Fluid transport in nanoporous materials: proceedings of the NATO advanced study institute, held in La Colle sur Loup, France, 16-28 June 2003 [1 ed.] 1402043783, 9781402043789

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Fluid Transport in Nanoporous Materials

NATO Science Series A Series presenting the results of scientific meetings supported under the NATO Science Programme. The Series is published by IOS Press, Amsterdam, and Springer (formerly Kluwer Academic Publishers) in conjunction with the NATO Public Diplomacy Division.

Sub-Series I. II. III. IV.

Life and Behavioural Sciences Mathematics, Physics and Chemistry Computer and Systems Science Earth and Environmental Sciences

IOS Press Springer (formerly Kluwer Academic Publishers) IOS Press Springer (formerly Kluwer Academic Publishers)

The NATO Science Series continues the series of books published formerly as the NATO ASI Series.

The NATO Science Programme offers support for collaboration in civil science between scientists of countries of the Euro-Atlantic Partnership Council. The types of scientific meeting generally supported are “Advanced Study Institutes” and “Advanced Research Workshops”, and the NATO Science Series collects together the results of these meetings. The meetings are co-organized by scientists from , NATO countries and scientists from NATO s Partner countries – countries of the CIS and Central and Eastern Europe. Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances in a field. Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and identification of directions for future action. As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science Series was re-organized to the four sub-series noted above. Please consult the following web sites for information on previous volumes published in the Series. http://www.nato.int/science http://www.springer.com http://www.iospress.nl

Series II: Mathematics, Physics and Chemistry – Vol. 219

Fluid Transport in Nanoporous Materials edited by

Wm. Curtis Conner Department of Chemical Engineering, University of Massachusetts, Amherst, MA, U.S.A. and

Jacques Fraissard Laboratoire de Physique Quantique, ESPCI, Université Pierre et Marie Curie, Paris, France

Published in cooperation with NATO Public Diplomacy Division

Proceedings of the NATO Advanced Study Institute on Fluid Transport in Nanoporous Materials La Colle sur Loup, France 16-28 June 2003 A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 1-4020-4380-5 (PB) ISBN-13 978-1-4020-4380-2 (PB) ISBN-10 1-4020-4378-3 (HB) ISBN-13 978-1-4020-4378-9 (HB) ISBN-10 1-4020-4382-1 (e-book) ISBN-13 978-1-4020-4382-6 (e-book)

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TABLE OF CONTENTS Preface .............................................................................................................. ix T. Q. GARDNER, D. M. RUTHVEN Nato-asi fluid transport in nanoporous materials course …a student’s perspective and explanations from a veteran................................. 1 D. M. RUTHVEN Transport in microporous solids: an historical perpective Part I: Fundamental Principles and Sorption Kinetics ........................................ 9 R. L. LAURENCE Measurement of diffusion in macromolecular systems: solute diffusion in polymers systems............................................................................................... 41 L. SARKISOV, K. F. CZAPLEWSKI, R. Q. SNURR Role of diffusion in applications of novel nanoporous materials and in novel uses of traditional materials ............................................................................... 69 H. RAMANAN, S. M. AUERBACH Modeling jump diffusion in zeolites. I. Principles and methods ....................... 93 L. V. C. REES, L. SONG Adsorption, thermodynamics and molecular simulations of cyclic hydrocarbons in silicalite-1 and alpo4-5 zeolites ............................................. 125 D. M. RUTHEN Transport in microporous solids Part II: Measurement of micropore diffusivities............................................. 151 S. VASENKOV, J. KARGER Structure-related anomalous diffusion in zeolites............................................ 187 W.C. CONNER The contribution of surface diffusion to transport in nanoporous solids ......... 195 R. KRISHNA The Maxwell-Stefan formulation of diffusion in zeolites................................ 211

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R. H. ACOSTA, PETER BLÜMLER, H. W. SPIESS Sensitivity and resolution in magnetic resonance imaging of diffusive materials ....................................................................................... 241 V. SKIRDA, A. FILIPPOV, A. SAGIDULLIN A. MUTINA, R. ARCHIPOV, G. PIMENOV Restricted diffusion and molecular exchange processes in porous media as studied by pulsed field gradient NMR ............................................................. 255 S. KENANE, C S. VASAM, P. P. KNOPS-GERRITS Vibrational spectroscopy to monitor synthesis, adsorption and diffusion in micro- and mesoporous metal phosphates ....................................................... 279 V.B. KAZANSKY Nitrogen – oxygen diffusion in zeolites studied by drift ................................. 299 M.-A. SPRINGUEL-HUET 129 Xe NMR for diffusion of hydrocarbons in zeolites and 1H NMR imaging for competitive diffusion of binary mixtures of hydrocarbons in zeolites....... 315 H. JOBIC Diffusion in zeolites measured by neutron scattering techniques................... 333 I.V. KOPTYUG, A.A. LYSOVA, A.V. MATVEEV, L.YU. ILYINA, R.Z. SAGDEEV, V.N. PARMON NMR imaging as a tool for studying mass transport in porous materials........ 353 S. VASENKOV, J. KARGER PFG NMR diffusion studies of nanoporous materials ..................................... 375 L. V. C. REES, L. SONG Diffusion of cyclic hydrocarbons in zeolites by frequency-response and molecular simulation methods ......................................................................... 383 J.-P. KORB Surface diffusion of liquids in disordered nanopores and materials: a field cycling relaxometry approach.......................................................................... 415 A. M. MENDES, F. D. MAGALHÃES, C. A.V. COSTA New trends on membrane science.................................................................... 439

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V. I. VOLKOV, E. V. VOLKOV, S. L. VASILYAK, Y.S. HONG, C.H. CLEE The ionic and molecular transport in polymeric and biological membranes on magnetic resonance data.......................................................... 481 P. PULLUMBI Molecular modeling: A complement to experiment in material research for non cryogenic gas separation technologies ................................ 509 S. M. AUERBACH Modeling jump diffusion in zeolites. Part II: Applications ............................. 535

Posters Communications Y.I. ARISTOV, I.S. GLAZNEV, L.G. GORDEEVA, I.V. KOPTYUG, L.YU. ILYINA, J. KÄRGER, C. KRAUSE, B. DAWOUD Dynamics of water sorption on composites “cacl2 in silica”: single grain, granulated bed, consolidated layer................................................................... 553 F. BENALIOUCHE, Y. BOUCHEFFA, P. MAGNOUS Effect of carbonaceous compounds on diffusion of alkanes in 5A zeolite ...... 567 C. CHMELIK, E. LEHMANN, S. VASENKOV, B. STAUDTE, J. KÄRGER Application of interference and IR microscopy for studies of intracrystalline molecular transport in AFI type zeolites ............................ 575 G. Di FEDERICO, I. CAMPLONE, S. BRANDINI, J. BARKER Coal characterization for carbon dioxide sequestration purposes.................... 583 V. T. HOANG , Q. HUANG , M. EIC, T. O. DO , S. KALIAGUINE Effect of the intrawall microporosity on the diffusion characterization of bi-porous SBA-15 materials ........................................................................ 591 I.A. HADJIAGAPIOU, A. MALAKIS, S. S. MARTINOS Structure of a single-species-fluid in a spherical pore ..................................... 603 S. LAGORSSE PONTES, F. D. MAGALHÃES, A. MENDES Carbon molecular sieve membranes: Characterisation and application to xenon recycling .................................................................. 613

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D. MALDONADO, N. TANCHOUX, P. TRENS, F. DI RENZO, F. FAJULA An experimental study of the state of hexane in a confined geometry ............ 619 M. NORI, S. BRANDINI A model for sound propagation in the presence of microporous solids .......... 629 M. PETRYK, O. SHABLIY, M. LENIUK, P. VASYLUK Mathematical modelling and research for diffusion processes in multilayer and nanoporous media................................................................ 639 T. RUSU, C. IOJOIU, V. BULACOVSCHI Neuronal network used for investigation of water in polymer gels .............. 657 V.Ⱥ. SEVRIUGIN, V.V. LOSKUTOV, V.D. SKIRDA Dependence of self-diffusion coefficient on geometrical parameters of porous media .............................................................................................. 671 Index ............................................................................................................... 677

PREFACE The last several years have seen a dramatic increase in the synthesis of new nanoporous materials. The most promising include molecular sieves which are being developed as inorganic or polymeric systems with 0.3-30nm in pore dimensions. These nanoporous solids have a broad spectrum of applications in chemical and biochemical processes. The unique applications of molecular sieves are based on their sorption and transport selectivity. Yet, the transport processes in nanoporous systems are not understood well. At the same time, the theoretical capabilities have increased exponentially catalyzed by increases in computational capabilities. The interactions between a diffusing species and the host solid are being studied with increasing details and realism. Further, in situ experimental techniques have been developed which give an understanding of the interactions between diffusing species and nanoporous solids that was not available even a few years ago. The time was ripe to bring together these areas of common interest and study to understand what is known and what has yet to be determined concerning transport in nanoporous solids. Molecular sieves are playing an increasing role in a broad range of industrial petrochemical and biological processes. These include shape-selective separations and catalysis as well as sensors and drug delivery. Molecular sieves are made from inorganic as well as organic solids, e.g., polymers. They can be employed in packed beds, as membranes and as barrier materials. Initially, the applications of molecular sieves were dominated by the use of zeolites. Their molecular sieving is a consequence of the intra-crystalline pore network spanning from ~3->~13Å in dimensions. Thus, small to modest sized molecules can be selectively sorbed, separated and/or reacted in zeolites. Larger species are excluded from the pore network. The discovery of mesoporous molecular sieves in 1992 opened a whole new spectrum of possible applications and processes . The pore dimensions span a range from ~15Å to several hundred angstroms and the pores can have a narrow pore-size distribution. The accessible surface areas can exceed 1000 m2/g of substrate and many mesoporous solids have been developed. Concurrently, polymeric solids have been developed with morphologies including controlled pore structures from 0.4 to 20nm, crystallinity, tacticity and molecular weight distributions. They also can behave as molecular sieves and are already employed in membrane and biological processes. The larger pores of these materials admit the potential for reactions of larger molecules, including many pollutants and biological moieties excluded from the micropores of zeolites. In addition, the surfaces of these molecular sieves can be functionalized for selective adsorption and reaction, including catalysis . ix

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Preface

The transport processes in nanoporous systems are considerably more complex than diffusion in homogeneous phases or in macroporous systems. The pathways in nanoporous systems are, by design, similar to the dimensions of the diffusing species. Adsorbate-adsorbate and adsorbate-adsorbent interactions are enhanced. The energetics of transport differs from molecular theory often to reflect activated processes with apparent activation energies. Entropic factors also play an increasing role in the transport. Dissolution as well as diffusion between chains must be considered in polymers. Adsorbent flexibility is a factor of varying significance in both inorganic and organic solids. Simple extensions of homogeneous theories are inadequate, as these specific interactions need to be included. Fortunately, the capabilities of computational chemistry have increased rapidly in the last few years and these complex problems can now be analyzed with increasing detail. Concurrently, in situ experimental capabilities have increased to give much more insight into the interactions between the diffusing species and the solid medium. As an example nuclear magnetic resonance has developed multidimensional imaging and quantitative probe molecule analyses. Infrared, tomographic, scattering techniques and a variety of other techniques have been developed to probe the diffusing species and the nanoporous support. Measurement of the transport itself has been enhanced by frequency response, chromatographic, imaging, isotopic and a variety of recently developed dynamic techniques. It is apparent that these diverse approaches to analyze and to understand the transport within nanoporous solids have been greatly facilitated by interactions between these disciplines: materials science, theory, analysis and engineering. Further, it was opportune to document and to discuss the current state-of-the art in each of these areas. This NATO school has facilitated these interactions and have provided an opportunity for others in physics, materials science, chemistry and engineering to learn what is known and to contribute to future research in this timely area. A NATO Advanced Study Institute, an ASI, was conducted on this timely subject in France during June of 2003. This Book documents the lectures that were presented as well as it attempts to represent the extensive discussions that resulted from these presentations. Thus, we first choose to discuss the dialogue that occurred with respect to the definitions of the various processes involved in transport in nanoporous solids. This discussion occurred on the last day of the meeting and was voluntary. More than 90% of the attendees present at that time contributed to the discussion. The result of these discussions is summarized in the definitions presented as part of this summary. First we attempt to represent the atmosphere of the meeting through a poem written by the students:

Preface

xi

POEM We came for two weeks to France Excited that we had the chance To learn from the best Without any tests And allow our thoughts to advance The diffusion of knowledge was great And though it seemed as if we only ate We grew at this school And played in the pool And we took time to celebrate On the weekend some went to Canne And tried to get a tan(ne) Some of us burned And others they learned That playing Frisbee can be fanne(?) The lecturers all had their roles Curt helped Scott fill in holes So much NMR Drove us to the bar After which we all thought we saw trolls? Diffusion coefficients there are a LOT When to use which, we know not (yes we do) The temperature rose But somehow the blows Were confined to the fan that Curt bought We slept with no covers on Some of us woke up at dawn We all made some friends And all poems have their ends Now tell me, is this a pro…or a con? Many people have contributed to the success of the ASI on which this volume is based. We thank of course all participants for contributing to the intellectual dialogue. But the success of this ASI was primarily due to student participation in the discussions following the plenary presentations. These continued for hours, often exceeding the original presentations! It is a great pleasure to acknowledge the financial support provided by the Scientific Affairs Division of the North Atlantic Treaty Organization, the french “Centre National de la Recherche Scientifique-CNRS”, and the European Community (INTAS); also the National Science Foundation for the fellowships to American Students and post-doctorates fellows; and the

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International Science Foundation for the support it provided to Russian scientists. Finally, we acknowledge the efforts of all the people who contributed to organizational aspects of the Institute, and specially the secretariat assistance of Mrs Chantal Bonmkratz. Jacques Fraissard, Director of the ASI Vladimir D. Skirda, Co-director of the ASI Wm Curtis Conner.

NATO-ASI FLUID TRANSPORT IN NANOPOROUS MATERIALS COURSE …a student’s perspective and explanations from a veteran

T. Q. GARDNER* AND D. M. RUTHVEN** *Colorado School of Mines, Chemical Engineering Department Alderson Hall 441, Golden, CO 80401 **Department of Chemical and Biological Engineering University of Maine, Orono, ME 04469-5737

One of the first things mentioned in the introduction to many of the articles I’ve read about diffusion in zeolites, is that “the diffusivities reported in the literature vary by up to xxx orders of magnitude”. Although it is not always specifically stated, the implication is that these widely varying diffusivities have been measured or estimated for the same molecule/host system, and I’ve seen “xxx” as low as one and as high as four. (Interestingly enough, despite this, many authors, myself included, still report at least 2 or 3 significant figures in the diffusivities they presented in their own articles…but that is another issue.) Although most articles I’ve read have been about diffusion of relatively simple molecules in zeolites, we learned at the NATO course that large discrepancies exist between the reported diffusivities of species diffusing through nanoporous materials in general. While this can be useful to us relative newcomers to the field of diffusion, for instance when we are writing our first papers we can almost always say something like, “the diffusivities measured fell within the range reported in the literature” (if we can’t we are probably in serious trouble), but frankly it is disturbing and confusing to realize the magnitude of the discrepancies and to try to make sense of our measurements or estimates in this light. In the end, we as students are faced with the many obvious questions posed by the quote at the beginning of this paragraph: Why are there discrepancies? Why are they so large? What exactly do the different techniques for measuring diffusion actually measure and how are the diffusivities calculated from the measurements? All reported diffusivities are really estimates since no one has a digital diffusivity meter that reads out the error-free exact diffusivity of a species diffusing through a nanoporous material. Which values can we trust and to which should we compare our own estimates? What are the different diffusivities (self, transport, corrected, etc.) and how are they 1 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 1–7. © 2006 Springer. Printed in the Netherlands.

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T.Q. Gardner and D.M. Ruthven

related? Which techniques measure which diffusivities and how sensitive are the results to the operating conditions and assumptions made? What are the limitations of each technique? What exactly do Monte Carlo and Molecular Dynamics simulations approximate and how sensitive are they to the input force fields, dynamic vs. static lattice assumption, the flexibility of the guest molecules, etc.? The presentations and ensuing discussions at the NATO course addressed all of these issues (and more) in one form or another, answering many of the questions for the students and clearly outlining where disagreements still persist even among the leaders of the field (and hence what questions we should attempt to answer in our own work!). There are many answers to the question of why there are discrepancies in the diffusivities reported in the literature for diffusion in nanoporous materials, many of which are NOT related to errors in the measurements. This may seem surprising at first, but after a couple of weeks with many of the world’s experts on diffusion it is a bit more banal. The first and foremost answer to the question is simply that there are different diffusivities, and when the loading is finite, as it clearly must be in order for a guest molecule to be diffusing through a host, diffusivities are by definition not the same. Let’s first take a look at some definitions of and relationships between different diffusivities. 1. Transport Diffusivity The diffusivity of A in a binary mixture of A and B (DAB, m2/s) is defined in terms of the flux of A (JA, mol/m2 s) relative to the plane of no net molal flux:

JA

 DAB C

wx A wz

(1)

where C is the total concentration (mol/m3), xA is the mole fraction of A:

xA

CA C A  CB

(2)

and z is distance in the direction of the concentration gradient (m). Note that DAB can be a function of the concentration.

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When considering adsorption into a zeolite (or other solid adsorbent) we are interested in the flux (NA, mol/m2 s) relative to the fixed framework of the adsorbent. The relationship between NA and JA is (see Bird, Stewart, and Lightfoot, p. 323):

xA  N A  N B  J A

NA

(3)

In this case B is the fixed adsorbent, hence NB = 0 and combining Eq’s. (1) and (3) leads to:

N A 1  x A

JA

 DAB C

wx A wz

(4)

Since C = CA + CB and CB is constant, differentiation of Eq. (2) yields:

C

wx A wz

1  xA

wC A wz

(5)

Combining Eqs (4) and (5), we have for diffusion of a single species A in a stationary adsorbent where CB does not vary with z (such as pure component diffusion in a zeolite):

NA

 DAB

wC A dz

 DA

wC A dz

(6)

The B can be dropped from the diffusivity since B is the stationary adsorbent. The pure component diffusivity (DA) defined according to Eq. (6) is consistent with DAB defined according to Eq. (1). This diffusivity – that which is multiplied by the concentration gradient to get the flux, is called the transport diffusivity, and it can in general depend on concentration. Equation (6) is also a form of Fick’s Law, which states that the flux of a component is proportional to its concentration gradient, and the transport diffusivity is sometimes also called the Fickian diffusivity.

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T.Q. Gardner and D.M. Ruthven

2. Corrected Diffusivity The transport diffusivity in Eqs. (1) and (6) was defined assuming that the driving force for diffusion of component A is the concentration (or mole fraction) gradient of component A. Thermodynamics tells us, however, that the true driving force for diffusion is the chemical potential gradient. The chemical potential of an ideal gas component i (µi, J/mol) has the form:

Pi

0

P i  RTAn

Pi P0

(7)

where µi0 is the chemical potential at a reference state, R is the ideal gas constant (J/mol K), T is the temperature, Pi is the partial pressure of i (Pa), and P0 is the reference pressure (Pa). The same equation can be applied to nongaseous species since the equilibrium vapor pressure, however small, provides an accurate measure of the thermodynamic activity, the only assumption being that the equilibrium vapor phase behaves as an ideal gas. Taking the chemical potential gradient as the driving force for diffusion leads to the following relationship between the transport diffusivity (D in the following equation) and a corrected diffusivity, D0:

D

D0

w ln Pi w ln Ci

D0

Ci wPi Pi wCi

(8)

In this equation Pi is the partial pressure of component i and Ci is the concentration of i in the host through which i is diffusing in equilibrium with that partial pressure. For example, when considering adsorption on zeolites, Ci can be replaced by qi, which is the adsorbed phase concentration (mols i adsorbed/kg zeolite). This equation is sometimes referred to as the Darken equation although in fact its origins go back to Maxwell, Stefan and Einstein. In fact for a single component D0 is identical to the Maxwell-Stefan diffusivity. Though this thermodynamic correction takes into account some of the possible concentration dependence of the transport diffusivity, D0 can also be a function of concentration. However, since the strongly concentration dependent factor ( w lnPi/ w lnCi) is removed, the concentration dependence of D0 is often much weaker than that of D. Note that in the low concentration limit ( w lnPi/ w lnCi) ĺ 1.0 so DĺD0. The factor ( w lnPi/ w lnCi) is often called the “thermodynamic correction factor” and given the symbol ī.

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3. Self-Diffusivity and Tracer Diffusivity Equation (1) implies the existence of a concentration gradient in order for transport of component A to take place. However, even if there is not a total concentration gradient, molecules will move with respect to each other (Brownian motion). We can consider a situation in which there is a non-uniform distribution of marked molecules within a uniform total concentration (thus the total concentration gradient is zero) and define a tracer diffusivity (Dtr) to describe the movement of the marked molecules:

J A*

D tr

wC A*

CA = const.)

wz

(9)

Note that Dtr and DA are physically different quantities because of the difference in the way in which they are defined (presence or absence of a total concentration gradient). While Eq. (9) resembles a transport equation since it includes a concentration gradient (of the tracer), the tracer diffusivity can be thought of as a measure of how fast molecules move amongst themselves, and is therefore actually a “self-diffusivity” rather than a transport diffusivity (which implies an applied gradient). The self-diffusivity may be defined on the basis of the Einstein relation:

Dself

1 r2 6 t

(10)

where r2 is the mean squared displacement (m2) of a molecule over a given time, t (s). One may argue mechanistically that in the low concentration limit the transport and self diffusivities should be equal: Dself = D0 = D

(as C Æ 0)

(11)

In general, Dself is ” D0. This follows from the Maxwell-Stefan model [see Paschek and Krishna Chem. Phys. Letters 323 278-284 (2001)] which shows:

1 Dself

1 T  D0 D11

(12)

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where D11 represents the contribution from direct exchange of molecules at a given site. Clearly as the concentration, and consequently the loading (ș) approaches 0 or the rate of direct exchange approaches infinity, Dself ĺ D0. 4. Comparison of Experimental Measurements Most of the “microscopic” methods, such as pulsed-field gradient nuclear magnetic resonance (PFG-NMR) and quasi-elastic neutron scattering (QENS), measure the self-diffusivity (Dself). Molecular simulations such as Monte Carlo methods also estimate self-diffusivities. Molecular Dynamics calculations, where a concentration gradient is imposed and molecules are allowed to diffuse, currently require too much computer time to be able to predict transport diffusivities because of the length scales over which the simulated molecules would need to “move” (computational time is proportional to distance2 / D and D is on the order of ~10-8 to 10-18 m2/s). (However, strides are being made in this area and transport diffusivities may be calculable by MD simulations in the near future.) Most “macroscopic” techniques, such as membrane permeation, zero length chromatography (ZLC) and frequency-response (FR) methods, measure transport diffusivities (D). As described here, the transport and self diffusivities are not directly comparable except at zero coverage. The diffusivities obtained by macroscopic methods are therefore usually corrected to get D0 and those values are generally compared to the Dself values obtained from the microscopic methods. The corrected diffusivities are often calculated from the measured transport diffusivities and the thermodynamic correction factor applied at some concentration (for example using the feed side concentration or an average of the feed and permeate side concentrations for a zeolite membrane transport diffusivity measurement). This comparison removes the often large concentration dependence of the thermodynamic correction factor (inherent in the transport diffusivity, D) but as shown in Eq. (12), Dself and D0 are only expected to be equal to each other when the coverage is zero (which is never the case when measurements are being made). Therefore, the lower the coverage, the closer should be the correspondence between D0 and Dself, but these quantities should not necessarily be expected to be the same for any real measurements. Additionally, even in Eq. (12), not all of the possible molecule/molecule or molecule/host interactions that could be present have been taken into account. Further, the calculation of D0 requires estimation of the thermodynamic correction factor from the slope of the equilibrium isotherm. Even when accurate isotherm data are available this introduces considerable uncertainty. Therefore, while this short chapter serves to define and clarify some of the different diffusivities reported in the literature of diffusion through

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nanoporous materials and to explain some of the discrepancies in diffusivity measurements (and to alert the reader to be sure to compare apples with apples when thinking about diffusivities), there is still work to be done to understand all of the differences between diffusivities measured by different techniques. Inappropriate comparisons between different diffusivities can account for some but not all of the discrepancies in the reported diffusivity data. Most macroscopic measurements are subject to the intrusion of extracrystalline resistances to mass transfer and finite rates of heat dissipation. In some situations such effects can be rate controlling and if this is not recognized the values derived for the apparent intracrystalline diffusivity will be much too small. There is also mounting evidence from PFG NMR and various mesoscopic techniques such as IR and interference microscopy that the impact of surface resistance and intracrystalline defects may be much greater than has been generally assumed. As a result the measurements carried out over short distances (perhaps 10 – 100 unit cells) yield much higher diffusivities, approaching the values derived from molecular simulations, whereas on the scale of the entire crystal the apparent diffusivities are much smaller. This discussion gives just a smattering of the topics covered and discussed at the NATO course. The other chapters in this book give details about the relevant parameters, range of applicability, sources of error, etc. for each of the measurement and estimation techniques mentioned here, as well as comparisons of diffusivity and adsorption measurements, modeling, industrial applications, and emerging theories of diffusion in nanoporous materials. Defect analyses, window effects, intramolecular interactions, pot hole filling, diffusion through pores and erops, and other spell binding phenomena are discussed by the world’s leaders in the field. Read on, my friends…a fascinating world awaits you. And as for the remaining questions that lie in wait, be challenged…

TRANSPORT IN MICROPOROUS SOLIDS: AN HISTORICAL PERPECTIVE Part I: Fundamental Principles and Sorption Kinetics

D. M. RUTHVEN Department of Chemical and Biological Engineering University of Maine, Orono, ME 04469-5737 Abstract The fundamental principles governing diffusion and sorption kinetics in microporous materials are reviewed from an historical perspective with examples selected to illustrate some of the different patterns of behavior that have been observed experimentally. Adsorption/desorption kinetics in porous materials are generally controlled by mass transfer rather than by the intrinsic sorption kinetics which, at least for physical adsorption, are generally rapid. An understanding of diffusion theory, especially the special features associated with diffusion in very small pores, is therefore essential background for the study of sorption kinetics. This subject has been discussed in detail in a number of recent books and review articles[1-4]. The present article therefore provides only a brief introductory treatment at a level necessary to understand the different patterns of kinetic behavior which have been observed in experimental studies. 1. Fundamental Principles of Diffusion Fick’s first law:

J

 D(q)

wq wz

(1)

which is in essence a definition of the diffusivity [D(q)] provides a convenient basis for the quantitative analysis of transport in microporous solids. The diffusivity, defined in this way, is, in principle, a function of the adsorbed phase concentration and may be called, more precisely, the transport diffusivity since it measures the rate of transport of sorbate under a given concentration gradient. One may also define the tracer diffusivity [D(q)]: 9 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 9–39. © 2006 Springer. Printed in the Netherlands.

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D.M. Ruthven

J*

 D (q)

wq * wz

(2)

q

This coefficient measures the flux of marked molecules (J*) under the influence of a concentration gradient of marked molecules (q*) at constant total species concentration(q). In general D(q) and D(q) will not be the same although it may be argued by intuitive physical reasoning that the two coefficients should converge in the low concentration limit. Fick’s second law, which in one dimension, is:

wq wt

D

w 2q wz 2

(3)

follows directly from Eq. 1 provided that the diffusivity is independent of concentration. Since, in accordance with Eq. 2, the tracer diffusivity depends only on the total concentration (q) and is independent of the concentration of labeled molecules (q*), one may write, quite generally:

wq * wt

D (q ) div grad q *

(4)

which is simply a generalization of Eq. 3 to a three dimensional system. Eq. 4 may be solved to provide the distribution in space and time for a pulse of marked molecules introduced at t = 0, r = 0:

q*r, t

4S Dt

3 / 2

ª r 2 º exp « » ¬ 4Dt ¼

(5)

This function is sometimes referred to as the (Fickian) propagator. The mean square displacement of the labeled molecules after time t may be calculated directly by integration:

r 2 (t )

6 Dt

;

z 2 (t )

2 Dt

(6)

for three dimensional and one-dimensional systems respectively. An alternative and conceptually simpler approach to self-diffusion comes from the “random walk” representation. If a molecule moves by steps of average distance O at average time intervals W, with equal probability in all directions, the self diffusivity is given by:

D

1 O2 ˜ 2n W

(7)

Transport in Microporous Solids: an Historical Perpective

11

where n is the dimensionality of the structure. Since ¢r2² = NO2 and t = NW (where N is the number of steps) Eq. 7 is seen to be identical with Eq. 6. 2. Diffusion Regimes The diffusion mechanism depends on the pore diameter or, more specifically, on the ratio of the pore diameter to the mean free path and the collision diameter of the sorbate molecules. In larger pores (dp >> O) the dominant mechanisms are viscous (Poiseuille) flow and molecular diffusion. As the pore size decreases the importance of these mechanisms decreases and Knudsen and surface diffusion become dominant. The way in which these four mechanisms combine is shown, by analogy with a resistance network, in figure 1. This allows the a priori prediction of the effective diffusivity for a straight cylindrical pore, provided that the pore size and physical properties of the diffusing species are known. The greatest uncertainty arises in the estimation of the tortuosity factor (W) which allows for the difference in path length and driving force in going from a straight cylindrical pore to a real pore as well as for the impact of variations in pore diameter. Various methods of predicting tortuosity have been suggested [5] but the tortuosity factor is probably best thought of as an empirical constant characterizing the particular pore structure. A further difficulty arises from the form of the expression for the combination of Knudsen and molecular diffusion which, in its general form, involves the composition and the fluxes of both components. However, an important simplification is sometimes possible. For equimolar counterdiffusion, as in catalytic processes with no change in mole numbers, NA = -NB and the flux term drops out. This term also drops out for pure component systems (yA = 1, NB = 0) and for dilute systems (yA ĺ 0). The relative importance of surface diffusion increases as the pore diameter decreases. When the pore size becomes comparable with the effective diameter of the diffusing molecule the molecules never escape from the force field of the pore walls, even when in the center of the pore. Steric effects become important and diffusion becomes an activated process, showing an Arrhenius temperature dependence. This regime, which has been variously termed “micropore diffusion”, “configurational diffusion”, “intracrystalline diffusion” or even simply “surface diffusion” is the most interesting since steric and surface interaction effects play a dominant role. Despite extensive study many aspects are still poorly understood.

12

D.M. Ruthven

Figure 1. Electrical resistance analog showing how the various contributions to the pore diffusivity combine[6]. The quantitative expressions used for the calculation of Dp are also given[7].

Transport in Microporous Solids: an Historical Perpective

13

3. The Driving Force for Transport Diffusion Since the true driving force for the transport process is obviously the gradient of chemical potential, rather than the gradient of concentration, it is more logical to write, in place of Eq. 1:

 B (q ) ˜ q ˜

J

wP wz

(8)

where B(q) is the intrinsic mobility and P the chemical potential. The relationship between the mobility (B) and the Fickian diffusivity (D) can be easily derived by considering equilibrium between the adsorbed phase (concentration q) and an ideal vapor phase (pressure p, temperature T):

D(q)

B(q ) RT

d ln p d ln q

Do ( q )

d ln p d ln q

(9)

where Do = BRT is the “corrected” or “intrinsic” transport diffusivity. Eq. 9 is commonly referred to as Darken’s equation since it was used by Darken to describe the inter-diffusion of two metals to form an alloy [8]. However, the same relationship was introduced much earlier by Maxwell [9] and Stefan [10] as well as by Einstein. The validity of this relationship has been confirmed by both numerical simulations and experimental studies. It is evident from Eq. 9 that, in general, D  Do. The thermodynamic correction factor (dlnp/dlnq) depends on loading and can be quite large. For example, for a Langmuir system:

d ln p d ln q

1 1  q / qs

; D

Do 1  q / qs

(10)

so, according to this model, D will tend to infinity as q ĺ qs, the saturation limit. In practice a variation in transport diffusivity by more than an order of magnitude between low and high loadings is common. At low loadings, within the Henry’s Law region dlnp/dlnq ĺ 1.0 and D ĺ Do so, in the low concentration limit the transport and tracer diffusivities coincide D # Do # D. 4. The Stefan-Maxwell Equation Eqn. 8 neglects any mutual diffusion effects (interference between diffusing molecules) and must therefore be considered only as an approximation. A more rigorous formulation that includes such effects has been derived in terms of

14

D.M. Ruthven

irreversible thermodynamics or, equivalently and more conveniently, in terms of the Stefan-Maxwell equation. Originally formulated for diffusion in a homogeneous fluid in the form:



1 wP i ˜ RT wz

¦ D u n

j 1

xj

i

uj

(11)

ij

the driving force for transport (the gradient of chemical potential) is assumed to be balanced by the frictional drag arising from the transfer of momentum between species. In terms of the fluxes (Ni = xicui) Eq. 11 can be written:

 xi wP i RT wz

n

x j N i  xi N j

n 1

cDij

¦

(12)

For single component diffusion in a microporous solid component i is taken as the diffusing species and j the immobile framework:

J

D

Ni

w ln p dq ˜ w ln q dz

(13)

which, with D = Do is equivalent to Eq. 9. In multicomponent diffusion one must consider both the drag between the diffusant and the framework and between the diffusant and the other species.



Ti

wP i RT wz ˜

a

T j Ni Ti N j

j z1

q s Dij

¦



Ni q s Di

(14)

where Ti = qi/qs, the fractional loading. The development and application of this approach to diffusion in zeolites has been pioneered largely by Krishna and coworkers [11-13]. In this expression Di represents the interaction with the framework and is equivalent to Do (the “corrected” diffusivity) in Eq. 9 while Dij represents the mutual interference between diffusing molecules. For a binary Langmuir system:

Ti the flux is given by:

qi qs

bi pi 1  bi pi  b j p j

(15)

Transport in Microporous Solids: an Historical Perpective

Ni qs



Do1 >1  T 2  T 1 D02 / D12 @’T 1 1  T 1  T 2

 T 1 >1  D02 / D12 @’T 2 1  T 2 D01 / D12  D02 / D12

15

(16)

with a similar expression for N2. If there is negligible interference between the diffusing molecules Dij o f and Eq. 16 reduces to:

N1 qs

 D01 >1  T 2 ’T 1  T 1’T 2 @ 1  T1  T 2

(17)

the form originally suggested by Habgood [14,15] for diffusion in a binary adsorbed phase. Although the mutual diffusivity cannot be predicted a priori Krishna has proposed that it may be approximated by a logarithmic mean value: T1

T2

D12 # D10 T1 T 2 ˜ D20 T1 T 2

(18)

This is the familiar Vignes correlation [16], developed originally for diffusion in liquids. With this approximation the generalized Stefan-Maxwell model thus provides a way to predict the fluxes in a binary (or multicomponent) system knowing only the equilibrium isotherm and the corrected transport diffusivities for the pure components. The validity of this approach has been tested by comparison with MD simulations and by comparison with experimental data under both transient and steady state conditions. This formulation also provides useful insight into the self-diffusion behavior since, for self-diffusion, we have D10 D20 , T 1  T 2 T and

’T 1

’T 2 . This yields: 1 D

1 T  Do D11

(19)

for the general relationship between the self and corrected transport diffusivities [17]. In general one can therefore expect that the self diffusivity will be smaller than the corrected transport diffusivity. Only in the limits when Dij ĺ 0 or T ĺ 0 will the self-diffusivity and the corrected transport diffusivity coincide. The impact of mutual diffusion effects on transport in a binary adsorbed phase has been investigated in detail by Karimi and Farooq [18]. Some of their results are shown in figure 2 in which selectivities for a membrane system are shown as a function of composition C for two different diffusivity ratios. It is clear that the impact of the mutual diffusion effect is greatest when the diffusivity ratio is high and the loading of component 2 (O2) is also high.

16

D.M. Ruthven

Figure 2. Effect of mutual diffusion on selectivity in a membrane system showing the effect of the loading of both components (O1=q10/qs, O2=q20/qs) and the diffusivity ratios (a) D10/D20=10; (b) D10/D20=2. Open symbols, Habgood model; filled symbols, generalized Stefan-Maxwell model. From Karimi and Farooq[18].

As an example of the practical impact of mutual diffusion figure 3 shows experimental data for diffusion of a mixture of methane and ethane through a silicalite membrane [19]. The plots show the variation of the fluxes of each component and the selectivity as a function of the feed composition,

Transport in Microporous Solids: an Historical Perpective

17

together with the theoretical curves predicted from the generalized StefanMaxwell equation (Eq. 16) using the diffusivity values for the pure components (together with the equilibrium data). Also shown are the curves calculated from the Habgood model in which mutual diffusion effects are ignored (Eq. 17). For the more strongly adsorbed species (C2H6) the predictions of the two models are very similar but, for the weakly adsorbed species (CH4) the flux is greatly overpredicted by the Habgood model leading to a large error in the predicted selectivity.

Figure 3. Variation of flux and selectivity with feed composition for permeation of a CH4-C2H6 mixture through a silicalite membrane at 303K, showing the effect of mutual diffusion. - - - - - -, Habgood model; _________ Stefan-Maxwell model with mutual diffusion. From van de Graaf et al.[19].

5. Single File Diffusion [20] Normal diffusion processes described by Eqns. 1-6 provide an adequate description of transport in microporous solids provided that the pore diameter is large enough to allow molecules to pass one another. If this condition is not fulfilled an entirely different class of behavior called “string of pearls diffusion”

18

D.M. Ruthven

or “single file diffusion” results. In a sufficiently large array the mean square displacement, for a single file system, increases with the square root of time, rather than with the first power of time:

z 2 (t ) where

F

2F t

§1T · 1 / 2 ¸2SW © T ¹

O2 ¨

(20)

(20a)

In a finite single file system the effect of the boundary becomes important since only molecules at the surface are able to exchange with the ambient fluid. As a result the mean position of any molecule in a single file system is shifted. Subsequent shifts of this type are clearly uncorrelated since they depend on the elementary adsorption/desorption processes at each boundary. The corresponding effective diffusivity may be shown to be:

DA

§1T · DA1 ¨ ¸ © TN ¹

(21)

where N denotes the number of sites in the file and DA1 is the diffusivity of an isolated molecule. Comparison of this expression with Eq. 20 shows that for displacements larger than

2 §1T · ˜¨ ¸ Ld (L = file length, d = site-site S © T ¹

distance) i.e. much smaller than L, the second mechanism becomes dominant. The peculiarities of single file diffusion are related to the prohibition of mutual exchange between adjacent molecules. Confinements of this type are relevant when different molecular species (in multicomponent diffusion) or differently labeled molecules (in self-diffusion measurements) are considered. For transient sorption and permeation with only a single adsorbed species, a single file system is not expected to show any peculiarity of behavior since mutual exchange of molecules in this situation does not lead to a physically different situation. 6. Sorption Kinetics Sorption kinetics in microporous solids may be determined by a variety of different processes. The simplest situation arises for physical adsorption in an adsorbent with a narrow distribution of (micro) pore size such as a large zeolite

Transport in Microporous Solids: an Historical Perpective

19

crystal. When the micropores are of similar dimensions to the sorbate molecule diffusion is sterically hindered and therefore relatively slow and rate controlling. Under these conditions, at least for larger particles, thermal effects can generally be ignored and the uptake, for a set of spherical particles subjected to a differential step change in surface concentration is given by the familiar solution to the Fickian diffusion equation:

mt mf

1

6

S

2

¦ exp n S f

2

2

Dt / r 2



(22)

n 1

with equivalent expressions for different particle shapes. At short times this reduces to:

mt Dt #6 mf Sr 2

(23)

and at long times:

§ 6 · Dt ln 1  mt / mf ln¨ 2 ¸  2 ©S ¹ r

(24)

The diffusional time constant (r2/D) is easily found from plots of mt/m’ vs t or ln (1-mt /mf) vs t. Equivalent solutions for particles of other shapes are available but in general these differ only slightly from the solution for spherical particles with the same equivalent radius (defined by req = 3V/A). When diffusion is rapid (small particles/high diffusivity) this simple behavior may by substantially modified by heat effects. The impact of heat effects has been studied in considerable detail [4]. In extreme situations the uptake rate may be controlled entirely by the rate of heat dissipation. The transient sorption curve is then given by [21]:

mt mf

1

ª § ha exp « ¨¨ 1 E ¬ © UC s

E

· t º ¸¸ » ¹ 1  E ¼

(25)

and clearly contains no information on the intrinsic diffusional behavior. The intrusion of heat transfer effects is not always obvious from the shape of the sorption curve. However variation of the size and configuration of the adsorbent sample provides a straightforward experimental check. In commercial bi-porous adsorbents which are formed by pelletization of small microporous zeolite crystals, the situation is somewhat more complex since the sorption rate may be controlled by either micropore or macropore resistance (or both), depending on the ratio of the time constants for these

20

D.M. Ruthven

Figure 4. Mathematical model for diffusion in a biporous adsorbent under conditions of macropore diffusion control showing how the apparent diffusivity depends on temperature and sorbate loading in a manner similar to micropore diffusion.

Transport in Microporous Solids: an Historical Perpective

21

processes. When the isotherm is linear and micropore diffusion is rapid (Dmicro/r2 >> HpDp/KR2) the micro particles are always close to local equilibrium with the fluid in the macropores and the behavior can be represented by Eq. 1 with D replaced by the “effective” diffusivity HpDp/[Hp+(1-Hp)K] and r replaced by R. Figure 4 shows how the behavior of such a system mimics micropore diffusion control since the diffusion process appears to be activated and, beyond the Henry’s Law region, the apparent diffusivity increases strongly with concentration. 7. Anomalous Diffusion in Zeolites Figure 5 shows gravimetric uptake curves for benzene and p-xylene in large crystals of HZSM5, reported by Beschmann, Kokotailo and Riekert [22]. The behavior of benzene is well described by the simple Fickian diffusion model. The uptake curves for two different concentration steps are very similar suggesting that the diffusivity does not vary greatly with concentration at least over the relevant range. In contrast the uptake curves for p-xylene, measured under similar conditions, show pronounced deviation from the Fickian diffusion model. There appears to be a very slow first order process superimposed on the initial rapid diffusion process which conforms to the parabolic law (Eq. 23). This surprising difference in behavior between benzene and p-xylene is also revealed by other experimental techniques and is discussed in greater detail later in this paper. There is now convincing evidence that benzene can exchange freely between the straight and sinusoidal channels of the silicalite pore system whereas the longer p-xylene molecules can change direction only with difficulty. Benzene therefore shows normal diffusion behavior whereas with pxylene the straight channels equilibrate rapidly followed by much slower equilibration of the sinusoidal channels. The initial portion of the uptake curve therefore conforms to the diffusion model with much slower uptake in the longer time region. Another example of anomalous sorption kinetics has recently been reported by Bülow for CO2 in large crystals of NaX [23]. Piezometrically measured uptake curves for CO2 in the Ba++ exchanged form (BaX) conform fairly well to the isotherml diffusion model but, for NaX, we see a rapid initial diffusion controlled region followed by a very slow final approach to equilibrium. Infrared spectroscopy confirms the existence of substantial amounts of chemisorbed CO2 on NaX but not on BaX so the slow approach to equilibrium is attributed to slow chemisorption.

22

D.M. Ruthven

Figure 5. Gravimetric uptake curves for (a) benzene and (b) p-xylene in two different samples of silicalite at 298K. The change in fractional loading (T) for the experiments are indicated. Note the conformity to the diffusion model (Eq.22) for benzene but not for pxylene. From Beschman et al [22].

Transport in Microporous Solids: an Historical Perpective

23

Figure 6. Anomalous behavior of the benzene-NaX system (100 Pm crystals at 403K). (a) Apparent isotherms determined at different time scales. (1) Adsorption, 20 minute end point (Â); (2) Desorption, 20 minute end point (ɰ); (3) Adsorption, 48 hour end point ({); (4) Desorption, 48 hour end point (Ƅ). (b) Transient sorption curves (gravimetric) corresponding to figure 6 (a) showing the anomalous kinetics. From Tezel and Ruthven[24].

24

D.M. Ruthven

Anomalous behavior has also been observed for benzene on large crystals of NaX (figure 6)[24]. The isotherms show an unusual hysteresis loop in which the adsorption isotherm is higher than the desorption isotherm. At low loadings the difference is substantial. The kinetic curves show corresponding anomalies with normal diffusion curves obtained only initially or in the high loading region. In the intermediate region the response curves show a slow exponential approach to equilibrium superimposed on a rapid initial response. This behavior suggests slow relaxation of the structure in response to changes in the benzene loading, the most likely change being relocation of the cations, but it is possible that this behavior can also be explained by slow chemisorption. 8. Surface Resistance When access to the pores is controlled by a constriction at the pore mouths, a common situation in carbon molecular sieves and hydrothermally aged zeolite crystals, mass transfer resistance is concentrated at the particle surface. The uptake curve, for a step change in external concentration, then follows a simple exponential approach to equilibrium:

ª  3kt º 1  exp « ¬ r »¼

mt mf

(26)

When both surface resistance and internal diffusional resistance are significant the solution for the uptake curve is given by:

mt mf

f

1 ¦ u 1



6 L2 ˜ exp  E n2W 2 2 E n LL  1  E n

>

@



(27)

where En is given by the roots of

E n cot E u  L  1 0 and

(28)

W = Dt/R2, L = Rkc/D.

Curves calculated from these equations for a range of values of the parameter L are shown in figure 7. When L is small Eq. 27 reduces to Eq. 26 (surface resistance control) while for large L Eq. 27 reduces to Eq. 22 (diffusion control).

Transport in Microporous Solids: an Historical Perpective

25

Figure 7. Theoretical uptake curves (step response) for a dual resistance system of spherical particles with surface and internal diffusion resistances showing the change in shape of the response curve for eq. 32-36 with variation of the parameter L=Rkc/D.

26

D.M. Ruthven

9. Sorption Kinetics in Carbon Molecular Sieves Carbon molecular sieve adsorbents commonly show both surface resistance and internal diffusional resistance and it is not uncommon to find both types of behavior exhibited by different sorbates in the same CMS adsorbent or even by the same adsorbent in different ranges of temperature. The behavior of N2 and O2 in Bergbau Forschung CMS provides a good example of internal diffusion control [25]. As may be seen from figure 8 the uptake curves conform well to the diffusion model. The diffusional time constants (D/R2) increase with loading in conformity with the Darken equation (Eq. 9); the values of Do/R2 are essentially independent of concentration. In contrast CH4 and CO2 (at 273K and 298K) show clear evidence of surface resistance control (figure 9)[26]. Not only do the transient uptake curves conform to Eq. 26 but there is excellent agreement between the uptake curves measured from sorbate free and partially saturated initial states. The behavior of CO2 is especially interesting since, at lower temperatures, it shows surface resistance control but, at higher temperatures, diffusion control. Furthermore, the surface rate “constant” varies linearly with CO2 pressure (see figure 9). A simple model that explains the main features of the experimentally observed behavior is summarized in figure 10. Recent studies of the sorption kinetics of O2 and N2 in Tabeda CMS, carried out by Qinglin et al.[27] reveal a more complex behavior pattern. Both surface and internal diffusional resistances are significant but the concentration dependence of both the surface rate coefficient and the diffusivity is even stronger than predicted by the Darken expression. 10. Shrinking Core Behavior The “shrinking core” model has been widely used to account for diffusion and reaction in solid-gas systems (such as combustion of coal) as well as for adsorption in macroporous adsorbents when the equilibrium isotherm is highly favorable. Under these conditions all adsorption occurs at the adsorption front and the adsorbed phase concentration profile assumes the form of a shockwave penetrating into the adsorbent at a rate that depends on both the adsorbent capacity and the pore diffusivity. The progress of the shock wave profile is given by:

t

Slab (thickness 2 l):

Sphere (radius Rp):

W t

W

 z / A 2

1 1 §¨ R f  6 3 ¨© R p

2

· §R ¸  1¨ f ¸ 2 ¨© R p ¹

(29)

· ¸ ¸ ¹

3

(30)

Transport in Microporous Solids: an Historical Perpective

27

Figure 8. Experimental uptake curves for N1 and O2 in CMS (Bergbau Forschung) showing conformity with diffusion model. From Ruthven[25].

28

D.M. Ruthven

Figure 9. Experimental uptake curves for CO2 in CMS at 298K showing conformity with surface resistance model (a) response curves for various pressure steps; (b) shows conformity between response curves for clean and partially saturated initial conditions for similar pressure steps. From Liu and Ruthven[26].

Transport in Microporous Solids: an Historical Perpective

29

Figure 10. Suggested mechanism for CO2 sorption in carbon molecular sieve. From Liu and Ruthven [26].

30

D.M. Ruthven

where

W slab

q A2 ˜ s and W sphere 2H p D p co

R p2 6H p D p

˜

qs . co

The penetration distances are respectively z and (Rp-Rf) and the corresponding expression for the uptake curves are:

mt mf

Slab:

Sphere:

mt mf

t

(31)

W

1  2 1  mt / mf  31  mt / mf

2/3

(32)

Recent measurements of the ion exchange of Ca++ in a SR5 Ionic resin (highly favorable equilibrium) provide a clear illustration of the usefulness of this model (see figure 11)[28)]. 11. Magnetic Resonance Imaging: Direct Measurement of Transient Concentration Profiles Magnetic resonance imaging (MRI) provides a possible way to measure transient concentration profiles directly. This approach has been applied by several research groups during the last five years (29-31]. Unfortunately the resolution of such techniques is not sufficiently fine to allow measurements on the scale of an individual zeolite crystal. However, measurements on the scale of an adsorbent pellet (a few mm) are indeed possible and provide useful insight. Figure 12 shows the measured profiles for diffusion of water into a bed of small 4A zeolite crystals, closed at one end and exposed at the other end to humid air (at t=0)[31]. Following the initial transient period the profile assumes the approximate form of a shockwave and penetrates into the bed in accordance with Eq. 29. A simplified model that captures the main features of the observed behavior is summarized in figure 13. A different situation is shown in figure 14. A pellet formed from large (r ~ 20 Pm) crystals of silicalite, initially under vacuum, was exposed at time zero to a finite pressure of benzene vapor. The rectangularity of the profile shows that the distribution of benzene must have been essentially uniform through the bed so the sorption rate was controlled by intracrystalline diffusion. The time dependence of the signal intensity therefore yields the uptake curve from which the diffusivity may be extracted. A more complex situation is shown in figure 15. A commercial 13X zeolite extrudate (1 cm length) formed from small NaX crystals was wrapped in

Transport in Microporous Solids: an Historical Perpective

31

Shrinking Core Behavior (Exchange of CU++ on Ionac SR-5 Resin) Phelps and Ruthven, Adsorption 7, 221-229 (2001)

Figure 11. Uptake of Cu++ on SR-5 Ionac resin showing (a) shrinking core behavior; (b) variation of core radius and (c) uptake curve. From Phelps and Ruthven[28].

Teflon tape to prevent access through the cylindrical surface and exposed to a stream containing a small concentration of propene in He. Diffusion occurs through the ends of the extrudate with accumulation in the zeolite crystals. The concentration profile has the form expected for a system in which the effective diffusivity increases strongly with sorbate concentration.

Figure 12. MRI profiles for diffusion of water vapor into a bed of small 4A zeolite crystals exposed at the outer surface to humid air showing development of shockwave profile.

32 D.M. Ruthven

Figure 13. Simplified model for development of concentration profile assuming a rectangular adsorption isotherm.

Transport in Microporous Solids: an Historical Perpective 33

34

D.M. Ruthven

Figure 14. MRI concentration profile tor benzene—silicalite at 293K. (a) Loosely packed 20 Pm crystals. (b) Pellet formed from the same crystals. The corrective flow maintains a uniform sorbate concentration through the sample. Development of the signal intensity is controlled by intracrystalline diffusion. From Fraissard et al.[29].

Transport in Microporous Solids: an Historical Perpective

35

Figure 15. MRI profiles for propane in a 1 cm NaX extrudate (exposed at the ends) showing (a) adsorption and (b) desorption. Duration of experiment was one hour each for adsorption and desorption. From Bär et al.[31].

36

D.M. Ruthven

Figure 16. Time dependence of signal intensity during desorption of (a) propane and (b) water from 13X extrudate showing conformity with Eq. 33. From Bär et al.[31].

The pronounced difference between the adsorption and desorption profiles is clearly apparent. As a result of the very strong concentration dependence of the effective diffusivity the profile during desorption is essentially flat over the central high concentration region and decreases very sharply at the external surface. Under these conditions the desorption rate is controlled by diffusion through the low concentration surface layer and the desorption curve can be represented by the very simple expression:

qo qf

1

1 4O

Do t A2

Conformity with the expression is shown in figure 16.

(33)

Transport in Microporous Solids: an Historical Perpective Notation a b B c Cs d D Dp Do F h J k L n N p q qs r R t T

ratio of external area to volume ( = 3/R for spherical particles) Langmuir equilibrium constant intrinsic mobility (Eq. 8) total species concentration (Eq. 12); sorbate concentration in gas phase heat capacity site-site distance (in single file diffusion) transport diffusivity pore diffusivity corrected transport diffusivity tracer/self diffusivity Stefan-Maxwell diffusivity (Eq. 11) parameter for single file diffusion (Eq. 20) external heat transfer coefficient (Eq. 25) flux (relative to fixed frame of reference surface rate coefficient (Eq. 26) file length; parameter defined in Eq. 28 dimensionality of lattice flux (relative to center of mass of diffusing species) partial pressure adsorbed phase concentration saturation limit of q crystal radius gas constant; particle radius time absolute temperature

u

species velocities (Eq. 11)

x z l

mole fraction distance coordinate half-thickness of slab

D D

Greek letters E T P W O Hp U

parameter in Eq. 25 fractional loading (q/qs) chemical potential time interval (Eq. 7); time constant (Eq. 29-32); tortuosity factor step size porosity density

37

38

D.M. Ruthven

References 1. 2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

Kärger, J. and Ruthven, D.M. (1992) Diffusion in Zeolites and other Microporous Solids, John Wiley, New York. Kärger, J. and Ruthven, D.M. (2002) Diffusion and Adsorption in Porous Solids in Handbook of Porous Solids p. 2089-2173 F. Schüth, K.W. Sing, and J. Weitkamp eds., Wiley-VCH Weinheim, Germany. Ruthven, D.M. and Post, M. (2001) Introduction to Zeolite Science and Technology, p. 525-577 Studies in Surf. Sci. and Catalysis 133, H. van Bekkum, E.M. Flanigen, P.A. Jacobs and J.C. Jansen eds., Elsevier, Amsterdam. Ruthven, D.M. (1997) Physical Adsorption: Experiment, Theory and Applications, pp 241-259, J. Fraissard and C.W. Conner eds., Kluwer Academic Publishers, Dordrecht. Dullien, F.A.L. (1979). Porous Media: Fluid Transport and Pore Structure, p 223-230 Academic Press, New York. Krishna, R. (1993) Gas Separation and Purification I, 91. Scott, D.L. and Dullien, F.A.L. (1962) AIChE Jl 8, 113. Darken, L.S. (1948) Trans AIChE 175, 184 . Maxwell, J.C. (1860) Phil Mag 19, 19; 20, 21; (1867) Phil Trans Roy Soc. 157, 49-79. Stefan, J. (1872) Wien. Ber 65, 323. Krishna, R. (1990) Chem. Eng. Sci. 45, 1779-1791. Keil, F. Krishna, R. and Coppens, M-O (2000) Rev. Chem. Engg. 16, 71197. Krishna, R. (2000) Chem. Phys. Lett. 326, 477-484. Habgood, H.W. (1958) Can. J. Chem. 36, 1384-1397. Round, G.F. Habgood, H.W. and Newton, R. (1966) Sep. Sci. 1, 219-244. Vignes, A. (1966) Ind. Eng. Chem. Fund. 5, 189-199. Paschek, D. and Krishna, R. (2001) Chem. Phys. Lett. 333, 278-284. Karimi, I.A. and Farooq, S. (2000) Chem. Eng. Sci., 55, 3529-3541. van de Graff, J.M., Kapteiju, F. and Mouliju, J.A. (1999) AIChE Jl. 45, 497-511. Karger, J., “Single File Diffusion” in Molecular Sieves – Science and Technology 7, H.G. Karge and J. Weitkamp eds., Springer-Verlag (in press). Ruthven, D.M. (1984) Principles of Adsorption and Adsorption Processes p. 192, John Wiley, New York. Beschmann, K., Kokotailo, G.T. and Riekert L. (1987) Chem. Eng. Process 22, 223-229. Bülow, M. (2002) Adsorption 8, 9-14. Tezel, H. and Ruthven, D.M. (1990) J. Colloid Interface Sci. 139, 581-583.

