the f data. (3) Np, the number of * , the rate of nucleation at steady state precipitates per unit volume will be calculated from G. (4) Jss will be calculated from Np. (5) The interfacial energy calculated from J* and the homogeneous nucleation theory will be compared with what is independently calculated. (6) Revisions of (5) by Shiflet et al. [99] will be shown to improve the agreement between the gab calculated from the nucleation measurement and that obtained independently, thereby strengthening considerably the support that this investigation provides to the view that the Gibbsian nucleation theory is quantitatively applicable to solid–solid nucleation. Servi and Turnbull [97] used six Cu–Co alloys, containing from 1.00 to 2.69 wt.% Co. Specimens were fabricated into wires, solution annealed and then isothermally transformed at temperatures from 6508C to 1658C below the solvus temperatures. The transformation was followed by a resistometric method. The fraction of the precipitates formed, f, was taken to be given by f ¼
R0 R R0 R1
(2:536)
where R0 is the initial resistivity, R1 is the final, fully transformed resistivity. Subject to the assumptions made, an exact equation relating f to the isothermal reaction time, t, will be derived, as originally reported by Wert and Zener [100]. This equation, however, proved to be unintegrable. The following equation due to Wert [101] is semiempirical, but was shown by Wert and Zener to produce results nearly the same as the exact equation when f < ca. 0.5. In integrated form, this relationship is ( ) n
t 3=2 t (2:537) ¼ 1 exp f ¼ 1 exp G G where G is a constant. By rearranging and taking logarithms, we get n 1 t ln ¼ 1f G
(2:538)
Taking logarithms again 1 ¼ n ln t n ln G ln ln 1f
(2:539)
213
Diffusional Nucleation in Solid–Solid Transformations
°C 666 656 646 637 627 617 607 598
1
log
1–f
0.1 0.05
0.7 0.6 0.5 0.4 0.3 0.2 0.1
Slope = 1.45
0.05
Fraction transformed ( f )
G
0.5
Specimens 8C, 8D 1.11wt.% cobalt
0.01 0.005 0.1
1.0
0.01 100
10 Time (min)
FIGURE 2.101 Transformation kinetics of Cu-1.33 wt.% Co alloy. (From Servi, I.S. and Turnbull, D., Acta Metall., 14, 161, 1966; Errata, Acta Metall., 14, 908, 1966. With permission from Elsevier.)
2.69%
2.05%
1.33% 1.48%
Solvus
1.11%
1.0%
Thus, straight line plots of ln (ln 1=(1 f)) vs. ln t with slope n are anticipated. Figure 2.101 is a typical set of plots of this type. Equation 2.537 is based on the assumptions that growth is diffusion-limited and that the number of particles per unit volume, Np, is constant. Under these circumstances, n ¼ 1.5. Experimentally, however, n decreased from ca. 2 to ca. 1 as the bulk Co concentration increased from 1.00 to 2.69. Unpublished observations of Kinsman and Aaronson [102] indicate that an overlap of the diffusion fields of adjacent Co particles occurs very early in the transformation; this may be responsible for these unforecast values of n. But, the straight lines in the above plot and in equivalent plots for other Cu–Co alloys are very good. Figure 2.102 summarizes the values of G obtained. These values were secured by noting that when f ¼ 0.6321, t ¼ G (see Equation 2.539). They are seen to vary markedly with reaction temperature and alloy composition. G varies more rapidly with temperature as the pct. Co increases. This figure also shows that the critical undercooling required to initiate transformation
2.69%
1.48%
2.05%
1.0%
log G (s)
4
1.33%
1.11%
5
3
2 1 550
600
650
700 750 800 Temperature (°C)
850
900
FIGURE 2.102 Time constants as a function of temperature and composition. (From Servi, I.S. and Turnbull, D., Acta Metall., 14, 161, 1966; Errata, Acta Metall., 14, 908, 1966. With permission from Elsevier.)
214
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
decreases with increasing Co concentration. (In temperature ranges below the solvus from 9008C at 1.00% Co to ca. 6008C at 2.69% Co, there was no perceptible nucleation.) Both results follow from the shape of the solvus curve: both DGV and the solute concentration gradient driving growth rise more sharply with decreasing temperatures the higher the Co concentration in the alloy. A relationship will next be developed to calculate Np, the number of particles per unit volume from G. Following Wert and Zener [100], assume that the precipitates are growing as isolated, simultaneously nucleated spheres. The flux of solute ingested, F1, is given by ba dV F1 ¼ xab (2:540) a xb dt where a represents the precipitate, b represents the matrix, V is the volume of one sphere of precipitate, t is the isothermal growth time, ba xab a and xb are atom fractions of solute in a at the a=(a þ b) and in b at the b=(a þ b) phase boundaries, respectively. The flux of solute diffusing into the precipitate, F2, on Fick’s first law is qx 2 qx F2 ¼ AD ¼ 4pR D qr qr R
(2:541)
where A is area of the interphase boundary, D is the chemical interdiffusivity in substitutional solid solutions or interstitial diffusivity in interstitial solid solutions (as determined from the Boltzmann–Matano solution), R is precipitate radius, r is the radial distance from the center of the precipitate. To evaluate qx=qr at r ¼ R, a solution to the appropriate diffusion equation is needed. On the assumption that the matrix is a dilute solution, the precipitate can be considered to grow so slowly that the interphase boundary is effectively stationary. Hence, the diffusion equation to be solved is Fick’s second law under steady-state conditions, written in spherical coordinates: 2 qx q x 2 qx (2:542) ¼0¼D þ qt qr2 r qr As described by Shewmon [10] and as will be discussed in Section 3.3.2.1, the solution to this equation is
R (2:543) x ¼ xb xb xba b r
where xb is the atom fraction of solute in the alloy that is the atom fraction of solute remote from the interphase boundary prior to overlap of the diffusion fields of adjacent precipitate x is the atom fraction of solute in b at distance r.
By differentiating Equation 2.543 with respect to r and then replacing r with R in accordance with the requirements of Equation 2.541, we get xb xba qx b (2:544) ¼ R qr R
Diffusional Nucleation in Solid–Solid Transformations
215
By substituting this equation into Equation 2.541 and noting that F1 must equal F2 (i.e., equating Equations 2.540 and 2.541), we get xb xba b ba dV 2 ¼ 4pR xab D x a b R dt
(2:545)
For a sphere, dV=dt ¼ 4pR2dR=dt. Assuming dilute solution thermodynamics, so that activities may be replaced by concentrations, the fraction of the thermodynamically allowable precipitate that remains to occur is 1f ¼
xb (t) xba b
xb (0) xba b
(2:546)
where xb(0) ¼ xb as defined in conjunction with Equation 2.543 xb(t) is average solute concentration in the untransformed matrix at time t. Furthermore, assuming that all precipitates start to grow at the same time and that they are uniformly dispersed in an initially homogeneous matrix, f ¼
R Ro
3
(2:547)
where Ro is the final radius of the precipitates. Differentiating this equation with respect to time yields dR 1 2=3 df ¼ f Ro dt 3 dt
(2:548)
Substituting this relationship and also Equations 2.546 and 2.547 into Equation 2.545, recognizing that xb in Equation 2.545 is actually xb(t) under present circumstances, rearranging, and dividing both sides of the resulting equation by 2 (in order to produce a relationship parallel in form to the one that yielded Equation 2.537 after integration) leads to
ba R2o xab a xb df 3
(2:549) ¼ (1 f )f 1=3 ba dt 2 2D xb (0) x b
where
ba R2o xab x a b
¼G 2D xb (0) xba b
(2:550)
Forming now a mass balance (note that Equation 2.545 is a flux balance) xb (0) xba b ¼
4p 3 ab Ro xa xba Np b 3
(2:551)
The right-hand side of this equation represents the solute concentration incorporated in the array of precipitates, whereas the left-hand side gives the solute concentration available for this purpose. Rearranging this equation in terms of Np, rearranging Equation 2.550 in terms of R2o and then
216
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
rewriting on the basis of R3o , and substituting the expression for R3o into Equation 2.551, which eliminates Ro from further consideration, leads to 2 31=2 xab xba 3 a b 4 5 Np ¼ 8p(DG)3=2 2 xb (0) xba
(2:552)
df 3 ¼ (1 f )f dt 2
(2:553)
b
The foregoing derivation also yields the exact differential equation for the overall kinetics of the precipitation process. By substituting Equation 2.550 into Equation 2.549, we get G
1=3
This exact relationship may be contrasted with the semiempirical one due to Wert [101]: 1=2 df 3 t (2:554) G ¼ (1 f ) dt 2 G Straightforward integration of this equation yields Equation 2.537. A comparison of the f vs. t plot obtained from Equation 2.537 with that secured by numerically integrating Equation 2.553 was previously indicated to give closely comparable results until f is ca. 0.5. Hence, an evaluation of Np can be conducted on the basis of G values obtained by applying Equations 2.537 and 2.539 to the experimental data. Figure 2.103 summarizes the Np data obtained as a function of temperature for the various alloys used. Np is seen to rise swiftly with decreasing temperature. For example, in Cu-2.69% Co,
800
850
2.69%
2.05%
17
1.48%
1.0%
18
Temperature (°C) 700 750
650 1.33%
600 1.11%
550
16
log Np
15
14
13
2.69%
2.05%
1.48%
1.33%
1.11%
Solvus
1.0%
12
FIGURE 2.103 Calculated particle number densities as a function of temperature and composition. (From Servi, I.S. and Turnbull, D., Acta Metall., 14, 161, 1966; Errata, Acta Metall., 14, 908, 1966. With permission from Elsevier.)
Diffusional Nucleation in Solid–Solid Transformations
217
it increased by four orders of magnitude in less than 508C. The full range of Np is 1011.5 to 1017.5. The higher values, especially, are in excess of the concentration of heterogeneous nucleation sites provided by grain boundaries, dislocations, and inclusions. Only excess vacancies, quenched in from the solution annealing treatment, remain as possible heterogeneous nucleation sites. However, given a constant solution annealing temperature at least for a given alloy, the number of such sites would not depend sharply upon temperature, and hence a relatively weak temperature-dependence of Np is to be expected, as contrasted with the marked variation of Np with isothermal reaction temperature experimentally observed. Hence, it appears reasonably safe to conclude that the Co-rich precipitates did form homogeneously. N must now be connected to the nucleation rate, assumed to be steady state. More exactly, the * , which obtained prior to the diminution of xb below its initial value, xb(0), is sought initial value of Jss in order to facilitate a comparison with the nucleation theory. At a given reaction temperature, Np ¼
1 ð
J*dt
(2:555)
0
It would be simpler to assume that * Np ¼ aJss,o
(2:556)
* is the ‘‘initial’’ steady state nucleation rate. This will be where a is a constant and Jss,o done initially and then a second approximation will be made. For the nucleation rate, Servi and Turnbull [97] wrote 9 8 > > = < 3 2 No x2b D 16pg va * ¼ (2:557) exp Jss,o h i2 2 > > a ; : 3k3 T 3 ln xb =xba b
This equation should be compared with the steady-state component of Equation 2.114, also for homogeneously nucleated spheres. As Russell [1] pointed out: (1) This equation omits the Zeldovich factor; (2) the n*2=3 term in the relationship for b* is omitted, however, this item and the preceding one approximately cancel, and (3) x2b should be replaced by xb. Thus, by substituting rough estimates of the various components in the pre-exponential terms of Equation 2.557 and then of Equation 2.114, we get 23 4 No x2b D 10 10 D 1034 D 1015 a2 2Dxb g1=2 2 102 10 D 7 1035 D 1030 [1016 103 ]1=2 a4 (kT)1=2 Hence, the numerical effects of these discrepancies are not large by the standards and requirements of nucleation kinetics. The origin of the (correctly written) exponential term in Equation 2.557 should be briefly noted. The different appearing portion of this term is that expressing the volume free energy term. On the ideal solution approximation converted to a per atom basis, " # ba 1 xba kT ab xb b ab þ 1 xa ln x ln DGV ¼ xb 1 xb va a
(2:558)
218
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Since xab a is nearly unity in the Cu–Co system, DGV ffi
kT xb ln ba va xb
(2:559)
The first approximation used to compare theory and experiment was effected through rewriting Equation 2.557 as
ln
No x2b a Np ¼ ln D a2
b h
i2 T 3 ln xb =xba b
(2:560)
* , by aNp and b ¼ 16pg3 v2a =3k3 . * , rather than Jss where Equation 2.556 has been used to replace Jss,o h
i2 Appreciable scatter is said to have characterized the plot of ln(Np=D) vs. 1=T 3 ln xb =xba , b which the authors did not show. From the slope of this plot, b ¼ 6 109 and gab ¼ 188 mJ=m2. In the second approximation, an accounting was made of the diminution in supersaturation, and hence in DGV as precipitation proceeds. This approximation is begun by noting that f ¼
xb x
(2:561)
xb xba b
where x is the average atom fraction of solute remaining in the matrix. The term ln (xb =xba b ) in Equation 2.557 is then rewritten as ln (x=xba b ) to allow for less than full supersaturation. The latter term can be rewritten as
ln
x xba b
"
xba b ¼ ln ba þ ln 1 f 1 xb xb xb
!#
(2:562)
The following series expansion can be employed when z is small: z2 z3 ln (1 z) ¼ z þ þ þ 2 3
(2:563)
For small values of f, Equation 2.562 thus becomes
ln
x xba b
xba b ffi ln ba f 1 xb xb xb
!
(2:564)
By repeating Equation 2.557 in a condensed form, but now taking account less than full supersaturation, we get 8 >
=
b Jss* ¼ K exp h
i2 > > ; : T 3 ln x=xba b
(2:565)
219
Diffusional Nucleation in Solid–Solid Transformations
For the initial, i.e., full supersaturation value of the steady state nucleation rate 9 8 > > = < b * ¼ K exp Jss,o h i2 > > ; : T 3 ln xb =xba b
(2:566)
Taking the logarithm of the ratio of Equation 2.565 to Equation 2.566, we get ln
Jss* ¼ J* ss,o
b b h i2 þ h i2 3 ln x =xba T 3 ln x=xba T b b b 9 8 . ba > > = < 2bf x x xb b b b 1 1 ¼ 3 h i2 þ h i2 ffi h i3 > T > ; : ln xb =xba f 1 xba =xb ln xb =xba T ln xb =xba b b b b
(2:567)
where both Equation 2.564 and f 1 are used. Let 2b xb xba b F¼h . i3 T ln xb xba xb b
(2:568)
Hence,
ln
Jss* ¼ Ff J*
(2:569)
ss,o
Using the b value calculated from the first approximation, the relationship between Jss*=J*ss,o and f for each of the experimental conditions can be obtained from Equation 2.569. In more general terms, Figure 2.104 shows the relationship between Jss*=J*ss,o and f for selected values of F. Throughout the span of the experimental data, F varied between 8.5 and 64. Since the electrical resistivity measurements provide data on the variation of f with time, Equation 2.569 and the 1.0 F=
0.8 F = 10
J *ss J *ss,o
0.6
2b {T In (co /ce)3}
(1 – ce /co)
where b = 6 × 109
F = 20 0.4 F = 40 F = 60
0.2 0
0
0.01
0.02
0.03
0.04 0.05 0.06 0.07 Fraction tranformed ( f )
0.08
0.09
0.10
FIGURE 2.104 The relative nucleation rate as a function of the fraction transformed for selected experimental conditions. (From Servi, I.S. and Turnbull, D., Acta Metall., 14, 161, 1966; Errata, Acta Metall., 14, 908, 1966. With permission from Elsevier.)
220
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
calculations of F permit the evaluation of Jss*=J*ss,o as a function of time. Constructing plots of this relationship then permits graphical evaluation of the function: z¼
1 ð 0
Jss* dt J*ss,o
(2:570)
z varies from 36 down to 0.5 s and from 462 down to 38 s, as the reaction temperature decreases from 8438C to 5988C and from 6668C to 5988C, respectively, for the 1.11% Co alloy. By rewriting Equation 2.555 in an equivalent form, we get Np ¼ J*ss,o
1 ð 0
Jss* dt ¼ zJ*ss,o J*ss,o
(2:571)
Therefore, from Equation 2.560 Np ¼ J*ss,o ¼ z
No x2b D a2
9 8 > > = < b exp h i2 > > ; : T 3 ln xb =xba
(2:572)
b
Making a numerical estimate of the temperature-independent portion of the pre-exponential factor No x2b 1023 104 c2b ¼ 1034 c2b a2 1015 where the authors have chosen to replace xb with the weight percent, cb, of Co in the alloy. By substituting this result into Equation 2.572 and rearranging, we get ln
Np ¼ ln 1034 zc2b D
b h i2 T 3 ln cb =cba b
(2:573)
h
i2 Figure 2.105 plots log Np =zc2b D vs. 109 T 3 log cb =cba . The intercept is 32, in reasonable b
agreement with the theory. Once again, however, b ¼ 6 109. From the expression for b given in Equation 2.560, gab 200 mJ=m2. Using a curiously incorrect version of Equation 2.153, the twoplane discrete lattice relationship for gc ¼ gab, and by doing so incorrectly, a value of 230 mJ=m2 was obtained in very good agreement with the experimentally derived value. Shiflet et al. [99] have conducted a detailed reexamination of these important calculations. The calculation of gc ¼ gab from the nucleation rate data was first considered. Inasmuch as Servi and Turnbull [97] found b to be unchanging during their second approximation, only the first was employed. Co-rich precipitates are fully coherent with the Cu-rich matrix at diameters greater than 100 nm [102]. Servi and Turnbull took coherency into account by recalculating DGV using the coherent solvus (to be discussed in the section of precipitation) instead of the equilibrium or incoherent solvus. This was shown to reduce gab from 200 to 190 mJ=m2. Shiflet et al. used a different, though supposedly equivalent approach. They continued to use the incoherent solvus; the volume strain energy, W, was then algebraically added to DGV. The equation for a sphere given in Table 2.1 was employed, where both phases are assumed to be elastically isotropic but have different values of the shear modulus and Poisson’s ratio. From the Servi–Turnbull type of plot shown in Figure 2.106 and a least squares analysis, a value of gab ¼ 165 mJ=m2 was obtained. The independent calculation of gc ¼ gab was performed by the multi-plane discrete lattice treatment. Evaluation of DE ¼ HAB (HAA þ HBB )=2 was performed on the regular solution
221
Diffusional Nucleation in Solid–Solid Transformations 33 31 29 ζcβ2D
log
Np
Cobalt content % 27
1.00 1.11 1.33 1.48 2.05 2.69
25 23 21 19 0
5
10
15
20 109
T 3 log
Co Ce
25
30
35
40
2
h
i2 . (From Servi, I.S. FIGURE 2.105 The relationship between log Np =zc2b D and 109 T 3 log cb =cba b
and Turnbull, D., Acta Metall., 14, 161, 1966; Errata, Acta Metall., 14, 908, 1966. With permission from Elsevier.)
70 Cobalt content % 65
1.00 1.11 1.33 1.48 2.05 2.69
D
ℓn
Np
60
55
xx
50
45 0
1
2
3
1023 kT ℓn xβ + T – W Vα xββα
4
5
6
2
" ! #2 Np kT xb 23 FIGURE 2.106 The relationship between ln ln ba þ W . (Reprinted from and 10 =T D va xb Shiflet, G.J. et al., Scr. Metall., 15, 719, 1981. With permission from Elsevier.)
222
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
constant considerations. When the best value selected for the regular solution constant, V ¼ 33,300 J=mol, was employed, both g(111) ¼ 180 and g(100) ¼ 190 mJ=m2 were calculated at the average temperature of the nucleation rate measurements, ca. 0.5 T=Tc, where Tc ¼ 2010 K in this system. When the two-plane discrete lattice model, Equation 2.153, was utilized with the same regular solution constant, g(111) ¼ 265 and g(100) ¼ 310 mJ=m2. When the regular solution constant appropriate to the solubility method is used in conjunction with Equation 2.153, these two coherent interphase boundary energies are increased to 440 and 510 mJ=m2; these are the values which Servi and Turnbull should have secured; they are clearly in poor agreement with the value derived from the experimental measurements. However, using the multi-plane discrete lattice model and the excess free energy approach to calculate the regular solution constant restores the agreement between theory and experiment incorrectly claimed by Servi and Turnbull. This experiment thus remains the best proof available that classical homogeneous nucleation theory is appropriate for solid–solid nucleation. One remains concerned, however, about the highly indirect, albeit very clever, method through which the experimental values of Jss* were obtained. (Shiflet et al. found that steady-state nucleation is obtained quickly enough in this system so that the Servi–Turnbull implicit assumption of steadystate nucleation kinetics is justified.) The wide range of slopes in the plots of Figure 2.101, whereas only one slope, with a value of 3=2 was expected theoretically, is a specific point of concern. The extensive overlap of diffusion fields occurring early in the transformation and the failure of the second approximation to change the calculated gab are also worrisome. One needs to know how much coarsening affected the results. Finally, it would be desirable to check, via high-resolution TEM, the accurate sphericity of the nuclei assumed in the analysis and also forecast by the weak anisotropy shown in the T=Tc ¼ 0.5 plot of gab given in Figure 2.27. 2.18.1.1.2 LeGoues–Aaronson Investigation In this investigation of the homogeneous nucleation kinetics of Co-rich precipitates in Cu–Co matrices [103], the direct measurement of precipitate number densities was substituted for the indirect measurements used by Servi and Turnbull. Additionally, the problem of transformation during quenching to and=or from the intended isothermal reaction temperature was explicitly recognized and alloys were chosen in which transformation kinetics were neither too rapid nor too slow to permit reliable, convenient measurement of nucleation rates. Finally, comparisons between theory and experiment were made utilizing classical theory, Cahn–Hilliard continuum nonclassical theory, and Cook-de Fontaine-Hilliard discrete lattice point nonclassical theory. The times required to form 1011 critical nuclei=cm3 (a conservative estimate of the minimum observable rate of nucleation) were calculated from the classical homogeneous nucleation theory using the mathematical apparatus outlined for application of the general relationship J* ¼ Zb*N exp (DG*=kT) exp (t=t). Critical nucleus shapes were determined using the discrete lattice plane theory; results of the shape calculations are shown on Figure 2.62. The invariant field approximation for spherical, diffusion–controlled growth (Section 3.3.2.1) was used to estimate the time required for spherical precipitates. A good approximation of the shape of Co-rich precipitates used to reach radius R ¼ 5 nm beyond R* were calculated from t¼
(R R*)2 (1 2xc ) 2D(1 xa xc )
(2:574)
where xc is the composition of the critical point of the miscibility gap formed by the extension of the Cu-rich and Co-rich terminal solid solubilities, xa is the average composition of the alloy. The nucleation and growth times were added to produce the TTT-like curves for bulk alloy compositions ranging from x ¼ 0.005 to 0.04 as shown in Figure 2.107. Two different diffusivities, D, were employed. The solid curves were computed using diffusivities extrapolated from temperatures in the
223
Diffusional Nucleation in Solid–Solid Transformations x = 0.005
0.2
0.3
0.4 Tm/Tc T/Tc
0.3
x = 0.008
Tm/Tc T/Tc
T/Tc
Tm/Tc
0.2 0.1
0.1 0
2
4
6 8 log (t)
10 12
0
x = 0.01
0.3 0.2 0.1
2
4
6 8 log (t)
0
10 12
2
4
6 8 10 12 log (t)
Tm/Tc Tm/Tc
0.2 0.1 0
2
4
6 8 log (t)
10 12
0.3
0.3
0.2
0.2
0.1
0.1
0
2
4
6 8 log (t)
10 12
x = 0.03
0.4 T/Tc
0.3
x = 0.02
0.4
x = 0.015 T/Tc
T/Tc
0.4
0.5
0.5 Tm/Tc
0
2
4
6 8 10 12 log (t)
Tm/Tc 0.5
x = 0.04
T/Tc
0.4 0.3
Number of vacancies defined by annealing temperature Number of vacancies defined by reaction temperature
0.2 0.1 –2
0
2
4
6 8 log (t)
10 12
FIGURE 2.107 TTT curves: The solid curves indicate the vacancy concentration at the annealing temperature and the dashed curves indicate the vacancy concentration at the reaction temperature. (Reprinted from LeGoues, F.K. and Aaronson, H.I., Acta Metall., 32, 1855, 1984. With permission from Elsevier.)
solid solution region at which the data were conventionally measured. The dashed curves were determined on the assumption that the vacancy concentration present at the solution annealing temperature (8708C=20 min) was retained at the isothermal reaction temperatures employed. On this basis, alloys containing 0.005, 0.008, and 0.01 atom fraction of Co were prepared. It was decided to conduct the measurements of nucleation kinetics at temperatures in the vicinity of the nose of the TTT-curve in the various alloys employed so that small errors in temperature measurement and control would not cause drastic changes in the nucleation rate. TEM was used to count particle number density; a convergent beam electron diffraction technique was employed to determine the thickness of the electron-transparent foils prepared from the heat-treated specimens. Thus, the number of Co-rich particles per unit volume could be straightforwardly obtained. Figure 2.108, taken from a specimen of the l% Co alloy reacted for 10 min at 6208C (the ‘‘white’’ line bisecting each particle is the Ashby–Brown no-contrast line, indicating that these particles are fully coherent with their matrix), indicates that the precipitates were sufficiently wide spaced so that their diffusion fields did not significantly overlap and also the transformation kinetics were sufficiently slow so that there was no danger of precipitate formation occurring anisothermally. Figure 2.109 presents typical plots of precipitate number density vs. isothermal reaction time obtained from the 1 at.% Co (0.01) alloy. The incubation time, during which the nucleation rate is time-dependent, is evidently very short and lost in the scatter of the data. Hence, only steady-state nucleation rates were determined. The plot (a) of Figure 2.110 shows the variation of the steady-state nucleation temperature expressed as T=Tc (where Tc was evaluated as 2768 K) in the
224
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
1000 Å
FIGURE 2.108 Precipitation in a Cu-1 at.% Co alloy reacted 20 min at 6208C. The image under two beam conditions and g ¼ . (Reprinted from LeGoues, F.K. and Aaronson, H.I., Acta Metall., 32, 1855, 1984. With permission from Elsevier.) T = 600°C
T = 620°C 8
N/1015 (ppt/cm3)
N/1015 (ppt/cm3)
8 6
4 2
360 Time (s)
540
4
2
J = 4.22 × 1013 ppt/cm3/s τ = 100 s 180
(a)
6
J = 1.7 × 1012 ppt/cm3/s τ = 30 s 600
720 (b)
1200
1800
Time (s)
FIGURE 2.109 The number of precipitates as a function of time in a Cu-1 at.% Co alloy at T ¼ 6008C and T ¼ 6208C. (Reprinted from LeGoues, F.K. and Aaronson, H.I., Acta Metall., 32, 1855, 1984. With permission from Elsevier.)
1% Co alloy. The calculated curves were determined from the Cook-de Fontaine-Hilliard discrete lattice point theory, with strain energy incorporated in the style of Cook and de Fontaine; D was again calculated in the presence of quenched-in vacancies from T ¼ 8708C and also in the absence of quenched-in vacancies. Note a good agreement was obtained with the quenched-in vacancies curve. The plot (b) of Figure 2.110 is an equivalent one for t, the incubation time. Note that the experimental data closely lie on the curve obtained with the diffusivity of no-quenched-in vacancies. Evidently, the splendid levels of agreement shown in these two plots are partially fortuitous. It seems safe to conclude, however, that both the steady-state nucleation rate and the incubation time (calculated from classical theory, using the appropriate shape factors) have been successfully accounted for by theory within 1–2 orders of magnitude. Considering the extreme sensitivity of DG* to small variations in its components, these results uphold a level of reasonable agreement between theory and experiment. This is, in fact, the first fully reliable demonstration that nucleation theory is correct for a transformation in a condensed phase! Similar levels of agreement were obtained in the 0.8 and 0.5 at.% Co alloys. Figure 2.111 compares all three nucleation theories used with the data on the Cu-1 at.% Co alloy. Because strain energy is very difficult to incorporate in the continuum theory (though Cahn has
225
20
18
18
16
16
14
14
12
12
10
10
8
8
6
6
4
4
2
2
0
0 0.0
(a)
log (τ), s
log (J *), ppt/cm3/s
Diffusional Nucleation in Solid–Solid Transformations
0.1
0.2
–2
0.4
0.3
T/Tc
0.1
0.2
0.3
0.4
0.5
T/Tc
(b) c0 = 0.01
D defined by number of vacancies at the annealing temperature D defined by number of vacancies at the reaction temperature Experimental results
FIGURE 2.110 Experimental data on J* and t compared with the theoretical values for a Cu-1at.% Co alloy. (Reprinted from LeGoues, F.K. and Aaronson, H.I., Acta Metall., 32, 1855, 1984. With permission from Elsevier.)
20 18 16
log (J*), ppt/cm3/s
14 12 10 8 6
c0 = 0.01 Discrete lattice model
4
Continuum model Classical theory
2 0
Experimental results 0.0
0.1
0.2
0.3
0.4
T/Tc
FIGURE 2.111 A comparison of experimental data on J* in a Cu 1 at.% Co alloy with the values calculated from three theoretical models. (Reprinted from LeGoues, F.K. and Aaronson, H.I., Acta Metall., 32, 1855, 1984. With permission from Elsevier.)
226
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
1.0 0.8 0.6
0.6
0.4
0.4
0.2
0.2 1
(a)
2
1 (b)
R/a 1.0
2 R/a
C0 = 0.005 T = 447°C
0.8 C
C0 = 0.008 T = 585°C
0.8 C
C
1.0
C0 = 0.01 T = 613°C
0.6 0.4 0.2 1
(c)
2
3
R/a
FIGURE 2.112 Concentration profiles calculated at three different temperatures. (Reprinted from LeGoues, F.K. and Aaronson, H.I., Acta Metall., 32, 1855, 1984. With permission from Elsevier.)
made an attempt to do so), strain energy was excluded from all three theories in this set of calculations. Hence, agreement with the experimental data points is not as good. Note, however, that within the temperature range studied, the three theories are in excellent agreement with one another. This result was surprising, since Cahn–Hilliard III [66] predict that classical theory is appropriate only when the interface thickness is much less than the radius of the critical nucleus. Figure 2.112 presents the concentration profiles calculated at a representative temperature in each of the three alloys studied. Note that while R* is not so much larger than L, nonetheless there is a region near the center of the nucleus where the composition profile is nearly flat; hence the volume free energy change and the interfacial energy can be realistically taken separately into account in the style of classical theory. We thus conclude that the Cahn–Hilliard criterion for the applicability of classical nucleation theory may be a little too strict. Finally, Figure 2.113 compares the TEM data on nucleation kinetics in Cu-1 at.% Co with that obtained by Servi and Turnbull using an electrical resistivity technique and a complex, multilevel analysis with which to extract nucleation rates. The agreement is seen to be remarkably good, despite the various difficulties and uncertainties associated with the Servi–Turnbull analysis. 2.18.1.2 Homogeneous Nucleation of Ni3Al Precipitates in Ni-Rich Ni–Al Alloys Kirkwood [104] used TEM to determine the nucleation kinetics of g0 Ni3Al ordered fcc precipitates in Ni-13.23 at.% Al (6.55 wt.% Al) alloys at 8008C and at 7508C. As shown in Figure 2.114, however, the number of particles per unit volume, Np, decreased continuously with time. Thus, all experimental measurements must have been made in the coarsening or Ostwald ripening rather than in the nucleation-and-growth regime. West and Kirkwood [105] repeated these studies at higher reaction temperatures, i.e., at smaller undercoolings, namely at 8108C, 8008C, and 7908C, using much higher quenching rates from the reaction temperature to room temperature. Hirata and Kirkwood [106] extended these studies to a 6.05 wt.% Al alloy reacted at several temperatures in the
227
Diffusional Nucleation in Solid–Solid Transformations 20 18 16
log ( J*), ppt/cm3/s
14 12 10 8 c0 = 0.01
6
D defined by number of vacancies at the annealing temperature D defined by number of vacancies at the reaction temperature This work Servi and Turnbull
4 2 0
0.0
0.1
0.2 T/Tc
0.3
0.4
FIGURE 2.113 A comparison of the Servi–Turnbull J* data on a Cu-1 at.% Co alloy with those of the LeGoues–Aaronson data. (Reprinted from LeGoues, F.K. and Aaronson, H.I., Acta Metall., 32, 1855, 1984. With permission from Elsevier.)
18
log Np
750°C 17 Slope of –1 16
15
800°C
–1.0
0
1.0 log t (min)
2.0
3.0
FIGURE 2.114 A number of gamma prime precipitates vs. aging time at T ¼ 7508C and 8008C. (From Kirkwood, D.H. Acta Metall., 18, 563, 1970. With permission from Elsevier.)
range of 6958C–6708C. With the exception of the 8108C measurements of West and Kirkwood, however, the particle number density again decreased with time; even the 8108C measurement of West and Kirkwood showed only a small increase in the number density prior to the onset of coarsening. Although Kirkwood et al. attempted to secure more accurate particle number densities by extrapolation, and on the basis of their calculations, which are not easy to follow in detail, claimed some examples of good agreement between the calculated and measured homogeneous nucleation rates, attempts to make independent calculations of Jss* under the experimental conditions
228
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
that they employed came a cropper. The solid solubility of Al in Ni is yet to be accurately established. The combination of inadequate thermodynamic data availability and ordering of the compound with inadequate solubility data made the attempts to calculate DG* a complete failure. Kirkwood et al. used a coherent interphase boundary energy of 30 mJ=m2, taken from the coarsening measurements and analysis of Ardell and Nicholson [107]. However, as will be described in the section on precipitation, it is quite possible that Ni3Al coarsens by the ledge mechanism, in which case the value obtained from an analysis assuming continuous atomic attachment and detachment cannot be correct. Attempts to make an independent calculation of the boundary energy using the nearest neighbor, broken bond model also came to a bad end; depending upon how one cut the Ni3Al lattice for matching against the disordered fcc Ni–Al solid solution, the energy was calculated that was either too high or negative! At the present time, therefore, it appears that even if good nucleation rate data were available on this transformation, a successful accounting for Jss* could not readily be made because of serious inadequacies in the ancillary data. 2.18.1.3 Homogeneous Nucleation in Liquids Most efforts to effect quantitative comparisons of nucleation theory and experiment have been conducted by physicists and physical chemists on liquids. Usually, the systems employed ‘‘quasi-binaries’’ that exhibited a miscibility gap. Tc is often of the order of 1008C 2008C; hence measurements of ancillary data, such as interfacial energy and volume free energy change, are far more readily and accurately accomplished than in metals. On the other hand, no effort appears to have been made to quench partially transformed liquids (recall that both matrix and product are now liquid) into solids in order to measure nucleation kinetics with a microscopic technique. Instead, in situ measurements are made in a ‘‘cloud chamber.’’ The first observations of homogeneous nucleation were reported by Wilson [108,109]. These studies were conducted on nucleation of water from its vapor. A volume of inert gas was saturated with water vapor and then expanded in a piston cloud chamber. No precipitation was observed until a supersaturation ratio of four was achieved; supersaturation was varied by simply increasing the expansion furnished by the thrust of the piston. Above a supersaturation ratio of eight, a veritable fog or ‘‘cloud’’ of water droplets formed. The supersaturation at which the cloud first appears is often taken as that needed to produce extensive nucleation. For studies of nucleation in liquids, developed subsequently, the appearance of a ‘‘cloud’’ of 2nd phase liquid particles when the temperature of the liquid was changed a small amount (the order of 18C or less) was similarly interpreted. Changes in the light scattering properties of the liquid are a typical experimental technique employed to determine the ‘‘cloud point.’’ More recently, macrophotography, say at the magnification of fifty, of second phase droplets was employed to study nucleation more quantitatively. Systems on which studies of this type were made of liquid ! liquid nucleation include C7H14-C7F14, 2–6 lutidene-water and isobutyric acid-water. The results obtained from comparisons of theory and experiment were disappointing. The supersaturation at which the ‘‘cloud point’’ appears was significantly greater than that predicted by homogeneous nucleation theory. Hence, the effective nucleus:matrix interfacial energy must be higher than that independently measured or calculated. The reverse result could have been easily explained by heterogeneous nucleation; this result, however, seemed to indicate a disproof of nucleation theory. Binder and Stauffer [110] were the first to understand that the interpretation of the experiments rather than the theory may have been at fault. They pointed out that by the time the second phase particles were large enough to be detected, the system had not only undergone much growth as well as nucleation but also considerable coarsening. Hence, all three processes must be considered in analyzing the experimental observations. They did make a preliminary analysis combining all three. Langer and Schwartz [111] produced a more refined version of this analysis. In metallurgical terms, their treatment combined a Johnson–Mehl correlation between the fraction transformed, nucleation rate, and growth rate with a Wagner–Lifshitz–Slyozov analysis of coarsening. Typical results from the comparison of the treatment with the experiment are shown in Figure 2.115. The y-axis is the initial relative supersaturation of the system. The x-axis represents scaled temperature. Data points for various systems indicate the
229
Initial relative supersaturation y1
Diffusional Nucleation in Solid–Solid Transformations
100 s
0.5
10 s
1s
0.4 0.3 0.2 0.1 0
1 10–4
2 log10 εc
10–3 ε (Two-fluid systems)
3 10–2
4 10–3 ε (CO2)
FIGURE 2.115 Typical plots of initial relative supersaturation vs. scaled temperature in cloud point experiments. (From Langer, J.S. and Schwartz, A.J., Phys. Rev. A, 21, 948, 1980. With permission. ß 1980 by The American Physical Society.)
coordinates at which a ‘‘cloud point’’ was observed. The curves in this figure are ‘‘completion times’’ arbitrarily chosen for the transformation, including coarsening. Note that all of the data are rather well correlated by this analysis. However, Langer and Schwart were careful to point out that experiments of this type are a very poor vehicle for testing nucleation theory. Figure 2.116 compares the halfcompletion time for transformation with the nucleation time (the latter is the reciprocal of the nucleation rate, or the time required to form a nucleus) as a function of relative initial supersaturation. Note that only when the supersaturation becomes exceedingly small, probably too small to cause nucleation to occur at a practicable rate, do the two curves converge. 8
log10 τc
6
4
2
0
0.4 0.5 0.1 0.2 0.3 Initial relative supersaturation y1
FIGURE 2.116 Half completion time vs. initial relative supersaturation. (From Langer, J.S. and Schwartz, A.J., Phys. Rev. A, 21, 948, 1980. With permission ß 1980 by The American Physical Society.)
230
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Clearly, the data of Kirkwood et al. on Ni3Al precipitation in Ni–Al represent an equivalent situation in solid–solid transformations. In fact, Wendt and Haasen [112] studied the same transformation; using the field ion microscope atom probe, they again ‘‘landed’’ mostly in the coarsening regime and concluded that the Langer–Schwartz treatment explained their results. As a summary statement of the various comparisons between homogeneous nucleation theory and experiment, only those studies on Cu–Co appear to be valid—and they showed that within the envelope of conditions examined, all three nucleation theories we have examined are correct.
2.18.2 NUCLEATION
AT
GRAIN BOUNDARIES
2.18.2.1 Nucleation of Proeutectoid Ferrite at Austenite Grain Boundaries in Fe–C Alloys An obvious disadvantage to attempting a comparison of the nucleation theory and the experiment at grain boundaries in Fe–C alloys is that the untransformed austenite matrix is destroyed by the martensite transformation during quenching, thereby not only preventing TEM observation of the grain boundary structure at which nucleation took place but also largely inhibiting even the determination of the orientation relationships between the austenite grains forming a given grain boundary and the orientation of the grain boundary with respect to its bounding grains. The excessively rapid transformation kinetics of the proeutectoid ferrite reaction in high-purity Fe–C alloys are also disadvantageous in that they limit the accuracy with which kinetic measurements can be made. The factors dictating the choice of this transformation and set of alloys were the overwhelming superiority of the ancillary data available on this transformation relative to any other one and the exceedingly detailed information on the morphology of the proeutectoid ferrite reaction. Obviously, though, the effort now to be described represents more of a beginning than a definitive study of nucleation kinetics of a precipitation reaction at grain boundaries in the matrix phase. Previous attempts to study ferrite nucleation kinetics at austenite grain boundaries have been reviewed by several investigators. The study by Mazanec and Cadek [113] is particularly worthy of mention. However, none of these investigations took all of the necessary precautions in making the experimental measurements or dealt adequately with either the critical nucleus shape problem or with formulation of the pre-exponential factor in the nucleation rate equation [114]. The measurements of Lang III et al. [114,115] were conducted on grain boundary ferrite allotriomorphs of proeutectoid ferrite at austenite grain boundaries in high-purity Fe–C alloys containing 0.13, 0.32, and 0.63 wt.% C. Measurements were made as a function of isothermal reaction time at the first alloy and at two temperatures in each of three temperatures in each of the other two. The circumstance that all measurements were made on grain boundary allotriomorphs demonstrated that they were conducted on high-angle grain boundaries. A special effort was made to evolve an etch that would render all former austenite grain boundaries visible. At high reaction temperatures, precipitation occurs on only a minority of grain boundaries, hence, a significant error was introduced by failure to observe unoccupied grain boundaries. Two successive etchants were used: one was a saturated aqueous solution of picric acid to which a commercial wetting agent (‘‘Teepol 610’’) had been added; the other, also water-based, contained oxalic acid and hydrogen peroxide. This etching procedure also permitted a distinction to be made between nucleation at grain faces and at grain edges. The critical nucleus shape problem caused measurements to be restricted to grain faces. However, as discussed before, most nucleation at low undercoolings occurred at grain edges. (The grain boundary geometry produced nearly eliminated grain corners.) Exclusion of edge-nucleated ferrite crystals likely accounts for the 2 3 orders of magnitude lower nucleation rates observed in this study relative to predecessor investigations at comparable temperatures= undercoolings but at higher carbon and alloy contents. Specimens of 0.06 0.06 0.0018 cm were austenitized for 30 min at 13008C in an argon protected, graphite deoxidized bath of BaC12 in order to make the austenite grain boundaries perpendicular to the broad faces of the specimens. They were then isothermally reacted in stirred, graphite-deoxidized lead baths for times ranging from 2 to 40 s at
Diffusional Nucleation in Solid–Solid Transformations
231
temperatures. The highest temperature was that at which a sufficient number of face-nucleated allotriomorphs appeared so that meaningful nucleation rates could be measured, while the lowest one was that below which transformation kinetics were too rapid to permit accurate measurements to be made. This temperature range was only 258C in the 0.13% C alloy and less in the other two. Since nucleation is a highly statistical phenomenon, some embryos will have passed into the growth stage at some positions along a given grain boundary, while others are still struggling to reach critical nucleus size at other places along the same boundary even though the grain boundary orientation is precisely constant along the entire boundary. The diffusion field associated with a growing crystal will, however, decrease DGV for further nucleation within the grain boundary area falling under its composition shadow. How much adjacent area should be considered unavailable for nucleation as a result of this effect? Parker and Kirkwood [117] have treated this problem through a Johnson–Mehl type of approach. Here it was presented more directly. Grain boundary ferrite allotriomorphs are customarily modeled, with reasonable accuracy, as oblate ellipsoids. Atkinson [118] has solved numerically the diffusion equation for the growth of such morphology of the given aspect ratio when the diffusivity in the matrix phase is composition-dependent, as it pronouncedly is in the case of carbon in austenite. This solution was used to establish by computer techniques the concentration–penetration curve of carbon in austenite in the plane of the grain boundary (assuming only volume diffusion is operative) as a function of growth time. On this basis, the distance along the grain boundary at which DGV was reduced to the point where J* diminished fivefold was calculated. (This fivefold reduction, though reasonable, is arbitrary.) A specific model of the critical nucleus had to be invoked to make this calculation; this was done iteratively with the analysis of the experimental data on J*. As expected, the ratio of the radius of this distance to the radius of the allotriomorph in the grain boundary plane is independent of time. This ratio varies from 2 to 5, depending upon the carbon content of the alloy and the reaction temperature. The following procedure was employed to measure nucleation rates. Using optical microscopy, the length of grain boundary occupied by allotriomorphs was measured and multiplied by the diffusion field factor. When allotriomorphs were close enough so that their diffusion fields overlapped on the operational criterion employed, all of the grain boundary between them was considered to be ‘‘occupied.’’ The total length of grain boundary occupied by allotriomorphs and their associated diffusion fields was subtracted from the total length of grain boundary examined in the plane of polish. The total length was obtained by application of the following relationship [119]: p (2:575) LT ¼ LA A ¼ PL A 2 where LT is the total length of grain boundary examined, A is the total area of the plane of polish examined, LA is the grain boundary length per unit area on the plane of polish, PL is the number of grain boundary intercepts per unit length of randomly oriented test line. Determination of A and associated quantities was greatly facilitated by developing a grid of squares, 5m on edge, upon the plane of polish using a photosensitive, etch-resistant lacquer. Saltykov [120] has developed an analysis for extracting the number of spherical particles per unit volume from measurements of the number of particles observed per unit area on a random plane of polish. It was shown that this analysis is exactly applicable to the conversion of the number of particles of the circular cross-section (which the grain boundary allotriomorphs are assumed to be) per unit length of grain boundary observed on a random plane of polish to the number of particles per unit grain boundary area, when the grain boundaries are perpendicular to the plane of polish. The basic Saltykov relationship, rewritten in terms of the present problem, is NA ¼
k 1 X ai (NL )i D i¼1
(2:576)
232
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
where NA is the number of particles per unit area of unreacted grain boundary. As to the other terms in this equation, Saltykov divides the apparent diameter (which is the apparent length along the grain boundary) of particles observed on the plane of polish into k size groups (where k should not exceed 15 or fall below 7; 10 12 is entirely sufficient). The size interval, D, is the diameter of the largest (assumed to be the true size of the largest—a good assumption if enough particles are present) divided by the number of groups, k. (NL)i is the number of particles in a given size group divided by unreacted grain boundary length in the plane of polish. ai is a numerical coefficient, a different one for each size group, i. The values of NA thus determined are plotted as a function of the isothermal reaction time. Slopes of this plot yield the nucleation rate J*, also as a function of reaction time. This nucleation rate is the number of particles per unit unreacted (i.e., unoccupied and unaffected by diffusion fields) grain boundary area per unit time. In addition to permitting direct determination of nucleation rates at grain boundaries, the Saltykov analysis confers other important advantages. As DeHoff [121] pointed out, this analysis permits each size to be calculated independently of the others. Thus, errors made in enumerating one group are not carried through the calculations. Also, the numerical coefficients are applicable to a number of size groups up to 15. Most importantly, DeHoff noted, the number of particles in each size class is a function only of the apparent diameters in larger classes. Therefore, except in the smallest size group, errors will not be introduced by failure to resolve the smallest particles. Hence, the Saltykov analysis appears to be particularly useful for present purposes. Figure 2.117 shows typical plots of NA vs. time, presented for the Fe-0.13% C alloy. Their diminution with time at later reaction times is presumably due to the impingement of adjacent
12.0
800°C Fe—0.13 wt% C
8.0
815°C
6.0
Particle density
(
)
No. of particles × 10–3 Unreacted cm2
10.0
4.0
825°C
2.0
0
0
4
8 12 16 Reaction time (s)
20
24
FIGURE 2.117 Particle density per unit unreacted grain boundary area as a function of reaction time at three different temperatures in Fe-0.13 wt.% C. (With kind permission from Springer Science þ Business Media: From Lange III, W.F. et al., Metall. Trans., 19A, 427, 1988.)
233
Diffusional Nucleation in Solid–Solid Transformations 4.0
800°C 3.0
No. of particles
815°C 2.0
J*
(
Unreacted cm2 –s
× 10–3
)
Fe—0.13 wt.% C
1.0 825°C
0 0
4
8 12 16 Reaction time (s)
20
24
FIGURE 2.118 The nucleation rate as a function of time at three different temperatures in Fe-0.13 wt.% C. (With kind permission from Springer Science þ Business Media: From Lange III, W.F. et al., Metall. Trans., 19A, 427, 1988.)
ferrite allotriomorphs and the invisibility to optical microscopy of the ferrite–ferrite impingement boundaries (and=or to grain growth within the ferrite phase) at least as much as if not more than due to coarsening. Figure 2.118 shows the nucleation rates derived from the plots on Figure 2.117. Taking of slopes was halted at the maximum NA since negative nucleation rates were without physical significance. Note that a steady-state nucleation rate was achieved only at the highest temperature. But two temperatures per composition were usable in the 0.32% and 0.63% C alloys; a steady-state rate was again achieved at the highest reaction temperature in each. Table 2.7 summarizes the ancillary TABLE 2.7 Ancillary Data employed Temp. (8C)
ag 101 nm
aa 101 nmþ
0.13
825
3.6417
2.9009
0.13
815
3.6409
2.9005
wt.% C
0.13
800
3.6397
2.9000
0.32
780
3.6437
2.8991
0.32
770
3.6429
2.8987
0.63
735
3.6493
2.8973
0.63
725
3.6485
2.8968
0.63
710
3.6473
2.8963
DV cm2=sþþ 1.66 108 1.42 108 1.12 10
8
1.30 108 1.10 108 1.27 108 1.07 108 8.31 109
dJ ergs m3 cm3 5.71 107
DGV
7.03 107
dJ ergs m3 cm3 2.56 107
W
8
2.70 107
4.61 107
1.03 10
7.51 107 3.87 107 8.07 107 1.62 108
ggg
mJ ergs m2 cm2 715
gag
mJ ergs m2 cm2 620
715
620
2.98 107
715
620
1.23 107
640
540
1.32 107
640
540
1.81 105
545
460
2.44 105
545
460
5.39 105
545
460
Source: With kind permission from Springer Science þ Business Media: Lange III, W.F. et al., Metall. Trans., 19A, 427, 1988.
234
Mechanisms of Diffusional Phase Transformations in Metals and Alloys γαγ
γ ψ α
γγγ γαγ
(a)
γ
γαγ
γ
α ψ
γcαγ α
γγγ γαγ
(b)
γ γ
γcb αγ γγγ
ψ
α γ
γαγ
(c)
γ
γcb αγ γγγ (d)
ψ γαγ
γ cαγ α
α γ
FIGURE 2.119 Nucleus models at a g-phase planar grain boundary. (With kind permission from Springer Science þ Business Media: From Lange III, W.F. et al., Metall. Trans., 19A, 427, 1988.)
data employed for the calculations now to be described. These calculations were undertaken in a * (the latter is the maximum measured sequence of attempts to reproduce the measured Js* or Jmax nucleation rate; it is used as the best, albeit conservative approximation to Js* ) vs. temperature vs. % C data. In the first set of attempts, the critical nucleus shape models shown on Figure 2.119 were employed. (All shape-dependent factors in the J* equation for these shapes have either been derived already or can be readily deduced from the derivations given.) In seeking to match these models against the experimental data, the following restrictions were imposed: (1) the spherical cap component of a nucleus must be at least three atoms high in order to avoid so much empty space within the model that the model becomes nonrepresentative; and (2) in the case of model (d) where much of the model consists of two parallel planes, it was considered necessary that the nucleus be at least one atom high. (If the orientation relationships {111}fcc=={110}bcc are invoked, then in order to form a complete unit cell, and thereby to hope that a near bulk value of DGV might be applicable, the minimum height should be at least two atoms. Figure 2.120 shows the logarithm of the steady-state nucleation rate at 8258C in Fe-0.13% C calculated as a function of the interfacial energy assumed for the facet in part (b) of Figure 2.119 and for the facet lying in the plane of the grain boundary in parts (c) and (d). Several points should be noted in this figure: (1) Only a single data point is possible for the unfaceted nucleus, part (a), 106 * as contrasted with the at the energy where gcb ag ¼ gag . Note that this model yields Js 10 2 2 experimental rate of ca. 10 =cm s. (2) Since the case of (d) requires two facet energies, the second energy was calculated by setting the nucleus height at the minimum of one atom diameter, thereby giving the maximum Js* . (3) The dashed portion of the curve given for (d) represents the region where the facet length is smaller than the diameter of the particle by more than 2 nm, thereby giving an unacceptably large volume wherein only fractional atom heights are possible. (4) As the facet energy decreases, the model of (b) degenerates into that of (c); at still lower facet energies
235
Diffusional Nucleation in Solid–Solid Transformations –1
–101
Minimum acceptable nucleus height for
–102 (1 atom high)
log J *s
–103
–104
–105
–106
–107
Homogeneous J s* (no facets) 0
2.0
4.0 2 –2 γcb αγ (mJ/m × 10 )
6.0
FIGURE 2.120 The steady-state nucleation rate as a function of facet-free energy. (With kind permission from Springer Science þ Business Media: From Lange III, W.F. et al., Metall. Trans., 19A, 427, 1988.)
(d) becomes numerically equivalent to those of (b) and (c). (5) The minimum nucleus height for a spherical cap of three atom diameters terminated the (b)=(c) curve at an interfacial energy of ca. 97 mJ=m2. Even at minimum gcb ag , these models fail by several orders of magnitude to predict the experimental Js* . If the generally bruited-about (but not really known) value for the facet energy of 200 mJ=m2 is utilized, failure to match the experimental value occurs by far larger margins. To ascertain whether any of these nucleus models becomes practicable, at lower reaction temperatures, Js* was calculated as a function of temperature for Fe-0.13% C using models (c) and (d), which yield the highest nucleation rates under given conditions. To increase Js* further, the nuclei heights were fixed, and set as low as acceptable, i.e., a diameter of three atoms for (c) and a 2 * diameter of one atom for (d). A gcb ag of 115 mJ=m was used with (d) to permit the maximum Js while still maintaining the necessary facet length. As shown in Figure 2.121, at no temperature between the Ae3 and the lowest temperature (for which the Mf was used) does Js* come within several orders of magnitude of experimentally detectable rates. These failures are repeated for the 0.32 and 0.63% C alloys. The foregoing unfavorable result required recourse to the coherent pillbox model shown on Figure 2.122. Both the broad faces of the pillbox were taken to have a low energy and the edge energy was left as a ‘‘free variable’’ and back-calculated from the J* data. Using standard procedures for this model, 1=2
J* ¼
2Dxg va Ne a4 (3kT)1=2
0
2 1 ! 4p geag e 12kTa4 geag B C exp@ A exp Dxg v2a f2 t f2 kT
(2:577)
236
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 50
log J *s (calculated)
0 –50 –100 –150 –200 300
400
500 600 700 Reaction temperature (°C)
800
900
FIGURE 2.121 The calculated nucleation rate as a function of temperature for Fe-0.13 wt.% C with nucleus models of (c) and (d) in Figure 2.119. (With kind permission from Springer Science þ Business Media: From Lange III, W.F. et al., Metall. Trans., 19A, 427, 1988.) γcαγ γeαγ
α
γγγ
cb γαγ
FIGURE 2.122 A coherent ‘‘pillbox’’ nucleus model; all interfaces are fully or partially coherent. (With kind permission from Springer Science þ Business Media: From Lange III, W.F. et al., Metall. Trans., 19A, 427, 1988.)
where e ¼ gcag þ gcb ag ggg , 2geag , r* ¼ f 2e h* ¼ . f When the pillbox is taken as two atoms high, the calculated edge energy is 24 mJ=m2 in the presence of volume strain energy (calculated in the manner of Laszlo [25] for an intragranular plate, since no relationship for strain energy at a grain boundary is available) and 32 mJ=m2 in its absence. The latter value seems more likely since very little free volume in the grain boundary should be sufficient to absorb the misfit between the ferrite nucleus and the austenite matrix. These values, though wholly unsubstantiated by experiment, and only roughly confirmed by a crude calculation, seem not unreasonable since the difference in carbon content across the nucleus:matrix boundary is nearly zero. However, this approach introduces another problem. As shown in Figure 2.123, when agreement is forced between calculated and measured nucleation rates at 8258C in the 0.13% C alloy (similar results in the other two alloys), a discrepancy of ca. 6 orders of magnitude between the calculated and measured nucleation rates appears at 8008C. This difference is reasonably ascribed to severe geometric limitations upon the available nucleation sites resulting from the requirements that: (1) a habit plane in the matrix grain within which the nucleus develops be exactly parallel to the plane of the grain boundary and (2) the two austenite grains must have an orientation relationship that permits a low
237
Diffusional Nucleation in Solid–Solid Transformations 20
16
log J s*
12 Calculated 8
4
0 300
Experimental
400
500 600 700 Reaction temperature (°C)
800
900
FIGURE 2.123 The calculated nucleation rate as a function of temperature for Fe-0.13 wt.% C with a pillbox nucleus model. (With kind permission from Springer Science þ Business Media: From Lange III, W.F. et al., Metall. Trans., 19A, 427, 1988.)
energy habit plane of the adjacent austenite grain to lie exactly in the plane of the grain boundary. Hence, the N ¼ 1015 atomic nucleation sites=cm2 available at the grain boundary in the absence of these specifications were reduced, an order of magnitude at a time, to ascertain whether or not a lower * experimentally observed. Figure N would better reproduce the temperature dependence of Js* or Jmax 2.124 shows that when N is reduced to 108 1010, a better accounting is made for the experimental observations. Similar results were obtained on the other two alloys. 7 6
N = 1015
5
N = 1010
log J *s (normalized)
Fe—0.13 wt.% C
4 3
N = 109
Experimental
N = 108 2
N = 108
1 0 –1 800
N = 109 N = 1015
810 820 830 Reaction temperature (°C)
N = 1010 840
FIGURE 2.124 The calculated nucleation rate as a function of temperature for Fe-0.13 wt.% C with various nucleation site densities. (With kind permission from Springer Science þ Business Media: From Lange III, W. F. et al., Metall. Trans., 19A, 427, 1988.)
238
Mechanisms of Diffusional Phase Transformations in Metals and Alloys γαγ
γ eαγ
ψ γ eαγ
γ eαγ
α
α
γ γγγ (a)
γγγ
γ
γ cb αγ
(b)
ψ γαγ
γ γ
FIGURE 2.125 Two variants for the pillbox nucleus model. (With kind permission from Springer Science þ Business Media: From Lange III, W.F. et al., Metall. Trans., 19A, 427, 1988.)
