Studies in Surface Science and Catalysis 104 EQUILIBRIA AND DYNAMICS OF GAS ADSORPTION ON HETEROGENEOUS SOLID SURFACES
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Studies in Surface Science and Catalysis A d v i s o r y Editors: B. D e l m o n a n d J.T. Yates Vol. 104
EQUILIBRIA AND DYNAMICS OF GAS ADSORPTION ON HETEROGENEOUS SOLID SURFACES Editors
W. Rudzir~ski Department of Theoretical Chemistry, Maria Curie-Sklodowska University, Lub/in, Poland W. A. Steele Department of Chemistry, Pennsylvania State University, University Park, PA 16802, USA G. Zgrablich Department of Physics, University of San Luis, San Luis, Argentina
1997 ELSEVIER A m s t e r d a m m L a u s a n n e - - N e w York m O x f o r d m S h a n n o n m T o k y o
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ISBN 0-444-82243-7 91997 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A.- This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands
PREFACEThere was a period in the development of adsorption science when the theoretical descriptions of gas adsorption on solid surfaces were based on highly idealized models of solid surfaces. Most commonly, models of localized adsorption were used in which the gas molecule approaching a solid surface "saw" a regular array of the local minima in the gas-solid potential function - the adsorption sites. In the first introductory chapter of this book by Cerofolini and Rudzinski, that period is called the "pioneering age" of adsorption science. It began in 1917 when Langmuir published the theory of his famous isotherm equation. It was based on the model of one-site/one molecule - occupancy adsorption with no interactions between the adsorbed molecules. Until the Second World War, the main stream of the theoretical work was oriented toward generalizing that model to account for possible interactions between the adsorbed molecules. It started with the application of the mean-field approach to develop the Bragg-Williams isotherm. Next various forms of the Quasi-Chemical Approximation were introduced. That trend reached its peak by the publication of the exact solution for the isotherms of molecules adsorbed on two-dimensional regular arrays of adsorption sites in 1944 by Onsager. We could reasonably set this year to be the end of this "pioneering age" of adsorption science. Although most of the theoretical work was based on models of localized adsorption, mobile adsorption also received interest during that period. Here the most spectacular achievement was the application of the two-dimensional analogue of the Van der Waals model to develop isotherm equation. However, the most famous achievement of that time was probably the generalization of the Langmuir model by Brunauer, Emmet and Teller (BET) to account for the possibility of a secondary adsorption on the already adsorbed molecules. The BET equation became a standard in the determination of the surface area of adsorbents but was also criticized for various thermodynamic inconsistencies and shortcomings. An alternative well-known approach to multilayer adsorption was then proposed by Frankel, Halsey, and Hill, usually called the FHH slab theory. However, in spite of the advanced theoretical treatments seen in many papers published before the Second World War, adsorption experiments showed more and more discrepancies between theory and experiment. Attempts to fit experimental isotherms by various theoretical equations always showed negative deviations from experiment at low adsorbate pressures~ and positive deviations at higher adsorbate pressures. Deviations between theory and experiment were probably most dramatically demonstrated in measurements of the enthalpy changes upon adsorption. The theoretically predicted isosteric heats of adsorption should be increasing functions of the surface coverage. The reason is as follows: Although the solid phase must perturb to some extent the interactions between adsorbed molecules, the interactions should basically preserve their Lennard - Jones character. This means that there exists only a narrow range of distances at which adsorbed molecules interact via moderate repulsive forces. At still smaller separations they will not be adsorbed at all, and at all other distances they should interact via attractive potentials. The probability of finding a physical situation where adsorption sites are located at distances such
vi that admolecules interact via moderate repulsive forces is therefore small compared to the probability of finding the situation where adsorbed molecules interact via attractive forces. It means that, in general, the experimentally observed isosteric heat of adsorption should increase with increasing adsorbed amount due to the increase in attractive lateral interaction energy at high density. Meanwhile, almost all the reported experimental heats of adsorption showed an opposite trend. As the theoretical adsorption isotherms could not be successfully used to fit experimental data, various empirical equations were used for that purpose. At small adsorbate pressures, the Freundlich empirical isotherm was commonly applied as superior to Langmuir equation. (It is interesting to note that this isotherm equation was published in 1909 and was the first ever used.) In 1927 Bradley proposed a hybrid which preserved the superiority of Freundlich isotherm at low adsorbate pressures, but predicted also that the surface coverage will tend to unity at high pressures. Bradley's name was later forgotten and the hybrid isotherm is now generally called the Langmuir-Freundlich isotherm. Other empirical isotherm equations include Temkin's and Tdth's isotherms. The more and more frequent discrepancies between theory and experiment showed that there must exist another yet important physical factor that had not been taken into account in the theories. It was Roginsky in the former Soviet Union who, at the end of the thirties, launched the idea that the missing factor is the energetic heterogeneity of the actual solid surfaces which is crucial in governing the behaviour of actual adsorption systems. Roginsky's works marked the end of the "pioneering age" of adsorption science in which theoretical works were based on highly idealized models of solid surfaces. However, it should be emphasized that it was Langmuir himself who first introduced the idea of adsorption on heterogeneous solid surfaces. In his second theoretical paper, he emphasized that in the case of adsorption on real solid surfaces, one will have to deal with a number of distinct kinds of adsorption sites. His equation, with suitably chosen constants, should be used to describe adsorption on a certain kind of sites, and the experimentally observed isotherm will be a sum of these Langmuir terms, each multiplied by the fraction of a given kind of site on the solid surface. Roginski assumed that in the case of the real adsorption systems, the spectrum of adsorption energies characterizing various adsorption sites will be very dense so that the differential distribution of the number of sites over the values of the adsorption energy e can be represented by a continuous function X(e). That function is nowadays called commonly the adsorption energy distribution. Roginski also introduced another idea that appeared to be extremely fruitful in further theoretical studies. He postulated that, as the adsorbate pressure increases in the case of real heterogeneous solid surfaces, adsorption proceeds gradually on adsorption sites in the sequence of decreasing adsorption energies. At a given temperature T and pressure p, the adsorption sites having energy larger than a certain critical adsorption energy e~(p, T) are completely filled, whereas the others are totally empty. After the Second World War, the idea of adsorption on energetically heterogeneous solid surfaces received strong interest by Americans like Hill, Adamson, Halsey, Sips and Zettlemoyer who, in the forties and the fifties, made an enormous contribution to this area of research. The feeling that a certain era of adsorption science was closed and a new period was starting encouraged some authors to write monographs reporting on the current state of
vii the art. Thus, Young and Crowell published Physical Adsorption of Gases in 1962 and two years later Ross and Olivier published their book: On Physical Adsorption. In the latter case, adsorption theories based on idealized models of solid surfaces were reviewed, but some elementary principles of adsorption on energetically heterogeneous surfaces were also introduced. Ten years later Steele published his book The Interaction of Gases with Solid Surfaces, oriented already toward new trends in the theoretical studies of adsorption. It was the time when the nature of the gas/solid interactions was studied in more detail, as was the combined effects of admolecule interactions and of surface energetic heterogeneity. The so-called virial formalism was extensively used to study adsorption at low surface coverages. Adsorption at higher surface coverages was studied commonly by applying the so-called integral equalion for adsorption isotherm. The experimentally observed isotherm was assumed to be an average (integral) of the local adsorption isotherm, describing adsorption on adsorption sites characterized by an adsorption energy e, with the adsorption energy distribution X(c). As mentioned already, the idea of an integral isotherm equation was introduced by Roginski at the end of the thirties and was developed further by American scientists in the fifties, based almost exclusively on use of the Langmuir equation as the local adsorption isotherm. An explosive development of that research took place in the seventies and the eighties, and was marked by local isotherm equations that took account of the interactions between the adsorbed molecules. Simultaneous consideration of the effects arising from the interactions between adsorbed molecules and the effects of the energetic heterogeneity of adsorption sites involves another important factor yet to be taken into account. This is the way in which the various adsorption sites are distributed on an energetically heterogeneous surface. In 1949 Hill published the first fundamental paper concerning that problem. In his paper he studied the model of the solid surface in which various adsorption sites are randomly distributed on the solid surface, now commonly called - the random model. Some ten years later Ross and Olivier introduced another extreme model to represent the topological (topographical) distribution of different adsorption sites on partially graphitized carbons by assuming that identical adsorption sites are grouped into large patches. That model is now commonly called the patchwise model. Both these models were studied extensively by the group working in the Department of Theoretical Chemistry of Maria Curie-Sklodowska University in Lublin. Very advanced theoretical works were published by Tovbin in Russian journals. The explosive development in the seventies and in the eighties of the research based on the integral equation was accompanied by another vigorous trend of solving the reverse problem. That necessitated solving the integral equation to calculate the adsorption energy distribution from an experimental adsorption isotherm. This trend was initiated by the paper published by Sips in 1948, who used the inverse Stieltjes transform for that purpose, but it was not until the beginning of the seventies that this research started to develop vigorously. A variety of methods were proposed to solve the integral equation. The most stable and convenient of these seem to be the methods which were advanced versions of the Condensation Approximation approach introduced by Roginski. Here, the method proposed by Adamson and co-workers in 1966 was the first attempt of that kind. Later developments were based on the work of Cerofolini, Rudzinski and Jagiello.
VIII
Much attention was also devoted to the effects of surface energetic heterogeneity upon the adsorption of gas mixtures on real solid surfaces. In 1967 Hoory and Prausnitz published the first paper on the applicability of the integral equation approach to describe mixed-gas adsorption on solids. However, essential progress toward further development of this idea was due to the papers published by Jaroniec. At the beginning of the eighties, Myers and his co-workers showed how the other fundamental approach to mixed-gas adsorption- the Ideal Adsorbed Solution Approach- can be generalized further to take account of energetic surface heterogeneity. The new period of adsorption science which started after the Second World War reached its peak in the eighties. It can be characterized by: 9 consideration of the energetic heterogeneity of real solid surfaces, 9 looking for more and more advanced analytical solutions for the combined effect of surface energetic heterogeneity and of the interactions between adsorbed molecules. The beginning of the nineties seems to mark the beginning of a third era of development of adsorption science. In our opinion, the following two factors opened that new era: 9 the development of STM and AFM microscopies, which allow one to "see" the molecular structure of real solid surfaces, 9 the explosive development of computer simulations of adsorption processes, which take account of more detailed mechanistic models of solid surfaces. Again, the feeling that a certain era of adsorption science had closed encouraged some authors to review the state of art. Thus, Jaroniec and Madey published in 1989 their monograph: Physical Adsorption on Heterogeneous Solids, and 3 years later Rudzinski and Everett published their monograph: Adsorption of Gases on Heterogeneous Surfaces. There is a tendency to describe the beginning of every new era as "modern" so we will follow that tradition by calling the third era starting at the beginning of the nineties "the modern era of adsorption science". The judgment of every era at its beginning must always be imbalanced in many respects. Nevertheless, we believe that there has been a need for a book presenting the current trends in this new era of adsorption science. This was the idea behind our efforts to publish this monograph. So far we discussed adsorption on essentially fiat solid surfaces with no limits in either the direction parallel or perpendicular to the solid surface. Meanwhile, many of the adsorption systems of great technological importance are the systems with the so-called restricted geometry, where the dimensions of the adsorption system are limited. First of all, these are porous carbons and zeolites of various kinds. In the case of porous sorbents, one usually distinguishes between two adsorption mechanisms. One is the adsorption in micropores having dimensions less than 2 nm. This adsorption mechanism is frequently called - pore filling. The other adsorption mechanism is one occurring in mesopores having dimensions between 2 and 50 nm. Adsorption in macropores having dimensions bigger than 50 nm is essentially the same as on a fiat solid surface. Historically, the first theoretical works on adsorption in porous materials concerned adsorption in mesopores. A very special feature of this adsorption mechanism is the appearance of capillary phenomena, demonstrated by the hysteresis loops on the experimental adsorption isotherms.
They make it possible to calculate mesopore size distributions using Kelvin's equation and its further improvements and modifications. As to adsorption in micropores, the achievements of Dubinin's school of adsorption have for many years remained the most essential contribution in this area. The theory of micropore filling, developed by Dubinin and Radushkevich in the late 1940's, is a major contribution to adsorption science, alongside the Langmuir and BET isotherms. This approach was based on Polanyi's idea of adsorption potential, which was assumed to be different in different micropores. An implicit assumption was made that the adsorption proceeds in a stepwise fashion in micropores in the sequence of decreasing adsorption potential. This was the basis for the first attempts by Dubinin and co-workers to calculate the distribution of micropore sizes. The idea that the Dubinin-Raduskhevich (DR) equation is related to the geometric heterogeneity of porous solids has to be tested critically soon. First, it is a surprise that adsorption in hundreds of different carbon samples studied by various investigators obeyed the DR equation, i.e. had the same type of pore size distribution. Secondly, at the beginning of the sixties, Hobson in Canada published his surprising discovery that the DR equation developed for adsorption in porous carbons can be successfully used as a general isotherm equation to describe low-coverage adsorption on real flat solid surfaces. Hobson was first to consider the DR isotherm as a result of averaging the Langmuir isotherm with a certain adsorption energy distribution. He was also first to use the CA (Condensation Approzimation) approach to show that the energy distributions appeared to be gaussian-like, with a widened function of e on the high energy side. Later experimental studies showed that at higher surface coverages, adsorption can be correlated by Freundlich's equation, and at still higher surface coverages, by the Langmuir-Freundlich isotherm. This striking behaviour was studied theoretically by Cerofolini, who launched a well-documented hypothesis that this behaviour should be a universal feature of all adsorption systems, and is related to certain rules governing the formation of the real solid surfaces. As a result, the fact that the DR equation can describe low-coverage adsorption on flat solid surfaces must put into question the hypothesis that it is related to factors affecting the adsorption in micropores. Also, deviations between the DR equation and the experimentally observed behaviour of microporous adsorption systems were reported more and more frequently. The modification of the DR equation proposed by Astakhov and Dubinin did not solve the problem. Thus, at the beginning of the eighties Stoeckli and Dubinin launched the hypothesis that the DR equation should be used "locally" to describe adsorption in a certain class of micropores having the same dimensions. Their hypothesis received strong support from SAX experimental studies showing some relation between the size of micropores and the parameter in the DR equation. This experimental finding was the basis for developing a new method of determining the distribution of micropore sizes (Dubinin-Stoeckli, MacEnaney, Jaroniec). It seems, however, that an important conclusion that follows from these works has not received enough attention. Namely, if the DR equation describes adsorption in pores of the same geometric dimensions, it cannot be related to geometric surface heterogeneity. Two factors may be responsible for the successful application of the DR equation to systems (subclasses) of homogeneous micropores:
9 the energetic heterogeneity of micropore walls, 9 the mutual effects arising from energetic heterogeneity and those due to the restricted geometry of the system. Recent computer simulations seem to favour the first explanation. Simulated adsorption isotherms in micropores show behaviour far different from that predicted by the DR equation. These simulations were based on the assumption of a regular micropore shape with chemically homogeneous walls. Such idealized models of micropores were used to calculate the pore size distribution by applying certain analytical solutions (Everell-Powell, Horvath-Kawazoe), as well as by comparison with computer simulations. Recently, however, a certain feeling is spreading that more realistic models should be applied to represent the geometric - energetic features of real microporous solids. The "pioneering age" can be characterized by the attempts to apply Langmuir, BET, FHH and virial equations, along with certain specific approaches like t-plot, c~,-plot and others. They were accompanied by extensive studies of capillary phenomena in mesopores. That period was covered by the review by Gregg and Sing published in 1976 - The Adsorption of Gases on Porous Solids and by Dubinin in 1972 - The Adsorption and Porosity (in Russian). The book Adsorption, Surface Area and Porosity covering the period ending at the beginning of the nineties, was published by Gregg and Sing. In considering the time dependence of adsorption on heterogeneous solid surfaces, it can be seen that studies of the equilibrium and the time dependence of adsorption have proceeded on two somewhat separate paths. The lack of Langmuirian kinetics in real adsorption systems was primarily discussed during the "pioneering age" of adsorption science. As in adsorption equilibrium, various empirical equations were used to describe the kinetics of gas adsorption on the real solid surfaces. The most famous and commonly used is the Elovich equation proposed at the end of the thirties. Its appearance and common use marked more and more strongly the end of the "pioneering period" based on the use of idealized surface models to describe adsorption equilibria and kinetics. However, as in the equilibrium problem, these idealized models were the starting point for further generalizations that take account of surface energetic heterogeneity. Here, the most popular approach had been based on the application of Absolute Rate Theory, developed originally to describe the kinetics of chemical reactions in bulk systems. Soon after the first attempts were made to apply it to adsorption/desorption kinetics at the beginning of the seventies, serious discrepancies were reported between the theoretical predictions and experimental observations. This was especially seen in the coverage dependence of the kinetics. These reports led to the introduction of the concept of precursor states and of the sticking probability concept. One stream of research was to further improve the theoretical description of adsorption/desorption kinetics based on a model of the regular (homogeneous) solid surface. In 1986 Kreuzer and Gortel discussed this work in the exhaustive review Physisorption Kinetics. Another stream of work led to improving the Absolute Rate Theory approach by incorporating the idea of surface energetic heterogeneity. It was particularly popular among the scientists using TPD (Temperature Programmed Desorption) to study the thermal desorption of gases from solid surfaces. The principles of that method were published in 1963 by Amenomiya and Cvetanovic. Two years later papers began to appear suggesting that the activation energy for desorption should be considered to be a function of the surface coverage because of the existence
of various adsorption sites on a solid surface which should be characterized by different activation energies for desorption. The implicit assumption underlying that interpretation was that desorption proceeds in an ideally stepwise fashion in the sequence of increasing activation energies for desorption. As mentioned already, the inapplicability of the adsorption/desorption theories could be seen most clearly by comparing the predicted and observed coverage dependence of the kinetics. Very special, but also technologically extremely important, is the kinetics of adsorption in, and desorption from porous adsorbents. This is because the kinetics of molecula~r motion in restricted spaces may be predominantly governed by surface diffusion across pore walls. Diffusion of adsorbed particles is one of the most fascinating phenomena on solid surfaces and one of great importance in catalysis, metallurgy, microelectronics, material science, and many other scientific and technological applications. Beginning with Volmer and Eastermann's pioneering experiment in the early 20's, studies of surface diffusion have constantly expanded. From the early stages, the problem of surface heterogeneity was crucial. This can easily be understood in a figurative way since "the best way of feeling the landscape is to take a walk through the hills". For a long time, surface diffusion studies were mainly associated with the necessity of estimating the surface flux contribution to the total flux of a gas through a catalyst support (often, a random porous medium). This is a very complex problem and very limited tools were available at that time, such as the effective Arrhenius-type equations and the experimental technique of the diffusion cell. The development of field emission microscopy in the 40's and of field ion microscopy in the 60's opened a new horizon by enabling the observation of the migration of individual adatoms on more precisely characterized surfaces. With the help of lattice-gas theories, important advances were made in understanding surface self-diffusion for both physisorbed and chemisorbed species. The availability of many new surface spectroscopies and the utilization of computers starting in the late 70's allowed the possibility of studying new and exciting phenomena such as the behavior of surface diffusivity in the presence of two-dimensional phase transitions, the growth of thin films and clusters, and the effects of controlled defects (like steps) and other geometrical heterogeneities upon atomic motion on the surface. The invention of Scanning Tunneling Microscopy in the 80's and the continued growth of computer power produced a new revolution: detailed numerical simulations (either Monte Carlo or Molecular Dynamics) of diffusion of adsorbed species on heterogeneous surfaces and experiments performed on well-characterized surfaces became possible, fields which are now expanding rapidly. The old problem of diffusion on the surface of disordered porous media is still open and receiving new attention, while interesting problems such as the diffusion of chain molecules (k-mers) and polymers are continuously arising to make this field more exciting. Many adsorbents that are not porous in bulk are also not "flat". Their properties can frequently be described by using the concept of adsorption on fractal surfaces. The concept of the fractal nature of solid surfaces in relation to adsorption phenomena (Pfeifer, Avnir) is one of the new approaches marking the beginning of the modern era in the nineties. From this overview of the research on adsorption on heterogeneous solid surfaces, one can see that surface heterogeneity affects adsorption at the gas/solid interface in a variety of ways. For that reason, the papers treating various aspects of surface energetic hetero-
xii geneity were published in a variety of scientific journals addressed to various groups of scientists. No book has been published yet that would give an overview of all the effects of surface energetic heterogeneity of real solid surfaces. The present monograph is a first attempt of such a kind. It concerns both adsorption equilibria and the time dependence of adsorption phenomena. We have used the term dynamicsfor the aspects related to the time evolution of adsorption systems. This book was not aimed to provide the readers with the historical development of adsorption science. For that reason we decided to give a historical overview in this Preface. We have distinguished here basically three periods in the development of surface science. Our book was aimed to present the state of this art at the beginning of the third period which we call "modern", and which started at the beginning of the nineties. As mentioned already, this modern era is marked by the development of STM and AFM microscopies which provide the information about the mechanistic models to be accepted in computer simulations. The explosive development of the computer simulations seems to take the lead over the development of analytical approaches. The latter, of course, will follow that development as a necessary scientific synthesis of the information obtained from the true and simulated experiments. We are very aware that the panorama of the research presented here is incomplete. There are surely many excellent names to be mentioned and many adsorption problems to be discussed. Nevertheless, we believe, that this monograph is a substantial step toward presenting the present state of the art. We would like to express our warmest thanks to all the colleagues who contributed to this book.
Wladyslaw Rudzinski William A. Steele Giorgio Zgrablich
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AUTHOR INDEX
1. F. Bardot ~ Laboratoire Environnement et Mineralurgie, ENSG et URA 235 du CNRS BP 40, 54 501 Vandoeuvre les Nancy cedex, FRANCE 2. I. B e r e n d - Laboratoire Environnement et Mineralurgie, ENSG et URA 235 du CNRS BP 40, 54 501 Vandoeuvre les Nancy cedex, FRANCE 3. J . M . C a s e s - Laboratoire Environnement et Mineralurgie, ENSG et URA 235 du CNRS BP 40, 54 501 Vandoeuvre les Nancy cedex, FRANCE 4. G.F. Cerofolini ~ EniChem'- Instituto Guido Donegani, 28100 Novara, ITALY 5. YuDong C h e n - - The BOC Group, Inc., 100 Mountain Avenue, Murray Hill,
NJ 07974, U.S.A. 6. A.S.T. C h i a n g - Department of Chemical Engineering, National Central University, Chung-Li, Taiwan ROC 32054 7. J. C h o m a - Institute of Chemistry, Military Technical Academy, 01489 Warsaw, POLAND 8. D.D. D o - Department of Chemical Engineering, University of Queensland, Qld 4072, AUSTRALIA 9. J.A.W. Elliot o Thermodynamics and Kinetics Laboratory, Department of Mechanical Engineering, University of Toronto, CANADA M5S 1A4 10. M . F r a n c o i s - Laboratoire Environnement et Mineralurgie, ENSG et URA 235 du CNRS, BP 40, 54 501 Vandoeuvre les Nancy cedex, FRANCE 11. S.P. Friedman w Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, U.K. 12. G . G e r a r d - Laboratoire Environnement et Mineralurgie, ENSG et URA 235 du CNRS, BP 40, 54 501 Vandoeuvre les Nancy cedex, FRANCE
xiv 13. K.E. Gubbins ~ School of Chemical Engineering, Cornell University, Ithaca NY 14853, U.S.A. 14. M.Jaroniec ~ Separation and Surface Science Center, Department of Chemistry, Kent State University, Kent, Ohio 44242, U.S.A. 15. K . K a n e k o - Department of Chemistry, Faculty of Science, Chiba University Yayoi 1-33, Inage, Chiba 263, JAPAN 16. H.J. Kreuzer ~ Department of Physics, Dalhousie University, Halifax, N.S. B3H 3J5, CANADA 17. C . M . L a s t o s k i e - Department of Chemical Engineering, University of Michigan, Ann Arbor MI 48109, U.S.A. 18. C.K.Lee ~ Department of Environmental Engineering, Van-Nung Institute of Technology, Taiwan ROC 32054 19. Kuang-Yu Liu ~ Department of Physics and Astronomy, University of Missouri Columbia, MO 65211, U.S.A. 20. J.M.D. MacElroy m Department of Chemical Engineering, University College Dublin, Beldfield, Dublin 4, IRELAND 21. L.J. Michot - - Laboratoire Environnement et Mineralurgie, ENSG et URA 235 du CNRS, BP 40, 54 501 Vandoeuvre les Nancy cedex, FRANCE 22. J.Narkiewicz-Michalek o Department of Theoretical Chemistry, Maria Curie-Sklodowska University, 20-031 Lublin, POLAND 23. S.H. Payne m Department of Physics, Dalhousie University, Halifax, N.S. B3H 3J5, CANADA 24. P. Pfeifer - - Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, U.S.A.
XV
25. N.Quirke w Department of Chemistry, University of Wales at Bangor, Gwynedd LL57 2UW, U.K. 26. W. R u d z i n s k i - Department of Theoretical Chemistry, Maria Curie-Sklodowska University, 20-031 Lublin, POLAND 27. N.A.Seaton- Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, U.K. 28. W. A. S t e e l e - Department of Chemistry, The Pennsylvania State University, 152 Davey Laboratory, University Park, PA 16802, U.S.A. 29. P. Szabelski ~ Department of Theoretical Chemistry, Maria Curie-Sklodowska University, 20-031 Lublin, POLAND 30. Yu.K. Tovbin - - Karpov Institute of Physical Chemistry, Vorontsovo Pole Str. 10, 103064 Moscow, RUSSIA 31. F.Villieras- Laboratoire Environnement et Mineralurgie, ENSG et URA 235 du CNRS, BP 40, 54 501 Vandoeuvre les Nancy cedex, FRANCE 32. C.A. Ward m Thermodynamics and Kinetics Laboratory, Department of Mechanical Engineering, University of Toronto, CANADA M5S 1A4 33. T. Yang - - Department of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109, U.S.A. 34. J . Y v o n - Laboratoire Environnement et Mineralurgie, ENSG et URA 235 du CNRS, BP 40, 54 501 Vandoeuvre les Nancy cedex, FRANCE 35. G. Zgrablich- Departamento de Fisica, Universidad Nacional de San Luis, Chacabuco y Pedernera, 5700 San Luis, ARGENTINA
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XVII
CONTENTS
Preface Author Index Chapter 1. Theoretical Principles of Single- and Mixed-Gas Adsorption Equilibria on Heterogeneous Solid Surfaces G.F. Cerofolini and W. Rudzinski Chapter 2. Application of Lattice-Gas Models to Describe Mixed-Gas Adsorption Equilibria on Heterogeneous Solid Surfaces Yu.K. Tovbin
105
Chapter 3. Theories of the Adsorption-Desorption Kinetics on Homogeneous Surfaces H.J. Kreuzer and S.H. Payne
153
Chapter 4. Theory of Adsorption-Desorption Kinetics on Flat Heterogeneous Surfaces Yu.K. Tovbin
201
Chapter 5. Statistical Rate Theory and the Material Properties Controlling Adsorption Kinetics on Well Defined Surfaces J.A.W. Elliot and C.A. Ward
285
Chapter 6. A New Theoretical Approach to Adsorption-Desorption Kinetics on Energetically Heterogeneous Flat Solid Surfaces Based on Statistical Rate Theory of Interracial Transport W. Rudzinski
335
Chapter 7. Surface Diffusion of Adsorbates on Heterogeneous Substrates G. Zgrablich
373
~
XVIII
Chapter 8. Computer Simulation of Surface Diffusion in Adsorbed Phases W. A. Steele
451
Chapter 9. Multicomponent Diffusion in Zeolites and Multicomponent Surface Diffusion YuDong Chen and Ralph T. Yang
487
Chapter 10. Energy and Structure Heterogeneities for the Adsorption in Zeolites A.S.T. Chiang, C.K.Lee, W.Rudzinski, J.Narkiewicz-Michalek and P. Szabelski
519
Chapter 11. Static and Dynamic Studies of the Energetic Surface Heterogeneity of Clay Minerals F.Villieras, L.J. Michot, J.M.Cases, I. Berend, F. Bardot, M.Francois, G.Gerard and J.Yvon
573
Chapter 12. Multilayer Adsorption as a Tool to Investigate the Fractal Nature of Porous Adsorbents P. Pfeifer and Kuang-Yu Liu
625
Chapter 13: Heterogeneous Surface Structures of Adsorbents K.Kaneko
679
Chapter 14. Characterization of Geometrical and Energetic Heterogeneities of Active Carbons by Using Sorption Measurements M.Jaroniec and J. Choma
715
Chapter 15. Structure of Porous Adsorbents: Analysis Using Denstity Functional Theory and Molecular Simulation C.M.Lastoskie, N.Quirke and K.E. Gubbins
745
Chapter 16. Dynamics of Adsorpion in Heterogeneous Solids D.D. Do
777
Chapter 17. Sorption Rate Processes in Carbon Molecular Sieves J.M.D. MacElroy, N.A.Seaton and S.P. Friedman
837
W. Rudzitiski, W.A. Steele and G. Zgrablich (Eds.) Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces Studies in Surface Science and Catalysis, Vol. 104 9 1997 Elsevier Science B.V. All rights reserved.
Theoretical Principles of Single- and Mixed-Gas Adsorption Equilibria on Heterogeneous Solid Surfaces G. F. Cerofolini ~ and W. Rudzifiski b EniChem- Istituto Guido Donegani, 28100 Novara, Italy b Department of Theoretical Chemistry, Maria Curie-Sklodowska University, 20031 Lublin, Poland
1. I N T R O D U C T I O N
1.1. History and logic of adsorption Logic and history often do not run parallel to each other. Therefore, when a treatise describes the status of a discipline, it has to avoid loops and misconceptions, and often to anticipate later results. Dirac's Principles of Quantum Mechanics does not furnish us with any information on the history of this discipline; conversely, Jammer's The Conceptual Development of Quantum Mechanics can hardly be considered a treatise on quantum mechanics. The theory of adsorption, possibly because of its non-foundational character, does not suffer from this difficulty, so that its logical development is almost the same as its historical development. This will allow us to sketch the theory of adsorption simply by following its evolution. Though periodization is always a risky activity, we have divided the (history of) adsorption into three phases, referred to as Pioneering Age, Middle Age and Modern Age. The Pioneering Age is characterized by experimental or theoretical discoveries of new adsorption isotherms; this period is marked by the 'equation of ...'. The Middle Age is characterized by the attempts of explaining the most frequently observed experimental isotherms in terms of surface properties like the adsorption-energy distribution; this period is marked by the 'method of ...'. The Modern Age is characterized by the attempts toward understanding of specific behaviours of complex adsorbents; it is characterized by the absence of general methods or equations and by the extended use of large computational methods. The Pioneering and Middle Ages are adequately covered by the literature, like Steele's compact treatise on adsorption [1], Jaroniec and Madey's and Rudzifiski and Everett's books on adsorption on heterogeneous surfaces [2, 3], and Gregg and Sing's treatise on adsorption on porous surfaces [4]. There is no book covering adequately the Modern Age and it would not be an easy job to write such a book at present. This is because of the explosive research in this area, and especially of computer simulations. It is to be expected that, after collecting a sufficiently large body of computer simulation data, they will lead to a theoretical synthesis having a form of compact analytical expressions.
Computer simulations will surely take the lead in explaining fundamental features of various adsorption systems. The application of the Density Functional Theory to adsorption in micropores shows, on the other hand, that analytical approaches will always be competitive as far as the computational time is considered. Some analytical expressions developed at the beginning of the century, (the Langmuir equation for instance), are still used at the end of this century in a more or less modified form. Adsorption technologies create a large market (demand) for compact analytical expressions that could be calculated fast to control technological processes. The fascination by the new possibilities created by computer simulations is also accompanied by a growing nostalgy for having simple expressions that could be applied by anyone in research and practice. As this chapter is supposed to provide one with "theoretical principles", we will try to review some analytical expressions and approaches that have most frequently been used, and perspectives of their further development and generalizations. Then, although the energetic surface heterogeneity, i.e. the dispersion of gas-solid interactions, and the admolecule-admolecule interactions affect the behaviour of adsorption systems in a cooperative way, we will focus here our atention on the role of the energetic surface heterogeneity. 1.2. Establishing the t h e o r y of adsorption ~ Gibbs e q u a t i o n Gas adsorption is a way through which the unsaturated forces at the surface of a given system in a condensed phase (adsorbent) are partially saturated by the interaction with gas-phase molecules (adsorbate). Being a way to approach equilibrium, adsorption is a spontaneous process, usually exothermic in nature. When the adsorption energy is higher than approximately 0.5 eV per an adsorbed molecule, a true chemical bond is formed between the adsorbate and the adsorbent; this case is referred to as chemisorption. Though chemisorption is a relatively highly exothermic process, it may be hindered by the activation energy required to destroy the molecular structure of the gas-phase molecule or the bonds at the reconstructed surface. When the adsorption energy is lower than about 0.5 eV per a molecule, adsorption involves secondary (electrostatic or Van der Waals) forces and this case is referred to as physisorption. The forces responsible for physisorption are essentially the same as those responsible for the condensation of a vapour to the liquid state. When a new chemical bond is not involved in the adsorption process, activation energy is not required for the formation of the adsorbed phase, so that equilibrium is attained in a short time, thus allowing a direct experimental characterization of this state. The founding father of the theory of adsorption equilibrium was Gibbs, who at the end of the 18th century established his celebrated equation, r
b2
Odln
p
(1)
where k is the Boltzmann constant, T is the absolute temperature, b2 is the area occupied by one molecule in the surface, p is the equilibrium pressure, pt is a reference pressure, 0 is the fraction of surface covered by the adsorbate, and r is the chemical potential of the system relative to that in the absence of adsorbate
Equation (1) stands on purely thermodynamic considerations and is therefore unable to specify a functional relationship r = r (actually, it holds true whatever is this relationship) so that it is unable to specify the adsorption isotherm 8 = 8(p). An isotherm can be specified only by assuming a microscopic model of the adsorbate, which eventually allows the function r to be determined.
2. T H E P I O N E E R I N G
A G E OF A D S O R P T I O N
2.1. A d s o r p t i o n of s t r u c t u r e l e s s gases on ideal surfaces
This Age of adsorption was characterized by the search of physically plausible and mathematically simple microscopic models for the adsorbate, and by the discovery of experimental isotherms which could hardly be understood in the frame of the theoretical models. The first models considered the very ideal case of the adsorption of structureless gases on ideal surfaces. With 'structureless' molecule we intend here a molecule characterized exclusively by its covolume b, and with 'ideal' surface we intend a surface with an energetically homogeneous distribution of adsorption sites. Moreover, the volume above each site is assumed to allow the accomodation of one and only one molecule, whatever is the value of b. In a first approximation, the potential energy of a gas-phase molecule near a surface will be the sum, extended to all the atoms of the solid, of the pair potentials acting between the molecule and each atom of the solid [5, 6, 7, 8]. Describing the interaction potential by the 6-12 Lennard-Jones pair potential [9], u(r) = uo [(r0/r) 12- 2 (r0/r) s] (where r is the nuclear separation of the pair, r0 the equilibrium distance and u0 the depth of the minimum), the adsorption potential acting on a gas-phase molecule in the point r, r = (Xl,X2, x3), in the vicinity of a solid is therefore given by U(r) = ~-~zu ( I r - RI[), where Rx is the position of the I-th atom in the solid, and the sum is extended to all these atoms. This potential depends both on the plane vector xll = ( x l , x 2 ) lying in the surface plane and on the distance x3 from the surface. A summary of key data concerning adsorption potentials over 250 gas-surface systems is given in ref. [10]. Some general features of the adsorption potential, almost independent of the specific crystal lattice, can be pointed out. Low-index faces are characterized by a two dimensional lattice of minima that reflect the periodic structure of the underlying surface. High-index faces are characterized by regular lattices with several distinct minima, of different depths, separated by barriers of different heights. The zones of atomic size centred on the minima constitute the adsorption sites. These sites are separated from one another by saddle points, so that an activation energy is required for surface migration. If the potential well at the adsorption site is sufficiently deep to contain one or more quantum states for the adsorbed molecule and is separated from the saddle point by a high energy compared to the thermal energy kT, the adsorbed molecule will be localized in one or another of these sites. Otherwise, the adsorbed molecules, although still held near the surface by the vertical part of the adsorption potential, will be free to move along the surface, in which case one speaks of mobile adsorption. The thermodynamic properties of the adsorbed molecules, mainly the adsorption isotherm, can be obtained from the knowledge of the adsorption potential. In the most
general and used scheme the partition function Zx,adof Af adsorbed molecules is evaluated from the adsorption potential U(r) and the gas interatomic pair potential u(r~j): Zx,ad . . . J .
J
exp
,,~ ( --~--~ 1 [~i U(ri)+~E'u(rij)])dr~...drx,
(2)
where: rij -- Iri - rj[, the indices runs over all the adsorbed atoms, and the prime means that the case i = j is excluded. With the possible exception of helium, spin of adsorbed molecules is irrelevant so that classical statistical mechanics can be used. Considering the adsorbed molecules as forming a distinct phase (the adphase) at temperature T, the chemical potential of the adphase can be computed,
~t~d=-kT (OlnZx'~d) 0Af
(3) V,T
(V denotes the system volume), and when equated to the chemical potential of the gas considered as perfect,
~tg~= kTln(p/po) (where p0 = kT(mkT/2rli) 3/2, m is the mass of the adsorbed molecule, and
(4)
h is the reduced Planck constant), the adsorption isotherm is obtained [11, 1]. However, the difficulties encountered in computing the partition function for realistic potentials make it necessary the use of drastic approximations, usually related to simplified models of the physical situation.
2.2. Submonolayer adsorption The models are particularly simple if one stipulates that adsorption is exhausted when one monolayer is filled. This situation is expected to occur when the adsorbent temperature is above the critical temperature of the adsorbate.
2.2.1. Localized adsorption Localized adsorption is characterized by the existence of an activation energy for surface diffusion and by a temperature so low as not allow an appreciable diffusion. In modern language, the models for submonolayer localized adsorption are variants of the two-dimensional Ising models, either with no interaction between nearest neighbours (Langmuir model) or with an interaction described in the mean field approximation (Fowler-Guggenheim model). L a n g m u i r i s o t h e r m . The first, and still the most used, adsorption model was established by Langmuir in 1918 in a paper with the reductive title The adsorption of gases on plane surfaces of glass, mica and platinum [12]. The original derivation of the Langmuir isotherm was kinetic in character, but a rigorous statistico-mechanical derivation was soon found. The statistico-mechanical derivation of the Langmuir isotherm is based on the following assumptions:
9 the gas is perfect; 9 adsorbed molecules are classical objects localized on their adsorption sites; 9 the surface is characterized by Afs identical sites; 9 adsorbed molecules do not interact laterally; and 9 adsorption is exhausted after the formation of the first layer; from which a straightforward application of eqs. (3) and (4) leads to the celebrated Langmuir isotherm [12, 11]: 0=
P
p+pLexp(-r
(5)
where 0 = A/'/A/'s, PL is a characteristic pressure given by PL = po/z, z is the partition function of an adsorbed molecule in the adsorption field, and e is the binding energy of this molecule to the surface. F o w l e r - G u g g e n h e i m i s o t h e r m . This equation is based on the same assumptions as the Langmuir model, however improved to include nearest-neighbour interactions in the Bragg-Williams approximation. In this approach lateral interactions are taken into account in a simple way by assuming that the total interaction energy is the same for all the possible configurations of Af molecules on A/'s sites. The resulting adsorption isotherm is [13] e
P=PLi
'0exp
kT
kT
'
(6)
where: PL and e maintain the same meaning as in the Langmuir equation, w is the nearest neighbour interaction energy, and c is the site coordination number. This isotherm exhibits phase transition loops at temperatures lower than a critical temperature Tr with Tr =
cw/4k. 2.2.2. Mobile adsorption Mobile adsorption is characterized by an activation energy for surface diffusion so low as not allow localization. In modern language, the models for submonolayer localized adsorption are two-dimensional (2D) variants of the Van der Waals model. V o l m e r i s o t h e r m . The hypothesis, that the adsorbed phase behaves as a 2D Van der Waals gas with only co-area effects, led Volmer [14] to propose the following isotherm: o
P=Pvi
'0exp
1--0
'
(7)
where Pv is a characteristic pressure,
pv = (mkT/27rh2)l/2kT/b2z•
(8)
z• is the partition function for the motion of the admolecule in the direction perpendicular to the surface, r maintains the meaning of adsorption energy, and b2 is the co-area. The co-area is the 2D analogue of the covolume in the Van der Waals equation of state. Though co-area effects are expected to be very important in an adsorbed phase (because this phase may have a density of the same order as the bulk phase) the Volmer model ignores ignores that lateral interaction between adsorbed molecules may be equally, or even more, important.
H i l l - d e B o e r isotherm. This difficulty was solved by Hill [15] who described the adsorbate as a 2D Van der Waals gas held at the surface by the adsorption field. The resulting adsorption isotherm is given by:
0 p=pv 1
{" 0 /~exPkl_O
a20 kT
kT
'
(0)
where a2 takes into account the interaction energy among the adsorbed molecules. The qualitative behaviour of this isotherm is roughly similar to that of the Fowler-Guggenheim one. Even this equation exhibits phase transition below a critical temperature To, with Tr = 8a~/27b2k. The first evidence for systems behaving as 2D Van der Waals gases was provided by de Boer [16] and since then the above equation is usually referred to as Hill-de Boer isotherm.
2.3. Multilayer adsorption In the previous part we have considered only submonolayer adsorption, i.e. the formation of only one layer of adsorbed molecules, held on the surface by gas-solid forces. However, since gas-solid interaction energies which are responsible for adsorption are not very different from vapour-phase interaction energies which are responsible for condensation to liquid state, it is not surprising that multilayer adsorption (i.e., adsorption on the top of already adsorbed molecules) occurs. Actually, when the temperature of a solid surface is lower than the critical temperature of the adsorbate, multilayer adsorption is a common phenomenon, resulting in a gradual increase with pressure of the adsorbed molecules up to bulk condensation as the adsorption pressure approaches the bulk vapour pressure. The possibility of multilayer formation complicates considerably the treatment of adsorption on homogeneous surfaces. In principle, the general equations (3) and (4) take into account the multilayer formation when the rigorous potential is included in the expression (2) for the classical partition function Z~c,~d: this is no more true for the various simplified models which lead to the local isotherms (Langmuir, Fowler-Guggenheim, Hill-de Boer, etc.) reported in the previous section. Simplified models for multilayer adsorption are formulated by explicitly allowing the possibility of multilayer formation. Among these models we shall consider only the ones due to Brunauer, Emmet and Teller (BET), and to Frenkel, Halsey and Hill (FHH). They can be considered as complementary, since the BET isotherm gives generally a satisfactory description of the real systems at coverages lower than 2 - 3 layers, while the FHH isotherm becomes adequate only at coverages higher than 3 layers. 2.3.1. The B E T isotherm and its extensions The original derivation of the BET isotherm was based on kinetic arguments [17], although statistical derivations are known [1]. The hypotheses upon which the BET theory is built are the following: 9 the gas is perfect; 9 adsorbed molecules are classical objects localized on their adsorption sites; 9 the surface is characterized by A/'s identical sites;
9 adsorption takes place either on surface sites or on the top of molecules already adsorbed but not in in-between positions; 9 the first layer only interacts with the surface; all other layers have interparticle interaction with the same energy as would apply in the liquid state, and involving only nearest neighbours in the vertical stack of adsorbed atoms in each site; and 9 adsorbed molecules do not interact laterally. Since the first three assumptions are the same as for the Langmuir isotherm, the BET model is essentially an extension of Langmuir model to multilayer adsorption. The above assumptions lead to the following expression for the surface coverage: 1 Cx e(x) = 1 - x 1 + ( C -
1)x"
(10)
where x is the relative pressure (x := p i p . , ratio of the equilibrium pressure p to the saturation pressure ps), and C = z-~ exp ( ~ -- 81iq) Zr,n
kT
(11)
"
In eq. (11) z and znq are the partition functions for a molecule in the first layer and liquid phase, respectively, while ~ and enq are the adsorption energies in the first layer and higher layers, respectively. If C >> 1 (BET isotherm of II type), higher layers are occupied only when the first layer has been filled almost completely. If C ~< 1 (BET isotherm of III type), adsorption in the first layer occurs in competition with adsorption in higher layers. The BET isotherm can formally be reduced to the Langmuir isotherm by a simple variable transformation, a fact that is useful to develop a unified treatment of mono- and multi-layer adsorption. In fact, defining an 'enhanced pressure' P [18], p=
1
P
(12)
and a modified surface coverage OM(P, ~), 0(P, ~) 8M(P,e) = 1 + P/p~'
(13)
for the BET isotherm one has 8M(P, 8)
=
p +
PL
P exp ( - e / k T ) '
(14)
where PL = PL e x p ( - e r , q / k T ) . The BET assumptions can be modified by imposing that piles with a maximum of n molecules can be accomodated on the surface. In this case one gets C z 1 - (n + 1)z '~ + n z T M 6,.,(x) = i - x I + ( C - l ) x Cx n+l' -
of eq. (10). Form lly a (x) for n pressure with above properties can be defined even for the BET equation with n layers [19].
(15)
A simple inspection of the BET assumptions clearly shows that the BET equation cannot give an adequate physical description of multilayer formation. However, in the neighborhood of the B point (i.e., the point in the x - 6 plot where the experimental isotherm changes its concavity) the BET equation has provided a rationalization of so many experimental data as to have become a standard for the quantitation of surface areas. Many treatments have been developed in order to improve the BET theory [20, 21]. Among them we mention the model by Hill [20], essentially based on the same assumptions of BET theory, but accounting for lateral interactions among adatoms in the same layer (within the Bragg-Williams approximation). The obtained improved isotherm is considerably more complicated without leading to better agreement with experimental results [22]; for these reasons it is not frequently utilized in p r a c t i c e - that occurs for other seemingly 'improved' isotherms. 2.3.2. T h e a d s o r b e d phase as a liquid - - F H H isotherm In the FHH theory [23, 24, 25] the adphase is considered as a liquid phase subjected to an external potential generated the adsorbing solid surface: these attractive gas-solid interactions are responsible for a stabilization of the adphase with respect to the bulk liquid. The FHH isotherm is derived by assuming the adsorbate as a uniform thin layer of liquid on a planar, homogeneous, solid surface and considering the effect of the replacement of the solid by the liquid: a molecule in the adsorbed layer will feel different potentials in these two situations. Equating such a potential-energy difference to the difference of chemical potentials between the adsorbed layer and the bulk liquid, one obtains the following implicit isotherm kT ln(p/p~) = up(t),
(16)
where t is the thickness of the adsorbed layer and up(r), known as perturbation energy, is the difference between the actual potential U(r) acting on a point r of the adsorbed layer and the hypothetical potential acting on the same point if the solid adsorbent were substituted with liquid adsorbate: up(r) = [U(r)-Unq(r)]. Equation (16)is interpreted by stating that a liquid condenses in a volume within a distance t from the surface when the perturbation potential in that volume is less than, or equal to, k T l n ( p / p s ) . The quantity (17)
c = -kTln(p/ps)
is usually referred to as Polanyi potential. See Steele's treatise for a compact discussion the the physical bases of the FHH theory [1]. An explicit form for the FHH isotherm (16) is obtained by putting t = 0d~ (where d~ is the thickness of each layer), and assuming that the perturbation energy is attractive and varies as an inverse power ; of the distance, U(x3) cx x3 r With this assumption the FHH isotherm becomes in
=
O~fsn
kTd~
(18)
'
where aFHH is a proportionality constant. Considering the long range part of the gas-solid interaction and the intermolecular interaction in the bulk liquid as due to dispersion interactions, the exponent s should be equal to 3.
Of course, the assumption t cx 0 provides a 'continuum' description of the adsorption process; this description is realistic only for 0 high enough, say 0 > 3. 2.4. Classic empirical i s o t h e r m s for s u b m o n o l a y e r a d s o r p t i o n None of the isotherms considered above is observed frequently: Clear-cut evidence for the Fowler-Guggenheim or Hill-de Boer isotherms has been provided only for adsorption on highly homogeneous lamellar surfaces (see [26] and references therein quoted). The BET isotherm describes poorly both the submonolayer region and the high-coverage region; it provides a satisfactory description of adsorption only for C >> 1 and in the vicinity of the B point. The FHH isotherm provides an adequate description of multilayer adsorption for 0 > 3, but most experimental data can be fitted with r = 2.1 - 2 . 8 [27] rather than with the theoretical value r = 3. These anomalies cannot surprise, because in most cases the surfaces of adsorbents of practical interest are highly non-ideal due to numerous physico-chemical factors, like: the presence of different compounds, phases or crystalline faces, the absence of short-range order, or the complex topographic structure. Rather, what is surprising is that in spite of the plethora of ways through which non-ideality can be manifested, only three adsorption isotherms and their combination are frequently observed in submonolayer adsorption: the Freundlich (F), Dubinin-Radushkevich (DR), and Temkin (W) isotherms. 2.4.1. Freundlich i s o t h e r m The F isotherm has both a historical importance, because it is the oldest reported rationalization of adsorption data (it was extensively used by Freundlich [28] well before Langmuir's derivation of his isotherm [12]), and a practical importance because it is still largely used in the description of real systems. The F isotherm is given by 0F(p) = (p/pF) ~,
(19)
(where PF and s are at the moment unspecified parameters characteristic of the adsorbent-adsorbate system, with s < 1) and is defined for 0 _< p _< PF- However dOF/dp ~ +oc for p ~ 0, so that eq. (19) is not reduced to the Henry isotherm in the low pressure limit. A variant of the Freundlich isotherm was proposed by Sips [29, 30]" 0F,S(p)=
P +PP F
(20)
Equation (20) is defined even for p _> p~, but does not behave asymptotically as the Henry isotherm in the low-pressure limit. Another isotherm which is reminescent of the F equation was proposed by Tbth [31]" OF,T(p) = [ 1Is
P
~PT -4- pUS
)s.
This isotherm is defined for all p and behaves as the Henry isotherm in the low pressure limit.
l0 2.4.2. D u b i n i n - R a d u s h k e v i c h i s o t h e r m
The isotherm, originally proposed by Dubinin and Radushkevich in 1947 for adsorption on microporous solids [32], reads lnA/" = CDR- B[kTln(p/p~)] 2,
(22)
where CDR and B are suitable constants, depending on the considered system. This equation is defined for 0 _ p < p~. Only later Kaganer observed that the parameter CDR coincided with lnA/'s, where Af~ is the monolayer coverage as determined by the BET technique [33], so that eq. (22) could be written In 0DR(P) = -B[kT ln(p/ps)] 2.
(23)
It was with astonishment that Hobson [34] found that eq. (23), an equation manifestly in the realm of the Polanyi potential theory, could be applied to adsorption on non-porous surfaces. Soon after Hobson's discovery the DR equation, eq. (23), was found to describe several adsorption systems, though in some cases better fits could be obtained by replacing psat by the saturated pressure of the solid phase [35, 36]. The theoretical analysis of the DR isotherm carried out by Cerofolini [37] and the subsequent numerical analysis by Rudzifiski et al. [38] then showed that for adsorption on non-porous surfaces the DR equation (23) should actually be replaced by the modified DR (mDR) equation: In 0mDR(P) = -B[kT ln(pm/p)] 2,
(24)
where Pm is a suitable pressure related to the minimum adsorption energy. A large number of systems is known to be described by eq. (24) in the deep submonolayer range (8 < 0.1); a nonexhaustive list of systems obeying this equation is given in ref. [2]. A short history of the DR isotherm is sketched in ref. [40]. Equation (24) is defined only for 0 < p _~ pro; however, as suggested by Misra [39], the equation in 0mDR,M(P) = -B[kT ln(1 + pm/p)]2
(25)
is very close to the mDR equation for p << Pm and extends the domain of p to the entire physical region (0, +oo). However, dO~DR/dp ~ 0 for p ---+0 and so does dOmDR,M/dp~ even in these cases the expected Henry behaviour is not reproduced. 2.4.3. T e m k i n i s o t h e r m
An adsorption isotherm where 0 increases logarithmically with p was first found by Slygin and Frumkin in 1935 [41]. However, the pressure-logarithm isotherm is usually associated with the name of Temkin, who had already proposed such a law on theoretical grounds in 1933 [42] and later applied it to the kinetics of ammonia synthesis and decomposition [43]. We shall write the T isotherm in the form 0T(p) = 1 + CT ln(p/pm),
(26)
where CT and pin are characteristic constants of the system. Equation (26) has a meaning for - 1 _ fiT ln(p/pm) < O, but is known to provide accurate description of physical systems only in a more restricted interval {p" 0.2 < 0T(p) < 0.8}.
ll 2.5. Blends of classic empirical isotherms The study of actual adsorption isotherms over extended pressure ranges has shown that pure F, DR or T behaviours are not observed in most cases; usually the actual isotherm exhibits a gradual transition from one classic empirical isotherm to another. 2.5.1. Merging the F and D R isotherms The FDR isotherm was proposed by Cerofolini [44] in the frame of the condensation approximation (see section 3.7.1) assuming that the adsorbing surface was obtained by quenching an equilibrium configuration and that lateral interactions are not negligible. The FDR isotherm is defined for 0 < p < pm and reads in 8FDR(P)
=
1 kTF
(kTln ( ~ ) [ l + - - l nkT ~o
(~-~)])
(27)
where TF is a 'frozen temperature' and ~0 is a suitable energy. For exp(-no/kT) <
In O = - Z
Bj[kT In(pro~p)]j for 0 _< p _< Pro,
(28)
j=0
where the temperature-independent coefficients Bj characterize the adsorbent-adsorbate system. Equation (28) follows from an idea of Heer, according to which the expansion of the logarithm of fugacity gives a better description of the adsorption isotherm than the traditional virial expansion [47].
12 2.5.2. M e r g i n g the T and F isotherms In an analysis of published experimental data, Gottwald and Haul [48] found that isotherms, previously referred to as obeying the DR equation, are better described for 0 < p ~ p~ by an equation of the type ( p / p m ) s -- 8TF
exp(STF/8O),
(29)
where 80 is a characteristic coverage. Equation (29) cannot be inverted in terms of elementary transcendental functions. However, it behaves as the F equation in the low coverage limit, 8TF << 80 =~ 8TF(P) (p/pm) s, while behaves as the T equation in the high coverage limit, 8TF >> 80 ~ 8TF(P) -- in 80 +8oS ln(p/pm), provided that 8oS is identified with ~T. Because of these behaviours eq. (29) will be referred to as TF equation. 2.5.3. M e r g i n g the T, F, and D R isotherms We have now the instruments for finding an expression behaving as the mDR equation at low coverage, as the T equation at high coverage, and as the F equation in the intermediate coverage region. Such an equation, which is defined for 0 < p
(~)]})=STFDRexp\
(STFDR 8o ) "
(30)
At last we note that the change of argument which allows the F and mDR equations (defined for 0 _ p < Pro) to be transformed into the generahzed Freundlich and DR equations (eqs. (20)and (25)), i.e. Pm/P ~ Pm/P + 1,
can also be used to extend the domain of the blended isotherms (27), (29) and (30) from (0,pro) to (0, + ~ ) . 2.6. A d s o r p t i o n on heterogeneous surfaces Most of adsorbents of practical interest in physical chemistry are not energetically homogeneous, so that they are expected not to obey the isotherms described in section 2. The problem of describing adsorption on energetically heterogeneous surfaces was considered by Langmuir himself [12], who described the coverage St(p) of the entire surface as a weighted average of the coverages 8(p, ej) of sites of adsorption energy ej:
0~(p) = Z 0(p,~)xj,
(31)
J where Xj is the fraction of sites with energy Q, and the sum is extended to all different sites forming the surface. By definition
Vj : Xj >- 0
(32)
and
Z ~ = 1. J
(33)
13 The function 0t(p) will be referred to as overall isotherm, while the function O(p,e) will be referred to as local isotherm. Langmuir could ignore surface topography because he assumed his equation as local isotherm. When other models are considered, however, surface topography is no longer mute in the description of adsorption. In this case, two extreme situations are usually considered: patchwise heterogeneity and random heterogeneity.
2.6.1. Adsorption on patchwise heterogeneous surfaces The description of adsorption is significantly simplified if the adsorption sites having the same adsorption energy are assumed to be grouped in homotattic patches, each large with respect to the atomic size, so that they can be considered as indefinitely extended and lateral interactions between molecules adsorbed on different patches can be neglected. This model, known as homotattic patch approximation (HPA) [49], allows the use for each patch of the theory developed for the homogeneous surface. If the family of different patches is so densely distributed as to allow the finite set {Xj} to be replaced by a continuous distribution function X(e), the overall isotherm is given by
~
M (34)
where ~m and ~M represent the minimum and maximum adsorption energy, respectively. For reasons of mathematical convenience the lower limit of integral (34) is often put to 0, while the upper limit is put to + ~ : (35)
j~O+~ This substitution does not produce difficulties, provided that X(r in (0, +oc) and identically null for r < Cm and for e > eMIt is stressed that in the realm of the HPA, eq. (35) holds true isotherm describing the adsorption on each homogeneous part of the Langmuir isotherm as well as for the Fowler-Guggenheim or
is considered as defined irrespective of the local the surface: it holds for the Hill-de Boer ones.
2.6.2. Adsorption on random heterogeneous surfaces In the second extreme model, known as random approximation, the adsorption sites having the same adsorption energy are assumed to be distributed completely at random on the heterogeneous surface. Although considered as early as in the forties by Hill [50] and Tompkins [51], this model is employed much less than the HPA, because of its greater computational difficulties and (perhaps) of its lower adherence to the physical reality. The Hill-Tompkins model extends the Fowler-Guggenheim theory of localized adsorption on homogeneous surfaces to adsorption on heterogeneous surfaces with a random distribution of sites. This model is based on a canonical ensemble approach, in which only interactions between adsorbed molecules are taken into account. On the basis of this assumption, Hill derived the following expression for the partition function, In ZX,~d -
z
Afs,i In
3'
+
7 +~1
nlw + k--T-- '
(36)
14
(
W (0rL 1 ~ 7 = !zi exp a - -s ' N'si' is the number of sites of adwhere: a = kT kT ~,-~,]' sorption energy ei, ~t is the chemical potential of the adsorbate, Af is the total number of adsorbed molecules, zi is the partition function of the adsorbed molecule on the i-th site, w is the nearest-neighbour interaction energy, the configuration factor n l is defined through the relationship exp
(TZlW E g(nll)exp(nlxw/kT) =
(37)
Eg(nlx )
g(n11) is the number of configurations of the adsorbed molecules with rill nearest-neighbour pairs, and the sum is extended to all possible configurations. Making use of eqs. (3) and (4), and replacing the summation over Af~,~by an integration in eq. (36), Hill derived the following overall isotherm
j[o~176 0t(p) =
(p/pL ) exp( r / kT)
exp [ - w / k T (Onl/ON')] + (P/PL) exp(e/kT)
(3S)
Using the Bragg-Williams approximation, Hill showed that Onl/OAf = cOt(p), so that the overall isotherm is given by Ot(p) =
/o=[
1 + P__L_L exp
p
(
r kT
cwOt(p) kT
)]
-1
X(e)de.
(39)
The comparison of this equation with the one which describes Fowler-Guggenheim adsorption on patchwise heterogenous surfaces, i.e. Ot(p) =
/=[
1 -4- PL exp
p
(
e
kT
cwO(p, r kT
)]
-1
Z(e)dr
(40)
shows that the HPA expression holds also for random heterogeneity provided that 0 t ( p ) is substituted for O(p,e) - - a rather intuitive conclusion.
3. T H E M I D D L E AGE: E M E R G I N G S T U D I E S OF S U R F A C E H E T E R O GENEITY EFFECTS 3.1. T h e inverse problem A coarse-grain look at the log p-O plot of all the considered theoretical isotherms describing submonolayer adsorption on energetically homogeneous surfaces shows that they remain close to 0 for p << Pe while become close to 1 for p >> pc, where p~ = 0(1) exp(-e/kT). The transition region from 0 to 1 is quite narrow and extends over less than one order of magnitude. The classic empirical isotherms, by contrast, are characterized by a transition region usually extending over several orders of magnitude. The most direct explanation for this behaviour is to ascribe it to a distribution of adsorption energy along the surface, so distributed as to allow for the observed transition region. This interpretation originates the problem of finding such a distribution.
15 3.2. S t a t e m e n t of the problem Adsorption on patchwise or random surfaces is described by eq. (35), provided that the possible dependendence of 8(p,~) on 0t or 8 are considered (as in eq. (39) or (40), respectively). Relationship (35) is the source of three problems [52]:
9 the problem of the unknown kernel, in which the overall isotherm 0t(p) is experimentally determined in correspondence to a given distribution function X(e), and eq. (35) is considered as an equation for 0(p,e); 9 the direct problem, in which 0(p,e) and X(r are assumed to be known and 0t(p) is calculated by a quadrature; and 9 the inverse problem, in which 0(p,e) is assumed as theoretically known,/~t(p) as experimentally determined, and eq. (35) is considered as a Fredholm integral equation of first kind for X(r The problem of the unknown kernel is often ignored, but is of fundamental importance since its solution would give information on the actual adsorption mechanism. This problem would be solved if one were able to prepare energetically homogeneous adsorbents. Actually, the preparation of energetically homogeneous sorbents with surface area so high as to allow rehable adsorption measurements is a conceptual problem which has practically been solved in few cases only [26]. Only few works have been devoted to the problem of the unknown kernel; its mathematical nature is discussed in ref. [52]. The direct problem is not simply a quadrature problem, but rather aims at calculating the adsorption isotherm for an assigned surface; the specification of X(r is a part of this problem, and its solution requires the formulation of a model of the adsorbing surface. This problem will be considered in section 2.4 for equilibrium surfaces. The inverse problem is the one more frequently encountered in practice, because (i) the overall isotherm is directly accessible to experiments, and (ii) there are good reasons for choosing one or the other local isotherms. 3.3. Choice of the functional space In mathematical terms, the inverse problem is the problem of solving eq. (35) for X(e) when 0t(p)and O(p,c)are assigned functions. The functional space in which the distribution function is searched plays an important role in the discussion of the problem. Indeed, as discussed in ref. [53], the choice of the functional space in a physical problem is not straightforward and may affect the solution of the problem; because of this fact this choice must be decided on physical bases. Consider, for instance, the case 0t(p) -- 8(p, ~). It is easy to show that in this case there is no continuous function X(e) satisfying eq. (35). However, in the space of distributions eq. (35) admits the trivial solution X(e) = 5 ( c - ~), where ~(-) is the Dirac 5 distribution. Ignoring this trivial case (and its natural extension, for which /gt(p) is given by eq. (31) with few coefficients Xj, each of the order of unity), one may reasonably assume that X(r is a smooth (however not necessarily continuous) function. Because of the physical conditions of non-negativity,
> 0,
(41)
and of normalization,
f0 ~176
= 1,
(42)
16 the function X(r belongs therefore to the space LI(0, +oo) of Lebesgue-summable functions. Equations (41) and (42)transcribe conditions (32) and (33), respectively, from the discrete case to the continuous one. The overall isotherm can in turn be considered a smooth function (so are the classic empirical isotherms considered in section 2). Since 0t(p) is monotonically increasing with p, from a famous theorem due to Lebesgue its derivative Ott(p), O~(p):-- dOt(p)/dp, exists almost everywhere [54] and is non-negative: o
Vp" O~(p) >_ 0,
(43) o
where the symbol ' > ' means '>_ almost everywhere'. Assuming, quite realistically, that the monolayer is completed when the pressure is increased to +oc (i.e., p ~ +oo =~ 0t ---* 1), one has
fo ~176 O't (p)dp = 1.
(44)
The combination of conditions (43) and (44) guarantees that O~(p) too belongs to LI(0, +oc). The function O~(p) plays a special role in the theory and will be referred to with a new n a m e - disotherm. Since a natural functional space for O't has been identified, rather than eq. (35) it is convenient to consider the equivalent equation
O't (p) =
fo
O'(p, e)X(~)de,
(45)
O0(p,e)/Op. where O'(p, e) is the local disotherm O'(p,r Considered as an equation for X(r eq. (45) is linear only if the local disotherm does not depend on 0~(p). If we want to remain in the frame of linear integral equations, we have therefore to limit ourselves to the HPA. Unless otherwise explicitly mentioned, all the forthcoming considerations are limited to this case. Moreover, the attention will mainly be concentrated on O(p, ~) given by the Langmuir isotherm. The 'good reasons' for choosing 0(p,e) are reasons of simplicity, since other choices of local isotherms do not allow us even to express analytically O(p, r in terms of elementary transcendental functions. When O(p, ~) is given by eq. (5), the local disotherm is given by O'(p, e) =
p2 exp
[
(
(46)
1 + PL e p exp -~-~
This expression takes a more compact form on defining er
~(~r
kTln(pL/p), := 0(p(~),~), :=
(47) (48)
17 for which the local disotherm is given by 6"c
6'(r162r --
1
exp
-- ~I
kT
Cc
where O'(r162
-- C
(49) 2~
i)0(p(r162162
3.4. Ill-posedeness of t h e problem Putting
(O'x)(P)
O'(p,e)X(e)dr
:=
(50)
eq. (45) can be written in operator form as 0it ---~Or,)(""
(51)
The corresponding inverse problem reads: given 0~ E L 1, find the element X E L 1 satisfying eq. (51). According to Hadamard, a physico-mathematical problem is well posed when: (a) it admits at least one solution; (b) it is unique; and (c) it is stable. None of these conditions is trivially satisfied. To demonstrate that existence and uniqueness are not trivial, we shall construct specific counterexamples; for instability we shall use the property that if the kernel @'(p, c) is sufficiently regular, then the operator | is compact. 3.4.1. E x i s t e n c e Equation (51) admits solution only if 0't belongs to the image of O'. This condition is not trivial. For, let {r be a complete orthonormal system of L 2 belonging to L 1 too. An example is given by Laguerre functions which form a complete orthonormal system in L2(0, +or and belong to LI(0, +cr If the local disotherm is given by l
o'(;, ~) = ~ o'~jC~(p)~j(~),
(52)
i,j=l
then
O~(p)must
be of the form
l
0'~(p) = ~ b,r
(53)
i=l
with
b~ = ~ I % j--1
jfO§
~j(~)~(~)d~.
(54)
18 with j > l or 0't E LI\L2), eq. If O[(p) has not this form (for instance, if O~(p) = r (51) does not admit solution. However, we shall later see that for O(p, e) given by Langmuir isotherm, it is possible to modify slightly each of the classic empirical isotherms to such a form which allows a solution to be found.
3.4.2. Uniqueness Let ker O' := { f : O ' f = 0}.
(55)
The set ker O' is not void since the null element 0 of L 1 belongs to ker O'. If X0 is a solution of eq. (51), then any other element X0 + f, with f E ker t9', is a solution of this equation because of the linearity of 19'. Necessary and sufficient condition for a solution X0 of (51) be unique is that ker O ' = {0}.
(56)
Condition (56) is not trivially satisfied by all operators on L 1. Consider, for instance, a finite-rank kernel O'(p,e) of the form (52). Then any element (i with j > 1 of {~,} belongs to ker O'. The analysis of the next section will however show that the solution of eq. (51) is unique when O'(p, ~) is the Langmuir disotherm.
3.4.3. Stability Rather than the 'true' overall disotherm 0~(p), one knows an 'experimental' overall disotherm O~,~(p), i.e. a disotherm known in a certain pressure domain (pl, p~) (usually, a proper subset of (0, + ~ ) ) and differing from the true isotherm by less than a given quantity a (the experimental error). The best one can do is to perform a set of measurements, in which the pressure domain (pl, p2) is gradually extended and the error a is reduced. In so doing one determines a sequence {0't,~} of disotherms converging to the true isotherm for n ---, + ~ . Assuming that for each 0~,~ a solution exists and is unique, one can thus construct a sequence of functions {X~}. The solution is stable if 0~,n ---, 0't implies Xn --* X. This property, however, does not hold true for the Langmuir disotherm. In fact, since the kernel O'(p,e) of equation (45) is continuous and bounded, the associated operator O' is compact [55]. That means that for each bounded sequence {X~ }, one can extract from the the corresponding sequence {0't,~} a subsequence {0't,m} converging to the true disotherm 0't. Assume now that the measurements have actually produced just the sequence {0't,,~}. In this case the solution of the inverse problem generates a bounded, however non-convergent, sequence {X-~}The above difficulties is more fundamental than practical. In fact, though we cannot be sure that from a convergent sequence {0~,m} we can always calculate a convergent sequence {X-~}, in practice a convergent sequence may result (and often does). Even in this case, however, we run in difficulties. Indeed, the experimental and true overall disotherms satisfy the condition
I o',,.(p) 1
O,(p)
@
<
(57)
19 Taking the limit for an ~ 0 means that 0't,n tends to O't with the metric of L 2. Even assuming that xn converges to X, this tendency is however in the mean. A uniform convergence is not guaranteed, and it may happen that even small errors are responsible for large local deviations of X,(e) from X(e). This difficulty is made more dramatic by the fact that the n~ighbourhoods of p = 0 and p = +oo are physically inaccessible. That the convergence in the mean of X,(e) to X(e) does not guarantee the uniform convergence is immediately realized by observing that a piecewise continuous function X(e) having a finite number n of simple jump discontinuity points (say e l , ' " ,e,), may be approximated by a continuous function Xcont(e) obtained by replacing X(e) in ( e ~ /Xe, e~ + Ae), by means of the straight line joining the points ( ~ - / X e , X(e~- Ae)) and (c~ +/X~, X(r + Ar for all i = 1 , . . . , n. Indeed if M = sup [ X(e)I, one has ! X ( e ) - Xr
de < 8M2nAr
(58)
which can be made vanishing taking/Xe ~ 0 [56]. 3.5. B e y o n d l i n e a r i t y The analysis of section 3.4 is essentially based on the properties of linear integral operators, so that it cannot be applied to eq. (39). However, as shown by Cerofolini and Re [52], it is possible to reduce the problem of solving the integral equation (39) to the problem of solving (35) with the Langmuir local isotherm (5). This reduction is possible only when 0t(p) is a monotonically increasing continuous function of p. In this case the functions f ( p ) = exp(cwOt(p)/kT) and pf(p) are continuous and monotonically increasing. Hence, Vp E (0, +cxz) the equation p = pf(p) has a unique solution p = p(p). Defining v~(p) = 0t(p(p)), eq. (39) becomes ~(P) =
P X(e)de, P + PL e x p ( - e / k T ) jr0+~176
(59)
which is exactly eq. (35) with the Langmuir local isotherm as kernel. Equation (59) also allows the direct problem for the random approximation to be solved. In fact, if 0L(p) is the solution for the direct problem in the patchwise approximation with the Langmuir local isotherm corresponding to a given distribution function X(e), then /~t(P) = t~L(pexp(cwt?t(p)/kT))
(60)
is the solution of the direct problem in the random approximation for the same distribution function. The above relationship is an implicit equation for 0t(p) and its solution, supposedly existing, may be calculated iteratively, starting from 0~~ = 0L(p) and proceeding with the recursive relationship
3.6. Exact m e t h o d s Three exact methods have been proposed to solve the integral equation (35) with Langmuir kernel: The first method, essentially an application of Stieltjes transform theory,
20 was proposed by Sips in 1948 [29] and improved two years later [30]; the second method is an application of Wiener-Hopf technique and was proposed by Landman and Montroll in 1976 [57]; and the third method is a kind of infinite-order approximate method and was developed by Jagietto et al. [58, 59] in the period 1989- 1991 following early ideas of Rudzifiski et al. [60, 61]. 3.6.1. Sips method The method considered in this section is based on the theory of Stieltjes transform [62]. Putting
x := --PL/P-- 1
(--oo < x < --1)
y := e x p ( e / k T ) - 1
(0 <_y < +oo)
(62) (63)
Ot(z) := (1/kT)Ot(-PL/(1 + x))
(64)
2(v) := ~(kTl.(1 + v))
(65)
(the symbol x has not the same meaning as in section 2.3.1), eq. (35) becomes
~(~) = f0 ~176 y_~ 2(Y)du.