Transport in Microporous Solids: an Historical Perpective

39

25. Ruthven, D.M. (1992) Chem. Eng. Sci. 47, 4305-4308. 26. Liu, H. and Ruthven, D.M. in Fundamentals of Adsorption 7, p. 529-536, M.D. LeVan (ed) Kluwer Academic publisher, Boston, MA (1996). 27. Qinglin, H., Sundaram, S.M. and Farooq, S. in Fundamentals of Adsorption 7, p.779-786. K. Kaneko, H. Kanoh and Y. Hanzawa eds. I.K. International, Shinjuku, Japan (2002). 28. Phelps, D. and Ruthven, D.M. (2001) Adsorption 7, 221-229. 29. N’Gokoli-Kekele, P., Springuel-Huet, M-A., Bonardet, J-L., Dereppe, J-M. and Fraissard, J. (2001) Studies in Surf-Sci and Control 135, 93-102. A. Galarneau, F. DiRenzo, F. Fajula and J. Vedrine (eds)., Elsevier, New York. 30. Aarden, F.B., Valckenborg, R.M., Pel, L., Kerhof, J. and Kopinga, K. (2000) Proc. 2nd Pacific Conf. on Adsorption p. 26-30, D.D. Do ed., World Scientific, Singapore. 31. Bär, Nils-Karsten, Balcom, B.J. and Ruthven, D.M. (2002) Ind. Eng. Chem. Res. 41, 2320-2329.

MEASUREMENT OF DIFFUSION IN MACROMOLECULAR SYSTEMS: SOLUTE DIFFUSION IN POLYMERS SYSTEMS

R. L. LAURENCE Chemical Engineering Department University of Massachusetts Amherst MA 01003 USA e-mail - [email protected]

Abstract Diffusion is a macroscopic process by which relative motion takes place between the components of a mixture. Diffusion in polymers is relative motion, although the scale of diffusing molecules may be very different. Macromolecular systems present a spectrum of systems that range from a dilute solution of a polymer to rubbers swollen with a diluent to glassy solids to semi-crystalline solids to composite media. We will limit our discussions to a subset of the materials: solutions of solutes in polymers, and polymer-polymer systems. We recognize that a polymer is in itself a multi-component system, given that there is a distribution of molecular mass comprising the polymer. However, we treat the polymer as a single component of a mixture. The measurement of diffusion in any system is a compromise. Tradeoffs may be evaluated using the Fick ratio, Fi, a measure of experimental time to the time for diffusion over a distance L. An estimate of the time of an experiment can be made from a Fick ratio of 0.1

Fi =

Dtexp ≈ 0.1 L2exp

Depending on the system being studied, the feasible experimental times may vary from millisecond to years. The techniques range from gravimetry, permeation, spectrometry, interferometry to chromatography. In this paper, we ignore NMR techniques and refer you to the work of Speiss, Karger and others. We discuss measurements in binary systems usually of solutes in polymers and provide other examples of polymer-polymer diffusion and complex media. 41 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 41–68. © 2006 Springer. Printed in the Netherlands.

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1. Introduction

The experimental problems experienced in the study of diffusion in polymers stem largely from the range of diffusivities that exist for polymers. The diffusion coefficients range from 10-4 to 10-16 cm2/sec. The measurement techniques always require a compromise between the time and distance scales required for the measurements. Polymer systems also exhibit a very strong dependence concentration. We will see that there is appreciable size dependence as well. Finally, virtually all experimental systems require precision for slow processes. The problem was evident many years ago when in 1908, B.W. Clack wrote a paper in which he outlined a research program for steady-state diffusion measurement. His system was flowing water across the face of a pipe (10 cm. dia. & 30 cm. Long) fed by a stream containing KCl. Clack failed to estimate the time that it would take to reach a steady state. A world war interceded, and his results were not published until 1924[1]. A simple analysis would have shown that for Dtexp/L2 = 0.5 and D ~ 10-5 cm2/sec and L = 30 cm., the time to reach steady state texp was 5 107 sec ( 45 days). Clearly a steady-state experiment was not the way to go. The table listed below shows typical experimental times and distances for diffusion experiments. Table 1. Time and Distance Scales for Diffusion.

Phase

Diffusant

Diffusion coefficient

Experimental time

Experimental distance

gas

gas

10-1 cm2/sec

900 sec

30 cm

liquid

liquid

10-5 cm2/sec

104 sec

1 cm

liquid

polymer

10-6-10-7 cm2/sec

103 – 104 sec

1 mm

Polymer melt Polymer melt Polymer solid Polymer solid

liquid

10-6 -10-9 cm2/sec

1 – 102 sec

100 Pm

Polymer melt gas

10-8 -10-15 cm2/sec

102 – 108 sec

10 Pm

10-6 -10-9 cm2/sec

103 – 106 sec

1 mm

liquid

10-8 -10-13 cm2/sec

103 – 106 sec

0.1 mm

Measurement of Diffusion in Macromolecular Systems

43

2. Theories for Diffusional Transport in Concentrated Polymer Solutions At present, no theory that describes the diffusion process in polymer solutions through the entire concentration range (from bulk to infinite dilution). Instead, existing theories are applied to specific regions of the concentration interval. The three concentration regions usually considered are the infinitely dilute solution, the dilute solution, and the concentrated solution. This review discusses only the theories describing the diffusion process in concentrated polymer solutions consisting of an amorphous, monodisperse, uncross-linked polymer and a low molecular weight penetrant. Concentrated polymer solutions are solutions in which the polymer concentration is high enough so that there is a significant amount of entanglements formed by the polymer chains. The complex intra-molecular and intermolecular interactions present in concentrated polymer solutions make the study of diffusional transport in that concentration region the most difficult. In contrast to the infinitely dilute and dilute solutions, all attempts to derive theories for concentrated solutions using statistical mechanics have been unsuccessful. Consequently, all the theories developed for concentrated polymer solutions are more approximate in nature than the theories applicable to the infinitely dilute and dilute solutions [2-4]. Among these theories, the freevolume theory of diffusion is the only theory sufficiently well-developed to describe transport processes in concentrated polymer solutions. Before considering the free-volume theory, other theories will be discussed briefly. Barrer [5,6] developed a zone theory based on energy considerations. In his theory, the activation energy for diffusion is assumed to be distributed through the degrees of freedom in the system. An equation relating the diffusion coefficient to the thermal vibration frequency of the penetrant molecule, the jump distance, and the total energy of the activated zone was derived. Brandt [7] formulated a molecular model for the estimation of the total energy of the activated zone. The activation energies predicted with that model were 25% to 70% lower than experimentally determined values. DiBenedetto [8] developed a molecular model and DiBenedetto and Paul [9] formulated a volume fluctuation theory, but both theories predict only the activation energy of diffusion. 2.1. FREE-VOLUME THEORY OF TRANSPORT The original work on free volume was developed by Doolittle [10-12] who formulated an empirical relation using free volume to describe the temperature dependence of the viscosity of simple liquids. Using the theory of fluctuations [13,14] they developed an analysis of polymer segmental mobility. Meares [15] applied Bueche's theory to describe the dependence of segmental mobility on the free volume. By assuming that the diffusion coefficient was

44

R.L. Laurence

proportional to the segmental mobility, Meares obtained an expression for the diffusion coefficient that he used to predict the data for allyl chloride in polyvinyl acetate at 40oC. Fujita et al. [16] used a Doolittle type expression to describe the dependence of the diffusant mobility on the average fractional free volume of the system. Cohen and Turnbull [17] and Turnbull and Cohen [18] gave the free volume theory its first theoretical basis by developing an expression relating the self-diffusion coefficient to the free volume for a liquid of hard spheres. Their theory is based on the concept that molecular diffusion occurs by the movement of molecules into voids formed by a random redistribution of the free volume within the material. Naghizadeh [19] proposed a modified version of the Cohen-Turnbull theory by considering a redistribution energy for the voids in the system. Macedo and Litovitz [20] broadened the Cohen-Turnbull theory by taking into account both attractive and repulsive forces. The Macedo-Litovitz expression was also derived by Chung [21] from statistical mechanical arguments. Finally, Turnbull and Cohen [22] improved their theory by considering variable magnitude of the diffusive displacement.

2.1.1. Application to Polymer-Solvent Systems Fujita [23] was the first to utilize the Cohen-Turnbull theory to derive a freevolume theory of diffusion for polymer-solvent systems. Vrentas and Duda [24,25] proposed a more general version of the theory formulated by Fujita. They demonstrated that the further restrictions needed for the theory of Fujita are responsible for the shortcomings of Fujita theory[23,26]. Another weakness of the Fujita theory is that diffusivity-temperature data at several solvent concentrations are needed to evaluate the parameters of the theory. Therefore, the Fujita theory has a more correlative character than the theory of Vrentas and Duda for which only a few diffusivity data at infinite solvent dilution are needed to evaluate the parameters of the theory. Finally, Paul[27] used the Cohen-Turnbull theory to develop a model for predicting solvent self-diffusion coefficient in polymer-solvent solutions. A major advantage of Paul theory is that it contains only three parameters that can be estimated without any diffusivity data. However, a drawback is that it is anticipated that this model predictions will be less accurate for solutions with polymer volume fractions higher than 0.9. A comparison [28,29] of the Vrentas-Duda free-volume theory for self-diffusion with Paul's, both conceptually and experimentally, and concluded in favor of the Vrentas-Duda theory. 2.1.2. Basis of the Free-Volume Theory The free-volume model for diffusive transport in liquids and glasses [17,21,22] is based on the following oversimplified view of the molecular process. A molecule is able to move if two conditions are fulfilled. The first condition is

45

Measurement of Diffusion in Macromolecular Systems

that, due to a fluctuation in the local density, a sufficiently large hole opens up next to the molecule. The second condition is that the molecule must have enough energy to break away from its neighbors. However, the diffusional transport will be completed only if another molecule jumps into the hole before the first molecule can return to its initial position. These authors developed the following expression for the self-diffusion coefficient of a pure simple liquid:

γV* D1 = D01 exp - E1 exp - 1 RT VFH

(1)

where D1 is the self-diffusion coefficient, D 01 is a pre-exponential factor, E1 is the energy per mole that a molecule needs to overcome attractive forces, R is the gas constant, T is the absolute temperature, V*1 is the critical free volume required for a molecule to jump, V FH is the average free volume per molecule in the liquid, and Jҏҏis an overlap factor (0.5 < J < 1) introduced because the same free volume is available to more than one molecule. Bueche [30] introduced the concept of a jumping unit in a polymer chain, and extended equation 1 to the self-diffusion of polymer molecule in a pure polymeric liquid: *

D2 =

N D02 exp - E2 exp - γV 2 RT M2 N* VFH2

(2)

where, D2 is the self-diffusion coefficient of the entire molecule, N is the number of freely orienting segments in the polymer molecule, N* is the effective number of segments in each polymer chain, M2 is the molecular ^ weight of the polymer, D 02 is a pre-exponential factor, V FH2 is the specific ^ average hole-free volume of the polymeric liquid, V*2 is the critical hole-free volume per gram of polymer required for a polymer jumping unit to jump, and E2 is the critical energy required for displacement of a jumping unit.

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R.L. Laurence

2.2. VRENTAS & DUDA FREE-VOLUME SELF-DIFFUSION THEORY POLYMER 2.2.1. Basic Assumptions Vrentas and Duda [24] used equations 1 and 2 as a starting point of their freevolume theory. In order to extend the theory to binary polymer-solvent systems, three aspects of the self-diffusion process must be considered [28]. The first one is to define how much of the volume is actually free in the sense that it is available to molecular transport. The second aspect is to define an average free volume for a binary mixture to replace V FH. Finally, the third aspect to consider is the effect of addition of polymer on the backscattering effect in the liquid [22]. A- Definition of the free volume: Vrentas and Duda [24] consider that ^ the specific volume of an equilibrium liquid i, V i, is composed of three ^ components, as shown on figure 1. The occupied volume, V oi, is the close packed volume of the liquid and is defined as the volume of the equilibrium ^ ^ liquid at 0oK, V0i (0) . The specific free volume, V Fi, is defined by: 0

VFi = Vi - V0i = Vi - Vi (0) =VFIi + VFHi

(3)

^ The interstitial free volume, V FIi, is the part of the free volume that is distributed uniformly among the molecules since its energy of redistribution is ^ large. The hole-free volume, V FHi, is the remaining part of the free volume and it is discontinuously distributed in the liquid. It is assumed that the hole-free volume is redistributed with no energy change and hence is available for molecular transport. ^ For polymeric liquid, the specific occupied volume, V oi, is assumed to be independent of molecular weight. Ueberreiter and Kanig [31] and Fox and Loshaek[32] offer some justifications for this assumption. The thermal expansion coefficient for the sum of the specific occupied volume and the specific interstitial free volume, DCi, is also assumed to be independent of the polymer molecular weight. This thermal expansion coefficient is defined by the equation:

47

Measurement of Diffusion in Macromolecular Systems

α Ci =

∂ VFIi + V0i

1 VFIi + V0i

∂T

(5)

Upon integration, T

VFIi +

0 Vi (0)

=

0 Vi (0)

α Ci dT

exp

(6)

0

or T

VFIi +

0 Vi (0)

= VFIi (Tgi ) +

0 Vi

α Ci dT

(0) exp Tgi

(7)

B- Average free volume for a binary mixture: Following Cohen and Turnbull[17], Vrentas and Duda [24] assume that the size of the jumping unit do not influence the distribution of the hole-free volume. The average free volume per molecule of the Cohen and Turnbull theory, V FH, is replaced in Vrentas and Duda theory by an average free volume per jumping unit defined as the total hole-free volume divided by the total number of polymer and solvent jumping units. For binary mixtures, Vrentas and Duda [24] assume that there is additivity of the sum of the specific occupied volume and the specific interstitial ^ free volume. Thus, the specific hole-free volume for a binary mixture, V FH, can be expressed as,

VFH = V - VFI + V0 M

(8)

with

VFI + V0

M

=ω1

0

0

VFI1 + V1 (0) + ω2 VFI2 + V2 (0)

(9)

^ ^ where (V FI + V 0)M represents the sum of the specific occupied volume and the specific interstitial free volume of the mixture, Zi is the mass fraction of ^ ^ component i, and V FIi and V0i (0) , the specific interstitial free volume and the specific volume at 0 oK of component i.

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R.L. Laurence

C- Backscattering effect in polymer-solvent systems: In Vrentas-Duda free-volume theory, the fluctuations in local hole-free volume, due to a recession process involving the molecules surrounding a particular molecule, create a hole next to that particular molecule. If the hole is large enough and if the molecule has enough energy, the molecule will jump in the hole leaving a hole at its original position. This hole is then diminished by a process similar to the one that created the original hole. The jump of the molecule is a successful diffusive motion only if the hole left behind closes before the molecule can jump back to its original position. Clearly, the critical volume required by a solvent molecule to jump does not depend on the size of the polymer jumping unit. In that model, where the formation and disappearance of the hole are governed by the same basic processes, the redistribution of the hole-free volume and any backscattering effects are independent of the polymer concentration. Thus, the probability of obtaining a successful diffusive motion can be absorbed into a constant pre-exponential factor [28]. By introducing the assumptions stated above, Vrentas and Duda [24] derived a modified version of equation 1 and equation 2 for the self-diffusion coefficients for a polymer-solvent system: *

*

γ (ω1 V1 + ω2 ξ V2 ) - E1 exp RT V

D1 = D01 exp

(10)

FH

D2 =

N M2 N*

D02

exp - E2 RT

*

exp -

*

γ ω1 V1 + ω2 ξ V2 ξ VFH

(11)

^ where V*i is the specific critical hole-free volume of component i, with i = 1 for ^ the solvent and i=2 for the polymer, and V FH is the average hole-free volume per gram of mixture. The quantity [is defined as the ratio of the critical volume ~ of solvent per mole of solvent, V1* , to the critical volume of polymer jumping ~ units per mole of jumping units, V2* : *

*

*

*

ξ = V1 /V2 = V1 M1 /V2 Mj2

(12)

49

Measurement of Diffusion in Macromolecular Systems

where, M1 is the molecular weight of the solvent and Mj2 is the molecular weight of a polymer jumping unit. For temperatures close to the glass transition temperature of a liquid, i.e., Tg2 to Tg2+100oC, the amount of hole-free volume in the liquid is relatively small and the self-diffusion process is free volume driven. Consequently, for temperature in that interval it is possible to absorb the energy term into the pre-exponential factor [24] and write equations 10 and 11 as, *

D1 = D01 exp -

*

γ ω1 V1 + ω2 ξ V2 VFH *

D2 =

(13)

*

γ ω1 V1 + ω2 ξ V2 N D02 exp M2 N* ξ VFH

Specific Volume

where the D01 and D02 can be treated as constant.

EQUILIBRIUM LIQUID VOLUME

HOLE FREE VOLUME

NON-EQUILIBRIUM LIQUID VOLUME EXTRA HOLE FREE VOLUME

INTERSTITIAL FREE VOLUME

OCCUPIED VOLUME

Tg2

Temperature

Figure 1. Volume –Temperature behavior of an amorphous polymer.

(14)

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R.L. Laurence

2.3. FREE-VOLUME THEORY DESCRIPTION OF THE SELF-DIFFUSION PROCESS IN GLASSY POLYMERS The following assumptions, in addition to those introduced earlier, are used to develop free-volume equations describing the transport in glassy polymers[33]: A- Local density fluctuations occur both above and below the glass transition temperature. The free-volume theory satisfactorily describes transport in glassy polymers (elastic diffusion). B- A polymeric material at temperatures below its glass transition temperature generally has a non-equilibrium liquid structure (figure 1). It is assumed that this non-equilibrium structure does not change during the self-diffusion process. C- The rapid change in the expansion coefficient of the polymer in the vicinity of the glass transition temperature is idealized as a step change from D2 to D2g at Tg2. It is also assumed that the thermal expansion coefficient of the glassy polymer, D2g, can be approximated by an average value over the temperature range of interest. Equation 13 can be used to describe the diffusion of a solvent in a ^ glassy polymer, if V FH, the specific average hole-free volume of the mixture in an equilibrium liquid structure, is replaced by the specific average hole-free ^ volume for the non-equilibrium liquid, V FHg. The difference between the specific average hole-free volume of the mixture in an equilibrium liquid structure and the specific average hole-free volume for the non-equilibrium liquid structure is the extra hole-free volume, which is frozen into the non-equilibrium liquid 0

0

0

VFH2g = VFH2 + ω2 (V2g - V2g - V2 )

(15)

^ ^ Here, V02 g is the specific volume of the pure glassy polymer and V02 is the specific volume of the equilibrium liquid polymer. In the limit of zero solvent concentration, Vrentas and Duda [33], using the same kind of considerations as before, derived the following expression: *

D1 = D01 exp -

γ V2 ξ

K 12 K22 + λ T - Tg2

(16)

Measurement of Diffusion in Macromolecular Systems

λ= with:

51

α 2g - 1 - ƒG H2 α C2 α 2 - 1 - ƒG H2 α C2

(17)

The parameter O represents the character of the change of the volume contraction which can be attributed to the glass transition. For O= 1, the material has an equilibrium liquid structure at all temperatures. For O= 0, the ^ specific hole-free volume of the glassy polymer equals to V FH2(Tg2) at any temperature below Tg2. Clearly, the actual value of O for the polymer will depend on the mechanical and thermal history of the material. 3. Experimental Tools Techniques for diffusion measurements can be classed as Direct or Indirect. Direct methods measure a concentration profile directly. An Indirect measurement obtains a result of diffusion process, and in analysis presumes a constitutive relation, e.g. Fick’s Law, to interpret a related measure, mass, for example. Few techniques are direct. In polymer systems, these are observable in solid diffusion. The following list includes techniques useful in the study of small molecules in polymers. Bubble Dissolution Spectrometry Microinterferometry Chromatography Sorption/Desorption Fluorescence quenching Nuclear Magnetic resonance The useful techniques differ as to the nature of the system. In dilute solutions (low solids concentrations), the preferred techniques are Interferometry, Bubble collapse, Spectrometry (Forced Rayleigh Scattering). In concentrated solutions, one would choose Sorption/Desorption, Chromatography or Spectrometry. In solids, the choices are Permeation, Infrared spectrometry, or X-ray spectrometry. Among the sorption/desorption techniques are Gravimetry, Barometry, and Volumetry (pycnometric) 3.1. GRAVIMETRIC MEASUREMENT Gravimetry is one of the oldest techniques useful in diffusion measurements. The technique is Indirect and is used in many systems, not only polymers, but in adsorption measurements as well. The classic experiment is the McBain balance[34], but newer technologies afford more rapid measurement, e.g.,

52

R.L. Laurence

piezoelectric crystals and the Cahn electrobalance. We will discuss the evolution of the technique and detail some of the results. In gravimetric measurements, one must take care that the diffusion measurements made over small concentration changes because of the strong concentration dependence found in polymer systems. Measurements made at long times provide the diffusion coefficient at the final concentration. The model is a straightforward one, where a volume average frame of reference [35] is preferred. For such a reference frame, the diffusion equation is given in terms of the specific mass density of the solute

∂ρα . + ∇ • (ρα .vϕ )= ∇ • (D∇ρα . ) ∂t

(18)

where D is the mutual diffusion coefficient. The reference frame is given by (19) The result is the appropriate diffusion equation and its boundary and initial conditions.

The total mass is any sample is M

For a system (a film of thickness L, the limiting long time solution for the mass pick-up yields a straight line solution with time, as below: (21) At short times, the solution is a function of the square root of time:

ADap ρα 2 dM =− dt 2 Dap t π

(22)

For virtually all polymer systems, the diffusion coefficient is a very strong function of solute concentration. One finds then that the measured diffusion coefficients at short times will differ substantially in sorption or desorption [36]. The means to circumvent the problem of large changes in diffusion coefficients is to do the sorption in small steps. The technique was perfected by Duda and Vrentas. [34].

53

Measurement of Diffusion in Macromolecular Systems

C D/A

TC G P

GD MFC

He

T

O

V MFC

H V2

TB

B

Figure 2. A typical flow sorption gravimetric apparatus is shown. It differs little from that used for porous solids.

The data shown in the adjacent figure are due to Frennsdorf [37] in which he made long-time measurements of halocarbons in ethylene-propylene rubber. One can observe the difference in diffusivity upon sorption and desorption.

Figure 3. Adsorption and desorption of halocarbons in EP rubber measured gravimetrically[37].

54

R.L. Laurence

3.2 VOLUMETRIC OR BAROMETRIC MEASUREMENTS The collapse of a bubble in a polymer solution can provide a measure of the diffusion of the gas in the liquid.

Pressure

Absence of sorbent

P0 Pf Presence of sorbent t=0

Time

Figure 4. Analysis of step changes in pressure followed by adsorption/diffusion.

An equivalent problem is the monitoring of a pressure decrease in a fixed volume. Each has analogue in the measurement of nanoporous materials that can be viewed in Rees’ work with frequency response in zeolites. The apparatus shown below is a barometric device that can be used in frequency or step response mode. One can make a step change in the volume of the system (say, 3%) and examine the pressure decay as the sorbent polymer seeks it equilibrium state. The decay curve can be analyzed exactly as one might with gravimetric experiments. Duda and his group [38] have developed another pair of barometric apparatus to measure diffusivities. 3.3 SPECTROMETRY Spectrometric measurements have been made with a broad range of electromagnetic waves from visible light to IR to neutron recoil spectrometry. We discuss some techniques useful in determination of solute diffusion in polymers and present some discussion relevant also to ymer-polymer diffusion.

Figure 5. Frequency response analysis system employing a metal-bellows pump.

Measurement of Diffusion in Macromolecular Systems 55

56

R.L. Laurence

Lavrentyev and Popov [39,40] studied diffusion of a solute through a solid polymer using attenuated infrared spectroscopy and were able to probe the depth of penetration of a solute as a function of time. A sample in an ATR cell can be probed by light incident at varying angles to change the depth of penetration of the light. The smaller the angle, the deeper is the penetration. One can also study the interpenetration of two compatible polymers as has been done by Fundakowski[41] and Lustig[42]. Lustig measured using FTIR spectroscopy, the diffusion of PMAA and PVME. Earlier work was done by Klein [43,44] using transmission Infra-red spectrometry to measure the concentration profile in a deuterated /hydrogenated polyethylenes. Some more recent work was done by Watkins [45] recently used a fiber optic probe in a high-pressure cell to study probe diffusion in an acceptor-receptor couple in polystyrene. Recently, Koenig [46] has developed new imaging techniques for FTIR microscopy that he has used in polymer dissolution studies that could also be used in polymer-polymer systems. The technique shows much promise. 3.4. CHROMATOGRAPHY Chromatographic techniques have Inverse Gas Chromatography was developed by Guillet in the 1970s. Early experiments were done on polymers deposited on porous supports. Pawlisch [47,48] improved the technique with the use of capillary columns. Below we discuss the technique in detail and present results from our group and the work of Danner and Duda [53,58-61] Inverse Gas Chromatography (IGC) is a technique that manages to circumvent many of the problems associated with gravimetric sorption experiments. Until the late 1980s, all reported applications of IGC to the measurement of diffusion coefficients used packed chromatographic columns in which the stationary phase is supported on a granular substrate. Equations similar to those developed by van Deemter et al. [47] were used to calculate the stationary phase diffusion coefficient from the spreading of the elution profile. The equation developed by van Deemter is commonly written as H = A + B/V + CV,

(23)

where H is the height equivalent to a theoretical plate (HETP), and V is the mean velocity of the carrier gas. The constants A, B, and C represent the contributions of axial dispersion, gas phase molecular diffusion, and stationary phase mass transfer resistances toward broadening of the peak. The equation is only valid for describing the elution of symmetric peaks, which requires that mass transfer resistances be small, but not negligible. From plate theory, it can

57

Measurement of Diffusion in Macromolecular Systems

be shown that for a column producing Gaussian-shaped peaks, the HETP is related to the peak width, or variance, by the following: H = L {Vt2/tr2}

= L/tr2{W1/2/2.335 }2

,

(24)

where L is the column length, Vt2 is the variance of the peak, tr is the retention time of the peak, and W1/2 is the width of the peak at half-height. For the case in which all mass transfer resistance is due to diffusion in the stationary phase and the stationary phase is uniformly distributed on the surface of a uniform spherical packing, the constant C is related to the solute diffusivity by C = (8/S2)(T2/Dp)(K/H)[1 + (K/H @ ,

(25)

Where, Dp is the diffusion coefficient in the stationary phase, t is the film thickness, K is the partition coefficient, and e is the ratio of the stationary phase volume to the gas phase volume. These results may be used to determine diffusivity from experimental data as follows: Solute elution curves are obtained for a range of flow rates. From measurements of peak width, a plot of H versus V is prepared. At sufficiently high flow rates, the second term on the right side of Equation 23 becomes negligible, and the plot is linear. From the measured slope, Dp is calculated using Equation 25. It is presumed in the analysis that diffusion in the stationary phase is Fickian and that the diffusion coefficient is concentration independent. This technique is well suited to the study of solutes with low diffusivities. When the solute diffusivity is small, diffusion within the stationary phase is the dominant process determining the shape of the elution profile. The contributions of other processes are neglected justifiably. Improvements in the reliability and accuracy of the IGC method depend on the development of more suitable columns to support the stationary phase. Several authors have speculated that the use of capillary columns, or open tube columns would eliminate some of the concerns cited above, and would be advantageous for IGC applications [33-35]. The principal attraction of a capillary column is the possibility of achieving more uniform dispersal of the polymeric phase. Ideally, the polymer would cover the wall as a uniform annular film. Such a geometrical configuration would simplify modeling of the transport processes within the column, and improve the inherent reliability and accuracy of IGC measurements.

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Following the ideas developed by Guillet and his coworkers, a method using Capillary Column Inverse Gas Chromatography (CCIGC) was developed by Pawlisch to measure diffusion coefficients in polymer-solvent systems at conditions approaching infinite dilution of the volatile component. The polymer is deposited as a uniform annular coating in a glass capillary column. A solute is injected into an inert carrier gas that flows through the column. The elution curve of the sample is then used with a model to determine the solute activity and diffusivity in the stationary phase. A detailed description of the equipment and the experimental procedure is given by Pawlisch[47]. The description of the model was provided and indicates how it was developed. Model . The assumptions are the following: (1) the column is a straight cylindrical tube; (2) the system is isothermal; (3) the carrier gas is treated as an incompressible fluid; (4) the carrier flow is steady laminar flow; (5) the polymer stationary phase is homogeneous and constant in thickness; (6) the polymer film thickness is much less than the radius of the column; (7) the axial diffusion in the stationary phase is negligible; (8) the carrier gas is insoluble in the polymer; (9) the absorption isotherm is linear; (10) no adsorption occurs at the surface interfaces; (11) no chemical reaction occurs between the sample gas and the polymer; (12) diffusion coefficients are concentration independent over the range of interest; (13) the inlet concentration profile is modeled as an impulse function. Many of these assumptions can be relaxed. A modified version of this model was developed for a nonuniform polymer film [48]. With these assumptions, the continuity equations for the gas and polymer phase may be written as

(26)

Measurement of Diffusion in Macromolecular Systems

59

and (27)

where, c and c' are the gas phase and stationary phase solute concentrations, Dg and Dp are the gas phase and stationary phase diffusion coefficients for the solute, z and r are the axial and radial coordinates, and V is the mean velocity of the carrier gas. Appropriate initial and boundary conditions for the problem are: (28a) (28b) (28c) (28d) (28e) (28f)

where G (t) is the Dirac delta function, co is the strength of the inlet impulse, K is the partition coefficient, R is the radius of the gas-polymer interface, and Wis the thickness of the polymer film. The problem stated above is sufficiently complex that a closed-form analytical solution in the time domain has not been found. For most purposes, the details of the radial distribution of solute are unimportant, and a description of the longitudinal dispersion of solute in terms of a local mean concentration (that is, radially averaged) suffices. The most mathematically convenient mean concentration is an area-averaged concentration, defined as

(29)

Application of this definition to Equations 26, and 27, making use of the boundary conditions given by Equations 28, yields the following:

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The equations derived from radial averaging still contain the local concentration as a variable. To proceed further, approximations are developed to relate the local concentration to the area-averaged concentration. The approach used in earlier models [39,40] was to define a new variable, c, which describes the deviation of the local concentration from the mean concentration, (32)

When the chromatographic peak is dispersed, the radial variation in the gas phase concentration is expected to be small, so that c  (1 T2  iZS  iJ G ˜ r ) t @ dr

(16)

The spatial distribution of these parameters, that is the image, can be obtained by Fourier transformation of eqn. (16),

S (Z) v

1

³ U(r ) >1 T2  i(ZS  Z  JG ˜ r )@ ³ U(r ) psf (Zc  JG ˜ r ) d r

dr , (17)

Thus, the NMR spectrum S(Ȧ) in the presence of a field gradient is a projection of a spatial distribution function U(r) convoluted with the NMR spectrum, psf(Z), in the absence of a field gradient. In general, psf(Z) is called the pointspread function. Due to that convolution it is clear that the spatial resolution (1/'r) achievable by direct frequency encoding is limited by the width of the

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NMR spectrum, or, in case of one line only, by the line width ' Ȧ = 2/T2. While the concept of spatial resolution is straightforward, meaning that one should be able to distinguish two neighbouring structures, a qualitative definition is much more difficult. Such a definition depends on the value the signal must drop in between two adjacent objects. The Rayleigh criterion [6] can be used for this purpose or other definitions [7]. However, we want to discuss here only the basic dependence. Other definitions may scale the derived equations.

Figure 1. Schematic representation of a simple MRI sequences: The first line shows the excitation of the spin system with a radio frequency (rf) pulse of (flip angle Į) and the NMR-signals in grey. The second line indicates where the signal is sampled. Inversion of the frequency encoding gradient is used in the third line to generate a gradient echo after a time 2IJ. The last line schematically displays the second dimension (phase encoding) of the experiment, where the gradient strength is stepped through and for each step an echo is acquired.

Hence, the minimum distance 'r, which can be resolved, is simply defined by the ratio of the line width or the width of the present chemical shifts 'Ȧҏ and the gradient strength G

Sensitivity and Resolution in Magnetic Resonance Imaging

'r

'Z JG

247 (18)

It is obvious that if the line width increases, the resolution decreases accordingly, unless the gradient strength is increased. This explains the difficulties of NMR imaging of solids, where line widths can be broader than in liquids by 5 or more orders of magnitude, because the dipole-dipole are no longer averaged out as a consequence of reduced mobility. 2.3.2. Phase Encoding Gradient amplitude and time dependence are adjustable during imaging experiments under control of the spectrometer. Therefore, space encoding can also be achieved in an evolution period prior to data acquisition and by varying the influence of the gradient in a second dimension of the experiment. Therefore, the gradient strength is incremented in steps 'G at constant time (cf. Fig. 1, last line). In analogy to two-dimensional NMR spectroscopy the evolution of the spins under the influence of the gradient is reflected in a phase shift 'ij of the signal acquired during the detection period. In this case, eqn. (15) has to be rewritten as

s (t , 'M) v exp > 'M@

³ ³ U(r ) exp >  (1 T2  iZS  i J G ˜ r ) t @ d T2 dZS dr (19)

with

'M = J 'G ˜ r W

Then, a two-dimensional Fourier transformation reveals the image. A closer look at eqn. (19) shows that no other interaction is present in the phase term that the gradient increment and the evolution time, IJ, which is a constant. Since all other NMR-interactions evolve with time, their influences on the phase after IJ are constant. They might damp the signal, but do not cause an additional spread of the spatial information. Hence, for phase encoding by this technique, the minimum resolvable distanceҏ 'r is no longer determined by the NMRspectrum, but exclusively by instrumental variables, the maximum gradient amplitude, Gmax, and the evolution time, IJ

'r

1 J Gmax W

(21)

Typically a combination of both, frequency and phase encoding techniques are used as demonstrated in Fig. 2, which clearly demonstrates the differences in spatial resolution by the frequency and phase encoding technique.

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Figure 2. a) A 2D MR-image of three vials (spatial arrangement is sketched in b) filled with different chemicals to demonstrate the difference between the frequency encoded, horizontal dimension and the phase encoded, vertical dimension. In the frequency encoded dimension chemical shift artifacts appear which correspond to the spectra of the chemicals as shown in c). The phase encoded dimension shows pure spatial information.

2.3.3. Point-Spread Function and Self-Diffusion As already introduced in eqn. (17), spatial resolution is conceptually easiest explained by discussing a point-spread function, PSF(r) (i.e. the Fouriertransform of psf(Ȧ)), which convolves the pure spatial information, ȡ(r). The width of this point-spread function in relation to the width of an image pixel then directly gives a measure of the spatial resolution. An NMR-image, I(r), is then described by the following convolution (cf. eqn. (17)): I(r) = PSF(r) … ȡ(r) + noise

(22)

The width of the point-spread function can be directly obtained by switching the ‘spatial term’ off, which means to measure at gradient strength zero and Fourier-transform the result. As already shown for the frequency encoded dimension, this is then simply the normal NMR-spectrum, and the maximum distance of chemical shifts or line width determines the width of the point-spread function and the blurring of the image. This is the reason why NMR-images of solids with their dipolar broadened lines are hard to resolve [8]. As seen, the case is quite different when the spatial information is obtained via phase-encoding with constant evolution time. Because the signal phase is only modulated by the strength of the gradient, the point-spread function is a delta-function. If the gradient is switched off, nothing is varied, hence resulting in a constant which Fourier-transforms into a delta-function. This is confusing, because then the spatial resolution (cf. eqn. (21)) depends only on the product of gradient strength and evolution time. One may

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be tempted to conclude that the latter can be made arbitrarily long, and infinitely high spatial resolution could be obtained. This is of course not true, because so far only the width of the pointspread function was discussed. However, it also has an amplitude which rather damps the signal than spreads it. Therefore, the concept of a point-spread function in eqn. (22) has to be extended to a resolution/sensitivity function. For frequency-encoding the amplitude of PSF(r) is simply given by the intensity of the NMR-lines, which is proportional to T2, and the same holds for the phase-encoded dimension. The idea of spatial resolution is therefore only valid until the signal vanishes in the noise of the measurement, or in other words, the dominating limitation for spatial resolution in NMR-imaging is the sensitivity. The spatial resolution then depends also on the number of experiments, which are repeated to increase the signal-to-noise. Therefore, it is difficult to quantify the possible spatial resolution, but for liquid samples a voxel volume of 104-105 µm3 is typically achieved with micro-imaging equipment. Similar considerations hold for the influence of self-diffusion on the resolution. The random walk of the observed molecules causes a spatial offset, which can blur the image in the frequency encoded dimension. This happens according to the Einstein-Smoluchowski equation,

'r

2 D 't

(23)

where D is the self-diffusion coefficient and ǻt a sampling interval with which the signal is recorded. Furthermore, self-diffusion also has a strong influence on the amplitude of the point-spread function, which is given by,

PSF (r )

§ mW3 · exp ¨ J 2 DG 2 ¸ ¨ ¸ 3 © ¹

(24)

where m = 2 for the frequency encoded dimension and where m = 1 for the phase encoded dimension in Fig. 1. The influence of self-diffusion is usually smaller than chemical shifts, dipolar couplings and other interactions as long as liquids are considered. For water as an example (with D = 2.3 ǜ 10-9 m2/s at 25°C) a typical experimental setup [9] gives ǻr | 0.3 µm and a PSF | 1-10-6 which is negligible. 3. MRI of Gases The low sensitivity of NMR makes investigations of gases very difficult due to their low density. Therefore, experiments have to be performed either at high

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pressures or with artificially polarized samples. The latter can be realized by hyperpolarization, which increases the polarization by 4-5 orders of magnitude (we typically find 60% polarization for 3He and 40% for 129Xe, cf. eqn. (7))[10,11]. Production of hyperpolarized gases and their application to problems in biomedicine and material sciences can be found in recent reviews [3-5]. Although, about 3 orders of magnitude are lost due to the lower density of gases when compared to liquids or solids, these polarizations match or even exceed the signals from thermally polarized water at medium fields (ca. 7 – 10 T). Therefore, hyperpolarized 3He and 129Xe were originally expected to overcome the sensitivity limited resolution in MRI of gases. However, self-diffusion also increases by 4-5 orders of magnitude (3He: D = 2 ǜ 10-4 m2/s and 129Xe: D = 4.80 ˜ 10-6 at 25°C and 1 bar). In difference to liquids the main cause of resolution limits may be expected from rapid Brownian motion of the gas atoms. The same experimental values in the example above [9] give for 3He ǻr | 90 µm and a PSF | 0.5 for frequency encoded dimension, which are both very substantial. However, this calculation assumes free, unrestricted diffusion. In a realistic sample, one will find walls, which restricts the diffusivity of the gas atoms close them. This effect is termed “edge enhancement” [12,13]. On the other hand pores will cause restricted diffusion of the gas inside their entire volume. In such situations the effective diffusion coefficient can also be estimated by the Einstein-Smoluchowski equation (23) when the pore size is smaller than a critical distance, rc

rc

2 D0 W

(25)

where D0 is the coefficient for free diffusion, so that the effective diffusion coefficient can be approximated as

D

­ r 2 for r  r ° 2W c ® °¯ D0 for r ! rc

(26)

The spatial restriction by pore walls therefore reduces the effective diffusion coefficient and increases the amplitude of the diffusion point-spread function in eqn. (24). From these facts it is expected, that MRI of hyperpolarized gases in porous media lead to better resolved images. Consequently it is of interest to investigate how the “coherent” resolution, ǻr, of the image in eqn. (21) is related to the size of a pore, r, and whether there is an optimum of resolution, respectively sensitivity. For such a relation one has to combine eqns. (21), (24) and (26) giving

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PSF (r )

§ r2 · exp ¨  ¸ ¨ 6'r 2 ¸ © ¹

(27)

A consequence of this equation is that if the resolution of the image is much better than the size of the pore, the signal is decaying exponentially. For instance if only 3 pixels per pore (3ǻr = r) should be resolved, the PSF drops to 22% of its maximal amplitude, and for 5ǻr = r it is already down to about 1%. This effect is experimentally demonstrated in Fig. 3, where for the better resolved images the effect of edge enhancement by unilateral restrictions can be clearly recognized. The relation between resolution and pore size is best studied at the central pore (2.5 mm diameter). Following the increasing resolution from image 3e to 3p the signal loss is clearly visible.

10

Figure 3. a) – p) 16 images of a resolution-phantom made from PTFE (sketch on the left, measures in mm) inside a larger tube and filled with hyperpolarized 3He. The resolution / gradient strengths were increased from left to right and top to bottom 64 u 64 points were acquired with a evolution time of W = 310 µs and a sampling interval of ǻt = 10 µs. The strengths of the frequency and phase encoding gradients were chosen to be equal and increased from image a) to p) according to the following list of values in mT/m: a) 8.9, b) 13.4, c) 17.8, d) 22.3, e) 26.7, f) 33.4, g) 35.6, h) 40.0, i) 44.5, j) 49.0, k) 55.6, l) 66.8, m) 77.9, n) 89.0, o) 100.1, p) 111.3.

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4. Summary The very strong influence of diffusion in MRI of gases plays an important role in designing an experiment. Although, the effective diffusion coefficient of liquids and especially gases in confined geometries can be greatly reduced, significant signal loss occurs, when the spatial resolution reaches the dimensions of the confinement. The simple question about optimal resolution cannot be answered, except for the trivial statement, that the lowest resolution which allows to distinguish the investigated features is the best. Principally two different types of experiments have to be distinguished, one which investigates the spins inside the pore volume and one which aims to resolve the pore geometry via the walls. In the latter case the necessary resolution is then determined rather by the wall thickness than the pore size (e.g. MRI of foams and lungs) and the gas is only used to render the structure. The appearance of the NMR image of such a sample depends on the timing and the resolution (gradient strength). The duration of the sequence determines where the influence of diffusion is reduced due to restrictions. On the other hand, the resolution of the image controls the signal contribution depending on the pore size. In samples with a large distribution of pore sizes the choice of the two parameters will have a strong influence on the result. However, eqn. (26) and (27) are a good rule of thumb to estimate and tailor diffusion contrast in MRI of porous media for problem specific contrast. Other mechanisms of signal losses, like relaxation and susceptibility differences, which are of great importance for hyperpolarized gases in porous media, were ignored in this discussion. Acknowledgements We would like to thank Luis Agulles Pédros for his assistance in sample preparation and Jörg Schmiedeskamp (Inst. of Physics, University of Mainz) for the polarized 3He. References 1. 2.

Chardairerivere, C., Roussel, J. C. (1992) Principles and Potential of NMR Applied to the Study of Fluids in Porous Media Revue de l’Institut Français du Pétrole 47(4) 503. Gladden, L.F.(1994) Nuclear Magnetic Resonance in Chemical Engineering – Principles and Applications, Chem. Eng. Sci. 49 3339.

Sensitivity and Resolution in Magnetic Resonance Imaging 3. 4. 5. 6.

7. 8. 9.

10. 11.

12. 13.

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Leawoods, J. C., Yablonskiy, D. A., Saam, B., Gierada, D. S., Conradi, M. S. (2001) Hyperpolarized 3He Gas Production and MR Imaging of the Lung, Conc. Magn. Reson. 13(5) 277. Goodson B. M. (2002) NMR of Laser-Polarized Noble Gases in Molecules, Materials and Organisms, J. Magn. Reson. 155 157. Mair, R. W., Walsworth, R. L. (2002) Novel MRI Applications of LaserPolarized Noble Gases, Appl. Magn. Reson. 22(2) 159. von Kienlin, M. and Pohmann, R. (1998) Spatial Resolution in Spectroscopic Imaging, P. Blümler, B. Blümich, R. Botto, and E. Fukushima (eds.): "Spatially Resolved Magnetic Resonance", Wiley-VCH Publisher, Weinheim 3. Callaghan P. T. (1991) Principles of Nuclear Magnetic Resonance Microscopy, Clarendon Press, Oxford. Blümler, P. and Blümich, B. (1994) NMR Imaging of Solids, NMR-Basic Principles and Progress 30 209. The estimations base on sample of water with a diameter of 10 cm, which is imaged with a gradient strength of 10 mT/m at 7 T (300 MHz for 1H). For the frequency encoded dimension an ADC interval of 20 µs is assumed and an evolution time of IJ = 3 ms for phase encoding. Appelt, S., Baranga, A. B., Erickson, C. J., Romalis, M. V., Young, A. R., Happer, W. (1998) Theory of spin-exchange optical pumping of 3He and 129 Xe Phys. Rev. A 58(2) 1412. Eckert, G., Heil, W., Meyerhoff, M., Otten, E. W., Surkau, R., Werner, M., Leduc, M., Nacher, P. J., Schearer, L. D. (1992) A dense polarized 3He target based on compression of optically pumped gas, Nucl. Inst. And Meth. A 320 53. Callaghan, P. T., Coy, A., Forde, L. C., and Rofe, C. J. (1993) Diffusive Relaxation and Edge Enhancement in NMR Microscopy J. Magn. Reson. A 101 347. de Swiet T. M. (1995) Diffusive Edge Enhancement in Imaging, J. Magn. Reson. B 109 12.

RESTRICTED DIFFUSION AND MOLECULAR EXCHANGE PROCESSES IN POROUS MEDIA AS STUDIED BY PULSED FIELD GRADIENT NMR V. SKIRDA, A. FILIPPOV, A. SAGIDULLIN, A. MUTINA, R. ARCHIPOV, G. PIMENOV Kazan State University Kremlevskaya 18, Kazan 420008, Russia [email protected]

1. Introduction The spatial and time resolution of the PFG NMR technique is directly proportional to the amplitude of the magnetic field gradient. In standard NMR equipment (for example, NMR scanners) the typical gradient amplitude is limited to a 1 T/m. Using such gradient pulses it is possible to investigate translational displacements of fluid molecules on the distances about a few microns only. Producing higher magnetic field gradients involves serious technical difficulties. Nevertheless, while in the 1980s the maximum amplitude of PFG did not exceed 10 T/m, at present magnetic field gradients of about 100 T/m and higher are used in several laboratories. For a long time we used a gradient with a maximum amplitude up to 50-100 T/m. With a special probehead, we have achieved real gradient values up to 500 T/m. This considerably broadens the scope of the method and its applications. It also opens up a new window to study porous media by PFG NMR in the range beyond the sub-micron level. Some preliminary measurements demonstrate the potential of this NMR technique to characterize the pore space over distances below 1 Pm. Such information is essential for the study of relationships between the pore geometry and transport properties of the porous media.

2. General Principles The use of the PFG NMR technique to study the translational dynamics of molecules is based on recording the loss of phase coherence of spins as a result of their translational motion in magnetic field gradients. Information about G diffusion processes can be obtained by analysing diffusion decay A(q, t ) , or the 255 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 255–278. © 2006 Springer. Printed in the Netherlands.

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dependence of the amplitudeGof the spin-echo signal on magnetic field gradient JG 1 parameters and time t. The q (2S ) JG g value, where J is the gyromagnetic G ratio of resonating nuclei, is directly related to the amplitude g and duration G of the magnetic field gradient; this value is an analogue of the waveGvector, for instance, in neutron scattering. It follows that diffusion damping A(q, t ) can be represented by the van Hove correlation function:

G A( q , t )

G G

G G

G

G G

³³ U (r ) P (r ; r c, t ) exp(i 2S q (r c  r ))drdr c s

(1)

G G where U (r ) is the initial density of spins and Ps (r ; r c, t ) is the probability for G observing a spin at a point with radius vector r c at time t, if at the initial time G G G the spin had radius vector r . Due to the direct relation between Ps (r ; r c, t ) and G A(q , t ) , the method has a wide range of applications. For free diffusion in a G G one-component system and times different from microscopic, Ps (r ; r c, t ) has the form of a Gaussian function: ­ rGc  rG 2 ½ ° ° exp ®  ¾ (4S D t )3 / 2 °¯ 4 Ds t ¿° s

(1a)

G G with the root-mean-square displacement [r c(t )  r (0)]2

6 Ds t ; Ds is the self-

G G P ( r , r c, t ) s

1

diffusion coefficient.

Figure 1. Stimulated echo pulse sequence.

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It follows that, for a single-phase system (from the point of view of NMR) with a single Ds value and provided G g !! W g 0 , where g 0 is the constant magnetic field gradient, diffusion decay for the most widely used stimulated spin-echo sequence (fig. 1) is: A(2W ,W 1 , g 2 )



A(2W ,W 1 ,0) exp  q 2 Ds t d



(2)

In the case of exponential relaxation, A(2W ,W 1 ,0)

§ 2W W 1 · A(0)  ¸¸ exp¨¨  2 © T2 T1 ¹

(2a)

where A(0) is the initial amplitude of the free induction decay after the first 90o pulse; T2 the spin-spin relaxation time; T1 the spin-lattice relaxation time; W and W 1 the time intervals between the first and second and between the second and third 90o radiofrequency pulses, respectively; ' the time between gradient pulses; and td ('  G / 3) the diffusion time. For a simple one-component G system, there is no difference in the way we obtain diffusional decay: ', g or G can be varied.

Figure 2. Dependence of the relation (Dseff/Ds0) for the mean self-diffusion coefficient of a monomer diffusant, confined in Vycor porous glass (pore diameter 40Å) to the mean diffusion coefficient of bulk diffusant on the molecular mass of the monomer diffusant in the case of water and carbohydrates.

3. Self-Diffusion in Porous Systems 3.1. RESTRICTED DIFFUSION PHENOMENA IN PFG NMR The behaviour of the mean self-diffusion coefficient of a liquid in a porous structure is indicative of the existence of three diffusion time ranges:

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(i) Short time range, when the root-mean-square displacements of molecules of a liquid are smaller than the linear pore dimensions, and there is no substantial effect of pore walls on the observed self-diffusion coefficient; that is, D* | D0 , where D0 is the self-diffusion coefficient of bulk liquid. (ii) Intermediate diffusion time range, when the root-mean-square displacements of molecules of a liquid are comparable to the linear pore dimensions and the effective coefficient D * depends on the diffusion time. (iii) Long diffusion time range. The behaviour of D * is then determined by the structure of the porous space. For closed pores completely constrained diffusion with D* D eff (t ) v t 1 can be observed at long diffusion time. Experimental data can then be used to determine linear constraint dimensions by the Einstein equation 2

R p | 6 D eff (t )t . For connected pores and long diffusion times, the root-mean-square displacements of molecules are larger than the pore dimensions, and the motion of the liquid becomes averaged over the space of the system. For porous systems with a random structure, diffusion decay is, as a rule, exponential and characterized by a diffusion-time-independent effective self-diffusion coefficient D* Df . It follows that the behaviour of the mean self-diffusion coefficient of molecules in a system of constraints is determined by three terms, namely, the self-diffusion coefficient of the pure liquid, the effect of constraints, and the permeability effect. According to [1, 2], these effects can be separated correctly by scaling equations for: ª º D0 D eff (t ) > D * (t )  Df @ ˜ « (3) ». ¬ D0  D * (t ) ¼ The D eff (t ) dependency can be used to obtain information about constraint dimensions (porous medium characteristics) by analysing the overall time dependence of the self-diffusion coefficient, including the three principal time ranges. 4. Effect of Molecular Size The self-diffusion of alkanes with molecular weights from 84 to 212, and water, confined in the controlled-pore glass "Vycor" with average pore diameter 40 Å, was measured in the diffusion time td interval from 3 to 1000 ms at 300C. We observed dependence Ds(td)~td0 for all liquids studied. These results suggest

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that the fractal structure of this glass is not displayed in the length scales from 0.1 Pm to 24 Pm. The second important result was the observation of power law dependence of Ds on the size of the liquid molecule (fig. 2). It was shown that this result should be taken into account for the estimation of tortuosity [ as well as the form-factor F of the porous media, according to the self-diffusion data. For the "Vycor"-water system we observed peculiarities which can be explained only by structure ordering caused by strong hydrogen bonds. We suggest that our results can be used to study the association phenomenon of low molecular weight liquids by the PFG NMR method. Thus, to obtain information about the porous substance (permeability, tortuosity, etc.) from the diffusion experiments in the slow diffusion time range, the experiment should be performed under conditions such that the ratio of molecular size to pore diameter is small. 5. Investigation of Structure of Porous Media by PFG NMR

A(g)/A(0)

Self-diffusion of liquids confined in porous media is observed to be nonGaussian. The shape of the diffusion echo decay (DD) is not exponential and the average diffusion coefficient D depends on the diffusion time. In recent years it was shown that analysis of the diffusion data for a liquid confined in a porous structure can be used to obtain structural information about the porous medium [3-5]. To this end the dependence of D on diffusion 1 Indiana time is usually used [4, 6]. The t=3 ɦɫ, = 6 Pm 20 14 dependence of the shape of DD 100 29 on the diffusion time contains 300 47 more complete information; 0,1 however, it has not been studied until now. Analytical equations for DDs were obtained only for systems with 0,01 0 1 2 3 4 rather simple structures, when 2 (JgG) *D(td)*td pores are randomly distributed in channels [7-9], layers [8, 10, Figure 3. Normalized DDs for water in pores of 11] and spherical or cylindrical Indiana limestone at diffusion times of 7-300 ms and corresponding mean square displacements. cavities [6, 8, 12]. Temperature is 30ɨC.