Finally, it was noted that the e term relating to Equation 2.577 is actually a composite of three interfacial energies. By making one of the broad face energies sufficiently small, the other can be large enough to correspond to a disordered a:g boundary and hence require modeling as a spherical cap. Figure 2.125 shows the two alternate morphologies of the critical nucleus thus generated. These result in somewhat lower, but still possible edge energies. The model, (b), is of special interest because it relaxes the requirement of a highly specific orientation relationship between the two austenite grains forming the grain boundary at which the nucleus appears. At the present time, knowledge of the orientation-dependence of interfacial energy even between austenite and ferrite is far too limited to distinguish among the three variants of the pillbox presented or even to ascertain whether some further refined, more complex variant is actually operative. Quite possibly, there are additional variations on this theme from one grain boundary to the next. The Kossel x-ray microdiffraction studies of King and Bell [122] on ferrite crystals nucleated at austenite grain boundaries in an Fe–C alloy provide experimental support for pillbox-type models that require specialized lattice orientation relationships with respect to both bounding austenite grains. In most cases examined, when planar facets (connoting low energy facets as proved in other alloys with TEM observations of misfit dislocation interfacial structures) were observed on both sides of a ferrite crystal, specific low energy-type orientation relationships were obtained with respect to both austenite grains. Even at those ferrite crystals that appeared to have such an orientation relationship with respect to only one austenite grain, facets appeared facing both, suggesting that an undetected or undiscovered orientation relationship capable of yielding low energy interfaces obtained with respect to the latter austenite grain. In a more general syle, the persistent finding of studies, that the measured growth kinetics and their comparison with calculated kinetics assuming no interfacial interfernce with growth tend strongly to yield unsatisfactory agreement (as will be discussed in Chapter 3), is consistent with the deduction of the pillox-type nuclei faceted with respect to both matrix grains forming the grain boundaries. Further studies on the influence of substitutional alloying elements on proeutectoid ferrite nucleation kinetics [124] yielded similar features, i.e., very low nucleus:matrix interfacial energies and nucleation site density much lower than the atomic density in the boundary plane, although both grain and nucleus:matrix boundary energies are likely to be influenced by segregation of the alloying elements.
2.18.3 NUCLEATION
AT
GRAIN FACES
VS.
GRAIN EDGES
References are Lange III and Aaronson [125] and Cahn [89] for theoretical basis. Several investigations, e.g., those of Middleton and Form [126] and Parker and Kirkwood [117], have reported isolated observations to the effect that nucleation occurs on grain edges, before it takes place on grain faces. The above study is the first in which a systematic examination of the relative nucleation kinetics at grain faces and grain edges was undertaken. Even here, the test of Cahn’s predictions is incomplete, since the number of ferrite crystals at the two types of sites was determined rather than the nucleation sites. The study was a by-product of the detailed investigation of nucleation rates at grain faces recounted earlier. Hence, the geometry of the grain boundaries in the specimens used largely precluded the presence of grain corners. The ratio of the numbers of edge- to face-nucleated grain boundary allotriomorphs as a function of reaction time at each reaction temperature employed
239
Diffusional Nucleation in Solid–Solid Transformations 4.0
16
Edge nucleated particles Face nucleated particles
Edge nucleated particles Face nucleated particles
20 Fe—0.13wt.% C
12
825°C
8 815°C 4
Fe—0.32wt.% C
3.0 780°C 2.0
1.0
770°C
800°C 0 (a)
0
4
20
8 12 16 Reaction time (s)
0
24
0
4
8 12 16 Reaction time (s)
(b)
20
24
Edge nucleated particles Face nucleated particles
2.0 1.6
Fe—0.32wt.% C 725°C
1.2 0.8 0.4 0
710°C 0
4
(c)
8 12 16 Reaction time (s)
20
24
FIGURE 2.126 A variation of the ratio of the number of edge-nucleated allotriomorphs to the number of face-nucleated allotriomorphs as a function of temperature. (With kind permission from Springer Science þ Business Media: From Lange III, W.F. et al., Metall. Trans., 19A, 427, 1988.)
in all three alloys (0.13, 0.32, and 0.63% C) is shown in Figure 2.126. It is evident that as the reaction temperature decreased, i.e., with increased undercooling, the edge=face nucleated allotriomorph ratio diminished in all three alloys. At least until the diffusion field impingement becomes important, this ratio is largely a direct reflection of the relative, time-integrated nucleation kinetics at the two types of sites. The dependence of this ratio is just that predicted by the Cahn analysis. The initial decrease in the ratio with increasing isothermal reaction time at each temperature may result from a possibly longer incubation time for face-nucleated allotriomorphs, slower growth rates of these crystals to detectable size, and=or continued nucleation at grain faces after the available edge nucleation sites have been saturated. The rise in the ratio at later reaction times is attributed to erroneous counting as edge-nucleation those face-nucleation allotriomorphs that had sufficient time to grow into contact with a grain edge. Nucleation rates of ferrite allotriomorphs were measured in Fe–C–X alloys, where X is a substitutional alloying element [115]. Comparison with theory using a pill-box type ciritcal nucleus model showed that the influence of alloying elements follow a similar pattern to that of face nucleation.
2.18.4 NUCLEATION
AT
DISLOCATIONS
In this situation, the ‘‘data’’ are much more meager than those available for the grain faces vs. edge problem considered in the preceding section. With the exception of an occasional precipitate number per unit length of dislocation, the ‘‘data’’ on nucleation at dislocations consist of qualitative
240
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
morphological observations. However, this information permits useful preliminary tests to be made on a number of the predictions made or which can be deduced from the theoretical considerations presented earlier. As will be explained in Section 4.5.4 on precipitation, precipitates that nucleate at dislocations tend to have lattices exhibiting particularly good matching with respect to the lattice of the matrix phase. On the size scale of nucleation, it is thus a fair presumption that most, if not all, observations to be summarized deal with fully or nearly completely coherent nucleation. The morphological observations to be presented are almost entirely inconsistent with the incoherent nucleation. Despite this circumstance, the parameter a in Figure 2.93, devised for incoherent nucleation by Cahn [91], is a particularly useful parameter upon which to base the understanding of the following observations. The usefulness of this parameter may be based in part upon the circumstance that the only strain energy contribution, which the Cahn theory takes into account, is that of the strain field of the dislocation in the matrix; core energy is not included. This is the upper limiting contribution of the dislocation during coherent nucleation provided that the volume strain energy associated with nucleation is added to volume free energy change attending nucleation. In a complete form for nucleation at an edge dislocation, a¼
mb2 DGV v)
(2:578)
mb2 DGV 2p2 g2ab
(2:579)
2p2 g2ab (1
and for nucleation at a screw dislocation, a¼
The observations to be presented are summarized in a review paper by Aaron and Aaronson [127]. In some cases, the reader is asked to refer to this paper for references to the original papers. (1) Edge dislocations are much more effective nucleation sites than screw dislocations: typically, precipitates are observed on edge dislocations but not on screw dislocations in a given specimen reacted=aged at a relatively low supersaturation a is larger at edge dislocations. Therefore, DG* is smaller and thus J* is larger at edge dislocations. (2) The longer the Burgers vector of a dislocation, the more potent is the dislocation as a catalyst for nucleation. Amelinckx [128] has shown that silver particles precipitated from KCl form more rapidly at a than at a =2 dislocations. Note that a is proportional to b2 and thus DG* is smaller at dislocations with a longer Burgers vector (Table 2.8). (3) Nucleation occurs more readily at dislocation nodes and at jogs in dislocations. Dislocation nodes have been observed to be preferred nucleation sites in a fcc Al–Zn–Mg–Cu alloy and in a bcc Fe–P alloy. From the work of Amelinckx, Figure 2.127 shows that Ag precipitation from KCl has occurred almost entirely at nodes. As was judged from qualitative observations on a metallic and an TABLE 2.8 Observations of a Lesser Tendency for Precipitation at Screws than at the Edge Dislocation Alloy Al-4pct Cu Al-7pct Mg KCl-0.75pct AgCl AgCl Si-trace Cua
Dislocation Arrangement
u (complex tet.) b0 (complex, ?) Ag (fcc) Ag (fcc) Cu (fcc)
Small-angle boundary Prismatic loop Networks Helices Loops
With kind permission from Springer Science þ Business Media: Aaron, H.B. and Aaronson, H.I., Metall. Trans., 2, 23, 1971. Diamond-cubic matrix; all other matrices are fcc.
Source: a
Precipitate 0
Diffusional Nucleation in Solid–Solid Transformations
241
FIGURE 2.127 Precipitates of Ag from KCl þ AgCl preferentially at nodes. (From Amelinckx, S., Acta Metall., 6, 34, 1958. With permission from Elsevier.)
ionic alloy, the nucleation rate per unit length of isolated dislocations is higher in deformed specimens than in annealed specimens. The average jog density should, of course, be higher in deformed materials. Although not specifically dealt with in Cahn’s theory of precipitation on (straight) dislocations, one may consider the addition of a jog or a node to be tantamount to lengthening the Burgers vector of the dislocation locally, hence, diminishing DG* thereat. (4) Nucleation is more rapid at isolated dislocations than at sub-boundary dislocations. This has been observed in bcc Fe–P [129] and bcc Fe–N [130] alloys, and is illustrated in the latter system in Figure 2.128. Whereas the strain fields associated at isolated dislocations rather than at sub-boundary dislocations extends, in principle, to infinity, that attending a dislocation in a sub-boundary with an equilibrium configuration is largely damped out at a dislocation distance to 0.2 μ
FIGURE 2.128 Fe16N2 precipitates in an Fe–N alloy. (From Keh, A.S. and Wriedt, H.A., Trans. TMS-AIME, 224, 560, 1962. With permission from Elsevier.)
242
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
the order of the inter-dislocation spacing. Hence, the dislocation strain field contribution to DG* is less for a dislocation in a sub-boundary. It was also shown that nucleation is more rapid at nonequilibrium or extrinsic dislocations in a grain boundary than at equilibrium or intrinsic dislocations in the same boundary, in further confirmation of this effect and its explanation. (5) At a given small-angle or sub-boundary, only one or two of n crystallographically equivalent habit planes is normally operative when the boundary orientation is constant. A change in boundary orientation alters the favored habit plane(s) to others of the same crystallographic form. These effects are illustrated for u0 at a small angle grain boundary in Al-4% Cu in the set of TEM pictures in Figure 2.129, taken of three different areas along this extensively curved boundary [131]. u0 plates form parallel to {100} habit planes in the a matrix. Note that different {100} habit planes are operative in (a) and (c) and that both of these habit planes are present at an intermediate orientation of the boundary, (b). Hence, two equivalent habit planes are inoperative in (a) and (c) and one is missing in (b). Table 2.9 summarizes the observations of this type that were reported. Thomas and (a)
(b)
(c) 1μ
A
B
FIGURE 2.129 Three different locations along a curved grain boundary in a Ag-4 wt.% Cu alloy. (From Vaughan, D.A., Acta Metall., 16, 563, 1968. With permission from Elsevier.)
TABLE 2.9 Habit Plane Selectivity at Edge-Type Dislocations
Alloy Al-4 pct Cu Al-2.5pct Cu-1.2 pct Mg Al-25pct Fe-15 pct Cr-5 pct Nb Siþtrace Cua Fe-0.02 pct Nb
Precipitate
Habit Plane
Does the m ==b Rule Predict the Observed Selectivity?
u0 (complex tet.) S0 (fc ortho.) g*(bct) Cu (fcc) Fe16N2(bct)
{100} {210} {100} ? {100}
Yes Yes Yes Yes Yes
Source: With kind permission from Springer Science þ Business Media: Aaron, H.B. and Aaronson, H.I., Metall. Trans., 2, 23, 1971. a Diamond-cubic matrix. b Bcc matrix; all other matrices are fcc.
243
Diffusional Nucleation in Solid–Solid Transformations Misfit vector aM Matrix
aM cp
Precipitate
ap cp ≠ aM : Large misfit ap ≈ aM : Small misfit
FIGURE 2.130 A misfit vector parallel to the direction of the largest misfit strain and also normal to the broad face of the precipitate. (With kind permission from Springer Science þ Business Media: From Aaron, H.B. and Aaronson, H.I., Metall. Trans., 2, 23, 1971.)
Nutting [132] have provided the following explanation of this phenomenon. As illustrated schematically in Figure 2.130 [127,133], since good lattice matching customarily obtains across the broad faces of a plate, most of the misfit between precipitate and matrix lattices might be considered to take place normal to the broad faces. The vector describing the latter mismatch was termed the ‘‘misfit vector.’’ Thomas and Nutting [132] proposed that the particular crystallographically equivalent habit plane of n habbit planes would have the highest nucleation rate (and thus would likely be the only one present) that is most nearly parallel to the Burgers vector of the sub-boundary dislocations. On the Cahn and especially the Larché views, this is equivalent to maximizing the elastic interaction between the matrix and the nucleus through permitting the maximum physically possible relief of the transformation strain. The careful TEM studies of Vaughan accurately confirmed the habit plane forecast by the Thomas–Nutting mechanism. More likely, the presence of plates with two different habit planes along a given sub-boundary means that two sets of dislocations are present in the boundary. Nearly equal nucleation rates for plates parallel to two different habit planes at a single array of dislocations is possible but not likely because of the potent influence of even small changes in DG* upon J*. (6) As supersaturation, i.e., undercooling, increases, the preceding habit plane selectivity and also the preference for nucleation at edge rather than at screw dislocations become less pronounced and finally disappears. Dash [134] observed that Cu precipitates from Si preferentially upon edge dislocations at low supersaturations but impartially upon both edge and screw dislocations when the supersaturation is high. Keh and Wriedt [130] observed that the habit plane selectivity of Fe16N2 plates formed in a Fe–N alloys becomes less pronounced with increasing supersaturation. From Equations 2.578 and 2.579, since a is proportional to jDGVj, the increased a and hence diminished DG* for the nucleation of all types under consideration will decrease the advantage of edges over screws and of the preferred habit plane over equivalent habit planes. J* may still be higher in the former instances but will not be detectable unless microstructures are observed at the beginning of their evolution, or better, unless the time dependent J* is measured for the competing nucleation processes. (7) At small-angle grain boundaries in fcc matrices, observations on many alloy systems with a variety of precipitate crystal structures indicate that only primary sideplates or primary sideneedles form at dislocations and at small-angle grain boundaries and sub-boundaries. Observations of this type are summarized in Table 2.10. Two exceptions are noted at the bottom of this table. In the case of Au-Co, since the matrix and precipitates have the same crystal structure, the anisotropy of growth is not expected. The Al–Zn–Mg observation is a well-confirmed anomaly, not yet explained. As also will be discussed in more detail in Chapter 3, the absence of a high-energy, high-diffusivity area between dislocations presumably prevents the development of allotriomorphic growth (preferentially along the grain boundary) and permits an extension of a platelike critical nucleus shape into a platelike growth shape (with a different aspect ratio and for a different reason—anisotropy of growth kinetics rather than of interfacial energy) essentially by default.
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Mechanisms of Diffusional Phase Transformations in Metals and Alloys
TABLE 2.10 Morphological Effects of Dislocation Boundaries in the fcc Matrix Alloy
Precipitate
Angular Range
Morphology Small Angle
Intermediate Angle — Primary sawteeth Primary sawteeth(g?) Primary sawteeth ?
Al-4 pct Cu Fe-0.29 pct C Al-18 pct Ag Ag-5.6 pct Al Co-20 pct Fe
u0 (complex tet.) a (bcc) g0 (hcp) b (bcc) a (bcc)
08 to 128 Small-angle 08 to 178 08 to 148
Primary sideplates Primary sideplates Primary sideplates Primary sideplates Primary sideplates
Al-6-7 pct Zn 2.5-3 pct Mg Au-5 pct Co
M0 (hcp)
Small-angle
(fcc)
Small-angle
Primary sideplates þallotriomorphs Allotriomorphs
?
Source: With kind permission from Springer Science þ Business Media: Aaron, H.B. and Aaronson, H.I., Metall. Trans., 2, 23, 1971.
For reasons not at all understood, a variety of precipitate morphologies can form at small-angle grain boundaries when the matrix has a bcc structure. This has been observed during the ferrite-toaustenite transformation in silicon–iron bicrystals, an Alnico alloy, and in Fe-8% Mo—though not in Fe-4% Au. The micrograph of Alnico 2 in Figure 2.131 shows that precipitate morphology is ca. constant along a given sub-boundary with a constant orientation, but often varies from one subboundary to the next. The entire area shown is a single crystal. Not only primary sideplates but also grain boundary allotriomorphs and also the intermediate morphology of primary sawteeth are seen. (Primary sideplates and primary sawteeth grow directly from a boundary; secondary sideplates and secondary sawteeth evolve from grain boundary allotriomorphs.)
2.18.5 SECONDARY SIDEPLATE SELECTIVITY The references are Aaronson’s two reviews [136,137]. The evolution of secondary sideplates from a ‘‘base layer’’ of grain boundary allotriomorphs is clearly a growth phenomenon. Allotriomorphs
FIGURE 2.131 Different grain boundary allotriomorph morphologies at various subboundaries in Alnico 2. (From Vaughan, D.A., Acta Metall., 16, 563, 1968. With permission from Elsevier.)
Diffusional Nucleation in Solid–Solid Transformations
245
form at large-angle grain boundaries. Dislocations are presumably present at many of these boundaries, but as a consequence of their smaller Burgers vector and=or closer spacing are unable to produce primary sideplates even in fcc matrices (unless the lattice and boundary orientations are close to an energy cusp situation). It seems a plausible, if not proved, presumption that the dislocations present may not be able to control the operative habit plane. However, secondary sideplates formed from the allotriomorphs precipitated along a grain boundary with a constant orientation also tend to form parallel to only one, or sometimes two of n crystallographically equivalent habit planes. Dube0 [138] has shown that for the Kurdjumov and Sachs orientation relationships, which often relate austenite and ferrite crystals, a given ferrite crystal can provide a {110}a habit plane parallel to only one {111}g habit plane. Hence, parallel secondary sideplates mean that the grain boundary allotriomorphs developed from nuclei with the same unique pair of conjugate habit planes. (As will be discussed in Section 3.4.5, the apparent habit planes of ferrite plates often deviate appreciably from {111}g; however, recent work has shown that the atomic habit planes are as specified here.) The influence of the tilt angle, f, upon DG* at a disordered grain boundary shown in Figures 2.84 and 2.86 provide a possible explanation for the habit plane selectivity observed at grain boundaries of this type. Limited experimental observations, which need to be considerably extended, indicate that the habit plane of grain boundary allotriomorphs tends to be parallel to the grain boundary plane; if this parallelism is sufficiently accurate, the evolution of secondary sawteeth or sideplates is considerably inhibited.
REFERENCES 1. K. C. Russell, in Phase Transformations, H. I. Aaronson, ed., ASM, Metals Park, OH, 1970, p. 291. 2. H. I. Aaronson and J. K. Lee, in Lectures on the Theory of Phase of Transformations, H. I. Aaronson, ed., TMS-AIME, Warrendale, PA, 1975, p. 83, and 2nd ed., 1999, p. 165. 3. H. I. Aaronson, J. K. Lee, and K. C. Russell, in Precipitation Processes in Solids, K. C. Russell and H. I. Aaronson, eds., TMS-AIME, Warrendale, PA, 1978, p. 31. 4. K. C. Russell, Adv. Colloid Interface Sci., 13, 199 (1980). 5. J. W. Christian, Theory of Transformations in Metals and Alloys, 1st ed., Pergamon, New York, 1965, p. 377; 2nd ed., Pergamon, New York, 1975, p. 418. 6. R. Becker and W. Döring, Ann. Phys., 24, 719 (1935). 7. J. B. Zeldovich, Acta Physicochim, URSS, 18, 1 (1943). 8. L. Farkas, Z. Phys. Chem., 125, 239 (1927). 9. J. W. Gibbs, On the Equilibrium of Heterogeneous Substances, Collected Works, Vol. I, Longmans, Green and Co., New York, 1928 reprinted by Dover, New York, 1993. 10. P. Shewmon, Diffusion in Solids, TMS, Warrendale, PA, 1989, p. 66. 11. D. Turnbull, Trans. AIME, 175, 774, (1948). 12. F. F. Abraham, J. Chem. Phys., 51, 1632 (1969). 13. K. C. Russell and D. H. Hall, in Defects and Defect Clusters in BCC Metals and Their Alloys, R. Arsenault, ed., National Bureau of Standards, Washington, DC, 1973, p. 545. 14. H. Wakeshima, J. Phys. Soc. of Jpn., 10, 374 (1955). 15. A. Kantrowitz, J. Chem. Phys., 19, 1097 (1951). 16. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Clarendon, Oxford, U.K., 1959, p. 92. 17. R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience, New York, 1953, p. 76. 18. J. Feder, K. C. Russell, J. Lothe, and G. M. Pound, Advances in Physics (Phil. Mag. Supplement), 15, 111 (1966). 19. K. C. Russell, Acta Metall., 17, 1123 (1969). 20. M. Hillert, in Lectures on the Theory of Phase Transformations, H. I. Aaronson, ed., TMS-AIME, Warrendale, PA, 1975, p. 1, and 2nd ed., 1999, p. 1. 21. J. K. Lee and H. I. Aaronson, Scr. Metall., 8, 1451 (1974). 22. C. Herring, in Structure and Properties of Solid Surfaces, R. Gomer and C. S. Smith, eds., University of Chicago Press, Chicago, IL, 1953, p. 5. 23. S. E. Offerman, N. H. van Dijk, J. Sietsma, S. van der Zwaag, E. M. Lauridsen, L. Margulies, S. Grigull, and H. F. Poulsen, Science, 298, 1003 (2002). 24. M. Enomoto and J. B. Yang, Metall. Mater. Trans., 39A, 994 (2008).
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J. K. Lee and H. I. Aaronson, Acta Metall., 23, 799 (1975). J. K. Lee and H. I. Aaronson, Acta Metall., 23, 809 (1975). D. W. Hoffman and J. W. Cahn, Surf. Sci., 31, 368 (1972). J. W. Cahn and D. W. Hoffman, Acta Metall., 22, 1205 (1974). W. L. Winterbottom, Acta Metall., 15, 303 (1967). G. P. Vander Velde, J. A. Velasco, K. C. Russell, and H. I. Aaronson, Metall. Trans., 7A, 1472 (1976). N. A. Gjostein, in Diffusion, ASM, Metals park, OH, 1974, p. 272. Y. Adachi, K. Hakata, and K. Tsuzaki, Mater. Sci. Eng., A412, 252, (2005). R. Weinstock, Calculus of Variations, McGraw-Hill, New York, 1952, p. 53. G. Salmon, A Treatise on the Analytic Geometry of Three Dimensions, 4th ed., Hodges, Figgs and Co., Dublin, Ireland, 1882, p. 276. J. W. Cahn, Acta Metall., 4, 449 (1956). P. J. Clemm and J. C. Fisher, Acta Metall., 3, 70 (1955). J. W. Cahn, Acta Metall., 5, 169 (1957). R. Gomez-Ramirez and G. M. Pound, Metall. Trans., 4, 1563 (1973). B. Ya. Lyubov and V. A. Solv’Yev, Fiz. Metal. Metalloved, 19, 333 (1965). C. C. Dollins, Acta Metall., 18, 12 (1970). D. M. Barnett, Scr. Metall., 5, 261 (1971). F. Larché, in Dislocation Theory: A Collective Treatise, F. R. N. Nabarro (ed.), Dekker, New York, 1979. I. S. Servi and D. Turnbull, Acta Metall., 14, 161 (1966); Errata, Acta Metall., 14, 908 (1966). L. E. Tanner and I. S. Servi, Acta Metall., 14, 231 (1966). G. J. Shiflet, Y. W. Lee, H. I. Aaronson, and K. C. Russell, Scripta Metall., 15, 719 (1981). C. A. Wert and C. Zener, J. App. Phys., 21, 5 (1950). C. A. Wert, J. App. Phys., 20, 943 (1949). K. R. Kinsman and H. I. Aaronson, unpublished work, Ford Motor Co., 1965. F. K. LeGoues and H. I. Aaronson, Acta Metall., 32, 1855 (1984). D. H. Kirkwood, Acta Metall., 18, 563 (1970). A. W. West and D. H. Kirkwood, Scr. Metall., 10, 681 (1976). T. Hirata and D. H. Kirkwood, Acta Metall., 25, 1425 (1977). A. J. Ardell and R. B. Nicholson, Acta Metall., 14, 1295 (1966). C. T. R. Wilson, Phil. Trans. Roy. Soc., A189, 265 (1897). C. T. R. Wilson, Phil. Trans. Roy. Soc., A193, 289, (1900). K. Binder and D. Stauffer, Adv. Phys., 25, 343 (1976). J. S. Langer and A. J. Schwartz, Phys. Rev. A, 21, 948 (1980). H. Wendt and P. Haasen, Acta Metall., 31, 1649 (1983). K. Mazanec and J. Cadek, Rev. de Metallurgie, 212, 501 (1958). W. F. Lange III, M. Enomoto, and H. I. Aaronson, Int. Mater. Rev., 34, 501 (1958). W. F. Lange III, M. Enomoto, and H. I. Aaronson, Metall. Trans., 19A, 427 (1988). M. Enomoto, W. F. Lange III, and H. I. Aaronson, Metall. Trans., 17A, 1399 (1986). G. W. Parker and D. H. Kirkwood, Chemical Metallurgy in Iron and Steel, Iron and Steel Institute, London, U.K., 1973, p. 248. C. Atkinson, Trans. AIME, 245, 801 (1969). E. E. Underwood, Quantitative Stereology, Addison-Wesley, Reading, MA, 1970. S. A. Saltykov, Stereometric Metallography, 1st ed., Metallurgizdat, Moscow, 1950. R. T. DeHoff, Trans. AIME, 224, 474 (1962). A. D. King and T. Bell, Metall. Trans., 6A, 1428 (1975). M. Enomoto and H. I. Aaronson, Metall. Trans., Mater. Trans., 17A, 1385 (1986). T. Tanaka, H. I. Aaronson, and M. Enomoto, Metall. Trans., Mater. Trans., 26A, 547 (1995). W. F. Lange III and H. I. Aaronson, Metall. Trans., 10A, 1951 (1979). C. J. Middleton and G. W. Form, Metall. Sci., 9, 521 (1975). H. B. Aaron and H. I. Aaronson, Metall. Trans., 2, 23 (1971). S. Amelinckx, Acta Metall., 6, 34 (1958). E. Hornbogen, Trans. ASM, 55, 719 (1962). A. S. Keh and H. A. Wriedt, Trans. TMS-AIME, 224, 560 (1962). D. A. Vaughan, Acta Metall., 16 563 (1968). G. Thomas and J. Nutting, in The Mechanism of Phase Transformations in Metals, Institute of Metals, H. K. Hardy, ed., London, U.K., 1956, p. 57. A. Kelly and R. B. Nicolson, Prog. Met. Sci., 10, 151 (1963).
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134. W. C. Dash, in Dislocations and Mechanical Properties of Crystals, J. C. Fisher, W. G. Johnston, R. Thomson and T. Vreeland, Jr., ed., Wiley, New York, 1957, p. 57. 135. A. H. Geisler, Trans. ASM, 44A, 269 (1952). 136. H. I. Aaronson, in The Mechanism of Phase Transformations in Metals, Institute of Metals, H. K. Hardy, ed., London, U.K., 1956, p. 4. 137. H. I. Aaronson, in Decomposition of Austenite by Diffusional Processes, V. F. Zackay and H. I. Aaronson, eds., Wiley, New York, 1962, p. 387. 138. C. A. Dube0 , PhD thesis, Carnegie Institute of Technology, Pittsburgh, PA, 1948. 139. D. R. Gaskell, Introduction to the Thermodynamics of Materials, 3rd ed., Taylor & Francis, Washington, DC, 1995, p. 278. 140. K. F. Kelton, Solid State Phys., 45, 75 (1991). 141. D. T. Wu, Solid State Phys., 50, 37 (1996). 142. W. C. Johnson, in Lectures on the Theory of Phase Transformations, TMS-AIME, H. I. Aaronson, 2nd ed., Warrendale, PA, 1999, p. 35. 143. D. R. Gaskell, Introduction to the thermodynamics of Materials, 2nd ed., McGraw Hill, New York, 1981, p. 392.
3 Diffusional Growth 3.1 BASIC DIFFERENCES BETWEEN DIFFUSIONAL NUCLEATION AND DIFFUSIONAL GROWTH Once the size of the critical nucleus exceeds that corresponding to n* þ d=2, at DG8 ¼ DG* kT (see Figures 2.1 and 2.9), the probability that a chance thermal fluctuation will cause the loss of an atom that will initiate dissolution of the nucleus becomes sufficiently small so that the diffusional growth stage can be considered to have been entered. When growth has proceeded a little further, to the point where DG8 < 0, the dominating influence that interfacial energy minimization exerts upon precipitate shape during the nucleation stage can be considered to have been largely overthrown. The characteristics of further growth should now rapidly approach those of the ‘‘macroscopic,’’ that is, supra-micron size regime. Whereas nucleation in a matrix free of structural heterogeneities is a statistical fluctuation process that takes place in a matrix of constant average composition, growth is diffusional mass transport down a concentration gradient. This statement is exact in this form when growth takes places to yield a precipitate with a composition different from that of the matrix. When the precipitate and matrix have the same composition, as during a massive transformation, the term ‘‘composition’’ need only be replaced by ‘‘solute partial molar free energy.’’ Hence the solute concentration in the layer of matrix atoms in contact with the critical nucleus is xb, whereas in the absence of (a) growth barriers, (b) growth accelerators, and (c) capillarity effects, the solute concentration in this layer during growth is xba b , the atom fraction of solute in b at the b=(a þ b) phase boundary. The shape of the critical nucleus is largely determined by the interfacial energy minimization. During growth on the other hand, once the ratio of the total interfacial energy to the product of DGV and the volume of a precipitate crystal become much less than unity, the shape of the crystal is determined by the boundary orientation dependence of growth kinetics. When DGV ¼ 0, further shape changes occur, in the ‘‘coarsening’’ or ‘‘Ostwald Ripening’’ regime and are once again controlled by interfacial energy minimization, but now taking place interactively among an ensemble of particles, normally resulting in changes in both precipitate size and shape. Stated a little differently, capillarity is the dominant factor during nucleation. In the growth process, when concurrent coarsening is of negligible importance and particle size has entered the micron range, capillarity influences the growth process only when growth barriers result in the development of finely curved interfaces, such as the edge of a plate or the tip of a needle, whose curvature does not diminish with growth time. In addition, whereas capillarity acts to diminish the concentration gradient driving the displacement of these finely curved interfaces, the markedly more favorable diffusion geometry that these interfaces provide for the rejection or ingestion of solute by volume diffusion results in much higher growth kinetics than can be achieved by a planar interphase boundary. Hence, capillarity is the major barrier to nucleation but can indirectly, though markedly, expedite growth.
3.2 A GENERAL THEORY OF PRECIPITATE MORPHOLOGY This section draws on Aaronson [1] and Aaronson et al. [2]. The theory of precipitate morphology provides a convenient framework upon which to organize this chapter. As a direct consequence of the predominant role that DG* plays in determining nucleation kinetics, and of the circumstance that the critical nucleus morphology with the lowest DG* is normally that with the lowest total interfacial energy, low energy interfaces tend to occupy an appreciable proportion of the area of 249
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Mechanisms of Diffusional Phase Transformations in Metals and Alloys
nucleus:matrix boundaries. When the crystal structures of the nucleus and matrix phase are different, especially formation of such interfaces requires that the nucleus lattice bear a quite specific and special orientation relationship with respect to that of the matrix phase. When nucleation occurs at a face, edge, or corner of a matrix grain, there will be an almost equally strong tendency to develop such relationships with respect to all matrix grains involved, if the misorientations of the matrix grain with respect to each other make this physically possible. The low-energy lattice-orientation relationships thus formed must remain operative during growth unless some form of recrystallization and=or deformation process destroys them. Most likely, some or all of the low-energy interfaces present on the critical nucleus will continue to appear during growth and others may also appear. Physically, these interfaces will correspond to close matching of atom patterns and spacings across the interphase boundary. Unlike the situation during nucleation, however, unless the matching is virtually perfect, the fully coherent structure of such interfaces presumed to obtain during nucleation will be replaced by a partially coherent structure during growth. As originally described theoretically by Frank and van der Merwe [3], the misfit between the two lattices is largely taken up by the presence of extra half-planes at regular intervals. These half-planes and their environs constitute misfit edge dislocations. Between them, the interphase boundary is fully coherent and differs from perfect continuity of lattice planes through both phases by elastic distortions with exact maintenance of planarity. An interphase boundary of this type is schematically illustrated in Figure 3.1 [4]. High-energy interfaces with a disordered or incoherent structure may also be present at the nucleus:matrix interface. If such interfaces survive during growth, the development of a higher degree of order within them is most unlikely. After having described two different types of interphase boundary structures, it is now appropriate to consider how each affects the atomic mechanism and thus the kinetics of growth. Although the following considerations are applicable with fundamentally minor (though kinetically major) modifications to the massive transformation, they will be developed within the easier-to-visualize context of precipitation reactions, where solute concentrations in the matrix and the precipitate are different. Examining first disordered interphase boundaries, the time required for an atom to cross such a boundary should be very short at reduced temperatures wherein diffusional processes proceed at appreciable rates. Only two or three atomic jumps seem likely to be required (or at most a handful more), and the operative diffusivity will be one appropriate to diffusion across an interface rather than a volume diffusivity. Since the precipitate and matrix are assumed from the outset to differ in D
C
B
A
D΄
C΄
FIGURE 3.1 Interphase boundary containing misfit dislocations. (Reprinted from Christian, J.W., The Theory of Transformations in Metals and Alloys: Part I Equilibrium and General Kinetic Theory, 2nd ed., Pergamon, Oxford, NY, 1975, p. 359. With permission from Elsevier.)
Diffusional Growth
251
composition, volume diffusion in the case of interstitial alloys or volume interdiffusion in the case of substitutional alloys within the matrix phase must attend the growth process. Such diffusion takes place over distances the order of the half-thickness of the precipitate crystal. Even at dimensions of the crystal whose energy exceeds thermal energy (kT) over that of the critical nucleus, this distance is larger than that for trans-interphase boundary diffusion. Usually volume diffusion distances of the order of microns, or much more, are required. Volume diffusivities are many orders of magnitude smaller than grain boundary (and hence presumably interphase boundary) diffusivities, particularly at lower values of the homologous temperature (¼T=TM, where TM is the melting point). Recalling that the diffusion distance is s ¼ (2Dt)1=2, where D is the operative diffusivity, and t is the diffusion time, transport across an interphase boundary involves a much smaller s and a much larger D than does the attendant long-range volume diffusion. Hence, unless a special solute segregation effect is involved, transport across a disordered interphase boundary should not be the rate-controlling factor during the growth of a precipitate crystal. One would anticipate instead that the migration kinetics of a disordered boundary should be controlled by non-interfacial structural factors, such as the concentration or activity difference, which would be set by the difference between xba b and xb, and the curvature of the a:b boundary, and the appropriate diffusivity in the matrix phase. As will be seen later (Section 3.3), this expectation is not often fulfilled, because the assumptions upon which it is based are not applicable under certain experimental conditions. When a precipitate with appreciable areas of disordered boundary forms at a high-angle grain boundary along which diffusivity is rapid, relative to volume diffusion, the interphase and grain boundaries can serve as conduits for the accelerated mass transport of solute. A second circumstance leading to different results obtains when two or more solutes are present, one solute is an interstitial and the other(s) is=are substitutional, and a substitutional species not only has a strong binding free energy to the interphase boundary but also affects the activity of interstitial solute in contact with the boundary. In this situation, volume diffusion of the substitutional solute species is usually largely or entirely absent. Hence, transformation to a product with a nonequilibrium composition normally occurs. The substitutional species influences growth kinetics by altering the mass transport rates of the kinetically controlling interstitial species. Both of these effects will be discussed later in more detail. In the case of dislocation interphase boundaries, the theory proposes that these boundaries are wholly immobile in the direction normal to themselves when the crystal structures of the matrix and precipitate are substantially different and=or there is a marked difference in the long-range order across the boundary [5]. The origin of the immobility lies in the coherent regions rather than in the dislocations. In order to displace these regions by individual, essentially uncoordinated diffusional jumps (rather than by a cooperative shearing process), it is necessary to place substitutional atoms in what are temporary interstitial sites. For more or less close-packed crystal lattices, the energetics of this should be most unfavorable and hence except for an occasional atom briefly occupying such a site, and the ledges shortly to be introduced (Section 3.4), these boundaries should be atomically flat. An example of what must be accomplished in order to induce boundary migration in the normal direction on an atom-by-atom basis, consider the stacking sequence normal to the often excellently matching interphase boundary formed by {0001}hcp=={111}fcc, hcp==fcc. The lefthand column of Figure 3.2 shows the stacking sequence at a given stage of growth while the right-hand column displays the sequence after the passage of a Shockley partial misfit dislocation has displaced the boundary ‘‘upward’’ by two atomic layers. Conversion of a C-type layer to an A-type layer requires the insertion of a substitutional atom into an already close-packed atomic plane, virtually an energetic impossibility. The mechanism of migration proposed for such boundaries is the ledge mechanism, originally devised by Gibbs for the migration of close-packed solid:liquid and solid:vapor interfaces. As shown in Figure 3.3, the broad faces of the ledges are taken to have a partially or fully coherent
252
Mechanisms of Diffusional Phase Transformations in Metals and Alloys A B C A B C B A B A
FIGURE 3.2 dislocation.
fcc
hcp
A B C A B A B A B A
Growth direction
Migration of a coherent fcc:hcp interphase boundary by the passage of a Shockley partial
structure and the edges or risers of the ledges are considered to have β a disordered structure. Migration of the boundary in the direction indicated by the hollow arrow takes place by the formation and lateral movement of the ledges. As long as the ledges are suffiα ciently widely spaced, this indirect mode of growth will result in an overall growth rate less than that allowed by volume diffusion FIGURE 3.3 Migration of an directly toward or away from a planar, disordered boundary at the interphase boundary by the latsame temperature and alloy composition. eral movement of ledges. Finally consider the consequences of having a partially or fully coherent boundary with widely separated ledges at one boundary orientation and disordered boundaries and=or partially coherent boundaries with closely spaced ledges at the other orientations of the precipitate:matrix interface of a single precipitate crystal embedded within an otherwise structurally homogeneous matrix phase. The boundary with the large inter-ledge spacing migrates more slowly than that of any of the others enclosing the precipitate crystal. Hence, the other boundary orientations will tend to ‘‘grow out’’ or at least diminish in extent insofar as their growth kinetics relative to those of the slow-growing interface will allow—or in the limit, insofar as capillarity will permit—and result in a plate or disc-shaped precipitate morphology. If good matching between the two lattices is one- rather than two-dimensional, the slow-growing partially coherent boundary may take the form of a cylinder, resulting in a needle-shaped morphology. Following the lead of this theory, in Section 3.3, the characteristics of the disordered boundaries, of which growth kinetics is almost the only one presently accessible, will be considered first theoretically and then experimentally. In Section 3.4, partially coherent interphase boundaries will be considered, first theoretically and then experimentally from the standpoints of structure, energy, and migrational characteristics. In addition, in Section 3.5, the effects of the simultaneous presence of disordered and partially coherent boundaries upon the evolution of precipitate morphology will be examined. Throughout this chapter, emphasis will be placed largely upon simple binary alloy systems and upon growth kinetics occurring under carefully chosen, particularly simple, conditions. Consideration of nucleation and growth under more complicated conditions will be delayed until Chapter 4.
3.3 DISORDERED INTERPHASE BOUNDARIES 3.3.1 INTRODUCTION The structure and energy of disordered interphase boundaries have been discussed in Section 2.11.4 in connection with nucleation kinetics. Knowledge of these vital aspects of such boundaries is yet in a primitive state—and doubts have already been expressed as to whether or not this type of interfacial structure even exists. Even if atomic resolution TEM eventually shows that all interphase boundaries are discretely structured in some manner, however, from the standpoint of growth kinetics, it will be necessary to continue to assume that they exist: the rationale is that, in an
253
Diffusional Growth
operational sense, a disordered interphase boundary is simply one that does not detectably accelerate or decelerate growth kinetics. Hence, such boundaries provide a ‘‘baseline’’ for diffusion–controlled growth kinetics. When careful experimental measurements of the growth kinetics of an interphase boundary assumed to have a disordered structure exhibit significant differences with respect to the kinetics calculated under the assumption of volume diffusion control, then one may conclude that the structure and=or the chemistry of the interphase boundary deviate from those characterizing our idealized non-interfering boundary. Then an important clue is provided as to the presence and possible nature of the deviation.
3.3.2 VOLUME DIFFUSION–CONTROLLED GROWTH KINETICS 3.3.2.1
Mathematics for Diffusion and Flux Equations
3.3.2.1.1 Planar Boundary of Infinite Extent 3.3.2.1.1.1 Linearized Gradient Approximation This is due to Zener [6]. The physics of growth can be understood much more clearly if the assumption is first made that the concentrationpenetration curve of solute in the matrix phase is linear, rather than curved, as is actually the case. In Figure 3.4a, the precipitate is assumed solute poor, and in Figure 3.4b, it is taken to be solute rich. The two cases are mathematically equivalent. The solute-poor case is arbitrarily selected for use unless otherwise specified. The concentration–penetration curve appropriate to the solute-poor precipitate is redrawn in Figure 3.5, with additional information supplied. The flux of solute atoms diffusing away from the interphase boundary per unit area of the boundary is given by Fick’s law: qx (3:1) F1 ¼ D qs b:a where D is the diffusivity or interdiffusion coefficient in the matrix phase, here assumed independent of composition (qx=qs)b:a is the concentration gradient of solute in the matrix at the a:b boundary
The flux of solute atoms swept into b by the growth of solute-poor a is given by ab dS F2 ¼ xba x a b dt
(3:2)
βα
β α xβ
Concentration
Concentration
xβ
xααβ
xααβ α
β
xβ
βα
xβ (a)
0
Distance
(b)
0
Distance
FIGURE 3.4 Solute concentration profile within and near the a:b interphase boundary for (a) solute-poor precipitate and (b) solute-rich a precipitate growing into the b matrix.
254
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Solute concentration
βα
xβ
Δx A2
xβ
ΔS A1
xααβ 0
S Growth distance
FIGURE 3.5 Linearized gradient approximation for the growth of a solute-poor precipitate.
where ab xba b and xa are again the atom fractions of solute in b and in a at the boundaries of the (a þ b) phase field, S is the position of the a:b boundary, dS=dt is the rate of growth of the boundary. Since dS=dt is taken to occur as rapidly as the solute can diffuse away from the boundary, it is necessary that F1 ¼ F2: qx ba ab dS x b xa ¼ D : (3:3) dt qs b:a
Let DS be the extent of the matrix affected by the solute pile up at time t. Replacing (qx=qs)b:a with the linearized gradient taken from the concentration–penetration plot, Dx ab dS ffiD xba b xa dt DS
(3:4)
1 ba xb xb DS xb xab a S ¼ 2
(3:5)
2 dS D xba xb b ab ¼ xba b xa dt 2S xb xab a
(3:6)
(Dt)1=2 xba x b b S¼ 1=2 1=2 ab xba xb xab a b xa
(3:7)
Since the amount of solute removed from the matrix by the growth of a (A1) must equal the amount piled up in b in front of the advancing b:a boundary (A2):
Solving for DS and substituting into Equation 3.4, and noting that Dx ¼ xba b xb , yields
Integrating,
255
Diffusional Growth 7 6 5
Growth rate constant, α1
4 3 2
1.0 .9 .8 .7 .6 .5 .4 .3
.3
.4 .5 .6 .7 .8 .9 1 2 3 Growth rate constant obtained from lineaged approximation α*1
4
5 6
FIGURE 3.6 Relationship between the growth rate constant, a1, obtained from the transcendental equation and that from Zener’s linearized approximation a*1 . (From Zener, C., J. Appl. Phys., 20, 950, 1949. American Institute of Physics. With permission.)
Rewriting, S ¼ a*1 (Dt)1=2
where
(3:8)
xba x b b a*1 ¼ 1=2 1=2 ab ab x x x xba a a b b
In the next section, an exact solution to this problem will be obtained and the approximate a*1 will be replaced by the correct a1. Figure 3.6 plots a1 (the ordinate) as a function of a*1 (the abscissa). Note that over a rather wide range, the error introduced by using a*1 is not very large. This information is of interest when one wishes to incorporate a growth rate equation into a more complex treatment. Since the exact solution is a transcendental equation, further analytic use of this result is not possible. Two results are needed when a diffusional growth kinetics problem is solved, the parabolic rate constant, here a*1 and a relationship for the concentration–penetration curve. To obtain the latter result, area A2 is reproduced in Figure 3.7 and further annotated. Utilizing similar triangles,
βα
xβ
x
xβ
(3:9)
Δs
ΔS
FIGURE 3.7 Linearized approximation of concentration-penetration curve in the matrix.
xba b x
xba b
xb
¼
Ds DS
where Ds is distance into b normal to b:a boundary, x is the atom fraction of solute at S þ Ds.
(3:10)
256
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Substituting Equation 3.5 for DS, xba b x ba xb x b
Ds xba x b b ¼ ab 2S xb xa
(3:11)
Rearranging and substituting Equation 3.8 for S,
2 Ds xba x b b x ¼ xba b 1=2 2a*1 (Dt) xb xab a
(3:12)
1=2 Ds xba xb xba xab b b a x ¼ xba 1=2 b 2(Dt)1=2 xb xab a
(3:13)
a1** ¼ a*1 D1=2
(3:14)
Substituting Equation 3.9 for a*1 ,
In order to inspect the temperature dependence of growth kinetics, a new term is defined:
Thus, S ¼ a1**t 1=2 and the rate of growth, G, is given by G
dS a1** a*1 D1=2 ¼ ¼ dt 2t 1=2 2t 1=2
(3:15)
Figure 3.8 shows schematically the variation of a1** with reaction temperature, from the temperature of the b=(a þ b) phase boundary in the alloy utilized down to a temperature at which D has become vanishingly small (see Figure 1.17). As the temperature of transformation approaches that ** of the b=(a þ b) phase boundary, xba b ! xb and a1 ! 0. When the temperature of transformation ** ** becomes very low, D becomes ba very small and a1 ! 0. Hence, a1 must pass through a maximum. However, since xb xb increases with decreasing temperature at rates which are usually less than the exponential rate at which D decreases, the maximum in a1** tends to fall at a temperature fairly close to the b=(a þ b) temperature.
α1**
3.3.2.1.1.2 Exact Solution Dubé [7] and Mehl and Dubé [8] first achieved the development given here. This solution was also reported by Zener [6] as part of a more general solution to the problem of diffusional growth of simpler closed shapes. The exact solution also begins with
0
β/(α + β) Reaction temperature
FIGURE 3.8 Parabolic growth rate constant (including diffusivity) as a function of reaction temperature.
257
Diffusional Growth
Equations 3.1 and 3.2, leading to Equation 3.3, the flux balance equation. However, instead of approximating (qx=qs)b:a, the indicated differentiation is performed on the correct solution to the diffusion equation for the diffusion of solute in the b matrix. This equation is Fick’s second law: 2 qx q x ¼D qt qs2
(3:16)
The boundary conditions for the solution of this equation are as follows: x ¼ xb
at
s¼1
(3:17a)
x ¼ xba b
at
s ¼ S,
(3:17b)
and
where S is the position of the a:b boundary at time t. The initial condition is x ¼ xb at all s and t ¼ 0:
(3:17c)
The solution to this equation is given by
s x ¼ A þ B erf pffiffiffiffiffi 2 Dt
where
erf
s pffiffiffiffiffi 2 Dt
2 ¼ pffiffiffiffi p
pffiffiffiffi s=2ð Dt
(3:18)
2
eh dh
(3:19)
0
where p erf(s=2 Dt) is the well-known error function A and B are constants h is an integration variable
Equations 3.17a and b are now used to evaluate A and B. Applying Equation 3.17a to Equation 3.18, xb ¼ A þ B erf (1) ¼ A þ B:
(3:20)
Applying Equation 3.17b in a similar manner, xba b ¼ A þ B erf
S pffiffiffiffiffi 2 Dt
(3:21)
1=2 Since xba must also be a constant. Hence, one can set b , A and B are constants, the ratio S=t
S t 1=2
¼ a1
(3:22)
(Following Zener, the subscript ‘‘1’’ applied to a denotes isotropic planar growth, the subscript ‘‘2’’ will indicate isotropic two-dimensional growth, that is, cylindrical growth and the subscript ‘‘3’’
258
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
describes isotropic three-dimensional growth, that is, spherical growth.) Substituting Equation 3.22 into Equation 3.21, a1 ba (3:23) xb ¼ A þ B erf pffiffiffiffi 2 D Subtracting Equation 3.20 from Equation 3.23 and rearranging, xb xba b B¼ a1 1 erf pffiffiffiffi 2 D
(3:24)
Substituting this relationship into Equation 3.20 and rearranging, a1 ba xb xb erf pffiffiffiffi D 2 A¼ a1 1 erf pffiffiffiffi 2 D
(3:25)
Returning now to Equation 3.3, substitution of the relationship for ds=dt obtained by rearranging and differentiating Equation 3.22 as dS a1 ¼ 1=2 dt 2t
(3:26)
a 1 ab dS ab xba ¼ xba b xa b xa 1=2 dt 2t
(3:27)
yields for the left-hand side of Equation 3.3
The right-hand side of Equation 3.3 is obtained by differentiating Equation 3.18 with Equations 3.24 and 3.25 substituted for A and B: 2 6 qx D ¼ D6 4 qs s¼S
xba b
3
2
7 q xb 2 7 6 4pffiffiffiffi a1 5 qs p 1 erf pffiffiffiffi 2 D 2 3 rffiffiffiffiffi ba 7 a2 =4D xb xb D6 6 7 e 1 ¼ 4 a1 5 pt 1 erf pffiffiffiffi 2 D
ffiffiffi ð
pS 2 Dt
0
3
2 7 eh dh5
(3:28)
where the condition s ¼ S at t has been applied in the last line and Equation 3.22 has been applied to the exponential. Equating Equations 3.27 and 3.28, a1 ba a1 1 erf 2pffiffiDffi 1 xb xb pffiffiffiffi p ffiffiffi ffi ¼ 2 ab p xba 2 D ea1 =4D b xa
(3:29)
This is a transcendental equation and cannot be solved analytically. A trial-and-error solution is achieved instead on a computer, yielding an exact value of a1 appropriate to the solute concentrations and D employed.
259
Diffusional Growth
A note is inserted here on how to carry out the indicated differentiation of an integral in Equation 3.28. When the integration limits are a function of a variable with respect to which the differentiation is being made, as in the present situation, proceed as follows. In the following prototype relationship, let the integration limits, m1 and m0, be functions of a:
f(a) ¼
m1ð {a}
f (x, a)dx
(3:30)
m0{a}
This equation is differentiated through application of the formula df dm dm ¼ f (m1 , a) 1 f (m0 , a) 0 þ da da da
m1ð {a}
qf (x, a) dx qa
(3:31)
m0{a}
This formula is applied here by setting a ¼ s and x ¼ h. To obtain the equation for the solute concentration x in b as a function of s (at distances greater than S), it is merely necessary to substitute the relationships for A and B, Equations 3.25 and 3.24, into the solution to the diffusion equation, Equation 3.18: s a1 ba ba xb xb erf pffiffiffiffi þ xb xb erf pffiffiffiffiffi 2 Dt 2 D x¼ (3:32) a1 p ffiffiffi ffi 1 erf 2 D pffiffiffiffiffi Note that Equation 3.32 predicts x ¼ xb only when s goes to infinity, where erf s=2 Dt 1
3.3.2.1.2 Spherical Growth This problem was solved by Rieck [9] in 1924, by Zener [6] and by Frank [10]. The solution of Frank is followed here. Equation 3.16 is a one-dimensional statement of Fick’s second law. In three dimensions, this equation becomes 2 qc q c q2 c q2 c þ þ ¼D qt qx2 qy2 qz2
(3:33)
where c temporarily replaces x as the atom fraction of solute, x, y and z represent the usual Cartesian coordinates. This equation may be rewritten in spherical coordinates r, u, and f through application of the transformation equations (following Barrer [11], see also Crank [12]): x ¼ r sin u cos f
(3:34a)
y ¼ r sin u sin f
(3:34b)
z ¼ r cos u
(3:34c)
qc D q 1 q qc 1 q2 c 2 qc ¼ r þ sin u þ 2 qt r 2 qr qr sin u qu qu sin u qf2
(3:35)
This procedure yields the relationship
260
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
For radially symmetric diffusion, c does not vary with either u or f, thereby eliminating the second and third terms in brackets. Carrying out the indicated differentiation in the first term and now replacing c by x: 2 qx q x 2 qx (3:36) ¼D þ qt qr 2 r qr where r is the distance from the center of the sphere, not the radius of the sphere. The solution of this equation is initiated by converting it into reduced coordinates. The dimensionless quantity, u, the reduced radius, is employed for this purpose r u ¼ pffiffiffiffiffi Dt
(3:37)
Application of this relationship to Equation 3.36 permits x to be made a function only of u, rather than of both r and t. The three derivatives in Equation 3.36 are successively transformed as follows: qx qx qu qx r u qx ¼ ¼ ¼ qt qu qt qu 2(Dt)1=2 t 2t qu
(3:38)
qx qx qu qx ¼ ¼ (Dt)1=2 qr qu qr qu
(3:39)
and q2 x q qx q qx qu q2 x qu qx q2 u ¼ ¼ ¼ þ qr2 qr qr qr qu qr qrqu qr qu qr2 q2 x qu qr qu q2 x qu 2 q2 x ¼ 2 (Dt)1 ¼ ¼ 2 qrqu qr qu qr qu qr qu
(3:40)
Substituting Equations 3.38 through 3.40 into Equation 3.36, u qx 1 q2 x 2 qx 1 ¼D þ 2t qu Dt qu2 r qu (Dt) 1=2 1 q2 x qx u 2 D 1=2 ¼ þ t qu2 qu 2t r t 1=2 q2 x qx u 2 þ ¼ qu2 qu 2 u Making the substitution p ¼ qx=qu, the above equation becomes qp u 2 ¼ p þ qu 2 u
(3:41)
(3:42)
Integrating, ln p ¼
u2 2 ln u þ A1 4
(3:43)
Rearranging, p
qx u2 ¼ Au2 e 4 : qu
(3:44)
261
Diffusional Growth
Integrating again, xðb x
dx ¼ A
1 ð
u2
u2 e 4 du
(3:45)
u
This equation is integrated by parts in accordance with the usual formula 1 ð
2
2 u4
u e
u
u2
e 4 1 du ¼ u 2
1 ð
u2
e 4 du
(3:46)
u
Note that, as a general relationship involving the error function, 2 pffiffiffiffi p
1 ð
e
h2
z
2 dh ¼ pffiffiffiffi p
1 ð
e
h2
0
Let h ¼ u=2 and thus 2dh ¼ du. Hence, 1 2
1 ð
e
2
u4
u
1 du ¼ 2 2
1 ð
h¼u2
e
2 dh pffiffiffiffi p
h2
ðz 0
2
eh dh ¼ 1 erf (z)
pffiffiffiffi h ui p 1 erf dh ¼ 2 2
(3:47)
(3:48)
Substituting into Equation 3.46 and incorporating the resulting relationship into Equation 3.45, " u2 pffiffiffiffi # uo pn e 4 1 erf (3:49) x xb ¼ A u 2 2 To evaluate A, the flux balance equation is derived for a sphere of radius R ¼ U(Dt)1=2. Consider first the flux diffusing away from the sphere. From Equations 3.39 and 3.44, F1 ¼ 4pR2 D
qx qx u2 ¼ 4pU 2 D2 t(Dt)1=2 ¼ 4pAD3=2 t 1=2 e 4 qr qu
The flux of solute displaced by the advancing a:b interface is given by ab d 4p 3 ab ¼ 2p xba x R U 3 D3=2 t 1=2 F2 ¼ xba b b xa a dt 3 Balancing of fluxes requires that F1 ¼ F2, which yields 1 ba U2 xb xab U3e 4 A¼ a 2
At u ¼ U, x ¼ xba b . Substituting Equation 3.52 into Equation 3.49 on this basis,
pffiffiffiffi xba p U2 U2 U b xb 4 1 Ue 1 erf ¼ ba ab 2 2 2 xb xa
(3:50)
(3:51)
(3:52)
(3:53)
Following Zener, R ¼ U(Dt)1=2 can be rewritten as R ¼ a3t1=2 where a3 is the parabolic rate constant for spherical growth. Hence, U ¼ a3=D1=2. Making this substitution in Equation 3.53, rffiffiffiffi
xba a23 a3 p a2 =4D a3 b xb (3:54) 1 1 erf pffiffiffiffi ¼ e 3 ab 2D 2 D 2 D xba b xa
262
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
This is the transcendental equation for the parabolic rate constant a3, for the diffusion–controlled growth of a sphere when the diffusivity in the matrix phase is independent of composition. To obtain the concentration-penetration curve of solute concentration in the matrix phase as a function of distance, r, from the center of the sphere, it is only necessary to substitute Equation 3.52 into Equation 3.49 and again to replace U by a3=D1=2, x ¼ xb þ
1=2 pffiffiffiffi 1 ba r p 3 a23 =4D t r2 =4Dt p ffiffiffiffiffi e xb xab e 1 erf a 3 a Dr 2 2D3=2 2 Dt
(3:55)
This equation yields a meaningless result when r < R (¼a3t1=2). At larger values of r, inserting successively larger values of r at a fixed t permits the concentration-penetration curve to be constructed. Equations 3.54 and 3.55 are again best handled by computer. For use in these relationships and in the corresponding equations for planar growth, Equations 3.29 and 3.32, a relationship for the error function is useful. See Abramowitz and Stegun [13, p. 299], see Philip [14] for an equation appropriate to the complement of the error function when the argument of the error function is greater than 3. Both equations are reproduced in Atkinson et al. [15]. 3.3.2.1.3 Approximations for Planar and Spherical Growth Reference is Aaron et al. [16]. The intractable form of the transcendental equations that provide the exact solutions for the parabolic rate constants for planar and spherical growth has encouraged searches for approximate solutions that can be cast in a closed form. The linearized gradient approximation, already described for planar growth, represents one successful approach to achieving such a result. Following a brief summary of this approach, descriptions will be given of two others. Numerical comparisons will then be made among approximate and the exact solutions for both planar and spherical growth. The linearized gradient approximation simplifies the diffusion field in the matrix by assuming it to be linear with s or r. Conservation of mass determines the extent of this field. Substitution of the diffusion field relationship thus obtained into the flux balance equation yields dS=dt or dR=dt. The invariant field approximation assumes that in the generalized form of Fick’s law for planar (Equation 3.16) or for spherical (Equation 3.36) growth qx=qt ¼ 0. Hence, the diffusion equation for both cases can be written as r2 x ¼ 0, known as the Laplace equation. Assuming for the present purposes only that the precipitate is solute rich, this approximation is useful only when ba xab a xb xb xb . The precipitate size then varies with time slowly enough so that the diffusion problem can be taken as independent of growth time. From the solution to the Laplace equation for spherical growth, a relationship between x and r is obtained. This relationship is differentiated with respect to r at r ¼ R and substituted into the flux balance equation. Rearrangement yields a relationship for dR=dt. This approximation cannot be used for planar growth since one cannot simultaneously satisfy q2x=qs2 ¼ 0 and the boundary conditions. The invariant size approximation considers the growth rate to be identically zero. This approximation applies under the same composition conditions as the invariant field approximation. The assumption of constant size permits the diffusion field to be taken as that which would obtain if the interface had been fixed at position R or S ab initio. The solution to the diffusion equation that satisfies the boundary conditions defined by Equations 3.17 and the added condition that the boundary is stationary, i.e., dR=dt or dS=dt ¼ 0 is then substituted into the flux balance equation which again yields dR=dt or dS=dt by rearrangement. The three approximations were compared on the basis of the parameter k defined by ab ba = x 2(CI CM )=(CP CI ). This term is seen to appear in the equation x x k ¼ 2 xba b a b b for the parabolic rate constant for both the planar (Equation 3.29) and the spherical (Equation 3.54) cases. In Figures 3.9 and 3.10 the parabolic rate constant is denoted as lj, with the various j’s
263
Diffusional Growth C M = Cp Cp 2 CM 100
∞ λ4
10
S = λj (Dt)1/2
λj
1.0
λ1 λ2
0.1
0.01
λ4
0.001 0.001
0.01
0.1 –2(CI – CM) –k = (Cp – CI)
1.0 |–k|Max
FIGURE 3.9 Parabolic growth rate constants of a planar interphase boundary calculated using various approximations: l1, exact solution; l2, invariant size approximation; and l4, linearized gradient approximation. Invariant field approximation (l3) is not applicable. (From Aaron, H.B. et al., J. Appl. Phys., 41, 4404, 1970. American Institute of Physics. With permission.)