(66)
Consider now the analytic extension 0t(() of 0t(x) in the whole complex plane; for 0t(() the Cauchy theorem holds 0~(r
c
-
where (0 is any internal point of any simply connected set C, and 0C is the boundary of C. If (a) [([ ~ +oo =~ 0t(()---* 0, and (b) the singular points of 0t(() lie only on the positive semiaxis, eq. (67) gives 1 [+oo 0t(y + ir - 0t(y - ir dy, v-~
0t(~) = ~i ~o
(68)
where r is an arbitrarily small positive number. Comparing (66)with (68)one eventually gets 1 :~(y) - ~i[0t(y + i~b)- 0t(y - i r
(69)
The uniqueness of this solution is ensured by the uniqueness of the analytic extension. This solution was applied to physically interesting cases by Sips himself [29, 30] (generalized Freundlich isotherm), Misra [39, 63] and Sokotowski [64] (generalized DR isotherm). Here, rather than to apply eq. (69) to real cases, we are interested in discussing the difficulties it hinders.
21 First of all, the conditions (a) and (b) are suggested (but not ensured) by the behaviour of St(p) for p ---, 0 and by expression (66), respectively. Moreover, ~Jt(()is experimentally known only in a portion of the segment P = ( - o o , - 1 ) (the 'physical region'), while eq. (69) gives ;~(y) when ~Jt(() is known in the small strip E = {( 9Re ( E (0,+oo) A Im ( E ( - r 1 6 2
(70)
Hence the evaluation of ;~(y) requires the knoledge of the analytic extension of ~Jt(x). But (a) and (b), is known only in an approximate form ~Jt~(x), whose analytical expression is chosen in order to satisfy (a) and (b). The maximum one may assume about the knowledge of 0t(x) is that
~t(Z), supposedly satisfying conditions
w
--oo
<
9<
-1 ~ 16.(~)-
#t(~).
I< o-,
(71)
where (r is the precision of the measurement and a physical meaning can be given only to ~t~(x) for a ~ 0 and no to 0t (x) itself. This means that we can extract from {St~(x)} a sequence of functions converging uniformly in the physical region to the true function as O" --'4 0 .
No difficulties would occur if the uniform convergence of {/t~(z)} to^~t(x) for x < - 1 ensured the uniform convergence of the analytic extension {8t~ (() } to 8t (r in the whole complex plane or, at least, in the small strip across the positive serniaxis, but this fact usually does not hold true [65]. 3.6.2. L a n d m a n - M o n t r o l l
method Through the change of variables
r162
:=
~(cr
kTln(pL/p)
:= O(p(~r
g,(~r
=
1 + exp
:= 0,(p(~r
( "kT )]_1 g'c --
5"
(72) (73) (74)
eq. (35) is transformed into an integral equation whose kernel depends only the difference e r ~, rather than on r162and e separately. This occurrence makes the corresponding integral a convolution integral: t~t(~r =
K(er ~0+~176
-
r
(75)
given by eq. (73). with K(er ~ ) = 0(er To solve this equation assume X(r be analytic, consider its analytic extension to (-oc, +oc), and define f+(er :=
K(c,: ~0+~176
1_(~r :=
F (x)
~)X(e)de,
K(~r - ~)x(~)a~.
(76)
(77)
22 Summing these equations member to member,
f+(r
+ f+(r162= ~.oo g(r162- e)X(~)de,
(78)
taking the Fourier transform of both members, 9v[f+ + f _ ] ( x ) = 3:[K](x).T'[X](x )
(79)
solving for .7-[X](x), and calculating the inverse Fourier transform, one eventually gets X(e) = $--1[$-[/+ + f_]/J:[g]](e).
(80)
Since f+(er experimentally known (f+(er = 0t(ec)), the problem of finding X(e) would be reduced to the calculation of Fourier transforms if one knew f_(x). In the absence of this knowledge, however, the previous calculations are only a formal development. The Wiener-Hopf technique is a method which allows the calculation of X(e) even without knowledge of f_(x). This technique is based on suitable decomposition of K(ec - e) and requires that each decomposition is analytically continued [66]. Its practical use is difficult, but Landman and Montroll succeeded in this job and applied the Wiener-Hopf technique to the generalized Freundlich and DR equations, and to any other overall isotherm admitting a virial expansion [57]. 3.6.3. dagietto m e t h o d Though both Sips's and Landman-Montroll's methods work well for equations which strictly resemble the F and DR equations, they however suffer from a fundamental diff i c u l t y - both methods require indeed that both 0t(p) and X(~) are analytic functions. In general this is false: the support of X(~) is compact (that excludes the possibility that this function is analytic) and the overall isotherm is known only in an interval (pl, p2) and is raccorded at the boundaries in a piecewise manner (see, for instance, the high pressure behaviour of the classic overall isotherms, or Hobson-and-Armstrong's low-pressure extension of the DR isotherm [67]) rather than by analytic continuation. A solution is said to be local when a one-to-one correspondence ~ ~ p exists such that the unknown function X(~) is completely determined by the behaviour of the given function 0t(p) in the neighborhood of p. While naive considerations suggest that the adsorption behaviour at pressure p is related to the energy distribution function X(~) in the neighborhood of ~ = ~c(p) (thus suggesting the possibility of local solutions), this fact is recognized neither in the Sips method nor in Landman-Montroll one. In other words, the solution is expected to be local, but the described methods provide global solutions [65]. A local method was first proposed by Jagietto in different collaborations [58, 59] and will be referred to as Jagietto method. A similar method was independently proposed by Chen for the mathematically equivalent problem of finding the density of states of Fermi systems [68]. It is however noted that this method is based on a formula already known to Widder in 1938 [69] and that a similar result was also obtained by Brs et al. in 1985
[7o]. The derivation is obtained from the following representation of the Dirac ~ distribution: +o~ )~7r2~ 02~+1 t~(x- y ) = ~ (-1 ~ ,=0 (2n + 1)! 0y 2n+1 1 + e~-y"
(81)
23 The proof of this relationship is obtained in two steps. First, the cases y < x and x < y, are considered separately, (1 + e=-Y) -1 is expanded in powers of e -Ix-yl, and the resulting series are summed and found to be null. Second, the integral of the rhs of eq. (81) is calculated and found equal to 1, that completes the proof. To apply this formula, we first define the local and overall nisotherms:
~,(=c,~) := i - O(~c, ~)
(82)
i(er
(83)
:= 1 - 0 t ( c r
for which
i(~o) =
(84)
A(s~,s)x(~)&
because of the normalization condition of X(e). Then we observe that for Langmuir adsorption A(sc, e ) =
(
l+exp
e-er kT
(85)
so that we can apply eq. (81): +~
(-1)n(,rkr) 2n o2n+l
~ ( ~ - ~ c ) = n=o
(2n + I)!
O~cn+,A(~c,e).
(86)
Inserting this relationship into the identity
x(~c) =
f0
(8;)
~(~ - ~c)X(~)&
one has x(er
(2n + r~=O
n=o +c~
L n=o
(2n + 1)1
1)!
n+--------~(ec'e)X(e)de Oe~
Oe~n+l
fo
A(ec, e)X(e)de
(-1)n(~kT) ~n O~n+' ~(~o) (2= + 1)!
b-~c~:;~
(88)
Equation (88) gives X(er as a function series; its first two terms are
X(~c) =
o~(~c) &c
(89)
and
oh(~c) X(~c)=
&c
(~kT) = O~h(=c) 6 &~ "
(90)
24 Since the series (88) involves (odd) derivatives of the experimental nisotherm in ~ = ~r it gives a local solution, thus satisfying the initial need which motivated Rudzifiski and Jagietto. It is however noted that the_ Jagietto method allows an exact solution to be calculated only if all the derivatives of )~(~r are known. On another side, if all derivatives of an analytic function are known, the function is known everywhere by analytic extension, so that the described method is equivalent to any other non-local methods. One serious advantage of this method, however, is that it does not assume that X(r is analytic (though the first derivation was carried out under this restrictive condition [58, 59, 60]) so that the method seems suitable for application to practical systems where X(r is not analytic, the support of X(r being compact.
3.7. Approximate methods The methods described in this section are mely, the condensation approximation and the considered as special cases of eq. (88) obtained second term, respectively, we shall nevertheless aiming at clarifying their physical bases.
local methods. Though two of them (naRudzifiski-Jagietto approximation) can be by truncation of the series to the first and propose for them independent derivations,
3.7.1. Condensation approximation The condensation approximation (CA) was first proposed by Roginsky [71] and later better formalized by Harris [72] and Cerofolini [73]. This method is based on the replacement of the the true local isotherm by a step function, 0r
J" 0 for 0 < p < pr l ] for pr < p,
~)
(91)
where the pressure pc(~) can be interpreted as a condensation pressure, i.e. the pressure at which the gas suddenly condenses on the patch with adsorption energy ~. A convenient criterion for choosing the condensation pressure pc(~) is the minimization of the distance between O(p,e) and Or [73]. Using whether the Lagrangian distance,
dc..[O,Or =
sup I O(p, ~) - Or pe(0,+oo)
~) I,
(92)
or the L2(0, +oo) distance, dL2[O,/~c] =
{/0
I /~(p,r
Oc(p,e)! 2 dp
,
(93)
one obtains O(pr
= 89
(94)
which for the Langmuir isotherm gives
pc(c) = PL e x p ( - c / k T ) .
(95)
The same result is obtained with Harris's criterion, which states that pc(c) must be chosen as the pressure at which the local isotherm changes concavity in a log p - / 9 plot.
25 Operating as in eq. (48), i.e. using er = - k T l n ( p / p L ) , as independent variable (obtained by inverting eq. (95)), expressing p as a function of er and putting 0t(er Ot(p(er the integral equation (35) becomes t~t(ec) =
Xc(e)de,
(96)
where Xr is the distribution function corresponding to the condensation approximation. Equation (96) can easily be solved by a differentiation with respect to ~r (97)
Xc(ec) "-- --l~#t(ec)/l~ec.
Interesting consequences of eq. (97) are: (a) since er is defined on the real axis (-c~, +co) while only positive values of adsorption energy have physical meaning, X:r162 must be 0 for er < 0, i.e. for p > PL; and (b) Xr is temperature independent, as physically required for hard adsorbents (see section 4.2), only if 0t(er does not depend on T, i.e. if 0t(p) does not depend on p and T separately, but only on kT ln(pL/p). The errors occurring in the CA may be estimated by defining 0(ec, e) as in (48) and writing the integral equation (35) in the form 0t(er =
f0+= 0(er
(98)
e)X(e)de.
Differentiating eq. (98) with respect to er and taking eq. (97) into account, one has Xr162 = -
f
+~ 0~(~r e) O-e-cf
X(e)&-
Equation (99) shows that the more -0t}(er162 resembles 6 ( e - er similar to :~(er For the Langmuir local isotherm one has 0#(So, e) _ i Oer kT
exp
-
1 + exp
(
kT ec - e kT
(99) the more Xr162
(i00)
This function has a bell-shaped form, with a maximum centred on er = e, and becomes sharper and sharper as T decreases, tending to a Dirac distribution when T --, 0. It may hence be assumed that the CA gives reliable results when applied to surfaces whose distributions have a spread much greater than kT. In the opposite case, i.e. for surfaces whose distribution functions present peaks with width smaller than kT, the CA gives distributions which are smoother than the true ones.
3.7.1.1. Application to classic overall isotherms The application of CA to the classic overall isotherms and their blends specifies the meaning of p~ - - the characteristic pressure pm is given by p~ = PL e x p ( - e ~ / k T ) , where r is the minimum binding energy. This physical meaning specifies how pm depends on T (at least when Cm is temperature-independent, as it happens for hard adsorbents) and states that p~ < PL.
26 The application of CA to the classic empirical isotherms is straightforward: the F isotherm is associated with an exponential energy distribution function; the mDR isotherm with a Rayleigh distribution, and the T isotherm with a uniform distribution. 3.7.1.2. E x t e n s i o n to o t h e r local isotherms
The condensation approximation can easily be extended to other local isotherm, like the Fowler-Guggenheim and the Hill-de Boer ones. While in the first case Harris's and Cerofolini's criteria predict the same condensation pressure [74], in the second case they lead to slightly different choices [75]. This small difference affects only marginally the resulting distribution function [75]. 3.7.2. Asymptotically correct approximation
The asymptotically correct approximation and modified to the present form by Cerofolini true local isotherm by an approximate kernel behaviours both at low pressure (Henry law)
(ACA) was first introduced by Hobson [76] [73, 78]; it is based on the replacement of the O~(p,s) which shows the correct asymptotic and at high pressure (saturation):
{ (p/pL)exp(s/kT)l forf~ Pa(~)0 _ p _~
(101)
where p~(e) is a suitable pressure. If this pressure is so chosen as to minimize the distance
bCtw~r 0(p,~) ~.d 0~(p, c), o,r h~s p.(~)= p~ ~xp(-~/kT). Since p~(e) - pc(e), the arguments of the previous subsection can be repeated here. In particular, expressing p as a function of ~c and putting 0t(~r = Ot(p(ec)), the integral equation (35) becomes #t(sc) = exp ( - ~ T )
/o
(k-~)X,(s)ds +
Xa(s)ds,
(102)
where Xa(e) is the distribution function corresponding to the condensation approximation. Equation (102) can be solved by two differentiations with respect to sr and gives 00t(er x~(~r
=
0~r
kTO20t(er 0~r~
(103) '
to which all the general considerations made in the previous subsection for the CA distribution Xr apply again. In order to evaluate the errors occurring in the ACA consider eq. (98) with its first and second derivatives: 0~t(er
= rl +or a~(~r ~)x(~)d~
a~r -_ a~0~(~o)
o~ x(~)d~. f0+~176 a:0(~o,~)
Multiplying both members of eq. (105) by eq. (103) into account, one has
(104) (lo5)
kT, adding the result to eq. (104) and taking
0_ kT0~r ) 9(~r ~)x(~)d6
(~06)
27 Equation (106) shows that the more its kernel resembles 6 ( r r the more Xa(Cc) is similar to X(r For example, for the Langmuir local isotherm, one has 1-exp( 0 0ec
0(~r162
kT-~r
ec-e)kT
1+
kT
l+expk,
kT
(107) l+exp
kT
This function is bell-shaped and tends to a Dirac 6 distribution when T --, 0 K. Furthermore, a comparison may be performed between the CA and ACA approximations. In fact, it is easy to verify that the width at half maximum of the ACA kernel (107) is about 3 k T compared with 3.5 k T of the CA kernel (100). However, this advantage of the ACA is neutralized by the fact that the in this case the kernel is not centred on cc but on cr + k T In 2 [37].
3.7.2.1. Extension to other local isotherms Like the CA, the ACA can be extended to other local isotherms, namely the Hill-de Boer one (for which it was developed [76]) and the Fowler-Guggenheim one [78]. The mathematical machinery employed in these two cases is the same. We do not intend to describe here the details of the calculations; rather we call the attention on the following mathematical aspect. Any method (like the CA and ACA) in which the local isotherm is approximated by an isotherm which is identically equal to 1 for p greater than a suitable pressure p'(~) transforms the Fredholm integral equation of the first kind (35) into a Volterra integral equation. This process removes the instability. While solving the resulting Volterra equation for the CA is essentially reduced to the calculation of the derivative of the overall isotherm, in the ACA the procedure is somewhat more complex. While the solution of the Volterra equation for the Langmuir local isotherm is immediately reduced to the calculation of two derivatives, for the Hill-de Boer and Fowler-Guggenheim isotherms it requires the method of solution proper of the Volterra equation of the second kind [73, 78]. 3.7.3. Rudzifiski-Jagietto approximation This approximation was first developed by Hsu et al. [79], Rudzifiski et al. [61] and Jagietto et al. [58] without assuming any local isotherm approximation. Their derivation is based on equation (99) which can be considered as an alternative formulation of the integral equation (35); its kernel is the derivative of the local isotherm with respect to ~c which in the case of the local isotherm is, as already seen, a bell-shaped function. The true distribution function is then expanded in power series of ~ around r = ec, so that eq. (99) may be written as
0r
=
\
j=o
(lo81
) ~=~c
where Cj(~r = k T
~r
' O~c
kT
d
--~--]
"
(109)
28 When only the first term of the series in eq. (108) is retained, the resulting distribution coincides with that given by the CA, while when also the second term is retained eq. (108) becomes O~c
= -X(e,:) -
( kT) 6
(110)
0~
which is a second order linear differential equation for X(ec). If the second term in eq. (110) can be considered as a correction, this equation can be solved in an iterative way, giving the CA distribution function at the first step, and the RJ distribution function, )~RJ (gc) -----
~
0~c
t (~kT) ~ 0 ~ t (~r 6
(111)
0g 3
at the second step. Rudzifiski et al. [61] have shown that, when applied to real systems, eq. (111) gives results remarkably better than those obtained by the CA or ACA. 3.7.4. B e y o n d the R J a p p r o x i m a t i o n In view of what wassaid in section 3.6.3, the CA and RJ approximations can be seen as the first- and second-order aproximations of general formula (88). A somewhat different procedure, the logarithmic symmetrical local isotherm approximation (LOGA), was developed by Nederlof et al. [80]. Approximating the local isotherm with the kernel ON(p,e) =
-~ (p/pL) ~ e x p ( ~ e / k T ) 1 - ~1 (pL/p) ~ e x p ( - ~ e / k T )
for 0 _< p < pN(e), for PN(r _< P,
(112)
(with PN(r = PL exp(--e/kT)), changing the independent variable from p to r162= k T ln(pL/p), inserting the local isotherm (112) into eq. (48), taking three successive derivatives with respect to r162and noting that in the equation for the third derivative enter the same integrals which appear in the equation for the first derivative and which can therefore be eliminated, one obtains XN(gc) = (~0t(gc_____~)_[_ ( k T ) 2 030t(er 0ec ~2 (~ec3
(113) "
For ~ = x/~/~r = 0.780 this equation coincides with the RJ approximation, so that this procedure can be considered as a way to deduce the RJ formula in a scheme in which the local isotherm is suitably approximated. Nederlof et a/., however, optimized /3 by minimizing the distance between the Langmuir isotherm and its approximating form (112). They obtained fl = 0.7 wheter using the Lagrangian distance or using the L2(0, +oo) metric, and showed that with this choice eq. (113) works slightly better than the RJ approximation. Although both the RJ and LOGA approximations lead to good results in a wide experimental range of pressures, the previous methods present a common drawback - - they are not reduced to the Henry isotherm at low pressure. Consequently, both methods may lead to physically meaningless values of X(r at high adsorption energies. Accordingly, the
29 analysis of the adsorption isotherm in this regime is performed more conveniently within the ACA. A general method for the determination of the distribution function approximated to the n-th order, in which the local isotherm is approximated consistently with its asymptotic behaviours, has been proposed by Re [81]. 3.8. N u m e r i c a l m e t h o d s
The particular nature of Fredholm integral equation of the first kind poses severe difficulties to its solution and strict limits to the range of numerical methods that can be utilized for its solution. Instability in particular poses the most strict limitations from a computational standpoint. Several numerical algorithms have been developed in order to solve the Fredholm integral equation of the first kind (35) [82, 83] and many of them have been applied to the determination of the adsorption energy distribution function from experimental adsorption isotherm [84]. The most frequently employed codes are described in refs. I85, 2]; the methods upon which they are based are shortly discussed in ref. [52]. A list of them includes: 9 discretization methods, in which the solution of the integral equation is reduced to the inversion of a square matrix; 9 regularization methods, in which the ill-posed problem is replaced by a stable minimization problem in various ways; 9 iterative methods, which use various iterative algorithms in order to improve an initial guess for the distribution function; 9 ezpansion methods, which are based on the expansion of the functions A(p), X(~) and A(p,r in series of a complete orthonormal set (the concept of nisotherm is especially important for this class of methods); 9 integral transform methods, which are based on analytical methods in which the experimental isotherm is approximated by a suitable interpolating expression (e.g., a cubic spline) and the analytical inversion formula is expressed in an easily computable form; and 9 optimization methods, based on the choice of an analytical form for the distribution function containing some parameters which are subsequently determined by a best fit of the calculated global isotherm (through a simple numerical integration) to the experimental isotherm.
The last class is briefly considered here because these methods can be considered to solve the direct problem. Without pretending to be complete, among the numerous attempts we mention: 1. The description of heterogeneous adsorbents in terms of exponential energy distribution was probably considered first by Zeldovich [86] and later by Temkin and Levich [87]. Halsey used the exponential distribution for explaining the Freundlich behaviour [88], while Sparnaay used the discrete (rather than continuous) exponential distribution in eq. (31) with Fowler-Guggenheim local isotherm to explain the Dubinin-Radushkevich behaviour [89].
30 2. The possibility of using a Gaussian distribution function for X(e) and the Hill-de Boer isotherm for 0(p,r was explored by Ross and Olivier [90]. For a long time Ross and Olivier's work remained the unique systematic analysis of experimental submonolayer isotherms in terms of surface heterogenity, and created great interest on this topic. 3. Skew distributions with several maxima were described by van Dongen with exponentials of higher-degree polynomials X(r = exp j=0 cjr j , where Co, .-. , ct are adjustable parameters, l of them being independent and the remaining one being obtained with the normalization condition [91]. van Dongen's choice is a natural extension of the choice l = 2 of Ross and Olivier. 4. Wojciechowski and his coworkers described most of classic empirical adsorption isotherms in terms of Langmuir local isotherm and Maxwell-Boltzmann energy distribution function [92, 93].
)
In the absence of 'good reasons' for chosing one or the other energy distribution function and in view of the variety of possible choices, the above approaches must considered essentially as fitting schemes. 3.9. Direct p r o b l e m Ideally homogeneous surfaces can hardly be obtained. In general, they can be prepared either by exfoliating Van der Waals lamellar crystals (like mica or graphite) or by cleaving crystals along low-index planes (e.g., the (111) plane of diamond-cubic silicon). In the latter case, however, the cleavage of true chemical bonds results in appreciable relaxation of subsurface planes and reconstruction of the surface lattice. Once these really few cases are excluded, real surfaces are usually characterized by a complex situation: The surface of crystalline solids will contain several crystalline faces with different extension; each face will be relaxed and reconstructed, and will contain point defects (like vacancies or self-interstitials) and line defects (like emerging dislocations and grain borders). The surface of dispersed amorphous solids will be characterized by an even more complex distribution, since dispersion is often based on processes (like milling) responsible for phenomena (fracture, plastic deformation, etc.) where a macroscopic energy is imparted to few degrees of freedom. The concentration of a large energy on few degrees of freedom, with the consequent creation of highly defective sites, is unavoidable during the preparation of highly dispersed solids even when the process is formed by close-to-equilibrium steps (like supercritical extraction of embedded liquids in sol-gel preparation techniques). What is truly surprising is that in spite of the complexity and variety of formation modes of real heterogenous surfaces, they exhibit striking similarities in their adsorption behaviours, usually obeying one or the other classic empirical isotherm in various pressure range. 3.9.1. E q u i l i b r i u m surfaces The problem of explaining why so few experimental isotherms describe so many adsorbent-adsorbate systems was raised by Cerofolini [94], who suggested that the classic empirical isotherms are observed when the surface was formed in equilibrium conditions
31 (at a certain temperature Tf) and was quenched at room temperature preserving the previous equilibrium structure. Equilibrium surfaces are neither geometrically plane nor energetically homogeneous. The problem of determining the equilibrium shape of a body is a classic problem of surface science, solved within a continuum approximation. Considering first crystalline solids and assuming that the faces with (hkl) Miller indices can be described by a surface tension 7hkt, the equilibrium extension Ahkt of the face (hkl) of a body at assigned temperature T is determined by the condition E h k i AhklThkl
--
V =
min, const.
(114)
If Vhkl : 7hkt = 7 (this situation is characteristic of amorphous bodies) the above condition gives A = min for assigned V, so that condition (114) is strong enough to specify the equilibrium shape - - the sphere. Wulff showed that even when "~hkt is not constant (that occurs in general for crystalline bodies), condition (114) is sufficient to specify the equilibrium shape of the body m the polyhedron obtained via the Wulff's construction [95, 96]. Though very powerful and elegant, the continuum description of the surface of real bodies is inadequate to account for their adsorption properties. Indeed, while the surface of an isotropic amorphous adsorbent is described in a continuum approximation by a unique parameter (the surface tension), it is characterized by a distribution of surface configurations which in turn result in a distribution of adsorption energy. For the reasons discussed at the beginning of this section this conclusion is also true for crystalline adsorbents. The first elementary description of the energetic heterogeneity of equilibrium surfaces was proposed by Cerofolini, who postulated that the number N~ of surface atoms with excess energy/Xi relative to the ground state energy is given by a Boltzmann distribution N~/No = exp(-U~/kTf), where No is the number of atoms in the ground state (with null excess energy//). Assuming a dense distribution of excess energy, the following distribution function was then derived: 1
exp(-U/kTf)
(115)
where b/M is the maximum value of/4;//M is not greater than the energy of sublimation. Two limiting cases of eq. (115) are of interest: weak heterogeneity: strong heterogeneity:
> 1 =~ Xs(//)~- k-~exp
/~M 1 ~ << 1 =~ Xs(//) ~ ~MM"
(116)
3.9.2. Adsorption on equilibrium surfaces For a description of adsorption on equilibrium surfaces it is necessary to specify the adsorption energy e as a function of the excess energy//. Since sites w i t h / / = 0 are those for which surface atoms are most fully coordinated, they are expected to have the minimum tendency to attract gas-phase molecules; since
32 sites with higher excess energies are less strongly bonded to the substratel they are expected to manifest a higher tendency to attract gas-phase molecules. The combination of these occurrences strongly supports the idea that, in the absence of steric or electronic factors,/4 is an increasing function of 6. At the present stage it is not possible to derive on physical considerations the function / / = / / ( 6 ) . However, a Taylor expansion with positive coefficients truncated to the second power, ~I/ --- C~1(6
-- ~ m )
"IF C~2(6
(i17)
-- ~ m ) 2
(with ai, a= positive coefficients and null constant term, because b/ = 0 for e = ~m), is sufficient to explain why the FDR and Temkin are so frequently observed on real adsorbents: the FDR isotherm is associated with adsorption on weakly heterogeneous equilibrium surfaces [94], while the Temkin isotherm corresponds to adsorption on strongly heterogeneous equilibrium surfaces [97]. See ref. [98] for a discussion of this matter. For, it is necessary to calculate the adsorption-energy ditribution function X~q(e) of an equilibrium surface,
Xoq(e)
=
x~(U(e)) I OU/a~ I.
(118)
The first conclusion follows from a comparison of the energy distribution function calculated within the CA for the FDR equation (27) with the energy distribution X~q(e) : l:ul ted from er Conclusion follow from Comparison of the energy distribution function calculated within the CA for the T equation (26) with the energy distribution X~q(e) calculated from eqs. (116) and (117), and ignoring the quadratic term. It is also noted that the case blM/kTf ,~ 1 is expected to originate the TFDR blended iso;therm (30).
4. T H E C U R R E N T SOLIDS
TRENDS
IN T H E S T U D I E S OF A D S O R P T I O N
ON
4.1. A d s o r p t i o n of s t r u c t u r e l e s s molecules on rough surfaces As already observed, the formation of the first layer is particularly interesting when observed in log p - 0 plot. If the attention is limited to homogeneous surfaces, any adsorption isotherm is defined by a characteristic pressure p~ below which (say, for p/p~ << 1) 0 is negligible and above which (say, for p/p~ >> 1) 0 is close to unity. Since adsorption on different homogeneous surfaces is in general defined by different characteristic pressures, p~ is a free (external) parameter. Plotting these isotherms vs. log(p/p~), all of them have the same shape so that p~ is a scale parameter. When the constraint of surface homogenity is removed, the pressure region in which 0 varies from 0 to 1 extends over several orders of magnitude. The attempts to account for this situation have originated the theory of
adsorption on heterogeneous surfaces. The above remark seems to suggest that the simplest adsorption systems (homogeneous adsorbents) can be described by a unique scale parameter. This conclusion, however, ignores that experiments do not provide the isotherm 0t(p), but rather the isotherm JV'(p). The calculation of 0t(p) requires one additional (scale) parameter - - the number of sites
33 As. This would not be a great complication if Af~ were simply related to the number of sites Afs,~at of a geometrically flat surface (Af~,nat = 1014-1015cm-2). This however occurs in very special conditions only, and almost never for adsorbents of interest for chemistry. Even for systems for which a geometric surface area can be defined, the ratio Afs/Af~,flat may exceed 1 by several orders of magnitude and varies from one system to another. The attempts to account for this situation have originated the theory of adsorption on rough surfaces. Of course heterogeneous surfaces can hardly be flat and, conversely, rough surfaces can hardly be homogeneous. There is always a certain correlation between heterogeneity and roughness, so that any theory of adsorption on real surfaces should take both them simultaneously into account. Fortunately, the problems can however be decoupled since the adsorption isotherm is sensitive to heterogenity mainly in the submonolayer region, while is sensitive to roughness mainly in the multilayer region. The theory of adsorption on rough surfaces is complicated by the fact that the surfaces of chemical interest usually cannot be described by a smooth function z3 = x3(xl,x2), because they often exhibit very pathological behaviours. In most of the theoretical treatments of adsorption on rough surfaces, the adsorbent is assumed not to vary appreciably during adsorption itself. Adsorbents with this property will be referred to as 'hard'. In general, an adsorbent is expected to be hard when the energy which stabilizes its configuration is much higher than the adsorption energy. Typical examples of hard adsorbents are: ionic or covalent crystals with high melting point, and highly reticulate polymers like amorphous carbon or silica. Very rough surfaces are possible for hard adsorbents only. This conclusion is based on the fact that highly rough surfaces are non-equilibrium configurations, and they can be preserved only when an appreciable activation energy is required to bring the adsorbent to its equilibrium configuration. Hard adsorbents are usually characterized by X(e) independent of T. 4.2. P o r o u s solids Porous solids are known by a long time and are very familiar in chemical practice. In spite of this, modelling adsorption on porous surfaces is a very difficult task, because of the enormous number of possibilities in which porosity can be manifested. Assuming that the complex topography can be represented in terms of distribution of voids in an otherwise bulk adsorbent, these voids will be referred to as 'pores'. Pores can be defined by a characteristic shape and parametrized with one or more characteristic length (like radius for spherical pores, diameter and height for cylindrical pores, etc.). According to the International Union for Pure and Applied Chemistry (IUPAC), pores are classified as in Table 1.
34 Table 1: IUPAC classification of pores in relation to their size
type
width (nm)
micropores mesopores macropores
< 2.0 2 . 0 - 50 > 50
The adsorption properties of pores are very different in relation to their sizes. In fact, if the adsorbed layer is thinner than approximately 3 monolayers, its description in terms of properties of bulk phase (possibly perturbed by the vicinity of a surface) is manifestly meaningless. Assuming that adsorbed molecules have diameter around 3/~, the typical pore size below which the adsorbate cannot manifest the properties of the bulk phase is around 20/~. This might be the ultimate reason supporting IUPAC classification.
4.2.1. Adsorption in micropores The description of adsorption on adsorbents with pore size below 2 nm is expected to require both the volume available to adsorption and the special nature of the adsorption potential inside the pore (where the potentials generated by different walls can superimpose). Adsorption in micropores is described with two methods: In the first method one assumes that inside each pore adsorption is described by a BET behaviour, with the additional stipulation that only a finite number of site can be filled. In the second method (essentially due to Dubinin and Stoeckli) one postulates a special law for adsorption in micropores and stipulates a scaling law for adsorption in pores with different size T h e B E T equation with finite n. The adsorption in a restricted geometry can be described in terms of BET model by expicitly assuming that a finite number of layers can be formed at the surface. This assumption leads to eq. (15). By construction, On(x) remains finite even for x = 1, where attains a maximum coverage 0M given by OM=n+l 2
Cn
1 +Cn"
(119)
Of course, a real porous surface is expected to be characterized by pores with a given distribution w(n) of n, so that this scheme has the possibility to describe the complexity of real system. A method for the determination of w(n) is discussed in ref. [52]. This method is however unpractical, since requires the knowledge close to x = I of experimental isotherms at different temperatures. In most cases micropores are studied via the semiempirical Dubinin-Stoeckli method. T h e D R equation and Stoeckli extension. The description commonly employed for adsorption on structurally heterogeneous solids (i.e., solids possessing pores of different sizes) has been developed essentially by Dubinin and Stoeckli [99, 100] and recently extended by Jaroniec and Madey [101,102]. This description is semiphenomenological and
35 essentially based on the experimental obvervation that adsorption on homogeneous microporous solids (mainly activated carbons) may be described by the DR equation [103,104], usually written in the form (120)
O(e, B) = exp[-B(e/eo)2],
where: the Polanyi potential e, defined by eq. (17), is used as independent variable in place of the equilibrium pressure p; e0 is a similarity coefficient depending on the chemical nature of the adsorbate; and B is a structural parameter that depends on the adsorbent properties only. Various experimental studies on carbonaceous adsorbents with cylindrical pores showed that B is related to the micropore size d by the relationship
B = add 2,
(121)
where O~D is a proportionality constant [105]. Representing the structural heterogeneity with the distribution function F(B) of the parameter B, the characteristic adsorption curve 0t(e) for a structurally heterogeneous solid may be written Or(el =
= exp
-B
~
F(B)dB,
(122)
with the normalization condition .fsM F(B)dB = 1, where Bm and BM are the minimum and maximum values of B, respectively. Although eq. (122) has the same form as the fundamental equation (35) for adsorption on heterogeneous surfaces and poses essentially the same problems faced in sections from 3.1 to 3.9, only the direct problem is usually considered for it. A gGussian [99] or, better, a gamma type [101, 102] distribution for F(B) leads in fact to an adsorption isotherm 0t(e) which fits satisfactorily almost all experimental data; the use of free parameters in the distribution function and their optimization enables a quantitative characterization of F(B). Because of (121), the distribution function J(d) for the micropore size d is eventually obtained
J(d) = 2aDd F(B(d))
(123)
This treatment does not explicitely consider the energy heterogeneity, which is implicitly contained in the kernel (120) of eq. (122). Jaroniec and Madey have developed a simple model for evaluating the adsorption energy distribution function X(e) for porous solids with a given structural parameter distribution function F(B) [101].
4.2.2. Adsorption in mesopores Multilayer-type isotherms are usually exhibited by adsorbates below their critical temperature. When the pore size is higher than 20/~ the adsorption at high relative pressure produces therefore the formation of films which can, with a good deal of approximation, be regarded as liquid. Liquid layers in a concave region can manifest a lot phenomena usually referred to as capillary condensation. As these phenomena involve the formation of menisci, they are usually analyzed in terms Laplace equation (describing the mechanical stability of the vapour-fluid interface) or Kelvin equation (thermodynamic stability).
36 Adsorption in mesopores is always associated with the existence of hysteresis phenomena in adsorption-desorption cycles. Hysteresis can be explained in terms of capillary condensation. Hysteresis is a complex phenomenon and several explanations have been proposed. For instance, it can be understood by assuming that (a) during adsorption the vapour is held at the surface with adsorptive filling until capillary condensation occurs; (b) once this has occurred the liquid-vapour interface area is drastically reduced so that the adsorption branch of the isotherm has a plateau; and (c) in the desorption branch vapour is gradually desorbed from filled pores by preserving a bulk filling [1]. For any assumed geometrical model of pores and for any mechanism supposedly responsible for hysteresis, one can calculate from the isotherms describing a complete adsorption-desorption cycle a pore size distribution. This distribution is however strongly model-dependent and should be substantiated by more structural analysis, like x-ray diffraction, electron microscopy, etc. Porous adsorbents either are queer systems or have fractal surfaces.
4.3. Queer s y s t e m s The area A of bodies with regular shape (such as the cube, sphere, regular polyhedra, etc.) increases with volume as V 2/3. For bodies with a regular shape the effect of the surface on their intensive thermodynamic properties (such as specific heat, magnetic susceptibility, etc.) disappears in the thermodynamic limit (infinite volume, constant density). A body whose area increases with V faster than V 2/3 is said to have a queer shape. The area of a queer system is a well defined quantity; its effect on intensive thermodynamic properties, however, can persist even for V ~ ~ . The effect surely persists in the thermodynamic limit for bodies with A cx V. Though it may appear that the concept of queer system is an extravagant concept, Nature however abounds in bodies with queer shape (the first systematic analysis for queer systems was given in ref. [106]). Among the most interesting queer systems we mention: zeolites (in which queerness is due to a lattice of void cages connected by tubes, regularly arranged in the system), biological systems (because reproduction is a way which allows the overall area to increase in proportion to the volume), and films obtained by low temperature physical vapour deposition (where the condensed film grows in such a way that the with an exposed sites increase in proportion to the average thickness [107]). 4.4. Fractal surfaces When the geometric irregularities in the surface have spatial extensions which are comparable with the size of the adsobed molecules, new phenomena are expected to occur. In particular, several surfaces observed at a length scale 3 - 15 ~ give evidence for fractality. For the observation of geometric irregularities at a certain length scale, one needs probes with the same length. Molecules with diameter betwen 3 and 15 ~ have areain the range 7-180/~2. Assuming a local density of site given by Afs,~at (Afs,~at ~ 1015 cm -2 = 0.1/~-2), according to its size a molecule can cover from just one site to approximately 18 sites. In this case an adsorption measurement can provide the monolayer coverage A/'m (number of adsorbed molecules per unit area at 0 = 1) rather than the number of sites Afs. The surface area of a solid can be determined by choosing a probe gas, with effective cross-section b~ (known, for instance, from the molar volume of liquid adsorbate), de-
37 termining with adsorption techniques the number N ~ of molecules forming a monolayer (e.g., from a BET plot), and taking the product A ~ = A/'~ The area determined by the BET plot of the N2 adsorption isotherm in the BET confidence range has become a widely accepted standard. This procedure would not produce ambiguities if such a product for a different gas, A = A/'mb2, were independent of the gas itself. Actually, the relationship
is found to be satisfied only for certain adsorbents [108]. Adsorbents which are not described by eq. (124) are usually found to obey the equation
Arm = Nm(b /b )
(125)
with 2 < D < 3 [108]. Surfaces whose surface areas vary with the size of the probe as in eq. (125) are said fractal, and D is their fractal dimension [109]. For fractal surfaces the concept of surface area loses its original absolute meaning and becomes relative to the probe through which the area is determined. Defining the characteristic length ~ of the probe as ~0= v~2, the surface area varies with ~0as A cx ~2-D,
(126)
which diverges for D > 2. From a mathematical point-of-view, a fractal set exhibits the property that the 'whole' can be represented as the collection of several parts, each one obtainable from the 'whole' by a contracting similitude [110]. A typical fractal object is self-similar, i.e. a magnified portion of it appears identical to the entire object observed under lower resolution: from this point of view it is said to be invariant under scale trasformation. A fractal object can usually be defined through an iteration process in which an initiator is contracted with a similarity ratio 1/~ and put v(~) times in a given arrangement called generator, the same operation being then repeated without end. The area of a fractal set varies with the probe yardstick ~ as in (126), the fractal dimension D of the set being given by D = - / slim I ln t,(~)/ln(1/~) i-
(127)
The identification of an irregular surface with a fractal set poses strict constraints on the surface characteristics: it implies the recurrence of the same irregularity details when the surface is magnified successively. Although this seems to be a very limitative condition, evidence for the fractal behaviour of a few adsorbing surfaces has been provided [108, 111, 112, 113]. The fractal nature of an adsorbing surface has important consequences on its adsorption behaviour [114, 115, 116, 117, 118, 119]. Indeed, any adsorbed molecule has a number of available adsorption sites between that for two and three dimensions, and this fact affects their statistico-mechanical behaviour. Such an effect is even more pronounced when lateral interactions between adsorbed molecules are taken into account. Several attempts have recently been performed in order to understand the influence of fractal geometry of the adsorbent on the adsorption isotherm. We can distinguish two general approaches: a statistical mechanical one which is based on the general relations (3) and (4) with the fractal nature of the surface accounted for in the partition function Z~r
38 through different geometrical considerations [114]; and a kinetic approach which considers how the fractal nature of the surface affects the rates of adsorption and desorption processes [116]. An ultimate understanding of adsorption on fractal surfaces has not been reached yet and such studies constitute an active area of research. In particular, tittle is known as regards the adsorption on simultaneous geometrically (fractal) and energetically heterogeneous surfaces. A fractal surface is a typical non-equilibrium configuration, so that the occurrence of fractal surfaces may seem surprising. Fractal surfaces are typically obtained under strong non-equilibrium conditions, such as those typical of electrochemistry (anodic oxidation leading to dendritic corrosion) or sol-gel technology [120] (SiO2 gels, especially via acid catalysis, where filamentary, weakly branched, structures are produced [121]). The difference D - 2 can be seen as a generalized driving force toward equilibrium. Since adsorption is a way to restore equilibrium at unsatured surface bonds, adsorption is presumably a way to reduce dimensionality. Surface defractalization by adsorption was actually observed [115]. Particularly intriguing is the situation encountered when a surface displays a fractal behaviour in the size scale characteristic of microporosity, so that each pore could be described in terms of DR isotherm. Reminding that for cylindrical pores the pore size distribution function J(d) is related to the fractal dimension D by J(d) c( d 2-D [109], Avnir and Jaroniec [118, 119] have shown that inserting this distribution function in the integral (122) and taking into account eq. (123) one obtains a global isotherm which, in the limiting case of a very wide range of possible size values, may be approximated by an isotherm with the same functional form as FHH isotherm, i.e.
(i2s) Reminding the definition (17) of the Polanyi potential, the comparison of the Avnir-Jaroniec isotherm, eq. (128), with the FHH isotherm, eq. (18), shows that they have the same functional dependence of 0 on p, 0 oc (ln(psat/p)) -1/~. In the FHH theory, however, the exponent q is determined by the adsorption potential (r = 3 for the 6-12 Lennard-Jones potential), while in the Avnir-Jaroniec model 1/~ = D - 3. The currently known values of ~, in the interval 2.1 - 2.8 [27], interpreted in the realm of Avnir-Jaroniec model give D in the interval 2.5 - 2.7, which is expected for fractal surfaces. The question, whether or not the FHH behavior is due to a fractal nature of the surface combined with the Dubinin-Radushkevich isotherm, rather than to a special behaviour of the adsorption potential, remains unanswered. 4.5. A d s o r p t i o n of s t r u c t u r e l e s s molecules on soft a d s o r b e n t s An adsorbent which is not hard will be referred to as 'soft'. Not only do not hard adsorbents exhaust the class of adsorbents, but also soft adsorbents have practical and conceptual interest. Though adsorbent softness-hardness results from a comparison of the configuration energy with the adsorption energy, the weakness of the adsorption field suggests that soft adsorbents belong to the class of soft matter (e.g., polymers with low reticulation degree, elastomers, Langmuir-Blodgett films, etc.), whose interest has dramatically increased in
39 last years [122] and which plays a fundamental role in biological phenomena. Other examples of soft adsobents are solids with non-directional bonds (like metals) and low melting point. In general, soft matter is characterized by an energy landscape with several minima, separated by low barriers. Because of either thermal excitation or external stimuli, the system can therefore undergo even large structural transformations. Roughness is a non-equilibrium property, and in most cases rough-to-smooth transitions are exothermic. If the energy landscape of the adsorbent is as described above, any rough configuration of the system will tend toward a smoother configuration. An example, showing how the rougness is progressively lost with time, was presented by Endow and Pasternak, who determined the monolayer volume of molybdenum films deposited and kept at 77 K. They showed that the monolayer volume decreased monotonically with time during the first half day, at which time it was reduced by a factor of 2 with repect to the value measured 2 h after sample preparation [123]. Since different adsorbent conformations are characterized by different energy distribution functions or topographic configurations, adsorption will consequently feel these structural changes. Conversely, the adsorption isotherm can be used as a probe to test the softness of any adsorbate. Since an adsorbent is softer the higher is the adsorption energy or the adsorption temperature, it is not surprising that softness is particularly evident for adsorption of highly polar molecules (like H20) at room temperature. 4.6. R e c o n s t r u c t a b l e surfaces Surface reconstruction has ever been well known to occur in chemisorption, in situations where the adsorption energy is of the same order as the binding energy of surface atoms. In chemisorption (and especially in oxidation) the surface reconstruction is expected to affect the process rate. Landsberg's explanation [124] of the Elovich equation (a time-logarithm law which plays in kinetics a role as important as the ones of the the classic empirical isotherms in equilibrium [125, 126]) is an example showing how surface reconstruction can affect kinetics. While surface reconstruction of hard adsorbents can occur only after chemisorption, an appreciable surface reconstruction may occur during physisorption on soft adsorbents. Large substrate modifications are indeed known to occur during the uptake of small polar molecules like H~O or CH3OH by biological macromolecules like proteins or cellulose. Adsorption in polymers, originally described either as adsorption on rigid porous surfaces or as absorption in adsorbent bulk with the formation of a solid solution, was based on a picture of hard adsorbent. Limited attempts to overcome these limits were based on a description of the polymer as a flexible linear chain [127], though interchain bridges could be taken into account [128]. These methods were however based on a specific model of adsorbent, rather than on the details of the adsorption process. A model for surface reconstruction resulting after adsorption has recently been proposed by Cerofolini [129] along the line indicated by Landsberg in kinetics. For adsorption on unreconstructable surfaces one has
d N / d p = A/'~dO/dp,
(129)
40 with N's = constant. Even though this constraint is no longer valid for adsorption on reconstructable surfaces, one can reasonably admit that eq. (129) continues to hold, provided that one considers the actual number of sites N's resulting from surface reconstruction. The following assumption is a way to consider a self-similar reconstruction: dp
=
77
A/'~
(130)
=
(131)
where 77 is the available area destroyed (7/> 0) or generated (7? < 0) after adsorption of one molecule. Equation (131) can be solved by separation of variables 1
N'~(p) = N'~(0) 1 +
rlAf~(O)O(p),
(132)
where Afs(0) is the number of exposed sites at p = 0. Inserting eq. (132) into eq. (129) one has the following differential equation dO dAf = Aft(0) 1 + r/Aft(0)0"
(133)
For r / = 0 eq. (133) gives Af(p) = Af~(0)0(p), so that the standard theory of equilibrium adsorption on unrecontructable surfaces is reproduced. Otherwise, i.e. for r/ 7~ 0, the solution of eq. (129) is 1 Af(p) = ~ In (1 + r/Af~(0)0(p)).
(134)
Equation (134) shows that, irrespective of 0(p), adsorption on reconstructable surfaces occurs in the low coverage limit with the same law as on unreconstructable surfaces, 0 << 1/[~IX.(0 ) ~ Zr(p)_~
~(O)O(p).
(135)
At higher coverage, the adsorption isotherm on the reconstructable surface is immediately calculated provided that the isotherm O(p) is otherwise known. For instance, if O(p) is the Langmuir isotherm, adsorption on the reconstructable surface is described by the isotherm
1
N(p)=~ln
(l+p/p'~) 1+p/p,
(136)
and p'~ := p~/[1 + bN~(0)]. Formally speaking, this isotherm is where p~ = PL exp(-r just the Temkin equation [3, sect. 5.3]. However, the appearance of the Temkin behaviour (i.e., Af increasing with In p) requires !
<< p', >> p~
(i.e.,
>> l) for > 0, 1 + r/A/'~(0) = o(1)) for 77 < 0.
(137)
41 4.7. Space-filling surfaces The above model applies to surface shrinkage (77 > 0) as well as to surface magnification (r/< 0). While no problem are met for 77 > 0, for r/< 0 a divergence of A/'(p) may occur for Af~(0)r/ < - 1 . When this condition is satisfied, as soon as O(p) approaches a value 0. = -1/Af~(0)b < 1, A/'~(p) goes to infinite. At a first glance, one is tempted to reject this conclusion and to advocate the existence of physical factors (like the amount of available matter) which limit the otherwise unlimited increase of Af~ with p. However, an infinite surface area is not absurd, provided that space-filling surfaces are allowed. Curves filling a space are quite common: linear polymers are good examples of space-filling curves. Any globular protein "is essentially a one dimensional system folded into a three dimensional structure" [130]. Less familiar are space-filling surfaces: an example might be given by dendromers (highly branched polymers like the 94-met Cl134Hl146 and 127-met C139sH1278 described in ref. [131]); this work suggests that space-filling surfaces can also be produced by physisorption on highly reconstructable (hence soft) adsorbents. Of course, a space-filling surface is a mathematical, rather than physical, concept because of the limitations imposed by the atomistic structure of matter. These limitations hold true in other situations too, like for fractal surfaces, whose scale invariance is usually valid in a quite restricted scale range. In view of the atomistic nature of matter the maximum allowed value of Afs is given by the condition (1 + u)A/', ~ ~bL,
(138)
where u is the number of atoms in each adsorbed molecule, ~b is a typical atomic density in bulk condensed matter (~b ,~ 1023 cm-3), and L is the thickness of the adsorption region. On another side, the complete exposure of the adsorbent, with atomic density to the gas atmosphere implies A/'s = ~L. Combining this equation with (138), one gets the density conditions for space-filling surfaces: ~ = ~b/(1 + u). Typically u = 1 - 3, so that space-filling surfaces cannot be obtained under extreme dispersion. Picturing the space-filling surface as a porous adsorbent, it should be microporous. It is worthwhile noting that, as follows from the analysis of carbon blacks [108], even fractality tends to appear only for relatively little dispersed adsorbents. 4.8. Allosteric surfaces An allosteric molecule E is characterized by n equivalent sites, which modify their structure (binding energy) when the sites are progressively filled. Allosteric transitions play an important role in biology: the transport of oxygen by haemoglobin is a classic and important example [132]. The theory of allostericity has been the matter of one of the seminal works in molecular biology [133]. The theory of allostericity has been developed for molecules formed by a small number of large subunits each containing an active site [133]. Usually the conformation of each subunit is ignored except for the change it brings to the adsorption energy on the active site. This approach has been found extremely interesting for the description of phenomena like 02 adsorption on hameoglobin, etc. Allostericity plays also a major role in phenomena like the adsorption of polar molecules on proteins, or protein denaturation in suitable solvents. The description in this case is complicated by the wide variety of adsorption sites and of protein configurations resulting after adsorption. Allostericity and heterogeneity combine together in a truly intriguing case in the adsorption of polar molecules on proteins [134]. In this case, the adsorption energy is comparable to the energy stabilizing the secondary structure of the proetein, so that a kind of surface reconstrunction takes place after the adsorption; accordingly, the adsorbent and
42 adsorbate cannot be considered as slightly coupled entities, but rather as a new molecular system. The adsorbing properties of the protein are due to its exposed polar sites: the back-bone groups of the peptide bonds, )CO and )NH, and the side-chain groups,-COOH, -NH2, -OH, - S H and - S - S - , in neutral or ionized forms. The complete distribution of the adsorbing sites is determined both by the primary structure of the protein, which specifies the site number and types, and from the secondary and tertiary structure, which specify the relative spatial arrangement of the sites. The heterogeneity of the protein surface is due both to the different chemical nature of the various explored polar sites and to the splitting of each adsorbing energy produced by the particular position of the site in the protein arrangement. In this context the protein surface could be classified as a random heterogeneous surface. However, the protein presents an additional particular adsorbing behaviour. The folding of protein chains, described by the secondary and tertiary structure, can mask several sites to the adsorbate. This masking is due both to steric constraints to the motion of adsorbate molecules and to interactions between polar sites: more polar sites have a higher probability to be unexposed. When small polar molecules like water are adsorbed on the dried protein, the long range interaction between polar sites is reduced by protein swelling and chain folding is consequently reduced. The result of such an unfolding is an increase in the number of the exposed sites. As the adsorption process continues, the straightening of protein chains increases: this straightening renders available new sites which, in turn, adsorb more water, thereby reducing further protein folding. The whole process is concerned with heterogeneity in the first stages and with allostericity later: both concur to make its description a complicated problem. 4.9. Mobile surfaces An extreme case of softness is represented by mobile adsorbents. A physically interesting situation of mobile adsorbent is represented by an ordered Langmuir-Blodgett film adsorbing a multisite molecule. The description of adsorption of structured molecules on solid surfaces requires complicated statistico-mechanical analysis and will be the matter of the next section. In the case of mobile adsorbents, however, the comparison of the following energies: 9 Ea, energy required to deform the adsorbate to a configuration which allows it to be accomodated on the adsorbent in a given configuration (Ea > 0); 9 E~os, energy gained in forming adsorptive bonds between all site of the adsorbate in its original configuration and the site of the deformed surface (E~os < 0); 9 E~0, energy gained in forming adsorptive bonds between all site of the adsorbate in its modified configuration and the site of the unperturbed surface (E~.,0 < 0); 9 E~ energy required to deform the surface to a configuration which allows the adsorbate in its original configuration to be accomodated on the adsorbent (E~ > 0); the following inequality chain is easily established: E~ > ]E~01 _'2 [E~0~I > E~. This comparison suggests that the configuration resulting after adsorption is an almost unchanged adsorbate, whose polar sites are bonded to molecules of the substrate; the enthalpy change resulting after adsorption is expected to be around E~o. + E~(< 0). Though the described situation may appear quite extravagant, a lot of cases of large biological relevance runs in it. Reminding that the surface of eukaryotic cells is essentially constituted by a phospholipidic Langmuir-Blodgett vescicle, phenomena like antigen attack or pinocytosis probably run in the considered situation. The adsorption of proteins on Langmuir-Blodgett films is a case of large practical importance and extensively considered in the literature (see, for instance, the reviews [135, 136]).
43 The description of adsorption of a hard adsorbate on a soft adsorbent is dramatically simplified if one assumes that the adsorbent is undeformable and the surface sites are mobile. In a statistico-mechanical approach, the hypothesis of site mobility allows the partition function to be specified. This produces a description of adsorption in terms of Langmuir isotherm with modified PL [137]. The statistico-mechanical description ignores however the complicated temporal pattern characterizing the kinetics of adsorption and desorption, probably associated with conformational changes of the adsorbate [135, 136]. 4.10. Multi-Site O c c u p a n c y A d s o r p t i o n on Rigid H e t e r o g e n e o u s Surfaces In many cases, the adsorbed molecule can be considered as consisting of a number of quite distinct chemical fragments (mers). The adsorption characteristics show then, to a good approximation, additivity of various features. In other words, one may assume, that the adsorption is affected in a more or less independent way, by the adsorption of these chemical fragments. It is, therefore, much more realistic to consider the whole adsorption process as an adsorption of interconnected fragments of the minima of the fragment (mer)-surface interaction potential. Because the various mers are connected in some way by chemical bonds, a simple analogy emerges to collective adsorption of molecules interacting via Van der Waals forces, for instance. So, there is no surprise, that even in the absence of interactions between the molecules composed of distinct mers, their adsorption will be influenced by surface topography, like in the case of collective adsorption of simple molecules. Until the surface has a patchwise topography, the problem is relatively simple. One can apply directly one of the already existing approaches to multi-site adsorption on homogeneous surfaces, which are now the patches. In the case of random topography, the problem becomes much more complicated, and it was only a decade ago, when a first solution of that problem was proposed by Nitta[138]. However, Nitta's original approach could be applied only to surfaces characterized by discrete distributions of adsorption energy. Some few years ago, Rudzinski and Everett have developed further Nitta's approach to apply also continuous adsorption energy distributions. Let )r be a continuous distribution of the possible occupancies of n sites on the surface by an adsorbed molecule, among the values of its total adsorption energy cn corresponding to a particular occupancy of sites. For a simple molecule occupying one adsorption site the term "surface" occupancy means occupancy a certain surface esite. To calculate the overall (total) adsorption isotherm St, one canthen use anyone of the methods discussed in the previous sections, based on employing the "condensation function", co. Rudzinski and Everett[3], have developed the form of the condensation function for an adsorbed molecule occupying n sites, cc~. For surfaces for patchwise topography, con takes the following form,
e~ = - k T l n
( nl+n/2(nl/2 +
1)n_l g'p
)
(139)
whereas for surfaces with random topography
e~ = - k T l n ( n g ' p ) - ( n - 1)kTln ~t
(140)
where K ~ is a temperature dependent Langmuir-like constant. Marczewski et. a1.[139] published a paper which is of a crucial importance for the application of Nitta's approach. Namely, they have shown how the number of mers in an adsorbed molecule affects the multi-site-adsorption energy distribution X~(e~). The multi-site-occupancy adsorption on heterogeneous solid surfaces is becoming rapidly a hot topic in the recent studies of adsorption at gas/solid interface.
44 5. M I X E D - G A S A D S O R P T I O N E Q U I L I B R I A : T H E C U R R E N T STATE OF THE RESEARCH In the fundamental studies of mixed-gas adsorption the knowledge of the mechanism of adsorption and a possibly best agreement between experiment and theory have the highest priority. The experimental problems and computational time are of a secondary importance. In the numerical programs developed for engineering purposes the priorities are different. At first the computation of mixed-gas adsorption isotherm must be fast, as it is only a part of a large computer program. Then the data necessary to predict mixed-gas adsorption should not require time-consuming experiments on single and mixed-gas adsorption. In the early stage of theories of gas adsorption on solids, theoretical considerations were based on the idealized models of a solid surface. This sometimes led to dramatic discrepancies between theory and experiment. Nowadays the role of the energetic heterogeneity of the actual solid surfaces is almost commonly taken into account by the authors proposing various theoretical approaches to the mixed-gas adsorption. They may be classified into three groups: 1. the molecular approachesemploying methods of statistical thermodynamics[140-210]; 2. the thermodynamic approaches based on the methods of phenomenological thermodynamics [211-260]. 3. the computer simulations of mixed-gas adsorption[304-318] The molecular approaches attracted many scientists working on fundamental problems of mixed-gas adsorption, but they were rarely used by the scientists working on gas-separation processes. The complexity of the accompanying theoretical considerations based on the methods of statistical thermodynamics seems to be the main reason for that. The second group of approaches was also very popular in the fundamental studies of mixed-gas adsorption, but was rarely applied in the theoretical description of gas separation by adsorption. The main reasons for that were the time-consuming computations necessary to predict mixed-gas adsorption from the pure gas isotherms. The exception was the so-called Potential Theory (PT) approach which required very simple computations and knowledge of a small number of data for the adsorption of single components. Recently, a new group of works based on computer simulations of mixed-gas adsorption has emerged rapidly. There are obviously two main factors affecting predominantly the mixed-gas adsorption on solids: 1. the gas-solid interactions, 2. the interactions between adsorbed molecules. As in the case of single-gas adsorption equilibria, we will focus our attention on how the dispersion of the gas-solid interactions across a solid surface, called surface energetic heterogeneity, affects mixed-gas adsorption on solids. We do it with the purpose in mind to present in a possibly simple and clear way the theoretical approaches which were used, or could be used to describe mixed-gas adsorption equilibria. The role of the interactions between the.adsorbed molecules in the mixed-gas adsorption was considered in the earlier
45 review by Rudzinski et. a1.[262] Fundamentals of Mixed-Gas Adsorption on Heterogeneous Solid Surfaces. At the same time the analysis of the energetic surface heterogeneity was limited there only to the Gaussian-like symmetrical adsorption energy dispersion, which in the case of single-gas adsorption leads to the Langmuir-Freundlich isotherm. Although we give here a pretty exhaustive list of references of the papers dealing with the theories of mixed-gas adsorption equilibria, we are not going to present a typical balanced review. We will focus our attention on the two approaches that have most commonly been used in the hithero theoretical studies of mixed-gas adsorption. One of them is the Integral Equation approach introduced into literature by Hoory and Prausnitz[143] at the beginning of the seventies. That approach was later developed by the adsorption group in the Department of Theoretical Chemistry, Maria Curie--Sklodowska University in Lublin. The other fundamental approach is the Ideal Adsorbed Solution approach, also introduced into literature at the beginning of the seventies by Myers and Prausnitz[211]. That approach was used and developed further in numerous papers on mixed-gas adsorption. In the next chapter of this book, Tovbin will review the achievements of the Russian school of adsorption, which are less known as most of them were published only in Russian, and in Russian journals. The newly emerging computer simulations of mixed-gas adsorption will be briefly reviewed at the end.
6. T H E I N T E G R A L E Q U A T I O N A P P R O A C H 6.1. T h e o r e t i c a l principles The first approach ever used to predict mixed-gas isotherms from the single component data is known as the integral-equation (IE) approach and is based on the integral representation for Oti [143,262]:
0ti(p,T ) -- [
1
..-~_ J~n
Oi(E,p,T)X(n)(lf.)d6l...ds
(141)
where: 8ti(p, T) is the total surface coverage by the component i at the set of the partial pressures p -- {p~,p2, ...,p,}, 0~(e, p, T) is the fractional coverage by the component i (i --- 1, 2, ...,n) of a certain class of adsorption sites, characterized by a set of the adsorpt ion energies e = { o , ~ 2 , . . . , ~,} for the single components; X(,)(e) is the n-dimensional normalized differential distribution of the number of the adsorption sites among various sets e,
9" ~ 1
X(n)(e)dQ"'den--1,
(142)
n
and f~i is the n-dimensional physical domain of ei. For the adsorption isotherms of single components, we have /gti(p, T) = f Oi(ei, p, T)xi(ei)dei, f2i
(143)
46 where
Xi(s163163163
(144)
The integral (143) can be easily evaluated using the Rudzifiski-Jagietto method, oo
Ot(p,T) = - E
(kT)l l~ Cl
1=0
where:
x(~) = f
(146)
x(e)de,
Ct-s are the temperature-dependent coefficients given by,
[
(~M-er Cz - J(~-~r
tte t (1 + e')2dt'
the function er
T) is found from the condition,
(147)
(0~0/0e~),_,~ = 0,
(148)
and 0 is the local isotherm O(e,p,T) under the integral in eq. (143) When T ~ 0, all the terms under the sum in eq. (145) vanish, except the leading one. It is also true when the variance of X(e) is much larger than that of the derivative O0/Oe. Then, Ot,(p,T) = -X,(er
(149)
The features of the adsorption model are coded in the function er is the Langmuir equation,
T). When 0(e, p, T) (150)
gpexp(e/kT) O(e,p,T) = 1 + K p e x p ( e / k T ) '
condition (148)is fulfilled when 0(e = er
x Then 7"
(151)
ec(p,T) = - k T l n K p .
If Xi(ei) is the following bell-shaped adsorption-energy distribution 1 exp x~(~,) =
c,
(152)
~,
l+exp(e/-e~
2'
then
-Xi(eei)= [l+ exp ( eci-
-I
(153)
47 so we arrive at the Langmuir-Freundlich (LF) isotherm for the single-component adsorption isotherm 0ti,
Oti(pi, T)=
[Kipiexp(e~ 1 + [Kipi exp(eoi/kT)] kT/c'
(154)
The experimentally measured adsorbed amount, ~ti(P, T), is equal to A/'~Oi(p,T), where N's~ is the number of the adsorption sites on a solid surface, expressed in the same units aS J~fti(P, T). A convenient way to analyze an experimental adsorption isotherm in terms of eq. (154) is to use the following linear regression
~,IN~,
_
In 1 - Xti/Afsi -
kT l n K + c--~-
eoi ) k-T
+ -kT -lnp. c,
(155)
The only adjustable parameter is the monolayer capacity Af~i, which is chosen in such a way as to make the 1.h.s. of eq. (155) a possibly best linear function of In p. The heterogeneity parameter kT/ci is then the tangent of that linear plot, and K exp (eo~/kT) is found from the intercept multiplied by c~/kT. The actual adsorption energy distributions are much more complicated functions, so Xi(e~) in eq. (152) is a 'smoothed' form of an actual energy distribution, the third central moment of which is equal to zero. In many cases, the actual energy distribution may be better represented by the following non-symmetrical function
X,(e,) = r , ( c - e~,) r'-I E~'
exp
-
e, - e~i
Ei
,
for e, > e~,
(156)
This is a right-hand-widened Gaussian-like function when r < 3, a pretty symmetrical function for r = 3, and a left-hand-widened one for r > 3. The parameter Ei is the variance of that function. Then --X(er
(--(er Eem) ),
(157)
or
Ot(p, T) -- exp (-[~-~--T In ( p ~ ) ] r )
,
( 58)
where ln pm = - l n K em/kT. Equation (158) is the well-known Dubinin-Astakhov (DA) equation. While analyzing an experimental isotherm in terms of this equation, it is convenient to make the following linear regression In Aft = lnAfs - (~--~T) r [ln ( ~ )
]~ 9
(159)
The adjustable parameters are pln and r which are now chosen so as to make lnAft a possibly best linear function of [ln(pm/p)] r. It is a frequent practice to choose p~ to be saturated vapour pressure, but it is not correct. The DA isotherm was first used to correlate the experimental isotherms of adsorption in porous carbons. Such a choice of Pm reflects the classical view on adsorption in porous materials as filling micropores by a bulk liquid-like adsorbate phase. In this picture of adsorption, e = kTln(p~/p) is the
48 value of the adsorption potential that causes the liquefaction of adsorbate molecules in an empty pore at the pressure p. The present molecular simulations show that the condensation pressure depends on the pore dimension, and may be several orders of magnitude lower than Pm- When r = 2, eq. (159) becomes the Dubinin-Radushkevich (DR) isotherm equation. In many theoretical works the following rectangular (constant) function is accepted to represent X(e): I
for
eM--em
x(e) =
0
em < e < --
eM,
(160)
elsewhere.
The corresponding isotherm equation Ot(p, T) takes then the form of Temkin's isotherm:
kTlnK
8t(p,T) = ~ s
kT
+
~
(161)
~lnp.
s
~M ~
s
The linear relationships between Aft(p, T) and In p were reported in literature for strongly heterogeneous solid surfaces. The theoretical background for that is following: On every heterogeneous solid surface there will exist minimum and maximum values of adsorption energy Cm and eM- The assumption that r varies from a minimum value s to plus infinity, like in eq. (156), is due to a mathematical convenience. The consequence of making this assumption is that the corresponding isotherm equations do not reduce to the Henry's isotherm when p ~ 0. Meanwhile, for some fundamental thermodynamic reasons such reduction must take place in every adsorption system, provided that the adsorbent structure is fully rigid, i.e., it is not affected by the presence of adsorbate. The latter assumption seems to be a good approximation for the majority of the adsorption systems. The assumption that e varies within infinite energy limits has no large impact on the behaviour of the calculated adsorption isotherm, except for the regions of small (St ~ 0) or high (8t --~ 1) surface coverages. For most of the industrial applications of mixed-gas adsorption, the mediate surface coverages are of interest. While using the IE approach to predict the mixed-gas adsorption, it is sufficient to know the behaviour of the single adsorption isotherms at mediate surface coverages. Therefore the problem of infinite energy limits, accepted for mathematical convenience, is not essential for that theoretical approach. On the contrary, it is essential when one uses the ideal adsorbed solution (IAS) approach to predict mixed-gas adsorption from pure component isotherms. Therefore we shift the discussion of that problem to the section on the application of IAS for predicting mixed-gas adsorption equilibria. Further, it is known that the rectangular function (160) applies only to strongly heterogeneous surfaces. For that purpose we consider that X(r is represented by the function (152) defined in the interval (era, eM), so, it has to be written as follows, 1
c
c "
where FN is a normalization factor, FN=
[
l+exp
(
era. --co c
[
-- l + e x p
. (2
(163)
49 For the purpose of illustration, we assume that s = 0 , G0 = 5kT, s = 10kT, and make the heterogeneity parameter c to accept higher and higher values, corresponding to stronger and stronger surface heterogeneity. The result of that model investigation is shown in Fig. 2 of the chapter 6 by Rudzinski. One can see in Fig. 2 that as the heterogeneity parameter c increases, X(e) defined in eq. (162) becomes more and more similar to a rectangular adsorption energy distribution. One fundamental problem of mixed-gas adsorption occurs, when the monolayer capacities estimated from single-gas adsorption isotherms are different for different components. One faces that problem even in the case of mixed-gas adsorption on a hypothetical homogeneous solid surface. In such a case even the generalization of Langmuir equation for mixed-gas adsorption is difficult. So far it has been done only for the case when Vj(j -# i): .N'sj = A/'~,[140]:
Ki pi exp Oi(e,p,T) =
-ff-f ,
n
(164)
1 + ZK p exp j=l
Provided that N'.~ and A/'sj(j # i) are not much different, the next problem is to define the multidimensional adsorption energy distribution X(,0(e), which would reduce to X~(e~) after n - 1 integration steps, as outlined in eq. (144). This, however, is not a trivial problem at all. Then, the multidimensional integrals would have, for sure, to beevaluated numerically. Although such computer calculations could be carried out, there is still a strong demand for relatively simple analytical expressions for 0t(p,T). Firstly, because carrying out necessary computer calculations would not be convenient in many cases. Secondly, these might be time-consuming too much for certain purposes. Finally, having analytical expressions for 0t(p,T) is convenient for carrying out further mathematical operations leading to other thermodynamic quantities of interest. The success of using the 'integral equation' (IE) approach depends much on making proper assumptions about the nature of an adsorption system under investigation. A general strategy is to reduce the nD integral (141) to a 1D integral by using various physical arguments. Most commonly, it is done by considering correlations between the adsorption energies e~ and ej(j # i), with i,j = 1, 2 , . . . , n. Two physical situations have been considered so far: 1. The adsorption energies ei and ej are not correlated at all for j -# i; 2. Functional relationships exist:
ei = fij(ej),
i r j = 1,2,.-.,n.