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We investigated the dependence of the shape of DD obtained for water in hydrocarbonates, in porous sedimentary rocks (Fontainebleau, Berea, Indiana Limestone, Austin Chalk) and in artificial porous media formed during gypsum solidification . It should be noted that in previous work some peculiarities of the sedimentary rocks structure [13] and NMR relaxation of confined liquids [14] were observed. Our the most important result (fig.3.) is that, in the length scale 5÷50 Pm, the DD’s shape is independent on diffusion time and these curves obtained for different diffusants coincides (if DD’s curves are plotted in the G G G G coordinates log ª¬ S  q , t º¼ vs q 2 D  t t , where q JG g is the wave vector and D(t) is the average measured D at time t). The observed effect reflects the selfsimilarity of the molecular diffusion in the samples studied. Our investigation showed that there are also some general characteristics for the self-diffusion of confined liquid in rocks. It was proposed that the fractality of self-diffusion in porous media can be conditioned by the fractality of the structure [15]. We consider that any conclusion of fractal structure requires additional investigations for the all samples. 6. Cryodiffusometry The possibility of obtaining information about separate parts of a porous medium is restricted, particularly due to molecular exchange between these parts. For a selective study of diffusion in the separate parts of porous space, the signal from other parts of the pores must be suppressed, for example by freezing the interporous liquid [16]. The melting temperature of a sample inside pores with a diameter ɚ is decreased, together with a decrease in the pore size, according to the Gibbs-Thomson relation [17, 18]:

'Tm

4 ˜ V sl ˜ Tm /(a ˜ 'H f ) ,

(4)

where Tm is the equilibrium melting temperature, 'T = Ɍm-T(a); T(a) the melting temperature of the crystal with a diameter a; ısl the free surface energy of the solid-liquid interface; 'Hf the enthalpy of fusion. After complete solidification of the liquid, subsequent heating to temperature Tm(dɤɪ) < Tm leads to melting of the liquid in the pores with sizes d d dɤɪ. This effect was studied by NMR relaxation [19] for samples with known pore sizes, and it was shown that relation (4) was satisfied with high precision. This allowed us to use this technique as a method for estimating pore size distribution ("cryoporometry").

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The applicability of relation (4) is restricted 0.1 3 because of: (i) in small b) pores (< 200 Å) the values of ısl and 'Hf can be different from the bulk values [20]; (ii) the shape of the crystal phase in 0.0 most cases is different 100 200 300 400 500 600 700 800 900 1000 1100 from spherical, cylindrical d,A or rectangular; (iii) there is an unfrozen layer of liquid Figure 4. Differential pore size distribution curve for about 5 Å thick between gypsum stone obtained by melting cyclohexane in pores. Ɋ is the ratio of pores with diameter d. 1, 2 and the pore surface and the 3 shows dɤɪ, for temperatures 0ɨɋ (1); 2ɨɋ (2); 4ɨɋ (3). crystal down to very low temperatures [21]; (iv) the signal from the liquid phase can be distorted because of spin diffusion between the liquid and crystal phases [22]. However, for mesoporous systems some of these effects can be corrected by calibration [23, 24]. 2

P xd 3

1

10

D, ɦ 2 /ɫ ɯ10 -11

3

2 -1

1 0.01

1

0.1

t ,c

1

d

Figure 5. The dependence of D on the diffusion time for cyclohexane in pores of gypsum stone at temperatures 0ɨɋ (1); 2ɨɋ (2); 4ɨɋ (3). Straight line shows the slope -1, which corresponds to completely restricted diffusion.

Freezing a liquid in pores opens a new possibility for investigating selfdiffusion in pore areas containing a liquid phase. Crystal and liquid phases are characterized by completely different transverse relaxation times, Ɍ2, so for a

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given set of pulse sequence parameters we can obtain the signal just from the liquid. This allowed us to investigate self-diffusion in the liquid phase, restricted by the pore walls and the crystallized diffusant. The usual methods can be used to analyze restricted diffusion (for example, [6]). The change in Ɍ and td allows us to estimate simultaneously both the pore size and the mean square distances for restricted diffusion in areas d d dkp and also the connectivity of the areas in pores. The differential curve of the pore size distribution, obtained by stepwise melting of cyclohexane in pores of gypsum stone, is shown in fig. 4. Pore sizes for this sample range from ~150 to more than 1000 Å. Temperatures 0ɨɋ, 2ɨɋ and 4ɨɋ are related to the partially melted cyclohexane in pores and are shown in fig. 5 as 1, 2 and 3, respectively. These melting temperatures are related with the higher limits of dɤɪ, which is 240 Å, 360 Å and 600 Å, respectively. NMR diffusion measurements at these temperatures showed that the shapes of DD and D depend on td. The slopes of the dependence for D (fig. 5) in all cases are less than -1. We used relation (4) to separate the effects of permeability and restriction. The results is that (In the result) Deff(td) and Df were determined for each Tm and it was found that Rp/dɤɪ is at all temperatures essentially greater than 1 and decreases as with Tm increasing. This means that the areas of the porous space, containing liquid, are essentially anisotropic; the degree of anisotropy is higher for the areas with the smaller diameter. 7. Diffusion of Liquid Molecules with Internal Magnetic Field Gradients (IMFG) When the molecular mobility is studied by NMR diffusometry and NMR relaxometry in objects with a different magnetic susceptibility , 'F between parts of the system (for instance, between porous matrix and liquid), the G presence of internal magnetic field gradients (IMFG), ’H v 'F , has to be taken into account to interpret the experimental results correctly. For porous materials filled with a liquid, IMFG values and their distribution over the G sample depend on the external magnetic field, H 0 , on the morphology of the porous medium and on the magnitude of 'F. Thus, the IMFG corresponds to the geometry and properties of the porous medium. However, for most real porous materials it is extremely difficult both to define the characteristics analytically and to take into account these effects on the results of the NMR experiment. In particular, the problem of IMFG becomes very difficult in cases when the applied external magnetic field gradients are not high enough. One of the methods of avoiding the effects of IMFG on a result is to use special pulse sequences which decrease these effects (for example, the so-called

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"13 interval sequence"). Nevertheless, it turns out that these sequences do not fully compensate the IMFG contribution; moreover, the mechanisms of this compensation have not yet been clarified. Another way is detail investigation of IMFG itself, as they reflect porous material properties. 7.1. EFFECTS OF INTERNAL MAGNETIC FIELD GRADIENTS STUDIED BY THE STIMULATED SPIN ECHO METHOD IMFG effects have been studied and described in [25-27]. Notwithstanding the results obtained by these groups, this problem remains. A (W) 10

10

0

1 2 3 4 5

-1

0

J

2

G

2

4

t

d

8

D

S

[ x 10

12

,m

2

12

/T

2

]

Figure 6. Diffusion decays A(W) obtained for tridecane in sand at ': 5 ms (1); 8 ms (2); 10 ms (3); 20 (4) and 320 ms (5).

The effect of internal gradients was studied for a sand sample with particle size 300-400 Pm using a stimulated spin-echo sequence, on a spectrometer operating with a superconductor magnet at ɇ0 = 7 T. The sand samples were filled with tridecane (diffusion coefficient of tridecane is DS = 0.68·10 -9 m2/s). Pulsed field gradient was not applied in these experiments. The diffusion decay curves were obtained as functions of W: A

W

v

¦

td

{ i

p

A i

W

ex p

'  W

,W

1

, g

 J

i in t 2

G

v g

2

i in t



2

td D

s



3

(5)

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V. Skirda et al.

Fig. 6 shows the typical shape of diffusion decay A(W) and illustrates the procedure for extracting the IMFG value from the slope of the curve shown, in accordance with the equations:

g

2 i int

tgD i , D i

' ln A W

' ln  J 2G 2td DS

(6)

.

A family of curves was obtained for different diffusion times, ', varied from 5 to 320 ms. Each curve of the family was analyzed in accordance with the procedure described earlier(6), and the values of the internal gradients were calculated in this way. The IMFG distribution is presented as a histogram, pi 1 ) and column height reflects where the column width reflects share pi (

¦

g

i int

value. Location of columns on the histogram was chosen consecutively,

i value. For example, such histogram obtained for according to gint sand/tridecane system at ' = 80 ms is shown in fig. 7. It should be noted that the shape of the IMFG distribution obtained differs significantly from the parabolic magnetic field model, ( H  z v z 2 , g g  z ), which is usually used in theoretical work concerning the problem of taking into account the IMFG effect [28]. As seen in fig. 7, the proportion of high IMFG values is rather low (less than 20%). Thus, the result obtained needs to be more accurately related to modern theoretical work and models describing the effects of IMFG in porous media.

Distribution function of IMFG in parabolic magnetic field

1,0

i

IMFG, g int, T / m

1,5

0,5

0,0 0,0

0,2

0,4 0,6 population, p

0,8

1,0

Figure 7. Distribution function of IMFG calculated for sand sample with tridecane.

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12 10 8

,p

io n

la t

'=5 ms '=8 ms '=10 ms Di ffu '=20 ms tim sion e '=320 ms

1,0 0,8 0,6 0,4 0,2 0,0 pu

4 2

i

6

po

int

IMFG, g

i

,T/m

i with increase in Fig. 8 reveals a decrease of internal field gradients gint '. This effect is conditioned by translational mobility of diffusant molecules. According to the experimental data (fig. 8) the maximum calculated gradient decreases from 11.6 T/m to 0.7 T/m as the diffusion time increases from 5 to 320 ms. Thus, the study of the internal magnetic field gradient, which was performed with stimulated spin-echo, showed that: (i) for the natural sand/tridecane system studied, the distribution of IMFG values may not be described within the framework of the parabolic field model; (ii) increase in the mean square displacement of the diffusant molecules leads to change of the i value internal field gradient distribution function. The maximum measured g int decreases due to the effect of averaging the internal field gradient by the diffusing liquid.

Figure 8. Distribution function of IMFG, calculated from experimental decays.

7.2. EFFECTS OF INTERNAL MAGNETIC FIELD GRADIENTS STUDIED BY NMR RELAXOMETRY The transverse (T2) NMR-relaxation of hexane in sand with 300-400 µm particles has been investigated using an NMR spectrometer equipped with a magnet with ɇ0 = 0.5 T. The methods of Hahn and Carr-Parcel-Meiboom-Gill (CPMG) were used to obtain the dependence of T2 on the evolution time IJ (delay between the 90° and 180° radio-frequency pulses). The existence of this dependence is conditioned by the internal magnetic field gradients inside the porous matrix. Time IJ increases the misphasing value of the proton spins of

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V. Skirda et al.

hexane, 'ij v Ȗgint IJ ('ij is the change in the spin precession phase). This effect leads to acceleration of NMR signal attenuation. As much as the relaxation attenuation shape was not exponential, we characterized the relaxation processes through the values of times T2 e and

T2 e2 when the NMR signal decreased by e and e2 times, respectively. These time moments may be also defined by the relations: A T2( e )

e 1 ,

A(0)



A T2( e2 )



A(0)

e 2

For the relaxation curves, the deviation of the shape from exponential for different W values can be characterized by a function: T2 e2 W 2



T2 e W

1,3 1,2

( ( W ) = ( 1 / 2 T 2(e

2

)

) / T 2(e)

( W

1,1 1,0 0,9 0,8

0

2

4 6 W, ms

8

10

Figure 9. Function ((W) obtained for hexane in native sand.

Fig. 9 shows the function ((W) obtained for hexane. In the range of small W the shape of the relaxation decay is close to exponential. At some intermediate value of time Wm, the function ((W) shows a maximum, and after that ((W) begins to decrease again, approaching 1. As a rule, it is assumed that pore space can be conventionally presented as the set of NMR phases characterized by different IMFG values. The

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267

Hexane Tridecane

gradient, T m

-1

characteristic spatial size of these phases is lm. In the limit of short times the diffusant molecules displace only small distances so that is impossible to feel the IMFG’s effects. Consequently, these effects cannot be measured; the shape of the relaxation decay is very close to exponential, the values of ((W) § 1. As the IJ increases, the diffusion contribution to the relaxation decay increases but this effect is different for areas with different values of IMFG; the shape of relaxation decays is nonexponential and the information about IMFG’s distribution function can be obtained by analyzing the decays. The maximum deviation of E(IJ) from 1 is observed when the molecule displacements reach the diameter of the NMR-phase, lm. In the range of long IJ, when the molecule displacement exceeds lm, the IMFG’s effects are averaged out; the relaxation curve becomes again exponential and the function E(IJ) tends to 1.

10 0

1 0 -1

1 0 -2

1,

0

0, p

8

0,

6

0,

4

0,

2

0,

0

Figure 10. The distribution function of IMFG obtained for native sand impregnated with hexane and tridecane.

The characteristic length lm may be estimated by times Wm. In the case discussed, for the native sand/hexane system, the characteristic size of the part of the pore space where the IMFG can be assumed to be constant, lm, is:

lm = D hexaneτ mhexane = 4 ⋅10−9 ⋅1,05 ⋅10−3 ≅ 2 ( µm)

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V. Skirda et al.

To define the distribution function of IMFG values and to analyze the effect of the diffusion averaging of IMFG described above, the transverse NMR relaxation study has been performed by the Hahn method for native sand impregnated with hexane and tridecane. Assuming that the relaxation curves may be described as the sum of exponential functions by the relation: A  J 2 DSW 3 A0

¦

i

2· § 2 i pi exp ¨  J 2 DSW 3  gint ¸ © 3 ¹

i the experimental curves were decomposed and the values of g int and pi were then calculated. Fig. 10 presents the results of this decomposition. As seen in fig. 6, a "more resolved" spectrum of IMFG was recorded for the sand/tridecane system. This can be explained by the smaller translational mobility of tridecane as compared to hexane, and, as a consequence, the lower effect of averaging IMFG due to self-diffusion of liquid molecules. The results presented show that it is more advantageous to use liquids with comparably low diffusion coefficients as a NMR sensor to study the internal gradients in porous media, because this permits one to record a wider spectrum of IMFG values. Taking into account the experimental results of the study of IMFG effects performed by different NMR methods, it should be concluded that in principal it is possible to obtain information about the effective values of IMFG and their distribution as well as some data concerning the characteristic sizes of porous media.

8. Water Flows Through Porous Media 8.1. INVESTIGATION OF WATER MOBILITY THROUGH POROUS MEDIA UNDER CONDITION OF FLOW PFG NMR permits one to estimate both the pore and the particle sizes of the porous media. However, a disadvantage of this method is the comparably narrow range of measured sizes corresponding to the porous matrix. Here the limiting factors are relaxation times and insufficiently high values of fluid diffusion coefficients. Nevertheless, the liquid mobility can be increased by the flow. In the present part of this work we report the use of PFG NMR in the presence of fluid flow through native sand to determine the mean size of the particles of the porous medium. We examined the mobility of water through samples of sand with particle size of 120-180, 180-250, 250-355 and 355-500 Pm (porosity 37%) and

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269

glass beads with a narrow particle size distribution, 78-88 Pm (porosity 40%). The experiments were performed with an NMR diffusometer operating at a resonance frequency 1ɇ 64 MHz; the maximum PFG amplitude was 30 T/m. Diffusion decays (DD) characterizing fluid mobility through porous media were obtained for the mean velocities of the fluid flow (8, 15, 17 mm/s) at two values of the diffusion time, 7 and 25 ms. To describe the shape of the diffusion decay, when there is laminar fluid flow through porous media (Re < 1), the following expression is used [29, 30]: ­° 6Deff ' ª sin( 2 S qb) º½° 2 u 1 exp(2S 2q2[ 2 ) E(q, ') So (q) exp( i2S qVavg ) uexp® 2 »¾ 2 « 2S qb ¼¿° °¯ b  3[ ¬

(8)

where ~S0(q)~ is the structural factor of the unit pore and connecting channel; b the distance between the centers of neighboring pores; [ the standard deviation -1 from b; q = (2S) JGg the wave vector; Deff the effective coefficient of the pore connectivity; and Vavg the mean flow velocity through the porous medium. Under certain experimental conditions and time parameters, coherency peaks (minima and maxima) arise on the diffusion decay curves [7, 31]. These peaks are conditioned by the periodicity of the porous medium. The appearance of the first peak is anticipated at q corresponding to inverse value of b that is close to the particle size for the model porous medium, q = b–1 [29, 30]. In the general case the maximum position on the diffusion decay curve is determined by the relation: qb = m, where m is an integer. As a rule, it is experimentally possible to obtain the first diffraction maximum (m = 1). The mean particle size may be calculated from the position of this maximum and the formula: q = b–1.

10

qb = 2 b = 71.4 Pm

0

qb = 1 b = 76.92 Pm

A/A 0 10

-1

0,0

4

2,0x10 4,0x10 -1 q, ( rad ɦ)

4

6,0x10

4

Figure 11. Diffusion decay obtained for water in glass beads at flow rate 3 mm/s.

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V. Skirda et al.

For water flow (3 mm/s) in glass beads the coherency peaks are observed on the diffusion decays at position qmax (fig. 11). The mean particle size was calculated by the relation [29, 30]: b = 1/qmax,

(9)

It should be noted that particle diameters calculated with (9) are very close to the real size of the glass beads used (see Table 1). Table 1. Particle size distribution, Pm

Particle diameter, calculated from the first peak, Pm

Particle diameter, calculated from the second peak, Pm

74-88

77

72

The typical diffusion decay obtained for natural sand has a more complex shape. Nevertheless, in few cases only (e.g., for narrowly fractionated sand) the position of the diffraction peaks can be obtained and sizes of particles can be calculated. The estimation of the mean size of particles performed with (9) is consistent with the real particle size (Table 2). Table 2. Particle size distribution, Pm

Particle diameter, calculated from (2), Pm

250–350

251

For samples with a wider particle size distribution (350-500 Pm), diffraction peaks were not observed under any experimental conditions. These results on the study of fluid mobility inside natural porous systems shows the potential of PFG NMR to obtain the mean particle size of narrowly fractionated sands only. 8.2. FILTRATION THROUGH POROUS MEDIA: “STAGNANT” AREAS OF FLUID AND THEIR CHARACTERISTICS In a static regime (without fluid flow), as a rule, it is impossible to separate the contributions from liquid in stagnant zones and in closed pores from the liquid in penetrable pores by diffusion decay analysis. This can be explained by the very similar values of the diffusion coefficients of liquid corresponding to these

271

Restricted Diffusion and Molecular Exchange Processes in Porous Media

"types". A difference in mobility can be achieved by the creation of a fluid flow inside the porous media studied. In this case one part of the liquid molecules will be involved in the flow and, thus, will be characterized by an extra (induced) diffusion coefficient. Another part of the molecules located in "stagnant" zones (including the locked pores) will be characterized by lower values of the diffusion coefficient. The detailed analysis of diffusion decays obtained for systems with fluid flow permits one to estimate the fraction of molecules located in "stagnant" zones, and, in some cases, to define the fraction corresponding to each pore type. In order to define the characteristics of "stagnant" zones the translational mobility of the water involved in flow was studied for sands with particle size 300-400 Pm. The diffusion decays obtained by the stimulated spinecho sequence for the system without flow and with fluid flow, velocity v = 13 mm/s, are depicted in fig. 12. This indicates that flow leads to two main effects: increase in the initial slope of the diffusion decays, and decrease in the initial amplitude of the spin-echo signal as compared with the amplitude at v = 0 mm/s. Nevertheless, the tails of the diffusion decays have not changed. This means that some of the liquid molecules in the porous medium were not involved in the flow and, consequently, are characteristic of so-called "stagnant" zones. The initial amplitude decrease of the spin-echo signal is related to extra misphasing due to internal gradient effects. These effects increase as the diffusion time increases.

A / A (0)

diffusion time (td) = 25 ms, W = 2.2 ms

A / A (0)

diffusion time (td) = 50 ms, W = 2.2 ms

0

10

0

10

V = 0 mm/s

V = 0 mm/s -1

10

V = 13 mm/s

-1

10

V = 13 mm/s

-2

10 -2

10

0,0

0,0

9

2,0x10

9

4,0x10

9

6,0x10

9

8,0x10

5,0x10

9

10

1,0x10

k2 td , m-2 s

k2 td , m-2 s

Figure 12. Diffusion decays obtained for water molecules in sand.

10

1,5x10

272

V. Skirda et al.

Comparing the diffusion decays obtained for samples with and without flow it is easy to obtain the fraction (Pst) of molecules not changing the translational characteristics. In particular, for the sand/water system studied the fraction Pst decreases in accord with an exponential law as the diffusion time increases (fig. 13). This Pst(td) dependence shows that as the diffusion time increases most of the water molecules are involved in flow and their transport properties are sufficiently changed.

p st ( t d )

0,4

0,1

0,0

0

50

100 150 td, ms

200

Figure 13. Dependence of population corresponding to water not involved in flow on the diffusion time at flow velocity 13 mm/s. 120 100

m,mg

80 60 40 20 0

0

4

4

4

4

4

4

1x10 2x10 3x10 4x10 5x10 6x10 7x10

4

t,min

Figure 14. The kinetic curve of hexane adsorption in “Vycor-220”.

The molecules involved in flow can exchange with molecules located in "stagnant" zones due to self-diffusion. Applying the approach developed to study the exchange processes, it is easy to determine the mean life-time (W h ) of

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273

molecules in the "stagnant" zones from the data presented in fig. 13 and the relation:

p st  t d



§ t · p st  0 ˜ exp ¨  d ¸ . © Wh ¹

The calculated time (W h ) was equal to 105±5 ms; it allows one to obtain the mean size of "stagnant" zones by the Stokes-Einstein formula. The value obtained was 30r3 Pm, in agreement with the pore structure of sand and its particle size (300-400 Pm). 9. Kinetics of Hexane Adsorption It is known [33, 34] that the process of vapor adsorption by liquids and solids is the consequence of three stages: sorbate supply to the surface of the adsorbent; diffusion of molecules into pores through the holes on the surfaces of grains; and the sorbate condensation on the internal pore surface. It is often proposed that the first stage is conditioned by environmental factors, so it is not interesting for the adsorption process [34]. This opinion is valid in the case of forced supply of the sorbate. In our work, the adsorption process was carried out in such a way that all three stages are present. "Vycor" porous glasses were used as a model porous medium. The primary porosity of these glasses (pore sizes r) was 20–400 Å. The secondary porosity was characterized by the grain size R (0.05-0.3 mm). The kinetics of adsorption was studied at 21°C. We used a ɉ-shaped adsorption ampoule. One parts of this ampoule contained the porous glass, the other was filled with liquid hexane. The tube containing porous glass was placed in the probe of the NMR relaxometer. The mass of adsorbed hexane m was determined from the amplitude of the free induction decay (FID). From the kinetic curve (fig. 14) it was seen that the adsorption process is the consequence of two stages. The first one is related to monomolecular adsorption because it is completed at m close or equal to monomolecular covering; secondly, for this stage the kinetic equation for monomolecular adsorption was satisfied: m / mM = 1 – exp (-kt ), where m is the hexane mass in the time t ; mM the mass of hexane for monolayer; k the rate constant for monomolecular adsorption. The FID in the first stage of adsorption can be presented in the form: A(t ) ɪɚ ɟ  t / T2 ɚ  ɪɜ ɟ  t / T2 ȼ ,

274

V. Skirda et al.

where ɪɚ and ɪɜ are the populations of protons with transverse relaxation times Ɍ2ɚ and Ɍ2ɜ, respectively. The shortest time Ɍ2ɜ is about 50 µs and does not dependent on the hexane concentration at long adsorption time, and was related to the hexane molecules in micropores, which were characterized by low mobility. The equilibrium amount mɜ = 10.5 mg/g of these molecules is the same for all the porous glasses studied. This amount was reached after an adsorption time of about 190 minutes for "Vycor-220" (r = 220 Å) and 28 minutes for "Vycor-55". Ɍ2ɚ was related to the more mobile hexane molecules. In Table 3 the rate constants for the first stage of adsorption K1 (adsorbate flow in 1 minute) are presented. The second stage was related to capillary condensation. This stage occurs from one day in "Vycor-20" up to 1.5 months in "Vycor-220". For most of the used glasses this stage was finished when glasses were completely saturated with the liquid. Partial filling was obtained only for "Vycor-400". This stage is linear and can be described by the relation: m = m0 + K2 t, where Ʉ2 is the rate constant of the second stage. For "Vycor-20" the stages of monomolecular adsorption and capillary filling were indiscernible and experimental values for the second stage of adsorption are presented in Table 4. Vp is the micropores volume; Mc and Me the calculated and experimental quantities of hexane after completion of the process of the pores filling. Table 3. Masses mC of porous adsorption constants k and K1. Vycor mC , mM, mM, mg mg, mg type exp. calc. 20 349 25 25.5 55 170 15.4 17 160 145 7.2 7.6 220 140 5.9 5.9 400 271 2.43 3.5

glass, corresponding values mM, S, SS and S,m2

SS,cm2

69.8 43 19.,9 16.5 6.8

7 47.3 21.2 18.1 27.1

k .103, K1.103, mg/min min-1 . 1.6 36 3.1 102 5.9 32 5.5 25 4.0 16

Table 4. Parameters for the second stage of adsorption. Vycor Vp*102, Ɇc, mg Ɇe, mg Ʉ2.103, 3 cm mg/min type 20 7 46 45 7.23 55 11.9 78 77 9.84 160 15.9 105 122 2.27 220 18.1 119 110 1.66 400 13.6 89 58 1.38

j.103,mg/ min 40.,6 14.7 5.08 3.69 2.03

S. SS m2cm2 490 2030 420 300 180

SS /r, cm2/Å 0.35 0.86 0.13 0.082 0.068

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From Tables 3 and 4 we see that experimental values coincide with the calculated ones. Some difference between Ɇc and Ɇe was found for "Vycor400" with maximum pore sizes. This effect can be related to the pore size distribution; larger pores were not filled by capillary forces. In the second stage Ʉ2 are equal to the flow j of hexane vapor during one second. According to the kinetic theory of diffusion [35] we have:



j

1 f

dP

cS dx

(10)

where f is the friction coefficient; c the concentration; µ the chemical potential of diffusant; x the diffusion direction. The independence of Ʉ2 on time means that values in eq. (10) are also time-independent. In particular, independence of the moving force for diffusion from t is possible only for capillary condensation, but not for polymolecular adsorption where the effective radius decreases monotonically. The motive force for capillary adsorption is the difference in chemical potential:

'P

V 2V / r ,

where V and V are the molar volume and the surface tension of hexane. For the ɉ-shape tube kinetics of adsorption is determined primarily by the hexane Vapor transport over a distance x between the surfaces of liquid hexane and the porous glass. This was checked using the relation for flow:

j

D V 2V cS . RT rx

For this purpose we took D = 7.7*10-6m2/s; Ɍ = 293Ʉ ɫ = 6.53 mol/m3; V = 130.1*10-6 m3/mol; V = 18.41*10-3 J/m2; the ampoule cross-section S = 0.49*10-4m2; x = 0.3 m. Calculated values for j coincide rather well with the experimental Ʉ2, (Table 4), excluding "Vycor-20" (in this case the pore diameter is comparable to the size of the hexane molecule). Additionally, for "Vycor-20" the holes on the surfaces of grains are minimal SS = 7 cm2. SS /r values in Table 4 correlate better with Ʉ2 , because the flow of Vapor into pores also depends on SS. The rate constants K1 from Table 3 are about ten times higher than Ʉ2 because the driving force of adsorption is also higher and equal to the difference between the heat of adsorption 36.8 kJ/mol and the heat of condensation of hexane 31.8 kJ/mol; the flow calculated taking into account 'P 'H is equal to 84.6*10-3 mg/minute.

276

V. Skirda et al.

Thus, our results show that hexane supply to the surface is the limiting stage for the adsorption of hexane vapors. The rate of adsorption depends on the pore characteristics and the parameters of the adsorption ampoule. 10. Anomalous Dynamic Properties of Low Molecular Weight Liquids Confined in Porous Media. Molecular Exchange Processes In addition to the changes in translational mobility of liquids, conditioned by obstacles, in a number of papers an anomalous behavior of the self-diffusion coefficient was found [1, 36, 37]. Decrease in the concentration of liquid resulted in an increase in the molecular mobility. Sometimes measured selfdiffusion coefficients exceeded Do by a factor of ten more. A hypothesis was outlined about the fast (from the NMR point of view) exchange between the molecules of liquid and saturated vapor. As the result of a number of investigations [38, 39] and [1] we can now establish the mechanism of influence of saturated vapor on the diffusion of confined liquid is the result of fast molecular exchange between the vapor and the liquid phases. For the first time this exchange was measured experimentally by the PFG NMR technique in [1] and the form of the function for the lifetime distribution in the liquid phase was determined. It was shown that the form of this function is very sensitive to the interaction of liquid molecules with the surface. In the case of partial saturation, the surface affects the character of the liquid distribution in the porous space. From these data the need for careful preparation of the experiments is clear, as well as the analysis of experimental data itself for the self-diffusion of smallsize (particularly fugitive) liquids in porous media. For the first time, the problem of measurements of diffusion in the presence of exchange between phases was considered for an example of a twophase system [40]. It was carried out on the basis of the assumption of exponential distribution functions of the phase lifetimes. Intermediate exchange is most interesting. For instance, diffusion decay is described by a continuous spectrum of self-diffusion coefficients even in the case of the two-phase system. The most important result [32] is that the spectrum is unambiguously bounded by Dsa and Dsb values and the component populations pa (t d ) and pb (t d ) , which are decreasing functions of diffusion time td . To summarize, the analysis of

pa ,b (td ) dependence provides the possibility of determining the distribution function \ (W ) for lifetimes in phases. Acknowledgements Authors would like to thank Schlumberger Research & Development for the support of this research.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Valiullin, R.R., Skirda, V.D., Kimmich, R., Stapf, S. (1997) Phys. Rev. E, 55 2664. Valiullin, R.R., Skirda, V.D. (2001) J. Chem. Phys. 114 452. Fordham, E.G., Gibbs, S.J., Hall, L.D. (1994) Magn. Res. Imaging 12 279. Hurlimann, M.D., Helmer, K.G., Latour L.L., Sotak, C.H. (1994) J. Magn. Reson. 111 A 169. Latour, L.L., Mitra, P.P., Kleinberg, R.L., Sotak, C.H. (1993) J. Magn. Reson. 101 A 342. Mitra, P.P., Sen, P.N., Swartz, L.M. (1993) Phys. Rev. B 47 8565. Callaghan, P.T. (1994) J. Magn. Reson. 113 A 53. Linse, P., Soderman; O. (1995) J. Magn. Reson. 116 A 77. Neuman, C.H. (1974) J. Chem. Phys. 60 4508. Blees, M.H. (1994) J. Magn. Reson.109 A 203. Tanner, J.E. (1978) J. Chem. Phys. 69 1748. Balinov, B., Jonsson, B., Linse, P., Soderman, O. (1993) J. Magn. Reson. 104 A 17. Katz, A.J., Thompson, A.H. (1985) Phys. Rev. Lett. 54 1325. Borgia, G.C., Brown, R.J.S., Fantazzini, P. (1995) Phys. Rev. E 51 2104. Chachaty, C., Korb, J.-P., Van der Maarel, J.-P.C., Bras, W., Quinn, P. (1991) Phys. Rev. B. 44 4778. Filippov, A.V., Skirda, V.D. (2000) Colloid J. 62 759. Gibbs, J.W. (1928) The Collected Works of J. Willard Gibbs. New York: Academic Press. Thomson, (Lord Kelvin) W. (1871) Philosoph. Magazin 42 448. Strange, J.H. (1994) Nondestr. Test. Eval. 11 261. Jackson, C.L., McKenna, G.B. (1990) J. Chem. Phys. 93 9002. Kimmich, R., Stapf, S., Maklakov, A.I., Skirda, V.D., Khozina, E.V. (1996) Magn. Reson. Imaging 14 793. Valiullin, R.R., Furo, I., Skirda, V.D., Kortunov, P. (2003) Magn. Reson. Imaging 21 299. Alnaimi, S.M., Strange, J.H., Smith, E.G. (1994) Magn. Reson. Imaging 12 257. Hansen, E.W., Stocker, M., Schmidt, R. (1996) J. Phys. Chem. 100 2195. Song, Y.-Q. (2000) Phys. Rev. Lett. 85 3878. Song, Y.-Q. (2003) Concepts in Magn. Reson. 18A 97. Song, Y.-Q., Lisitza, N.V., Allen, D.F., Kenyon, W.E. (2002) Petrophysics 43 420. Le Doussal, P., Sen, P.N. (1992) Phys. Rev. B. 46 3465. Seymour, J.D. (1997) J. of AIChE 43 2096. Callaghan, P.T. (1999) J. Magn. Reson. 11 181.

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31. Callaghan, P.T., Coy, A., Macgowan, D., Packer, K.J. (1992) J. Molec. Liq. 54 239. 32. Maklakov, A.I., Skirda, V.D., Fatkullin, N.F. (1990) Self-Diffusion in Polymer Systems, Encyclopedia of Fluid Mechanics, N. Cheremisinoff, Editor., Gulf Publ. Co.: Houston. p. 705. 33. Timofeev, D.P. (1962) Kinetics of adsorption, Moscow, Academy of Sci. USSR. 252. 34. Alexandrov, G.G., Larionov, O.G., Chmutov, K.B. (1973) Kinetics and Dynamics of Physical Adsorption. Moscow, Nauka. 200. 35. Hartly, G.S., Crank, J. (1949) Trans. Faraday Soc. 45 801. 36. Dvoyashkin, N.K., Skirda, V.D., Maklakov, A.I., Belousova, M.V., Valiullin, R.R. (1991) Appl. Magn. Reson. 2 83. 37. Orazio, F.D., Bhattacharja, S., Halperin, W.P. (1989) Phys. Rev. Lett. 63 43. 38. Maklakov, A.I., Dvoyashkin, N.K., Khozina, E.V., Skirda, V.D. (1995) Colloid J. 57 55. 39. Dvoyashkin, N.K., Maklakov, A.I. (1993) Colloid J. 55 96. 40. Kaerger, J. (1969) Annalien der Physik 24 1.

VIBRATIONAL SPECTROSCOPY TO MONITOR SYNTHESIS, ADSORPTION AND DIFFUSION IN MICRO- AND MESOPOROUS METAL PHOSPHATES S. KENANE, C. S. VASAM and P. P. KNOPS-GERRITS* Département de Chimie, Université Catholique de Louvain (UCL), Batiment LAVOISIER, Place L. Pasteur n°1, B-1348 Louvain-laNeuve, Belgium

Abstract Diffusion of poly-atomic or mono-atomic guest species within confined media has been understood to a good degree due to investigations carried out during the past decades. Ordered mesoporosity (porosity) offers good adsorption and diffusive transport properties for guest species present in the porous materials. Here, we report two methods of making ordered mesoporous aluminophosphates (AlPO4s) and iron phosphates (FePO4s) materials. The methods comprising the steps of: batch preparation, conventional hydrothermal or microwave crystallization, product recovery and calcinations. The effect of microwave irradiation on AlPO4s and FePO4s has been studied in comparison with classical heat treatment. The thermal behaviour and the structural transformations of the AlPO4s and FePO4s were investigated by X-ray powder diffraction and the adsorption-desorption properties of these samples have been analysed. The sample morphology was carried using scanning electron microscopy (SEM). The crystallization process of these metal phosphates has been observed by collecting the micrographs using confocal Optical and Raman microscopy. 1. Introduction The study of diffusion of organic molecules confined porous materials is of interest in petroleum industry and in the field of catalysis in particular and also in other relevant areas[1-4]. Significant efforts have been devoted to understand diffusion in zeolites, since transport properties play a central role in catalytic and separation process using ordered porous zeolites. A broad variety of microporous and mesoporous materials are used in catalytic process in which the selectivity and activity can depend significantly on the diffusion properties. 279 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 279–298. © 2006 Springer. Printed in the Netherlands.

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The mesoporous materials are very promising model systems for fundamental adsorption studies of organic molecules confined in regular pore structure. These materials are present in unidimensional and uniform pore systems in the range of 16-200 Å.

Figure 1. SEM photographs of lamellar AlPO4s obtained by conventional hydrothermal synthesis.

In the development of zeolite science, infrared and Raman spectroscopy are major tools for structure and reactivity characterization [5, 6]. In comparison with this vast amount of IR work, the field of zeolite and alumino-phosphate Raman spectroscopy seems rather underdeveloped. Reasons for the modest use may be the considerable difficulty obtaining Raman spectra with acceptable signal-to-noise ratios from highly dispersed materials such as zeolites. The Raman effect is intrinsically a weak phenomenon, and spectra are often obscured by fluorescence [7, 8], which can be overcome by varying the wavelength of the laser used for excitation and the generally lower sensitivity compared to IR-spectroscopy. Nevertheless, it appears that Raman microscopy

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analysis has not yet been exploited in this context to its full extent. The number of applications reported in the literature is presently only about few per year [9].

Figure 2. SEM photographs of lamellar AlPO4s obtained by microwave heating.

However, with the advent of micro-Raman spectroscopy (or Raman microspectroscopy) and due to recent major advances in low-light detection systems, in monochromators and in lasers tunability, Raman spectra may now be recorded with adequate sensitivity to allow the definitive characterization of different molecular compounds present within a microscopic-sized sample. Furthermore, this sensitivity can be greatly improved by coupling it to confocal microscopy, so as to visualize the spatial distribution of a target species [10-12]. Contrary to conventional microscopy, where the entire field of view is illuminated and observed uniformly, confocal microscopy allows focusing with

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a resolution close to the diffraction limit through spatial filtering by optically conjugate pinhole diaphragms. This provides a way to isolate a micrometricsized region of a sample, coincident with the illuminated spot, so that the Raman spectra originating from this precise region can be taken without significant contributions from out-of-focus zones. Thus transparent samples can be sectioned optically and a given species can be tracked within micrometric sized volumes, so that the spatial distribution of several molecular compounds present in a whole sample may be identified simultaneously and their concentration profiles mapped individually at the micrometric scale. The sensitivity of confocal Raman microspectrometry can be enhanced greatly by using resonance Raman scattering (RRS) so that even very dilute species concentrations can be mapped with a micrometric resolution. Indeed, though scattering cross sections for usual Raman are usually very small, they may be increased up to a million times by adjusting the wavelength of the incident laser light close to an electronic absorption band of a species, so that the vibrations bands of a target molecule most closely associated to this electronic transition are enhanced. The objective of this research is to give an overview of the existing Raman microscopy literature on MePO4s (Me: Al, Fe) framework vibrations, MePO4s synthesis (Hydrothermal or microwave), adsorption on MePO4s characterized by Raman microscopy. Our investigation follows the controling of the ordered mesoporosity in mesostructured materials. We study precisely the effect of the surfactant packing parameters on crystallization, shape and pore diameter of the pores. These systems are important for fundamental adsorption studies.

2. Experimental 2.1. MATERIALS AND METHODS 2.1.1. Aluminophosphate synthesis The synthesis mixture has the following molar ratio: 0.5 Al2O3 : H3PO4 : Q: 2TMAOH : 25ROH : 65 H2O. The source materials are distilled water, isopropanol or isobutanol (ROH :15 ml), Al2O3 Aluminum oxide) ALDRICH, TMAOH (Tetramethylamonium hydroxide pentahydrate) 97% ALDRICH, Q: di-[N,N dimethyldodecyl] diamino-para-xylene and H3PO4 (phosphoric acid) 85% ACROS. The synthesis procedure is as follows: di-[N,N dimethyldodecyl] diamino-para-xylene (synthesized in our laboratory referred as Q in the later sections), tetramethylammonium hydroxide (TMAOH, 97% ALDRICH), 85% H3PO4 and water were mixed for several hours until a clear solution was obtained. Aluminium oxide was dissolved in isopropanol (15 ml) and was added to this clear solution under vigorous stirring, and the stirring was

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continued up to 24 h. The starting mixture was synthesized at various temperatures using classic hydrothermal synthesis or microwaves heating. 2.1.2. Ironphosphate synthesis Fe(NO3)3. 9H2O (Acros-Organics) was used as iron source, H3PO4 (85%) (Acros-Organics) and P2O5 (Merck) were used as phosphorous sources. Q: di[N,N dimethyldodecyl] diamino-para-xylene, Tetramethylammonium iodide and Tetramethylammonium hydroxide (Aldrich) were used as structure directing agents. Mesostructured iron phosphate was obtained from the molar gel compositions by conventional hydrothermal method: Fe(NO3) 3: H3PO4: 0.125 Q: 0.5TMAOH: 100H2O In a typical synthesis 4.04g of iron(III)nitrate (10mmol) was combined with 1.15g of ortho phosphoric acid (10mmol) diluted with deionised water. The mixture was continuously stirred for 1 hour to obtain a solution A. solution B was obtained by dissolving 0.75g of Q (1.25mmol) along with 0.91 g of TMAOH (5mmol) in deionised water under stirring for 1 hour. Finally solution A and B were mixed and stirred for 6 hrs at room temperature to get a homogeneous mixture. 2.1.3. Conventional hydrothermal synthesis The starting mixtures (AlPOs and FePOs) were synthesized at various temperatures using classic hydrothermal method in the range of 50¯200°C for 1¯15 days. The reaction mixture was transferred in to PTTF container, filled to a maximum of 80%, sealed and placed into a hot air furnace. 2.1.4. Microwaves heating The same recipe of AlPOs, which is used for conventional hydrothermal synthesis, has been crystallized under microwave heating. A total processing time of 10-20 mn by exposing the reaction mixture to a microwave electromagnetic field under atmospheric pressure was found to be sufficient for the completion of the synthesis. In the synthesis of FePOs, phosphorous pentoxide was used as a phosphorous source in the place of phosphoric acid. The strong effect of microwave radiation has been used to progressively reduce the water content in the formulation, up to 86.9 mol%, with a 30% (in weight) yield in dried product. 2.1.5. Solid and solution separation After the conventional or microwave reaction, the solid and solution products were separated by centrifugation. The solid reaction products were washed with deionised water several times to remove all entrained salts, and then dried in an oven at 50°C. The dried solids were subsequently characterized by powder XRD, SEM and adsorption-desorption techniques.

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2.1.6. Synthesis of microstructured Iron phosphate Micro structured iron phosphate was obtained from the molar gel compositions by hydrothermal method: Fe(NO3) 3: H3PO4: TMAI: 4TMAOH. In a typical synthesis 4.04g of iron(III)nitrate (10mmol) was combined with 1.15g of ortho phosphoric acid (10mmol) diluted with deionised water. The mixture was continuously stirred for 1 hour to obtain a solution A. solution B was obtained by dissolving 2g of TMAI(10mmol) along with 7.2 g of TMAOH (40mmol) in deionised water under stirring for 1 hour. Finally solution A and B were mixed together by the slow addition and stirred for 6 hrs on a hot plate. 2.2. INSTRUMENTATION The prepared samples were characterized by several instrumental analysis techniques. 2.2.1. X ray diffraction and scanning electron microscopy (SEM) X-ray powder diffraction (XRD) patterns were obtained using a Philips 1710 powder diffractometer with Cu-KD radiation (40 kV, 40 mA), 0,01° step size and 3-5 s counting time by step. Direct observation of the particle morphology was made possible using a high-resolution scanning electron microscopy (SEM). The images were obtained on Hitachi-8100A operated at 3 KV. 2.2.2. Adsorption-desorption technique Nitrogen adsorption-desorption measurements were carried out using the volumetric method with a Coulter SA3100 sorptometer at –196°C. Before analysis, calcined samples were dried at 120°C and evacuated for 8 h at 200°C. The surface area was calculated on the basis of the adsorption branch in the 0.05-0.2 partial pressure (p/p0) range. Pore volume was determined from the amount of N2 adsorbed at p/p0=0.99 while the pore diameter (DBJH) was estimated from the peak position of a BJH pore size distribution curve from the desorption isotherm. The mesoporous volumes (V0.99) were estimated from the amount adsorbed at the relative pressure of 0.99, assuming that the mesopores had been fully filled at this relative pressure. 2.2.3. Raman microscopy The Raman spectra and micrographs are recorded on a Renishaw Raman Microscope. The samples (MePOs), in powder form, were placed on a glass microscope slide. The power on the sample was about 2 mW/mm2. The collection time varied from sample to sample and was between 15 and 20 minutes. Raman micrographs of FePO-TMA crystal growth was collected by 3 magnifications (x5, x10x and x10) and are shown in the following pages.

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Figure 3. Optical micrographs of FePO4s crystals growth with tetramethylammonium ions (magnification u5, u10, u50) obtained by confocal Raman microscope. (a-p) The initial crystal growth comprised of the development of small and long thin needles.

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Figure 4. Optical micrographs of FePO4s crystals growth with tetramethylammonium ions (magnification u5, u10, u50) obtained by confocal Raman microscope. (a-h) The crystal growth, after 10 days, shows that crystals have fused together to form large well shaped octahedral type particles.

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Figure 5. Optical micrographs of FePO4s crystals growth with tetramethylammonium ions (magnification u10, u50) obtained by confocal Raman microscope. (a-f) The final crystal growth, after 14 days, shows a change in to either much larger crystals or a change in the morphology from the second stage to obtain aggregates.

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3. Results and Discussion 3.1. CRYSTAL GROWTH EXAMINATION OF FePO4-TETRAMETYLAMMONIUM by RAMAN MICROSCOPY The characterization of iron(III) phosphate crystals formed with Tetramethylammonium ion was studied by using Raman Microscope. After the heat treatment at 80O C for 6 hours, the crystal growth was measured in three stages. During the measurements in all the cases the soft material hardened upon cooling. The crystal morphologies studied in three different stages under confocal Raman microscope are shown in the Fig. 3, Fig. 4 and Fig. 5, respectively. The initial growth (Fig. 3) was measured after 6 days of the synthetic reaction. The second (Fig. 4) and final growth (Fig. 5) was measured after 10 and 14 days, respectively. The micrograph data show that the material undergoes a series of crystal transitions. The initial growth resulted in the formation of both small and long thin needle type crystals. Where as the micrographs collected for the crystal growth in the second and final stage show that the crystals grown in the first stage are got fuse together to give a dominant morphology, forming well-shaped octahedral type crystals. It has been anticipated that these transitions must have occurred due to the basic pH and high ratio between the structure directing agent and iron source. And consequently, giving a change in the packing parameter in the crystal growth. The involvement of the organic template in the building of microstructured iron phosphate is evidenced from the appearance of strong band corresponds to Q (C-N) and Q(C-H) along with the P-O vibrations in the Raman spectra. It is further supported by X-ray data corresponding to the formation of microstructured iron phosphate by the templating action tetramethyl ammonium cation. 3.2. X RAY DIFFRACTION 3.2.1. XRD of AlPOs The XRD patterns of the aluminophosphate product synthesized under hydrothermal heating is shown in Fig 6. Lamellar mesophase as described by Beck et al. [13] was identified by XRD as shown in Fig. 6, which also contains the d-spacing of the reflection peaks. It has seen that the as-synthesized product can be indexed according to a lamellar lattice [14]. The appearance of a small peak with a d-spacing of 1.1 nm shows the existence of a small portion of hexagonal phase, indicating that the lamellar phase was probably transferred from the hexagonal phase as in the case of aluminosilicate system [14]. We have synthesized aluminophosphate and ironphosphate under microwaves heating using the same recipe followed in the case of

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Vibrational Spectroscopy to Monitor Synthesis, Adsorption

aluminophosphate and ironphosphates synthesized hydro thermally. The XRD patterns of the as-synthesized product Fig. 2 indicate a lamellar structure of the as-synthesized product, with interlamellar spacing of the peaks at d-spacing of ~ 12 nm. The first three intense (00l) peaks (l=1, 2, 3) are evident and the phase gives no higher angle peaks. 3.2.2. XRD of FePOs A sequence of reactions was performed to study the influence of surfactant as a structure-directing agent. The preliminary indexed powder XRD patterns of assynthesized FePOs prepared by conventional hydrothermal and microwave assisted novel dry synthetic method is shown in Fig.7. The as-synthesized compounds templated by Gemini surfactant [Q] along with TMAOH have been displayed low-diffraction angles with interlamellar spacing of the peaks at dspacing of ~3.6(HT) and 2.9(MW) nm, which reflects the formation.

Intensity (a.u.)

2.4 nm

10 nm

1.2 nm

1.1 nm

0

2

4

6

8

10

2*theta Figure 6. XRD pattern of lamellar aluminophosphate – Hydrothermal synthesis.

Intensity (a.u.)

12 nm

3,2 nm 2,4 nm

0

1

2

3

4

5

2*theta

Figure 7. XRD pattern of an iron phosphate – Hydrothermal synthesisof meso-lamellar structured materials. The first three intense (001) peaks (1=1,2,3) are evident and phase gives no higher angle peaks. From XRD data it is clear that the surfactant is playing a crucial role in the construction of FePO4s framework of the present report.

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In the present case Gemini surfactants, with two quaternary ammonium head groups separated by a P-xylene methylene chain attached to hydrophobic tail, can be used to control organic charge sitting relative to the bivariable hydrophobic tail configurations. This approach has led to the synthesis of mesophases of several frameworks. The tetramethylammonium hydroxide does influence the crystallinity in terms of its basic nature. The mechanism of surfactant action in the construction of FePO4s framework is not clearly understood. To our surprise, removal of surfactant from the FePO4s by extraction followed by calcinations at 250°C, resulting in the phase transformation from lamellar to hexagonal. 3.3. MICROSCOPY ANALYSIS Direct observation of the morphology of the aluminophosphates product was made possible using a high-resolution scanning electron microscopy. Depending on the synthesis conditions, various morphologies occurred, the samples consisting mostly of polycrystalline aggregates. The SEM pictures of the samples are shown in Fig. 5. The AlPOs prepared by conventional heating have a sheet-like morphology and are aggregated as “sand rose”. The AlPOs crystals obtained under microwave heating also showed a sheet-like morphology, but the crystals were separated, forming small cotton-ball-like aggregates as shown in Fig. 6. It appears that the microwave heating gives rise to homogeneous nucleation, and leads to a different aggregation mode of AlPOs has a sheet-like morphology. AlPOs synthesized using microwaves was obtained as spherical aggregates of platelet crystals of about 2-4 µm, as shown in Fig. 6. Similar type of morphology is observed for ironphosphate. 3.4. ADSORPTION PROPERTIES The nitrogen adsorption–desorption isotherms of AlPOs and FePOs have been measured at liquid nitrogen temperature. The isotherm for both aluminophosphates and ironphosphate was of typical Type IV (Langmuir type) for mesoporous solids Fig. 8, and the Langmuir surface area was determined to be 150 m2/g for AlPOs and 116 m2/g for FePOs. Both of MePOs synthesized hydrothermally and crystallized using microwave heating showed a small specific surface area of Dib , i.e. molecule-molecule collisions dominate).

DiK and Dib can be presented in terms of a tortuosity factor, which accounts for

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the differences between these diffusivities and the corresponding reference diffusivities

1

DiK

D0 K ,

KK

Dib

1

Kb

D0 b ,

(10)

where K k , D0 K and K b , D0b are the tortuosity factor and the reference diffusivities in the Knudsen and in the bulk regimes, respectively. The reference diffusivity in the Knudsen regime is usually defined as that in straight parallel non-overlapping cylindrical pores of infinite length with a diameter equal to the mean intercept length ( d ) in the considered real pore structure

1 ud , 3

D0 K where u

8RT SM

1/ 2

(11)

is a mean thermal speed, R is the gas constant, M is the

molar mass and T is the absolute temperature. The reference diffusivity in the bulk regime is given by

D0 b

1 uO , 3

(12)

where O is the molecular mean free path. Most recent experimental results have shown that the effective tortuosity factor in the Knudsen regime may be significantly larger than that in the bulk regime for one and the same porous medium [11]. This finding may be rationalized by noting that, in contrast to the bulk regime, the momentum exchange in the Knudsen regime proceeds for the most part between the gas molecules and the pore walls. Therefore the diffusion resistance imposed by the walls may also be larger in the Knudsen regime. The results of a number of recent dynamic Monte Carlo simulations of gas diffusion in porous media [12,13] have revealed the same tendency as observed experimentally in [11]. References 1. 2.

Kärger, J., Ruthven, D.M. (1992) Diffusion in Zeolites and Other Microporous Solids; Wiley & Sons, New York. Kimmich, R. (1997) NMR Tomography, Diffusometry, Relaxometry; Springer, Berlin.

PFG NMR Diffusion Studies of Nanoporous Materials 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

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Kärger, J., Heitjans, P., Haberlandt, R. (1998) Diffusion in Condensed Matter; Vieweg, Braunschweig/Wiesbaden. Kärger, J., Vasenkov, S., Auerbach, S. (2003) Handbook of Zeolite Science and Technology, S.M. Auerbach, K.A. Carrado, P.K. Dutta (Eds.), Diffusion in Zeolites, Marcel Dekker, New York. Callaghan, P.T. (1991) Principles of NMR Microscopy; Clarendon Press, Oxford. Cotts, R.M., Hoch, M.J.R., Sun, T., Markert, J.T. (1989) J. Magn. Reson. 83 252. Vasenkov, S., Galvosas P., Geier, O., Nestle, N., Stallmach, F., Kärger, J. (2001) J. Magn. Reson. 149 228. Kärger J., Pfeifer H. (1992) Zeolites 12 872. Kärger, J. (1991) J. Phys. Chem. 95 5558. Kärger, J., Kocirik, M., Zikanova, A. (1981) J. Colloid Interface Sci. 84 240. Geier, O., Vasenkov, S., Kärger, J. (2002) J. Chem. Phys. 117 1935. Burganos, V.N. (1998) J. Chem. Phys. 109 6772. Vignoles, G.L. (1995) J. Phys. IV 5 159.

DIFFUSION OF CYCLIC HYDROCARBONS IN ZEOLITES BY FREQUENCY-RESPONSE AND MOLECULAR SIMULATION METHODS L. V. C. REES AND L. SONG School of Chemistry, University of Edinburgh, West Mains Road, Edinburgh EH9 3JJ, UK, [email protected]

1. Introduction The Frequency Response (FR) technique is a quasi-steady state relaxation technique in which a parameter influencing the equilibrium state of the system is perturbed periodically at a particular frequency. The response of a parameter characteristic of the state of the system depends upon the time scale of the dynamic processes affecting the parameter relative to the period of perturbation, the type of perturbation and physical characteristics of the system. The response of the system to the frequency spectrum thus allows the determination of the dynamic parameters. This technique was firstly used by Yasuda [1] to measure diffusion coefficients in gas-zeolite systems by applying a sinusoidal-wave perturbation to the equilibrium gas phase volume of the system. The present paper shows that the technique has been improved by the use of a square-wave volume perturbation, by the reduction of the response time of the pressure transducer measuring the changes in pressure which arise from the volume modulation, by automation of the apparatus and, finally, by an expansion of the frequencyrange [2]. The FR technique has proved to be a very effective and a very powerful method for determining inter- and intra-crystalline diffusivities of sorbate molecules in zeolites. An outstanding advantage of the FR method is its ability to distinguish multi-kinetic processes in an FR spectrum, i.e., various ‘independent’ rate processes which occur simultaneously can be investigated by this technique. 2. Theoretical In our frequency response method an equilibrium adsorption state is perturbed by applying a small square-wave modulation to the volume of a gas phase. The 383 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 383–413. © 2006 Springer. Printed in the Netherlands.