Cp 2 CM
CM = Cp
10 λ1 R = λj
λj
1.0
λ2
(Dt)1/2 λ3
0.1 λ4 0.01 0.0001
0.001
0.01 –k =
0.1
–2(CI – CM)
1.0 |–k|Max
10
(Cp – CI)
FIGURE 3.10 Parabolic growth rate constants of a spherical interphase boundary calculated using various approximations: l1, exact solution; l2, invariant size approximation; l3, invariant field approximation; and l4, linearized gradient approximation. (From Aaron, H.B. et al., J. Appl. Phys., 41, 4404, 1970. American Institute of Physics. With permission.)
264
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
signifying the following: (1) exact solution, (2) invariant size approximation, (3) invariant field approximation, and (4) linearized gradient approximation. Figure 3.9 compares the rate constants for planar growth and Figure 3.10 compares the various rate constants for spherical growth. Inspection of these figures shows that in all cases the rate constant for the exact solution is the largest. The viable approximations, l2 and l4, for planar growth are seen to yield a result closely comparable to that for the exact solution over a wide range of k. The invariant field approximation, l3, is undefined for planar growth. In the case of spherical growth, the linearized gradient approximation, l4, is quite inaccurate in a large region of k values, but the invariant size approximation, l2, is very good when k < 0.7; at larger values of k, the linearized gradient approximation, l4, becomes quite reasonable.* When k < 0.3, the invariant size approximation, l2, is best for analyses of both planar and spherical growth. The invariant field approximation, l3, is not optimal for any situation during spherical growth; however, it is clearly a useful approximation for spherical growth at low supersaturations. 3.3.2.1.4 Cylindrical Growth The growth process described in this section is the radial growth of a cylinder (or needle) of effectively infinite length. This development was reported by Zener [6] and Frank [10]. The derivation given here is an elaboration of that presented by Sekerka et al. [18]. In terms of polar coordinates, Fick’s second law for cylindrical growth is written as 2 qc q x 1 qx ¼D þ qt qr 2 r qr
(3:56)
where the 2=r in the second term of Equation 3.36 is replaced by 1=r. In Cartesian coordinates, with c again replacing x as the symbol for the atom fraction of solute, 2 qc q c q2 c þ ¼D qt qx2 qy2
(3:57)
where x and y designate the distance from the longitudinal axis of the cylinder in the orthogonal directions. Equation 3.57 will be used here. In order to express c in terms of a single variable, designated as w, instead of the two variables x and y, the ‘‘reduced radius’’ is introduced as w¼
x2 þ y2 4Dt
1=2
(3:58)
In the following derivations, the various differentials in Equation 3.57 are converted into their reduced variable counterparts: 1=2 qc qc qw qc 1 x2 þ y2 w qc ¼ ¼ t 3=2 ¼ 4Dt qt qw qt qw 2 2t qw 1=2 qc qc qw qc 1 x2 þ y2 2x x qc ¼ ¼ ¼ 4Dt qx qw qx qw 2 4Dt w(4Dt) qw
(3:59)
(3:60)
* A more rigorous treatment of linearized approximation in three dimensions results in a slightly smaller rate constant at small supersaturations [17].
265
Diffusional Growth
q2 c q qc qw qc q2 w qw q2 c ¼ ¼ þ qx2 qx qw qx qw qx2 qx qwqx
(3:61)
Deriving qw=qx and q2w=qx2 separately, qw x ¼ qx 4Dtw
(3:62)
q2 w 1 x2 ¼ qx2 4Dtw w3 (4Dt)2
(3:63)
and
Continuing the manipulation of Equation 3.61, q2 c qc q2 w qw q2 c qw qx qc q2 w qw q2 c ¼ þ þ ¼ qx2 qw qx2 qx qwqx qx qw qw qx2 qx qw2
(3:64)
Substituting Equations 3.62 and 3.63 into Equation 3.64,
q2 c qc 1 x2 q2 c x2 þ ¼ qx2 qw w(4Dt) w3 (4Dt)2 qw2 w2 (4Dt)2
(3:65)
q2 c qc 1 y2 q2 c y2 ¼ þ qy2 qw w(4Dt) w3 (4Dt)2 qw2 w2 (4Dt)2
(3:66)
Equivalently,
Substituting Equations 3.59, 3.65 and 3.66, into Equation 3.57, 9 8
2 2 2 > > qc 1 x q c x > > > > þ 2 2 3 > > 2 2 = < qw w(4Dt) qw w (4Dt) w (4Dt) w qc ¼D
> > 2t qw > > qc 1 y2 q2 c y2 > > > > þ 2 2 3 ; :þ 2 2 qw w(4Dt) w (4Dt) qw w (4Dt) ¼
1 qc 1 q2 c þ w4t qw 4t qw2
(3:67)
Rearranging and multiplying by 4t, q2 c qc 1 þ 2w ¼0 þ qw2 qw w
(3:68)
Let p ¼ qc=qw; hence, qp=qw ¼ q2c=qw2. Substituting into Equation 3.68, rearranging and integrating as in the derivation of the concentration–penetration curve for the sphere, pw ¼ Aew
2
(3:69)
266
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Recalling that p ¼ qc=qw, xðb x
dc ¼ A
1 ð w
2
ew dw ¼ AE1 {w2 } A w
1 ð
el
2
dl l
(3:70)
w2
where E1(w2) is an exponential integral. Hence, x ¼ xb þ AE1 {w2 }
(3:71)
To evaluate the integration constant A, invoke the boundary condition that at the interphase boundary x ¼ xba b , with the boundary being located at w ¼ W, 2 xba b ¼ xb þ AE1 (W )
A¼
xba b xb E1 (W 2 )
(3:72) (3:73)
Substituting into Equation 3.71 yields the concentration-penetration curve equation for the atom fraction of solute as a function of the distance from the longitudinal axis of the cylinder, beginning at w ¼ W and extending to w ¼ 1, x ¼ xb
xb xba E1 (w2 ) b E1 (W 2 )
(3:74)
Development of a relationship for W is accomplished based on the flux balance equation as written in cylindrical coordinates, ( )1=2
qc qc 2 qc 2 ab dR þ ¼ xba ¼D D b xa qr W qx qy dt
(3:75)
where R is the radius of the cylindrical particle. Equation 3.58 is rearranged to R ¼ W(4Dt)1=2. Substituting Equation 3.60 for qc=qx and the equivalent relationship for qc=qy into the left-hand side of Equation 3.75 and differentiating the equation just obtained for R with respect to t and substituting it for dR=dt on the right-hand side of Equation 3.75, 1=2 pffiffiffiffiffiffi x2 y2 qc ba ab 1 þ ¼ x x W 4Dt 1=2 a b 2 2 2 2 qw W 2 W (4Dt) W (4Dt) r ffiffiffi ffi D qc D ab ¼ xba W b xa 1=2 qw t (4Dt) W
D
Thus,
qc ab ¼ 2 xba W b xa qw W
(3:76)
(3:77)
267
Diffusional Growth
The left-hand side of Equation 3.77 is obtained by differentiating Equation 3.74, 21 3 ð xba x b b q 4 l2 dl5 ba ab 2 xb xa W ¼ e E1 (W 2 ) qw l w2
(3:78)
W
Thus,
2 W2
W e
2
E1 (W ) ¼
xb xba b
ab xba b xa
(3:79)
W is found from this equation by trial and error and substituted into Equation 3.74 to complete the evaluation of the concentration–penetration curve. From the equation for the radius of a cylinder, r, Equation 3.75 is recast in terms of the appropriate parabolic rate constant as R ¼ a2 t 1=2
(3:80)
a2 ¼ 2WD1=2
(3:81)
where
3.3.2.1.5 Ellipsoidal Growth Ham [19] and Horvay and Cahn [20] solved this problem. The solution presented here is a more detailed version that was summarized by Sekerka et al. [18]. The diffusion field associated with a growing ellipsoidal precipitate is given by
h2e
x2 y2 z2 þ 2 þ 2 ¼ 4Dt 2 2 K1 he K2 he
(3:82)
where he is the reduced variable used to describe growth, K1 and K2 are parameters describing deviations from sphericity, which are thus characteristic of the ‘‘eccentricity’’ of the ellipsoid, x, y, and z represent distances in Cartesian space, C will again be used to designate the atom fraction of solute. This equation may be rewritten as x2 y2 z2 þ þ ¼1 A2 B2 C2
(3:83)
1=2 pffiffiffiffiffiffiffiffi A ¼ h2e K12 4Dt ,
(3:84a)
where the semi-axes of the ellipsoid are
and
1=2 pffiffiffiffiffiffiffiffi 4Dt , B ¼ h2e K22
(3:84b)
pffiffiffiffiffiffiffiffi C ¼ he 4Dt
(3:84c)
268
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
At a particular value of he, designated Ke, Equation 3.82 describes the surface or interphase boundary of the ellipsoid. In three dimensions, Fick’s second law is 2 qc q c q2 c q2 c þ þ ¼D qt qx2 qy2 qz2
(3:85)
See Equation 3.33. This equation, in which c is a function of four variables, x, y, z, and t will now be replaced with another in which c is a function of only one variable, he. Beginning with qc=qt, qc dc qhe ¼ qt dhe qt
(3:86)
Differentiating Equation 3.82 with respect to t, and rearranging, " # qhe 2D x2 y2 z2 2D ¼ þ 2 þ 4 2 2 h qt he h 2 K 2 2 h ef h K e 1
e
(3:87)
2
e
Considering next qc=qx, Equation 3.82 is differentiated with respect to x,
(3:88)
Similarly,
2x " #1 qhe x2 y2 z2 x h2e K12 ¼ þ þ 4 2 2 2 2he he qx he he K12 f h2 K 2 h2 K 2
(3:89)
and
2y " #1 qhe x2 y2 z2 y h2e K12 ¼ 2 2 þ 2 þ 4 2 2 2he he qy he he K22 f h2 K h2 K
(3:90)
where
2z " #1 qhe x2 y2 z2 z h2e ¼ 3 2 þ 2 þ 4 2 2 2 2 he qz 2he h K he f h K
Hence,
x2 y2 z2 f þ þ 4 2 2 he h2e K12 h2e K22
Turning next to q2c=qx2,
qc dc qhe x dc ¼ ¼ 2 qx dhe qx he he K12 f dhe
1
e
1
e
e
2
e
2
e
1
e
2
" # q2 c q x dc ¼ qx2 qx he h2e K12 f dhe
(3:91)
(3:92)
(3:93)
269
Diffusional Growth
Multiplying the numerator and denominator of the r.h.s. by g utilizing Equation 3.88 for qhe=qx,
1 h2e K12 h2e K22 2 , and
" # q2 c q dc x ¼ g qx2 qx dhe he g h2e K12 f
" # q dc x dc q 1 x þg ¼ g qx dhe he gf h2e K12 dhe qx ghe h2e K12 f
" # q qx qhe dc x dc q 1 x þg g ¼ qx qhe qx dhe he gf h2e K12 dhe qx ghe h2e K12 f
! " # d dc x2 dc x q 1 1 q x þ g ¼ þg dhe dhe h2 gf2 h2 K 2 2 dhe f qx ghe h2e K12 ghe h2e K12 qx f e e 1
(3:94)
Similarly,
and
q2 c d dc y2 ¼ g qy2 dhe dhe h2 gf2 h2 K 2 2 e e 1 " ! # dc y q 1 1 q y þ þg dhe f qy ghe h2e K22 ghe h2e K22 qy f
q2 c d dc z2 dc z q 1 1 q z þ 3 ¼ g þg qz2 dhe dhe h2 gf2 h2 2 dhe f qy gh3e ghe qz f e
(3:95)
(3:96)
e
Hence, r2c is the sum of Equations 3.94 through 3.96. Summing initially the first terms on the r.h.s. of each of these three equations, # " d dc x2 y2 z2 g 2 þ 2 þ 2 dhe dhe h2e gf2 h2e K12 h2e gf2 h2e K12 h2e gf2 h2e
d dc 1 2 g ¼ dhe dhe he gf
(3:97)
Now, the indicated operations in the first term in the square brackets on the r.h.s. of the last line of Equation 3.93 are performed. Replacing qhe=qx from Equation 3.88, ! ! x q 1 x d 1 qhe ¼ qx f qx ghe h2e K12 f dhe ghe h2e K12 " # x x 1 d 1 1 2he ¼ þ f he h2e K12 f h2e K12 dhe ghe ghe h2e K12 x2 d 1 2x2 ¼ 3 2 he f2 h2e K12 dhe ghe he f2 g h2e K12
(3:98)
270
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Similarly,
and
y q 1 2 f qy ghe he K22
!
d 1 2y2 ¼ 3 2 he f2 h2e K22 dhe ghe he f2 g h2e K22 y2
z q 1 z2 d 1 2z2 ¼ 2 3 4 f qz ghe he f he dhe ghe he f2 gh6e
(3:99)
(3:100)
Adding Equations 3.98 through 3.100, ! ! x q 1 y q 1 z q 1 þ þ f qx ghe h2e K12 f qy ghe h2e K22 f qz gh3e 1 d 1 2u ¼ fhe dhe ghe he gf2
(3:101)
where Equation 3.91 was used to simplify the first term on the r.h.s of Equations 3.98 through 3.100. Here, u
x2
þ 2 3
h2e K1
y2
z2 þ 3 3 h2e K22 h2e
(3:102)
Performing differentiation, the r.h.s. of Equation 3.101 becomes
he 2h2e K12 K22 1 1 2u þ (r:h:s of Equation 3:101) ¼ fhe gh2e he g3 he gf2 2 h K12 h2e K12 þ 2h4e h2e K12 h2e K22 2u ¼ e fh3e g3 he gf2 3h4e 2h2e K12 þ K22 þ K12 K22 2u ¼ fh3e g3 he gf2
(3:103)
Next expanding the second term in the square brackets on the r.h.s. of the last line of Equation 3.94,
1
2
ghe h2e K1 From Equation 3.88,
q x 1 1 x qf ¼ 2 qx f f f2 qx ghe h2 K 2 e
(3:104)
1
qf 2x qhe 2x 4ux ¼ ¼ 2 4he u 2 2 2 2 2 2 2 qx qx f h he K1 he K 1 e K1
Substituting in Equation 3.104,
1 q x 1 2 x2 4u x2 þ ¼ ghe f h2e K12 qx f ghf h2e K12 ghe f2 h2e K12 3 ghe f3 h2e K12 2
(3:105)
271
Diffusional Growth
Similarly, q y 1 2 y2 4u y2 ¼ þ ghe h2e K22 qy f ghe f h2e K22 ghe f2 h2e K22 3 ghe f3 h2e K22 2 1
1 q z 1 2 z2 4u z2 ¼ þ ghe fh2e ghe f2 h6e ghe f3 h4e ghe h2e qz f
(3:106)
(3:107)
Adding Equations 3.105 through 3.107,
" # 1 1 q x 1 q y 1 q z þ 2 þ 2 ghe h2e K12 qx f he qz f he K22 qy f
1 1 1 1 2u 4uf ¼ þ þ þ ghe f h2e K12 h2e K22 h2e ghe f2 ghe f3 3h4e 2h2e K12 þ K22 þ K12 K22 2u þ ¼ fh3e g3 ghe f2
(3:108)
As noted earlier, q2c=qx2 þ q2c=qy2 þ q2c=qz2 equals the sum of Equations 3.94 through 3.96. Calculations subsequently performed have replaced these three equations with Equations 3.97, 3.101, and 3.108. Summing the latter three equations,
q2 c q2 c q2 c 1 d dc 2 þ þ ¼r c¼ 2 g qx2 qy2 qz2 he gf dhe dhe
(3:109)
Substituting this result and Equations 3.86 and 3.87 into Equation 3.85,
dc 2D D d dc ¼ 2 g dhe he f he gf dhe dhe
(3:110)
d dc dc þ 2he g g ¼0 dhe dhe dhe
(3:111)
Rearranging,
To integrate this equation, first let p¼g
dc dhe
(3:112)
Substituting into Equation 3.111, dp þ 2he p ¼ 0 dhe
(3:113)
Hence, p ¼ A exp h2e
(3:114)
272
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Substituting into Equation 3.112, cðb c
dc ¼ A
1 ð
he
where
cb c ¼ Afe (l),
2
el 2 1=2 dl l K12 l2 K22
and
fe (l) ¼
1 ð
he
(3:115)
2
el 2 1=2 dl l K12 l2 K22
(3:116)
To evaluate the constant A, note that when c ¼ cba b , he is replaced by Ke, corresponding to the ellipsoid congruent with the surface of the precipitate. Hence, A¼
cb c fe (Ke )
(3:117)
Substituting in Equation 3.116, the equation for the concentration-penetration curve equation for an ellipsoid is obtained as
c ¼ cb
cb cba b fe (he ) fe (Ke )
(3:118)
Finally, Ke is to be evaluated. This is again accomplished by means of the flux balance equation, recast in terms of he at Ke. However, the flux balance equation must now be written in more general form, because in the case of an ellipsoid, the normal to the advancing interphase boundary, which gives the direction of growth, usually does not pass through the origin (the normal does pass through the origin in the cases of sphere and cylinder). This equation may be written ab dRe ¼ D(rc ~ n) c cba a b dt
(3:119)
where ~ n is the local unit normal, directed toward the b matrix and Re is the ‘‘reduced radius’’ of the ellipsoid. Re may be expressed as Re ¼
1 2
ð
dt re t
(3:120)
where re ¼
x2 y2 z2 þ 4þ 4 4 A B C
1=2
(3:121)
Differentiating Equation 3.120, dRe 1 ¼ : dt 2re t
(3:122)
273
Diffusional Growth
For an ellipsoid, x2 y2 z2 þ 2 þ 2 ¼ 1: 2 A B C
(3:123)
Defining f as some function, ~ n may be written,
qf qf qf , , rf qx qy qz ~ n¼ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 : jrf j qf qf qf þ þ qx qy qz
(3:124)
2x 2x 2 2 x A A ~ ¼ n ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 4x2 4y2 4z2 2re re A þ þ A4 B4 C4
(3:125)
Hence, in the x direction,
Similarly, ~ n¼
y z and ~ n¼ in the y and z directions. The unit normal vector is expressed by, re B 2 re C 2 x y z ~ (3:126) , , n¼ re A2 re B2 re C 2
In Equation 3.119, rc ¼
qc qc qc , , qx qy qz
(3:127)
Substituting Equation 3.92 for qc=qx and equivalent relationships for the other terms, and Ke for he, ! x y z dc , , Ke Ke2 K12 f Ke Ke2 K22 f Ke3 f dhe he ¼Ke
rc ¼
(3:128)
Forming the scalar product of Equation 3.119, "
# x2 y2 z2 1 dc þ þ ~ n rc ¼ 2 Ke K12 A2 Ke2 K22 B2 Ke2 C 2 re Ke f dhe he ¼Ke f 1 dc ¼ 4Dt re Ke f dhe he ¼Ke
(3:129)
Substituting Equations 3.122 and 3.129 into Equation 3.119,
1 f 1 dc ab ¼ D c cba a b 2re t 4Dt re Ke f dhe he ¼Ke dc ab ¼ 2 cba Ke b ca dhe he ¼Ke
(3:130)
(3:131)
274
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Differentiating Equation 3.118 with respect to h at he ¼ Ke, and substituting into Equation 3.131, " # 2 cb cba b eKe ba ab 0 2 cb ca Ke ¼ (3:132) 1=2 fe {Ke } Ke2 K12 Ke2 K22
Rearranging,
ba c c b 2 b 2 1=2 K 2Ke Ke K12 Ke2 K22 e e fe (Ke ) ¼ ab cba b ca
(3:133)
The integral fe{Ke} is given by Equation 3.116, with Ke replacing he. This integral cannot be evaluated analytically. Numerical methods or graphical ones must be employed. However, for the special cases of oblate and prolate ellipsoids of revolution, Horvay and Cahn (in the GE Research Report only) have given series expansions of the integrals that allow approximate evaluations at low- and high-aspect ratios. For the oblate ellipsoid case, at low-aspect ratios, fe {Ke } ¼
pffiffiffiffi p p K2 1 3 RKe e erfc(K) 2 1 2 þ 4 erf (RKe ) 4 K K K 2K 4K 1 0 3 þ R2 Ke2 C R2 Ke2 B @1 2 Ae K2
(3:134)
When the aspect ratio, R, of the ellipsoid approaches unity,
fe {Ke } ffi
R2 Ke2
2e RKe
0 1 13 1 1 3 4 2K B C 4K B C7 6 @1 22 2 A þ @1 22 2 þ 44 4 A5 41 þ R Ke R K e R Ke 3 15 2
2
0
pffiffiffiffi 2K 2 4K 4 þ 2 perfc(RKe ) 1 þ 3 15
(3:135)
For an oblate ellipsoid of revolution take K2 ¼ 0 and set K1 ¼ K. Hence, from Equations 3.84 B ¼ C and both are larger than A. The thickening of the oblate ellipsoid may thus be written as,
where
1=2 pffiffiffiffiffiffiffiffi 4Dt ¼ at 1=2 A ¼ Ke2 K 2
(3:136)
1=2 pffiffiffiffiffiffi a ¼ Ke2 K 2 4D:
while lengthening of the oblate ellipsoid is described by,
where
pffiffiffiffiffiffiffiffi B ¼ C ¼ Ke 4Dt ¼ bt 1=2 pffiffiffiffiffiffi b ¼ Ke 4D
(3:137)
275
Diffusional Growth
Rearranging, Ke is obtained in terms of measurable quantities, b Ke ¼ pffiffiffiffi 2 D
thus,
1=2 pffiffiffiffi b2 2 K a¼2 D 4D
(3:138)
(3:139)
Rearranging, K2 ¼
b 2 a2 4D
(3:140)
Let the aspect ratio, R ¼ a=b. Hence, K¼
b(1 R2 )1=2 pffiffiffiffi 2 D
(3:141)
Hence, Ke and K are now expressed in terms of measurable quantities. Knowing D, and taking R from the experiment (this quantity is not calculable), b can thus be obtained from a trial-and-error solution of Equation 3.133. To calculate the concentration–penetration curve along the B (or C) and the A directions of an oblate ellipsoid of revolution in the Cartesian coordinates, and also to use Equation 3.133 conveniently for this morphology, substitute Equation 3.138 into Equation 3.141, obtaining, K ¼ Ke (1 R2 )1=2
(3:142)
With K2 ¼ 0 and Kl ¼ K Equation 3.133 can be solved by trial and error for one variable, Ke. Now, select a value of he larger than Ke. Obtain the corresponding value of K from Equation 3.133. Use Equation 3.118 in conjunction with either Equation 3.134 or 3.135 for the term fe(he), to compute the concentration c at this value of he. From Equations 3.84a and 3.84b obtain the distances along the A and B (or C) at which this concentration applies. 3.3.2.1.6 Lengthening of Plates The first treatment of this problem was by Zener [21]. A more comprehensive and rigorous approach was reported by Hillert [22]. A particularly attractive treatment was presented by Kaufman et al. (Ref. [27] of Chapter 1). The thermodynamic treatment of interstitial solid solutions presented in this paper was recounted in Section 1.3. This material will be utilized here in developing the KRC analysis of plate-lengthening kinetics. The most sophisticated treatment presently available is that of Trivedi [23]. Each of these analyses has something different to offer in respect of understanding the rates at which precipitate plates lengthen. The possibilities for still further refinements will be also briefly considered. 3.3.2.1.6.1 The Zener–Hillert Equation The approximate approach developed here offers several important advantages in the context of an introduction to the plate-lengthening problem. A simple, closed-form growth rate equation is obtained. The physics of plate lengthening are clearly delineated. The optimization principle problem is introduced, and the question of whether the growth rate is best described by a flux balance or a form of mass balance is raised. Zener
276
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
proposed that a flux balance is appropriate, when modified for the presence of capillarity introduced by the small radius of the plate edge, qx ba ab dS xb,r xa ¼ D dt qs a:b
(3:143)
ba where xba b,r is xb as modified for an interface of radius r, and this relationship differs from Equation ba ba 3.3 only through replacement of xba b by xb,r . The qx term is approximated as xb,r xb . qs is considered to be the diffusion field length and is taken to be proportional to the radius of the plate edge, r,
xba qx b,r xb ffiD D ar qs a:b
(3:144)
where a is here the constant of proportionality. Rearranging, and writing the lengthening rate, dS=dt, as G ‘ , D xba b,r xb G‘ ¼ ab ar xba b,r xa
(3:145)
Hillert, on the other hand, noting that the lengthening rate is experimentally found to be constant, concludes the amount of solute pile up or depletion in front of the advancing edge remains constant. He therefore uses a mass balance approach. Hillert also evaluated a as 2. Hence, D xba b,r xb G‘ ¼ 2r xb xab a
(3:146)
A more detailed consideration of the problem by Trivedi and Pound [24] supports Equation 3.146, the Hillert approach, as the better approximation. In particular, Trivedi and Pound [24] note that on the Zener approach the growth rate remains finite, rather than going to infinity, as xba b,r xb = ba xb,r xab a ! 1. xba b,r may be evaluated from Equation 1.78 by assuming that the edge of a plate is a hemicylinder. Hence, one of the two principal radii is infinite and thus, the factor of 2 in the exponential of Equation 1.78 is replaced by unity,
xba b,r
9 8 = < gV a 1 xba b ¼ xba b exp ba ; :RTr xab a xb
(3:147)
Defining Dxr and Dx, respectively, as the difference in solute concentrations in the matrix at the b:a boundary and far away from the boundary for the curved interface and for a planar (disordered) interface at the same temperature and xb, ba ba Dxr ¼ xba b,r xb ¼ xb,r xb,rc
(3:148a)
ba ba Dx ¼ xba b xb ¼ xb xb,rc ,
(3:148b)
277
Diffusional Growth
where rc is the critical radius whereat xba b,r becomes identical to xb. This is the growth counterpart to r* in nucleation. The ratio of the two Dx’s is expressed by K ba K Dxr xb e r erc ¼ ba K Dx x b 1 e rc
where
Recalling that ey 1 þ y when y 1,
(3:149)
gV a 1 xba b : K¼ ab RT xa xba b Dxr rc ¼1 Dx r
(3:150)
Substituting Equation 3.150 into Equation 3.146, rc D xba x 1 b b,r r G‘ ¼ 2r xb xab a
(3:151)
This relationship, however, permits an infinite number of choices for r, and thus for G ‘ . What determines r? On an essentially intuitive basis, Zener proposed that r assumes the value that maximizes G ‘ . Taking the derivative of G ‘ with respect to r, setting it equal to zero and solving for r yields the following result, qG ‘ A rc rc A ¼0¼ þ 1 2 qr r r r2 r A rc A 1 ¼ 3 rc 2 r r r
(3:152)
where D xba b xb A¼ 2 xb xab a Hence, r ¼ 2rc
(3:153)
Substituting Equation 3.153 into Equation 3.151 yields the Zener–Hillert equation: D xba b xb G‘ ¼ 4r xb xab a
(3:154)
278
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
ba If xba b,r in Equation 3.147 is replaced by xb,rc ¼ xb , this relationship can be rearranged to yield a calculated value of r, 2gV a 1 xba b ! r ¼ 2rc ¼ (3:155) xb ab ba RT ln ba xa xb xb
Hence, G ‘ can be calculated from the bulk composition of the alloy, xb, the equilibrium compositions at the phase boundaries and knowledge of g, the interfacial energy of the plate edge. Conversely, g can be calculated from this information and measured G ‘ values. 3.3.2.1.6.2 KRC Approach This treatment is essentially a refined Zener–Hillert approach incorporating the framework of interstitial solid solution thermodynamics developed and utilized in Section 1.3. Again using Zener’s maximum growth rate hypothesis, Kaufman, Radcliffe, and Cohen (KRC) carry the optimization process further and in the course of doing so introduce the problem of the influence of a composition-dependent diffusivity upon growth kinetics. The KRC treatment was developed for the proeutectoid ferrite reaction in Fe–C alloys. Hence, sub- and superscripting appropriate to that reaction will be used here. From the KRC interstitial thermodynamics, recall, ¼ DGg!a Fe
RT 1 6xga g ln 1 xga 5 g
(3:156)
Assigning all interfacial energy at the edge of a ferrite plate to the ferrite phase, as was done earlier, and again bearing in mind that one principal radius is infinite, KRC write, a ¼ a G G Fe,r Fe
a a Fe gag V gag VFe r r
(3:157)
a Fe where V is the partial molar volume of iron in ferrite. Comparison with Equation 3.147 shows that KRC have replaced the molar by the partial molar volume of iron. However, as indicated above, the numerical discrepancy thereby generated is minor [25]. A considerably more serious error is introducedby the omission the ag of ga composition terms; rewritten for the proeutectoid ferrite reaction, 27 at the eutectoid temperature and approximately 4 at these are 1 xga g = xa x g 2008C. Since Equation 3.157, is an integral part of the KRC treatment, it will be retained throughout the following derivation with the understanding that the necessary correction can be made to the final results simply through multiplying by the composition ratio omitted. In the derivation of a Equation 1.111, GFe GaFe is thus replaced by Equation 3.157,
GaFe þ
a gag VFe RT 1 6xga g,r ¼ ln þ GgFe 1 xga r 5 g,r
(3:158)
Thus, g!a þ DGFe
a gag VFe RT 1 6xga g,r ¼ ln r 1 xga 5 g,r
g!a Substituting Equation 1.111 rewritten in terms of DGFe and rearranging, 1 6xga 1 xga g,r g 1 RT ¼ a ln r 5gag VFe 1 xga 1 6xga g,r g
(3:159)
(3:160)
279
Diffusional Growth
KRC write the Zener–Hillert equation in the form, DgC xga xga g,r xg g,r G‘ ¼ 2rxg
(3:161)
ag where DgC xga is the diffusivity of carbon in austenite at composition xga g,r g,r , and xa has been
omitted from the denominator as negligible. The diffusivity of carbon in austenite varies exponentially with carbon content; on the approximation of Wagner [26], this was taken into account by using the diffusivity applicable at the curved edge of a ferrite plate. On the Wagner approximation, this diffusivity is considered to be the most influential since it is the one at which the concentration gradient is determined. Substituting Equation 3.160 into Equation 3.161, 2 3 ga ga x xga 1 6x 1 x g g,r g,r0 g 4 RT ln 5DgC xga G‘ ¼ (3:162) g,r : a 5gag VFe 2xg 1 6xga 1 xga g,r0
g
Using a combination of the experimental data of Wells et al. [27] and growth kinetics data, KRC obtained the following empirical equation for the temperature and composition dependence of DgC over a wide range of temperature and carbon content up to 16 at.%,
38,300 1:9 105 x þ 5:5 105 x2 (3:163) DgC ¼ 0:5 exp (30x) exp RT Substituting this equation into Equation 3.162, the maximum growth rate hypothesis is applied in the form, qG ‘ ¼0 qxga g,r
ga when xga g,r ¼ xg,r0
since xga g,r is the only explicit ‘‘disposable’’ variable in Equation 3.162. This yields
1:9 11 xga 5 xga g,r0 xg g,r0 ¼ 1 þ xga 30 x g g,r0 105 RT 1 6xga 1 xga g,r0 g,r0 1 6xga 1 xga g,r0 g ln 1 xga 1 6xga g,r0 g
(3:164)
(3:165)
where r0 denotes the radius of the plate edge yielding the maximum growth rate. This equation is ga solved for xga g,r0 by trial and error. Substitution of xg,r0 into Equation 3.162 together with Equation 3.163 yields the maximum G ‘ . To calculate r0 substitute xga g,r0 into Equation 3.160. Hence G ‘ and r0, both of which are experimentally measurable, have been calculated by another route without introducing any arbitrary parameters. 3.3.2.1.6.3 More Sophisticated Approach to the Diffusion Problem—The Trivedi Treatment Although, a constant growth rate is customarily taken as indicative of interfacial reaction control of growth kinetics, Papapetrou [28] showed long ago that if the portion of a precipitate under consideration is either a parabolic cylinder (appropriate to the edge of a plate) or a paraboloid of revolution (suitable as the tip of a needle), a constant growth rate will obtain even though growth kinetics is entirely controlled by volume diffusion of the solute toward or away from
280
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
the advancing interface. A constant growth rate means that a steady-state solution exists to the diffusion equation. Evidently, this can occur if the radius of curvature remains fixed. A rough physical analogy to this process is found in the prow wave of a ship. Dispersal of most of the displaced water to the sides of the ship permits the buildup of water in front of the prow, which must be continuously pushed to be restricted to a steady-state wave of fixed extent. Ivantsov [29] and Horvay and Cahn [20] have obtained exact steady-state solutions to the growth of a parabolic cylinder and of a paraboloid of revolution. Although realizing their importance, they did not take into account the effects either of capillarity or of infinite interface reaction kinetics upon the growth process. Trivedi incorporated both and modified his predecessor’s solutions to the parabolic cylinder [23] and to the paraboloid of revolution [30]. His solution to the parabolic cylinder morphology will be sketched here. In all of these treatments, the influences of elastic strain energy and of the anisotropy of interfacial energy and structure are neglected. The capillarity of dilute solutions is assumed applicable and both the diffusion coefficient and the partial molar volume of solute are taken to be independent of composition. The interface barrier to migration is assumed to be uniform, rather than of the more likely ledge type; mathematical difficulties enforce this restriction. In developing the Trivedi approach, the atom fraction of solute is once again designated as c rather than as x for the usual reason. Another concept, not needed in the previous derivations of volume diffusion–controlled growth kinetics (of the plane, sphere, cylinder, and ellipsoid), is introduced here, namely that of transforming the coordinate system in which the diffusion process is treated from the usual fixed Cartesian coordinates to another coordinate system. Here, two successive coordinate transformations will be made. For the parabolic cylinder, the two-dimensional form of Fick’s second law for diffusion in the matrix is appropriate, 2 qc q c q2 c ¼D þ qt qX 2 qY 2
(3:166)
This equation differs from Equation 3.56, only in the formal sense that X and Y replace x and y, the capital letters being introduced to denote fixed Cartesian coordinates; the lower case letters are here reserved for moving, dimensionless Cartesian coordinates. The changeover is accomplished by the use of two transformation equations, x¼
X Gt , r
(3:167)
Y r
(3:168)
and y¼
where t is growth time r is the radius of the leading edge of the parabolic cylinder The transformation is accomplished through the application of the chain rule for partial differentiation. In the case of a total differential, one simply writes, for example, dy=dx ¼ dy=dz dz=dx. Partial differentiation must be chain-ruled through the application of the following more complicated relationship, here written in the form needed for the first application, q q qx q qy ¼ þ qX qx qX qy qX
(3:169)
281
Diffusional Growth
Applying this relationship to transformation of qc=qX to moving coordinates, qc qc qx qc qy qc 1 ¼ þ ¼ qX qx qX qy qX qx r
(3:170)
Proceeding to the remaining transformations required, q2 c q qc 1 q2 c ¼ ¼ , qX 2 qX qX r 2 qx2
(3:171)
since qc 1 qc ¼ , qY r qy q2 c q 1 qc 1 q2 c ¼ ¼ , qY 2 qY r qy r 2 qy2
(3:172)
(3:173)
thus, qc qc qx qc qy G qc ¼ þ ¼ qt qx qt qy qt r qx
(3:174)
Substituting Equations 3.171, 3.173, and 3.174 into Equation 3.166, G qc 1 q2 c 1 q2 c ¼D 2 2þ 2 2 r qx r qx r qy
(3:175)
q2 c q2 c Gr qc þ þ ¼0 qx2 qy2 D qx
(3:176)
Rearranging,
Noting that the dimensionless quantity known as the Peclet number is p ¼ Gr=2D, q2 c q 2 c qc þ 2 þ 2p ¼ 0 2 qx qy qx
(3:177)
The next step is to transform this equation to moving parabolic coordinates, j and h through the relationships, j2 ¼ and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 þ x,
(3:178)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ y2 x
(3:179)
h2 ¼
The parabolic coordinate system is illustrated in Figure 3.11 [20]. In this system, j ¼ constant and h ¼ constant constitute two orthogonal families of the confocal parabolas, that is, parabolas with the same focus whose focus is at the origin and whose principal axis is the original x-axis, that is, the
282
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 3 8
–1 –16
–24
0
–2
–8 –3
1 2
8 3
16
x
24
–4 –8
–5
5
–6
ξ=4
6
ξ=4
–16 3
3 –7
2
2 1
0 –24
7
1
η=8
η = –8 –32
FIGURE 3.11 Parabolic coordinate system. (From Horvay, G. and Cahn, J.W., Acta Metall., 9, 695, 1961. With permission, Elsevier.)
lengthening direction of the plate. The parabola j ¼ 1 defines the interphase boundary. Larger values of j represent iso-j parabolas, which turn out to be iso-concentration curves in the matrix, and the smaller values of j lie within the precipitate plate. For convenience, Equations 3.178 and 3.179 are rewritten as, 1 x ¼ (j2 h2 ), 2
(3:180)
y ¼ jh
(3:181)
and
Substitution of Equations 3.178 and 3.179 proves these equalities. In the first coordinate transformation, qy=qx and qx=qy were equal to zero, that is, x and y were noninteractive. This appreciably simplified the algebra. In the transformation to be developed now, this situation does not obtain. In the following applications of the transformation equations, qc=qx, q2c=qx2 and q2c=qy2 are successively converted to their equivalent in moving parabolic coordinates. First, qc qc qj qc qh ¼ þ : qx qj qx qh qx
(3:182)
Differentiating Equation 3.178 with respect to x, 2j Thus,
qj x j2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi qx x2 þ y 2 x2 þ y2 qj j ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi qx 2 x2 þ y2
(3:183)
(3:184)
283
Diffusional Growth
Squaring Equations 3.180 and 3.181 and adding, 1 x2 þ y2 ¼ (j2 þ h2 )2 4
(3:185)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi j2 þ h2 x2 þ y2 ¼ 2
(3:186)
qj j ¼ 2 qx j þ h2
(3:187)
Substituting this relationship into Equation 3.184,
Differentiating Equation 3.179 with respect to x, 2h Thus,
qh x h2 2h2 : ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 qx j þ h2 x 2 þ y2 x2 þ y2 qh h ¼ 2 qx j þ h2
(3:188)
(3:189)
Substituting Equations 3.183 and 3.189 into Equation 3.182, qc j qc h qc ¼ , qx j2 þ h2 qj j2 þ h2 qh
(3:190)
from which, q2 c q j qc h qc ¼ qx2 qx j2 þ h2 qj j2 þ h2 qh q j qc h qc qj q j qc h qc qh þ ¼ qj j2 þ h2 qj j2 þ h2 qh qx qh j2 þ h2 qj j2 þ h2 qh qx 2 j þ h2 2j2 qc j q2 c h 2j qc h q2 c j þ þ ¼ 2 2 2 2 2 2 2 2 2 2 2 2 (j þ h ) qh j þ h qhqj j þ h2 (j þ h ) qj j þ h qj j 2h qc j q2 c j2 þ h2 2h2 qc h q2 c h þ þ (j2 þ h2 )2 qh j2 þ h2 qh2 j2 þ h2 (j2 þ h2 )2 qj j2 þ h2 qjqh
(3:191)
Similarly, qc qc qj qc qh ¼ þ qy qj qy qh qy
(3:192)
Differentiating Equation 3.178 with respect to y, 2j
qj y 2jh ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 2 2 qy j þ h2 x þy
(3:193)
284
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Thus, qj h ¼ qy j2 þ h2
(3:194)
Differentiating Equation 3.179 with respect to y, 2h
qh y 2jh ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2 2 2 qy j þ h2 x þy qh 2j ¼ qy j2 þ h2
(3:195)
(3:196)
Substituting Equations 3.194 and 3.196 into Equation 3.192, qc h qc j qc ¼ 2 þ 2 2 2 qy j þ h qj j þ h qh
(3:197)
The second derivative becomes q2 c q h qc j qc ¼ þ qy2 qy j2 þ h2 qj j2 þ h2 qh q h qc j qc qj q h qc j qc qh þ þ þ ¼ qj j2 þ h2 qj j2 þ h2 qh qy qh j2 þ h2 qj j2 þ h2 qh qy
2jh qc h q2 c j2 þ h2 2j2 qc j q2 c h þ ¼ þ 2 þ 2 2 2 2 2 2 qj 2 2 2 2 2 qh qjqh (j þ h ) (j þ h ) j þ h qj j þh j þ h2 2
j þ h2 2h2 qc h q2 c 2jh qc j q2 c j þ þ þ þ 2 2 2 2 2 2 2 2 2 2 2 2 qj j þ h qjqh (j þ h ) qh j þ h qh j þ h2 (j þ h )
(3:198)
Substituting Equations 3.190, 3.192, and 3.198 into Equation 3.177 and sorting terms, 2 qc j(h j2 ) 2h2 j 2h2 j j(j2 h2 ) 2pj þ þ þ qj (j2 þ h2 )3 (j2 þ h2 )3 (j2 þ h2 )3 (j2 þ h2 )3 j2 þ h2
qc 2j2 h h(j2 h2 ) h(h2 j2 ) 2h2 j 2ph þ þ þ qh (j2 þ h2 )3 (j2 þ h2 )3 (j2 þ h2 )3 (j2 þ h2 )3 j2 þ h2 2
2
qc j2 h2 q c h2 j2 þ þ þ þ qh2 (j2 þ h2 )2 (j2 þ h2 )2 qj2 (j2 þ h2 )2 (j2 þ h2 )2 2
qc jh jh jh jh ¼0 þ þ þ þ qhqj (j2 þ h2 )2 (j2 þ h2 )2 (j2 þ h2 )2 (j2 þ h2 )2
(3:199)
Collecting terms, 2pj qc 2ph qc q2 c 1 q2 c 1 þ ¼0 þ 2 2 2 2 2 2 2 2 2 qj qh qh j þh j þh qj j þ h j þ h2
(3:200)
285
Diffusional Growth
Multiplying by j2 þ h2, q2 c q2 c qc qc þ 2 þ 2pj 2ph ¼0 2 qh qj qh qj
(3:201)
q2 c q2 c qc qc þ þ 2p j h ¼0 qj qh qj2 qh2
(3:202)
Hence,
This equation represents Equation 3.166, Fick’s second law in fixed two-dimensional Cartesian coordinates, transformed into moving parabolic coordinates. The solution to Equation 3.202 is now sought subject to the boundary condition that along the j ¼ 1 parabola the solute concentration is given by a
c ¼ cba b
cba gab V A G b n ba ab m0 RTr xb xa
(3:203)
In this equation, the first two terms are the Friedel and Mullins [31] expression for cba b,r (with the usual conversion of an exponential to the first two terms of the series, ey ¼ 1 þ y þ ) for the case of an a interface between two dilute interstitial solid solutions, V A is the partial molar volume of solvent in the precipitate phase and the various c’s have the same meaning as the similarly subscripted x’s previously employed. In the third term on the r.h.s. of this equation, G n is the growth velocity normal to the interphase boundary. Since the normal at the leading edge of the plate forms an angle, say u, with respect to a normal elsewhere along the parabolic cylinder, defining G ‘ as the lengthening rate at the leading edge, G n ¼ G ‘ cos u. Also in the third term, m0 ¼ interface kinetic coefficient, equivalent to an interfacial mobility. Rewriting Equation 3.203 in parabolic coordinates after substituting the expressions in the parabolic coordinate system for cos u and for r, a
c ¼ cba b
cba g V G b ab A (1 þ h2 )3=2 n (1 þ h2 )1=2 ba ab m0 RTr cb ca
(3:204)
Solution of Equation 3.202 subject to the boundary condition of Equation 3.204 requires further substitutions that enable the separation of variables to be achieved. A constant coefficient to the general solution (such as has been previously encountered for the sphere) was evaluated by means of Equation 3.204. G ‘ was determined by applying the flux balance equation at the leading edge of the parabolic cylinder, 2
3 a ba c g V G qc b ab A ba ‘ ab 4 5 ¼ 2p cb ca ab m0 qj j¼1, h¼0 RTr cba b ca
(3:205)
Substituting in the r.h.s. of this equation the partial differential with respect to j of the solution to Equation 3.202 and rearranging yields
pffiffiffi G‘ rc pffiffiffiffiffiffi p V0 ¼ ppe 1 erf p 1 þ V0 S1 {p} þ V0 S2 {p} Gc r
(3:206)
286
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
where ba ab (all at r ¼ 1 is the supersaturation), V0 ¼ cba b c b = cb c a rc is the critical radius at which cba b,r ¼ cb , G c is the velocity of a planar interface controlled only by interface kinetics. To evaluate rc, use the Friedel–Mullins capillarity expression, and note that at r ¼ rc, cba b,rc ¼ cb . Rearranging after the usual series expansion, a
cba b gab V A rc ¼ RTr cba c b b
(3:207)
From the standard relationship, that growth velocity is the product of the mobility and the driving force, G c ¼ m0 cba b cb
(3:208)
In Equation 3.206 the term outside the brackets is the Ivantsov solution to the growth kinetics of a parabolic cylinder in which both capillarity and interface kinetics are ignored. The second term inside the square bracket is the correction for interface mobility and the third term is the correction for capillarity. S1{p} and S2{p} are complex functions of p, involving G (gamma) functions, confluent hypergeometric functions and the integral error function; see the original paper for details and for plots of these quantities as a function of p. Yost and Trivedi [32] subsequently showed that S1 1=pp and S2 2=pp. Again applying the Zener maximum growth rate hypothesis in order to secure single values of G ‘ and r, Equation 3.206 is differentiated with respect to r and set equal to zero, qV0 V0 V0 pffiffiffi ¼0¼ (2p þ 1) pffiffiffiffiffiffi p qr 2p ppe 1 erf p
pffiffiffi G ‘ 0 rc 0 rc pffiffiffiffiffiffi p þ ppe 1 erf p S {p} þ S2 {p} S2 {p} Gc 1 r r
(3:209)
where S01 {p} and S02 {p} are graphed as a function of p in Trivedi’s paper. Simultaneous solutions of Equations 3.206 and 3.209 yields the maximum value of G ‘ and the radius at which this lengthening rate obtains. The dimensionless parameter
q¼
m0 xba x b rc b 2D
(3:210)
indicates the relative magnitudes of the interface kinetics effect and capillarity-affected diffusion. When q ¼ 1, the interface kinetics effect does not exist and G ‘ is controlled by the capillarityaffected diffusion. When q is small, interface kinetics dominate G ‘ . Figure 3.12 gives the relationship between p and V0 for various values of q under the condition of maximum G ‘ except the Ivantsov solution. Figure 3.13 gives the ratio of r at the maximum G ‘ to rc as a function of V0, again for various values of q. Knowing gab and m0, these figures can be used to calculate the
287
Diffusional Growth 10.0 Parabolic cylinder growth Ivanstov 1 1.0
3
2
p
4
0.1
0.01
Curve 1 2 3 4 0
0.2
0.4
0.6
q ∞ 10 10–1 10–2
0.8
1.0
Ω0
FIGURE 3.12 Variation of Peclet number with the normalized supersaturation V0. (With kind permission from Springer Science þ Business Media: From Trivedi, R., Metall. Trans., 1, 921, 1970.) 103 Plate growth
q = 0.01 2
r/rc
10
q = 0.1 101
q = 1.0
q = .∞
100 0.2
0.4
0.6 Ω0
0.8
1.0
FIGURE 3.13 Variation of r=rc, which corresponds to the maximum growth rate with the normalized supersaturation V0 for different values of the parameter q. (With kind permission from Springer Science þ Business Media: From Trivedi, R., Metall. Trans., 1, 921, 1970.)
maximum G ‘ and the corresponding r. More usually, G ‘ and r would be measured experimentally and then employed in the foregoing manner to calculate gab and m0, and thereby ascertain the mechanism of plate lengthening.*
* According to Hillert et al. [33] (Acta Mater., 51(2003), p. 2089), the treatment of Trivedi [23,34] causes a large error at low supersaturations because of the assumption that the plate shape (parabolic cylinder) is not affected by capillarity. They proposed that the use of the equations be confined to very high supersaturations.
288
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
3.3.2.1.7 Lengthening of Needles: Comparison with Plate-Lengthening Kinetics Trivedi [30] has also investigated the lengthening kinetics of needles and has compared kinetics with those of plates [23]. By utilizing moving parabolic coordinates, Trivedi obtained the following relationships as counterparts to Equations 3.203 and 3.206:
G‘ rc0 p (3:211) V0 ¼ pe Ei(p) 1 þ V0 R1 ( p) þ V0 R2 ( p) Gc r
qV0 V0 V0 G‘ 0 rc0 0 rc p ¼0¼ (2p þ 2) p þ pe Ei( p) R ( p) þ R2 ( p) S2 ( p) (3:212) r Gc 1 qr 2p pe Ei( p) r where Ei ( p) ¼
ðp
eu du u
(3:213)
1
is the exponential integral function. rc0 ¼ 2rc is the critical radius for a plate. Rl( p), R2( p), R01 ( p) and R02 ( p) are complicated functions which are graphed vs. p in Ref. [23]. These equations are solved simultaneously for the maximum lengthening rate and for the corresponding needle tip radius. Figures 3.14 and 3.15 are plots corresponding to those presented for plates, p vs. V0 for the lvantsov solution and for the Trivedi solution with various values of q0 , the dimensionless parameter which indicates the relative magnitude of the interface kinetics effect for needle, and r=rc vs. V0 for the same values of q0 . q0 ¼ 2q where q is defined by Equation 3.207; the doubled value of q0 arises because the critical radius of a needle is twice that for a plate as a result of the 2=r instead of 1=r contribution to needle capillarity. Both behaviors are seen to be qualitatively similar to those for the plates. Figure 3.16 compares needle and plate lengthening kinetics in terms of the parameter G ‘ rc0 =2D, where rc0 is used for both so that this plot serves as a direct comparison of lengthening rates. Needles are seen to lengthen more rapidly under all circumstances. Qualitatively this obtains because the point effect of diffusion is even more favorable to the lengthening of the needles than of plates. At low supersaturations, the ratio of the two rates is seen to approach an order of magnitude. On this basis, one would expect to observe needles more often than plates. This expectation is fulfilled 10.0
Needle growth Ivanstov 1 2 3 4
p
1.0
0.1
0.01
Curve 1 2 3 4 0
0.4
0.6
0.8
q΄ ∞ 2.0 0.2 0.02 1.0
Ω0
FIGURE 3.14 Variation of the Peclet number with the normalized supersaturation V0, which corresponds to the maximum growth rate for different values of the parameter q0 . (With kind permission from Springer Science þ Business Media: From Trivedi, R., Metall. Trans., 1, 921, 1970.)
289
Diffusional Growth 103 Needle growth
102 r/rc
q΄= 0.02
q΄= 0.2
101
q΄= 2.0 q΄= ∞ 100 0
0.2
0.4
0.6
0.8
1.0
Ω0
FIGURE 3.15 Variation of r=rc0 , which corresponds to the maximum growth rate with the normalized supersaturation V0 for different values of the parameter q0 . (With kind permission from Springer Science þ Business Media: From Trivedi, R., Metall. Trans., 1, 921, 1970.) 10.0 Needle Plate 1
Curve q΄ ∞ 1 2 2.0 3 0.2
2
Gi r΄c/ 2D
1.0
0.1
0.01
3
0
0.2
0.4
0.6
0.8
1.0
Ω0
FIGURE 3.16 Plot of Vi rc0 =2D vs. V0 for plate and needle growth. The subscript i stands for a plate or a needle. (With kind permission from Springer Science þ Business Media: From Trivedi, R., Metall. Trans., 1, 921, 1970.)
during solidification, but not during solid–solid phase transformations. Trivedi advances two arguments to explain this disagreement with theory. One is that the volume strain energy associated with a plate is lower. However, as shown in Figure 2.16 when the shear modulus of the precipitate is greater than that of the matrix, under the condition of isotropic elasticity in both phases the volume strain energy associated with a plate is much higher than that with a needle. The other derives from the circumstance that because xba b,r is smaller for needles than for plates, thereby resulting in a smaller driving force for growth; when diffusivity increases with solute concentration in the matrix phase the smaller xba b,r associated with the needles will further diminish the advantage in respect to diffusion geometry which they enjoy. In aAl–Cu alloys, however (to quote only one example), the diffusion coefficient is approximately
290
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
independent of Cu concentration and yet two transition phases u00 and u0 form as plates with very small aspect ratios. The equilibrium u phase, on the other hand, forms as plates, rods (tantamount in the present context to needles) and idiomorphs (roughly intragranular shapes). As discussed elsewhere, it seems more likely that when one crystalline solid forms within another, the determinant of morphology of anisotropic precipitates developed in the interiors of matrix grains is the number of different orientations at which crystallographic barriers to growth develop. When a given barrier is one dimensional, it becomes possible to have this barrier appear in cylindrically symmetric form provided that a planar or two-dimensional barrier does not develop and interfere at another orientation. In the case of u, many different, though discrete, lattice orientation relationships develop; this provides a basis for explaining the formation of many different morphologies. 3.3.2.1.8 Effects of a Composition-Dependent Diffusivity When the diffusivity in the matrix phase, D, varies with composition, one uses the original form of the Fick’s second law, qx ¼ r (Drx) qt For unidirectional mass transport Equation 3.214 is specialized to qx q qx ¼ D qt qs qs
(3:214)
(3:215)
where s is the distance. An obvious way in which to avoid the additional mathematical complexities introduced by a composition-dependent D is to select a single value of D, which might provide a numerically adequate representation of the variation that D undergoes between the matrix composition in contact with the moving interphase boundary and the matrix composition prior to the transformation in case that the diffusion fields of adjacent precipitate crystals do not perceptibly overlap. The Wagner approximation used by KRC to treat a composition-dependent diffusivity of ba carbon in austenite during the lengthening of the ferrite plates uses the diffusivity at xba b or xb,r on the ground, that qx=qs or qx=qr is calculated at this composition. This approximation, however, cannot be a very good one since the concentration-penetration curve in the matrix in front of a growing precipitate crystal is a single interactive entity. Events happening in one portion of the curve necessarily affect all others. Hence, if a single value of D is to be utilized, this value should be affected by the variation of D with s throughout the entire range of the concentration-penetration curve. Trivedi and Pound [24] used a weighted average diffusivity of carbon in austenite during a treatment of ferrite plate lengthening kinetics, xba b
D¼
Ð
D(x)dx
xb xba b
xb
:
(3:216)
Atkinson [35], on the other hand, prefers a second moment type of weighted average diffusivity in which, the emphasis is given to the diffusivities in the vicinity of the interphase boundary that is somewhat reduced, xba b
Ð
xb
D¼
D(x)(x xb ) xba b xb
2 dx:
(3:217)
291
Diffusional Growth
Trivedi and Pound [24] evaluated the lengthening rate of the ferrite plates using a KRC-like approach. The maximization of the growth rate incorporated adjustment of the austenite:ferrite boundary compositions that were calculated on the basis of the KRC treatment of the thermodynamics of austenite=ferrite equilibria (see Section 1.3), the use of the KRC equation for the variation of DgC with x (Equation 3.163) and a Zener–Hillert type equation for G ‘ , but with the diffusion distance in austenite estimated on the basis of a hemicylindrical plate edge. When DgC was assumed to be constant, Equation 3.163 was used; DgC was successively assumed to be that corresponding to g Equation 3.216, to xb and to xba b,r :G ‘ was also computed for a variable DC . In this situation, an exponential variation of the diffusivity with carbon content in the austenite was utilized and the diffusion equation was solved by means of the finite difference method. Figure 3.17 shows the variation of G ‘ with temperature in Fe-0.24 wt.% C, computed on all four bases. The variable D results, denoted ‘‘D ¼ f (c),’’ were calculated for only a limited range of temperatures because of the large amounts of computer time required. These are shown with a dashed curve. Comparison among the four curves shows that the weighted average D yields growth rates nearly the same as those obtained from the composition-dependent diffusivity growth rates. On the other hand, assuming that D is constant and equal to that at either xb or xba b introduces substantial errors, particularly at low ba temperatures where xba b is high and D varies more rapidly with composition as xb is approached. Atkinson [35,36] has developed a more general approach to the problem of growth kinetics when D is composition dependent. He first applied it to the growth of an infinite planar boundary and then to the growth of spheres, cylinders, and ellipsoids. Additional details of the method, with application of both the planar and the oblate ellipsoidal analyses to the growth kinetics of ferrite have been reported by Atkinson et al. [15]. The basic mathematical apparatus is sketched here for the infinite plane case. Carrying out the indicated differentiation in Equation 3.215, qx q2 x qx qD ¼D 2þ qt qS qS qS
(3:218)
where s has been replaced by S to denote a fixed coordinate reference system, as in Section 3.3.2.1.6. A transformation from fixed to moving Cartesian coordinates is then made through a relationship that is basically the same as Equation 3.167a, s ¼ S a(DI t)1=2
(3:219)
800 700
D = f(c)
Temperature (°C)
600 500 D(c∞) 400 Dav 300 10–8
10–7
10–6
D(cβγ)
10–5 10–4 Gl (cm/s)
10–3
10–2
10–1
FIGURE 3.17 Temperature dependence of growth rate G ‘ calculated incorporating concentration-dependent diffusivity. (From Trivedi, R. and Pound, G.M., J. Appl. Phys., 38, 3569, 1967; J. Appl. Phys., 40, 4293 1969. American Institute of Physics. With permission.)