(165)
6.2. The case w h e n adsorption energies are strongly correlated, and the nature of the gas-solid interactions is similar for various components In such a case one can expect that the local minima of the gas-solid potential function for one of the components, will simultaneously be the local minima for other components. Then, until the sizes of the adsorbed molecules are not much different,we m a y assume that all the components are adsorbed on the same lattice of energetically different adsorption
50 sites. Next, it is to be expected that the higher will be the adsorption energy for a certain component on one of the adsorption sites, the higher will also be its adsorption energy for other components. It means, the function c~ = f~j(ej) in eq. (165) will be one-to-one function. Now, let us consider the extreme case of high correlations existing between the adsorption energies of various components, represented by the condition on every adsorption site:
Aji, j,i = 1 , 2 , . . . , n ,
ej = ci +
(166)
where Aji-s are the constants. Then, 0i(c~, p, T) can be rewritten to the following form:
Kipi
Oi(p, ei, T ) =
exp (~--~)
E Kjpj exp kT ] J
./
,
(167)
j
and the averaged function
0ti(P, T)
is given by the 1D integral
E Kjpjexp
9 1 + E Kjpj exp kT]exp ( ~ T ) J J The above integral can also be evaluated by using the RJ approach. The function ec is defined now as follows: ~r
T) =
-kT In
Kjpj exp kT ]
"
Equation (168) is again a kind of a master equation from which various isotherm equations can be developed corresponding to various adsorption energy distributions X(ei). For the Gaussian-like function (152) leading to the LF isotherm for single gas adsorption, 0ti takes the following form
( kT eo, ~ Kip~exp ~, ]
0ti(p, T) =
Kjpj exp k kT] J (170)
because (e0i + Aji) is the most probable value of ej. (The numerator and denominator before the integral in eq. (168) have been multiplied by exp (eoi/kT).) When the adsorption isotherms of single components are correlated well by the DA equation (158), the equations for mixed-gas adsorption take the following form ri
0ti(P, T) = ~
Pi/Pmi exp Pi/Pm,
j=l
~-ln
1
~ P.i/Pmj j=l
(171) "
51 Eq. (171) has not been tested in the literature yet, but both eq. (170) and (171) seem to be promising for correlating mixed-gas adsorption data for the components having a similar chemical character. Another obvious condition is that the molecules of the different components should have similar sizes. However, even in the case of such chemically similar molecules, the functional relationship (165) may have a form more complicated than that in eq. (166). It can be deduced from the low-temperature adsorption isotherms of single components. At low temperatures, the adsorption will proceed in a fairly stepwise fashion, and the experimentally measured isotherm Ot~(p,T) will be given by eq. (149). At the same coverage of surface by two components i and j, the following relation will hold:
-x,(~r
= -xj(~r
(172)
Now let us assume that the adsorption isotherms of both components obey the LF behaviour originating from the fully symmetrical, Gaussian-like energy distribution (152). Then, according to eq. (172) we have:
[
l+exp
(
er
)]' [ =
ci
l+exp
(
ecj-eoj cj
))'
.
(173)
From eq. (172) we obtain the following linear relation
cj
~j=--ei+ Ci
(r
cj ) ,
(174)
Ci
where we have already omitted the superscript 'c' in cr When the single-gas adsorption isotherms of all the components are described by the DA isotherm (157), then, we obtain from eqs. (157) and (172) the following interrelation ri/r 3
(175) As the function X(e) is called the 'cumulative adsorption-energy distribution', eq. (172) is called 'seeking for the adsorption energy interrelations through the cumulative distributions'. That idea has been proposed independently by Valenzuela et a/.[243] and by Jaroniec and coworkers[198], in the same year (1988). One difficult fundamental problem in the application of lattice formalism to describe mixed-gas adsorption is when the values of .Ms, (the total number of sites on a solid surface), estimated either from eq. (155) or eq. (159) is different for different components. Moreover, the value of A/'s is temperature dependent as a rule. There might be various reasons for that. One possible reason is that the linear regression (155) or (159) is made only for a certain region of pressures, i.e. surface coverages, corresponding roughly to a certain region of the adsorption energies. The functions (152) or (156) approximate the true function X(e), in that region of energies. Beyond that region, X(e) may not be described by the function (152) or (156), or at least not by the same sets of parameters {e0,, c~} or {em~,Ei, ri}. Thus, such local approximations may not give the true value of The commonly observed temperature dependence of A/'s could mean that some weakly adsorbing sites simply disappear. Of course, the number of the local minima in the
52 gas-solid potential function will not be affected by temperature. It will rather happen that a part of the adsorbed molecules is no longer in the localized states. Going into 2D or 3D mobile states means switching to another isotherm equation predicting weaker adsorption at the same pressure and temperature. If, in spite of that, one uses the same (Langmuir) isotherm, valid for localized adsorption, to correlate the experimental data, then the changing mobility of a part of the adsorbed molecules may simulate decreasing a number of adsorption sites. Assuming that the substantial part of the adsorbed molecules is still in the localized states, and remembering that this is the total number of adsorbed molecules essential for the seeking correlations through the cumulative distributions, eq. (172) should be considered rather in the following form: (176) For the same reason, while calculating the experimentally monitored quantities Nti(p, T) one is likely to use the relation Nt~(p, T) - Afu0u(p, T). It should, however, be remembered that this is a kind of approximate relation. Before we approach the other extreme case when no correlations exist between the adsorption energies of various components, we would like to mention the first paper on mixed-gas localized adsorption on a heterogeneous solid surface, published in 1975 by Jaroniec and Rudzinski[150]. As the local i s o t h e r m - the Langmuir-Markham-Benton[140] equation for mixed-gas adsorption was used, whereas X2(el,e2) for the (ethane + ethylene) coadsorption on an activated carbon was represented by a two-dimensional Gaussian function. That function was introduced in the adsorption literature by Hoory and Prausnitz [143] who used an extension of Hill-de Boer isotherm, to represent the local isotherm $(E,p,T)in eq. (141) along with patchwise model of surface topography. The 2D Gaussian function includes a parameter $ E [0,1], describing the degree of correlations between el and e2. When $ - 0, no correlation exists between s and e2, whereas for ~ - 1, perfect correlation exists. This kind of intermediate correlations will be discussed in more detail in the forthcoming section 8. 6.3. The case when the adsorption energies of various c o m p o n e n t s are not correlated at all This is the case of coadsorption of components exhibiting a much different character of interactions with the same solid surface. These differences are usually due to different chemical nature of the coadsorbed components. There is a certain tendency in literature to apply again eq. (141) to such cases, and to consider X(~)(E) to be the following product: n
X(=)(E) - 1-I X,(e,).
(177)
i=l
This seems to be an improper strategy. Even if the molecules of different components have similar sizes, their different nature of the gas-solid interactions will result in various depths and positions of the local minima in the gas-solid potential function. It means, we
53 have to consider various non-overlapping and even not correlated lattices of adsorption sites for each of the coadsorbing components. The case when ei and ej (with j ~t i) are not correlated at all was considered by Wojciechowski et al.[192]. They used the following argument: When the energies e~ and ej#~ are not correlated at all, there should also be no spatial correlations between the local minima (adsorption sites) of the components i and j. So, when the adsorption energies of components i and j, e~ and ej (with i,j = 1 , 2 . . . ,n,i # j), are not correlated at all, the presence of other components will affect the adsorption of i only by random blocking of adsorption sites, proportional to their total coverages 8tj-s. (The probability that the molecule j will be adsorbed on a certain site does not depend on ei.) Thus, 0,~(p,T) = -
1- ~
0,~(p,r)
X~(e~),
(17a)
where the functions e~i-s are found from the conditions
(028,/0e~)~,=~., = O,
i = 1,2,-.-,n,
(179)
When 8~-s are Langmuir isotherm equations, one arrives at the same definition of er as in eq. (178). er = - k T l n ( g , p , ) . Equation (178) is another kind of a master equation from which various expressions for the mixed-gas isotherm can be derived by assuming various adsorption energy distributions X(e). The equation system (178) is linear with respect to 8ti(p, T), so, it can be solved easily, to express 8t~(p,T) by X'~(er For practical purposes it is sufficient to consider the adsorption from a ternary gaseous mixture (1 + 2 + 3). Let r denote -X'~(er then, the solution takes the following form: - x ~ - x ~ x 2 x 3 + x~x2 + XlX3
t~tl(pl,p2,Pa, T) = 1 - 2X1X2X3 + XIX 2 -~- X I X 3 -4- X 2 X 3 ' - X2 - X1X2 X3 + X~ X2 + X2 X3
0t2(pl, p2, P3, T)
=
1 - 2,u
-[- X 1X2 + ,'~'1r
(~80)
(181)
-[- X 2 X 3 '
- Z 3 - Z~X2X3 + XlX3 + X2X3 8t3(p~,p2,p3,T) = 1 - 2X~X2X3 + X1X2 + X~X3 + X2X3"
(182)
Generally if all -k'i-s are the LF isotherms for single component adsorption, the solution of the equation system (180-182) yields the generalized LF mixed-gas adsorption isotherm
[Kipi exp ( e~ ) ] kT/c' 8t,(p,T) =
,
1--b ~
i = 1,2,-.-,n.
[Kjpjexp (-~T)] kT/~'
j-'l
m decade ago Ruthven wrote[263] (See page 108 of his monograph): "Although not thermodynamically consistent, these expressions, have been shown to provide a reasonably good empirical correlation of binary equilibrium
(183)
54 data for a number of simple gases on molecular sieve adsorbents, and are widely used for design purposes. However, because of the lack of a proper theoretical foundation this approach should be treated with caution". The theoretical origin of that isotherm had remained a mystery until Wojciechowski et al. published its derivation[192]. A convenient way to check the thermodynamic consistency of a system of equations for mixed-gas adsorption is to use the criterion proposed by Keller[264]" 1 (0Afti~
=1
(OAftj)
(lS4)
In the case of the generalized LF isotherm, eq. (183), that criterion is fulfilled when ci = cj even for j :/: i. The theoretical derivations based on the mechanistic models will always contain a number of simplifying assumptions so the lack of the thermodynamic consistency is to be expected here as a rule rather than an exception. The only problem is, how much serious impact on predicting mixed-gas adsorption that thermodynamic inconsistency may have in a particular case. The IAS approach always thermodynamically correct may lead in many cases to a less accurate prediction of mixed-gas adsorption equilibria than a thermodynamically inconsistent system of equations developed by using the IE approach. In view of the wide applicability of DR and DA isotherm equations, it should be interesting to investigate the behaviour of the corresponding equations for mixed-gas adsorption. Thus, we will consider the adsorption from a binary gaseous mixture, when the adsorption of single gases follows the DA isotherm (158). For adsorption from a binary gas mixture, the equation system (180-181) reduces to the following one 0.(p~,p~,T)
=
8t2(px,p2,T)
=
- & + x~& l+&& -&+&& I + X~X2
(lS5)
.
(186)
So, when A'i is given by equation, eq. (157), we arrive at the following explicit expressions:
O.(pl,p~,T)
=
1-exp (-- [~21n (Pm2~]"2)]exp P2 / (- [~~~ln \ Pi / kT (pm2 Pro1~ rl 1 ex (
T1) ,
(lST)
"~2 In x,-~2 /
and
(Pml"~"~) ] exp (-[~-T-T2In ( ~ )
1 - e x p (-[~---~Tlln k - ~ / ] 1 - exp
-
~ l In Pml \ Pl /
_
~
r2)
In Pro2 \ P2 /
This equation is to be compared with eq. (171) which is the extension of the DA isotherm for the case of mixed-gas adsorption, when the adsorption energies of various components are strongly correlated. Equations (171) and (188) should also be compared
55 with the semiempirical extension of the DR equation for mixed-gas adsorption proposed by Bering et. a1.[266,267], at the begining of the seventies. By using the master equations (180-182) or (185-186) one can generate a still larger variety of the theoretical expressions for mixed-gas adsorption, by putting for X/any of the expressions for single gas adsorption. Now, there still remains the problem of relating the theoretically calculated functions 0ti(P, T), to the experimentally monitored quantities Aft~(p, T). Applying the multicomponent LF equation, eq. (183), most of the authors assumed that Aft~ = N'si0ti. This statement is not obvious and deserves further theoretical studies. We summarize our consideration in this section as follows: For a mixture of components the single-gas adsorption isotherms of which are known, one may develop a variety of expressions for the adsorption from their mixture. Thus, there is no universal guideline how to apply the IE approach. For every adsorption system a suitable method is to be chosen on some rational basis. This, however, requires certain experience in the theoretical description of adsorption at all. The attractive side is the compact analytical form of the obtained expressions, which makes computer calculations of mixed-gas adsorption equilibria fast. The rapidly growing industry of gas separations by adsorption creates, however, a large market for such theoretical expressions. Having such expressions, one can also develop easily compact analytical expressions for the heat effects accompanying mixed-gas adsorption, and mixed-gas surface diffusion characteristics. This is because their calculation involves only calculation of derivatives from a certain system of isotherm equations.
7. T H E I D E A L A D S O R B E D S O L U T I O N A P P R O A C H
(IAS)
7.1. Theoretical principles The IAS approach was launched first in the work by Myers and Prausnitz[211] and reexamined recently by Rudisil and Le Van[251]. The IAS theory is based on the assumption that the adsorbed phase can be treated as an ideal solution of the adsorbed components. The reduced spreading pressures H~, p~
II; = ri,A = f ~dp,, RT pi
(189)
0
of the components i = 1, 2 , - . . , n, in their single-gas standard states, are equal to the reduced spreading pressure of the adsorbed mixture, H'" n~ = n~ . . . .
= n: = n'.
(190)
The function A/'~(p,,T) denotes the adsorbed amount, (moles per unit mass of adsorbent), at the pressure pi and temperature T, and A is the specific surface area. A/'i(pi, T) = A/'s~0~(p~,T), where 0~ is the fractional coverage by component i of the surface, and Afs~ is the maximum adsorbed amount when 8ti = 1. It is assumed in the present considerations that Af~ and Afu are expressed in moles (per unit area or unit weight).
56 The relation between the mole fraction in the gaseous phase Yi, and the mole fraction in the adsorbed phase Xi is described by Raoult's law for ideal solutions, (191)
PY~ = p~(II*)X,,
where p~ is the pressure of the single component i in its standard state which is fixed by the spreading pressure of the mixture according to eq. (190). In addition, the mole fraction constraints are ~] Xi = 1 and i----1
Y~ = 1. z'-I
The total amount of the adsorbed gases Af* depends on the amounts A/'i* adsorbed in the standard state:
1 _ ~-, X!, N'*~ z',"
(192)
"__
where A/'~ is the value A;~ for the pure component i, at p~ = p;(II*). The actual amount of each component adsorbed is Afi = A/'*X~. There is one fundamental problem accompanying the application of IAS to predict the mixed-gas adsorption equilibria on heterogeneous solid surfaces. That problem has never been explicitly discussed in the papers reporting on application of IAS, except for the recent paper by Rudzifiski et a/[262]. The term ideal solution refers to the systems of interacting molecules in which the 'interchange energy' W is equal to zero: W = Wi, + Wjj - 2Wj, = O,
i , j = 1,2,.-. ,n,
(193)
were Wij is the interaction energy between two molecules i and j adsorbed on two neighbouring adsorption sites. As discussed in Sect. 4, when the adsorption of interacting molecules (collective adsorption) on a heterogeneous surface is considered, one must take also into account the surface topography. So far two extreme models of surface topography have been commonly considered: The patchwise model assuming that the surface is composed of arge homogeneous surface domains - - 'patches', grouping identical adsorption sites. Thus, the adsorption systems have to be considered as composed of macroscopically different subsystems being only in thermal and material contacts. At a certain adsorbate pressure p and temperature T the adsorbed amount Afi and the corresponding spreading pressure II~ are to be referred to a certain patch and will be a function of the adsorption energy e on this patch. Thus, the correct use of IAS to predict the mixed-gas adsorption equifibria will require the following sequence of operations. First, IAS has to be applied to every homogeneous patch to calculate {X i}. They are, of course, a function of the energy set e. Next, the calculated Xi-s values have to be averaged over all possible sets e as in the IE approach. The random model assuming that adsorption sites characterized by different adsorption energies are distributed on a heterogeneous surface completely at random. As a result, the local 'microscopic' composition {Xi} of the surface phase is the same across the surface. The adsorbed phase is a thermodynamic entity, the spreading pressure of which is the same across the surface, and has to be calculated by replacing Afi by Afti, in eq. (189).
57 One course, one may ask the question what the condition (193) for the mixtures of molecules exhibiting quite different character of interactions with the solid surface means. In the model of localized adsorption they will be adsorbed on different incompatible lattices of adsorption sites. It is clearly stated that when considering interactions between adsorbed molecules, the attice is treated as a kind of an abstract theoretical tool as in the lattice theories of liquid mixtures. 7.2. A p p l i c a t i o n of IAS to surfaces with r a n d o m t o p o g r a p h y Myers who was the first to consider the extension of IAS for the case of heterogeneous solid surfaces did it for the surfaces with random topography first[229]. He accepted the rectangular adsorption energy distribution to represent Xi(ei). Then, the problem was considered more extensively in his second paper coauthored by Richter and Schfitz[248]. Three isotherm equations were considered in their work: Freundlich, Langmuir, and DR. For the first two, the integration in eq. (189) can be performed analytically, whereas in the case of DR equation it must be carried out numerically. Then, the single and mixed isotherms of methane and ethane adsorption on an activated carbon were used for illustration. Of all the equations the DR isotherm fitted the adsorption isotherms of single components best. The DR equation was also superior to predict mixed-gas adsorption equilibria. Since DR does not reduce to Henry's isotherm as p ---, 0, a certain error in calculating the spreading pressure is surely involved. Even then DR appeared, to be superior over the Langmuir isotherm which reduces correctly to Henry's Law. That problem was more extensively analyzed in the paper by Talu and Myers[242]. They took into consideration the following three equations used frequently to correlate the single adsorption data: 1. The Langmuir-Freundlich equation, eq. (154), 2. The Dubinin-Radushkevich equation, eq. (158), in which p~ was taken to be the saturation pressure ps, 3. The T6th equation (21),
p---,0
The limiting slope, lim dAft/dp, is equal to: +c~ for the LF equation, 0 for the DR equation, and a finite value for the T6th equation. Here, the behaviour of the function Af/p is of crucial importance. That study led Talu and Myers to the following conclusions. The limit +c~ for the LF isotherm makes it unsuitable for calculating spreading pressure under any conditions. The limit of the T6th equation is too large because the slope of the Aft/p curve goes to - ~ at the origin, and the limit of the DR equation is too small (zero), because the slope of the A/'t/p curve goes to +or at the origin. However, the error in the integral (189) introduced by the incorrect limits of the T6th and DR equations is in fact small unless the pressure is very low. For the DR equation, the maximum of .N't/p is given by
/J~t/ T
J~Sexp(/ E /2)
max = P--/
.
(194)
58 The error in the spreading pressure/kII*, introduced by this false maximum, is /xm=N~exp
-
2-~
"
and the pressure p ~ x at which p~x = psexp
(195)
3/'t/p attains its maximum value is given by,
-~
(196)
According to Talu and Myers, this error is usually smaller than the experimental error attains its maximum value and below which in II* and the pressure Pm~ at which the calculated ratio deviates significantly from the experiment, is usually very low. In view of that positive judgement concerning the applicability of the DR equation, the readers may appreciate using the analytical approximation for IF(p), developed by Myers and coworkers for the case when Aft(p) is the DR isotherm,
A/'t/p
II* = IIakT= 2Ns(II)1/2 (k__~~In (~)) (kT/E--------~erfc
.
(197)
As the error function erfc(-) is the standard in computers now, eq. (197) may therefore be considered as the analytical expression for II*. The conclusion by Talu and Myers that Pmax is small, is likely to be true in the case of adsorption by activated carbons. But in the case of adsorption in zeolites, for instance, the adsorption isotherms which quite frequently obey the DR or DA equation at mediate coverages, reduce clearly to Henry's law at measurable pressures. The attempts to modify the DR equation toward an expression having Henry's limit have a long history, and were described in the monograph by Rudzifiski and Everett[3]. Most recent attempts of that kind have been published by Sundaram[268], who used a modified version of the DR equation to calculate spreading pressure II*[261]. All these modifications were made on an empirical basis. Sundaram used the following modification of DR isotherm
po
k-T
i=a
i
(198)
where the term within the square bracket in eq. (198) is the truncated series expansion,
-ln0t = ~ i=1
(1 -at)' ,
0 < 0, < 2
(199)
z
and where D is an additional temperature independent parameter. The problem of the transition to Henry's Law of the equations used to calculate II" in IAS is of primary importance for a successful application of IAS, but a good fit of the experimental isotherms at mediate coverages is another essential condition too. In the mediate coverage region a good fit is usually obtained by using isotherm equations which do not reduce to Henry's Law (LF, DR., DA, ... ). This is a serious difficulty accompanying the use of the IAS approach to predict mixed-gas adsorption equilibria.
59 Recently Rudzifiski et a/.[262] have proposed a general solution for that problem, by using the RJ approach. As already discussed, the failure of the LF, DR, DA, and other isotherm equations to reduce correctly to Henry's Law, has its source in assuming the infinite energy limits (-oc, +cr or (0, + ~ ) while developing these isotherm equations. For obvious physical reasons, there must exist finite integration limits, (era, eM). When RJ expansion is used to represent Oti(pi, T), the function eci(pi, T ) i s defined in the interval (-oo, +oo), and x(e) is to be defined as follows[3]:
X(e) =
0 forO_<e<e=, X for em_<e< eM, 0 for eM <_ e.
(200)
Accordingly, the function X(e) is to be defined now as follows: ,u
forO<e<em,
X(s
for ~M ~< s
(201) and the normafization condition states thatX(eM) -- X(em) = 1. In the case of finite energy limits (era, eM), the expansion (145) takes the more general form
Ot(p,T) = [O(eM)X(ei) - O(~m)X(s
oo ( k r ) ' l! el ~'-~cl /
]- E l=O
(202) s163
While carrying out our considerations we take the function (162) for illustration. In the limit of a strongly heterogeneous surface, when k T / c ---, 0 (practically when k T / c < 0.9), eq. (202) reduces to:
Ot = O(eM)[X(eM) -- X(er
+ O(e=)[X(e~) - X(em)].
(203)
At very low adsorbate pressures such that er > s from eqs. (201) and (203) it follows that 0t = 0(era). It means that adsorption is occurring as on a homogeneous solid surface characterized by the adsorption energy e = era. That apparently strange result was predicted even earlier in earlier theoretical works. Let us suppose that half of the surface is covered, and that function (162) is still symmetrical. Then, IX(~M)- X(,r
1 = IX(~c) -- X(*m)l = 5,
(204)
and eq. (203) reduces to (205)
0t = 0(eM)[X(~M)- X(~c) ] because 0(eM) >> 0(era). When the surface is strongly heterogeneous, i.e. when and eM ~ +cr then FN ~ 1, 0(ei)"-* 1, and X ( e i ) ~
0,
~m ~
-cr
(206)
60 as can be deduced from eq. (162). Then eq. (205) reduces to eq. (153), i.e. to the LF isotherm. When eM is large, the reduction takes place even at very small surface coverages (adsorbate pressures). This is why linear log 0t-vs.-logp plots are commonly observed at low adsorbate pressures. (The Freundlich isotherm is the low-pressure limit of the function -k'(cc) when X(e)is given by (162).) This also means that there is a transition within a certain range of pressures, around p = PM = ( 1 / K ) e x p ( - c M / k T ) , from the LF isotherm to the Langmuir isotherm. As expected at low adsorbate pressures, Langmuir behaviour means simply Henry's Law, p < PM =*" 8t " p K e x p ( e M / k T ) . This means that there is a transition region, where the tangent of the plot In 8t vs In p increases from a value smaller than unity (typically between 0.5 and 0.9) to values close to unity when p ~ 0. In view of what was said above, the integral (189) should be written as follows PM
P*
f
n-=
p
0
- f
PM
p
where
(2o8)
PM = ( l / K ) e x p ( - c M / k T ) .
Equation (207) is quite general and any function k'(~c) i.e. any X(C) can be accepted. It is convenient to carry out the integration by writing the second integral on the r.h.s. of eq. (207) in the following form p"
ec(p*)
(209) PM
ec (PM)
For the particular case of the function (162), we have
/
X(cr
_
c...._~_~In
c 1 -I- exp c
1 + (Kp" exp (~TT)) ~
c
(2~o)
FNkT
Then, the first integral on the r.h.s, of eq. (207) takes the form ,~o(pM)
PM
r 0
1 + exp
e(,,,,) dp =
p
de~
+oo
= I n (1 + e x p ( e = ' - - ; T ( P M ) ) )
kT
(211)
61 Finally H * = HM +
FNkT In 1 +
(~-~
,
(212)
where
1 + exp (era-Co H M = .hfs In
s -- s
kT kT [l + exp (e~ -- eM)]
"
(213)
For the purpose of further considerations we will rewrite eq. (212) to the following form HA I_[M c kT = H" = - Z ' ~ In (1 - 0t).
(214)
After solving eq. (212) for p*, we have
p. 1
[
) lc/T
where K exp(co/kT) is found from the linear regression (155) for the experimental adsorption isotherm. Let us consider the simplest case of a binary adsorbed mixture. Then
pl = p~Xx and p2 = p~(1 - X~).
(216)
The quantity p* defined in eq. (215) is a function of H* and of the parameters K exp {eo/kT}, kT/c, (eo - em)/C, and (eM -- eo)/C. The first two parameters are easily found from linear regression (155) of the adsorption data measured at moderate surface coverages for a single adsorptive. The other two parameters: (e0 - em)/c and (eM -- e0)/c can be found by an appropriate numerical analysis of low-coverage data, based on more general equation (203). Such low-coverage data are not commonly available, so the parameters (e0 - em)c and (eM -- eo)/C are likely to be treated as best-fit ones. Another general solution of the problem of Henry's Law in IAS was proposed by Morbidelli and coworkers[258]. The main idea is somewhat similar to that described above. It does not involve introducing additional best-fit parameters, but creates strong criteria for the accuracy of adsorption experiments. Recently, Rudzinski et a/.[269] have shown, that the PT (Potential Theory) approach to mixed gas adsorption is nothing else, but a special edition of IAS approach which is applicable only to strongly heterogeneous surfaces, characterized by random surface topography. The potential theory of adsorption, introduced by Polanyi, has been widely accepted as the basis for correlating the effect of temperature on the adsorption isotherms of pure gases. A number of authors have modified Polanyi's original method of treating pure-gas adsorption isotherms. These modifications attempt both to improve the temperature correlation and to account for the effect of the nature of the adsorbate on the adsorption isotherm for a given adsorbent. The milestones of the use of the PT for estimating adsorption equilibrium of gas mixtures are: (a) proposal by Lewis et a1.[270];
62 (b) formalization based on the thermodynamic arguments by Grant and Manes[214]; (c) improvements by Reich et a/.[223] and Mehta and Danner[233]. The most recent theoretical achievements along these lines have been reviewed by Moon and Tien[257]. The PT method for predicting mixed-gas adsorption equilibria is based on the assumption that the adsorption data for all pure components fall on the same line when plotted on appropriate coordinates. A number of methods based on the PT have been advocated for correlating pure-gas adsorption isotherms. The most recent extensive experimental and theoretical study by Mehta and Danner[233] showed that the method of Lewis et a/.[270] appeared to be the most useful one. It consists in expressing the adsorption isotherm of a pure component i in the following functional form
Nt,'ViS=F
(kTw (f,i'~)fi] v-;:,.In
(217)
,
where F is the temperature independent function of (kT/V~s) ln(f~,/fi), V~s is the saturated liquid molar volume of the i-th pure adsorbate at a pressure equal to the adsorption pressure pi, fsi is the saturation fugacity of pure i at the adsorption temperature T, fi is the fugacity at the pressure pi and the temperature T. Moon and Tien have shown that in most cases one may replace the bulk fugacities by pressures[257]. For surfaces having random topography the integral for the spreading pressure H* takes the form, Pt
'r
ii. = f N,idp__j_= ~i / Pi
o
kT
X(Qi)d6.ci.
(218)
+oo
Assuming H~ = II~#i, we have,
*r
r
.Afsi f Xi(Qi)deei =.A['sj J +oo +oo
Xj(ecj)decj.
(219)
The above equation will be important for our further consideration. Meanwhile we use the following transformation of the variables:
ti = e~i+ kT in(Kip~i) = v:
kT
----7In vi
(Psi ~ -- .
(220)
Then, eq. (219) takes the form (kr/vt)
~(,,,,/pr )
f +oo
(kr/~,) ~ (,,,, #,.; )
Vi'Nti(ti)dti=
f
V/Nt,(t,)dtj.
(221)
+o0
Thusl if it happens that V~SNti plotted as a function of ti coalesces with V/Ntj plotted as a function of tj, then the upper integration limits must be equal, i.e.
63 Taking into account the Raoult Law (191), we can rewrite eq. (222) to the following form Vi---;In
~i
= ~
Vjs
in
\
Pj
,
(223)
Pi
pi
where means the partial pressure of the adsorbate i. Usually, well approximates fi, so, eq. (223) becomes essentially the Grant-Manes equation. Thus, it should be clearly stated that the PT approach is nothing else, but a special form of IAS theory, valid only for the case of heterogeneous solid surfaces. An extensive study of a large body of experimental data brought Mehta and Danner to the conclusion that the potential curves of different adsorbates have, however, similar shapes and can be coalesced by an appropriate modification of abscissa. So, they have modified the functional relationship (217) as follows:
(kT kov/sln k(f,i'~) ,Z]
NtiVis = F
(224)
'
where k~ is the coalescing factor for adsorbate i relative to a standard adsorbate. a/.[269] have shown that their theoretical development of the PT approach Rudzifiski suggests a possibility of two coalescing factors to be introduced. These authors also give certain recommendations as to the way in which one can use the PT approach to predict mixed-gas adsortpion equilibria better than by applying the traditional 'coalescence' operation.
et
7.3. Application of IAS to the surfaces with patchwise topography In their review, Rudzinski et a/.[262] first emphasized it clearly, that the authors proposing various extensions of IAS for energetically heterogeneous surfaces did not address explicitly their approaches to a certain topographical model of surface topography. To illustrate the problem of surface topography Rudzinski et. al. took extensions of the Langmuir-Freundlich equation into consideration. Their reasults are briefly reported belOW. We start with calculating the spreading pressure l'I~ on a homogeneous surface patch characterized by the adsorption energy e,
or, in another form, II~A kT = n~ = - a r ~ In (1 - 0,).
(226)
After solving it for p~, we have P~=E
exp
~
-1
exp -~-~
.
(227)
For two-component adsorption, from eqs. (191) and (227) we have, X-exp (H'/N',x) - 1, P'
=
' K,
tl
X ' exp
(22S)
'
- 1
54 After solving the above equation system with respect to Xl and X2, we obtain
[(en./~2_l)/(eri./~_l)]P_zK1 XI =
P2 ~ exp 1+ [(en'/Ar'2- 1 ) / ( e n'/~a - 1)]
(el - e2) . .kT.
(229)
p--LKl ~ p2exp (el e2)kT -
When Afsa = A/'s2, on a local patch characterized by the adsorption energies el and e2, the molar functions X1, and X2 are not functions of the independent variables el and e~, but rather of their difference (Ca - e2). Of course, it is difficult to know whether Aft1 = A/'~2 on a certain microscopic patch, but it seems reasonable to assume it until their values found for the whole heterogeneous surface are not much different. Therefore, in order to arrive at the average values Xtl , Xt2 for the whole heterogeneous surface, the following averaging is to be done /
Xlt
Xl(eX - e2)Xa2(ea - e2)d(ex - e2)
(230)
f2(q _,~)
where Xa2(ex- e2) is the differential distribution of the number of adsorption sites among corresponding values of ( c a - e2). Trying to relate Xzx2 to Xi(el) and Xx(e2), one has to consider again the correlations between el and e2. For the case of the highest correlations represented by the condition in eq. (166), Xa2 in eq. (230) is a Dirac delta function. This, for the LF single-gas isotherms, leads to the following expression for Xu:
[(r
(eo, - Co2) ~22/~2 exp
Xtl
"-
1 + [(e:~,'2- 1) / (e:~i,'~ --1)]
p--2.E1Klpexp 2 (r
kT
(231)
' -k Tr
which is reduces to
plK1 Xu =
(e01 - e02)
P2 K2 exp 1 + P-22K-~2exp
kT kT
(232)
when Afsl = A/'s2, One can see that the above expression fulfills the condition (Xu + Xt2) = 1. Coming to the practical side of calculating Xu, Xt2 the problem is as follows: From linear regression (155) applied to the experimental isotherms of single components Kiexp(eoi/kT)values are found. Next, for a certain pair of values of pl and p2 appearing in eq. (231), the corresponding value of H* is calculated from equation system (228) in which the K~exp(e,/kT)are the values K~exp(eo,/kT)found from linear regression (155). In a similar way, one can calculate X1, X2 for the systems where single-gas isotherms are correlated by DA isotherms. (In eqs. (231), (232), eo~ has to be replaced by emi).
65 Thus, we are facing an interesting physical situation. When the adsorption energies of components are highly correlated, and Afsi ~- Afsj for j # i, then for the patchwise surfaces the adsorption from their mixture proceeds as on a homogeneous solid surfaces, although this surface may be strongly heterogeneous for single-gas adsorption. When el and e2 are not correlated at all, the function Xa is found from the following relation 1 exp (e2 - Co2)
x,', (t) = /
J n, 2
[
1 + exp
(
C2
C2
t + e2 - eol
(233)
Cl
where t = (el - e2) and fl,2 is the domain of e2. Our numerical exercises showed, that for the infinite energy limits, e 1 E (--OO, +(X)) and e2 E (-oo, +oo), the function xa(t) is still Gaussian-like, and may be well approximated by the following expression, 1 exp
(/o) t
C
(234)
[l+exp(t-t~ where to = e l 0 - e20, and ca is given by
(CA) 2 = V/( ~C1 ) 2 + (C2) 2+ ~
0.05.
(235)
For the special case when A/'sl = AYe2,Xtl is now given by
Xt, =
Pl K1 ~2 K-'22exp
Col - s kT
Contrary to eq. (232) the above eq. (236) shows the influence of the dispersions of the adsorption energies of components on the adsorption of their mixture. It means, that the lack of correlations between the adsorption energies of components on various patches will increase the effect of surface heterogeneity of the adsorption from their mixture. That conclusion must be true also when Af~i # Af~j for j -# i. The IAS theory is just one of the possible approaches which must lead to at least generally similar conclusions. Thus, the conclusion which we have drawn above must be of a general validity. Valenzuela et al.[243] were first to combine IAS with the concept of seeking the correlations between ei and ej for j # i through their cumulative distributions. They wrote the integral equation for Nti in the following form :
...
1
p,
n
(237)
56 where Afi(e, p,T) is the 'local' isotherm describing the mixed-gas adsorption on a patch characterized by the set of the adsorption energies e = el, e2,..., en. That isotherm was calculated by applying IAS to every patch, so, for the jth patch,
PY~= p~jXij,
i = 1,2,-.., n,
*
nxj(;,) =
:~
.
(238) $
n j(p j)= ri;,
,
(239)
and Pb 1-I:.j = f Afi(eij, p, T). dpp
(240)
0
The total adsorbed amount Afi(e, p, T) is obtained from
1 _- ~ X!j.,
(241)
where Af~ = Afi(eij, p, T)v=v;i(n;)
(242)
and Aft(E, p, T ) =
X~Aft*(e,p,T).
(243)
The integral (237) is reduced to a 1D one by considering the correlations between ei and Qr They seek these interrelations through the cumulative adsorption energy distribution,
F(e) = / X(e)de.
(244)
o j s
Thus, they express the function X(n)(e) as follows: n
X(,)(e) = H Xkx (ek/el) Xl (~1),
(245)
k-2
where Xka is the density function for the conditional probablity that a particular adsorption site has the energy ek, provided that it is characterized by the adsorption energy el for the component number 1. If every site on the surface, irrespective of the component chosen as a reference, has the same higher and lower energy neighbours, then the conditional probablity can be written: Xkx (ek/ex) = ~lFk(ek) -- Fx(ex)l,
(246)
67 where 5 is the Dirac delta distribution. Physically, eq. (246) means that the ordering of the sites from low to high energy is the same for each adsorbate; it is called 'perfect positive correlation'. Insertion of eq. (244) and (246)into eq. (237) yields Xt~(p, T) = [ ~ ( p , T, el, c-2,e~,..., (.n)Xl(s163
(247)
t /
where ek is found from the relation
Fk(Ck) = Fl(el).
(248)
For the purpose of practical calculation they use the binomial distribution,
X(s
= (~)uJ(1-- u) k-j,
(O < j <_ k),
(249)
which gives the 'probablity' of energy ej for (k + 1) sites, u is a skewness parameter, (0 < u < 1), and for a symmetric distribution u = 0.5. Valenzuela et al. took k = 40 which they believe is large enough to represent any unimodal adsorption energy distribution, but small enough not to be a computational burden. For the discrete adsorption energy distribution, the integral (237) is replaced by a summation, k
Aft,(p, T) = ~ Afi(p, T, elk, e2k,'--, e~k)AFk, j=O
(250)
where AFk is the fraction of the distribution for which each adsorbate has a constant energy eik. Of course ~ AFk = 1. Valenzuela et. al. took for illustration the experimental data reported by Talu and Zwiebel on the adsorption of the binary and ternary mixtures of H2S, CO2, and C3Hs on H-modernite at 303.15 K[237]. While predicting mixed-gas adsorption equilibria, they carried out their calculations twice. First they used the single-gas adsorption isotherms to calculate mixed-gas equilibria in terms of the ordinary IAS theory. Next, they calculated the mixed-gas isotherms by using the theoretical approach described above, and called by them the heterogeneous ideal adsorbed solution (HIAS) theory. The ideal-adsorbed-solution calculations show that (CO2+C3Hs) and (H2 +C3Hs) form highly nonideal mixtures, as can also be inferred from the azeotropic behaviour exhibited by these systems. For (CO2 + C3Hs), HIAS correctly predicts an azeotrope but does not predict correctly azeotropic composition. For (H2 + C3Hs), HIAS does not predict the azeotrope but the prediction of HIAS is better than IAS. The mixture H2S + C02 forms a nearly ideal solution; therefore IAS agrees fairly well with experiment. HIAS calculation is almost in the same agreement. Generally, the improvement in agreement obtained by adopting HIAS is only moderate when compared to that achieved by IAS, so, Valenzuela et al. concluded that still "something was missing in the theory". Just at the same time (1987) the concept of the 'site matching' was elaborated independently by Moon and Tien[244]. They made, however, one step forward by generalizing it through the assumption that instead of eq. (248) we will have in general
68
Fk((k) = F I ( s Skl, and called that approach the regular site-matching (RSM) model. When Ski = 0, the above model reduces to what Valenzuela et al. c a l l e d - the perfect positive correlation. In fact, these authors realized that it may not be true for some binary mixtures of molecules exhibiting much different interactions to a solid surface, so, in their work referred above they wrote that "heterogeneity is an important factor, for correlation of adsorption energies". That obvious physical fact was taken into consideration by Moon and Tien, who assumed another model when, Fk(e-k) = 1--FI(s S~: 1, which they called the complementary site-matching (CSM) model. For the purpose of mathematical convenience they accepted the rectangular function (160) to represent the adsorption energy distribution X1(q)- For this particular function ek -- emk ,
Fk(ck) = Ak = s
Cmk _< ~k _< ~Mk-
(251 )
- - ~mk
For regular site matching Ai(ei)
-
A/(ej)+ Sij,
[Sij[ < 1" i 7~ j,
(252)
whereas for complementary site matching A,(ei) = 1 - Aj(ej)+ S~i,
ISSsl _< x, i # j.
(253)
Then, it can be shown that the following identity requirements for Sii and S~ and the combination rules must be obeyed. The identity requirements are: Sij
=
--Sji,
S~j = S~s - S~k = s~j - s ~ =
Sji , Sks, s~,
s~k - s ~
sis.
=
(254)
The combination rules are: 1. If the match of the i - j pair and that of the i - k pair are of the same type (either regular or complementary), then the match of the j - k pair is regular. 2. If the matching of the i - j pair and that of the i - k pair are dissimilar (namely, one is regular and the other complementary), then the match of the j - k pair is complementary. The physical significance of the two types of matching conditions differs as follows. The RSM condition implies that if all the adsorption sites are arranged in an ascending order of their adsorption energy with respect to the i-th adsorbate, e-i, then these sites also fall into the ascending order of the jth adsorbate's adsorption energy. On the other hand, the CSM condition means that an arrangement in the ascending order of ei is an arrangement in the descending order of ej. As the high-energy sites permit a greater degree of adsorption, matching according to RSM, means that the sites which favour the adsorption of the ith adsorbate also favour that of the j-th adsorbate. The opposite is true with the CSM condition. Since the overall adsorption is the sum of results occurring at the individual
69 sites, the presence of two different ways to describe the competitive effect among the adsorbates makes it possible to provide a better or more accurate description of adsorption equilibrium. Determining which of the conditions will be used and the evaluations of S~j (or S~j) are done by comparing binary gas mixture data with predictions. Furthermore, the values of S~j (or S~j) determined from binary gas mixture data must obey the identity requirements. The satisfaction of the identity requirements ensures that the proposed site-matching criteria are self-consistent. With the identity requirements, the criteria used for an n-components system, in the limiting situation when one of the n-components vanishes, reduce automatically to those of the corresponding (n - 1)-component system. It should be stated clearly, that the HIAS approach used by Valenzuela et al. and its more general version - - the site matching (SM) approach used by Moon and Tien are nothing else but the application of IAS theory accompanied by the assumption that the solid surface has patchwise topography. It is important to emphasize it, because all the authors do not express this assumption explicitly, and use sometimes such confusing expressions as 'the solution at some point', instead of the fully rigorous expression - - 'the solution on some patch'. Like Valenzuela et al. Moon and Tien took for analysis the adsorption data for H~S, COs, and C3Hs adsorption on H-modernite reported by Talu and Zwiebel. When Sij (or S~j) becomes zero the Moon-Tien SM approach reduces to the Valenzuela et al. 'perfect positive correlation' HIAS approach. Then, both these groups of authors conclude that the improvement over the (classical) IAS approach is only modest or none. The assumption S~j ~ 0 (or S'zj -fi 0) does not improve it much. A much better agreement was, later on, achieved by Karavias and Myers[252] who additionally considered the non-ideality of the adsorbed phase due to the interactions between the adsorbed molecules. Their theoretical treatment, however, involves introducing not one but two additional best-fit parameters. Eiden and Schlfinder (ES)[222] have proposed another extension of the IAS approach to surfaces characterized by patchwise topography. As their theory was taylored to describe mixed-gas adsorption by activated carbons, the DA equation was assumed to represent the adsorption of single components. They prefer to consider it in terms of 'volume filling' rather than 'site filling', so, they write the DA equation in the following form:
(255)
V - Vs exp ( - (e~ EeSm)~ ) where
es=-kTln(p/p~),
e~m = - k T l n
(P~-~)
(256)
and esm is the lowest value of es below which the DA isotherm does not apply any longer. The esm value is that corresponding to the beginning of the hysteresis loop, which means the end of the filling of micropores, and the start of the adsorption in mesopores. The adsorption in mesopores is described by other than DA equation. This adsorption results into capillary condensation at the end. The value of V~ in eq. (256) is taken as equal to V(es = esm), i.e., is lower than that estimated as the maximum adsorbed amount, found in experiment at p---, p~.
70 Eiden and Schliinder studied experimentally the adsorption of three species: benzene, DM (dichloromethane), and HMD (hexamethyldisiloxane), on two kinds of activated carbons: Supersorbon and Kontisorbon. While considering the spreading pressure integral (189) ES introduced a new integration variable es, through the relation (256). They operated also with the spreading pressure r = HA. Then, in the case of adsorption of ith component, they wrote: oo
if
(257)
where In (P~ /Ps,) ,
c:~ = - k T
(258)
and ~ is the molar volume of the ith adsorbed species. (~ defined by ES is a constant, and should be close to the volume Vi~ defined in the Potential Theory approach). For the temperatures below the boiling point of adsorbate, ES take Q to be the molar volume of the bulk liquid. The Raoult's Law (191) takes then the following form pi = X i p s i e x p (_e~,(r
= r
....
= r
(259)
kT
Next ES divide the whole micropore volume into z subvolumes. The fiUing of these subvolumes is assumed to proceed in a perfectly gradual way in the sequence of decreasing values of e~ characterizing these subvolumes. After filling the j th subvolume, the adsorption is running in the micropores characterized by es = esj, so that e~ij = ~mi ~- Ei [ I n ( z / j ) ] 1/'' ,
i=
1,2,...
, n,
j = 1, 2, . . . , z.
(260)
The spreading pressure is defined as a local property for every subvolume, so r
= r
= ....
r
= Cj,
(261)
where (262)
=
cs*O and = - k T In
.
(263)
The above statements do not leave any doubt that ES introduce the concept of patchwise topography. The so-called 'subvolumes' are simply classes of identical micropores. The essential concept of patchwise topography is not necessarily related to adsorption on planar surfaces. We have to deal with patchwise topography every time when the adsorption system can be considered as a collection of independent subsystems. Then the adsorption properties must be the same in every part of a subsystem under consideration. And now some confusing statements appear in the theoretical considerations by ES, who
71 do not relate DA isotherm with the concept of adsorption on a heterogeneous surface. This immediately creates the fundamental difficulty in their consideration. Namely, how one should define the function V~j(e~) under the integral in eq. (262). Valenzuela et al. and Tien and Moon assumed V~j(e~/Q) to be simply the Langmuir isotherm. In order to overcome that difficulty ES launched the following tricky statement: "Because adsorption in phase j is only possible for the adsorption potentials
qi > esij and the adsorbed volume in phase j is always A V = ~z ' integration in eq. (257) is possible, and one obtains AV
= ~
r
(esij - el*j)
(")
pi=Xi.ip~iexp
-
e~iJ(r kT
(264)
, i=l,
2,
..
9 n j = l , ,
,
2,
..
9 z ,
(265)
o.ol,
While analyzing that tricky statement it is difficult to understand why the function V/j(~si) is replaced by the constant A V. However, all that can be true by adopting the following assumption: The Langmuir isotherm, or any other describing the adsorption in a sub-system of identical micropores is replaced by a step function, V/j(esi) =
0
AV
for esi < ~ j , for esi _> e~ij.
(266)
In other words the condensation approximation is used implicitly to describe the local adsorption in the sub-system of identical micropores. What should also be clearly said is, that ES introduce implicitly the same concept of the "perfect positive correlation" which was first applied by Valenzuela et al. ES combine equations (264), (265) and the condition ~lj = r . . . . . ~j. For the particular case of a binary gas mixture, they arrive then at the following equation:
(
(Xlj)~/V1 -- ~(Prl)V2/V1 e x p 1 - X~j pr2
(Y2/Yl)s
kT
- s
)
(267)
'
where pri is the relative pressure Pi/P~. As the mixed phase composition on jth patch (in the jth subvolume) is the function of qlj and q2j values, it means, that after filling the volume ( j - 1)V~/z, the adsorption of both 1 and 2 occurs on the same patch (subvolume). To calculate the amounts adsorbed in phase j, ES apply the equation, AV J~fJ -- X l j Yl -t- X2j ~r2
(268)
which is the analogue of the IAS equation (192). Then, A/'lj = X~jAfj. For the whole surface (all micropores) Vesij(> ei*j)" N" = Y'~Afj J Ve~ij(> e2ij) : .Af/= ~ A f / j ./
,
(269)
, i = 1,2
(270)
72 The overall (average) composition in the adsorbed phase is calculated as X1 = Af~/Af. ES have done interesting model calculations based on their approach. They showed that different exponents rl and r2 can lead to a strong 'real' azeotropic behaviour, even if there are no interactions between adsorbed molecules, i.e. the adsorbed mixture can be treated as an ideal solution. The above described approach developed by ES turned out to be very successful in correlating mixed-gas adsorption in activated carbons. ES argue that the above procedure for calculating mixed-gas adsorption can be used only when V~ has the same value for both components. If not, the sieving effects must take place, and they also proposed a special procedure for such cases.
8. T H E A P P R O A C H E S B A S E D ON T H E M O D E L S OF M O B I L E ADSORPTION 8.1. E x t e n s i o n s of Hill-de B o e r approach. A first treatment of mixed-gas adsorption, based on a model of mobile adsorbed phase was proposed in 1948 by Kemball et. a1.[142], who extended Volmer's equation for that purpose. This isotherm equation takes into account, and in a crude way, only repulsive interactions between adsorbed molecules, so, this equation seldom provides a good representation of single-gas isotherms. This approach is, therefore, of a limited value for predicting mixed-gas adsorption. Twenty years later Hoory and Prausnitz[143] extended the Hill-de Boer isotherm equation for a two-dimensional mobile monolayer of a gas mixture on a solid surface. To obtain the equation of equilibrium between the two phases, they introduced the concept of surface fugacity; for a component designated by subscript 1, the two-dimensional fugacity in the adsorbed layer is designated by f~'
d#~ = R T d In f~
(271)
and lim f~
l-I--.o X----'~ -
(272)
1"
where H is the spreading pressure,/z~ is the chemical potential and X1 is the mole fraction of 1, all in the adsorbed layer. The surface fugacity of component 1 in the adsorbed mixture is then calculated from a two-dimensional equation of state by the thermodynamic relation,
R T In f~ =
0II A
-
A--A-dA - R T In A/'IRT
(eTa)
T,A,A/'2 ....
where A is the total surface area available for adsorption and A/'I is the number of moles of 1 in the adsorbed layer. Let fl stand for the fugacity of 1 in the gas phase. It can then be shown that the equation of equilibrium is
f~=
KH1RT NA--.---..~.f~
(274)
73 where NA is the Avogadro's number, KH1 is the Henry's constant for pure component 1 and fll is the surface occupied by one molecule of component 1 when 01, the fraction of a monolayer covered by 1, is equal to unity. The parameters KH1 and fll are obtained from a suitable analysis of pure component data, and K m fulfills the following condition, 9
K m - l : m~ \ f ~ ]
(275)
A/'2 . . . .
(275)
0
Next, the two-dimensional analogue of van der Waals' equation of state was accepted by Hoory and Prausnitz to represent H II =
kT O" ~
a /~
(276)
0 .2
where k is the Boltzmann's constant, a and ~ are the characteristic constants (analogous to van der Waals' a and b) and a is defined by A (277)
o" = N A./V'F_,
where A/'z is the total number of moles adsorbed. Hoory and Prausnitz assumed further, that for a binary mixture, the constants a and are related to those of the pure components by (278) and a -- a1X21 + 2 a 1 2 X i X 2 + a2X 2
(279)
where a12 is a constant characteristic for the 1-2 interaction. They assumed that a12 =
x/ala2. Substitution into Eq. (273) yields then for the surface fugacity of component 1 X1 k T In f~ = In cr " / 3 -+
~1 a-~
2 akT
(miX1 + a12X2)
(280)
Further substitution into the equation of phase equilibrium, and the assumption that the bulk gas phase is ideal yields X1~1 ~1 In (a - ~)KH1PY1 -~ a - - ~
2 o'kT ( a l X 1 + a12X2) = 0
(281)
A similar equation can be obtained for component 2. The two equations of equilibrium predict the adsorption isotherm for a binary mixture using only the data obtained from the pure-component adsorption isotherms. Given the temperature, the total pressure, the composition of the gas phase and the parameters Ks, a and 3 for each pure component, the two equations of equilibrium can be solved to yield
74 cr and X. For binary mixture, the number of moles of component 1 which are adsorbed is found from All = X1/~IAf~
(282)
o"
where Aft1 is the number of moles of 1 adsorbed at the full monolayer coverage of pure 1. One has a similar relation for component 2. Since the area of the solid is a constant, it follows that (/3Afs)l = (/3A/',)2
(283)
To illustrate their equations, Hoory and Prausnitz performed numerical calculations for the adsorption isotherms of several mixtures on homogeneous graphitized carbon. In these calculations we have used pure-component parameters reported by Ross and Olivier[271]. These parameters are given in Table 2. X 10 30
Gas
(ergcmZl molecule z)
/3 • 10 TM
(cm21 T molecule)(~
Nitrogen Argon Benzene
33.85 43.98 112.0
15.55 13.60 30.60
Chloroform
283.0
29.40
CFC13
297.0
31.20
90.1 90.1 273.2 322.6 286.2 322.6 273.2 286.2
KH 1
6.30 13.20 0.71 9.93 15.10 87.63 15.20 28.80
Table 2. Parameters for pure-gas adsorption on graphitized carbon P-33 (2700~
(From Ross and Olivier[271]).
Unfortunately, they could not make any comparison with experimental results because no data for adsorption of these gas mixtures on a nearly homogeneous graphitized carbon, were available at that time. The application of the two-dimensional analogue of Van der Waals' equation (276) also makes it possible to describe the adsorption systems showing considerable departures from ideality in the two-dimensional adsorbed phase. The activity coefficient of component 1 is then given by, f~(X1, n, T)
~/1 = X~fp% l (H, T)
(284)
75 where f;~(II, T) is the surface fugacity of pure component 1 in the adsorbed layer at the same spreading pressure II and temperature T as those of the mixture. One can calculate 3' from the two-dimensional equation of state. For component 1 in a binary mixture /~I In 71 = In 0"1 -- /~1 a - ~ I a /3 -
a~
/~1 -
~1
2
akT(a~Xl+X~2X2)+
2C~I alkT
(285)
and there is a corresponding equation for component 2. In Eq. (285), al is the area occupied by one molecule of component 1 when pure component 1 is adsorbed at the same temperature and spreading pressure as those of the mixture. In the limiting case, when al = a2 and fll = f12, one obtains a = o" 1 - - a 2 and hence 71 = ")'2 = 1. If a l and a2 are not zero, the necessary (and sufficient) condition for the mixture to be ideal is that al = a2 and ~1 =/32. An insight into Table 2 must bring one to the conclusion, that the two-dimensional surface mixture (CFC13 + CHC13) is nearly ideal. On the contrary, the surface mixture (CFCI3 + C6H6) should exhibit considerable departures from an ideal behaviour. Hoory and Prausnitz demonstrated next that this surface phase non-ideality does not affect much the ( X - Y) diagram which is so frequently used to compare theory with experiment. At the same time, however, that surface nonideality affects strongly the relative volatility 7"]21defined as, r/2~ = I/1/X~
(286)
As the relative volatility is the reciprocal of the selectivity coefficient which is an important characteristics for gas separation, one may conclude it as follows. Small departures of a theoretically predicted (X - Y) diagram from the experimentally observed one may suggest a satisfactory agreement between theory of experiment, but this may mean serious errors in predicting separation factors. The graphitized carbon P33 is nearly homogeneous, but not perfectly homogeneous. Thus, Ross and Olivier studied the adsorption on this material by assuming patchwise topography, and mobile adsorption on every homogeneous patch. The local adsorption on n-th patch was described by Hill-de Boer isotherm, P=
(1) ~
0 (0 J : - O exp 1 - 0
2aO'~ kT~]
(287)
where o= - , Vr
= --exp Pv
(288)
Eq. (287) is the particular form of eq. (281) when X = 1, and Y = 1. (Please compare eq. (287) with its slightly different form (9)). It was assumed next that the distribution of the adsorption energy e can be represented by the gaussian function, g(e), =
exp
76 where w 2 is the variance of that function. Table 7.2 collects the values of the parameters ~, w, pv, and Afs obtained by fitting eqs. (287) and (289) to experimental adsorption isotherms of nitrogen and argon on P33.
N2 -g kcal/mole w kcal/mole pv(90-l~ mmHg Z's cm3(STP)/g
Ar
2.150 0.289 6.79x105 3.30
2.066 0.242 11.32• 3.76
a and f~ as in Table 7.1
Table 3. Adsorption parameters for N2 and Ar on P-33 (1000~
(From Ross and Olivier[271]).
Hoory and Prausnitz have generalized that treatment for the case of adsorption of a binary gaseous mixture by assuming that the local adsorption on every patch is described by their generalized Hill-de Boer isotherm (281) and that the two-dimensional adsorption energy distribution X2(el, e2) is the generalized Gaussian function, 1 ( X2(el,e2) = 2Hwlw2V/1 _ ~2 exp -
~I/12-- 2~I/1 ~2 + ~1/22) ~
(290)
2: e2)
where ~1 -- el -- s el
~i/2
e2 -- e2 W2
(291)
and where ~ois the correlation coefficient = cov(e,, e2)
(292)
WlW2
This correlation coefficient may be positive or negative; its absolute value must lie some-
where in the interval zero to unity. Let the probability that for a patch ij the adsorption energies lie in the intervals (el + Ael), and (e2 + Ae2) be designated by 5ij,
~ij
-- X(s
s163163
E E ~ij = 1 {
(293)
(294)
j
The parameters ~'1, ~'2, Wl, W2 can be found from the pure--component adsorption data, but the correlation coefficient ~ can be found only by the analysis of mixed-gas isotherm. For the components which are chemically similar ~0should take positive values and close to unity.
77 The mixed-gas isotherm is obtained by applying eq. (281) to each homogeneous patch, and then summing over all patches with the weighting factor 6ij. Upon applying the equation for homogeneous surfaces to a patch ij, one calculates for 6~j and the mole fractions Xlij and X2~j. Let F1 and F2 be the overall number of moles of components 1 and 2 respectively, adsorbed per unit area of the heterogeneous surface. To find F1 and F2 the summations are used,
Xlij i
(295)
j
F~ = ~ ~ ~y X2~j i
j
(296)
NAaij
The overall mole fractions of the adsorbed film on the heterogeneous surface are then given by F1 X1 = F1 + F2
and
X2 =
F2 F1 + F~
(297)
Experimental data for the adsorption of the (N2+ Ar) mixture on P33 were not available at the time when Hoory and Prausnitz worked on their publication. For this reason they presented only some model calculations. To be able to confront their theoretical predictions with the experimental data, Hoory and Prausnitz took into consideration the experimental isotherms of adsorption of the (ethane+ethylene) mixture on the Nuxit-A1 active carbon, reported by Szepesy and I1les[272]. These authors also reported the experimental isotherms of the pure components which were fitted first by Hoory and Prausnitz to estimate the single-gas adsorption parameters; ~, w, p~, ~, a, and A/'s. The values of these parameters found by Hoory and Prausnitz are collected in Table 4.
C.H6 -~ kcal/mole w kcal/mole
4.144 1.430 pv(90.1~ mmHg 3.84x106 x 1016 cm2/molecule 24.1 123.9 a x 1030 erg cm2/molecule 2 Afs cm3(STP)/g 218.0
C2H4 3.923 1.430 4.05x106 22.2 88.7 237.0
Table 4. Adsorption parameters for C2H4 and C2tt6 on Nuxit-Al active carbon (From Hoory and Prausnitz[143]). The calculations of mixed-gas adsorption were made by applying these single-gas parameters, pv, fl and a found from single-gas adsorption of C2H4and C2H6 on P33 to (ethane + ethylene) mixture adsorbed on Nuxit-A1 active carbon. Two values of the correlation
78 coefficient; ~0= 0, and ~o= 1, were assumed in this calculation. Because of the chemical similarity of C2H4 and C2H6, the value g = 1 seems to be a more reasonable assumption. This was confirmed by the calculation of the corresponding mixed-gas isotherms. The results calculated with full correlation (g = 1) gave an excellent agreement with the experimental mixture data. The hypothesis that the adsorption energies of hydrocarbons are strongly correlated received further support in the theoretical work by Nakahara. Nakahara et a/.[226,227] studied the adsorption of hydrocarbons by the carbon molecular sieve MSC-5A and showed failure in fitting the mixed adsorption data for these systems by either the semiempirical approach by Cook and Basmadjian[212], or the classical edition of IAS theory, as well as by VST approach proposed by Suwanayuen and Danner[222]. After proving the failure of fitting these data by theoretical approaches based on the model of homogeneous surface, Nakahara[273] turned to the approach developed by Hoory and Prausnitz. However, the procedure of fitting single-gas isotherms used by Nakahara was somewhat different from the procedure of Hoory and Prausnitz, because no data for a reference homogeneous carbonaceous adsorbent were used. While fitting mixed-gas adsorption isotherms, Nakahara employed the generalized Hill-de Boer isotherm (281) developed by Hoory and Prausnitz, but again the fitting procedure was different. Nakahara did not employ the generalized gaussian distribution (290), but introduced in an implicit way the assumption about high correlations between the adsorption energies of the adsorbates. In the calculation of probability densities for a model isotherm he used a normalized Gaussian distribution whose variables were continuous. But in the calculation of adsorption of binary gaseous mixtures he treated the number of patches as countable and discrete variables. The number of patches which have the same adsorption potential was calculated by the Gaussian equation multiplying the normalizing factor by 1000 and counting the fractions of 0.5 and over as a whole number and disregarding the rest. The important point of the extension of pure isotherms to describe the adsorption of mixtures was how to combine the two isotherms. In the first step of this study Nakahara set the 1000 patches in a series with respect to the potential values, being numbered consecutively by integer i from 1 to 1000 in highest potential, and increasing i values corresponds to either the same or higher potential. While considering adsorption of binary mixtures, the patch which has the highest potential in the adsorption of pure component 1 was considered to be the same patch which has the highest potential in the adsorption of pure component 2. The second patch which has the potential next to the highest for component 1 is also the patch which has the potential next to the highest for component 2. The same rule was applied to the third, fourth, ..., and to the thousandth patch. Thus, although it was not stated explicitly by Nakahara, the above procedure was equivalent to the assumption of high correlations between the adsorption energies of the adsorbates. The average surface coverage of each component 8tj was calculated as follows,
Zj
1
1000
Otj = 1000 i=1 ai = 1000 ~
Oii
(298)
i--1
The amount adsorbed of each component, to be compared to the experimental data in
79 the unit of rag/g-adsorbent, Aftj is calculated as follows .Aftj = .M'~jOtj
(299)
All the figures presented in the Nakahara's work must be viewed as an impressive evidence for the success of Nakahara's theoretical method. Also, they must be viewed as an evidence of the importance of the surface energetic heterogeneity of MSC-5A in mixed-gas adsorption. Nakahara's method can be applied only for the systems in which high correlations exist between the adsorption energies of the components of the adsorbed mixture. This, for instance, should be the large class of the adsorption systems in which a mixture of hydrocarbons is adsorbed on a carbonaceous adsorbent. In the systems where the adsorption has a mobile character rather, but the adsorption energies of the various components are not strongly correlated, using the classical Hoory-Prausnitz approach allowing to consider existence of limited correlations may be necessary. Danner and coworkers[220,221] attempted to use the Hoory-Prausnitz eq. (281) to describe the adsorption of (ethane + ethylene) mixture in 13X molecular sieve, which was treated as creating a homogeneous gas-solid interface. This led them to a limited success, and it turned out, that even the simplest edition of IAS worked better. In the case of (ethane + ethylene) mixture adsorbed on the nearly homogeneous carbon adsorbent Sterling FTG-DS, Lee and Connel[274] used successfully a statistical description based on a model of partially localized and partially mobile adsorption. The Hill-de Boer isotherm is known for that it does not describe well the repulsive (excluded area) interactions. So, Findenegg and coworkers[275,276] used the more accurate treatment offered by the Scaled Particle Theory to develop an isotherm equation for the adsorption of a binary gas mixture. That equation turned out to provide a good description of hydrocarbon mixtures on graphitized carbons which are known for having relatively homogeneous surfaces. Nitta et. a1.[204] developed an isotherm equation for mobile mixed-gas adsorption, where repulsive interactions were described in terms of the Scaled Particle Theory. Next, patchwise topography was accepted, along with Gaussian energy distribution. However, it is also essential to account properly for attractive interactions between adsorbed molecules. In the case of an adsorbed mixture which may be highly non-ideal, due to attractive interactions, the attractive terms in the Hoory-Prausnitz' extension of Hill-de Boer isotherm may not describe well the complicated nature of attractive interactions. The Hill-de Boer (Van der Waals) isotherm was developed statistically by using the simple mean field approximation to represent effects of attractive interactions. Thus, Patrykiejew et. al.[174] and next Konno et. a1.[193,194] used more accurate 2D equations of state to develop isotherm equations for adsorption of binary gas mixtures. In their second paper Konno et. a1.[193] used their mixed-gas isotherm to describe adsorption on patchwise heterogeneous surfaces. While describing the adsorption of (methanol + n-hexane), (methanol + acetone), and (acetone + n-hexane) mixtures on the sieving carbon MSC-5A, they assumed that its surface could be modelled as composed of two kinds of patches. Very recently, Zhou et al.[208] have developed a variety of mixed-gas isotherms corresponding to five 2D equations of state.
80 8.2. Low p r e s s u r e formalism.
adsorption
and
the
application
of virial
description
When the total bulk pressure of the adsorbed gas mixture is relatively low, the virial description formalism may become a very convenient tool to deal with mixed-gas adsorption. It has been used extensively to describe single-gas adsorption on both homogeneous and heterogeneous solid surfaces. A detailed exhaustive description of the use of the virial formalism to describe single-gas adsorption has been given by Rudzinski and Everett in Chapter (7) of their monograph[3]. Attempts to use the virial formalism in the theoretical description of mixed-gas adsorption were started in 1955 by Kwan, Freeman and Halsey[277], but these concerned adsorption on a homogeneous solid surface. Jaroniec and Sokolowski [278] were the first to use the virial formalism to describe mixed-gas adsorption on heterogeneous surfaces. For that reason we refer to their paper in a more detailed way. Let us consider a two-component gaseous mixture of volume V and temperature T in contact with a solid surface. The activities of gaseous molecules are aA and aB. The solid is assumed to be completely inpenetrable for gaseous molecules. The quantity adsorbed is defined as the difference between the average number of gaseous molecules in the system and the average number of molecules in a hypothetical system, so-called "calibration system" of equal volume but with no gas-solid interactions. The grand canonical function for the real adsorption system is
_NA _NB
-"
E
uA uB Z(NA, NB) NA~NB! NA,NB>O
(300)
whereas for the calibration system is
--*-":
aNAA NB E NA!NB!aB Z*(NA,NB) NA,NB~O
where Z(NA, Ns)and calibration systems:
(301)
Z'(NA, Ns)are the configurational integrals for the adsorption and
Z(NA, NB) = / drNAdrNB exp ( U(NA,NB)
(302)
V
Z'(NA, NB) =
drNAdrNB exp
--
kT
(303)
V In eqs. (302-303) U(NA,Ns) and U'(NA, Ns) denote the potential energies of the indicated set of molecules in the adsorption and calibration systems, respectively. The adsorption isotherm is obtained from the relation -
Nr = 01n(~.) + l n ( ~ . ) = -NA + -Ns cOIn aA In as -
(304)
81 where Nz, NA, NB are the average numbers of the molecules adsorbed. Using the expansion of In (~.), it is possible to express Nr. as an infinite series in powers of aA and aB. While retaining only the first power and quadratic terms in this expansion, we have
N--a= aABAs ~- aBBBs -+-aAaBCABs -F a~CAAs q- a~CBBs
(305)
where BAS and Bss are the second gas-solid virial coefficients, which in the case of a non-reactive, non-swelling solid considered here are given by,
B~s=/(g,-1)dri=/[exp(~'(ri)) v
- 11 dr i
kT
'
i = A, B
(306)
v
Cijs (i,j = A, B) are the gas-solid coefficients,
(307)
Cqs = j j(gigjfij - f~)dridrj v
where
f,j=exp(
wq~rj))_l
*
( uij(ri' rj) ) - i kT
f~j = exp
(308)
The function wij is an "effective pair potential" of interaction between two adsorbate molecules, one of species i, one of j, in the presence of solid surface, and is a quantity averaged over all allowable configurations on the adsorbent. The uij is the interaction potential between two gaseous adsorbate molecules under calibration conditions. The potential 0i(ri) is the average energy of interactions of i-th type of adsorbate with the adsorbent, averaged over all configurations on the adsorbent. As the adsorption on heterogeneous solid surfaces is usually considered in terms of localized adsorption, Jaroniec and Sokolowski accepted that model to arrive at a more detailed form of their gas-solid virial coefficients. They introduced next the simplifying assumption that adsorption sites are labelled only by their values of ei (i = A,B), i.e. by the local minimum of the gas-solid potential 0i(ri). While accepting that simplifying assumption, w
-ff~.
de, ,
i = A, B
(309)
12i
and
B,s = f B~
,
i = A,B
(310)
fti
where B~ is the expression for the second gas-solid virial coefficient for a homogeneous solid surface, on which the adsorption sites have adsorption energy equal to c,.
82 While B~s depends only on the energy distribution of the sites, the presence of lateral interactions introduces an additional dependence on the spatial distribution of sites, X(2) O (%i, e~,, Rni~~), giving a number of pairs of sites, having energies %, and e~j which are separated by a distance P~i~j. The function XI~) has the following features"
/
X~)(%,,e,,,,t?..,..,)dR,7,,r, = Xi(%,)X,(e,,,)
(311)
R,Ti ,,i >0
(312) where 6 is the Kronecker's symbol, the (77,x) label the sites, and species, On the other hand, we have
(i,j = A,B).