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theoretical solutions of the frequency-response technique have been comprehensively developed over the past decade [1-5]. The full FR parameters (phase lag and amplitude) are experimentally derived from a Fourier transformation of the volume and pressure square-waves. The phase lag ) Z-B = )Z - )B is obtained, where )Z and )B are the phase lags determined in the presence and the absence of zeolites, respectively. The amplitude is embodied in the ratio PB/PZ, where PB and PZ are the pressures response to the r1% volume perturbations in the absence and presence of sorbents, respectively. From the solution of Fick's second law for the diffusion of a single diffusant in a solid subjected to a periodic, sinusoidal surface concentration modulation, the following equations in-phase: out-of-phase:

( PB / PZ ) cos ) Z  B  1 KGin  S ( PB / PZ ) sin ) Z  B KGout

(1) (2)

can be obtained. K is a constant related to the gradient of the adsorption isotherm at the equilibrium pressure, S is a constant that represents a very rapid adsorption/desorption process, which may co-exist with the diffusion process being measured, Gin and Gout are the overall in-phase and out-of-phase characteristic functions, respectively, which depend on the theoretical models describing the overall kinetic processes of a system. The models available in the literature and used in this work are as follows. 2.1. SINGLE DIFFUSION PROCESS MODEL When only a single intracrystalline diffusion process occurs in a system, the characteristic functions are [1,4,5]

KGin

in-phase:

KGout

out-of-phase:

RTVs K P Gc Ve RTVs K P Gs Ve

(3) (4)

where R is the gas constant, T is the isotherm temperature, Vs is volume occupied by the sorbent, Ve the mean volume of sorbate outside the sorbent and Kp is the equilibrium constant based on pressure. For crystals of slab shape, Gc and Gs are given by [1]

Gc

1 sinh K  sin K ) ( K cosh K  cos K

(5)

Diffusion of Cyclic Hydrocarbons in Zeolites by Frequency-Response

Gs

1 sinh K  sin K ( ) K cosh K  cos K

385 (6)

For diffusion in spherical crystals, Gc and Gs are given by [1]

Gc Gs

3 sinh K  sin K ( ) K cosh K  cos K 3 sinh K  sin K 2 (  ) K cosh K  cos K K

(7) (8)

Where K (2Zl 2 D) 1 2 , Z is the angular frequency, f = Z / 2S = frequency, l is the half thickness of the slab or the radius of the sphere, and D the transport intracrystalline diffusion coefficient. The diffusion coefficient is obtained by a least-squares curve-fitting of the experimental characteristic functions versus frequency data, with the diffusion coefficient being the only adjustable parameter. 2.2. TWO INDEPENDENT DIFFUSION PROCESSES MODEL When two diffusion processes occur simultaneously, provided they are independent of each other, the theoretical treatment can be expanded to give [1,6] in-phase: KGin K1Gc ,1  K 2Gc ,2 (9)

KGout

out-of-phase:

K1Gs,1  K 2Gs,2

(10)

where subscripts 1 and 2 indicate the two separate kinetic processes. The characteristic functions Gc and Gs are also generated by Equations (5) - (8). 2.3. NON-ISOTHERMAL DIFFUSION MODEL Periodic adsorption and desorption inside adsorbent particles, induced by the volume modulation, may lead to a heat of sorption effect which is dissipated through a heat exchange between the sorbent and the surroundings. When the heat exchange rate is comparable with the diffusion rate, a bimodal form for the frequency response characteristic curves is found [7-9]. The overall characteristic functions Gin and Gout for this model are given by in-phase:

KGin

out-of-phase: KGout

RTVs K P Ve

§ Gc (1  Z 2 t h2 )  J (Gc2  Gs2 )Z 2 t h2 · ¨ ¸ (11) © (1  JGsZt h ) 2  (1  JGc ) 2 Z 2 t h2 ¹

RTVs K P § Gs (1  Z 2 t h2 )  J (Gc2  Gs2 )Zt h · ¸ (12) ¨ Ve © (1  JGsZt h ) 2  (1  JGc ) 2 Z 2 t h2 ¹

where th is the time constant for heat exchange between the sorbent and its

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surroundings, and

J

(13) is a measure of the non-isothermalicity of the system. Cs is the volumetric heat capacity of the sorbent, Qst is the isosteric heat of adsorption and KT is derived from the adsorption isotherm with respect to the temperature and defined by

KT

PQst RT 2

KT Qst Cs

(14)

KP

Gc and Gs are also given by Equations (5) and (6) respectively for slab-shaped crystals and Equations (7) and (8) respectively for spherical crystals. 2.4. DIFFUSION-REARRANGEMENT MODEL When Fickian diffusion occurs in transport channels, for example, the straight channels of silicalite-1 framework structure, and sorbate is immobilised in storage channels, e.g., the sinusoidal channels of silicalite-1, and a finite-rate mass exchange between these two kinds of channels exists [10] the overall characteristic functions derived by Sun et al. [7] are in-phase:

Gin

out-of-phase:

G out

G c  G s Zt R 1  Z 2 t R2 G G Zt G s  K : s c 2 2R 1 Z tR

Gc  K :

(15) (16)

For diffusion in slab geometry, Gc and Gs are given by

Oc sinh 2Oc  Os sin 2Os

Gc

Oc2  Os2 cosh 2Oc  cos 2Os

Gs



Os sinh 2Oc  Oc sin 2Os

 Os2 cosh 2Oc  cos 2Os For diffusion in spherical geometry, Gc and Gs are given by 3 § Oc sinh 2Oc  Os sin 2Os Oc2  Os2 · ¨ ¸  2 Gc Oc2  Os2 © cosh 2Oc  cos 2Os Oc  Os2 ¹

Gs

Oc2

§ Os sinh 2Oc  Oc sin 2Os 2Oc Os · ¸ ¨  Oc2  Os2 © cosh 2Oc  cos 2Os Oc2  Os2 ¹ 3

with

Oc  iOs

>Q

s

 iQ c Zl 2 D

@

12

(17) (18)

(19)

(20)

(21)

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387

and

Qc

K:

1

(22)

1  Z 2 t R2 K : Zt R

Qs

(23)

1  Z 2 t R2

tR is the time constant of mass exchange between the transport and storage channels and K: is the ratio of mass hold-up in the storage and transport channels. A bimodal FR spectrum is displayed with this model. 2.5. DIFFUSION WITH SURFACE-RESISTANCE OR SURFACE-BARRIER MODEL When surface-barriers or surface-resistance to sorbate gases or ‘skin’ effect occurs, the overall characteristic functions can be indicated by [4,11,12]

ak  A Z 2 a  cGc T ak  A Z >1  ak  A Z ^ak  A Z  cG s ` T @ Gin

in-phase: out-of-phase: G out

(24) (25)

where k  A is the rate constant for the resistance, Gc and Gs are as same as those in Equations (5) - (8) for slab and spherical crystals respectively, and

T

^ak

A

c { dC dP e

Z  cG s

`  ^a  cG ` 2

c

2

^d  A  C dP` e { 1  a

(26) (27)

Here A is the adspecies on the external surface, C is the adspecies within the

porous adsorbent and P the equilibrium pressure. a | 102 and c | 1 for most zeolites. The surface resistance can be demonstrated by the area of the intersection between the in-phase and the out-of-phase characteristic function curves as presented in Figure 1, which depends on the ratio of [ defined by

[ ak  A

D l 2

(28)

suggesting that the ‘skin’ effects depend not only on the ratio of the rate constants between the surface resistance and diffusion processes, but also on the size of crystals. A large value of [ means small ‘skin’ effects [12,13]. The limiting values of the in-phase component as Zo0 can be experimentally used to determined the equilibrium constants [14] since

KGin

Z 0

K

RTVs K P Ve

(29)

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If the experimental range of frequency is wide enough to cover all the rate processes occurring in a system, the asymptotes of the in-phase and out-ofphase characteristic curves should satisfy the following two relations [4]

lim  PB PZ cos ) Z  B  1 0 and lim  PB PZ sin ) Z  B

Z of

Z of

lim  PB PZ cos ) Z  B  1 K

Z o0

and

lim  PB PZ sin ) Z  B

0

(30)

RTVs K P Ve

0

Z o0

(31)

ξ=1 ξ=10 ξ=100

0.16

0.12

0.08

0.04

0.00 1E-3

0.01

0.1

1

10

100

Frequency / Hz Figure 1. Theoretical FR spectra for diffusion with a surface resistance, [ (see Eqn 28), and spherical crystal geometry with K = 0.15 and

D r2

01 . s-1.

The diffusion coefficients measured by the FR method are transport diffusivities, D. By applying a correction factor, the Darken factor, the selfdiffusivity, D0 can be determined by the Darken equation [15,16]

D0 Dw ln q w ln P (32) The Darken factor, w ln q w ln P , can be calculated from the models describing the adsorption isotherms. By applying the Arrhenius equation,

D ' exp  E a RT the activation energy of diffusion, Ea , can be determined. D0

(33)

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389

3. Experimental and Computational Procedures 3.1. FR METHOD The principal features of the FR apparatus developed by Rees et al. are shown in Figure 2. An accurately known amount of sorbent sample is scattered in a plug of glass wool and outgassed at a pressure of < 10-3 Pa and 623 K overnight by rotary and turbo molecular drag pumps (6). The temperature was raised to 623 K at 2 Kmin-1 using a programmable tube furnace. A dose of purified sorbate is brought into sorption equilibrium with the sorbent in the sorption chamber (8) at the chosen pressure and temperature.

Figure 2. Schematic diagram of the FR apparatus.1: Sorbate inlet; 2: Valve; 3: Electromagnets; 4: Moving disc; 5: Bellows; 6: Rotary and turbo drag pumps; 7: Computer with A/D and D/A card; 8: Adsorption vessel with zeolite in glass-wool; 9: Vacuum connectors; 10: Differential baratron; 11: Signal conditioner; 12: Reference pressure side

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A square-wave modulation of r 1% was then applied to the gas phase equilibrium volume, Ve. The modulation was effected by applying a current to each of the two electromagnets (3) in turn, which moves the disc (4) between the electromagnets rapidly (< 10 ms) and periodically. The brass bellows (5) attached to the disc, which is part of the sorption gas phase volume, was expanded and compressed to produce the r 1% change in volume. A frequency range of 0.001 to 10Hz was scanned over some 30 increments. The pressure response to the volume perturbation was recorded with a high-accuracy differential Baratron pressure transducer (MKS 698A11TRC) (10) at each step over three to five square-wave cycles (256 reading points per cycle) after the periodic steady-state had been established. The frequency was controlled by an on-line computer (7), which was also used for the acquisition of the pressure data from the Baratron transducer. The conversion rate of the analogue-to-digital converter in the interface unit must be fast enough to cope with the 1 to 4 ms response time of the pressure transducer. The pressure response to the volume change over the whole frequency range was measured in the absence (blank experiment) and presence of sorbent samples to eliminate time constants associated with the apparatus. The FR spectra were derived from the equivalent fundamental sine-wave perturbation by a Fourier transformation of the volume and pressure square-wave forms as discussed above. The silicalite-1 zeolite samples used in this study, silicalite-1 (A) and silicalite-1 (B), have been described in another paper presented to this conference in the adsorption section [17]. These samples were calcined, initially, at 823 K for 10 hours in an oven to remove the templating material. The crystals were heated from room temperature to 823 K at 2 K min-1. X-ray diffraction patterns and SEM micrographs showed that silicalite-1 (A) was highly crystalline and of near spherical shape. The spherical shape arose from the intergrowths of the more common coffin shaped crystals. Silicalite-1 (B) was comprised of small 4x3x4 µm3 cubes. The details of AlPO4-5 can be also found in the adsorption paper [17]. Benzene and cyclohexane were obtained from the National Physical Laboratory, UK; toluene, p-xylene, ethylbenzene and p-dichlorobenzene (pDCB) were supplied by Aldrich Chemical Company, Inc.; and cyclopentane, and cis-1,4- and trans-1,4-dimethylcyclohexane (c- and t-DMCH) were supplied by Fluka Chemie AG. All chemicals have purity of 99+%. 3.2. SIMULATIONS The temperature dependence of the equilibrium configurations of the sorbed molecules inside the channel framework is very helpful for a better understanding of the diffusivities of sorbate molecules. To obtain this information, the simulations of a single p-xylene molecule adsorbed in siliceous MFI zeolite, silicalite-1, were performed at different temperatures using the

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391

rapid Monte Carlo statistical simulation, the canonical ensemble Monte Carlo (fixed loading) simulation in which the Metropolis scheme was used. The simulations were started by choosing the initial coordinates of the sorbate molecule in a sinusoidal channel segment, a straight channel segment and an intersection of these two channels, respectively. Accelrys’ Cerius2 4.2 software was used to carry out the simulations on a Silicon Graphics workstation. Initially, high-energy configurations were rejected to save computational time during the simulations. These configurations are those in which the sorbate molecules and the framework are very close together; i.e., the distance between the atoms of the sorbate and the framework is less than half their van der Waals radii. The van der Waals energies between the sorbate and the framework were calculated by summing all pair interactions within a specific volume, in which the radius is determined by a cut-off distance. The van der Waals energy term within the zeolite framework is restricted to the minimum image convention, in which an atom is considered to interact with its closest neighbour atoms in a periodic box around it. In the case of sorbate-sorbate energy, the interactions are not limited to the atoms within the minimum image border but to the molecules whose centres of mass are within it. Both the aromatic molecules and the zeolite framework were treated as rigid units. However, introduction of framework flexibility into the calculations has been shown to affect the diffusivity. This effect, however, does not alter the equilibrium configurations calculated for the sorbed molecules in the system [18-21]. The only permitted degrees of freedom are the three translational and three rotational variables associated with the sorbate molecule. Electrostatic interactions were not included in all of the simulations due to the very low loading and the pure silica framework involved. The unit cell of silicalite-1 used has an orthorhombic Pnma space group with a = 20.022 Å, b = 19.899 Å, c = 13.383 Å containing 96 silicon and 192 oxygen atoms. The simulation box was defined as two crystallographic unit cells to which periodic boundary conditions were applied in order to simulate the infinite zeolite structure. The various Monte Carlo step sizes for the simulations were adjusted in order to obtain a fifty percent acceptance probability. The interaction cutoff distance was fixed at 9.9 Å, which is slightly less than half the smallest parameter for the simulation cell, so it accounts for all of the necessary interactions. At least 4 million simulation steps were performed and the equilibration of the system was monitored by measuring the configuration energy as a function of Monte Carlo steps, which should exhibit a small fluctuation around a central value after sufficiently long runs. The Sorption Demontis [22], Burchart-Dreiding [23,24] and PCFF [25] Force Fields were applied in the simulations, respectively. For the Sorption Demontis forcefield, the sorbate-sorbate interactions are described using Buckingham potentials. Partial charges are applied to the hydrogen and carbon

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L.V.C. Rees and L. Song

atoms of the aromatic molecules to calculate the electrostatic interactions. The sorbate-zeolite interactions take into account the short-range atom-atom interactions between the aromatic compound and the oxygen atoms of the zeolite. The short range interactions with the silicon atoms are neglected because they are well shielded by the oxygen atoms of the SiO4 framework tetrahedra. The Burchart-Dreiding forcefield combines the Burchart forcefield which prescribes the host framework and the Dreiding II forcefield which treats the guest molecule. The parameters for the framework-molecule interactions are derived from the parameters of both forcefields combined by the arithmetic combination rule. The van der Waals interactions in the Burchart forcefield are expressed by an exponential-6 term and the electrostatic interactions by partial atomic charges and a screened Coulombic term. In the Drieding II forcefield, the Lennard-Jones potential is applied to describe the van der Waals interactions and the electrostatic interactions are described by atomic monopoles and a screened Coulombic term. PCFF was developed based on CFF91 with parameterisation including the functional groups of zeolites. The van der Waals interactions in this forcefield were calculated using an inverse 9th-power term for the repulsive part rather than the more conventional Lennard-Jones model. In order to understand the significant influence of the sorbate molecules on the diffusivity in silicalite-1, the energy minimisation and molecular dynamics methods supplied in the Cerius2 4.2 software were applied to calculate the minimum energies of benzene, p-xylene, cyclohexane, c- and tDMCH down the straight channels of silicalite-1 by using the Burchart-Dreiding forcefield. The initial position of these sorbate molecules were in the middle of one straight channel segment. The framework structure and the dimension of the simulation box are the same as those used in the sorption calculations described above. During the simulations, the framework is assumed to be fixed, whereas all the atoms of the sorbate molecules are flexible. The sorbate molecule is forced to diffuse stepwise along the straight channel axis at steps of 0.2 Å. At each step, the energy minimisation was followed by a quenched molecular dynamics in which 5 short dynamic runs (0.1 ps, 300 K) were applied and the system was reminimised after each dynamic run. The minimum energies at each step were recorded to reflect on different orientations and the internal degrees of freedom of the sorbate molecules inside the channels. The periodic boundary conditions were also applied for these simulations. Molecular dynamics (MD) was also used to simulate the diffusivities of benzene, p-xylene and cyclopentane in silicalite-1 framework. These simulations were carried out by using the Materials Studio software supplied by Accelrys. For benzene and p-xylene, the same framework structure and simulation box as described above were, again, applied. During the simulations, however, both the framework and the sorbate molecules were allowed to

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393

vibrate. Only one benzene or one p-xylene molecule was loaded in the simulation box and, in the case of p-xylene, initially located at one of the sinusoidal channel segments. For cyclopentane, eight crystallographic unit cells were employed with a loading of 2 molecules per unit cell (m./u.c.). During the simulation the framework was assumed rigid. PCFF Force Field was selected and both van der Waals interactions and electrostatic interactions were included in non-bond interactions with summation methods of atom-based and Ewald for these two kinds of interactions respectively. The interaction cutoff value was set to 12.5 Å with a spline width of 3.0 Å and a buffer width of 1.0 Å. For the coulomb term calculations, a dielectric constant of 1.0 was used. The constanttemperature, constant-volume ensemble (NVT), i.e., the canonical ensemble, was specified in the simulations. All the simulations had the periodic boundary conditions applied. The time step used in MD runs was 1.0 fs with total simulation time of 1.0 to 1.8 ns. Prior to the MD runs, an energy minimisation and an equilibration stage with a simulation time of 10 ps were utilised. 4. Results and Discussion The influence of subtle differences in molecular shape and size on the diffusivities of the sorbate molecules in silicalite-1 can be clearly demonstrated in Figures 3 to 8 and Table 1 which present the FR diffusivities of all the sorbate molecules, involved in this study. It can be seen that the mass transfer of benzene and ethylbenzene in MFI framework is mainly controlled by a pure, single diffusion process at loadings < 4 m./u.c.. At high loadings, however, a pronounced bimodal FR spectrum was observed for benzene in silicalite-1 with a new peak appearing in the higher frequency range than that at low loadings. Analogous to the diffusion behaviour of benzene, two distinct peaks were also found in the FR spectra of cyclopentane in silicalite-1 at loadings higher than 4 m./u.c.. Nevertheless, unlike the FR spectra for benzene, the FR spectra for cyclopentane at high temperatures also display a bimodal behaviour even though the two peaks are not as apparent as those found at low temperatures. For p-xylene and toluene, the FR results presented in Figures 4 and 5 exhibit a similar interesting phenomenon which is difficult to detect by other techniques. At low temperatures, single-peak spectra are observed at loadings generally > 4 m./u.c., while at low loadings a bimodal behaviour appears. At high temperatures, however, only a pure, single diffusion process can be, once again, detected. The diffusivities of the four aromatics decrease in the order of p-xylene > toluene > benzene > ethylbenzene as presented in Figure 9 and Table 1. The diffusion coefficients of cyclopentane in silicalite-1 are in the same order of magnitude as those for benzene. The saturated hydrocarbons diffuse much more slowly within the channel framework of silicalite-1 than their aromatic equivalents and only one single

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L.V.C. Rees and L. Song

diffusion process can be observed. As found between p-xylene and benzene, tDMCH diffuses at least one order of magnitude faster than cyclohexane. The cDMCH, on the other hand, diffuses more slowly than cyclohexane. These findings are in agreement with the results obtained from the simultaneous thermal analyzer and the Zero Length Chromatography techniques [26]. With molecular shape and size close to p-xylene, a long rigid molecule, p-DCB displays a FR behaviour in silicalite-1 very close to that of p-xylene. All these Table 1. Diffusion coefficients of the cyclic hydrocarbons in silicalite-1 measured by the FR method. Silicalite-1 (A) Silicalite-1 (B) 

Sorbate

T (K)

benzene

273 303 373 323 373 415 373 415 323 373 473 253 273 373 423 398 398 448

toluene c

EB

P-xylene d

e

CP

CH c-DMCH g t-DMCH

f

a

13 b

n D01u10 (m./u.c.) (m2s-1) a

7.1 1.1 4.9 3.9 -

7.4 160 210 -



13

a n D02u10 D01u1013 bD02u1013 (m2s-1) (m./u.c.) (m2s-1) (m2s-1)

1.56 2.72 15 -

5.8 5.6 1.1 4.1 2.0 0.5 2.9 0.81 4.0 3.3 3.8 3.6 0.64 -

19.4 14.6 0.35 9.6 13.4 1.02 7.04 10.9 37.2 122 14.3 9.4 0.06 activated diffusion.

c)

d)

Figure 8. Sketch of the selective layer on carbon molecular sieve membranes produced by Carbon Membranes Lda. a) after pyrolysis of the precursor; b) after CVD; c) after activation with oxygen and d) regions of ultra-microporosity.

New Trends on Membrane Science

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Figure 9. Sketch of a honeycomb-like carbon molecular sieve membrane module (adapted from an original by Blue Membranes Gmbh).

CMSM in general show very promising characteristics. The most commonly mentioned is the combination of high permeabilities and selectivities. For example, an O2/N2 selectivity of 30 and a oxygen permeability of 1.19x10-15 mol m Pa-1 m-2 s-1 has been reported for lab-made membranes [Shiflett and Foley, 1999]. In addition, the pore aperture can be nearly continuously tuned [Koresh and Soffer, 1980], so that a membrane module can be specifically designed for a given gas separation. These membranes can also operate with highly aggressive chemicals, such as hydrogen chloride, and at high temperatures. The CMSM do not swell such as happen in many cases with polymeric membranes and the selectivity is more stable with temperature. Some unresolved problems for this type of membranes are the deactivation that they suffer with time in the presence of oxygen and the blockage effect observed when a larger molecule with higher adsorption strength blocks the diffusion of smaller ones. For example, by adsorbing on a hydrophilic site (e.g. with a chemically adsorbed oxygen atom), water vapor can block a pore. The performance characteristics of CMSM are also difficult to reproduce when consecutive batches of membranes are produced. Finally, even though CMSM can withstand high-pressure differences, they are also brittle, which complicates the assembly of commercial modules. Up to now, most authors have concentrated their efforts on the development of new strategies for producing CMSM, performing most of the times basic characterization work. This involves measurement of monocomponent permeabilities and ideal selectivities and a simple morphologic characterization. In same cases, authors also provide experimental data on adsorption uptake curves and densities. Few attempts were made for modeling the monocomponent mass transfer and no attempts were made to model multicomponent transport. There is nearly no multicomponent permeability data

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A.M. Mendes, F.D. Magalhães and C.A.V. Costa

available. This portrait evidences how the development and characterization of this new family of promising membranes is yet in its infancy. Koresh and Soffer were among the first researchers to study the CMSM. They performed a series of experiments with membranes activated by degassing at progressively higher temperature under vacuum or oxidation at 400 to 450°C in presence of air. The CVD procedure was not used in these experiments. They realized that the mass lost in the activation should imply larger pores than the actually observed from the size of the largest molecule adsorbed [Koresh and Soffer, 1980]. It was therefore inferred that, even though the activation process widens the pore throughout its length, localized constrictions of lower diameter remain. This series of “necks” along the pore actually determines the molecular sieving characteristics of a particular membrane. This picture is not too different from the cage/window structure that characterizes zeolite channels, except for the fact that the spatial regularity that exists in crystalline lattices is absent in CMSM. An added complexity is introduced when the CVD procedure is used. The subsequent activation affects the thin CVD adlayer (opening a system of ultramicropores across it) and the pore system present in the precursor (resulting in the aforementioned “pore with constrictions” morphology). The implications of these proposed pore structures in the modeling of mass transfer will be discussed later. Five issues are essential in the characterization of carbon molecular sieve membranes: permeability/selectivity, long time stability, rigidity/brittleness, blockage effects and membrane module configurations. These will be discussed next. 8.1. PERMEABILITY/SELECTIVITY During the pyrolysis or carbonization process, when a polymeric precursor is transformed in a carbon membrane, heteroatoms are progressively removed leaving a carbon skeleton. This frame determines the mechanical properties of the membrane and, if a selective layer is not added in a subsequent step, it is also responsible for the separation performance. Some authors suggested that a faster removal of the pyrolysis gases improves the permeability and selectivity [Ismail and David, 2001]. Others claim that oxygen may be used for improving the permeability, leaving the selectivity untouched [Koresh and Soffer, 1980, Ismail and David, 2001]. A reductive atmosphere of hydrogen at a controlled high temperature can also be used during the activation of the CMSM or during a possible re-activation. The permeability of a carbon membrane should be determined by the total number of connecting pores (pores that connect the two external faces of the membrane) and the diameter and frequency of the constrictions along these

New Trends on Membrane Science

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pores. The selectivity, on the other hand, should be related not only to the diameter and frequency of constrictions, but also to their size distribution. In principle, the overall mass transfer properties of a CMSM can be varied nearly continuously through the activation process. A better structural characterization of the pore system present in these materials and a better understanding of how this microporosity is related to the activation process and to the nature of the membrane precursor are needed in order to guarantee efficient membrane synthesis procedures. 8.2. LONG TIME STABILITY UNDER MILD OXIDATIVE ASTMOSPHERES One of the most foreseen applications for the CMSM is air separation, mainly for producing enriched nitrogen streams [Ismail and David, 2001]. Some authors reported extremely high O2/N2 selectivities, α O2 N 2 = 42.7 , with significant O2 permeability, 5.32x10-15 mol m Pa-1 m-2 s-1, and even N2/Ar selectivity, α N 2 Ar = 2.0 [Wang et al., 2003], while others report no N2/Ar selectivity [Koresh and Soffer, 1980]. These good performances allow us to dream with very efficient membrane modules and cheap in situ production of high purity nitrogen. However, a fundamental question is raised: are the membranes stable with time? Some authors claim that they are [Ismail and David, 2001] while others report significant deactivations after exposure to oxygen [Pontes et al., 2003]. On the other hand, the carbon molecular sieve granular adsorbents produced, for example, by CarboTech (www.carbotech.de) are suitable for nitrogen production from air and have long life-time stability. It was observed by authors of this work that CMSM stability seems to increase after some temperature induced sintering occurs. This phenomenon may be related to structural rearrangements in the carbon skeleton, making it more stable towards oxidation. Even though, to the authors’ best knowledge there are no sound experimental evidences of this, further research should be done in this area. Eventual adjustments in the activation process during membrane preparation may yield higher stability. 8.3. RIGIDITY/BRITTLENESS CMSM are fragile and may break easily. This fragility was reported to increase with the degree of activation [Koresh and Soffer, 1980]. When a CMSM it activated under vacuum to high temperatures the carbon atoms start sintering, approaching the graphite arrangement, which minimizes energy. This is also, however, a rigid arrangement and a trade off should be identified between rigidity, long-life stability and permeability/selectivity.

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A.M. Mendes, F.D. Magalhães and C.A.V. Costa

8.4. BLOCKAGE EFFECTS When a molecule adsorbs strongly at a constriction entrance, it might block, total or partially, the passage of other species. This happens with water molecules at sites where oxygen is chemisorbed. Water tends to adsorb very energetically at these sites, even in multilayer arrangements, eventually blocking the pore. The same can happen with heavier hydrocarbons [Ismail and David, 2001]. The blockage effect can be reduced working at higher temperatures, when solutes are less strongly adsorbed. In the case of water, this effect can be tremendously reduced if the CMSM internal surface has no chemisorbed oxygen atoms. This condition happens for stable carbon skeleton arrangements, when high temperature activation was performed and highly energetic sites, prone to chemisorption, are no longer available. Another form of reducing water sorption, as suggested by Jones and Koros, 1995, is coating the CMSM with a very thin film of a highly porous and hydrophobic material such as Teflon.

8.5. MEMBRANE AND MEMBRANE MODULE CONFIGURATIONS The CMSM can be classified into supported or unsupported. The supported membranes can be flat or tubular while the unsupported membranes can be flat, capillary or hollow fibers [Ismail and David, 2001]. Commercially, only hollow fiber CMSM membranes were available, from Carbon Membranes. Currently, Blue Membranes is developing waved flat membranes forming a honey-comb structure (figure 9), which behave mostly as hollow fiber membranes. The unsupported membranes can be symmetrical or asymmetrical. The carbon layer deposition technique (adlayer CVD) yields an asymmetrical membrane. This process produces a very thin selective coating, with very homogeneous pores whose dimensions can be almost continuously tuned. In these cases, the support should essentially provide the selective layer with the necessary mechanical stability. 8.6. CHARACTERIZATION The characterization work is essential in order to optimize the material’s performance. An overview of the most important properties to characterize and the techniques to be used is given next. The first characterization should consider the morphology and chemistry of the membranes. Let us assume that the materials to characterize are asymmetric hollow fiber membranes. The geometrical dimensions can be easily obtained from SEM imaging. Then an elemental analysis should be performed mainly to evaluate the content in carbon, oxygen and hydrogen,

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New Trends on Membrane Science

using for example a FTIR linked to a furnace or performing an X-ray microanalysis, as long as this measurement has the desirable precision. Wide angle X-ray diffraction can be used for evaluating the degree of crystallinity of the material. Carbon Membranes evaluated the elastic modulus of their carbon molecular sieve hollow fiber membranes by making a ring with the fibers. The ring was then squeezed until it breaks. The minimum diameter is related with the hollow fiber elasticity modulus. The apparent density of a membrane, dSF6 , can be obtained using a probe molecule, like SF6 (øSF6 = 0.502 nm, [Koresh and Soffer, 1980]), that does not penetrate the pores. The accessible structural density, d He , is obtained using helium. The internal porosity of the CMSM material can be estimated next from:

ε=

d He − dSF6 d He

(5)

It is also possible to compare the apparent density of the CMSM with the graphite density (dgraphite = 2.2, Koresh and Soffer, 1987), for obtaining the porosity which is not accessible even to the smallest atomic particles. The pore structure can be inspected using BET analysis with a range of gas molecules of different sizes, which only penetrate partially in the pores, e.g. nitrogen, argon, krypton or xenon. This analysis provides relevant information on the pore size distribution [see e.g. Gregg and Sing, 1982]. A more sophisticate technique, the positron annihilation lifetime spectroscopy (PALS), can be used to measure the size and concentration of free volume elements inside the CMSM [Merkel et al., 2002]. To describe the adsorption in the microporosity of a CMSM, carbon dioxide, a smaller molecule than nitrogen, is normally used together with the Dubinin-Radushkevich model [Pontes et al., 2003]. This method allows for estimating the average micropore diameter. Uptake curves, mono- and multicomponent adsorption isotherms, diffusivities and permeabilities are important also for learning more about the pore structure, for obtaining data necessary to modeling the separation process and for obtaining a direct insight on the membrane’s performance. Monocomponent uptake curves with gas or vapor molecules of different sizes and at different temperatures can be performed to evaluate the dimension of the pore constrictions. These adsorbents can include “flat” molecules such as benzene to evaluate the shape of the constrictions: round or slit-shaped [Koresh and Soffer, 1980].

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Mono- and multicomponent adsorption isotherms should be obtained at different temperatures. The gravimetric method using a magnetic suspension balance (e.g. by Rubotherm, www.rubotherm.de) is suitable for mono and bicomponent measurements. Multicomponent adsorption can be predicted by appropriate models, such as the Ideal Adsorbed Solution Theory (IAST) [Ruthven, 1984]. The model predictions should always be validated by comparison to some experimental points. The most used monocomponent equilibrium models are the Langmuir and Sips [Do, 1998]. Monocomponent permeabilities are easy to obtain at different temperatures provided that a membrane module is available. The permeate flow rate can be measured with a mass flowmeter. If the flow rate is too small, a pressure sensor together with a calibrated volume permeate chamber can be used, after the steady state has been reached. To obtain multicomponent permeabilities, an appropriate concentration analyzer should be introduced in the permeate stream. Usually a large feed flow rate is used to guarantee homogeneous concentration on the retentate side. Monocomponent diffusivities can be experimentally accessed using the so-called zero length column (ZLC) method [Ruthven, 1984]. The ZLC method was used for characterizing CMSM by Pontes et al., 2003. 8.7. MODELING A mathematical model is based on a set of assumptions and contains a series of parameters (obtained or not from direct experimental results). The way models fit or predict the experimental results indicate us about the reasonability of the assumptions. Mass transfer models are important in membrane science for: 1) Improving fundamental knowledge and developing support theories; 2) Optimizing the performance of the membrane material; 3) Designing and optimizing and controlling a separation unit. Until now it is not possible to have a direct visualization of the internal pore structure of a CMSM. The use of tunneling effect microscopes, or other tools, might however provide some valuable information in the near future, helping to elucidate how the mass transfer occurs inside the CMSM micropores. So far, one has to rely on assumptions based on indirect observations in order to formulate models able to describe this phenomenon. The complexity of the pore system in a CMSM has been described above. Two regions of distinct morphologies can be pointed out. One consists of the precursor pore system, made of irregularly shaped micropores with a series of localized constrictions along their length. The other is located in the CVD layer (if present) and is made of ultramicropores opened in the final activation stage. It is not possible to a priori conclude whether the limiting mass

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transfer resistance will be located at the CVD layer or not. Depending of the manufacture conditions, the precursor may just act as a non-selective porous support, or actually offer a significant resistance and contribute to the overall membrane selectivity. This type of structural heterogeneity is often modeled in terms of a combination in series of two patches with different mass transfer resistances [Do et al., 2001]. This, however, implies independent knowledge of the sorption equilibrium data for each region, so that the proper thermodynamic correction can be applied to the diffusional driving force (see formal treatment below). Since the pore system in a CMSM is essentially within the nanoporous range, one may expect that the mass transfer follows an activated diffusion mechanism (also called molecular sieving or configurational diffusion). However, other types may have to be considered, as discussed next. So far, few models have been specifically proposed for representing the mass transfer in a CMSM. Depending on the pore size different diffusion mechanisms will dominate the mass transfer. Gilron and Soffer, 2002, studied a CMSM prepared without carbon deposition (CVD). Based on the fitting of experimental data, they suggested a simplified pore model involving an association in series of two uniform pore sections of different diameters. Mass transfer in the narrower portion was assumed as being controlled by an activated diffusion process. For the wider section, on the other hand, the Knudsen diffusion mechanism was proposed. This model is attractive as it attempts to represent the heterogeneity of the pore structure by combining two separated resistances in series: one corresponding to the wider pore chambers and the other to the narrower constrictions. However, it is debatable whether, at moderate temperatures, the transport mechanism in the wider section actually follows straight Knudsen diffusion, specially considering the pore diameters suggested (0.58 nm for the wider section and 0.43-0.48 nm for the narrower). Figure 10 schematizes the relative potential of a molecule within a pore. For pores large enough there is no overlap of the potential well associated to the opposing walls, but as the pore diameter decreases, the potential energy of the sorbate crosses a minimum for increasing afterwards, until passage becomes impossible (case IV). The constriction shown in figure 10b may act as a preferred sorption site (case II) if the combined potential field from the surrounding walls creates a potential well that is deeper than for sorption onto a single wall surface. If the constriction is narrower, however, repulsion dominates and it becomes a barrier that a diffusing molecule must transpose (case III). This is a molecular sieving situation, usually described in terms of an activated diffusion formulation. In case I, on the other hand, the pore walls are separated enough to allow for the existence of minimum in the potential field in the neighborhood of the pore walls and a region of higher potential in the center of the pore. In this later region molecules can be assumed as being in a

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“compressed” gas phase state. Mass transfer in these conditions will, according to Burggraaf, 1999, involve two combined mechanisms: a) jumps between adjacent sites in the pore wall (i.e. surface diffusion) and b) “desorption” from the pore wall to the center of the pore, followed by transport to another sorption site. Burggraaf calls mechanism b) “activated Knudsen diffusion” and, somewhat ambiguously, describes it using the “gas translation” model developed by Xiao and Wei, 1992, referred by Burggraaf, 1999, for diffusion in zeolites. A more appropriate formulation might be the one referred by Do et al., 2001, for the combination of surface diffusion with Knudsen diffusion when there is an interaction potential between the molecules in the central portion of the pore and the surface, the so called “Knudsen diffusion in a potential field”. In any case, this transport regime must be recognized as being distinct from simple Knudsen diffusion, which would only occur if the surrounding potential field were zero. Mass transfer in carbon molecular sieve membranes is therefore a many-sided problem, involving a heterogeneous pore structure and different transport mechanisms. The validity of mass transfer models, assuming single or combined mechanisms (surface, Knudsen, activated) or associating, in series or parallel, regions of different diffusional resistance, has yet to be thoroughly tested with a variety of experimental data. The formal treatment of a simplified model, assuming only activated diffusion throughout the membrane length is now exemplified. The molar flux through a CMSM of one component diffusing along one direction is given by [Krishna and Paschek, 2000, Burggraaf, 1999]:

N i = −ρεDiΓ

∂qi ∂z

(6)

Γ is the so called thermodynamic correction factor:

Γ=

∂ln(pi ) ∂ln(qi )

(7)

where ρ is density of the membrane, ε is the porosity, Di is the MaxwellStefan diffusivity of component i, qi is the adsorbed concentration of component i, z is the diffusion coordinate and pi is the partial pressure of component i.

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II

III

IV

E

0 d

ε a) 2ε

b)

Figure 10. Graphical interpretation of the molecule/wall interaction within a micropore. a) Potential energy E of a molecule as a function of its distance d to the wall for different cross-sectional pore dimensions ( ε is the minimum potential for d → ∞ ); b) sketch of the particle in a hypothetical pore (adapted from Koresk and Soffer, 1980b).

The Maxwell-Stefan diffusivity Di may be related to the number of nearest neighbor sites, ζ , the jump length, λ , and the jump frequency, υ , which can be dependent on the total coverage [Krishna and Paschek, 2000]:

Di =

1

ζ

λ2υ

(8)

Further assuming that transport occurs through an activated molecular sieving mechanism, the Maxwell-Stefan diffusivity for zero loading should depend on the temperature as follows: (9)

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It was verified that for many cases the Maxwell-Stefan diffusivity dependences on the adsorption load, θ ( θ i = qi qi,sat and qi,sat is the adsorbed saturation concentration of component i), especially when θ > 0.5 . For multicomponent diffusing along one direction, the molar flux through a CMSM pore is given by [Krishna and Paschek, 2000]: (10)

where n is the number of components and,

The molar flux can be written explicitly as: −1

N = −ρε [qsat][B] [Γ ]

∂θ θ ∂z

(11)

Among others, Pontes et al., 2003, and Gilron and Soffer, 2002, verified that the Langmuir or Sips equations [Do, 1998] are the ones that give a better fit of the equilibrium experimental results. 9. Conclusions Membrane science and technology is not much past its infancy. Completely new membrane applications are being developed and leaving the research laboratories. Some of these new applications are briefly described in this paper: drug delivery systems, gas concentration sensors, proton exchange membranes and membrane reactors. Special emphasis was given here to a novel class of membranes that is gathering considerable attention: ultrananoporous (molecular sieve) membranes. These show simultaneously very high selectivities and permeabilities. So far, zeolite membranes are the only materials of this class to be commercially available. Carbon molecular sieve membranes are expected to undergo steady commercial production in the near future.

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Acknowledgements The authors would like to thank their Ph.D. students José Sousa, Sarah Pontes, Vasco Silva and Rosa Rêgo for helping gathering some of the information and making useful suggestions and comments. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13.

Adrian, S. and Meusinger, J. (2000) Process of Liquid-feed Direct Methanol Fuel Cell System, J. Power Sources 91 193-201. Balachandran, U., Dusek, J., Maiya, P., Mieville, R., Kleefisch, M. and Udovich, C. (1997) Ceramic Membrane Reactor for Converting Methane to Syngas, Catal. Today 36 265-272. Bernardi, D. and Verbrugee, M. (1991) Mathematical Model of a Gas Diffusion Electrode Bonded to a Polymer Electrolyte, AIChE J. 37 11511163. Burggraaf, A. (1999) Single Gas Permeation on Thin Zeolite (MFI) Membranes: Theory and Analysis of Experimental Observations, J. Membr. Sci. 155 45-65. Caruso, F., Caruso, R.A., Möhwald, H. (1998) Nanoengineering of Inorganic and Hybrid Hollow Spheres by Colloidal Templating, Science 282 1111-1114. Costamagna, P. (2001) Transport Phenomena in Polymeric Membrane Fuel Cells, Chem. Eng. Sci. 56 323-332. Dagani, R. (1997) Intelligent Gels, Chemical & Engineering News, June. Decher, G. (1997) Fuzzy Nanoassemblies: Toward Layered Polymeric Multicomposites, Science 277 1232-1237. Diaspro, A., Silvano, D., Krol, S., Cavalleri, O., Gliozzi, A. (2002) Single Living Cell Encapsulation in Nano-organized Polyelectrolyte Shells, Langmuir 18 5047-5050. Dittmeyer, R., Höllein, V., Quicker, P., Emig, G., Hausinger, G. and Schmidt, F. (1999) Factors Controlling the Performance of Catalytic Dehydrogenation of Ethylbenzene in Palladium Composite Membrane Reactor, Chem. Eng. Sci. 54 1431-1439. Do, D. (1998) Adsorption Analysis: Equilibria and Kinetics, Imperial College Press,. Do, H., Do, D. and Prasetyo, I. (2001) On the Surface Diffusion of Hydrocarbons in Microporous Activated Carbon, Chemical Engineering Science 56 4351-4368. Dohle, H., Divisek, J. and Jung, R. (2000) Process Engineering of the Direct Methanol Fuel Cell, J. Power Sources 86 469-477.

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29. Kaliaguine, S., Bouchard, C., Wu, S. and Shu, J. (1998) Use of Catalytic Membranes as Interphase Contactors for Multiphase Reactions, Canadian patent no 2,206,626. 30. Kerres, J. (2001) Development of ionomer membranes for fuel cells, J. Memb. Sci. 185 3-27. 31. Kordesh, K. and Simader, G. (1996) Fuel Cells and Their Applications, VCH Publishers. 32. Koresh, J. and Soffer, A. (1980) Study of Molecular Sieve Carbons. Part 1: Pore Structure, Gradual Pore Opening and Mechanism of Molecular Sieving, J.C.S. Faraday I 76 2457-2471. 33. Koresh, J. and Soffer, A. (1987) The Carbon Molecular Sieve Membranes. General Properties and the Permeability of CH4/H2 Mixture”, Sep. Sci. & Techn. 22 973-982. 34. Krasemann, L., Toutianoush, A. and Tieke, B. (2001) Self-Assembled Polyelectrolyte Multilayer Membranes with Highly Improved Pervaporation Separation of Ethanol/Water Mixtures, J. Membrane Sci. 181 221-228. 35. Kreuer, K. (2001) On the development of proton conducting polymer membranes for hydrogen and methanol fuel cells, J. Memb. Sci. 185 2939. 36. Krishna, R. and Paschek, D. (2000) Separation of Hydrocarbon Mixtures using Zeolite Membranes: a Modeling Approach Combining Molecular Simulations with the Maxwell-Stefan Theory, Sep. & Puri. Techn. 21 111136,. 37. Kumar, M. (2000) Nano and Microparticles as Controlled Drug Delivery Devices, J. Pharm. Pharmacet. Sci. 3 234-258. 38. Lai, Z., Bonilla, G., Diaz, I., Nery, J., Sujaoti, K., Amat, M., Kokkoli, E., Terasaki, O., Thompson, R., Tsapatsis, M. and Vlachos, D. (2003) Microstructural Optimization of a Zeolite Membrane for Organic Vapor Separation, Science 300 456-460. 39. Lee, S., Mitchell, D., Trofin, L., Nevanen, T., Söderlund, H. and Martin, C. (2002) Antibody-based bio-nanotube membranes for enantiomeric separations, Science 296 2198-2200. 40. Libby, B., Smyrl, W. and Cussler. E., "Polymer-zeolite Composite Membranes for Direct Methanol Fuel Cells", AICHE J., 49, 991-1001, 2003. 41. Lu, Y., Dixon, A., Moser, W., Ma, Y. and Balachandran, U. (2000) Oxygen-permeable Dense Membrane Reactor for the Oxidative Coupling of Methane, J. Membrane Sci. 170 27-34,. 42. Mallada, R., Menéndez, M. and Santamaría, J. (2000) Use of Membrane Reactors for the Oxidation of Butane to Maleic Anhydride under high Butane Concentrations, Catal. Today 56 191-197.

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THE IONIC AND MOLECULAR TRANSPORT IN POLYMERIC AND BIOLOGICAL MEMBRANES ON MAGNETIC RESONANCE DATA V. I. VOLKOV*, E. V. VOLKOV**, S. L. VASILYAK*, Y.S. HONG***, AND C. H. LEE*** * Center of Science and High Technology, Karpov Institute of Physical Chemistry, 10,Vorontsovo Pole. Moscow, 103064, Russia ** Physics Department, Moscow State University, Moscow 117234, Russia *** Graduate School of Biotechnology, Korea University, Seoul 136-701, Korea

Abstract Ionic and molecular transport mechanism in nanostructure systems could be understood on the basis of interconnection transport channel structure; ionic and molecular state; and translational mobility in different spatial scales. This information may be obtained from magnetic resonance data. We have investigated ion-exchange polymeric membranes and biological membranes. 1. Ion-exchange Membranes The ion exchange membranes contain acid or basic (charged) groups, which together with mobile ions and water molecules organize the network of transport channels. The well-known ion-exchange system, which has a regular structure, is the perfluorinated ion-exchange membrane Nafion, produced by Dupont Company. We investigated Russian perfluorinated sulfonated exchange membrane MF-HSK, which is similar to Nafion. This membrane contain a main polymeric perfluorinated chain with side ethereal fragment containing the sulfate groups, which can exchange their positive charge counter ions. It may be H+ ion or alkaline ions or other metals [1,2]. The structure of the polymeric chains and the structure of the transport channels are shown in Fig. l. According to numerous investigations including referent porometry, X-ray scattering data, Mössbauer spectroscopy, EPR, ENDOR, NMR-relaxation and PFG NMR data, the structure of transport channels is understood. The volume of transport channels is about 1/4 of the polymeric volume. The transport 481 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 481–507. © 2006 Springer. Printed in the Netherlands.

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channels structure is very regular. Their width is about 3-5 nm, the distance between sulfonic groups is about 0.7 nm, the charged groups are distributed rather homogeneously.

L2

l1

l3 L1

l2

1

2

3

Figure 1. Schematic representation of a fragment in the amorphous part of a sulfonated cation exchange membrane. 1 – main chain of the polymer; 2 – hydrated counterions and ionogenic groups at low water content (the condition of chlor-alkal electrolysis); 3 – transfer “channels” for ions and water at high water content; L1=4nm from small-angle X-ray scattering data; L2=10nm from Müssbauer spectra; l1 and l2 from ENDOR and NMR relaxation date; l3 from porometry and ENDOR data.

The ionogenic group’s diameter, including hydrated metal ions, is about 1.0 nm. These structural characteristics strongly depend on the water content, sulfur group concentration and the polymeric pretreatment. First of all, the power of the magnetic resonance technique is demonstrated and applied to solve some problem. 1.1 EPR [3 – 5] If counter ions are paramagnetic ions or if we want to use them as paramagnetic probes, the EPR is an ideal technique for the investigation. To analyze the EPR spectrum, it is possible to investigate the charge group distributions, because of the possibilities of these groups to organize due to the coordination bonding between charge groups and metallic ions. Using the ENDOR technique, it is also

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possible to investigate other coordination spheres not only nearest neighbors. The combination of EPR and ENDOR enabled us to obtain the detail information about the paramagnetic center to within 1-1.5 nm. 1.2. HIGH RESOLUTION NMR. [6 – 8] We studied the NMR of the internal water and also the NMR of Li+, Na+, Cs+ cations which was sorbed by Nafion membrane. We have measured the chemical shift and also the line width, which gave us the opportunity to investigate the water and ionic behavior. We investigate the humidity dependences of the chemical shift and line width, which is inversely proportional to the spin-spin relaxation times. So the line width is proportional to the residence times of ions on the sulphonic group. From the proton high resolution NMR, we could received information about the concentration of water molecules, h0 , in the first hydration shell of the ions. The value of h0 was less compare to the water solution of the acid and salts containing the same cations. For H+, h0 is always 2 which means that the acid proton in solution usually exists as H5O2+, but not as H3O+. The h0 value has a maximum for Na+ due to the dependence of h0 on the crystallography radius [6] (Fig. 2). It is also possible to obtain the fraction of broken H bonds compare to that for pure water. The possibility of breaking the hydrogen bond decrease from Li+ to Cs+ for alkaline ions and from Mg2+ to Ba2+ for alkaline earth ions [6] (Fig. 3).

h0

Na+(4,8)

4 3 2

K+(3,5)+ Rb (3,3) Cs+(3,0)

Li+(3,3) H+(2) 0.5

0.76

1.0

1.3 1.49 1.6

o

r, A

Figure 2. The hydrated numbers h0 for H+ and alkaline metal ions for Nafion-type membranes [6].

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From the high resolution data for the nuclei of alkaline ions, it is possible to obtain the mechanism of ionic interaction to the ionogenic sulfogroup. This is very important for the understanding of the membrane selectivity [7, 8].

D 0,6

Li+, Mg2+ Na+, Ca2+ Rb++ K , Ba2+

0,4

Cs+

0,2

1.0

1.5

o

2.0 r, A

Figure 3. The fraction of broken H bonds, D , for the alkaline and alkaline earth metal ions in perfluorinated Nafion type membranes [6].

The type of cation-sulphogroup interaction depends on the water contents. When the amount of water per one charge group n is less than h0, the cation and charge group interact directly and the contact ionic pair is organized. If n > h0, the ionic pair is separated. The concentration of contact ionic pairs and also the ionic diffusion mobility, which is determined by the residence time Wd, depend on n. From the 23 Na and 133Cs NMR it was possible to calculate the contribution of contact ionic pairs and some value, which is proportional Wd [7]. It was shown that for Cs+, the contact ionic pairs contribution is more than for Na+, but the residence time of ions on the SO3- group is less for Cs+ compare to Na+. This is due to the different crystallographic radii, which makes the Na+ positively hydrated but Cs+ negatively hydrated. These data explain the fact that the Cs+ ions penetrate through membrane easier than Na+ ions: The Cs+ ion interact more strongly with SO3- groups compare to Na+ and its sorption is better, but the residence time for Cs+ is less and Cs+ is moving faster compare to Na+.

485

The Ionic and Molecular Transport in Polymeric 1.3. T1, T2 RELAXATION MEASUREMENTS. SELF-DIFFUSION COEFFICIENT MEASUREMENTS [9, 20, 26-29]

We investigate the spin-lattice and spin-spin relaxation processes for 1H of water molecules and for 7Li of Li+ ions in the Li+ ionic form of the Nafion membranes. The main task was to obtain the characteristics of water and Li+ mobilities which are characterized by the correlation times W ( W = 1/ Q, where v is the frequency of motion). The correlation times and the motion activation energies were obtained from the temperature dependencies of relaxation rates T-1, T-2. As it is shown in Fig.4, these dependencies were described by the theoretical models based on the Bloembergen, Purcell, Pound theory and the correlation time, W calculated. The relaxation rates, inverse of relaxation time T1 and T2 for Li+ and H+ in ionic form membranes, were plotted as a function of temperature, as shown in Fig. 4 where the prepared sample conditions and experimental settings are summarized in Table 1 for convenience.

2' 10

4

1' -1

10

1

10

3

10

2

-1

2

T1 , T2 (ɫ )

10

-1

2

-1

-1

2'

1

-1

T1 , T2 (ɫ )

103

1 2

3

100 10

1'

104

101

3

-1

3

4 3

5 -1

10 / T ( K ) (ɚ)

6

3

4 3

5 -1

10 / T ( K ) (ɛ)

Figure 4. (left) Temperature dependence of the spin lattice T1-1 (unprimed numbers) and spin-spin T2-1 (primed ones) relaxation rates for 1H and 7Li nuclei in a Li+ form MF-4SK membrane. (right) Temperature dependence of 1H spin-lattice T1-1 (unprimed) and spin-spin T2-1 (primed) in a Li+ form (curves 1, 1' and 2, 2') with different content of waters and a H+ form (curve 3) MF-4SK membrane. Refer to Table 1 for other experimental conditions.

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Curves of 1 and 1' in Fig. 4(a) are approximation curves of lognormal distribution of correlation times. The distribution parameter G is 2 and the mean value of activation energy is 44 kJ/mol. The mean value of correlation time at 250 K is 1.8u10-10s. Curves of 2 and 2' are approximation curves of rectangular distribution of correlation times. The mean value of activation energy is 50 kJ/mol and IJc = (b-a)/ln(b/a) at 250 K is 1.6u10-10 s where b/a = 300. Here a and b are minimal and maximal values of correlation times [9]. In Fig. 4(b) the distribution parameter G is 2.4, the mean value of activation energy and of correlation time for 1 and 1’ curves are 46 kJ/mol, 1.0u10-9 s at 250 K, respectively. Table 1. Samples adopted in the measurements and experimental resonance frequencies. The last column indicates the curves used in Fig. 5. The unprimed curves are T1-1 and the primed curves are T2-1 values in Fig. 4. Figure

a

b

Curve

Nucleus 1

Qn (MHz)

n

Ionic form

9.83 90 34.97

21 21 21

Li+ Li+ Li+

Curve in Fig. 5

1(1’) 2(2’) 3

H H 7 Li

1(1’)

1

H

9.83

5.4

Li+

2(2’)

1

H

34.98

5.4

Li+

4

3

1

H

300

4.7

H+

1

1

3 2 4

The relaxation rates seen in Fig. 4 contain significant features as following: (i) the exponential behavior of longitudinal and transverse magnetization kinetics, (ii) T1-1(T) has two maxima in the samples with higher water-content (n=21). One of them is located in the low temperature range between 200 K and 250 K. This T1-1 maximum shifts to the lower temperature as NMR frequency decreases while water content increases ( see curves 1, 2 in Fig.4(a) and (b) ). Other maximum of T1-1(T) curves were observed at room temperature region. This maximum do not depend on NMR frequencies (curves 1, 2 in Fig.4(a)). (iii) at low temperature region, T2-1 shows an Arrhenius behavior. In the high temperature region, however, T2-1(T) behaves as T1-1(T), showing a maximum near room temperature in samples with high water-content (curves 1, 2 and 1’, 2’ in Fig 4(a)). (iv) at low water-content T1-1(T) and T2-1(T) shapes at temperatures are similar to those at high water-content (curves 1, 2 and 1’, 2’ in Fig 4(a) and (b)). Near room

The Ionic and Molecular Transport in Polymeric

487

temperature, the shoulder-shaped curves of T1-1(T) and T2-1(T) were observed (curves 1, 1’ and 2, 2’ in Fig. 4(b)). The similar temperature behavior of 1H relaxation rates was observed for other types of perfluorinated sulfonated exchange membranes [10], but the absolute values were one order of magnitude larger than our measurements. This is possibly due to the existence of paramagnetic impurities in the membranes. Thus, we may suggest that the present measurements yield more reliable results because we carefully eliminated paramagnetic impurities in the samples with HCl aqueous solution. The hydrogen relaxation processes in this system may be understood by utilizing the Bloembergen, Purcell and Pound (BPP) theory and its applications developed to describe the 1H relaxation of absorbed water in heterogeneous systems, including ion-exchangers [11 – 19]. In the simplest situation, the diffusional motion of water molecules is characterized by a single correlation time, and T1 and T2 can be described by Eqs. (1) and (2), respectively.