292
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
where DI is the value of D at xba b and a is the rate constant. Conversion to a reduced variable, so that x is a function of only this variable rather than of s and t, is then accomplished through the substitution, s h ¼ pffiffiffiffiffiffiffi 2 DI t
This leads to
(3:220)
dx d D dx ¼ (2h þ a) dh dh DI dh
(3:221)
Multiplying by dh=dx and integrating yields D dx ¼ DI dh
ðx
xb
(2h þ a)dx
(3:222)
The flux balance equation is
dx ba x xab a ¼ a b dh h¼0
(3:223)
Solution of Equation 3.222 is then obtained, subject to the boundary conditions, x ¼ xb x ¼ xba b
at at
h¼1
(3:224a)
h¼0
(3:224b)
by a numerical technique based upon the finite difference method. Although this computerized technique is a little easier to use when an equation can be written for D as a function of x, this is not necessary. As long as data on D vs. x are available this technique can be applied. 3.3.2.1.9 Morphological Instability of a Disordered Interphase Boundary Ham [37] has made highly sophisticated analyses of this problem and has concluded that, as long as growth is controlled purely by volume diffusion, once a precipitate morphology is established it cannot be changed from diffusion considerations alone. For example, the aspect ratio of an ellipsoid of revolution will remain constant during growth. Suppose, however, that a perturbation appears, for whatever reason, in a disordered interphase boundary during growth. Bear in mind that, during growth, interfacial energy minimization per se will exert but a minor driving force to eliminate the perturbation when the driving force for growth is large relative to that for coarsening. For convenience, let the perturbation take the form of a sine wave as illustrated in Figure 3.18. Because of the point effect of
β Unperturbed interface α
FIGURE 3.18
Interface perturbed in the form of a sine wave.
293
Diffusional Growth
diffusion, the protuberances will tend to grow more rapidly than the other areas of the boundary. However, because of capillarity, growth of the protuberances will tend to be restrained, thereby allowing the other areas of the boundary the opportunity to ‘‘catch up.’’ The question thus arises: under what conditions will the protuberances be able to evolve without limit? Mullins and Sekerka [38] have elegantly investigated this question. Shewmon [39] has summarized the essentials of their calculations and extended them to include the effects of surface diffusion, interfacial reaction control, strain energy, and third elements. The approach of Shewmon will be followed here. Assume the y-axis to be normal to the unperturbed boundary, the x-axis to lie in the plane of the unperturbed boundary and also the atomic fraction of the solute to be c. The flux balance equation for this boundary is G¼
D ba cab a cb
qc qy y¼0
(3:225)
where the a precipitate is taken to be solute rich and a moving coordinate system is used (hence y ¼ 0 lies at the boundary). Now superimpose a sinusoidal perturbation on the moving a:b boundary of the form, y ¼ e sin (wx)
(3:226)
where e is the amplitude of the perturbation (which may vary with time), l ¼ 2p=v is the wave length, and v is the wave number. This interface gives rise to a variation in solute concentration in the matrix in contact with the interface as a function of x due to capillarity. At a particular position along the interface, gag V a ba K ffi c 1 þ cba b,r b RT
(3:227)
where the assumption of a solute-rich precipitate and cba b 1 permits omission of the concentration ratio in the capillarity term. The curvature of the interface, K, is expressed as, d2 y dx2 K¼" 2 #3=2 dy 1þ dx
(3:228)
When the amplitude of the perturbation is small, so also is dy=dx and thus K ffi d2y=dx2. From Equation 3.226, dy=dx ¼ evcosvx and hence d2y=dx2 ¼ ev2sinvx. Therefore, Equation 3.227 may be rewritten as ba 2 cba b,r ffi cb 1 þ Gev sin (vx)
(3:229)
ab ba
1, the diffusion field can be considered where G ¼ gabV a=RT. When cb cba b = ca c b invariant and Fick’s second law can be safely replaced by the Laplace equation, r2c ¼ 0, as solved for a particular shape and with Equation 3.229 as a boundary condition. This approach also requires that l < 2(Dt)1=2, the mean diffusion distance, thereby permitting interaction between the protuberances and the adjacent depressions in the interface.
294
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
The general solution to the Laplace equation for a sinusoidal interface in the presence of a given solute gradient, g, in b at the interface is c ¼ A þ Bevy e sin (vx) þ gy
(3:230)
Expanding evy as 1 vy and taking the e2 term later encountered is insignificantly small, c ¼ A þ B(1 vy)e sin (vx) þ gy ffi A þ (B þ g)e sin (vx)
(3:231)
To evaluate A and B, boundary conditions are utilized. When l ¼ 1, v ¼ 0 this situation obtains at ba c ¼ cba b,r¼1 ¼ cb , cba b ¼ A
(3:232)
For a finite wavelength and at y ¼ 0, Equation 3.229 is substituted into Equation 3.231, ba ba ba 2 2 cba b,r ffi cb þ cb Gev sin (vx) ¼ cb þ (B þ g)ev sin (vx)
(3:233a)
2 B ¼ cba b þv g
(3:233b)
Thus, yielding,
Substituting Equations 3.232 and 3.233b into Equation 3.230, ba 2 c ¼ cba þ c Gv g e sin (vx)evy þ gy b b
(3:234)
Differentiating Equation 3.234 with respect to y at y ¼ 0 and substituting into Equation 3.225, G¼
D ba cab a cb
h
i 3 vy g þ gv cba b Gv e sin (vx)e
(3:235)
The velocity of the mean position of the interface, G 0 , is that at which l ¼ 1 or v ¼ 0, G0 ¼
D ba cab a cb
g
(3:236)
The maximum velocity of the interface, G max, is that at which sin(vx) ¼ l, G max ¼
D cab a
cba b
h
i 3 g þ gv cba b Gv e
(3:237)
Hence, the velocity or growth rate of the peak or maximum amplitude relative to the mean position of the interface is given by Equation 3.237 minus Equation 3.236, e_ ¼ G max G 0 ¼
D ba cab a cb
2 gv cba Gv e b
(3:238)
295
Diffusional Growth
Expressing the peak growth rate in reduced form, as the ratio of this velocity to the mean velocity of the interface Equation 3.236, ! 2 cba e_ b Gv ve (3:239) ¼ 1 g G0 This equation may be physically interpreted as follows. The first term on the r.h.s. represents a gradient of G þ G ve ¼ G(1 þ ve), as may be better seen from Equation 3.237 and as viewed on an absolute rather than on a relative basis. This gradient is applicable at the protuberances. In the depressions or valleys of the interface, the gradient becomes G(1 ve). The resulting variation in the gradient along the interface increases the amplitude, e, of the sine wave with time. Conversely, the second term on the r.h.s. of Equation 3.239 arises from the curvature variation along the interface and acts to cause solute transport that will smooth out the sine curve. These two effects are equal when the term in parenthesis in Equation 3.239 is zero, that is, when, v v0 ¼
g cba b G
!1=2
(3:240)
When v > vc, e_ < 0 and protuberances disappear. When v < v0, e_ > 0 and protuberances can grow. Following the argument that Mullins [40] used in the analysis of the various possible paths operative during thermal grooving, Shewmon wrote by analogy an equivalent generalization of Equation 3.238, ! " # ab Db D c D d a ab ba a gv cb Gv3 1 þ v4 G e (3:241) e_ ¼ ab Db Db cba ca cba b b The second term in square brackets includes a term in parentheses that takes account of the possibility that the product cD, where D is the volume diffusivity in the specified phase and c is the solute concentration in contact with the a:b boundary, is comparable in the two phases, thus requiring that diffusion in both be considered. The third term in square brackets includes the product of the interphase boundary diffusivity, Dab and the thickness of the boundary, d, and thus represent the contribution of interphase boundary diffusion to interface stability. Like the ev3 term derived from volume diffusion, the ev4 term inhibits protuberance evolution by transporting solute, along the a:b boundary, so as to smooth out protuberances that are attempting to develop. In Equation 3.241, the D’s, c’s and G are fixed. g, however, approaches infinity as t approaches zero, and conversely approaches zero as t goes to infinity. On Equation 3.240, v0 will thus approach infinity at short times; since l0 ¼ 2pv0, l0 will approach zero in this situation. Recalling that the protuberances will grow when v < v0, they will thus do so when l > l0. Thus, any protuberance whose radius exceeds the critical radius will at some stage of growth be able to undergo preferential development in the style of Equations 3.238 and 3.239. Further, at a particular value of v, that is, of l, the growth rate of the protuberance relative to the mean growth rate will bepa maximum. Designating such values as vm and lm, when volume diffusion is dominant vm ¼ v0= 3 and when interphase boundary diffusion is dominant vm ¼ v0=41=3. These results are obtained as follows. To secure a further developed expression for v0, though assuming a negligible contribution from interphase boundary diffusion, thus ignoring the third term in the square brackets of Equation 3.241, set the first and second terms in the square bracketed group equal, as in the derivation of Equation 3.240 one obtains !1=2 g (3:242) v0 ¼ cba b GA
296
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
where A¼1þ
Da cab a Db cba b
(3:243)
Next, combine Equations 3.241 and 3.236 under the same assumption, 3 cba e_ b Gv ¼v g G0e
(3:244)
Taking derivatives with respect to v,
e_ q G0e qv
¼0¼1
2 3cba b Gv g
(3:245)
Hence,
vm ¼
g 3cba b G
!1=2
(3:246)
Combining Equations 3.242 and 3.246, v0 vm ¼ pffiffiffi 3
(3:247)
Taking the second derivative of e_ =G 0 e yields a negative value and thus demonstrates that Equation 3.246 obtained from the first differentiation, is indeed the maximum. The same type of procedure is employed to show that vm ¼ v0=41=3 in the situation where interphase boundary diffusion predominates. The circumstances under which the protuberances are viable will now be deduced. An approximate upper limit is first developed. The extent of solute diffusion in the matrix in time t is approximated by 2(Dt)1=2. Hence, when l > 2(Dt)1=2 capillarity will still tend to smooth out the protuberances but the enhanced solute concentration gradient in the matrix, between G(1 þ ve) in front of the protuberances and G(1 ve) in front of the valleys, will be ‘‘disconnected.’’ Hence, e_ =G 0 < 1 and the protuberances will not be amplified. At the lower limit, to be safe assume that protuberance growth does not occur unless l > 2l0 instead of the usual l0; hence the lower limit is l > 4p=v0. The limits within which protuberance growth takes place are thus,
Substituting Equation 3.240,
pffiffiffiffiffi 4p 2 Dt > l > v0 pffiffiffiffiffi cba b 2 Dt > 4p g
!1=2
(3:248)
(3:249)
Using Zener’s linearized gradient treatment for the concentration gradient in front of a planar, ba ab disordered boundary and assuming that cba b cb and cb ca , a simple approximation will now
297
Diffusional Growth
be developed for cba b =g. In Zenerian terms, G ¼ Dx=DS. First rewriting Equation 3.5 in terms of DS and then substituting Equations 3.8 and 3.9 for S, with appropriate reversals in the order of terms, because a solute rich precipitate is now considered. Substituting this approximation into Equation 3.249, squaring both sides of the equation and rearranging, 2S cab a cb
1=2 2(Dt)1=2 cab a cb ¼ Ds ¼ 1=2 ba cba b cb cab a cb 1=2 1=2 ba cab cb cba a cb b Dx g¼ ¼ 1=2 DS 2(Dt)1=2 cab a cb
(3:250)
! 1=2 ba 1=2 cba c c 2(Dt)1=2 cab b b b a 1=2 ¼ 2(Dt)1=2 1=2 2(Dt) g cb ab ba ba cb cb ca cb Thus, pffiffiffiffiffi Dt ¼ 8p2 G
(3:251)
pffiffiffiffiffi Dt 105 cm
(3:252)
Taking typical values for the components, that is, gab ¼ 500 ergs=cm2 ( ¼ 500 mJ=m2), Va ¼ 10 cm3=mol and T ¼ 1000K, G ffi l07. Hence,
Hence a planar, disordered interphase boundary whose growth is volume diffusion–controlled is stable as long as the mean diffusion distance is less than approximately 105 cm ¼ l07 m ¼ l01 mm. Typical precipitates contained within a thin foil suitable for TEM should be stable against protuberance formation. As indicated by Equation 3.248, however, protuberances will be able to develop in larger precipitates with wavelengths of up to 2(Dt)1=2. 3.3.2.1.10 Dissolution of Planar and Spherical Particles In this section, the mathematics of dissolution is described, though it was not included in the original notes. The dissolution process is of practical importance because one often needs to know the appropriate heat treatment required for homogenization (or reversion) of aged alloys. Conversely, one needs to prolong the precipitate life to maintain mechanical properties during high temperature services of alloys. Analytical solutions to the diffusion equation for dissolving particles are limited compared to those for growth because of non-zero initial radius. Hence, approximation is commonly employed [16,41]. 3.3.2.1.10.1 Planar Precipitates The precipitate is assumed to be initially in equilibrium with the depleted matrix. We use the same notation as used in Section 3.3.3, see Figures 3.9 and 3.10. The alloy is aged at a temperature TG in the (a þ u) two-phase region and is raised to a temperature TD in the a single phase region, see Figure 3.19. The diffusion field of a dissolving u precipitate is shown schematically in Figure 3.20. The flux balance equation for dissolution is identical to that of growth, that is, dS qc (cP cI ) ¼ D dt qs s¼S
(3:253)
298
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Temperature
θ
α TD α+θ TG cM
cI
cP Concentration
FIGURE 3.19 Binary phase diagram for dissolution at temperature TD of particles initially in equilibrium with the matrix at temperature TG. (With kind permission from Springer Science þ Business Media: From Aaron, H.B. and Kotler, G.R., Metall. Trans., 2, 393, 1971.)
Concentration
cP
t=0
cI
cM S
S0 Distance
FIGURE 3.20 Solute concentration profile around a dissolving precipitate. (With kind permission from Springer Science þ Business Media: From Aaron, H.B. and Kotler, G.R., Metall. Trans., 2, 393, 1971.)
In case of the planar precipitate an analytical solution for the concentration penetration curve can be obtained as, s S0 erfc pffiffiffiffiffi 2 Dt (3:254) c cM ¼ (cI cM ) erfc( l) where
2 erfc(l) ¼ 1 erf (l) ¼ 1 pffiffiffiffi p
l ð 0
e
u2
2 du ¼ 1 þ pffiffiffiffi p
ð0
l
2
eu du ¼ 1 þ erf (l)
(3:255)
299
Diffusional Growth
And l is the growth rate constant, l satisfies the transcendental equation, pffiffiffiffi l2 cI cM ple erfc(l) ¼ cP cI
(3:256)
Note that the r.h.s. of this equation has a value from zero to infinity for dissolution and the sign is opposite to supersaturation for growth that takes a value from zero to unity. The precipitate size varies with time as pffiffiffiffiffi (3:257) S ¼ S0 2l Dt Following the same procedure as for growth, the parabolic growth rate constant can be obtained by the linearized gradient approximation [42], see Figure 3.21. The extent of diffusion field, DS, can be calculated from the equation, 1 (cI cM )DS ¼ (S0 S)(cP cM ) 2
(3:258)
Noting that the concentration gradient is equal to (cI cM)=DS, Equation 3.253 becomes (S0 S)
dS D (cI cM )2 ¼ dt 2 (cP cI )(cP cM )
(3:259)
Integrating, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dt S ¼ S0 (cI cM ) (cP cI )(cP cM )
(3:260)
The readers are referred to Ref. [16] for the solution of stationary-interface approximation for dissolution of a planar precipitate. 3.3.2.1.10.2 Spherical Precipitates Analytical solution is not available for dissolution of a spherical precipitate. The linearized gradient approximation was applied to spherical precipitates by Aaron [42]. Here, we start with the diffusion equation for spherically symmetric growth, Equation 3.36. Let u ¼ cr
(3:261)
Concentration
cP
t=0
A1
cI A2
cM
ΔS S S0 Distance
FIGURE 3.21
Linearized gradient approximation for dissolving particles.
300
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
This converts Equation 3.36 to an equation for linear flow in one dimension, qu q2 u ¼D 2 qt qr
(3:262)
In the invariant-field approximation the term qu=qt ¼ 0 in Equation 3.262. The solution to the Laplace equation is c cM ¼ (cI cM )
R r
(3:263)
where R is the radius of precipitate [16]. Differentiation yields, qc R cI cM ¼ (cI cM ) 2 ¼ R qr r r¼R
(3:264)
Substituting Equation 3.264 into Equation 3.253 with replacement of S by R, (cP cI )
dR cI cM ¼ D cP c I dt
(3:265)
Integrating, R2 ¼ R20 2D
cI cM t cP cI
(3:266)
where R0 is the initial radius of the precipitate. In a stationary-interface approximation [43], the diffusion field around the precipitate is assumed to be the same as that which would exist if the precipitate–matrix interface had been fixed there from the beginning. Thus, when u is defined to be (c cM )r, the boundary conditions are* u ¼ (cI cM )R u¼0
for
at
r¼R
r!1
(3:267) (3:268)
The solution to Equation 3.262 is u ¼ A þ B erfc
rR pffiffiffiffiffi 2 Dt
(3:269)
From Equations 3.267 and 3.268 one can determine A and B, and Equation 3.269 becomes R rR c cM ¼ (cI cM ) erfc pffiffiffiffiffi (3:270) r 2 Dt Differentiation yields
qc 1 1 p ffiffiffiffiffiffiffiffi ffi ¼ D(c c ) I M qr r¼R R pDt * u is defined by u ¼ (c cM )r.
(3:271)
301
Diffusional Growth
Substituting into Equation 3.253, " rffiffiffiffiffi# dR cI cM D D ¼ þ cP cI R dt pt
(3:272)
According to Whelan [43], this equation is integrated as follows. First, reduced variables of R and t are defined R R0
(3:273)
2Dt cI cM R20 cP cI
(3:274)
y¼ t¼
With another dimensionless variable p and w defined by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 cI cM p¼ 2p cP cM
(3:275)
and y2 ¼ w 2 t Equation 3.272 becomes dy 1 p ¼ pffiffiffi dt 2y t
Multiplying 2y,
(3:276)
d(w2 t) dw ¼ w2 þ 2wt ¼ 1 2pw dt dt
(3:277)
2wdw dt ¼ w2 þ 2pw þ 1 t
(3:278)
Hence,
The l.h.s. is integrated as ð ð ð 2wdw 2(w þ p)dw 2pdw ¼ (w þ p)2 þ 1 p2 (w þ p)2 þ 1 p2 (w þ p)2 þ 1 p2
2
¼ ln (w þ p) þ 1 p
2
! 2p wþp 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi tan 1 p2 1 p2
(3:279)
The r.h.s. is integrated as ln t þ const. Since t(w þ p)2 þ t(1 p2 ) ¼ tw2 þ 2twp þ t, Equation 3.279 becomes, pffiffiffi 2p ln y2 þ 2py t þ t pffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 1 p2
wþp pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p2
!
¼ const:
(3:280)
302
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
When t ! 0, y ¼ 1 (hence, w ! 1) and tan
1
Thus,
wþp pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p2
!
!
p 2
2p p const: ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p2 2
(3:281)
In view of the relationship, tan (a p=2) ¼ cota, substitution into Equation 3.280 yields 1 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 pffiffiffiffiffiffiffiffiffiffiffi 2p B 1p C (3:282) ln y þ 2py t þ t ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi tan1 @ y A 1 p2 pffiffiffi þ p t Figure 3.22 shows the variation of the square of the precipitate radius with the dimensionless time for a few values of supersaturation. For u precipitates in Al-4wt.% Cu, p 0.04.
3.3.2.1.10.3 Influence of Curvature on the Dissolution Kinetics In the foregoing analysis, the curvature of precipitate particle is not taken into account. It is expected that the curvature increases the solute concentration in the matrix at the precipitate=matrix interface. Hence, the concentration gradient increases and thereby dissolution of particles is expedited. This effect may become progressively larger as the precipitate radius decreases. Aaron and Kotler [44] incorporated this in the stationary-interface approximation by Whelan [43]. The solute concentration in the matrix at the curved interface, crI varies with particle radius, r, as crI
2gau Vu , ¼ cI exp RTrcP
(3:283)
1.0
(R/R0)2
0.8 0.6
P= 0.4
0
0.
04
0.2
0
P=
0.2
P=
0.2
0.4
0.6
0.8
1.10
q2t 2
R0
FIGURE 3.22 Variation of the radius of dissolving particle with time calculated from the stationary-interface approximation. (From Whelan, M.J., Met. Sci., 3, 95, 1969 (online at www.maney.co.uk=journals=mst and www.ingestaconnect.com=content=maney=mst). With permission from Maney Publishing.)
303
Diffusional Growth 1.0 0.7 0.5
1.0
0.3
0.547 0.447
0.2 R0
R(t)
2
β = 10–6
β=
0.03 0.02
P = 7.2 × 10–4
0.01 10–3
2 3
5 × 10–8
5 × 10–7 10–7
0.1 0.07 0.05
2γαθVθ RTCθ
5 7 10–2
2 3
10–8 0
0.316 in cm
R(t) in μm
0.706
0.224 0.173
5 7 10–1
0.1 2 3
5 7 1.0
τ = α2t/R2o
FIGURE 3.23 Variation of the radius of dissolving particle with time calculated from the stationary-interface approximation incorporating surface curvature. (From Aaron, H.B. and Kotler, G.R., Met. Sci., 4, 222, 1970. American Institute of Physics. With permission.)
where cI is the solute concentration at the flat precipitate=matrix interface, Vb is the molar volume of the precipitate phase. This equation can be obtained from Equation 1.79. Substituting crI for cI, Equation 3.253 becomes G rffiffiffiffiffi! cI exp cM dR D D r (3:284) ¼ þ G dt R pt cP cI exp r where G¼
2gau Vu RTrcP
The effects of curvature on the dissolution kinetics vary considerably with p and G values, and are larger at a smaller p value. Results at p ¼ 7.2 104 are shown for various G values in Figure 3.23. The effects are, however, rather small for p ¼ 0.088 and 0 G 5 106 cm which are typical of u precipitates in Al-4wt.% Cu which have grown below 2008C and dissolved at temperatures less than 5508C. It was shown that stationary-interface approximation yields a considerable amount of error because at the end of dissolution the dissolution kinetics become progressively faster [45]. 3.3.2.2
Comparisons with Experiment
3.3.2.2.1 Introduction These comparisons are effected by comparing the growth kinetics of a disordered interphase boundary of a well-defined shape, whose diffusion field in the matrix phase minimally overlaps that associated with other interphase boundaries, with the growth kinetics calculated from the parabolic rate constant equation for that shape as presented in one of the preceding sections. At least three different approaches to securing such boundaries come to mind. Perhaps, others will be devised in the future: 1. ‘‘Synthetic interphase boundaries’’ usually of approximately planar shape. They are present within the diffusion couples produced by welding, large-scale mass transport of solute into or out of an initially homogeneous single-phase alloy, vapor deposition upon a specimen of such an alloy, etc. Each phase may be monocrystalline or polycrystalline. Avoidance of the
304
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
nucleation step, such as would be present in the vapor deposition case, is probably essential if a large area of the disordered boundary is to be secured. Welding of mono- or polycrystalline a to equivalent b is the obvious way to secure a disordered-type boundary since lattice orientation relationships can be arbitrarily predetermined. The mass transport approach, for example, carburization or decarburization of an Fe–C alloy in the (g þ a) region, will doubtless be troubled by the low energy orientation relationships established during nucleation, if both phases are monocrystalline. However, growth of an ferrite or an austenite layer of a polycrystalline structure into the similarly structured companion phase may permit this orientation relationship problem to be surmounted. 2. ‘‘Recrystallized interphase boundaries’’ produced by the method of Smith [46]. These interfaces are the most likely to be disordered. The experiment would be conducted by forming the two-phase structure at one temperature in the (a þ b) region and then abruptly raising or lowering the temperature and following the progress of growth or dissolution at the second temperature. Morphologies made available in this manner are the sphere in the interiors of the matrix grains and double spherical caps at grain boundaries, approximating oblate ellipsoids. This technique is yet to be utilized. 3. ‘‘As-transformed interphase boundaries.’’ These are the interfaces enclosing precipitates developed at grain boundaries and grain interiors during isothermal transformation. The ‘‘oriented, transformable bicrystals’’ have been grown and successfully used. Further by careful control of specimen size and shape and heat treatment can produce, grain boundaries perpendicular to the specimen surface and sufficiently widely spaced and few precipitates at the grain boundaries or in the grain interiors, in conventional polycrystalline specimens, so that each precipitate crystal may be safely considered to be growing in isolation from its neighbors. The principal drawback to this conventional approach to the problem of measuring the growth kinetics of disordered interphase boundaries is that minimization of DG* during nucleation appears to limit seriously the proportion of a:b boundaries, which have this structure even during the growth process. In the next section, the principal types of precipitate morphology observed will be described and the possibilities that they offer for the measurement of the growth kinetics of partially (or fully) coherent as well as of disordered interphase boundaries will be discussed. 3.3.2.2.2 Dubé Morphological Classification System The references for this classification system are Dubé [7], Dubé et al. [47], and Aaronson [48]. Dubé originally devised this system for the morphologies of proeutectoid ferrite formed from austenite in plain carbon steels. Subsequent research showed that this system is applicable to single-phase precipitation in a very wide variety of alloy systems, independently of the crystal structures of the matrix and precipitate phases (as long as they are significantly different), the type of solid solution (interstitial or substitutional) involved, the predominant type of interatomic bonding operative in each phase, and the identities of the solute and solvent atoms. Figure 3.24 shows schematically the principal components of the Dubé system. (a) Grain boundary allotriomorphs: crystals that nucleate at grain boundaries in the matrix phase and grow preferentially and more or less smoothly along them. (b) Widmanstätten sideplates or sideneedles develop into the interior of a matrix grain from the vicinity of the matrix grain boundaries: b1, primary sideplates that grow directly from grain boundaries; b2, secondary sideplates that evolve from crystals of a different morphology, usually grain boundary allotriomorphs, though sometimes primary or second Widmanstätten saw-teeth. (c) Widmanstätten saw-teeth have a triangular crosssection in the plane of polish and develop from the region of the grain boundaries: c1, primary sawteeth that grow directly from the grain boundaries; c2, secondary saw-teeth: develop from grain boundary allotriomorphs. (d) Idiomorphs are roughly equiaxed crystals: d1, Intragranular idiomorphs that develop in the interiors of the matrix grains; d2, grain boundary idiomorphs growing
305
Diffusional Growth
(a) (1)
(2)
(b) (1)
(2)
(c) (1)
(2)
(d) (e)
(f )
FIGURE 3.24 Dubé morphological classification system. (From Dubé, C.A., PhD thesis, Carnegie Tech, Pittsburgh, PA, 1948; Dubé, C.A. et al., Rev. Met., 55, 201, 1958. With permission.)
directly from grain boundaries. (e) Intragranular plates or intragranular needles: form in the interiors of the matrix grains. (f) Massive structure that is composed of impinged crystals of other morphologies, with grain growth within the impinged aggregate often further complicating the situation; a bulky, roughly equiaxed, and polycrystalline mass results. This is not a fundamental morphology. The term used to describe it, though apt in an immediate sense, invites confusion with the product of the massive transformation.* 3.3.2.2.3 Problem of Selecting Suitable Alloy Systems and Phase Transformations The discussions that have been presented on the theories of solid–solid nucleation and volume diffusion–controlled growth make clear that comparisons between the theory and experiment for growth, and particularly for nucleation, must be conducted on simple alloys if the necessary ancillary data are available and if serious perturbing effects from multiple alloying elements and=or trace impurities is minimized. In the case of growth, accurate calculation of and=or experimental data on phase boundary compositions is basic to the use of the relationships for the parabolic rate constant. This is far more readily done in binary alloys than in higher order systems. The appropriate diffusivity must also be accurately known. During growth, when volume diffusion controls the process in a substitutional alloy, the chemical interdiffusion coefficient is appropriate; the Boltzmann–Matano diffusivity is correct when only an interstitial species is mobile. Either these data or self-diffusivities (whose use will not necessarily introduce a large error) are often available for binary alloys or at least can be measured with adequate accuracy without undue labor. In ternary systems on the other hand, both diffusivity and phase diagram data are usually lacking. Hence, most of the comparisons to be presented will be taken from studies performed on binary alloy systems. Further, the overwhelmingly * Kral and Spanos (Metall. Mater. Trans. A, 36A, 1199, 2005) proposed a revised classification system based on threedimensionally reconstructed morphology of proeutectoid ferrites in iron alloys.
306
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
superior knowledge of morphology, overall reaction kinetics and especially of the ancillary parameters for the proeutectoid ferrite reaction in Fe–C alloys has made this transformation and alloy system the combination of choice for most experimental studies of this type. 3.3.2.2.4 Planar, Disordered Boundaries of Effectively Infinite Extent As will be shown in Section 3.3.2.2.5, grain boundary allotriomorphs have, at least in the proeutectoid ferrite reaction, too high an aspect ratio to permit their broad faces to be modeled as infinite, planar boundaries. Hence, a resort is made to ‘‘synthetic interphase boundaries,’’ also of the austenite: ferrite type in Fe–C alloys, of which the portion of the phase diagram is shown in Figure 3.25 [49]. Purdy and Kirkaldy [50] reported the described experiment. They utilized synthetic austenite:ferrite boundaries for two reasons. One is to avoid the stereological problems involved in measuring lengthening and thickening kinetics of grain boundary allotriomorphs. The other is to destroy specific, low-energy lattice orientation relationships between austenite and ferrite by utilizing a polycrystalline layer of ferrite. This layer is developed completely across a diffusion couple and grown through a polycrystalline austenite matrix until the particular austenite grain through which a given ferrite crystal is growing is many austenite grains away from the austenite grain in which the ferrite crystal was nucleated. A high-purity Fe-0.57 wt.% C alloy was subjected to a brief, partial decarburization treatment in wet hydrogen at 9208C. The H2 atmosphere was then replaced by that of argon, that is, a neutral or non-decarburizing atmosphere and the specimen was slowly cooled at 208C=min to 7558C. As shown in the Fe-Fe3C phase diagram in Figure 3.25, the original composition of the alloy lies barely within the austenite region. The decarburized volumes of the specimen lie within the (g þ a) or the pure ferrite region. Since the specimen surfaces should have essentially zero carbon, during cooling from the austenite region the volume free energy change for ferrite nucleation should first become negative at the specimen surfaces. Nucleation will likely begin where the austenite grain boundaries meet the surfaces. Although in the absence of decarburization ferrite neither nucleates nor grows preferentially on the surfaces of Fe–C specimens, the large relative driving force for both processes in the immediate vicinity of the surfaces encourages growth preferentially along these
940
912°
Temperature (°C)
900
(γ-Fe) Austenite
860 820 780 740
(α-Fe) Ferrite 0.0206
770° (Curie temperature)
0.68
0.0218
0.77
700 660 Fe 0.1 John Chipman
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Weight percent carbon
FIGURE 3.25 Fe-Fe3C phase diagram. (From ASM, Metals Handbook, 8th ed., ASM, Metals Park, OH, 1973, vol. 8, p. 276. Reprinted with permission of ASM International1. All rights reserved.)
307
Diffusional Growth
surfaces and quite possibly additional nucleation. A solid polycrystalline layer of ferrite rapidly develops along each decarburized surface. These layers then grow toward the interior of the specimens as the constantly diminishing specimen temperature provides a usable driving force for ferrite growth in progressively larger volumes of the specimen. The carbon diffusing away from the advancing austenite:ferrite boundary, however, discourages ferrite nucleation at austenite grain boundaries (or grain interiors) ahead of the boundary by diminishing the driving force for such nucleation. The slow cooling rate enables ferrite growth to keep up with the cooling down of the specimen sufficiently well so that little, if any, austenite outside of the carbon diffusion zone has sufficient driving force available for the nucleation and growth of new ferrite crystals. Once the specimen has reached 7558C, it is equilibrated at this temperature for 36 h, a time precalculated as sufficient to remove carbon concentration gradients remaining in both phases. The specimen is then quenched to room temperature. The resulting double diffusion couple is pictured in Figure 3.26. The thickness of the ferrite layers was measured metallographically. The growth normal to the specimen surfaces and the approximately planar character of the austenite:ferrite boundaries made these measurements accurate and ensured that apparent and true layer thicknesses were essentially identical. The specimen was then reheated many times at 7928C. At this temperature, the austenite grew back into the ferrite as shown in Figure 3.27. After each annealing, approximately 1 mm. was ground off the plane of the polish and the thickness of the ferrite layers was again measured (the material removed ensured that any change in carbon content during heat treatment would not be
FIGURE 3.26 Macrograph of a binary ferrite–austenite diffusion couple. 50. Enlarged approximately 9 pct for reproduction. (From Purdy, G.R. and Kirkaldy, J.S., Trans. TMS-AIME, 227, 1255, 1963. With permission.)
xγγα at 755°C xγγα at 792°C
x
α
xααγ
FIGURE 3.27
γ
s
Carbon concentration profile in the diffusion couple of Figure 3.26.
308
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
40
x΄ (μ)
30
Calculated Experimental, side 1 Experimental, side 2
20
10
0
1
2
3
4
5
6
7
√t (min1/2)
FIGURE 3.28 Growth of austenite into ferrite at 7928C in the double diffusion couple of Figure 3.26. (From Purdy, G.R. and Kirkaldy, J.S., Trans. TMS-AIME, 227, 1255, 1963. With permission.)
able to penetrate deep enough to affect the measured layer thickness). The plot in Figure 3.28 indicates that austenite:ferrite boundary migration occurred in accurately parabolic fashion. The slope of this plot, a1, is the parabolic rate constant. Since the details of the growth process here are a little different than those that are obtained during the growth of ferrite into austenite, Equation 3.29, leading to a slightly different transcendental equation for a1, a1 1 þ erf pffiffiffiffi a1 1 xg xga g 2 D
p ffiffiffiffi ga pffiffiffiffi ¼ where xg ¼ xga (3:285) ag g at 755 C a2 x x p 1 2 D a g 4D e
Calculation of a1 from this relationship using the Wagner approximation for D (which should not be much in error here because the range of carbon content encompassed is relatively small, and especially because the maximum carbon content involved is relatively low) yields 7.32 l05 cm=s1=2 in good agreement with the measured value of 7.1 l05 cm=s1=2. This result appears to be the best evidence to date that interphase boundaries that are likely to be of the disordered type exhibit the expected volume diffusion controlled growth kinetics. It would be desirable to extend these measurements to lower reaction temperatures so that a wider range of carbon contents could be encompassed. Why is a large extension of the temperature range below 7558C in this type of specimen likely to be infeasible? Over a long period, a number of studies have appeared in the literature on the migration kinetics of the essentially planar interphase boundaries in multiphase diffusion couples produced in binary alloy systems. When more than two phases are present, non-semi-infinite boundary conditions are inevitable, requiring the use of numerical methods to solve the applicable diffusion equations. Particularly when the solute is continuously added to the system through a free surface, significant volume changes occur during the diffusion process. In substitutional systems, the unequal diffusivities of the two species can complicate the diffusion process by giving rise to Kirkendall porosity and substructure formation. Variation of the interdiffusion coefficient with composition can be very rapid where a phase involved is a defect intermetallic compound with an appreciable existence range. The studies of Shankar and Seigle [51] on ‘‘pack aluminization’’ of Ni, in which Ni–Al solid solution (j), Ni3Al (e) and d phases (NiAl,
Diffusional Growth
309
a defect intermetallic compound with an existence range of up to approximately 25 at.%) form a good example of this research genre. Boundary migration kinetics appears to be those expected from volume diffusion control. A clear proof of departure from such kinetics would be given by interphase boundary compositions differing significantly from reliable phase diagram values. Eifert et al. [52] have reported that the concentrations of Al in the a(fcc) and b(bcc) phases at the a:b interface in diffusion couples formed between Cu and Cu-12.5wt.% Al are greater than those given by the phase diagram. However, Langer and Sekerka [53] note that the microprobe spot size used was too large to permit reliable extrapolations to the phase boundary composition applicable. Langer and Sekerka treated theoretically the problem of deviations from equilibrium interface compositions. Deviations from equilibrium were found to be proportional to an assumed interfacial resistance to transformation. A term proportional to the interface velocity and another term proportional to the flux of one atomic species through the interface contribute to the interface resistance. Powell and Schuhmann [54] have offered an alternate explanation for deviations from equilibrium interface compositions, based upon a dynamic equilibrium among A atoms, B atoms, and nonequilibrium vacancy concentrations. Although some slight deviations from equilibrium interface compositions are required in order to drive the countercurrent diffusion of solute and solvent across the interface, whether or not significant deviations are present at interphase boundaries which can be legitimately considered to have a disordered structure remains to be established, and thus the theories of such deviations remain to be tested. 3.3.2.2.5 Oblate Ellipsoids The precipitate morphology modeled as oblate ellipsoids is that of grain boundary allotriomorphs. Perhaps the closest idealized model of an allotriomorph is a double spherical cap. As a consequence of sectioning allotriomorphs at random angles by the plane of polish and of the diversity of allotriomorph morphology in three dimensions due at least in part to the essentially infinite range of angles between the plane of the grain boundary and those of the facets on allotriomorphs, a more precise modeling of the allotriomorph morphology is desirable.* Figure 3.29 [55] shows allotriomorphs of the equilibrium hcp phase at grain boundaries in an a Al–Ag alloy; although precipitation is far advanced, the allotriomorphs have not impinged to a significant extent. By contrast, as seen in Figure 3.30 [56], the proeutectoid cementite in a Fe-1.20% C alloy, reacted at 8008C for 2 h, displays a typical microstructure resulting nearly complete impingement of allotriomorphs. TEM photographs of u (CuA12) allotriomorphs in Al-4% Cu are shown in Figure 3.31 [57]. Although these allotriomorphs will be seen in Section 3.3.2.2 to have formed by interfacial diffusion-aided growth, their morphology is not significantly different from that of the ferrite allotriomorphs in which growth of this mechanism does not play a significant part. Even if one assumes that a grain boundary allotriomorph is, say, a double spherical cap in threedimensions, measurement of its growth kinetics presents important stereological problems. These can be mitigated if specimen thickness and solution annealing or austenitizing treatment are controlled so as to make the grain boundaries more or less perpendicular to the plane of polish established on the specimen broad faces (the specimen is a rectangular parallelepiped whose thickness is much less than its length or width). Difficulties, however, remain. Hawbolt and Brown [58] modeled allotriomorphs in the present manner and on this basis deduced the twodimensional shapes produced by a plane of polish that is assumed to be perpendicular to the grain boundary. They obtained expressions for the length and thickness as a function of distance of the plane of polish from the center of the allotriomorph. Speich and Cohen [59] similarly modeled precipitate plates and derived an expression for the lengthening rate as a function of the distance between the plane of polish and the center of the plate, and of the angle between the plane of polish * Recall that facets appear on critical nuclei due to interfacial energy minimization, while during growth they are present because they correspond to interfaces whose structure either markedly slows or prohibits migration in the normal direction and requires that migration take place by the less direct and slower agency of the ledge mechanism.
310
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
FIGURE 3.29 Widmanstätten g0 and g plates in Al-20.2% Ag aged for 25 h at 3878C. 2000. (From Barret, C.S. et al., Trans. AIME, 143, 134, 1941. With permission.)
FIGURE 3.30 Complete impingement of proeutectoid cementite allotriomorphs in an Fe-1.20% C alloy, reacted at 8008C for 2 h. 150. (From Heckel, R.W. and Paxton, H.W., Trans. ASM, 53, 539, 1961. Reprinted with permission of ASM International1. All rights reserved.)
and the plane of the plate. Bradley and Aaronson [60] considered the effects of both factors together in an examination of both two-dimensional and apparent shapes of allotriomorphs and of their lengthening and thickening kinetics. With the latter investigators, first consider the case of a double spherical cap allotriomorph formed at a grain boundary exactly perpendicular to the plane of polish; further assume that the plane of polish has passed through the center of the volume of the
311
Diffusional Growth
.1 μ
.1 μ
(a)
(b)
.1 μ
.1 μ (c)
(d)
FIGURE 3.31 u allotriomorphs in Al-4%Cu aged for (a) 10 min at 2758C, (b) 2 min at 2508C, (c) 15 min at 2508C, and (d) 2325 min at 2008C. (Reprinted from Aaron, H.B. and Aaronson, H.I., Acta Metall., 16, 789, 1968. With permission from Elsevier.)
r
S/2 d
GB
L/2
r
FIGURE 3.32 Allotriomorph sectioned through its center perpendicular to the grain boundary. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. and Aaronson, H.I., Metall. Trans., 8A, 317, 1977.)
allotriomorph. A sketch of this geometry is shown in Figure 3.32. For the resulting two-dimensional trace observed on the plane of polish, r 1 ¼ d 2 cos (f=2)
(3:286)
where r is the radius of the spheres used to construct the two spherical caps (r is taken as the same for both) d is the distance between their centers f is the dihedral angle formed at the edge of the allotriomorph (this construction is analogous to that of the modified g plot described in Section 2.14.1)
312
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
The true thickness, S, is given by S ¼ 2r d
(3:287)
L ¼ (4r2 d2 )1=2
(3:288)
and the true length is given by
The aspect ratio, K, defined as the ratio of thickness to length, is expressed as the ratio of Equation 3.287 to Equation 3.288, K¼
1 cos (f=2) 1=2 1 þ cos (f=2)
(3:289)
Now consider the case in which (a) the plane of polish is displaced from the center of the precipitate by a distance x, and (b) the normal to the plane of polish makes an angle of 908—c with respect to the normal to the grain boundary plane (c 6¼ 08). Both situations should normally be encountered almost invariably when experimental measurements are made. This situation is sketched in Figure 3.33: (a) view parallel to the grain boundary plane and (b) view orthogonal to a plane of polish. The apparent length measured along the grain boundary, which is not a function of c, is now given by 1=2 d2 2 2 L ¼2 r x 4 0
(3:290)
The trace of the precipitate on the plane of polish is symmetrical about the grain boundary for any value of x only when c ¼ 08 and only when x ¼ 0 when c 6¼ 08. A representative asymmetric case is shown in Figure 3.33b. This figure also shows that, at the large values of x and c employed, the radius of curvature of one side of the trace of the allotriomorph can be less than the apparent halfthickness of that side. The same figure further shows that, in this instance, the maximum apparent length does not lie along the trace of the grain boundary. Sphere 2 r
χ
a2
GB
b
d
S1
a1 Plane of polish
r1 S2
GB
θ2
1
θ1
r2 d΄ 2
Sphere 1 (a)
ψ (b)
FIGURE 3.33 Idealized allotriomorph (a) sectioned by a plane of polish with x 6¼ 0 and c 6¼ 0, (b) section of the allotriomorph as viewed on plane of polish. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. and Aaronson, H.I., Metall. Trans., 8A, 317, 1977.)
313
Diffusional Growth
The radius of curvature of the allotriomorph trace on either side of the grain boundary can be calculated from the intersection of the plane of polish with the two circles in Figure 3.33a, 1 r1 ¼ r 2 a21 2 1 r2 ¼ r 2 a22 2
(3:291)
where a1 and a2 are the perpendicular distances of the plane of polish from the centers of sphere 1 and sphere 2, respectively. In terms of b, the vertical distances between the center of sphere 1 and the plane of polish are given by a1 ¼ b sin c,
(3:292a)
a2 ¼ (d þ b) sin c
(3:292b)
where b ¼ x cot c
d 2
(3:292c)
Hence, r1 ¼ (r 2 b2 sin2 c)1=2 2
2
2
(3:293a)
r2 ¼ [r (d þ b) sin c]
1=2
(3:293b)
The apparent distance between the centers of curvature of the two sides of the allotriomorph trace, d0 , will be less than d for c 6¼ 0, thus, d 0 ¼ d cos c
(3:294)
The apparent thickness of the allotriomorph can now be expressed as s 0 ¼ r1 þ r2 d 0
(3:295)
For non-zero values of c and x, S1 ¼ r1 þ b cos c
c sin c
S2 ¼ r2 (d þ b) cos c þ
(3:296a) c sin c
(3:296b)
Thus, the two components of S0 ¼ S1 þ S2 must be separately calculated. From Figure 3.33 the apparent dihedral angle fapp (¼ f1 þ f2) may be evaluated. Since L0 2r1
(3:297a)
L0 , 2r2
(3:297b)
cos u1 ¼ cos u2 ¼
314
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
p u1 2 p f1 ¼ þ u 1 2 f1 ¼
for
r 1 S1
(3:298a)
for
r 1 S1
(3:298b)
for
r 2 S2
(3:299a)
for
r 2 < S2
(3:299b)
and p u2 2 p f2 ¼ þ u 2 2
f2 ¼
From the considerations of the section on ellipsoidal growth, see Section 3.3.2.1.5, when x and c 6¼ 0,
and
pffiffiffiffiffi S1 ¼ a1 Dt pffiffiffiffiffi S2 ¼ a2 Dt pffiffiffiffiffi L0 ¼ b0 Dt , 2
(3:300a) (3:300b)
(3:301)
where from Equations 3.296a, b, and 3.290,
and
x r1 þ b cos c sin c pffiffiffiffiffi a1 ¼ Dt x r2 (d þ b) cos c þ sin c pffiffiffiffiffi a2 ¼ Dt
(3:302a)
(3:302b)
1=2 d2 r 2 x2 4 pffiffiffiffiffi b0 ¼ Dt
(3:303)
2 Kx t0 ¼ a
(3:304)
and Dt ¼ t t0 where t is the true growth time and t0, the time at which the precipitate is first visible on the plane of polish, is evaluated as
The effects of non-zero values x and c upon a were evaluated as follows. Data on ferrite allotriomorph thickening kinetics in an Fe-0.11% C alloy, to be presented shortly, were utilized. From these data, K ¼ 1=3. S1 and S2 were calculated for allotriomorphs nucleated at a distance, x, of 0 to 100 mm beneath the plane of polish as a function of the apparent growth time, Dt, for c ¼ 0, 30 and 458. To compute S1 and S2 the true total thickness, S, as given by Equation 3.287, is set equal,
315
Diffusional Growth
on the theoretical considerations of the ellipsoidal growth, to two times the product of the true parabolic rate constant for thickening, a, and the square root of the growth time. Thus, S ¼ 2r d ¼ 2at 1=2
(3:305)
The true aspect ratio, K, is the ratio of Equation 3.287 to Equation 3.288, that is, K¼
2r d 1=2 2r þ d
(3:306)
Solving Equations 3.305 and 3.306 simultaneously, 1 1=2 1 þ1 r ¼ at 2 K2
(3:307)
(3:308)
and d ¼ at
1=2
1 1 K2
Equations 3.307 and 3.308 with the experimental values of a and K are then substituted into Equations 3.293a and b to evaluate r1 and r2. Substituting these values into Equations 3.296a and b, together with b as determined from Equation 3.292c and d from Equation 3.294, yields S1 and S2. Plots of 1=2(S1 þ S2) vs. Dt1=2 were then constructed. A least-squares slope was then obtained to evaluate an apparent thickening constant, aapp. These calculations were repeated for the various conditions of x and c as a function of reaction temperature. Figures 3.34 and 3.35 show sample results of aapp=atrue as a function of x for various values of c. It is seen that aapp=atrue varies from approximately 1.9 to approximately 0.1 for c ¼ 458. Even at c ¼ 0 a substantial error is possible due to the effect of sectioning away from the center of the allotriomorph (x 6¼ 0). The implications of these results will become apparent when experimental data are analyzed. 2.00 1.80 1.60 S2, ψ = 45°
αapp/αtrue
1.40 1.20 1.00 0.80 0.60
S1, ψ = 0° S1, ψ = 45°
0.40 0.20 0
S2, ψ = 30°
S1, ψ = 30° 0
10
20
30
40
50 χ (μm)
60
70
80
90
100
FIGURE 3.34 aapp=atrue vs. x at 8408C. atrue ¼ 0.15 104 cm per s1=2, K ¼ 1=3. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. and Aaronson, H.I., Metall. Trans., 8A, 317, 1977.)
316
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 2.00 1.80 S2, ψ = 45°
1.60 αapp/αtrue
1.40
S2, ψ = 30°
1.20 1.00 S1, ψ = 30°
0.80
S1, ψ = 0°
S1, ψ = 45°
0.60 0.40 0.20 0
0
10
20
30
40
50 χ (μm)
60
70
80
90
100
FIGURE 3.35 aapp=atrue vs. x at 7458C. atrue ¼ 2.31 104 cm per s1=2, K ¼ 1=3. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. and Aaronson, H.I., Metall. Trans., 8A, 317, 1977.)
In an attempt to solve directly the stereological problems associated with measurement of the thickening kinetics of grain boundary allotriomorphs, Kinsman and Aaronson [61] applied thermionic electron emission microscopy (THEEM) to the measurement of ferrite allotriomorph thickness vs. time in an Fe-0.11% C alloy. Specimens were austenitized to make the austenite grain boundaries essentially perpendicular to the broad faces. As to the technique itself, a flat and relatively thick specimen is encouraged to emit electrons by the application of a potential of approximately 50 kV, elevated temperatures and an ‘‘activator’’—an insoluble substance such as Ba or Sr that lowers the work function for the emission of electrons from iron. Since the work function is also affected by the crystal plane exposed on the specimen surface, focusing, projection, and photomultiplication of the emitted electrons onto a fluorescent screen yields an optical metallographic-type of image corresponding to that produced by a good grain contrast etch. Both ferrite crystals and the austenite crystals that they consume can be continuously observed at the temperature of transformation. Motion pictures taken of the fluorescent screen allow measurements to be made of the kinetics of growth with the assistance of a suitable projection and measurement device. Figure 3.36 shows representative data from an Fe-0.11% C alloy, which are indicative of the precision that can be obtained from this technique [15]. Data from both lengthening and thickening are included in this plot. (Only a small fraction of the number of available picture frames, exposed at the rate of 8=s, were used to construct these plots.) It was found, however, that significantly different values of a were obtained on different ferrite allotriomorphs measured not only at the same reaction temperature but also in the same area of the specimen imaged during a given ‘‘run.’’ The above plots indicate that little of this scatter was due to errors inherent in the measured a’s. Figure 3.37 suggests that variations in x and c from one allotriomorph to the next could be responsible for the scatter. (This figure was constructed by plotting the upper and lower limits of the range of possible stereological scatter for each value of c as a function of temperature taken from the plots of appt=atrue vs. x.) However, it is not likely, on the basis of observations made on grain boundaries on a plane of polish prepared normal to the measurement plane of polish, that such large values of c often occurred. A more mechanistic source of scatter that presumably supplements the stereological one will shortly be suggested. There is, on the other hand, little doubt that the stereological problem was the source of an appreciable fraction of this error. The latter statement is supported by the results obtained from the following approach to the measurements of allotriomorph growth kinetics [62]. Specimens were again austenitized so that the grain boundaries were essentially perpendicular to the intended plane of polish. Following
317
Diffusional Growth
24 22 20 18 (R0)
R0, Z0 : cm × 104
16 14 12 800°C
10
β = 6.5 × 10–4cm/s1/2 α = 1.0 × 10–4cm/s1/2 K = 0.15
8 6 4
(Z0)
2 0
0
1
4 5 2 3 (Growth time)1/2 s1/2
6
FIGURE 3.36 Typical plots of allotriomorph half-length, R0, and of allotriomorph half-thickness, Z0, as a function of the square root of the growth time. (With kind permission from Springer Science þ Business Media: From Atkinson, C. et al., Metall. Trans., 4, 783, 1973.)
7.0 6.0
ψ = 45° ψ = 30° ψ = 0°
α × 104 (cm/s1/2)
5.0 4.0 3.0 2.0 1.0 0 700
720
740 760 780 800 Reaction temperature (°C)
820
840
FIGURE 3.37 Calculated stereological scatter bands applied to the THEEM data for the thickening of allotriomorphs in an Fe-0.11wt.% C alloy. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. and Aaronson, H.I., Metall. Trans., 8A, 317, 1977.)
318
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 9 8 Fe – 0.23%C 775°C
7
α = 0.30 β
S/2 × 104 cm L/2 × 104 cm
6
β = 1.92 × 10–4 cm/s1/2
5 4 3 2
α = 0.58 × 10–4 cm/s1/2
1 0
0
1
2
3
Reaction time
4
5
6
(s1/2)
FIGURE 3.38 Typical plot of allotriomorph half-length, L=2, and half-thickness, S=2, as a function of the square root of the reaction time. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. et al., Metall. Trans., 8A, 323, 1977.)
isothermal reaction of a series of specimens for successively increasing times at a given reaction temperature, the specimens were quenched into iced brine, and measurements were made on the plane of polish prepared on each. In each specimen, the length of the longest and the thickness of the widest allotriomorph were sought. Any allotriomorphs suspected of having nucleated at a grain edge were not included in the measurements. Severely faceted and pronouncedly lopsided allotriomorphs were also excluded. Sample plots of the variation of the half-thickness, S=2, and the half-length, L=2, with the square root of the isothermal reaction time are shown in Figure 3.38. Although the ‘‘density’’ of data points is far less than in the counterpart plots obtained with THEEM, adequate precision is nonetheless obtained. As shown in the plots of a and b vs. temperature reproduced in Figure 3.39, the single data point for each at a given temperature results in adequately smooth and well defined plots of the variation of the parabolic rate constants with temperature, such as will permit the growth kinetics of the ferrite allotriomorphs in different alloys to be compared in a meaningful way. The circumstance that the ‘‘room temperature’’ method used to construct these plots ensured that x ! 0 and tends to minimize c (through avoidance of lopsided allotriomorphs) is probably an important factor in reducing the scatter attending the THEEM plots. Figures 3.40 through 3.42 show plots of a(exp)=a(calc) and b(exp)=b(calc) vs. temperature for all three Fe–C alloys used in this investigation. Calculation of a was performed by means of the Atkinson analysis for oblate ellipsoids, described earlier. These calculations took full account of the variation of DgC with xg. The plots in Figures 3.43 through 3.45 show that the aspect ratio of ferrite allotriomorphs in these alloys is independent of isothermal reaction time, temperature, and carbon content, and is approximately one-third under all circumstances investigated. The parabolic rate constant results make clear that the measured thickening kinetics and lengthening kinetics are significantly slower than those calculated assuming direct volume
319
Diffusional Growth 5
Present investigation Kinsman and Aaronson 0.39
α × 104 cm/s1/2
4
Fe-0.11% C Atkinson ellipsoid K = 0.3
3 0.30 0.37 2
0.36 0.25
1
0 700
0.24
0.34 0.27
0.30 720
(a)
800 740 760 780 Reaction temperature (°C)
0.34 840
820
12.0 Fe-0.11% C
10.0 Atkinson ellipsoid K = 0.33
β × 104 cm/s1/2
8.0
6.0
4.0
2.0
0 700 (b)
720
740 760 780 800 Reaction temperature (°C)
820
840
FIGURE 3.39 (a) Comparison of experimental and calculated values of a for Fe-0.11wt.% C as a function of reaction temperature. (b) Comparison of experimental and calculated values of b for Fe-0.11wt.% C. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. et al., Metall. Trans., 8A, 323, 1997.)
diffusion of carbon in the austenite away from the advancing austenite:ferrite boundaries as the rate controlling process in growth. SEM and TEM observations of facets on the broad faces of the allotriomorphs support the view that the interphase boundaries of these crystals consist of a mixture of partially coherent and disordered areas. As illustrated schematically in Figure 3.46, migration of the disordered (curved) areas of the boundary is interfered with by the need to grow around the partially coherent areas. The latter, migrating by the ledge mechanism, grow more slowly. Hence, the overall growth kinetics is slower than those predicted for simple ellipsoidal growth are. The constancy of the aspect ratio at K1=3 remains not understood. Measurement of the dihedral angle at the edges of ferrite allotriomorphs yielded a value of 100 58, significantly less than the equilibrium value [63] of approximately 1158, but the equilibrium value corresponds to an aspect ratio of 0.55 and the kinetic value of K ¼ 0.47. Both are well above the experimental value. Evidently, faceting is preferentially inhibiting the thickening process.