(i, j) label the adsorbed (313)
As usual, one may distinguish the following cases of site pair distributions: (i) patchwise distribution, distribution with finite correlation length, random distribution. In the case of single-gas adsorption the problem of surface topography was elaborated in the theoretical works by Steele[279], and by Pierotti and Thomas[280]. The case of the finite correlation length was elaborated by Ripa and Zgrablich[281]. In their study of mixed-gas adsorption, Jaroniec and Sokolowski limited their interest to only patchwise and random site distribution. For patchwise surface topography, (p), the third gas-solid virial coefficients take the following form,
(ii)
(iii)
exp
(2tgi(rl;ei)) kT dei] . fii- f~} -
drzdr2
(314)
and
X(2)(eA'eS)exp(--VqA(rA;eA)--~B(rB;eB))deAdeB] " k T
BS-V-V*
A XI2B
fAS 9 -- f~S } drAdrB
(315)
whereas in the case of random site distribution, (r), =
exp
kT
" (f~i +
f Xi(e')exp (-2tg'(rz; ei)) de' - f~]} t2i
1)-
(3 6)
83 and CA(r) BS
/ ~V-V*
[{j (
}
" {a/BgS(eB)exp(--tgS(rB;eB))deB}
/
X(2)(eA,eB)exp (--OA(rA;
(317)
eA)kT~S(rB; - eB)) deAdeB -- f~B] drAdrB
~A X 12B
The above considerations by Jaroniec and Sokolowski were kept on a purely theoretical level. The case of partial correlations has, more recently, been considered in the paper published by Zgrablich and coworkers[187]. They applied, however, another kind of virial expansion based on the assumption that the adsorbed phase can be considered as a strictly 2-dimensional one. The 2-dimensional compressibility factor Z of that adsorbed phase was represented in their work by the following expansion:
Z = A/'r~RT =
1 + B,(T)
-~
where A/'r~is the total number of moles in the adsorbed phase, a,~ = ~A) , and Bn is the 2-dimensional nth virial coefficient, which is the function of temperature. It can be shown, that the second virial coefficient of an adsorbed mixture is a quadratic function of the molar fractions Xi's of the components in the adsorbed phase composed of i and j.
B2 = x, 2 ii + 2X XjB ' + x,2Bjj
(319)
and the third coefficient B3m is the following cubic function, = Xi B3 +
~3 +
+ Xj B 3
(320)
The coefficients/3~i and -3 fqiii have the following form,
B~i= / / d r l d r 2 f 1 2 exp (e~(rl) ei(r2))kT +
(321)
A
Biii f f/drldr2drsf12f23f31exp (ei(rl) + ei(r2) + ei(rs)) a = kT A
(322)
84 The interaction potential uij(r), between two adsorbed molecules being at the distance r is approximated by the "square well" function, oc
uij(r) =
for r < too
-kTgg for roo < r < rgg 0
(323)
for r > rgg
The crossed 2D coefficients can easily be written by expressing properly f12 functions. In the case of B~~, for instance, u12 in fij is the potential function between two molecules i and j. Zgrablich and coworkers still used the square--well function (323) to describe the interaction potential between molecule "i" and molecule "j", along with the crossed parameters, . .
=
1 [(roo)i-t-(rcc)j]
(324)
[(T,,),.
(325)
The expression (318) was used in the relation (273) to carry out the integration leading to the isotherm equation for mixed-gas adsorption. For that purpose Zgrablich and coworkers wrote the Gibbs' relation in the following form,
In
RTKH, Y~P = / fli
[0(AfZ) -10am X, RT + In L (gAfi T,A,JV, am am
(326)
O'rn
where P is the total bulk pressure of the adsorbed mixture. Eqs. (318-325) yield the following expressions for the mixed-gas adsorption,
2 (X.B" 3 \.~,,i~.~3 (y2Riii In XifliO k ' 2 + XjB~j) 0 + + I~KHiYiP ~ In
~iij 2~ 2
- } - 2 X i X j ~'3
"Jl- X.Bijj ' 3 )
02
= 0
Xj~jO 2 (XjB~ j + X,B~j) 0 3 (X~B~ j~ + 2XiXjB~ jj + X}B~ 'j) 02 + + = 0 ~KHjYjP ~ 21~~
(327)
(328)
The virial coefficients were calculated by means of the Ripa-Zgrablich generalized Gaussian model[281]. While calculating the 2D coefficients for a heterogeneous solid surface, Zgrablich and coworkers replaced exp ( ~ ~ , e(ri)) under the integral sign by these functions averaged with the following multivariate Gaussian distribution X~(e) Xn(E, T1 -- T2,..., Tn--I -- Tn) =
[(2r) ~ det HI-' exp
-~ ~ i,j=l
(e(Ti) -- ~) (H~ 1) (e(rj) - ~)
)
(329)
where
Hii = w2C(ri- Tj)
(330)
85 is the covariance matrix, and C(Ti - rj) is a certain correlation function. In the simplest case n = 1 function (329) reduces to the ordinary Gaussian function (289). Thus, for a heterogeneous solid surface, the coefficients B~i and ~-3~" take the following form,
drldr2f12exp (~--~)C(rl-r2)
B~i=--~
(33~)
A 3 = -'~
drldr2drsfx2f23fm
exp
I w 2
(~-~)[C(rl-r2)+
A
\
+ C(r~ - ~ ) + c ( ~ - ~,)])
(332)
As for the correlation function C(r), it was expressed by the formula C(r) = exp
-~
,
rc e [0,1]
(333)
where rc was called the "correlation distance" between two points on the surface. When r~ = 1 the points are totally correlated. Then, when rc = 0 they are completely independent. The case rc = 1 means patchwise topography. For two different molecules / and j, the crossed correlation parameter (%)ij was approximated by the formula
(r~),j = [(rr189
(334)
Thus,
B2= --~
where ~ = exp
--~
- 1
(337)
The second virial coefficient can be evaluated analytically in terms of the integral exponential function E~(x), (Abramowitz and Stegun[2821),
(33S) rc
w
2
1
roo
86
[
w
2
{
1 (rgg~ 2
l(roo)2}]}
(339)
;:0 The i}3) integrals were calculated by using Simpson's tridimensional integration subroutine with 1% accuracy. The calculated values were next expressed in terms of the dimensionless coefficients; B~,, B],
B2
B; = (H~L/2)
'
B~ =
Ba
(n~L/2)
(340)
After solving the equation system (327-328) one obtains the values of the individual adsorption isotherms. The adsorption isotherms calculated in that way by Zgrablich and coworkers were tested against a variety of experimental mixed-gas adsorption isotherms already reported in literature, (Szepesy and Illes[272], Lewis at. a1.[270], Markham and Benton[140]), and mixed-gas isotherms measured in Zgrablich's laboratory in San Luis.
9. T H E STATE OF ART A N D T R E N D S IN A N A L Y T I C A L A P P R O A C H E S The term 'analytical' approaches is used here for all the works which are not purely computer simulations. They include, therefore, both the 'thermodynamic' and 'molecular' approaches. Of course, the analytical approaches also involve carrying out more or less complicated and time consuming computer calculations. Most of the papers using either molecular or thermodynamic approaches were based on the models of localized adsorption. Therefore, the papers based on the models of mobile adsorption have been discussed separately in the previous section. It seems reasonable to assume that in the real adsorption systems the mixed adsorbed phase is neither perfectly mobile or localized[144,173]. The choice of localized adsorption models in some cases may be viewed as adopting a convenient t o o l - the lattice description formalism, in a similar sense as it is made in the theories of liquid state. The works in which the thermodynamic approaches were used created in literature a certain tendency of verifying various approaches and corresponding mixed-gas adsorption isotherms, through the thermodynamic consistency test, which may be expressed in a number of ways. It should, however, be realized that this test is related to the sense of macroscopic thermodynamic phase, defined in the phenomenological thermodynamics. So, while applying that test one has to consider to which extent, the adsorbed molecules in a certain adsorption system can still be considered in that way. Various approaches involve accepting various simplifying assumptions, so, the most essential test would be that stressed by the simple question - how does the approach work. In other words, how effective a particular approach is, i.e. corresponding isotherm equations, in predicting mixed-gas adsorption equilibria. The VST ( Vacancy Solution Theory) which was so popular for some time, has recently been abandoned as it does not predict correctly selectivity in adsorption of a binary mixture at very low surface coverages[283]. At the same time using VST to predict mixed-gas adsorption equilibria is easier than using other thermodynamic approaches m IAS, NIAS and HAS, for instance. The VST approach does not involve necessity of studying carefully
87 low-coverage behaviour of single gas isotherms, what may be difficult in many cases. As it was emphasized by Talu and Knabel, VST "has a built-in flexibility" due to adopting powerful though semiempirical expressions for the surface activity coefficients. These are expressions which have been used in the theories of highly non-ideal bulk solutions (Wilson, Flory-Huggins). In VST bulk concentrations were replaced by surface concentrations (coverages). Danner an coworkers used their VST approach also to adsorption systems with strongly heterogeneous surfaces. It is, therefore, obvious that the parameters in the surface activity coefficients, found by computer, reflected not only effects due to interactions between adsorbed molecules, but also simulated effects arising from surface energetic heterogeneity. As far as the behaviour of adsaorption isotherms is concerned, these two effects can simulate each other to some extent. This has been discussed recently by Koopal et. a1.[207]. That mutual simulation, however, becomes much worse in the case of enthalpic effects accompanying single and mixed-gas adsorption. Thus, Talu and Knabel showed[240] that using VST equations to fit single-gas isotherms in various adsorption systems leads to wrong predictions of accompanying heats of adsorption. Rudzinski et. a1.[262] have shown that the VST approach can still be modified further to take separately into account the surface energetic heterogeneity, and the interactions between the adsorbed molecules. Their considerations were kept on a purely theoretical level, and the applicability of that modified VST approach has not yet been prooved. As far as adsorption at one temperature is concerned the test "how does it work" may justify using some usefull analytical expressions. In addition to the classical VST approach, another interesting example of that kind may be the NICA equation proposed by Koopal et. a1.[207]. Instead of Markham-Benton (Langmuir) equation (164) these authors put the generalized Langmuir-Freundlich isotherm 0~=
(/s
"~'
(341)
1 + E ([fiP') m' under the integral in eq. (141). The values m~ < 1 may arise from either repulsive interactions between adsorbed molecules or from energetic surface heterogeneity. The values mi < 1 may also account for both of them. On the contrary, the values mi > 1 may simulate attractive interactions between adsorbed molecules. In other words, one parameter mi is used to describe the combined effect of surface energetic heterogeneity and of the interactions between the adsorbed molecules. Koopal et. al. argue that this combined effect will be different on different adsorption sites so eq. (341) should, again, be averaged with the function describing the dispersion of the parameter (/~)m~. Provided that this parameter can be expressed a s / ~ 0 e x p \-~--/, where/s is the same for all adsorption sites, and the new variable e~~ changes from one site to another, one 9may assume a function X~(e~~ e~om,---, C~om)to exist, and next carry out the averaging in a similar way as it is outlined in eq. (141). Koopal et. al. assume, that c~~ are highly correlated like e~'s in eq. (~66), and that X~(~~~ is the Gaussian-like distribution function (152). Then, the result of averaging 0~ in eq. (341) with X(er176 takes
88 the following form,
8ti(p,T)=
([ffipi)m' j
(K'pi)s ~j (Rjpj) ms
1+
(342)
Koopal et al. called the above equation NICA (Nonideal Competitive Adsorption). There was a time, (the late seventies), when the Jovanovic isotherm equation was extensively used to describe mixed-gas adsorption on heterogeneous surfaces. For single-gas adsorption on homogeneous solid surface the Javanovic equation reads,
{
O(e, p, T) = 1 - exp - K p e x p
(343)
~-~
It was kineticaUy developed by Jovanovic[284], who also took into account the collisions between the adsorbed and bulk molecules while considering localized monolayer adsorption. (The kinetic derivation of Langmuir isotherm ignores these collisions). Jaroniec et. a1.[147] proposed then the following generalization of the Jovanovic equation for mixed-gas adsorption:
n
i) 8({e}, {p}, T) = 1 - e x p { - Z KiPi exp ( e~~
}
(344)
i'-1
Next, they applied the IE approach to extend it for adsorption on heterogeneous surfaces. A good correlation was obtained in that way for a variety of adsorption systems, by accepting some additional assumptions making analytical integration in eq. (8) possi-
b1~[151,175]. There are, however, certain problems related to the statistical derivation of the Jovanovic equation for which it was later abandoned. Still, some other approaches were used to describe mixed-gas adsorption on heterogeneous solid surfaces. Thus, Nikitas et. a1.[188] applied the whole theory of liquid state to describe localized and non-localized adsorption on heterogeneous surfaces. They developed appropriate theoretical expressions for both patchwise and random topography, but their considerations were kept on a purely theoretical level. Dubinin et. a1.[182] showed that the osmotic theory of adsorption might be useful in the studies of mixed-gas adsorption on heterogeneous surfaces. Jaroniec[173] and Patrykiejew et. a1.[174] attempted to extend the Kiselev's approach for collective adsorption based on the picture of associating molecules[285-289], to describe mixed-gas adsorption equilibria. Of course, considering the "associates" is only a theoretical tool to take into account interactions between the adsorbed molecules. The idea is similar to the graphs techniques used to describe nonideality effects in real bulk gaseous phases. Like in the case of single-gas adsorption, one may expect tendencies to exist toward multilayer adsorption in mixed adsorbed phases. Attempts to describe mixed-gas multilayer adsorption were started by Hill's extension of BET model in 1946. Gonzales and Holland[290] introduced in 1977 a number of simplifying assumptions to that model. A
89 year later Jaroniec and T6th[177] proposed certain generalizations of Freundlich and T6th equations for the case of mixed-gas adsorption. Jaroniec et. a1.[168] and Jaroniec[173] also attempted to use the Kiselev's approach, based on the picture of "associating molecules" to describe mixed-gas multilayer adsorption. Nowadays, there is widely spreading view that the differences in the dimensions of adsorbed molecules are an important factor affecting mixed-gas adsorption equilibria. When thermodynamic approaches are used, this factor is a source of strong deviations from ideality of the "adsorbed solution". The success of Danner's VST approach in some cases comes from using Wilson's type activity coefficients which have turned out to be very powerful in describing non-ideality effects in bulk liquid mixtures composed of molecules of much different sizes[291]. Effects arising from different sizes of adsorbed molecules are becoming now a hot topic in the works employing molecular approaches. Here not only the size, but also the structure of adsorbed molecules is very important. For molecules having a linear chain-like structure, the application of the ideas underlying the Guggenheim's theory of polymer solutions seems to be strightforward. Nitta et. a1.[190] derived an adsorption isotherm for a multicomponent gas mixture in which each molecule can occupy any number of adsorption sites on a uniform lattice of sites. Interactions among the homogeneous molecules were treated by using the Bragg-Williams approximation. Nitta et. al. also derived equations[138] for adsorption of molecules composed of different segments. Most recent attempts along these lines have been published by Russel and LeVan[292], and by Rudzifiski et. a1.[293]. Russel and LeVan have adopted Guggenheim's lattice theory of solutions[294], along with the Bragg-Williams approximation to describe the interactions between adsorbed molecules. The Guggenheim's lattice model was reduced to a 2D lattice of (adsorption) sites, but it was assumed, that adsorbed molecule may be composed of chemically different segments. One segment was assumed to occupy one lattice site, and the vacant sites were treated in the same way as monomer-solvent molecules in the original Guggenheim's theory[294]. Russel and LeVan assumed further, that different chemical segments may have their own adsorption energy distributions. If a certain segment is common for two or more different molecules in an adsorbed mixture, that fragment has still the same adsorption energy distribution. The integral equation (141) takes then the following form 0t~(p)
/ ' ' " / Oi(p,T, ~1,s ..., s163163 fh
(345)
~m
where n means as before the number of components in the mixture, ei is the adsorption energy of ith segment, and m is the number of different segments in the adsorbed mixture. Like in the case of eq. (141) Russel and LeVan considered the possibilities of reducing the m-th dimensional integral (345) to one-dimensional, by considering functional relationships between the adsorption energies ei and ejr of various segments, ei = ei(e i). For the purpose of illustration Russel and LeVan analyzed the Nakahara's experimental data for (ethene + propene) mixture on the carbon molecular sieve MSC-5A[292]. Two kinds of segments, -CH3 and = CH~ were considered in this case by Russel and LeVan. As the segments are similar with respect to the gas-solid interaction, a perfect positive correlation between e-OH3 and e=CH~ was assumed. So, the relation (172) was used to reduce the
90 integral (345) to one dimensional. Good agreement was obtained between the theoretical predictions and the experimental data. The estimated adsorption energy distributions X(eCH3), and X(e=CH,) turned out to be Gaussian-like but had different variances. There was found a broader distribution of adsorption energies for the = CH, segments. One limitation of this very interesting approach by Russel and LeVan is that it can be applied only to surfaces with patchwise topography. Multi-site-occupancy adsorption on heterogeneous surfaces having random topography has been considered very recently by Rudzinski et. a1.[292]. The starting point in their consideration was the paper by Nitta et. a1.[138] on adsorption of molecules composed of different mers, on heterogeneous solid surfaces. Rudzinski and Everett modified further Nitta's approach toward using continuous adsorption energy distributions. Then, the essential point in Rudzinski-Everett approach was to apply the theoretical considerations by Marczewski et. a1.[139] concerning the relations between one mer adsorption energy distribution Xj(ej), j = 1,2,...,m and Xm(el,e2,...,em). In the simplest case of a homogeneous molecule composed of m identical mers, the variance ~,~ of X,~(em) is related to the variance 01 of Xl(e) in a way which depends on surface topography. For patchwise topography, ~,~ = m01
(346)
whereas for random topography 0m = v/mOt
(347)
The variance 0 is related to second central moment #2 by, #2 = ~2
(348)
Marczewski et. a1.[139] also developed relations between 01 and 0,.,., for the cases when the adsorbed molecule is composed of chemically different mers. Marczewski et. al. showed that surface topography also affects the relations between higher central moments of Xm(em) and Xx(e). However, surface energetic heterogeneity manifests itself in adsorption through the second central moment (variance) first. It is related to the width of the adsorption energy distribution, which is the most crucial factor. Higher central moments describe the shape of that function, which is of a secondary importance. Thus, while considering the multi-site-occupancy adsorption on heterogeneous solid surfaces, Rudzinski et. al. took into account only the effect of the number of mers rn on the variance of Xm(em). While considering single-gas adsorption, Rudzinski et. a1.[3,294,295] took the following functions X~ (em) into consideration, . l e x p (~=-~=o) X,~ (em) =
~"
\
~"
(349)
where 0.~ = ~-Z-c~ v~ ' and (Era) ~
r) exp
-
E,~
(35o)
91 where zgm = Era. The effect of surface topography was, thus, coded in the relation 0m = zgm(m), and in the form of the condensation function ecru. Now m denotes the number of identical mers in the adsorbed molecule, and em,m is the minimum value of era. For patchwise surface topography em is given by eq. (139), whereas for random topography ec,~ is given by eq. (140). So, let us consider, for example, isotherm equations corresponding to the quasi-Gaussian adsorption energy distribution (348). Then, from eqs. (139), (149) and (348)we arrive at the following equation for single-gas adsorption on surfaces characterized by patchwise topography, (p), kT
O~P)(p,T)=
K(~')P~11 1 + K(p)p~
(351) kT
K(P) =
(m i]5 7
-~),.,.,-xK'
exp
Cl ]
(352)
where cx is the value of c for one mer. For random surface topography, (r), we have, kT
=
K(")[o~rn-1)p] K<")-(rnK')~r~lexp(e~
(353)
1+
\r
(354)
]
Depending on the correlations between the adsorpion energies emi and emj of the components i and j, one can have a variety of equations for mixed-gas adsorption, being generalizations of eqs. (351) and (353). In their first paper on that problem, Rudzinski et. a1.[291] have considered the Gaussian-like function (349). Then, for patchwise surface topography, we have, kT
e~ ) (p , T) =
K~P)P~"=~'~
(355)
1 + ~ Kj(p).~,j-,jca~ j----1
whereas for random topography, one arrives at the following equation, kT
up) C' "t,A(")(P,T) 9 =
1+
'~ r , , ( ~ - l )
i <'<j
1
kr
(356)
PJ
Eq. (356) turned out to describe well the adsorption of (oxygen + carbon monoxide) mixture in the molecular sieve 10X, studied experimentally by Nolan et. a1.[298]. Nitta's approach[138] which was the starting point for Rudzinski et. al.[3, 293], follows essentially the ideas underlying Guggenheim's theory of solutions. Thus, as in the case of the paper by Russel and LeVan[292], all the equations developed by Rudzinski et al.
92 should apply to chain-like molecules rather, composed of well-defined chemical fragments - - mers. The developed equations may not, therefore, be a good tool to describe adsorption of large molecules having a compact structure. For such molecules Scaled Particle Theory (SPT) should be a better tool to describe the effects arising from different sizes of adsorbed molecules. The STP approach is essentially related to mobile adsorption model, but it seems that surface phase mobility may be of a secondary importance, compared to the effects arising from different sizes of large adsorbate molecules. Interesting remarks concerning this problem can be found in the recent paper by Frances et. a1.[210]. There is one intriguing problem related to the Langmuir-Freundlich isotherm equations for single and mixed-gas adsorption. In all the considerations presented so far, the exponent k__T < 1 was related to the dic spersion of adsorption energies on a solid surface. Keller and coworkers[265, 299, 300] have shown, that Langmuir-Freundlich equations also may be deduced by considering fractal properties of the actual solid surfaces. Theoretical studies along these lines are being continued by Giona and Giustiniani[301-303]. This would suggest that certain relations exist between fractal nature, and surface energetic heterogeneity of the real solid surfaces.
10. C O M P U T E R S I M U L A T I O N S OF M I X E D - G A S A D S O R P T I O N As in the case of single-gas adsorption, computer simulations of mixed-gas adsorption have been used primarily to study adsorption in the systems having a limited geometry. These were mainly activated carbons and zeolites of various kinds. Computer simulations of mixed-gas adsorption were actually started in 1990, by Sokolowski and Fisher[304] using the methods of molecular dynamics, and by Karavias and Myers[305] using Monte Carlo calculations. The latter method has, almost exclusively, been used in the studies of mixed-gas adsorption. Karavias and Myers studied the adsorption of (C02 + C2H4), (C02 -4-CH4), and ( C4H10 q- C2H4) mixtures in zeolite 13X. Their attention was focused on the heat effects accompanying the mixed-gas adsorption. The theoretical predictions were compared with those given by the simplified IAS theory, but no comparison with the experimental data was presented. Rasmus and Hall[306] used MC (Monte Carlo) simulations in the grand cannonical ensemble to study the adsorption of the (N2 + 02) gas mixture in the zeolite 5A. Their studies of mixed-gas adsorption were proceeded by the studies of the single-gas adsorption. Good agreement was found between the experimental and the theoretical heats of 02 adsorption, and a dramatic discrepancy between the simulated and the experimental heats of N2 adsorption reported for this system by Miller et a/.[307]. At the same time the experimental heat of nitrogen adsorption showed a behaviour predicted by the Langmuir-Freundlich isotherm. The results for the multicomponent adsorption isotherms were quantitatively correct; however, their simulation was not able to predict the mixture data quantitatively. The failure of simulation was ascribed by these authors to inaccuracies in the adsorbate-zeolite potential used in the simulation. Such an explanation can also be found in the later studies of mixed-gas adsorption.
93 Maddox and Rowlinson[308] used the grand-cannonical MC simulations to study the adsorption of (N2 q- CH4) mixture in zeolite Y. In a first step, the adsorption of the single gases was simulated[309]. A dramatic discrepancy was observed between the simulated heat of N2 adsorption, and the experimental heat of adsorption measured for this system by Hampson and Rees (personal communication). Maddox and Rowlinson ascribed it to "the omission of the quadrupolar interaction of the nitrogen molecules with the zeolite". However, an inspection into Figure 2 of their paper also shows a disagreement between the simulated and experimentally observed heats of methane adsorption. The latter heats plotted versus the adsorbed amount show a behaviour typical for that predicted by Langmuir-Freundlich isotherm. The aim of the paper by Maddox and Rowlinson was to test the predictions of IAS theory versus the computer simulations of mixed-gas adsorption. Thus, it is interesting to note that for this particular adsorption systems both these theoretical methods led to similar results. Cracknell and Nicholson[310,311] used the grand cannonical MC simulations to study the adsorption of fluid mixtures in the slit-shaped micropores which are often assumed in modelling the adsorption in active carbons. The pore walls were assumed to have a graphitic structure, and the adsorbed fluid was assumed to be a (methane § ethane) mixture. The results of computer simulations were compared with the IAS predictions considering the selectivity isotherms. In a first step the simulated single-gas isotherms were approximated by either the Langmuir-Freundlich equation, or by the Temkin isotherm equation obtained by averaging the Langmuir isotherm with the rectangular adsorption energy distribution. These isotherm equations were next used to calculate the spreading - pressure integrals. The selectivity isotherms calculated by the subsequent application of IAS were practically the same for both the Langmuir-Freundlich and Temkin equations used to approximate the simulated single-gas isotherms. These selectivity isotherms were also close to those given by the direct computer simulations. All the calculations (simulations) were done only for the temperature 296.3 K. For that reason a certain inconsistency did not affect that comparison. Namely, the heat of adsorption curves calculated for both the Langmuir-Freundlich and Temkin equations will be decreasing functions when plotted versus the adsorbed amount. Meanwhile, the heats of adsorption of the single gases predicted by the molecular simulations were increasing functions of the adsorbed amount (Figure 3 in the paper by Cracknell and Nicholson[311]). It has been known for a long time[3], that finite geometry of an adsorption system affects adsorption isotherms (pressure dependence of adsorption) in a similar way to that of the surface energetic heterogeneity. So, this is why the Langmuir-Freundlich and Temkin equations could be successfully used by Cracknell and Nicholson to fit their simulated isotherms of adsorption in the slit micropores. At the same time, however, the finite system geometry must affect the temperature dependence of adsorption isotherms, (heats of adsorption), in a much different way from that the surface energetic heterogeneity does. As has already been mentioned already, the picture of slit pores is frequently used in modelling the adsorption in the carbonaceous adsorbents. Recently a new research has emerged rigorously. This is the determination of the micropore size distribution based on computer simulations of adsorption in pores having different dimensions. Most frequently
94 the slit pore model is accepted for that purpose. The experimentally observed adsorption isotherm is assumed to be the function obtained by the following averaging HM
Ot(p, T) = / O(p,T, H)~(H)dH
(357)
Hm
where O(p,T, H) is the simulated isotherm of adsorption in a micropore the dimensions of which (the width of slit pore, for instance), are given by the parameter H. Hm and HM are the minimum and maximum values of H found in the investigated adsorption system. Next, numerical methods are used to solve the integral equation (357) to recover the pore size distribution ~r The literature concerning that problem will be reviewed in detail in the forthcoming chapters by Lastoskie, Quirke and Gubbins, and by MacElroy, Seaton and Friedman. In most of the computer simulations reported so far, it was assumed that pore walls have a regular molecular structure. This picture may be too much idealized in many cases. Recently Chmiel et a/.[312] have used MC simulations to study the single--gas adsorption in the slit micropores having energetically heterogeneous walls. Gubbins [313] has started simulations of binary gas mixtures in micropores of such a type. Bojan and Steele[314] have performed simulations of single-gas adsorption in the cylindrical micropores with an irregular molecuIar structure of pore wails. That irregular structure created energetic surface heterogeneity. MacElroy and Raghavan[315] have used another model of heterogeneous porous structures while modelling gas adsorption in the silica xerogels. They assumed that the structure of silica gel may be taken to a first approximation as uniform spheres of silica arranged to correspond to an equilibrium configuration of hard spheres. That model has been elaborated further by Kaminsky and Monson[316], who simulated also the adsorption of (CH4 + Ar) mixture in such model systems. It is frequently emphasized in the papers on computer simulations, that a proper choice of gas-solid interaction potentials is of a fundamental importance. The failure of simulation of gas adsorption in zeolites in some cases was almost exclusively ascribed to inaccuracies in the definition of gas-solid potential function and its parameters. This has also been emphasized in the works by Van Tassel et a/.[317] who studied (simulated) the adsorption of Xe, Ar, CH4 and their binary mixtures in zeolite NaA. It seems, however, that one important factor has not been yet sufficiently recognized in the computer simulations of adsorption in zeolites. Namely, zeolites are probably less regular structures than it is commonly assumed. This has, recently, been emphasized by Golden and Sircar[318]. They draw attention to the fact that many common zeolites such as A, X, Y, modernite, etc., may exhibit a substantial degree of physicochemical energetic heterogeneity for gas adsorption caused by (a) structural defects, (b) presence of trace water in the cages, (c) nonuniform distribution of silica-alumina in the framework, (d) location and distribution of one or more cations of different charge densities in the framework, (e) presence of bare and hydrolized cations etc. These factors, which can drastically influence the polarity of the adsorbent and, subsequently, the adsorption characteristics of a gas in the zeolite, may not be quantitatively known or easily modelled. Consequently, gas adsorption isotherms generated for these zeolites by simplified models of gas-solid molecular interactions can be ambiguous.
95 While finishing our consideration in this chapter, we would like to mention that as in the case of single gas adsorption, Density Functional Theory appears to be a powerful tool to study the mixed-gas adsorption in porous systems[304,319-322]. It may be competitive to computer simulations in some cases, because of its advantage of being computationally faster than full molecular simulations.
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(~993) [297] W. Rudzinski, K. Nieszporek, J.M. Cases, L.I. Michot, and F. Villieras, Langmuir 12, 170 (1996) [298] J. W. Nolan, T.W. McKeehan, and R. P. Danner, J. Chem. Eng. Data 26, 112 (1981) [299] J.V. Keller, Ber. Buuserges. Phys. Chem., 1510, 92 (1988) [300] J.U. Keller, Physica, A166 (1990). [301] M. Giona, M. Giustiniani, Ind. Eng. Chem. Res. (in press) (1994) [302] M. Giona, M. Giustiniani, D. K. Ludlow, L. M. Sperle, Lecture delivered at the AIChE Annual Meeting, San Francisco, November 13-18 (1994) [303] M. Giustiniani, M. Giona, L. Marelli, A. Viola, "Fractal Adsorption Isotherms: A Critical Comparison of Energetic and Geometrical Descriptions of Heterogeneity in Adsorption", published in Chaos and Fractals in Chemical Engineering, First Italian Conference, G. Biaroli, M. Giona, A. R. Giona Eds; World Scientific: Singapore (1995), pp. 49-65 [304] S. Sokolowski and J. Fisher, Mol. Phys. 71,393 (1990) [305] F. Karavias and A. L. Myers, Mol. Sim. 8, 23 (1990), 8, 51 (1991) [306] D. M. Razmus and C.K. Hall, AIChE J. 37, 5 (1991) [307] G. W. Miller, K.S. Knaebel and K.G. Ikels, AIChE J. 33, ~94 (1987) [308] M.M. Maddox ~ a J.S. Rowli~so~, J.Chem. Soc. Faraday Trans. 89, 3619 (1993) [309] G.B. Woods, and J.S. Rowlinson, J. Chem. Soc. Faraday Trans. 85, 765 (1989) [310] R. F. Cracknell, D. Nicholson, and N. Quirke, Mol. Phys. 90, 885 (1993) [311] R.F. Cracknell and D. Nicholson, J. Chem. Soc. Faraday Trans. 90, 1487 (1994) [312] G. Chmiel, L. Lajtar, and S., Sokolowski, J. Chem. Soc. Faraday Trans. 90, 1153 (1994) [313] K. Gubbins, Lecture delivered at the NATO Summer School on Physical Adsorption held in Nice (France), May 20-31 (1996) [314] W.A. Steele, Lecture delievered at the Fundamentals of Adsorption conference, in Asilomar Center, California, May (1995) [315] J. M. D. MacElroy and K. Raghavan, J. Chem. Phys. 93, 2068 (1990) [316] R.D. Kaminsky and P.A. Monson, Langmuir 10, 530 (1994), see also J. Chem. Phys. 95, 2936 (1991) [317] P.R. Van Tassel, H. T. Davis and n. V. McCormick, Langmuir 10, 1257 (1994), see also Mol. Phys. 76, 411 (1992) and J. Chem. Phys. 98, 8919 (1993) [318] T.C. Golden and s. Sircar, J. Coll Interface Sci. 162, 182 (1994) [319] E. Kierlik, M. Rosinberg, J. E. Finn and P.A. Monson, Mol. Phys 75, 1435 (1992) [320] E. Kierlik and M.L. Rosinberg, Plays Rev. A 44, 5025 (1991) [321] Z. Tan, U.M.B. Marconi, F. Van Swoll and K.E. Gubbins, J. Chem. Phys. 90, 3704 (1989) [322] Z. Tan and K.E. Gubbins, J.Phys. Chem. 96,845 (1992)
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W. Rudziriski, W.A. Steele and G. Zgrablich (Eds.) Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces
Studies in Surface Science and Catalysis, Vol. 104 9 1997 Elsevier Science B.V. All rights reserved.
Application of Lattice-Gas Models to Describe Mixed-Gas Equilibria on Heterogeneous Solid Surfaces
105
Adsorption
Yu. K. Tovbin Karpov Institute of Physical Chemistry Vorontsovo Pole Str. 10, 103064 Moscow, RUSSIA
In order to describe the separation and purification of gases and liquids by adsorption [1,2], heterogeneous catalysis [3] and many similar processes, it is indispensable to describe the adsorption equilibria of mixtures of components on real solid surfaces. Early experimental data on the adsorption of mixtures were reported in papers [4,5]. Recent papers discuss a large number of binary adsorption systems [6,7]. Most of those data relate to the adsorption of hydrocarbon mixtures by activated carbons, silica gels and zeolites. A generalization of the Langmuir model for the adsorption of mixtures on homogeneous surfaces was given by Markham and Benton [8] (see also [9-11]). The energetic heterogeneity of real surfaces and the lateral interactions between adsorbed molecules can result in considerable deviations from strictly ideal behaviour of adsorbed phases on homogeneous surfaces. Calculations of adsorption equilibria are connected with a search for the optimum conditions for carrying out separation processes over a wide range of temperatures, total pressures and partial pressures of various components. In a similar way as for one-component systems, an increase in surface coverage results in a corresponding increase of the role of the cooperative effects and of the related processes of local clustering and macroscopic ordering of the adsorbed molecules. To describe real adsorption systems it is necessary to simultaneously calculate the cooperative effects and the heterogeneity of the adsorbent surface. At present, the theory of adsorption equilibrium for gas mixtures based on the lattice-gas model is elaborated at the same level as for one component systems. The equations obtained for adsorption equilibria can serve as a basis for the verification of the applicability of the simplified equations which are widely used in the description of experimental data [12,13]. Below, we will discuss the basic results obtained for the case of approximately equal sizes of the components of the mixture. This is the most commonly used assumption (explicitly or otherwise) in most papers on the adsorption of mixtures. Rejection of this assumption highly complicates the description of the adsorption equilibria. This problem is briefly considered in Section 6. Special emphasis is laid on the molecular basis of the theories and on the physical meaning of the simplifications used. Before considering the influence of surface heterogeneity and lateral interactions between adsorbed particles (Section 3), peculiarities of the adsorption of mixtures on heterogeneous surfaces are considered in the absence of lateral interactions (Section 1) and on homogeneous surfaces but including lateral interactions (Section 2). Multilayer adsorption of mixtures on non-porous adsorbents, and in
106 porous materials is also considered in Section 4. The equations for the partial adsorption isotherms are used to calculate the mixture separation coefficient (Section 5).
1. T H E O R Y OF M I X E D G A S A D S O R P T I O N O N H E T E R O G E N E O U S FACES IN A B S E N C E OF L A T E R A L I N T E R A C T I O N S
SUR-
1.1. T h e lattice gas m o d e l for adsorption of a m i x t u r e While describing surface processes, their equilibrium and kinetic characteristics usually are defined for unit surface . It means that we are interested in an average of various physical quantities which are obtained by averaging over all the local fragments of the surface. In the lattice model, the local fragments are surface centers (sites). In the monolayer lattice model, the surface of the adsorbent A can be divided into sites M ah, where h is the site number of the surface. For a homogeneous of area ah: A = ~--,h=l surface, ah = A / M , where M is the number of sites on unit surface. For a heterogeneous surface the area of a site can change from one to another. Let the site's volume be Vh = o'hd where d is the linear size of a molecule. Also, let the number of components of the mixture be s. We will limit our considerations to the case of localized adsorption. Every adsorption site can be occupied by one molecule of the mixture or it can be empty; i.e., the number of occupation states of every site is equal to s' = s + 1. In the case of adsorption of component i on a homogeneous surface, the sites of the lattice structure represent a periodically repeating area for localization of molecules, in their minima of the potential energy. On a heterogeneous solid surface, the dependence of the potential energies Ui(r) on the coordinates r (r = (x, y , z ) ) is more complicated. A wide spectrum of areas of localization will exist for adsorbate molecules having different sizes and different binding energies. The positions of these potential energy minima are, generally speaking, nonperiodic. The situation gets still more complicated when the adsorption of mixtures is considered. Even for a homogeneous surface, the potential Ui(r) for various kinds of molecules i differ, and can have different positions of the individual points: minima, maxima and saddles. Those particularities are connected with differences in the interactions of molecules with the adsorbent. On a homogeneous surface, however, the periodic character of Ui(r) is kept and the partition of the surface into sites (when the sizes of the mixture's components are equal), can be done for molecules of any kind. Then, at the center of every such point there is an energy minimum for a molecule i (for example), and other molecules j there may have other coordinates for their minima. In any case, the positions of the local minima shall not differ more than by half a lattice constant. Every site h can be characterized with a set of local Langmuir constants a~, 1 < i < s, 1 < h < M. For a homogeneous surface, Aij = a~ - a ~ , does not depend on the number of the site h. On a heterogeneous surface, the potential Ui(r) can be different for various molecules not only by the current meaning of energy and by the position of the particular points, but also by the number of the local minima N i. As a consequence, if the partitioning of the surface into sites is made according to such a rule as for a single component (the center of a lattice site ~ the minimum U~(r)), then, for various molecules i, various kinds of fragmentation can be done which may result in treating various functions by various kinds of lattice sites. In particular, the sites of the lattice for the kind i can be without
107
the local minima of energy for molecules j. In order to use the advantages of the lattice model - those connected with a discrete change of coordinates - it is necessary to formulate some more formal rules of partitioning the surface into sites. Let us remark that all the sites have more or less similar sizes though some differences cannot be excluded. This limits their differences in such a way that the main condition for the admittivity of the model could be assured, i.e. every site can be occupied or free, and every molecule can occupies one and only one site. The procedure of partitioning the surface into sites turns out to be common for all the molecules of a mixture. It requires the presence of one value of M, such that N i < M, 1 < i < s. Of course, as the support for a chosen kind of molecules g, such a kind of molecules should be used for which the condition N t = M can be fulfilled. If there are several such kinds, any of them can be used as the supporting one. To the supporting kind g corresponds a partitioning of the surface into sites with area a~ (as in the case of adsorption of one component g), as well as the local constants of the Langmuir equation. For molecules of any other kind, formally, local Langmuir constants are introduced which correspond to their volumes v~. Let us denote those constant values as a~ (g). In consequence, s meanings of the local constants of Langmuir correspond to every center h"
(i)
4(e) = f f exp [-#,d drda/F~
where ei is the binding energy of molecule i to the surface. The integrals are over the coordinate of the molecule's mass centre r, inside the site volume v~, and over the orientations of molecules ft. The formula (1) can be presented in the form [6,11]
a~(g) = F~(g)/3exp [/3Qh(g)]/F i ~= s
i [flQh(g)]
(2)
where Q~(g) is the energy of binding of molecule i to the site h, F;,(g) is a statistical sum for the molecule i on the site h, F ~ is the same quantity for the molecule i in the gas phase, and a ~ ( { ) = F;,(g)~/F ~ is the preexponential factor, l~ = (ksT) -1. On a heterogeneous surface, adsorption sites are characterized by a different adsorption capability and their coverage depends on the value of that capability. We can introduce the concept of a local coverage of the site h characterized by a~(g)" e~(g) --
a~(g)pi
,
1 + k a~(g)pi
0h(g)= ~ e~(g)
(3)
i=1
i=l
where 0},(g) is the local coverage of site h by molecules of i, Oh(g) is the local coverage of the site h by all the molecules of the adsorption mixture, and pi is the partial pressure of component i. The macroscopic coverage of the surface can be calculated from the local coverages (3), by taking the sum for all the sites M
MO~=~ O~(e), h='
M or
M
o~= M-1 E O~(e)-- M-1 E h=l
h=l 1 +
9
ai,(e)p~ fi a~(e) i=1
(4)
108 M
and similarly, 0 = ~] Oh(g)/M. h=l
Equations (4) describe isotherms of adsorption (partial ones for molecules i, as well as the full one) on a heterogeneous surface in absence of lateral interactions between adsorbed molecules. 1.2. D i s t r i b u t i o n f u n c t i o n s for various kinds of a d s o r p t i o n sites For large M, the calculation of macroscopic values by eq. (4) is not suitable. To simplify the calculations of averages, one introduces distribution functions for sites of various kinds. For the one component system, such a procedure is simply the partition of all the sites of the surface into t groups of sites having similar adsorption features. This allows us to apply a simpler procedure of calculating averages: instead of averaging over all the centers, we average over the groups of identical centers (t << M). Such a procedure means that one does not average over the space coordinates, but energetic coordinates because in the first approximation, the adsorption capability depends on the energetic binding between the adsorbent and the adsorbate. In the case of mixture adsorption, to every surface site h there corresponds s values of the Langmuir constants, and for every kind of molecule, such a partition results in its own distribution function. If one wants to keep the analogy with one-component adsorption, it is recommended that one carry out the partition with the help of molecules of supporting kind g defined in the preceding subsection. Thus, we introduce the discrete distribution function for sites of various kinds according to their adsorption capability fq(g) defined by the following relations: fa(e) = Na(g ) M '
t(O E fq(g) = 1, q=l
aq(g)in _< a~(g) _< aq(g)r~ aeq(g)in ~ atq(g)~ ,
for
1 <_ i(# g) < s,
(5)
for i = g,
where Nq(g) and t(g) are the number of sites of kind q and the number of types of surface sites with respect to molecules g. In the relations (5), it is assumed, that in the group of sites q are gathered all the sites h for which we have ath(l) = atq(1). The constants a~(1) are defined in eq. (2) where the index h is replaced by the index q for the constants depending on the molecular properties of the sites, and not on the assigned number. On the other hand, in a general mixture, the classification of sites according to identical values of the constants ath(g) leads to an absence of relations between the local Langmuir constants for other kinds of molecules aih(g). These differences depend on the nature of the interactions between the adsorbent and the adsorbate as well as on the lattice structure. The differences between a~ (g) become more important as the differences of potentials of interaction between the adsorbent atoms i and the adsorbed molecules i and g grow. In consequence, the values ah(g ) can change i(g)lin, where i(g)fin and i(g)in and the smallest in a certain range from a~(g)in to aq aq aq values of a~(g) correspond to a given kind of sites q. The last relation (5) results from the condition that the sites are partitioned into groups.
109 If the condition analogous to the second condition of (5) is fulfilled for molecule i -# g and
aiq(e)in ~ aq(e)fm ,
(6)
we can talk about a definite correspondence of the kinds of sites and the adsorption i capabilities of the molecules of various kinds. In such a case, A~,(h) = ah(ig ) - a~(g) does not depend on the number of sites h relative to the sites of kind q. The last case is usually assumed in the majority of the papers dealing with the theory of mixture adsorption . The expression for the adsorption isotherm can then be written as: t(s
Oi = ~
t(s
fq(/)Oq(/) = ~
q=l
q=l
fq(gl aq(/)pi 1 + s 4(g)pj j=l
t{l)
e = ~_. q(e)Oq(g)
,
oq(e) =
q=l
o~(e),
(7)
i=1
where 8~(g) is the coverage of the sites of type q with molecules i for supporting kind of molecules g and ~q(g) is the full coverage of the same sites. In the general case of eq. (5), it is necessary to use the multidimensional energy distribution functions f(qz,-.., q~) = N(ql, ..., q~) M '
t(1) ~""
ql=l
t(~~ cb=l
f(q~, ..., qs) = 1 ,
(8)
where N(ql, ..., qs) is the number of surface sites possessing the adsorption capabilities aq1(1) for molecules of the first kind, aq2(2) is for molecules of the second kind, etc, up to a~(s) for molecules of type s (the dimension of the function be defined by the number of components of the mixture s); 1 < qk < t(k), 1 < k < s; qk is the type of site for the adsorption of molecules of kind k, and t(k) is the number of types of surface sites for adsorption of molecules of kind k. The second equation (8) defines the normalized distribution for this distribution function. Using the distribution function (8), we can express the partial adsorption isotherms as"
0i
=
t(1)
t(s)
f(ql,---,; qslaq(g)pi qs=l 1% E aJq(g)PJ j=l
~---~ ... ~ q1=l
(9)
Eq. (9) is a symmetrical picture (contrary to eq. (7)) that takes account of the adsorption capabilities of all kinds of molecules on various surface sites, but with the initial partition of the surface into sites done with respect to the molecules of the selected kind g. The use of another kind of molecule k for the partition of the surface into sites is given by some 9 ) and corresponding aq~(k) . other distribution function f(ql, ...,q,) and local constant a'h(k On the other hand, the calculation of 0i should not depend on the kind of the selected
II0 molecules at their identical sizes if they do not bind the position of the local minimum Ui(r) with the center of the site, and if one uses the formal partition of the space into areas having the sizes characteristic of the diameter of the molecules. Let us observe that the application of the idea of grouping lattice sites q causes the disapearance of the information about the spatial distribution of the lattice sites, belonging to this group. This can be done accurately only in absence of lateral interactions. When one takes into account the lateral interactions (and especially in kinetic considerations) the spatial distribution of the lattice sites of various types plays an important part and it is necessary to consider it. 1.3. T h e integral f o r m of the a d s o r p t i o n i s o t h e r m If the number of types of lattice sites t is large (1 < < t < < M), then we can integrate instead of summing in eq. (7):
#i = / f l ( x ' ) O i ( x ' ) d x ' , t xi n
~i(Xl)=
O = f f~(xe) i xitn
ai(xt)pi i+ aj(x')pj j=l
(I0)
0i(x')dx'
i=l
n
/ f (x )dx = 1 x.tx n
where x e defines the adsorption capability of molecule ~, zi,t ~ and x~i~ are the initial and the final values of x t, f t ( x l) is the continuous density distribution function of sites according to their adsorption capability, defined for molecules of kind ~. It characterizes the lattice sites existing on the unit area of the surface and having the values x t in the interval from x e to x l + d x l. The differential distribution function f t ( x t) usually normalized to unity. The a i ( x l) is usually the local Langmuir's constant for molecule i that corresponds to lattice sites h which have a given adsorption capability x e for molecules ~. Strictly speaking, instead of x t in the expression f t ( x t) one should use a t ( x t ) . Usually, however, ^ -- aq ^e = 5t ,~ c o n s t , and the basic contribution to the change of the one assumes that a~h local Langmuir constants comes from the energy of binding of a molecule to the surface Qtq = Q~h. In this case, x corresponds to this binding energy, and the upper and the lower intervals of integration are the smallest and the largest energy of binding of the molecule to the adsorbent surface. In the case of a large number of types of sites, eq. (9) can be written in the form: xL
x~o
-/.../,/xi,..., xs ax , /--'/f(xl,...,xs)dxi---dxs=l
(11)
,
Oi(x = j--1
111 where f(xl, ...,xs) is the multidimensional continuous distribution function of the lattice sites of various types having the adsorption capability from x~ to xi + dxi, 1 < i < s; the function f(zl, ...,xs) is a continuous analogue of the discrete function of eq. (8). Formally, when ones chooses the integral form of equation (10), the description of the adsorption isotherm becomes simpler. It involves a considerably smaller number of molecular parameters than in eq. (7). If we use eq. (7) we must know the t 9s Langmuir constants, where t = t(/), and ( t - 1)is the number of h(g). Even for a simple where case s = 2 and t = 10, we need 20 Langmuir constants and 9 values of fq(g). When we use eq. (10) we must know x~,~, z~i ~, two parameters of the Gaussian distribution function fe(x ~) (for example), and the correlation between the values of ae(x l) and ai(xe). A considerable simplification can be achieved when the following conditions are fulfilled: 1) the definition of fq(g), 1 < q < t for a given adsorbent surface composition, can be approximated to a good accuracy by the distribution function fe(J) and, 2) there exist some simple relations between ae(x ~) and ai(xe). To fulfill the first condition is not obviously a difficult task but the realization of the second one is more problematic. If we assume that a simple relation between he(Z.e) and a~(x ~) does not exist, passing over to the integral form of the equation does not give any advantage in comparison to eq. (7). The reason is that in both cases (for any concrete q in (7) or for any current meaning x t in (10)), a full list of all the local Langmuir constants aiq(g) or a,(xe), 1 _< i <_ s is necessary. That is why looking for simple relations between ae(z ~) and a~(x e) is a central question in the attempt to simplify the description of the adsorption of mixtures on heterogeneous surfaces. In order to find the relation between ae(z t) and a~(x e) one should consider the molecular interpretation of the binding between molecules and the adsorbent surface. 1.4. Physics of the adsorption of mixtures on h e t e r o g e n e o u s surfaces The fundamentals of the theory of adsorption of mixtures on heterogeneous surfaces were presented by Roginski and Todes [14-17]. Those fundamentals have remained practically unchanged till the present. As a measure of the adsorption capability of the lattice sites of various kinds, one needs the energy of binding for various kinds of molecules with a certain adsorption site. The preexponential factors of the local Langmuir constants ^~(g) = 5~(x~) ~ hi are considered to be independent of the kind of lattice site. In ~h(g) = aq this case, the binary relations between the local Langmuir constants for molecules i and j on various sites x ~ were found as a functional dependence between the binding energies of those molecules:
Qj(x') =
[Qi(x')]
(12)
In fact, in most cases, the shape of the function ~2ij depends on the molecular nature of the system. Roginski and Todes separated three kinds of molecules that differ by their potentials of adsorbate-adsorbent interaction. The dominant factors are: 1) nonspecific dispersive forces, 2) electrostatic forces of interaction between the charged particles of the surface and the permanent or induced dipoles of the adsorbate, and, 3) chemical adsorbent-adsorbate interactions. 1. In the case of non-specific interactions between adsorbate molecules and the surface, their potential energy of attraction can be represented as: V = -CasoZaa/d 3, where C is a constant, depending on the ionization potential of the adsorbate molecules and of
112 the atoms of the adsorbent, aa is the polarizability of the adsorbate molecule, asol is the polarizability of the surface atoms, and d is the molecular diameter. It allows one to "separate" the contributions of the surface and the adsorbate. We may consider in the first approximation, that if we equate the adsorption heats with the potentials of interaction Qi "~ Vi "~ asotgi, gi = C o l i / d i 3, then, when passing from one part of the surface to another, the adsorption heats should also change proportionally to the local polarizabilities between the studied parts ol~ot(xl). In such a case, the ratio of the adsorption heats of two different gases should be constant to a first approximation: Qj(x e) = flijQi(x ~) ,
~ij = gi , gj
(13)
for all points of the surface, even though the heats can change according to that part. Changes in Qi and Qj for various x l, take place in the same direction (likeness, see Fig.l). If we neglect the changes of the polarizability of the molecules of adsorbate ai and aj for various x ~, the proportionality constant can be estimated from the relation ~ij = aiaj ~ T~,r ~ ~ , where Ti,c is the gas-liquid critical temperature of gas i, and bi is the Van der Waals constant for gas i.
Q2
1
3
~ ~2
Q1 Figure 1. The relation between adsorption heats Q1 and Q2 of the adsorbed mixed species: 1 - changes in the same direction; 2 - changes in the opposite direction; 3 - noncorrelated dependence (black dots)[16]. 2. In the first approximation, the electrostatic forces of interaction also result in the relationship (13) where the proportionality constant depends on the nature of the potential. In the case of adsorption of polar molecules on metallic surfaces, the mirror image potential is proportional to #2, where #i is the dipole moment of molecule i. That is why when the mirror image forces prevail in adsorption, ~ij '~
~
. In the case of adsorption
of slightly polarizable molecules on ionic lattices, the basic role (as in case of dispersive
113 interactions) is played by the polarizability of the molecules and therefore/3~j ~ ~. In aS the case of the adsorption of permanent molecular dipoles on an ionic lattice,/3ij ~ ttj u~. Thus, in the case of the simultaneous adsorption of various molecules characterized by identical kinds of interactions (dispersive or electrostatic), one should expect roughly likeness relations between all the adsorption heats. 3. Chemical adsorption. Chemical interactions are characterized by a strong specificity of binding and the character of the relations between the adsorption heats Qi and Qj for various parts of the surface can be quite different. 3-a. If one considers the adsorption of adsorbates which of a similar chemical nature, e.g., various halogens or various olefins, then one can expect likeness changes of adsorption heats while passing from one part of the surface to another. At the same time one can expect that the relationship (12) will be linear, Qj = AijQi + Cij ,
Aij, Cij -
const.,
Aij > 0
(14)
or more complicated. 3-b. When one considers the adsorption of substances of opposite functions, e.g., acids and bases, then it can be expected an antilikeness will exist in the heats. In such cases the condition (14) is fulfilled, but with Aij < O. 3-c. Finally, some situations are possible when, for two chemisorbing molecules, correlations don't exist between the adsorption heats Qi and Qj. In Fig.l, a set of points corresponds to such a case, (Qi, Qj) for various parts of the surface. Cases 3-b and 3-c should be more probable when examining gas mixtures characterized by potentials of interactions of various types. These will also depend on the peculiarities of the surface topography. When the adsorption heats of the components of a mixture are related to each other by the function (12), then if we know them and if we know the distribution function for one of the kinds of molecule, e.g. i, we can obtain the expression for the distribution function for the other kind of molecules. It follows from the fact that a group of lattice sites of surface dM can be characterized by the adsorption heats of various molecules [17] dM = fi(Qi)dQi = fj(Qj)dQj ,
or
fj(Qj) = fi(Qi)
dQi
(15)
where dQ,/dQj follows from the relation (12). Let eq mean an inverse function to ~,j(Q,). Then dQ~/dQj = e~j(Qj). The sign of the modulus concerns to the antihkeness run of the curves. The sign of eij is positive for the surface fragments dM. In a particular case, for the likeness relation (14), eq. (16) takes the form: fJ(QJ)=~~-jfi( Qj-CijAij )
(16)
When A~j = 1, the functions f~ and fj are identical. A change from Qi to Qj produces a displacement of the whole distribution by a constant amount toward increasing Q (at the positive value of Cij) or toward the opposite direction in the case of negative Cij (Fig.2a). When Aij > 0, the function fjcan be calculated from f~ by a simultaneous extension of the scales along the axis of the abscissa in Aij, one time, and diminishing of the scale
114 along the axis of the ordinates by the same amount of times. In other words, the curves
fj(Qj) and f~(Q~) can be considered as similar to each another. The displacement of the whole curve along the abscissa by the value ~AO corresponds in this way to Cij # O. (see the curves on Fig.2b).
a
b
f, I
I
I
C12
2
,
I
I I
~-~ C12
Q
C12
I
C fl
I
I
I
I
f2
I I I
Q
I I I
I I l
I I i
Q
C12/2
Figure 2. Distributions of the adsorption energies of the components of a binary mixture: (a) identical distribution functions, A12 = 1; (b) similar distribution functions, A12 > 0; (c) opposite dependence, A12 < 0 [17]. For Aij < 0 there is an antilikeness change of the adsorption heats. According to (15), eq. (16) can be rewritten in the form: fj(Qj) = f~
(Co-Qj) IA,~l /IA,ji. In such a case,
the distribution fj(Qj) can be obtained from the distribution f~(Q~), with additional reflection around the vertical line Qj = -q-~, as shown in Fig.2c at A~j = - 1 . It should be noted that these conceptions about a likeness and an antilikeness of the changes in the adsorption heats of the gas mixture mean that, generally speaking, a whole range of functional relationships exist, instead the linear relation (14). These equations are partially expressible: Qj = AijQ'~ and Qj = C~j exp [-AqQi], where Aij, Cij and n are constants. In Fig.3, relationships are shown between f(Q2) and the uniform distribution function f(Q~) = const, for n > 1 (a) and between f(Q2) and the exponential function
a
f
I I I
', I I ,
b
"
fl --i
fl I I I
I I I I
"-4.. ! I !
I I I I I
!
Q
I
I I I I / ~
I I
I i
I
I
Q
Figure 3. Examples of the relations between the distributions of adsorption energy for the components of a binary mixture: (a) power-law relation, Q2 = A12Q~2. For the first component, a constant distribution is shown; (b) exponential relation between adsorption energies: Q2 = C12exp(-A12Q1), C12, A12 > 0; for the first component, an exponential distribution function is used, f(Q1) = con,st, exp(gQ1), g > 0 [17].
ll5
f(Q1) = const, exp[TQl] when 7 > 0 (b). In other words, monotonic relationships are admissible: OQj/OQi > 0 in the first case and OQj/OQi < 0 in the second case, for any z I. The case 3-c corresponds to the absence of correlations between the adsorption heats Qi and Qj on various parts of the surface. That means the presence of more complicated relationships (12) than monotonic, as well as fullfilling the conditions (5). In the last case, the distribution function turns out to be multidimensional and the expression (15) is no longer true. If one uses the multidimensional functions then the calculation of the thermodynamic characteristics of the adsorption mixture becomes complicated. In order to do it, some additional suppositions must be introduced. One of the simpler ones is the statistical independence of each of the functions, x~
s
f(x,,...,xs)= Hf(Xm)
,
/f(xm)dxm=l
,
(17)
d x.m
m=l
in
This case was considered to be fundamental by Roginski and Todes when considering the adsorption of a binary mixture of gases, one of which is chemisorbed and the other physically adsorbed. In Fig.4a one can see the section of a two dimensional distribution function in the absence of correlations between the distribution functions of the individual components. To any value of the adsorption energy of the second component from Q~ to Q fin 2 , there may correspond any value of the adsorption energy of the first component Q~.
~
f--r-f2 ~,
'Q1 ~,
~ -9-
"
I |
f
S ' Q1
J ' Q1 I
Q
Figure 4. Range and constant energy contours of a two-dimensional distribution function for binary mixed adsorption: (a) with no correlation in adsorption energies; (b) with correlation; (c) with an unambiguous correlation between adsorption energies. In the case of a partial correlation between the adsorption energies, a limited range of [')fin , values from Qi2"(QT) to ~2 (QT) corresponds to O~(Fig.4b) (i.e. not the whole interval (-}fin from Q~ to , ~ corresponds to a particular energy Q~). This situation corresponds
116 to condition (5). This case of course, corresponds best to the real physical nature of adsorption of mixtures but it is the most complicated. This is because in equations (9), (11) the order of integration (summing) is very important, according to the "internal" variables from the dependence of their initial and final meanings on the current fixed meanings of the other "exterior" variables. Functional relationships between the adsorption heats (12) mean strong correlations: to every value of Q~ there corresponds a certain single value of Q2(Q~). Such relations correspond to the conditions (6) (see Fig.4c). They are fundamental for practical calculations and they are going to be considered in more detail further on. General expressions for the partial adsorption isotherms with multidimentional energy distribution (for any number of components mixture) were given first in a paper by Balandin [18]. The author used, however, only the case of the complete similarity of the individual distribution functions. Jaroniec and Rudzinski [19-21] gave a deeper consideration of the case of multidimensional energy distributions. A practical application of that idea meets many difficulties and cumbersome calculations are necessary. In the paper [20] the isotherms of two binary adsorption systems were described: ethane-ethylene/carbon (Nuxit AL) at 333~ (experimental data by Syepesy and Illes [22]) and CO-N2/CsI at 83.56~ (experimental dat~ by Tompkins and Young [23]). Assumption (17) was widely used in the first papers by Roginski and Todes. It somewhat simplifies the calculation which still remains complicated. The unavoidability of calculating the effect of competition for various kinds of molecules does not allow one to simplify the expressions for the local adsorption isotherms, even for the case described by eq. (9). That is why a calculation of the isotherms is indispensable. This assumption was used in paper [20] to describe the adsorption data for the mixture (CO + N2)/CsI [23]. Jaroniec[21] has pointed out that the use of the extended empirical equation of Jovanovic for gas mixtures on a homogeneous surface has the form: 0({p}) = 1 - exp [- ~i~__1Kipi] for the local isotherm. Then using the approximation (17), one obtains the following expression for a heterogeneous surface:
0({p}) = 1 - 12i f exp [-Kipi] f(xi)dxi = 1 - 11I (1 - 0i(pi)) i=l
9 xl.
(iSa)
i--1
where the 0i(pi) are adsorption isotherms of the individual components. When s = 2, a similar equation was obtained by Roginski and Todes [14,15] who postulated that it is valid for any kind of adsorption isotherm for the pure components. Misra [24] proved that the result of averaging a local isotherm of Jovanovic with an exponential distribution function can be effectively approximated by the classical Freundlich isotherm. Jaroniec then obtained the following general form of Freundlich's isotherm for gas mixtures [21] ..F ~2 -lxlr~2Pl T.F..F ~1P2~2 , 0 = K F p ~ l+ix2p2
(18b)
where K F and 0i are the parameters of the Freundlich equation for the pure gases. This equation was used for the description of experimental data for the binary systems (CO +
117 N2) and (At + 02)[23]. Other results of this approach have been discussed in a survey by Jaroniec and Madey [12]. 1.5. Simplified m e t h o d s of describing the a d s o r p t i o n of gas m i x t u r e s In the subsection, the partial adsorption isotherms for mixtures with functional relationships between the adsorption heats of various molecules are discussed for ideal adsorption.
1.5.1. Quasi-Homogeneous Heterogeneous surface The term "quasi-homogeneous surface" to describe the adsorption of mixtures on heterogeneous surfaces had been introduced and applied by Balandin in his theoretical paper on catalytic processes [18,25]. The expression means a full similarity of the adsorption heats of the components - a case where the change of adsorption heats of various components remains constant (see Fig.2a) when passing from one part of the surface to another. This is the simplest form of relationship (14): Aij = 1, and it leads to the following relationship between the Langmuir constants for various molecules: ai(X i) -- aj(Xi)~ij
,
~ij =
~ / ~ j exp{-~Cij}
,
(19)
(For simplicity, the index g is omitted below). In fact, the condition (19) removes the concentration differences of kinds of molecules in relation to sites of various kinds. In such a case, eq. (10) has the form: Xfin
0= / Xin
f(x)a,(x)p~f dx , 1 + a~(x)pef
0i = 0~ig p---Li Per
(20)
Per = ~ 6ijpj j=l
It can be seen from the above equations that the total coverage of the surface 0 with the molecules of a mixture appears as if it were concerned with one a kind of molecule g at the effective pressure p~f. Balandin applied in his papers a constant and an exponential distribution function. For the first distribution function, he arrived at a generalized form of the quasilogarithmic isotherm of Temkin for a gas mixture:
0i =
Pi . in (1 + a ~ A i ) Aifl(xfm - Xin) 1 + a~Ai
Ai=~ '
s j=a ai
(21) '
_fin and a iis are the Langmuir constants for the lattice site characterized by the where ui greatest and smallest adsorption heats, respectively. His equation was repeated many times in other papers (see e.g. [26-28]). Temkin has shown [27] that the condition (19) allows us to write a partial adsorption isotherm for the components of a binary mixture of A and B, provided that we know the adsorption isotherm of the pure substance 0A = F(0A), where F is a functional relation such as: OA = p A F (r)/~r where r = PA + 5ABPS. In particular, when the pure component
118
0 <_ 7 ~ 1, then the partial isotherm is described by Freundlich's isotherm 0A = I'(~pA, F of a mixture of A and B can be expressed as: OA = KFAPA/(PA + 8ASPS) 1-'Y. In the paper [26], the condition (19) had been used to obtain equations for partial isotherms by generalizing the quasilogarithmic (for dissociating molecules), negative power and Freundlich's isotherms for an arbitrary number of components , and also can be adopted to express configurational entropy and enthalpy. In particular, in the middle coverage range, for negative power distribution functions, the system of equations for the $ partial isotherms can be expressed as: a~ = D (1 - 8vDS), D = y~/~-~j=~ yj, where 0v is a portion of free surface (6v = 1 - 0). Here, yi is related to the sites characterized by the maximum adsorption heats with 1 > ~ > 0 by y~ = (aipi) 1/m~, where m is the degree of dissociation of the molecule i in the gas phase. The value rni is usually equal 1 or 2. Relations (19,20) make it possible to use the results obtained for one component systems. Thus, the isotherms of pure substances can be extended to mixtures. In this way, in paper [29] a total adsorption isotherm was obtained which turned out to be in the general form of the empirical four-parameter equations of Marczewski and Jaroniec [30] and of the equation by Dubinin and Astakhov [31]. In the first case, the total isotherm has the following form:
0=
( ( a ~ p e f ) k ) ~/k 1 + (a~pef)k
where a~ is the Langmuir constant of the species ~, for the sites having the maximum adsorption heats. From the above equation, other empirical equations follow. They are generalizations of the Langmuir-Freundlich isotherm when 0 = k (the Toth isotherm when = 1, and the generalized Freundlich isotherm when k = 1). In the second case, the total isotherm can be expressed as: 0 = exp { - B ~
j (~-~f) }
where B ~ is a parameter related to the surface heterogeneity, and pat is the energetic parameter for the l'th component. From this formula, at j = 1, one obtains the Freundlich isotherm, and with j = 2, the Dubinin-Radushkevich isotherm. From the partial isotherms (20), we can obtain the linear relationship:
Oi ~il Pi In ~j = In ~jl + in--pj
(22)
The slope of the expression is equal to 1 for both a homogeneous surface and for the quasihomogeneous heterogeneous surfaces. Its deviation from unity means that condition (19) was not fulfilled.
1.5.2. Other Relationships (14) Tompkins and Young [23] and Glueskauf [32] considered the adsorption of a binary mixture on a heterogeneous surface with an exponential site energy distribution function and with C~j = 0 (14). The authors found two characteristic cases: c~' < c~' + R T and c~' > d~ + RT, where c~ is the heat of adsorption of molecule i on a completely covered surface. In the first case, the adsorption isotherm at medium coverage of molecules of
119 the 1st kind, in the presence of molecules of the second kind, is similar to the Langmuir isotherm: 01 = Kip1/
_(<7'1<7') }~ x-21-,2 pl + ~,-
,
a = (e~ -
RT) 1~7,
r
e~ > RT
(23)
At small values of pl the relationship (23) is linear, and at large values of pl the relationship (23) saturates. In the second case, in the range of medium total coverage, the molecules of the first kind - even in presence of a large excess of the second component - follow the Freundlich isotherm" 01 = KI{pl/[p~ ~'/<7) + g2p2]}RT/(<7-~7). There is, however, a tendency toward saturation at larger coverage. If the surface is practically saturated and it contains an excess of the more strongly adsorbed component, then in both cases, the condition (22) is fulfilled. Some more convenient correlational relationships than formula (22) were suggested by Jaroniec [33] based on the following generalization of the Langmuir-Freundlich equation for gas mixtures: 0= ~
1 +
(a;Pi)~
'
oi =
O(a;pi)e
(24)
E(arpi) '~
i=l
i=l
where ~ is a parameter of heterogeneity which characterizes the adsorption of all the components on a given surface, a~ corresponds to the Langmuir constant for the component with the maximum adsorption heat. According to formula (24), there is a relationship between the partial isotherms:
Oi
a~'
Pi
&~'
pj
ln~ = ~ln-- +~ln--
(25)
A complete survey of the application of correlations was given in [12].
2. A D S O R P T I O N O F A M I X T U R E O N A H O M O G E N E O U S WITH LATERAL INTERACTIONS BETWEEN ADSORBATE LES.
SURFACE MOLECU-
The body of experimental data on the adsorption of gas mixtures on homogeneous surfaces is significantly smaller than for the adsorption of single components. Nevertheless, the available data suggest strong mutual interaction effects to exist between the components of an adsorbed mixture. For example, from the experiments on the diffraction of slow electrons by single crystals of platinum metal, data were obtained suggesting that in CO and 02 systems there are present either as stratified and ordered phases of individual components or mixed ordered phases, coexisting with an ordered stratified phase [34]. Mutual attractions between adsorbed K atoms and CO molecules on Ru (0001) faces accelerates the desorption process and modifies significantly the nature of the thermal desorption curves in comparison to analogous curves obtained for the individual components [35,36]. These data emphasize the importance of the lateral interactions which determine
120 the non-ideality of adsorption systems. For description of these interactions, various thermodynamic approaches were used. The first attempts were presented in [37,38] as well as by Bering and Serpinski [39,40,41]. Further development of this trend was reflected in the numerous papers published by different authors e.g. Myers and Prausnitz [42], Hoory and Prausnitz [43,44], Bering and Serpinski [45,46] (more detailed information is given in [47]; discussion of these trends is not a subject of interest here.) Lateral interactions in mixtures of adsorbed gases were considered using virial expansions[48]. Calculations made on such a basis are, hovever, time-consuming. At present semi-phenomenological models based on the concept of local composition are widely used [49-51]. Lattice-gas models allow one to check the competency and accuracy of different semi-phenomenological models. We shall discuss lattice-gas model with the pair potential interactions between adsorbate - adsorbate. At first the equations for partial isotherms adsorption of binary mixtures have been obtained by Temkin [52]. This equations take into account a lateral interaction in the mean field approximation between the nearest neighbors. Later such equations for multicomponent mixtures have been considered by Jaroniec et al [53] and by Tovbin [54]. For the adsorption of binary mixtures the quasichemical approximation was introduced first by Barter and Klinowski[55], whereas for multicomponent gas mixtures it appeared first in [54]. Vicinal neighbors make a major contribution to the energy of lateral interactions amounting to about 70-90% of the total energy. The remaining energy is connected with contribution of more and more distant neighbors. As in the analysis of single gases [56], these contributions significantly influence the equiliibrium and kinetic characteristics by changing either the numerical values of the estimated quantities or the conditions of realization of a condensed state (and therefore the transitions between different condensed states). A generalization of the theories of the adsorption equilibria of mixtures by considering contributions from the second and more distant neighbours was given by Tovbin [57,58]. Closed equations were obtained in the mean field approximation, in the quasichemical approximation, and in the superpositional approximation (the last one is more accurate approximation than quasichemical). Let us limit the discussion to the first two approximations. We measure the distances between molecules by the numbers of the coordination sphere; z(r) is the number of sites in the r-th coordinational sphere (z(1) = z). Let s be the interaction of molecules i and j at the distance r; the interaction of the molecules i with the free sites is equal to zero ei~(r) = 0. The potential of every pair of molecules ij is characterized by its radius of interaction Rij. Let us choose the largest Rij and denote it by R. The equations of partial adsorption isotherms for multicomponent mixtures have the following form [58] Oi =
aipiSi [1"~t]j--'l ~" "pj'~j
(26)
The structure of the equations for the partial adsorption isotherms (26) is maintained at any radius of interaction R. The expressions for the function Si (which is used to consider the non-ideality of the adsorption system) change for different approximations.