§ · Wc 4W c ¨¨ ¸  2 2 ¸ 1 ( ZW ) 1 ( 2 ZW )   c c © ¹ § · 5W c 2W c 1 2 ¸  J 'H 2 ¨¨ 3W c  2 2 ¸   1 ( ) 1 ( 2 ) 3 ZW ZW c c © ¹

1 T1

2 2 J 'H 2 3

1 T2 'H where

J

2

9J 2 = 2 20r 6

(1) (2) (3)

is the gyromagnetic ratio of 1H, r is the distance between two hydrogens,

ƫ is the Planck’s constant divided by 2ʌ, Z is the resonance frequency (Z = 2ʌ Ȟ = J H0), and H0 is the magnetic field. The temperature dependence of the correlation time is given by IJc = IJc0 exp [Ec/RT]

(4)

where Ec is the activation energy, R is the gas constant, and T is the absolute temperature. The theoretical values of T1-1 maximum in the low temperature region obtained from the BPP theory (T1-1)cmax and those determined from experimental curves (T1-1)emax are summarized in Table 2. There is a good agreement between the experimental values and the calculated ones. This result implies that the main magnetic relaxation mechanism may be due to the intermolecular 1H dipole-dipole interaction in water molecules with a distance of 0.156 nm. Eqs. (1)

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V.I. Volkov et al.

and (2) describe nicely the experimental temperature dependencies in the high temperature region, where the short correlation time approximation (Z IJc r (0)  r (t ) @ ! 2

lim

D

6t

t of

(7)

f

D

1  v(0)v(t ) ! 3 ³0

(8)

The solubility coefficient was calculated via simulations in the canonical ensemble in which the chemical potential is calculated using the Widom particle insertion method. The interaction energy of a gas particle inserted within the accessible free volume of the polymer matrix is calculated and the excess thermodynamic potential µex can be estimated from (9)

P

RT ln  exp( E int/ kt ) !

ex

(9)

The solubility S is then obtained from relation (10)

S

exp( P ex / RT )

(10)

The D and S coefficients can also be determined through TST calculations [53]. This approach assumes that the gases diffuse through the polymer matrix by a series of activated jumps. Their diffusion doesn’t depend on structural relaxation of the polymer but only on the thermal (elastic) isotropic motion of the matrix characterized by the smearing factor (analogous to the “DebyeWaller” factor in solid state physics) representing the fluctuations of the polymer atoms around their equilibrium positions. The isotropic approximation of elastic motion reduces the calculation of the solute distribution function to a multiplication of terms corresponding to pair interactions between the solute and the polymer atoms at their averaged equilibrium positions. while the smearing factors indicate the stiffness of the springs holding the polymer atoms together. Short-time-scale mol. dynamics runs (20 ps) are used to determine the smearing factors for the atoms of the polymer matrix. These factors are then used for a stochastic simulation of solute dynamics. This simulation consists in determining the solute distribution function U (r) in the polymer matrix by localizing its free volume sites as well as the penetrant jump probabilities between different sites. For a detailed description of the method the reader is invited to consult the literature [53]. The TST approach permits the

Molecular Modeling: A Complement to Experiment in Material Research

521

calculation of D by using Einstein’s relation (7) only and the solubility through relation (11)

S

1 U (r )dV kTV V³

(11)

Molecular dynamics (MD) is a powerful method for investigating detailed atomic-scale behavior on time scales of a nanosecond or less. Study of diffusion of small gas molecules in high free volume rubber polymers is feasible. For slower, infrequent-event processes, transition state theory (TST) should be employed [5, 28]. 4. Results and Discussion The AASBU method has been systematically used for screening over the first 70 (resp. over 20) space groups using one sodalite (resp. one D4R) cage per asymmetric-unit for each calculation. The results obtained have recently been reported in the literatur and apparently new structures reported in [18] have been discerned. Several known structures have been obtained: for example, sodalite (SOD) was obtained using the sodalite cage as SBU, ACO and AFY were obtained using D4R cage, while LTA structure was obtained either with sodalite or D4R SBUs. It is to notice that not only the space groups but also other predicted parameters (unit cell parameters, atomic coordinates) coincide well with experimentally observed ones. In order to evaluate the relative stabilities of the all-silica frameworks their lattice energies per tetrahedral unit are compared to the energy of the SiO2 quartz. An as-synthesized structure usually contains templating agents, water molecules, bridging hydroxy groups as well as fluorine atoms that are included in the framework. In order to predict the structure of the template-free structure all the species known to be removed upon calcinations are deleted from the experimentally defined as-synthesized structure generating a highly distorted structure. Upon constant pressure lattice energy minimization in the space group of the original structure, the distorted framework converges into a zeolitic topology. Its energy is then compared to that of existing structures of the same class to evaluate its relative stability. The prediction of the calcined form of the as-synthesized aluminophosphate MIL34, [Al4(PO4)4OH·C4H10N] illustrates this approach. Knowing the crystal structure of the as-synthesized MIL-34, the template-free one, namely AlPO4, has been anticipated using appropriate interatomic potentials, before it was obtained experimentally. The predicted open-framework compound showed an unknown zeolitic topology (Figure 2). This simulation method has also been successfully applied to a series of gallophosphates. The experimental removal of the SDA without breaking down the framework is not always an easy task because the stability of the as-synthesized

522

P. Pullumbi

structure often depends on the stabilizing template-framework interactions as well as on the experimental procedure employed during calcinations. For these reasons it is sometimes very helpful to estimate the adsorption properties of the final material prior to its calcination. GCMC simulations can be used to predict gas adsorption isotherms if we dispose of reliable force fields describing gas-solid and gas-gas interactions. The prediction of N2 and O2 adsorption in Lithium and Sodium Low Silica Xzeolite (LSX) is an example but the approach is quite general and could be applied to all crystalline porous adsorbents. Typical runs of 2*106 Monte Carlo steps from which the first 200000 are used for equilibration and not included in the averaging are sufficient to sample configuration space. The predicted adsorption isotherms of single component simulations in NaLSX and

Figure 2. a) Predicted structure of calcined MIL-34 by lattice energy minimization along [110]. b) Comparison between experimental (up) and simulated (down) XRD powder pattern.

LiLSX are reported in Figure 3 and compared to their experimental counterparts. The general trends are well reproduced with a systematic tendency to overestimate the adsorption of nitrogen for low loadings and underestimate that of oxygen. On the contrary the N2 loading is underestimated at higher pressures. This indicates that the balance between van der Waals and Coulombic interactions could still be improved by slightly modifying the adopted point charges at atomic positions. It is possible in principle, to improve the description of sorbent-sorbate interactions by using sophisticated expressions of the potential functions with higher order dispersion and induction terms recently discussed in [25] but it still requires the calculation of the electrostatic field inside the zeolite, which in its turn depends on the choice

Molecular Modeling: A Complement to Experiment in Material Research

523

of the point charges assigned to the atoms. It seems [54] that parameterization of polarization contribution to the potential energy by choosing partial charges cannot be carried out without resorting to a fitting procedure of some type.

Figure 3. Predicted (C) and experimental (E) single component isotherms for nitrogen an oxygen in NaLSX and LiLSX zeolites.

Figure 4. Predicted adsorption isotherms for different gases on a given activated carbon.

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P. Pullumbi

Prediction of gas adsorption isotherms on amorphous adsorbents is possible if the integral in (5) is inverted and the PSD f(H) has been obtained. To illustrate this approach we consider the activated carbons modelled as polydisperse slitpores. After experimental measurement of the adsorption isotherm of CO2 at room temperature, the f(H) is determined by solving (5) and than used to predict the adsorption isotherms of other gases on the same carbon at various temperatures. The quality of the databases is essential for correctly predicting the adsorption isotherms. In Figure 4 the predicted adsorption isotherms for several gases on a given activated carbon adsorbent are in good agreement with the experimentally measured ones validating the quality of the databases. The adsorption properties of an adsorbent determine most of the characteristics of a separation by adsorption process. As the PSD of an activated carbon strongly affects its adsorption properties, as adsorption capacity, selectivity, adsorption kinetics, exclusion phenomenon and adsorption competition, the ability to tailor it for a given separation offers an excellent opportunity to improve the performance of industrial PSA units.

Figure 5. Amorphous PDMS cell.

Figure 6. Trajectories of N2 mol. during MD simulations (1ns).

The mechanism of diffusion in rubbery polymers is different from the glassy ones. The diffusion coefficients for small gas molecules in rubbery polymer membranes do not depend on concentration while in glassy polymer membranes they do depend and reach a constant value at relatively high concentrations. This is mainly due to the fact that glassy polymers are not in a thermodynamic equilibrium state. For these polymers the final “metastable” chain configuration depends on the processing history of the membrane. This detail makes even more difficult the modelling of glassy polymer membranes

Molecular Modeling: A Complement to Experiment in Material Research

525

due to lack of experimental structural data for validating computational approaches. The prediction of self-diffusion coefficients for nonpolar small gas molecules in amorphous rubber polymer matrices is normally done through MD calculations [53,55,56]. After construction of the amorphous cell using the method of Theodorou-Suter and geometric free volume analysis of the cell several (4-6) penetrant molecules to improve sampling are inserted at the free volume positions. The cell is further relaxed by 100ps NPT MD simulation at 1 bar and room temperature before starting a longer (ns) NVT dynamics. The recorded trajectories (1ps) of each penetrant gas molecule are analysed and the diffusion coefficient is determined by means of relation (7) In figures 5 and 6 the packed cell model of PDMS (Polydimethylsiloxane) and the trajectory of N2 molecules in the PDMS matrix is reported. The MD simulations show two types of motions of the N2 molecules: jumps between cavities and local motion inside cavities. The predicted self-diffusion coefficients depend principally on the quality of the force fields used to model the interactions not only between the penetrant and polymer matrix, but also intramolecular interactions between polymer chains. These last ones, strongly affect the quality of the amorphous polymer cell and in particular the total free volume its distribution and dynamics which in their turn affect the predicted values of diffusion coefficients. The role of chain relaxation and matrix fluctuations in explaining the diffusion mechanism of small gas penetrants as N2 in rubber polymer membranes has been clearly demonstrated through MD calculations in which the polymer matrix has been kept fixed [53,56]. MD simulation of gas diffusion in polymer membranes generates a wealth of information on the mechanism of 3 pi4p1 pi4p2 pi4p3

2.5

t6p1 t6p2

2

t6p3

1.5

1

0.5

0

-0.5

-1

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Figure 7. Variation of the gradient of the ratio (accessible volume/accessible surface) with the probe radius for 6 amorphous cells of 2 different polymers.

526

P. Pullumbi

gas transport but its use is limited to high free-volume rubber matrices and small gas molecules due to prohibitive CPU times for diffusion coefficients smaller than 4x10-7 cm2/s [53] For this reason the technique is not adequate for systematic screening of a large number of polymer candidates and generating data to be used in a materials design approach.

Figure 8. Variation of the permeability as a function of H of the penetrant molecule.

Figure 9. Variation of the permeability as a function of V of the penetrant molecule.

Molecular Modeling: A Complement to Experiment in Material Research

527

The importance of understanding the factors that control gas transport and selectivity in glassy polymer membranes stems from the fact that most of industrially relevant gas separations or purifications using this technology utilize glassy polymers [3]. From a computational point of view the simulation of these membranes is more difficult than the rubber ones. The generation of the realistic amorphous cells is not a straightforward procedure and as it has been reported recently it represents a CPU intensive calculation [52]. The packing of aromatic polyimides is difficult not only because of the rigidity of the repeat units but also of the possible con-catenations of phenylene rings or spearing of side groups or backbone chains through ring sub-structures. In order to avoid this as well as to generate packings with “realistic” chain configurations the Theodorou-Suter approach has been used by first introducing about 500 “solvent” molecules in the 3D box followed by packing the polymer at a very low density. Next the “solvent” molecules are removed in a stage wise procedure followed by chain relaxation and cell compression up to near experimental densities. Several simulation steps detailed in [52] are needed to generate the final amorphous cells. The quality of the packings can be guessed through the calculation of the free volume and its distribution. In figure 7 the analysis of the quality of the amorphous cells made of 2 different repeat units (pi4 and t6) [52] is illustrated. The variation of the gradient of the ratio (Accessible Volume/Accessible Surface) with the radius of the probe used to geometrically measure the free volume of all the packings is used to measure the quality of the cell. Looking at the variation between 1.5 and 1.7 Å which correspond to the size of small gas molecules (inside the ellipsoid in figure 6) one can conclude that the packing 2 of t6 polymer is different from with the two other t6 ones indicates the others. The comparison of the TST predicted diffusion and solubility coefficients of this packing that this criteria can be used for rapid evaluation of the quality of the packed cells. The TST method can be used to determine equilibrium and transport properties of small molecules in glassy polymer membranes. In a recent work [52] this method has been used to study diffusion and solubility of small gas molecules in 10 different stiff chain polyimide membranes. The free energy to insert a gas molecule at all orthogonal fine grid points of the packed polymer cell is calculated first. From this calculation minimum sites are identified as well as the site-to-site transition probabilities. It is to notice that in the commercial code [57] the polymer is described at an atomistic level with the compass force field while the penetrant gas molecules are defined as united atoms and their interaction with a simple Lennard-Jones potential. The electrostatic interactions are neglected. The Lorentz-Berthelot rules are used for generating the LJ potential parameters for the penetrant-“packed polymer” and penetrant-penetrant interactions. From the interaction energy values on the grid,

528

P. Pullumbi

the solubility of the permeate molecules is calculated. In addition, the energybased free volume of the cell is determined and the connectivity of the energetically favorable transition paths between adjacent holes is identified. After calculating the transition probabilities between sites a Monte Carlo of gas diffusion by a ‘hopping’ mechanism between sites is carried out. The TST predicted S and D coefficients for N2, O2, CH4, and CO2 in the selected polyimide membranes are in good agreement with experiment for the first two penetrant molecules, acceptable for CH4 and unacceptable for CO2 indicating that some of the approximations used in the TST [57] need be modified. The obtained S and D strongly depend on the force field used to represent the interactions. In a series of TST calculations in which the Lennard-Jones potentials s and e of the penetrant molecule (O2, N2) is systematically varied the predicted permeabilities vary over orders of magnitude as reported in figures 8 and 9. This results show that the permeability is very sensitive to the LennardJones parameters of penetrant molecules. A detailed analysis of these calculations has revealed that the diffusion coefficient D depends principally on the size of the penetrant (ı)җ while the solubility S depends principally on the energetic parameter (İ). Recently a multidimensional transition-state approach has been developed [28,31,58-63], in which polymer degrees of freedom are incorporated explicitly in the reaction coordinate of the infrequent events whereby diffusion takes place and transition paths connect minima in polymer and penetrant configuration space through saddle points in that space. The penetrant molecules are described at an atomistic level and the electrostatic interactions are taken into account explicitly. Predictions of gas transport properties of N2, O2, CH4, and CO2 on some polyimide membranes carried out in the framework of the PERMOD project [27] show an overall improvement. The application of this approach together with the coarse grained methodology developed by the same group of authors for realistic simulations of rubbery and glassy polymers at modest computational cost would open the way for the rational design of polymer materials with desired properties [29,30]. The generation of practical design rules for new polymer membranes to expand the use of membrane based gas separation technology needs the development of reliable QSAR/QSPR methodologies. To accomplish this goal, molecular descriptors that mirror fundamental physico-chemical factors that in some way relate to gas solubility and diffusion are needed. It is desirable that the description is reversible, so that the model interpretation leads forward to an understanding of how the modification of chemical structure influences gas transport and equilibrium properties. Many literature examples that use the QSAR approach associated with polymer permeability are based on the use of group contribution methods to establish a correlation between the structure of the repeat unit and some physical property of the polymer membrane such as the free volume, the mean segment distance or dielectric constant polarizability,

Molecular Modeling: A Complement to Experiment in Material Research

529

which in its turn is used to predict permeation properties [64]. As a result of such studies some practical criteria have emerged to guide synthetic researchers in improving the permselectivity of membranes which have evolved through extensive experimentation: (i)inhibition of inter-segmental packing while simultaneously inhibiting intra-segmental (backbone) mobility. (ii)weakening of inter-chain interactions (reduction of charge transfer complexes). These design rules are based on phenomenological paradigms that provide guidelines for polymer selection. The QSAR approach under development integrates data at different length scales including the processing history of the polymer membrane in order to detect patterns of behaviour that could lead to new criteria for materials improvement. From a previous QSAR study [65] carried out in Air Liquide the following conditions are of fundamental importance for building reliable correlations: (i) the selection of the training set, (ii) the selection of adequate descriptors, (iii) the validity of experimental data used, and (iv) the manner in which the information dimensionality is reduced and the model validated. In figure 10 a schematic diagram of the QSAR methodology is reported. The generation of the molecular models for a large number of polymer membranes as well as the collection of reliable experimental measurements of S and D for several gas molecules in these membranes is the first step towards building a useful QSAR. Generation of appropriate physically meaningful descriptors based on molecular information not only of the repeat unit but also of amorphous cells is key to developing and searching for associations between apparently disparate or disjointed datasets. Generation of different kind of descriptors (topological, geometrical and electronic) is followed by PCA (Principal Component Analysis) which is a multivariate statistical factor analysis technique. This leads to data reduction since it points to the possibility

Figure 10. Schematic diagram of the applied QSAR methodology.

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P. Pullumbi

that only a limited number of properties have to be measured or calculated in order to explain the major part of the information concerning gas permeation through the membranes. Next genetic forms of statistical regression methods such as GFA or G/PLS are used to generate multiple QSAR models for the collected experimental gas permeation data. The use of this methodology for building reliable correlations between the structure of the polymer, the characteristics of the amorphous cell and the properties of the penetrant molecule with gas transport properties through the polymer membrane is under progress. The development of rapid algorithms that would permit rapid construction and estimation of S and D for different gas molecules paralleled by accurate experimental characterization of well-defined polymer membranes is needed to accelerate this process. New potential descriptors that may indicate not only what polymeric materials may be worthwhile investigating, but also those connecting the processing history of the polymer membrane to its morphology and by consequence to its separating properties would be essential for deriving useful design rules.

5. Conclusion The rate of scientific discovery of novel superior adsorbents and membrane materials could be significantly accelerated through a judicious combination of experiment with computation strategies. The successful development of novel materials lies in their rational design and can be achieved through an understanding of fundamental interactions at the molecular level. A multifaceted modelling approach even far from being complete can help in focussing the experimental effort in the search of new materials to be used in noncryogenic separation technologies and complement experimental studies. Acknowledgments I acknowledge Air Liquide S.A for the permission to publish this work. It is a pleasure to thank a large group of internal and external collaborators, Dr. S. Girard, I. Roman, M. Heuchel, D. Hofmann, C. Mellot-Draznieks and M. Sweatman as well as Pr. D. N. Theodorou and N. Quirke for their helpful discussions. References 1. 2.

Bulow, M. et al. (1998) Conference on Fundamentals of Adsorption], 6th, Giens, Fr., May 24-28, 47-50. Barton, T. J. et al. (1999) Chemistry of Materials 2633-2656.

Molecular Modeling: A Complement to Experiment in Material Research 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

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Roman, I. C., Ubersax, R. W. & Fleming, G. K. (2001) Chimica e l'Industria (Milan, Italy). Stern, S. A. & Koros, W. J. (2000) Chimie Nouvelle 3201-3215. Theodorou, D. N. (2002) Challenges in Molecular Simulations: Bridging the time-scale and length-scale gap (SIMU) 19-40 (The European Science Foundation programme). Mathias, P. M., Cheng, H., Cook, S. J., Klotz, H. C. & Parekh, V. S. (1996) Fluid Phase Equilibria 116 225-236. Rajan, K., Suh, C., Rajagopalan, A. & Li, X. (2002) Materials Research Society Symposium Proceedings 223-232. Catlow, C.R.A. (ed.) (1997) Computer Modelling in Inorganic Crystallography. Panagiotopoulos, A. Z., Quirke, N., Stapleton, M. & Tildesley, D. J. (1988) Mol Physics 527-45. Ravikovitch, P. I., Vishnyakov, A. & Neimark, A. V. (2001) Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 011602/1011602/20. Sweatman, M. B. & Quirke, N. (2001) Molecular Simulation 295-321. Uhlherr, A. et al. (2002) Computer Physics Communications 1-22. Baschnagel, J. et al. (2000) Advances in Polymer Science 41-156. Kremer, K. (2000) Proceedings of the Scottish Universities Summer School in Physics 145-184. Boisen, M. B., Jr., Gibbs, G. V. & Mukowinski, M. S. T. (1994) Physics and Chemistry of Minerals 269-84. Boisen, M. B., Jr., Gibbs, G. V., O'Keeffe, M. & Bartelmehs, K. L. (1999) Microporous and Mesoporous Materials 219-266. Mellot-Draznieks, C., Newsam, J. M., Gorman, A. M., M., F. C. & Ferey, G. (2000) Angew. Chem. Int. Ed. 2270-2275. Girard, S., Mellot-Draznieks, C., Ferey, G. & Pullumbi, P. (2002) Studies in Surface Science and Catalysis 1907-1914. Gale, J. D. (1997) Journal of the Chemical Society, Faraday Transactions 629-637. Lewis, D. W. et al. (1997) Angew. Chem. Intl. Ed. Eng. 36, 2675-2677. Zhao, X. S., Lu, G. Q. & Millar, G. J. (1996) Industrial & Engineering Chemistry Research 2075-2090. Girard, S., Draznieks, C. M., Ferey, G. & Gale, J. D. (2000) Chemical Communications (Cambridge) 1161-1162. Loiseau, T. et al. (2001) Journal of the American Chemical Society 96429651. Pullumbi, P., Lignieres, J., Arbouznikov, A. & Goursot, A. (2000) NATO Science Series, Series C: Mathematical and Physical Sciences 393-415.

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25. Fuchs, A. H. & Cheetham, A. K. (2001) Journal of Physical Chemistry B 7375-7383. 26. Robeson, L. M. (2003) Abstracts of Papers, 225th ACS National Meeting, New Orleans, LA, United States, March 23-27, POLY-459. 27. EU Project (2000-2003): FP5-GROWTH-1999 PERMOD Molecular modelling for the competitive molecular design of polymer materials with controlled permeability properties. 28. Theodorou, D. N. (1996) Plastics Engineering (New York) 67-142. 29. Theodorou, D. N. (2003) Lecture Notes in Physics 67-127. 30. Greenfield, M. L. & Theodorou, D. N. (2001) Macromolecules 34, 85418553. 31. Greenfield, M. L. & Theodorou, D. N. (2001) Macromolecules 85418553. 32. Mueller-Plathe, F. (2003) Soft Materials 1-31. 33. Uhlherr, A., Mavrantzas, V. G., Doxastakis, M. & Theodorou, D. N. (2001) Macromolecules 8554-8568. 34. Uhlherr, A. et al. (2002) Computer Physics Communications 144, 1- 22. 35. Freeman, B. D. (1999) American chemical society 32, 375-380. 36. Park & Paul. (1996) Journal of Membrane Science 125, 23-39. 37. Hofmann, D., Heuchel, M., Yampolskii, Y., Khotimskii, V. & Shantarovich, V. (2002) Macromolecules 2129-2140. 38. Wilks, B. R. (2002) Thesis,Georgia Institute of Technology, Atlanta, USA Free volume and free volume distribution impact on transport properties in amorphous glassy polymers 140 pp. 39. Bajorath, J. (2001) Journal of Chemical Information and Computer Sciences 233-245. 40. Gale, J. D. & Henson, N. J. (1994) Journal of the Chemical Society, Faraday Transactions 3175-9. 41. Henson, N. J., Cheetham, A. K. & Gale, J. D. (1996) Chemistry of Materials 664-70. 42. Gorman, A. M., Freeman, C. M., Kolmel, C. M. & Newsam, J. M. (1997) Faraday Discussions 489-494. 43. Buttefey, S., Boutin, A., Mellot-Draznieks, C. & Fuchs, A. H. (2001) Journal of Physical Chemistry B 9569-9575. 44. Buttefey, S., Boutin, A. & Fuchs, A. H. (2002) Molecular Simulation 1049-1062. 45. Watanabe, K., Austin, N. & Stapleton, M. R. (1995) Molecular Simulation 197-221. 46. Lignieres, J. & Pullumbi, P. (1998) Conference on Fundamentals of Adsorption], 6th, Giens, Fr., May 24-28, 719-725. 47. Seaton, N. A., Walton, J. P. R. B. & Quirke, N. (1989) Carbon 853-61.

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48. Scaife, S., Kluson, P. & Quirke, N. (2000) Journal of Physical Chemistry B 313-318. 49. Ravikovitch, P. I., Vishnyakov, A., Russo, R. & Neimark, A. V. (2000) Langmuir 2311-2320. 50. Gusev, V. Y., O'Brien, J. A. & Seaton, N. A. (1997) Langmuir 2815-2821. 51. Theodorou, D. N. & Suter, U. W. (1985) Macromolecules 1467-78. 52. Heuchel, M., Hofmann, D. & Pullumbi, P. (2003) Macromolecules in press. 53. Gusev, A. A., Muller-Plathe, F., Van Gunsteren, W. F. & Suter, U. W. (1994) Advances in Polymer Science 207-47. 54. Jameson, C. J., Lim, H.-M. & Jameson, A. K. (1997) Solid State Nuclear Magnetic Resonance 277-301. 55. Muller-Plathe, F. (1991) Journal of Chemical Physics 3192-9. 56. Sok, Berendsen & Gunsteren. (1992) Journal Chemistry Physical 96, 4699-4704. 57. Accelrys, (2000) Inc (Gsnet/Gsdiff implemented programs in InsightII /Discover). 58. Gray-Weale, A. A., Henchman, R. H., Gilbert, R. G., Greenfield, M. L. & Theodorou, D. N. (1997) Macromolecules 7296-7306. 59. Greenfield, M. L. & Theodorou, D. N. (1993) Macromolecules 5461-72. 60. Greenfield, M. L. & Theodorou, D. N. (1995) Polymer Preprints (American Chemical Society, Division of Polymer Chemistry) 687-8. 61. Greenfield, M. L. & Theodorou, D. N. (1997) Molecular Simulation 329361. 62. Greenfield, M. L. & Theodorou, D. N. (1997) Book of Abstracts, 213th ACS National Meeting, San Francisco, April 13-17 PMSE-261. 63. Greenfield, M. L. & Theodorou, D. N. (1998) Macromolecules 70687090. 64. Tokarski, J. S., Hopfinger, A. J., Hobbs, J. D., Ford, D. M. & Faulon, J.-L. M. (1998) Computational and Theoretical Polymer Science 199-214. 65. Pullumbi, P. & Milosavljevic, I. (1999) Organohalogen Compounds 553556.

MODELING JUMP DIFFUSION IN ZEOLITES: II. APPLICATIONS S. M. AUERBACH Department of Chemistry and Department of Chemical Engineering University of Massachusetts, Amherst, MA 01003 USA

Abstract We review recent applications of jump models for diffusion in zeolites. We describe the results of a coarse-grained model of the interplay between zeolite anisotropy and disorder, finding that certain disorder patterns can change how anisotropy controls membrane permeation. We show the results of a lattice model for single-file diffusion in zeolite membranes, demonstrating how singlefile motion is manifested in anomalous mean-square displacements at short times, and in non-intensive Fickian self-diffusion coefficients at later times. We discuss a normal-mode analysis approach for treating framework flexibility for tight-fitting zeolite-guest systems, showing that simulations allowing for framework flexibility can converge is less CPU time than those that keep the framework rigid. We then explore models of the loading dependence of selfdiffusion in zeolites, with emphasis on benzene in NaX and NaY. We enumerate the decisions that need to be made when modeling such systems, and indicate the choices/approximations we have made for modeling benzene in NaX and NaY. We report kinetic Monte Carlo results for the loading dependence of benzene diffusion in NaX, which is found in reasonable agreement with NMR data, but in poor agreement with tracer ZLC results. We then speculate on the possibility of having a subcritical fluid adsorbed in a nanoporous material, and how such a thermodynamic state would impact diffusion in such a system. We close with a review of outstanding problems in modeling jump diffusion in zeolites. 1. Introduction In our preceding chapter, we introduced the main ideas necessary to understand jump diffusion in zeolites. Here we bring these ideas to life with some examples studied by Auerbach and coworkers. 535 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 535–551. © 2006 Springer. Printed in the Netherlands.

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2. Modeling the Interplay between Non-Zeolitic Voids and Anisotropy Zeolite membranes exhibit two structural features that can conspire to produce novel transport properties; these are anisotropy and non-zeolitic voids. Structural anisotropy is particularly important for zeolites, and takes on even more importance when considering, e.g., para-xylene motion in an MFI-type zeolite (Figure 1), where motion along the straight channel is expected to be much faster than in other directions. Unfortunately, most MFI membranes are not as “simple” as that. Figure 2 shows a scanning electron micrograph (SEM) of an MFI membrane cross-section, showing clear evidence of grain boundaries and non-zeolitic voids in the heart of the membrane [1]. The question is: what is the interplay between such anisotropy and non-zeolitic voids on fluid transport through nanoporous membranes?

Figure 1. MFI structure cartoon.

Figure 2. SEM of C-oriented MFI.

To address this issue, Nelson et al. developed a finite-difference formulation of Fick’s diffusion equation for use with the model membrane system shown in Figure 3 [1]. Details of the calculations can be found in Ref. [1]. We wish to determine how steady-state fluxes vary with anisotropy and void properties. In particular, it is interesting to wonder whether transport through the void in Figure 3 (i.e. “short-cut” flux) can compete with transport that avoids the void. We define an anisotropy parameter Ș = ky/kx = Dy/Dx, which is the ratio of site-to-site jump rates in the in-plane (ky) and transmembrane (kx) directions, and is also the ratio of self-diffusivities in these directions. In general, the anisotropy depends on temperature because kx and ky are determined by different activation energies. In terms of these microscopic inputs, the finite-difference equations take the form:

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GT x , y Gt

Dx D (T x 1, y  T x , y )  x 2 (T x 1, y  T x , y ) 2 (Gx) (Gx) Dy Dy  ( T  T )  (T x , y 1  T x , y ) x , y 1 x, y (Gy ) 2 (Gy ) 2

where įx and įy are grid spacings. Edge-site fluxes are replaced with terms involving Ȟ, the insertion-attempt frequency per edge site, and kd, the rate coefficient controlling activated desorption from edge sites.

Figure 3. Model with pore, void and erop, and with straight and short-cut pathways.

We seek to determine how steady-state flux depends on anisotropy for membranes with defects. The flux also depends on temperature; here we quote the maximum possible flux for a given set of input parameters. Figure 4 shows the peak flux, Jmax, as a function of diffusion anisotropy for two values of the distance between defects, ǻy. Because the diffusion anisotropy varies with temperature, we plot Jmax against the value of Ș at the temperature Tmax, corresponding to the maximum flux. Three regimes can be identified in Figure 4: (i) diffusion-limited along the x-direction for low Ș values; (ii) diffusion-limited along the y-direction for intermediate Ș values; (iii) sorption-limited along the y-direction for large Ș values. Indeed, at low values of the anisotropy, diffusion in the y-direction is slowed down dramatically and transport through the shortcut becomes negligible. As a result, the peak flux approaches a constant value for small Ș. This constant flux is related to the average length of defects along the xdirection. At intermediate values of Ș, membrane permeation through the short

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cut begins to dominate, producing peak flux that grows with Ș. This indicates that membrane permeation is controlled by the rate of diffusion from pore to void to erop, i.e., along the y-direction. When Ș becomes even larger, motion from pore to void to erop is no longer limited by diffusion, but rather by the rate of desorption into the void. In this case, the peak flux becomes independent of Ș, precisely because membrane permeation is no longer limited by diffusion along the y-direction.

Figure 4. Peak flux (Jmax) vs. diffusion anisotropy (Ș) for membranes with voids.

As a concrete example to illustrate the importance of these results, several research groups have endeavored to synthesize b-oriented silicalite membranes with the expectation that permeating molecules will diffuse primarily down the straight channels. While there has been remarkable success reported in synthesizing oriented silicalite membranes, most of these membranes still suffer from many defects such as grain boundaries and nonzeolitic voids. Our results show that, with certain defect patterns and diffusion anisotropies, it is entirely possible that membrane permeation can be controlled by motion along the zig-zag channels of silicalite, even when the membrane is oriented with the straight channels parallel to the trans-membrane axis. That is because motion through the zig-zag channels may carry molecules from pore to void to erop faster than that possible by motion exclusively through straight channels. Zeolite scientists who make membranes may have to strike a balance

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between synthesizing membranes that are sufficiently oriented and sufficiently defect-free. 3. Modeling Single-File Diffusion in Zeolites of Finite Length In the limit of vanishing diffusion anisotropy, Ș ĺ 0, one-dimensional or socalled single-file diffusion dominates transport. For transport down a gradient, single-file diffusion obeys the usual equations of Fickian theory. On the other hand, for self-diffusion under equilibrium conditions, single-file diffusion suffers from such strong vacancy correlations (see previous chapter by Ramanan and Auerbach) that the phenomenology changes. In particular, the mean-square displacement (MSD) for single-file self-diffusion takes the anomalous form ‹x2› = 2Ft1/2, as opposed to the normal linear dependence in the Einstein equation; here F is denoted the single-file mobility. This t1/2 dependence was predicted by theory for infinitely long files [2]. However, all real files are finite in extent. Thus, we wonder how single-file self-diffusion is influenced by the finite extent of real files. To explore this, Nelson and Auerbach performed open-system kinetic Monte Carlo (KMC) simulations on the system pictured in Figure 5 [3]. For details regarding these simulations, we refer the reader to Ref. [3]. Because self-diffusion involves the stochastic motion of tagged particles, we used KMC to evolve the motions of tagged particles in files with untagged particles. The entire system (tagged plus untagged particles) is at equilibrium, and all particles have identical diffusion and sorption properties.

Figure 5. Simulation set-up to study single-file self-diffusion in files of finite extent.

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Figure 6 shows a log-log plot of the resulting MSD. In such a log-log plot, the slope gives the exponent of time in the MSD, e.g., a slope of 1 indicates normal Fickian behavior. Here we see three regimes: (i) mean-field diffusion at very short times; (ii) anomalous t1/2 behavior at intermediate times; (iii) at longer times a surprising return to Fickian behavior even though none of the tagged particles has desorbed from their files. The time in Figure 6 is measured in units of the average site-residence time, IJ. At very short times, t < IJ, a particle is unlikely to feel its neighbor, and thus exhibits normal diffusion in the mean-field limit where the vacancy correlation factor is near unity. At longer times, t > IJ, highly-correlated collisions between neighboring sorbates confined in the single file yields anomalous diffusion. One might naively presume that such behavior will persist until tagged particles desorb from their files. However, Figure 6 shows a surprising return to Fickian behavior well before particles leave their files. We and others have shown that this cross-over occurs at the time IJc = L2/ʌD0, where L is the file length and D0 is the infinitedilution self-diffusion coefficient. This cross-over time is essentially the time required for vacancies to diffuse from one end of the file to the other. After this time, the file edges strongly influence motion, which can be pictured as normal diffusion of the center-of-mass of particles in each file [4]. We and others have shown that the self-diffusion coefficient for this compound motion scales inversely with file length, which can be viewed as a new kind of anomaly.

Figure 6. Log-log plot of MSD for tagged particles in finite single files.

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This recurrence of Fickian motion with an anomalous self-diffusion coefficient alters somewhat the picture of single-file diffusion. We predict that the signature t1/2 time-dependence of the MSD lasts for a relatively short time. Indeed, for a great majority of the lifetime inside a single-file, the tagged particles are found to exhibit Fickian transport with the anomalous selfdiffusion coefficient. It is our contention that this provides the proper signature of single-file self-diffusion. Observing this predicted recurrence of Fickian motion may be challenging for modern pulsed field gradient (PFG) NMR methods. PFG NMR is limited by spin-lattice relaxation, which significantly reduces signal-to-noise ratios. PFG NMR can usually measure motion for up to ca. 0.1 seconds. Assuming L = 10 µm and D0 = 10-5 cm2/sec, the cross-over time falls right around the maximum observation time for PFG NMR. Thus, testing the above simulation results may require the development of longer-time microscopic methods for measuring self-diffusion in zeolites. 4. Modeling Zeolite Flexibility in Rare-Event Dynamics When guest molecules fit tightly in zeolite nanopores, transport through zeolites is dominated by jump diffusion. As shown below in Figure 7, benzene in silicalite provides an excellent example of such strong confinement. Snurr et al. applied harmonic transition state theory (TST) to benzene diffusion in silicalite, assuming that benzene and silicalite remain rigid [5]. As a consequence of this assumption, their results underestimate experimental diffusivities by one to two orders of magnitude. Forester and Smith subsequently applied TST to benzene in silicalite using constrained reaction-coordinate dynamics on both rigid and flexible lattices [6]. Lattice flexibility was found to have a very strong influence on the jump rates. Diffusivities obtained from these (computationally demanding) flexible framework simulations are in excellent agreement with experiment, overestimating the measured room temperature diffusivity (2.2u 14 2 10– m /s) by only about 50%. These studies establish benzene in silicalite as an important benchmark system for which including framework flexibility is crucial for describing guest diffusion. Computing the fundamental site-to-site rate coefficients for such confined diffusion is challenging for the following reasons. First, guest diffusion is likely facilitated by peristaltic framework vibrations; simulating such vibrations requires the significant computational expense of a flexibleframework model. Second, peristaltic framework vibrations are highly cooperative motions, which are conveniently simulated using molecular dynamics (MD). Third, straightforward MD cannot be used to model the motions of strongly confined guests, whose site residence times are typically

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much longer than MD run times. These facts taken together make it challenging to find an efficient method for simulating strongly confined guest diffusion.

Figure 7. Benzene in silicalite’s straight channel; example of tight fit.

A solution to this problem was suggested by Turaga and Auerbach [7], following the normal-mode analyses of zeolite vibrations reported by Iyer and Singer [8]. They found that zeolite normal modes often correspond to breathing motions of rings and channels, suggesting that these coordinates can efficiently sample framework distortions during molecular jumps. What's more, a remarkable speedup can be obtained by exploiting the fact that zeolite vibrations are nearly harmonic, which has been established by Turaga and Auerbach. As such, after computing the normal modes, sampling lattice flexibility costs essentially no CPU time because the zeolite force constants are known. Thus, we use normal-mode coordinates for natural sampling of zeolite vibrations, and normal-mode force constants for efficient energy calculations. Below we show free energy surfaces for benzene jumping in silicalite's straight channel, finding excellent agreement with the results of Forester and Smith. However, in contrast with their calculations, the flexible-lattice simulations reported below converged in less CPU time than that required for fixed-lattice simulations. The free energy landscape for benzene in silicalite is now reasonably well known, with relatively flat minima at intersection sites and corrugated regions of high free energy in channels. This landscape arises from a balance between the host-guest potential energy, host distortion energy and guest configurational entropy. Using the methods outlined above, Turaga and Auerbach calculated benzene's free energy surface (FES) along the

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crystallographic y-axis describing the jump between intersection sites, which are separated by about 10 Å. Results for flexible and rigid lattices are compared below in Figure 8. Both curves show the qualitative features indicated above. However, the rigid-lattice barrier is much higher than the flexible-lattice one, because the zeolite is allowed to distort during the latter simulations. Our flexible lattice FES is in excellent agreement with results of Forester and Smith. In agreement with their results, we find three shallow free-energy minima in the channel. Our barrier, 20 kJ/mol, is in very good agreement with their result, 25 kJ/mol, considering that slightly different forcefields were used. These results confirm that our local normal-mode Monte Carlo approach can faithfully represent molecular motion in tight-fitting zeolite-guest systems.

Figure 8. Free energy surface for benzene jumping in silicalite’s straight channel.

For each flexible-lattice free energy in Figure 8, Turaga and Auerbach performed two Monte Carlo runs of length 106 steps (attempted moves). On the other hand, for each rigid-lattice free energy we performed two Monte Carlo runs of length ca. 107 steps. We note that the rigid-lattice FES does not reflect silicalite's symmetry along the reaction coordinate, while the flexible-lattice FES does. This indicates that, despite the longer Monte Carlo runs, the rigidlattice FES remains more poorly converged than the flexible-lattice FES. This slow convergence occurs because of the decreased likelihood of jumping through a rigid lattice. A more efficient window sampling method might speed up the rigid-lattice FES convergence. Nonetheless, because the normal-mode

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algorithm makes rigid - and flexible-lattice calculations equally fast step for step, and our flexible calculations converged in fewer steps, we have shown that flexible-lattice calculations can actually be faster than rigid-lattice ones. This algorithm will facilitate simulations of adsorption and diffusion in tight-fitting host-guest systems for hosts that behave as multi-dimensional harmonic oscillators during guest diffusion. This class of hosts includes most siliceous zeolites, many carbon nanotubes, and possibly the selectivity filters of biological ion channels. Exceptions include zeolites that undergo phase transitions upon guest adsorption, zeolites with exchangeable cations that diffuse alongside guests, biological ion pumps, and any host that executes large amplitude motion during guest diffusion. 5. Loading Dependence of Benzene Diffusion in FAU-type Zeolites Several experimental and theoretical diffusion studies have been reported for benzene in NaX and NaY to help resolve persistent, qualitative discrepancies between experimental probes of the coverage dependence of self-diffusion. In particular, PFG NMR diffusivities decrease monotonically with loading [9], while tracer zero-length column (TZLC) data increase monotonically with loading [10]. The discrepancy between PFG NMR and TZLC casts doubt on using experimental self-diffusivities for designing processes in zeolites. Atomistic simulations, lattice models and field theories have been reported for this transport system. The simulations performed and methods employed have been reviewed in Refs. [11-13]. This discrepancy points to a larger problem: in general, we lack qualitative understanding how host-guest and guest-guest interactions conspire with thermal energies to produce different loading dependencies of selfdiffusion. By analyzing PFG NMR diffusivities for many zeolite-guest systems, Kärger and Pfeifer reported five typical loading dependencies of selfdiffusion [14], in analogy with the IUPAC designations for adsorption isotherms. Kärger and Pfeifer’s results are shown schematically below in Figure 9. In 1999, Saravanan and Auerbach reported a lattice model of benzene diffusion in NaX, which yields diffusivities in agreement with PFG NMR data [15]. In addition, by varying fundamental energy parameters in their model, they found four of the five loading dependencies reported by Kärger and Pfeifer. Here we review the essential aspects of this work.

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Figure 9. Kärger and Pfeifer’s schematic loading dependencies of self-diffusion in zeolites.

When building a model of diffusion in zeolites at finite loadings, a variety of decisions must be made that impact the accuracy of the results and the efficiency of the computations. Below in Figure 10 we outline the typical decisions that need to be made, and underline the choices we have made in modeling benzene in NaX. Our model for benzene diffusion assumes that benzene molecules jump among SII and W sites, located near Na+ ions in supercages, and in 12-ring windows separating adjacent supercages, respectively. Simulation details can be found in Ref. [15].

Figure 10. Model-building decisions for diffusion in zeolites; our choices underlined.

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Below we give three main results: (i) three qualitative loading dependencies exhibited by the lattice model; (ii) the predicted loading dependence for benzene in NaX; (iii) novel adsorption and diffusion phenomena that arise when guest-guest attractions are large compared to thermal energies. The three loading dependencies that arise from our model when temperatures are relatively high are shown below in Figure 11. Alongside these loading dependencies are schematic cartoons that depict the physics of these transport systems. When the W and SII site energies are thermally degenerate, adding additional molecules blocks sites and slows diffusion, hence giving Kärger and Pfeifer’s type I dependence. In the other extreme, adding additional molecules fills SII site traps, which actually speeds up diffusion, giving Kärger and Pfeifer’s type IV dependence. At room temperature, we predict that type I will be found for benzene in NaX, while type IV will be found for benzene in NaY. As discussed below, the former prediction agrees with PFG NMR data. However, the latter prediction is at odds with recent quasi-elastic neutron scattering data, which find a type I dependence for benzene in NaY [16].

Figure 11. Loading dependencies of self-diffusion from lattice model, with corresponding cartoons.

Figure 12 shows simulation for benzene in NaX at T = 393 and 468 K, compared to PFG NMR data at the same temperatures (uniformly scaled by a factor of 5) [9], and TZLC diffusivities at T = 468 K (uniformly scaled by 100) [10]. Figure 12 shows that our model is in excellent qualitative agreement with PFG NMR, and in qualitative disagreement with TZLC. We suggest that high-

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temperature TZLC experiments should be performed, to test whether their type IV becomes a type I or II, as our simulations predict should happen.

Figure 12. Comparing simulation, PFG NMR and TZLC for benzene in NaX.

We pause from these diffusion studies to wonder about the thermodynamic state of the confined fluid when guest-guest attractions become large compared to thermal energies. It is interesting to wonder whether fluids confined in nanopores can exhibit the analog of vapor-liquid equilibrium (VLE). The terms “vapor” and “liquid” are less meaningful for the adsorbed phase. Instead, we imagine a densification transition, whereupon a small increase in external sorbate pressure produces a precipitous increase in the sorbate loading. In adsorption experiments, the signature of this transition is hysteresis and capillary condensation, which are routinely observed for fluids confined in mesopores. However, for molecules in zeolites, these phenomena are much less common. This is not completely unexpected, since the critical temperature(s) for hysteresis and capillary condensation are expected to plummet as pore sizes approach molecular dimensions. The question is: can we find a zeolite-guest system for which this densification transition survives? The likelihood of survival is increased when considering a guest phase with a large bulk VLE critical temperature, and a zeolite with relatively large cages. Here we consider benzene in NaX. Benzene has a bulk critical temperature of 562 K, and the NaX supercage is among the largest in zeolites. Saravanan and Auerbach performed grand canonical Monte Carlo simulations for benzene in NaX, using the lattice model derived for treating diffusion [17]. Figure 13 shows the resulting “coexistence curve,” alongside a cartoon depicting the transition from subcritical to supercritical states. We predict a densification critical temperature for benzene in NaX of ca. 370 K. This is

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lower than benzene’s bulk critical temperature, as expected, but is high enough to suggest that adsorption experiments may be able to observe hysteresis upon adsorption of benzene in NaX. (We note that hysteresis has been observed in adsorption isotherms measured for benzene in NaX [18]. However, this observation must arise from a structural transformation of the zeolite rather than from cooperative interactions among guests, because the measured densities in the adsorption branch exceed those in the desorption branch).

Figure 13. Simulation and schematic showing subcritical benzene in NaX.

We close this section by wondering what loading dependence of selfdiffusion is expected for a subcritical adsorbed phase. Saravanan and Auerbach explored this by performing KMC simulations for analogs of benzene in NaX with various strengths of guest-guest attraction [15], all at T = 468 K. The results are shown in Figure 14. The systems with guest-guest attraction parameter J = –2 and –4 kJ/mol are supercritical, both showing a broadly decreasing loading dependence. However, the system with J = –7 kJ/mol is in a subcritical state, and shows a qualitatively different loading dependence of self-diffusion. In particular, cluster formation in subcritical systems (see Figure 13) suggests a diffusion mechanism involving “evaporation” of particles from clusters. Increasing the loading in subcritical systems increases mean cluster sizes, and smoothes cluster interfaces. Once these interfaces become sufficiently smooth, we surmise that evaporation dynamics remain essentially unchanged by further increases in loading. As such, we expect the subcritical diffusivity to obtain its full loading value at low loadings, and then remain roughly constant up to full loading, as seen in Figure 14. This type of loading dependence, involving broad regions of constant diffusivity, is surprising, since isotherms for interacting sorbates are expected to

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decrease with loading when site blocking dominates. This loading dependence is essentially the same as Kärger and Pfeifer’s type III, which is observed for strongly associating fluids such as water and ammonia in NaX. This analysis suggests that Kärger and Pfeifer's type III loading dependence may be characteristic of a cluster-forming, subcritical adsorbed phase.

Figure 14. Predicted loading dependencies of subcritical and supercritical fluids in zeolites.

6. Concluding Remarks Jump diffusion models have significantly elucidated transport in zeolites. Here we have seen how defects in zeolites, single-file diffusion, zeolite flexibility, host-guest interactions and phase transitions influence the temperature and loading dependencies of diffusion in zeolites. Despite this initial progress, much work remains. In particular, recent PFG NMR data suggests strongly that zeolites are not the ordered frameworks we surmise them to be [19]. A major challenge for future zeolite scientists is thus developing models that retain the beautiful atomic structure of zeolites, while incorporating realistic representations of disorder present in real materials. Progress along these lines should feature prominently at future NATO Advanced Study Institutes on Fluid Transport in Nanopores. Acknowledgments I thank Profs. J. Fraissard, W.C. Conner, Jr., and V. Skirda for the invitation to speak at and write for the NATO Advanced Study Institute. I thank them also for partially funding my trip. I thank all my previous and present coworkers for

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their invaluable efforts toward revealing the fascinating world of fluid transport in nanopores. But mostly, I thank my wife Sarah for taking care of the kids while I write this. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15.

Nelson, P.H., Tsapatsis, M., and Auerbach, S.M. (2001) Modeling Permeation Through Anisotropic Zeolite Membranes with Nanoscopic Defects. J. Membrane Sci. 184 245-255. Levitt, D.G. (1973) Phys. Rev.. A8 3050. Nelson, P.H. and Auerbach, S.M. (1999) Self Diffusion in Single-File Zeolite Membranes is Fickian at Long Times. J. Chem. Phys. 110 92359243. Hahn, K. and Karger, J. (1998) J. Phys. Chem. B. 102 5766. Snurr, R.Q., Bell, A.T. and Theodorou, D.N. (1994) J. Phys. Chem. 98 11948. Forester, T.R. and Smith, W. (1997) Bluemoon Simulations of Benzene in Silicalite-1: Prediction of Free Energies and Diffusion Coefficients. J. Chem. Soc. Faraday Trans. 93 3249-3257. Turaga, S.C. and Auerbach, S.M. (2003) Calculating Free Energies for Diffusion in Tight-Fitting Zeolite-Guest Systems: Local Normal-Mode Monte Carlo. J. Chem. Phys. 118 6512-6517. Iyer, K.A. and Singer, S.J. (1994) Local-Mode Analysis of Complex Zeolite Vibrations: Zeolite A. J. Phys. Chem. 98 12679-12686. Germanus, A., et al. (1985) Zeolites 5 91. Brandani, S., Z. Xu, and Ruthven, D. (1996) Microporous Materials 7 323-331. Auerbach, S.M. (2000) Theory and Simulation of Jump Dynamics, Diffusion and Phase Equilibrium in Nanopores. Rev. Phys. Chem. 19 155198. Auerbach, S.M., Jousse, F. and Vercauteren, D.P. (2003) Dynamics of Sorbed Molecules in Zeolites, Computer Modelling of Microporous and Mesoporous Materials, B. Smit, Editor., Academic Press, London. Karger, J., Vasenkov, S. and Auerbach, S.M. (2003) Diffusion in Zeolites, in Handbook of Zeolite Science and Technology, S.M. Auerbach, K.A. Carrado, and P.K. Dutta, Editors, Marcel Dekker, Inc.: New York. 341422. Karger, J. and Pfeifer, H. (1987) Zeolites. 7 90. Saravanan, C. and Auerbach, S.M. (1999) Theory and Simulation of Cohesive Diffusion in Nanopores: Transport in Subcritical and Supercritical Regimes. J. Chem. Phys. 110 11000-11010.

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16. Jobic, H., Fitch, A.N. and Combet, J. (2000) Diffusion of benzene in NaX and NaY zeolites studied by quasi-elastic neutron scattering. J. Phys. Chem. B. 104 8491-8497. 17. Saravanan, C. and Auerbach, S.M. (1998) Simulations of High Tc VaporLiquid Phase Transitions in Nanoporous Materials. J. Chem. Phys. 109 8755-8758. 18. Tezel, O.H. and Ruthven, D.M. J. Coll. Inter. Sci. (1990) 139 581. 19. Vasenkov, S., et al. (2001) PFG NMR Study of Diffusion in MFI-type Zeolites: Evidence of the Existence of Intracrystalline Transport Barriers. J. Phys. Chem. B 105 5922-5927.

DYNAMICS OF WATER SORPTION ON COMPOSITES “CaCl2 IN SILICA”: SINGLE GRAIN, GRANULATED BED, CONSOLIDATED LAYER Y.I. ARISTOV, I.S. GLAZNEV, L.G. GORDEEVA Boreskov Institute of Catalysis, Pr.Lavrentieva 5, Novosibirsk, 630090 Russia I.V. KOPTYUG, L.YU. ILYINA International Tomography Centre, Institutskaya 3a, Novosibirsk, 630090 Russia J. KÄRGER, C. KRAUSE University of Leipzig, Linnéstraße 5, D-04103 Leipzig, Germany B. DAWOUD Aachen University (RWTH–Aachen), Schinkelstrasse 8, D-52062 Aachen, Germany

1. Introduction Development and study of novel nanoporous adsorbents is a very challenging goal of current material science research. Among these materials are new alumosilicates, alumo- and iron phosphates, MCMs, SBAs, MOFs, carbon sieves, etc., with the pore size of few nanometers. Less progress is made in developing new adsorbents with relatively large pores of 7-20 nm. The idea to modify a common mesoporous adsorbent (silica, alumina, etc.) by introducing inside its pores a hygroscopic salt (CaCl2, LiBr, etc.) was known at least since 1929 [1]. Such materials were used as adsorbents in gas masks and then were almost forgotten. New impact has recently been done in a set of papers [2-6] which presented the result of systematic study of adsorption and thermophysical properties of a family of composite materials “salt inside porous host matrix” which were called “Selective Water Sorbents” (SWSs). Some of the SWSs synthesised and studied are displayed in Table 1. Despite of the short renaissance period these materials have already met practical applications as gas drying agents and materials for active heat insulation in aerocraft “black boxes”. For these and other possible applications that are under development now (freonless solar driven chillers [7] and fresh water production from the atmosphere [8]) the dynamics of water sorption is of high interest. 553 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 553–565. © 2006 Springer. Printed in the Netherlands.