320
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 1.2
α (exp)/α (calc)
1.0 0.8 0.6 0.4 Fe-0.11% C
0.2 0 720
740
(a)
760 780 800 Reaction temperature (°C)
820
840
820
840
1.2
β (exp)/β (calc)
1.0 0.8 0.6 0.4 Fe-0.11% C
0.2 0 720
740
(b)
760
780
800
Reaction temperature (°C)
FIGURE 3.40 Plots of experimental and calculated parabolic growth rate constants for (a) thickening and (b) lengthening as a function of reaction temperature for Fe-0.11wt.% C. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. et al., Metall. Trans., 8A, 323, 1977.) 1.2 α (exp)/α (calc)
1.0 0.8
Fe-0.23% C
0.6 0.4 0.2 0 700
720
(a)
760 780 800 740 Reaction temperature (°C)
820
1.2 β (exp)/β (calc)
1.0 0.8
Fe-0.23% C
0.6 0.4 0.2 0 700
(b)
720
740 760 780 800 Reaction temperature (°C)
820
FIGURE 3.41 Plots of experimental and calculated parabolic growth rate constants for (a) thickening and (b) lengthening as a function of reaction temperature for Fe-0.23wt.% C. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. et al., Metall. Trans., 8A, 323, 1977.)
321
Diffusional Growth 1.2 α (exp)/α (calc)
1.0 0.8 0.6 0.4
Fe-0.42% C
0.2 0 680
740 760 700 720 Reaction temperature (°C)
(a)
780
1.2 β (exp)/β (calc)
1.0 0.8 0.6 0.4 Fe-0.42% C
0.2 0 680
700 720 740 760 Reaction temperature (°C)
(b)
780
FIGURE 3.42 Plots of experimental and calculated parabolic growth rate constants for (a) thickening and (b) lengthening as a function of reaction temperature for Fe-0.42wt.% C. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. et al., Metall. Trans., 8A, 323, 1977.) 0.60 0.50
Aspect ratio
0.40 0.30 0.20
Fe-0.23% °C 775°C
0.10 0
0
10 20 Reaction time (s)
30
FIGURE 3.43 Representative plot of average aspect ratio as a function of reaction time in Fe-0.23wt.% C. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. et al., Metall. Trans., 8A, 323, 1977.)
The ratio of partially coherent to disordered interphase boundary area and the distribution of the partially coherent areas should vary from a grain boundary to the next, because of the concomitant variation of the angle(s) between the partially coherent facet(s) and the grain boundary plane. Thus, the average parabolic rate constants should vary in a similar manner. This crystallographic factor probably supplements to a considerable extent the stereological one in engendering the scatter in a found during THEEM measurements of allotriomorph growth kinetics (Figure 3.37).
322
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 0.50
Aspect ratio
0.40 0.30 0.20
Fe-0.11% C
0.10 0 720
740
760
780
800
820
840
860
Reaction temperature (°C)
FIGURE 3.44 Representative plot of average aspect ratio as a function of reaction time in Fe-0.11wt.% C. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. et al., Metall. Trans., 8A, 323, 1977.) 0.50
Aspect ratio
0.40
0.30
0.20
0.10
0
0
0.10 0.20 0.30 0.40 Weight percent carbon
0.50
FIGURE 3.45 Average aspect ratio vs. wt.% C. Error bars are 80% confidence limits. (With kind permission from Springer Science þ Business Media: From Bradley, J.R. et al., Metall. Trans., 8A, 323, 1977.)
3.3.2.2.6 Plates and Needles The view originally held about the structure of the edges of precipitate plates was that this is of the disordered type. This idea arose simply because optimum matching between the matrix and precipitate lattices normally obtains at the broad faces of the precipitate plates. Considerably poorer matching might thus be anticipated at the interphase boundaries perpendicular to these faces. Subsequently, misfit dislocation structures were observed at the edges of u00 Al–Cu and gAl–Ag plates and in the latter case hot-stage TEM showed that lengthening takes place by the ledge mechanism. In phase transformations taking place in matrices wherein the interdiffusion coefficient is high, for example, the proeutectoid ferrite reaction in Fe–C base alloys and the proeutectoid reaction in Ti-base alloys, individual precipitate plates are from one to two (or
t3 t2 t1
γ
α
α t3 > t2 > t1
FIGURE 3.46 Elimination of a partially coherent facet on an allotriomorph. (From Aaronson, H.I. et al., Phase Transformations, ASM, Metals Park, OH, 1970, p. 313. With permission.)
Diffusional Growth
323
more) orders of magnitude longer than the thickness of an electron-transparent thin foil, and the probability that the edge of a plate will be found within the foil is rather small. In such transformations, deductions about the structure of plate edges must be made through comparison of calculated and measured lengthening kinetics and radii of the edges. There are two basic types of technique for measuring plate lengthening kinetics. One is the room temperature technique, wherein the length of the longest plate in each of a series of specimens reacted for successively increasing times at a given reaction temperature is determined. The other type is the high temperature technique, in which the growth of individual plates is followed continuously at the transformation temperature. Motion pictures of the growth process are helpful when the growth kinetics is rapid. Otherwise, conventional ‘‘still’’ micrographs, taken at appropriate intervals, suffice for measurement of the lengthening rate. When diffusivities are low and nucleation kinetics are relatively rapid, suitably oriented precipitate plates will grow for useful intervals of time entirely within a thin foil. If the precipitate plates are fortunately positioned relative to the foil surfaces and the material parameters of the alloy are such that the diffusion field associated with the plate edges is not too many times greater than the edge radius, the field, and thus the lengthening kinetics, will not be altered by the presence of the foil surfaces. On the other hand, when diffusivities are high, growth kinetics is so rapid that the thin foil technique is of limited value. Hot-stage optical microscopy is employed instead in such circumstances. This technique is dependent upon the rather angular surface relief effect associated with the precipitate plates in order to make the plate visible. Should the relief effect be insufficiently pronounced, polarized light will make the precipitates visible if either they or the surrounding matrix (or the thin film of oxide formed upon either) exhibit sufficient optical anisotropy. Thermionic electron emission microscopy has also been employed to measure precipitate lengthening (and thickening) kinetics, essentially in the manner described earlier. To date, THEEM appears to have been employed only to measure growth kinetics of precipitates in steel. The problems of securing a suitable activator, of the relatively rapid loss of activator through evaporation (thereby restricting use of time to a few minutes or less) and of accurate temperature measurement and control, tend to make the THEEM technique of limited value. At present, no THEEM instruments are being produced commercially. Photoelectron emission microscopy, in which electron emission is produced through intense bombardment of the specimen surface with high-intensity ultra-violet light, is a more promising technique for growth kinetics measurements in alloys wherein precipitate sizes are too large for TEM’s limitations. No activator is required and in principle, adequate intensity of electron emission can even be obtained at room temperature. A commercial instrument based upon this technique is available, but is not equipped for adequate temperature measurement or control. Furthermore, it does not permit sufficiently rapid cooling from the austenitizing to the isothermal reaction temperature and is very expensive. Application of photoelectron emission microscopy to quantitative studies of growth kinetics has accordingly been quite limited. When high temperature measurement techniques are employed, it is essential that precautions be taken to prevent changes in specimen chemistry during transformation, particularly in the vicinity of the specimen surface. Preferential vaporization of one species is an obvious source of difficulty in this regard; dezincification of brass thus seriously handicaps high temperature studies in this otherwise convenient material. A change in surface chemistry through reaction of an atomic species in the alloy with a constituent of the microscope atmosphere is another problem often encountered. Decarburization of steel by oxygen or carburization through reaction with hydrocarbons introduced by an oil diffusion pump is an example of this type of difficulty. Dissolution of a species present in the microscope atmosphere can drastically affect not only transformation kinetics but also the transformation product formed. Dissolution of oxygen, and probably nitrogen in titanium-base alloys is an excellent example of this problem. Similar difficulties are to be expected in Zr- and Hf-base alloys. Effects of the specimen surface per se ought also to be noted. One such effect, the truncation of the diffusion field in the matrix associated with a precipitate, has already been mentioned. A more insidious effect is the use of the surface with which a precipitate comes into
324
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
contact as a diffusion short circuit. Significant acceleration of plate growth due to this effect has been noted during the formation of gAl–Ag plates. Thermal grooving of grain boundaries at a free surface markedly inhibits grain growth [64,65]. The much higher driving force typical of phase transformations, usually some 3–5 orders of magnitude greater than that for grain growth, should minimize such difficulties. A much more serious effect arises when nucleation takes place on the specimen surface and results in lattice orientation relationships, morphologies, and even precipitation sequence that are not characteristic of transformation within the bulk alloy. Thomas and Whelan [66] have observed such effects when studying precipitation in an Al-4% Cu alloy with hot-stage TEM. In the discussion of the stereology of grain-boundary allotriomorph growth kinetics, the influence upon the measured kinetics of the distance, below the plane of polish at which the allotriomorph nucleated, and the angle between the grain boundary plane and the plane of polish were quantitatively characterized. By substituting, the habit plane of a plate for grain boundary plane along which an allotriomorph grows makes these considerations equally applicable to plate lengthening measurements. Hence, once again hot-stage growth kinetics data tend to be very precise but to exhibit differences from one plate to the next formed in the same specimen that are likely to be explicable at least in part by stereological rather than real, for example, interfacial structure effects. Hence, the room temperature technique ought again to be superior for collecting data useful for theoretical analysis. Figure 3.47 shows typical plots of a plate length vs. time and needle length vs. time in Cu-40 at.% Zn and in Cu-43 at.% Zn, respectively [67]. Both rates are seen to be constant, in agreement with expectation from theory that neither exhibits a dependence upon growth time. These data are representative of those obtained in a wide variety of alloy systems with respect to the constancy of lengthening rate. Again, most of the available data on plate lengthening kinetics have been reported for the proeutectoid ferrite and the bainite reactions in Fe–C alloys and in variously alloyed steels. In general, the kinetics of plate lengthening in these materials is within an order of magnitude of those calculated using even the simpler theories. Such results are illustrated in Figure 3.48, in which KRC compare lengthening rates calculated from Equations 3.162, 3.163, and 3.165 with unpublished data determined by Hillert [68] through the room temperature technique on Fe–C alloys of both relatively high and moderate purity levels (data on 8.7%Ni and on 1.8%Ni, 0.8%Mo steels are included in the figure [69]). Small amounts of Mn, Si, etc. were found to have little effect upon G ‘ . Characteristically, the calculated lengthening rates are higher than those measured experimentally. The calculations were performed on the assumption that gab ¼ 200 ergs=cm2, that is, that the plate edges are partially coherent. If the gab value appropriate to a disordered austenite:ferrite
Length cm × 104
20
Plate length 250°C Needle half length 300°C
16 12
V = 3.7 × 10–6 cm/s
8 4
V = 9.9 × 10–8 cm/s 1
2
3
6 4 5 Time s × 10–3
7
8
9
FIGURE 3.47 Plots of plate length vs. time in Cu-40.0 at.% Zn and needle half-length vs. time in Cu-43.0 at. % Zn. (From Simonen, E.P. and Trivedi, R., Edgewise Growth Kinetics of Widmanstätten Precipitates, Ames lab., USAEC, Iowa State University, Ames, IA, 1972, p. 27. With permission from Elsevier.)
325
Diffusional Growth 700 (b) (1.8 Ni, 0.8 Mo)
1.2
600 0.69 0.39
(b) (8.7 Ni) 0.50 0.10
1.4
500
1.6
°C
T (K)
103
400
1.8 (a)
2.0
0.21 0.35 0.42 0.59 0.81
300 250
Calculated, 0.96
200
Calculated, 0.44
10–5
10–2 10–4 10–3 Edgewise growth rate (mm/s)
10–1
FIGURE 3.48 Comparison of calculated rates of edgewise growth for iron-carbon alloys with experimental rates. Data (a) are Widmanstätten ferrite and bainite in plain-carbon steels by Hillert [68] and data (b) are bainite in alloy steels by Goodenow et al. [69]. The numbers are carbon contents given in weight percent of each curve [Ref. 27 in Chapter 1].
boundary, approximately 750 ergs=cm2, were employed, instead the discrepancy would be reduced from a factor of 5–10 to 1.3–2.7. Simonen et al. [70] obtained additional G ‘ data on high-purity Fe–C alloys, utilized some of Hillert’s data, measured the radii of plate edges and employed Trivedi’s relationships Equations 3.203 and 3.206 to examine in more detail the mechanism through which ferrite plates lengthen. The basic result is shown in Figure 3.49. In both of these plots, the experimental data are shown by means of individual data points, coded to indicate the reaction temperature at which they were obtained. In the lower plots of Figure 3.49, the interfacial energy of the plate edge was taken to be that for a disordered austenite:ferrite boundary, here estimated as 800 ergs=cm2. The interface kinetics coefficient, m0, was set equal to infinity; hence, no interface resistance to growth is assumed. Note that at 7008C and 6508C the calculated curves lie below the data points. If the equations and their ancillary data are correct, this is not permissible. When gab is reduced to 200 ergs=cm2, in Figure 3.49a, this anomaly is removed, but now theory and experiment are in agreement only at 7008C, and at all lower temperatures the calculated growth rates exceed the measured ones by increasing margins. This type of discrepancy is theoretically acceptable, since it can be eliminated by assigning finite values of m0. This is in Figure 3.50 and reasonable agreement is seen to have been obtained between calculated and measured lengthening rates; gab was accepted as 200 ergs=cm2 independently of temperature. m0 thus determined varies from 0.60 cm=smole fraction at 7008C to 0.07 cm=smole fraction at 4508C. However, the activation energy for m0 turns out to be meaninglessly low. Hence, it is concluded that the edges of these plates have a partially coherent structure, and that instead of a uniform barrier to migration as assumed in the Trivedi analysis, lengthening probably takes place by a ledge mechanism. Using a multiple sectioning technique, Eichen et al. [71] obtained optical microscopy evidence of coarse ledges on the edges of ferrite plates. Davenport [72], and Kinsman et al. [73] made similar observations with TEM and replication electron microscopy. As will become apparent when widely separated ledges of infinite
326
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
v × 103 cm/s
550°C 3
700°C
700°C 650°C 550°C 450°C
2 σ = 200 ergs/cm2 μ0 = ∞
1
(a)
v × 103 cm/s
450°C
650°C
0.04
0.08
0.12
0.16
3 2 σ = 800 ergs/cm2 μ0 = ∞
1
0.04 (b)
0.08 0.12 (C0 – C∞) Mole fraction carbon
0.16
v × 103 cm/s
FIGURE 3.49 Comparison of experimental and predicted velocity as a function of driving force for infinite interface kinetics and an assumed surface energy, (a) 200 ergs=cm2 and, (b) 800 ergs=cm2. (With kind permission from Springer Science þ Business Media: From Simonen, E.P. et al., Metall. Trans., 4, 1239, 1973.)
700°C 650°C 550°C 450°C
3 2 1
σ = 200 ergs/cm2 μ0 – finite 0.04
0.08 0.12 (C0 – C∞) Mole fraction carbon
0.16
FIGURE 3.50 Comparison of experimental and predicted velocity as a function of driving force for finite interface kinetics. (With kind permission from Springer Science þ Business Media: From Simonen, E.P. et al., Metall. Trans., 4, 1239, 1973.)
extent are treated in a later section, the ledge growth problem is intrinsically difficult even in this basic form. Modifying the treatment to take account of three-dimensional diffusion at the edges of the ledges, arranging the ledges round a parabolic cylinder, and incorporating the overlap of the diffusion fields of adjacent ledges (which has been done for straight ledges) would be a truly formidable undertaking.* Radii of curvature of the plate edges were determined through replication electron microscopy. At each alloy composition, 0.24%, 0.33%, and 0.43% C, and at each reaction temperature a frequency histogram was constructed from 50 to 100 measurements. The radius corresponding to * The motion of a single step and a train of steps have been analyzed by an analytical method as well as computer simulation. See Further Reading for references.
327
Diffusional Growth
the maximum in the histogram for a given composition and temperature was accepted as the true value. In an Fe-0.24% C alloy, where most of the data were collected, the radii varied from 180 Å at 7508C down to 60–65 Å at 5008C–3608C. At a given temperature, radius increased with carbon content. In Figure 3.51, experimental and calculated radii are compared. Reasonable agreement is seen to have been achieved. The measured radii are found to place an upper limit on gab of approximately 650 ergs=cm2, hence providing a little further support for the non-disordered nature of ferrite plate edges. Equation 3.203 allows the separation of the diffusion, mobility, and capillarity contributions to growth, and the results are shown in Figure 3.52, where 80% or more of the supersaturation available I II III 280
Alloy I Alloy II
Radius of curvature (Å)
240
Alloy III
200 160 120 80 40
800
700
600 500 400 Temperature (°C)
FIGURE 3.51 Experimental and theoretical radii vs. temperature for 0.24% C (I), 0.33% C (II) and 0.43% C (III) alloys. (With kind permission from Springer Science þ Business Media: From Simonen, E.P. et al., Metall. Trans., 4, 1239, 1973.) 100 Difusion 700°C
60 40 20 0
(a)
Difusion
80
Capillarity
% Supersaturation
% Supersaturation
100
80 60 40 20
Mobility 0.02 0.03 0.04 (C0 – C∞) Mole fraction carbon
0 (b)
Fe-0.24 % C
Capillarity Mobility 700
500 600 Temperature (°C)
FIGURE 3.52 Percent total supersaturation consumed by diffusion, capillarity, and interface mobility (kinetics) as a function of (a) driving force at 7008C and (b) temperature in the 0.24% C alloy. (With kind permission from Springer Science þ Business Media: From Simonen, E.P. et al., Metall. Trans., 4, 1239, 1973.)
328
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
are consumed in driving diffusion. Capillarity and mobility are relatively minor factors from this viewpoint, though they markedly affect the lengthening rate. Simonen and Trivedi [67] also investigated the lengthening kinetics of plates and needles in a number of other transformations. They concluded from literature data that a(hcp) plates in b(bcc) Ti–Cr, like those of ferrite and bainite, also grow more slowly than volume diffusion control allows and that plates of u0 Al–Cu and proeutectoid Fe3C and needles of g-Fe4N grow more rapidly than when controlled by volume diffusion. Only needles of Mo2C precipitated from ferrite in an Fe–C-Mo alloy grow at the rate predicted by theory, see Equations 3.211 and 3.212. This result was taken to show that the tips of these needles have a disordered structure. Such an interpretation is plausible, though unproved, since large and continuous changes in boundary orientation are required in order to form a needle tip. Growth of plates, too slow for volume diffusion control, is readily hypothesized to result from a partially coherent structure at the plate edges, requiring the services of the ledge mechanism for lengthening. However, the lengthening that is faster than allowed by volume diffusion control seems particularly difficult to understand in the case of g-Fe4N needles. Very rapid plate lengthening may result from closely spaced ledges in systems wherein the material parameters make the maximum diffusion distance in front of a ledge quite short, as discussed further. A deficiency in the theory, not yet even qualitatively understood, seems a possible explanation for at least some of these discrepancies. Simonen and Trivedi [74] also made their own measurements on plate and needle lengthening in Cu-40 and 42 at.% Zn alloys which are worth independent presentation for several reasons. One is that concentrated solid solutions are involved, and hence dilute solution thermodynamics no longer suffice. Hence, a modification is required in the capillarity correction at plate and needle tips. This will be done here for needles. Conversion to the plate case is accomplished with a simple modification. Equation 1.76 giving the increase in the partial molar free energy of solute, B, due to capillarity is first recapitulated, that is,
ba
ba
DGB,r ¼ GB b,r GB b
2gV a 1 xba b ¼ ba r xab a xb
(3:309)
The definitional relationship for the partial molar free energy of the ith species is Gi ¼ Gi þ RT ln ai ¼ Gi þ RT( ln xi þ ln gi )
(3:310)
where Gi is the free energy of pure i in a given crystal structure, ai is the activity of the ith species in the substitutional solid solution with the same crystal structure, gi is the activity coefficient of the ith species, xi is the concentration of this species. Differentiating with respect to ln xi, dGi d ln gi RTei ¼ RT 1 þ d ln xi d ln xi
(3:311)
where ei represents the expression in parenthesis, known as Darken’s thermodynamic factor from the classic treatment of diffusion by Darken [75]. This relationship can be readily integrated if it is assumed that the concentration range is small enough so that ei remains substantially constant,
329
Diffusional Growth ba
ð
xba b,r
GB b,r
dGi
ð
dGb ¼ RTeba Bb
ð
d ln xB
(3:312)
xba B
ba
GB b
Thus, ba
ba
GB b,r GBb ¼ RTeba Bb ln
xba B, r xba B
:
(3:313)
Substituting into Equation 1.76 yields the equation for capillarity correction for concentrated solid solution as 2gV a 1 xba B xba ba ba GB b,r GB b ¼ RT ln B,r ¼ (3:314) ab ba xba reba B B b xa xB
from which,
xba B,r
3 2gV a 1 xba B 4 5 ¼ xba B exp ba ab reb xa xba B 2
(3:315)
where the restriction of e to species B is omitted (Darken has shown from the Gibbs Duhem equation that eB ¼ eA). This relationship is due to Purdy [76]. Subsequently, Simonen and Trivedi [77] treated the problem more rigorously by simultaneous solution of equations in which the equality of the partial molar free energy of each species across a planar interphase boundary is a a modified by the introduction of a capillarity term, gV i =r, where V i is the partial molar volume of ith component in the precipitate phase, here a. Numerical relationships between partial molar free energy and composition for Cu and for Zn in both a and b are then obtained from thermodynamic data compiled by Hultgren et al. [Ref 2 in Chapter 1]. They reported a value of eb ¼ 4.4 very close to the value, 4.41, computed from Equation 3.311. Purdy [76] reported 4.5 for ea and 13 eb0 of CsCl ordered b0 phase. The appropriate diffusivity for the interdiffusion coefficient is available for both b and aCu–Zn, but they scatter considerably. Instead, both Purdy [76], and Simonen and Trivedi [77] elected to use e b , they wrote published measurements of tracer diffusivities. However, for the interdiffusivity, D e b ¼ eb D*b D
(3:316)
where D*b is simply described as ‘‘a tracer coefficient.’’ According to Darken, however, e b ¼ (1 xb )DB, *b þ xb DA, *b eb D
(3:317)
where the two D*b ’s are the diffusivities in an alloy of composition xb in the b crystal structure. Since DCu* and DZn* can differ by several times, a possibly significant error may have been introduced. Simonen and Trivedi [74] combined their own measurements with those of Purdy [76] and Repas and Hehemann [78] on the lengthening kinetics of a needles in both b0 and bCu–Zn. For consideration of the lengthening kinetics of plates in b0 in the same and slightly more dilute alloys, they used their own data, those of Repas and Hehemann and also data reported by Hornbogen and Warlimont [79]. The portion of the Cu–Zn diagram (Figure 3.53) includes a dashed
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Temperature (°C)
330
β
α+β
600
α 400
Needle Plate
200 35
40
45
Composition (at.% Zn)
FIGURE 3.53 Cu–Zn diagram showing the temperature-composition regions in which plates and needles are the dominant Widmanstätten morphology. (From Repas, P.E. and Hehemann, R.F., unpublished research, Case-Western Reserve University, Cleveland, OH, 1967, written up as an ONR report.)
curve, obtained by Repas and Hehemann [78] separating the temperature-composition regions in which plates and needles, respectively, are the predominant Widmanstätten morphology. Subsequent research has shown that this phase diagram does not predict the phase in which precipitate plates appear. Cornelis and Wayman [80] have shown that the supposedly fcc a plates are actually face centered orthorhombic in structure. They designated this phase as a1 and further confirmed the previous observation of Hornbogen and Warlimont [79] and Flewitt and Towner [81] that, unlike Hence, a1 should be regarded, at least tentatively, aCu–Zn, the a1 phase is initially heavily faulted. 0 0 as a transition phase. Whereas both xaa11 b and xbb0 a1 are needed in the data analysis, repeated attempts to determine the former value (particularly) have led to serious discrepancies, as will be discussed in the section on bainite. Simonen and Trivedi [74] used the values for aCu–Zn and possibly introduced errors into their analyses. Measurement of neither needle nor plate tip radii is strictly necessary on the Trivedi analyses of plate and needle lengthening. However, since both lengthening rate and radii can be calculated when the maximum growth rate hypothesis is employed, Purdy [76] did make such measurements, on a needles or rods, in the most attractive fashion, using a deep etch followed by three-dimensional observation of the tips thus exposed with scanning electron microscopy. Using a single reaction temperature, 4008C, in a 44.1 at.% Zn alloy (V0 ¼ 0.2), r ¼ 1.3 2.5 105 cm for the long rods and in a 41.1 at.% alloy (V0 ¼ 0.6) r < 5 l06 cm. All investigators appear to have measured lengthening kinetics as the slope of a plot of the length of the longest plate or needle as a function of isothermal reaction time at a particular reaction temperature. When a particular interface was assumed to have a disordered structure, its interfacial energy was assumed 500 ergs=cm2. For a partially coherent structure, 250 ergs=cm2 was employed. Figures 3.54 through 3.56 compare the predictions of the Trivedi theory for needles (Equations 3.211 and 3.212) and for plates (Equations 3.203 and 3.206), shown as solid curves, with the experimental data. Figure 3.54 shows that calculated aCu–Zn needle lengthening kinetics in b0 are in good agreement with the experimental measurements when the needle tips are assumed to have a disordered structure. Recall that Simonen and Trivedi [74] reached the same conclusion for the lengthening kinetics of Mo2C needles in an a Fe–C-Mo alloy reported by Hall et al. [82]. On the other hand, Figure 3.55 shows that measured lengthening rates of a needles in b (the order=disorder transition occurs between 4548C and 4688C) are several-fold more rapid than predicted by the theory. The single data point at 5208C can be rationalized by reducing the
331
Diffusional Growth 10–3 450°C
400°C 350°C
Velocity (cm/s)
10
–4
10–5
300°C
10–6 250°C 10–7
10–8
0.01
0.02 0.03 0.04 0.05 0.06 (Co–C∞) Mole fraction Zinc
FIGURE 3.54 Growth kinetics of a needles in ordered beta. The theoretical results are for s ¼ 500 erg=cm2 and for local equilibrium condition at the a:b interface. (From Simonen, E.P. and Trivedi, R., Acta Metall., 25, 945, 1977. With permission from Elsevier.)
10–2
Velocity (cm/s)
520°C 10–3
10–4
10–5
500°C
0.01 0.02 (Co–C∞) Mole fraction Zinc
0.03
FIGURE 3.55 Growth kinetics of a needles in disordered beta brass at 5008C and 5208C. The theoretical calculations assume g ¼ 500 erg=cm2. (From Simonen, E.P. and Trivedi, R., Acta Metall., 25, 945, 1977. With permission from Elsevier.)
332
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 10–2
Velocity (cm/s)
10–3
350°C
10–4
300°C 10–5
10–6 250°C 10–7 0.050 0.054 0.058 0.062 0.066 0.070 0.074 (Co–C∞) Mole fraction Zinc
FIGURE 3.56 Growth kinetics of a plate precipitates in ordered beta brass. (From Simonen, E.P. and Trivedi, R., Acta Metall., 25, 945, 1977. With permission from Elsevier.)
interfacial energy to 250 ergs=cm2, that is, to that for a partially coherent boundary, but the two points at 5008C do not appear to be rationalized. Similarly, Figure 3.56 indicates that measured al plate lengthening in b0 are up to an order of magnitude more rapid than those calculated at 2508C. Agreement is better at 3008C, though the calculated values are still too low. Simonen and Trivedi rationalized the two cases of disagreement by postulating higher than equilibrium concentrations of Zn in the Widmanstätten precipitates involved. They attempted to support these deviations through published electron probe data. As already noted, however, these data are all controversial. The disagreement shown in Figure 3.55 for the growth of a needles in the disordered b, is particularly disturbing. A reexamination of the diffusivities used seems necessary, as well as independent measurement and calculation of the radius of the needle tips. In the case of Figure 3.56, the lengthening of a1 plates in b0 , agreement between calculated and measured lengthening rate is not to be expected on the reasons already given. Further research on the lengthening kinetics of plates and needles is obviously required. Interfacial structure determination for the tips of these crystals, though obviously very difficult, is most urgently required. Both the validity of the maximum growth rate hypothesis and the suggestion that it be replaced by one based upon ‘‘the ragged edge of Mullins–Sekerka instability’’ are needed for careful reexamination. Most difficult of all, equations need to be devised for the lengthening of plates and needles by the ledge mechanism.
3.3.3 GROWTH FASTER
THAN
VOLUME DIFFUSION CONTROL ALLOWS
3.3.3.1 Grain Boundary Allotriomorphs Aaron and Aaronson [57] measured the growth kinetics of u, that is, CuAl2 allotriomorphs in an Al-4% Cu alloy, by means of transmission electron microscopy, of specimens isothermally reacted for various times at given temperatures and then quenched to room temperature. The thin foils were
333
Diffusional Growth Temperature (°C) 500
400
250
200 700
Dapp (Horvay–Cahn)
600
10–10
500
10–11
400
10–12
DV (Murphy)
DV
10–9
D (cm2/s)
300
Dapp
10–8
300
10–13
200
10–14
Dapp/DV
10–15 1.2
1.4
100
1.6
1.8
2.0
2.2
0
103/T (K–1)
FIGURE 3.57 Arrhenius plots of DV and Dapparent, and the temperature dependence of Dapp=DV. (With kind permission from Springer Science þ Business Media: From Goldman, J. et al., Metall. Trans., 1, 1805, 1970.)
tilted until the grain boundary plane was parallel to the electron beam; hence, true thickness and true length (both maximum values) could be directly measured. By comparing the measured thickening kinetics with those calculated from the oblate ellipsoid model (Equation 3.133, etc.) they found that the growth kinetics measured were significantly more rapid than those calculated. Their measurements were performed on specimens reacted at 2008C–3258C. Subsequently. Goldman et al. [83] extended these measurements to 4508C, and prepared Figure 3.57 in which this point is made for both investigations. The line labeled ‘‘DV Murphy’’ represents the volume interdiffusivity extrapolated from measurements made by a conventional technique at higher temperatures on aAl–Cu by Murphy [84]. In the figure, Dapp is the apparent interdiffusivity calculated from experimental measurements of a and b and Equation 3.133 and the following factors. The dashed curve in this figure shows that Dapp=DV is nearly unity at 4508C, but rises at an increasing rate with decreasing temperature, reaching 7008C at 2008C. Taking the homologous temperature as, the ratio of the absolute reaction temperature to the absolute temperature of the solidus, it was thus found that a mechanism other than volume diffusion is either a significant or a predominant contributor to growth kinetics up to a homologous temperature of 0.84. Hawbolt and Brown [58] studied the growth kinetics of bcc allotriomorphs precipitated from an fcc matrix in a Ag-5.6 wt.% Al alloy. A similar comparison indicated that only after a homologous temperature of 0.93 is reached does volume diffusion directly to the allotriomorphs, becomes effectively the sole contributor to growth. The mechanism envisaged to explain these results is one of interfacial diffusion-aided volume diffusion, as illustrated in Figure 3.58. Solute is taken to diffuse in the matrix to a grain boundary. The boundary acts as a ‘‘collector plate’’ for the solute. The solute then diffuses along the grain boundary to the allotriomorph. Deposition of some solute then takes place on the edges of the allotriomorph. Most of the solute, however, diffuses along the disordered areas of the interphase boundaries of the allotriomorph and deposits on their broad faces. The two interfacial diffusivities are taken to be high enough relative to DV, so that, despite their minute thicknesses (approximately
334
Mechanisms of Diffusional Phase Transformations in Metals and Alloys α
α
θ
S
θ
α (a)
R
(b)
a
FIGURE 3.58 Models for (a) the lengthening and (b) the thickening of grain boundary allotriomorph by interfacial diffusion. (Reprinted from Aaron, H.B. and Aaronson, H.I., Acta Metall., 16, 789, 1968. With permission from Elsevier.)
one lattice parameter), the much larger area of the collector plate relative to that of the broad faces— with the effective area of the collector plate being double to account for diffusion to it from both sides of the grain boundary—permits interfacial diffusion aided DV to make a much larger contribution to growth, when DB=DV is sufficiently higher than volume diffusion directly to the broad faces. Because the activation enthalpy of boundary diffusion, DB, is considerably less than that for volume diffusion, the collector plate mechanism should, and does make a more important contribution to growth at the lower homologous temperature of aging. In their simple initial treatment of the collector plate mechanism, Aaron and Aaronson [57] arbitrarily divided it into two nominally independent processes, lengthening and thickening. Considering first the lengthening process, this was further simplified by assuming that volume diffusion of solute to the grain boundaries is the rate-controlling process and that the subsequent diffusion along the grain boundaries takes place rapidly enough to be ignored. Hence, the flux of solute to the grain boundaries, next to be derived, could be equated directly to the flux converted into lengthening of the allotriomorph. The flux of solute to one side of a grain boundary is given by Fick’s first law as qx F ¼ ADV qs s¼0
(3:318)
where e symbol, DV, the volume interdiffusivity, replaces for convenience the usual D A is the area of the ‘‘collector plate’’ assigned to a particular allotriomorph, S is the distance coordinate normal to the boundary. From Crank [85] the solute concentration as a function of distance from the boundary of a semiinfinite medium, where the initial concentration in the medium was uniformly xa, the atom fraction of Cu in the alloy, is given by au x ¼ xau a,r þ xa xa,r erf
s pffiffiffiffiffiffiffiffi 2 DV t
(3:319)
where xau a,r is the solute concentration in the boundary. Since this concentration is taken to be effectively that of the finely curved edge of the allotriomorph toward which the solute is diffusing, the solute concentration at a planar, disordered a:u boundary is modified from xau a , the atom fraction of Cu in a at the a=(a þ u) phase boundary, to xau a,r . On Equation 1.78, modified to take account of the circumstance that only one principal radius is small,
xau a,r
¼
xau a
"
#
gau Vu 1 xau gau Vu a au xa exp exp au RTrxua RTr xua u u xa
(3:320)
335
Diffusional Growth
where Vu is the molar volume of u, gau is the energy of a disordered a:u boundary, xua u is the atom fraction of Cu in u at the u=(u þ a) phase boundary. ua Since in the Al–Cu system, xau a ¼ 1 and xu xu , the atom fraction of Cu in u, this relationship can be simplified as shown with little loss of accuracy. Differentiating Equation 3.319,
at s ¼ 0,
qx xa xau s2 a,r ¼ pffiffiffiffiffiffiffiffiffi exp 4Dt qs pDt qx xa xau ffi ¼ pffiffiffiffiffiffiffiffia,r qs pDt
(3:321)
(3:322)
Substituting Equations 3.320 and 3.322 into Equation 3.318, 1
F¼
AD2 1
(pt)2
xa xau a,r exp
gau Vu RTrxu
(3:323)
This equation gives the flux of Cu to one side of a grain boundary ‘‘collector plate.’’ The flux of Cu converted to u is F¼
dm dm dV ¼ dt dV dt
(3:324)
where V is the half-volume of an allotriomorph. Modeling the allotriomorph as a disc of halfthickness S, taken as equal to the radius of the edge of the allotriomorph, r, and decoupling lengthening and thickening by allowing only the radius, R, of the allotriomorph to vary with time: V ¼ pR2 S
(3:325)
dV ¼ 2pSR dt
(3:326)
dm ¼ xu xau a,r dV
(3:327)
Hence,
and
Substituting Equations 3.326 and 3.327 into Equation 3.324, F ¼ 2p SR
dR gau Vu xu xau exp a RTrxu dt
(3:328)
336
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Equating the two fluxes given by Equations 3.323 and 3.328 and rearranging yields,
gau Vu 1 exp AD2 xa xau a dR RTrxu
¼ gau Vu dt 3 1 au 2p2 St 2 xu xa exp RTrxu
(3:329)
where dR=dt is the lengthening rate of the allotriomorph along the grain boundary. Taking S to be an experimentally determined constant, this equation integrates to R ¼ kt 1=4
(3:330)
where
9 8 1 gau Vu >1=2 > au 2 > > =
> gau Vu > > ; : p32s xu xau a exp RTrxu
(3:331)
At temperatures from 2008C–3508C, the experimentally observed time laws ranged from t0.22 to t0.34, averaging t0.27, in excellent agreement with Equation 3.330. Taking the experimentally determined value of r, approximately 1.1 105 cm independently of reaction time and temperature under the conditions studied, gau ¼ 300 ergs=cm2, Vu ¼ 10 cm3=mol and using the experimentally evaluated time laws, the calculated values of A are of the same order of magnitude as those experimentally estimated to be the square of the average inter-allotriomorph spacing, assuming each plate to be a square. The assumption that the diffusivity along a grain boundaries is infinite, though appearing reasonable in view of the experimental data, did prevent an important piece of information from being extracted from the growth kinetics data. This information is, of course, the grain boundary diffusivity, Daa. This diffusivity can be independently measured, though it is yet to be accomplished in the present system, and would thus allow an independent check to be made of the analysis. Brailsford and Aaron [86] have made a sophisticated and quite detailed analysis of the collector plate mechanism, in which lengthening and thickening are treated together. The results of this analysis will be briefly noted later. For present purposes, we incorporate in the foregoing analysis a preliminary, approximate expression that they derived for the flux of Cu along the grain boundary at the edges of an allotriomorph. Equations 3.318 through 3.323 are replaced by 2pDaa d xa xau a,r F¼ aa d ln jD DV R
(3:332)
where d is the width of the grain boundary, ln j ffi 0.116. Equating (3.332) and (3.328), rearranging and integrating, 91=2 > > = 2Daa d xa xau a,r
t 1=2 R¼ > > D d aa > > :r xu xau þ 0:366 ; a,r ln DR 8 > >
dislocation loops arrayed along the length of the laths (Figure 3.71d) compensate primarily the former misfit. The Burgers vector of these loops makes a 458 angle with respect to the broad faces. Since [100]S==[100]p a in the 0.012 misfit direction, that part of the vector that is able topcompensate this misfit is 1= 2. 2]a and thus only 1= 10 of the Burgers vector is so In the 0.014 misfit direction [100]S==[01
361
Diffusional Growth
TABLE 3.4 Structure of Some of the Simpler Eutectic Interfaces Phases and Crystal Structures
Interface Crystallography
Cr (bcc)-Ni–Al (ordered bcc, CsCl) Cr-Mo (bcc)-NiAl (see above)
(100)Cr==(100)NiAl [100]Cr==[100]NiAl (100)Cr==(100)NiAl [112]Cr==[112]NiAl
Ni (fcc)-Ni3Nb (ordered orthorhombic, b0 Cu2Ti) Ni3Al(ordered fcc, Cu3Au)-Ni3Nb (see above)
(110)Ni==(010)Ni3Nb [110]Ni==[100] Ni3Nb
(111)Ni3Al==(010)Ni3Nb Ni Nb==[100]Ni Nb [110] 3 3 [112]Ni3Nb==[001]Ni3Nb (113)Ni3Nb==(013)Ni3Nb [110]Ni3Nb==[100]Ni3Nb
Dislocation Structure
Investigators
a, in square and hexagonal networks; d ¼ 70–400 nm
a, often not in interface plane; networks sometimes rotated about Cr-rich rods; d ¼ 13–200 nm a=2 (?) in hexagonal networks, slightly dissociated; d ¼ 45nm
a=2; d ¼ 40 nm a=2; d ¼ 15 nm
Walter et al. Cline et al.
Annarumma and Turpin
Nakagawa and Weatherly
a=2; d ¼ 35 nm
Source: Aaronson, H.I., J. Microsc., 102, 275, 1974. With permission.
]
[001
]
[010
0.2 μ
FIGURE 3.72 Typical dislocation interphase boundary structure formed by Cr rod and NiAl matrix in directionally solidified Cr-NiAl eutectic. (Reproduced from Cline, H.E., Acta Metall., 19, 405, 1971. With permission from Elsevier.)
employed. The slow and complicated mechanics of generation of misfit dislocation arrays around S laths to be described in a later section explain this inefficient and incomplete compensation of misfit on purely kinetic grounds. Matthews and his colleagues [100,119–124] have reported a number of careful studies of coherent and misfit-dislocation interphase boundaries formed in the following manner.
362
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
A monocrystalline substrate, in the form of a thin film, is prepared by vapor deposition upon an oriented monocrystal of, for example, NaCl. A thin film of another substance is then epitaxially deposited, often with a systematically varied thickness, upon the substrate. When both the misfit and the thickness of the second film, the ‘‘overgrowth,’’ are sufficiently large, misfit dislocations form at these ‘‘synthetic’’ interphase boundaries. Most of the systems were studied with a boundary plane of the form {100}; the Burgers vector of the dislocations comprising the misfitcompensating array was of the type a=2 . In all but Au on Pd [119,120], however, the particular a=2 Burgers vectors observed made an angle of 458 with respect to the boundary [120–123]. Their misfit-compensation efficiency was only 50%. Most of these dislocations dissociated into a=6 partials in the Co on Cu [120] bicrystals. Apparently, all misfit dislocations in the Co on Ni [124] bicrystals were of this type; however, this increased only slightly their efficiency of misfit compensation. The formation of these inefficient misfit dislocations was again ascribed to the kinetics of the dislocation generation process. 3.4.2.2.5 Misfit Dislocations with a Burgers Vector Stable Only at an Interphase Boundary Such dislocations, previously reported at grain boundaries [125], have now been found on the broad faces of u0 Al–Cu and hAl-Au plates by Sankaran and Laird [126], both diffraction contrast experiments and plots of d vs. d for each of the Burgers vectors under consideration led to the conclusion that a=2 dislocations exist at these faces. The singly imaged dislocations at A in Figure 3.73 are of this type. Figure 3.74 shows that a=2 dislocations do not distort seriously either lattice at a:u or a:h boundaries; the only disturbance which they produce is ‘‘chemical,’’ in the form of a fault plane of solute atoms, marked A, having the wrong relationship with respect to Al atoms. This opportunity for a new type of dislocation was provided by a difference in both chemistry and long-range order across the interphase boundary. One might also anticipate that, as in the grain boundary case, a structural difference across the boundary would provide another type of origin for a non-bulk type of misfit dislocation [125,126].
200 W>0
A
0.3 μ
FIGURE 3.73 a=2 dislocation at u0 plate in Al-4%Cu aged 9 h at 3008C. (From Sankaran, R. and Laird, C., Philos. Mag., 29, 179, 1974. Reprinted with permission of Taylor & Francis Group, http:==informaworld.com)
363
Diffusional Growth
α
A
A
A
β
FIGURE 3.74 a=2 dislocations at a:u0 or a:h interface. Atoms marked A are at wrong sites. Al atoms are denoted and ., and solute atoms are and . In the case of u0 , . and are on (100) and and are on (200). (From Sankaran, R. and Laird, C., Philos. Mag., 29, 179, 1974. Reprinted with permission of Taylor & Francis Group, http:==informaworld.com)
3.4.2.2.6 Misfit Dislocation Structures as Minimum Energy Array As is implicit in the theoretical studies previously recounted, the array of misfit dislocations formed is the one yielding the low interfacial energy per unit area. The Burgers vector(s) of the dislocations, the geometry of the array, and the spacing(s) between parallel dislocations are the disposable variables determining the energy of an array under otherwise constant conditions. The observations of Burgers vectors not parallel to the plane of the overgrowth-substrate boundary and the broad faces of laths noted in the Section 3.4.2.2.5 provided one indication that the minimum energy condition may not always be attained in practice. Even if consideration is restricted to misfit dislocations in edge orientation with Burgers vector lying in the interphase boundary, it is often possible to compensate the misfit across a given boundary in more than one way, with the various possible arrays differing in either two or three of the disposable variables. For example, Figure 3.75 shows that misfit at the {100}u0=={100}a boundary at the broad faces of u0 can be compensated by arrays of either a or a=2 dislocations. Consider first an approximate, but simple approach to determining which of these configurations has the lowest energy. The energy of an individual dislocation is proportional to the square of its Burgers vector [108]. Assuming that the constant of proportionality is unity (since one is interested here only in the ratios of energies of various configurations), the energy per unit area of an array was assessed as the
[110]
[100]
b = a b=
a
2
FIGURE 3.75 Schematic of two misfit-compensating dislocation arrays possible on the broad faces of u0 plates in an Al-4%Cu alloy. (From Aaronson, H.I. et al., Phase Transformations, ASM, Metals Park, OH, 1970, p. 313. With permission.)
364
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
product of b2 (energy=length) and the dislocation line length per unit area (cm=cm2). On this basis, thepratio of the energy=unit area of the a array to that of the a=2 array in Figure 3.75 is 2, yet Table 3.3 indicates that the misfit dislocations found by means of transmission electron microscopy at the broad faces of u0 plates are of the a type. Since this energy ratio is applicable to the equivalent choice at the broad faces of UC2 plates (Table 3.3), the same disagreement is found between the minimum energy criterion and direct experimental observation in this situation. Application of the Frank-van der Merwe relationship for g (Equation 3.361) to Figure 3.75, with b substituted for c and the result doubled to take account of the second set of dislocations in each array, yields (necessarily) identical ratios, and interesting absolute values. For the UC2=UC boundary, ga ¼ 410 ergs=cm2 (0.41 J=m2), whereas ga=2 ¼ 290 ergs=cm2 (0.29 J=m2). In the case of u0 plates, these values are 37.5 and 26.5 ergs=cm2 (0.038 and 0.027 J=m2), respectively. The interfacial energy of the broad faces of u0 plates has recently been the subject of a theoretical study [120], performed just before the presence of misfit dislocations at these interfaces became known. ga is nearly the same as the chemical component of the interfacial energy estimated for these faces (40 ergs=cm2). Since the estimated sum of the structural and the chemical components of the energy of a disordered a:u0 boundary is 445 ergs=cm2 (0.445 J=m2) [127], the result obtained from Equation 3.361 is certainly of the correct order of magnitude, and is likely somewhat better. The misfit-dislocation structure of the broad faces of g plates precipitated from a Al–Ag offer a particularly convincing example of interfacial structures not of the minimum energy configuration [128]. Although all of the misfit dislocations are a=6 partials, they are diversely arrayed, as indicated by Figure 3.71b. Within a given type of array, the distance between parallel dislocations can often be seen to vary markedly, even within a quite small area of a given g plate. The basic types of array found after prolonged aging, when the arrangements become less complex, are sketched in Table 3.3. The presence of such multiplicity, and particularly the appearance of the obviously nonequilibrium one-Burgers-vector array, makes it clear that the minimum energy criterion has failed again. Taken in conjunction with the previously cited failures, this unsatisfactory result indicates that this criterion is usually not useful, at least during the early stages of transformation. From studies on the structure of interfaces in regularly formed eutectics, an unusual example has been found of a situation wherein at least the inter-dislocation spacing matched very well with that anticipated from lattice parameter measurements. (The question of whether or not another set of dislocations would have provided a lower energy interface was not investigated.) Cline et al. [118] found that the mismatch calculated from the measured inter-dislocation spacing at the interfaces of the Cr=NiAl eutectic (see Figure 3.72) agrees very well with that computed from lattice parameter data when the mismatch varies from 0 0.03 by additions of up to 7% Mo, as shown in Figure 3.76. A typical interfacial structure, formed on the Cr-rich eutectic rods is illustrated in Figure 3.72 (Mo dissolves primarily in this phase). This type of result was indicated to be unusual in the previous review [2] as a consequence of the difficulty so often encountered in acquiring and appropriately distributing the necessary misfit dislocations along the interphase boundary. In the Cr(Mo)=NiAl case, however, the facts that the misfit dislocations also serve as glide dislocations in NiAl and that the interface plane is usually the glide plane allow addition or elimination of misfit dislocations to be readily accomplished. On the other hand, when it is feasible to anneal precipitates for sufficient lengths of time, a definite tendency is found for the interfacial structure to seek a lower energy, if not equilibrium, configuration, as illustrated by the following two examples. (a) During precipitation, Sankaran and Laird [126] found that when two a a dislocations intersect on the broad faces of u0 Al–Cu and hAl-Au plates, they dissociate according to the reaction, a[100]a ! a=2[110]a þ a=2[110]a
365
Diffusional Growth
Percent mismatch calculated from dislocation spacing
3.0
2.0
1.0
–10
0
1.0 2.0 Percent lattice mismatch
–1.0
FIGURE 3.76 Comparison of mismatch calculated from inter-dislocation spacing with that obtained from lattice parameter data on Cr (Mo)=NiAl eutectic interfaces. (From Cline, H.E., Acta Metall., 19, 405, 1971. With permission from Elsevier.)
a[110]α
a[110]α a/2[110]α a/2[110]α a[010]α
a[010]α (a)
(b)
FIGURE 3.77 Interactions of intersecting a dislocations to produce a=2 dislocations, as observed on the broad faces of u0 Al–Cu and h Al-Au plates. (From Sankaran, R. and Laird, C., Philos. Mag., 29, 179, 1974. Reprinted with permission of Taylor & Francis Group, http:==informaworld.com)
The observed p geometry of this reaction is illustrated in Figure 3.77. The g of the final structure is 1= 2 that of the original. Interaction of pairs of a=2 a dislocations occurs as, a=2[110]a ! a=2[100]a þ a=2[010]a through the more complex rearrangements described by Figure 3.78 to yield a structure whose g is 0.53 that of the initial structure. Misfit at the broad faces of u0 plates is less than 0.01 [109], whereas that at h plates is approximately 0.05. As expected, the kinetics of both the initial acquisition and the rearrangement of misfit dislocations are much more rapid at the broad faces of the h plates. (b) During spinodal decomposition of Cu-Ni-Fe alloys, Butler and Thomas [129] and Livak and Thomas [130] found that a=2 dislocations are spontaneously nucleated at the interfaces of the plates, approximately 500 nm thick and 12 mm long with approximately {100} habit planes. The Burgers vector b of these dislocations initially lies at a 458 angle to
366
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
A a/2[110]
– a/2[1 10]
a/2[110]
a/2[100]
a/2[110] (a)
(b)
(c)
a/2[110] a/2[100]
(d)
(e)
(f )
FIGURE 3.78 Interaction of a=2 dislocations to form a=2 dislocations near the edges of u0 Al–Cu and h Al-Au plates. (From Sankaran, R. and Laird, C., Philos. Mag., 29, 179, 1974. Reprinted with permission of Taylor & Francis Group, http:==informaworld.com)
g 200 0.5 μ
FIGURE 3.79 Rotation of a spinodal decomposition interface from {100} toward {110} where misfit dislocations have formed (see arrowhead) in a Cu-Ni-Fe alloy. (From Livak, R.J. and Thomas, G., Acta Metall., 22, 589, 1974. With permission from Elsevier.)
the interphase boundary. The interphase boundary itself then rotates toward a {110} plane, as illustrated in Figure 3.79, thereby allowing b to lie in the interface and hence be fully utilized to compensate misfit. On the theory of precipitate morphology, this rotation is ‘‘allowed’’ because the crystal structure and orientation of the two phases are the same. 3.4.2.2.7 Substitution of Coherence Strains for Misfit Dislocations Consider first the small d situation in which misfit dislocations can be wholly replaced by full coherency between the oriented overgrowth or precipitate and the substrate or matrix. Equation 3.369 is appropriate to the situation in which there is misfit in only one direction, whereas in the
367
Diffusional Growth
specific examples to be considered equal misfits obtain in two orthogonal directions. Revision of the relationship ghom ¼ g (Equation 3.361), upon which Equation 3.369 is based, to take account of this difference is readily accomplished. As previously justified, g (Equation 3.361), is simply doubled. The ghom given by Equation 3.368, however, must be multiplied by 2=(1 n), since the Poisson’s ratio contraction, accompanying the strain in one direction must be counterbalanced by a higher stress in the orthogonal direction [131]. Equation 3.369, thus needs to be modified only by changing (1 n)2 in the second term to (1 n). Sufficient data on lattice parameters, ma and mb are available with which to test the modified relationship on several thin-film bicrystals prepared by Matthews [120] and Jesser and Matthews [121,124,132]. In each of these studies, the thickness of the overgrowth was deliberately varied as a function of position in the bicrystal. The critical thickness, hc, at which misfit dislocations appeared, could thus be readily determined. Table 3.5 summarizes the key ancillary parameters employed in these calculations and compares the experimental and the calculated values of hc under two conditions. The first is from the direct use of Equation 3.369. The second takes into account the misfit p=2 greater than the critical value that is needed to provide the necessary activation energy for the formation of misfit dislocations [3]. In both calculations, the Burgers vector used is that of the dislocations shown to be present experimentally. The experimental values of hc range from 1.5 to 3 times greater than those calculated, doubtless at least in part, because of the difficulty of nucleating misfit dislocations. Both less complete experimental observations and greater complexity make the application of the modified form of Equation 3.369 to the broad faces of Widmanstätten plates less decisive. The calculations and results for g plates in aAl–Ag and for k plates in aCu-Si are also summarized in Table 3.5. In the case of Cu-Si, extensive observations indicate that the hc (calculated) is correct at least in the sense of a lower limit. Almost no misfit dislocations were observed at the broad faces of k plates despite an intensive search. These plates attained a maximum half-thickness of about 1.5 105 cm, as compared with a calculated hc, of 6.1 106 cm. The thickness at which coherency is lost at the broad faces of the g plates is a more complicated question. If an x-ray criterion
TABLE 3.5 Comparison of Theory and Experiment for Critical Thickness (hc) for Generation of Misfit Dislocationsa Overgrowths Item 0
dc navg a1(beff) A_ m (film or ppt) 1011 dyn=cm2 m (substrate or matrix) 1011 dyn=cm2 Calculation (1) Calculation (2)b Experimental
Precipitates
Au on Ag
Co on Cu
Fe on Cu
Co on Ni
Cu-Si, a!k
Al–Ag, a!g
Parameters 0.00179 0.40 1.44 2.78
0.0201 0.33 1.25 8.15
0.0169 0.32 0.25 8.3
0.00574 0.31 0.8375 8.15
0.00055 0.35 1.476 4.85
0.0083 0.34 1.647 3.42
3.0
4.85
4.85
7.45
4.85
3.42
6.7 8.9 17
24.2 33.6 50
610 1030 1500
27 48
Critical thickness (hc), A_ 110 5.1 157 6.7 250–300 20
Source: Aaronson, H.I. et al., Phase Transformations, ASM, Metals Park, OH, 1970, p. 313. a Data on n, a1, and m from Friedel, Dislocations, Addison Wesley Publishing Company, Reading MA, 1964, p. 454. b Includes an activation energy for the generation of misfit dislocation.
368
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
is used for the loss of coherency, the in situ transformation of coherent g0 to semi-coherent g occurs when the parameters of the hcp lattice undergo the appropriate changes [133,134]. On this basis, the electron micrographs of Hren and Thomas [135] indicate that the calculated hc of 28 Å is far too low, as some of the g0 plates shown by these investigators appear to have a half-thickness of about 150 Å. On the other hand, if the presence of a few misfit dislocations is taken to constitute loss of coherency—one possible electron microscopic definition of this event—then the calculated hc is too high. Hren and Thomas have shown that g0 thickens by the ledge-wise migration of these dislocations, which accordingly must have been present when the one-parameter-thick ‘‘nucleus’’ [135] began to thicken. Quite likely a few misfit dislocations can form on g0 broad faces even at very small values of h under the influence of local stress concentrations. The presence of these dislocations may delay the onset of a rate of nucleation of misfit dislocations sufficiently high to accomplish the conversion of g0 to g by reducing the driving force for such nucleation below the critical threshold. Turning now to the problem of the partial retention of coherency, Matthews [100] and van der Merwe [136] have critically tested Equation 3.371, the relationship for the strain, em, at which the energy of the system is a minimum under the condition of partial coherency. Matthews deposited, a monocrystalline film of PbSe atop one of PbS so as to form a {100} boundary; both films were about 450 Å thick. Dislocations of Burgers vector a=2 were found to accommodate 96% of the misfit, the remainder being taken up elastically. Moiré fringe spacings showed that the lattice spacings in each crystal were changed by 0.06% because of the residual coherency. Setting ma ¼ mb ¼ m and n ¼ 0.3, Equation 3.371 yields em ¼ 0.06%. However, an {001} overgrowth of Au on Pd was found experimentally to have accommodated about 13% of the misfit elastically, whereas Equation 3.371 predicts em ¼ 0.065%, and thus (alem)=(al a2) ¼ 1.4% of the misfit taken up in this manner [120]. The recurrent problem of nucleating a sufficient density of misfit dislocations is probably especially troublesome in respect of attempts to compare theory and experiment on the partial retention of coherency. On the other hand, there is some additional support of a qualitative nature available for the theory’s predictions in this area. It has been repeatedly shown that the density of the misfit dislocations increases with the thickness of the overgrowth [120–124]. Similarly, observations of inter-dislocation spacings larger than those called for by the operative misfit have been found at the broad faces of S laths in Al–Cu-Mg [109], g plates in Al–Ag [128], and probably the UC2 plates in U-C [102], and are indicative of the partial retention of coherency.