121 In the quasichemical approximation, the functions of non-ideality of an adsorption system have the form: R
Si "- H[Si(r)]-z(r)
,
Si(r) = 1 +
r=l
xij(r)tij(r)
(27)
j=l
xij(r) = exp {-/3eij(r)} -- 1, where tij(r) = Oij(r)/Oi, and 0ij(r)is the probability of finding molecules i and j at a distance r. (Thus the function tij(r) is a conditional probability). For the functions 0ij(r), we have the following system of equations
(2Sa)
0ii(r)0jj(r) = 0~(r)exp {/3 [en(r) + ejj(r) -- 2eij(r)]}
s+l
Z 0,j(r) =
(2Sb)
j=l
which has a well-known form for R = 1 resulting from the theory of solutions [59,60] (but here R > 1). The system of equations (28) can be solved with respect to the unknown values 0ij(r) at fixed values of the partial coverages 0i. The dimension of the system of equations (28a)is equal to Rs(s + 1)/2, and increases rapidly with an increase in the number of kinds of molecules s. But, as previously noted [61] (for R = 1), the system of equations (28) can be reduced into an identity 1 - 1, when new variables X~(r) are introduced with the help of the relationships:
O,j(r) =
ij(r)X (r)Xj(r),
aij(r) = exp {fleij(r)}
(29)
After that, the functions 8~ and S~ in equations (27) and (28b) must be re-written in terms of the new variables: s+l s+l Si(r) -- E X j ( r ) / E Xj(r)aij(r) ' j--1 j--1
s+l 0i -" xi(r) E Xj(r)t2qj(r) j--I
(30)
This sharply reduces the dimension of the system (28) [561. The solution of the system (28) when surface coverages 8i are small, can be written as: 8i~(r) = 8,8j exp[~e,j(r)] : for every distance, the average fraction of neighbouring pairs of molecules i and j depends only upon their direct interaction. The attraction increases and the repulsion diminishes the number of pairs ij. In the mean field approximation, we have the following expression for
[
Si = exp / 3 E r=l j=l
z(rl%eij(r)
I
= exp ;~z
]
eijSj j=l
,
122 R ~ij = Z Z(~)'~j(~)/Z, r=l
In the first equality, the contributions of the second and further neighbours enter additively to the index of the exponential function. They can be re-grouped and, as can be seen from the second equality, give the concentration dependence of the function Si upon 0i. Thus, the calculation of the long range contribution to the interaction results in a change of the short range eij(1) into the effective parameter ~ij. Its value can be determined by the shape of the cij(r) and by the structure of the surface (as shown by At small interactions both approximations give similar results. But for an increase of ~eij, the difference between them is related to the fact that Oij(r ) differs from the product 0~0j which defines the function 8~*j(r) = 0~0j in the mean field approximation.
0.5
~
01,02
0.3
3
7 / /
///4
01156 3 9 15 21 lnP1 Figure 5. Partial isotherms for the first (curves 1 - 4) and the second (curves 5 - 8) components of a binary mixture with z=4,/3ell = -4, and/3e22 = -8. ~ex2 = -1 for curves (1, 3, 5, 7); -4 for (2, 4, 6, 8). aa > a2 for curves (1, 2, 5, 6) and aa < a2 for (3, 4, 7, 8). Fig. 5 shows some typical binary chemisorption isotherms for a wide range of surface coverages. The initial and the final values of the partial pressure pl change by 5.104 at a fixed p2. Curves 4 and 8 qualitatively correspond to the behaviour of C O and 02 on the platinum metals [61-63]. At small pressures, two variants are possible, depending on the relation between Ya and Y2, yi = (aipi) 1/m~. When Yl > Y2, the coverage of the surface by the first component is larger than the coverage by the second component and vice versa, when Ya < y2. When the pressure Pa increases, the lateral interaction of the adsorbed particles starts to play its role. In such cases, the sign of the parameter e12 is very important. The curves 1 and 2 correspond to the case al > a2. The value of 81 for attraction between particles 1 and 2 is greater than in the case of their mutual repulsion. The fraction of the second component increases if there is attraction between particles 1 and 2 and decreases for repulsive interactions (the curves 5 and 6). The same shape of the curves, only in a more explicit form, is observed for al < a2 (the curves 7 and 8). It should be noted here, that 7 passes through the maximum and diminishes when pl continues to grow. This is connected with the substitution of particles of the second kind
123 for adsorption of the first component. A comparison of curves 1 and 3 shows, that with growing coverage by the second component, the surface coverage by the first component increases more quickly than their attraction. The mutual repulsion of molecules of various kinds causes a decrease in the adsorption of both components. A detailed comparison of the physical adsorption isotherms of a binary mixture with their individual isotherms was carried out by Barrer and Klinowski [55] for various values of the lateral interaction parameters and relations between the Langmuir constants. The calculations were carried out for an equimolecular gas mixture with increasing total pressure of the system P = pl + p2, and with Pl/P~ = const. Fig.6 shows the characteristic curves for the adsorption isotherms calculated in this paper. The full lines represent the partial isotherms, the dotted lines correspond to the pure components (the parameters wij of the paper [55] are connected with the parameters used here by % = -2w~j/z).
a
b
c
1.0 0"8 !0.6
B
0.4 tZD
< tZD
B ~
-
B.
0.2
0.8 -
,
0.6-
/
A
y a ~ """ -
B
0.4-
-
tt
,
-
i
I
0.2 00
A,' " ~
.~
40
80 120 160
d
0
40
- BtfB t A
r~
-
I~ tla I
-
t
80 120 160 0
e
40
/
80 120 160 200 pressure/Tort
f
Figure 6. The dashed lines show isotherms of pure A and B. The full lines denote partial isotherms of A and B in their 1:1 mixture. Here, z=4, ai = (p.)-I with i=A,B; P~, = 760 Torr, P~ = 380 Torr. Values of ~wAA: (a) through (c), 0; (d) through (f),-2.303. Values of ~WAB: 1.151 in (a), (e), and (f,);-2.303 in (b);-3.354 in (c);-1,151 in (d); ~WBB: 0 in (a) through (e);-2,303 in (f)
[~5].
Weak repulsion between molecules A and B results in a decrease of adsorption of both components, for zero parameters of the pure gas interactions (a). Their attraction increases the adsorption of both components (b). A strong attraction of various components (c) results in a stepped change of the adsorption of both components. This case can be considered as the formation of dimers AB, for the relation 0A : 0B .w. 1 : 1. In their shape,
124 such isotherms are similar to the experimental isotherms [64] for the equimolecular mixture of NH3 and HCI sorbed in zeolites. The calculations confirm the existence of a strong interaction between the sorbed pair N H 3 . . . HCl. At high temperatures (245-315~ each gas taken separately hardly adsorbs. 1.0
-
Oi
0.5
--
2 .,.
m
.
.
.
.
.
1 m..-.
.
.
.
.
.
.
.
s s
"
"
"
""
3
s
I
- 16
-
-
I
I
I
-9
-2
5
In P
Figure 7. The effects of second neighbors on the partial chemisorption isotherms of a binary mixture. The quasi-chemical approimation is used with z=4, XA=0.1, T=450 K, QA=30, QB=27 kcal/mol, a~4= a~=104 Torr, /3eAA(1) = --2 kcal/mol, eBB(r) = 1.2EAA(r), EAB(r) = (eAA + s r-'l a n d 2; e2 = 7eAA(1), with 7 = 0 (curve 1), 1/3 (curve 2),-1/3 (curve 3). As the total pressure P rises, a surface is fined faster when attraction of the second neighbors is included and slower with repulsion, relative to absence of the second neighbor contribution. The genera] influence of lateral interactions remains the same when one allows for the contributions of the second neighbors; however, as attractions with the second neighbors increases, the tendency to form regular adsorbed films becomes larger. Hence, one has to take account of these effects upon monolayer ordering. For the systems with a strong enough attraction between A and B, and an absence of interactions between molecules of B (d, e), the attraction or the repulsion between the molecules A and B changes their partial isotherms in various ways. In the first case, the attraction between A and B keeps them together and the divergence between the partial isotherms is considerably smaller than in the second case. In both cases, there can be observed an adsorption substitution of the component B simultaneous with an increase of
125 component A. When the molecules repel each other, the quantity of displaced molecules of B increases and the maximum on the curves 8B(P) is moved toward smaller pressures (e). In such a case, one can observe a loop on the curve 8A(P) which is a witness of an instability in the adsorption system. Depending on the correlation between the parameters of the intermolecular interaction and the partial coverages and temperature of the system, its one-phase disordered state can become energetically unfavorable and condensation can take place. The concentrations of the co-existing phases can be determined from the material balance as well as from the equality of the chemical potentials in the system [65]. The number of phases which originate in the many-component adsorption system is considerable and should be treated separately. This question has been investigated only slightly (see ref.[56,66,67]). Changing the lateral interactions between molecules B (f) at constant values of the other parameters produces a change in the kind of gas removed from the surface. In this way, the lateral interactions can essentially influence the equilibrium characteristic of an adsorption mixture. The effect of contributions of the second neighbours on partial adsorption isotherms are shown on Fig.7. While concluding this chapter, we shall compare the equations obtained using the quasichemical approximation with the concept of local composition introduced by Wilson [49]. The importance of this concept is contained in the fact, that the ratios of the molar concentrations of the components j and k in the vicinity of can be expressed with the help of the Boltzmann approximation xJ---Li= 0j exp [flCjk]/Ok,
Cjk = eji -- eva ,
(32)
Xki
(A positive sign of parameters corresponde to attraction of molecules - another sign is used in papers [49,50]). According to the definition of the above pair-functions t~j (here R = 1), they are identical to the xj~ [49]. At small coverage, it can be seen that the relationship (32) is fulfilled. But in the general case, equation (32) is not accurate and any further considerations based upon it result in half-phenomenological models. Their parameters have no clear molecular interpretation and they can be treated only as empirical. For example,the application of equation (32) does not reflect condensation in the solutions. In order to describe this effect, one should introduce additional parameters or modify the models [50,51].
3. D E S C R I P T I O N OF A G A S M I X T U R E ON A H E T E R O G E N E O U S FACE T A K I N G A C C O U N T O F L A T E R A L I N T E R A C T I O N S
SUR-
3.1. General remarks The most important problem in the theory of adsorption is the simultaneous description of the effects of the surface heterogeneity and lateral interaction upon thermodynamic properties. In this case, one must consider a joint realization of all the previously considered effects: adsorption substitution of the molecules of various kinds (competitive adsorption) and cooperative behaviour in the adsorption system, as well as redistribution of molecules on the various types of sites on a heterogeneous surface. The complex character of the
126 joint effects of heterogeneity and lateral interactions has stimulated the development of various thermodynamic approaches and semi-phenomenological models. The best known were presented in the papers by Hoover and Prausnitz [43,44], Myers [68], Nakahara [69], Bering et al. [70-72], Danner and co workers [73-75] and their modifications. For small coverages it was Jaroniec [76] who used a virial approach to describe these adsorption processes. Various empirical equations are applicable only for narrow ranges of concentrations and temperatures and their forecasts should be verified by more rigorous theories. They also can help to determine the ranges of concentrations and temperatures where empirical relationships can be considered as well-grounded scientifically. Theories considering both heterogeneity and lateral interactions, based on the model of a lattice gas started more than ten years ago [77-80] (see also [56,62,81]). In these theories, the surface of the adsorbent is considered to be practically unchanged during the adsorption of any component of the mixture and at any concentration. It allows one to consider as stable all the distribution functions used for the description of the composition and for the structure of the adsorbent surface. To describe the surface composition the function used is the distribution of sites according to their adsorption capability (see section 1). Surface structure can be depicted in various ways that differ only in the accuracy of their descriptions. The most exact is obtained by considering a given arrangement of a set of sites on a certain rather small fragment of the surface. If translational operations on the surface repeat such treatment, then the given fragment approximation turns out to be suitable. Otherwise, in more complicated situations, it may be necessary either to increase the size of the fragment or to simplify the description of the surface structure. In the latter case, a cluster approximation can be applied, or the even simpler pair approximation. In the cluster approximation, the surface structure is described in terms of a cluster site distribution function d(q{m}R). It characterizes the conditional probability of finding a cluster consisting of a central site of type q and of its zq(r) neighbours among which there are mqp(r) sites of type p and for which the following relationship is fulfilled: ~-]~p=~ t mqp(r) = zq(r), 1 < r _< R. Here R is a radius of interaction equal to the greatest value of all the radii of the pair potentials of interaction between molecules i and j, where 1 < i,j < s, and s is the number of components of the gas mixture. In the pair approximation, the structure of the surface is described in terms of the distribution of pairs of sites fqp(r), characterizing the probability of finding a pair of sites of type qp on the surface separated by distance r. Those functions obey the relations: ~-']~p=lfqp(r) = fq, l
127 3.2. A d s o r p t i o n i s o t h e r m s The equations which describe the distribution of adsorbed molecules on a heterogeneous surface can be considered from the viewpoint of the accuracy of calculating the correlation effects (the mean field and quasichemical approximations) as well as the accuracy of calculating the surface structure. The most detailed description of the interacting molecules can be obtained by a "site-by-site" consideration of a certain fragment of the surface (a fragment approximation). In such a case, one investigates the partial coverages of every site of the fragment. Such a detailed description allows one to analyze the distribution of the molecules within the fragment, at any displacement of the sites of various types, i.e. the description is carried out in the real coordinate space and not in an energy space as done when one uses surface composition distribution functions. Cluster and pair approximations are intermediate approaches. Let us illustrate this by an example of application of the average description of the surface structure by the pair distribution functions . The equations representing a more exact description of the surface structure were published in [56,80,81]. In all the approximations, the contributions from every coordination sphere can be calculated independently. Differences are connected with keeping or abandoning the correlation effects. This leads to various forms of the equations. The coverage of the surface by molecules of i (partial isotherm) is given by the expression: t(0
0i = E
t(g)
fq(g)0iq(g) ' E
q= 1
fq(g) = 1,
(33)
q= 1
where the local coverages of the sites of various types are calculated from the following equation system (0~ = 0~(g))" R i v = eqi H %pieq
[Siq(r)]zq(r) ,
~"~ eqi eqv = 1 - a..,
r= 1
(34)
i= 1
For the quasichemical approximation: siq(r) = [1 + ~
p=l
dqp(r) ~I~ tqp(r)Xqp(r) ij ij ] , j=l
xiJqp(r)= exp (--/3eiJqp)-- I,
E
t
p=l
t~(r) = O~(r) , Oiq
(35)
dqp(r)= 1,
where dqp(r) = fqp(r)/fq is the conditional probability of finding a site of type p at a distance r from site of type q, 1 < r < R, and the functions 0~J(r) give the probability of finding molecules i and j on the sites of type q and p on distance r. For every value of r the pair functions 0qp(r) ij can be calculated from the system of non-linear equations"
O~(r)t~J~(r)
"i ii e i ~ ( r ) - e"~ ( r ) e- i ~ ( r ) ] } = O~(r)t~qp(r) exp {~ [eqp(r)+
s+l
E0~(r) j=l
s+l i
=/gq,
E0~(r) i=l
= ~ ,
(36)
128
0.5-
/~2
b
Oi
1.0-
1.0- Oi
3/ 2
0.5-
1
3~ .... /
. . . .
. - _ -_ _ 2--_3
1
I
I
I
I
i
-1
2
5
8
11
Oi
l.O"
C
1
s-
i
1
lnP
~
1.0"
//I
2
-1
5
8
I
11
lnP
d
0A
/
2
1
0.5
0.5"
4
3 I
i
i
4
2
. . . . . . . . . . . :::::::-==-::1 i
7
i
!
!
!
!
2 5 8 11 lnP -1 Figure 8. Partial physical adsorption isotherms of a binary mixture in the quasi-chemical approximation, with similar dependence of adsorption energies and with heterogeneous surfaces consisting of two types of sites: Q1A = 3.0, Q~ = 2.5, QA = 2.0, Q~ = 1.5 (all in kcal/mol), with z=4, ~ E . A A - - 1, e B B "- 1.2eAA, ~-AB = (~-AA "Jr e B B ) ~ 2 . Here, T=78 K, a~ = a~ = 10 -2 tort. (a) The effect of random surface composition upon the partial adsorption isotherms at XA =0.2 with fl = 0.25 (curve 1), 0.50 (curve 2), 0.75 (curve 3). Lateral attraction decreases the total pressure for the nearly full surface, relative to the case of zero lateral interaction (curve 1, Fig. 8d). As the fraction of sites with stronger adsorption capability rises at constant pressure, the quantity of both adsorbed components increases. (b) The effect of bulk composition upon the partial isotherms of adsorption with XA = 0.2 (curve 1), 0.5 (curve 2), 0.8 (curve 3), all for s = 0.5 (a random surface). As the fraction of component A in the bulk increases, its adsorbed amount increases and the amount of the other component decreases. (c) The effect of heterogeneous surface structure on the partial adsorption isotherms at XA = 0.5 and s = 0.5 for a regular (curve 1), random (curve 2), and a patch-wise surface (curve 3). Curve 1 corresponds to an ideal adsorbed solution with all eij - 0 and is given for comparison. The structure of a heterogeneous surface plays an important role in determining surface composition. On the patch-wise surface, two steps are clearly shown because each patch is filled independently. On the regular surface, the curve is smooth (without steps) as on a homogeneous surface, but its slope is smaller. The joint influence of a regular structure and lateral attraction results in the same effects as lateral repulsion on a homogeneous surface. At medium coverages, curve 2c can be described by a straight line, i.e., by Tem-kin's isotherm. Curve 3 for the random structure is intermediate; and two slight steps are manifested. (d) The effect of regular (2, 4) and patch-wise (3, 5) structure at various fractions of sites of the first type is shown: fl = 0.25 (curves 2,3) and 0.50 (curves 4,5) at XA = 0.5. The partial isotherms of component A are presented but partial isotherms of component B are omitted due to their small differences on the given scale. Curve 1 is for an ideal adsorption system at fl = 0.25. As the fraction of the sites of either the first or the second type decreases in comparison with the case fl = 0.5, the effect of the structure becomes smaller for the sites that are more numerous. -1
2
5
8
11
lnP
129 A reduction of the dimension of the equation system (35), (36) by replacement of the changeable values can be done for every coordination sphere exactly in the same way as for a homogeneous surface (see above). For the mean field approximation [78]: i v aqPiOq
-
"
i Oq exp --/3
Zq(r) r=l
t i
-- 8q exp{-flz ~
ij " eqp(r)
dqp(r) p=l
=
(37)
j--1
s ~'j
dqp
p=l
"
~qp~p}, j=l
R
-_ij
ij
% =
4, = 4,(1)
r----1
The second equality can be obtained by replacing the sums and regrouping the items. It shows that for any radius of the interaction potential, the concentrational dependence of the index of the degree of the exponential can be reduced to the case R = 1. The "long range" contributions change the value of the effective parameters of the "short range" parameter eqv. "ij For physical adsorption, the energetic parameters hardly depend on the types of site and eqv -~j ~ ~j. For longrange potentials and chaotic surfaces, there takes place a sharp reduction of the dimension of the algebraic equations (37) where the effective constant values aqi are equal to: aq " The resulting dimension of -~ = aqi exp ~Zq E j = I s such equation system is equal to s. The joint effects of heterogeneity of the surfaces with sites of the two types, and the lateral interactions (R=I) are shown on Fig.8. 3.3. C o n t i n u o u s distribution functions The transition to a larger number of sites on the surface does not change the physical basis of the distribution functions, and the procedure is limited to a change of discrete variables into continuous ones. Let us limit our considerations to the use of the pair approximation to describe the surface structure. An increase in the number of the neighbouring sites increases the number of variables, so the transition to continuous functions corresponding to the cluster and fragment approximations can be done in an analogous way. In case when relations exist between the adsorption capabihties of various molecules, the continuous pair distribution function of types of sites f t ( z ~, ytlr) characterizes the probability of finding a pair of sites with adsorption capabilities within the intervals from z ~ to x l + dx l and from y~ to ye + dy l at a separation distance r. In absence of functional relationships between the adsorption capabilities of various molecules, the continuous pair distribution function of types of sites f(5,Y I r) characterizes the probability of finding a pair of sites with adsorptional capabilities which are given by the indexes varying from to ~ + d~ (5 = xl,...,x,) for the first and from ~ to ~ + d-ff (~ = yl,...,y,) for the second site occurring at a separation distance r. For both situations the differential pair
130 distribution functions are normalized to the single distribution functions (10) and (11)'. Y~n
Y-~n
/ fe(xe, S [ r)dy = fe(xe) ,
/ f(~, y [ r)dy = f(~),
#b
q]
Y~n
(38)
Y-in
where 5i,~, Yi,~ and ~f~n, Yfi~ are the vectors of the initial and final values of the adsorption capabilities of the sites on the heterogeneous surface for various molecules. The expressions describing the stucture of the heterogeneous surface (38) and the thermodynamic functions, are similar in form, and that is why below, in order not to double them, we use below only one vector descriptor 5. Aside from the functions (38) one can determine distributions d(5, ~[ r) characterizing the probability of finding some sites with adsorption capability ~ at distance r from the site for which the adsorption capability is 5:
/
Yfin
d(~, Y lr) =
f(~, y I r)/f(~),
(39)
d(2,ylr)dy = 1
Yin
Using the distribution functions describing composition and structure of a heterogeneous surface, we can write expressions for the partial adsorption isotherms which incorporate interactions between the adsorbate molecules,
0i(X)-" ai(x)piSi(x)/[1 -[-s aj(x)pjSj(x)] ,
(40)
j=l
The form of the functions Si(x) describing the non-ideality of an adsorption system depends upon the approximation used to calculate the effects of lateral interactions. For the mean field approximation: Si(x)
=
i
Yfin
R
exp fl E z(~l r) r=l
d(2, Y lr)
-
Yin
s ~, Y l rlOj(Yld
=
j=l
YfinM
exp
f a( ,y I Yin
(41) j=l
R
,-~j(~,y) = ~ z(~ I r)d(~,y) I r)~j(~,Y I r)/[z(~ I 1)d(~,Y I 1)1,
(42)
r=l
where z(5 [ r) - is the number of nearest neighbours in the r-th coordinational sphere around the site characterized by the adsorption capability 5, z(5) = z(S[ 1); eij(5,~[ r) is a parameter of interaction between the molecules i and j, occurring on the sites of type 5 and ~ at distance r in the second equality (41), a regrouping of the items took place that
131 allowed us to introduce the effective energetic parameters ~ij(x, y) The use of expressions (11), (40) is connected with the preliminary solution of the integral equation (40), (41). For the quasichemical approximation:
R Si(x) = I I [ s i ( ~ l r=l
r)]-z(xlr)
Xij(Y, Y lr) = exp [-/3eij(Y,y I r ) ] - 1 ,
s Si(~ I r) = 1 + / d(~, y [ r ) E Xij(~, y ! r)tij(x, y I r)dy, J j=l Yin Yfin
(43)
where tij(5,~ I r ) = 0,j(5,ff I r)/0i(5), 0~j(5,ff I r ) is a pair function, characterizing the probability of finding a molecule i on a site of type 5 and a molecule j on a site of type at a distance r. These functions are subject to equations which are analogoues of the equations (36)
0ii(x,y I r)0jj(x, ff I r ) -- 0ij(x,y [ r)0ji(x,y ! r)exp {fl [eii(~,y I r ) + ejj(~,y [ r) --s
s+l s+l if[r) -- eji(~,y [ r)]} , E 0ij(~' Y lr) = 0i(x) , E 0ij(x, Ylr) = Oj(y). j=l i=l
(44)
The dimension of the equation system (43), (44) can be lowered by changing the variable values as explained earlier. The above system of integral equations represents a correlated arrangement of sites of various types. In the general case, their solution is complicated. In order to make this easier, it is necessary to use admissible simplifications. As a rule, such simplifications are connected with certain kinds of heterogeneous surface. Traditionally, the greatest interest is connected with two kinds of surfaces" patchwise and chaotic. For such surfaces, the equation systems are simplified. 3.3.1. P a t c h w i s e surface This case is the simplest and it has frequently been used to describe the adsorption of pure components [82]. The structure of the surface is described by the function d(5,~ ] r) = 5 ( 5 - ~) for any value of r, where 5 is here a delta-function. That means we neglect the contributions from the borders between various surface fragments. As a result, the coverage of every fragment is described by the equations of section 2: For the quasichemical approximation
R Si(x) --
II[si(~ ] r)]-z(~l~) , r=l
Si(~ I r)
= 1 -1t- ~ Xij(x , ~ ] r)tij(~,~ I r) , j=l
(45)
where the function X,j(5,~ ! r) and t,j(5,~ ! r ) are given by eq. (43). For the mean field approximation
Si(x) = exp [flz(x)~ ~ij(x,x)Oi(x)dx] , j=l
(46)
132 where parameter ~d(5, 5) can be determined from eq. (42).
3.3.2. Chaotic surface The condition of a chaotic distribution of the sites of various types on the surface (and in the case of the patchwise surface) makes it possible to use the information about the surface composition of adsorbent which is given by the distribution functions f(g), since for these surfaces we have: d(~, ~ [ r ) = f(~) for any value of r. In this case, the functions which enable us to incorporate the non-ideality of the adsorption system ~5i(5), are the expressions (41) and (43). These can be written as: For the quasichemical approximation: Yfin
Si(K, [r) - 1 + f fly) ~
(47)
Xii(K, y [ r)tij(K, Ylr)dy,
j=l
Yin
where the interpretation Si(5 1r) and S/(5) is given by eq. (43). For the mean field approximation" Yfin
Si(K) = exp flz(~)
f(y) Y--in
(48)
~ij(g, y)Oj(y)dy j=l
Eqs. (47), (48) along with eq. (40) determine the partial adsorption isotherm. Their use is not much simpler than calculations made using equations (40), (41), (42). The evaluation of the integral equations involves integration along the values ~, appearing in eqs. (47), (48). It order to simplify the task, some additional assumption must be made. In case of physical adsorption c~j(5,~ ] r) = e~j(r). Such an assumption fundamentally simplifies the equations for the mean field approximation. Eq. (48) can be rewritten as
Si(K) = exp [flz(K) ~ c-ij/gjl
(49)
j=l
where the partial coverages of the surface, 0j, appear in the exponent. The form of eq. (49) is essentially that of eq. (27) for a homogeneous surface. Hence we have the system of equations having the dimension s, for the partial adsorption isotherms
0i = -~'in
Ai(x)pi/ 1
d~, j--1
Ai(~) = ai(~)exp [ / 3 z ( ~ ) ~ e"ij0j] (50) j=l
Thus, instead of looking for the local coverages 0,(5) (or 8~(s for the discrete distribution functions) it is necessary now to solve a nonlinear system of equations with respect to the 0i's. For such cardinal simplification in the quasi-chemical approximation, one must introduce an assumption about the additivity of the contributions from surface heterogeneity
133 and from the lateral interactions. Then, by analogy to eq. (49), we change the function of non-ideality of the adsorption system into the function for a homogeneous surface (28)
Si(g I r) = 1 + ~
tij(r)xij(r) ,
(51)
j=l
So the expressions for the partial coverages have the form as in eq. (50) where the coefficients Ai(5) are given by: Ai(~) = ai(~)l-I
1+
r=l
(52)
tij(r)xij(r) j--1
Equation (52) represents the generalization of the results obtained by Hill [83] for chaotic surfaces with mixtures of the adsorbed molecules. From equation (50) for the corresponding distribution functions f(5), one obtains the results of Jaroniec et al [55,84] which generalize the Langmuir-Freundlich equation for gas mixture on chaotic surfaces. This gives a possibility of molecular interpretation of the parameters in the models [55,84]. It should be noted that the hypothesis of additivity of the effects due to surface heterogeneity and to lateral interactions is in general in contradiction to the idea of calculating direct correlations between the neighbouring molecules in the quasi-chemical approximation for which the local coverages of the neighbouring sites are important and not just the surface coverage. Its applications should be controlled by calculations using more exact formulas such as (47). The hypothesis brings us to expressions for the partial adsorption isotherms of the components of a mixture which are analogous to those of section 1. In a series of situations, the integrals in those expressions can have simple analytical forms. This possibility is related to the correlations between the heats of adsorption of various components, and to the shape of the distribution function f(5). For an illustration we discuss the case of a uniformly heterogeneous surface for which expressions can be developed for the whole range of surface coverages. 3.3.3. U n i f o r m l y h e t e r o g e n e o u s surface We consider examples of the relations (14), corresponding to the likeness and antilikeness correlations of the energies of the components of the adsorbed mixtures. Let us write y(x) = (ai(x)p~) 1/m' as yi(x) = Ciexp(/~x), Ci = (fzip,) I/m'. Then, the expression (14) can be represented as yi(x) = [yj(x)]~ crij = Ci(Cj)-~'3. If aij > 0, we have likeness behaviour of the adsorption coefficients of molecules i and j. If on the contrary, aij < 0, the antilikeness; aij is the shift of the values of the adsorption coefficients for molecules i and j. To the assumption of a similar likeness behaviour of the adsorption coefficients for various kinds of molecules, there corresponds the expression yi(x) = yj(x)a~j. To the assumption of a strongly antilikeness behavior of the adsorption heats, there corresponds the expression yi(x) = C, Cj/yj(x). Using the tables [85] of the natural and rational values of aij, we can rewrite the system of equations by introducing elementary functions. With these limitations, these equations can be written as:
Oi
Xin) -1
- - f(x) Xin
exp [fl~ix]/
+
exp [tic,ix] j=l
dx,
(53)
134
1.0-
Oi
a
3
1.O-
4
Oi
b
0.5-
0.51
32
.--"-4-2
2222--f 3
-1
2
5
8
11 1.0
14 Oi
0.5
lnP
- '1
2
5
8
11
14
InP
C
3
4
3
Jd
_7, ~_-- .~ -_--- _ ~ ~ ' - = ~ " "
-'1
;1 ;4
ln e
Figure 9. Partial physical adsorption isotherms of a binary mixture in the quasi-chemical approximation', with similar dependence of adsorption energies and with heterogeneous surfaces having continuous distribution functions f(x): QB(X) = QA(X) = 0.5 kcal/mol, Q~A? = 1.2, Q~OH = 3.2 kcal/mol, z=4, t~E.AA "" 1, e B B - " 1.2eAA, i C . A B - " ( ~ . A A + eBB)~2, a~t = a~ = 10 -2 tort, T=78 K. (a) The case of a uniform distribution function: f(x) -" (QAoH -- Q HAA ? ) -1 , ZA = 0.2 (curves 1,4), 0.5 (curve 2), 0.8 (curve 3); for curve 4, eBB = 0.7EAA. As the fraction of one component in the bulk increases, its adsorbed fraction is also increased, as shown in curves 1 - 3. Decreasing the lateral attraction between molecules of B causes the quantity of the adsorbed molecules A to increase, and that of molecules B to decrease. (b) The gaussian distribution function f ( z ) = e x p [ - ( z - ~.)2/2a2]/n, where n is a normalization factor and a =1, ~' = ( Q A o H - Q~A?)/2 = 2.2 (curves 1,3) and 2.7 (curves 2,4) kcal/mol, ZA =0.5, for curves 1 and 2; all eij =0. When the lateral interaction is larger, more rapid filling A of the monolayer is present (compare curves 1 and 3 with 2 and 4). As the range ( QA li~ -Qi,~) increases, the region of monolayer filling becomes larger (compare curves 1 and 3 with 2 and 4). (c) Comparison of partial adsorption isotherms for various distribution functions f(x) at XA =0.2, QAoH --QHA? A = 2 kcal/mol. The uniform (curve 1), gaussian a = 1, ~, = 2.2 kcal/mol (curve 2), and exponential functions f ( z ) = exp(Tx)/n, 7 = 1 (curve 3) and 7 = -1 (curve 4). The uniform and symmetric gauss distribution functions are characterized by similar isotherms. As the fraction of the sites with strong adsorption capability increases, the surface is filled faster; a decrease in such sites results in the opposite effect.
135 where the coefficient ai characterizes the behaviour of the heat of adsorption of molecule i relative to the heat of adsorption of the selected type of molecule I (at = 1), for the sake of simplicity given the index I is omitted. For example, for a mixture of molecules with a similar likeness behavior for the adsorption coefficients of all components of the mixture, a~ = 1, we have Oi = CiSi In {(1 + C exp[~x~])/(1 + C exp[~xin])}/(~C(xf~ - Xin)) ,
c =
(54) i--1
In absence of lateral interactions, the functions S = 1 and eq. (54) reduces to the well-known expression [18]. In this way, one can arrive at a closed systems of equations which make it possible to account for the lateral interactions of the adsorbed molecules, as well as for the heterogeneity of the surface. Their dimension is equal to the number of the components in the mixture. These equations are nonlinear with respect to the values of 8~ in the functions Si. The simplifications which we have made while developing these equations are, however, sufficient that their applicability can be controlled with the help of some still more accurate equations. Curves on Fig.9. are illustrated the joint effects both the lateral interactions and the heterogeneity of the surfaces describing by continuous distribution functions. 4. M U L T I L A Y E R A D S O R P T I O N
OF MIXTURES
The description of the multilayer adsorption of gas mixtures appears in many practical calculations [1,2]. The first theoretical work by Hill [86] (see also studies by Bussey [87]) are a generalization of the BET model to gas mixtures. Hill's results were used by Arnold [88] for description of experimental data on the adsorption of a O~, N2 mixture on anatase for various compositions of the gas phase. Similar to the BET equation for the pure components, the equation of Hill predicts the quicker increase of the full coverage of the surface which was observed in the experiments. The principles of the BET model have been considered in ref. [89] for one component systems. In order to describe multilayer adsorption, phenomenological approaches have been used. Thus, Sircar and Myers [90] used a combination of the potential theory of adsorption and the theory of the ideal adsorbed solution, while Gonzalez and Holland [91] proposed a kinetic method to obtain the equation for bilayer adsorption.The main idea of their work reflects one of the postulates of the BET theory, that the influence of the surface does not concern the second and the further layers, so the isotherms for two component mixtures can be represented by: = g0(1) ,
A
B
g = 1 + a 2ph + % P s ,
(55)
where 0(1) is the total adsorption isotherm of a mixture in the first surface layer, and the quantity g expresses the coverage in the second layer. The form of the factor g was chosen by considering the two following limiting conditions: coverage should be equal to one at decreasing pressure of both components of the gas mixture, and it should increase without limit with increasing pressure in the adsorption system. Jaroniec [92] generalized that approach [91] for heterogeneous surfaces. Surface heterogeneity changes the function
136 in the expression for coverage in the first layer, although the general structure of equation (55) remains unchanged. As for the function g, any function can be chosen, provided that it fulfills the two limiting conditions. In the paper [93] dealing with many component S ai2pi) was proposed. mixtures , the function g = exp (~-~i=1 The description of the multilayer adsorption of a mixture is basically similar to that of calculating a concentration profile of non- electrolyte solutions in contact with a solid surface. That is why, for the description of mixture adsorption, it is necessary to use the same theories. With vacancies, the number of components s of the adsorbed mixture corresponds to the (s + 1)-component solution. The equations developed using the lattice model for surface regular solutions with an arbitrary number of components were proposed in the papers by Smirnova [94-96], as well as in the papers [97,98]. In all those works the quasichemical approximation was used to calculate the effects of the interactions between the nearest neighbours for specific interactions at various orientations of the adsorbed molecules [94,95]. It was assumed that only the molecules of the first layer interact with the surface. In paper [99] one considered more distant neighbour contributions as well as the possibility of changing the internal degrees of freedom of the molecules and their parameters of lateral interaction in various layers of the surface region. In paper [100], equations were obtained for the concentration profiles of molecules in the presence of the pair interaction, as well as for the three- and many-particle contributions to the short range potentials. These papers consider adsorption on homogeneous surfaces. The theory of adsorption of a mixture of molecules on a heterogeneous surfaces was elaborated by Tovbin [101,102]. He obtained equation systems for the cluster distribution functions by considering the pair interaction potential between molecules at any terminal distance, and obtained closed expressions for the quasichemical and for the mean field approximations.
4.1. A lattice model for mixture multilayer adsorption In adsorbed systems, in addition to the heterogeneity of the surface we also must deal with the nonuniformity of the distribution of molecules in the surface region. In the lattice model, this is reflected by the fact that the sites in various layers turn out to be sites of various types. The description of the surface region should reflect a strict sequence of the layers, starting from the first (surface) one up to the final layer of this region ~ where the density of molecules is practically the same as in the bulk phase. Here ~ = L / d , where L is the width of the surface region, d is an average diameter of the molecules of the mixture having more or less the same size. The layers are considered to be situated parallel to the gas-solid interface. The strict alternation of the layers can be calculated with the help of cluster distribution functions, which is why the structure of the whole surface region should be described by these functions. We shall assume that for the multilayer adsorption of a mixture, the same procedure can be used which was discussed in the subsection 1.1. This is based on the partitioning of the surface into separate sites of area 6~. The volume of the first monolayer is composed of the volumes of the separate sites v~. For a flat separation border, the second layer and all the following are divided into sites with respect to a molecule of the reference type I. For a rough interface, , there can occur atoms of adsorbent in various layers of the surface region. In such a case, the region which is not occupied by the adsorbent is also separated into sites v~ with the the molecules of reference type ~, when the partitioning of the surface of adsorbent is made into areas $~. For the surface region, the index h denotes
137 the sites of a volume structure and not those of the surface, just as in subsection 1.1. We shall also assume that the same way of grouping the sites according to their adsorption capability is kept as in subsection 1.2. The type of sites in the the surface region is determined by the layer number k and the value of the interaction of molecule i with the adsorbent Q~q(k) which is connected with a local Langmuir constant aq~(k) = aq Ai (k) is the preexponential ^~(k)exp[~Q~(k)], where aq factor of the local Langmuir constant. Its form is similar to that given by equation (2). Let tk be the number of types of sites in layer k. Then a general number of types of sites in the system is equal to t = ~]k=l tk. In a similar way, for every layer we can introduce its own distribution functions, characterizing the composition of a layer fq(k) and its structure d(q{m}R)k: tk
tk
Z fq(k) = 1 ,
Z
d(q{m}R)k = 1, Z
q=l
6q(k)
mqp(k Jr) = zq(k J r ) ,
(56)
p--1
where rnqp(k I r) is the number of the neighbours of type p at a distance r from the central site q in the layer k and zq(k I r) is the number of neighbours at a distance r around the site of type q in layer k ; 5q(k) is the number of a different clusters with a central site of type q in the layer k. Let the mole fraction of the coverage of the sites of type q with molecules i in layer k be expressed by O~(k). Then the partial coverages in layer k by these molecules and the partial coverage by these molecules throughout the surface region can be written
tk 0i(k) = ~
,` fq(k)0q(k) ,
s
0i = Z
q=l
0i(k)Lk/L,
k=l
L=
Lk,
(57)
k=l
where L is the relation of the sites in layer k to those in layer x (in the bulk gas phase) which can differ in their structural imperfection (for example as in their roughness) and in the structure of the adsorbed solution close to the surface relative to the bulk phase. The total coverages in layer k and throughout the surface region can be determined in the following way: O(k) =
•
Oi(k)
,
0
--"
i--1
i--1
The total numbers are given by Ni = LOIN,,,
•
Ni
Oi =
•
O(k)Lk/L,
(58)
k=l
of molecules i and their excess contents I'i in the surface region
ri = N,` s
- O~(,~)]L.,
(59)
k=l
where N,` is the number of sites in a homogeneous layer of bulk phase. For the calculation of the distribution of the values (58) and (59), it is necessary to know O~(k) which can be found from the solutions of the corresponding algebraic equations. As previously, we shall limit our considerations to interactions between neighboring molecules in the quasichemical and in the mean field approximations.
138 4.2. D i s t r i b u t i o n s of molecules in the surface region The distributions of molecules in the surface region can be described by the system of equations: aiq(k)piO;(k) = Oiq(k)Siq(k) ,
s Oiq(k) + Oq(k) = 1, i=l
(60)
where the function of non-ideality S~(k) help in the calculations of the contributions from various components of the adsorption mixture. In the quasichemical approximation, the system of algebraic equations consists of two groups. The first group of equations describes the local distribution of molecules of a mixture on various types of sites (60), where (for simplicity we assume the case R = 1) Sq(k) = E d(g{m}l) 6q(k) (61)
II H 1 + 0~(kn [1){exp (-fle~(kn I1)) - 1}/0iq(k) n=k-1 p=l j=l
The second group are equations for the local portion of the average amounts of pairs of various molecules occurring on sites of various types at distances r, where 1 < r < R. The equations of the second type resemble the analogous equations (36) and are omitted here. The difference consists in the fact that for the multilayer model, the type of site is given by two indexes: the number of the layer k and the number of the type of site q occurring in a given layer. In the mean field approximation, the function of non-ideality can be written as (R = 1):
[
sin(k) = exp
,,o,
s
--flZq(1) E E dqp(kn [ 1) n=k-1 p=l j=l
]
ij (kn [ 1)Oip(n) eqp
(62)
where dqp(kn [ r) is a conditional probability of finding the site of type p in the layer n at distance r from the site of type q in layer k. In the majority of tasks where the interest is focused on multilayer adsorption, it can be assumed that the parameters of interaction do not depend on the types of sites:
4.3. Simplified e q u a t i o n for the full i s o t h e r m For one component systems, the expression (55) has its motivation within the lattice gas model to a degree that makes it possible to formulate assumptions resulting in such formulas. Thus, the approximate isotherm of multilayer adsorption [103] results in a universal form of the factor g with arbitrary assumptions for the properties of the first monolayer in an adsorption system: the surface can either be homogeneous or heterogeneous, and lateral interactions can be present or not. In fact, there arises the possibility of obtaining some analogous approximate equations for the multilayer adsorption of mixtures. In this case, one should use the equations from section 4.2 and limit the consideration
139
300[ V
V
._-
240
2
.-'''
.-''- - "
180
00/160V 600
0
V
0
x
0.03
0.2
0.06
0.09
0.4
0.6
_.
0.8
x
Figure 10. Comparison of one component multilayer adsorption isotherms for oxygen (curve 1) and nitrogen (curve 2) on anatase. At low coverages, the approximate equation (63) was used to find Langmuir constants. The calculated (solid lines) and experimental data of Arnold [88] (dashed lines) are shown. Curve 1 is shifted by unity on the X axis. Curve 3 (50.2% 02 and 49.8% N~) was fitted by the parameters of the approximate equation with the help of the total adsorption isotherm. Thereupon the fitted parameters were used to calculate the total isotherms at other bulk composition without refitting. V is the volume of the adsorbed gases, x = P/P0. to the case where one considers the interactions between nearest neighbours on flat heterogeneous surfaces and a short range potential of adsorbent - adsorbate interaction. To arrive at the approximate equations one should eliminate the partial pressures (by dividing of all the equations of the system into the equation for a volume gas phase) and analyze the systems of equations describing the distribution of molecules of the mixture in the volume close to the surface according to the small parameter/3eij << 1. The details of these considerations result in a final expression that agrees with those for the approximate adsorption isotherm of a one component system [103]. Finally we obtain: t
0 = g0(1),
0(1) = E f q A ~ P / { 1
+ AXqP} ,
(63)
q=l
g = {(1 - 5P/Po)/(1 - P/Po)} ~/2
1 s
Aq =
i= 1
a;(1)xi
140
],'
V
Jf
2280
,'/:1
,'1:1 ,7,7
1710
1140 1
f 570
f 0
0.2
I
I
I
0.4
0.6
0.8
x
Figure 11. The predicted and experimental (dash-dotted lines [88]) total multilayer adsorption isotherms for the N2 - 02/mixture on anatase at 14.9 (curve 1) and 85.3 % 02 (curve 2). The same fair agreement was obtained at other bulk compositions. Curve 2 is shifted by unity on the Y axis. where 6 = ( z k k - 1)/z, x~ is the molar fraction of component i in the gas phase, P is the total pressure in the gas phase, and Pi = Pxi. Here 0(1) represents total isotherm for a heterogeneous surface in the absence of lateral interactions between molecules in the surface monolayer. At small pressures, equation (63) reduces to equation (7). If the number of the site types on the surface is large, then one can use continuous distribution functions and one arrives at equation (10) for 0(1). If one considers lateral interactions, then the factor 0(1) becomes expression (33)-(35)in the quasichemical approximation or (33),(34),(37) in mean field. Finally, for a homogeneous surface, the factor 0(1) is expressed by equations (26)-(28) and (26), (31), correspondingly. Thus, the lattice gas model gives universal expressions for the factor g and equation (55). This expression is essentially different from the empirical functions proposed previously. The expression for g (63) has the same form as that for one component systems. There we had P = p whereas P = pa + . . . p s for the many component systems. The saturated vapour pressure P0 depends on temperature and on the composition of the gas mixture. Figs. 10- 12 illustrate how the simplified equations apply for describing the multilayer adsorption isotherms for the system N2 + 02/anatase. In conclusion, while developing equation (63), we introduced some greatly simplified assumptions. Although (63) is considerably better for describing the full isotherm of multicomponent adsorption over a wider range of pressure than the various modifications based
141 on the BET model, its parameters are determined by the adjustment of the calculated curves to experimental data and have no obvious molecular interpretation. 5
........
-
3 2
0.5 1
~N2
1.0 -
""~-~--_
0.5-
- ~ --
"
1
""-~ 9 3 "
~
4
- 5 0
I
0.1
I
0.5
i
I
0.9
x
Figure 12. The values (i = Vi/V, ii=N2 and 02, found for various x with use of the experimental data [88], and their linear approximation obtained from the experimental points Xl =0.1 and X2 = 0.9: (i = [(i(X~)(X2- z ) + ( i ( z - X1)](X2- X1 ). This procedure was applied to extract the partial contributions from the total isotherm, since the distinction between mixed components vanishes upon constructing the approximate isotherm in this case (63). Therefore, one needs additional information to find the partial isotherms: 0~(x) = O(z)(~(z). As a result, the use of equation (63) together with any points X1 and X2 enables one to determine partial adsorption isotherms in the entire range of pressure.
4.4. Adsorption
of a mixture
in p o r e s
The adsorption of mixtures in pores can be formally described by the same equations used in the case of multilayer adsorption discussed above (subsections 4.1 and 4.2). The difference consists in the fact that the molecules can adsorb in the field of action of both walls of a pore. This can provoke an essential increase of the adsorption value. For wide pores one can separate such volumes in a pore where the molecule is not under any influence from the walls of the pore. In such a region, one can use equations (60), (61) or (62). The solution of the equations for all regions can be done together. Another peculiarity of adsorption of mixture of molecules in pores consists in the fact that the critical conditions of condensation inside the pore differ from the analogous conditions in the bulk gas phase. They may also depend on the relations between the
142 components of a mixture. It was deduced in paper [105] that the conditions of condensation of one component systems in pores with heterogeneous walls give critical temperatures and coverages that depend not only on the sizes of the pores, but also on the character of heterogeneity of the walls. Similar effects should also be observed for the adsorption of mixtures. Application of the lattice-gas model for describing adsorption Ar+Kr mixture in slit-like carbon micropore is given on Fig.13 as well as a comparison with the analogous molecular dynamic calculation [106].
16
l~J~
~
ff~\
Figure 13. Partial Ar adsorption isotherms for an Ar - Kr mixture in a slit-like graphite pore. (a) shows the results obtained in [106] by density functional theory and molecular dynamics simulation; and (b) shows the same curve obtained from the lattice model using molecular parameters from [106]. Both methods give the similar results.
5. T H E S E P A R A T I O N
COEFFICIENT
OF A M I X T U R E
The separation of mixtures by adsorption is characterized by the degree of purification of any particular gas from some other gases. For the quantitative evaluation of the degree of purification one can apply the double separation coefficients aij of components i and j for an adsorbent. The separation coefficient is determined [2,107] as follows: aij = 0iPj/0jPi 9
(64)
An important characteristic of adsorption separation is the separation coefficient concerning pairs of components, one of which occurs in a microquantitative amount and the other one is a "macrocomponent" as it determines the effectiveness of the purification of the special component by adsorption. It is a hard task to find experimentally such separation coefficients and therefore much effort is dedicated to its theoretical calculation. From the definition of the separation coefficients it follows that, in order to calculate them it is necessary to know the partial isotherms of adsorption which relate the partial coverages of the surface 0i with the pressure of the components in the gas phase. Let us limit our
143 considerations to binary mixtures only and let the microcomponent be denoted by A and the macrocomponent, by B. Let us study only the case when all the components of mixture occupy only one site of the surface. To calculate the separation coefficient a = aAB, one can use the previously considered partial isotherms of adsorption determined by the molecular features of the adsorption system. Let us study the relationship between the equilibrium coefficient of competitive adsorption in a given model with the separation coefficient. Competitive adsorption can be formally expressed as Agas + B~d, ~ A~,ds + Bg,~,. The thermodynamic equilibrium constant KAB for this process has the following form: /(AB = OAZ'/APBTB/(OBzysPATA) = KABZ'/A/ZYB, where 7/ and ~i are the activity coefficients of component i = A and B in the gas phase and in the adsorbed phase, and KAB is the equilibrium coefficient of the process. If the gas phase is ideal, then we have 7i = 1 and a = KAB. A calculation of the effects of nearest neighbor lateral interactions in the quasi-chemical approximation for heterogeneous surfaces described by the pair approximation leads to the partial isotherms (33)-(35). These equations give the separation coefficient as: t
t
=
fqqOq(SJ
/Z q= 1
,
q= 1
where the function S~ is given by eq. (35). For the binary systems microcomponent A macrocomponent B, this expression can be simplified, since 0A --+ 0. Then, the separation coefficient A depends on the local Langmuir constants of both components for various types of sites, on their parameters of lateral interactions, and on the composition and structure of the heterogeneous surface [108,109]. For many adsorption separations of mixtures, zeolites are used. Below we present the results of the calculations [110,111] of the separation coefficient of a microadmixture macrocomponent for the systems X e - K r and UH4 - N2 (the microcomponent is given first) using the lattice model. The calculations were carried out employing the information about the potential energy of molecule of every component in the zeolite cavity and about the potential of the intermolecular interactions. To calculate the potential energy, the data on the structure of zeolite were used (its model is discussed in papers [110,111]). The potential of interaction of molecules CH4 and N~ with the atoms of adsorbent and with the other molecules was first based on the approximation of effective spheres for these molecules. The zeolite cavity was broken into sites along the three coordinates according to the parallel planes to calculate the real spatial arrangement of the sites. The cluster approximation was also used. In this case, the functions of non-ideality of the adsorption system & in formula (65) have the form R
Siq = E
t
d ( q { m } R ) H I I [Si~(r)] mqp(~) '
6
r=l p=l
S~(r) = 1 + ~
8ijqp [exp {-/ge~(r)} - 1]/Siq,
(66)
j=l
To calculate the separation coefficient from equations (65) and (66), in addition to the parameters of adsorpion of the individual components (local Langmuir constants aiqai),
144
i~KrKr
SKrXe
SXeXe
SN2N2
8N2CH4
8CH4CH4
0.52 0.21 0.06 0.03
0.62 0.29
0.70 0.38 0.16
0.19 '0.10 0.05
0.25 0.13
0.33 0.18
0.02
0.06 0.02
0.09
0.04
O.11 0.03
0.03
Table i. Lateral interaction parameters (in kJ/mol)
t'~
~'1
0
9 ee
0.8
~
9
0.6
0.6
0.4
0.4
0.2
0.2 4
o ~
0.8
6
lg P
3-
2
4
61g p
In
b @Q I
I
I
I
0.2
0.4
0.6
0.8
0Kr
Figure 14. Theoretical adsorption isotherms for Kr and Xe and the separation coefficient a of their mixture. The parameters used are given in Table 1. (a) shows Kr isotherms at T=200 (1), 252 (2), and 303 (3) K; (b) shows Xe isotherms at 210 (1), 240 (2), 280 (3), and 303 (4) K; (c) shows the changes in In(a) for the Xe - g r mixture at T=200 (1), 252 (2), and 303 (3) K, [110, 111]. the interaction parameters of the molecules eqp(r), ii and the cluster functions d(q{m}R) describing the structure of the lattice) one also needs the parameters of the intermolecular interactions e~(r). In our calculations we assumed that e~p(r)= eij(r). The parameters c j ( r ) were calculated using the combination rules for the individual potentials u~(r) of the Lennard-Jones type. These were of the type e~j = (e~ejj) 1/2 and aij = (cr~q-ajj)/2. Finding the average value according to the volumes of the separate sites vq and vv occurring at
145
0
om 9
0.8
0.8 9
0.6
0.6
0.4
0.4
0.2
0.2 I ~
I
4
I
6
I
lg P
2
-'-
4
6 lg P
t.
0[ t" i,
I
I
I
0.2
0.4
0.6
I
0.8 0
Figure 15. Theoretical adsorption isotherms for nitrogen ((a), T=253 K), methane ((b), T=100 (1), 210 (2), 253 (3), and 303 (4) g), and the separation coefficient of their mixture ((c), T=253 K), [110, 111]. The values of the parameters used are given in Table 1. distance r from one another:
eli(r) = -~ uij(r) ~- = / /uij exp [-~uij(rij)] dridrj///exp [-~uii(rij] dridrj Vq Vp
,
(67)
Vq Vp
where rij is the distance between molecule i, on the site of volume vq of the type q, and j, occurring on the site of volume vp of type p. The results of the calculations for four coordination spheres are shown in Tab. 1. Figs. 14 and 15 show the comparison of the computed curves with the experimental data for the separation coefficients of the mixtures X e - Kr and C H 4 - N2. The small difference in the sizes of these pairs of molecules allows us to use the average values of the volumes of various molecules vi. In this case, the formula for the separation coefficient takes the form t t -zq / V i E fqaqOq(SJq) -z" , C~ij : Vi E fqaqOq(Sq) i v j v q=l q=l
(68)
One obtains satisfactory agreement between the theory and the experiments. Thus, the lattice model gives a possibility to calculate the separation coefficient of microcomponent macrocomponent mixtures, based on the data for the adsorption of individual components.
146 6. A D S O R P T I O N
OF A M I X T U R E
O F M O L E C U L E S O F V A R I O U S SIZES
All the theoretical considerations of the adsorption of a mixture of molecules presented in this chapter were based on the assumption that every molecule occupies one site of the lattice and that in every site there can occur not more than one molecule. This limitation allowed us to consider only mixtures of more or less similar molecular sizes. If the molecules can be greater than the dimension of the sites, we have to do with the "multisite adsorption" problem. The lattice model uses the concept of a site, the size of which is fixed. In confrontation of two molecules two problems arise: 1- the determination of the number of sites occupied by the molecules of each kind, and 2 - calculation of the deviation of the real sizes of molecules from the strictly discrete relations of their sizes given by the lattice structure. The first aspect has been very well elaborated in theory of the lattice models of solutions. As an example we should remember the Flory-Huggins theory [112-114] as well as its generalizations [59,60] such as that proposed by Nitta [115]. This kind of equation is well known in the theories of the surface tension of liquid solutions, as well as in the theories of adsorption of polymers on homogeneous solid surfaces. Such equations can also be found in papers [116-119], and their generalizations both for flexible chains and hard rods, in [94,98,120]. The second aspect of the problem concerns more precise calculations of the sizes of molecules than in a situation when as the unit distance of the calculation, there occurs a discrete site size. This aspect was discussed in papers by Prigogine and co-workers [121,122] on molecular solutions, and even earlier in paper by Wilson [123] and by Aptekar [124] for metallic alloys. The conclusion is that the lattice is considered to be "soft" and its states are determined from the condition of minimum free energy of the system. This assumption both allows to one to calculate some small deviations of molecules from the fixed site size, and to preserve all the convenience of the lattice models connected with (a) the simplicity of solving the algebraic equations for molecular distributions and (b) calculations of the peculiarities of the interaction potentials of the components of a mixture. In the papers [121-124], this approach was applied to homogeneous volume phases while calculating effects of lateral interactions in the mean field approximation. In paper [125], that approach was generalized and applied to heterogeneous lattices, with both monolayer and multilayer adsorption of components of a mixture, when calculating the effects of lateral interaction in the quasi-chemical approximation. So in principle, the formalism of the lattice model makes it possible to consider mixtures of components of molecules of various sizes and to calculate the thermodynamic characteristics of an adsorption system which are of interest for research.
Acknowledgement Athour wishes to thank Profs. W.Rudzinski, W.A.Steele and V.A.Bakaev for preparing the English version of this manuscript.
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152 122. I.Prigogine, A. Bellemans and A.Englert-Chowles, J. Chem. Phys., 24 (1956) 518 123. A.H.Wilson, Proc. Cambr. Phil. Soc., 34 (1939) 81 124. I.L.Aptekar, Zh. Eksp. Teor. Fiz., 21 (1951) 910 125. Yu.K.Tovbin, Zh. Fiz. Khim., 69 (1991) 104
W. Rudzifiski, W.A. Steele and G. Zgrablich (Eds.)
Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces Studies in Surface Science and Catalysis, Vol. 104 9 1997 Elsevier Science B.V. All rights reserved.
153
Theories of the Adsorption-Desorption Kinetics on Homogeneous Surfaces H.J. Kreuzer and S.H. Payne Department of Physics, Dalhousie University Halifax, N.S. B3H 3J5, Canada 1. I N T R ODU C T I O N Whereas the equilibrium properties of a large system are controlled by the minimum of its free energy, the kinetics involve questions of energy transfer. To establish the relevant time scales for the adsorption and desorption processes, let us follow a gas particle approaching the surface of a solid. If it rids itself of enough energy within the attractive region of the surface potential, it will get trapped. However, even if it descends all the way to the bottom of the surface potential well, it will eventually evaporate again. For times to required for a particle to traverse the attractive potential well, the particle will remain close to the top of the well within an energy of kB T. In this time there is a fair chance that the particle acquires enough energy from the heat bath of the solid to escape again. If this escape, which we can identify with inelastic scattering, has not happened within a few round trips, the particle will begin its descent to the bottom of the potential well, which, in a quantum picture, corresponds to a cascade of transitions between the bound states of the surface potential, each downward transition accompanied by the emission of phonons into the solid and each upward transition with the absorption of phonons. This adsorption process, characterized by a time scale ta is, of course, more likely at low temperatures. After it has happened, the particle will try again and again to climb back out of the potential well through a sequence of phonon absorption and emission processes. It will eventually succeed in doing so after a desorption time ta. If ta is much shorter than td, then adsorption and desorption are statistically independent, and the processes of sticking, energy accommodation (i.e. thermalization) and desorption can be well separated. This is most likely the case if the thermal energy kB T is much less than the depth of the surface potential. The energy necessary to desorb a particle from the adsorbate can either come from the solid substrate or from some external source. As examples of the latter, lasers or other sources of electromagnetic radiation have been used in photodesorption and photon stimulated desorption [1,2]. Likewise, electron and ion beams are employed to cause electron and ion-stimulated des0rption, respectively ~]. Strong electric fields at field emission tips cause field desorption and evaporation, sometimes used in conjunction with lasers or electrons to produce photon- and electron-stimulated field desorption [4]. If the solid itself acts as the reservoir from which the desorption energy is taken, we speak of thermal desorption. Lennard-Jones and co-workers [5] argued that the thermal motion of the lattice should act as a time-dependent perturbation on the surface potential, with which the adsorbate is bound to the surface, and can hence supply the desorption energy. It has been shown in recent years that this picture, in modern parlance called phonon-mediated desorption is, by and large, correct, although coupling to
154 the electronic excitations in the case of metal substrates may also be important. Kinetic theories of adsorption, desorption and surface diffusion can be grouped into three categories: (i) At the macroscopic level one proceeds to write down kinetic equations for macroscopic variables, in particular rate equations for the coverage or for partial coverages. This can be done in a heuristic manner, much akin to procedures in gas phase kinetics or, in a rigorous approach, using the framework of nonequilibrium thermodynamics [7-10]. (ii) If it cannot be guaranteed that the adsorbate remains in local equilibrium throughout desorption, then a set of macroscopic variables is not sufficient and an approach based on nonequilibrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory. It is derived from a markovian master equation, but is not totally microscopic in that it is based on a phenomenological hamiltonian and on postulated transition probabilities that are subject to the principle of detailed balance. For the kinetics in chemisorbed systems, this is as far as theory has progressed. (iii) Lastly, we realize that a proper theory of the time evolution of adsorption and desorption must start from a microscopic hamiltonian of the coupled gas-solid system. A master equation must then be derived from first principles with the benefit that transition probabilities are calculated explicitly involving microscopic parameters only. So far this programme has only been completed for phonon-mediated physisorption kinetics, as reviewed in a monograph by Kreuzer and Gortel [6]. This review is structured as follows: In the next section we briefly sketch experimental methods and procedures of data analysis to the extent they are needed in the ensuing discussions. Next, in section 3 we present the theory for adsorbates that remain in quasi-equilibrium throughout the desorption process in which case a single macroscopic variable, the coverage 0, and its rate equation are needed. We introduce the lattice gas model and discuss results ranging from non-interacting adsorbates to systems with multiple interactions treated essentially exactly with the transfer matrix method. Examples of the accuracy possible in the modelling of experimental data using this theory, from our own work, are presented for such diverse systems as rare gases and CO on metals and multilayers of metals on metals. Dissociative adsorption and the role of diffusion are also discussed. This is followed, in section 4, by a review of the macroscopic theory based on Onsager's approach to nonequilibrium thermodynamics, as adapted in recent years to surface kinetics by the authors. In section 5 we then p r o c e ~ to the semi-microscopic level by outlining recent advances in the theory of the kinetic lattice gas model. In section 6 we very briefly mention the few advances made in physisorption kinetics since 1985. The review concludes with a statement of policy, and the identification of some directions for progress. Recent reviews of adsorption-desorption kinetics since Menzel's articles in 1975 and 1982 [11,12] are by Rendulic [13], mostly on experimental advances, by Lombardo and Bell [14], with emphasis on Monte Carlo simulation, by Brivio and Grimley [15] on dynamics, and by the present authors [16]. 2. E X P E R I M E N T A L ANALYSIS
PRELIMINARIES, PHENOMENOLOGY
AND D A T A
Experimental studies of thermal desorption kinetics can be grouped into three approaches: (i) isothermal desorption, (ii) temperature jump desorption, and (iii) temperature programmed desorption. In an isothermal desorption experiment one removes the gas phase above the surface, either by reducing the initial pressure P~ to a final pres-
155 sure Pf <
(1)
as a balance between the adsorption rate, Ro, and the desorption rate, R~. Here the coverage, 0, is defined as the total number of particles in the adsorbate normalized either with the maximum number in a monolayer or with the number of adsorption sites; both conventions are used. The rate of adsorption is given by Ro = S(O, 7") Pa,/ ( 2 m n k B T ) 1/ 2
(2)
as the flux of particles of mass m hitting the area, as, of one adsorption site from the gas phase at pressure P and temperature T. The sticking coefficient, S(O, T ) , is a phenomenologieal transport coefficient that accounts for the efficiency of energy transfer between the substrate and the adsorbate. Two major assumptions have gone into writing down the rate of adsorption in the form (2): (i) we have assumed that the gas phase above the adsorbent surface remains homogeneous, i.e. that the process of adsorption does not deplete the gas phase. Depletion may occur in adsorption on small particles, or if fast reactions take place on the surface. An example for the latter is the oxidation of CO on P d ( 1 1 0 ) in the oscillatory mode. If depletion occurs we must couple the rate of adsorption to a gas phase diffu-
156 sion equation as done elsewhere in this volume. (ii) Adsorption can depend on the past history of the system. E.g. in dissociative adsorption at low temperature where surface diffusion is much slower than adsorption, single sites left over between occupied sites become ineffective such that the maximum coverage, for situations of negligible diffusion over the time scale of the experiment, is 0.907 for dimer dissociation [22]. Such situations can sometimes be accounted for through the temperature dependence of the sticking coefficient. Without further knowledge about the system, little can be said about the rate of desorption, R e , short of the fact that it describes an activated process. One frequently parametrizes the desorption rate Ra = rd (o, r)o ~
(3)
as a process with desorption order x. For atomic adsorption in the low coverage regime (where lateral interactions within the adsorbate are negligible) we have x = l , for dissociative adsorption in the low coverage regime we have x =2; but note that the coverage dependence of the rate constant in general precludes such simple interpretations. It is, indeed, the coverage dependence of the rate constant that contains all the interesting physics and is the reason why desorption experiments are done in the first place. We point out that the fact the rate of desorption as written in (3) is a function of temperature and coverage only is based on a major assumption, namely that the local configuration around a desorbing particle is completely determined by the overall coverage. This is clearly only the case if the adsorbate remains in quasi-equilibrium throughout the desorption process. Let us next implement the fact that desorption is a thermally activated process by writing the desorption rate constant in the Arrhenius parametrization, in surface science more frequently referred to as the Wigner-Polanyi equation, as ra = v exp(-E a / k B T )
(4)
in terms of an attempt frequency, J,, to escape the surface potential well, and an activation or desorption energy, E a . We will see below that both parameters are in general temperature- and coverage-dependent. To describe a TPD experiment we assume a linear temperature ramp, T(t) = T O + #t, which inserted in (1) with the adsorption term dropped, gives with (3), e.g. for x = l , dO dO dt v d--'T = - dt d T = # 0 exp(-E d / kBT)
(5)
If, and only if, g and E a are coverage and temperature independent, the temperature at the peak is given by the Redhead formula [23] Ed
-# = kB T exp( E,~/ kB T)
(6)
Conversely, determining the peak temperature experimentally, (6) provides one equation for two unknowns, i.e. p and Ed . To get a second equation, one can run TPD with a second heating rate differing by one or two orders of magnitude, and solve for p and Ea from the respective peak temperatures. We stress again that this analysis is only valid for systems in which v and Ed are c,overage independent, which is rarely the case (and uninteresting). Moreover, Redhead s analysis is invalid if two or more inequivalent adsorption sites are present on the surface, in which case a single variable 0 does
157 not suffice to characterize the adsorbate, and a rate equation like (1) must be written down for each partial coverage. In case particles can convert from one kind of adsorption site to another, additional diffusional terms must be added to the rate equations. This will be discussed below. To finish our discussion of phenomenology, we comment on methods of extraction of the kinetic parameters Ea and J,. First, we point out that inspection of TPD data can frequently lead to some (albeit semi-quantitative) insight into the structure and kinetics of the adsorbate. Let us assume that the TPD data show one peak. We can then extract a value for E a assuming that the prefactor r has a "standard" value of the order of 1013s -1 , characteristic of the vibrational frequency of an atom in a potential well with a spatial extent of an angstrom or so. The assumption of a "standard" frequency is rather dubious as r may vary anywhere from 10 ~0 to 1020s -~ from system to system and even over several orders of magnitude within one system as a function of coverage. Even so, with the assumption of a standard frequency and a heating rate of 5 Ks -1 one gets, approximately, from (6) that Ea = 31kBT p
(7)
If the TPD trace shows several (hopefully well-separated) peaks, this simple formula can be used to estimate different adsorption states, although great caution must be exercised and further analysis is required to substantiate such claims. Fortunately, this Redhead-type analysis, and variations on it [24], has been largely displaced by the "complete" analysis of a set of TPD curves, yet one still finds attempts at refinements [25], based on the coverage-independence of the parameters. A proper analysis of TPD data starts from the fact that in the desorption rate (5) the prefactor and the desorption energy are coverage (and particularly at low temperatures, also temperature) dependent quantities. The method proposed by several groups [26-28] consists of the construction of a series of isosteric (constant coverage) Arrhenius plots of enRd(O) (or en(ra) if the reaction order x is assumed) against 1/T, as interpolated from a family of TPD traces generated by varying initial coverages and/or heating rates. If the Arrhenius plots are straight lines or consist of sections of straight fines, the 0-dependence of E a and 1, can be deduced from the variations of slope and intercept, respectively, on these plots. Deviations from straight fines must be expected if (i) more than one adsorption state contributes to a TPD peak, or (ii) if the adsorbate does not remain in quasi-equilibrium throughout desorption, or (iii) if quantum effects produce a strong temperature dependence in the prefactor, e.g. in physisorbed systems. As for the interpretation of the parameters Ea(O) and r(0), we note that their zero coverage values refer to an isolated particle on the substrate, and their coverage dependence reflects lateral interactions between and different adsorption sites of adsorbed particles. A third method of determining such variations is based on the technique of threshold TPD [29,30]. Here the desorption rate is accurately measured at initial desorption for small changes in coverage and temperature, the initial coverage is then varied and the results fitted to (5). Compared to the second method, this extracts the kinetic parameters more directly, and is of advantage for systems away from equilibrium. Any temperature dependence of E d and J,, however, will not be as easily observed because the desorbing range is so short at each coverage value. We shall return to several points mentioned above in the following sections. Difficulties associated with the various methods of analyzing desorption data and remedies to overcome them are discussed at length elsewhere [31 ].
158
3. DESORPTION UNDER QUASI-EQUILIBRIUM CONDITIONS We consider desorption from an adsorbate where surface processes is so fast (on the time scale of desorption) that the adsorbate is maintained in equilibrium throughout the desorption process. That is to say that, at the remaining coverage 0(0 at temperature T(t), all correlation functions attain their equilibrium values. Thus the adsorbate can be characterized by its chemical potential,/z(0(t),T(t)). E.g., if we were to look at desorption of an adsorbate that shows coexistence of a dense and a dilute two-dimensional phase in a certain range of coverage and temperature, then the assumption of quasiequilibrium implies that at the remaining coverage, 0(t), the distribution of particles among the two phases is identical to that in equilibrium at coverage 0. In such situations a purely macroscopic description of the desorption process in terms of a single variable, the coverage 0(t), is sufficient. Its time rate of change has been written down in (1) in terms of an adsorption rate, R,, and a desorption rate, Ra. _We now look at a situation where the gas phase pressure, P, is different from its value, P, to maintain an adsorbate at coverage 0. There is then an excess flux to establish equilibrium between gas phase and adsorbate so that we can write
dO/at = S(O,T) (P-rJ)a,I (2mnkBT) ~/ 2
(8)
Next we express the equilibrium pressure in terms of the gas phase chemical potential, #s,
= kBT~-haexp(#gl kBT)Zi,I
(9)
where
X,h = hi (2~nkBT)l/ 2
(10)
is the thermal wavelength and Z~n, is the intramolecular partition function accounting for rotations and vibrations. However, in equilibrium, the chemical potential in the gas phase is equal to that in the adsorbate, #a, so that we can write the desorption rate in (1) as
R d = S(O,T)a, kBT/hXt2hZ~,,, exp(#o/kBT )
(11)
This is our principal result for the rate of desorption from an adsorbate that remains in quasi-equilibirum throughout desorption. Noteworthy is the clear separation into a kinetic factor, the sticking coefficient S(O, T), and a thermodynamic factor involving single particle partition functions and the chemical potential of the adsorbate. Thus a theory of desorption must simultaneously be a theory of adsorption, explaining not only kinetic data on desorption but also all thermodynamic data such as phase diagrams and heat and entropy of adsorption. The sticking coefficient is a measure for the efficiency of energy transfer in adsorption. Since energy supply from the substrate is required for desorption, the sticking coefficient, albeit usually at a higher temperature, must appear in the desorption rate by the detailed balance argument. The sticking coefficient cannot be obtained from thermodynamic arguments but must be calculated from a microscopic theory or be postulated in a phenomenological approach, based on experimental evidence for a particular system or some simple arguments.