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Here we present the first results on kinetics of water sorption on composite “CaCl2 in mesoporous silica KSK” (SWS-1L) that at the moment is the most studied material of SWS-type [3,5]. Here we focus mainly on the dynamics of water adsorption under vacuum conditions when water vapour is the only component in the gas phase although few results are reported on water adsorption from air flux as it takes place in air drying units. Three adsorbent configurations will be considered, namely, a single grain, a granulated layer and a consolidated layer prepared with a binder and pore-forming additives. Three methods have been used to study the water transport and sorption: a) 1H NMR microimaging experiments on the spatial distribution of sorbed water in the sorbent and its temporal evolution, b) the PFG NMR method and c) the kinetics of water sorption under constant volume-variable pressure conditions. The first method is a direct one because it directly measures water concentration profiles and can give the water transport diffusivity. The second method is used for measuring water self-diffusivity, and the last one, in principle, allows to estimate the water diffusivity from the uptake curves [9]. Table 1. A list of some SWS materials synthesised and studied so far. Host Water sorption, matrix g/g Silica gel KSK 0.65 Silica gel KSM 0.25 aerogels 1.0-1.50 Carbon Sibunit 0.57 Al2O3 0.52 Silica gel KSK 0.67 Silica gel KSM 0.25 Carbon Sibunit 0.60 Al2O3 0.55 Silica gel KSK 0.60

Remarks SWS-1L SWS-1S SWS-1Aero SWS-1C SWS-1A SWS-2L SWS-2S SWS-2C SWS-2A SWS-3L

[3] [4] [5] [5] [5] [6] [6] [6] [6] [6]

LiCl

Silica gel KSK

0.60

SWS-4L

[6]

MgSO4

Silica gel KSK Al2O3

0.65 0.50

SWS-5L SWS-5A

[5]

NaSO4

Silica gel KSK

0.62

SWS-6L

[5]

CuSO4

Silica gel KSK

0.58

SWS-7L

[5]

Salt

CaCl2

LiBr

MgCl2

Reference

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2. Experimental 1

H NMR microimaging experiments were performed at 300 MHz on an Avance NMR spectrometer equipped with a microimaging accessory. A sample was placed in the rf coil of the NMR probe with its axis oriented along the magnetic field of the superconducting magnet, pumped down and connected to an evaporator to maintain a fixed vapour pressure over the sample during the water sorption process. More details can be found in [10,11]. The sample was either a single cylindrical grain (diameter 6 mm, length 6 mm) or a bed of spherical grains 2-3 and 5-6 mm in diameter. The bed diameter was 25 mm, the length was 30 mm. The pellets of the commercial silica KSK (the average pore diameter 15 nm, the specific surface 350 m2/g, the pore volume 1 cm3/g) were impregnated with a 38 wt.% CaCl2 aqueous solution and then dried at 1500C. Consolidated SWS layers were prepared using silica (KSK, Davisil 60 and Davisil 150) or alumina A1 powder together with a binder (pseudoboehmite). In order to analyse the relative importance of the diffusional resistance in macropores between the adsorbent particles and in mesopores inside the particles, we varied the size of powder particles (between 0.04 and 0.5 mm), the size of mesopores (between 6 and 15 nm) and the amount of the binder (0-30 wt.%). A carbon Sibunit and an ammonia bicarbonate were used as pore-forming additives. The silica gels and alumina were moulded with the binder as a cylindrical tablet of 16 mm diameter and 4-8 mm thickness and then impregnated with aqueous solution of CaCl2. Porous structure of the tablets was determined by SEM, BET and mercury porosimetry. The sample holder allows the water adsorption only through the upper flat surface of the tablet. The PFG NMR experiments were performed using the home-built PFG NMR spectrometer FEGRIS 400 NT operating at a 1H resonance frequency of 400 MHz [12]. The probe head equipment allows measurements at temperatures from ~120 to 470 K (1H NMR resonance) and gradients up to 35 Tm-1. The procedure of the kinetic measurement under constant volume-variable pressure conditions is described elsewhere [13]. 3. Results and Discussion 3.1. SINGLE GRAIN First experiments with a single SWS-1L grain were performed under the flow regime when the dry cylindrical grain (radius R = 3 mm) was placed in a flux of humid air [10]. 1D profiles of adsorbed water are presented in Fig. 1. Sharp front of water adsorption is observed. The front propagates inside the grain with a constant rate (shrinkage cylinder behaviour) so that the calculated uptake is

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mt/m = (2 /R) t – ( /R)2 t2 and the rate of adsorption decreases with the sorption time as (2 /R) – 2( /R)2 t. This decrease is in line with our experimental data on water breakthrough curves in a plug flow adsorber with SWS-1L.

Figure 1. Visualization of the dynamics of the water sorption by a cylindrical SWS-1L grain. Air flow FL=390 l/h, relative air humidity RH=55%. 2D SPI images; accumulation time per image was 13 min 39 s [10].

Figure 2. Visualization of the dynamics of the water sorption by a cylindrical SWS-1L grain. 1D profiles of water content along the diameter of the pellet, accumulation of each profile lasted 34 s, every 8th profile is shown [10].

The physical reason of the rate decreasing could be a mass transfer resistance in the liquid layer near the external surface of the grain (Fig. 2). Indeed, the surface transfer coefficient ks depends on the layer thickness and the mass transfer surface area S = 2 (R – ) L (where L is the grain length) and can be defined as ks = Dl S/ , where Dl is the water diffusivity in the liquid layer. This coefficient decreases during the water sorption due to gradual reduction of S with simultaneous raise of (Fig. 2). This model could explain the observed effect only if the water diffusivity in the liquid phase (Dl) is lower than in the gas phase (Dg). Indeed, for molecular diffusion in large transport pores of silica the water diffusivity in air at P=1 atm equals Dg = 0.23 cm2/s while in a 3M CaCl2 aqueous solution Dl = 1.27 10-5 cm2/s. Such low gas diffusivities are observed in small silica pores of 60 nm size where the Knudsen regime takes place. In general case, the diffusion through both gas and liquid phase can take place and for these parallel routes the effective diffusivity Deff can be

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calculated as 1/Deff = xl /( l Dl) + xg /( g Dg), where xl and xg are the pore volume fractions occupied by the liquid and gas phase, respectively (xl + xg = 1); l and g are the tortuosity factors for these phases. Thus, to analyse the water transport in SWS-composites at least two new effects should be taken into considered, namely, a) the surface resistance due to formation of the solutions liquid layer near the grain external surface, and b) the complex water diffusivity both through gas and liquid phase in this layer. This pattern is formally similar to the case of the gas and surface diffusion but it is more general as the surface diffusion occurs along the geometrical surface (that means mainly within the first monolayer, xl 183 min. almost coincide to give a “universal diffusion profile” (Fig.9). To obtain the mass diffusivity from the measured water concentration profiles this equation can be integrated with respect to x, yielding w

Deff

0.5

1 dw dw 0 d w

The thus calculated diffusivity is presented against the sorbed water uptake in Fig. 10. The effective water diffusivity shows the tendency to increase from 0.5 10-6 cm2/s to 3.4 10-6 cm2/s during water adsorption, which is in agreement with the increase of the water self-diffusivity measured by PFG NMR. 4. Kinetics of Water Sorption at Constant Volume-Variable Pressure Conditions The kinetics of water sorption on SWS-1L loose grains on an isothermal plate (at T = 50oC) has been measured over a water uptake range 0 – 0.4 g/g. Data

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obtained for various grain sizes, namely, 0.34-0.5; 0.71-1.0; 1.3-1.6, 3.0-3.2 mm, and various salt contents (0 – 33.6 wt.%) will be published elsewhere. Here we mainly focus our attention on the kinetics of water sorption on SWS1L(33.6 wt.%) at low water uptake where the formation of CaCl2 hydrates takes place. The adsorption kinetic curves are found to be non-exponential with an exponential tail so that (1 – mt/m ) ~ exp(-Kt) at long times t (Fig. 11). To describe the rate of water sorption we use the characteristic time of this exponential ( = 1/K) as well as the times 0.5 and 0.9 that correspond to the dimensionless water loadings of 0.5 and 0.9, respectively. All these times are presented in Fig 12 at various water loadings which correspond to appropriate points 1-11 of the water sorption isotherm. The slowest transformation is observed when the formation of the salt dihydrate CaCl2 2H2O takes place

CaCl2 + 2H2O

CaCl2 2H2O.

6

4

2

uptake mol/mol

0

6

4

2

0

6

4

2

0 0

2

4

6

D istance , m m

Figure 8. Water concentration profiles in the SWS-1A layers with various binder content B from 2.5 to 20 wt.%. The profiles are measured every 86 min. The first one was measured at time 11 min. Primary particle size is 0.25-0.5 mm. Vapour pressure is 7.2 mbar.

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-5.5

2

Log D , sm / s

Water content , g/g

0.2

0.1

0.0 0.000

0.002

-6.5 0.00

0.004

sm s

-6.0

0.05

0.10

0.15

0.20

uptake g/g

-0.5

Figure 9. Water sorption profiles presented as a function of . SWS-1A layer with 20 wt. of binder. Profiles at 3 h 03 min. < t < 7 h 20 min. are treated. (■) “universal diffusion profile”.Vapour pressure – 7.2 mbar.

Figure 10. Effective water diffusivity in SWS-1A layer (20 wt. of binder) against the moisture uptake calculated from the “universal diffusion profile” (Fig. 9). Vapour pressure 7.2 mbar.

A similar dependence (w) is observed for the desorption run which does not coincide with the adsorption one (Fig. 12). Thus, one more inportant feature of the materials “salt inside porous matrix” is a slow chemical transformation due to hydrate formation inside the pores, so that the diffusion with slow reaction and hysteresis has to be considered. The effective water diffusivity estimated from these data is changed from 4.2 10-11 m2/s to 1.8 10-10 m2/s for w = 0 – 0.11 g/g and from 8.7 10-11 m2/s to 5.6 10-10 m2/s at w = 0.11 – 0.47 g/g. 1 0.9 0.8 0.7 0.6

7000

0.5

0

2

6000

10

12

0.14 0.12

0.9

0.3

11

6

0.2

5

3

8

4000 3000

5

10

0.06

2400

3600

4800

6000

7200

Time [s]

Figure 11. Kinetic curves of water sorption on SWS-1L (33.6 wt.%). Curve numbers correspond to appropriate points of the water sorption isotherm (see Fig. 12).

0.04

3

1000 1200

0.08

6

4

2000

0

0.1

10

7

1 0.1 0.09 0.08 0.07 0.06

9

1

0.02 2

0 0

Water Uptake [g/g]

0.5

5000 Time [s]

1 - mt / m

8

Adsorption Isotherm, 50 oC Desorption Isotherm, 50 oC =1/K

0.4

0.05

Pressure [mbar] 6

4

0.02

0.04

0.06 0.08 Water Uptake [g/g]

0.1

0 0.12

Figure 12. Characteristic sorption times for SWS-1L (33.6 wt.% ) as a function of the water uptake as well as the water sorption and desorption isotherms(bold symbols).

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Acknowledgments The authors thank the Russian Foundation for Basic Researches (projects N 0203-32304 and 02-03-32770), the Integration Grant Program of the SB RAS (project N 166) and the NATO (grant PST.CLG.979051) for partial financial support of this work. References 1. 2.

US Patent N 1,740,351. (1929) Dehydrating substance, H.Isobe, Dec. 17. Levitskii E.A., Aristov Yu. I., Tokarev M.M., Parmon V.N. (1996) Sol. Energy Mater. Sol. Cells 44, N 3, pp.219-235. c. References 3. Aristov Yu.I., Tokarev M.M., Cacciola G., Restuccia G. (1996) Selective water sorbents for multiple applications: 1. CaCl2 confined in mesopores of the silica gel: sorption properties, React.Kinet.Cat.Lett. 59, N 2, pp.325334. 4. Aristov Yu.I., Tokarev M.M., Cacciola G., Restuccia G. (1996) Selective water sorbents for multiple applications: 2. CaCl2 confined in micropores of the silica gel: sorption properties, React.Kinet.Cat.Lett. 59, N 2, pp.335342. 5. Aristov Yu.I. (2003) Thermochemical energy storage: new processes and materials, Doctoral Thesis, Boreskov Institute of Catalysis, Novosibirsk, 375p. 6. Gordeeva L.G., (2000) New materials for thermochemical energy storage, Ph.D. thesis, Boreskov Institute of Catalysis, 146p. 7. Aristov Yu.I., Restuccia G., Cacciola G., Parmon V.N. (2002) A family of new working materials for solid sorption air conditioning systems, Appl.Therm.Engn. 22, N 2, pp.191-204. 8. Aristov Yu.I., Tokarev M.M., Gordeeva L.G., Snitnikov V.N., Parmon V.N. (1999) New composite sorbents for solar-driven technology of fresh water production from the atmosphere, Solar Energy 66, N 2, pp 165-168. 9. Kaerger J., Ruthven D.M. (1992) Diffusion in Zeolites and Other Microporous Solids, Wiley, N.Y. 10. Koptyug I.V., Khitrina L.Yu., Aristov Yu.I., Tokarev M.M., Iskakov K.T., Parmon V.N., Sagdeev R.Z. (2000) 1H NMR microimaging study of water vapor sorption by individual porous pellets, J.Phys.Chem. 104, pp.16951700. 11. Aristov Yu.I., Koptyug I.V., Glaznev I.S., Gordeeva L.G., Tokarev M.M., Ilyina L.Yu. (2002) 1H NMR microimaging for studying the water transport in an adsorption heat pump // Proc. Int.Conf.Sorption Heat Pumps, Sept. 23-27 Shanghai, China, pp.619-624.

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12. Galvosas P., Stallmach F., Seiffert G., Kärger J., Kaess U., Majer G. (2001) Generation and Application of Ultra-High-Intensity Magnetic Field Gradient Pulses for NMR Spectroscopy, J. Magn. Reson. 151, pp. 260268. 13. Dawoud B., Aristov Yu.I. (2003) Experimental study on the kinetics of water vapour sorption on selective water sorbents, silica gel and alumina under typical operating conditions of adsorption heat pumps, Int.J.Heat&Mass Transfer 46, pp. 273-281. 14. Crank J. (1975)The Mathematics of Diffusion, Oxford Univ. Press, pp.230238.

EFFECT OF CARBONACEOUS COMPOUNDS ON DIFFUSION OF ALKANES IN 5A ZEOLITE F. BENALIOUCHE*, Y. BOUCHEFFA*, P. MAGNOUS** * UER de Chimie Appliquée, EMP, BP 17 Bordj El-Bahri, Alger, Algérie ** UMR 6503, Laboratoire de Catalyse en Chimie Organique, 40 Avenue du Recteur Pineau, 86022, Poitiers, France

Abstract Adsorption of n-pentane, n-hexane and isopentane on 5A zeolite was studied using a microbalance at 100, 250 and 420°C (P = 27kPa). The analysis of the compounds trapped in 5A zeolite pores shows a relation between the formation of a carbonaceous compound and an inconsistency between experimental rates of uptake and theoretical curves using the Fick’s second law for diffusion.

1. Introduction Various adsorption processes using 5A zeolite adsorbents (at 150 to 350°C) have been developed for separation of n-alkanes and isoalkanes within the framework of gasoline reformation [1, 2]. As in the case of catalytic processes, one of the main problems is the deactivation of the molecular sieves resulting from the transformation of adsorbates to heavy compounds [3]. Theses products, formed on 5A zeolite during the separation process could result from the acid transformation of alkenes resulting from cracking alkanes [4, 5], either present in the feed or formed by cracking of alkanes or isoalkanes. Indeed, the 5A zeolite has a relatively large number of acid sites : calcium cations (Lewis sites) or protonic sites resulting from the partial dissociation of water molecules by calcium cations [6]. On the other hand and in the previous studies, the adsorption of isopentane in D cage of 5A zeolite is clearly established [7]. Using a microbalance, we will show in this work that in the n/iso-alkanes (nhexane, n-pentane and isopentane) sorption on 5A zeolite powder, the Fick’s 567 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 567–573. © 2006 Springer. Printed in the Netherlands.

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second law, applied for small time at constant pressure and volume serves to explain the effect of carbons compound formation on self-diffusion.

2. Experimental Part 5A zeolite (Na3.1Ca4.45Al12Si12O48), which corresponds to an exchange rate of Na+ by Ca2+ of about 74% was supplied by Procatalyse. The adsorption capacity of nitrogen adsorbed at –196°C is equal to 0.26 cm3/g. The adsorption of pentane, hexane and isopentane (purity > 99%, from fluka) was operated at 100, 250 and 420°C for 5 hours using Setaram thermobalance (V=3L) linked to a computer by way of Cobra interface. Before introduction of adsorbates at 27kPa, the zeolite sample (60mg, grain size = 1-2µm) was pretreated in vacuum at 420°C for 5 hours. When the valve from the liquid adsorbate to the volume sorption was opened a negligible period (10 s) elapsed before establishment of an equilibrium vapour pressure. To avoid alkane condensation, the thermogravimetric system was maintained at approximately 40°C. The method used for recovering the adsorbed phase was previously reported [8]. It consists of treating the samples with hydrofluoric acid solution in order to liberate the compounds trapped in the zeolite pores. The gas liberated during this operation was analysed by GC whereas the other compounds were recovered in CH2Cl2 and analysed by GC and GC/MS coupling. This analysis was performed on samples (1g) treated during 50 hours under various alkanes in order to recuperate significant quantities of heavy compounds.

3. Results 3.1 ADSORPTION of n-PENTANE, n-HEXANE and ISOPENTANE The effect of time on the increase in weight of the 5A zeolite in contact with npentane, n-hexane or isopentane was determined at 100, 250 and 420°C (fig. 1). After 5 hours of adsorption at 100, 250 and 420°C, the adsorbed quantities decrease with temperature. At 100°C, with n-pentane and n-hexane, the increase in weight is very fast during the first minutes and a plateau was obtained after 5 hours (12 and 10.1% for n-hexane and n-pentane respectively). However for isopentane, the plateau stabilizes to a value of 1.8%. When the temperature increases (250 and 420°C), the adsorbed quantities from alkanes decreases with temperature in agreement with the adsorption laws but for isopentane it decreases (at 250°C) then slightly increases (at 420°C). This behaviour was elucidated by analysis of adsorbed phases (table 1) which shows that from 250°C the adsorption of isopentane leads to the formation of small

569

Effect of Carbonaceous Compounds on Diffusion of Alkanes in 5A Zeolite

amounts of cracked products (olefin and paraffin) and branched (saturated and unsaturated) C15 to C24 hydrocarbons. At 420°C, whatever the alkane, C2-C6 aliphatic hydrocarbons and C8-C15 aromatic compounds (methylbenzenes and methyl naphthalenes) appear which result from the transformation of n-pentane, n-hexane and isopentane. a: n-pe ntane

Wt (%)

b: n-he xane

Wt (%) 14

14 12 10 8

10

6

6

4

4

420°C

1

2

3

4

420°C

2

0 0

250°C

8

250°C

2

100°C

12

100°C

5

Time (h)

0 0

1

2 3 Time (h)

4

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c: isope ntane

Wt (%) 2.5 2

100°C

1.5

420°C

1 250°C

0.5 0 0

1

2 3 Time (h)

4

5

Figure 1. Weight increase of 5A zeolite vs. time at 100, 250 and 420°C in contact with n-pentane (a), n-hexane (b) and isopentane (c), P = 27kPa. Isopentane is transformed faster than n-pentane and n-hexane (approximately 2 times at 420°C between n-pentane and isopentane, for example). The higher reactivity of isopentane can be explained by the higher stability of its carbenium ions. Most of these are tertiary whereas most of the carbenium ions involved in n-pentane or n-hexane transformation are secondary. The resulting alkenes could also undergo, oligomerization at 250°C or can be transformed into aromatics at 420°C involving hydrogen transfer and cyclization reactions [9].

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Table 1. Composition of adsorbed phase obtained from n-pentane, n-hexane and isopentane. Composition of adsorbed phases 100°C 250°C

n-pentane (nC5)

n-hexane (nC6)

Isopentane (iC5)

nC5 (100%)

nC6 (100%)

nC5 (100%)

nC6 (100%)

nC5 (90%) C2-C6 (5%),

nC6 (90%) C2-C6 (5%),

iC5 (100%) iC5 (90%) ; C2-C6 (5%) C15- C24 oligomers (5%) iC5 (85%) ; C2-C6 (5%)

(CH3)x (5%) 420°C

x = 2 to 5

(CH3)x (5%)

x = 2 to 5

(CH3)x (10%) x = 2 to 5

(CH3)x x = 1 to 5

3.2. DIFFUSION of n-PENTANE, n-HEXANE and ISOPENTANE

Using the second Fick’s law for determination of values of self-diffusion coefficient of alkanes, we will show that the curves plotted for small time at low temperatures (100 and 250°C): mt/mf = 6/S 1/2 (D t/r02)1/2 (where mt and mf are respectively the adsorbed amounts at time t and equilibrium, D is the diffusion coefficient, r the radius of the zeolite crystallite and t the sorption time) are perfectly in agreement with theoretical curves obtained from solution equation of second Fick’s law solution: 2 2 f mt 1 6 1 ˜exp§¨  D˜n ˜S ˜t ·¸ ¦ 2 2 ¨ ¸ S2 n 1n r0 mf © ¹

Indeed, in figure 2 for n-pentane, n-hexane and isopentane adsorbed at 100 and 250°C, the curves are the same. When the temperature increases, the theoretical and experimental curves diverge especially at long times. In the case of isopentane, the concordance was limited to values of mt/mf < 0.5. This was explaining by the formation of carbonaceous compounds such has been found

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Effect of Carbonaceous Compounds on Diffusion of Alkanes in 5A Zeolite

from adsorbed phase analysis. The thermal effect and surface barrier during the adsorption were disregarded because of the low quantity of sample treated during thermogravimetric measurements (60 mg) and a large diameter of the glass bucket containing the zeolite (10 mm). 250°C

100°C

n-pentane

m t/m oo 1 0.8

m t/m oo 1

__ The ore tical … Expe rime ntal

0.6 0.4

0.4 0.2 0

2

3

4

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n-hexane

1

2 3 4 Time (min 1/2 )

0.4

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m t/m oo 1

1

2 3 4 Time (min 1/2 )

0.4

isopentane

… Expe rime ntal

0.6

1

2

3

4

0

5

0.6

m t/m o 1

__ The ore tical

0.8

… Expe rime ntal

0.6 0.4

0.2

0.2

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0 5

3

4

5

Time (min )

isopentane

0.8

2

1/2

0.2 2 3 4 Time (min 1/2 )

1

Time (min 1/2 )

0.4

1

5

__ The ore tical … Expe rimental

0.4

0.4

0

4

0

m t/m o 1

__ Theore tical

0.8

0.6 0.2

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3

0.8

__ The ore tical … Expe rime ntal

0.6

0

0

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n-hexane

m t/m oo 1

0.2

0.2

1

Time (min 1/2 )

n-hexane

0.8 __ The ore tical … Expe rime ntal

0.6

__ The ore tical … Expe rime ntal

0.4 0

0

m t/m o 1

0.8

0.6 0.2

Time (min 1/2 )

m t/m oo 1

n-pentane

0.8 __ The ore tical … Expe rime ntal

0.6

0 1

m t/m oo 1

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420°C

n-pentane

isopentane __ The ore tical … Expe rime ntal

0

0

1

2

3

Time (min 1/2 )

4

5

0

1

2

3

4

5

Time (min 1/2 )

Figure 2. Sorption kinetic of n-pentane, n-hexane and isopentane on 5A zeolite at 100, 250 and 420°C (P=27 kPa). However there is considerable experimental noise for the isopentane uptake data, as is showed in figure 2. This is probably due to the slow uptake as isopentane has a kinetic diameter greater than window apertures of 5A zeolite. This phenomenon was also observed for isoalkanes adsorbed on the same zeolite [10]. Table 2 shows the values of apparent diffusion coefficient (Di/r02) obtained from the approximation of second Fick’s law (determined from the

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slope of the linear part of the curve). From these results, we can observe that the values are of the same order of magnitude for n-pentane and n-hexane. However the diffusion, at 100°C, of isopentane is approximately 500 times lower than that of n-pentane or n-hexane. Table 2. Apparent diffusion coefficient of n-pentane, n-hexane and isopentane in 5A zeolite. Adsorbates n-pentane n-hexane isopentane 100°C 2.4 2.3 0.005 Di (10-3) s-1 250°C 5.4 4.8 0.018 420°C 8.1 6.9 0.021

4. Conclusion Using thermogravimetric measurements, we show the validity of sorption kinetic of n-hexane, n-pentane and isopentane at constant pressure and volume with Fick second law model applied at small times. This was validity at 100°C for n-hexane and n-pentane up to saturation. However, this agreement was limited in the case of isopentane. Above 250°C, the formation of carbonaceous compounds identified by analysis of adsorbed phase explains the divergence between theoretical and experimental results. References 1. 2. 3. 4. 5. 6.

Johnson J. A. and Oroska A. R. (1989) Zeolites as Catalysis, Sorbents and Detergent Builders, Studies in Surface Science and Catalysis, 46, Karge, H. G., Weitkamp, J., Eds. 451. Jullian S., Mank, L., Minkkiner A. Fr. (1993) Patent 2.679.245; (1993) U.S. Patent 5.233. 120. Guisnet M., Magnoux P., (1994) Catalysts Deactivation; Studies in Surface Science and Catalysis, 88, Delmon B. and Froment, G. F., Eds. 53. Thomazeau C., Cartraud P., Magnoux P., Jullian S., Guisnet M. (1996) Microporous Materials 5, 337. Boucheffa Y., Thomazeau C., Cartraud P., Magnoux P., Guisnet M. and Jullian S., (1997) Ind. Eng. Chem. Res., 36, 3198. Mix H., Pfeifer H., Standte B. (1988) Chem. Phys. Lett. 146 6 541.

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Magnoux P., Boucheffa Y., Guisnet M., Joly G., Jullian S. (1998) Fundamentals of Adsorption VI, Meunier, F., Eds. 135. 8. Guisnet M., Magnoux P., (1989) Appl. Catal. 54 1. 9. Misk M., Joly G., Magnoux P., Jullian S., Guisnet M. (1996) Zeolites 16 265. 10. Benaliouche F., Boucheffa Y. and Magnoux P. (2003) J. Soc. Alger. Chim.,13 (2)

APPLICATION OF INTERFERENCE AND IR MICROSCOPY FOR STUDIES OF INTRACRYSTALLINE MOLECULAR TRANSPORT IN AFI TYPE ZEOLITES C. CHMELIK, E. LEHMANN, S. VASENKOV, B. STAUDTE, J. KÄRGER Universität Leipzig, Institut für Physik, Linnéstr. 5, D-04103, Leipzig, Germany Abstract Interference microscopy (IFM) and FTIR microscopy (IRM) are applied to study intracrystalline concentration profiles in SAPO-5 and CrAPO-5 zeolite crystals. By using both techniques, the high spatial resolution of interference microscopy is complemented by the ability of FTIR spectroscopy to pinpoint adsorbates by their characteristic IR bands. Intracrystalline concentration profiles of water, adsorbed in large crystals of CrAPO-5 and SAPO-5 under equilibrium with water vapor at 1 and 20 mbar, were determined with use of interference microscopy. At lower pressure, the profiles reveal highly inhomogeneous distributions of intracrystalline water in both types of crystal. This effect is attributed to structural heterogeneity of the crystals. The structural heterogeneity has been found to have a low or no influence on the final uptake level at 20 mbar of the crystals. Concentration profiles of methanol during its adsorption into the onedimensional channels of CrAPO-5 crystals are reported. The exceptionally high spatial resolution allowed us to obtain detailed information on the interplay of intracrystalline diffusion, the permeability of the crystal surface and the role of the internal structure on molecule uptake.

1. Introduction Due to their regular structure and a well-defined morphology zeolites are broadly used as catalysts and molecular sieves in different fields of applied chemistry and technology. Considering diffusion, due to simplicity of the framework topology, a zeolite consisting of a packing of oriented cylinders (viz. AFI type) represents an ideal model system for investigations of the intracrystalline transport [1]. While the ideal structure of zeolites is routinely used to elucidate their 575 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 575–581. © 2006 Springer. Printed in the Netherlands.

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adsorption and transport properties it was only recently appreciated that these properties can be influenced to a great extent by building defects of the crystals [2,3]. Investigation of intergrowth effects in zeolites is particularly important since these phenomena may strongly influence molecular uptake and intracrystalline diffusion. Such investigations are also important in view of the persistent differences between intra-crystalline diffusivities obtained for the same zeolites by various experimental techniques [2,3]. In the present work interference microscopy (IFM) and IR microscopy (IRM) are applied to study the internal structure of zeolite crystals as well as the influence of this structure on intracrystalline molecular transport.

2. Experimental Section The measurements were carried out by applying the only two techniques, which have been proved to be suitable for studies of intracrystalline concentration profiles of guests in zeolite crystals [4-8]. The first one is the interference microscopy technique (IFM), which has been recently introduced in our laboratory. It is based on following the change of the refractive index of a zeolite crystal during molecular adsorption or desorption. Due to the proportionality of the refractive index and the local concentration of guest molecules in the crystal it is possible to monitor the concentration integrals in the direction of observation. The concentration integrals were also monitored in a somewhat more direct way by using the IR microscopy method (IRM). Despite a poor spatial resolution, IR microscopy presents an extremely useful tool to study intracrystalline concentration profiles due to its ability to distinguish between different adsorbates. Thus, it opens the possibility for tracer-exchange measurements. For the measurements and activation the zeolite sample was introduced into a specially made optical or IR cell, which was connected to the vacuum system. The adsorption, desorption or tracer exchange was achieved by appropriate changes of adsorbate pressure in the surrounding gas phase. 3. Results and Discussion

3.1. METHANOL IN CrAPO-5: EQUILIBRIUM WITH THE VAPOR PHASE The IFM equilibrium concentration profiles at a pressure of 1 mbar are shown in Figure 1a. These non-homogenous profiles can not be explained by using the ideal textbook structure [1]. In view of the image obtained for an unloaded crystal under crossed Nicols we ascribe these non-homogeneous profiles to the

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influence of regular intergrowth effects. They render parts of the channel system to be inaccessible for methanol molecules. Based upon the concentration profiles we propose an internal structure schematically shown in Figure 1c [7]. To confirm the IFM results concentration profiles of methanol in CrAPO-5 crystals were recorded under the same measurement conditions by IRM. One-dimensional concentration profiles were compared. Figure 1b demonstrates the good agreement between the results obtained by both techniques [7].

Figure 1. (a) IFM equilibrium intracrystalline concentration profile of methanol in a CrAPO-5 crystal. The color intensity is proportional to the integrals of local concentration. (b) Comparison of mean concentration integrals I recorded by IRM and IFM. (c) Suggested internal structure. The channel direction coincides with the z direction.

3.2. METHANOL IN CrAPO-5: UPTAKE KINETICS The primary aim of the dynamic Monte Carlo simulations was to investigate quantitatively the influence of the intergrowth structure and the transport barriers on the crystal surface on the intracrystalline transport (figure 2). The used simulations are analogous to the numerical solution of Fick’s second-lawtype equations for diffusion in one-dimensional channels (accessible part only) with transport barriers at the channel edges. Comparing the simulations and experimental results we finally obtain the value of D=0.43x10-12 m2/s for the

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intracrystalline diffusivity of methanol in the limit of small loadings and can estimate the rate constant of barrier penetration to Į=0.35x10-7 m/s. (a)

(b)

Assumptions (MC): 1.one-dimensional random walk in the lattice 2. probability of diffusion step is independent of concentration

Figure 2. (b) Intracrystalline concentration of methanol, integrated along the y crystallographic direction in CrAPO-5 at different times after the start of the methanol adsorption. The profiles were obtained by IFM (black line) and by the dynamic MC method (broken line). (a) The internal structure of CrAPO-5 crystals (shown only for the lower part of the crystal). x, y and z are the crystallographic directions. The broken lines outline the observation plane of the IFM measurements of the transient profiles with the arrow indicating the direction of observation.

The reported results show that the estimated value of the diffusion coefficient D is around a factor of 2 smaller than that of L*Į (L - crystal size in z-direction). This indicates that the rate of methanol uptake is mainly determined by the intracrystalline diffusivity of methanol. However, this rate is somewhat reduced due to existence of transport barriers on the crystal surface [9].

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3.3. WATER IN CrAPO-5 AND SAPO-5: EQUILIBRIUM WITH THE VAPOR PHASE Figure 3 shows the intracrystalline concentration profiles of water in the CrAPO-5 and SAPO-5 measured by IFM. It is seen that the profiles obtained for low pressure of 1 mbar reveal highly inhomogeneous, but reproducible patterns [10]. Noteworthy is that the profile in CrAPO-5 (figure 3 a1, a3) resembles, to some extent, the methanol profile observed for these crystals (figure 1a). Apparently, the explanation might also be applied to water. The origin of the intergrowth effects may be derived from the progress of the crystal growth process. As has been shown in ref. [11] the dumbbell shape is characteristic of some AFI-type crystals in the intermediate stage of growth.

Figure 3. Intracrystalline concentration profiles of water in the CrAPO-5 (a1, a2, a3) and SAPO-5 (b1, b2, b3) crystals integrated along the y direction under equilibrium with water vapor at 1 mbar (a1, b1, a3, b3) and 20 mbar (a2, b2). The channels run along the z axis. Darker regions correspond to higher concentration integrals.

The increase in water vapor pressure to 20 mbar results in an essentially homogeneous profile in CrAPO-5 (figure 4 a2), i.g. all the crystal constituents are equally filled with liquid-like water.

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Imaginable structural factors which can influence the intracrystalline water concentration at low water vapor pressure are the content and the distribution of the Cr atoms and/or the presence of defect sites. The inhomogeneous profiles of water in SAPO-5 may result from a structural heterogeneity of the crystals as well. For SAPO-5 has been indicated by electron microprobe analysis that the silicon content of the central part of the crystals was lower by a factor of 2 to 3 than that at the crystal margin [12]. By studying the progress of crystal growth it was found that “pencil-like” crystals are formed initially, later the tips of the “pencils” flatten out proceeding with a much higher consumption of silicon than the initial one. At the high water vapor pressure the condensation of water and the volume filling of the pores of SAPO-5 occur, which are primarily determined by the accessible pore volume as justified by the homogeneous concentration profile observed at 20 mbar (figure 4 b2). It may be speculated that the crystal components which form at the earlier stages of the growth process, i.e. dumbbell core (CrAPO-5) or “pencillike” (SAPO-5) core, can adsorb more water than the other crystal parts under the low vapor pressure of water [10]. 4. Conclusion The obtained results show large potentials of the new IRM and IFM methods for elucidation of structural and transport properties of zeolite crystals, particularly when these techniques are combined with dynamic MC simulations. From the experimental evidence provided by these techniques, the real structure of crystals, which reveal perfect shape characteristic for single crystals, has been found to deviate decisively from the textbook structure of the given type of zeolite. Combination of the experimental methods with dynamic MC simulations allowed us to investigate separately the influences of (i) the intracrystalline diffusion, (ii) the transport barriers on the external crystal surface and (iii) the effects of the intergrowth structure on molecular uptake. Acknowledgement We acknowledge Prof. F. Schüth and Dr. Ö. Weiss as well as Dr. J. Kornatowski and Dr. G. Zadrozna for synthesizing and providing us the large zeolite crystals. Financial support by Deutsche Forschungsgemeinschaft (SFB 294 and Graduierten-kolleg “Physikalische Chemie der Grenzflächen”), Fonds der Chemischen Industrie and Max-Buchner-Forschungsstiftung is gratefully acknowledged.

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References 1.

Baerlocher Ch., Meier W.M., Olson D.H. (2001) Atlas of Zeolite Framework Types, Elsevier, Amsterdam, pp. 34-35. 2. Kärger J., Ruthven D.M. (2002) Handbook of Porous Solids, Schüth F., Sing K.S.W., Weitkamp J. (Eds.), Diffusion and Adsorption in Porous Solids, Wiley-VCH, Weinheim, p. 2089. 3. Kärger J., Vasenkov S., Auerbach S. (2003) Handbook of Zeolite Catalysts and Microporous Materials, Auerbach S.M., Carrado K.A.,. Dutta P.K (Eds.), Diffusion in Zeolites, Marcel Dekker, New York, 4. Schemmert U., Kärger J., Krause C., Rakoczy R.A., Weitkamp, J. (1999) Europhys. Lett., 46 204. 5. Schemmert U., Kärger J., Weitkamp J. (1999) Microporous and Mesoporous Materials, 32 101. 6. Geier O., Vasenkov S., Lehmann E., Kärger J., Schemmert U., Rakoczy R.A., Weitkamp J. (2001) J. Phys. Chem. B, 105 10217-10222. 7. Lehmann E., Chmelik C., Scheidt H., Vasenkov S., Staudte B., Kärger J., Kremer F., Zadrozna G., Kornatowski J. (2002) J. Am. Chem. Soc. 124 8690-8692. 8. Geier O., Vasenkov S., Lehmann E., Kärger J., Schemmert U., Rakoczy R.A., Weitkamp J. (2001) Stud. Surf. Sci. Catal., 135 154-160. 9. Lehmann E., Vasenkov S., Kärger J., Zadrozna G., Kornatowski J. (2003) J. Chem. Phys., 118, 6129-6132. 10. Lehmann E., Vasenkov S., Kärger J., Zadrozna G., Kornatowski J., Weiss Ö., Schüth F. (2003) J. Phys. Chem. B, 107, 4685-4687. 11. Klap G.J., Wübbenhorst M., Jansen J.C., van Konigsveld H., van Bekkum H., van Turnhout J. (1999) Chem. Mater. 11, 3497-3503. 12. Schunk S.A., Demuth D.G., Schulz-Dobrick B., Unger K.K., Schüth F. (1996) Microporous Mater., 6, 273-285.

COAL CHARACTERIZATION FOR CARBON DIOXIDE SEQUESTRATION PURPOSES G. Di FEDERICOI, I. CAMPLONEI, S. BRANDINII, J. BARKERII Centre for CO2 Technology, University College London I Department of Chemical Engineering, UCL, Torrington Place, WC1E 7JE, London, UK II Department of Earth Sciences, UCL, Gower Street, WC1E 6BT, London, UK

1. Introduction The greenhouse gas effect helps to regulate the temperature of our planet. It is the result of heat adsorption by a number of gases in the atmosphere called greenhouse gases because they effectively ‘trap’ heat in the lower atmosphere. Water vapour is the most abundant greenhouse gas, followed by carbon dioxide and methane. Without a natural greenhouse gas effect, the temperature of the Earth would be about –18°C instead of its present 14°C [1]. So, the concern is not with the fact that there is a greenhouse gas effect, but whether human activities are leading to an enhancement of this effect. Reduction in CO2 emissions is the most quoted solution for global warming in the short time. At the third Conference of Parties of the Framework Convention on Climate Change (FCCC) in Kyoto (1997), developed countries agreed to stabilise emissions of greenhouse gases through a 5.2% reduction from 1990 levels by 2008-12. In order to stabilise atmospheric carbon dioxide concentrations it will be necessary to reach more than 60% reduction by 2050 [2]. Table 1. CO2 emissions in 1996 and reductions agreed in Kyoto [3]. Nation USA Russian Federation Germany UK Canada Poland

Total emission (1996) million tonnes/year 5043 1500 924 571.4 507 373

Reduction required % 7 0 21 12.5 6 6

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2. CO2 Sequestration Carbon dioxide sequestration is one of the options that are being considered for the reduction of greenhouse gas emissions. It would enable the world to continue to use fossil fuels and at the same time reduce emissions from their combustion. It involves two main stages that are the gas capture and its storage in suitable geological structures. In the capture operation CO2 is separated from flue gases produced in combustion processes using adsorption, absorption, cryogenic or membrane systems. The storage is realised by injecting the carbon dioxide in the coal seams, depleted gas reservoirs, depleted oil reservoirs, etc [4]. The present work is focused on the characterisation of coal for CO2 sequestration and simultaneous production of methane. The anthropogenic methane contained in coal can be regarded as a significant source of energy. This methane is usually referred to as coal-bed methane (CBM) in the literature. CBM is retained in coal in three ways: first, as free gas within the pore space or fractures in coal; second, as adsorbed molecules on the organic surface of coal; and third, dissolved in groundwater within the coal. The depletion techniques used to recover CBM allow the recovery of 50% of the total amount contained in the reservoir. Carbon dioxide is more strongly adsorbed in coal than methane therefore the injection of carbon dioxide will enhance the production of methane with recoveries of up to 100% [5]. This process is often referred to as enhanced-CBM (ECBM). 2.1. GAZ TRANSPORT IN COAL The gas transport through the coal reservoir is governed by the permeabilitycontrolled flow in the macropore system and by the diffusion-controlled flow in the micropore system. The diffusion-controlled flow obeys Fick’s law. Clarkson and Bustin [6] estimated that the inverse of the CO2 effective diffusion time constant, D` D l 2 , for Cretaceous Gates Formation is between 6·10–3 s–1 and 3·10–2 s–1 in the macropores and between 2·10–3 s–1 and 4·10–3 s–1 in the micropores. The permeability-controlled flow is generally considered to be laminar, so it obeys D’Arcy’s law [7].

m



Uk ’P P

(1)

with U the gas density, k the permeability, P the pressure. Permeability is a measure of coal ability to transmit fluids therefore knowing the coal permeability the injection pressure can be estimated. Typical values for the permeability are between 0.1 and 250 md for US coals [8].

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The permeability of a coal seam varies with effective external stress, which can be defined as the difference between the external stresses applied on a rock and the internal pressure within the pore space of the rock. Increasing the external stress the permeability decreases because the system shrinks and the pressure drops in the matrix increase. Coal matrix shrinkage and the resulting change in cleat or fracture system porosity can have a profound effect on reservoir permeability and thus on production (or injection) performance. In fact, gas desorption causes a decrease in internal pressure, so it is expected that this would cause an increase in the effective stress and hence a decrease in permeability. Another important effect that occurs in the sequestration of CO2 in coal is the swelling of the structure. Coal swells upon exposure to carbon dioxide and the extent of expansion increases with increasing carbon dioxide pressure. The application of the external stress limits the extent of coal swelling. In this project the swelling is evaluated measuring the radial and the axial variation for the sample size under various stress conditions. It will be useful to develop a physical model to predict the swelling and the shrinkage in a coal reservoir and the associated changes in permeability in order to predict the maximum amount of CO2 that can be sequestered. Most available laboratory data, show that CO2 adsorption causes more strain and swelling than CH4 and N2 because it is adsorbed in larger concentration by coal and also suggest that CO2 causes more swelling on a unit of concentration basis [9]. 2.2. COAL ADSORPTION The sequestration process is controlled by the absorptive capacity of coal in terms of the maximum amount adsorbed and its rate of sorption. Data available in the literature is sparse but provides insight into the applicability of simple isotherm models. Experimental data for Fruitland Coal A [10] show favourable shaped isotherms with carbon dioxide more strongly adsorbed than methane and nitrogen (Figure 1). This behaviour is the driving force for the ECBM process in which CO2 replaces CH4. Figure 1 shows that the experimental data can be correlated using the Langmuir (eq. 2) and the truncated virial (eq. 3) isotherms.

q

bp q

qs

bP 1  bP

exp1  A1q

(2) (3)

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where q is the quantity sorbed, P the pressure. qs, b, A1 are the models’ parameters determined by regression of the single component data.

Amount adsorbed (mol/g-coal)

0.0012 0.001

Virial Langmuir

carbon dioxide

0.0008

methane 0.0006 0.0004

nitrogen

0.0002 0 0

20

40

60

80

100

120

140

Pressure (bar)

Figure 1. Single component sorption isotherms for Fruitland Coal A at 115ºF.

Amount adsorbed (mol/g-coal)

0.0012 0.001 0.0008 0.0006

methane carbon dioxide

0.0004 0.0002 0 0

0.1

0.2

0.3

0.4

0.5

0.6

EVP-scaled

Figure 2. Single component sorption isotherms for Fruitland Coal A at 115ºF with scaled pressures.

The Langmuir parameters obtained from the Fruitland Coal A experimental data show a molecule dependent saturation capacity which is thermodynamically inconsistent [11]. For nitrogen qs = 7.416·10–4 mol/g-coal,

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methane qs = 1.014·10–3 mol/g-coal and carbon dioxide qs = 1.327·10–3 mol/gcoal, therefore the Langmuir model should be considered only as a useful fitting function. It is interesting to note [10] that by plotting the data on a reduced pressure basis, p/p*, where p* is the extrapolated vapour pressure (EVP) using a simple two constant Clapeyron form, CO2 and CH4 isotherms are in reasonable agreement as shown in figure 2. Using the Ideal Adsorbed Solution theory it is possible to obtain a reasonable agreement with the experimental binary adsorption data as shown in Table 2. Table 2. CH4-N2 equilibrium data [10]. Pressure (bar) 25.77

Total sorption Total sorption Total sorption Gas IAS-Virial Composition Experimental IAS-Langmuir (mol/g) (mol/g) (mol/g) ymethane 0.6865 4.21E-04 4.56E-04 4.52E-04

55.44

0.7085

6.20E-04

6.34E-04

6.23E-04

83.87

0.7203

7.07E-04

7.16E-04

7.23E-04

27.48

0.0832

2.55E-04

2.43E-04

2.52E-04

57.26

0.0906

3.84E-04

3.72E-04

3.76E-04

82.56

0.0954

4.46E-04

4.38E-04

4.49E-04

25.57

0.1478

2.63E-04

2.62E-04

2.75E-04

58.62

0.1633

4.32E-04

4.21E-04

4.27E-04

83.94

0.1713

5.22E-04

4.90E-04

5.03E-04

3. Design of Test Cell With the aim of characterizing coal for sequestration purposes, a test cell has been designed and assembled. What makes this a challenging problem is that coal will undergo swelling as a consequence of CO2 adsorption, thus modifying the solid properties, in particular permeation [12]. The coal can swell by 1-4% in volume and it is function of the mechanical stress state [13,14] and this swelling can have a large effect on the coal permeability. Therefore it is necessary to reproduce, as close as possible, the stress conditions of the coal sample in the coal reservoir. For typical UK coal the pressure along the bedding plane is lower than the overburden pressure (i.e. along the vertical direction) by a factor of 0.8 to 1 [15].

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A schematic diagram of the test cell is shown in Figure 3. The different stresses are obtained by flowing water at different pressures in the two compartments, thus obtaining the required pressure ratios. The water is also used to maintain the cell at isothermal conditions, which may vary from 25 to 60 ºC. Two HPLC pumps are used to maintain the water pressure constant and a schematic diagram of the system is shown in Figure 4.

Figure 3. Test cell.

Figure 4. Experimental set-up.

To investigate the carbon dioxide adsorption process for a given coal sample, the gas is injected from 1 (Fig.3) and the outlet stream 2 is analysed. The gas injected can be pure carbon dioxide or its mixtures with nitrogen to reproduce streams coming from a power plant. The gas pressure can be varied from atmospheric to 120 atm. In order to quantify the extent of the swelling a measurement of the piston 5 displacement is used to evaluate the variation in the longitudinal length during the experiment. It is also possible to measure the diameter changes of the sample 6 using a strain gauge belt. The use of multiple strain gauge belts to follow the propagation of the swelling along the axis is being considered. When combined with unconstrained adsorption isotherm measurements this test cell should provide the necessary experimental information to characterize coal at the coal matrix scale. The results will be included in a large-scale reservoir simulator currently being developed at UCL’s Earth Sciences Department.

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4. Conclusions Experiments will be performed in order to evaluate the kinetics of adsorption/desorption, the effect of the stress state and swelling on permeability and how it affects the coal structure. They will be carried out with nitrogen, carbon dioxide and methane as pure components or mixtures. From the analysis of the results it will be possible to understand and predict the swelling phenomenon and lead to a physical model for the sequestration process that will consider adsorption, permeability and swelling. 5. Notation CBM ECBM EVP FCCC SCF A D` D l m k

P

P pout

'p pav

U

Coal Bed Methane Enhanced Coal Bed Methane Extrapolated Vapour Pressure Framework Convention on Climate Change Standard Cubic Feet area of cross-section of flow effective diffusivity diffusivity diffusion path length mass flow rate permeability (1 md = 9.8692 10–4 € m2) fluid viscosity pressure outlet pressure pressure drops along the coal sample the average pressure between the injection pressure and outlet pressure gas density

Acknowledgements Financial support from the Leverhulme Trust (Philip Leverhulme Prize), Royal Society (Royal Society Wolfson Research Merit Award) and EPSRC is gratefully acknowledged. References 1. 2.

National Data Climate Center web site: http://www.ncdc.noaa.gov Royal Commission on Environmental Pollution Report 22, June 2000. Energy – The Change in Climate

590 3. 4. 5.

6. 7. 8. 9.

10. 11. 12. 13. 14. 15.

G. Di Federico et al. European Monitoring and Evaluating Programme web site: http://www.emep.no National Energy Technology Laboratory website: www.netl.doe.gov Pashin J. C., Groshong Jr. R.H., Carrol R.E., Enhanced coalbed methane recovery through sequestration of carbon dioxide: potential for a marketbased environmental solution in the Black Warrior Basin of Alabama, in www.netl.doe.gov. Clarkson C.R. and Bustin R.M (1999) The effect of pore structure and gas pressure upon the transport properties of coal: a laboratory and modeling study. 2. Adsorption rate modeling, FUEL 78 1345-1362. Harpalani S. and Chen G. (1997) Influence of gas production induced volumetric strain on permeability of coal, Geotechnical and Geological Engineering 15, 303-325. Jones, S.C. (1992) The profile permeameter; a new fast accurate minipermeameter , Paper SPE 24757 presented at the 1992 SPE Technical Conference and Exhibition, Washington, Oct 4-7. Pekot L.J. and Reeves S.R. Modelling (November 2002) Coal matrix shrinkage and differential swelling with CO2 injection for enhanced coalbed methane recovery and carbon dioxide sequestration applications, U.S. Department of Energy. Chaback J.J., Morgan W.D. and Yee D. (1996) Sorption of nitrogen, methane, carbon dioxide and their mixtures on bituminous coals at in-situ conditions, Fluid Phase Equilibria 117, 289-296. D.B. Broughton(1948) Ind. And Eng. Chem. 40 (8) 1506. Harpalani S. and Schraufnagel R.A.(Sept 1990) Influence of matrix shrinkage and compressibility on gas production from coalbed methane reservoirs SPE 20729, Procs. 65th Ann. Tech. Conf., New Orleans, LA,. Reucroft P.J. and Patel H. (1986) Gas-induced swelling in coal, FUEL, 65, June. Reucroft P.J. and Sethuraman A. R. (1987) Effect of pressure on carbon dioxide induced coal swelling, Energy & Fuels, 1,72-75. Sinka I.C. (1997) An investigation into rock and reservoir properties of coal with special reference to stimulated coalbed methane well performance, PhD thesis.

EFFECT OF THE INTRAWALL MICROPOROSITY ON THE DIFFUSION CHARACTERIZATION OF BI-POROUS SBA-15 MATERIALS

VINH THANG HOANG a, QINGLIN HUANG b, MLADEN EIC b*, TRONG ON DO a, and SERGE KALIAGUINE a a Department of Chemical Engineering, Laval University, Ste-Foy, Québec, Canada, G1K 7P4 b Department of Chemical Engineering, University of New Brunswick, P.O Box 4400, Fredericton, N.B., Canada, E3B 5A3

Abstract In our recent study [1], a series of synthesized SBA-15 materials were characterized by nitrogen adsorption/desorption isotherms at 77K, and four of them (MMS-1-RT, MMS-1-60, MMS-1-80 and MMS-5-80) were investigated by the ZLC technique involving n-heptane diffusion at a low concentration level. SBA-15 materials proved to have a broad pore size distribution within the micropore/small-mesopore range in the walls of their main mesoporous channels. Desorption/diffusion of n-heptane took place inside the intrawall pores and depended on the relative content of micropores. In this paper, the four selected samples were further investigated with regard to diffusion of two other probe molecules having different kinetic diameter, i.e., cumene and mesitylene. The results show that the diffusivities and activation energies of the three different probe molecules in MMS-1-RT, MMS-1-60 and MMS-1-80 samples are of the same order, while they show an opposite trend relative to their size in MMS-5-80 sample (the lowest micropore content). The transport of these probe molecules is controlled by a combination of micro- and meso-pore diffusion resistances in the intrawall pores. The mesopores in main channels played more and more important role in diffusion processes where the relative micropore volume was decreased, and the critical size of probe molecules increased. The diffusivities in SBA-15 samples were significantly higher than in ZSM-5 zeolites for cumene and mesitylene. The activation energies, in contrast, were found to be much smaller than in corresponding MFI- zeolites. Keywords: Diffusion, micro/mesoporosity, SBA-15, MFI-zeolites, n-heptane, cumene, mesitylene, ZLC technique 591 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 591–602. © 2006 Springer. Printed in the Netherlands.

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1. Introduction In a previous paper [1], we reported the formation and structural characterization of a series of bi-porous SBA-15 materials. These materials were found to have not only an array of hexagonally ordered primary channels but also a certain amount of smaller pores with a broad pore size distribution in the micropore and small mesopore range within the mesoporous walls. The micropores and small mesopores constitute the intrawall porous structure of the SBA-15 materials. The microporosity could be controlled systematically by the synthesis conditions. These results are similar to those reported by several authors [2-9], who indicated the existence of micropores within the pore walls of mesopores. Furthermore, the diffusion of n-heptane in four selected SBA-15 samples (MMS-1-RT, MMS-1-60, MMS-1-80 and MMS-5-80), having different micropore volumes, has been studied by the Zero Length Column (ZLC) technique. Intrawall pores were found to play a major role in controlling diffusion at a low concentration level. The overall diffusion process of nheptane was controlled by a combination of micro- and mesopore-diffusion resistances in the mesoporous walls and depended on the relative content of micropores. In the samples that have relatively high content of micropores, nheptane diffusion process is similar to that of typical microporous adsorbents. As the micropore content is decreased, diffusion becomes more and more controlled by secondary mesopores of the intra-wall pore structure. Furthermore primary mesopores, comprising the main channel structure could also play a role in facilitating transport through the biporous structure of SBA-15 materials with low micropore content. Studies of the simultaneous influence of micropores and mesopores on the mechanisms as well as transport properties of sorbate molecules in the SBA15 mesoporous materials is an essential first step in developing novel materials for future applications [10-13]. The overall diffusion process in porous materials is often governed by contributions from different transport steps. These can include slow transport through the micropores, and relatively fast diffusion in mesoporous channels. In this regard, using probe molecules with different kinetic diameters in the diffusion study is a promising strategy. In continuation of our previous study introduced above, the diffusion of cumene and mesitylene in the four selected bi-porous SBA-15 samples was investigated using the ZLC method. The diffusivity data obtained for bi-porous SBA-15 materials are compared and interpreted. The specific effect of microporosity on the diffusion of probe molecules was investigated in some detail.