3.4.3 ACQUISITION OF THE MISFIT DISLOCATION STRUCTURE INTERPHASE BOUNDARIES
OF
PARTIALLY COHERENT
3.4.3.1 Theory Van der Merwe, Frank, and Brooks clearly understood that their theoretical studies furnished equilibrium descriptions of the structure of partially coherent or dislocation interphase boundaries. They were aware that the kinetics of generation of these dislocations might be slow enough to prevent such descriptions from being experimentally fulfilled. The comparison with experiment during relatively early stages of transformation already recounted indicate that this concern was well justified. Insofar as the crude comparisons that have been made between theory and the misfit dislocation structures developed after prolonged annealing at the isothermal reaction temperature allow valid conclusions to be drawn, it appears that the theoretical predictions made as to the equilibrium structure are correct. Hence, the present section represents the kinetic route taken from the fully coherent structures initially present to the equilibrium of partially coherent structures that eventually developed. Figures 3.67d and 3.69, and Equation 3.370 provide the thermodynamic basis for stating that, in general, the initial structure of the boundaries that later become partially coherent is one of full coherency. As long as the misfit at the conjugate habit planes is less than approximately 0.1, which is usually the case, though not always, and the ratio of the thickness, h, of the ‘‘thin’’ phase to the lattice parameter of that phase, al, is sufficiently small, the fully
369
Diffusional Growth
coherent interphase boundary will have a lower interfacial energy than the partially coherent one (Figure 3.69). Although, this figure shows that the h=al for reduction of interfacial energy by coherency loss does not become appreciable until the misfit, d0c , is less than approximately 0.04, the considerations of Chapter 2, Nucleation, indicated that very thin critical nuclei are to be expected if DG* is to be kept below 60kT. Hence, the replacement of full by partial coherency will usually occur sometime during the growth stage of a diffusional phase transformation. 3.4.3.1.1 Energetics of Dislocation Loops The relationships that Brown et al. [137] obtained for the line tension and for the interaction energy with the stress field of a spherical precipitate of a single prismatic dislocation loop were given as Equations 2.279 and 2.280. The minimum radius at which such a loop is stable at the interface of a spherical precipitate, rcrit, is that at which,
qgint qgself ¼ qr‘ qr‘
holds, where r‘ is the radius of the loop. Differentiating and rearranging, rcrit ¼
b 8rcrit 3 2n þ ln b 16pe(1 n) 4(1 n)
(3:373)
where r0, the dislocation core radius, has been set equal to b. Setting gint ¼ gself yields the radius r*,
b 8r* 3 2n ln 1þ r* ¼ 8pe(1 n) b 4(1 n)
(3:374)
When r > r*, semi-coherent growth (a precipitate with a loop) is energetically more favorable than coherent growth. Semi-coherent growth is prohibited when r < rcrit. However, even when r > r* coherent growth can continue indefinitely if no loop is generated. The r* of this theory is the spherical analogue of the Frank-van der Merwe dc for a planar boundary. There is, however, no counterpart for rcrit at planar boundaries. Although this theory is useful in its present form, it is incomplete, since three sets of orthogonal loops, with as many different Burgers vectors, or some rearrangement thereof, should normally be present at the interface of a semi-coherent sphere. The interaction energy among these loops must also be taken into account. However, rcrit is more realistic than r*, since the loss of coherency is often initiated by the formation of a single prismatic loop. Jesser [138] has extended this treatment to an approximation of the situation wherein a dislocation network is present around a spherical precipitate. In Equation 2.279, the line tension of a single dislocation loop is replaced by the sum of the energy of a fully coherent interphase boundary, treated in Chapter 2, and twice the energy of a planar array of parallel dislocations as given by Equation 3.361. The former energy is not defined further and the latter applies to a planar array consisting of two sets of parallel misfit dislocations. It was expected that the use of a relationship appropriate to a flat interface ‘‘will be adequate for many experimentally encountered situations.’’ In Equation 2.280, the interaction energy of the stress field of a spherical precipitate with a single prismatic dislocation loop, was not directly replaced. Instead, Jesser wrote just the total volume strain-energy associated with a misfitting spherical precipitate and the surrounding matrix W¼
8pr3 md2 (1 þ n) 3(1 n)
370
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
The energy per unit volume of precipitate is thus, WV ¼
2md2 (1 þ n) Ed2 Ee2 ¼ 1n 1n 1n
(3:375)
d is here defined in ‘‘macroscopic’’ terms as one-third of the fractional difference in the volume per atom in a unit cell of bulk matrix and the volume per atom in a unit cell of bulk precipitate, and R is the precipitate radius. Hence, the total energy of the system that contains one spherical precipitate is the product of the area of the sphere and the sum of the coherent and the misfit dislocation interfacial energies plus the strain energy. One may find the equilibrium misfit d0eq that minimizes the total energy from Equations 3.361 and 3.375 [138], 3b ln 2b(1 þ b2 )1=2 2b2 0 (3:376) deq ¼ 8p(1 þ n)r where b ¼ pd=(1 n). A critical radius, rc, also exists that is obtained by equating the misfit interfacial energy (Equation 3.361, doubled) to the elastic strain energy,
2b 2b2 3b ln e (3:377) rc ¼ 4p(1 þ n)d Note that this rc is essentially the r* of Equation 3.374. The total value of the interfacial energy and the elastic strain energy thus passes through a minimum as a function of d only when r > rc. Since the volume strain energy is proportional to d2, it rises slowly at first, whereas the misfit interfacial energy increases swiftly; as long as the two energies can be equal at some value of d, and thus the strain energy can exceed misfit interfacial energy at larger values, a minimum in the total energy can exist. As shown in Figure 3.80, once the critical radius is exceeded, the equilibrium misfit of Equation 3.376 decreases rapidly, approximately in proportion to l=r, as a consequence of the swift rise in volume strain energy with the precipitate radius. 3.4.3.1.2 Prismatic Loop Punching at Spherical Precipitates This mechanism consists of the formation of a pair of prismatic loops, one interstitial and the other of the vacancy type, at the interphase boundary of a misfitting precipitate crystal [104,140]. A prismatic dislocation loop is one whose Burgers vector lies out of the plane of the loop. The slip surface, defined by the dislocation line and its Burgers vector, is cylindrical. Hence, the loop can
δ΄eq
r rc
FIGURE 3.80 Equilibrium misfit, d0eq , schematically shown against precipitate radius r. (From Jesser, W.A., Philos. Mag., 19, 993, 1969. With permission from Taylor & Francis.)
371
Diffusional Growth
glide along the cylindrical surface, but expansion or contraction requires a climb. If the precipitate has a larger volume (or atom) than its matrix, the vacancy loop will be left at the boundary and the interstitial loop will be expelled into the matrix, and vice versa. Using Eshelby’s theory [141] of the misfitting inclusion, Weatherly [142] showed that when an ellipsoidal precipitate is elastically isotropic, the maximum stress at its surface is a function only of the misfit and of the elastic moduli of the precipitate and the matrix, and is not affected by the precipitate size. Unless the critical misfit is exceeded, however, prismatic punching cannot occur. On the other hand, as long as this requirement is fulfilled and r > rcrit, there should be no additional obstacle to the formation of loops. As the precipitate grows, the stress at its boundary rises until the theoretical shear stress is attained. A loop is then punched out, the stress falls, and the cycle begins again [109]. Applying Eshelby’s theory to various ellipsoidal shapes, Weatherly obtained the following relationships for the shear stress, ts, acting on a {111} slip plane, required to produce punching. For an ellipsoid, 2 ts ffi meT 3
(3:378)
where m is the shear modulus, eT is the total stress-free transformation strain that is 3d for a sphere. The equation of an ellipsoid can be written (x2 þ y2)=i2 þ (z2=j2) ¼ 1, where x, y, and z are distances along the three orthogonal coordinate axes, and i and j are constants. When j i, the crystal is a needle. Assuming the misfit to be almost entirely in the directions normal to the needle axis and negligible in the parallel direction, rffiffiffi 2 T me ts ffi 3
(3:379)
where eT d normal to the needle axis. For a plate, i j, and eT d at the edges, leading to the equation 3 ts ffi meT 2
(3:380)
These relationships for ts can be converted into approximate minimum values of eT on the following basis. Kelly [143] has shown that in typical fcc metals, the theoretical shear stress for a perfect crystal is m=20 at 300 K, m=25 at 600 K, and m=35 at 1200 K. At, say, 1200 K, for ellipsoids punching requires that 2=3meT > m=35, or that eT > 0.04. Similarly, eT > 0.01 for spheres and eT > 0.02 for needles and plates. The foregoing analysis by Weatherly requires that the shear stress exceed the theoretical shear strength of the matrix. It thus proposes criteria that should define an upper limit to the critical misfit for prismatic loop formation. The r* relationship, Equation 3.374, of Brown et al. [137] that can be directly rewritten into a critical misfit, ought to give a lower limit to this quantity, since it is simply the misfit that lowers the energy of the system at r* if the prismatic loop is generated at a misfitting spherical precipitate. Ashby and Johnson [144] analyzed the critical misfit and sphere radius for the generation of a shear loop bulging from the interface of a misfitting sphere. The shear loop may then transform, by cross-slip, into a prismatic loop. Nucleation of a shear loop, whose Burgers vector lies in the plane of the loop at the interphase boundary, seems a physically more realistic process. Figure 3.81 illustrates the model for loop formation and transformation. When the diameter of pthe segment of the shear loop, whose shape is roughly that of the segment of a circle, is roughly 2r,
372
Mechanisms of Diffusional Phase Transformations in Metals and Alloys b
(a)
Initial slip plane
Cross-slip plane (b)
(c)
(d)
FIGURE 3.81 Generation of prismatic loops at a particle. (From Ashby, M.F. and Johnson, L., Philos. Mag., 20, 1009, 1969. With permission from Taylor & Francis.)
where r is the particle radius, the screw sides of the segment of the loop lie parallel to the cross-slip planes on which the shear stress is greatest. The shear stress of the particle does the greatest amount of work if cross-slip then occurs, as shown in the figure. Further cross-slip generates a prismatic loop. If the cross-slip is difficult or obstructed by other dislocations, a less regular loop may form. A critical size of the dislocation loop segment exists. Below this size, the loop is unstable; at larger sizes, the loop is stable. The critical size occurs before the onset of the first cross-slip. Hence, the critical step is the generation of the stable segment, not the cross-slipping process. Figure 3.82 shows the dislocation loop during the nucleation stage. The work, W, done by the component txz of the stress field of the particle as the loop expands over the area dA is given by W¼
ð
txz bdA
(3:381)
loop
y
c
rl
Dislocation loop x Box
FIGURE 3.82 Early stage of nucleation of a loop of radius rl around a particle. (From Ashby, M.F. and Johnson, L., Philos. Mag., 20, 1009, 1969. With permission from Taylor & Francis.)
373
Diffusional Growth
where txz is the shear stress in plane xz b is the length of the Burgers vector The work is calculated approximately by replacing the dislocation loop by a rectangle of the same area, shown by the broken line in the figure. One must also consider the line tension or self-energy (in effect interfacial energy) required to create and expand the loop. Idealizing the segment as part of a circular loop, and bearing in mind that the portion of the particle’s cross section shown by a heavy dashed arc represents the portion of the loop left at the particle:matrix interface, one may roughly approximate the energy of the shear loop as that of a complete loop of radius r‘. The energy for such a loop is given by Chou and Eshelby [145] as E‘ ¼
mb2 r‘ 2 n 8r‘ 2 ln 4 1n b
(3:382)
Thus, the change in energy of the system, DE, as a function of the loop radius is simply DE ¼ E‘ W
(3:383)
Figure 3.83 plots DE vs. r‘ in units of the particle radius Rp for various values of eC11 , where this quantity is known as the ‘‘constrained strain,’’ following the terminology of Eshelby.* Examining Figure 3.83, it is seen that when eC11 ¼ 0, DE rises almost linearly with the radius of the loop. With increasing misfit, the DE vs. r‘ curve changes shape until at eC11 ¼ 0:0061 (curve B), the energy becomes just negative at r‘ ¼ 0.67 Rp. Hence, such a loop will lower the energy of the system. However, it is still necessary that the energy of the system first be increased by DE* ¼ 4.8 1010 ergs (4.8 1017 J), whereas kT at room temperature is 4.2 l014 ergs (4.2 l021 J). At still larger misfit the energy of the system passes through the lower minima, and finally at eC11 ffi 0:05 (curve A), DE* ¼ 0 and DE is negative from the outset. Note that this figure applies to a particular particle radius, here with its radius equal to 1000 Å. Figure 3.86 shows the variation of eC11 at which DE just goes negative as Rp increases. * On the constrained strain and the stress-free strain: Assume that a misfitting spherical particle of radius Rp is removed from its hole in the matrix and that any stresses due to its presence in the matrix are relaxed. If the size or shape of the particle differs from that of the relaxed hole, then the misfit exists. The ‘‘stress-free strain,’’ eT11 ¼ eT22 ¼ eT33 , is given by eT11 ¼
Rp RH RH
(3:384)
where RH is the radius of the relaxed hole, also assumed to be spherical. Now insert the particle into the hole and re-bond the interphase boundary. Strains are again imposed since the particle and the hole are forced to share the same radius. A second measure of the misfit, termed the ‘‘constrained strain,’’ describes the new state of strain of the particle and is related to eT11 by an expression derived by Mott and Nabarro [139]: eC11 ¼
3Kp eT 3Kp þ 4m 11
where Kp is the bulk modulus of the particle m is the shear modulus of the matrix Since the constrained strain is directly related to the stress field in the matrix, results are plotted in terms of it.
(3:385)
374
Mechanisms of Diffusional Phase Transformations in Metals and Alloys Rp = 1000Å
e C11= 0 .004 e C11= 0 .005
Energy ΔE (ergs) ×1010
20 18 16 14 12 10 8 6 4 2 0 –2 –4 –6 –8 –10 –12
e C11= 0
e C11= 0 .006
ΔE *
B
e C11= 0 .0061 e C11= 0 .0065
e C11= 0 .01 e C11= 0 .05
e C11= 0 .007
A 0
0.1
0.2
0.3
0.4
0.5 0.6 0.7 0.8 rloop in units of Rp
0.9
1.0
1.1
1.2
FIGURE 3.83 Energy change as a loop bulges out of the interface of a particle of radius 1000 Å. If the misfit is increased to 0.05, the energy barrier vanishes. (From Ashby, M.F. and Johnson, L., Philos. Mag., 20, 1009, 1969. With permission from Taylor & Francis.)
From Figure 3.83, two critical values of misfit for the nucleation of a shear loop may be identified. (1) An upper limit to the misfit for this process is that DE* ¼ 0. This leads to a particle size-independent critical misfit, eC11 ffi 0:05, as shown with curve A. Once the misfit reaches this level nucleation of a shear loop must occur. This criterion is essentially the same as that of Weatherly for a sphere (see discussion below Equation 3.380), though it is numerically higher because of a dislocation core problem. (2) The lower limit to the critical misfit is that the curve just goes negative, for example, curve B. Hence, a dislocation can form only if there exists a position for it that lowers the energy of the system. Under this circumstance, though, the ‘‘activation hump’’ must be surmounted by work obtained from a source other than that of the misfit, for example, an applied stress field or a supersaturation of point defects. Brown and Woolhouse [146] further extended the treatment of Brown et al. [137] to include nucleation by prismatic punching and nucleation by the condensation of point defects. Again, the misfitting spherical precipitate was studied. Additionally they noted specific examples to which r* and rcrit (Equations 3.374 and 3.373) apply and derived two other critical-type radii of the precipitate. A dislocation lying wholly within the matrix phase will be attracted to a precipitate that is fully coherent by the stress field surrounding the precipitate. However, if such a dislocation climbing and gliding to the precipitate finds that its radius does not exceed rcrit, the dislocation will simply move through the precipitate and into the matrix again. The r* must be exceeded if a dislocation loop, say formed by vacancy condensation, is to be captured by the coherent interphase boundary. Another critical radius must be calculated for the achievement of partial coherency by a single act of prismatic loop punching. If two loops of equal and opposite signs are formed and one is punched off to infinity, the energy of the system is changed by twice the self-energy of a single loop plus the interaction energy of one of them with the precipitate. This radius is
b 8r2 3 2n 1þ ln , r2 ¼ pffiffiffi b 4(1 n) 2 2pe(1 n)
(3:386)
375
Diffusional Growth
which is called r2 because it is a ‘‘two loop’’ criterion. The r2 is approximately six times larger than rcrit. If a matrix dislocation can maneuver itself into an interface dislocation loop plus another, external loop, the loops do not now have to lie on the glide cylinder with the maximum punching stress, but can be as large as possible. If they have the radius of the precipitate this process is feasible when r2 r0 ¼ pffiffiffi : 2
(3:387)
To examine the nucleation of the prismatic loops, Brown and Woolhouse [146] took a somewhat different view of the situation than Ashby and Johnson [144]. As shown in Figure 3.84, when the loop is very small it will exist as a shear loop on the most highly stressed slip plane passing through the interface of the precipitate. As the shear loop grows, it will tend to cross slip to form two prismatic loops. The assumption is made that the loop will slip on the cylinder whose generators pass through the final prismatic loops (Figure 3.84a). On this cylinder, the loop has screw segments of length rs and edge segments of length re. As the shear loop grows, the screw segments will travel farther around the glide cylinder until they mutually annihilate. The configuration then becomes that of Figure 3.84b, and the energy now depends upon the separation between the two loops, x. The energy of the configuration is represented as a function of rs and x. Assuming that the configuration will always take up its lowest form, the configuration with the lower energy at all points in the punching process has been chosen. The model of Figure 3.84a yields ecrit ¼
1 6pa
(3:388)
where a is the proportionality constant in the assumed expression, and the core radius of a dislocation is ab. When ecrit is exceeded, a prismatic dislocation loop will form. At smaller values of e, the precipitate cannot spontaneously acquire a prismatic dislocation loop. The configuration of Figure 3.84b yields ecrit ¼
1 12pa(1 n)
(3:389)
When n ¼ l=3, this value of ecrit differs from that of the preceding equation by only 30%. On the Peierls model of interatomic forces, a 1, in which case ecrit 0.05. These calculations also show that thermal activation cannot result in coherency loss unless the precipitate is extremely large.
re r0 r /√2 0 x
rs
(a)
(b)
FIGURE 3.84 (a) Shows a shear loop spreading on the glide cylinder whose axis is the direction of the Burgers vector and whose generators pass through the points of maximum shear stress at the interface of the particle, and (b) shows the loop after its screw segments have mutually annihilated, leaving two prismatic p edge dislocation loops of radius r0= 2 and a distance apart. (From Ashby, M.F. and Johnson, L., Philos. Mag., 20, 1009, 1969. With permission from Taylor & Francis.)
376
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
For example, a Co precipitate in a Cu–Co alloy, approximately 1000b in radius, still requires approximately 40 eV of thermal energy to generate a misfit dislocation. A similar analysis was made of the nucleation of dislocation loops by the condensation of vacancies and it was concluded that homogeneous nucleation is effectively impossible because the necessary supersaturations of vacancies cannot be generated. Heterogeneous nucleation of loops, on precipitates, could only occur when both the misfit and the vacancy supersaturation are quite large, for example, e ¼ 0.03 and a supersaturation ratio of approximately 15. 3.4.3.1.3 Adsorption of Matrix Dislocations at Interphase Boundaries The driving force for this mechanism is the excess elastic strain energy at interphase boundaries resulting from coherency. Any dislocation within the matrix can react with the coherency stress field of a sphere. When the precipitate is a needle or a plate, however, interaction can occur only at certain orientations of the dislocations relative to the precipitate [109]. For all of these morphologies, a critical dimension (the equivalent of r* for a sphere) must be exceeded before adsorption can occur.* At least in the case of flat plates, there is a region of thickness in which partial coherency is energetically favorable, see Section 3.4.2.1.2. Adsorption will cease under this circumstance prior to the attainment of the equilibrium complement of misfit dislocations. When the precipitate and=or the matrix is elastically anisotropic—as is usually the case—the driving force for matrix dislocation adsorption at the boundary of a coherent precipitate increases with precipitate size even for the ellipsoidal morphologies. In general, since the adsorption process involves climb as well as glide, it is markedly dependent upon temperature. This mechanism is a very disorderly means for the acquisition of misfit dislocations by coherent precipitates initially, as it makes the kinetics of acquisition dependent to a large degree upon the distribution and nature of the matrix phase dislocations. A somewhat more controlled aspect of the same mechanism can develop, however, when a matrix dislocation passes through a precipitate at a relatively sharp angle to the interphase boundary. Such a dislocation may have been the nucleation site of a precipitate, or have survived the growth of a precipitate through it, or it may have been produced by the transformation strains associated with the growth of a neighboring precipitate crystal. Such dislocations are often found, for example, in the bicrystals produced by the oriented overgrowth method [123]. In order that part of the dislocation be extended along the interphase boundary and thus accommodate misfit, the additional line tension (gself of Equation 2.279) must be counterbalanced by the decrease in coherency stresses. Jesser and Matthews [123] concluded, however, that such extension, even when it produces an ‘‘inefficient’’ misfit dislocation, could be a more rapid process than the nucleation of dislocations of optimal Burgers vector and orientation. They derived equations for the critical thickness, and overgrowth must attain (or, approximately, for the critical half-thickness which a precipitate plate must reach) for coherency stress relief to offset the added line tension for (complete) a=2 [132] and (partial) a=6 [121] dislocations. They are hcomplete ¼
mb(1 n) 4ma d(1 þ n) cos c
(3:390)
and hpartial ¼
mb(1 n) 4ma d(1 þ n) cos c {[2g(1 n)=b cos x]}
* A matrix dislocation forced into an interphase boundary by an applied stress can be retained as r > rcrit.
(3:391)
377
Diffusional Growth
where m is the shear modulus of the interphase boundary ma is the shear modulus of the overgrowth or precipitate g is the stacking fault energy c is the angle between the surface of the overgrowth and the normal to the slip plane x is the angle between the direction and the direction in the plane of the overgrowth that is perpendicular to the intersection line of the slip plane and the specimen surface 3.4.3.1.4 Nucleation of Misfit Dislocations at Interphase Boundaries Frank and van der Merwe [3] and Matthews have suggested that dislocations might nucleate where an interphase boundary produced by the oriented overgrowth method met the edges of the bicrystal. Laird and Aaronson [147], monitoring continuously the growth of g plates in Al-15% Ag by means of hot-stage transmission electron microscopy, observed directly what appeared to be the operation of the equivalent mechanism in a precipitation reaction. As illustrated in the sketch of Figure 3.85, the distance between a Shockley partial misfit dislocation and the edge of the g plate increased as the plate lengthened. When this distance reached a value roughly twice the equilibrium spacing, a new dislocation was seen to appear at the edge. This dislocation may have been nucleated at the junction between the broad face and the edge of the plate. The maximum shear strain relieved by a misfit dislocation at the equilibrium inter-dislocation spacing is the ratio of the misfit to the interplanar spacing ¼ b=(am=3) ¼ 3.56 103, where m is the dislocation repeat distance. At twice the equilibrium inter-dislocation spacing, this strain is doubled and is equivalent to a shear stress of m=140. This represents about one quarter of Kelly’s [143] value of the shear stress required to produce flow in a perfect fcc crystal at this homologous temperature, indicating that the edges of g plates may serve as stress concentrators and thereby aid materially the production of new misfit dislocations. This nucleation mechanism appears most likely to be operative when the interface planes are the slip planes in both phases, as in the a ! g transformation in Al–Ag. This geometry allows maximum advantage to be taken of the coherency stresses in the nucleation of the misfit dislocations, and virtually insures that the Burgers vector of the dislocations will lie in the plane of the boundary and, when the dislocations are complete, allows them to glide within the boundary plane and thus redistribute themselves more readily into equilibrium positions. When this geometric condition is not met, for example, the {100} habit plane of u0 plates and most of Matthews’ overgrowth bicrystals, the operation of another mechanism of dislocation generation probably becomes more likely. 3.4.3.1.5 Nucleation at Grain Boundaries and Free Surfaces This mechanism is most likely to operate in thin oriented overgrowths containing a relatively low density of single-phase dislocations. Matthews [120] suggested that dislocations nucleated at the free surface of Au upon Ag bicrystals and were then drawn into the boundary. However, a precipitate plate whose area is small relative to that of a grain boundary or the surface of a specimen
Equilibrium dislocation spacing (a)
A΄ A
Lengthening direction (b)
A΄ A˝ A
(c)
A΄ A˝ A
FIGURE 3.85 Mechanism through which a=6 dislocations are observed to be introduced onto a broad face during the lengthening of a g plate in Al-l5% Ag. (Reprinted from Laird, C. and Aaronson, H.I., Acta Metall., 17, 505, 1969. With permission from Elsevier.)
378
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
and which is formed a long distance from such an interface relative to its largest dimension is unlikely to exert a stress upon the interface sufficiently large to nucleate a dislocation at that location. 3.4.3.1.6 Nucleation of Dislocation Loops within Precipitates Weatherly and Nicholson [109] have suggested that dislocation loops observed to form within u0 plates do so by the condensation of vacancies ‘‘built into’’ the u0 lattice by reason of its slight departure from stoichiometry [148]. Such vacancies, however, are essential to the stability of a defect lattice. Unless the composition of u0 varies with time, therefore, these vacancies cannot be regarded as ‘‘disposable,’’ in the manner of quenched-in thermal vacancies. A more likely array of possibilities, particularized to fcc precipitates, has been described by Baker, Brandon, and Nutting [149]. They note that a loop formed within a precipitate by adsorption of either interstitials (if under tension) or vacancies (under compression) is expected to collapse promptly—if not actually while it forms—to produce a stacking fault. If the stacking fault energy, g, is low, the loop will survive and be circumscribed by an a=3 Frank sessile dislocation. If g is high, nucleation of a Shockley partial is anticipated, leading to the reaction a a a h111i þ h112i ! h110i 3 6 2 In either case, if the habit plane is close packed, introduction of an a=3 -enclosed stacking fault can relieve all of the stresses. Baker et al. [149] have considered some rather elaborate mechanisms by which the a loops appropriate to this situation may be able to form. 3.4.3.2 Comparisons of Theory with Experiment These comparisons will be largely limited to experimental results that provide a definite test of the mechanism or theory under consideration, or at least serve further to illuminate the topic. 3.4.3.2.1 Energetics of Dislocation Loops While r* is very difficult to check experimentally because of the problem of nucleating misfit dislocations, rcrit should be subject to a relatively accurate check because it is essentially a question of dislocation ‘‘thermodynamics’’ rather than kinetics. Substituting n ¼ 1=3, b ¼ 2.56 Å and e ¼ 0.013, Brown et al. [137] found rcrit for Co-rich precipitates (fcc) formed in an fcc Cu–Co alloy to be 32 Å. The initial attempts to test this prediction experimentally yielded discouraging results. The smallest precipitates made semi-coherent by either deformation [150] or neutron irradiation [137] (the latter followed by annealing at 4008C) had radii of about 125 Å. Recently, however, bombardment with 450 kV electrons was shown by Woolhouse [151] to make precipitates as small as r ¼ 50 Å semi-coherent. The electron irradiation produced small loops, usually within the matrix but sometimes evidently, at the interphase boundaries, which swiftly migrated to the precipitates and caused loss of coherency. On varied evidence, it appears that the loops were of the interstitial type, consistent with the fact that Co-rich precipitates form with a decrease in volume. Particularly considering the previously noted incompleteness of the theory, it appears that Woolhouse’s experiment provides a good support for the rcrit aspect of the theory of Brown et al. [137]. 3.4.3.2.2 Prismatic Loop Punching and Further Energetics Comparisons Table 3.6 shows that the Weatherly ‘‘upper bound’’ treatment of prismatic loop punching successfully predicts the absence of punching but not always its presence. Weatherly [142] ascribed the failure in the case of Al2O3 precipitates in a Cu-rich matrix to inaccuracy in Kelly’s [143] estimates of the deformation that a dislocation-free region of an fcc lattice will tolerate before a dislocation is nucleated. He also noted that if prismatic loops disappear in alloys as rapidly, as they do in pure Al, where the lifetime is only one minute at 2508C, unless the growth rate of a precipitate is high, the
379
Diffusional Growth
TABLE 3.6 Tests of the Prediction That Precipitates Punch Prismatic Loops When the Theoretical Shear Stress for Slip is Exceeded System
Precipitates
eT
ts
ttheor
Punching
Ni–Al Ni-Cr-Ti-Al Cu–Co Cu-MgO Cu-Al2O3.
Ni3Al Ni3(Ti, Cr)Al Co MgO Al2O3
0.02 0.008 0.054 >0.05 >0.05
m=75 m=187 m=28 m=3 m=7
m=35 m=35 m=30 m=35 m=35
No No No Yes No
Source: Weatherly, G.C., Philos. Mag., 17, 791, 1968. With permission from Taylor & Francis. Note: ts is the calculated shear stress on a {111} slip plane due to eT. ttheor is the theoretical shear stress.
probability of finding a loop in its vicinity in a quenched specimen is so small that such observations provide an unsatisfactory means of testing the theory. When used with care, high temperature transmission electron microscopy is probably the best technique for this purpose, as the experiments of Eikum and Thomas [152] imply. Jesser [138] compared the rc calculated from Equation 3.377 with the observed radius at which TEM evidence was found in the literature for loss of complete coherency. Note first the calculated values, 73 Å vs. 120–150 Å in Cu–Co, 1333 Å vs. 3000 Å in Nimonic 80 Å, 197 Å vs. 250–300 Å in Cu-Fe. In the case of Cu–Co, disappearance of the Ashby-Brown [153] line of no-contrast was the criterion used. Direct TEM observation was employed and the experimental results are doubtless valid. In Cu-Fe, a tangle of interfacial dislocations was reported. The Cu–Co results are somewhat uncertain and need further experimental study. Those on Cu-Fe also seem unclear. Nonetheless, the results are quite similar to those that Weatherly found, namely, that theory tends to underestimate the ability of an interphase boundary to retain full coherency. Ashby et al. [154] reported a set of carefully conceived experiments that have both encouraged and guided the development of more exact theory. Amorphous spheres of SiO2 were formed within Cu-Si single crystals by internal oxidation. TEM was then used to determine the minimum particle diameter as a function of hydrostatic pressure at which dislocations nucleate at the interfaces of the spheres. From Young’s and the shear moduli of the matrix and precipitate phases, the hydrostatic pressure was converted to the stress-free transformation strain, eT11 . Ashby and Johnson [144] tested their upper and lower limiting criteria for the misfit needed to nucleate a shear loop using the results of the foregoing experiments and whatever other relevant information they could find. Figure 3.86 shows that, as previously described the upper limiting criterion is a constant misfit of approximately 0.05 that is independent of precipitate particle radius. The lower limit, on the other hand, which is the loop, and thus the related particle radius at which the DE vs. radius curve just becomes negative as a function of misfit, is markedly dependent upon particle radius. In addition to the data of Ashby et al. on Cu-SiO2 [153], those of Weatherly [159], who induced misfit of NbC particles in Fe and of SiO2 particles in Cu-Si by quenching and of Philips [155] and of Brown et al. [137] who produced loss of full coherency in Cu–Co by plastic deformation, were incorporated. The cross-hatched ‘‘coherent particles’’ region is taken from the work of Weatherly [142] who made these estimates from observations on dislocation generation at coherent Al2O3 and MgO particles in Cu-rich matrices.
380
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 10–1
Criterion A upper limit
Critical misfit eC11
Criterion B lower limit
10–2
Coherent particles, (Weatherly 1968)
Cu-Co (Philips 1964, Brown et al. 1968)
Fe-NbC (Weatherly 1968) Cu-Sio2 (Weatherly 1968)
Cu-Sio2 (Ashby et al. 1969)
10–3 10
10,000
1,000
100
100,000
Particle diameter 2Rp (Å)
FIGURE 3.86 Two criteria, one defining an upper bound to the critical misfit required for dislocation nucleation (A), and the other defining a lower bound (B). (From Ashby, M.F. and Johnson, L., Philos. Mag., 20, 1009, 1969. With permission from Taylor & Francis.)
The experimental results are seen to fall into two groups. Fully coherent particles lose coherency by dislocation generation only when the misfit is very large, close to the upper bound. For particles with disordered interphase boundaries (Cu-SiO2) or partially coherent interface (presumably Fe-NbC), the data fall in the vicinity of the lower bound. How the data for Cu–Co are to be treated is not clear. Ashby and Johnson [144] concluded that the interfaces of the ‘‘lower bound’’ particles contain line defects that can bulge from the interface when the misfit is present. The configurations through which such a dislocation passes will differ from those assumed in the Ashby-Johnson calculation, leading to a lower DE*. However, other means of reducing this energy barrier are apparent. Internal stresses due to quenching or nearby particles and deviations from the spherical shape that locally intensify the stress field can also make dislocation nucleation easier. These explanations may be appropriate for understanding why in the case of the ‘‘lower bound’’ experiments the experimental data lie below the theoretical prediction, rather than the other way round as the data so far treated. This result also suggests that the ‘‘forced’’ nucleation of dislocations used for all of the data on this plot may make this information inadequate for the testing of the theory. Turning attention now to the comparisons between theory and experiment reported by Brown and Woolhouse [146], both the results of the theory and the experimental data employed are summarized in Figure 3.87. Consider first the ecrit ¼ 0.05 line, the ‘‘upper bound’’ criterion for the initiation of misfit dislocation formation. Weatherly has suggested that ecrit 0.1 on the basis of experimental data mostly based on internally oxidized Cu-Al2O3 alloys (actually, the metallic solid solution contains dissolved Al). The alumina particles have a mismatch of approximately 0.08 and appear to be coherent. Woolhouse and Brown [156] dispute this. They report that the alumina particles have a lattice parameter characteristic of unstrained alumina even when the particles are very small. The authors conclude that Cu-Al2O3 is a system in which dislocations can be generated by the particle stresses and that in Cu–Co they cannot. Hence, they proposed that the former system could be used to place an upper limit on ecrit and the latter a lower limit on it. From data by Ashby–Brown [153] and Woolhouse–Brown [156] 0:013 < ecrit < 0:077
or
m m < ecrit < : 26 4
381
Diffusional Growth D΄ C΄ B΄ A΄–1
εcrit = 0.05 32
–2
1
5
log10 ε
4
69 7 8
r2
–3
ΔE approach
r* rcrit –4
0
1
2
3
A
B C D
log10 r0/b
FIGURE 3.87 Criteria for stability of coherent precipitate and generation of prismatic loop. (From Brown, L.M. and Woolhouse, G.R., Philos. Mag., 21, 329, 1970. With permission from Taylor & Francis.)
The lower limit corresponds well to Kelly’s estimate of the theoretical strength, that is, the stress at which homogeneous sliding of the atomic planes occurs. In addition, the lower limit is approximately fivefold greater than the estimate from Kelly’s theory [143] for dislocation generation in Cu at 900 K. After considering various possibilities for the discrepancies, it is concluded that the strength of the perfect lattice is considerably greater than previously assumed. A similar conclusion was earlier reached by Weatherly in respect of the discrepancies between calculated criteria for prismatic loop punching and experimental observations on this phenomenon, as summarized in Table 3.6. Consider now the various size-dependent parameters for the generation of misfit dislocations, or in effect the ‘‘lower bound’’ type of criteria discussed by Brown and Woolhouse [146]. The following discussions refer to the numbered experimental points in Figure 3.87. (1) Bonar [157] and Phillips [155] find that cobalt particles in copper undergo deformation-induced coherency loss provided they have radii greater than about 100 25 Å. Here, e ¼ 0.013 and the point labeled No.1 may be placed on the graph. There is good agreement between theory and experiment. (2) Bonar [157] observed that when cobalt particles are very small, they did not decorate dislocation lines, but that when they were larger, they did. His data suggest that the critical mean radius for this decoration to occur is about 50 Å. This point is shown as No.2, and we may identify this radius as r*. (3) Woolhouse [151] observed coherency loss of the smallest observable precipitates after intensive electron irradiation in the 750 kV electron microscope. Thus, when interstitial loops are already present, even very small precipitates can lose coherency. (4) Easterling [158] observed deformation induced coherency loss in the Cu-Fe system. The critical size for this process is about 200 Å (estimated from his micrographs) and the misfit is 0.0085 (from his data). The agreement between theory and experiment is again good. (5), (6), (7), and (8) Ashby et al. [154], in their beautiful study of dislocation generation at pressurized silica particles in copper, observed the particle size-misfit relationship plotted here. (The relationship between e and their quoted quantity is e ¼ tmax=3m.) It will be seen that dislocations are generated when the two-loop criterion is satisfied, and that there is apparently no barrier to the production of dislocations.
382
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
(9) Weatherly [159] observed for silica particles in copper quenched to produce strains due to differential thermal expansion that a mismatch of 5.3 l03 produced no dislocations around particles of radius 1000 Å, but did produce dislocations around particles of radius 2000 Å. The data presented by Ashby et al. [162] for alumina particles in copper do not appear to fit the two-loop criterion. For a given particle size, the alumina particles require a value of e about eight times less than the silica particles to generate dislocations. The alumina particles are in the form of thin triangular plates, and until an analysis of the regions of maximum shear stress for this shape is carried out, it is not possible to write down the energy balance to derive the two-loop criterion. It appears from the data that the theory works quite well in explaining the observations on coherent precipitates. The surprising results are those for the incoherent precipitates (5, 6, 7, 8, and 9). In these cases, it would appear that there is no barrier to the production of dislocations, and the interface acts as an ideal source of dislocations. It can produce them when the energy of the system is lowered by their presence. Clearly, one can think of the dislocations as already existing in the interface [159], or of the atomic steps in the interface as producing sufficient stress concentration to remove the barrier in the curves of Figure 3.88. Vincent [160] has observed another example of an ideal source of dislocations. Here, the surface of an epitaxial island of tin appears to be able to produce dislocations in just such a number to minimize the energy of the system. One may perhaps note that if crystal surfaces of very large radius of curvature are similarly able to produce dislocations, then the origin of dislocations in the crystals may well be grain boundaries and external surfaces. The theory as it now stands is appropriate to the simplest type of model system: elastically homogeneous, and with a spherical source of misfit. It can clearly be extended to situations that are more complex. It is clear, however, that for each geometry of precipitate, both the necessary conditions for the generation of misfit dislocations and the critical value of the misfit can be derived. Accordingly, similar principles will govern the generation of dislocations and the loss of coherency in these systems, although the detailed values of the critical misfit and critical sizes will be different.
60
20 r0/b = 50
40 20
E/μb3
0
60 10
20
30
40
50
70
80
–20
90 70
–40
0
50
60
70
80 90
10 20
–80
–120
–120
40
–60
–80 100
30
–40
–100
ε = 0.015
20
–20
–60
–100
(a)
60
20
–140
ε = 0.05
r0/b = 50
–160
1000 x/b
(b)
x/b
FIGURE 3.88 Variation of energy during the punching process. In (a) the parts of the curves before the discontinuities refer to the energy of configuration of Figure 3.89a. In (b) the parts of the curves after the discontinuities refer to the energy of configuration of Figure 3.89b. (From Brown, L.M. and Woolhouse, G.R., Philos. Mag., 21, 329, 1970. With permission from Taylor & Francis.)
383
Diffusional Growth
A (a)
(b) A
(c)
B
B (d)
FIGURE 3.89 The four stages for the formation of a helical misfit dislocation at a spherical particle by cross-slip of a screw dislocation. (From Matthews, J.W., Scripta Metall., 5, 1053, 1971. With permission from Elsevier.)
3.4.3.2.3 Adsorption of Matrix Dislocations at Interphase Boundaries Weatherly and Nicholson [109] have made a number of observations on the progressive loss of coherency of g0 spheres in Ni–Al-Cr-Ti, S laths in Al–Cu-Mg, and b0 in Al-Si-Mg that provide convincing evidence of the general importance of this mechanism when the level of misfit is relatively low. Complex climb mechanisms are involved. For example, in the cases of both S laths and b0 needles, a matrix dislocation climbs to the precipitate and then either positions itself along one interface or loops around it. Individual loops are then punched off, with the spacing between them eventually becoming uniform. Matthews [161] provides the specific mechanism for introducing misfit dislocations at a spherical particle as shown in Figure 3.89. A screw dislocation approaching a particle undergoes cross-slip, wraps a helical misfit dislocation round the particle, forms prismatic loops in the adjacent matrix and passes on. The micrograph in Figure 3.90 gives an example of a related situation in which this mechanism may have operated, though the precipitates are plates, of a nitride in a Nb-N alloy, rather than spheres.
g –
0.1 μ
FIGURE 3.90 Misfitting nitride particles in Nb-0.21%N aged at 808K and associated dislocation structures quite possibly formed by the mechanism shown in Figure 3.85. (From Dahlstrom, N. and Eyre, B.L., Met. Sci., 6, 96, 1972. With permission.)
384
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Wheatherly and Nicholson [109] point out that the loss of coherency of g0 spheres by the matrix dislocation adsorption mechanism requires interaction with at least two dislocations of different Burgers vector. The formation of a third dislocation by interaction of the first two allows a mesh of hexagonal network to be formed. Numerous repetitions of this reaction are needed to complete the network, making this a very slow process, but evidently the only one available. Jesser and Matthews [121,132] have made experimental tests of Equations 3.390 and 3.391, and their relationships for the critical thickness that an overgrowth must attain to force a dislocation intercepting an interphase boundary to enter the boundary when it is a complete or a partial dislocation, respectively. For overgrowths of both Co on a Cu substrate and of Fe on Cu, the experimentally determined overgrowth thickness required to produce misfit dislocations was about 20 Å. The calculated thickness for a=2 dislocations was 36 and 55 Å, respectively. Application of the Frank-van der Merwe [3] activation energy factor of p=2 to the misfit results in good agreement only for the Co on Cu case. The calculated critical thickness for forcing a=6 dislocations into Co on Cu boundaries is 18 Å, but it seems possible that dissociation of the a=2 dislocations into a=6 ’s occurred after the former dislocations had been emplaced in the interphase boundary. Thus, the difficulty of generating misfit dislocations appears to extend even to the situation in which dislocations that already intercept the boundary are being utilized for this purpose. A further illustration of the difficulty of emplacing a dislocation absorbed from the matrix at an interphase boundary is provided by the overgrowth-on-substrate study of Snyman and Engelbrecht [163]. The problem illustrated is specific for the fcc crystal structure and for the crystallography employed in their experiment, and has a degree of material dependency too in respect of the tendency of dislocations to dissociate. However, the implication that the detailed mechanism of the adsorption of a matrix dislocation must be taken into account is quite clear. They deposited a monocrystalline thin film of Ag on thin Au single crystals, but with a {111}, rather than the usual {100} plane at the interphase boundary. The a=2 dislocations, presumably nucleated at surface sources that glided to the boundary were found to have dissociated into about equal numbers of extrinsic and intrinsic stacking faults as a result of the habit planes selected. The mechanism proposed for the formation of these faults is sketched in Figure 3.91. Since these dislocations cannot glide along the interphase boundary in response to the coherency strain driving force for this extension, the authors suggest that they may climb away from the interface for a short distance with the assistance of electron beam heating and then glide back. Manifold repetition of this sequence displaces the dislocation from A to B, as shown in Figure 3.91b. Deposition at the interface of atoms removed from the half-plane during climb builds up the extra {111} plane of an extrinsic stacking fault. As shown in Figure 3.91c, if the net pressure of the redistributive forces encourages passage of the dislocation in the reverse direction, an intrinsic stacking fault results. In both cases, the a=2 dislocation dissociates into a Frank and a Shockley partial. 3.4.3.2.4 Nucleation of Misfit Dislocations at Interphase Boundaries Nemoto et al. [164] and Sankaran and Laird [126] have observed nucleation of a dislocations at the edges of u0 plates in Al–Cu alloys. These dislocations glide along the {100}a interphase boundaries to appropriate positions in the misfit dislocation array. A similar situation obtains at the broad faces of h plates in an Al-Au alloy also investigated by Sankaran and Laird. The hot-stage TEM pictures (Figure 3.92) indicate that the misfit dislocations nucleate probably as small loops at a number of positions along the edges of the plates, at the lines of intersection with the broad faces. The loops then spread inward. Loop generation and migration continues until the misfit at the broad faces has been well (though likely not completely) compensated. Subsequently the a misfit dislocations are replaced by a=2 dislocations as described earlier. Figure 3.71e due to Weatherly and Nicholson [109] indicates that misfit dislocations may nucleate at the tips of b0 needles in an Al-Si-Mg alloy. The misfit should be concentrated at needle tips, both structurally and geometrically, and hence facilitate formation of a misfit dislocation. Matthew [165] presented a particularly striking mechanism for the nucleation of misfit dislocation at the largearea interphase boundaries formed by monocrystalline overgrowths atop monocrystalline substrates.
385
Diffusional Growth (111)
1 –– — 2 [ 1 10] A
(a)
B
(b)
B
A
(c)
FIGURE 3.91 Mechanism through which an a=2 dislocation at a {111} interphase boundary, (a) forms an extrinsic stacking fault, or (b) an intrinsic stacking fault, (c) during extension along the boundary. (From Snyman, H.C. and Engelbrecht, J.A., Acta Metall., 21, 479, 1973. With permission from Elsevier.)
B
020
020
A
0.2 μ (a)
(b)
FIGURE 3.92 An h precipitate (formed by isothermal ageing at 523 K for 24 h) in Al-Au alloy showing the initial stages in the formation of misfit dislocations by loops at the periphery of the broad face. (a) Multiple loop nucleation, (b) the loops link up and glide across the broad face. (From Sankaran, R. and Laird, C., Philos. Mag., 29, 179, 1974. With permission.)
386
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
The PbS was vapor deposited on cube-oriented PbSe. The misfit dislocations initially compensated only 15% of the misfit between the two phases. When the overgrowth of brittle PbS became sufficiently thick, the remaining uncompensated coherency stresses were strong enough to crack the PbS severely. The remaining misfit dislocations required were nucleated at the interphase boundary where cracks met along the intersecting cube directions. These dislocations were a=2 with their Burgers vector in the interface plane. 3.4.3.2.5 Nucleation of Dislocation Loops within Precipitates Sankaran and Laird [90,126] confirmed the observation of Weatherly and Nicholson [109] that the dislocation loops form within u0 plates in Al–Cu alloys. These loops expand to the interphase boundaries where they serve as misfit dislocations. Loop formation was more appropriately ascribed to vacancy absorption from the matrix than to the condensation of non-stoichiometric vacancies in the precipitate. Similar observations were made on h plates in Al-Au, except that this and other coherency loss mechanisms operated much more rapidly than at u0 plates in Al–Cu because the misfit is much larger.
3.4.4 GROWTH LEDGES
AT
PARTIALLY
AND
FULLY COHERENT INTERPHASE BOUNDARIES
3.4.4.1 Prevalence and Role in Interface Crystallography When the general theory of precipitate morphology described in Section 3.2 was first published in 1962, the only evidence available on growth ledges was a single observation relying more on growth kinetics than on direct observation that was made by a preliminary thermionic electron emission microscopy study of the thickening kinetics of proeutectoid ferrite plates in a Fe–C alloy. By the time ‘‘Phase Transformations’’ was published in 1970 (Ref. [2]), a small but nonetheless convincing body of evidence had been accumulated as to the role of the growth ledges in the migration of partially and fully coherent interphase boundaries between crystals differing in structure. The presence of such ledges had begun to appear essential, just as predicted by the theory of morphology. Because of the development of improved TEM techniques for observing ledges, however, when a review was published [4], it had become possible for Weatherly [166] to remark that ‘‘there is no reason to doubt that the ledge mechanism is the universal one for the growth or dissolution of all faceted precipitates.’’ A particularly dramatic illustration of a well-ledged interphase boundary is furnished by the reconstruction from many field ion microscopy micrographs of part of a Co2Ta (complex ordered cubic) precipitate in a Co-5wt.% Ta alloy [167] shown in Figure 3.93. As Hirth and Balluffi [168], and Garmong and Rhodes [169] have noted, ledges play an important role in the structure and
10 nm
FIGURE 3.93 Reconstruction from many field ion micrographs of a heavily ledged Co2Ta precipitate. (From Hildon, A. et al., J. Microsc., 99, 41, 1973. With permission from John Wiley & Sons.)
387
Diffusional Growth
crystallography of the interfaces on which they are present, as well as in the migration of these interfaces. A ledge whose edge, or step face, is disordered, fully coherent, or partially coherent through the agency of glide dislocations changes only the apparent or average orientation of the interphase boundary. On the other hand, when the edge of a ledge contains climb dislocations, the lattice orientation across the boundary as well as the average boundary orientation can be altered. 3.4.4.2 Visibility Conditions for Ledges This vital problem has been carefully examined by Weatherly and Sargent [170], and further by Weatherly [166,171], Weatherly and Mok [110], with results that have significantly accelerated the progress of research in this area. Two different sources of diffraction contrast by ledges were identified: the thickness difference and the lattice strain across the ledge. Considering first the thickness difference effects, Figure 3.94 shows that the intensity difference inside and outside
(a)
(b)
(c)
(d)
g
1μ (e)
FIGURE 3.94 Effect of tilting from the Bragg diffracting condition upon contrast associated with a circular ledge. (a) w 0; (b) w þ0:8; (c) w þ2:0; w þ3:5 x38,000. (From Weatherly, G.C. and Sargent, G.M., Philos. Mag., 22, 1049, 1970. With permission from Taylor & Francis.)
388
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
the circular ledge on a u0 plate varies with the deviation from the Bragg diffraction position (denoted w). A comparison with the calculated perfect crystal rocking curves indicates that the thickness of the region within this ledge is 0.3j(112)u0 and that without is 0.4j(112)u0 ; hence the ledge is a hole, that is a dissolution ledge, approximately 6.5 nm deep. A simpler means of using the thickness effect has been devised by Gleiter [172]. Thickness fringes are displaced as they cross a ledge (Figure 3.95). Gleiter has derived a simple equation relating the magnitude of the displacement to the ledge height. Its application to Figure 3.95 yields heights of 2–4 nm. Numerous contrast effects of lattice strain are observed. When ledges are observed in a dark-field precipitate reflection, the contrast changes from a wide black to a wide white band across the ledge (Figure 3.96a). This contrast is reversed when the foil is tilted from w ¼ 12 to (12), when the sense of g at a fixed w is reversed, and along a line perpendicular to g. When a matrix reflection is excited, a weak double image with a line of ‘‘no contrast’’ can be observed perpendicular to g (Figure 3.96b). These effects, particularly the latter one, suggest that a ledge behaves as a prismatic dislocation loop atop a plate, with b perpendicular to the broad faces. Weatherly and Sargent [170] confirmed this point quantitatively using the Howie–Whelan two-beam diffraction equations.
FIGURE 3.95 Displacement of thickness fringes upon crossing ledges on u0 plates; g ¼ (111)a. (From Weatherly, G.C. and Sargent, G.M., Philos. Mag., 22, 1049, 1970. With permission from Taylor & Francis.)
g
g
1μ (a)
1μ (b)
FIGURE 3.96 u0 Al–Cu plate with a spiral ledge. (a) Dark field g ¼ (112)u0 , showing black=white contrast across ledges. (b) Same area, but in bright field g ¼ (200)a. (From Weatherly, G.C. and Sargent, G.M., Philos. Mag., 22, 1049, 1970. With permission from Taylor & Francis.)
389
22
0
Diffusional Growth
132 θ
0.2 μ (a)
(b)
FIGURE 3.97 (a) Plate of u0 Al–Cu, showing apparent absence of ledges in bright field and (b) the presence of ledges in dark field. (From Sankaran, R. and Laird, C., Acta Metall., 22, 957, 1974. With permission from Taylor & Francis.)
These investigators estimated the minimum ledge height visible through both contrast mechanisms. For Ni3Al, where d ¼ 5.7 l03, both mechanisms make a ledge height of approximately 3 nm visible. However, the lattice strain approach allows visibility of ledges approximately 0.015 nm=d high, where 0.015 nm was calculated be the minimum effective Burgers vector of a ledge for visibility. Thus, when d ¼ 0.075, a ledge 0.2 nm high, or about a monatomic ledge, should be visible. Based on these considerations, rules were given for imaging ledges: use a low order superlattice reflection if the precipitate is ordered, emphasize reflections with a short extinction distance, avoid exciting matrix reflections, and use w 6¼ 0 to minimize background intensity. Figure 3.97 shows the absence of ledges on a u0 plate when viewed in a (220)a reflection and their presence when these rules are followed. Contrast effects associated with ledges differ appreciably from those due to related defects. Thus, strains within a coherent precipitate are uniform [141,153], whereas they vary rapidly with distance from a ledge and give sharp strain field contrast. Ledges have an effective b perpendicular to the broad faces of a plate, whereas the b of misfit dislocations is parallel to or at least makes an angle < 908 with respect to these faces. Unlike the misfit dislocations, the strain fields associated with adjacent ledges do not mutually cancel and hence the minimum inter-ledge spacing detectable ought to be smaller than that between the misfit dislocations. 3.4.4.3 Sources of Ledges As a direct result of the foregoing considerations, the amount of experimental evidence available on ledges has markedly increased. Table 3.7 lists the various sources of ledges that have been proposed and the systems, if any, in which each source has been confirmed. Figure 3.98 shows evidence for several sources on b0 plates [90]. The pattern, that had begun to emerge by 1968, of a wide variety of sources is now well confirmed. However, the ability to predict in advance the sources that are likely to be important for experiment remains limited. The first source in Table 3.7, that is, two-dimensional nucleation, is considered the most basic as well as the last resort. Since our theory of precipitate morphology considers growth to be completely halted by a partially or fully coherent boundary between two crystals differing in structure, in the absence of ledges, the solute concentration in the matrix at such an interface is that of the matrix
390
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
TABLE 3.7 Sources of Ledges Source
Experimental Evidence
Two-dimensional nucleation Dislocation with out-of-plane component emerging from precipitate Interphase boundary dislocation or intruder dislocation with b not parallel to interface Edges or corners of plates
u0 Al–Cu, hAl-Au Same
Stepped misfit dislocations Intruder dislocations which create an unhealable ledge Volume change distorts path of boundary, creating a ledge Junctions between grain boundary allotriomorphs and secondary sideplates Impurity precipitates in contact with interphase boundary Anti-phase boundaries in precipitate Hole in thin foil
None gAl–Ag; u0 Al–Cu during dissolution; Mg2Si; u0 Al–Cu gAl–Ag None aFe–C indirect aFe–C
u0 Al–Cu, hAl-Au u0 Al–Cu during dissolution gAl–Ag
Source: Aaronson, H.I., J. Microsc., 102, 275, 1974. With permission.
which is remote from the interface (assuming widely spaced precipitates). Hence, DGV for nucleation is as high at such a location as it is within the matrix. Further, the partially or fully coherent interphase boundary is available to reduce appreciably DG* unless this boundary is not only fully coherent but also separates two phases which, particularly under the circumstance of halted growth, exhibit a very small composition difference. As shown, most simply by Equation 2.153, this circumstance can yield a vanishing value of the coherent interphase boundary energy, which is proportional to the square of the composition difference across the boundary. Hence a ledge-free (or nearly so) broad face of a precipitate plate should be a preferred site for the nucleation of another crystal of the same phase. The minimization of DG* will virtually require that at an otherwise perfect interphase boundary (i.e., no additional defects), the nucleus has exactly the same spatial orientation as its ‘‘substrate,’’ and that there be no boundary between the nucleus and substrate. Since such a nucleus should quickly develop, during growth, a prominent facet parallel to the one upon which it is nucleated, the nucleation process has constructed a ledge. This process should, however, be considered as one of ‘‘last resort.’’ Processes that require only growth ought to proceed much more rapidly, and if they have a sufficient number of sources should soon make DGV for ‘‘two dimensional nucleation’’ less negative and hence improbable of occurrence. The second source listed in Table 3.7, a dislocation with an out-of-plane component, is the original and classic source of growth ledges for crystal growth from the vapor on a close-packed plane. Figure 3.99, taken from a work by Frank [173], shows that when a screw dislocation emerges from the surface of a simple cubic crystal, unhealable ledges less than the height of ‘‘the unit building block’’ are formed. Hence, the energetically easy addition of ‘‘additional blocks’’ is perpetually feasible as growth takes place ‘‘up a spiral staircase’’ without ever requiring fresh nucleation. The probability of atomic or molecular attachment to the ledges provided by the emerging screw dislocation is always much greater than to the low index broad faces of this crystal at sites far removed from the screw dislocation. In view of the great difficulty of achieving atomic
391
Diffusional Growth
132θ΄
2 θ΄
A
020
11
0.4 μ 0.2 μ
0.2 μ (a)
(c)
(b)
B
112θ΄
132
θ΄
132θ΄
0.2 μ (d)
0.2 μ (e)
0.2 μ (f )
FIGURE 3.98 Examples of ledges on u0 Al–Cu plates. (a) Two-dimensional nucleation at no visible defect on plate surface, (b) ledges evolved from screw component of dislocation entering the plate at A, (c) nucleation of ledges at impurity precipitate, (d) ledge nucleation at plate edges (note area B); (e), (f) nucleation of ledges where u0 plates parallel to different habit planes impinge. (From Sankaran, R. and Laird, C., Acta Metall., 22, 957, 1974. With permission from Taylor & Francis.)
FIGURE 3.99 An end-view of a screw dislocation. (Reproduced from Frank, F.C., Discuss. Faraday Soc., 5, 48, 1949. With permission of the Royal Society of Chemistry.)