159
3.1. Lattice gas model Most microscopic theories of adsorption and desorption are based on the lattice gas model. One assumes that the surface of a solid can be divided into two-dimensional cells, labelled i, for which one introduces microscopic variables n~ =1 or 0, depending on whether cell i is occupied by an adsorbed gas particle or not. The connection with magnetic systems is made by a transformation to spin variables a~ =2n~-1. In its simplest form a lattice gas model is restricted to the submonolayer regime and to gas-solid systems in which the surface structure and the adsorption sites do not change as a function of coverage. To introduce the dynamics of the system one writes down a model hamiltonian
H=E,
~ ni + ~ V2 E ninj + . . . i (ij)
(12)
Here Es is a single particle energy and V2 is the two-particle interaction between nearest neighbours (ij). Interactions between next-nearest neighbours etc. and many-particle interactions can be easily added to (12). As long as the number of particles in the adsorbate does not change, which is the case for systems in equilibrium, or for diffusion studies, the first term in (12) is constant and can be dropped from further consideration. However, if we want to study adsorption-desorption kinetics, the number of particles in the adsorbate changes as a function of time and a proper identification of E, is mandatory. Arguing that the lattice gas hamiltonian should give the same Helmholtz free energy as a microscopic hamiltonian (for noninteracting particles) one can show that the proper identification is given by
E, = - Vo - kBT en(q3qi,,, ) - kBT [en(X?hP/kBT ) - en(Z,,,,)]
(13)
where v 0 is the (positive) binding energy of an isolated particle on the surface. Moreover,
q3 = qzchy
(14)
is the vibrational partition function of an adsorbed particle with
qz = exp(hvz/ 2kBT)/ [exp(h~'z/ kBT)- 1]
(15)
its component for the motion perpendicular to the surface. Likewise, q~. is the partition function for the motion parallel to the surface. We have also made a'djustment for the fact that the internal partition function for rotations and vibrations of an adsorbed molecule might be changed from its free gas phase value Zi,,, to qi,,,, if some of the internal degrees of freedom get frozen out or frustrated.
3.2. Non-interacting adsorbates To elucidate our principal result (11) for the rate of desorption we first consider a noninteracting adsorbate in the submonolayer regime. Its chemical potential is given by
Izo = -V o + kBT[en(O/ ( 1 - 0 ) - en(q3qi,,,)]
(16)
If the adsorbate is not localized, i.e. if the corrugation of the surface potential parallel to the surface is negligible, we can treat it as a two-dimensional ideal gas and have
160
q~y = a~/ N2h
(17)
Thus (11) reads in the high temperature limit, i.e. for kBT>>h~,z, for a nonlocalized, noninteracting adsorbate R a = S(O,T) 0/(1-0) Z,.,/q~., ~ exp(-Vo/kBT)
(18)
In particular, if S(O,T)=So(1-O), then (18) is the familiar first order rate expression, with i, and E d identified. Let us next consider a noninteracting, localized adsorbate for which q~y has two factors like (15) with frequencies ~,~ and z,y. In this case we find in the high temperature limit Ra = S(O,T) 0/(1-0) Z~.,/q~., 2 m n a s / k B T g~PyPz exp(-Vo/kBT)
(19)
Since substrate vibrational frequencies ~,x, ~, ~z, are typically of order 1012-10~3s -1, the prefactor here can be larger than that in ~ 1~,j by a factor as great as 104, depending upon the system under study. Thus (zero coverage) prefactors up to 1017s-1 are to be expected even if Z~,,, = q~,,t [32]. Nevertheless, one sees statements in the literature that prefactors as large as 1020s- ~ are erroneous or unphysical. A word of caution about the high temperature limit: typical vibrational frequencies correspond to vibrational temperatures Tv =50-500K. Recall that desorption of chemisorbed species, with a heat of adsorption of a few eV, occurs at temperatures well above room temperature so that the high temperature approximation of the partition functions is justified. This is not the case for physisorbed systems with heats of adsorption of less, and frequently much less, than an electronvolt. In such cases the high temperature approximation is not justified. Indeed, for very weakly bound systems, such as rare gases on metals or even N2 on Ni(ll0), one might well be in the low temperature regime when desorption occurs. In such cases we write for (15) qz = exp(-ht'z / 2kB T)
(20)
and find, instead of (19) R a = S(O,T) O/(1-0) Zi.,/qi., a s kBT/hk,2hexp(-Ed/kB T)
(21)
Ea = Vo - h(vx +vy +Vz)/2
(22)
Thus the effective desorption energy is reduced from its high temperature value by the zero point energy in the surface potential, a reduction which is significant for some systems. Similarly, the prefactor in (21) is reduced (by orders of magnitude) from its value in (19) to a quantity independent of substrate vibrations. Expressions similar to (21) have been used frequently in the literature without reference to the fact that they are only valid in the low temperature regime [33]. While neither the classical nor the quantum limits of the vibrational (and internal) partition functions may apply to a system, they underscore the point that, across systems, there is no fixed relation between the desorption energies and prefactors, as determined by an Arrhenius analysis, and the microscopic binding energies and frequencies of the underlying dynamics. Whereas V0 and ~'i are parameters in a Hamiltonian, which gives the energy level structure of the substrate binding potential, Ed and v are strictly phenomenological (but physical) parameters. There is a considerable confusion in the literature on this point. It is useful at this stage to realize that for systems that remain in quasi-equilibrium throughout the desorption process, the desorption energy is more or less equal to the
161 isosteric heat of adsorption, as derived from the adsorption isotherms OenP aiso ( O, T) = k B T2 " ~
(23)
To see this we first define a differential desorption energy as
e,d (O, D = ksT2
Oen(Rd) Or
(24)
corresponding to the local slope of an isosteric Arrhenius plot. The desorption rate here can be identified from (8) in terms of an equivalent pressure that would be maintained if the adsorbate with the instantaneous coverage 0(t) were in equilibrium at temperature T. We can therefore write (24) equivalently as
Ea(O,T ) = Qiso(O,T) - ~ k B T + kBT2
OenS ( O, T) OT
(25)
Note that the second term is essentially irrelevant in the exponential. If the temperature dependence of the sticking coefficient is small (which is not the case for all systems), then the effective desorption energy is given by the heat of adsorption, provided that (i) surface processes are fast enough to keep the system in quasi-equilibrium throughout desorption, and (ii) that precursor states are absent. The generalised prefactor, v(O,T), is now defined via (3) and (24). In passing we note that a fit to the equilibrium data (isosteric heats, isobars) also provides a sensitive method of deducing V0 and {~'i}3.3. Nearest neighbour interactions in the quasi-chemical approximation Next we look at the effect of lateral interactions between adsorbed particles which are assumed localized so that, in the simplest model, we can think of the adsorbate as a lattice gas with a nearest neighbour interaction of strength V2. We illustrate some elementary but important results within the quasi-chemical approximation which gives [34] ].ta/ k n Z = - Vo/ kB Z + [1--~c] en[O/(1-O)] - en[q3qi,,, ]
+ cV2/2knT + ~ cen[(~ 1 +20)/(a+ 1.-20)]
(26)
where c is the site coordination number and a,2 = 1 - 4 0(1- O)(1- e x p ( - V2 / k B T) )
(27)
The desorption energy, via (11) and (24) can be expressed as Ea(O,T) = E a ( 0 , T ) + ~cV 2 [(1-20)/tx- 1]
(28)
For a large, repulsive (V2 >0) interaction, for example, Ed(O,T) exhibits two distinct
162 and essentially constant values for 0 ~: 1/2, implying two peaks in the TPD spectrum for initial coverages 00 _<1. The case of attractive interactions is more interesting, especially for temperatures below the critical value, To, for the phase transition. Here dilute and condensed phase regions coexist and (26) and (24) are replaced, within the coexistence region, by ~a = 89cV2- ksren(q3q~,)
(29)
z ~ ( o , r ) = E~(o,r~
(30)
- ~cv2
Inserting (29) into (11) we readily get zero order desorption kinetics within the coexistence region as a result of the coverage-independent chemical potential. As an example we look at the Ag/Mo(110) system for which the isosteric heat of adsorption at zero coverage, identified with Ea, has been measured to be 2.91eV [35]. A fit to the isothermal desorption data by choosing vz = 3.0x1012s -1, vx = p = 3.4x1012s-l and cVz = -0.41 eV produces the TPD traces of Figure 1, their p e ~ heights agreeing, within 10-20%, with those of Figure 4 of [35]. We note that these vibrational frequencies result in an effective prefactor in the Arrhenius parametrization of about 101Ss-1 as obtained in the evaluation of the experimental data, as discussed in [10]. The coverage dependence of the kinetic parameters is of particular note. At a fixed temperature, Ed(O,T) assumes a constant value, (30), if 0 lies within the coexistence region, but a quite different value, (28), otherwise. Figure 2 shows this dependence for three temperatures spanning Tc ( = 969K for V2 above), but as calculated from the local slopes of a series of isoteric Arrhenius plots, i.e. as if the data were generated experimentally. Note that, because we have imposed first-order kinetics on a zero-order process, the prefactor, u(0,13, is not constant in the coexistence region, but decreases like 0-1. This artifact is often contained in the evaluation of experimental data as well. The abrupt changes in E a and ~, mark the phase boundary at that temperature, and a complete temperature sequence will map out the phase diagram of the system. As an example we take the work of K olaczkiewicz and Bauer [36] on the desorption of metals from metals: having increased considerably the sensitivity of their measurements over earlier work [27], they noted that most Arrhenius plots did not yield one, but two, straight lines. We reproduce their resulting plots of Ed,(0) and ~,(0) for metals on ~(110), in Figure 3. For comparison purposes only, we show in Figure 4 the values of Ed and obtained by assuming two linear fits to the theoretical isosteres obtained from Figure 1. These curves are the temperature averages over the desorption range of a complete sequence of curves of Figure 2. The double-valuedness of Ed(O) and ~,(0) is clearly the indicator of desorption through the two-phase region. The results of Figure 3 must be considered as one of the examples of considerable progress in experiment [37]. There are many examples in the literature where similar features should have been seen, e.g. in [38]. Finally, we observe from Figures 3 and 4 the familiar compensation effect, namely an approximate proportionality between the single phase values of Ed and v, as a function of coverage. While this is observed experimentally for systems with attractive interactions (see e.g. ref.[39] for a further example), and is understood in this case by consideration of entropy change, the argument does not apply for repulsive interactions; nor is the proportionality well founded in lattice gas models. It has received much attention, however [40,41]. Any application of a compensation 'law" to the modelling of TPD spectra is suspect [42]. It is clear from the rate expression (11) that for systems that remain in quasi-equilibrium during desorption, one can extract the chemical potential as a function of cover-
163 [,~
o o
/'X
/ /
s"--
x 0
(.J (D 03
.
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.
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.
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.
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.
.
.
.
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.
1 0 0 0 90
Temperoture
.
.
.
I
1050. 0
(K)
Figure 1" Temperature programmed desorption calculated for A g / M o ( 1 1 0 ) with data from ref. 35. Solid lines: quasichemical approximation with Vo = 33760 K = 2.91 eV, cV 2 = - 4 7 1 5 = - 0 . 4 1 e V , c = 4 , Tc = 9 6 9 K , vz = 3 x 1 0 1 2 s -1,v x = v . = 3 . 4 x 1 0 1 2 s -1, a, = 5 A2, S~ = S2 = 1. Dashed curves: Bragg-Wilhams approximation for which Vo = 2.93 eV, cV 2 = -4725 K and Tc = 1074 K. Heating rate is 1.21 K s- 1, initial coverages (top to bottom): 00 = 0.75, 0.55, 0.35, 0.15. From ref.[10]. .
.
.
.
Y
age and temperature, i.e. exp(#~/kBT) = ra / [S(O, T)a.kBTZ,., / hX,2h]
(31)
provided the coverage dependence of the sticking coefficient is known. Thus isosteres can be constructed, isosteric heats of adsorption can be calculated, and phase diagrams can be deduced. This procedure to use thermal desorption data to extract equilibrium properties is complementary to direct equilibrium measurements because the latter can usually be performed at low temperatures whereas the former extend to much higher temperatures, e.g. in the case of metals on metals to temperatures above critical. If equilibrium data can be measured independently in addition to thermal desorption, then (31) can be used to extract the sticking coefficient, although nobody has attempted this so far.
164 o t.,o ',z" 0 0 0 v
k_ I13 t't.IJ
o
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f
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.
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t/
:J"
o I21
n r._.J
o w
mO.O
0.2
0.4
Coverage
0.6
I
08
(ML)
Figure 2" Upper triple of curves: differential desorption energies, Ed(O,T), as a function of coverage and for three temperatures, derived from the local slopes of isosteres as calculated from TPD spectra generated in the quasichemical approximation;temperatures are Tl = 1000 K (dash-dotted curve), T2 = 950 K (solid), T3 = 900 K (dashed). Lower triple of curves: (logarithm of) the corresponding effective prefactors. From ref.[10]. 3.4. 'Exact' results using transfer matrix methods The rate expression (11) allows one to set up as precise a kinetic theory as the equilibrium properties of the system can be modelled. One can extend the quasichemieal approximation to more general cluster methods [43]. To overcome the restriction to finite clusters one can employ transfer matrix methods [44,45] to calculate #a, with multiple lateral interactions included, on an M x oo lattice. The transfer matrix is constructed for interactions between particles in adjacent columns of M sites in the finite direction; its leading eigenvalue, X1, gives the grand partition function, ~ T , M, #) = XlM, and hence the coverage
165 o
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+
+
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o o
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9
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(ML)
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9
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38
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I 08
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=-
10
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1
z
02 04 06 COVERAGE 8 (ML)
i
~
08
10
Figure 3: Desorption energies Ed(O) and pre-exponential factors p(~ of Ni (a), Cu (b), Ag (c) and Au (d) on ~ 1 1 0 ) as a function of coverage O. The data are derived from the evaluation of 5,4,6 and 7 series of TPD spectra of Ni, Cu, Ag and Au, respectively, characterized by different symbols. O = 1 corresponds to 14.12x10 TM atoms/cm 2. The lines through the experimental points are only intended to guide the eye. From ref.[36].
n
kBT aen~ M a#
(32)
Exact results can be obtained by this method in the limit of large M. In practice, it suffices to consider only the totally symmetric subblock of the transfer matrix for this calculation; 0 is obtained from the corresponding eigenvector. One finds rapid conver-
166 0
',e"
o o o
f 0
!I3
J E
l.j o
.9 k.
\
0 m ~ Cl
\ \
o
.
0 u o ~ 12-
. m
E
f
f
..J
o
,....
mO.O
I
0.2
II
i
0.4
0.6
Coverage (ML)
i
0.8
Figure 4: ,Coverage dependence of desorption energies (upper pair of curves) and (logarithm of) prefactors in quasichemical approximation, obtained by fitting equ.(5) to generated isosteres in the one-phase and two-phase regions separately (solid and dashed curves, respectively). From ref.[10]. gence of the method; e.g. for a system with nearest neighbour interactions on a hexagonal lattice, M-9 is essentially exact (but M=3 is not good enough). The transfer matrix method has been used extensively to determine phase diagrams of adsorbates [46-48], where details of the method can be found. As a first demonstration of the richness of the structures that can be dealt with we reproduce model calculations [49] of the isosteric heat and TPD rates for systems with first and second neighbour and trio interactions on a hexagonal lattice. We begin with a simple system with repulsive nearest neighbour interactions only, Figure 5a. The isosteric heat, as calculated from (9,23), shows a sharp drop. at low temperature around coverage 1/3 reflecting the onset of ordering into the V"3x~Y'3 R300 structure. It results from the large increase in chemical potential as nearest neighbour sites become occupied above coverage 1/3. The feature of note is the appearance of local maxima and minima, most pronounced for temperatures less than the (reduced) ordering temperature, Tc =Tc I V2 - 0.34, for the system which appear as one cuts isothermally through
167
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i
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-.6
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-.9
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1
2
3
.4
5
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1
.7
.8
1
-13
.9
.~
0
.2
5
.4
(,~tL)
.5
Coverage
o
.6
.7
8
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(ML)
1
-.1
1
-.3 -.2 I -.4 -,5 >~
-.~
-~ 0 (5
-,7 -.8 -.9 <1 9-~ .0 -1.1 -I .2 -1.3
~
-1.4
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. . , i . , . 1 , .
0
.1
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,
i
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,
.
l
,
,
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,
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l
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,
Coverage (ML)
L
.7
p
8
,
,
,
t
,
,
,
.9
Figure 5: Normalised isosteric heat, AQ - Q(O,T') - Q(0,T), as a function of coverage for nearest neighbour repulsive interactions (V 2 > 0 ) between particles on a hexagonal lattice. The curves are labelled by reduced temperature, T = k B T / V ~ , with the values (top to bottom at O = 0.1) 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5, 0.6, ...7, 0.85. Calculated for a) 9-site approximation to the two-row transfer matrix, b) including next nearest neighbour repulsion, V~ = 0.1V 2, and c) including trio interactions, V~ = -0.1V 2, W =0-2112- From ref.[49]. the phase diagram [50]. Thus there is an effective attraction, AQis o >0, between particles for coverages 0 _<1/3 and T = T / V 2 = 0 . 1 8 - 0 . 3 2 ; and a diminished repulsion, sig-
168 nailed by the local maxima, for T =0.32-0.35. These features result from the tradeoff between the energy and the entropy of the adsorbate in the vicinity of the orderdisorder boundary. Similarly for 0 _>1/3 the repulsion is enhanced, relatively, for these temperature ranges. This behavior is repeated at coverage 2/3 due to particle-hole symmetry around coverage 1/2. These features are also observed for the square lattice as discussed elsewhere [51] where connection is made to the structure of the two-particle correlation function. For temperatures greater than the ordering temperature all these features wash out leading eventually to a more or less linear decrease in A Qiso, as expected from mean field theory. Turning next to weak next nearest neighbour repulsion (V2'=0.1V2) we note in Figure 5b that additional drops appear at low temperatures around coverages 1/4, 1/2 and 3/4. These correspond to the onset of orderings into 2x2 and 2xl phases. These sharp decreases also disappear above the ordering temperature. Even this weak next nearest neighbour repulsion begins to suppress the v"3xV"3 structure. As one increases the next nearest neighbour repulsion to V2'=0.2V2, this suppression is complete with only period two structures remaining [52]. On a hexagonal lattice there are two kinds of trio interactions, (i) between nearest neighbours only, V,, and (ii) involving two nearest neighbour and one next nearest neighbour pair, Vt '. As Vt preserves period three structures induced by nearest neighbour repulsion, the additional effect of this (weaker) trio interaction on the isosteric heat is effectively to move the plateau region of Figure 5a above coverage 2/3 up (down) for Vt <0 (>0). The same is true when attractive next nearest neighbour interactions (V2' <0) exist as well. The situation is different for the case v2'>0 and Vt ;~0: depending on the relative strengths of these interactions, there is a competition between adsorbate structures of periods two and three above coverage 2/3. A similar situation applies for combined next nearest neighbour pair and trio interactions. Essentially, the trio interactions enhance the orderings induced by the pair interactions of the same sign and compete with them if the signs are opposite, for coverages above 1/2. We show an example of this situation in Figure 5c. We next turn to thermal desorption showing in Figure 6 the three corresponding TPD spectra, plotted as a function of reduced temperature and for various initial coverages. The three broad desorption peaks of Figure 6a (for 00 >0.75) correspond to the three plateaus of Figure 5a over this temperature range, with the valleys in the rate occurring at coverages 2/3, 1/3, where the adsorbate repulsive energy reduces dramatically. We compare the trace for 00 = 0.70 with the variation of A Qis o in Figure 5a: the sharpness of the low temperature peak, occurring at 0=0.67, arises from the excess repulsion reflected in the local minimum in zXQ~,o just above this coverage; the next peak occurs at 0 = 0.56 and T = 0.23 following, again, an excess of repulsion, relative to lower temperatures, in AQ~ o ; similarly the shoulders at 0 = 0.40, T = 0.27, and 0 = .22, T = 0.32. The effects of the local maxima of A Qi~o are harder to discern here but, compared to more approximate calculations of tza , they appear as a steepening of the leading edge of the desorption trace out of the valleys. Figure 6b shows the desorption traces obtained corresponding to the ratio V 2 ' / V 2 = 0.1 of Figure 5b. Again, the existence of essentially six 'plateaus' in A Qi,o, for low to intermediate temperatures, implies six desorption 'peaks' for high initial coverage, with the positions of the local maxima of F ~ u r e 6b reflecting the local minima of Figure 5b occurring about the 2 x 2 and v"3 x v 3 orderings. Figure 6c contains the effects of competition between these two orderings induced by the interactions corresponding to Figure 5c. The sharp peaks at T = 0.20 for the traces 00 = 0.85, 0.70 clearly arise from the large excess of repulsion at this temperature for 0 >__2/3. The asymmetry of the two desorption peaks for the trace 00 =
169
. . . .
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. . . .
'
a
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b o
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0 0 •
x
3
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.
.15
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.25
.30
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.15
.35
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~
9 9
.20
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.
.
.
.
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Reduced Temperature
,
.
.
,
,
,
.
.
.
.
,
.
.
.
.
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.
.
.
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T/V 2
.
c
oo 3
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.
25 JO .20 Reduced TemperGture
.
3~ T/V2
Figure 6: Temperature programmed desorption rates as a function of reduced temperature (T/V2) and initial coverages, for three sets of adsorbate interactions. The site parameters assumed are typical for those of CO on (111) faces of transition metals: V0 = 156 k J / m o l , ,,., = ~'xy = 5 x 1012 s- ' , nearest neighbour repulsion V2 = 12.5 kJ/mol = 1500 K. Heating rate is 5 Ks -1. Initial coverages (left to right) 0o = 0.85, 0.7, 0.55, 0.40, 0.25, 0.1. (a) nearest neighbour repulsion only; (b) next nearest neighbour repulsion included, V~ = 0.1V 2; (c) next nearest neighbour attraction V~ = -0.1V 2, and next nearest neighbour trio repulsion 11; = 0.2V 2. (For this interaction set the lowest temperature desorption peak for 00 = 0 . 8 5 lies below T~ V 2 = 0 . 1 5 . This is suppressed to maintain the scale.) From ref.[49].
170 0.55 nicely displays a general effect of the local maxima/minima structure of ~Q~,o. Such asymmetries are absent in mean-field type calculations. Although multi-peaked desorption spectra will occur with nearest neighbour repulsion (on a triangular lattice) alone, two lessons should be taken from these examples: the presence of additional, and weaker, interactions easily gives rise to multi-peaked spectra of more complicated and varied shapes; and the local minima of the isosteric heat significantly affect this spectral structure. To directly associate such structures with different binding states, as frequently done in the literature, is too naive and often misleading. As an application of the above theory we now look at the adsorption of CO on Ru(O001). In a number of respects this system is an ideal one to model: Menzel's group has performed a complete set of measurements [53-55], producing superb data, on the low temperature structural properties of the adsorbate (LEED), its equilibrium properties, and adsorption and desorption kinetics (the latter in both isothermal and temperature programmed modes). The system displays a number of interesting features that demand detailed analysis for their modelling. There is convincing evidence that CO is mobile on Ru(0001) for temperatures above 200K, so that the adsorbate is in quasi-equilibrium throughout desorption. LEED data [55] suggests that the CO molecules move off their original binding sites above coverage 0 = 0.4. The simple lattice gas model with one type of adsorption site, as specified in (12), is therefore inapplicable above this value, but can be used to fit the experimental data below it, with a trio interaction introduced to mimic effects in the data above 0=0.4. One can estimate the nearest neighbour and next nearest neighbour interaction strengths by making use of the equilibrium isobaric data, Figure 7, and the (derived) isosteric heat, Figure 8a [54]. An alternate, and common procedure, is to use the structural data and compare with model phase diagrams. This is always much more difficult computationally. From Figure 8a we have the following estimates: Q~o(O = 0,T=700K) = 154 kJ/tool, Q~,o(O = 1/3, T <700K) = 118 kJ/mol implying V2 = 12 kJ/mol. A fit to the low coverage isobars yields the site parameters V0 = 154.6 kJ/mol, ~'z Pxy ~" Phr = 2.64 x 10 ~2 s-~. A hindered rotation, of frequency ~'hr, has been included as the additional degree of freeAom of the molecule. All frequencies are set equal for simplicity. The behaviour of the isobars elsewhere in the 0-T plane is now determined by particle interactions alone. Figure 7 shows the fit assuming a next nearest neighbour attraction, V2' = -0.07 V2 and a next nearest neighbour trio interaction, V,' = 0.07 V2. The fit is excellent for 0 _<1/3, but is poorer in the vicinity of 0 = 1/3 (the flatter portions) as reflected in the calculation of Q~,o(O,T), which is shown in Figure 8b for several temperatures in the range 300-700 K. With the parameters which produced Figure 7 one now generates TPD spectra according to eq. (11). With the "standard" assumption for the sticking coefficient, i.e., S(O,T) = S(0)(1-0) with S(0)=0.7, one gets the desorption traces as depicted in Figure 9a displaying several features not present in the experimental data. However, the sticking coefficient varies dramatically as a function of both temperature and coverage [53]. If one parametrizes the experimental sticking coefficient, approximately, according to the insert in Figure 9b and folds this information into the calculation of the desorption traces, we get Figure 9b. The agreement with the experimental data, Figure 9c, with regards to peak positions and relative peak heights is excellent for 0 _<1/3 and still good elsewhere. On performing an Arrhenius analysis of the desorption rates in the standard manner, i.e., 'plotting' (-1/0 dO~dO vs. 1/T, thus averaging Ea(O,T) over the desorption temperature range, we obtain the solid line in Figure 8c. The dashed line plots Q~,o(g), obtained by averaging the isotherms of Figure 8b over the temperature range of the isobars at coverage 0. The hump below coverage 1/3 can only be obtained in an infinite lattice calculation. An earlier attempt [56] based on the generalized quasichemical approximation to a lattice gas with nearest neighbour repulsion only is too simplistic. =
171
. . . . . . . .
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400
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600
650
Tempereture
Figure 7: Isobaric plots for CO/Ru(O001): comparison of theory (curves) and experiment [54] (points) using a 9-site approximation to the two-row transfer matrix. Interaction parameters: V2 = 12 kJ/mol = 1440 K, V~ =-0.07V2, V; =-V~, site binding parameters as in text. The isobars are P = 1.3 x 10 -6 mbar (left) and 1.30 x 10 -4 mbar (right). Coverage and change in work function, A~, are related by 0 = 7.58 x 10 -4 A~ + 8.01 x 10 -7 ( A 4 - 4 4 0 ) 2. From ref.[49]. The isothermal desorption kinetics of CO adsorbed on Rh(111) has also been analyzed employing a lattice gas with up to third neighbour interactions [57]. Here again the form of the sticking coefficient, varying as 1-30, is important for the proper modelling of the kinetics. The transfer matrix method can be extended to deal with adsorbates with more than one adsorption site or state per unit cell. As an example we refer to hydrogen adsorption and desorption from R h ( l l 0 ) [58]. A lattice gas model with first, second and third neighbour and various trio interactions between hydrogen atoms adsorbed on two inequivalent sites on the reconstructed surface is capable of reproducing all observed ordered structures, i.e. lx3 - H at 0=1/3, lx2 - H at 0=1/2, lx3 - 2H at 0=2/3, lx2 -2H at 0=1 and l x l - 2H at 0=2. It also reproduces the experimental TPD spectra, Figures 10 and 11. There are few other applications of the transfer matrix method to the modelling of desorption kinetics [59]. Much of the modelling is performed using Monte Carlo simulations [14,60-68]. While this method has obvious advantages for simulating effects such as fast diffusion and relaxation of the local environment of the adparticle , under quasi-equilibrium conditions the transfer matrix method of calculating the chemical potential (which yields both equilibrium and kinetic data) is definitely superior for the desorption rate. This is also evident by comparing the quality of TPD spectra produced by the two methods. This and the fact that the
172 Eeff or
EB
o
{KJ/rr~
~
o
150
~:.~
9
.
oo o
100
4
i
......
,
....
,
, . . . . . . .
9
o
. . . .
o
o
o:2 o3 o'.4 ols o.6
; ....
e-
; .... L .
i
.
.
.
'
9i
- -
~;
....
, ....
i
....
t
b
150
!50
>,
140
i
Q) 130
g
13C
. _
t~
110
-4,'
100
120 110 ~-
~oo F ............................... 0
.1
.2
=
Coveroge
.4 (ML)
.5
.6
0
"
2
3 Coverage
.4
1. .5
.6
(IdL)
Figure 8: a) Experimental data on heat of adsorption (full circles) and desorption energy from ref.54, b) Temperature and coverage dependent isoteric heat, Qi,,, (e, 7) for CO/Ru(O001) for the interaction and site parameters of Figure 7. The curves are labelled by temperature (top to bottom at (9 = 0.2) T = 300, 380, 460, 540, 620, 700 K. c) Temperature-averaged desorption energy, Ea(~ (solid curve), obtained from a linear Arrhenius analysis of the TPD spectra of Figure 9b; and isosteric heat, Qi,o(~) (dashed curve) obtained by proper averaging over the curves of Figure 8b. transfer matrix computation is relatively fast (and definitely faster than Monte Carlo) is important for an accurate fit of experimental desorption data. We stress again that in the analysis of the thermal desorption data, the role of the sticking coefficient is crucial. Here a caution regarding the simulation of desorption rates by the Monte Carlo tech-
173 I
.
.
.
.
I
.
.
.
.
.
.
.
t
.
.
.
.
t
"'
"
"
" ' t
.
.
.
.
d__ar
o •
dt
o r~ .
2
!
0
4.00
350
300
500
450
Tem,,~eraLure s,o~
.
.
.
.
.
.
.
.
.
.
:
.
.
.
.
"'
";
'
~
"
"
"
b f
o ~
"
\ . ~
g2
j 300
0
500
350
4-00
4.50
Temperature
500
t,00
500
--T(K)
600
~"
~
Figure 9" Temperature programmed desorption spectra for CO/Ru(O001) for two forms of the sticking coefficient (a) S(O,T) = 1-0, (b) S(0,7") given by the inset to this figure. Parameters as for Figure 7. Heating rate 5 Ks -]. Initial coverages (left to right): 00 = 0.66, 0.58, 0.49, 0.43, 0.38, 0.3, 0.25, 0.15, 0.05. c) Experimental spectra from ref. [54]. nique is in order: by the principle of detailed balance it implicitly assumes an adsorption rate with a sticking coefficient S(O)=So(1-0) (unless precursors are invoked). Thus in any system where the sticking coefficient does not have this simple form, simulation by standard Monte Carlo is inappropriate. For example, the recent attempts [63,67] to fit the desorption data of CO~Ru(O001) are unacceptable.
174
H/ah(l 0) Tad= 90K 1.36
d
0.57
j[3
0.74
~
@
0.36 026
'
100
i
150
I
i
'
'
'
!
. . . .
200 250 300 Temperature in K
I
. . . .
350
400
Figure 10: Temperature programmed desorption spectra of H 2 from gh(110) [58]. Initial coverages are indicated. Heating rate: 10Ks -1. Before closing this section, we mention a widely used phenomenological approach to adsorption-desorption kinetics based on transition state theory. In this approach one views the desorption process as a chemical reaction where a particle A bound to a surface site I:, forming a "molecule" (A+I:), is removed into the gas phase via a transition state or an activated complex (A +I:)* leaving an empty site behind. One then assumes that the transition state is in equilibrium with the reactants and calculates the corresponding equilibrium constant. Arguing next that the transition state vibrates along the reaction path, one can calculate its partition function in the high temperature limit to arrive, eventually, at a formula for the desorption rate constant (which no longer contains the vibrational partition function of the transition state). These remarks are not meant to discredit transition state theory as such, which no doubt has its role in chemical reaction kinetics, but to point out that for desorption kinetics simpler arguments, as presented e.g. from (8) to (11), lead to answers in a straightforward way. A case in point is the several pages of arguments needed in ref.[69] to arrive at the expression for zero order desorption, already contained in (11) for constant chemical potential.
175
LJ
-9
045
,
,
|
o81111.
040 035
1
i::
>-"(1)
||
030
~
i
0.6
.........
ii :."
~.~'~'. " ..
'~" t..:
I I I I i
025
0.4
I!
Co 020
""- . . . ~ ,,/ I ....~:
._
015 L
o s
. ....
,
,
c_
010
~
0.2
!.'" ~.i
{[I
005
t
x x
C)
03
0 100
150
Temper
200
& ture
250
300
350
o of ................ o.o
(K )
o.s
1.o
1.5
!i'i
2.0
Coverage
Figure 11: TPD spectra (left panel) and isosteric heat (right panel: dotted at T=100K, dashed at T=150K and dashed-dotted at T=200K) in the "best" inhomogeneous 2-site model [58]. Solid line in the right panel gives the desorption energy (up to 0=1.36) obtained by a linear Arrhenius analysis of the TPD data. a- and b-site binding and interaction parameters: V0(a)=2.50eV, V0(b)=2.45eV, VabI =0.09eV, Vboo=-0.013eV, Vba I -'Vaa 1 = 0 . 0 4 3 e V , Vob2 =0.013eV, Vo~2 =0.0043eV, Vba~ =Vo.b~2 =0.013eV, Vba.a2-----'O.OO86eV, Vo~a=0.0065eV, V'aa=V'bb=-O.OO86eV, V~a=-0.0021eV. Initial coverages as in Figure 10 with the addition of 00=1.36.
3.5. Multilayer adsorption and desorption Whereas adsorbates with mainly repulsive lateral interactions are generally restricted to the submonolayer regime, those with mainly attractive lateral interactions exhibit multilayer growth. Examples of the latter are most physisorbed systems such as rare gases and inert molecules on metals or insulators, but also strongly chemisorbed systems such as noble metals on transition metals and others [37]. Although such systems have been studied experimentally for many years and are obviously of great interest and importance, little progress has been made, until recently, on the theoretical side despite major advances in our understanding of adsorption and desorption in monolayer adsorbates. Early work [7,9] showed that even a small population of second layer particles can in.fluence the overall rate when desorption from the islands, on which" the particles reside, is restricted. A bilayer model for zero-order desorption has also been advanced by Asada and Masuda [70]. Some non-trivial lattice-gas models have been developed for the equilibrium properties of multilayers [71,72]. Recently we have presented a theory of multilayer adsorption and desorption based on a lattice gas [73,74]. The theory is based on a lattice gas with a hamiltonian
U-'Esl~rl
i dffEs2~Ilim
i
i
i "+ I Vii
~
n.n.
ni nj
l'lillj "+" ~ V'll n .n .n.
176
+
+
n.n.
+
n.n.
~"
niminjm J + ...
(33)
n.n.n.
Here the occupation number ni = 0 or 1 depending on whether site i is empty or has a particle adsorbed in the first layer with a single particle energy Es~ , c.f. (13). Likewise, m/ = 0 or 1 depending on whether the site i in the second layer is empty or occupied with an energy Es2. Here we assume on-top sites for the second layer for simplicity. In (33) we identify vl~ (V'l~) and V22 (V'22,) as the lateral interactions between two particles in nearest neighbour (next-nearest nelghbour) sites in the first and second layers, respectively. The interaction V~ is between next-nearest neighbours with one particle in the first layer and the second m the second layer. The nearest neighbour interaction between a particle in the first layer and another one on top of it in the second layer is accounted for by Es2 which also contains the residual interaction with the substrate. As an example we quote results for Cu on Mo(110). Experimental TPD traces [75] are reproduced in Fig.12a with a theoretical fit given in Fig.12b. Although isotropic nearest neighbour interactions alone with on-top adsorption for the second layer is assumed, the model reproduces the experimental data surprisingly well, with regard to both the shape of the desorption traces as well as the absolute magnitude of the rates. Lateral nearest neighbour interactions in the first and second layers are attractive and give a critical temperature of the first layer of Tc = 1125K in agreement with an estimate by Paunov and Bauer from TPD data [75]. Desorption from the coexistence region of the first layer is quite clearly displayed in the leading edge of the high temperature TPD peaks indicative of zero order desorption. The growth mode of the Cu bilayer is shown in Fig.12c as the partial coverages 01 and 02 of the first and second layer, respectively, as a function of total coverage for temperatures spanning the desorption range. Layer-by-layer growth is clearly in evidence. This has an interesting effect on the partial rates dO1~dt and dazl dt, Fig.12d in the intermediate temperature range of TPD where first and second layer peaks overlap: although the partial rates show apparent independent desorption of the two layers with the first layer trailing the second at higher temperatures, desorption from the first layer is now constrained by that of the second layer, resulting in a sudden onset of the former and a shift of the first layer peak for initial converages larger than one monolayer to higher temperatures to accomodate the total amount of material to be desorbed. Desorption of Au from Mo(110) is also analyzed in detail in [74]. Most of the applications of the lattice gas model to crystal growth have been concerned with calculating phase diagrams, critical properties and growth mechanisms within the solid-on-solid model. While the literature on crystal growth is enormous it has mainly been concerned with phenomenological descriptions, see e.g. [76]. The theory developed in [73] is distinct in that it considers the details of the adsorptiondesorption kinetics together with equilibrium properties of the first few layers where effects of the substrate still occur. In particular, the interplay of structure and kinetics can be studied within the one framework.
177 7.S r
7.5
o
!
i
~.so
6.0 d4.5 *o w
~3.0
.
0 ' 6 )0
g
~ 2OC
900
1000
1 1 O0
1200
Tempercture
(K)
TEMPER~TUR E [K}
75 i i,
.0
i . . . . .
, "
......
:
"
'
"
0
"
"
'
.
.
.
.
.
'
"
"
"
'
""
" ' "
"
'
'
"
"
"/i
III
/
/i
. - . 6.0
1
/il
i Id i ',I i ',e
3.0
6 ~,I !.5
O
.2
.4
.6
.8
:.0
1.2
', 4
Coveroge (ML)
~.6
!.8
2.0
I
0 t ...... 800
t
=~. 900
6 , ~'1~
~000
Tempercture
, ',,,S
11O0
1200
(K)
Figure 12" (a) TPD data of the first two monolayers of Cu on M o ( 1 1 0 ) obtained with a heating rate of 2.43Ks -1 [75]. The curves differ in initial coverage by approximately 0.07 monolayers between 0.07 and 2.2 monolayers. (b) Theoretical fit for initial coverages between 0.07 and 2 monolayers in steps of 0.14 with the following parameters: V0O) = 3.2eV, Vo(2) = 2.98eV, Vl~=-0.172eV, Vj2= 0 eV, V22=-0.129eV, vx = v = v z = 5x1012s -]. (c) Partial coverages for first (sohd lines) and second (dashed lines~ as a function of temperature. (d) Partial rates for first (solid lines) and second (dashed lines) as a function of temperature. From ref.[74].
178 3.6. Competition between atomic and molecular desorption A gas in equilibrium contains atoms and molecules at partial pressures that, at low densities and high temperatures, are controlled by the mass action law. Thus in a onecomponent gas of atoms and diatomic molecules we get for the respective pressures in terms of the chemical potentials #rag and #og of the molecular and atomic gas,
-Pa --- kBT 3 et~ag !tBr Nh
(34)
_ Pm-
(35)
kBT kBr kBT T e(2Uag +edis)1kB 1" X23 Zin, eumg! - h23 2T~ Qv (T)
edis is the (positive electronic) dissociation energy of the gas phase molecule and =X,h/~'2 is its thermal wavelength. Zi,,, is the partition function accounting for the rotations and vibrations of the molecule. Thus Tr is the rotational temperature of the molecule, and Q,, (79 is an expression like (15). A molecular gas adsorbs atomically if the energy released by binding its constitutent atoms separately is large enough to supply the dissociation energy. Entropy considerations also play a role because in equilibrium the Gibbs free energy of the atomic adsorbate must be lower than than of a molecular adsorbate. This brings in vibrational partition functions of the atoms or molecules with respect to the surface and also of hindered internal motion of the molecule. For practical reasons the released energy must also be enough to surmount a possible activation barrier in order to achieve equilibrium on an acceptable time scale. The question remains under which conditions an atomic adsorbate desorbs as either atoms or molecules. The theory has recently been developed for homonuclear diatomic molecules and applied to the desorption of Te from ~ 100) [77]. Two models have been considered: (i) In the direct dissociation model one assumes that the gas phase molecules become dissociated on impact with the surface, so that only atomic species are present. (ii) For precursor-mediated adsorption one assumes that molecules adsorb in a precursor state from which dissociation occurs into the atomic state. If fast surface processes maintain the adsorbate in quasi-equilibrium, the system is completely characterized by the two partial coverages, i.e. of the precursor and chemisorbed states, and the theory can be set up quite straightforwardly. We consider the case of direct dissociation only [58]. The atomic species has a coverage 0. Its time rate of change is due to an excess flux of molecules and atoms hitting the area, as, of an adsorption site and sticking with probabilities Sm (0,7") and S,, (0, T), i.e. .o
_
dt - S m
(0, T)
(Pro - Pro) -t- Sa(O,T ) (
)]k(P a - Pa)
(36)
Her~ Pm and Po are the instantaneous molecular and atomic pressures above the surface and Pm and Po are the equilibrium pressures needed to maintain a coverage Oof atoms at the temperature T. Next one uses (34) and (35) and the equality of the chemical potentials of the atomic gas above the surface, #,,g, and the atomic adsorbate, #,,(o), to write the rate equation (36) as
dO a-7 = R,,d - Rd~,
(37)
179 where the rates of adsorption from the atomic and molecular gas components are, respectively,
RaaO0 = S,, (0, T) a, -h P"
(38)
Rad (m) = S m (0, T~ 2a, -~ I'm
(39)
and the rates of desorption as atoms or molecules are
kBT (I.ta(a))/kBT
Rdes (a) = S a (0, ~ a, " ~ - e
(40)
kBT T (a) .t.~dis)lkB T Rdes (m) = Sm(O,T) 2a, hX22 2T r Qv(T) e (2ua
(41)
To complete the theory one needs to specify the coverage dependence of the sticking coefficients and the chemical potential, #o
e
0 1 <-Vo+l,ti)lkBT - 1-0 q3(T) e
(42)
so that (40) and (41) read 0 a~kBT (#i-Vo)IkBT Rd,~
0z
askBT T
Q~(T)
Re,,(m) = S m (O,T) (1_0)2 hk22 Tr (q3(T)) 2 e
(43) (2ui-2Vo+eais)/kBr
(44)
Whether an atomic adsorbate desorbs as atoms or molecules obviously depends on the relative magnitudes of the prefactors and the desorption energies. Lateral interactions can also play a major role as we now discuss for Te on W(100). From LEED data [78] it is known that this system has a 3x3 structure at 0=1/3 below room temperature, a c(2x2) and p(2xl) structure at 0=1/2, a (3x3) structure at 0=2/3 and some more complex structures close to monolayer coverage. To explain these structures adsorption at hollow and bridge sites is necessary. The necessary two binding site lattice gas model with first, second, third neighbour, trio and quarto interactions has been set up and the chemical potential has been calculated as a function of temperature and coverage using a 12-site transfer matrix. In Figures 13 and 14 we reproduce the experimental data and a theoretical fit that also explains the ordered structures. It is very gratifying that the molecular desorption peak is exactly in the right position and of the right magnitude, i.e. about an order of magnitude smaller than the lowest atomic peak. We stress that no additional parameters are available for the fit to the molecular peak. Considering the desorption energy for Te we note that V0 =5eV and qi, =2.38eV for Te~. Thus in the absence of lateral interactions the difference in desorption energies tor atomic and
180
I
|
|
[
|
1
1
i
1
a Is
z
Te~ {1001
/
...190
Tom :500K t =20Ks" i ~ m e t e r : Exposure time (s} Te. 9
,/~
,,-- ~ ~ 160
6o 6~
I~5
~s
Z'tO$! Z~Z
]OOs Z6.3 3'm~ Zt,O
b
Te/W (100) :500K ~mn ..2gKff~ l:~-Qmet~ : Exi)osure time (s) Te.
w
t,-
~S
t
r
A
<
II II
w
Z20
w
/ / --
//
I~mmeter: Ex~:msure time (s) lez
195
w
<5
~
Tetw (lOOl I,:~: SO0K f : ZOKs"
i~
6OO
18o
8OO
1000
/
1200 ~ TEMPERATURE (K)
!
1600 --
I
1800
20C0
Figure 13: TPD spectra for Te 1 (a,b) and Te 2 (c) from Te adsorbed on ~ 1 0 0 ) at 500K. Mass spectrometer/electrometer ranges 10 -s A (a), 3x10 -8 A (b), 3x10 -9 A (c). Heating rate 20Ks -1. Parameter: exposure (s). From ref.[78]. molecular desorption would be roughly 2.6eV, i.e. molecular desorption would occur at much higher temperature and would never be significant. However, strong lateral repulsion in this system amounting to about 2.5eV at full coverage reduces the molecular desorption energy to the same magnitude as the atomic one in the high coverage regime leading to the occurrence of atomic and molecular desorption at low temperature. Here the molecular prefactor is about 3 orders of magnitude larger then the atomic one, also helping in the competition. In closing this subsection we want to stress that competition between atomic and molecular desorption should be observable in a number of metal-on-metal systems [79] but has rarely been reported most likely for lack of looking. Of course, in such situa-
181
ij oo
.12 9
.10 9 .08
r6
.06
C 0 o--
i:).. L. o E]
.04
9 .02
I
600
800
1000
1200
T emperature
1400
160Z
(K)
1800
2000
Figure 14: Temperature programmed atomic (solid lines) and molecular (dashed lines) desorption of Te from V~100) with a 2-site model. From ref.[77]. tions a separate Arrhenius analysis of the atomic and molecular rates must be performed. 3.7. The role of diffusion in desorption The quasi-equilibrium theory presented so far can easily be extended to systems with more than one adsorbed species and several adsorption sites. If, however, precursors are present as in activated chemisorption, quasi-equilibrium may not obtain during desorption as intermediate steps in the desorption process may be rate limiting. These problems have been discussed very clearly by Cassuto and King [80] at the phenomenological level, and not much has been published since apart from an excellent review by Ehrlich [81]. We briefly comment on one aspect of diffusion in desorption kinetics, following an analysis of the system N2/Ni(430) [82]. We assume that there are two distinct adsorption sites on the surface, referred to for simplicity as step and terrace sites, with partial coverages 8, and 0,. The phenomenological rate equations in the absence of precursors and lateral interactions are assumed to be
182 dOs/ dt = SsasP/ (27rmkBT) l/ 2(1-Os)-rsO s +rs._ t Ot (1-Os)-rt._sOs (1-Ot )
(45)
dOt / dt = St at P/ ( 2 mnkB T) l / 2 (1- Ot) - rt Ot- rs._ t Ot (1- Os) + rt.. s Os(1- Ot)
(46)
Here Si for i=s,t are the sticking coefficients at zero coverage with a simple linear decrease assumed as a function of coverage. The rate constants can be obtained from as
(19)
l"i = SiOiZint/ qin, 2mnail kBT vxiuyi~zi exp(-Voi/ kBT)
(47)
r ~ , = J,~., exp(-Q~_,/k B73
(48)
where Q ~ , is the activation energy for hopping from the terrace to the step site. Detailed balance demands that rz.. s = r~_,r,/ rt
(49)
implying in particular that the activation energy for hopping from the more deeply bound step site to the terrace site is enhanced by the difference in binding energies. To calculate TPD curves we assume that all parameters are such as to produce two distinct .10
F-"
.09
'
"
'
,
,..
9
,
9
,
9 ,
;
j
a
,
~
.
~
~
,
9
,
.
,
,
170
180
.'
',
b
~.o7
~.07 v
.10 .0'9
.06
v
.06
g -~
g
~ .o3 ~ .o2
-~ .o3 ~ .ca
0 1.90
0
'
17,'0
140
1,50
.~
160
Temperoture
170
180 (K)
190
200
120
,/I
"
,
I~
!~.(3
!I \,,,, t
-"
150 Temperature
t
160
190
.
200
(K)
Figure 15" Temperature programmed desorption spectra for the phenomenological model of equ.(45-49): (a) Parameters from a fitting to equilibrium data of experimental data of N 2 / N i ( 4 3 0 ) (M. Grunze, private communication); no hopping ( ~ , =0), initial coverage 00 = 2/3, heating rate 3Ks -1. (b) As for (a) except u~_, =10 s s -1, Q~.,=lO kJ/mol. The solid curve is the sum of the rates from the terrace (long-dashed curve) and step (short-dashed) sites; the net diffusive rate (step--, terrace) is the dash-dotted curve. These three components have been scaled down by 0.25. desorption peaks in the absence of diffusion, see Figure 15a. Conventional wisdom then has it that, in the presence of fast diffusion, particles trapped in the deeper-bound step sites will diffuse into the terrace sites as the latter desorb. One therefore expects a lowering of the high temperature peak and a shift to lower temperatures, thus filling up
183 the valley between the peaks, as borne out in Figure 15b. We note, however, that the transfer from steps to terraces only sets in at temperatures above the maximum of the lower peak, i.e. when the blocking factor l-O, approaches one, and this for all values of J'~ t, and for all heating rates. Indeed, the effect saturates for hopping prefactors us._t larger than 10Ss-~. In Figure 15b the solid curve represents total desorption; the shortdashed curve is desorption from the step site, i.e. the term -rsO, in (45). Diffusion replenishes the terrace sites maintaining desorption from the terrace sites even at higher temperature, i.e. due to the last two terms in (45), the sum of which is shown as the dash-dotted curve in Figure 15b. Indeed, this is the fastest route for depleting the population of the step site, as indicated by the terrace site desorption rate, -r, O, (long-dash curve). The total desorption rate thus shows the high temperature peak reduced and shifted to a slightly lower temperature with considerable filling of the valley. We condude: if system parameters are such that in the absence of diffusion the TPD curves would have two separated peaks, then the two-peak structure remains in the presence of diffusion, no matter how fast it is. 4. N O N E Q U I L I B R I U M T H E R M O D Y N A M I C S OF S U R F A C E PROCESSES So far we have assumed that quasi-equilibrium is maintained during the desorption process. This assumption breaks down when the time constant of surface diffusion is not appreciably faster than that for desorption, as it happens e.g. for rare gases on metals, in which case nonequilibrium effects show up in desorption experiments. To deal with such situations it is advantageous to formulate the kinetics of adsorption, desorption and diffusion using the methods of nonequilibrium thermodynamics. We will demonstrate the approach by considering, in more detail, gas-solid systems in which the adsorbate exhibits coexistence of a dilute, gas-like phase together with islands of a condensed phase, both of course, being two-dimensional (2-d). The 2-d gas phase consists of two parts, adsorbed, respectively, on the bare surface or on top of the condensed islands. Taking the system out of equilibrium by, e.g., changing the temperature of the substrate or the ambient gas pressure, will induce surface processes such as adsorption, desorption and diffusion. Looking at desorption from such a two-phase adsorbate, it can proceed via several channels, namely (i) out of the 2-d gas phase on the bare surface with a time constant ta, (ii) out of the 2-d gas phase on top of the condensed islands with a time constant ta', and (iii) directly out of the condensed phase with a time constant t~, a process particularly important around monolayer coverage. On the other hand, as long as desorption proceeds predominantly via the 2-d gas phase, the depletion of the condensate islands takes place by evaporating into the 2-d gas phase, with a time constant tev, from where particles then desorb into the 3-d gas phase. This will result in density gradients leading to surface diffusion. Two rather different situations may obtain: (i) if tev '~ ta, evaporation is so fast that during the desorption process a quasi-equilibrium is maintained between the adsorbed phases. In such a situation there will be a coverage regime where the desorption kinetics is roughly zero order. (ii) If t,v >) td then evaporation is the slowest process in a chain and thus rate determining. With evaporation proceeding via the rim of the condensed islands one expects roughly half order kinetics. These ideas were first put forward many years ago in a number of papers [83-88].) From the above discussion it should be obvious that adsorbates under nonequilibrium conditions can no longer be described by a single macroscopic variable, i.e. the coverage, but that, as a minimum, one must consider the partial coverages of the various phases and components. To set up a macroscopic theory of surface processes, it is then expeditious to follow the Onsager approach to nonequilibrium thermodynamics [89]. Our first task is to identify the proper set of macroscopic variables to describe the
184 adsorbate. To describe typical experiments involving adsorption, desorption and surface reactions, it is usually not necessary to introduce local densities at the macroscopic level. If the adsorbate forms a single phase, it will remain homogeneous throughout so that diffusion plays no role. If, on the other hand, the adsorbate shows coexistence of two phases, mass transport is limited to particle exchange between the two phases which can be described as 2-d condensation and evaporation. It is then rather dubious to introduce density gradients in the dilute phase, because, e.g., at half coverage, the number of molecules in a typical patch of dilute gas is at most a hundred, so that number fluctuations amount to at least 10%, and the number of particles per patch of dilute gas, and even more so the local particle density, can hardly be taken as macroscopic variables in the two-phase region. To set up the balance equations controlling energy and mass exchange in a two-phase adsorbate, we consider the total system, consisting of gas phase, adsorbate, and substrate, as closed in a volume V, and isolated with total energy U,. As extensive variables we consider the particle numbers X 1=N1 in the condensed 2-d phase, X2 =N 2 in the dilute 2-d phase on the bare surface, X3 =N2,in the dilute 2-d phase on top of the condensed islands, X4 = N 3 in the 3-d ambient gas, X 5 =N, in the substrate, and the respective energy variables X6 =U1, .... Xa0 =Us. Following Onsager we can write the macroscopm balance equations as 10
dX i -- Z dt
OS
L,j ~ j
Ut , Vt ,N, Ns
j=l
(50)
Such a description is valid (i) in the linear regime, and (ii) as long as local equilibrium pertains. The latter condition in particular implies that the entropy of the system is given by
S(Ut,Vt,N, Ns) = SI(U1,A1,NI) + S2(Uz,A2,N2) + S2,(U2,,A2,,N2, ) + ss(v3, V, N3) + S,(U,, V,,N,)
(51)
Here A~ are the areas occupied by the respective phases. Thus we get in (50) OS
vj,Aj(
=-
~rj
OS
1 m
(52)
(53)
introducing the chemical potentials, #j, and the temperatures, Tj, in the various phases and components. Noting mass conservation in the system dNs/dt = O, ( d / d t ) ( N 1+N2+N2,+N3) = O, and energy conservation d U t / d t = 0, one gets 30 conditions on the 100 phenomenological coefficients Lq, because the coefficients in front of each thermodynamic force have to vanish independently. Onsager's reciprocity relations Lq =Lji eliminate another 43 coefficients.
185 Under isothermal situations the equations simplify considerably and can be written as L12
dN1 _ at
-
-
T
L12' (~-~")
- -'T
L13 (~'-#~)
- 7"
(~-~")
dN 2 LI2 L23 L22' at - - r (#l-g,2) - " ~ (P,3-/z2) - " ~ (/z2'-P'2) dN2' dt - -
L12' T
(#l-~
d N 3 -__ d ( N l dt dt
,) L22' , ) - L2'3 ,) - " ~ (V'2-V2 " ~ (~-V'2
W N2 + N2' )
(54)
(55) (56)
(57)
Note that equilibrium conditions, i.e. v~ =t~j, imply vanishing fluxes. The next task is to relate the 60nsager coefficients in (54)-(56) to experimentally accessible quantities like sticking coefficients and activation energies. This has been done for the individual processes, such as adsorption, desorption, and 2-d evaporation and condensation [7,9]. The resulting equations are dN 1 dt - R13 4- R12 + R12,
(58)
(59)
dN2 dt -- R23 - R12 + R22' d~,
(60)
- R2'3 - R12, - R22'
dt
Here Ri3 = SiAi
[:
kBT
~h - " - ~ ~h
2
e~i
/kBr]
(61)
are the net rates, i.e. adsorption minus desorption, of mass exchange between the 3-d gas phase above the surface at temperature T and pressure P and the adsorbed components,/=1,2,2'. The chemical potentials, v~, are those apropriate for the instantaneous values of Ni(t) and Si is the sticking probability, to the i-th phase. There is some arbitrariness in deciding whether particles impinging from the 3-d phase onto the condensed islands become instantly part of the condensed phase (sticking coefficient S 1) or equilibrate first with the 2'-gas phase on top of the islands (sticking coefficient $2,). This seems irrelevant because adding (58) and (60) implies adding (61) for i = l and 2', which, with ~ ' = t ~ , results in an overall sticking coefficient S1 +S2,, since A~ ----'A2,. The rate of mass exchange between the condensed phase and the 2-d gas. phase (2-d condensation minus 2-d evaporation), summarizing diffusive processes, is gwen, within the coexistence region, by v RI2
=
1
S c f AI~ -Tr A-A 1 IN2- ~V2(Na'T)]O(N1)O(N2)
(62)
186 as an excess flux of particles, moving with an average speed v and sticking, with probability So, to the rim of_ the condensed islands, the boundary length of which we approximate by fair. Here N2(Na, 7) is the number of particles that would be in the 2-d phase if the adsorbate were at equilibrium with a total number of particles, Na=N ~+N 2 +N2,, and O(N)=I for N>I and zero otherwise. Although the phenomenological geometric quantities of f, and ~', must change as the coexistence region is traversed, approximations are possible, e.g. if the islands are n in number and circular then f -- 2V'mr, ~" = 1/2; for irregular perimeters f = fo n~-~, 0.5 < g" < 1, fo >2~r. Likewise, particles in the 2'-phase can be incorporated into the condensed phase at the island rim according to R12' = Sc'f'AI
t.,v 1 ~r Z IN2' - N2'(Na'T)IO(N1)O(N2')
(63)
Lastly, we get for the rate of equilibration between the 2'- and the 2-phase w
v 1 R22 , = (1-Sc'Rc)fAl~ -TrA-A l [N2 - N2(Na, T)IO(N~)O(N2)
(64)
Here it is assumed that particles skipping along the bare surface will either be reflected back with a probability R,, or stick to the rim of the condensed islands with a condensation coefficient Sr or hop on top of them with a probability 1-S,-R,. As is evident from the rate equations (54)-(57), the thermodynamic forces responsible for mass transport are differences in chemical potentials. To specify the latter for a two-phase adsorbate one can resort to analytic models, e.g. in simplest form, by treating a mobile adsorbate as a 2-d van der Waals gas, and a localized adsorbate as a lattice gas with nearest neighbour interactions within the Bragg-WiUiams and the quasichemical approximations [7-9]. This being specified, the macroscopic theory is complete. It has been used successfully to describe adsorption-desorption kinetics in a number of gassolid systems, such as metals on metals and rare gases on metals [7-10]. We comment briefly on a few of the results. One example of the new insights gained from the rate equations (58-60) that was not contained in (11), is the desorption from the coexistence region for a system in which the sticking probability on the bare surface, $2, is different from that on top of the adsorbate, S~ +$2,=S~,, both being assumed constant, for simplicity. This appears to be the case, e.g., for Xe on Ni and Ru where S 1=0.6 and $2=1.0 [90]. In quasi-equilibrium the chemical potentials of the various adsorbate phases are equal, i.e. ~1 = ~ =tza in (61), so that after summing (58-60) we get dNa/ dt = (SItA 1 + S2A2)[P~h/ h - kBT/ hXt2hexp(#a/ kBT)]
(65)
Within the coexistence region one can use the lever rule and re-express (65) under desorption conditions, i.e for P=0, as
dO 1 kBT 2 dt =-[($1'-$2)0+$201~-$It02~] 0~-02~ as "--~,~ exp(#a/ kB T)
(66)
The new variables here are the lower and upper boundaries to the two-phase region, 02c(T) <0it(T) We note that, within the coexistence region and at constant temperature, there .~..o coverage dependence outside the square brackets. This implies zero order kinetics, provided that the sticking coefficients are the same for the dilute and the dense phases. However, if this is not the case, quasi-equilibrium in the coexistence region does not imply zero order kinetics. As an illustration we present in Figure 16
187
0
!
-
_J
o
/1 0
9
~
|
9
I I/"
-
.
~
|
I i "i
-
(D~ 0 _..j
/"
! I
! t/3
-
03 I
--
-
0.0
o. 2
o. 4
o.s
Coveroge (ML)
o. 8
1. o
Figure 16: Isothermal desorption of X e / N i ( 1 1 1 ) for temperatures (top to bottom) T = 82,80,78,76 K and a two-component adsorbate in quasi-equilibrium (solid lines); and T = 80,76 K for a nonequilibrium situation (dash-dotted lines). Parameters for the Bragg-Williams approximation: V0 = 3200 K, cV2 = -375 K, Vz = 5xl011s-l, vx = v. = 1013S -1, a s = 5 . 0 A 2, $1, = 1 . 0 , S 2 = 0 . 6 . D a s h e d lines s h o w coexistence boundaries m quasi-equilibrium. From ref.[9]. 9
9
9
)'
an approximate fit to data on Xe desorbing from Ni(111) [90], calculated in the BraggWilliams approximation 9 Nonequilibrium effects are introduced by increasing the size of condensed-phase islands and decreasing the hopping rate away from the perimeters of these islands, such that Rl2 .~R23, R13. The effect on (two of) the curves of Figure 16 is indicated by the dash-dotted lines, corresponding to imposing the extreme condition,a~2 < R23, R13. In this case coexistence continues down to 0 = 0, i.e. it is desorption from the islands that contributes predominantly to the total rate. Such isothermal modelling also shows that the concept of a fixed desorption order (x in (3)) is found wanting: in the quasiequilibrium case, x may not be fixed at zero during coexistence, c.f.(66) and cannot be unity outside coexistence due to adparticle interactions; in nonequilibrium situations x varies markedly with temperature, and it does not necessarily approach the value of ~'. Even the utility of the Arrhenius parametrisation (4) is questionable when desorption is not a quasi-equilibrium process - isosteric Arrhenius plots are no longer one or more straight lines, but smooth curves. Figure 6 of ref. [90] contains an excellent example of
188 this; its significance was overlooked by all. This sounds acaution against the 'blind" fitting of a single line through scattered isosteric data points, so common in the literature. Now the quantity Ed(0, T), defined by (24) is a necessity if one insists on a parametric analysis of the data. Again, though, the temperature dependence of Ea (0, T) can be so great as to render that analysis almost useless unless a detailed comparison is also made with specific models of the adsorbate. In such cases it is far better to model the o
O O
x,._
o
~
t c;-i 0.0
",,\
J'/ / 0.2
0.4
0.6
Coverage (ML)
0.8
1.0
Figure 17: Desorption at T = 80 K of a two-component adsorbate with desorption from condensed phase suppressed (S~ =0); quasi- and nonequilibrium situations otherwise as Figure 16 Initial coverage 6(t=0) = 0.8. From ref.[8]. rate data directly. As a final example, we show in Figure 17, the result of setting $1 = 0, N2, = 0 in the rate equations (58-60). For quasi-equilibrium, c.f. (66), there is a linear growth of the rate as 0 decreases, as far as 0zc(7) then a nonlinear decrease as the dilute phase desorbs. The extreme nonequilibrium cor~dition produces a variable order process, however (dash-dotted line). Since the first phenomenological kinetic models describing zero-order desorption and the effects of islanding of the adsorbate were advanced [83-88], surprisingly little improvement of the theory has occurred despite the large number of experiments exhibiting these effects. Of the few groups that have modelled desorption trajectories through a phase coexistence region, Nagai and co-workers [91] have been restrictive in their use of transition-state theory (quasi-equilibrium), with a constant prefactor, and no distinction between the coexisting phases, e.g. differing sticking coefficients. That
189 thermal desorption experiments can detect unequal sticking coefficients, as described by refs. [7-10], has been shown recently by Meyer et a/.[92] in their modelling of H/Mo(110). The characteristics of the sticking coefficient at the two-phase interface can also be probed [93]. Desorption experiments clearly illustrating nonequilibrium effects are more scarce. Arthur and Cho [83] observed half-order kinetics, long ago, for Cu and Au desorbing from graphite. Their Arrhenius plots are not linear but, as has occurred with other experimental analyses t94] since, they derive a desorption energy from the straightest portion of the isosteres. Such fitting procedures, coupled with experimental uncertainties, must be partially responsible for this scarcity. As many systems are in the quasi-equilibrium regime at the approach to peak desorption temperatures only a careful examination of the low temperature-high coverage rates will illustrate lack of equilibration [17-18,95]. 5. K I N E T I C L A T T I C E GAS M O D E L To go beyond a macroscopic description of time-dependent phenomena, we explore a mesoscopxc approach entailing the kinetic lattice gas model. It was originally set up in close analogy to the kinetic Ising model for magnetic systems [96]. To set up the kinetic lattice gas model we restrict ourselves to gas-solid systems in which all relevant processes, like diffusion, adsorption, desorption etc, are markovian. We introduce a function P(n;t) which gives the probability that a given microscopic configuration n=(nl,n2,...,nt) is realized at time t, where I=N s is the total number of adsorption sites on the surface. It satisfies a master equation
dP(n;t)/dt = Z
[B(n;n')P(n';t)- B(n';n)P(n;t)]
(67)
1l'
where B(n';n) is the transition probability that the microstate n changes into n' per unit time. It satisfies detailed balance
B(n';n)Po(n ) = B(n;n')Po(n 9
(68)
where
Po(n) = Z -1 exp(-H(n)/ kBT)
(69)
is the equilibrium probability. In principle, B(n';n) must be calculated from a hamiltonian that includes, in addition to (12), coupling terms to the gas phase and the solid that mediate mass and energy exchange; this has been done for phonon-mediated adsorption-desorption kinetics [6]. Rather, we follow the procedure initiated by Glauber [97] and guess an appropriate form of B(n';n). We briefly indicate possible forms below, and their consequences for the solution of (67). Assuming that the residence time of an adsorbed particle in an adsorption site is much longer than the transition time into or out of that site, we can write B(n';n) as a sum of terms accounting for individual processes of adsorption, desorption and diffusion, i.e. ~,~,(n';n)= Wa.d(n';n ) + Wd/f(n';n)
(70)
The simplest choice has been termed Langmuir kinetics [98] for which one assumes that adsorption into a site i takes place provided that site is empty, irrespective of whether
190 neighbouring sites are occupied. We then get
nn,= oZIlni+ onlX+ Zni a + ania n + ] *~ni ', l-hi) -[-[iS(hi',nt) l ;ai
(71)
where a and a' enumerate the neighbours of site i. Similarly we write for diffusion
1
Waif(n';n) =J0 ~ ni(1-ni~) 1+C1 ~ ni.o, + ' " l b~a j i,a * ~ n i ', 1 - n i ) ~ n i + a ',
1-ni.~)
]-[
~("l',nl)
(72)
l~i,i+a Inserting (72) into (70) one gets
Co = exp(E,/kBr)
(73)
Cr = [exp(VJ k s r ) - 11r
(74)
Langmuir kinetics, in particular, results in a sticking coefficient
S(0, I9 = S0(1-0)
(75)
Other choices, and their physical or unphysical implications, have been discussed in ref.[98]. As long as the master equation (67) is used in Monte Carlo simulations solely to determine the equilibrium properties of the system, it does not matter what choices are made for the transition probabilities ~n';n) as long as they satisfy detailed balance (68). However, if we are interested in the adsorption, desorption and diffusion kinetics of a particular physical system, the choice of transition probabilities l~(n';n) is rather narrowed [98]. To solve the master equation (67), Monte Carlo techniques have been invoked [99-102]. The renormalisation group has also been used successfully [102,103]. Recently, a perturbative approach has been suggested that constitutes a time dependent generalisation of Kikuchi's method [104] in equilibrium statistical mechanics. In a systematic approach one can derive a hierarchy of coupled equations of motion for average occupations and correlation functions [97,98,105-115]. To this end we define
(ni) = ~ niP(n;t) n
(76)
191
(ninj) = ~ ninjP(n;t)
(77)
n
and obtain from (67)
d(n,)ldt=Wo(1-(n,))-WoC0[(n,)
(78)
+ C~ Z ( n , n , + = ) + ...]
I.
a
J
and similar equations for the higher order correlation functions. Comparing (78) to (1-2) for the case of a homogeneous adsorbate with (n~) = 0, allows one to identify W0 as
(79)
Wo = So pas / ( 2 7vnkB T)1/ 2
implying, with (73) and (13), that the desorption rate constant is given by
rd(o=O)
= woco
(80)
= SO as kB T/hXt2h61 Zi,,, / qi,,, exp(- Vo/k B T)
The infinite hierarchy (78) etc., is obviously exact. To solve it, approximations must be made; in particular, the hierarchy must be truncated, e.g. by invoking the 2-site or Kirkwood closure approximation, which states that (n irlj...n m
) =
(ninj )(rljn k )...(nIn m )/(n i)(rlj)(nk )...(nl)(nm
)
(81)
In higher, n-site closure approximations the right hand side would involve n-site correlation functions in the numerator and (n-1)-site functions in the denominator. In the simplest (Kirkwood) approximation, and for a homogeneous adsorbate, one is left with two coupled equations for 0 and for the average correlation function ~k = (nini+a), which read for a square lattice [98] (82)
dO/dt = W0[1 - 0- CoO(1 + C1~//0)4] d~k/dt = 2Wo[O- ~k- Co(1 + C~)r + 2./o(1 + C~r
+
q~/e)31
3(1 + C~)~/+ 6"107/+ 5C~r
q~(1
+ 2r
(83)
We note that the equilibrium solution of these equations yields, at least in the absence of the diffusional terms, the quasichemical approximation to the isotherms. Other treatments of the closure, in 2-d, correspond to quasichemical, Kirkwood or maximum entropy approximations [109,116,117]. In one dimension the truncation of the equations of motion has been worked out and examined in detail [112,115,118]. This has allowed an accurate examination of the role of diffusion on desorption and implications for the Arrhenius analysis in non equilibrium situations. The largest deviations from the desorption kinetics of a mobile adsorbate obviously occur for an immobile adsorbate where surface diffusion is negligible throughout desorption. If the time scale of surface diffusion is of the order of that
192 of desorption we expect results intermediate between those of fast and negligible diffusion.
~'-5
............
4.0
'
I . . . . . . . . .