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Effect of the Intrawall Microporosity on the Diffusion Characterization 2. Materials and Diffusion Measurements

Four hexagonal SBA-15 mesoporous silica samples (MMS-1-RT, MMS-1-60, MMS-1-80 and MMS-5-80) having different microporosities described in the previous paper [1] were used. The physico-chemical properties of these materials are summarized in Table 1. n-Heptane, cumene and mesitylene were used as probe molecules. n-Heptane has a kinetic diameter of 0.43 nm about twice smaller than mesitylene (0.87 nm), while, cumene or isopropyl benzene has a kinetic diameter of 0.67 nm [14]. Diffusion measurements were carried out using the standard ZLC method. Details of the method are given elsewhere [1,15]. Temperatures investigated in this study were in the range of 30 to 70oC. Sorbate gas-phase concentrations were adjusted and maintained within the linear region of the adsorption isotherm as required by the ZLC theory. Separate blank experiments were carried out to determine a cut-off point for the analysis of ZLC curves and then desorption data were fitted with the ZLC model [1]. 3. Results and Discussion The effect of purge gas flow rate on ZLC desorption curves for mesitylene diffusion measurements in MMS-1-60 sample of a bi-porous micro/mesoporous structure is displayed in Figure 1. 1

1

(b)

(a)

50 ml/min

C/Co

C/Co

100 ml/min

200 ml/min 0.1

150 ml/min 0.1 0

100

200

300

Ft (ml)

400

500

600

0

100

200

300

400

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Figure 1. Effect of flow rate on experimental (symbols) and theoretical (solid lines) ZLC curves of mesitylene over MMS-1-60 sample at 50oC: F (ml/min) = 50 ( ); 100 (V); 150 (U); and 200 (c).

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V.T. Hoang et al. Table 1. Textural properties of mesoporous SBA-15 silica*.. Sample

MMS-1RT 1 RT 680 362 0.447

MMS-160 1 60 750 415 0.579

Heating time (day) Synthesis temperature (oC) SBET (m2/g) Smeso (m2/g) Total pore volume, Vt (cm3/g) Total mesopore volume, Vmeso 0.324 0.459 (cm3/g) Primary mesopore diameter 4.6 5.5 (nm) 0.121 0.119 Micropore volume (cm3/g) Total intrawall pore volume 0.187 0.220 (cm3/g) % Micro-porosity (percentage 64.7 54.1 of micropore volume in the total intrawall volume) * Data taken from Vinh Thang et al. [1]

MMS-180 1 80 860 578 0.854

MMS5-80 5 80 920 733 1.106

0.747

1.053

7.1

7.8

0.105 0.332

0.059 0.318

31.6

18.6

Figure 1a shows ZLC desorption Table 2. Diffusivity data of mesitylene in curves at 50oC and different flow MMS-1-60 sample at 50oC at various flow rates. In this diagram normalized rates. sorbate effluent concentrations Flow rate Temperature are plotted against the total purge L Deff/R2 (s-1) (ml/min) (oC) volume, Ft. At the short time 50 4.2 0.298u10-3 asymptotes (Ft < 100ml) the 100 5.3 0.300u10-3 50 ZLC desorptions curves are 150 8.0 0.301u10-3 similar, indicating that the 200 10.0 0.300u10-3 transport is approaching the equilibrium control limit. However, at the long time region (Ft > 200ml) the curves clearly diverge, thus confirming all of them being in kinetically (diffusion) controlled regime regardless of the purge flow rate [16]. The same results indicating diffusion control regime were obtained for the other two sorbates investigated in this study, i.e., n-heptane and cumene. The values of effective diffusional time constant (Deff/R2) derived from the model fitting with experimental ZLC curves at different purge flow rates as shown in the Figure 1b are listed in Table 2. The fitted Deff/R2 values were found to be practically the same for all flow rates in conformation with the summary curves shown in

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Figure 1a. This analysis also generally confirms the diffusion-controlled mechanism in this mesoporous solid that possesses distinctive micropores in the walls of its principal mesoporous structure. Representative experimental and theoretical ZLC curves for mesitylene and cumene in four SBA-15 samples at different temperatures are presented in Figures 2. A good agreement between experimental results and theoretical fittings is observed for these two sorbates as well. Summary of effective diffusional time constants (Deff/R2) extracted from the model fittings of the experimental ZLC curves together with corresponding parameter L values for all sorbates are summarized in Table 3. The table also includes the values of effective diffusivity and activation energy, which were obtained from particle size analysis using SEM micrographs [1], and Arrhenius-type relationship, respectively. As can be seen from Table 3, the effective diffusivity values of molecules in the four SBA-15 samples were found to be in the order of mesitylene < cumene < n-heptane, which, as expected, shows the opposite trend with respect to their critical molecular sizes, thus indicating the important role of this geometrical factor regarding diffusion rate. Furthermore, the effective diffusivities for all sorbates increase with the decrease of the relative content of micropore volume in these samples, e.g., the diffusion rate of each sorbate is in the following order: MMS-5-80 > MMS-1-80 > MMS-1-60 > MMS-1-RT. In comparison to MFI-zeolites, i.e., all silica zeolite silicalite, diffusion of cumene and mesitylene in SBA-15 samples is generally much faster. For example, values of the effective diffusivity for mesitylene in SBA-15 samples are about six orders of magnitude higher than in ZSM-5 zeolite measured from the liquid phase using volumetric method as reported by Choudhary et al. [14,17]. For cumene, the values of the effective diffusivities are about four orders of magnitude higher than those reported by Choudhary et al. [18] also measured from the liquid phase. It is very unlikely that relatively large organic molecules, in particular mesitylene, can penetrate the internal structure of silicalite (micropores of the main channels are only about 0.6 nm in diameter). It is plausible that diffusion in this case occurs at the external surface of silicalite crystals. On the other hand it is obvious that the mesoporous structure of SBA-15 samples has a very strong effect on diffusion. The introduction of main and small mesopores is equivalent to an increase of external surface area and makes probe molecules more easily accessible to micropores. However, diffusivity of relatively small n-heptane

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molecule in SBA-15 samples is roughly equal to the diffusivity in silicalite (values extrapolated from the ZLC data presented in Kärger and Ruthven [19]). 1

1

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0.01 0

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Mesitylene

Cumene

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Figure 2. Effect of temperature on experimental (symbols) and theoretical (solid lines) ZLC curves for mesitylene and cumene, lnc/co versus t, at flow rate of 200ml/min: (a) MMS-1-RT; (b) MMS-1-60; (c) MMS-1-80; and (d) MMS-5-80.

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Table 3. Diffusivity data of n-heptane*, cumene and mesitylene in MMS-x-y samples. T (oC) L 10 8.0 n-Heptane 20 8.0 30 8.0 30 6.8 MMS-1-RT 50 Cumene 7.3 (12 Pm) 70 7.9 30 7.3 50 Mesitylene 7.8 70 8.5 10 11.5 n-Heptane 20 11.5 30 12.3 30 8.0 MMS-1-60 50 9.0 Cumene (12 Pm) 70 10.2 30 7.0 50 7.5 Mesitylene 70 9.8 10 12.0 n-Heptane 20 13.0 30 14.5 30 9.4 MMS-1-80 50 10.7 Cumene (15 Pm) 70 11.7 30 8.0 50 9.0 Mesitylene 70 11.2 10 12.0 20 n-Heptane 16.0 30 20.0 30 12.0 MMS-5-80 50 12.6 Cumene (15 Pm) 70 15.0 30 10.0 50 12.0 Mesitylene 70 14.0 * Data taken from Vinh Thang et al. [1] Sample

Sorbate

Deff/R2 (s-1.103) 0.506u10-3 0.692u10-3 0.944u10-3 0.342u10-3 0.520u10-3 0.759u10-3 0.202u10-3 0.303u10-3 0.418u10-3 0.625u10-3 0.837u10-3 1.102u10-3 0.396u10-3 0.570u10-3 0.788u10-3 0.240u10-3 0.339u10-3 0.456u10-3 0.961u10-3 1.150u10-3 1.375u10-3 0.519u10-3 0.699u10-3 0.913u10-3 0.303u10-3 0.400u10-3 0.515u10-3 1.388u10-3 1.526u10-3 1.659u10-3 0.707u10-3 0.852u10-3 1.002u10-3 0.386u10-3 0.481u10-3 0.583u10-3

Deff (m2.s-1) 1.822u10-14 2.491u10-14 3.398u10-14 1.231u10-14 1.872u10-14 2.732u10-14 0.727u10-14 1.091u10-14 1.505u10-14 2.250u10-14 3.013u10-14 3.967u10-14 1.425u10-14 2.052u10-14 2.827u10-14 0.864u10-14 1.220u10-14 1.642u10-14 5.406u10-14 6.469u10-14 7.734u10-14 2.919u10-14 3.932u10-14 5.135u10-14 1.704u10-14 2.250u10-14 2.897u10-14 7.808u10-14 8.584u10-14 9.332u10-14 3.977u10-14 4.792u10-14 5.636u10-14 2.171u10-14 2.705u10-14 3.279u10-14

E (kJ/mol) 22.2

17.2

15.8

20.2

14.8

13.9

12.8

12.2

11.4

6.5

7.5

8.9

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Deff (m2.sec-1x1014)

The variation of the effective diffusivity for n-heptane, cumene and mesitylene with regard to the primary mesopore size of SBA-15 materials at 30 o C is plotted in Figure 3. The figure displays similar trends for all sorbates. Apparently, the effective diffusivity increases only marginally as mesopore size increases from 4.6 to 5.5 nm, while it increases more 12 significantly as mesopore size MMS-5-80 10 increases from 5.5 to 7.8 nm. MMS-1-80 This observation indicates that 8 primary mesopores do not 6 MMS-1-60 influence the diffusion process MMS-1-RT in MMS-1-60 and MMS-1-RT 4 samples. However, it further 2 points out that those mesopores may play a significant role in 0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 facilitating transport through Mesopore diameter (nm) the biporous structure of SBA15 materials with low Figure 3. Dependence of the effective micropore content such as diffusivity (D ) on the mesopore diameter at eff MMS-1-80 and MMS-5-80. 30oC for: (V) n-heptane; (c) cumene; and Knudsen diffusion values of the (U) mesitylene. probe molecules investigated in this study in pores of 4-5 nm in diameter are 6-7 orders of magnitude larger than effective diffusivities determined experimentally here. It is therefore anticipated that sorbate molecules effectively slide along the surface rather than execute long trajectories with significant radial displacement in the primary mesopores. The activation energy of desorption of mesitylene and cumene in SBA15 samples having different micropore volume is in the following order: MMS1-RT > MMS-1-60 > MMS-1-80 > MMS-5-80. The trend is the same as that of n-heptane and has been explained in detail in the previous paper [1]. Satterfield et al. [20] reported activation energy of 53.1-66.5 kJ/mol for cumene diffusion in H-mordenite and Choudhary et al. [18] obtained values of 27.5-50.2 kJ/mol for the same species diffusion in ZSM-5. Our activation energy values for diffusion of cumene in SBA-15 samples are much lower in comparison with the above literature values. This conforms to a general pattern of less activated process involving diffusion in mesoporous adsorbents. The existence of micropores and small mesopores has been verified in our previous communication [1]. Imperor-Clerc et al. [8] have reported the existence of micropores corona surrounding the mesopores of SBA-15 materials (pore diameter < 2 nm). Ryoo et al. [4] have proposed a micropore-mesopore

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network in SBA-15 silica by TEM imaging of the inverse platinum replica. Recently, Galarneau et al. [9] have clearly suggested the presence of two kinds

Intrawall pores

Mesopore channel Micropores

Porous wall between two straight mesopore channels

Small mesopores

Mesopore channel

Figure 4. Schematic structural model of SBA-15 material.

of complementary porosity, an ultramicroporosity (pore size < 1 nm), and a secondary porosity with a very broad pore size distribution between 0.15 nm and 3-5 nm (depending on synthesis temperature). They also suggested that structural mesopores are connected through micropores. Based on our study and literature results, a schematic diagram of SBA-15 materials is shown in Figure 4. These materials have an array of hexagonally ordered primary mesopore channels and a certain amount of intrawall pores within the mesoporous walls. The intrawall pores include micropores ranging from ultra- to super-micropores and small mesopores, which may open at both ends, i.e., interconnecting with

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main mesopore channels, or closed at one end. The structure of SBA-15 materials changes with different synthesis conditions. When the synthesis temperature and length of heating time are increased, the mesopore size increases, the wall thickness decreases, and the collapse of the ultramicroporosity follows the formation of secondary porosity, spanning from supermicropores to small mesopores which represent bridges between adjacent mesopores. Generally, if entering a pore is an activated process for a sorbate, the energy barrier is determined by the size of the sorbate molecule and the pore structure. The order of kinetic diameters of sorbates is n-heptane < cumene < mesitylene. It is thus expected that the activation energy of n-heptane is the smallest and that of mesitylene the largest. However, it is interesting to observe that the activation energy in MMS-1-RT, MMS-1-60 and MMS-1-80 is in the opposite order, e.g., n-heptane > cumene > mesitylene, while that in MMS-5-80 is as expected n-heptane < cumene < mesitylene. These results logically suggest that the overall diffusion process in the micro-mesoporous SBA-15 materials is controlled by the micropores/small mesopores (intrawall pores) in the walls of the main channels. By using Horvath-Kawazoe model to analyze nitrogen desorption/adsorption isotherm data of MMS-1-RT, MMS-1-60 and MMS-1-80 samples, we estimated the pore size of microporous channels. When the synthesis temperature increases from room temperature to 80oC, the micropore sizes span from about 0.4 to 0.7 nm, close to the kinetic diameter of n-heptane and cumene, respectively. This provides evidence that n-heptane, and to a lesser extent cumene, can penetrate the microporous channels within the mesoporous wall of these materials. On the other hand, mesitylene having a kinetic diameter of 0.87 nm is very likely excluded from these micropores. Therefore, the adsorbed amount of sorbates in micropores is in the decreasing sequence of nheptane, cumene and mesitylene. This in turn allows to conclude that the diffusion of smaller molecules, such as n-heptane, is more strongly, if not entirely, controlled by the diffusion resistance in micropores, while the transport of the larger molecules, i.e., cumene and mesitylene, is more controlled by mesopore resistances in intrawall pores, which can explain the trend of activation energy in the three SBA-15 samples. In the case of MMS-5-80 sample, which was synthesized at 80oC for 5 days, all of its ultra-micropores collapsed and only supermicropores and small mesopores having pore size > 1 nm remained in the mesoporous walls. In this case, all molecules can be transported through the intrawall pores and the diffusion involving this material is predominantly controlled by resistance in the secondary and possibly in primary mesopores. This has been confirmed by the

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lowest values of activation energies for this sample. Moreover the activation energy in MMS-5-80 was in the expected increasing order for n-heptane, cumene and mesitylene, although their differences were not significant. 4. Conclusion In this work, SBA-15 mesoporous materials were further characterized with probe molecules, i.e., cumene and mesitylene using the ZLC technique. Diffusivity values for cumene and mesitylene in SBA-15 samples are much higher than those in conventional microporous zeolites, such as ZSM-5. It was established that diffusion of the probe molecules takes place entirely through the intrawall porous structure and is controlled by the combination of micropore and small mesopore resistances. The primary mesopores may also play a role in facilitating diffusion through the biporous structure of SBA-15 materials with low micropore content. With a decrease of the relative micropore volume, the diffusivity increases and activation energy decreases, and diffusion in small mesopores play a more and more important role. For MMS-1-RT, MMS-1-60 and MMS-1-80 samples, with the increase of molecular sizes, less and less probe molecules can enter the micropores yielding a more mesopore diffusion controlled process and a lower activation energy. For MMS-5-80, the diffusion is similar to the diffusion through a pure mesoporous material, and the trend of activation energy is opposite to that of diffusivity and the same as that of molecular size. These observations seem to imply that there is a large fraction of micropores with size less than 0.7 nm in MMS-1-RT, MMS-1-60 and MMS1-80 samples, and most of micropores in MMS-5-80 sample are larger than 1 nm. By using probe molecules of different critical size, the ZLC method proves to be a simple and reliable technique to study the diffusion mechanism and relate it to structure of the materials. Acknowledgments This project was supported by the Natural Science and Engineering Research Council of Canada (NSERC) through the Industrial Chair in Nanoporous Materials at Laval University.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

H. Vinh Thang, Q. Huang, M. Eiü, D. Trong On, S. Kaliaguine, (2004).Langmuir, in press. K. Miyazawa, S. Inagaki, (2000) Chem. Commun. 2121. M. Kruk, M. Jaroniec, C.H. Ko, R. Ryoo, (2000) Chem. Mater. 12, 1961. R. Ryoo, C.H. Ko, M. Kruk, V. Antochshuk, M. Jaroniec, (2000) J. Phys. Chem. B 104, 11465. Y. Sun, Y. Han, L. Yuan, S. Ma, D. Jiang, F.S. Xiao, (2003) J. Phys. Chem. B 107, 1853. M. Choi, W. Heo, F. Kleitz, R. Ryoo, . (2003) Chem. Commun1340. C.M. Yang, B. Zibrowius, W. Schmidt, F. Schuth, (2003) Chem. Mater. 15, 3739. M. Imperor-Clerc, P. Davidson, A. Davidson, (2000) J. Am. Chem. Soc. 122, 11925. A. Galarneau, H. Cambon, Di F. Renzo, R. Ryoo, M. Choi, F. Fajula, (2003) New J. Chem. 27, 73. D. Trong On, S. Kaliaguine, (2003).US Patent 6,669,924 B1 D. Trong On, D. Desplantier-Giscard, D. Danumah, S. Kaliaguine, (2001) Appl. Catal. A: General 222, 299. Y. Liu, W. Zhang, T.J. Pinnavaia, (2000) J. Am. Chem. Soc. 122, 8791. H. Vinh Thang, A. Malekian, M. Eiü, D. Trong On, S. Kaliaguine, (2002) Proc. 3th Internat. Mesostructured Mater. Symp. and (2003) Stud. Surf. Sci. Catal. 146, 145. V.R. Choudhary, V.S. Nayak, T.V. Choudhary, (1997) Ing. Eng. Chem. Res. 36, 1812. M. Jiang, M. Eiü, (2003) Adsorption 9, 225. S. Brandani, M. Jama, D.M. Ruthven, (2000) Chem. Eng. Sci. 55, 1205. V.R. Choudhary, A.P. Singh, (1986) Zeolites 6, 206. V.R. Choudhary, A.S. Mamman, V.S. Nayak, (1989) Ind. Eng. Chem. Res. 28, 1241. J. Kärger, D.M. Ruthven, (1992).Diffusion in Zeolites and Other Microporous Materials, John Wiley and Sons, Inc.: New York C.N. Satterfield, J.R. Katzer, W.R. Vieth, (1971) Ind. Eng. Chem. Fundam. 10, 478.

STRUCTURE OF A SINGLE-SPECIES-FLUID IN A SPHERICAL PORE I.A. HADJIAGAPIOU*, A. MALAKIS, S. S. MARTINOS Solid State Physics Section, Dept. of Physics, University of Athens Panepistimiopolis, Zografos GR 157-84, GREECE

1. Theory The wetting of solid substrates (planar, curved) or porous materials by classical fluids is of practical (tertiary oil recovery, foams, phase and film growth, chromatography, dyeing, membrane transport) and fundamental importance as an application of statistical mechanics of inhomogeneous systems and has received considerable attention in recent years. Initially, it was studied through the square gradient approximation (SGA), but because of the implied severe restrictions of SGA, density functional theory (DFT), in both versions, local density approximation (LDA) and weighted density approximation (WDA), has been applied. The key element in the DFT approach is the grand potential density functional ȍV[ȡ] written through the intrinsic free energy of the inhomogeneous system, which is considered as a functional of system density. The assumed one-component gas phase with constant density, consisting of N spherical particles of diameter d, is enclosed in a spherical pore of radius R

 R t d 2

that can vary. The pore is represented by an external potential

G Vext (r ) with which the confined particles interact. In such a finite system, its thermodynamic properties (bulk phase transitions, fluid phase boundaries) are altered in comparison to those of a similar non-confined system and, because of the strong confinement (especially in the case of small pores), the packing constraints are very pronounced. The boundary of the cavity forms a continuum of constant density nS , which is impenetrable and the total interaction between the pore boundary and a particle is given by, G Vext (r ) ns

³)

WF

S

 rG  rGc d rGc

603 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 603–611. © 2006 Springer. Printed in the Netherlands.

(1)

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I.A. Hadjiagapiou, A. Malakis and S.S. Martinos

G G

where )WF  r  r c is

the spherically

symmetric

G

pairwise

G

attractive

interaction between a fluid molecule at r and a wall molecule at r c and S is the area of the pore boundary. When the radius R of the spherical pore is infinitely large in comparison to the radius (d/2) of the fluid molecules, the system becomes equivalent to a planar wall in the presence of a bulk fluid. The key point in DFT is the proper choice of the grand potential density functional ȍV[ȡ], which is taken to be of the form,

G :V ª¬U r º¼

G 1 G G G G G G G½ G ­ ® fhs ª¬U r º¼  U r U rc )FF  r  rc drcVext  r  P U r ¾dr (2) 2 V¯ V ¿

³

³

where µ is the bulk gas chemical potential and V the volume of the system. The repulsive force contribution to the Helmholtz free energy is treated in the LDA,

G

in that f hs ª¬ U  r º¼ is the Helmholtz free energy density of a uniform hard-

G

G

sphere fluid at density U  r , r = r the distance to the center of the cavity. Although LDA fails to describe the oscillatory behavior of the density profile at strongly attractive walls, it gives reasonable results for the surface tension, adsorption and the density profile of the free liquid-vapor interface. The longrange attractive forces between fluid molecules are treated in the mean field G approximation: ĭFF( r ) is the attractive part of the pairwise potential between two fluid molecules, r distant apart; the attractive forces are treated as a perturbation in the initial hard-sphere system. The equilibrium G density U o (r ) of the inhomogeneous fluid is obtained by minimizing (2) via the variational principle

G

G

G :V ª¬ U  r º¼ GU  r 0 which leads to the corresponding Euler-Lagrange integral equation,

G

G

P Vext  r  P hs ª¬ U  r º¼ 

³

V

G

G G

where

G

G

U  r c ) FF  r  r c dr c

G

G

P hs ª¬ U  r º¼ w f hs ª¬ U  r º¼ wU  r

(3)

605

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is the chemical potential of a uniform hard-sphere fluid of density

G

G

U (r ) .

Choosing the potentials Vext ( r ) and ) FF ( r ) properly, the integral equation (3) can be converted to a functional non-linear second-order differential equation with appropriate boundary conditions. The fluid-fluid potential is,

) FF  r  DO 3 4S e  Or (Or ) ,

(4)

where Ȝ is an inverserange parameter and Į is given by

D



³

V

G ) FF (r )dr

(5)

For the sake of simplicity, all subsequent quantities and equations are transformed to dimensionless units, µ* { ȕµ, p* { ȕd3p, R* { ȜR, r* { Ȝr, ȡ* { ȡd3 Į* { ȕĮ/d3 = 11.102/ȉ*, V*ext { EVext

(6)

although the asterisks will be suppressed; p is the pressure, E k BT , kB Boltzmann's constant. If a wall molecule interacts with a fluid molecule via a potential of the form 1

)WF (r ) Ce  O

WF

r

(OWF r ) ,

where ȜWF =Ȝ (ȜWF inverserange parameter of the wallfluid attractive interaction) and C are positive constants, then (1) gives,

Vext  r HW  OR e where now

HW

 sinh  Or , Or

 OR

(7a)

is a new parameter characterizing the substrate, proportional to

n s and C; a measure of the well depth for the wallfluid interaction. A feature

of the potential (7a) is that its strength (the part separated from the distance

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I.A. Hadjiagapiou, A. Malakis and S.S. Martinos



dependence, HW  O R e

 O R

depends

strongly on substrate’s radius R. In

the limit r o, (7a) takes the form,

lim Vext  r HW (OR )e  OR

(7b)

r o0

while on the cavity boundary, r = R,

Vext  R 

HW

ª1  e 2 OR ¼º 2 ¬

(7c)

which, for large R, takes the form: Vext ( R ) |  HW 2 , i.e. is independent of R, and the spherical substrate behaves as a planar one. Substituting potentials (4) and (7a) into (3) and differentiating the resulting equation twice with respect to r yields,

2 r

P hscc >U r @  P hsc >U r @  P hs >U r @  P

DU r

(8)

the prime denoting derivative with respect to r. The final equation (8) depends only on the radial distance r, since both potentials are spherically symmetric. This is a functional differential equation, depending on ȡ(r); its solution is uniquely defined if supplemented by appropriate boundary conditions. In the limit r o 0, the solution is less well behaved, since at the origin r = 0 it is singular unless P chs(r) vanishes in this limit, lim r o0

P chs r r

lim r o0

dP chs r dr dr dr

lim P chsc r r o0

(9)

according to the de l’ Hopital rule, therefore, in the neighborhood of the origin (8) becomes lim P chsc (r ) r o0

1 P r  P  DU r 3 hs

(10)

Thus, the first boundary condition is,

P chs(0) 0

(11)

607

Structure of a Single-Species-Fluid in a Spherical Pore while the other on the substrate is,

§ ©

1·

P chs ( R) HW  ¨1  ¸  P hs ( R)  P , R ¹

(12)

resulting by differentiating (3) once with respect to r and evaluating it at r=R. The differential equation (8) in conjunction with the boundary conditions (11,12) constitute the problem under consideration, which will be solved numerically since Eq. (8) is an implicit non-linear second-order differential equation and cannot be solved analytically. The hard-sphere reference fluid is described by the Carnahan-Starling approximation,

E phs  U U

1  K  K 2  K3

1  K

3

, EPhs  U lnK 

8K  9K 2  3K 3

1  K

3

(13)

where Ș=ʌȡd3/6 is the packing fraction, which will be used as the dependent variable instead of the density ȡ(r). Substituting (13) into (8) yields the new equation, 2 r

K ccr  K cr  B1 K K c2 r  B2 K  B3 K K r ,

(14)

subject to the new boundary conditions resulting from (11,12), K c0 0

­

§ ©

1· ¹

(15a)

½

K c  R ®HW  ¨1  ¸  P hs K  R  P ¾ A1 K  R R ¯

¿

(15b)

where A1 K

B 1 K

w  EPhs 1 8  2K , wA1 K 1 30  6K A K   2 wK K 1K 4 2 wK K 1K 5 A 2 (K ) , B 2 K A1 (K )

E P  E P h s K , B 3 K A1 K

6D E S A1 K

(16a)

(16b)

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I.A. Hadjiagapiou, A. Malakis and S.S. Martinos

2. Results and Discussion The equilibrium density U o (r ) was evaluated by solving the boundary value problem (14,15) numerically for a wide range of values for R and

HW

at T=0.8.

Depending on the values of R and HW the confined system exhibits many growth modes. Firstly, the equilibrium density profile for a pore with constant R=4.5 is studied. The density profiles exhibit phase transitions between thin and thick films, liquid-full pore and empty, strong and weak wetting, and v.v., formation of a liquid drop at the origin and layers at intermediate points. For H W d 12.15 the formation of a thin liquid drop at the origin of the pore (of constant density Uo (0) ) is favored, while the remaining volume is practically empty and the inner

wall

exhibits

weak

wetting

(wall

for H W but

density UW , U o (0) ! UW );

12.16 an additional thin liquid film appears at the inner boundary, now U o (0)  UW , strong wetting, weak/strong transition. For

12.17 d HW d 12.21 , the equilibrium structure of the density consists now of a

thick liquid drop at the origin (thin/thick transition) while the inner wall exhibits weak wetting, U o (0) ! UW . For H W 12.22 , the thick liquid drop at the center persists, but now the remaining branch of the density profile is increasing towards the inner wall so that U o (0)  UW , the pore is now liquid-full, capillary condensation. For H W ! 12.22 , repetition of the previous four situations appears. The formation of the liquid drops at the origin is mainly due to the strong potential field appearing at pore center; from (7b)

Vext (0)

HW Re-R , while at the wall Vext ( R )



 HW 2 1  e2 R



resulting in a weaker field especially for larger R values for which Vext ( R )

becomes constant, Vext ( R ) |  HW 2 and the system becomes equivalent to a planar wall, Fig. 1. Other characteristic structures for the density profile appear for various 1.3, the pore is liquid-full almost values of R and HW . For R=1.3 and H W of the same density except at the origin where a thin liquid drop grows, see Fig. 2a. In addition, more complex structures can appear due to packing constraints, 1.3 , where two zones of strong as in the case for R=1.25 and H W localization appear at intermediate points, one close to the wall and the other at

609

Structure of a Single-Species-Fluid in a Spherical Pore

the origin; these zones are connected through a long “bridge” forming a film between them, while the wall exhibits weak wetting [5], Fig. 2b. 0 .2

0 .0 0

K

K

0 .1

(a )

-0 .0 5

(b )

0 .0 -0 .1

-0 .1 0

-0 .2 -0 .3

-0 .1 5

-0 .4 -0 .2 0 0

K

2

-0 .5

r

4

0 .0

0

K

2

4

r

4

r

0 .3 0 0 .2 5

-0 .1

(d )

0 .2 0

(c )

-0 .2

0 .1 5 0 .1 0

-0 .3

0 .0 5 -0 .4

0 .0 0

-0 .5 0

2

r

4

-0 .0 5 0

2

Figure 1. Density profile K1 ( r ) { K ( r )  K (0) vs. radial distance r for a pore with radius R=4.5, labeled by the parameter H W , (a) H W 12 , (b) HW 12.16 , (c) HW 12.21 , (d) HW 12.22 . 0.14

K 0.12 0.10 0.08 0.06 0.04 0.02 0.00

(a)

0.0

0.5

0.010 K 0.005 0.000 -0.005 -0.010 -0.015 -0.020 -0.025 -0.030

1.0

r

1.5

1.0

r

1.5

(b)

0.0

0.5

Figure 2. Density profile K1 ( r ) { K ( r )  K (0) vs. radial distance r for a pore 1.3 . with radius R=1.05(a) and R=1.25(b). Both cases correspond to H W

This investigation has pointed out the existence of significant structure inside a pore, monotonic or non-monotonic profiles, however small the pore is.

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In some cases, the confined particles tend to be organized into layers forming spherical shells about the pore center and at intermediate positions, Fig. 3; this is reflected in the oscillations of the mean density with radial distance. These oscillations can be captured better if a more sophisticated theory is invoked (WDA) but the structure found in the current description through the LDA is an indication that this approximation can describe the structure satisfactorily [6]. Another important case is that for which the pore is so small that it can enclose only one particle: quasi-zero-dimension (0D); this notion can be invoked for the description of the freezing transition. The bounding surface of the drop formed at the center of the pore is not sharp but undergoes thermally excited surface oscillations of various modes, so that the drop can wander throughout the pore and can collide with the pore boundary as long as there is not a stabilizing field. The surface tension (mechanical and equimolar) and the respective dividing surfaces cannot be defined rigorously as long as the system consists of a small number of particles; also the wetting transitions as they appear in the case of a flat wall cannot be defined since the system is finite [1].

K

0.3 0.2 0.1 0.0

-0.1 -0.2 -0.3 0.0

0.2

0.4

0.6

0.8

1.0

1.2

r

Figure 3. Density profile vs. radial distance for a pore with radius R=1.1 and HW=1.9. A thick liquid drop (density U(0)) grows at the origin and a layer with density higher than U(0) grows at an intermediate position, but the wall exhibits weak wetting.

Acknowledgement The research was supported by the Special Account for Research Grants of the University of Athens (SARG) under Grants Nos. 70/4/4071 and 70/4/4096.

Structure of a Single-Species-Fluid in a Spherical Pore

611

References 1.

2. 3. 4. 5. 6.

Rowlinson J. S. and Widom B. (1982) Molecular Theory of Capillarity (Oxford University). Evans R. (1992) Fundamentals of Inhomogeneous Fluids, edited by D. Henderson (Dekker, New York), H. T. Davis, (1996). Statistical Mechanics of Phases, Interfaces and Thin Films (VCH, New York). Hadjiagapiou I. and Evans R.(1985) Mol. Phys. 54, 383. Hadjiagapiou I. (1994) (1995) J. Phys. Condens. Matter 6, 5303 ; 7, 547. Hadjiagapiou I. (1996) J. Chem. Phys. 105, 2927; (1997) J. Phys. Chem. B 101, 8990. Gonzalez A. et al. (1998) J. Chem. Phys. 109, 3637. Tarazona P. and Evans R. (1983) Mol. Phys. 48, 799; Teletzke G. F., Scriven L. E. and Davis H. T. (1982) J. Chem. Phys. 77, 5794; (1983) 78, 143.

CARBON MOLECULAR SIEVE MEMBRANES: CHARACTERISATION AND APPLICATION TO XENON RECYCLING S. LAGORSSE PONTES, F. D. MAGALHÃES, A. MENDES LEPAE, Departamento de Engenharia Química Faculdade de Engenharia da Universidade do Porto

1. Introduction Carbon Molecular Sieve Membranes (CMS membranes) are a very recent and promising material. This paper deals with the characterisation of the performance of these membranes, in terms of sorption equilibrium and permeation properties, for Xe, He, CO2, O2 and N2. CMS membranes tested were supplied by Carbon Membranes Ltd. (Israel). The motivation for this work has to do with studying the suitability of these membranes for Xenon recycling within an anaesthetic closed loop. Xenon is an extremely expensive gas (about €10 to €13 per normal litre) that is known as a suitable inhalation anaesthetic agent, offering many advantages over the nowadays-used nitrous oxide. Its economical implementation on surgical theatres has implied research on automated, fully closed delivery systems that reuse xenon but remove the carbon dioxide and nitrogen exhaled by the patients. 2. Results and Discussion The CMS membranes used in this work resulted from the pyrolysis of a cellulosic precursor under controlled conditions followed by a carbon chemical vapour deposition (CVD) process and further controlled activation procedure. These treatments, CVD and activation, influences decisively the membrane’s sieving properties and can be used to tune its performance towards a desired separation. The main purpose of the present analysis is to find the most suitable membrane for Xenon recycling in an anaesthetic closed loop. Table 1 shows single gas permeabilities at 20ºC in three CMS membranes with different CVD/activation treatments. Large differences are found for the permeability values of different gases, because of the molecular sieving character of these 613 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 613–618. © 2006 Springer. Printed in the Netherlands.

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S.L. Pontes, F.D. Magalhães and A. Mendes

membranes (pores sizes very close to the dimensions of the diffusing gases). A small change in the effective micropore size can significantly modify the permeation and selectivity properties of the membranes. Ideal selectivities are shown in Table 2. For manufactured modules MS1b and MS2, CO2/Xe and N2/Xe selectivities were found to be quite promising. Table 1 – Single gas permeabilities at 20ºC for a feed pressure of 2 bar and a permeate pressure of 1 bar. N2 O2 CO2 Xe

Mod. MS1 10 100 270 1

Mod. MS1b 9 80 301 6000 9 33

Mod. MS2 >40 >1000 5 25

Results for bore side feed permeation through the MS2 membrane, measured at 30ºC and 50ºC and up to 4 bar for He, N2, O2, CO2, SF6 and Xe are shown in figure 1. Gas molecules with a diameter larger than 3.9Å do not exhibit measurable permeation. He, N2 and O2 permeabilities were found to be weakly dependent on the feed pressure, however, for strongly adsorbed gases like CO2, the permeability decreases with the feed pressure. This is partly related to the adsorption isotherm, that is of type I, which implies a strong concentration dependence of the diffusion coefficient. However, the Darken relation alone, designated by model 1 in the figure, fails to describe the CMS permeability towards CO2. Assuming a more general form for the diffusivity concentration dependence, given by equation (1), and further incorporating it into the steady state flux equation (2), one obtains an alternative model (model 2 in figure 1) that gives better results [1].

615

Carbon Molecular Sieve Membranes

Dc

q2

z2

³

Dc0 (1  T ) n

Jdz

 ³ Dc (q ) q1

z1

(1)

w ln p dq w ln q

(2)

All permeation rates through the membrane increased with temperature; the gas transport occurs according to an activated mechanism. 80

Sample MS2 Permeate pressure= 60 mbar

Experimental data at 50ºC Model 1 at 50ºC

70

Model 2 at 50ºC

60

Model 1 at 30ºC

-2

-1

Permeability / LN.m .bar .hr

-1

Experimental data at 30ºC

Model 2 at 30ºC

CO 2

50 40

He 30 20

O2 10

N2 0 0

0.5

1

1.5

2

2.5

3

3.5

4

Feed Pressure / bara

Figure 1. Single gas permeability as a function of pressure at 30ºC and 50ºC.

The N2, O2, CO2, Xe and SF6 equilibrium adsorption isotherms at three temperatures and up to 7 bar were measured using the volumetric and gravimetric methods for two CMS fibre samples MS1 and MS2 (figures 2 and 3). The data were fitted to the Langmuir (dashed-curve) and Sips-type (solidcurve) correlations.

616

S.L. Pontes, F.D. Magalhães and A. Mendes 8 10ºC

Sample MS1 7

20ºC

Carbon Dioxide Xenon at 20ºC

6

40ºC

Oxygen Nitrogen

q / mol.L-1

5 4 3 10ºC

2 40ºC

1 0 0

1

2

3

4

5

6

Pressure / atm

Figure 2. Equilibrium adsorption isotherms in MS1.

6

CMS samples: MS1, MS2 MS1, Carbon Dioxide

MS1, Nitrogen

5

MS2, Carbon Dioxide

MS2, Nitrogen MS2, Sulphur hexafluoride

q / mol.L-1

4

3

2

1

T= 30ºC

0 0

1

2

3 Pressure / atm

4

5

6

Figure 3. Equilibrium adsorption isotherms at 30ºC in two samples: MS1 and MS2.

Carbon Molecular Sieve Membranes

617

MS2 has lower adsorption capacity, resulting from higher carbon deposition (CVD) and lower activation. Xe uptake was very slow in this membrane, so that it was difficult to establish when equilibrium had been achieved. Its uptake is ~ 50 times lower than in MS1 [1]. From equilibrium isotherms it is possible to qualitatively predict competitive adsorption effects in multicomponent permeation. CO2 real selectivity will be enhanced by its strong affinity to CMS. In contrast, although Xe permeates very little through the tested membranes, it is preferentially adsorbed on the micropores. This behaviour will cause a blockage effect, having a negative influence on the multicomponent permeabilities, especially for the less adsorbables gases unable to displace the adsorbed xenon. Adsorption isotherm analysis is also a useful method for gathering information on the membrane’s ultramicroporosity. MS2 membrane pores are narrower than 5.02 Å (SF6 molecular diameter) and a fraction is larger than 3.94 Å (Xe molecular diameter) [2]. Pores in the MS1 membrane are larger than in MS2. The study of the structure of the CMS membranes was complemented with other methods. From electron microscopy it was found that these membranes have a uniform membrane thickness of 9 Pm and an external diameter of 170 Pm [3]. SEM pictures also showed a dense, apparently symmetric, crack-free structure. Wide Angle X-ray diffraction revealed a highly amorphous structure with an average micropore size of about 4 Å. A density of 1.7g.cm-3 (against the limiting value of 2.3g/cm3 for graphite) was measured by helium picnometry [1]. 3. Conclusions Recycling of xenon from an anaesthetic closed loop using CMS membrane modules is a promising application for this new material. Even though some ideal perm-selectivities seem quite good, these values may become useless when dealing with multi-component separation of absorbable gases in CMS; multi-component measurements are therefore essential and are currently under way. References 1. 2.

Pontes, S., Magalhães, F. and Mendes, A. (2003) “Carbon Molecular Sieve Membranes. Part 1: Gas Transport Characterization”, J. Membr. Sci., (submitted). Koresh J.; Soffer A. (1980) Study of Molecular Sieve Carbons J.C.S. Faraday I., 76, 2472.

618 3.

S.L. Pontes, F.D. Magalhães and A. Mendes Pontes S., Magalhães F., M. Adélio, (2002) “Carbon Molecular Sieve: Characterisation and Application”, International Congress of Membranes– ICOM, Toulouse, France.

AN EXPERIMENTAL STUDY OF THE STATE OF HEXANE IN A CONFINED GEOMETRY D. MALDONADO, N. TANCHOUX, P. TRENS, F. DI RENZO and F. FAJULA Laboratoire de Matériaux Catalytiques et Catalyse en Chimie Organique, UMR5618, CNRS / ENSCM, Institut Gerhardt FR 1878 8, rue de l'Ecole Normale 34296 Montpellier cedex 3

Abstract Literature data about hysteresis closure points of sorption isotherms has been summarized and it was shown that these points are located in one half of the (1-T/Tc, P/P°) plan. Below the limiting straight line, no hysteresis can be found. A study of the transition between the reversible and the irreversible (hysteretic) regime for n-hexane adsorbed on MCM-41 or similar mesoporous materials has been carried out. The dependence on pressure, temperature and pore size has been investigated and confirmed the literature data. Enthalpies of adsorption have also been derived from the isotherms using the isosteric method. The enthalpies showed a dependence on pore size: when the pore size decreases, the adsorptive is stabilised and the enthalpy of adsorption becomes more negative. 1. Introduction Although adsorption is of primary interest for catalysis applications, most studies of this phenomenon have been conducted at very low temperatures on gases such as N2 and Ar[1,2]. In this study, hexane adsorption at room temperature has been investigated, extending the scope of those studies to more realistic catalytic conditions. Moreover, hysteresis loops usually found for mesoporous materials strongly suggests that a branch of the isotherm is not at the thermodynamical equilibrium. In other experimental conditions, small pores or high temperatures, this hysteresis loop vanishes. The difference between these two situations in terms of driving force is yet to be completely understood. Some information can be found by deriving the adsorption enthalpies of hexane in mesoporous materials of different pore sizes, using the isosteric method. 619 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 619–627. © 2006 Springer. Printed in the Netherlands.

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D. Maldonado et al.

2. Behaviour of Vapours in Mesopores

2

CH4

C2H4

1.8

H2

Compressibility Z

1.6 1.4 1.2 1

Ideal Gas

0.8 0.6 0.4 0.2

NH3

0 0

250

500

750

1000

Pressure/ atm

Figure 1. Compressibility dependence on pressure of selected vapours and gasses.

1 VR = 2 VR = 1

Compressibility Z

0.8

0.6 VR = 0.9

0.4 Nitrogen Ethylene 0.2

Propane Methane

0 0

1

2

3

4

5

6

7

pR

Figure 2. Reduced compressibility of selected vapours and gasses (the reduced volume VR being fixed as a parameter).

At low pressure, gases and vapours can be accurately described by the model of the ideal gas. As pressure increases, though, this model is no longer valid and vapours behave very differently from each other, as shown in figure 1.

An Experimental Study of the State of Hexane in a Confined Geometry

621

0.7 0.6

Benzene c-alcanes n-alcanes i-alcanes CCl4

0.5

1-TR

0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

P/P°

Figure 3. Hysteresis closure points for some organic vapours.

In order to study the general phenomenon of hysteresis vanishing, and therefore to compare the properties of various gaseous species, the principle of corresponding states is used, and we carry out our study with reduced parameters (pressure, volume and temperature), obtained by dividing the corresponding parameter by its value at the critical point. Figure 2 shows that in this system of coordinates, the various gases exhibit the same behaviour. Adsorption data for various compounds in conditions where they show a hysteretic behaviour were taken from the literature[3]. The hysteresis closure points were plotted in a (1-TR, PR) plot (figure 3). It appears that all representative points are located in a half plan, the other half plan being the region of reversibility, and the data allows us to determine precisely the limiting line below which no hysteresis can be found. This existence of two distinct regimes of adsorption for all gases and vapours is a clear evidence for a general effect of confinement in porous materials. Nevertheless, the only apparent is temperature while pore size dependence still remains to be investigated. 3. Experimental Study 3.1. ADSORPTION ISOTHERM. DETERMINATION APPARATUS The apparatus (figure 4) is based on volumetric measurements and the adsorption is followed by two pressure gauges. The sample cell can be disconnected from the system to undergo a thermal treatment (up to 250°C)

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A primary pump coupled with a turbo molecular pump (PT 50, supplied by Leybold) is linked to the apparatus through two different inlets allowing rough or fine control of the depression rate in the system, protecting it from any solid contamination. Three thermal Pt 100 probes (provided by Serv'Instrumentation) are placed on the cell, the adsorptive cell and the main connecting pipe, used as calibrated reservoir, to record the temperature of the system and check the thermal stability of the whole system.

0-10 torr

0-1000 torr

p1

p2 V1

V4 Vacuum

V3

V2 Secondary Valve

Filter Valve

Climatic Chamber adsorptive adsorbent

Figure 4. Scheme of the Adsorption Isotherm Determination Apparatus (AIDA).

The apparatus is placed in a climatic chamber (VT200, provided by Vötsch) allowing a thermal stability better than 0.1 K from 250 K up to 350 K. The whole system (electromagnetic valves, climatic chamber, pressure gauges and thermal probes) is interfaced to a computer and controlled by a programme especially designed for it. The different volumes were calibrated by weighing the sample cell filled with mercury and then by expanding hexanes from this volume into the rest of the apparatus, monitoring pressure. The system was tested for the adsorption of n-hexane on 3.6 nm pore size MCM-41 material and the isotherm was found to be consistent with data available from the literature[4]. 3.2. TEMPERATURE DEPENDENCE OF THE ISOTHERMS Sorption isotherms of n-hexane on a 3.6 nm pore size material were determined for temperatures ranging from 1°C to 60°C. Figure 5 shows the adsorbed

An Experimental Study of the State of Hexane in a Confined Geometry

623

amount as an equivalent volume for standard pressure and temperature against relative pressure i.e. the pressure divided by saturated vapour pressure.

100

80

3

-1

Adsorbed Amount /cm .g (STP)

120

60

1°C 30°C 40°C 50°C 60°C

40

20

0 0

0.2

0.4

0.6

0.8

1

p/p°

Figure 5. n-Hexane sorption isotherms for a 3.6 nm pore size MCM-41 material (Filled symbols and crosses stand for adsorption, hollow symbols and underscores for Desorption).

All of the isotherms are reversible in this temperature range and clear trends appear as temperature increases. The adsorption step shifts towards high relative pressure, as can be predicted by the Kelvin equation relating the equilibrium pressure p to the pore radius for a spherical meniscus (equation 1). The slope of the step decreases for the same reason (if we consider a given pore size distribution and differentiate equation 1 with respect to rp, the step width depends on temperature). Finally, the maximal adsorbed amount decreases, as density of the fluid increases. (1)  2 JVl ln( p 0 ) p rp ˜ RT 3.3. PORE SIZE DEPENDENCE A series of sorption isotherms was then determined for materials with pore sizes of 4.4 nm and 5.9 nm (figure 6 and 7, respectively) for the same temperature range as before. For each studied pore size, the same trends as before can be shown. For the intermediate pore size (4.4 nm), an hysteresis loop appears at

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low temperature (0°C) and for the larger pore size (5.6 nm), hysteretic behaviour occurs up to 50°C, which is the highest temperature studied for this material. These results show that pore size and temperature play a role in the transition between hysteretic and reversible regime, and a complete study of this transition would requires not a (PR,TR) plot for the closure point but a three dimensional (PR, TR, pore size (Ɏ)) plot, and the determination of a limiting surface between the two observed regimes.

(STP)

200

160

160

-1

Adsorbed Amount / cm .g

3

3

Adsorbed Amount /cm .g

-1

(STP)

200

120

80 0°C 30°C 40°C 50°C

40

120

80

1°C 30°C 40°C 50°C

40

0

0 0

0.2

0.4

0.6

0.8

1

p/p°

Figure 6. n-Hexane sorption isotherms for a 4.4 nm pore size MCM-41 material (Filled symbols stand for adsorption, hollow symbols for Desorption)

0

0.2

0.4

0.6

0.8

1

p/p°

Figure 7. n-Hexane sorption isotherms for a 5.9 nm pore size MCM-41 material (Filled symbols stand for adsorption, hollow symbols for Desorption)

4. Hysteresis Vanishing A plot of the closure pressure against temperature when hysteresis is detected and of the inflexion point against temperature for reversible systems (figure 8) shows that for a given solid these points are located on a straight line. Moreover, if we report the literature data already discussed (dotted line), we find a good correspondence between the three series discussed above and literature data. The systems corresponding to the line located above the dotted line are fully reversible for this temperature domain (for instance for a 3.6 nm pore size), those located below the line are reversible for these temperatures (5.9 nm) and those crossing the line show a transition between the two regimes. This plot gives us precious hints at how to study the effects of this change on

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An Experimental Study of the State of Hexane in a Confined Geometry

reactivity: using a model reaction, we can vary the pore size of the catalyst in order to cross the line and monitor a selected property of the reaction such as conversion or selectivity in order to detect a change between the regimes.

p/p°

0.5

3,6nm 5,9nm Limit 4,2nm 4,4nm

0.25

0 -20

-10

0

10

20

30 T/ °C

40

50

60

70

80

Figure 8. Hysteresis closure pressure for the adsorption of n-hexane on MCM-41 type materials of different pore sizes (the dotted line figures the limit derived from the literature for other organic compounds). -25

Isosteric Adsorption Enthalpy /kJ.mol-1

0

0.2

0.4

0.6

0.8

1

1.2

-30 9.9 nm

5.9 nm 3.6 nm

-35

-40

Pore Filling

Figure 9. Isosteric enthalpy of adsorption for n-hexane over materials of different pore sizes.

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5. Isosteric Adsorption Enthalpy Isotherms also convey information about the thermodynamics of adsorption. Using the isosteric method[5], It was possible to derive adsorption enthalpies and plot them against pore filling for systems with various pore sizes. For any given pore size the general trend is the following (figure 9) : at very low coverage, the adsorptive molecules interact with geometrical defects of the structure leading to a strong adsorption and therefore to a very negative adsorption enthalpy. As the pore filling increases, the enthalpy increases towards a value which is predicted to be the bulk condensation enthalpy by the Brunauer-Emmett-Teller model and which is indeed attained for the 9.9 nm pore size material. According to this theory, it should then be constant and equal to the bulk condensation value during pore condensation. Our system however, behaves differently. In the first step, the enthalpy raises during multilayer adsorption to reach a maximum, the value of which depends on pore size (the smaller the pores, the more negative the maximum) and which is inferior or equal to bulk condensation enthalpy. The enthalpy then reaches a plateau during pore condensation. The value at the plateau becomes, again, more negative as pore size decreases, but it can be noted that even for large pores such as 9.9 nm, it is more negative than for bulk condensation, indicating some kind of stabilisation. Of course, at the onset of condensation on the outer surface of the particles, the enthalpy raises to the bulk condensation value. This dependence of condensation enthalpy on pore size is another effect of confinement and hints at a thermodynamic explanation for the existence of two different regimes for the corresponding isotherms. 6. Conclusion This work shows that confinement effects arise for pores of sufficiently small sizes, leading to reversible phase transition and to a stabilisation of adsorbed molecules. The magnitude of these effects increases as pore size decreases. As in porous materials, for gaseous bulk reactants, reactions usually occur in a liquid like state, these aspects are of major interest for catalysis. A precise cartography of hysteresis loop vanishing in a (T,P,Ɏ) 3-dimensional space will allow a catalytic study of reactivity using hexane as gas phase "solvent" and with a well chosen parameter f(T,P,Ɏ) varying in order to cross the reversible/irreversible border. The change of physical properties of hexane should then result in a change of rate or selectivity of the model reaction. This aspect is currently investigated in our group.

An Experimental Study of the State of Hexane in a Confined Geometry

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References 1. 2. 3. 4. 5.

Branton P.J., Hall P.G., Sing S.W., Reichert H., Schüth F., Unger K.K. (1990) J. Chem. Soc. Faraday Trans., 90, 2965. Morishige K., Shikimi M. (1998) J. Chem. Phys., 108, 7821. Branton P. J., Sing K. S. W., White J. W. (1997) J. Chem Soc. Farady Trans., 93, 2337-2340, Burgess C. G. V., Everett D. H., (1970) J. Colloid. Interf. Sci., 33, 611. Jänchen J., Stach H., Busio M. and van Wolput J.H.M.C, (1998) Thermochimica Acta., 312, 33. Rees L.V.C., Shen D. (2001) Stud. Surf. Sci. Catal.,137, 579.

A MODEL FOR SOUND PROPAGATION IN THE PRESENCE OF MICROPOROUS SOLIDS M. NORI AND S. BRANDINI Department of Chemical Engineering, University College London Torrington Place, London WC1E 7JE, UK

1. Introduction The diffusivity of sorbates in microporous solids is a fundamental physical property that needs to be measured for the development and the design of adsorption and catalytic processes. Microscopic techniques (NMR, QENS, etc.) or macroscopic techniques (gravimetric techniques, Zero Length Column, Frequency Response (FR), etc.) are used for the determination of these data. The microscopic techniques allow the measurement of fast kinetics, while the macroscopic methods can be used for kinetics with longer characteristic times. When the measurement has been possible with both types of techniques, often there have been discrepancies [1]. There is the need to develop a macroscopic technique that allows kinetic measurements of fast diffusing strongly adsorbed systems. Among the macroscopic techniques, the FR has the fundamental feature of being able to discriminate between different rate-limiting mechanisms. The current upper limit of the FR technique is approximately 10-30 Hz (Bourdin et al. [2] ) as a result of mechanical limitations of volume modulators. This upper limit of the FR coincides with the lower limit of the acoustic range, 20-20,000 Hz (Everest [3]). Therefore to extend measurements to faster systems a theoretical investigation of sound propagation in presence of a microporous solid is presented. 2. Model The key issues that the mathematical model should address are: a) The extent of the influence of an adsorbent material on sound propagation for the development of a new technique based on acoustic measurements, b) The validation of the basic assumptions of the FR technique in the region above 10 Hz. 629 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 629–637. © 2006 Springer. Printed in the Netherlands.