392
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
attachment to the broad face of a partially or fully coherent boundary between two crystals of different structure, this mechanism must be far more efficacious during the migration of such a boundary. Figure 3.98b shows an example of this mechanism of ledge formation. The third source in Table 3.7, an interphase boundary dislocation or an intruder dislocation with its Burgers vector not parallel to the interface represents the same kind of situation. An example of this source of ledge formation has not yet been identified. The fifth source, intruder dislocations creating an unhealable ledge, is also similar. Edges or corners of plates, can provide sources of ledges for both the lengthening of a plate with partially coherent edges and for the growth (or dissolution) of the broad faces of the plate. The origin of such a ledge may be nucleation that could be assisted by the simultaneous availability of two interfaces, either both on the edge and the other on a broad face, whose interfacial energy could be locally contributed to the nucleation process. Irregularities at the sometimes less well organized edges of plates might also provide an occasional source for ledges. The concentration of volumetric transformation strain at the edges of a precipitate plate could provide further assistance to the nucleation, but presumably not to the growth process. The set of four micrographs of Figure 3.100, secured by means of hot-stage TEM on a g Al–Ag plate [147], illustrates ledges at the plate edges providing the agency for plate lengthening. Laird and Aaronson [174] have used the same experimental technique to show lengthening of u0 plates in Al-4% Cu thin by the formation of ledges at their edges and the lateral movement of these ledges across the broad faces. Since the ledges are made visible by an a dislocation in their
A C
H – [1 10] (a)
(b)
0.5 μ H
(c)
(d)
FIGURE 3.100 Lengthening of a g plate enclosed within a thin foil by the ledge mechanism. Reaction temperature approx. 3508C. (a) time 0, (b) 35 min, (c) 65 min, (d) 146 min. (Reprinted from Laird, C. and Aaronson, H.I., Acta Metall., 17, 505, 1969. With permission from Elsevier.)
393
Diffusional Growth
edges, they must be four u0 tetragonal lattice parameters high, or approximately 23 Å. It was postulated that the misfit dislocation nearest to the upper or lower broad faces of a u0 plate might simply begin to migrate inward under the impetus of the concentration gradient driving dissolution. This mechanism, though, is likely to be incomplete—the edges of u0 plates are faceted—it seems that a discrete discontinuity (or in the limits of a nucleation process) need to cause the local disruption of the facet’s interfacial structure needed to make atom transfer across the interphase boundary feasible in either direction. The fourth mechanism listed – stepped misfit dislocations – seems so far to be peculiar to fcc=hcp transformations. As Laird and Aaronson showed by hot-stage TEM [147], this mechanism can develop when a nonequilibrium interfacial structure consisting of an area of parallel a=6 Shockley partials instead of arrays of two or three crossing sets of partials, see the Al–Ag entry in the Table 3.3. As shown in Figure 3.101a, the stacking order required in the thickening g (hcp) precipitate can only be secured by passing a stepped sequence of diatomic ledges through a given volume of a (fcc) matrix. Movement of another Shockley partial through the same volume would disrupt this order. Thus, the Shockley partials must be stepped with respect to their neighbors and accordingly, we have the unusual situation of misfit dislocations serving simultaneously as growth ledges. The hot-stage TEM has actually shown these dislocation-ledges migrating toward the edges of a lengthening and thickening plate; hence, local regaining of coherency is accompanying growth. The attendant rise in the interfacial energy is tolerable because at the precipitate sizes involved the resulting addition is minor relative to the volume free energy change made available by the volume that the precipitate has achieved. The hot stage studies showed that ledge formation and movement by this mechanism is very erratic. Overall, though, it tends to be minimal initially while the hcp precipitate is still largely coherent with the fcc matrix. Then it becomes intense and finally almost dies out as most of the Shockley partials find two- or three-dislocation configurations
α 2 1
γ
0
1
2
B C A B C A B A B A
A B C A B A B A B A
C A B A B A B A B A
(a) α
γ
α θ΄ (b)
(c)
FIGURE 3.101 Schematic illustration of three different experimentally observed ledge mechanisms: (a) Shockley partial dislocations provide natural conversion of fcc to hcp in Al-15%Ag [128]; (b) the edge of ledges that operate during dissolution of u0 in Al-4%Cu contain misfit dislocations [174]; (c) super-ledges operate during thickening of Widmanstätten ferrite [73]. (From Aaronson, H.I. et al., in Phase Transformations, ASM, Metals Park, OH, 1970, p. 313. With permission.)
394
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
wherein the dislocation-ledges oppose each other’s lateral growth by being on the same plane, hence causing the dislocation-ledge structure ultimately to become reasonably quiescent. The sixth mechanism visualizes a combination of elastic and γ plastic deformation driven by the volume strain energy attending a transformation causing local rotation of the broad faces of a plate out of the boundary orientation which permits it to have a predominantly misfit dislocation structure. Hence, partially coherent areas of the boundary might be connected by small areas that become disorα dered-type areas during further growth, and hence automatically serve as ledges. Observations on the broad faces of ferrite plates suggest that this mechanism is operative but are not sufficiently detailed to provide firm proof. The seventh mechanism, junctions between grain boundary allo- FIGURE 3.102 Schematic illustration of a junction triomorphs and secondary sideplates, has been fairly well confirmed between grain boundary allofor proeutectoid ferrite sideplates (Figure 3.102). As the orientation of triomorph and secondary the interphase boundary changes continuously (as seen at low mag- sideplate. nification) from those roughly parallel to the austenite grain boundary to that of a broad face of a sideplate, the proportion of the interfacial structure that is partially coherent increases. This structural arrangement should lead to a sequence of ledges. Replication electron microscopy observations of such a sequence of ledges have been reported, though very high resolution TEM confirmation of this mechanism, which is virtually impossible to achieve in ordinary steels, but may be possible in duplex stainless steels, is still required. Figure 3.98c shows an example of the eighth mechanism, nucleation of a ledge by an impurity precipitate on the broad faces of a u0 plate. The usual argument as to the catalytic effects of preexisting interfaces on DG* for nucleation applies. Weatherly [166] observed ledges on u0 plates associated with an antiphase boundary within these plates. He notes that a ledge is associated with the antiphase boundary at the periphery of the plate. The antiphase boundaries are formed by passage of a=2 dislocations through a u0 crystal; a low energy antiphase boundary, across which only the second nearest neighboring Cu atoms are in wrong positions, can be produced thereby. Evolution of the ledge would thus be expected to begin at the plate edge, a point that has not yet been confirmed. Broadening briefly our present scope, we summarize the observations of Garmong and Rhodes [169] who observed and thoroughly characterized the ledge structure on the broad faces of the Al-CuAl2 eutectic. As illustrated in Figure 3.103, the density of these ledges is relatively high and their arrangement is quite complex. Their origin is yet to be elucidated. However, that they are ledges and not some other defect was fully proved: (1) their b was shown to be normal to the interface plane (2) displacements were observed where thickness fringes cross the ledges, whose height was found to be approximately 3 nm. (3) Hot-stage microscopy demonstrated lateral movement of the ledges, and (4) when the interface is viewed at a glancing angle the ledges could be directly resolved. It thus appears likely that the displacement of eutectic interfaces in general must also be accomplished by means of the ledge mechanism. 3.4.4.4 Ledge Heights The common view, inherited primarily from the assumption of theories, is that ledges are, or at least should be of monatomic height. In solid-solid phase transformations, experiment has not verified this expectation in the alloy systems so far examined. As already noted, on hcp AlAg2 plates, the ledges appear to be two-atom planes high. Weatherly [166] searched carefully for monatomic ledges on u0 and Mg2Si plates and found none. Instead, ledge heights ranging from 110 nm were observed. As to whether or not such ledges may actually be closely spaced monatomic ledges, Weatherly and Mok [110] concluded that this question often could not be answered with the TEM
395
Diffusional Growth
2000 Å
g
FIGURE 3.103 Ledges on Al=CuAl2 eutectic interface. (From Garmong, G. and Rhodes, C.G., Acta Metall., 22, 1373, 1974. With permission from Elsevier.)
techniques then available. Since the width of a ledge image is 510-fold greater than the ledge height when d ¼ 0.05, the monatomic ledges would have to be spaced farther apart than this in order to be resolvable at this level of misfit. In Section 3.4.5, recent studies based on weak-beam dark-field TEM, will be noted to have resolved ledges three atom planes high that were not bounded by a misfit dislocation. At the other extreme, the ledges on the edges of gAlAg2 plates (Figure 3.100) are 20–50 nm high. Kinsman et al. [175] found ledges ranging in height from scores to thousands of nm on the broad faces of the proeutectoid ferrite plates. The latter ledges were often ill formed and have distinctly sloping rather than squared-off edges. It would be interesting to ascertain, if the martensite reaction makes this possible, whether such huge super-ledges are actually composed of large numbers of much smaller ledges. In summary, a very large range of ledge heights has been experimentally observed and there seems no obvious reason for why a particular height range obtains in any transformation other than fcc=hcp, that is, in Al–Ag. 3.4.4.5 Inter-Ledge Spacings This turns out to be a particularly crucial feature of interphase boundary structure, as will become apparent when the question of the migration of partially and fully coherent interphase boundaries by the ledge mechanism is discussed in Section 3.4.6. The average spacing between the ledges on a given type of interphase boundary at a given stage of reaction varies widely, as the figures in the preceding subsections indicate. Apparently whatever the mechanism(s) of ledge formation at play, the average spacing between the ledges on the broad faces of plates seems to be very large at early times, thereby ensuring that the plate morphology is generated. Then it becomes relatively small and finally turns quite large as the generating mechanisms become less active even though the driving force for the remaining growth may still be quite high. The plot in Figure 3.104 is illustrative of this trend. On the broad faces of hcp AlAg2 plates, ledges were initially scarce when the plates were fully coherent with the fcc matrix; later spacings of the order of 2040 nm were observed, and at late reaction times the spacing tended to rise to the order of the plate diameter. The edges of these plates, whose lengthening tended to occur by super-ledges, exhibited 40100 nm spacings under some circumstances. At both types of location, though, irregularity and marked variation with time were characteristic. From Kinsman et al. [175] the values plotted in Figure 3.105 were back calculated from growth kinetics measurements. Mostly replication TEM measurements of inter-ledge spacings, which are
396
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 500
Inter-ledge spacing (nm)
400
300
200
100
0
0
1000
3000 2000 Aging time (s)
4000
5000
FIGURE 3.104 Variation of spacing between ledges with aging time on Al-4%Cu u0 plates formed at 3008C. (From Sankaran, R. and Laird, C., Acta Metall., 22, 957, 1974. With permission from Taylor & Francis.)
not very certain because of the interference of the martensite structure and the irregularity of the ledges themselves, suggested values between a few tenths and ten microns. Possibly a number of ledges became immobile through growing into a predominantly partially coherent boundary orientation at late reaction times, thereby suggesting that the ‘‘Late Stage’’ spacings shown in Figure 3.105 represent the average effective spacing between the ledges. 100
100 Fe-0.11 wt.% C
80
80
70
70
60
60
50 40
Late stage
30 20 10 0 (a)
Fe-0.22 wt.% C
90
b (cm×104)
b (cm×104)
90
50 Late stage
40 30 20
Initial stage
Initial stage
10 700
720 740 760 780 800 Reaction temperature (°C)
0
820 (b)
680
700 720 740 760 780 Reaction temperature (°C)
800
FIGURE 3.105 Average inter-ledge spacing, b, calculated from the half-thickening rate data for: (a) 0.11 wt.% C, (b) 0.22 wt.% C,
397
Diffusional Growth 100
100
90
80
80
70
70
60
60 b (cm×104)
b (cm×104)
Fe-0.31 wt.% C 90
50 Late stage
40
Fe-0.41 wt.% C
50 40 30
30
Late stage 20
20 Initial stage
10 0 (c)
680
700
720
740
760
Reaction temperature (°C)
780
Initial stage
10 0
800 (d)
680
700
720
740
760
780
800
Reaction temperature (°C)
FIGURE 3.105 (continued) (c) 0.31 wt.% C and (d) 0.41 wt.% C alloys. (With kind permission from Springer Science þ Business Media: From Kinsman, K.R. et al., Metall. Trans., 6A, 303, 1975.)
3.4.5 STRUCTURAL LEDGES
AT
PARTIALLY COHERENT INTERPHASE BOUNDARIES
The ledges that are to be described now probably cannot contribute to the growth process. They are present as a supplementary means of increasing the level of coherency between two rather badly matching crystal lattices, in particular fcc and bcc, to the point where the remaining mismatch can be compensated by the periodic introduction of misfit dislocations. Because there is extensive evidence for barriers to growth at many different boundary orientations between a wide variety of crystal structures, it seems plausible that structural ledges play a large role as described below, or that nature has still other mechanisms available for improving misfit compensation. As previously noted, Brooks [104] has concluded that the homogeneous deformation required to interconvert the fcc and bcc crystal structures is too large to make a misfit dislocation structure survivable at an fcc: bcc boundary. The inter-dislocation spacing should be so small that adjacent dislocations would mutually disrupt each other. Nonetheless, the presence of interface barriers to migration during transformations of this type, as evidenced by the development of needle and plate precipitate morphologies, does raise the possibility that a misfit dislocation or some other type of partially coherent interfacial structure (as postulated by the general theory of precipitate morphology) is present at the boundary orientations corresponding to the broad faces of the plates and the cylindrical faces of the needles. In the case of needles, unpublished research by Hawbolt and Massalski has established that a misfit dislocation structure (which the authors were unable to characterize, perhaps because of severe elastic anisotropy of the matrix phase) is present at the cylindrical face of a needles precipitated from bCu–Zn. It is qualitatively easy to understand why a misfit dislocation structure develops in this situation. In Figure 3.106, the {111} plane in
398
Mechanisms of Diffusional Phase Transformations in Metals and Alloys – [0 11]γ
– [ 111]α
– [ 101]γ
– [ 110]γ – [11 1]α
FIGURE 3.106 Superposed plots of the atomic configurations in the Kurdjumov–Sachs related {111}g and {110}a planes.
austenite and a {110} plane in ferrite are superimposed in the Kurdjumov and Sachs orientation relationship [177], (1 11)g ==(101)a , [110]g ==[111]a Note that in the vicinity of the parallel directions, [110]g ==[111]a, matching of atom patterns and spacings between the two conjugate habit planes is very good. Constructing a cylindrical interface primarily upon the basis of these two conjugate directions would thus seem to ensure sufficiently good matching over the entire interphase boundary to permit the formation of a stable misfit dislocation structure. In the case of plate-shaped precipitates, however, in agreement with the prediction of Brooks, lattice matching is too poor to permit the entire interface to be described in terms of the array of misfit dislocations separated by coherent regions. However, sectioning of individual Widmanstätten ferrite crystals on two orthogonal planes of polish has shown that they tend to have the needle morphology at high temperatures but to be plates at temperatures in the lower bainite region, below approximately 3508C, more or less independent of carbon content. On the view that the broad faces of plates do constitute a barrier to growth, what then is the nature of this barrier in an fcc ! bcc transformation? Hall et al. [177], using transmission electron microscopy, have investigated this problem. They were unable to conduct their investigation on steel, however, since the austenite matrix is destroyed during quenching to room temperature by the martensite reaction in hypoeutectoid steels. They therefore employed a Cu-0.33 wt.% Cr alloy, in which nearly pure bcc Cr precipitates from an fcc Cu-rich matrix and the ratio of the lattice parameter of fcc to that of bcc is 1.253, nearly the same as the ratio in the proeutectoid ferrite reaction (1.255), see Figure 3.107. Unfortunately, the precipitates were too small for the purposes of this investigation when they were in the growth stage of their development. Therefore, aging times had to be extended until the precipitates had coarsened (Ostwald ripened) appreciably. However, the morphologies of the precipitates did not appear to be significantly changed during prolonged aging, and thus the results summarized here should be applicable to the growth stage as well. A wide variety of precipitate morphologies was found in the interiors of the fcc matrix grains. However, wherever planar facets were found, misfit dislocations were observed. A range of lattice orientation relationships was found. These relationships varied from Kurdjumov and Sachs (K-S) to Nishiyama. Both of these relationships are based upon {111}fcc=={110}bcc. 110]bcc. As shown in Figure 3.108, this difference, However, in the Nishiyama case, [ 211]fcc==[
399
Diffusional Growth
0.5 μm
(a)
0.1 μm (b)
0.1 μm
(c)
0.1 μm
(d)
FIGURE 3.107 Transmission electron micrographs of Cr-rich bcc precipitates, showing morphology and interfacial dislocation structure. (a) Portion of a lath precipitate after 5 days at 7008C. (b) Idiomorph, 5 days at 7008C. (c) Idiomorph 1000 min at 7008C. (d) Lath precipitate at an early stage of growth, 10 min at 8008C. (Reprinted from Hall, M.G. et al., Surf. Sci., 31, 257, 1972. With permission from Elsevier.)
which is equivalent to a small rotation of the two conjugate habit planes relative to each other, is actually quite small. Defining the angle u as that between [110]fcc and [100]bcc, when u ¼ 08, the relationship is Nishiyama (N), when u ¼ 514 8, the relationship is K-S. Values of u ranging from 08 to 58 were obtained experimentally. The habit plane situation turned out to be even more complex. Figure 3.109a is a stereographic projection of the poles of the habit planes of the planar facets on the fcc matrix, while Figure 3.109b similarly plots the habit planes forming the bcc portion of the planar interphase boundaries. The important item to understand from these plots is that, with one exception, all of the habit plane pairs are well removed from the {111}g=={110}a pair which make up the not very good matching interface portrayed in Figure 3.113. Despite the foregoing complexities, Table 3.8 indicates that the habit plane relationships are by no means random. Expectedly, the solution to the problem is a question in interfacial structure. The irrational and scattered habit planes are explained on the following basis. Figure 3.110a and b show atomic matching resulting from the Nishiyama (sometimes termed Nishiyama–Wassermann, N-W) and the K-S relationships, respectively. The region of good fit obtained in each case is outlined
400
Mechanisms of Diffusional Phase Transformations in Metals and Alloys – [ 202]Cu
– [ 110]Cr
– [0 22]Cu
– [11 2]Cr
– [00 2]Cr
– [ 220]Cu
–– [1 1 2]Cr
– [ 112]Cr θ
– [02 2]Cu
– [1 10]Cr
– [2 20]Cu
[002]Cr – [1 12]Cr
– [20 2]Cu
FIGURE 3.108 Orientation relationships of the precipitates. Foil has been tilted to a [111]fcc zone axis, showing a superposed [110]bcc diffraction pattern and irrational orientation relationship. u is the angle between the [110]fcc and [100]bcc directions. If u ¼ 0, the orientation relationship is a Nishiyama–Wasserman; u ¼ 5–1=48 gives the Kurdjumov–Sachs relationship. (Reprinted from Hall, M.G. et al., Surf. Sci., 31, 257, 1972. With permission from Elsevier.) {111} Interface normals, Experimental data 6 ± 2° (i) (ii) (iii)
θ = 0° heory
{335}
{111} Interface normals, Experimental data
(i) (ii)
θ = 3°
1 5a 4
3 θ = 0°
θ = 5 1/4°
heory {112}
θ = 3°
5b 2
θ = 3° θ = 0°
2 5b 4
θ = 0°
θ = 3°
7 {100} (a)
3 5a
fcc
θ = 3°
{110}
1
{100} (b)
bcc
6.7 {110}
θ = 0°
FIGURE 3.109 Unit stereographic triangle showing experimental and theoretical interface normals with respect to (a) the fcc crystal and (b) the bcc crystal. (Reprinted from Hall, M.G. et al., Surf. Sci., 31, 257, 1972. With permission from Elsevier.)
with heavy dashed lines. These constructions were made for a lattice parameter ratio, bcc afcc 0 =a0 ¼ 1:25. The K-S region of good fit contains approximately 30 atoms in each habit plane and the N-W region contains 9 atoms. In both cases, only approximately 8% of the atoms in the bcc habit plane can be considered to fit well with their counterparts in the fcc habit plane. The displacement of less than 15% of the interatomic spacing is regarded as a good fit. In order to define the proportionate area of good fit, inspect the shapes enclosed by solid lines in Figure 3.111a through c. These shapes lie on the same plane. In each case, they form a two-dimensional ‘‘superlattice’’ of regions of good fit. At u ¼ 08 (N-W), the dimensions of the unit cell are 22 55Å, as measured from equivalent positions on adjacent ‘‘points’’ (actually diamonds). At u ¼ 28, the unit cell is similar in shape, but larger and distorted. At u ¼ 514 8, the K-S situation, the unit cell is 36 109 and only the nearest neighboring regions are shown.
401
Diffusional Growth
TABLE 3.8 Morphology and Orientation Relationships Morphology
Orientation Relationship
Morphology a
(111)F==(110)B u < 18
b
{110}
{110}
{110}
Orientation Relationship (111)F==(110)B u ¼ 28
{110} (111)F==(110)B u ¼ 18
{110}
(111)F==(110)B u ¼ 18
(111)F==(110)B u ¼ 3–58 (100)F==(110)B (011)F==(111)B
{100}
(111)F==(110)B u ¼ 28 {110} Source: Reprinted from Hall, M.G. et al., Surf. Sci., 31, 257, 1972. With permission from Elsevier.
bcc
bcc
fcc
fcc
fcc
fcc
bcc bcc
fcc
fcc bcc
(a)
bcc
(b)
FIGURE 3.110 Superposed plots of the atomic configurations in the {111}fcc and {110}bcc planes. (a) Nishiyama–Wasserman orientation relationship. (b) Kurdjumov–Sachs orientation relationship. (Reprinted from Hall, M.G. et al., Surf. Sci., 31, 257, 1972. With permission from Elsevier.)
Now consider the distribution of ‘‘good fit’’ areas if a step of one atomic layer high is introduced. These areas are found to have the same size and shape as those shown above, but to be displaced horizontally. At u ¼ 08, for example, the displacement is 9 Å. The areas of good fit on the step are outlined by dashes in the figures. Since successive steps make the same displacement, only the first one is shown. In this manner, the proportionate area of good fit is increased to approximately 25% in each case. Note, however, that in no case can all the areas of good fit be utilized in constructing an interface. For example, if the areas aa0 bb0 are considered part of the interface, cc0 , dd0 , ee0 , etc. will lie solely within either phase. Because of this situation, several good interfaces, each described by a different pair of conjugate habit planes, could be described for each value of u studied. On this basis, the four best-stepped interfaces of good fit were constructed graphically at u ¼ 0, l, 2, 3, and
402
Mechanisms of Diffusional Phase Transformations in Metals and Alloys bcc fcc
θ = 0° c΄
fcc bcc
c d΄ d
a΄ a
e΄ e
b΄
10 Å
b
(a) θ = 5° 16΄
bcc fcc
bcc
fcc
c΄
bcc c a
fcc
a΄
d b
(b)
θ = 2°
b΄
bcc fcc
d΄ e
e΄
(c)
FIGURE 3.111 Superlattices of ‘‘good fit’’ formed by {111}fcc, {110}bcc conjugate planes. The solid lines outline the regions of ‘‘good fit’’ formed by the {111}fcc and {110}bcc planes. The dashed outlines show similar regions in the adjacent atomic planes. (Reprinted from Hall, M.G. et al., Surf. Sci., 31, 257, 1972. With permission from Elsevier.)
514 8. (At u ¼ 48, the errors of location of the good fit regions became large when graphical construction was used.) It was found that, as u was varied, the habit plane normals generated curves on the stereograms (Figure 3.109). At u ¼ 514 8, only one pair of conjugate habit planes of good fit materialized. The solid curve in the fcc plot, for example, connects habit planes with a 25% good fit; the dashed curves connect habit planes with a 16% good fit. Comparison of the calculated and experimental habit planes in Figures 3.109a and b shows reasonably good accord between theory and experiment. The major surprise is the finding of a habit plane pair which is exactly (111)fcc==(110)bcc. The interface was not only linear, however, but exhibited widely spaced growth ledges, approximately 80 Å high, indicating that it was an effective barrier to growth, despite the fact that it exhibited only about l=3 the density of good fitting atoms of the stepped interfaces. The misfit dislocations were concluded to serve only to compensate residual mismatch within the systems of steps. Figure 3.112a and b show two versions of the (335) habit plane. In Figure 3.112a, bcc ratio is such that the spacing between parallel (110) planes in bcc is equal to the the afcc 0 =a0 ‘‘resolved’’ spacing between (111) planes in fcc where they join. In Figure 3.112b, the lattice parameter ratio is such that there is a 10% difference between the two inter-planar distances. Increasing the angle of rotation of the apparent habit plane from 108 in the first example to 238 in the second example removes the mismatch in the faces of the steps. However, in the perpendicular direction an extra (110)bcc plane is needed for every 10 (111)fcc planes. In Cu-Cr the mismatch is approximately 2.5%, implying an extra bcc plane for every 40 (111)fcc planes (87 Å). Thus the effective inter-dislocation spacing is 87 Å=sin(138) ¼ 390 Å. This is in reasonable agreement with the experimentally measured values of 100–1000 Å. In this case of comparison with theory, it should be noted that only one habit plane has been treated whereas many are present. Thus, the usual problem of nucleating misfit dislocations, however, is likely to be present. Russell et al. [178] re-examined the structure of the disordered regions. Figure 3.113 shows the atomic details of an enlargement of a portion of Figure 3.111a, that is, of the N-W (u ¼ 08) partially
403
Diffusional Growth afcc 0 abcc 0
afcc 0
= 1.246
abcc 0
= 1.34 {111}
{111}
fcc
2.0 Å
fcc
2.0 Å
2.2 Å
2.0 Å
(a)
{110}
bcc
(b)
bcc
{110}
FIGURE 3.112 Steps and dislocations of a (335) habit plane between fcc and bcc crystals with the lattice bcc fcc bcc parameter ratio (a) afcc 0 =a0 ¼ 1:246 and (b) a0 =a0 ¼ 1:34. (Reprinted from Hall, M.G. et al., Surf. Sci., 31, 257, 1972. With permission from Elsevier.) Line defect (misit dislocation)
Step
Step
Step
FIGURE 3.113 Atomic configuration of N-W related partially coherent fcc=bcc interface. (With kind permission from Springer Science þ Business Media: From Russell, K.C. et al., Metall. Trans., 5, 1503, 1974.)
coherent interface. Inspection of this figure shows that in the region between the columns of diamond-shaped, nearly coherent regions of the fcc lattice contribute one more atom than does the bcc lattice to each row of atoms. The resulting extra ‘‘half-plane’’ is located along the indicated vertical lines, and the Burgers vector of the dislocations thus formed lies in the plane of the boundary. In this drawing, the atoms are shown in their unrelaxed positions. Allowing relaxation, however, would make the regions between the diamonds also coherent, with the misfit being classically concentrated in the dislocations. Such a structure would then provide the conventional barrier to the migration of dislocation interphase boundaries. Rigsbee and Aaronson [114] consolidated the considerations of Hall et al. [177] and Russell et al. [178] and extended them to a wider range of relative rotation of the fcc and bcc lattices and relative lattice parameters by means of a computer modeling study. Simultaneously, they tested these interfacial structure predictions experimentally by a TEM study of the broad faces of ferrite plates in a Fe-0.63 wt.% C-2.05 wt.% Si alloy. The modeling study will be considered first. Figure 3.114 is an isometric sketch of the stepped, partially coherent fcc: bcc boundary. The angle u is here the angle between the atomic habit planes, {111}fcc and {110}bcc, and the apparent habit planes, drawn so as to be in contact with the edges of the ledges. To construct such a boundary, the following procedure was employed. The basis of the modeling technique is the definition of Hall et al. [177] that an interphase boundary atom is coherent with both phases when its position
404
Mechanisms of Diffusional Phase Transformations in Metals and Alloys Coherent “patch” Misit dislocation – b Direction – – – a – [211]γ, [110]α —[011]γ, a [001]α 2 β fcc bcc
a
Structural ledge direction – – [011]γ, [001]α
θ
b
10 Å
a = Structural ledge height b = Structural ledge spacing θ = tan–1 (a/b)
FIGURE 3.114 Isometric sketch of N-W related partially coherent fcc=bcc interface with structural ledges and misfit dislocations. (Reprinted from Rigsbee, J.M. and Aaronson, H.I., Acta Metall., 27, 351, 1979. With permission from Elsevier.)
differs from a normal lattice site for both phases by a total of less than 15% of the average nearest neighbor distance. The analysis is begun by instructing the computer to make two-dimensional plots of each plane in the ABCABC stacking sequence normal to a direction in the fcc lattice. Then, these are projected along the [111]fcc direction onto the (111)fcc plane. Similarly, a suitably scaled two-dimensional plot is formed of the DEDE stacking sequence in the bcc lattice normal to a [110]bcc direction, projected along this direction onto the (110)bcc plane. Coherent atom pairs are next determined, for each of the six possible combinations of fcc and bcc planes in the interface, by calculating the distance between each bcc atom and its nearest neighboring fcc atom and applying the coherency criterion. On this basis, the coherent regions are determined for each atom layer combination. Figure 3.115 is a computer plot of an A:D interface having a N-W orientation relationship and a lattice parameter ratio of 1.254.
–– [1 1 2]γ
– [1 1 0]γ C
A
0°
B 20 Å
FIGURE 3.115 Computer plot of coherent atom pairs for N-W oriented (111)fcc and (110)bcc planes. (Reprinted from Rigsbee, J.M. and Aaronson, H.I., Acta Metall., 27, 351, 365, 1979. With permission from Elsevier.)
405
Diffusional Growth
slo c
n atio
Mo nat om ic
led ge
Mi
di sf it
20 Å
0°
FIGURE 3.116 Computer plot of misfit dislocations and monatomic structural ledges of N-W oriented (111)fcc and (110)bcc planes. (Reprinted from Rigsbee, J.M. and Aaronson, H.I., Acta Metall., 27, 351, 365, 1979. With permission from Elsevier.)
Following the directive to maximize the percent coherent area in the interface, the computer switches to another pair of interface planes, one-atom layer either ‘‘up’’ or ‘‘down.’’ The spacing between the ledges is determined so as to allow a contribution to interfacial coherency by all six combinations of interfacial layers, when the ledges are monatomic. When biatomic or triatomic ledges are introduced, only appropriate combinations of the six sets of layers are allowed. As shown in Figure 3.116 the ledges are parallel to the line connecting the centers of coherent areas nearest to the A:D interface. Knowing the ledge height, spacing and direction, the apparent habit planes are readily calculated. The misfit dislocations are then inserted. The dislocation lines are parallel to the line between the centers of adjacent coherent regions on successive ledge broad faces. The Burgers vector is determined by a Burgers circuit through the centers of adjacent coherent regions on the broad face of a single structural ledge. The inter-dislocation spacing is the product of the distance between the centers of adjacent coherent regions on the broad face of a single structural ledge and the sine of the acute angle between the ledge and the dislocation directions. A typical set of misfit dislocations thus emplaced is shown in the figure. When construction of an interface is completed, the bcc lattice is rotated through a small angle relative to the fcc lattice and the entire procedure is repeated. Calculations were made at small intervals from N-W to beyond K-S, and were then repeated for lattice parameter ratio ranging from 1.15 to 1.35. It was found that all interfaces studied are partially coherent. The coherent regions ranged in size from two to hundreds of atoms. A more restrictive definition of coherency was found to exert no fundamental effect upon the results reported. Triatomic ledges were in general observed to make more use of the coherent regions than the monatomic regions and are accordingly preferred. Spacings between ledges are tripled for the triatomic ledges. The Burgers vectors of the misfit dislocations were shown to be always of the lattice type, a=2 fcc, and a=2 bcc or a bcc. These vectors always lie in the atomic habit planes, {111}fcc and {110}bcc. The dislocations are usually of a mixed type, but in rare instances are either ‘‘pure edges’’ or ‘‘pure screws.’’ The nonequivalent (though in detail only) types of partially coherent interfaces can be constructed, depending upon which pair of coherent areas the construction of the interface is based. The pair O-A and O-B in Figure 3.115 results in different interfaces, whereas the pair O-C gives results identical to those from O-A. Figures 3.117 and 3.118 give the spacing between the monatomic ledges and between the misfit dislocations as a function of orientation relationship (rotation angle) at different levels of lattice parameter ratio.
406
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 40 ˝O–A˝ Type interface 30 1.32
1.32
20 10
1.254 1.22
1.254 1.22
0
–5 –4 –3 –2 –1 0 1 K-S N-W
(a)
2
3
4
˝O–B˝ Type interface
Monatomic ledge spacing (Å)
Monatomic ledge spacing (Å)
40
30 20 1.22 10 0
5 K-S
–5 –4 –3 –2 –1 0 1 K-S N-W
(b)
Rotation angle (°)
1.254 1.32 2
3
4
5 K-S
Rotation angle (°)
FIGURE 3.117 Monatomic structural ledge spacing as a function of orientation for lattice parameter ratios from 1.22 to 1.32. (Reprinted from Rigsbee, J.M. and Aaronson, H.I., Acta Metall., 27, 351, 1979. With permission from Elsevier.)
50
Misfit dislocation spacing (Å)
40
˝O–A˝ Type interface
30
20
1.32 1.254
10
0
1.22
–5 –4 –3 –2 –1 0 1 K-S N-W
(a)
Misfit dislocation spacing (Å)
˝O–B˝ Type interface
2
Rotation angle (°)
3
4
40
30 1.254 20
1.22
10
0
5 K-S
1.32
–5 –4 –3 –2 –1 0 1 K-S N-W
(b)
2
3
4
5 K-S
Rotation angle (°)
FIGURE 3.118 Misfit dislocation spacing as a function of orientation for lattice parameter ratios from 1.22 to 1.32. (Reprinted from Rigsbee, J.M. and Aaronson, H.I., Acta Metall., 27, 351, 1979. With permission from Elsevier.)
Figure 3.119a and b show how the fcc apparent habit plane, caused by the presence of structural ledges, varies with orientation, that is, with the rotation of bcc over fcc parallel to the atomic habit planes. The former includes both monatomic and triatomic ledges. Because of geometrical constraints, the monatomic ledges can be stepped either all ‘‘up’’ or all ‘‘down.’’ Triatomic ledges, however, can go both way, and shows the results when both types of ledges are stepped in the same direction. The latter shows the changes in apparent habit plane that result when the triatomic ledges are all stepped in the opposite direction. Figure 3.120 shows that the experimental results on apparent habit plane obtained from several fcc=bcc transformations can be explained by suitable combinations of the present results. This comparison should also serve to devalue the importance of the habit plane data. As to the experimental study on an Fe–C-Si alloy, the key to its feasibility was the use of a high Si steel. The Si inhibits Fe3C precipitation, though not that of epsilon carbide.
407
Diffusional Growth 1.254
1.30
1: K-S, O-A 2: N-W, O-A 3: K-S, O-B 4: N-W, O-B
5: K-S, O-A 6: N-W, O-A 7: K-S, O-B 8: N-W, O-B
{111} 533 4 1 211 8
{111}
5
211 3 221
2 6
3
7
533
5 4 1 221 8
7 6
{100}
2
{100} {110}
(a)
{110}
(b)
FIGURE 3.119 A stereographic triangle plot showing the fcc apparent habit plane as a function of orientation (from N-W to K-S) for lattice parameter ratios from 1.254 to 1.30: (a) monatomic through triatomic structural ledges; (b) inverse rotated triatomic structural ledges only. (Reprinted from Rigsbee, J.M. and Aaronson, H.I., Acta Metall., 27, 351, 1979. With permission from Elsevier.)
{111}
Wid. Ferrite
533
Bainite Cu–Cr
{100}
{110}
FIGURE 3.120 A stereographic triangle plot of experimental habit plane data compiled for Cu-Cr [177], proeutectoid ferrite [179], and ferrous bainite (T > 2008C) [180]. (Reprinted from Rigsbee, J.M. and Aaronson, H.I., Acta Metall., 27, 351, 1979. With permission from Elsevier.)
At the reaction temperatures used, 4258C4758C, the inhibition of cementite precipitation was evidently more important. This circumstance allowed the accumulation of levels of carbon in the austenite trapped between adjacent ferrite plates approaching (though not attaining) that of the extrapolated Ae3. The compressive stresses exerted on this austenite by the adjacent ferrite plates then prevented the formation of martensite upon quenching in regions in the order of a micron wide. This was sufficient to permit the necessary TEM experiments to be performed. The structures observed were so fine that they cannot be reproduced here. Suffice it to say, then, that both ledges and misfit dislocations were observed for the first time at the broad faces of plates formed during an fcc to bcc transformation, and that the agreement between the predictions of the model and the experiment was usually very good, as indicated in Table 3.9. The Burgers vector determination was accomplished by the g . b ¼ 0 method. The ledge heights were determined by the Gleiter thickness fringe displacement method and were found to be invariably triatomic. The inter-dislocation and interledge spacings were measured directly. Hence, the predictions of Hall et al. [177] and Russell et al. [178] and made quantitative and complete by Rigsbee et al. [114], are now confirmed experimentally.
408
TABLE 3.9 TEM and Computer Modeling Interfacial Structure Analyses Structural Ledges Spacing, Å
Burgers Vector
Direction
Height Triatomic
Incomplete Parallel to [111]g==[110]a
" " " " " " " " " " " "
" " " " " " " " " " " "
(1) TEM Model
37 37.3
188[101]g 148 "
(2) TEM Model (3) TEM Model (4) TEM Model (5) TEM Model (6) TEM Model (7) TEM Model
22 23 35 31.8 44 39.5 36 32.5 36 32.5 36 39.5
78[101]g 158 " 48[101]g 58 " 68[112]g 158 " 138[112]g 158 " 08[112]g 158 " 138[011]g 138 "
Habit Plane Information Apparent fcc Habit Plane
Angle from (111)g
Angle (TEM hp, Model hp)
Orientation Relationship {111}g =={110}a u
Direction
Burgers Vector
Angle (Ledge, Dislocation)
15 13
g 488[101] 508 "
Incomplete a 2 [110]g
308 368
(5, 8, 6) (11, 16, 12)
118 9.58
0
(7:28)
This approach leads to an expression for G not very different from that of Zener’s original treatment: gb Dp3 f a f b xga x g g 1 S0 G¼ (7:29) 1 S S B xba xab a b
where
B¼
1 X ln S pnSa 2 sin S 2pn4 n¼1
(7:30)
7.3.4.2 Boundary Diffusion–Controlled Growth Referring to Figure 7.24b, the basic equation in this situation is þ1 ð 1
Db
d2 x dy ¼ dz2
Db
d2 x B ¼ G xg 2 dz
xag a
(7:31)
where xB is the solute concentration in the boundary. Integration of this equation to obtain the boundary composition as a function of position and balancing free energy losses as before leads to the result 12KDb d xga xgb g g 1 S0 (7:32) 1 G¼ ab S S2 x f a f b xba a b which differs by a factor of only 1.5 from that approximate result of Equation 7.22.
595
Pearlite Reaction
7.3.5 EXPERIMENTAL MEASUREMENTS
FOR
GROWTH KINETICS STUDIES
7.3.5.1 Interlamellar Spacing Pellisier et al. [5] analyzed the interlamellar spacing, S, of colonies of pearlite sectioned at random angles. Assuming the true value of S to be constant, with the observed variation in the apparent value of S arising from sectioning at angles other than the one perpendicular to the broad faces of the lamellae, they calculated that the fraction of the area of the plane of polish occupied by spacing ranging from So (the true spacing) up to a given value of Sa (the apparent spacing) is given by So : (7:33) f ¼ cos sin 1 Sa Figure 7.25 plots the experimentally determined values of f vs. Sa, and also the values of f calculated from Equation 7.33 on the assumptions that So are 1.00 mm and 1.65 mm at 2500 magnification, respectively, where the first value was the smallest S measured experimentally. Neither is seen to accurately reproduce the experimental curve. Now assuming that the mean value of So is 1.65 mm and applying the method of Scheil and Lange-Weise [39] to the experimental and to the 1.65 mm, the calculated curve yields the distribution curve shown in Figure 7.26. Using this curve to re-calculate the cumulative distribution curve gives the much improved agreement with the experiment shown as seen in Figure 7.27. Brown and Ridley [6] have objected to the use of this spacing as the most appropriate description of S on the ground that the mean S is, among other things, time dependent, whereas G is not. They prefer the better behaved and more straightforward minimum value of S experimentally observed and have used Smin ¼ Strue in their careful studies of G and S in Fe–C and alloy steels. 7.3.5.2 Rate of Growth Measurement of the diameter of the largest nodule as a function of time in successively isothermally reacted specimens has long been the accepted method of measuring G, the rate of nodule growth. 100
Experimental
Cumulative area, %
80
60 heoretical So = 1.00 40
20 heoretical So = 1.65 0
1
2
3
4
5 6 7 8 S/So or S × 2500/mm
9
10
11
FIGURE 7.25 Comparison of the experimental curve with theoretical curves based on the assumption of a constant spacing. (From Pellisiev, G.E. et al., Trans. ASM, 29, 1049, 1942. Reprinted with permission of ASM International1. All rights reserved.)
596
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 100 Experimental values
Cumulative area, %
80
60 heoretical curve 40
20
0
1
2
3
4
5 6 7 S × 2500/mm
8
9
10
11
FIGURE 7.26 Comparison of experimental values with a theoretical curve. (From Pellisiev, G.E. et al., Trans. ASM, 29, 1049, 1942. Reprinted with permission of ASM International1. All rights reserved.)
%
20
16
P50
12 8 4 0 1.0
1.2
1.4 1.6 1.8 True spacing
2.0
2.2
2.4
2.6 2.8 So × 2500/mm
FIGURE 7.27 Probable distribution curve of true spacings. (From Pellisiev, G.E. et al., Trans. ASM, 29, 1049, 1942. Reprinted with permission of ASM International1. All rights reserved.)
Typical results are shown in Figure 7.28. Steel C had an ASTM grain size of 3, and Steel D had a grain size of 7. G was unaffected, as anticipated from the ability of pearlite to grow through austenite grain boundaries. Brown and Ridley [40] have compared this method with two other, more complicated and supposedly more rigorous methods, but concluded that this method was as good as any and more convenient.
7.3.6 COMPARISONS
OF
THEORY
AND
EXPERIMENT
AND THE
PROBLEM
OF
S
Puls and Kirkaldy [41] have recently made a careful comparison between the calculated and the measured growth rates basically using Equation 7.16. The particular version employed was from
597
Pearlite Reaction mm 0.25
Module radius
0.20
C-730°C D-875 C-875 D-1025 C-925
0.15
0.10
0.05
0
25
50
75 s
100
125
150
FIGURE 7.28 Maximum nodule radius vs. reaction time. (From Mehl, R.F. and Hagel, W.C., Prog. Met. Phys. 6, 74, 1956. With permission from Elsevier.)
a more accurate treatment version published by Hillert in 1957 [21], in which the factor of 2 in the numerator is replaced by 1.4. However, instead of the generally accepted relationship obtained from the maximum growth rate hypothesis (S ¼ 2S0), they used S ¼ 3 S0, obtained from the application of the maximum entropy production hypothesis. Numerically, though, this increases G by a factor of only l=6. They also made a special effort to take account of the variation of the diffusivity of carbon in austenite with carbon content, using an average D, which gives preference to carbon contents in front of the ferrite lamellae, which are sevenfold wider than the carbide lamellae. This source of possible error is probably larger than any other. The solid curve in Figure 7.29 is the one calculated for G in this manner. A comparison with the selected experimental data indicates agreement to 720
Temperature (°C)
700 680 660 640 620 600 10–5
Hull et al., 0.93C Brown and Ridley, 0.81C Frye et al., 0.78C Calculated, 0.76C 10–4
10–3
10–2
Velocity (cm/s)
FIGURE 7.29 Temperature vs. velocity plots for plain carbon steels. (With kind permission from Springer Science þ Business Media: From Puls, M.P. and Kirkaldy, J.S., Metall. Trans. 3, 2777, 1972.)
598
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
within a factor of 2, a considerable improvement over the results of previous investigators who found discrepancies between theory (based on volume diffusion control) and experiment ranging as high as 100 or more, though usually in the range of 2–15. However, the finding that the experimental results usually lay at higher values of G than those calculated suggested the possibility of a limited involvement of carbon diffusion along the austenite:pearlite interface. Puls and Kirkaldy [41] made a crude assessment of the possibility of a DB contribution on the following basis. To estimate DB vs. temperature, they tried values of the volume diffusion of carbon in ferrite and for diffusion in liquids (which tend to be standard and are a useful limiting approximation of the boundary diffusivity in a solid) and found them to be similar. They then compared DBd with its counterpart for volume diffusion, namely DgV multiplied by the area (from Fick’s first law) divided by h, taken crudely as DgV =G—the forward diffusion length. At 7008C, DgV =G was appreciably higher than DBd. At 6448C (0.54Tm), however, the situation had perceptibly reversed, indicating that boundary diffusion ought to be a significant contributor under these circumstances. Using the maximum growth rate rather than the entropy production rate criterion and the interlamellar spacing data for carbon steel shown in Figure 7.11, with the relationship, following G max and Equation 7.6, we get g¼
SDG 4V
(7:34)
One obtains g ¼ 900 ergs=cm2 for the lamellar boundary energy. Kramer et al. [42] obtained an interfacial energy from calorimetric studies of 0.70 0.30 J=m2. However, from the crystallographic studies of Ohmori et al. [2] and the predecessor previously recounted, the value of the interfacial energy obtained from the growth kinetics and S data may well be too high by a factor of 3 or more. This is not as bad as in the cellular reaction case, but can hardly be called good agreement. Cahn and Hagel [13] used a cruder approach to estimate apparent D in nonferrous (substitutional) pearlites and compared it with the volume diffusivity. Their central results are shown in Table 7.1. Clearly, these pearlites grew with kinetics that were controlled by DB rather than by DV, consistent with the results on the cellular reaction cited in Section 6.4.4. The interlamellar spacing problem continues to incite lively controversy. Both the systematic variation of spacing with temperature and the replacement of a coarse spacing by a fine spacing when the transformation temperature is abruptly lowered in mid-reaction indicate the existence of some control(s) upon the spacing. Particularly when spacing is developed and adjusted by means of branching, what is it that sets the spacing? Zener’s maximum growth rate, Cahn’s maximum rate
TABLE 7.1 Apparent Diffusivities in Nonferrous Pearlite System Ag-50Cd Cu-6Be Cu-6Be Cu-31In Cu-12Al
Reaction Temperature (8C) 200 592 550 570 500
Dapp (cm2=s) 1.1 10 2.4 10 6.4 10 8.8 10 5.0 10
11 9 9 8 10
Dexpl (cm2=s) 1 10 2 10 1 10 1 10 2 10
13 12 12 12 12
Source: Cahn, J.W. and Hagel, W.C., in Decomposition of Austenite by Diffusional Processes, Zackay, V.F. and Aaronson, H.I. (eds.), John Wiley, New York, 1962, p. 131. With permission.
599
Pearlite Reaction
e
f
Stability point
(
) (
Stability point
1 1– Sc = 1 1– Sc ~v S/2 S S/2 S d or S = 3Sc
)
ΔT
v
c
c
d
as + 2b = as + b ~ΔT s 2 s or S=√2 Sm
3Sc/2 (a)
Sc
2Sc 3Sc S
(b)
(√2/2)Sm Sm √2 Sm S
FIGURE 7.30 Illustration of a perturbation argument that uniquely defines the stability point of (a) isothermal and (b) forced velocity pearlite. (With kind permission from Springer Science þ Business Media: From Puls, M.P. and Kirkaldy, J.S., Metall. Trans. 3, 2777, 1972.)
of decrease of DG, and Kirkaldy’s minimum entropy production rate all seem deficient; The first two lack firm theoretical justification and the third is an approach applicable, if at all, only at small deviations from equilibrium, such as are encountered in diffusion. Perturbation analyses have been carried out analytically by introducing a ‘‘fault’’ and ascertaining its ‘‘stability’’ with respect to the balance of the lamellar structure during further growth. These shed useful, albeit indirect, light on the interlamellar spacing problem, though they don’t solve it. As shown in Figure 7.30, the following considerations are discussed by Puls and Kirkaldy on the basis of a contribution by Kirkaldy [43]. In Figure 7.30a, consider the left-hand part of the schematic of pearlite in conjunction with the left-hand velocity (V or G) vs. S diagram. A fault of spacing less than 3Sc=2 (where Sc S0) introduced in a structure whose spacing is otherwise ‘‘c’’ will soon disappear because it yields a lower G than average. Conversely, a fault of spacing greater than 3Sc=2 introduced in a structure whose spacing is otherwise ‘‘d’’ will become the dominant spacing because it has a higher G than average. However, according to Kirkaldy’s analysis, when a fault with a spacing of 3Sc=2 is introduced in a structure of spacing 3Sc, the growth rates of the two spacings will be equal, and the structure will be stable. Spacing 3Sc, i.e., 1.5 times the spacing at G max, is thus identified as a stability point. But the basic question of what sets S remains unanswered.
REFERENCES 1. H. Modin and S. Modin, Jernkont. Ann., 139, 481 (1955). 2. Y. Ohmori, A. T. Davenport, and R. W. K. Honeycombe, Trans. Iron Steel Inst. Jpn., 12, 128 (1972). 3. R. F. Mehl and A. Dubé, in Phase Transformations in Solids, R. Smoluchowski, J. E. Mayer and W. A. Weyl (eds.), John Wiley & Sons, New York, 1951, p. 545. 4. G. W. Rathenau and G. Baas, Acta Metall., 2, 875 (1954). 5. G. E. Pellisier, M. F. Hawkes, W. A. Johnson, and R. F. Mehl, Trans. ASM, 29, 1049 (1942). 6. D. Brown and N. Ridley, JISI, 207, 1232 (1969). 7. M. Hillert, in Decomposition of Austenite by Diffusional Processes, V. F. Zackay and H. I. Aaronson (eds.), John Wiley, New York, 1962, p. 197.
600
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
8. L. S. Darken and R. M. Fisher, Decomposition of Austenite by Diffusional Processes, V. F. Zackay and H. I. Aaronson (eds.), John Wiley, New York, 1962, p. 249. 9. F. C. Hull and R. F. Mehl, Trans. ASM, 30, 381 (1942). 10. R. F. Mehl and D. W. Smith, Trans. AIME, 113, 203 (1934); 116, 330 (1935). 11. G. V. Smith and R. F. Mehl, Trans. AIME, 150, 211 (1942). 12. R. F. Mehl and W. C. Hagel, Prog. Met. Phys. 6, 74 (1956). 13. J. W. Cahn and W. C. Hagel, in Decomposition of Austenite by Diffusional Processes, V. F. Zackay and H. I. Aaronson (eds.), John Wiley, New York, 1962, p. 131. 14. M. E. Nicholson, Trans. AIME, 200, 1071 (1954). 15. S. Modin, Jernkont. Ann., 135, 169 (1951). 16. C. S. Smith, Trans. ASM, 45, 533 (1953). 17. M. Hillert, Varmlandska Bergsm. Foren. Ann., 1958, p. 29. 18. A. Hultgren and H. Ohlin, Jernkont. Ann., 144, 356 (1960). 19. S. Modin, Jernkont. Ann., 142, 37 (1958). 20. C. Benedicks, JISI, 2, 352 (1905). 21. M. Hillert, Jernkont. Ann., 141, 757 (1957). 22. F. C. Frank and K. E. Puttick, Acta Metall., 4, 206 (1956). 23. M. Hillert, Discussion to Ref. [9]. 24. A. Bagaryatski, Dok. Akad. Nauk, SSSR, 73, 1l61 (1950). 25. N. J. Petch, Acta Cryst., 6, 96 (1953). 26. K. W. Andrews, Acta Metall., 11, 939 (1963); 12, 921 (1964). 27. I. V. Isaichev, Zhur. Tekhn Fiziki, 17, 835 (1947). 28. H. I. Aaronson, C. Laird, and K. R. Kinsman, Scr. Metall., 2, 259 (1968). 29. S. A. Hackney and G. J. Shiflet, Acta Metall., 35, 1007(1987); 35, 1019 (1987). 30. J. H. Beatty, S. A. Hackney, and G. J. Shiflet, Philos. Mag., 57, 457(1988). 31. R. E. Hoffman, Unpublished research at General Electric Co. 32. J. W. Cahn, Acta Metall., 7, 18 (1959). 33. J. W. Cahn, Acta Metall., 4, 449 (1956). 34. C. Zener, Trans. AIME, 167, 550 (1946). 35. M. Hillert, Metall. Trans., 3, 2729 (1972). 36. W. H. Brandt, J. Appl. Phys., 16, 139 (1945); Trans. AIME, 167, 405 (1946). 37. E. Scheil, Z. Metall., 37, 123 (1946). 38. M. Hillert, Jernkont. Ann., 141, 757 (1957); in The Mechanism of Phase Transformations in Crystalline Solids, Institute of Metals, London, U.K., 1969, p. 231. 39. E. Scheil ard A. Lange-Weise, Arch. Eisenhutt., 11, 93 (1937). 40. D. Brown and N. Ridley, JISI, 204, 811 (1966). 41. M. P. Puls and J. S. Kirkaldy, Metall. Trans., 3, 2777 (1972). 42. J. J. Kramer, G. M. Pound, and R. F. Mehl, Acta Metall., 6, 763 (1958). 43. J. S. Kirkaldy, Scr. Metall., 2, 565 (1968).
8 Martensitic Transformations 8.1 DEFINITION A martensitic transformation is one which takes place by shear, a cooperative, nondiffusional, nonthermally activated process, which permits such a transformation to accomplish a change in crystal structure but not in composition or in degree of ordering (for substitutional solid solutions). Some references for this chapter are Cohen in Phase Transformations in Solids, Wiley, New York, 1951, p. 588 [1]; for thermodynamics and kinetics, Kaufman and Cohen, Prog. Met. Phys., 7, 165 (1958) [2]; for nucleation, Magee in Phase Transformations, ASM, Metals Park, OH, 1970, p. 115 [3]; for general aspects with emphasis on crystallography, Christian, The Theory of Transformations in Metals and Alloys, Wiley, New York, 1962 [4]; for crystallography, Lieberman, in Phase Transformations, ASM, Metals Park, OH, 1970, p. l. [5]; Wayman, Adv. Mater. Res., 3, 147 (1968) [6]; Wayman, Introduction to the Crystallography of Martensitic Transformations, MacMillan, New York, 1964 [7].
8.2 SALIENT CHARACTERISTICS (DESCRIBED BRIEFLY) 8.2.1 CRYSTALLOGRAPHY A martensitic transformation may be formally visualized as consisting of two successive steps: the first is a homogeneous distortion that accomplishes the basic change in the crystal structure (this is also known as lattice deformation, the Bain strain, etc.), and the second is a lattice-invariant deformation, occurring by slip, by twinning, or both, which, in effect, bring the martensite and the matrix lattices together again at the habit plane. Only in the case of a fcc=hcp transformation is this second step unnecessary. In addition, if the homogeneous distortion cannot bring all of the atoms to the correct positions needed to establish accurately the product phase crystal structure, very short-range (less than a diffusional jump distance), nondiffusional atomic movements, known as ‘‘shuffles,’’ complete the job. In Figure 8.1, (a) represents the austenite prior to transformation (we shall term the matrix ‘‘austenite,’’ irrespective of the alloy system involved), (b) is the lattice deformation, (c) represents the lattice-invariant deformation, considered in isolation from the first step and here having occurred by slip, and (d) is the product of both steps, with the habit plane (indicated by two vertical lines, dashed) being rough or stepped only on an atomic scale. In reality, this two-step process is likely to take place as one, that is, both probably occur simultaneously. The crystallography of martensitic transformations is evidently governed by the minimization of the shear–strain energy. This requires that the habit plane be unrotated, and if possible undistorted. Possibly a small (few percent maximum) uniform distortion or dilatation of the habit plane does occur, but rotation does not. Fixed lattice orientation relationships and habit planes always characterize martensitic transformations. The orientation relationship may or may not be rational; except in fcc=hcp transformations, the habit plane is normally quite irrational because of the lattice-invariant deformation.
8.2.2 MORPHOLOGY Martensite crystals, on the foregoing crystallographic description (and on the discussion of the surface relief effects, Section 8.2.3), should be plates. Normally they are. Sometimes needles are found, but usually at a free surface where the strain energy constraint can be relaxed. Internally, 601
602
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
(a)
(b)
(d)
(c)
(e)
FIGURE 8.1 Types of deformation in martensitic transformations. (a) Undeformed crystal, (b) Lattice deformation, (c) Lattice-invariant deformation by slip, (d) Lattice deformation and Lattice-invariant deformation combined to give no total shape deformation and (e) Shape deformation casued by varying lattice deformation in different regions. (After Bilby, B.A. and Christian, J.W., The Mechanisms of Phase Transformations in Metals, Institute of Metals, London, U.K., 1956, p. 121. With permission.)
martensite crystals are supposed to be either twinned or slipped because of the lattice-invariant deformation. The release of the strain energy accompanying the transformation, however, usually causes accommodation deformation of the matrix and=or the martensite. In the case of slip, it is difficult to separate the two contributions to the observed dislocation density.
8.2.3 SURFACE RELIEF EFFECTS Interferometric observations of the austenitic specimen surface that is polished flat (with a few scratches being retained as fiducial marks) and then cooled to a lower temperature to allow just a few plates to form, show that the transformed portion of the surface associated with each martensite plate has undergone a tilt about one line of intersection with the free surface. The tilted region, however, remains flat. This is illustrated in Figure 8.2. The macroscopic tilt experimentally observed is the shape deformation, which is the sum of the lattice and the lattice-invariant deformations. The nature of this tilt and the failure of the habit plane to rotate as indicated by the straightness of the scratches, led to the description of the habit plane as an invariant plane of the shape deformation, and to the transformation (as a whole) as an invariant plane strain transformation. As illustrated in Figure 8.3a, the ideal shape change and scratch displacements are really to be expected only when a martensite plate completely penetrates a thin single crystal [10]. Figure 8.3b shows the effect upon the surface relief when accommodation of the shape change is elastic. The remaining components, Figure 8.3c through 8.3f [10] illustrate various plastic deformation modes of accommodating the shape change.