(3
"-~
3.0
~:2.5
7
~
61
~
5
~(
s
i
9[
,
:'/
4
t
)
!
1
o 90
.
'i'f 0
10(3
110
120
130
Temperoture
1( 0
150
. . . . . . . . . .
90
~9(:;
: 10
120
130
140
150
Temperoture
Figure 18:(a) Temperature-programmed desorption rate for a 1-d immobile adsorbate with repulsion V2 =400K. Solid lines: 3-site closure. Initial coverages (left to right) 00=0.9, 0.7, 0.5, 0.3, 0.1. Dashed lines: 2-site closure (for initial coverages 00=0.9, 0.5; for initial coverages less than 0.3 2-site and 3-site closures coincide). (b) Corresponding evolution rates d <-->/dt (solid lines) and d < . - - > / d t (dashed lines) in 3-site closure for 00=0.9, 0.7, 0.5. From ref.[115]. Figure 18a presents temperature programmed desorption spectra for an immobile adsorbate with nearest neighbour repulsion for varying initial coverages [115]. The dashed lines are obtained in the 2-site closure, while the solid lines result from the 3-site closure which, for an immobile adsorbate, is exact. Clearly the 2-site approximation is inadequate. Evans et al. [118] have interpreted the three peaks in the desorption spectra as "staged" desorption, with the low temperature peak reflecting desorption of particles from an environment of two occupied neighbouring sites and the smaller middle peak as desorption from the ends of chains. This picture is nicely confirmed by looking at the time evolution of <-- > = (nini+~) and <... > = (nini+ln~+2), Figure 18b. One can perform an Arrhenius analysis of the desorption spectra in Figure 18a by plotting isosteric rates vs. T-1. Although, for an immobile adsorbate these curves are not expected to be straight lines, one does a linear fit for simplicity, anyway. The resulting (temperature averaged) desorption energy and prefaetor are plotted in Figure 19a,b for the 2-site closure and the exact immobile and mobile cases. The mobile case correponds to the quasichemical approximation, c.f. (28). Although Figure 18a shows a two-peaked desorption spectrum for an immobile adsorbate calculated in the 2-site approximation, similar to the mobile case, the desorption energy looks quite different with the appearance of a peak at half coverage. The exact (3-site) calculation exhibits two peaks in the desorption energy. In the analysis of experimental desorption data, such features might be interpreted as arising from attractive interactions. This is clearly not the case here. The corresponding prefactors also show sharp peaks, which would be interpreted as an ordering above that for equilibrium. These features are an artifact of the Arrhenius analysis, and their form depends on the strength of the interaction,
193 ~4.5 ~. 0
39
[i
E~
"~ v
i
14.0
37 36
:3.5
c:~ 3 5
o
I,i
~
0 _.J
t 41
L
~
i L
,
4 i
:
i
125
31 3o
b
(3. ~.3.0
c 33 0 ".7, 3 2 O. m
,.i ' ,'
r L i
:
I
29 ~
0
.
.
.
.
.
.~
.
.
.
.
.
.2
.
.
.
.
.
.
.3
.
.
.
.
.4
.
.
.
.
.5
Coveroge
.
.
.
.
.
.
.
.6
.
.
.
.7
.
.
.
.
.
.8
.
.
.
.
.
.
:2.0
.
i
ij
0
.9
L'\
;
.2
.3
.4
.5
"Coveroge
(ML)
.6
.7
.8
.9
(ML)
Figure 19: (a) Temperature-averaged desorption energy, Ed((~), for adsorbate with V2=400K for cases" immobile, 3-site closure (solid); immobile, 2-site closure (dash); mobile (long/short dash). (b) Corresponding prefactors v(~). From ref.[115]. the initial conditions and the range of desorption temperatures over which the analysis is performed. Clearly, the Arrhenius analysis should be reserved for systems that remain in quasi-equilibrium where the interpretation of the desorption energy as the heat of adsorption and the prefactor as a measure for the entropy is rigorous. Figure 20 exhibits the transition from an immobile to a mobile adsorbate in the TPD spectra for a system with repulsive interactions. The variation of J0 between the immobile and mobile limits is approximately five orders of magnitude. Relaxation of correlation functions to equilibrium in the absence of desorption has also been examined [113,115]. In the remainder of this section we want to briefly survey other applications of the kinetic lattice gas model. The effects of adatom clustering on desorption rates,[106] and desorption of dimers [107] have also been considered. A specific application, in the one-dimensional case, has been to N2/Ni(430) [82], including multi-particle interactions and diffusive terms completely. Recently the kinetic lattice gas model has been used to find a time-dependent generalization of the grandcanonical ensemble to describe inhomogeneous adsorbates within the coexistence region [119]. If the adsorptiondesorption kinetics are precursor-mediated, one can extend the kinetic lattice gas model by introducing three sets of occupation numbers per lattice cell, namely ni =0 or 1, mi =0 or 1 and li =0 or 1, depending on whether the chemisorbed state, the intrinsic precursor and the extrinsic precursor are empty or occupied, respectively. The hamiltonian of the system is then [120]
H=Es~'~ ni + e ~ Z mi + ~ ' i
i
Ii + ~ V2 ~ ninJ i
(V)
For the transition probability in (70) we then write
(84)
194
o (2) -9 3 x
0
c
0
2
CL 0 r~
f
o 90
I
L
IO0
110
120
.
I
130
.
.
.
.
1
.
140
.
.
.
.
.
.
150
Temperature.
Figure 20: Temperature-programmed desorption rate for a 1-d adsorbate with repulsion, V2---400K, for 00=0.9 and varying hopping rates Jo=uoexp(-Q/kBT). Hopping barrier Q=0, and po=10-4s -l (solid, three-peaked) (effectively immobile), 10-3s -1 (- -), 10-2s -1 ( . . . . . . ), 10-1s -1 ( . . . . . . ), ls -1 (-), and 10s-1 (solid) (effectively mobile). From ref.[115].
T~n',m',l';n,m,l) = Wt + We + Wc + W c t + Wce + Wee + Wit + Wet
(85)
The significance of the individual terms is illustrated in Figure 21. As an example, we can specify adsorption into, and desorption from, the intrinsic precursor state with
Wt(n,m',l;n,m,l ) = wt ~ (1-ni)(1-mi +Dtmi)8(m,', 1-mi) ]-[ 6(ink',ink) i k~i
(86) where the first factor ensures that the intrinsic precursor only exists over an empty site. The kinetic lattice gas model with precursors has been used recently to study the temperature and coverage dependence of the sticking coefficient as a consequence of lateral interactions [121,122]. A similar formalism is also appropriate for the kinetics of rnultilayers.
195
W~
~
Q "i~ ~
Figure 21: Schematic illustration of the various processes between extrinsic precursor (E), intrinsic precursor (I), and chemisorbed (C) states included in the kinetic lattice gas model. 6. PHYSISORPTION KINETICS A monograph on physisorption kinetics was published by Kreuzer and Gortel [6] which covers the field up to 1985. The book starts with a review of the gas-solid interaction of physisorption. Next the master equation is derived and the transition probabilities are calculated for phonon-mediated adsorption and desorption processes. On this basis desorption times are calculated as well as time of flight spectra of the desorption flux, and is followed by a discussion of sticking and energy accommodation. In the final chapter the Kramers equation is derived from which the macroscopic laws are obtained, together with microscopic expressions of the friction coefficient. Thus the complete programme of nonequilibrium statistical mechanics, as envisaged by Boltzmann, has been completed for physisorption namely, starting from a microscopic hamiltonian, one derives kinetic equations and, finally, calculates macroscopic transport coefficients. The only other example where this programme has been completed before is the theory of dilute gases. This is not to say that there does not remain a large amount of work still to be done. In particular, the above theory is largely restricted to the low coverage regime, except for helium where the kinetics has been worked out for multilayer adsorbates. Although nonequilibrium thermodynamics and the kinetic lattice gas model are being applied to study the kinetics of physisorbed multilayers, little progress has been made on its microscopic foundations. Brenig [123] has reviewed the kinetics and dynamics of the gas-surface interaction in the zero coverage limit stressing the importance of the principles of detailed balance and unitarity. Jack et al. [124] have derived the FokkerPlanck equation for physisorption kinetics, again in the zero coverage limit, and presented a general formulation for its solution under isothermal desorption conditions. The good news in physisorption kinetics comes from Menzel's group where an exhaustive experimental study of rare gases on Ruthenium has been completed [17-19].
196 7. C O N C L U D I N G
REMARKS
In this review we have tried to give a concise overview of the physical processes underlying adsorption and desorption. We have reviewed the phenomenological approach and methods of evaluation of experimental data. For systems that remain in quasiequilibrium throughout desorption an essentially exact theory now exists, and accurate fitting of data is now possible. We have also outlined the current status of kinetic lattice gas models. We have only made passing reference to experimental aspects of thermal deso~tion, but mention some recent work as exemplifying a new standard. We note the margin by which such good experimental technique leads the many over-simplified theoretical models and analyses of data in the literature. It is time the simple but fundamental phenomenological concepts and results outlined in Sections 2 and 3, were understood by all interpreters of TPD data. Specific examples bear re-emphasis: the assumption of a constant prefactor is no longer tolerable, nor must high temperature limits be presumed; nor should a peak temperature analysis of experimental TPD data be considered sufficient - a complete analysis is mandatory if sense is to be made of the phenomenological parameters, E d and p, including their coverage and temperature dependence [31]. The complete theory at the macroscopic level, including non.equilibrium effects, has now been formulated for the submonolayer and multilayer regimes: apart from restrictions on one's knowledge of the chemical potential of the adsorbate, there is no need for further oversimplifed modelling of TPD spectra. In the quasiequilibrium situation transfer matrix techniques can give essentially exact results (apart from sticking coefficients), as long as the adsorbate can be described by a lattice gas model. Although originally conceived to deal with adsorbates on a fixed lattice of adsorption sites and simple two-body interactions, the model is also capable of dealing with reconstruction which is mimicked through the introduction of multisite interactions [125-127]. Employing these techniques one can now also model coadsorption accurately [128,129]. To make contact with microscopic concepts, the kinetic lattice gas model provides an alternate, and perhaps more satisfactory description. One hopes for further analytic work on this model in the monolayer and multilayer regime along the lines of the closure approximations. The application to heterogeneous surfaces holds great promise. This review has addressed developments in the theory of adsorption and desorption. Only cursory mention has been made of Monte Carlo simulations. They can, in principle, produce the exact kinetics from well-defined transition probabilities. It is, however, our view that such '~umerical experiments", although very useful in providing "data ~, are not a substitute for a statistical theory in which one first derives a small set of kinetic equations from which observable quantities and relationships are calculated. Even for very complex systems such as adsorbates with precursors and multi-particle interactions for which Monte Carlo simulations were deemed to be the only practical tool, progress has now been made in the statistical theory that promises a better understanding of their kinetics. ACKNOWLEDGMENT Funding for this research was originally provided by the Network of Centres of Excellence in Molecular and Interfacial Dynamics, one of fifteen Networks of Centres of Excellence no longer supported by the Government of Canada.
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W. Rudzifiski, W.A. Steele and G. Zgrablich (Eds.) Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces Studies in Surface Science and Catalysis, Vol. 104 9 1997 Elsevier Science B.V. All rights reserved.
Theory of adsorption-desorption on flat heterogeneous surfaces
201
kinetics
Yu. K. Tovbin Karpov Institute of Physical Chemistry Vorontsovo Pole Str. 10, 103064 Moscow, RUSSIA
1. I N T R O D U C T I O N Adsorption processes in gas - solid systems turn out to be fundamental to technologies of purification and separation of gas mixtures as well as being stages of more complex catalytic processes, membrane separation of gases, corrosion of solids in chemical gas mixtures, etc. In many cases, the time evolution of those processes is determined by the rates of adsorption and desorption. The investigation of those processes has been given, of course, quite a lot of attention. In view of the intensive experimental research, it appears necessary to develop a theory of adsorption rates. The simple theory of Langmuir describes the kinetics of adsorption on homogeneous surfaces, based on the law of mass action. One assumes that every adsorbed particle occupies one and only one site on the surface. The change of the surface coverage in time T can be described by the kinetic equation [1-4] dO dr - UA -- UD, UA = K A P ( 1 - 0) TM ,
00
=
0(~- = 0)
(1)
UD = KD0 TM ,
(2)
where UA and UD are rates of adsorption and desorption; KA and KD are rate constants for adsorption and desorption; P is the pressure of the adsorbing gas; O = N A / N is the degree of coverage of the adsorbent surface, 0 < 0 < 1, NA is the number of the adsorbed particles, N is the number of sites per unit area of surface, m is the adsorption order with m - 1 corresponding to adsorption without dissociation, m = 2, to adsorption with dissociation of the adsorbed molecules into two identical particles. The solution of equation (1) for the adsorption of non-dissociating molecules at a constant pressure in the gas has the form:
O(r) = 0o e x p ( - k r ) + 0o~[1 - e x p ( - k r ) ] ,
k = KAP + K D ,
0o~ = K A P / k ,
(3)
and for dissociating molecules in cases when the process of desorption prevails over adsorption (and the adsorption can be neglected) or vice versa, we have: Oo 0(~-) = 1 + 0oKDr '
0(T) = 1 -- (1 -- 00)/[1 + rKAP(1 - 0o)].
(4)
202 The kinetic relationships (3) and (4), however, do not describe well most of the experimental data [1-5]. It is not by accident that in the course of chemisorption, two stages can be separated, fast and slow [2]. Often enough, the slow adsorption process also can be observed for physical adsorption. It was Titov [6] who attempted to describe the slow course of the physical adsorption of gases in coal. McBain [7] described preliminary research concerning the sorption of iodine by activated coal, where the absorption did not stop after several years. The factors determining slow penetration in adsorption are related to the system, but the most important are considered to be the high heterogeneity of the adsorbent [1-5] and the strong non-ideality of the adsorbate [8,9]. To describe adsorption kinetics, one frequently applies empirical equations. The course of many sorption processes can be satisfactorily described by [10,11]
(5)
0(~) = a7 ~/~
where a and n are parameters. One discovers that the value n is analogous to the parameter in the Freundlich adsorption isotherm [1]. It seems that equation (5) was presented for the first time by Bangham and Bart [12]. A differential form of equation (5) is the empirical power law UA = KAP8 -~ ,
UD = KD~9k ,
n,k - const.
(6)
The equation for the rate of desorption corresponds to the above equation. These equations were used in papers [13-15]. Experimental data show a wide spread of the exponential dependence of the adsorption and desorption rates upon the coverage. A relationship of the type UD = KD exp(h~) ,
h = const ,
(7)
for the desorption of Th and Cs from the surface of tungsten was observed by Langmuir [8,9]. In order to describe the kinetics of adsorption in the system Co/Mn02, the following equation [16] was proposed UA = KA exp(--g~) ,
g = const ,
(8)
Methods of the determination of the constants in a given equation were considered in papers [5,17]. The integral form of the Roginski-Zeldovich equation (8) has the following logarithmic form [18,19]
O(T) = k'ln(w + ~'0),
k', r0 -
const ,
(9)
Equation (9) is often called the Elovich equation because it was he who gave its the first interpretation. It was shown in [19] that this equation is connected with a linear change of the energy of activation with surface coverage: EA(O) = EA(O = O)+gO. This equation describes the slow processes which often follow a fast initial stage of chemisorption. Equation (9) was confirmed many times. Its application and the methods for the determination of its constants are discussed in [20-26]. The necessity of explaining the origin of the empirical equations stimulated the development of the theory. In the first place, it was necessary to consider the fundamental physical factors affecting adsorption systems: the heterogeneity of the surface and the
203 lateral interactions between the adsorbed particles. This resulted in development of the theory in two directions. In some papers the main part was allotted to surface heterogeneity and the adsorbed particles were considered not to interact with one another [1-4, 27-32]. In some other papers, on the contrary, lateral interactions of the adsorbed particles were considered on homogeneous surfaces [9,27,33-38]. (A more detailed review of all those papers is given in [39,40]). The task of incorporating the influence of lateral interactions upon the kinetics of adsorption was found to be complex because non-ideal adsorption systems are characterized by co-operative behaviour. So it was necessary to use the methods of statistical physics of the condensed state in order to study this problem. That is why, at the very beginning, the theories of the first group were most widely studied and developed. In absence of lateral interactions, the description of kinetics was based on the law of mass action which essentially does not need a new method in spite of the procedure of averaging over the energy distribution function which is well elaborated in the theory of probability. Intensive development of the work of the second group started in 1973-1974. A new step in the development of the theory of kinetics of the adsorption processes was presented in 1982-1984, after elaborating equations describing the rates of adsorption and desorption on heterogeneous surfaces with lateral interactions taken into consideration [41-45]. In the present paper a review is given of the development of the theory of kinetics of the adsorption processes on heterogeneous flat surfaces. At first, we consider the theories of the first group, then we consider those of the second group. Then, we present the theories which take account of the simultaneous influences of the heterogeneity of the adsorbent surface and the lateral interactions. Because of the limited length of this review, we consider only the case of the equilibrated distribution of the adsorbed particles on a surface for all models. The state of the surface layer of an adsorbent is assumed to be unchangeable during adsorption-desorption processes (absence of reconstruction of the surface layer); also not considered is the possible solution of the adsorbed particles in the solid substance, and the possible occurrence of a precursor state. All these limitations can be calculated with the help of the kinetic equations, elaborated in terms of the lattice gas in papers [45, 47] (see also [40, 48, 49]). While considering also the moments which are obtained in current experimental investigations by the method of thermo-programmed desorption and the necessity of the molecular interpretation of thermo-desorption spectra (TDS), special attention shall be given to the application of the theory in the calculation of TDS in this review. The values of the rate constants are connected with the energy of activation are given by the Arrhenius law: =
exp(-ZE
),
KD = K~ exp(--/3ED) ,
/~ = (ksT) -1 ,
(10)
where K~4,Dare preexponential factors of the rate constants, and EA,Dis the energy of ac0 tivation of adsorption or desorption. While calculating TDS it always assumed that KA,D hardly depends on temperature. This results from rate theory of elementary processes [28,50,51] 9 In the molecular interpretation of K A~, D for surface processes, one most often uses the collisional model and transition state theory [1-4, 28, 52-55]. In the collisonal model, the rate of reaction is proportional to the probability of collisions of the reagent hard spheres which can be calculated from kinetic theory. The model of the transition
204 state [56], aside from a geometrical factor, also includes the redistribution of energy within the internal degrees of freedom of the reagents. This difference is important for the description of the kinetics of the non-ideal adsorption systems. In such systems the reagents constantly occur in the "field of action" of neighbouring particles so that lateral interactions can influence the energy of the transition state.
2. I D E A L A D S O R P T I O N
ON H E T E R O G E N E O U S
SURFACES
In his first work on the theory of adsorption, Langmuir pointed out the possibility of the existence of crystalline and amorphous surfaces in places which are characterized by various numbers of residual valences. Energetic heterogeneity produces differences in the adsorption and kinetic capabilities of various fragments of such surfaces. The sources of the energetic heterogeneity can also be conditioned by the chemical nature of the surface atoms of the solid (for example, by the occurrence of various atoms in alloys or by the occurrence of mixtures), by different crystallographic faces of the solid and their structural imperfection, etc. Such heterogeneity depends on the history of the formation of the surface of adsorbent. Expressions for the macroscopic rates of adsorption and desorption of nondissociating molecules on unit surface of adsorbent can be written as: t
t
UA = E
fqU~,
t
UD -- E
q--1
fquD'
q--1
E
t
fq = 1,
q--1
8
"~ E fqeq, q--1
(11)
where the UA(D are local rates of adsorption (desorption) on the sites of type q, 1 < q < t, t is the number of types of surface sites, fq is the fraction of sites of type q, and 8 is the macroscopic fractional coverage of the surface. The function fq (the discrete surface site distribution of the adsorption energy) characterizes the surface composition of the adsorbent. The local rates of adsorption-desorption have an analogous form to eq. (2), UqA
A =Ka(1-Oq),
D =KqOq, D Uq
KqA,D = K qA,Voexp(_~E~,D)
(12)
where the K A'D~ are the pre-exponential constants of the rate K {'D for a site of type q, E A'D is the energy of activation of the process for the site q; 0q is the probability of coverage of the sites of type q (i.e., the local coverage). The local rate constants are connected with the local Langmuir constants aq = aq0 exp(~Qq), where Qq is the adsorption energy on a site of type q, by aq = K A / K D, E D = E A + Qq, aqo: K{o/KDo If the number of types of sites is large, then the sums in (11) can be changed into integrals. It is usually also assumed that the pre-exponential factors hardly depend on the type of site so the change of the adsorption capability while passing from one fragment of the surface to another is determined by the differences in the adsorption energies on those fragments. Let x be the value of the adsorption energy. Then, formulas (11) and (12) can be re-written as: X2
UA,D = / f(x)Uh,D(X)dx, Xl
X2
0 : / f(x)0(x)dx, Xl
x2
/f(x)dx : 1 Xl
(13)
205 UA(X)
--"
KA(X)[1 0(x)] --
~
UD(X) KD(x)0(x) "--
~
KA,D(X)= K ~
A,D
exp[--BEA,D(X)]
(14)
where xl and x2 are the smallest and the largest adsorption energy. The first statistical description of the rate of reaction on heterogeneous surfaces was given by Constable [58]. He assumed the site distribution function to be an exponential function of the activation energy of the reaction: f(x) = exp(~/z)/n, where "7 is the width parameter of the distribution and n is a normalizing factor. He also assumed that the probability of a local coverage does not change while passing from one fragment of the surface to another. Later on, the statistical description of the rates of the surface processes was also treated by Cremer and Schwab [59,60]. To calculate the rates of adsorption-desorption, it is necessary to estimate the character of the changes of the energies of activation with the change of the adsorption capability of various fragments of the surface. In other words, it is necessary to know how the values KA,D(X) depend upon the value of x which in turn determines the relationship cwbetweenof KA,D(X) and 0(x)). In order to construct the function EA,D(X) (14), Temkin proposed to use an analogous relationship by Br<nsted [63] for homogeneous catalysis. Br<nsted proved that for the reaction of acid catalysis in solutions of various monovalent acids, we have the relationship ln a = a ln k + C, where g i is a rate constant which characterizes the catalytic reactivity of those acids, k is a dissociation constant for those acids, cr and C are constants. An analogous relationship was also observed for basic catalysis. Temkin proposed that one should consider various fragments of the surface as similar to compositions. Then the process of adsorption (desorption) on various fragments of the surface can be considered as a reaction, and the values of the rate constants KA,D(X) and those of the adsorption coefficients a(x), are linearly related to each other" AEA(x) = - ~ , A x ,
AED(x) = a A x .
(15)
A graphical interpretation of the Bronsted relations, with the help of the potential energy diagram, was given by Horiuti and Polanyi [64] (see Fig.l). If two monovalent reactions are characterized by an increase of the potential, then the change of the activation energy A E consists of a certain fraction of the change in the energy. At the same time, if- over a small distance- the curves can be considered as straight lines, these values turn out to be proportional. In this way, ~, < 1, and the change of the adsorption activation energy is a certain fraction of the change of adsorption energy. An analogous relationship is also valid for desorption. Then a + "7 = 1, because ED(x) = x + EA (x). The relationships (15) state that the fragments of the surface with the greatest adsorption capability possess the smallest adsorption activation energy and the greatest desorption activation energy. A decrease in the adsorption capability increases the energy of activation of adsorption and decreases the energy of activation for desorption. At "r = 0, the energy of activation for adsorption hardly depends upon the type of sites on the surface, and at a - 0, the desorption activation energies do not depend on the type of the surface fragment. In many adsorption, catalytic and electrode processes, the coefficients a and ~, turn out to be close to 0.5 [3, 65, 66]. Formulas (13)- (15) can be reduced to the following expressions for the rate of adsorption" X2
X2
/ .
Xl
Xl
206
A{__ Q ~ .~
AQ' --
AQ"
Distance Figure 1. A graphical illustration of the relation between the changes of the activation energy AE and the adsorption energy AQ (-= Ax)[64]. These expressions allow us to obtain some simple analytical formulas if one assumes a site distribution function. The most simple distribution function is the regular function f ( x ) = (x2 - xl) -~ [27]. If we assume that -y = 1/2, the following expression can be obtained"
UA = 2 K ~ P { a r c t g ( u 2 ) - arctg(ul)}/[a(aop)X/2] , ul,2 = (al,2P) 1/2 ,
al,2 = ao exp(/gXl,2 ,
a =/9(x2 -- Xl).
(17)
The value of the equilibrium pressure P, corresponding to the given coverage of the surface O, can be written with the help of the quasi- logarithmic isotherm of Temkin: 8 = ln{(1 + a2P)/(1 + a l P ) } / a .
(lS)
In this way, equations (17), (18) determine UA(O). In a similar way, we arrive at an equation for Up(O). If we limit these considerations to the range of middle coverages a l P << 1, a 2 P >> 1, then empirical equations (7), (8) can be found and we obtain a molecular interpretation of the constants in these equations [27]: rg~ KA = sin(Tr)er '
g = 7a,
7rK~) KD = sin(a~r)a '
h = ha.
(19)
The molecular interpretations of the empirical equations (7), (8) are not equivalent in meaning. Considering the lateral interactions on a homogeneous surface leads to the same kind of the concentration relationships [32,27]. The interpretation by Langmuir [32] corresponds to the values of the parameters h = 1 and g = 0 in equations (7), (8) (i.e. the model of collisions has been used). The interpretation by Temkin [27] was made with the help of the transition state model which accounts for the interaction of the activated complex with the environment. Equation (8) can also be obtained if we assume that the
207 number of sites on the surface changes during adsorption [67-69, 21]. In a similar way, with the help of exponential distribution functions (with positive and negative values of the exponent), one can explain various power dependencies of the rates of adsorption in equations (5), (6), [3,30,31]. Expressions (16) allow us to formulate the task of finding a distribution function f(z) from measured adsorption data. The first such work was carried out by Roginski, Lewin and others [1, 70-74]. There are many methods of solving the integral equations (16). The current state of this field is given in the monographs [75,76]). Fig.2 shows a distribution function obtained in [72] from data [72-74] on adsorption of hydrogen on sugar coal at 557 and 669K. Such work, based on application of the methods discussed in [1,70-72] was carried out by analysis of the kinetic data for the decomposition of methanol on zinc oxide. It was found that the distribution function obtained depended on the temperature of the experiment [72]. Based on these results, together with the data on isotopic exchange [73,74], the authors concluded that isothermal measurements are not effective for strongly heterogeneous surfaces, because at T = const., only a small part of the surface "works". This problem can be avoided when the method of programmed heating of the surface is applied.
I
I
'
9
16 18 20 x, kcal Figure 2. Site energy distribution function for hydrogen adsorption on the surface of an active carbon from sugar as obtained by Levin [72] from the processing of experimental data on the kinetics of adsorption [72-74]: f(x) = 640 exp[A-9.85-l0 s exp(a)], a = -7.19-lO-4x, x=energy in kcal/mole.
3. T H E R M A L D E S O R P T I O N S P E C T R A 3.1. Principles of t h e a n a l y s i s o f t h e r m a l d e s o r p t i o n s p e c t r a The method of thermal desorption was originally proposed in papers by Becker and Hartman [77] and Ehrlich [78]. It was widely developed and is presently used in many studies of adsorption processes. Its essence is contained in the fact that an adsorbent containing a certain amount of adsorbate is heated according to a given time-law of changing temperature T(r). The corresponding changes of pressure are measured in the experimental cell. The dependence of the pressure change (or of the number of molecules which desorb) on temperature is called the thermal desorption spectrum. Contrary to the lengthy measurements at a fixed temperature, here, within a short period of time, one can follow the process of desorption throughout the whole range of coverages from the full surface to practically zero. The mass material balance in the cell can be described in
208 the following way [79,80]:
(20)
an/dr = -AdNa/dr - nS/V - L,
where n is the number of molecules in the volume of the thermal desorption system V, N~ is the surface concentration of molecules, A is the area of the adsorbent, S is the speed of pumping the gas from the cell (liter/sec.), L is is the flow of gas into the cell from all the sources of molecules with exception of the sample itself. One assumes that at the beginning of the experiment: 1) the molecules in the cell are at thermal equilibrium with its walls, having a temperature To; 2) unadsorbed molecules obey the equation of the state of an ideal gas n = PV/RTo; 3) a stationary state occurs in the cell (dR~dr = O) for the gas pressure P equal Po = LRTo/S. If we write A P = P - P0, then the equation (20) can be re-written as:
(21)
d A P / d r = ARToN(-dO/dT)/V- S A P / V ,
where N is the number of adsorption sites on unit surface. For dO~dr one can use any equation of adsorption. Heating of the sample can be carried out according to any law T(r). The most widely used is linear T(r) = To + b r ,
(22)
where b is the velocity of heating (degrees/sec.). Other relationships considered include: 1/T = l/T0 +bT; a sequence of stepped increments of temperature; or a sequence of linear dependencies (22) with various velocities of heating. The description of the experimental data requires a simultaneous solution of equations (21) and (22).
a
b
~3
(
f
\ rl
__
t
t
Figure 3. (a) Desorption of nitrogen from tungsten after adsorption at T = 115~ Gas desorbed is proportional to the nitrogen ion current ip. 1 c m = l . 10 -9 amp. Time scale: 1 cm = 100 msec. Adsorption interval At = 10 min. Surface concentrations: n.y = 50.1012 molecules cm -2, n~ = 11.1012, nz = 136.1012. (b) Desorption of CO from W. Adsorption at 336~ At = 35 min. Time scale of desorption: 1 cm = 100 msec. Surface concentrations: nzl = 206- 1012 molecules cm -2, n& = 202.1012, n~3 = 62.1012 [81].
209 i
I
I
I
I
I
I
i
I
I
I
I
.,,--4
V
< "~ "0
10
_
8
6
r162
4 o E
2
<
0
-
!11
D
-_
!
__ _
(~1.Zt- ,~?.~r ,: ~ '
.......
.
I
1O0
~2
-
I
!
I
300
I
700
900
I
I
1100
i
: ............... ~,, I
I
I
1300
Temperature (K) Figure 4. Desorption spectrum of the nitrogen/tungsten system for a full virgin layer deposited at 770 K. (o) Heating time per step is 5 sec. A's represent alpha layer adsorbed at 77~ K after heating the full virgin layer to 600 ~ K. For comparison, a spectrum from polycrystalline tungsten is shown as dotted lines [82]. One can distinguish between two extreme cases of the outflow of gas from the cell. In the first case, the velocity of the outflow is small (S ~ 0) and the velocity of desorption can be calculated as the first derivative of pressure. On the curve AP(T) one can observe the leaps (Fig.3) which are connected with the occurrence of various phases of the adsorbate. With increasing temperature, the phase which is most weakly attached desorbs first and the phase desorbs next which has a stronger binding, etc. until the whole surface is free. In the second case, the velocity of the outflow is large and the velocity of desorption is proportional do the difference of pressures AP. The curve AP(T) is formed of several peaks, each of which corresponds to an adsorbed phase (Fig.4). In the real situation, the experiment can be carried out over a wide range of change of the velocity of the outflow [78-81]. (Discussion of the influence of the temporary delay upon the shape of TDS can also found in the papers [83,84]). Redhead [85] (see also Carter [86]) has presented a method of analysis of the TDS peaks to determine the molecular properties of the adsorbent-adsorbate system. Let us limit our considerations to conditions of heating (22) such that the temperature dependence of the the surface coverage due to desorption of molecules from a homogeneous surface in absence of re-adsorption has the form:
dO/dT = -K~O exp(-ED/RT)/b. TM
(23)
Let us observe that the pre-exponential K~ and the energy of activation ED correspond to every value of m. When these values are independent of the surface coverage. Redhead related the temperature Tp corresponding to the peak maximum to the energy of activation
210
a, u
dO
dO dT
a, u !
! !
Tp T,K T,K Figure 5. The TDS curves for different initial coverages 00 of the ideal adsorption system, m = 1 (a) and 2 (b): 00=0.1(1), 0.2(2), 0.4(3), 0.6(4), 0.8(5), 0.99(6). Dashed lines show the half-widths of the curves. of desorption by: ED/RT~ = K~ e x p ( - E D / R T p ) f m / b ,
fl = 1 ,
f2 = 20p ~ 00,
(24)
where 0p corresponds to the peak maximum. For a qualitative evaluation, one uses the fact that the peak is roughly symmetrical for dissociative adsorption (see Fig. 5) so that 0p ~ 0o/2 and f2 ~ 00. The value 00 can be determined by the measurements of the area under the TDS curve 0o = fo(dO/d'r)d'r. It follows from the formula (24) that the position of the peak for first order desorption does not depend on the initial coverage. For second order desorption the position of the peak moves toward the range of lower temperatures as the value of 0o increases (Fig. 5). For non-dissociating molecules, the position of the maximum can be found by considering the meaning of K~, ED and b. For them, the half-width of the peak AT1/2 ' - T~/2, " (Ti/2 ' > T"a/2,,~ is the width of the peak at half the peak's (where AT1~2 = T;/2 height) does not depend on the value of 00 [85]. Kislyuk et al. [87] obtained an approximate relationship between the half-width of the peak and the energy of activation for desorption: ATI/2
=
1117
ln(01/02) ,
7 = RTp/ED ,
(25)
where 01 and 02 are surface coverages corresponding to temperatures T~/2 and T~)2. Hence the simple relationship AT1~2 = 3.2. IO-3ED/R. More nearly exact is the relationship given by Edwards [88] ATx/2/T = 2.44647(1 - 1.40057- 3.532572...),
(26)
All values appearing in the above formula are dimensionless. Equation (26) can be accurate to within several percent over a wide range of model parameters 1011 < K~ < 1015, 102 < Tp < 2.103, 10 -2 < b < 102 degree/sec. The TDS curves for non-dissociative desorption are asymmetric with respect to their maxima but for dissociative desorption, they are symmetric. This criterion is used for the
211 determination of the order of desorption if the peaks do not overlap. Usually, the order of desorption can be determined graphically from the relation of the experimental data to the coordinates ln((mT]/b)- 1/Tp. For this purpose, equation (24) must be re-written in the form: ln(~mT2p/b) = ED/RTp + ln(ED/K~R)
(27)
The slope of the left-hand-side of equation (27) gives an energy of activation ED at fixed values of b and K~9. Fig. 6 shows the results of such an analysis for the desorption of //2 molecules from the phase i /~2 on the (100) face of tungsten [88]. One can avoid the preliminary assumptions concerning the value of K~9 in the determination of ED if one varies the velocity of heating b and shows the experimental data in the coordinates ln(~mTp/b)- 1/Tp [90,91]. The exactness of this method can be satisfactory if the velocity of the heating changes by more than one or two orders of magnitude. If we know ED, the value K~9 can be found from relationship (24). 1020 0
[-. 0
o
0
1019
E = 32.3 kcal 2
v2 =4.2*10
1018
-2
cm
molecule * sec
!
!
I
1.60
1.70
1.80
1 9103 (deg K) "1 T Figure 6. Plot to determine rate parameters for the ~2 state of H2 on the (110) face of W at 300 K. The coverage varies from 3.9- 1012 mol/cm 2 to saturation, iwhich is 2.5- 1014 mol/cm 2. The straight line gives ED = 135 kJ/mol and g~9 = 4.2.10 -2 cm2/(molecules, sec)[89].
In spite of the movement of the peak maximum toward lower temperatures (for second order desorption) with an increase of 80, its half-width remains practically constant. The interpretation is connected with ED through the approximate relationship AT1~2= 4.8. IO-3ED/R [87] or by the relationship, analogous to equation (26)[88]:
AT1/2/Wp = 3.52557(1 - 2 . 0 0 0 7 - 6.94172 + . . . )
(28)
212 It was shown in [85], that 7 in eq. (25) hardly depends on temperature and, if ED is given in units of Kcal/mole, then 7 is found in the interval from 30 to 34. With the help of this approximation, equation (24) can be changed into a more concenient expression for an approximate calculation [92]: ED/RTp
=
ln(K~(mTp/bT)
(29)
Quantitative analysis of the theoretical spectra proved [92] that one can construct some practically linear correlations for the half-width of the peaks: ATx/2/Tp = C1 - C2 l n ( K ~ m T p / b ) ,
(30)
where C1 = 0.20 and C2 = 0.00845 for m = 1 and C1 - 0.233 and C2 = 0.0083 for m = 2 (Fig.7). 0 -
1st ORDER
-
3
3
1 109 2 10 l~
40 ATI 2 80 -
21
3
- 120 -
2
100 l
l 0 ll
80
0
-
~
120 -
-
-
~
~
60 o ,.~
-
40
160 200
-b
m
t 20
i
0
_o
I
,
i
400
,
l
i
]
I
l
,
I
,
I
|
I
,
I
|
0
800
1200 1600 2000 Tp, K = Figure 7. Computer-generated graphs of the flash-desorption peak temperature Tp and peak width ATx/2 vs ED for a binding state with first-order desorption kinetics [92]. The majority of experimental TDS curves have a complex shape and cannot be described by the simple methods described here. The simple model of desorption which is the basis for these methods does not express the real properties of the experimental systems such as the heterogeneity of the surface and the lateral interaction between adsorbate molecules. Many adsorption spectra that are constituted of several peaks have been investigated - every peak was considered to belong to a separate phase and was described independently [92]. Such an approach turns out to be justified when it concerns various mechanisms of adsorption (with dissociation and without it) as, for example, for the system nitrogen/tungsten [79,81]. In some other situations, an even more detailed analysis is necessary. relationships for a h e t e r o g e n e o u s surface For the calculation of the TDS of non-dissociating molecules from a heterogeneous surface, one must replace the expressions for UD in section 2, by equation (20). The shapes of the TDS curves depends on the number of types of sites and the fractions of sites of various types on the surface. The desorption of molecules from the sites of each
3.2 Correlation
213 type can give a peak in the TDS. If the energetic features of the sites of various types are much different, the individual peaks do not overlap, or they hardly do. Then the TDS curve consists of several maxima and minima. If the energies of the sites of various types do not differ much from one another, the individual peaks overlap and one can observe a widened peak. The first case is considered as corresponding to a discrete distribution for fq, and the second case, to the continuous distribution function f ( x ) . In the first case, the main question is the relationship between the number of types of sites and the number of TDS peaks. In the second case, the relationship between the characteristics of the peak and the parameters of the distribution function is studied. The influence of the various kinds of distribution functions on the shape of TDS for discrete distribution functions was considered in the papers [86-87,93-96] and in the papers [79-81,86,87,92,97-101]. For the continuous distribution function f ( z ) = ( z 2 - Zl) -1, this was done by determining the doncentration dependence of the energy of activation of desorption; ED(0) =
E~ -
E~ =
~:0,
ED(O : 0 ) .
(31)
The simplest way of interpreting TDS curves is to seek for the correlation between the observed characteristics of a peak and the energetic parameters of an adsorption system. Carter [86] obtained a relationship between the differences of the activation energies of i neighbouring peaks AED (discrete distribution function) and their energy for first order desorption: AED = 3ED6/(0.91 -- 1.86), where 6--~ ( 1 . 5 - 1.7). 10 -2. While considering the conditions for peak splitting, it was assumed that the maximum of the smaller of the two neighbouring peaks should surpass e times the value of the minimum between them. This condition turns out to be analogous to that for the splitting of optical spectra. Kislyuk et al [87] obtained as a condition of splitting AED = ED/14 for non-dissociating molecules and AED = ED/9 i for dissociating ones. As a criterion for splitting, one used the condition of the occurrence of a minimum in the TDS at the increase of AED. The difference between the two evaluations is related to different criteria and to the degree of the mobility of the adsorbed molecules (see below). a
u, a.u.
U, a.u. 1
500
2
700
900
T,K
I
I
I
300
500
700
T, K
Figure 8. (a) TDS curves for atomic nitrogen on platinum: curve (al) is for the a form with N,n~ = 3-1013 atom/cm 2, b = 30 K/s; curve (a2) is the fl form with Nm~, = 5" 1013 atom/cm 2, b = 5 K/s. Part (b) is the TDS curve of CO on platinum with N,naz = 7.1014 molecules/cm 2, b = 38 K/s [87]. The same authors studied the correlation for an uniformly heterogeneous surface which is characterized by equation (31). In paper [86], a relationship was obtained between the
214 half-width of the peak and the temperature of the maximum in the TDS" ATll2 = 0.37Tp. 1/2, where In paper [87] yet another correlation was obtained: AT1~2 = [(~rT)2 + (AT~ 800
45
(2) VO no
13
700
11
1
= 1o10 deg 40 35
600 Tv (K) 500 25 400
300
200
100 !
I
I
I
I
I
I
25
50
75
100
125
150
175
AT, (K) 2
Figure 9. Computer-generated graphs of Tp azld AT1~2 for flash desorption from a second-order binding state with ED of the form ED - E~9 - aO [92]. AT~ is the peak half-width for a homogeneous surface. The value of a T characterizes the interval of the energetic heterogeneity of the surface given by the equation a E = aTT, cr ,-, 30 (25). Kislyuk et al applied this relationship to the experimental data on the atomic adsorption of nitrogen [102] and CO [103] on platinum (Fig.8). While analyzing these experimental data [104] and assuming that ln(0x/02) = 2.6 for m = 1 and 1.6 - 1.9 for m = 2, they obtained AT~ = 1050 from equation (25) for the low-temperature phase a and 900 for the high temperature phase 13 of nitrogen on platinum. The experiment gives the value AT1~2 ,,, 950 (a) and 630 (~). These authors conclude that for the a phase the agreement is satisfactory, and for the j5 phase the deviations are possibly due to non-linearity of the heating. This is why the surface of platinum is assumed to be roughly homogeneous (within every phase) with respect to the adsorption of nitrogen. For the system C O / P t , calculations based on equation (25) yield AT1/2 = 60 ~ compared to the experimental value AT1~2 = 190~ This means that the surface is nonuniform. The interval of heterogeneity evaluated by the above mentioned formulas as AED = 7 kcal/mole, agrees very well with the data on the adsorption heats [105].
215 The correlation of AT1~2 with the temperature of the peak for various meanings of E~ and a in equation (31) was studied in the paper by Schmidt [92]. The results of this analysis for second order desorption are shown in Fig. 9. In papers [106,107], attention was focused on the fact that a knowledge of only the value of the half-width of the spectrum is not enough to determine the molecular parameters of the model. This view was popular even previously, see, e.g. the paper [95], where one observed the indispensability of performing a preliminary comparison of the calculated and experimental data. In fact, it is not a complicated task to elucidate additional information from the TDS curves. In the paper [106], the width of the spectrum AT3~4 was obtained from two points where the rate were 3/4 of the maximum rate of desorption. In the paper [107], as additional information, the derivative dUD/dT, taken at two points of the spectrum where the rate of desorption was equal to 1/2 of the maximum gave another measure of peak-width. Drawing more detailed information from insufficiently separated spectra using simple relationships seems to be problematic. In paper [108] it was shown that a large number of parameters makes such an interpretation dubious (not having one meaning only). In order to better analyze the molecular nature of an adsorption system, more attention is paid to o b t a i n - from the TDS curves - data on the concentration dependencies of the effective energies of activation and on the pre-exponential factors of the rate of desorption.
3.3. Effective energies of activation and the pre-exponential constant of the desorption rate equation The equation for the rate of desorption can be represented by : UD = K0(O)O~ exp[--f~ED(0)] ,
(32)
where the pre-exponential coefficient Ko(O) and the desorption activation energies ED(O) are functions of the concentration of the adsorbed molecules. The technical possibilities to determine ED(O), if the order of desorption is known as well as Ko(O), were elaborated by Ehrlich [81]. His method has been applied to a whole series of systems [109-112]. King [112] proposed the so-called "complete analysis" for the determination of the coverage dependence, ED(O). For this purpose, one uses a sequence of TDS curves corresponding to various initial coverages 00. For every curve, an Arrhenius plot is constructed In UD -- l/T, from which one determines the value of E,I, related to the given initial coverage. The intercept of the Arrhenius plot with the coordinate axis gives the value B(O) = l n K 0 ( 0 ) + m In 0. From the plot of B(O)versus In 0, one estimates the order of the desorption and the value of Ko(O). If this plot is linear, i then the order can be determined from its slope, and Ko(O) is found from its intercept (independent of 0). If this plot is not linear, then the order m should be determined independently, which allows one to obtain the function K0(0). The method needs high quality measurements at high values of 0. It was applied successfully in numerical studies [113] for the range of coverage 0.04 < 0 < 0.3. The authors of [113] proved also that even a weak dependence of ED(O) (~ 6% on ED) provokes a considerable deformation of the concentration dependence of the isothermal rates of desorption which are obtained by calculation of the rates from the TDS curves while using the coordinates In UD - In 6 at T = const. The analysis of such isothermal rates of desorption does not require a knowledge of the order of desorption. That method
216
b a
I
I
0.s
4
hl %
I
I
I
x\5
3
02
"~e ~
~
I"
0-02 I
.~//
~
.~/"
]
0 01 ~ 0.01 0.02 0.05 0.10 0.20 0.50 Relative coverage
I
o.1
~-~ 0.02 0.01 2.15
(d) 00:24 (e, 0.6 i
i
2.25
\ i
,
2.35 103/T(K)
I
2.45
Figure 10. (a) Experimental desorption rate isotherms for CO desorption from clean Ni(110). (b) In N versus lIT for CO desorption. Data points obtained from the desorption rate isotherms in Fig. 10a. The curves correspond to relative coverages in Fig. 10a of 0.07, 0.10, 0.20, 0.40, 0.60
[113].
was applied to the analysis of the desorption of CO from the surface of Ni(110) (Fig. 10) and for the system N2/hydrazine. An improvement of this method of determining the functions ED(O) and KD(O) at various velocities of heating for fixed initial coverage was presented by Taylor and Weinberg [114]. In their method, the temporal law of heating T(r) can be arbitrary. In this way, one excludes mistakes connected with deviations from the chosen regime of heating. The method is applicable for any coverage, but it requires that the change of the velocities of heating be not less than two orders of magnitude. The authors applied their method for the description of desorption of CO from It(110). The concentration-dependent functions ED(O) and KD(O) obtained are shown in Fig. 11. They show a compensation effect. This kind of relationship can also be observed for the system 02/W(llO), for which a modified expression for the rate of desorption was used: Ud = K00exp{[-ED(0)/R](1/T- 1/Ts)} ,
(33)
where T, is a parameter. For the system 02/W(llO) it was estimated that K0 = 7.10ss -1 and T, = 1230K; the dependence ED(O) was approximately described by equation (31) with ED(O) = 35 and x = 16 kcal/mole, i The problems accompanying the determination of the desorption parameter T~ were discussed in paper [115]. A simpler way of describing TDS curves and estimating the function ED(0) is presented in paper [116]. Instead of analysing a family of curves, one proposes to use the phenomenological description of the energy of activation approximated by the polynomial
M ED(O) = ~ a k ( 1 - O)k, where the coefficients ak have no physical meaning, and M is k=l
the number of terms which can be determined from adjustment of the calculated to the
217
40l if,
30
=~
20
1015 ~ 10
13
10
o ~ t~
10
109
10 7 !
0
I
0.2
I
I
0.4
I
!
0.6
I
i
0.8
k.q
10 5
N
0
o=
!
1
Fractional coverage Figure 11. ED(8) and K(O) profiles obtained from analysis of TDS curves. Broken lines are Taylor and Weinberg's results [114]. Full lines are the results of Forzatti et al. [117]. experimental spectra using the method of least squares. After presenting the eight types of dependencies of ED(O) (Fig. (12)), the authors considered them as a combination of the contributions from the homogeneous and uniformly heterogeneous (31) surfaces. This approach was applied to describe of desorption of 02 molecules from the surface of ZnO and from a silver catalyst. In Fig. 13, the TDS curves are shown along with the constructed functions ED(O). The main problems which were studied by the authors were to how to eliminate the necessity of a large number of TDS measurements and how to select parameters of the model from one of r several curves. A method for determining ED(O) and K0(0) from one TDS curve was presented in paper [117]. The curve is normalized with respect to the maximum of the peak and one constructs a parabolic approximation around its maximum. This allows one to obtain a linear correlation between certain values which can be determined from the experimental curve. From this correlation, one can evaluate the energy of activation of desorption corresponding to the point of the maximum, and determine - approximately - all the functions ED(8). As a result one can calculate both m and K0(0). This method was been tested on the curves shown in Fig. 12; it gave an accuracy of about ~5%. The results of its application to the desorption of CO from the face of I r ( l l 0 ) are shown in Fig. 11. Habenschaden and Kuppers [118] proposed a method where the parameters may depend on temperature (though the importance of this factor is not great [119]). It was suggested that one choose a small fragment of the curves in the low-temperature range of the spectrum (leading edge analysis), so that the changes of 6 and T were insignificant.
218
3
40
,_q m
o
E r
\9
-.-.-
,,
gd
30
_
\ ~
1
_ _-~.~_,__ _-,,,.
\
" . . ~ ---~ .
\
4 I
0
"7
0.5
.7
1,4: l/2scale Il
r
o
I I
31
/I
4,1
_
i i
!,,, "~ !
i:
I
i
9
[
f ,6,
# i
"~x
~
.
I
,
i i
~
' l " . i "~.
_
-~
-" I.
~ ~
!
I
373
]l
I
I,
],"i: 1
0
I
8,
r~
473
573
673
373
473 573 673 T/K Figure 12. Upper graph: Distribution profiles of ED(e). Lower graphs: Thermal desorption curves simulated for first-order kinetics with K ~ = 1013 s -1, b = 10 K/s [116]. For this chosen fragment, a graphical analysis was carried out as in method [112]. In the papers [120,122], the role of compensation effects was studied and the prospects of using the various methods of analysis of the TDS curves to find the molecular parameters of an adsorption system were discussed. Fig. 14 shows the influence of the parameter Ts in equation (33) on the shape of the TDS curves in a system with a linear change of energy of activation of desorption (31). If one considers the coefficient x as a contribution of the lateral interactions, then the variation of Ts leads to a change of the sign of the interaction parameter (such a situation was observed in the system Au/W(llO) [109] and in a series of other systems). A detailed analysis of the various methods of determination of m, ED(O) and Ko(O) has been presented in paper [121]. Testing was carried out for desorption of the first and second order with calculation of the linear dependence (31) and of the compensational effect (33). The conclusions drawn by these authors is that the methods of [112] and [118] turn out to be preferable.
219
a
13/K/min
eq
o
_A
1;
6
13/K/min
4 _
9
1.0
0
1~/~ ^
2
1;
4 9
0
20.5 1-
0 373
473
573
I 373
473
573 T/K
T/K "7
13/K/min
O
"~ 35
b
13/K/min
A / s "1
E o
11.1
~ I\
l;
4
~--"~ 30
10
E
A/s
-1 10.5
35
10.4
' ~
10
3,4; 10
10 11.4 10 11.1 10
30
--t----.
25
ols
|
t 0
t\ l 0.5
Figure 13. Part (a): Thermal desorption curves for oxygen on zinc oxide. Part (b): Results obtained by the analysis of thermal desorption curves of oxygen on zinc oxide. Part (c): Thermal desorption curves of oxygen on silver catalysts. Part (d): Results obtained by the analysis of thermal desorption curves of oxygen on silver catalysts [116]. Equation (32) allows us to carry out a numerical analysis of the rates of desorption from the TDS curves, but does not yield much information about the molecular nature of an adsorption system. It is not by coincidence that this approach is used for the description of the process on single crystals where a large part is played by the lateral interactions as well as for processes on poly-crystalline and amorphous adsorbents. The influence of a certain molecular property of an adsorption system on the shape of the TDS curve will be considered in the following chapters of this survey: lateral interactions in Section 4, and the common influence of the heterogeneity of adsorbent and lateral interactions in Section 5. Right now we shall illustrate briefly the influence of the molecular characteristics of heterogeneous adsorbents on the shape of TDS in absence of lateral interaction.
220
r .s o. -o t_
Attractive interaction~ EO=300kJlmol J~
ld 3Is ~11 JP~t W= -25kJ/mol ~ \ 1 riO=
Repulsive interactions
%= 300kJ/tool no= ld3/s w= +25kJ/mol
t-
O
I,
To= 2000
To=
100~
Tc= 600K
1000 1100
attractive interactions
"i~88
"
T (K)
~" 115oL r~'--.~~
o
0.5
~
To= 1 0 0 0 ~
Tc= 600K
900
Tc= 2
--
1000
1
150
900
~.~!I~iI
1000 1100 TIK repulsive interactions
11501.-
lOO%
o'.s
150
9
i oo
.
Q (Mr.)
Q (ME)
Figure 14. Part (a): Simulated TDS for first-order kinetics and attractive (left) and repulsive (right) lateral interactions, and different values of the compensation effect at temperature To. The desorption energy varies with coverage as ED(O) = Eg - wO, and the preexponential factor varies according to the compensation effect as ln K = -wO/RTc + I n K ~ Parameter values: E~ = 300 kJ/mol, w = +25 kJ/mol, K~ = 1013s -1. The coverages for each set of spectra vary from 0.1 to 1 M1 in steps of 0.1 M1. (b) Dependence of the peak maximum temperature Tp and the full width at half maximum (s upon initial coverage for TDS corresponding to first-order kinetics and attractive (left) and repulsive (right) lateral interactions. Curves are for different values of the compensation effect temperature Tc with parameter values as in part (a) [120].
221 3.4. D e s o r p t i o n of n o n - d i s s o c i a t i n g molecules While investigating TDS for a surface characterized by a discrete site energy distribution function, the main problem is the explanation of the conditions for total splitting of the TDS peaks. This is related to the number of peaks Np and the number of the types of sites t. Examples of such splitting are shown in Figs. 15 and 16. These curves were calculated from equations (11) and (12). To the equilibrium distribution of the adsorbed molecules corresponds the following "chaining" of the local coverages aq/% = Oq(1-Op)/[Ov(1- 0q)]. Fig. 15 shows the influence of the change of the surface composition on the shape of the TDS for t = 2 and a = 1. The positions of the two peaks do not change but the height of every peak changes in proportion to the contribution of the sites of every type. Fig. 16 shows the simplest TDS curves for a discrete uniform distribution fq = t -1. The first peaks are the most distinct. With increasing temperature, the maxima decrease whereas the the minima increase. Such a form of the curves is connected with surface migration of the adsorbed molecules. Fig. 17 gives a comparison of the TDS curves for an equilibrated distribution of the adsorbed molecules and for the case of no surface migration [122]. (The methods of describing the surface migration of molecules are discussed in paper [45-48, 122, 123] for heterogeneous surfaces and in papers [124-126, 122] for homogeneous ones.) a.Ll.
dO dT
I
Figure 15. TDS curves for the inhomogeneous surface with sites of two types (t=2): Curve 1 is for fl = f~, f~ < 0.5; 2, for fl = 0.5, 3, for fl = 1 - f~, 00 = 0.99 [40].
222
dO dT
a.u.
I
I !
!
T Figure 16. Completely split TDS curves for an inhomogeneous surface with fq = t -1, t = 2(curve 1), 3(curve 2), and 5(curve 3); here, 0o = 0.99 [40].
dO dT 0.6
2 0.3
-
0 500
! 600
i 700
T
Figure 17. TDS curves for the inhomog;eneous surface with random distributions of the different type sites, t = 2: Curve 1 - adspecles are immobile, Curve 2 - equilibrium distribution of admolecules; K1D~ = K D~ = 1012 s -1, E ~ = 168 kJ/mol, E ~ = 126 kJ/mol, 0o = 0.99, b = 50
K/~ [122].
223 Numerical studies of the conditions for the total splitting of the TDS curves for heterogeneous surfaces are reported in paper [127]. There, the case AQ = Q q - Qq+l = const. is assumed, where Qq is the adsorption heat on site q, 1 < q < t.
AQ* kJ/mol
~Q* 20J
40,
I0i
i
i
I
l
3
I
I
6
l~t
9
20
0
I
I
0.2
0.4
,,
I
I
I
0.6
0.8
1.0
f2 Figure 18. Condition for the complete splitting of the TDS curve for nondissociating molecules on a heterogeneous surface (t=2) for Q1 = 126 kJ/mol, a = 1 (1,3) and 0.5 (2,4). Inset: dependence of the complete splitting upon the number of the site types t for a = 1 (curve 3) and 0.5 (curve 4) [127], calculated using the uniform site distribution function (h = t-l) [127]. The decrease of AQ leads to a situation where the high-temperature peak disappears first, below a certain value of AQ*. If AQ = AQ* is the critical value for the total splitting of the TDS, then at AQ _> AQ*, the TDS consists of t peaks, and when AQ _< AQ* there will be (t - 1) peaks, i.e. the case of partial splitting (t > 2), or of no splitting at all (t = 2). Instead of the high-temperature peak, there will appear a "shoulder" with a characteristic curvature. Any further decrease of AQ leads to the situation in which the shoulder becomes only weakly visible, and later on, the high-temperature peak disappears, etc. The determination of the conditions for total splitting is possible for arbitrary mobilities of the adsorbed molecules, and for arbitrary values of AQq :~ const. In the ge-
224 neral case, any local maximum can disappear, and not necessarily the high-temperature peak (the dependence for the condition of a total splitting on the shape of the distribution function). Diminishing the initial coverage of the surface results in a decrease of the low temperature maxima of TDS and in their complete disappearance. That is why, in order to show the total splitting of TDS, it is necessary to have the initial coverage 00 --~ 1.
2
2 2 09 1
""
22.0
'x~,,
2
29"4 l ~ X 9
3
29.4~ 37"8 V
a
b
Figure 19. C~176 for the complete splitting of the TDS curves for t=3, Q1=126 kJ/mol, a = 1 (part a) and 0.5 (part b). The cross sections are given for fq = 0.03, q= 1,2,3. The contour lines are given for AQ = 14.7(3), 16.8(4), 18.9(5), 21.7(6), 25.2(7), 29.4(8), 33.6 (9), all in kJ/mol [127]. Searching for AQ* can be done by numerical variation of AQ value and following the number of the local maxima of the curve dS/dT. That procedure permits us to investigate the lower limit of the differences in the adsorption energies of the various nodes that lead to total splitting of the TDS, and the dependence of this lower limit upon the molecular parameters of the adsorption system (a generalization of the criteria of paper [87]). The "optical" criterion of Carter [86] allows one to determine the conditions for a strong splitting of the peaks. Note the difference in the "optical" criterion and those obtained in paper [86] for AED (see Subsection 3.2) for the case when no surface mobility of the adsorbed molecules is possible [127]. The analogy of the TDS curves with the optical spectra in the case of a equilibrium distribution of the adsorbed molecules is not complete: it does not take into consideration the redistribution of the molecules between sites of various types. For a surface with two types of sites an approximate interpretation of 6E has been developed that relates it to the energy of activation of desorption from the sites of the first type: SE = aE + b, where a, b are constant values which depend on the surface composition and on the coefficient a (see Table. 1). That relationship holds with an
225 accuracy of 0.8 k J / m o l e for the range 4 < E12 < 500 KJ/mole. Fig. 18 shows how the conditions for the total splitting of the TDS are related to the surface composition. The minima of the curves are situated in the vicinity of fl "~ 0.5. When the number of sites of the first or second type increases, the value of AQ* increases too. W h e n a = 1 AQ* increases faster than f2 increases, and when a = 1/2, AQ* increases more than fl.
Composition
Parameters
1.0
0.75
0~50
0.25
0.15
0.10
0.206
0.162
0. I 16
0.072
0.055
0.048
0.641
0.536
0.406
0.272
0.242
0.222
0.161
0.124
0.088
0.053
0.038
0.030
0.515
0.440
0.306
0.204
0.155
0.138
O.120
0.089
0.060
0.042
0.032
0.026
0.398
0.297
0.201
0.168
0.130
0.113
0.7
0.128
0.108
0.086
0.062
0.052
0.047
0.490
0.222
0.264
0.226
0.201
0.268
0.9
0.170
0.145
0.119
0.096
0.091
0.090
0.649
0.113
0.302
0.327
0.377
0.394
0.1
0.3 0.5
,,
Table 1. Parameters of the approximate equation for
6ED(ED) [127].
Analogous relationships can be observed while investigating the conditions for a total splitting of TDS when t = 3 (the results for the splitting are shown in Fig. 19). The composition of any point in the concentration triangle (fa, f2, ]'3) can be determined by the usual rules describing three component diagrams of state: if a line is drawn across the point, parallel to any of the side of the triangle (let it be the side of the triangle which unites the corners f2 = 1 and f3 = 1), then on the other two sides of the triangle some fragments are cut out (they are along the triangles f2 = 1 and f3 = 1) which determine the part of the component occurring in the third angle (here it is q = 1). The value A Q ' , in the considered point, can be determined by the values which are on the isoline passing through the given point. The minimum value of AQ* can be determined in the vicinity of the point fq ,~ 1/3. The most rapid increase of AQ* value takes place when the value f2 increases (if a = 1), or at f~ (if a = 1/2). The conditions for the total splitting of the TDS for a uniform site distribution fq = t -a are shown in the supplement to Fig. 18. The total splitting of TDS takes place at practically one and the same value of AQ* (some differences can be observed at t = 2).
226 dO dT
1.0-
I
I
I
I
300
400
500
600 T,K
Figure 20. TDS for uniformly heterogeneous surfaces Qx=30 kcal/mol, Ax = 5(1,4), 10(2,5), 15(3,6) kcal/mol; ~ = l(Curves 1-3) and 0.5(4-6)[128].
dO
a
1.0
dO dT
1.0
6
4
4
1
I
200
I
400
I
I
600 T,K
I
200
400
600 T,K
Figure 21. TDS for heterogeneous surfaces with exponential distribution function f(z) = Here, n is a normalization coefficient, 7 = 0.33(part a) and -0.33(part b); Q1=30 kcal/mol, Ax= 5(Curves 1,4), 10(2,5), 15(3,6); a = 1(1-3) and 0.5(4-6) [128].
exp(Tz)/n.
227
b 1.0-
d_9_O
d.__OO dT
1.0-
dT
1
!
200
400
200
600
T,K T,K Figure 22. TDS for heterogeneous surfaces with the gaussian destribution function f(z) = exp[-(x - 5c)2/2a]/n, Q2 = Q1 - Ax, ~ = Q2 + $Ax, Q1 = 30 kcal/mol, (part a) a = 1, a = 2, 5 = 0.25 (Curves 1-3) and 0.75 (4-6); Az = 5 (1,4), 10 (2,5), 15 (3,6) kcal/mol; (part b) = 0.5, a = 2 (1-6) and 0.5 (7), a = 1(1-3,7) and 0.5(4-6); Az = 5 (1,4), 10 (2,5), 15 (3,6,7) kcal/mol [128].
Eef (O)
lg Kef(O )
30
12.8
~
5
1
4
6
12
18.7 0
0.5
I 11.3 l 1.0 0
! 0.5
1 ~
1.0 O O Figure 23. (a) Effective activation energies for desorption E~I(O) = -d{ln Uo(O)/O}/d~ for Q1 = 30 kcal/mol, Az = 12 kcal/mol, a = 1(1,3-5) and 0.5(2). The distribution functionis are: uniform (1,2), exponential with 7 = 1(3), and symmetric gaussian with a = 1(4) and 2(5); homogeneous surface(6). (Part b) Effective constants of the desorption rate K~s(0) = UD(0)exp[~E~s(0)]/0, corresponding to the curves of part a [128].
228 A relationship between the various conditions for the splitting of the TDS can be given in the form AQo = AQ* + nQ1, where AQ0 is a difference of the energies of desorption from the sites of the first and the second type, corresponding to the "optical" criterion. Here Q1 is the adsorption heat on the "strong" sites. Such a correlation holds within an accuracy of 20%, but the values ~ depend on a: ~(1) = 0.085, a(0.75) = 0.1, ~(0.5) = 0.13 and a(0.25) = 0.3. The most significant deviation from that correlation can be observed in the vicinity fl = 0.5 - 0.7. This correlation can be used for a qualitative evaluation with the largest number of types of sites. Thus, for the surface with three types of sites and a = 1, the conditions for the splitting of the TDS into two peaks yield AQ0 = 23 KJ/Mole and AQ* = 12.4 KJ/Mole and the conditions for the splitting into three peaks yield AQ0 = 27.1 KJ/mole and AQ* = 14.7 KJ/mole. Let us focus attention on some peculiarities of the use of continuous distribution functions f(z) to calculate TDS curves. Fig. 20 shows the spectra for a uniformly heterogeneous surface. The calculation was carried out by using equation (16), but not using the effective activation energy of desorption (31). An increase of Ax = x2 - xl makes the spectra wider and flatter. However, at a certain critical value of Ax* there appears a sharp low-temperature peak. Its appearance depends also on the value of the parameter a. Figs. 21 and 22 show the TDS for exponential and Gaussian distributions. Depending on the parameters of the distributions, there will also be possible various variants for the peak widening, and in some cases sharp low-temperature peaks can also appear. This can be related to the part of area of the comparatively weak adsorption sites which can be considered as analogous to the weak sites for a discrete distribution function, and by the redistribution of the adsorbed molecules due to surface migration. This migration causes the main stream of the desorbing molecules to go through the weak sites. Fig. 23 shows some concentration dependencies of the effective desorption activation energies for TDS curves calculated with continuous distribution functions. At a small value of Ax, the dependence of ED(O) for the uniform and the Gauss functions does not differ much from the linear ones which can well be approximated by the formula (31). In all the case one can observe the similarity of changes in ED(O) and KD(0), which usually is considered as the presence of a compensation effect. 3.5. D i s s o c i a t i v e a d s o r p t i o n The elementary processes of dissociative adsorption take place on neighbouring sites of the surface. On homogeneous surfaces the adsorption process is the same for any pair of sites whereas on a heterogeneous surface, the pairs of the sites differ from one another. In consequence, depending on the type of the pair of sites, the rate of adsorption will be different. A statistical description of the contributions of the elementary processes on various pairs of sites of the heterogeneous surface will involve introducing the distribution function of pairs f~q, 1 ~_ q, p <_t, t
t
fqp = 1 , q,p----1
~ q--1
fqp = fp,
(34)
229
dO dT
1.2
0.8
0.4
300
500
700 T,K
Figure 24. The TDS curves for dissociating molecules on heterogeneous surfaces with fq = 1/t, t = 2, Ell = 126, Qa = 84, AQ = 14.7 kJ/mol, a = 1. The surfaces are: regular(i), chaotic (2), patchwise(3) [130]. With these distribution functions one can express the rates of adsorption of dissociating molecules [129] as: t
U ~ = K ~ P ( 1 - 8n)(1 - 0p), q,p=l U D = K~0q0r, D ,
A,D = Kqp A,Do exp(--/~E~D), Kw
(35)
where t h e VqA'D are rates of t h e elementary processes on the pairs of sites qp, K A'D~ are the pre-exponential factors, and E ~ 'D are energies of activation.
230
AQ*, kJ/mol
AQ, kJ/mol 16.8
m
33.6 12.6
~
7
I
!
I
2
4
6
t
4
21.0
8.4
6
0.5
Figure 25. The conditions for complete splitting of the peaks for dissociating molecules on patchwise (1,4), chaotic (2,5), and regular (3,6) surfaces; Ell = 126, Q1 = 84 kJ/mol, a = 1 (1-3,7) and 0.5 (4-6,8)[130]. The function fq~ characterizes the structure of the surface. In general, it describes the correlation of various sites, with a given energy on the surface. If identical sites are situated together, we have to deal with a homogeneous or patchwise surface. Then, fqp = 1 for q = p and fqp = O for q :~ p, fqp = fqh. These two cases are very well known in the literature [3,27,72]. To describe them, the equations (35) should be re-written as: t
UA,D = ~fqqUqAr D , q= 1
t
UA,D= ~
fqfpU~;D ,
(36)
q,p= 1
The second equation (36) can be simplified if one introduces a multiplicative approxitrA'DAn'A'D mation for the rate constant of the elementary process K~ 'D = K~ 'D. "A ---'*J'ql "**~pl "
231 The essence of this approximation is that the contributions AK~ 'D and Agp4 'D to the change of the rate on various sites is observed independently on every site of the pair qp (relative to a pair of sites on the homogeneous surface of type "1"). Such an expression, for example, is obtained if we assume independence of the pre-exponential constants on the types of pairs qp and additivity of the contributions of the energy of activation on every site of the pair. This assumption was presented first in paper [27] as a consequence of the linear relationships (15), and it was frequently used in recent work [3,29-32, 72]. Determining the energies of activation using this assumption seems to be rough, but it greatly simplifies the description of the reaction rates. In consequence, the expression for the rate of desorption for a chaotic surface, for example, can be written as: UD = U1D~
fqAK~0 n
,
AKqD1= exp[-~(E~ - E~)]
(37)
q=l
In the case of patchwise and chaotic surfaces, the macroscopic rate of dissociative adsorption can be calculated, knowing only the function fq characterizing the surface composition of the adsorbent. Generally, this information is not complete [129]. For a large number of types of sites, expressions (34), (35) can be re-written by introducing two-dimensional distribution functions f(z, y): X2
Xl
Y2
Yi
ll 'x'y' xdy:'' X2
Y2
Xl Yl
where xi = yi, i = 1 and 2. Let us limit ourselves to analysing the conditions for the total splitting generated by a discrete distribution function. The shape of the TDS curves for dissociative adsorption is similar to that for the shape of the curves for non-dissociative adsorption discussed above [130]. The influence of the construction of the surface on the shape of TDS and on the conditions of the total splitting for the case t = 2 is shown in Fig. 24 (f~ = f2) and Fig. 25. For surfaces having ordered structure 1 (d12 = 1), the TDS begins at higher temperatures, and for a surface with a patchwise structure (d12 = 0), the TDS begins at lower temperatures. For the surface with a chaotic structure the TDS curves occupy an intermediate position, nearer however to the TDS for patchwise surfaces. On a strongly ordered surface, the desorption of molecules takes place when a pair of molecules occupies two similar sites, which is the reason that, for such surfaces with a given value of AQ, splitting does not take place (curve 1). Fig. 25 shows the conditions for total splitting depend on the surface composition. The most comfortable conditions for the splitting are realized on patchwise surfaces, and the least comfortable ones, on an ordered surface. At a fixed structure of the surface, the value of AQ*(f2) increases with a decrease in the coefficient a. No matter what the structure of the surface, all the curves of AQ*(f2) sharply increase as the number of sites of the first or second type diminish to zero. In the medium composition range 0.2 < ]'2 _< 0.8, the curves AQ'(f2) strongly depend on the structure of the surface and on the value of the parameter a. Thus, the conditions for total splitting can be determined both by the surface composition and by the surface structure (at a fixed surface composition). The conditions for total splitting of the TDS for the uniform distribution function fq = 1/t
232 depend on the number of types of sites on the surface and are given in the supplement to Fig. 25. An increase of a leads to a complete splitting of the spectrum, and the increase of the number of the types of sites t at t > 3 is practically constant. In general (also for non-dissociative adsorption), all the molecular characteristics of the adsorption system influence the shape of the TDS. A more detailed discussion of the influence of these parameters on the shape of the TDS curves is presented in paper [130].