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The model is based on the theory of sound propagation in tubes (Rayleigh [4], Tiejdeman [5] ) and the linearized adsorption-diffusion problem (Sun et al.[6]). To simplify the problem, the geometry considered is that of two parallel semi-infinite slabs. The Navier-Stokes equations in the vertical and axial directions are used to describe the motion. The fluid continuity equation and the ideal gas equation of state combined with the energy balance equation complete the set of equations needed to describe the gas phase [7]. Fickian diffusion in the solid slab combined with the differential mass balance and the energy balance in the solid phase are used to describe the kinetics of the adsorbent phase. These are linearised [6] since the pressure waves associated with sound propagation are of very small amplitude. For the solution, one has to solve the wave problem in the fluid phase and the diffusion problem in the solid phase, and make them meet the boundary conditions at the interface gas-microporous solid. The full set of differential equations is summarised as follows. GAS-PHASE Momentum balances horizontal and vertical directions: ª wu wu º wu u U«  v » wx ¼» wz ¬« wt



­° w 2 u w 2 u 1 w ª w u w v º ½° wp  P® 2  2  «  »¾ 3 wx ¬« wx wz »¼ °¿ wx wz °¯ wx

0

(1)

ª wv wv wv º U«  v u » wz wx »¼ «¬ wt



2 2 wp °­ w v w v 1 w ª w u w v º ½°  P® 2  2   »¾ « wz 3 wz «¬ wx wz »¼ °¿ wz °¯ wx

0

(2)

Continuity equation ª wu w v º wU wU wU u v  U«  » wt wx wz «¬ wx wz »¼

0

(3)

Equation of state p

UR o T

(4)

Energy balance ª wT wT wT º UC p « u v » wx wz ¼» ¬« wt

ª w 2 T w 2 T º wp wp wp O« 2  2 »   P) v u wz wx wz ¼» wt ¬« wx

(5)

A Model for Sound Propagation in the Presence of Microporous Solids ª§ w u · 2 § w v · 2 º § w v w u · 2 2 § w u w v · 2 ¸ ¸  ¨ 2 «¨¨ ¸¸  ¨¨ ¸¸ »  ¨¨   3 ¨© wx wz ¸¹ «© wx ¹ © wz ¹ » © wx wz ¸¹ ¼ ¬

)v

631 (5’)

MICROPOROUS SOLIDS Continuity equation wq wt

§ w2 q w2 q · D¨¨ 2  2 ¸¸ wx ¹ © wz

(6)

Energy balance U hm Cp hm

w Thm wt

§ w 2 Thm w Thm  O hm ¨ ¨ wz 2 wx 2 ©

· ¸ ¸ ¹

(homogeneous

medium

assumption) (7)

2.1. ACOUSTIC APPROXIMATION For a planar wave propagating confined between two layers of a microporous solid, with gas in the channel able to exchange gas molecules with the solid layers through an adsorption-desorption process, the following simplifying assumptions are introduced: (a) homogeneous medium, which means that the wave length and the distance between the slab and the microporous layer must be large in comparison with the mean free path; for air of normal atmospheric temperature and pressure, this condition breaks down for f>108 Hz and 2h

@

1 S1 ( E 2 ) r S 2 ( E 2 )  0 and p 2

f.

Here:

S1

E k (1  J k )  Kk2  Drk E 2 ;

S2

( E k J k  Kk2  Drk E 2 ) 2  E k E k 1  2J k  2 Kk2  Drk E 2 ! 0.

>



@

652

M. Petryk et al.

Thus, in the area Re p t 0 we can determine the following original analytical functions [4]:

1 if  * 2 pt ³ W1k ( p, E ,z )e dp 2S i if

W1k ( t, E 2 ,z ) 1

~ R j1,k (t , E 2 , z )

S

>

f

S

0

@

~* 2 ist ³ Re R j1,k (is, E , z )e ds; 0

Re>H S³ 1

~ H kj (t , E 2 , z )

1 f ª * 2 ist ³ Re ¬W1k ( is, E ,z )e º¼ ds;

f

~ kj

@

(is , E 2 , z , ] )e ist ds;

0

Zkj ( t, E 2 ,z,] )

^



1f E jJ j t 2 ist ³ Re H kj ( is, E ,z,] ) e  e

S

0

 E jJ j  is ` G 2 j jE 2zjJ 2 ds, E J D1

1 1

Returning to the original solutions of the problem (20)-(22) and after mathematical transformations, we obtain the exact solution of the problem (14)(19): t f

Ck (t , r , z )

³ ³W

1k

0 0

f

(t  W , r , U , z )Z 0 (W , U ) UdUdW  ³ W1k (t , r , U , z )Z 01 ( U ) UdU 0

2 f

t f l1

 ¦ ³ R j1,k (t , r , U , z )Z j1 ( U ) UdU  ³ ³ ³ (t  W , r , U , z , ] ) Dz11[ f1 (W , U , ] ) j 1 0

0 0 l0

t ff

 C01 ( U , ] )G t (W )]UdUd]dW  ³ ³ ³ H k 2 (t  W , r , U , z , ] ) Dz21[ f 2 (W , U , ] ) 0 0 l1

f l1

 C02 ( U , ] )G t (W ) UdUd]dW  ³ ³ Z k1 (t , r , U , z , ] )a01 ( U , ] )Ud]dU 0 l0

ff

 ³ ³ Z k 2 (t , r , U , z, ] )a02 ( U , ] ) Ud]dU ; k

1,2

(24)

0 l1

t

ak (t , r , z ) e  E kJ k t a0k (r , z )  E k ³ Ck (t  W , r , z )e  E kJ kW dW 0

Here, the main solutions of the boundary problem (14)-(19) are:

(25)

653

Mathematical Modelling and Research for Diffusion Processes f

2 ³ W1k ( t , E ,z )J 0 ( EU ) E d E , k

W1k ( t ,r , U ,z ,] ) R j1,k ( t,r, U ,] ) H kj ( t ,r , E ,z ,] ) Z kj ( t ,r , U ,z ,] )

1, 2 ;

(26)

0

f

2 ³ R j1,k ( t, E ,z )J 0 ( E r )J 0 ( EU )E d E ;

j,k 1,2

(27)

0

f

2 ³ H kj ( t , E ,z ,] )J 0 ( E r )J 0 ( EU ) E d E ;

k, j

1, 2 (28)

0 f

2 ³ Z kj ( t , E ,z ,] )J 0 ( E r )J 0 ( EU ) E d E ;

k, j

1, 2 (29)

0

5. Conclusions

The models developed and exact solutions obtained allow a more general view, which enables one to analyze and investigate different variants of the schemes and regimes of diffusion and adsorption process. Using the obtained analytical solutions of the proposed models as well as experimental concentration profiles and solving the inverse problems, the kinetic parameters of internal processes (the effective diffusion coefficients, the adsorption constants, and the mass transfer coefficients as functions of geometrical coordinates of layer media) are defined. The methodology of the process model solution can be further developed and applied to: - Simulation of diffusion processes in multilayered media and adsorption processes in n-component media ( n ! 2 ) and media of complicated structure; - Models which consider mass transfer by diffusion and convection/ adsorption processes as well as non-linear models. References 1. 2. 3.

Lykov A.V., Mykhaylov Y.A. (1963) The theory of mass transport, (State Energy Publishing, Moscow) 535. Ufliand Y. (1967) Integral transformations in problems of theory of elasticity, (Nauka, Saint Petersburg). Sergienko I., Skopetsky, Deineka B. (1991) The mathematical modelling and research of process in no regular medias, Kyiv, Naukova Dumka (Academic Publishers) 432.

654 4.

5. 6. 7. 8. 9. 10.

11. 12. 13. 14. 15. 16. 17. 18.

M. Petryk et al. Lenyuk Ɇ.P., Petryk M. (2000) Fourier, Bessel integral transformations with spectral parameter in mathematical modelling problems of mass transfer in non regular media, Kyiv, Naukova Dumka (Academic Publishers) 372. Stepanov V.V. (1959) Course of differential equations, (Physical and Mathematical Publ., Moscow), 468. N’Gokoli-Kekele P., Springuel-Huet M.-A., Fraissard J. (2002) An Analytical Study of Molecular Transport in Zeolite Bed. Adsorption, (Kluwer Academic Publishers, Dordrecht) 8, 35-44. Bonardet J.-L., Fraissard J., Gédéon A., Springuel-Huet M.-A. (1999) Catal. Rev.-Sci. Eng., 41, 115-225. Chen, N.Y., T.F. Degnan, M.C. Smith (1994) Molecular Transport and Reaction in Zeolites: Design and Application of Shape Selective Catalysis, (V.C.H. Weinheim, New York). Kärger, J., Ruthven D. (1992) Diffusion in Zeolites and Other Microporous Solids (John Wiley & Sons, New York). Magalhães, F.D., Laurence R.L., Conner W.C., Springuel-Huet M.-A., Nosov A., Fraissard J. (1997) Study of molecular transport in beds of zeolite crystallites: semi-quantitative modelling of 129Xe NMR experiments, J. Phys. Chem. B, 101, 2277-2284. Springuel-Huet, M.-A., Nosov A., Kärger J., Fraissard J., 129Xe NMR study of bed resitance to molecular transport in assemblages of zeolite crystallites (1996) J. Phys. Chem., 100, 7200-7203. Mathiu-Blaster, Sicard J. (1999) Thermodynamics of irreversible processes applied to solute transport in nonsaturated porous media, J. Non Equilibrium Thermodyn., 24, 107-122. Zapolsky H., Pareige C., Marteau L., Blavette D., Chen L.Q. (2001) Atom probe analyses and numerical calculation of ternary phase diagram in NiAl-V, Calphad, 25, 125-134. de Boor R. (2000) Contemporary progress in porous theory, Appl. Mech. Rev., 53 (12), 323-369. Mongiovi M.S. (2002) Thermo mechanical Effects in the Flow of Fluid in Porous Media. Math. And Comp. Modelling, 35, 111-117. Petryk M., Babiyk M. (2000) Mathematical Modelling, Saint Petersburg, 1, 133-137. Krimitsas N., Vorobiev E., Petryk M.,(2000) Continuous solid/liquid expression in screw press: process analysis and modelling, Proceedings of 8th World Filtration Congress FC-8, Brighton (UK), Vol. 2, 763-766. Petryk M. (2002) Modelling and investigation of mass exchange of biological materials processes, Thermoynamics, Microstructures and Plasticity. NATO Advanced Study Institute, Frejus (France), 74.

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19. Petryk M., Petryk N., Matiega V. (2002) The Intensification of Process of Vibrofiltration in non Regular Cylinder Media, Proceedings ICEST-2002, Compiegne (France) 1, 143-149. 20. Petryk M., (1998) Mathematical model of the filtration process halflimited mediums in conical canals. Ternopil State Technical University Bulletin, 3, 20-29. 21. Shabliy O., Petryk M., Vasylyuk P., Katerynyuk ȱ., (1999) Diffusion processes in oxide layer of Fe-Cr alloys, Ternopil State Technical University Bulletin, 1, 5-13. 22. Vasylyuk P.M., Butenko L.I., (1989) Increase of the Fe-Cr alloys resistance, Metals. ȺS USSR, 1, 154-156.

NEURONAL NETWORK USED FOR INVESTIGATION OF WATER IN POLYMER GELS

*T. RUSU *, *C. IOJOIU and **V. BULACOVSCHI * P. Poni Institute of Macromolecular Chemistry, Ghica Voda Street, 41A, Iasi 6600 Romania, [email protected] ** Gh. Asachi Technical University, Iasi, 6600, [email protected]

Abstract The paper deals with the synthesis and characterization of some copolymer networks build up from sequences of polydimethyilsiloxane (PDMS) and poly(methacrylic acid) (PMAA). The water diffusion properties of the networks are also investigated by using a neuronal network (NN) algorithm. The proposed algorithm is an empirical one, which simultaneously retrieves several physical parameters over the copolymer network properties as a relation between the molecular ratios of copolymer sequences. Comparison with other NN algorithms is presented. 1. Introduction Modern chemistry has made important efforts to develop and to test theoretical concepts for providing new macromolecular compounds with the structure related to imposed properties for new areas of applications. Hybrid materials with different molecular architectures have attracted an increased interest in recent years. This paper focuses on copolymers containing hydrophilic and hydrophobic sequences that confer them special transport properties. The molecular ratio between the two sequences has an important role for the properties of such copolymers like shape-selective separation and water (small size particles) delivery. It is known from literature that hydrogels reveal phase transitions making swelling-deflation changes in response to environmental modifications, e.g. solvent composition [1, 2], ionic strength [3], temperature [2-5], electronic field [6] and light [7]. Such stimuli-responsive polymers have been investigated 657 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 657–670. © 2006 Springer. Printed in the Netherlands.

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with application to artificial muscles [8], immobilization of enzymes [9], concentration of dilute solutions [10, 11] and chemical valves [12]. Using stimuli-responsive amphoteric gels there have been attempts for temporal control of water release. Thus, a swollen gel in ethanol immersed in water undergoes immediately spontaneous translation and rotational motions with two-thirds of its volume immersed in water. In the course of its motion the gel gradually sinks and finally settles at the bottom of the vessel due to contraction and cease of motion [13]. The speed, duration and mode of gel motions are associated with its size, shape and chemical nature. Such polymer gels, exhibiting motion in water, have potential applications as soft-touch manipulators, target drug-delivery devices, micro-agitators, and micro generators. Their study is under way. Networks prepared by reacting functionalized polydimethylsiloxanes (PDMA) with different cross-linking agents are considered ideal models for studying network formation and properties [14]. PDMS is highly hydrophobic, very flexible and shows little variation of its properties with temperature. Since its mechanical properties are poor, for their improvement one should synthesize cross-linked copolymers based on PDMS [15]. This paper deals with the synthesis and characterization of amphoteric networks (gels) based on PDMS–co–PMAA sequences. The water-diffusion properties of the obtained gels are investigated by using some neuronal network algorithms. 2. Experimental The polymer gels were obtained by radical copolymerization of polydimethylsiloxane macro-initiators with methacrylic acid in presence of ethyleneglycol dimethacrylate as cross-linking agent [16]. The synthesis of the azoester macroinitiators (AzoPDMS), with different molecular weights of siloxane sequences and different contents of azo groups, was realized according to Scheme 1, and the experimental data are summarized in Table 1. The synthesis of PDMS-poly(methacrylic acid) (PMAA) hydrophobic– hydrophilic gels was realised according to Scheme 2 [17], their characteristics being summarized in Table 2.

Neuronal Network Used for Investigation of Water in Polymer Gels

Table 1. Characteristic data for Azo-PDMS.

659

660

T. Rusu, C. Iojoiu and V. Bulacovschi CH3 CH3 CH3 Si O R O OC (CH2)2 C N N C (CH2)2 COO R Si O CH3 CN CN + +

MAA EGDMA

PDMS - co - PMAA

Scheme 2

Table 2. Characteristic data for PDMS–co-PMAA obtained by radical polymerization of MMA in presence of AzoPDMS. Initial mixture* Sample Network SiO/MAA Yield, SiO/MAA** Aspect MnPDMS (from Molar % Molar ratio AEPS) ratio 5040 0.5 I.1 85.9 0.42 Gel 6620 0.5 I.2 87.6 0.44 Gel 14060 0.5 I.3 86.9 0.43 Gel 5040 2.0 II.1 89.4 1.63 Gel 6620 2.0 II.2 87.1 1.74 Dense gel 14060 2.0 II.3 90.4 1.81 Dense gel * Polymerization in sealed ampoules; 80qC; 20 hours; solvent, toluene (total concentration 25%); EGDMA, 1% molar against MAA ** Determined from elemental analysis (Si content) 3. Characterization Data 3.1. MACRO-INITIATOR CHARACTERIZATION The characterisation of macro-initiators was made by IR (SPECORD M80 IR) and NMR (Bruker AC 200). The IR spectra revealed the following absorptions: 1265 cm-1 - Si-CH3; 800, 1110, 1020 cm-1 - Si-O-Si; 1789 cm-1 -> 1738 cm-1 CACV inclusion in the PDMS sequence. In the 1NMR spectra (CDCl3) (Fig. 1) one finds: 0.01ppm (6H, CH3-Si, s), 0.43-0.47 ppm (2H, -CH2-Si, m; E isomer), 0.81-0.88 ppm (3H, CH3-CH, d;

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D isomer), 1.00-1.08 ppm (1H, CH3-CH, q; D isomer), 1.40–1.47 ppm (2H, CH2 -CH2 -CH2 -, m; E isomer), 2.73-2.85 ppm (2H, CH2-NH2, m). For the AzoA-PDMS: 0.01ppm (6nH, CH3Si, s), 0.43-0.47 ppm (2H, CH2Si, m; E isomer), 0.80-0.88 ppm (3H, CH3-CH, d; D isomer), 1.00-1.08 ppm (1H, CH3-CH, q; D isomer), 1.20 ppm (3H, CH3-C(CN), s), 1.40–1.47 ppm (2H, -CH2 -CH2 -CH2 -, m; E isomer), 1.60-1.66 ppm (2H, CH2-C(CN), m), 2.26-2.43 ppm (2H, -CH2-CO, m), 3.11-3.19 ppm (2H, CH2-NHCO, m).

Figure 1. 1H-RMN spectra for AzoA-PDMS.

3.2. CHARACTERIZATION DATA FOR THE PDMS–co–PMAA NETWORKS The chemical structure and the thermal behavior of the cross-linked copolymers were established by IR spectroscopy and differential scanning calorimetry (DSC) (Perkin Elmer DSC-7 device), respectively (Fig. 2). Scanning Electron Microscopy (SEM) (Phillips XL 300 microscope) analysed the microstructure of the resulting networks (Fig. 3).

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Figure 2. Typical IR and DSC curves.

Figure 3. The SEM image of the network.

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663

4. Water Diffusion Tests The synthesized copolymers were submitted to the water delivery test. Equal amounts of samples from each of the six copolymers were allowed to achieve the swelling equilibrium by keeping them in an excess of water for three days. Then, the excess of water was removed by filtration and the swollen samples were submitted to control dryness thermosetting at 100qC and weighed at every two hours [18]. The results are presented in Figure 4 and Table 2. The content of residual water was followed as a function of time and molar ratio of the two polymeric blocks.

Figure 4. Residual water from the macromolecular network vs. time and molecular ratio.

The water delivery tests were subsequently used in conceiving a Multilayer Perceptron Neuronal Network. A comparison of this algorithm with other global diffusion speed retrieval algorithms for amphoteric copolymers/ polymers is presented. We also examined different diffusion conditions related to different forms of polymer-soil moisture presenting various cross-sections of evaporation speed retrieval error analysis.

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Neural networks (NNs) are well suited for a very broad class of non-linear approximations and mappings. Neural networks consist of layers of uniform processing elements, nodes, units, or neurons. The neurons and layers are connected according to a specific architecture or topology. The number of input neurons, n, in the input layer is equal to the dimension of the input vector X. The number of output neurons, m, in the output layer is equal to the dimension of the output vector Y. A Multilayer Perception NN always has at least one hidden layer with k generic neurons. The neuron is a non-linear element because its output zj is a non-linear function of its inputs X. So, problem which can be mathematically reduced to a non-linear mapping can be solved using the NN represented. NNs are robust with respect to random noise and sensitive to systematic, regular signals. From a mathematical point of view, the multi-parameter retrieval algorithm corresponds to a continuous mapping [19]. It maps a vector of ECs, T, onto a vector of retrieved physical parameters, g. NNs are well suited for performing a wide variety of continuous mappings. The architecture of the NN that we used is the OMBNN3 algorithm (Scheme 3). The NN which represents this algorithm has n = 4 inputs, m = 3 outputs, and one hidden layer with k = 12 neurons. This NN can be also written explicitly as:

gq

k

n

j 1

j 1

bq  aq tanh{¦ wqj [tanh(¦ : jiTi  B j )]  E q },.....q

1,..., m

(1)

where the matrix :ji and the vector %j represent weights and biases in the neurons of the hidden layer; Zqj and the Eq represent weights and biases in the neurons of the output layer; the aq and bq are scaling parameters.

Scheme 3

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However, as shown in Scheme 3, even such an approximate separation of signals allows the reducing of the random and systematic errors in diffusion speed and to reduce the dependence of the diffusion speed bias on V and L. For comparison were selected three alghoritms: - the original global operational (cal/val) algorithm developed by Goodberlet [20] (GSW); - the current operational algorithm (GSWP) which is the same as the GSW algorithm but corrected for water vapor by Petty [21]; - a physically-based (PB) retrieval algorithm developed by Wentz [22]. Table 3 shows that the NN algorithm provides significant improvement in retrieval accuracy at high diffusion speeds. Error budgets (m/s) for different diffusion speed algorithms and separately for higher diffusion speeds are also considered. Table 3. Performances of NN algorithm. Algorithm

Bias

GSW GSWP PB OMBNN3

-0.2 (-0.5) -0.1 (-0.3) 0.1 (-0.1) -0.1 (-0.2)

Algorithm RMSE 1.4 (1.8) 1.3 (1.6) 1.3 (1.8) 1.0 (1.3)

Total RMSE 1.8 (2.1) 1.7 (1.9) 1.7 (2.1) 1.5 (1.7)

W>15 m/s RMSE (2.7) (2.6) (2.6) (2.3)

5. Results and Discussion The OMBNN3 algorithm has the ability to retrieve not only diffusion speed but also three other parameters: columnar water vapor V and columnar liquid water L as shown in Figure 5. Figure 6 presents diffusion speed related to molecular ratio of PDMS/PMAA and the number of matches for the OMBNN3 algorithm.

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Figure 5. Water diffusion speed retrieved by four different algorithms as functions of columnar water vapor (Solid line - OMBNN3, dotted line - GSWP, dashed line - GSW, and dash-dotted line the Wentz algorithm).

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Figure 6. Water diffusion speed related to molecular ratio of PDMS/PMAA.

6. Conclusion One of the challenges of modern chemistry deals with the design of macromolecules with imposed properties. With this idea in mind we have used/tested an empirical neural network (NN) algorithm, which simultaneously retrieves several physical parameters over some polymer network properties from evaporation conditions (EC), primary emphasizing on water diffusion speed in a copolymer gel. This NN-based algorithm, which retrieves diffusion speed, columnar water vapor, and columnar liquid water, has been developed recently. Conditional simulation is one of the most rational approaches for modeling unsaturated flow and transport. The presented NN algorithm is a multi-parameter one that includes some inputs, one hidden layer, and corresponding outputs. The combined analytical-numerical multilayer Perceptron NN and OMBNN3 approach was developed for the analysis of flow and diffusion transport processes in porous media. The OMBNN3 and the multi-layer Perceptron algorithms, both employing the simultaneous multiparameter retrieval approach, reduce the bias, and the dependence of the bias,

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669

on both water vapor and columnar liquid water concentrations. The OMBNN3 algorithm demonstrates the best performances. The random errors for the OMBNN3 algorithm are significantly smaller and less dependent on the other related medium parameters than those for the other algorithms. A comparison of this algorithm with other algorithms is also presented. The simulation parameters can be used for optimization of the molecular ratio of the polymer sequences to build up networks with imposed diffusion properties according to the desired areas of application. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Tanaka F. and Ishida M. (1994) Physica A, 204 660. Katayama S., Hirokawa Y. and Tanaka T. (1984) Macromolecules, 17, 2462-2463. Hirokawa Y. and Tanaka T.(1984) J. Chem. Phys., 81, 6379-6380. Ohmine I. and Tanaka T., (1982) J. Chem. Phys., 77, 5725-5729. Tanaka T., Fillmore D., Sun S. T., Nishino I., Swislow G. and Shah A. (1980) Phase Transition in Ionic Gels. Phys. Rev. Lett., 45, 1636-1639. Tanaka T. (1981) Gel Sci. Am., 244, 124-138. Tanaka T., Nishio I., Sun S. T. and Ueno-Nishhio S. (1982) Collapse of Gels in an Electric Field, Science, 218, 467-469. Irie M. and Kunwatchakun D. (1986) Macromolecules, 19, 2476-2480. Suzuki M. (1991) Polymer Gels, Ed. D. DeRossi, Plenum Press, New York, pp. 221-236. Hoffman A. S. (1987) J. Control. Release, 6, 297-305. Freitas R. F. S. and Cussler E. L. (1987) Chem. Eng. Sci., 42, 97-103. Trank S. J., Johnson D. W. and Cussler E. L. (1989) Food Technol., 43, 78-83. Osada Y. and Hasebe M. (1985) Chem. Lett., 9, 1285-1288. Hamurcu E. E. and Bahattin M. (1995) Macromol. Chem. Phys., 196, 1261. He X. W., Windmaier J. M., Herz J. E. and Magers G. C. (1992) Polymer, 33, 866. Rusu T., Pinteala M., Iojoiu C., Harabagiu V., Cotzur C., Simionescu B. C., Blagodatskikh I. and Shchegolikhina O., (1998) Synth. Polym., J. 5, 29. Harabagiu V., Hamciuc V. and Giurgiu D., (1990) Makromol. Chem., Rapid Commun., 11. Rusu T., Ioan S. and Buraga S. C., (2002) Euro. Polym. J., 37, 2005, 2001. Rusu T. and Gogan O. M., Proceedings of Franco-Romanian Symposium on Applied Chemistry CoFrRoCA, 129-132.

670

T. Rusu, C. Iojoiu and V. Bulacovschi

20. Goodberlet M. A., Swift C. T. and Wilkerson J. C., (1989) Remote sensing of ocean surface winds with the special sensor microwave imager. JGR, 94, 14574–14555. 21. Petty G. W., A (1993) Comparison of SSM/I algorithms for the estimation of surface wind, Proceedings Shared Processing Network DMSP SSM/I Algorithm Symposium, 8-10 June. 22. Wentz F. J., (1997) A well-calibrated ocean algorithm for special sensor microwave / imager, JGR, 102, 8703-8718.

DEPENDENCE OF SELF-DIFFUSION COEFFICIENT ON GEOMETRICAL PARAMETERS OF POROUS MEDIA *V.Ⱥ. Sevriugin, **V.V. Loskutov, *V.D. Skirda *Kazan State University 420000, Kazan, Kremlyovskaya st., 18. **Mari State Pedagogical Institute 424002, Yoshkar-Ola, Communisticheskaya st., 44.

The self-diffusion of liquids in porous media is a general problem of molecular physics with a wide range of practical applications. In [1,2] analytical expressions for the effective self-diffusion coefficient D (t ) of a liquid in a closed pore have been reported for the shorttime regime. It has been shown there that D(t ) v t  0.5 and is defined by the ratio of surface area to pore volume S . With t increase D(t ) changes in V

1

accordance to D(t ) v t , which corresponds to a regime of completely restricted particle diffusion. However, theoretical approaches describing diffusion in porous media in the long-time regime t o f , taking into account geometry, pore connectivity and the sizes of molecules of liquids are have not been presented, though there are some empirical expressions. The dependence of the selfdiffusion coefficient Ds on the pore observable in a regime t o f for cyclohexane and cyclodecane in porous Vycor glass were described by [3]: Ds Do

1 1

1 Ud

(1)

where Do - self-diffusion coefficient of a bulk liquid, U - a "permeability" of the medium, d – an average pore diameter. In [4] it was remarked that dependence (1) fails for experimental data at small d values. In the same way, attempts have been made to take into account the size of molecules in expression (1) by replacing d by d  a , where a is an molecular effective diameter.

671 W.C. Conner and J. Fraissard (eds.), Fluid Transport in Nanoporous Materials, 671–676. © 2006 Springer. Printed in the Netherlands.

672

V.Ⱥ. Sevriugin, V.V. Loskutov and V.D. Skirda

The aim of this work is to find the dependence of the molecule selfdiffusion coefficient in a liquid in a porous medium on the geometrical characteristics of the porous medium in a regime t o f . Let the molecule of a liquid in the porous media diffuse during a long time t. The value L2 represents the sum of the squares of diffusion transitions

A 2i

¦i A2i . Then:

of the molecule in all pores: L2

¢ L2 ²

k ¢A 2 ( R )²

6 Ds ( R )t

(2)

where k – an average number of diffusion transitions of the molecule from pore to pore during the time t , Ds (R ) – the measured self-diffusion

coefficient, ¢A 2 ( R )² – an average square diffusion replacement of molecules for the time between two consecutive inputs to the neighboring pore. In (2) the average number of transitions k of molecules is defined by the average value of the "life"-time ¢W ( R )² of molecules in pore:

t ¢W (R )² .

In accordance with (2) it is possible to write for the selfdiffusion coefficient

k

¢A 2 ( R )² . 6¢W ( R )²

Ds ( R )

(3)

Let there be no specific interactions between molecules of a liquid in a pore and molecules of a material of the porous media, and the translational mobility of molecules of a liquid in the medium will be characterized by the 2 same characteristic sizes of length o and time o as in the bulk liquid. Generally, for any porous medium with any geometry, it is possible to consider the characteristic time ¢W A ² determining the frequency of "interactions" of molecules of a liquid with the walls of the pore. Thus the average life-time ¢W ( R )² of molecules in a pore can be defined as

W

A

¢W  R ²

¢W A ²¢ n R ² , where ¢ n R ² – an average number of "interactions"

of a molecule with walls for a time from the moment a molecule enters the pore to the moment it exits. The number of "interactions" of any molecule with the walls of a pore for some interval of time t is a random variable and, it is assumed, follows the Poisson distribution Pn Ot e Ot

n!

where

Pn

– the probability for any molecule to interact with pore walls n times

during t and

O 1 ¢W A ² . The probability Pn

is a function of the geometry of

673

Dependence of Self-Diffusion Coefficient on Geometrical Parameters the (

pore:

t !! ¢W A ² ):

Pn ( R, t ). The

Pn

n  R,t

average

n  R, t

will be defined as

f

¦ nP  R, t . n

n 0

There are two possibilities for the molecule in the pore to interact with a boundary surface under diffusion, namely: to appear on an element of a boundary surface and to pass into the next pore, or to appear on an element of a – total effective area of surface S and to remain in the same pore (where channels). Two different probabilities correspond to "interaction" with a surface Pst and "interaction" with the channel PV . It is necessary to note that both probabilities are functions of the pore size. Then the expression for an average number of "interactions" with

V

V

boundary surfaces for a macroscopically long time

¢n R, k ² where

k

f

t !! ¢W (R )² is:

f

f

n

n

¦nPn  R, k n

0

¦nPn  R, k Pst  R, k  ¦nPn  R, k PV  R

t

¢W (R )²

k 1

. At

0

(4)

0

expression (4) may be written in the time-

independent form:

¢ n  R,1 ²

f

f

¦ nPn  R,1 Pst  R  ¦ nPn  R,1 PV  R . n

n

0

0

In this expression the second sum is a constant close to unity. Then it is possible to write the differential equation for n  R in the form:

w n R wR

w f [¦ nPn  R Pst  R ] wR n 0 wPst  R ½ ­  n P R P R P R P R  ª º     ® ¾. ¦ n n ¬ n1 ¼ st wR ¿ n 0 ¯ f

For high n values probabilities therefore:

Pn1  R

and

Pn  R

(5)

in (5) are equal and

674

V.Ⱥ. Sevriugin, V.V. Loskutov and V.D. Skirda w n R wR

f

¦ nP  R n

n 0

and

n R

The value

no

Then,

W  R

wPst  R wR

n R

wPst  R wR

no exp  Pst  R .

(6)

1 in (6) is easily defined from the condition Pst  R =0. according

to

(3)

and

taking

into

account

that

n  R , we obtain the expression for the self-diffusion A coefficient of a liquid in the pore:

W

Ds ( R) where

Do

¢A 2 ( R)² exp[ Pst  R ] Do exp[ Pst  R ] , 6¢W A ²

– a self-diffusion coefficient of a bulk liquid.

D s/D 0 1

exp(-11.04d 0/d) 0,000

0,005

0,010

0,015

0,020

4

1/d, 10 A

-1

Figure 1. Dependence of the relative self-diffusion coefficient Ds / D0 for benzene in porous media on the glass sphere diameter: straight line – theoretical values, symbol – experimental data.

Dependence of Self-Diffusion Coefficient on Geometrical Parameters

Pst

675

is the probability for a molecule to appear near the wall at the

distance of the effective diameter of a molecule d 0 . Hence, this probability is equal to the ratio of the number of molecules in the layers near the walls to the total number of molecules in the pore or, otherwise, to the ratio of the volume of layers near the walls to the pore volume, that is: Pst do S / V . Finally, it is possible to write an expression for the self-diffusion coefficient of molecules of a liquid in a pore in long-time regime t o f as:

Ds

­ 6 Sd o ½ Do exp ®  ¾ ¯ V ¿

(7)

Figure 1 shows the experimentally observed dependences of the selfdiffusion coefficient diameter for benzene in porous media, formatted by glass spheres (the sphere diameters are: 44-53 mkm, 53-63 mkm, 63-74 mkm, 74-88 mkm). The diffusion measurement were made by the PFG-NMR method with a proton resonance frequency of 60 MHz. For this system the dependence of the area of a pore surface and its volume on the diameter of a glass sphere can be calculated: S=0.90675d2;

V=0.08215d3;

S/V=11.04

and (7) can write:

Ds D0 where

d 0 =4.96

­ 11,04d 0 ½ exp ®  ¾ d ¿ ¯

(8)

A – is an effective diameter of benzene molecule. Fig.1

illustrates the theoretical values (8) Ds / D0 (straight line) and experimental data for glass spheres of different sizes. Acknowledgments This work was performed within the framework of projects: CRDF REC 007; CRDF RPO 1331; References 1. 2.

Mitra P.P., Sen P.N., Schwartz L.M. (1993) Phys. Rev. 47, N.14, P. 8565. Fatkullin N.F. (1992) ɀɗɌɎ. 101, ʋ 5, P. 1561,

676 3. 4.

V.Ⱥ. Sevriugin, V.V. Loskutov and V.D. Skirda Mitzithras A., J.H. Strange. (1994) Magn. Res. Imagina. 12, N.2, P. 261. Skirda M.V., Skirda V.D. (1997) Structure and dynamics of molecular systems./ Yoshkar-Ɉɥɚ, Moscow, Kazan,1. P. 186.

INDEX 129

Xe NMR spectra, 315, 320 Xe resonance of adsorbed xenon, 315 13 interval sequence, 263 13X zeolite, 30 1 H NMR imaging, 315 2D (two dimensional), 353 2-methylhexane (2MH), 218 4A zeolite, 32 5A zeolite, 567 5A zeolite, 567 7 Li spin-lattice relaxation, 491 AASBU (Automated Assembly of Secondary Building Units) method, 510, 512, 515 absorption cross-sections, 336 acetone, 421 acoustic wave propagation, 355 activated carbon, 523, 524 activation energies, 189 activation energy of desorption, 598 addresses, 353 adsorption, 130 adsorption from nitrogen–oxygen mixtures, 299 adsorption into a zeolite, 3 adsorption isotherms, 312, 523 adsorption of benzene, 317 adsorption of paraxylene, 322 adsorption of pure gas, 326 AFM, 82 alkanes, 567 AlPO4, 279, 292 AlPO4 adsorption–desorption, 291 ALPO4–5, 139, 409 ALPO4–5 zeolites, 125 AlPOs, 283, 288 aluminophosphate, 289 aluminophosphate synthesis, 282 analyses of the diffusive multilayers, 644 angiography of blood flow, 355 anisotropy, 536 anisotropy effects for diffusion, 223 anomalous behavior of the self-diffusion coefficient, 276 129

anomalous diffusion, 187 anomalous diffusion in zeolites, 20 anomalous dynamic properties, 276 approach, 214 aromatics, 166 automotive, 70 bed, 559 Belousov-Zhabotinsky reaction, 355 benzene, 24, 30, 83, 137, 141, 326, 348, 541, 674 benzene in NaX, 546, 547 benzene/silicalite, 34 benzene–toluene, 84 binary gas sensor, 449 binary mixture in a zeolite, 229 biological membranes, 481, 496 biphasic fast exchange mode, 424 blockage effects, 465 Bloembergen, Purcell and Pound (BPP) theory, 485, 487 Boltzmann distribution, 243 Brownian motion, 250 bulk diffusivities, 379 Burchart-Dreiding forcefield, 392 C6–C8 aromatics, 130 CaCl2 IN, 553 calibration curves, 354 capillary column inverse gas chromatography (CCIGC), 58 capillary condensation, 274 carbon, 27 carbon membranes Lda, 462 carbon molecular sieves, 29 carbon molecular sieve membranes (CMSM), 460, 462 catalyst stabilization, 458 catalytic membrane reactor, 454 catalytic polymeric membranes, 455 c-DMCH, 141 cement-based material, 428 CH4 in MFI, 221 chain length, 173 677

678 characteristic time: inter, 316 characteristic time: intra, 316 characterization of amorphous adsorbents, 513 characterization of porous amorphous adsorbents, 518 chemical composition of the alloy, 640 chemically specific MRI experiments, 355 chemisorbed species, 203 chlorella cells, 497 chromatographic measurements, 155 chromatography (IGC) polymers, 56 Chudley and Elliott, 343 CMS membranes, 613 CO2, 29 CO2/Xe, 614 coal, 583 coal-bed methane, 584 coherent cross section, 336 coherent elastic scattering, 335 coherent scattering, 339 coherent scattering cross section, 335 coherent scattering function, 340 comparison, 168 competitive adsorption, 328 competitive diffusion, 315 computational methodologies, 512 configurational, 202 confocal optical Raman microscopy, 279 contact ionic pairs, 484 contributions, 198 controlled-pore glass, 258 corrected diffusivity, 4 correlation functions, 112 correlation time, 485, 489 counter-current ZLC, 168 Cr and Si diffusion coefficient, 646 Cranck equation, 325 CrAPO-5 crystals, 189, 575 cross-sections, 334 cryodiffusometry, 260 crystal growth, 288 crystals of slab shape, 384 Cs+ cations, 483 cumene, 592 CVD/activation, 613

Index cyclic hydrocarbon, 125, 139, 383 cycling relaxometry, 415 cyclohexane, 83, 141 cyclopentane, 137 D’Arcy’s law, 584 Darken equation, 4, 13 demontis forcefield, 391 density of a membrane, 467 DFT (Density Functional Theory), 513 differential adsorption bed, 164 diffusion, 11, 17, 41, 74, 85, 100, 151, 165, 166, 189, 211, 299, 302, 305, 567, 592, 595, 671 diffusion anisotropy, 537, 538 diffusion coefficient, 209, 385 diffusion coefficient Dinter, 316 diffusion coefficients of liquid, 270 diffusion coefficients of the cyclic hydrocarbons, 394 diffusion decay, 264, 269 diffusion decays water molecules in sand, 271 diffusion equation, 361 diffusion in applications, 69 diffusion in MFI-type zeolites, 378 diffusion in spherical crystals, 385 diffusion in zeolites, 279, 333 diffusion of benzene, 345 diffusion of binary mixtures of hydrocarbons, 324 diffusion of D2O, 359 diffusion of hydrocarbons, 315 diffusion of liquid molecules, 262 diffusion of organic molecules, 279 diffusion of paramagnetic tracers, 360 diffusion of pure hydrocarbons, 324 diffusion of Xe in a xenon-soluble, 366 diffusion with surface-barrier model, 387 diffusion with surface-resistance, 387 diffusional permeability, 499 diffusion-limited nitrogen, 303 diffusion-rearrangement model, 386 diffusivity, 102, 171, 173, 348 diffusivities in zeolites, 333 diffusivities of benzene in the MFI, 349

Index diffusivities of para- and meta-xylene, 350 diffusivities of the sorbate molecules in silicalite-1, 393 diffusivities of xylenes, 350 diffusivity measurements with PFG NMR, 356 discrepancies in the diffusivities, 2 disordered nanopores, 415 dissolution of polymers, 362 distribution function of IMFG, 265, 267, 268 drift, 299, 300, 302, 304 drift N=N stretching band, 309 drift study of adsorption of molecular hydrogen, 300 drying of alumina pellets, 360 dynamic Monte Carlo simulations, 193, 577 edge enhancement, 250 effect of carbonaceous, 567 effect of molecular size, 258 i for the effective diffusion coefficients D elements Al, Cr and Si, 641 Einstein, 4 Einstein’s equation, 490 Einstein’s relation, 5, 520, 521 Einstein-Smoluchowski equation, 249, 250 emissions, 70 ENDOR technique, 482 energy minimisation calculations, 403 energy profile, 205 enthalpy of adsorption, 625 entropy of adsorbed phase, 136 EPR, 482 ethylbenzene/silicalite-1, 131 exchange processes, 276 experimental measurements, 6

FAU-type zeolites, 544 Fe-Cr multilayer systems, 640 FePO4, 279, 288 FePO4S crystals growth, 285, 286, 287 Fick ratio, 41, 102

679 Fick, Onsager and Maxwell-Stefan, 229 Fick’s diffusion equation, 536 Fick’s first law, 9 Fick’s law, 3, 214 Fick’s second law, 10 Fickian self-diffusion coefficients, 535 field gradient, 244 filtration through porous media, 270 flow imaging of water, 354 fluid flow, 355 foams, 242 forbidden IR bands, 300 FR apparatus, 389 FR method, 389 FR parameters, 384 FR spectra for diffusion with a surface resistance, 388 FR spectra of benzene in silicalite-1, 395 FR spectra of c-DMCH, 400 FR spectra of cyclic hydrocarbons in AlPO4–5, 410 FR spectra of cyclohexane, 400 FR spectra of cyclopentane in silicalite-1 (B), 399 FR spectra of ethylbenzene in silicalite-1 (B), 398 FR spectra of in silicalite-1 (B), 400 FR spectra of p-DCB, 400 FR spectra of p-xylene in silicalite-(A), 396 FR spectra of t-DMCH, 400 FR spectra of toluene in silicalite-1 (B), 397 fractal dimension of the porous space, 360 free volume theory, 65 free-volume self-diffusion theory polymer, 46 free-volume theory, 50 free-volume theory, polymer, 43 frequency encoding, 245 frequency-encoding, 249 frequency response (FR) technique, 383, 629 frequency response measurements, 159 frequency response, 55, 168 FTIR microscopy (IRM), 575

680 fuel cell unit, 450 GAI (Generalized Adsorption Isotherm), 513 gas, 165 gas adsorption in zeolites, 517 gas adsorption isotherms, 522 gas concentration sensors, 439, 448 gas phase transport, 198 gas separation by adsorption permeation technologies, 509 gas transport properties, 519 gaseous diffusion, 202 GCMC (Grand Canonical Monte Carlo), 510, 517 geometrical, 671 geometrical factor, 595 glassy polymers, 50, 524 glassy polymer membranes, 514, 527 glucose-sensitive hydrogels, 445 granular systems, 242 gravimetric measurement polymers, 51 Green-Kubo relation, 520 GULP (General Utility Lattice Program), 512 gyromagnetic ratio, 242 Habgood model, 17 Hall and Ross, 343 Heaviside's theorem, 318 heterogeneity of nanoporous samples, 355 hexane, 267, 273 hexane adsorption in “Vycor-220”, 272 human lungs, 242 hybrid membranes, 459 hydrocarbon trapping, 70 hydrogen, 335 hydrogen relaxation processes, 487 hydrogen/oxygen fuel cells, 452 hydrothermal synthesis, 283 hyperpolarization of nuclear spins of noble gases, 366 hyperpolarized (HP) gases, 242, 250, 366 hysteresis closure, 625 hysteresis closure points, 621 hysteresis loops, 137

Index HZSM-5, 322 HZSM-5 zeolite, 315, 324 IGA (Intelligent Gravimetric Analyzer), 126 IMFG, 263, 264 in silicalite-1, 135, 137, 141 incoherent cross section, 336 incoherent scattering, 338, 340 incoherent scattering cross section, 335 industrial applications of membranes, 439 inelastic neutron scattering (INS), 333 inelastic scattering, 342 influence of repulsions, 222 infrared, 280 inorganic crystalline materials, 512 inorganic membranes, 454 intelligent membranes, 443 intercrystallite diffusion, 316, 383 interference, 576 interference microscopy (IFM), 7, 155, 189, 545 intergrowth effects, 576 internal field inhomogeneities, 377 Internal Magnetic Field Gradients (IMFG), 262 interphase contactors, 457 intracrystalline diffusion process, 316, 384 intracrystalline diffusivities, 378, 383 intracrystalline transport, 575 intracrystallite concentration profiles, 320 intracrystallite diffusion of paraxylene, 322 ion exchange membranes, 481 ion-exchangers, 492 IR microscopy (IRM), 7, 576 iron phosphate, 289 iron phosphate synthesis, 283 isopentane, 567 isoreticular metal-organic frameworks (IRMOF), 76, 85 isosteric heats of adsorption, 135 isotherms, 139, 312 isotherms of nitrogen adsorption from mixtures with oxygen, 312

Index jump diffusion models, 93, 343 jump models for diffusion in zeolites, 535 Kelvin equation, 623 kinetic curve, 273 kinetic Monte Carlo (KMC) simulations, 120, 219, 221, 539 kinetics, 554, 561 kinetics of hexane adsorption, 273 KMC simulations, 222 Knudsen, 11, 202 Knudsen diffusion, 200, 379 Knudsen regime, 345 Larmor frequency, 243 lattice energy minimization GULP, 516 lattice flexibility, 541 LbL synthesis technique, 446 LiLSX zeolites, 300, 523 liquid, 165 loading dependence of benzene diffusion, 544 loading dependencies of self-diffusion, 546 long time stability of membranes, 464 long-range diffusivities, 379 long-time regime, 671 Lorentz-Berthelot rules, 527 magnetic resonance imaging, 241 mapping of electric current, 355 mass transport in porous materials, 353 mass transport of liquids, 361 mass transport processes, 359 mathematical modeling, 639 mathematical modeling of mass transfer processes, 640 Maxwell-Stefan, 102, 211, 214 Maxwell's field equations, 244 Maxwell-Stefan diffusivity, 4, 471 Maxwell-Stefan formulations, 100 Maxwell-Stefan-Krishna approach, 102 MCM-41, 619, 625 MD (molecular dynamics), 510 MD simulations, 224 MD vs KMC, 228 Me-Al2O3-FeCr2O4, 640

681 measurements, 154 measurements of nitrogen adsorption, 300 Me-FeCr2O4-Al2O3, 640 membrane, 439 membrane MF-HSK, 481 membrane module configurations ;465 membrane permeation, 162, 538 membrane reactor, 454, 456 membranes CMSM, 466 membranes in fuel cells, 449 mesitylene, 592 mesoporous materials, 279 metal phosphates, 279 metallic membranes, 458 metal-organic, 74 methane, 188 methanol, 189 methanol adsorption, 366, 578 methanol fuel cell, 451 methanol in CrAPO-5: uptake kinetics, 577 method of Theodorou-Suter, 525 MF-4SK membranes, 491 MFI membrane cross-section, 536 MFI zeolite, 218 MFI-type membrane, 94 MFI-type zeolite, 378, 595 microbalance, 567 microcapsules, 439 microcapsule and nanocapsules, 445 microporous, 151, 279 microporous solids, 629 micro-Raman spectroscopy, 281 microreactors, 460 microscopy (IFM), 576 microwaves, 283 microwave crystallization, 279 mixtures, 300 model of diffusion, 316 modeling, 85 modeling jump diffusion, 535 modeling membranes, 468 modeling single-file diffusion, 539 modeling zeolite flexibility, 541 molecular dynamics simulations, 86 molecular exchange, 255

682 molecular mobility, 262 molecular modeling, 512 molecular shape, 171 molecular sieve, 27 molecular simulations, 6 molecular square membranes, 82 monomolecular adsorption, 273 Monte Carlo method, 86 Monte Carlo simulations, 517 MRI, 243 MRI of gases, 249 MRI of hyperpolarized gases, 250 MRI sequences, 246 M-S diffusivity, 219 M-S theory, 217, 226 multicomponent diffusion, 14 multilayer perceptron neuronal network, 663 multilayer porous media, 648 multilayer structure of an oxide medium, 642 N2 and O2 adsorption in lithium and sodium low silica X-zeolite (LSX), 522 N2/Xe selectivities, 614 Na+, 483 NaA, 300, 303 NaA grains, 302 Nafion, 481 Nafion-type membranes, 483 NaLSX, 300, 304 NaLSX zeolites, 523 Na-mordenite (MOR), 72 nanocapsules, 439 nanoporous materials, 69, 375 nanoporous media, 639 nanoporous membranes, 458 nanoporous solids, 195 NaX zeolite, 24 NaZSM, 304 NaZSM-5, 72, 300 n-butane, 188 neuronal network, 657 neutron, 333, 341 neutron scattering (QENS), 333 neutron spin-echo (NSE), 341

Index n-heptane diffusion, 591 n-hexane, 327, 567, 619, 625 nitrogen, 27, 299 nitrogen adsorption, 302, 304, 312, 591 nitrogen adsorption by NaLSX zeolite, 306 nitrogen adsorption from air, 307 nitrogen diffusion in N2 + O2 mixtures, 306 nitrogen–oxygen mixtures, 312 nitrogen–oxygen mixtures by drift, 305 NMR cryoporometry, 356, 358 NMR diffusometry, 262 NMR imaging, 324, 353 NMR microimaging, 554 NMR of Li+, 483 NMR relaxation porosimetry, 358 NMR signal, 355 NMR spectroscopy, 353, 375 non cryogenic, 509 non uniform model of diffusion, 317 non-isothermal diffusion model, 385 non-zeolitic voids, 536 novel adsorbent design, 512 novel crystalline framework, 515 n-pentane, 567 nuclear magnetic relaxation dispersion (NMRD), 416, 428 nuclear paramagnetic relaxation, 417 nuclear spin relaxation times, 357 null point technique, 360 Onsager, 100 Onsager approach, 214 oxygen, 27, 299 PAA + PSF composite membrane, 495 packed bed membrane reacto, 454 parabolic magnetic field model, 264 PCFF force fields, 391 PDMS (polydimethylsiloxane)525 pellet diffusion control, 559 perfluorinated sulfocarboxy membranes;494 permeability, 499 permeability/selectivity of membranes, 464

Index PFG NMR, 6, 188, 192, 255, 257, 268, 270, 497, 547 PFG NMR 13-interval sequence, 377 PFG NMR diffusion, 375 PFG NMR method, 554 PFG NMR stimulated echo sequence, 376 phase, 576 phase encoding, 247 photography of chlorella cell, 503 pH-sensitive hydrogel, 444 physiosorbed, 196 point-spread function, 248, 249 poly(methacrylic acid) (PMAA), 657 polydimethyilsiloxane (PDMS), 657 polyimide membranes, 528 polymer, 366, 657 polymer film, 65 polymer membrane models, 519 polymer membranes, 454, 481, 513, 519 polymeric, 458 polymers, 41 polymer-solvent, 44 pore blockage, 71 pore size dependence, 623 pore size distribution, 261 pores of SAPO-5 occur, 580 porous, 671 porous glasses, 421 porous systems, 242 prediction of gas adsorption isotherms, 524 prediction of self-diffusion coefficients, 525 process optimization methodology, 511 propagator, 376 propane and toluene, 71 propane, 35 propene, 31 pulsed field gradient, 375 pulsed field gradient NMR, 353 pure nitrogen, 302 p-xylene, 137, 141, 166 p-xylene and toluene, 393 QENS, 192

683 QSAR (Quantitative Structure Activity Relationship), 511, 515 QSAR/QSPR methodologies, 528 quasi-elastic, 333 quasi-elastic scattering, 342 Raman analysis, 292 Raman microscopy analysis, 280, 288 Raman spectra of ironphosphate (FePO4), 294, 295 Raman spectroscopy, 280 rapidly diffusing molecules, 242 rare-event dynamics, 541 reaction pathway, 457 relationship, 102 relaxation decay, 266 relaxation time, 341 resolution, 241 resonance imaging, 30 resonance Raman scattering (RRS), 282 restricted diffusion, 255, 257 rigidity/brittleness, 465 rubbery polymers, 514, 524 Saccharomyces cerevisea, 500 saturated hydrocarbons, 393 SBA-15, 591, 592, 595, 598 scanning electrochemical microscopy (SECM), 81 scattering cross-section, 334 scattering techniques, 333 scattering theory, 334 SDA (Structure-Directing Agent), 512 second Fick’s law, 571 selective dense ceramic membrane, 457 selective removal of reaction products, 456 selective water sorbents (SWSs), 553 self diffusion coefficient, 335 self- or tracer diffusion in zeolite, 226 self-diffusion, 224, 248, 249, 375 self-diffusion and self-exchange, 225 self-diffusion coefficient, 249, 379, 671, 674 self-diffusion coefficient measurements, 485

684 self-diffusion coefficients of benzene, 347 self-diffusion coefficients of water, 490, 495 self-diffusion in porous systems, 257 self-diffusion of alkanes, 258 self-diffusion of organic molecules in membranes, 494 self-diffusivity, 5, 6 sensitivity, 241 separation processes, 442 separation technologies, 509 sequestration, 583 shrinking core model, 26, 322 silica, 553 silicalite membrane, 17 silicalite, 30, 166, 541 silicalite/HZSM, 166 silicalite-1, 125, 130, 137, 188 simulation, 141, 644 single diffusion process model, 384 single file diffusion, 17 single-component diffusion, 217 single-file diffusion, 70, 192, 535 Singwi and Sjölander, 343 ‘skin’ effect, 387 solid and solution separation, 283 solubility coefficient, 520 sorbed benzene in silicalite-1, 406 sorbed phase entropies, 137 sorption, 391, 554 sorption kinetics, 18 sound propagation, 629 spatial resolution, 245, 249, 354, 355 spectrometry FTIR polymers, 54 spherical pore DFT approach, 603 spillover, 205 spillover hydrogen, 209 spilt over species, 204 spin-lattice relaxation, 485 spin-polarised, 341 spin-spin relaxation, 485 SPRITE, 358 steady-state flux, 537 Stefan-Maxwell equation, 13 stimulated echo pulse sequence, 256, 498 stimulated spin echo method, 263, 271

Index structural defects, 175 structural defects in zeolite crystals, 187 structural factor, 269, 580 structure of porous media by PFG NMR, 259 subcritical benzene in NaX, 548 subcritical fluids, 549 subcritical systems, 548 sulfonated cation exchange membrane, 482 supercritical fluids, 549 surface, 198, 202 surface area, 428 surface diffusion, 11, 195, 203, 204, 415 surface resistance, 23 surface transport, 196 swelling, 362 synthesis, 283 synthesis of microstructured iron phosphate, 284 T1, T2 relaxation measurements, 485 T2 relaxation times, 377 t-DMCH, 141 techniques, 151 temperature, 202 temperature and loading dependencies of diffusion coefficients, 108 temporal analysis of products (TAP), 165 thermogravimetry, 145 toluene, 141 toluene and propane, 72 tortuosity, 12, 360 tortuosity factor, 380 tracer diffusivity, 5, 9 tracer ZLC (TZLC), 159 transition state theory, 118 transport diffusion coefficient, 335 transport diffusivities, 2, 3, 6, 388 transport processes during sorption, 356 transport properties of small molecules, 527 transverse relaxation times, 2, 261 tridecane, 267 TST (Transition State Theory Approach), 511 TST calculations, 520

685

Index two independent diffusion processes model, 385 two-phase model, 499 uptake, 154 vapor adsorption, 273 vibrational spectroscopy, 279 volumetric or barometric measurements polymers, 54 voxel, 353 Vycor porous glass, 257 water, 32, 657 water diffusion, 657 water flows, 268 water in CrAPO-5 and SAPO-5, 579 water mobility, 268 water molecules in chlorella, 500, 502 water sorption, 553, 554, 561 water transport, 554

Xe NMR, 193 xenon diffusivity, 366 xenon recycling, 613 X-ray diffraction, 288 XRD of FePOs, 289 zeolites, 70, 93, 187, 211, 299, 300, 375, 575 zeolite membranes, 459, 460, 536 zeolite normal modes, 542 zeolite vibrations, 542 zero length chromatography techniques, 394 zero length column (ZLC), 157, 168, 591, 592 ZSM-5, 188