8.2.4 TIME-DEPENDENCE
OF
MARTENSITE FORMATION
The original view of this was that the amount of martensite characteristic of a given temperature in a given alloy is achieved virtually instantaneously. Thus, Bunshah and Mehl [11] found by an electrical resistivity method that individual martensite plates form in ca. 3 10 7 s, growing at ca. 105 cm=s, independent of temperature in the range of 2008C to 208C in an Fe-Ni alloy. The martensite reaction itself was taken to be effectively insuppressible. However, Kurdjumov and Maximova [12] found that
603
Martensitic Transformations C
G
F B S T΄ T S΄
D
E A
H
N M
O L
FIGURE 8.2 Shape deformation produced by a martensitic plate. (After Bilby, B.A. and Christian, J.W., JISI, 197, 122, 1961. With permission.)
β α
α
β
β
α
α
α
α Kink boundary Slip planes in α
(a)
(b)
(c)
Slip in β
α α
Kink boundary (d)
β
α
β
Slip in α
α
Slip in α
Slip planes in α (e)
(f )
FIGURE 8.3 Shape changes resulting from transformation or accompanying plastic deformation. (a) Unconstrained transformation is a single crystal, (b) lenticular or tapering plate accomodated elastically in the matrix, (c) and (d) alternative possible configurations of tapering plate accomodated by kink planes in the matrix, (e) accomodation by slip in the matrix, and (f) accomodation by slip in both plate and matrix. (From Christian, J.W., Decomposition of Austenite by Diffusional Processes, John Wiley, New York, 1962, p. 371. With permission.)
604
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
–40 –60 1%M
Temperature (°C)
–80
5%M 10%M 15%M 20%M
–100
25%M
30%M
–120 –140 –160 –180 –200
50 100 Isothermal holding time (t + l) min
500
1000
FIGURE 8.4 Isothermal transformation curves for martensitic formation in an iron-nickel manganese alloy. (After Cech, R.E. and Hollomon, J.H., Trans. AIME, 197, 685, 1953. With permission.)
in some alloys martensite can exhibit the C-curve behavior characteristic of diffusional transformations and can thus be suppressed. This is shown in Figure 8.4, taken from the corroboratory work of Cech and Hollomon [13]. Even a casual observation of the occurrence of martensite in such alloys, however, indicates that the transformation is slow, although, to the eye, the growth is instantaneous. In some other martensite transformations, however, growth also appears to occur slowly, for example, in some Fe-Ni alloys [14], U-Cr [15], and in Au-Cd [16]. Private discussions with Holden and Read elicited the view, however, that the growth process was one of short, high-velocity shears, punctuated by comparatively long rest periods. However, in so-called thermoelastic martensite, which is highly reversible, Hull (private communication) indicated that when viewed at high resolution in a transmission electron microscope these martensites can grow slowly, and likely do on an atomic scale. Hence, high-velocity growth is a usual, but not a general characteristic of martensite.
8.2.5 TEMPERATURE DEPENDENCE
OF
MARTENSITE FORMATION
When martensite formation occurs in a two-phase field, as in Fe–C, Fe-Ni and many other alloys, the ultimate upper limiting temperature at which martensite can occur is the T0 temperature. However, there is an appreciable temperature interval in most systems between the temperature and the highest temperature at which martensite forms, known as the martensite-start or Ms temperature. Figure 8.5 is the result of one set of calculations of T0 vs. carbon concentration x (mole fraction of C in g) for Fe–C, showing T0 (and the intersection of the free energy vs. 0 temperature curves with the line of DGg!a ¼ 0) and Ms for a series of Fe–C alloys. The temperature interval between T0 and Ms is likely due to the elastic shear strain energy, primarily associated with the martensitic of transformation (as discussed in a subsequent Section 8.3.2). Figure 8.6, based upon quantitative metallography, shows the percentage of martensite formed as a function of temperature. Although thermodynamically there should be no need for any temperature interval at all below Ms, it is probable that the elastic (and possibly plastic) strains created by the martensite plates initially formed requires an additional driving force to be overcome by successor plates. This driving force can only be acquired, of course, by additional undercooling. Customarily, a number of martensite crystals form within each grain, each very rapidly. Further cooling results in
605
Martensitic Transformations
.%
wt
3000
wt
0.4
0.0
3500
Ca
.%
on
.% Ca
on
.%
1000
Ca
(J/mol)
rb
wt o rb n
ΔG γ
α΄
Ca
on
.%
rb
wt
1.2
1500
on
rb
wt
1.0
2000
rb
Ca
0.8
2500
500
Ms
0
1.1wt.% Carbon
–500 –1000 –1500 –2000 200
400
600
800
1000
1200
Temperature (K) 0
FIGURE 8.5 DGg ! a for Fe–C alloys as a function of temperature and carbon content. Solid and open circles indicate Ms and As temperatures, respectively. (From Kaufman, L. and Cohen, M., Prog. Met. Phys., 7, 165, 1958. With permission from Elsevier.) 100 Austenitized at 845°C Austenitized at 925°C Austenitized at 1040°C
Percent martensite
80
Highest temperature at which traces of martensite were detected
60
40
20
0 –20
80 0
160 50
240 320 150 100 Temperature
400 200
480 (°F) 250 (°C)
FIGURE 8.6 Martensitic transformation curves for 1.1% C–2.8% Cr steel. (From Harris, W.J. and Cohen, M., Trans. AIME, 180, 447, 1949. With permission.)
606
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
the nucleation and swift propagation of additional martensite plates, not in the further development of those previously formed. A process that markedly condenses the temperature range in which martensite plates form is the ‘‘burst phenomenon,’’ in which many martensite plates form in swift, interlocked succession. The Mf or martensite-finish temperature is that at which the fraction transformed to martensite first reaches 100%. In some systems, however, the complete transformation to martensite does not occur at liquid nitrogen or even at lower temperatures; the plot of percent martensite vs. temperature simply flattens out and remains so. The austenitizing temperature has two effects upon Ms in steel. The first is exerted through the increase in austenite grain size with temperature. The Ms tends to rise with increasing grain size. In an Fe-29.5% Ni alloy, Scheil found that increasing the area of the grain cross section more than an order of magnitude raised the Ms from 658C to 308C. There is also an effect of austenitizing temperature per se; the lower the austenitizing temperature the higher is the Ms and this effect, not presently understood, has been shown to be reversible [17].
8.2.6 REVERSIBILITY In principle, martensitic transformations are reversible. When several martensite crystals form within a single matrix grain, reheating to a temperature above the As or austenite-start temperature should cause the original matrix grain to be fully restored as the martensite plates reverse-transform martensitically to austenite. When the reversibility is good, the martensite plates will appear repeatedly in the same place upon recycling through the Ms and As temperatures. The As lies above T0 for just the same reason(s) that the Ms falls below this temperature. Two factors tend to inhibit reversibility. One is the obvious factor of the decomposition of the martensite to more stable phases during reheating toward the As temperature, for example, tempering of martensite in Fe-Cbase alloys. Another factor is plastic deformation damage to the austenite matrix accompanying the martensite formation, which, unless fully annealed out after reversal, will cause difficulties when one attempts to form martensite again. As their name implies, thermoelastic martensites are the type most readily reversible.
8.2.7 INFLUENCE OF APPLIED STRESS Martensitic transformations are usually far more sensitive to this factor than those proceeding by nucleation and growth. Deformation usually increases the amount of martensite formed and also raises the Ms to the Md temperature, and lowers the As to the Ad temperature, both being markedly closer to the T0 temperature in consequence, as illustrated in Figures 8.7 [18] and 8.8 [19]. However, these effects are as described only, however, when the deformation is primarily plastic and is acting principally to aid nucleation. When an elastic stress is applied, it will raise the Ms, if it aids the shape deformation associated with a given habit plane, and lower the Ms if it opposes the deformation. Since normally some variants of the habit plane will be favored, the Ms will usually be raised. However, Machlin and Weinig [60] have showed the habit plane favored, in a Ti-9% Mo alloy, to be that along which the favorable resolved shear stress is greatest. Kulin, Cohen, and Averbach [61] demonstrated that hydrostatic stresses lower the Ms (this is a LeChatelier effect; martensite forms with a volume increase). Tensile elastic stresses and shear stresses acting along a potential habit plane aid the transformation, but compressive stresses hinder it. The latter observation, which is puzzling, was not supported by a follow-on work by Patel and Cohen [22]. As shown in Figure 8.9, compression raises the Ms, though less than tension; hydrostatic compression lowers the Ms. Patel and Cohen [22] accounted for these results by resolving both shear and normal stresses onto the habit plane. In compression, the normal component opposes the transformation, whereas in tension both help it. In hydrostatic compression, there is no shear component to promote martensitic transformation.
607
Martensitic Transformations
400
Temperature (°C)
300
T0
200 Md Ad
100
0
–100
–200
0
10
20 %Ni
30
40
FIGURE 8.7 Bracketing of T0 in cobalt-nickel system by plastic deformation. (After Hess, J.B. and Barrett, C.S., Trans. AIME, 185, 599, 1949. With permission.) 550 500 450 400 As
Temperature (°C)
350 300
Ad
250
Ad Ad
200 150
1 (M + A ) d d 2
100
0
Md
Md
50
Md
Ms
–50 –100
27
28
29 Nickel at.%
30
31
FIGURE 8.8 Bracketing of T0 in iron-nickel alloys by plastic deformation. (From Kaufman, L. and Cohen, M., Trans. AIME, 206, 1393, 1956; J. Met., 8, 1393, 1956. With permission.)
608
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Change in Ms temperature (°C)
30
20
0.5% C, 20% Ni Steel
Uniaxial tension
10
Uniaxial compression
0 Hydrostatic compression
–10 70% Fe, 30% Ni alloy –20
–30 0
0.5
1.0 1.5 Stress or pressure ×10–8 Pa
2.0
FIGURE 8.9 Change in Ms temperature as a function of loading condition. (From Patel, J.R. and Cohen, M., Acta Metall., 1, 531, 1953. With permission from Elsevier.)
8.2.8 THERMAL STABILIZATION If one cools steel partway down to the ‘‘final’’ temperature, stopping either above (less certain) or below the Ms, less martensite will form at the final temperature than would have developed had one cooled directly to this temperature at the same rate. This phenomenon is known as stabilization. It is attributed currently to the attraction of carbon to whatever serves as the nucleating agent of martensite and the interference of this carbon with the nucleating mechanism. Another hypothesis is that carbon is attracted to dislocations in the matrix; these dislocations then become more difficult to move and so resist further martensite formation. Thermal stabilization is not observed in steel in the absence of carbon and nitrogen; see the works by Glover [23] and by Kinsman and Shyne [24].
8.3 THERMODYNAMICS OF MARTENSITE TRANSFORMATION This consists of two parts: the calculation of the T0 temperature and evaluation of the difference between the T0 and Ms temperatures in terms of free energy difference.
8.3.1 T0 TEMPERATURE As discussed earlier, this is the temperature at which two phases of the same composition are in unstable equilibrium, both of the phases being stress free. Normally, these phases have different equilibrium compositions bounding a common two-phase region. In a system of n components, the composition corresponding to a given T0 temperature is given by n X i¼1
g
xi G i ¼
n X i¼1
a
xi G i
(8:1)
609
Martensitic Transformations
where xi is the mole fraction of the ith component, Gi is the partial molar free energy of this component in the a or g phase, as designated. When the standard states are pure i in the crystal structures of a and g, RT
n X i¼1
xi ln agi ¼ RT
n X i¼1
xi ln aai þ
n X
xi DGig!a
(8:2)
i¼1
where ai is the activity of i in the designated phase, DGig!a is the free energy change associated with the transformation of pure i from g to a. If the activities of component i are determined with respect to the same standard state, that is, pure i in either g or a form, then the second term on the r.h.s. is eliminated. When ai vs. xi can be expressed as an equation, solving Equation 8.2 either graphically or analytically yields the desired composition. Normally, the activity data will be available at temperatures other than those of the T0–x curve. However, from the relationship q ln ai =q(1=T) ¼ DH i =R, where DH i is the partial molar heat of solution of i in the designated phase, the activity data are readily extrapolated to the temperature range of interest, assuming that no unusual changes intervene such as the onset of ferromagnetism. Although the necessary activity data will eventually become available at least in a few multicomponent alloy systems of greatest industrial interest, and is already known for a number of binary systems, in many instances this information has not been obtained and therefore the T0–x curve must be estimated by approximate methods. The simplest method is to construct a curve from the ga composition midpoints, xag a þ xg =2 of the two-phase region. An experimental method of determining T0, applicable when secondary reactions such as tempering of martensite do not interfere, follows from, 1 1 T0 (Ms þ As ) (Md þ Ad ) 2 2
(8:3)
Since the difference between Md and Ad is less than that between Ms and As (see Figrue 8.8), this approach yields the more accurate result. Most of the efforts exerted to date in respect of calculating T0 have been in connection with Fe–C, Fe-X, and Fe–C–X alloys. In Chapter 1 an expression for T0 was derived for the austenite-to-ferrite reaction in Fe–C alloys, through the statistical thermodynamic treatment of Kaufman, Radcliffe, and Cohen [27] in Chapter 1, see Equation 1.114. Other developments of T0–x equations are given by Aaronson et al. [25] and by Bell and Owen [26]. Figure 8.10 presents the plots of T0 vs. xg from a number of equations. The curves that cluster tightly in the middle of these plots are the most accurate ones; Equation 1.114 is included in this group. Bell and Owen [26] favored an equation developed by Fisher [27]; its derivation, however, entails a number of crude approximations that appear to introduce significant numerical errors. Kaufman and Cohen [19] calculated T0 in Fe-Ni alloys using the regular solution approximation. Zener [28] used a stimulating approach based upon statistical thermodynamics to calculate T0 and phase boundaries in Fe–C, Fe-X, and Fe–C–X alloys. These results are, however, in disagreement with experiments in a significant number of instances. Subsequently, Zener [29] developed a more detailed approach to such calculations for Fe-X that yields results that are more satisfactory. This approach will be recounted here and then combined with the KRC approach for Fe–C in an elementary fashion. (for pure Fe) vs. Following a concept due to Johannson [30], Zener separated the plot of DGa!g Fe temperature, that is, curve A in Figure 8.11, into two components. Component B represents the
610
Mechanisms of Diffusional Phase Transformations in Metals and Alloys 1000 900
Temperature (°C)
800 700 Microstructural Bs
600
Zener
500 Zγ = 5 Ellis et al. constants; KRC
400
Zγ = 6.7, Darken & Gurry; KRC
Zγ = 6, Sheil constants; KRC KRC Ref. [22]
300 200
Wγ = 1.5 kcal/mol Darken-Smith
0
01
02
LFG
03
04
06
05
07
08
09
Xγ
FIGURE 8.10 T0 for transformation of austenite to ferrite of the same composition as a function of mole fraction of carbon. See the original paper for equations. (From Aaronson, H.I. et al., Trans. AIME, 236, 753, 1966. With permission.) J/mol +4000 +3000 1600
α
+2000
+1000
0
–1000
γ
ΔGFe(Mag)
1400 α
Temperature (°C)
–2000
ΔG Fe
C
1200
γ
B
A
1000 800
α
γ
ΔGFe(NM)
600 400 200 +1000
+600
+200 α
ΔGFe
0
–200
–600
γ
(cal/mol)
FIGURE 8.11 Separation of DGa!g into magnetic and nonmagnetic terms. (From Zener, C., Trans. AIME, 203, 619, 1955. With permission.)
611
Martensitic Transformations
nonmagnetic component, the only one in the low temperature range wherein the uncoupling of the magnetic moments of the Fe atoms that presages the Curie temperature at higher temperatures has not yet begun. Component C, obtained by subtracting component B from curve A, is the magnetic portion of the free energy change. The magnetic component is taken to arise from the uncoupling of moments. Zener proposed that when an alloying element is added to Fe, it can affect the two components of the free energy change associated with the g=a transformation independently. Consider first the nonmagnetic component. The variation of this component with temperature can be represented by the empirical equation for curve B: DGg!a Fe(NM) ¼ 1:41(T
1013)
(8:4)
where T is the temperature in K. Noting that the interval between T0 and Ms is nearly independent of composition, Zener estimated that the influence of an alloying element upon Ms is equivalent to its effect upon T0. Assuming that this effect results only in a displacement of curve B parallel to the temperature axis, the nonmagnetic component of the free energy change associated with the transformation of austenite to ferrite of the same composition becomes 0
DGg!a Fe(NM) ¼ 1:41(T
Y DTNM
1013)
(8:5)
where Y is at.%X DTNM is the displacement in the temperature of curve B per at.%X, which is equal to the displacement of Ms per at.%X Treating the magnetic component of the free energy change similarly, the total free energy change in an Fe-X alloy becomes 0
g!a {T DGg!a ¼ DGFe(NM)
g!a Y DTNM } þ GFe(Mag) {T
g!a {T1 } DGFe(NM)
þ
Y DTMag }
g!a DGFe(Mag) {T2 }
(8:6)
where the braces denote the indicated free energy change at temperature T minus the product of Y and the appropriate DT. The free energy change actually dealt with is the free energy change in pure Fe, not in Fe-X. This free energy change in pure iron is at temperature T displaced by the amount of YDT. 0 To evaluate DTMag, use the relationship that DGg!a ¼ 0 at T0. Then, from Equations 8.5 and 8.6, 0
DGg!a Fe(Mag) {T2 } ¼ 1:41(T0 Y DTNM
1013)
(8:7)
Substituting T0 obtained from the phase diagram, usually via Equation 8.3, and knowing DGg!a,* 0 the T2 that yields a value of DGg!a that balances Equation 8.7 is determined by trial and error. From the definition of T2, DTMag ¼
T0
T2 Y
(8:8)
Table 8.1 lists available values of DTNM and DTMag.
* From a table constructed by Fisher [25]. Equations (17) and (18) of Ref. [30] represent this table empirically; Kaufman et al. [31] have developed an improved table of values of this DG in pure iron, but the differences with respect to Fisher’s collation are neither large nor systematic.
612
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
TABLE 8.1 DT NM and DT Mag for Alloying Elements Alloying Element
DT NM per at.% X
Si Mn Ni Co Mo Al Cu
8C per at.% X
0 39 12 18 16 17 15 11 12
0 40 17 15 10 30 10
DT Mag per at.% X
8C per at.% X
3 3712 6 19 12 26 8 4 12
6 38 5 12 19 15 17 4
Source: Aaronson, H.I. et al., Trans. AIME, 236, 768, 1966. With permission.
g!a Equation 8.6 is placed in more useful form by subtracting and adding DGFe(NM) {T2 }: 0
g!a {T1 } DGg!a ¼ DGFe(NM)
g!a g!a g!a DGFe(NM) {T2 } þ DGFe(Mag) {T2 } þ DGFe(NM) {T2 }
(8:9)
g!a g!a Substituting Equation 8.5 and the relationship DGg!a Fe(Mag) {T2 } þ DGFe(NM) {T2 } ¼ DGFe {T2 }: 0
¼ 1:41(DTMag DGg!a S
g!a DTNM ) þ DGFe {T2 }
(8:10)
where the subscript ‘‘S’’ is introduced to indicate a binary substitutional alloy. To obtain T0 for 0 0 0 , and set DGg!a (now for an Fe–C–X alloy, it is only necessary to replace DGg!a , with DGg!a S Fe–C–X) equal to zero, and solve by trial and error. Equation 8.10 was tested against the Kaufman 0 and Cohen regular solution relationship for DGg!a in Fe-Ni [19]. As shown in Figure 5.7, agreement between the T0s calculated from these two equations is effectively perfect until 10 wt % Ni and remains reasonably good at the higher nickel contents (up to 32 wt%) investigated.
8.3.2 DIFFERENCE
BETWEEN
T0 AND MS
Zener [27] was the first to determine that Ms lies below T0. He calculated this temperature difference to be equivalent to a free energy difference of 1200 J=mol in Fe–C alloys, independent of carbon content. KRC, using Equation 1.114, found this difference equivalent to 1100 J=mol, again independent of composition. Fisher [27] obtained larger values, which increased with carbon content, and so did Bell and Owen [26], relying primarily upon Fisher’s treatment. As already noted, however, the Fisher treatment is particularly crude. Zener ascribed the difference between T0 and Ms to the strain energy associated with martensite formation. Kaufman and Cohen [2] have summarized other views as to the origins of this difference. The undercooling needed to produce martensite nucleation is an obvious possibility, but difficult to deal with, since, as we shall see later, the nature of martensite nucleation remains uncertain (other than that it is not thermally activated in the sense of the nucleation process in diffusional transformations). Interfacial energy is another possibility, but seems unlikely unless it is somehow involved in the nucleation stage because the volume free energy change released ought again to be large enough to make this factor unimportant at larger sizes. Plastic deformation of austenite (and self-deformation of martensite) and the ‘‘clicks’’ readily audible to the unaided ear associated with the martensite transformation in some alloy systems have also been advanced as possible
613
Martensitic Transformations
sources of this free energy loss. However, both ought logically to be incorporated in Zener’s original suggestion of the volume strain-energy loss. Attempts to estimate this loss, by Fisher and by Knapp and Dehlinger [33], yielded values much in excess of the free energy difference made available by the temperature interval between T0 and Ms. The calculations to be noted here, by Eshelby [34] will be seen to have yielded a more satisfactory result. Another factor involved in this free energy difference turns out to increase the difference. This is the Zener ordering energy, which is the difference in the free energy between the ferrite containing randomly distributed carbon atoms and martensite of the same composition in which the carbon atoms are present only in certain sites. It is our tentative contention that it may be possible to account for the Ms temperature on purely thermodynamic grounds as 0
DGg!Ms ¼ DGg!a þ DG{ordering} þ DG{elastic strain energy}
(8:11)
Consider first Eshelby’s strain energy treatment. As rough estimates of the physical quantities involved during the formation of ferrous martensites, Eshelby suggested E ¼ 80 GPa, Poisson’s ratio, n ¼ 1=4, a 5% volume expansion, and a 108 shear (equivalent to eT13 ¼ 0:009). Two contributions to the volume strain energy are identified. One is the dilatational contribution resulting from the volume expansion. This is given by Equation 2.137 when Young’s modulus and Poisson’s ratio are the same in both phases and is ca. 105 J=mol with the given physical constants. Employing Zener’s value of 1200 J=mol for the difference between T0 and Ms in terms of the free energy change and the assumption that this difference is due entirely to the strain energy, then the shear strain energy must be 1100 J=mol. From Equation 2.140, the shear strain energy will be of this amount when the ratio of the martensite plate thickness to diameter is 0.08, which is in general agreement with the experiment. Clearly, it would be of value to redo this calculation, using the best available experimental values of the various physical constants. The Zener ordering energy arises in the following manner. Only one-third of the interstitial positions in ferrite correspond to interstitial sites in austenite. As long as the growth of martensite proceeds at velocities too high to allow even single interstitial jumps to occur during transformation, these will be the only positions occupied by carbon atoms in martensite. Further, the interstitial positions in ferrite have tetragonal rather than cubic symmetry; of the six Fe atoms nearest to each interstitial site, two are closer than the other four. The line passing through the two closest positions is termed the tetragonal axis of this position. The interstitial sites in ferrite that are occupied because of the austenite-to-martensite transformation are those whose tetragonal axes are parallel to the cube axis of compression that produces the fcc to bcc transformation by shear. Since the tetragonal axes of the carbon atoms are parallel, the martensite as a whole will be tetragonal. Since martensite remains tetragonal at room temperature for long times despite the fact that carbon atoms can change their sites about 1010 times per second, the free energy of ordered tetragonal martensite must be lower than that associated with disordered cubic martensite. Hence, the carbon ordering in martensite that automatically attends the transformation makes the free energy change associated with this transformation more negative. Zener’s analysis of this ordering free energy was extended by Fisher; the latter’s result was cast into the following form by KRC: DGordering ¼
50,000
x 2 p2 þ 0:7xT f{p} cal=mol 1 x
(8:12)
where x is the mole fraction of carbon in austenite and in martensite, p is the order parameter, which is equal to 3=2{(nt=n) (1=3)}, where nt is the number of carbon atoms in tetragonal sites, and n is the total number of carbon atoms present, and f{p} is a numerical function of p, tabulated by Fisher in terms of T=Tc, where Tc is the critical temperature for ordering ¼ 28,000x=(1 x). Figure 8.12 shows the variation of Tc with wt.% C. Figure 8.13 due to Bell and Owen [26] shows the contribution of Zener ordering energy as a function of mole fraction of carbon at different temperatures. It is seen
614
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Critical temperature (°C)
300
200
100
0
–100 0.1
FIGURE 8.12 permission.)
0.2 0.3 0.4 Weight percent carbon
0.5
Critical Temperature of Zener ordering. (From Zener, C., Trans. AIME, 167, 550, 1946. With
2500 T (K) –0 – 200 – 400
2000
–ΔG* (J/mol)
– 600 – 800
1500
– 1000
1000
500
0
0
0.02
0.04
0.06
0.08
0.10
Atom fraction interstitial solute, X
FIGURE 8.13 Variation of the free energy due to Zener ordering, DG* (¼ DGordering) with either carbon or nitrogen concentration. (From Fisher, J.C., Trans. AIME, 185, 688, 1949. With permission.)
that in hypoeutectoid steels (less than 0.036 mole fraction of carbon), the Zener ordering free energy is less than about 50 cal=mol ( ¼ 200 J=mol). At higher carbon contents, however, this factor is of rapidly increasing importance. In support of the general correctness of the ordering concept, Owen, et al. [35] found that Zener’s equation for Tc successfully predicts the temperature-composition region in which martensite is cubic (though a detailed check does not appear to have been made).
615
Martensitic Transformations
8.4 OVERALL KINETICS OF MARTENSITE TRANSFORMATION Because of the high velocity with which martensite often forms, it is necessary to resort to studies of this type in order to extract additional information as to the transformation mechanism. The considerations of this and Section 8.5 on the nucleation of martensite are based on Ref. [3]. All of the martensites considered in this review are of an Fe-base.
8.4.1 QUALITATIVE KINETICS The diverse kinetic behaviors of martensite thus described can be rationalized on the following basis, and accordingly grouped into two. (1) Dynamically stabilized alloys (typically Fe–C): The fraction transformed to martensite, f, is independent of reaction time, but faster quenching to the transformation temperature from the austenitizing temperature will increase f. The basic prerequisite for the presence of this behavior is the presence of C or N. Philibert [36] has shown that thermal stabilization of martensite does not occur in the absence of these interstitials. Magee explained this behavior as abnormal, that is, as not a basic characteristic of martensite, but resulting from stabilization during the quench to the reaction temperature. (2) Nonstabilized alloys: this group can be subdivided into two. The first is (259)g habit plane operative (typically Fe-31Ni); martensite forms predominantly during quenching, but more forms isothermally, often as a ‘‘burst’’ in which many plates are formed in swift, interactive succession. The second is other habit planes, for example, (225)g operative – typically Fe-24% Ni-3% Mn. Martensite forms predominantly in isothermal transformation. Mn, Cr, and higher transformation temperatures favor this type of martensite. In the early years of research on martensite formation, carbon was always present in the alloys studied and Ms was high enough to encourage stabilization giving the conclusion that f is timeindependent. When martensite forms isothermally, increasing the cooling rate to the reaction temperature decreases the amount of martensite present (anisothermally formed martensite, even in small quantities, stimulates the nucleation of isothermal martensite). The difference in the isothermal behavior of the two groups of martensites noted evidently arises from differences in their autocatalytic behavior. When this behavior is strong, bursts appear, and the small amounts of martensite formed during the quench from the virgin austenite are thereby greatly multiplied.
8.4.2 QUANTITATIVE KINETICS 8.4.2.1 Anisothermal Martensite Formation As Zener originally pointed out, more martensite forms at lower temperatures because the volume 0 , is larger. Magee began this analysis by assuming the simplest free energy change, DGg!a v relationship between the number of martensite plates per unit volume of austenite, N, and the volume free energy change: dN ¼
0 f d DGvg!a
(8:13)
where f is a proportionality constant. The relationship between the change in f, that is, df, and the change in the number of plates per unit volume of specimen, dNv, is df ¼ V dNv
(8:14)
where V is the average volume of a martensite plate. dNv and dN are related by dNV ¼ (1
f )dN
(8:15)
616
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
0 0 ¼ dDGg!a =dT dT, Substituting Equations 8.14 and 8.15 into Equation 8.13, and using dDGg!a v v and rearranging, V dNv ¼
df ¼
V(1
f )f
0 d DGg!a v dT
dT
(8:16)
Integrating Equation 8.16 from the Ms temperature (at which f ¼ 0) to the reaction temperature, Tq, 0 =dT are constants, assuming V, f, and dDGg!a v
ln (1
f ) ¼ Vf
0 d DGg!a v dT
(Ms
Tq )
(8:17)
3
(8:18)
Thus,
1
2
f ¼ exp4Vf
0 d DGg!a v dT
(Ms
Tq )5
When f is much less than unity, since ey is ca. 1 þ y with y 1,
f ¼ Vf
0 d DGg!a v dT
(Ms Tq )
(8:19)
Figure 8.14 provides direct support for Equation 8.18, showing that a plot of ln (1 f ) vs. (Ms Tq) yields a straight line [36]. The finding indicated on this plot that the coefficient of (Ms Tq) is independent of alloy composition, however, is not general. In further support of this picture, 0 0 Brook, Entwisle, and Ibrahim [38] showed that d DFvg!a =dT ¼ DSvg!a correlates much of the 1.0
Fe–1.1C
Harris and Cohen
Fe–1.1C
0.50
Fe–0.18C Fe–0.50C Fe–0.37C
Koistinen and Marburger
l–f
0.10 0.05
l – f = e–.011 (Ms–Tq)
0.01 0.005 0
100
200
300 Ms – Tq (°C)
400
500
FIGURE 8.14 Plot of the volume fraction of retained austenite, 1 f, against Ms Tq. (After Koistinen, D.P. and Marburger, R.E., Acta Metall., 7, 59, 1959. With permission from Elsevier.)
617
Martensitic Transformations
(a)
(b)
(c)
(d)
FIGURE 8.15 (a) and (b) random transformation and partitioning of austenite into smaller regions, (c) and (d) the progress of transformation (formation of clusters of martensite plates) due to autocatalysis. (From Magee, C.L., Phase Transformations, ASM, Metals Park, OH, 1970, p. 115. Reprinted with permission of ASM International1. All rights reserved.)
difference in the temperature dependence of f among different iron-base alloys. An important difference between this and previous analyses is that V is assumed constant. Subsequent research directly supports this view and contradicts that of Fisher, who assumed that martensite plates nucleate at random, and partitioning the austenite into successively smaller regions, themselves become smaller with increasing f as illustrated in Figure 8.15a and b. Experimentally, however, when one plate forms, many do through autocatalysis; other regions of the austenite remain untransformed until ‘‘clusters’’ of martensite plates appear there, that is, the ‘‘clusters’’ seem to spread, as illustrated in Figure 8.15c and d. 8.4.2.2 Isothermal Martensite Formation This analysis, again due to Magee, is similar to that of previous analyses by Raghavan and Entwisle [39] and by Pati and Cohen [40]. The main improvement incorporated by Magee is the assumption that V is constant. The key assumption made in this and in the preceding analyses is that the rate of nucleation from a fixed number of nucleation sites is independent of time. The basic equation is dNv ¼ nt n exp dt
Q RT
where nt is the number of nucleation sites per unit volume of specimen n is the attempt frequency Q is the activation energy
(8:20)
618
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
nt is not constant because each martensite plate formed uses up its nucleation site, and more importantly, austenite:martensite interfaces themselves (or more strictly, perhaps, the vicinity of martensite plates) are by far the most prolific nucleation sites. So, nt ¼ (ni þ pf
Nv )(1
f)
(8:21)
where ni is the initial number of nucleation sites per unit volume p is the autocatalytic factor pf is the number of nucleation sites provided by the already formed martensite plates Since df=dt ¼ V dNv=dt, substituting Equation 8.21 into Equation 8.20, df ¼ V(ni þ pf dt
Nv )(1
f )n exp
Q RT
¼
Rn þ f
f (1 Vp
f )Vpn exp
Q RT
(8:22)
where Rn is the ratio of initial to autocatalytic sites. Integration is possible, but is so messy as to be useless. Instead, assume that pV 1, Rn 1 and f 1. Thus, 1 f 1 and f=Vp 0. Hence, df ffi (Rn þ f )Z dt
(8:23)
f ¼ Rn (ezt 1)
(8:24)
f ln þ 1 ¼ zt Rn
(8:25)
where z ¼ pV neQ=RT. Integrating,
or
If the correct value of Rn is chosen, plotting ln ( f=Rn þ 1) vs. t should yield a straight line. Figure 8.16 shows that this is obtained. Values of Rn were found to range from 102 to 104 indicating that autocatalysis is much more important than nucleation in the virgin austenite. Equation 8.25 shows that f is proportional in a linear fashion to the number of nucleation sites initially present, but depends exponentially upon p, the autocatalytic factor. Thus, Cech and Hollomon [13] found that stabilization is far more effective when performed on partially transformed than on wholly untransformed austenite. In evaluating Q, one is handicapped by the lack of good estimates of p and n. Therefore, a range of plausible values of p was employed. The results shown in Figure 8.17 indicate the order of magnitude of Q (5,000–15,000 cal=mol or 20,900–62,700 J=mol) and that Q decreases about linearly with increasing driving force. These results will be used in Section 8.5 to evaluate various nucleation theories.
619
Martensitic Transformations 100
f/Rn + 1
ln (f/Rn + 1) = zt
10
–105°C, Rn = 10–3 –115°C, Rn = 10–3 –115°C, Rn = 5 × 10–4
1
0
1
2
3
4
5
Time (min)
FIGURE 8.16 Plots used to analyze isothermal transformation curves by means of Equation 8.24 [3]. The data are for Fe-24Ni-3Mn. (From Pati, S.R. and Cohen, M., Acta Metall., 17, 189, 1969. Reprinted with permission of ASM International1. All rights reserved.)
23.2 Ni-3.62 Mn, Shih et al. 24 Ni-3 Mn, Pati and Cohen 25.9 Ni-1.94 Mn, Raghavan and Entwisle
14
20,000
11
15,000
pν = 1026 s–1 8
Q cal/mol
Activation energy, Q (10–20 J/event)
17
10,000
pν = 1023 s–1 5 pν = 1020 s–1
5,000
2 –2.0
–2.5 Driving force,
–3.0 ΔG vγ
α΄(102
–3.5 J/cc)
FIGURE 8.17 Plot of the activation energy derived from isothermal transformation results against the driving force for martensite nucleation. (From Magee, C.L., Phase Transformations, ASM, Metals Park, OH, 1970, p. 115. Reprinted with permission of ASM International1. All rights reserved.)
620
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
8.5 NUCLEATION OF MARTENSITE Kaufman and Cohen [2] and Magee [3] presented useful references on this very difficult topic. Kaufman and Cohen attempted to apply conventional homogeneous nucleation theory to the martensite reaction and found the calculated activation free energy ca. 104 greater than the range of experimental Qs. Heterogeneous nucleation theory failed to solve the problem, yielding an incorrect dependence of DG* upon DGv, namely (DGv) 4 instead of (DGv) 2. The circumstance that the unit atomic process of nucleation theory is diffusional jumps, which in Fe-base alloys at temperatures often of interest in connection with martensite formation almost never occur, seems to have escaped the attention of these investigators. In view of this failure, Kaufman and Cohen suggested that, viewing the martensite nucleus (with Frank [41] and Knapp and Dehlinger [33]) as a thin oblate spheroid surrounded by loops of dislocations, see Figure 8.18, the rate-controlling step might be the creation of the additional misfit dislocations needed to accommodate growth. The oblate spheroids themselves were taken to be present in supracritical size and inherited from the austenite, becoming viable below the Ms by a dislocation adsorption type of mechanism (which would have to operate at a phenomenal rate in an athermal martensite transformation). Magee, on the other hand, proposed that the rate-controlling step in martensite nucleation is the propagation or motion of the misfit dislocations. Again, where the nucleus came from is a somewhat begged question that is discussed further below. For both mechanisms, the activation energy should be similar to that for plastic flow. Motion of the dislocations should be hindered by interaction with the lattice, impure atoms, and dislocations within the austenite matrix. The activation energies for plastic flow as a function of the driving force are in fact much akin to those shown in Figure 8.17, thus supporting the overall approach of the Kaufman–Cohen and the Magee mechanisms, but not distinguishing between them. Considering nucleation in a mechanistic fashion from the standpoint of the assembly of the critical nucleus, this has been done most successfully so far in the case of fcc=hcp transformations.
– [1 10]γ 2c
Negative screw 2r
S d
– [55 4]
Positive screw
[225]γ
Dislocation loops
FIGURE 8.18 Knapp and Dehlinger’s model of the martensite embryo. (From Knapp, H. and Dehlinger, U., Acta Metall., 4, 289, 1956. With permission from Elsevier.)
Martensitic Transformations
621
Christian [42] was the first to point out that the splitting of an a=2 dislocation into two Shockley partials produced a bilayer of hcp lattice that ought to serve as the nucleus for martensite. This is the same mechanism as was earlier described for the formation of hcp plates from an fcc Al–Ag matrix. The differences between the martensitic and the diffusional versions of these mechanisms are the following. In the case of martensite, the solute diffusion toward or away from the whole dislocation originally present is not necessary to make splitting energetically feasible. As long as the transformation temperature is below the Ms, the free energy of the system will be decreased by the formation of the hcp region without a change in composition. Further, migration of the Shockley partials need not be accompanied by diffusion. The Shockley partials accomplish the transformation by glide, without a change in composition. As in the Al–Ag case, misfit dislocations have been observed as the broad face of an hcp e-martensite plate formed in an fcc stainless steel matrix [43]. Votava [44] has observed dislocation splitting via hot stage transmission electron microscopy in the case of cobalt. Olson and Cohen [45] have considered nucleation of the fcc ! hcp martensite transformation from the standpoint of stacking faults. As we have discussed in connection to precipitation in Al–Ag, the dissociation of an a=2 dislocation into two Shockley partials, a=6 , yields the hcp structure. Olson and Cohen noted both the volume free energy change and the interfacial energy contributions to nucleation associated with such dissociation; in the absence of a composition change, they concluded that in Fe-Cr-Ni alloys, for which data on stacking fault energies are available, a ‘‘finite’’ tilt boundary, 4 or 5 dislocations high with a dislocation on every other (parallel) {111} plane, will provide sufficient DGv to balance both the transformation strain energy and the interfacial energy at the experimentally measured Ms temperatures. Because a lattice-invariant deformation is not needed in the case of the fcc=hcp transformation, the atomic mechanism can be simple. When an fcc=bcc transformation takes place martensitically, on the other hand, if the reaction is to occur as an invariant plane strain the Bain distortion (or lattice deformation) is, by itself, insufficient to accomplish this, and thereby minimize the volume shearstrain energy. A lattice-invariant deformation is needed as well, and this considerably complicates the atomic mechanism of transformation, as is particularly well reflected in the nucleation process. Bogers and Burgers [46] suggested consecutive shears on two different {111}g planes; Jaswon [47] pointed out that a half-twinning shear on a {111}g plane leads to a fault which is nearly bcc. Transmission electron microscopy observations may lend some support to the latter mechanism. Dash and Brown [48] and Magee subsequently found faults of some type parallel to {111}g planes. They are neither stacking faults nor twins, and have complex diffraction contrast effects not fully interpreted. In an Fe-24% Ni-3% Mn alloy, Magee watched these faults or plates grow at 1008C in two instances, observing them to do so at rapid, but nonetheless readily observable rates. However, martensite in bulk alloys does not have a {111}g habit plane. In addition, the crystallography of martensite is known to be different in thin foils, doubtless because of the much-reduced external constraint, greatly reducing the need to adjust crystallography to minimize volume strain-energy. The significance of these observations is thus doubtful. The question of the dislocation mechanism through which martensite nucleates remains open. A variety of postulates has been offered in the past as to both the atomic details of the nucleation mechanism and the nature of the heterogeneous sites at which nucleation occurs. Theoretical ideas, however, remain unsatisfying and the experimental observation is puzzling. Several workers have shown that coherent twin boundaries are particularly favorable sites for martensite nucleation. Magee attempted to check the obvious possibility that steps in these boundaries are the active sites, but without success. Dislocations were shown, by plastic deformation experiments, to facilitate nucleation of martensite in fcc. Fe precipitates from a Cu-rich Cu-Fe matrix, but no special arrays seemed to be involved [49]. At lesser undercooling, Breedis [50] found that coplanar pileups of dislocations are particularly effective in Fe-Ni-Cr alloys. Previously formed martensite plates, as already noted, are by far the most effective nucleation sites. However, mechanistically, how any of the foregoing sites, especially coherent twin boundaries, operate to nucleate martensite is not at all clear.
622
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Some discrimination amongst possible nucleating defects can be achieved on the following basis. Modifying slightly the view of Magee on this point, one can adapt the result of Cahn on the nucleation in dislocations (see Chapter 2), wherein, under some circumstances DG decreases continuously with increasing r. This requires no diffusion; it is only necessary that the strain and interfacial free energies acquired from the nucleating defects offset the interfacial free energy needed to expand the embryo to the point where further enlargement is compensated by the release of volume free energy. Thus, the energy of the nucleating defect must be greater than the experimentally measured Q. Magee calculated that, for Fe-30% Ni at Ms, this stricture excludes loops of single dislocations, vacancy aggregates, and inclusions with diameters less than l=2 mm that have undergone a 10% volume change. As Pati and Cohen [40] pointed out, the chances of seeing a martensite nucleus in virgin austenite is exceedingly small because the nucleation rate therein is so low. Future experimental research on this subject would be most profitably concentrated on nucleation at the interfaces of martensite plates formed in bulk alloys prior to thinning for transmission electron microscopy. It is at martensite:austenite interfaces, many investigators have shown, that nearly all martensite nucleation occurs.
8.6 CRYSTALLOGRAPHY AND GROWTH (OR PROPAGATION) OF MARTENSITE This work began with the original concept of ‘‘Bain straining’’ the fcc ! bcc transformation via a directed homogeneous distortion [51] and reached fruition with the near simultaneous publication of two essentially equivalent treatments which phenomenologically describe the crystallography of the martensite transformation: Wechsler, Lieberman, and Read (WLR) [52], and Bowles and MacKenzie (BM) [59].
8.6.1 PHYSICS
OF
PHENOMENOLOGICAL THEORY
OF
MARTENSITE CRYSTALLOGRAPHY
We have given a brief preliminary description of this theory earlier, and elaborate further here. As described before, the martensite transformation may be quite accurately described as an invariant plane-strain transformation. Figure 8.19 gives schematic examples of such transformations; the lower line in each figure represents the habit plane that is unrotated and undistorted by the transformation [54]. The theory shows that the necessary and sufficient conditions that a plane be undistorted during a transformation is that one of the principal distortions be unity, one greater and the third less than unity. Figure 8.20 illustrates this situation through the transformation of a sphere to an ellipsoid. It is assumed that there is no distortion perpendicular to the plane of the figure. In the case shown in Figure 8.20a all planes are changed in length by the transformation. In Figure 8.20b, however, AB and CD, in the sphere, are transformed to A0 B0 and C0 D0 in the ellipsoid with a rotation, but without a change in length, that is, with zero distortion. This introduces the third component of a martensitic transformation as viewed by the phenomenological theory. The three are (1) the lattice deformation
Simple extension
Simple shear
General invariant plane strain
FIGURE 8.19 Invariant-plane strains. The dotted outline is deformed into the solid outline. (From Mackenzie, J.K., J. Aust. Inst. Met., 6, 90, 1961. With permission.)
623
Martensitic Transformations X3
X3 C΄ η1 = I +
1
B΄
C
B
I η2 = I +
X2 2
X2 D
A A΄ (a)
D΄
(b)
FIGURE 8.20 (a) Section of a unit sphere and the ellipsoid into which it is transformed. (b) Traces of the undistorted planes in their initial positions AB, CD, and their final positions A0 B0 , C0 D0 . (From Bilby, B.A. and Christian, J.W., The Mechanisms of Phase Transformations in Metals, Institute of Metals, London, U.K., 1956, p. 121. With permission.)
or Bain strain that accomplishes the transformation of crystal structure, (2) the lattice-invariant deformation that brings the two lattices together again at the undistorted habit plane, and (3) the rigid body rotation that ensures that the habit plane is unrotated as well as undistorted. The Bain strain is shown in more detail for the austenite to tetragonal martensite transformation in Figure 8.21a. The inscribed tetragonal lattice is the martensite; the pair of fcc crystals is the austenite. A contraction of ca. 20% along the Z axis, and expansions of ca. 12% along the X and Y axes are seen to yield the bct structure. Figure 8.21b shows a plane and a direction in austenite, which become differently designated planes and directions in martensite. These equivalences are known as ‘‘lattice correspondences’’ and describe the Bain strain. The Bain strain that successfully describes the crystallography of the transformation is that which entails the minimum strain. For a given transformation, it is possible to describe other lattice correspondences that yield a bigger strain; but these are evidently not operative in practice. The minimization of the transformation strain energy is evidently the principle determining the crystallography of a martensite transformation. The lattice-invariant deformation also serves to minimize the strain energy attending the transformation, but in a different manner. In Figure 8.22a, the square lattice initially present has transformed into a tetragonal lattice. In adjacent regions, however, the transformation yields the two differently oriented, but completely equivalent rectangles, ABCD and A0 B0 C0 D0 . The rotation of the two rectangles toward each other so that C and C0 coincide makes the rectangles twin related. By alternating the two orientations of the rectangle, the expansions and contractions accompanying the transformation are localized and minimized by being, in effect, played off against each other, and thus the strain energy is held to a minimum. This is seen in more detail in Figure 8.22b. The transformation has taken place in different orientations in alternating regions. The proportion of each present, x and l x, respectively, is adjusted so that the martensite habit plane, AB’CD’E. . . . has the same projected length as a mathematically planar habit plane, ABCDE. . . . The prediction that martensite could be twinned was made well in advance of electron microscopic discovery that closely spaced twins actually exist in some martensite plates. The proportion of the twinned region also turned out to have been correctly predicted by the theory. However, the theory can also be formulated in an equivalent manner in terms of slip, or a combination of twinning and slip, as the lattice-invariant deformation. Which mode of deformation actually occurs depends upon physical conditions.
624
~20% Contraction
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
a
/√ a0
a 0/√2
Z΄, Z
2 a0 c
a
Y
a0 sion
Expan
2% ns
pa Ex
Y΄
[ 1–0
1]
f
Z , Z΄
b
ion
X
(a)
[ 1–1– 1]
~12%
a0
~1
X΄
(101)f
(112)b
Y
X΄ (b)
X
Y΄
FIGURE 8.21 (a) The Bain correspondence and distortion for the fcc ! bct (bcc) transformation. (b) The correspondence between the parent and product lattices uniquely associates points, planes, and directions. (From Bain, E.C., Trans. AIME, 70, 25, 1924. With permission.)
Although the phenomenological theory considers the separate operations described by a mathematical device, with only the initial and final position of the lattice points and planes being rigorously described, there has been an inevitable tendency to ascertain whether or not at least the lattice-invariant deformation portion of the theory corresponds to ‘‘real-world’’ operations. The WLR and the BM theories are differently formulated, but have been shown to be mathematically equivalent. They differ only in one significant respect. The BM theory allows a uniform dilatation,
625
Martensitic Transformations D
C Mirror plane C΄
B, B΄
A
A΄
B΄
C΄
A, A΄ D
B
Single crystal parent
Mirror plane
F΄
E
1–
D
D΄
Twinned product
X
D΄
C X B
B΄
X
Twin C, C΄ plane
1–
A, A΄
F
D΄
C
D
G
X
A 1–
D΄
X
(a)
(b)
FIGURE 8.22 (a) Two-dimensional analog of crystallographically equivalent transformation distortions in two adjacent regions, leading a twinned product. (b) The equal segmented line in the parent ‘‘austenite’’ ABCD . . . G goes into the broken line AB0 CD0 EF0 G during this two-dimensional analog of the transformation to a twinned product. The habit plane is on the average undistorted. (From Lieberman, D.S., Phase Transformations, ASM, Metals Park, OH, 1970, p.1. Reprinted with permission of ASM International1. All rights reserved.)
d, of the habit plane, which amounts to a uniform increase of the martensite lattice parameters. In the WLR theory, d ¼ l. The WLR and the BM theories are formally written in terms of matrix algebra, or alternatively, of stereographic projections. The irrational orientation relationships and habit plane which the theory yields, when executed on a computer, turn out to be the most useful one. The theory and the necessary matrix algebra background are developed in detail in Wayman’s text [7]. Bullough and Bilby [55], Bilby and Frank [56], and Crocker and Bilby [57] have developed another body of mathematically equivalent theory. In this approach, however, the habit plane is constructed ab initio as an array of glissile misfit dislocations. The glide of these dislocations, in the direction normal to the interphase boundary, accomplishes the transformation by shear. Only one array, rather than a grid, will be present for this purpose.
8.6.2 COMPARISONS
WITH
EXPERIMENT
We first summarize the basic experimental information available on habit planes and orientation relationships, confining ourselves to the austenite-to-martensite transformation in steel, which provides the most revere test of the theory. Later, results will be briefly compared with theory from other alloy systems.
626
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
(1 1 2) (2 2 5)
1 (2 5 9)
2
3 4 (1 3 5) (3, 10, 15)
FIGURE 8.23 Habit plane determinations for Fe-Ni alloys of varying Ni content. Regions 1, 2, 3, and 4 correspond respectively to 30.9, 31.9, 33.1, and 34.8% Ni. (From Reed, R.P., U.S. Department of Commerce, National Bureau of Standards, Report No. 9256, Aug. 22, 1966.)
Among Fe–C alloys, at Fe-1.78 wt.% C, the habit plane is ca. {259}g. At Fe-1.4 and Fe-0.92 wt.% C, experiment shows that the habit plane is ca. {225}g. Recent experimental results, done with care, indicate that at low carbon contents the habit plane is very close to {111}g, this result, however, is considered questionable in part because theory cannot account for it at all. In Fe-Ni and Fe-Pt alloys, {3, 10, 15}g often appears. Figure 8.23 shows the locations of these planes. Lattice orientation relationships are summarized in Table 8.2. The first three references are from the 1930s; these indicate rational lattice-orientation relationships. The other references are considerably more recent. Note that the nearly parallel planes and directions are rational. In each case, however, there is a small deviation from exact rationality, indicating that the lattice orientation relationships for martensite are also irrational. Table 8.3 compares the calculated and measured lattice orientation relationships in the case of a (3, 15, 10)g habit plane. Figure 8.24 shows this habit plane and the habit plane calculated as a function of dilatation parameter, d, from the BM theory. Good agreement is achieved when d is ca. unity; agreement is also very satisfactory in the case of the orientation relationships. The same results are achieved with the WLR theory. In the case of Figure 8.24, note that d can divert the habit plane calculated only in the sidewise direction. However, the calculated habit plane is seen to be quite sensitive to the particular value of d chosen. Figure 8.25 presents a different breed of results. The 1.78% C data spread refers to {259}g. The highest axial ratio curve applies in this case, but agreement with the {259}g data is not very good. Agreement is much better for the {225}g habit plane (0.92, 1.40% C), but the orientation relationship is incorrectly predicted. The {225}g difficulty is the more serious one and is presently an active subject of research. In some alloys, habit planes fall in the region between {225}g and {259}g and these are not predicted either.
Martensitic Transformations
TABLE 8.2 Experimental Determinations of the Austenite–Martensite Orientation Relationship
Material
Transformation Type
Fe 1.4%C
fcc–bct
Fe 30%Ni
fcc–bcc
Fe30%
fcc–bcc
Fe 30%Ni
fcc–bcc
Fe 31%Ni
fcc–bcc
Experimental Relations (111)f ==(011)b [101]f ==[111]b (Kurdjumov–Sachs relation) (111)f ==(011)b [112]f ==[011]b Nishiyama relation KS relation for transformation near room temperature; N relation for transformation at low temperatures. (111)f (011)b 2.48 apart [1 12]f [011]b 1.08 apart (111)f (011)b 0.38 apart [101]f [111]b 2.48 apart [0 11]f [111]b 7.88 apart [112]f [011]b 2.28 apart
Direction Cosines of Plane Normals and Directions Referred to Austenite Axes (If Reported)
Method X-ray pole figure
X-ray pole figure
X-ray pole figure
X-ray
Microbeam Laue back-reflection
Microbeam Laue back-reflection
(continued)
627
628
TABLE 8.2 (continued) Experimental Determinations of the Austenite–Martensite Orientation Relationship
Material
Transformation Type fcc–bcc
Fe 2.8% Cr 1.5%C
fcc–bct
Fe 3.09%Cr 1.51%C
fcc–bct
Fe 22%Ni 0.8%
fcc–bct
(111)f (011)b 18 apart [101]f [111]b 41=4 apart [011]f [111]b 68 apart (111)f (011)b 18 apart [101]f [111]b 38 apart [011]f [111]b 78 apart (111)f (011)b 0.38 apart [101]f [111]b 2.88 apart [011]f [111]b 6.18 apart (111)f (011)b 18 apart [112]f [011]b 28 apart [101]f [111]b 21=2 apart [011]f [111]b 61=2 apart
Direction Cosines of Plane Normals and Directions Referred to Austenite Axes (If Reported)
Method
Axis of rotation and magnitude given to bring martensite unit cell axes into coincidence with those of austenite
Axis of rotation and magnitude given to bring martensite unit cell axes into coincidence with those of austenite
Microbeam Laue back-reflection.
Microbeam Laue back-reflection.
Laue back-reflection.
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
Fe 32%Ni
Experimental Relations
fcc–bct
Fe 7.9%Cr 1.11%C
fcc–bct
Fe 7.9%Cr 1.11%C
fcc–bct
Fe 24.5at.%Pt
Fe1%C Fe1.5%N
fcc–bcc
fcc–bct Thin foils
(111)f (011)b 0.58 apart [101]f [111]b Parallel [011]f [111]b 108 apart (111)f (011)b 0.418 apart (111)f (011)b 0.458 apart [101]f [111]b 0.538 apart [211]f [311]b 0.248 apart [110]f [100]b 4.658 apart (111)f (011)b 0.888 apart [101]f [111]b 4.588 apart [001]f [001]b 9.008 apart (111)f (011)b 0.838 apart [101]f [111]b 4.268 apart [001]f [001]b 9.208 apart (101)f ==(112)b [101]f ==[111]b
Microbeam Laue back-reflection
Microbeam Laue back-reflection. (011)b ==[0:57402, 0:58351, 0:57447]f
Microbeam Laue back-reflection.
Martensitic Transformations
Fe 7.9% Cr 1.11%C
[111]b ==[ 0:71111, 0:00739, 0:70304]f [311]b ==[ 0:81789, 0:40439, 0:40930]f [100]b ==[ 0:74042, 0:66939, 0:06076]f (011)b ==[0:56622, 0:57780, 0:58783]f
Microbeam Laue back-reflection, two determinations
[111]b ==[ 0:74563, 0:05522, 0:66407]f [001]b ==[0:11323, 0:10803, 0:98768]f (011)b ==[0:56818, 0:57540, 0:58829]f [111]b ==[
0:74359, 0:05065, 0:66672]f
[001]b ==[0:11844, 0:10456, 0:98744]f Transmission electron diffraction.
629
(continued)
630
Material Fe30%Ni Fe 30.2% Ni
Transformation Type fcc–bcc thin foils fcc–bcc Strain induced
Experimental Relations (101)f ==(112)b (101)f ==(112)b [101]f ==[111]b for compression along h110if (001)f ==(001)b [100]f ==[100]b for compression along h001if
Source: Wayman, C.M., Adv. Mater. Res., 3, 147, 1968. With permission. Note: For references of each entry, readers are referred to Ref. [5].
Direction Cosines of Plane Normals and Directions Referred to Austenite Axes (If Reported)
Method Transmission electron diffraction. X-ray pole figure.
X-ray pole figure and transmission electron diffraction.
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
TABLE 8.2 (continued) Experimental Determinations of the Austenite–Martensite Orientation Relationship
631
Martensitic Transformations
TABLE 8.3 Comparison of Predicted and Measured Orientation Relationships for Fe-22% Ni-0.8% C [habit plane (3, 15, 10)f] Planes and Directions
Measured Angle between, deg.
Predicted Angle between, deg.
(011)b [111]b [111]b [011]b
1 2.5 6.5 2
0.2 2.7 6.6 1.9
(111)f [ 101]f [011]f [ 112]f
0.99
4
02 1.0
1.0
10
Source: Bowles, J.S. and MacKenzie, J.K., Acta Met., 2, 129, 138, 224, 1954. With permission.
FIGURE 8.24 Variation of the predicted habit plane with the dilatation parameter d (the values are shown I the figure) for Fe-22% Ni-0.8% C. The encircled point is the mean experimental result of Greninger and Troiano [60]. (From Wayman, C.M., Adv. Mater. Res., 3, 147, 1968. With permission.)
08
0.92, 1.40% C
14
0.99
1.0
2
1.0
1.015
6
8
0.99
6
1.00
1.78% C
γ = 1.0 γ = 1.045 γ = 1.08
FIGURE 8.25 Predicted habit-plane variations with d for carbon steels of varying axial ratio g. The experimental results are those of Greninger and Troiano [60] for carbon steels. (From Wayman, C.M., Adv. Mater. Res., 3, 147, 1968. With permission.)
632
Mechanisms of Diffusional Phase Transformations in Metals and Alloys
TABLE 8.4 Comparison of Experimental and Calculated (WLR Theory) Crystallographic Features of Martensitic Transformations in Au-Cd, In-Tl, and Fe-22Ni-0.8C Feature
Experimental
Theoretical
Discrepancy
Au-Cd (47.5 at%Cd) CsCl!twinned orthorhombic Interface plane normal, n
Orientation relationship Macroscopic shear direction Angle macroscopic shear
1 0:696 @ 0:686 A 0:123 b
1 0:6968 @ 0:6810 A 0:2250 b
0
0
(001)b== to (001)b [111]b== to [011]b 0 1 0:660 @ 0:729 A 0:183 b
(001)b 28400 from (001)b [111]b 180 from [011]b 0 1 0:6510 @ 0:7322 A 0:2001 b
2.948