4. L A T E R A L I N T E R A C T I O N
ON A HOMOGENEOUS
SURFACE.
4.1. P e c u l i a r i t i e s of n o n - i d e a l m o d e l s In addition to the heterogeneity of an adsorbent surface, the lateral interactions of the adsorbed particles influence the kinetics of adsorption. It was Langmuir who considered this for the first time [33, 131,132] while studying the desorption of thorium, cesium, and hydrogen from the surface of tungsten. Assuming the presence of dipole-dipole interactions between the adsorbate atoms, he described desorption with the help of an effective rate constant of desorption which depends on the surface coverage 0: UD = --KD(0)0,
KD(0) = K~ exp[--/3ED(0)],
ED(0) = E~ - H0
(39)
where ED(O) is the effective activation energy of desorption, H is a constant depending on the potential of interaction, and K~9 is the pre-exponential factor of the rate equation. It was assumed in papers [33, 131, 132] that the effective energy of activation coincides with the energy of adsorption ED(O) = Q(O). In the theory of the kinetics of surface processes, the model of the elementary event plays an important part. As already mentioned, in most cases one uses the model of collisions together with transition state theory. In calculations of the effects of lateral interaction, the differences between these models appear as different dependencies of the energy of activation on the surface coverage. This is due to the effect of lateral interactions upon the energetic states of neighbouring particles. Fig. 26 shows the differences of the effective adsorption potentials in the models based on collisions and on transition states for various surface coverages. These curves represent the averaged values of the potential curves corresponding to the local environments of the particles participating in the elementary process. Fig. 27 shows the local configurations of the neighbouring particles around a site where adsorption takes place without dissociation on lattices with z = 4 and 6 nearest neighbours. While describing the adsorption equilibrium when particles are in their ground states, this produces a change of the adsorption heat Q with surface coverage. This is due both to the lateral interactions and to the energetics of the transition state ED(O) (see Fig. 26a). If one assumes that the lateral interactions do not influence the transition state, one arrives then at another dependence ED(O) (see Fig. 26 b). Such an assumption corresponds to the conditions of the model of collisions, since then the adsorption activation energy EA(O) does not depend on the surface coverage 0. In transition state theory, EA(O) turns out to be a function of the surface coverage even
if EA(O= O) = O.
233 v(e)
a
E D (0)
A (0) e
Q(1)
/
O(O),
/ .
.
.
.
.
.
.
.
.
.
.
Figure 26. Potential curves for the model transition state (part a) and the impact model (part b). EA and ED are the adsorption and desorption activation energies, Q is the heat of adsorption, and I is the distance from the surface (horizontal axis of the figure).
4.2. Principles of t h e calculation of the a d s o r p t i o n rate Most often, the effects of lateral interactions are treated in terms of the lattice gas model. The particles localized on the sites of the lattice interact with each other if their separation does not exceed the radius R of the interaction potential. The energetic parameters of the adsorbed particles, being in the ground state, can be expressed as e(r) where r is the number of the coordination circle (in two dimensions) around any site (Fig.28). In the lattice model, the distance can be conveniently measured by the numbers denoting the coordination circles, 1 _< r _< R. The interaction of an activated adsorption-desorption complex (adsorption and desorption have the same activated complex) with the neighbouring particles in their ground state at a distance r, can be expressed as e(r). In consequence, the effective energy of activation of adsorption is a function of the difference =
_
234 n=O
1
2
3
4
1
2
3
4
5
~=1
n=O
6
Figure 27. The local configurations of neighboring admolecules around any central site (re=l) for the planar lattices z = 4 and 6 [40,49]. The theoretical description of adsorption, taking account of the interaction of nearest neighbours and based on the model of collision, was given by Roberts [133]. (The first calculation of TDS was carried out by Toya [134]). Analogous expressions for the model of the transition states were obtained much later independently in 1974 by Adams [135] and by Tovbin and Fedyanin [136-138]. In paper [135], the interactions between the nearest neighbours are considered. In papers [136-138], a more complex potential was discussed: a direct interaction between nearest neighbours and a collective interaction with the remaining adsorbed particles. A detailed review of the papers from 1937 to 1990 dealing with the kinetics of surface processes (adsorption, desorption and surface migration) and based on the lattice gas model, was given in paper [40]. Most of these papers are concerned with homogeneous surfaces. Here, we consider the basic factors influencing the kinetics of adsorption processes on homogeneous surfaces. These factors can be divided into two groups. The first group of factors characterize the physical properties of an adsorbent-adsorbate system calculated from kinetic equations. These properties include the lateral interaction potentials, the aggregated (phase) state of the adsorbed layer, and the surface mobility of the adsorbed particles. The second group of factors characterizes the accuracy of the calculated physical parameters. The most difficult problem in describing non-ideal adsorption systems lies in the fact that these systems possess cooperative properties whose calculation is a traditional many-body problem [109, 110]. Such problems seldom have an exact solution. For practical purposes one has to use approximate methods. One of the main difficulties of the theory of
235 non-ideal adsorption systems consists in the choice of an approximate method and in the estimated accuracy of the results obtained. We shall discuss this problem here in detail because it is quite important for the description of kinetics on heterogeneous surfaces. The many-body problem dscussed above concerns adsorption not only on homogeneous surfaces but also heterogeneous ones. In order to understand such complicated situations, one has to analyze the advantages and disadvantages of various methods of calculating the effects of lateral interactions for simple cases.
gl
1
2
3
r~ fxJ
r~ gxJ
g3
9
4
7 6 5 10 Figure 28. The neighboring sites belonging to various (r) coordination spheres g~ are indicated for a central site f. For the central pair of sites f,g, the sites (1-6) belong to the nearest neighbors of the unified first coordination sphere, and the sites (7-10) are second neighbors. The rate of desorption of non-dissociating molecules on a homogeneous surface can be expressed as:
UD=--KDOS(O i R , z ) ,
KD = K~ exp[-/~E~] ,
(40)
where KD is the desorption rate constant defined in (10), the function S(0 I R,z) describes the non-ideality of an adsorbed system by taking account of: 1) the structure of the surface (z is the number of the nearest neighbours); 2)the radius of the pair potential for lateral interactions R: 3) the model of the elementary process (for the model of collisions e*(r) = 0, whereas for transition state theory, e*(r) ~: 0); and 4) the calculation of the probabilities of the various configurations of neighbouring particle. If n is the number of the configuration, calculation of the probabilities 8z+x(n) introduces certain assumptions that make it possible to close the systems of equations with respect to the probability of the probable single, binary etc. configurations. Fig. 29 shows schematically the result of applying approximate methods of calculating the probability 0z+l(n) for four approximations: chaotic, polynomial, quasi-chemical and an approximate calculation via indirect correlations. The latter is the best, followed by the quasi-chemical approximation. The method of calculating the probability of triplet configurations can be related either to the superposition approximation [141] or to the approximation proposed by Hill [142]. In the
236 chaotic approximation the probabilities of the many-particles configurations are expressed by the singlet probabilities 8q (= 8). Effects of the correlations between the interacting particles are absent. However, this situation can be treated by different approximations: the mean field approximation [143-145] or a special chaotic approximation [146, 147]. In the polynomial approximation [148] one calculates the effects of the interactions between nearest neighbour particles though equations that are closed with respect to the function 81. In the quasi-chemical approximation [149-151], the values 8z+l(n) are closed through 81 and 82, which is why we then calculate the effects of the direct correlations between the interacting particles.
oj iO
On
io--o i o--Ok
Ok
J
3..........
~ i
e
k
~
oj
e
iO
0
Ok
k
Oe jO
io
Ok
% 0 P
o~ P
O
in
In
i p
rn ~ j O
Ok Oe
iO 0 P
O m
/ ~
e p
Figure 29. The schemes of the many-particles configuration probabilities for admolecules on planar lattices with z = 4 and 6 in various approximations: 1 is for the mean field and chaotic approximations, 2 is for the polynomial approximation, 3, for the quasi-chemical approximation, and 4 takes account of indirect correlations; o denotes 81, o-o, 62, triangles denote 83 [40,49]. For nearest neighbours (R = 1) the function accounting for non-ideality of an adsorption system S(8 [ R,z) takes, in the chaotic approximation (CA), the mean field approxi-
237 mation (MFA), and the quasi chemical approximation (QCA), the following forms: S(0 I R,z) =
(1 + xO) z , x = exp(/~Ee)- 1 , (CA) exp(/~zEeO) , ~ = [(1 - 20) 9. - 40(1 - 0)exp(fle))] 1/2 ( M F A ) (1 + xt) ~ , t = 1 - 2(1 - 0)/(1 + or), (QCA)
(41)
In the polynomial approximation, the function S(O ] R, z) has a more complicated form [135, 138, 49] which is not discussed here. However, we shall discuss certain peculiarities of the TDS curves that are connected with the effects of correlations and the application of the lattice gas models to analyze experimental systems. A more detailed discussion is given in [40]. 0
0.5
1
1.0
12
0.5
r 1.0 Figure 30. The concentration dependencies of the heats of chemisorption in various approximations for z = 4, r = [Q(O)-Q(O)]/[Q(O)-Q(1)]; 1 denotes PA, 2 denotes QCA, 3 denotes MFA, 4 denotes CA [147]. The differences in the methods of calculating the effects of correlations are reflected in the concentration dependencies of the adsorption heats (Fig. 30) and of the rates of desorption (Fig. 31). These figures show the relation between Q(0) and In Up(O) for the various approximations. The existence of correlations results in the appearance of points of inflection in those curves (if there are no correlations, then there are no points of inflection). The TDS curves calculated using these approximations are shown in Fig. 32 [118]. The main difference between the approximations considered here and those where the effects of correlation are ignored, consists in the existence of split TDS. The absence of correlations leads to curves with a maximum of one peak. One should observe significant
238 differences in the positions and heights of the maxima in the chaotic approximation relative to the MFA. The existence of correlations affects the splitting of the spectrum and the number of peaks is equal to (k + 1), where k is the number of the points of bending of the functions Q(0) and in UD(O).This is the main reason that, in the chaotic approximation, the coverage dependence of the adsorption heat Q(0) is strongly non-linear (curve 4). The linear dependence predicted by MFA (curve 3). (Curve 1 in the polynomial approximation) is situated nearer to curve 3 than to curve 4, but anyhow there is a splitting in it. 0
20
]
0.5 ,
!
I
1.0 I
!
O
12
-4 -!I In V 2
Figure 31. The influence of the lateral interaction of admolecules upon the concentration dependence of the desorption rate when ~e = -5, ~Se = 5:1 is MFA, 2 is PA, 3 is QCA, 4 is CA
[147].
The non-ideality of an adsorption system S depends upon the difference 6e. In equation (3), repulsion between molecules corresponds to a negative e(1). If the interaction of an activated complex with the neighbouring molecules is the same as that between adsorbed particles in the ground state, then S = 1 and the kinetics of desorption is equivalent to an ideal adsorption system, regardless of the method of approximation. Fig. 33 shows TDS curves calculated in the QCA for various values of the parameters of the lateral interaction and the energy of activation for desorption at zero coverage. The width of the TDS turns out to be proportional to the value of 6e, with an increase of 6e producing
239 a stronger splitting of the TDS, and a decrease in the temperature of the initial TDS, with a corresponding increase of the share of the particles desorbing at lower temperature values.
d_..~e dT 0.4
0.2
300
400
500
600
T
Figure 32. TDS curves for 80 = 0.9 in the approximations-polynomial (1), quasi-chemical (2), chaotic (3), and mean field (4) for g~9 = 1012 s -z, E D = 126 kJ/mol, e = -16.8 kJ/mol, e* = -8.4 kJ/mol, b = 50 K/s, z = 4 [147]. Repulsion between nearest neighbours is typical for chemisorption systems and results in a decrease of Q(0) with increasing coverage 0). In such systems the condition [ e~ I__0. To physical adsorption there corresponds attraction to nearest neighbours which leads to an increase of the heat of adsorption with increasing surface coverage. In such a case, e, e* > 0. However the effects of the repolarization of the binding (decrease of
240 the binding with the surface coverage and increase of the binding with the neighbouring particle) can lead to the situation where e > e* and. vice versus. An increase of the parameter e at *e = const, provokes an increase of the splitting of the spectrum (Curves 4-6), but its width remains constant. Curves 4 and 8 correspond to e* = 0. In this case the width of the spectrum is maximized. The diminishing energy of activation E~9 moves the spectrum toward lower temperatures and makes it possible to increase the splitting of the spectrum due to an increase of the contribution of the lateral interactions to the energy of adsorbent-adsorbate binding z~e/E~9 , E~9 ,~ Q(O = 0)). dO 0.4~- dT
0.3
0.2
0.1
0 300 600 900 1200 Figure 33. TDS in the quasi-chemical approximation, z = 4, b = 50 K/s, 00 = 0.9. 21]ae energy perameters are given in Kelvins. Curves 1 and 2: ED = 15- 103, ~e = 103, e = - 2 . 1 0 3 (1) and -103 (2). Curve 3: ED = 20.103, ~e = 103, e = -2-103. For the other curves (4-8), ED = 30-103. For curves 4-6, ~e = 10 3, e = --103 (4), --2.103 (5), -3-103 (6). Curve 7 is obtained with ~e = 0 at any E. Curve 8: e = -3-103, ~e = 3.103 [147]. Curve 5 qualitatively reproduces features of the experimental TDS curve for the C O / W system[152]. (This fact was not mentioned in paper [134].) By coincidence, the TDS curves calculated in [147] correspond to the experimental data for the systems C o / M o ( l O 0 ) [153], and on a series of metals [154], C s / W [156], etc.
241
4.3. P r a c t i c a l a p p l i c a b i l i t y of t h e cluster m e t h o d s A detailed analysis of the papers published in the years 1970-1980 [40] shows that the calculation of the interaction between nearest neighbours can assure only a qualitative description of the experimental TDS curves. For quantitative agreement, it is indispensable to use the full set of lateral interactions. This conclusion is physically obvious, because the contributions of the nearest neighbours amount to 60 to 80% of the total energy of binding of a particle with its neighbours in any condensed phase. For the first time, a quantitative description of the kinetics of the desorption of atoms from the surface of tungsten [156-158] was given by considering both the direct interactions between nearest neighbours and the collective interaction with all the remaining adsorbed molecules [136,137]. The rates of desorption (Fig.34) obtained from the experimental TDS curves were estimated for 840 K and compared with calculations done using three approximations at f l s - - -3.6. The contribution from the collective interactions was equal to 30% of the total and the marked areas in Fig. 34 correspond to the permitted values of the parameter ~c~. g which range from -0,38 to 0. That comparison shows that the approximations, after taking account of the effects of correlations, lead to considerably better agreement with the experimental data than those ignoring the correlations.
0.5
1.0
0
..
0
-8
I-.-!
I--4 1
-24 "
fp'Ar 3
In V
Figure 34. The desorption rates K/W for T=840 K, calculated for the chaotic (1), polynomial (2), and quasi-chemical (3) approximations [137], V = U ( ~ ) / U ( O = 1); symbols denote experimental values of the desorption rates with their errors [156-158].
0
~"
0
I~
~
,.~"
,-,-
'
0
---.;'l
0
,
~
~
0
v
0
0
0
....
0
.-'I.
i l
0
v
,_..
~-- ~
,..,
~.
',I.,
~
'_.
-,
i,0
--,
~
v
~
~
i
0
"
o
,---, 0
o -' ~--'
~
EE
~
(D~cl
~ 0
L,O
o
0
o
243
rl
dO dT 1.0
O.
0
200
1
2
350
3
4
5
400
450
500
550
Figure 36. Experimental (1) and calculated (2-4) TDS curves for C O / P t ( l l l ) , corresponding to the versions 2-4 of the interaction parameters in Table 2. Inset" the concentration dependence of the sticking coefficient r/(0): curve (p) - calculation, the another curve - experiment,[159]. A satisfactory qualitative agreement with experiment is obtained when one includes interactions with the first and second neighbours [159] in the description of the data for desorption of Co~It(Ill) [160], 02/Ir(lll) [160] and 02/Pt(lll) [161] (Fig. 35). The values of the lateral interaction parameters that lead to a quantitative description of the TDS have been collected for a series of systems in Table 2. These parameters correspond to the initial values 00 < 8*, where 8" is the maximum measured value of the initial surface coverage. Calculations [159, 168] have been carried out without applying special optimization procedures (see for example [169]). The search for the parameters was ended when the deviation between the experimental and calculated curves was less than 3-5%. The accuracy of the experimental data does not allow one to ask for still more accuracy. The procedure used for estimating the parameters is not precise enough. For the system Co/Pt(lll), several sets of parameters were obtained, all corresponding to a given accuracy in the description of the experimental TDS curves, but the procedure of carrying out such calculations proves that for a quantitative description of the experimental data one needs, as a minimum, calculations of the contributions for the first and second neighbours. The indispensability of including the contributions not only from the nearest neighbours, but also from more distant neighbours, has been shown in the description of
244 the rate of adsorption [159]. The concentration dependencies of the rate of adsorption are even more sensitive to contributions from further neighbours. For the system Co/Pt(111), a comparison of two procedures for searching for the lateral parameters was carried out. In the first case, the parameters e(r) and e'(r) were determined from the TDS curves and accordingly, the concentration dependencies of the sticking coefficient was evaluated (variants 1-3). In the second case, these parameters were determined from the sticking coefficients, and the TDS curves were calculated (variant 4). In the first case, the calculated sticking coefficients were in poor quantitative agreement with the experimental data. In the second case the agreement was much better (see. curve 3 in Fig. 36). In the supplement in Fig. 36 the experimental and calculated concentration dependencies of the sticking coefficients are given. -1
dO dT
0.8 0.4
0.5
1
~..~
0.1
!
I
I
I
400
500
600
700 T,K
Figure 37. TDS calculated with (solid lines) and without (broken lines) taking account of ordering, R = 1, e =-8.4 (Curves 1) and-12.6 (Curves 2) kJ/mol with e* = e/2. The phase diagrams (broken line -exact calculation) and the trajectory 0(T) in the thermodesorption process [174] are given in the inset. From general physical ideas it follows that the contribution from the more distant neighbours can not surpass the contribution of the nearest neighbours. Anyhow, their role in the kinetics of the adsorption processes is essential. The analysis of of their influence on the TDS curves [168] showed that: 1) the width of the spectrum is proportional to the value ~r=lnz(r)~e(r); 2) a consideration of the attraction with the second neighbours for chemisorption systems brings about an increase in the the splitting of the spectrum; and 3)
245 the corresponding changes in the contribution from the second neighbours (i10-15%) can considerably shift the positions of the maxima and minima of TDS to higher temperatures. At the same time, the contributions from the second and following neighbours do not split the spectrum additionally, so that the number of peaks in the spectrum is determined by the interactions with the nearest neighbours. Strictly speaking this concerns the curves with 60 -~ 1 for values of the parameters which are distant from their critical values for the splitting of the TDS. These conclusions, of great importance for the interpretation of TDS, were drawn not only in the QCA but also in the calculation of the indirect correlations [170], as well as in all recent calculations [40,171]. Finally, it should be pointed out that contributions from the second and following neighbours show a strong influence on the state of the multilayer film. The phase transformations connected with the ordered phases in these films and also with the stratification of the ordered or disordered phases, lead to changes in the character of the distribution of adsorbed particles on a homogeneous surface. Instead of a homogeneous distribution of the particles on the surface, there is a locally heterogeneous distribution in the presence of ordered phases and macro-heterogeneous in the presence of the stratification of the disordered phases. As a consequence of these changes of state, there are also changes in the character of the local surroundings of the particles in the phases and, correspondingly, of the rate of adsorption. A direct calculation of the rate should precede the calculation of the phase diagram of the adsorbed particles in order to determine the number and amounts of the phases present at a given temperature, density of the monolayer film (0) and potential of lateral interaction [172-174]. Fig.37 shows TDS curves obtained using the quasi-chemical approximation without calculation of ordering of the type C(2 x 2) on the (100) crystal face for non-dissociating particles. In the supplement, the phase diagram of this ordered system is shown, as calculated in the quasi-chemical approximation, and in an exact calculation [139] taking account of only nearest neighbours interactions (R = 1). The ordering phase transition significantly changes the shape of the TDS curves (the arrows show the disorder-order and order-disorder transitions occurring during thermal desorption). The splitting of the spectrum considerably increases (the minimum decreases), the high-temperature peak becomes sharp, and the low-temperature peak can have shoulders, the nature of which is not connected with the existence of an additional state. The attraction with second neighbours increases the range of existence of the ordered state and the shoulders cannot be observed on the rapidly increasing front of the low-temperature peak [40,174]. The indispensability of the calculation of the phase changes while considering the kinetics of adsorption was illustrated by the description of the desorption of Hg atoms from the (100) surface of W [175, 176]. That system is characterized by the presence of a common edge for a family of the curves with different initial values of 00, which indicates two-dimensional condensation of the adsorbed atoms. Previously, the existence of such a common edge was explained on the basis of double-layer models [40, 177, 178]. The calculation of a condensed phase allows to one explain the existence of a common edge, if equations (40), (41) for a monolayer model along with the quasi-chemical approximation at c*(1) = ~(1)/2 [179]. The authors [175, 176] described their experimental data using equation (39) where ED(8)(= Q(0))is the adsorption heat, calculated in the quasi-chemical approximation [180]. As a result of that description, these authors arrived at the values: K~ = 10x6 cek-Xr; ~Hg-Hg(1) = 5.85KJ/mole; and E~ = 172KJ/mole. However, their in-
246
7
I !
/ /
I I I I I I I I |
,t1,I10 I
I i I !
3
, I
I
I
II I
50
630
590 T,K
Figure 38. The TDS curves for the system Hg/W(100) coincide with the experimental data (broken curves)when calculated with 801 < 8 < 802 (1-6), 8o1 ,,~ 0.45 and 8~) ,,~ 0.8 [175]; they deviate from the experimental data at 80 = 0.99 (7) [179]. teraction parameters did not assure the creation of islands of Hg atoms[176], the existence of which was found in Monte-Carlo simulations. In this way, the thermodynamic and kinetic characteristics of the adsorption system were not selfconsistent. The calculation of the condensation of the Hg atoms lead us to a satisfactory agreement with the experimental data [175, 176] (Fig.38) at other values of the energetic parameters E~ = 136.5K J/mole, eHg-Hg(1) = lO.4gJ/mole, K g = 1013 cek -1, e*Hg_Hg(1)/eHg_Hg = 0.55 [179]. The parameters obtained have reasonable values: an increase of the attraction eHg-gg results in condensation in the range of temperatures investigated and allows the anomalously great value of K~ to decrease by three orders of magnitude and consequently, a considerable decrease in the estimated value of the energy of activation for desorption E~ was observed. Thus, the values of the parameters obtained lead to consistency between thermodynamic and kinetic adsorption properties. The deviation of the calculations from experiment can be observed only at 80 ~- 1. It is possible that this is connected with the formation of the second layer of adsorbed atoms which is ignored in the model.
4.4. Calculations using cluster methods Cluster methods of calculation of the equilibrated distribution of the particles have some well-known limitations [181-183]: in the critical areas of phase transitions they give only qualitative descriptions, with differences in predictions of critical temperatures from the exact values by 15-30%. (e.g. for the (100) face, the exact value (~e) = 1.78, and the
247 quasi-chemical approximation gives the value 1.38). In adsorption, the concentration of the adsorbed particles changes over a broad range and in many situations, the trajectory of thermal desorption O(T) crosses areas with various states of aggregation. Thus, there appears a question: how well can the cluster methods describe the kinetics of adsorption? A review should give a comparison with analogous calculations carried out with the help of other methods that assure a more precise calculation of the effect of correlation. Monte Carlo simulations belongs to such methods as well as the matrix method. Monte Carlo simulation is at present a fundamental method of calculation in statistical physics (see for example [184, 185]). The algorithm is simple enough and the computations allow one to investigate lattices, containing up to 106 sites. In many situations the difference between the solution obtained in this way and the exact value is insignificant. The matrix method of investigating lattice systems, allows to one obtain an exact solution for a periodically repeating representation of the lattice [186, 187, 139]. In general, with the help of those methods one can investigate a slab of sites of width L, where L is the number of lattice parameters. In certain situations,one can investigate the thermodynamic limit at L ~ ~ [188, 139, 140]. As mentioned in paper [189], in the absence of phase transitions the numerical analysis of a small slab when L > 4 gives practically the same results as for the macroscopic L. This was a good reason to apply this method to the construction of a scaling procedure for values of L corresponding to the critical range of phase transitions. Such an approach increases possibilities of calculating effects of correlation in studies of surface phenomena and for the construction of phase diagrams [172, 173]. While applying the two methods to investigate the kinetics of adsorption, one can basically evaluate the reliability of the results obtained by cluster methods. For the first time, the calculation of the kinetics of desorption by the Monte Carlo method was reported in ref. [190], and the matrix method for L > 1, in ref. [191]. For a comparison of the calculations of the adsorption kinetics carried out with the help of cluster methods and with matrix and Monte Carlo method, it is indispensable to view the differences in two ways. The first one is connected with the influence of the neighbouring particles (through the non-ideality of the adsorption system SD(O[R,z)) upon the local rate of the elementary process. The second aspect is connected with the influence of the state of aggregation of the adsorbed monolayer. Usually, the influence of these two factors cannot be separated. Equation (41) shows that in the quasi-chemical approximation, the influence of the neighbouring particles is manifested through/92 = /?It which represents the probability of finding two neighbouring particles together. Depending on the level of accuracy of calculation of this function, the same accuracy is required to calculate the contribution from the lateral interaction. The exact solution for/92 is known only for 19 = 0.5 [139, 140]. Fig. 39 shows the comparison of the exact dependence 09(~e) for interactions of nearest neighbours with an analogous dependence obtained by the approximate cluster methods. With increasing interaction, the difference between curves 2 and 3 increases. The calculation of indirect correlations considerably diminishes this difference and the curves become similar, though the difference still remains observable. This is obvious because phase transformations occur with increasing interactions, and the character of the distribution of the particles changes. In consequence, the expressions for/92 and/93 calculated without taking into account the change of the aggregate state of the system are not exact. Considering the ordering of the particles (with the formation of a C(2 • 2)
248 structure) even without taking into account indirect correlations (curve 4)) considerably improves the agreement between 82 calculated in the quasi-chemical approximation with the exact solution. It was observed [49], that considering both the ordering of particles and the indirect correlations leads to an approximate relation 82(fie) which differs from the exact solution by only several %. An analogous influence on the accuracy of calculation of 62(~e) by the cluster method shows the effect of stratification of the particles in the case of attraction between adsorbed particles.
O2
0.5- -
5 1
0.3-
I
-3
I
I
-l
i
1
I
I
3
Figure 39. The dependence of 02 on ~e for 0 = 0.5 on a planar square lattice, calculated by various approximate methods: 1, with indirect correlations in t-he quasi-chemical approximation, 2, exact solution (after Onsager), 3, quasi-chemical approximation, 4, with ordering in quasi-chemical approximation, 5, taking account of two-dimensional condensation in the quasi-chemical approximation [40,49]. The duster methods assure a satisfactory accuracy of the description of the local distribution of the interacting particles with any interaction. It is just this fact which determines the non-ideality SD(8[R,z) and in consequence, an accurate local description of the kinetics of adsorption. The solution of the problem of calculating the aggregate state of the adsorbed layer is more complicated. For a correct description of the critical region, it is indispensable to decrease the phase transition region in comparison to the exact solutions (see the supplement for Fig. 37). This effect of the cluster methods is usually avoided by using phase diagrams obtained by applying some other theoretical methods or determined from
249 experimental data while applying the law of corresponding states (see the example in [195]). An analogous problem exists, however, in the matrix methods. These methods cannot basically describe the change of aggregation even when the calculation is made for large (but finite) values of L [140]. Scaling procedures on L give approximate results, even if they are additionally controlled by the fluctuations in the correlation functions 0~(r) [172]. In Monte Carlo simulations, the problem lies in searching a compromise between the necessity of increasing the number of the sites of the lattice (in order to allow "renormalization-group" transformations) with the assurance of attaining the equilibrium state on the lattice. If we consider that while calculating TDS curves such a procedure should be applied at every comparatively small temporary change of temperature, keeping these conditions correctly demands much care even with contemporary computers. I
I
I
I
Noool
1
_=
i 400
500
400
50O
400
500
T,K
Figure 40. Thermal desorption spectra calculated o using a combination of quasi-chemical and mean-field approximations (curves 1), the Monte Carlo method (curves 2) and the transfer-matrix technique (curves 3) for g ~ = 10x5 s-x, E~9 = 35 kcal/mol and different laterm interactions: ex = -1.4 and e2 = -0.6 (a), ex = -2.0 and e2 = 0 (b),ex = -2.6 and e2 = 0.6 kcal/mol (c). The initial coverage is equal to unity and the heating rate is 40 K/s. The inset (upper right) shows adsorbed particles (dark circles) on a square lattice [191]. Comparison of the results obtained using cluster methods and the Monte Carlo method was carried out in a large number of studies. Looking at the results (see [40]), one can conclude that in the majority of cases, the calculations basically agree with one another. This agreements has a quantitative character when the calculations are carried out for the entire range of a phase transition, while they have only a qualitative character in the cases where phase transitions occur. It is, however, necessary to emphasize that in all cases the Monte Carlo calculations are comparable with calculations done using the
250 analytical expression of type (40), (41) where the aggregate state of the adsorbed particles is considered even when the comparison was carried out outside of the expected range of applicability of the expressions (40) and (41)).
a
b 35 o
2
~
400
3o
25
500
T,K
0.2
0.6
O
Figure 41. Part (a)" thermal desorption spectra for a square lattice calculated for K~9 = 1014 s -1, ED = 35 kcal/mol, b = 50 K/s, and E = - 2 kcal/mol (see fig. 5c [197]). (b) The coverage dependence of the desorption activation energy for (fig. 5f [197]). Curve "c" corresponds to the repulsion of only nearest neighbors (the case under consideration). The other curves reflect the contribution of triple interactions.
Q 1
23-
21-
19-
0.1
0.2
0.3
0.4
O
Figure 42. The heat of adsorption of repulsive molecules (z = 4, e = - 2 , Q(0 = 0) = 23 kcal/mol) in the quasi-chemical approximation taking account of ordering (1) [198] and without taking ordering into consideration (2); curve 3 - calculation by the matrix method [199].
251 In order to illustrate the current state of the problem, we limit our considerations to the calculation of the characteristics of the ordering of type c(2 x 2). Paper [196] reported on the calculation of the TDS curves by Monte Carlo and then in the quasi-chemical approximation without considering of the effect of ordering. The main conclusion of the paper consists in the statement that in the range beyond the phase transitions, both methods give practically the same results, but in the range where the phase transitions occur, their results differ. Moreover, in the Monte Carlo method one can see the a qualitatively new effect which is the presence of an intermediate peak that cannot be obtained when one uses the quasi-chemical approximation. However, in paper [191] these calculations were also compared with calculations based on the matrix method (see Fig. 40). The results show that the intermediate peak is not observed, and that it was due to an incorrect Monte Carlo calculation. In spite of the qualitative agreement of the matrix method with the quasi-chemical approximation, one can observe considerable differences; in the matrix method, the minimum is more profound and the peaks are sharper. Such a result (for strongly separated peaks) can be obtained from the quasi-chemical approximation with explicit considering of ordering [159]. The peaks, however, are a bit more diffuse than in the matrix method, and the low-temperature peak has a jagged structure as shown in Fig. 37. The absence of the jagged structure in the matrix method means that the phase transition of the ordering has not been taken into account. In paper [197], the Monte Carlo calculation no longer contains the additional peak and the TDS curve is similar to the TDS of paper [191]. The ED(0), however, (see Fig. 41), strongly differs from the isosteric heat of adsorption Q(8) (Fig. 42) [198]. Fig. 42 also shows the curve of Q(0) calculated by the matrix method [199]. In these calculations [191,196,197], the collision model was applied. In this model either ED(6) = Q(O) or the curve of ED(0) is similar to the function In UD(O). (The difference is related to the two possible methods of determination of ED(O), see [39,40]). The function In Up(O) has break at the points of the phase transition [159] which also is absent in Fig. 41. Thus, these Monte Carlo and matrix calculations do not give the ordered state. In the fundamentals of the matrix method there is assumed the impossibility of describing the ordered state for finite L (calculations [191] executed at L = 4). That is why instead of a sharp jump at the point of the phase transition, a bend in the line appears in the function Q(0). With increasing L, Q(8) does not change its character. Finally, both methods give artificial effects. The cluster method assures the appearance of a jump in the heat of adsorption at the point of the phase transition, but after that jump we have QCM(~) > Q(~ - 0). The matrix method not only does not give the jump Q(O) at the phase transition, but also leads to the value QMM(O)> Q ( 0 - 0) (an increase of L to 12 keeps these artificial effects [200]), even though QcM(O) > QMM(O). The flattened character of the curves of Q(O) is expressed in the shape of the TDS: the jagged structure of the low-temperature peak disappears and the high-temperature peak has a flattened maximum. These differences are mainly observed near the critical temperatures ([ /~e ]> 1.7). With increasing repulsions between the molecules ([ f~e ]> 3), the cluster method and the matrix calculation give practically the same results. Fig.43 shows the rates of desorption, calculated using both methods [199] for the transition state. The generalization of the matrix method assures the inclusion of the interactions of the activated complex with its neighbours, and was presented in paper [201].
252 Thus, the description of even a simple adsorption system having c(2 x 2) ordering on a homogeneous surface leads to difficulties when applying any method. Outside the critical range, the cluster method is not inferior in its accuracy in comparison to the matrix method and to Monte Carlo. The calculations carried by the cluster method are 2 to 3 orders of magnitude faster than either of the other two. Of course, within the critical ranges it is better to use the combined methods (see [40]) but this is a separate question not considered here. For interactions on heterogeneous surfaces, the problem becomes much more complicated, and the use of the cluster methods [40-49] gives a possibility of considering lateral interactions with the same degree of accuracy as they provide for homogeneous surfaces. lg
S
65-
6
43-
f
2b-
-1 -2 -3 ~4 i
0
I
I
I
I
I
0.5
I
I
I
I
O
1.0
Figure 43. Concentration dependence of the desorption rate for z = 4. ~e(1) --2 (1-4) and -5 (5-6); ~(2) - 0 (~,4-6) ~ d - ~ / 3 ~ ( ~ ) (2,3); c(~) - 0 (~,2,5) and ~(~)/2 (3,4,6), r-X ~nd 2. The solid lines are given by the quasi-chemical approximation, and the broken lines by the matrix method [199].
253 5. T H E T H E O R Y O F A D S O R P T I O N ON HETEROGENEOUS SURFACES
WITH LATERAL INTERACTIONS
5.1. The problem of calculating lateral interactions Real adsorbent surfaces are characterized by a large variety of compositions and structures which depend on the preparation of these materials. Their surfaces are heterogeneous and in many cases, the description of adsorption on them over a wide range of gas pressures and temperatures, will require that one consider the effects of lateral interaction. Even between nearest neighbours, this involves a complex analysis of the whole adsorption system. The state of occupancy of any lattice site (denoted as "f') depends on the type of the site, as well as on the occupancy of its nearest neighbours, which depend in turn on the type and state of occupancy of the more distant neighbours h, including also the lattice site f, from which the consideration began. As a result, the probability of localization of a molecule on a particular site depends not only on the type of this site, but also on the types of the vicinal sites. Then, depending on the required accuracy of the evaluation of the effect of the distribution of vicinal sites upon the local filling at each lattice site, different approaches to the description of the equilibrium distribution of the particles and the kinetics of adsorption will be required. On a homogeneous surface, the distribution of the of adparticles depends on the structure of the surface and the potential of intermolecular interaction. On heterogeneous surfaces an additional effect is observed that is connected with different adsorption and kinetic features for different lattice sites as well as their spatial distribution on the surface. Changing the spatial character of the distribution of different lattice sites (at a fixed number), yields surfaces differing in their adsorption and kinetic characteristics. Obviously for very low coverages where the contribution of lateral interactions can be neglected, such surfaces will be indistinguishable in adsorption experiments. However, at higher coverages where the contribution of lateral interactions cannot be neglected, these interactions will influence the course of adsorption. The theory of adsorption taking simultaneous account of the effects of surface heterogeneity and of lateral interactions on the equilibrium and kinetic characteristics of adsorption has been discussed in [41-45]. This theory makes it possible to consider the local properties of individual surface lattice sites, and leads to macroscopic expressions
,I ,1 i f a] hi
hi
h2
gl
fig 4
g3
f
g31 l~
h 4, g4: h 3 14
gl
=~
h2 f
g3 ha
h2
g2 f
ti 4
g2
11 81]
12
f 13
ha
gJi 14
Figure 44. The sites g, h and 1 are first, second, and third neighbors of the central site f. The mapping of a fragment o~ the surface onto a cluster consisting ot one central site and its nearest neighbors ( R=I )[202].
254 for properties such as isotherms, heats, adsorption rates, desorption, surface migration etc. by considering the local coverages of surface sites. In order to estimate these local coverages, there were constructed systems of algebraic (for equilibrium) and kinetic (for nonequilibrium distribution of molecules) equations. Several levels of the description of adsorption exist for heterogeneous surfaces. These describe the probability of coverage of individual surface sites. Although the considered fragment of the surface is small, it allows for a precise estimate of the effect of the spatial distribution of sites on the adsorption processes. At the next level, which is a cluster description of the surface, it allows for recognition of the local surface structure. In this case, only nearest neighbor correlations are retained in the distributions of sites of different types. A third level coupled description of the surface is produced by mapping in terms of site pairs. Such description is least exact when the surface is expressed in terms of individual sites, where lateral interactions of vicinal molecules can become impossible. The minimum dimensions of the fragments and clusters as well as the distances between individual site pairs cannot be smaller than the range R of the lateral interaction. In order to estimate macroscopic characteristics of adsorption, one must average the local characteristics for different fragments, clusters or site pairs of different types. For this purpose, it is necessary to introduce approximate distribution functions of fragments, clusters, and site pairs that characterize the structure of a heterogeneous surface. The binary distribution function fqp(r) for r = 1 was already discussed when the adsorption kinetics of dissociating molecules on heterogeneous surfaces in the absence of lateral interactions was described. Analogous functions were introduced also for distances r < R. The second level of the models introduces the cluster distribution functions f(qmR) describing the probability of finding a cluster on the surface containing the central site around which there are locahzed at the distance r mqp(r) sites of p-type, 1 _< q p >_ t where t is the number of site types; the symbol mR denotes the totality mqv(r) 1 > p > t 1 > q > t 1 >_ r > R. The condition of normalization of the cluster function and the relations between the choices of mqp(r) are given by: t
f(q{m}R) = fq bq
~
rr~(r) = 2q(r)
(42)
p=l
where bq is the total number of different choices of mqp(r) for central site q for different surface clusters, 2q(r) is i the number of vicinal sites in the r - th coordination shell of site q. A detailed description of different distribution functions and examples of the construction of adsorption isotherms and heats is given in [49]. In the following sections of this article we will review the results of the applications of this theory, mainly by presenting examples of the estimation of desorption processes.
5.2. Kinetic equations of adsorption. In the case of heterogeneous surfaces, the parameters of the lateral interactions between different adsorbed molecules may depend on the type of sites on which these molecules are locahzed. For this reason we will denote these parameters for the pairs of molecules localized on q and p sites at distance r by %~(r). The interaction of an activated complex localized above q site with vicinal adsorbed molecules localized above q site at distance r will be denoted by %~(r). For physical adsorption, the dependence on the site type is
255 significantly smaller than for chemisorption. In the final expressions we will assume that e~(r) = e(r). Macroscopic values of adsorption and adsorption rates for the case of non-dissociating molecules localized on a heterogeneous surface are expressed by the equations (11) in which the local rates of elementary processes including lateral interactions are given by A = K ~ P ( 1 -- 8q) Sq A Uq
D = KqD 8qSq D Uq
(43)
where therate constants K {'D are defined by equation (12), and the s~'D function includes the effect of the non-ideality of the adsorption system on the rate of elementary processes. In the absence of lateral interactions, this function is unity and equation (43) reduces to equation (12). For homogenous surfaces, the equation defining UD transforms into equation (40). Apart from the four factors mentioned above and referring to a homogenous surface, the functions of non-ideality derived for heterogenous surfaces also take account of the surface structure and its effect on the distribution of adsorbed molecules. We will limit our consideration here to the third level of the binary distribution function fqp(r) for the description of this structure. In this case, the non-ideality function derived for desorption rate including interactions of the vicinal neighbours has been derived using different approximations in chapter 4. It may be written as: t
[1 + ~ d~x~8,] =~
x;~ =
exp(fl&~)- 1
(CA)
p=l t
Sq
=
exp[flZq E
dqp&qpOp]
&qp = Qo - eqp
(MFA)
(44)
p--1 t
dqpxqptqp]
t ~ = 2Qp/(A~ + bqp) (QCA)
p--1
A ~ = 1 + x~(1 -- 0q-- 8p)
Xqp = exp(-fleqp) - 1 bqp =
where dqp = fqp/fq characterizes the conditional probability of localization on a q-site at a distance r from a p-type site; tqp characterizes the equilibrium conditional probability (in the quasi-chemical approximation) of localization of an adsorbed molecule on a q-type adsorption site when the p-type is occupied. It was shown in ref. [147] that the disordered approximation is not correct, because in this approximation the symmetry relating to the sequence of selection of molecules of different types is disturbed. For this reason this approximation is presented only to illustrate the fact that the cluster methods developed for a homogenous surface can be adapted fully to a heterogenous surface without loss in correctness. As in the case of homogenous surfaces, the mean field approximation includes the mean energy of interaction of vicinal molecules. In the quasi-chemical approximation, short ranged correlations are also considered. Estimation of the rates of adsorption on the basis of the equations (43) and (44) requires knowledge of the local coverage 8q. These values are obtained from solution of the equations aq(1 -8q)SpSp = ap(1 -8p)SqSq, where Sq includes the effect of the non-ideality of the adsorption system on the coverage of q-type
256 sites in the absence of Sq = 1 interactions (the form of Sq depends on the approximation). For example in the case of the quasi-chemical approximation we have t
Sq = [1 + ~
dqpxqptqp]~q
(45)
p----1
where aq is the Langmuir constant for a q-type site. In following parts of this paper the main results of the complex consideration of the effect of surface heterogeneity and of lateral interactions using the quasi-chemical approximation will be considered.
5.3. Effect of vicinal neighbours. In numerous instances, the expressions for desorption rates derived by considering the conditional distribution functions of clusters and site pairs cover each other and the analysis of the complex effects of the factors affecting the form of the TDC simplifies. Especially in the case of the square lattice (2g=4) such superposition takes place for fl = f2 = 0.5 and for the three following surface structures: a) ordered (chess-board) b) disordered c) patchwise. In the case of this last structure, the contribution of the boundary existing between different homogeneous fragments is neglected. In the first case, each site of the first type is surrounded by sites of the second type and vice versa: d(1 [0.4]) - d12 = d21 = 1; and the remaining functions d(qm) and dqp are equal to zero. In the second case, d(q[mql, rnq2]) = c2q mql f~ql f~q2 and dqp = fp, where c2q mq~ is the number of connections derived from zq in relation to mq~. In the third case dn = d22 = d(l[4,0]) = d(2[0.4]) = 1 and the remaining distribution functions are equal to zero. One of the major problems considered in TPD analysis is the number of peaks obtained. Surface heterogeneity, repulsions between adsorbed molecules, as well as porosity may lead to splitting of the TPD into two peaks. Their common effect may be illustrated by the model calculations presented in Fig. 45 for the three surfaces mentioned above. (Curves 1 correspond to an ideal adsorption system [203].) On the ordered surface (Fig. 45a), an increase of the repulsion between molecules does not change the number of the peaks in comparison to an ideal adsorption system but rather leads to an increase of the height of the low-temperature peak and to a shift of this peak toward lower temperature. Such behaviour also takes place with increase of the differences between Q1 and Q2 on a heterogeneous surface even without considering lateral interactions [127], and on a homogeneous surface where these interactions exist [147]. In the latter case, either the magnitude of the TDC minimum or the widths of the low and high temperature peaks are significantly larger than for the ideal case. Significant qualitative differences are observed for heterogeneous ordered surfaces. With increasing repulsions, the maximum of this peak increases and shifts toward higher temperatures. This is related to the fact that with stronger repulsions, larger site coverage is characterized by lower bond energies compared to that in the absence of lateral interactions at 0 = const in the range of 0 < 0.5. Such behaviour of heterogenous ordered surfaces leads to an "anomalous" dependence of adsorption heat on concentration [49,205]; in the case oflow i coverages and repulsive forces between admolecules, there will exist a range of coverage starting from 00 = 0 in which the adsorption heat increases from Q(0 = 0) to Q1.
257 4
a
dO dT
1
dO dT
1
2 1
,
360
I
300
660
450
450
600 T,K
T,K 1 dO
C
dT
2
tI / 250
.I 400
I
550
T,K Figure 45. TDS for heterogeneous surfaces with ordered (part a), chaotic (part b), and patch-wise (part c) arrangements of different types of sites; t = 2, fl = 0.5, Q1 = 113.4, Q2 = 84 kJ/mol, a = 1, g D~ = 1012 s -1, 80 = 0.99, b = 50 K/s, eqp = e, e~p = e/2;e = 0 (curve 1),-4.2(2), -8.4(3),-12.6(4) kJ/mol [203]. On heterogeneous disordered and patchwise surfaces (Fig.45b and 45c), increasing repulsions between admolecules leads to formation of a more jagged structure for the TDC (an increase in the number of peaks). Similar to the case of ordered surfaces, an increase of the repulsions affects the width of the TDC peak and shifts its initial part toward lower temperature. Simultaneously, the heights of the high temperature peaks decrease but their position remains practically constant. The positions and heights of the low temperature and intermediate peaks depend in the same way on surface heterogeneity and on lateral interactions. On disordered surfaces, the number of peaks for fixed model parameters increases to 4. This fact can be interpreted as meaning that on the different homogenous domains of the surface, there occurs a splitting determined by lateral repulsions between adsorbed particles. Such interpretation must, however, be treated with caution since patchwise surfaces characterized by the same parameters, can produce not 4 but 3 peaks.
258 Assuming, thus, that the height of the intermediate peak exceeds significantly both the low and the high temperature peaks, one can conclude that this is due to some superposition of contributions to the "high temperature" peak for surface fragments characterized by a low energy bond, and the "low temperature" peak for the surface fragments characterized by a high energy bond. In order to clarify such an interpretation one must analyze the dependences of the local contributions dOq/dT and dS/dT. The calculations show that, except for the high temperature region in which 8 < 0.1, the relation dQ2/dT > dQ~/dT holds. That means, that during the whole process, the main stream of the desorbed molecules flows through the sites characterized by a lower bond energy. This permanent "leaking" of the molecules from the sites of the second type is assured by the condition of their equilibrium distribution. For this reason we cannot say anything about the effects of the splitting of the TDC on different fractions of a homogeneous surface. Analogous changes of the number of peaks may also be caused by changes of other molecular properties of an adsorption system. The effect of the surface composition of a disordered surface on the form of the TDC, is shown in Fig. 46 [205]. The decrease of the fraction of strong sites leads to an increase of the number of peaks from 3 to 4.
dO dT
1
290
390
490
590 T Figure 46. The influence of the composition of a heterogeneous surface with a chaotic distribution of sites of different kinds upon the shape of the TDS. E - -8.4 kJ/mol, fl = 0.2(curve 1), 0.5(2), 0.8(3). The other the molecular parameters are the same as in Fig. 45 [205]. Fig.47 shows a TDC picture for the case of attractive interactions between adsorbed molecules [203]. In the case of weak interactions, differences in the sites of different types prevent the molecules on an ordered surface from getting closer because for these molecules it is more comfortable to be localized in the regions situated between second neighbours and thus not to interact with other molecules. However, when the attractions increase, the contribution of the lateral interactions increases. This in turn decreases the splitting of the TDC. At stronger interactions, the form of the TDC can be superposed
259 "structurally" with the TDCs for homogenous surfaces. Similar effects will also be found for other heterogeneous surface structures. dO
dT
4
1.2-
0.8-
0.4-
300
I
I
!
400
500
600
T Figure 47. TDS of attracting molecules on a strictly-ordered surface fl = 0.5, e = 0(curve 1), 4.2(2), 8.4(3), 12.6(4) kcal/mol, with the rest of the parameters taken to be as in Fig. 45 [203]. It should be emphasised that in the general case, the fragment, cluster or pair method of describing the structure of a heterogeneous surface are not equivalent. Different modes of desorption may change the form of the TDC curve. As a simple example, we will consider the surface of a model ordered alloy, assuming, that on the usual "chess-board" (t=2), each fourth series consists of the sites of a type characterized by a lower adsorption energy. Surface composition is represented by fl = 3/8 and f2 = 5/8 functions. Calculations of the TDC for that system (Fig.48) were done using the various methods of description
260 of surface structure. Curve 1 corresponds to the first level method: an elementary cell (fragment) consists of eight sites (see the remark to Fig.48). Such a description of local coverage of any fragment leads to 8 local coverages. Curve 4 corresponds to the pair method of the description of surface structure (dqv functions are then employed). Curves 2 and 3 correspond to cluster descriptions of the surface. For curve 2, the number of types t = 2, as for the pair method of description, whereas for curve 3, it was assumed that the sites situated in the 4th series can be characterized by different coverages. Such situations are due to the fact that different coverages of vicinal sites in the first and third series can, owing to lateral interactions, influence the coverage of the sites localized in the fourth series in different ways. In the last case the number t is equal to 3. (adsorption energies for the sites of first and third type are the same). dO dT 1,3
1 x
20
3 x
40
x
50
6x
70
80
0
x
O
x
O
x
O
x
O
O
O
1,3
I
I
I
I
300
400
500
600
Figure 48. The influence of the structure of a heterogeneous surface on the shape of TDS curves for R = 1, e = -8.4 kJ/mol. Distributed model (curve 1), cluster models (2,3), averaged model (4). The rest of the molecular parameters are as in Fig. 45. The fragment of the surface under consideration is shown in the inset.
261 Curves 1 and 4 show different results; curve 4 does not contain the intermediate peak and there can be seen only a thin structure in the low temperature peak. This is related to an incorrect application of the third level method. Cluster methods occupy an intermediate position between the fragment and the pair methods of description of surface structure. Due to differences in the coverages of the sites characterized by the same adsorption energy (Fig. 3), one obtains an almost exact solution. Such methods of introducing sites of competing types permits one to solve successively for the correct distribution of adsorbed molecules on a heterogeneous surface [202] while simultaneously retaining the small dimensions of the system of type (45). (For curve 3 this dimension is 3 and for curve 1, it is 8.) Thus, the structure of a heterogeneous surface influences significantly the form of the TDC and the number of TDC peaks. This number increases for chemisorption systems (repulsions between neighbouring molecules) and decreases for physical adsorption (attractions between neighbouring molecules). These circumstances must be taken into account in the interpretation of experimental data. The results obtained agree with Monte Carlo calculations [206,207]; the nature of splitting as well as the form of TDC are dependent in typical ways on the surface heterogeneity and on the lateral interactions. In the case where t=2, similar changes are observed in the number of TDC peaks, depending on the sign of the parameter describing the interactions between adsorbed molecules. Note also that in ref. [208], the nature of the TDC splitting is determined only by the term resulting from site heterogeneity (t=2), and the lateral interactions influence the peak width and the position of minima without changing the nature of the splitting. This is related to the application of the disordered approximation while including the lateral interactions because that approximation does not include correlation effects [147]. 5.4. P e a k s p l i t t i n g . Including lateral interactions increases the number of TDC peaks and requires very precise definition of the conditions for full splitting [127]. The problems which appear in the investigation of the conditions for full splitting will be analyzed for the special case of surfaces consisting of two types of sites (t+2). In this case AQ* denotes a value of AQ for which (at AQ < AQ*) corresponds to n = l (lack of splitting) and at AQ > AQ* n/2 (instead of n=2 because of the absence of interactions). Such a definition involves the possibility of the appearance of a "thin" TDC structure. In order to explain this definition, we will consider the TDC for the disordered surface shown in Fig. 49. The increase of AQ (curves 1-3) or the repulsion parameter modulus [ e [(curve 4-6) leads to successive n values equal to 2, 1 4. The splitting conditions are determined by the presence of local minima in the TDC curve because when these minima are not deep, there appears a plateau (curve 3) or weakly outlined (curve 4). Note also that on a homogenous surface in the absence of interactions, only one TDC peak will appear, whereas in the presence of interactions, two peaks may appear which under certain circumstances can be considered as jagged structures. For a homogenous surface the appearance of a local minimum on a TDC curve depends on the e and e* parameters. The values of ~ such that, f o r [ e [<[ ~ [, the local minimum in the TDC curve is absent, are given in Table 3. The importance of the value of e* is also related to the fact that, at e* = e, no splitting occurs (for any e values) on a homogenous surface.
262
dO dT 0.5
3 ..._/_//'kTP~
2
0.1
~'~
~
~"
J'~-~
# #
I
300
I
~th
I
500
700 T,K
Figure 49. TDS for a chaotic surface, t = 2, fl = 0.5, Q1 = 126, E1 = 139 kJ/mol, eqp = e, e~p = e/2, zl = z2 = 4, b = 50 K/s, 00 = 0.99, Q2 = Q1 - AQ. The values of e and AQ are given by the same numbering scheme as that used in Fig. 51 [205].
6 2
3
,
4
6
,.
0
-8.7
-7.3
-6.3
'5.1
0.25
-8.8
-7.4
-6.5
-5.4
-7.0
-5.8
0.50 .
0.75
-8.9 .
.
12.3
-7.7 .
- 10.2
.
.
-8.9
-7. I
Table 3. The condition of complete splitting of TDS on homogeneous surfaces [205]. T h e dependences of AQ*(fl/e) presented in Fig. 50 permit one to determine the effect of surface composition on the conditions for full splitting. These regularities are different for different surface structures. On a regular surface, m u t u a l localization of the sites and repulsions between molecules "occur" in the same direction. This permits one to determine the contribution of molecules desorbed from the sites of different types, decreasing in
263 AQ*, kJ/mol 50.4
33.6
15.8
!
0.5 fl Figure 50. The condition of the complete splitting of TDS curves for various compositions of a regular (curves 1-5), chaotic (6,7), and patch-wise (8,9) surfaces with e =-4.2 (4,6,8)-2.1 (3), 0 (1), 4.2 (5,7,9) kJ/mol [205]. '
'
~
this way the magnitude of AQ*. Increased attractions favours the appearance of vicinal molecules on sites of different types, which leads to an increase of AQ*. Such a trend is observed for any fx value. A more complex dependence for AQ*(fx/e > 0) is connected with competitive contributions to Eq originating from other pairs (11 and 22) of vicinal sites when fl values deviate in both directions from 0.5. On patchwise surfaces, the increase of attractions (repulsions) between admolecules causes ~Q* to decrease (increase). The different nature of the contributions to a TDC curve originating from different sites localized on regular and pathwise surfaces should be emphasized. In the first case the individual peaks are related to isolated sites of different type. (We assume that F1 = 0.5 and d12 = 1.) In the second case, the peaks are related to the contributions of individual macroregions. In the latter case the attractions between molecules favour appearance of admolecules in the "homogenous" phase, and show more precisely the differences between individual macroregions. The repulsion (less than g) promotes the transfer of molecules between vicinal macrophases and causes the suppression of TDC peaks. On a disordered surface, the effect of interactions between admolecules on the full splitting is slight. This, however, does not mean that the form of the TDC is due to the absence of interactions.
264 A more impressive picture of the AQ*(e/fl = 0.5) dependence is presented in Fig. 51. For other ]'1 values these interrelations, are qualitatively similar. With a common effect of surface heterogeneity and of lateral interactions, the splitting of the TDC depends on AQ, e, and on e*. All curves presented in Fig. 51 start from e* typical for a homogenous surface (AQ = 0). Fig. 51 shows the function ?.(ED) (ED is the activation energy of desorption on homogenous surface) at e* = e/2. The simplest situation is realized on regular surfaces: AQ* values increase monotonically with the increase of e, although the form of this function can change slightly depending on fl. In region A there are always two peaks, and in region B, one. 168 I
AQ, kJ/mol
33.6
-
64 I
I
Ed, k J / m o l
-2.1
-6.3
a
^
e
4-/]~"
b
"
A
" 16.8
I
A
~
.
A
B
I
-8.4
7.
"1~
0.0
IB B
" I
~
-8.4
6 C
3.
0.0
7 I
111
I
l
-8.4
l
I
0.0
I
e, kJ/mol Figure 51. The dependence of AQ* upon e on regular (part a), chaotic (b), and patch-wise (c) surfaces. In the region A, the TDS have two peaks. There is one peak in region B (splitting is absent). The coordinates of the points correspond to the parameters AQ and e for the TDS curves in Figs. 49 and 52 with the same numbers. The inset shows the dependence of ~ upon ED [205]. On patchwise and disordered surfaces the situation is more complex. In the case of a patchwise surface, the AQ* curve consists of three easily distinguishable fragments. For a disordered surface, these fragments are less distinguishable. For low AQ, the TDC curves differ slightly from those observed for a homogenous surface in that they have two peaks connected with lateral interactions. The increase of AQ leads to a decrease of the TDC splitting (curves 1 and 2, Fig. 49 and 52). At weak repulsions between admolecules the splitting of the TDC is determined only by surface heterogeneity (curves 5 and 6 in Fig. 49, curve 6 in Fig. 52). In region B there is one maximum (curves 2 and 4, Fig. 49, curves 2 and 5, Fig. 52); such curves are described usually by continous distribution functions defined for different surface compositions. While considering only the form of these functions, it is difficult to draw any conclusion about the number of different surface sites. The analysis presented here shows that for a representative description of the spectrum, a knowledge of the two types of sites assumed in order to estimate the lateral interactions between admolecules is sufficient.
265
d._.~O 0.5
dT
4
0.1-
,
300
I
500
I
I
700
T,K Figure 52. TDS curves for a patch-wise surface. The parameters axe as for Figs. 49 and 51 [205]. Further increase of AQ or c (transition from curve 2 to curve 3 in Figs. 49 and 52, as well as from curve 5 to curve 4 in Fig. 52), leads again to splitting in the TDC. This is connected with the cooperative contribution of both factors (surface heterogeneity and lateral interactions). At the same time these curves can contain typical (or "suspicious") fragments: a "shoulder" plateau (curve 3, Fig. 49) or a high temperature "tail" (curve 3-5, Fig. 52). The TDC curves can have a jagged structure with three or four peaks (curves 7 in Fig. 49 and 52). The true nature of these peaks becomes clear by analysis of the contributions of individual components to the whole stream of desorbed molecules [203]. In particular the differences in the maximum number of peaks on patchwise and on disordered surfaces may be explained by the equilibrium distribution of molecules between sites with strong and weak bonds. [203] 5.5. Effect of m o r e d i s t a n t n e i g h b o u r s . Figs. 53 and 54 show the effect of the second neighbours on the TDC for the case of attractive and repulsive interactions betweeen the adsorbed atoms. In both cases, increasing second neighbour repulsions (curves 3, 5, 9, 11, Fig. 53, curves 4,5 Fig. 54) leads to a shift of the initial part of the spectrum toward lower temperature (plus an increase of the width of the TDC) and to decreases of the peak maxima. At the same time, for disordered surfaces characterized by c(1) < 0, the low temperature (fourth) peak is more distinct (curve 9, Fig. 53). An increasing second neighbour attraction leads to a decrease of the width of the TDC, and to an increase of the spectral maxima. The effects due to
266
dO
12
~Y 1.6 5
61
8
2 11 ~ "
4
3
400
5
1
~
600
400
600
T,K Figure 53. The influence of second neighbor interactions upon the TDS for patch-wise (1-6) and chaotic (7-12) surfaces with a = 1, t = 2, fl = 0.5, zl = z2 = 4, b = 50 K/s, 00 = 0.99, g D~ = 1012 s - l , Q1 -- 113.4, Q2 = 84, E, = 126 kJ/mol, eqp(r) = e(r), e;~(r) = e(r)/2, e(1) = -8.4 (curves 1,3,4,7,9,10) and 8.4 (2,5,6,8,11,12) kJ/mol, 7 = e(2)/e(1) = -1/3 (4,5,10,11), 0 (1,2,7,8), and 1/3 (3,6,9,12)[209]. interactions with second neighbors were considered at I" ~ I< 1/3 where 7 = e(2)/e(1). This is probably reliable for most real adsorption systems, because with increasing distance, the interactions between adsorbed atoms decrease. It was concluded that the nature of the TDC splitting is determined mainly by the interactions with nearest neighbours, but the positions and heights of the maxima depend strongly on the interactions with the second neighbors. Similar conclusions concerning the role of the second neighbours were drawn earlier for homogenous surfaces [168]. Nevertheless situations are also possible where interactions with second neighbors lead to some qualitative changes such as change of number of peaks for values of the parameters close to those obtained for the abovementioned conditions for full TDC splitting [127]. In Fig. 54 one can see the TDC curves corresponding to different values of the Br<nsted-Temkin parameter (curve 7 and 8) which characterizes the changes of the activation energy of desorption from different types of sites due to the changes of their bond energy. If a < 1, the changes in the desorption activation energy are smaller than the changes of bond energy. In other words the lower a, the lower the difference between this and the activation energy of the strongest sites. That means an increase of the activation energy in comparison to the case where a = 1 which leads to a shift of the whole spectrum toward higher temperatures. The other conclusions
267
dO dT
6
_
3 7\
1 300
600
Figure 54. The influence of second neighbor interactions upon the TDS shape for the ordered surface with fl = 0.5, e(1)= -8.4 (curves 1,3,4,7) and 8.4 (2,5,6,8) kJ/mol, a = 0.5 (1-6) and 1.0 (7,8); 7 = -1/3 (1,2,7,8) and 1/3 (4,6). The rest of the parameters as in Fig. 53 [209]. about the role of the composition and structure of the surface, as well as of the lateral interactions and heating rate, remain practically unchanged. The effect of the anisotropy of the interaction with first neighbours and of the averaged desorption due to interactions with second and more distant neighbours were obtained using a simpler version of the R approximation according to which ~qp = ~=R1+1 zq(r)dQp(r)eqp(r)/zq(1). Here R1 is the distance at which the interactions are considered in the quasi-chemical approximation ; parameter used during the exchange of %p(r) for e;p(r)0 shown (the same form has a e~p in Fig. 55. Such an analysis was made for real surface structure in the description of experimental data obtained for K on Si(100)-2xl [210-212]. The heterogeneity of the surface of the Si(100)-2xl edge is caused by the dimerization of surface silicon atoms. This leads to the formation of two types of sites shifted with respect to the other by a lattice semiperiod [210]: fl = f2 = 0.5 m11(1) = m22(1) = 2, m21(1) = 4 (cluster distribution functions); zx (1) = z2(1) = 6. The variations of the energetic parameters were considered for curve 1. The parameters of this curve were selected on the basis of experimental TDC curves
268
dO
1.4
d-T
1
2
3
0
620
860
T,K
Figure 55. The influence of the anisotropy of the parameters of the nearest neighbor interactions (curves 2-5) and of the averaged description of distant neighbors upon the shape of the TDS. The parameters with respect to those of curve 1 are :2 - e~l(1) = -6.3, 3 - e22(1) = -12.4, 4 el2(1) --4.2, 5 - el2(1) - - -8.4, 6 - (eqp - f.qp) 0.7, 7 - (~qp - f.qp) 2.3 (all energies in kJ/mol) [209]. -~
:
obtained for different starting coverages [211] (Fig. 56): K ~ = 1013c-1, Qo = 0.99 b = 7 . 7 K / c E 1 = 2 3 5 , 2 E 2 = 2 0 7 . 9 , A Q = 2 7 . 3 e 1 1 ( 1 ) = - 1 0 . 5 e~1(1)=8.4 e~2(1)= e2~(1) = -12.6, e~2(1) = e;,(1) = -6.4 e22(1) = ei~(1) = 0, e ; p - eqp = 4.6 for all qp pairs (all quantities are expressed in kJ/mole). The increase of repulsions between adsorbed atoms localized in the series consisting of the sites characterized by a strong bond (Fig. 55, curves 1 and 2) at az~ = const, where aqv = e;p(1)/eqp(1) leads to a significant increase of the width of the high temperature part of the TDC, and to a slightly less significant increase of the width of the whole spectrum. However, the minimum separating the low and high temperature parts of the spectrum decreases. An increase of the repulsion between atoms localized in the series of sites characterized by a weak bond (Fig. 55, curves 1 and 3) at a22 = const does not cause any changes in the high temperature part, and leads to spliting and expansion of the low temperature part of the spectrum. Increasing repulsion between adsorbed atoms localized on vicinal sites (Fig. 55, curves 1 and 4) at Dd12 = const increases the total width of the central part of the spectrum. One can assume that each eqp(1) parameter is "responsible" for its TDC part. The role
269 of these parameters is determined by the nature of the surface s t r u c t u r e - in this case all contributions are mutually commensurate. In general, the variation of each parameter of the anisotropic interaction %p(1) and r causes a partial change of the TDC curve analogous to the changes caused by these parameters in the case of a homogenous surface [147]. The effect of the averaged interactions of more distant neighbours is illustrated by the curves 1,6,7 in Fig. 55. An increase of their contribution broadens the TDC uniformly but does not introduce qualitative changes. This results from the specificity of the molecular field approximation [47]. Its use on a homogenous surface does not lead to spectrum splitting, although increasing the repulsions does cause anincrease of the TDC width. dO dT
I
!
I \
I
!
\ / /
J 3 ~, / /
2,,/
I
350
I
I
550
I
]
750
950
T Figure 56. The TDS curves for K on Si(100)(2xl). Broken lines are experimental data [211] for 00= 0.03 (curve 1), 0.11 (2), 0.32 (3), 0.56 (4), 0.90 (5) and 0.99 (6)[209]. Fig. 56 illustrates the correlation of experimental and calculated TDC curves. A model taking account of the anisotropy of interactions between atoms localized on vicinal sites and the averaged description of the interactions between more distant neighbours gives satisfactory correlations. It reflects the basic relations between the curves near their maxima and minima, the width of individual parts of whole spectrum, and the presence of an inflection at 0o ~ 0.5. The discussion of the "resistant" curve 1 was not included in this paper because of the necessity of verification of other hypotheses: the possibility of a change
270 in the aggregate state of the potassium atoms as well as the possibility of reconstructing the silica surface owing to adsorption of potassium atoms. Neverthless, the parameter analysis permits one to conclude that an adsorption model for K on (100)Si-2xl [212] that accounts for the existence of the anisotropic structure with one type of site is not realistic because it does not permit one to arrive even at a qualitatively correct description of the experimental data [211].
5.6. Dissociative adsorption. Dissociative adsorption occurs on two vicinal surface sites. To describe the rates of the elementary processes in the absence of lateral interactions on homogenous surfaces, one must pass from the distribution function fq (or f(x) ) to fqp (or f(x,y) ), functions characterized by larger dimensions. The same increase of dimensions of the distribution functions is necessary for description of this process on heterogeneous surfaces when lateral interactions are taken into account. (This increase takes place in the first level model). In the cluster description of surface structure, the f(q{m}R) with one central site of q-type and R coordination shell corresponding to this site, is replaced by the f(qpAR) function with two central sites and R coordination shells [44,45]. In the same way the binary Probable location of hydrogen on Ir (110)-(1 x2)
dO
rx
dT
1/ ~ , \
b
a
(
)
(
~
[
,'ll
,'1_,1
,,'1,7
,t
,'/ /
',~/I ." I ./ 200
il
-;'/I
.Y" J
~"~
300
z
-,\/
l
Z"
!
Top view
,Y
'\
~~.
)
~'~
.
400
Sideview
I
500 T,K
Figure 57. (Part a) Probable locations of hydrogen on Ir(110)2x1. Part (b) shows the TDS curves for H2on Ir(l10)2xl, calculated for various initial coverages [214]. Broken lines- experiment [213] with e0 = 0.99 (curve 1), 0.86 (2), 0.61 (3), 0.48 (4), 0.37 (5), 0.27 (6), 0.068 (7). It was assumed that adsorption occurs at position B (sites with strong adsorption bonding) and C (sites with weak bonding).
271 distribution function fqp(r) is replaced by the triplet distribution function fqpo(rl,r2) in which rl and r2 are the distances between vicinal r/ site and central q and p sites respectively. These distribution functions were discussed in detail in [40,44,45,49]. An example of an application of the theory taking into account both lateral interactions and surface heterogeneity is the TDC for D 2 o n ( l l O ) I r - 2xl described in [213]. The elementary lattice consists of three sites (Fig.57a) - one "strong" (site of the first type) and two "weak" (sites of the second and third type t=3). The sites of the third type are separated only formally for an easier presentation of the distribution function - the molecular characteristics of these sites are the same as those of the sites of second type. The lateral interactions considered are limited to three coordination shells R = 3, and the related distribution functions have the form: fq = 1/3 Zq(r) = 4 li < q < t 1 < r < R [214]. In order to present kqAA, a multiplicative representation is used, leading to the conclusion that e;p(r) = e;q(r) and %p(r) = %q(r). It implies also equality of the parameters relating to the sites of the second and third type. All parameters related to lateral interactions for the third coordination shell are assumed to be zero except e11(3) and e~1(3). Calculations show that the presence of these two factors is sufficient to produce first order desorption for the high temperature peak (see below). Fig.57 shows a comparison of the experimental and the calculated TDC curves. The calculations were done using"/~AA = 2-10~3c -~, E~ A = 71,4; AQ = 17,2 k J / m o l {ElA - E A } / A Q = 0.8 [fA/KIA = 0.07, en(1) = e22(1) = e23(1) = --2.1 e12(1) = --0.2 e12(2) = e 2 2 ( 2 ) - - - 0.2 e11(3) = 2.1 a11(1) = 0.9, a23(1) = 0.4 a~2(1) = 0.5; eqv(r) and %p(r) not mentioned here are equal to zero (all energy quantities are expressed in kJ/mol), hap(r) = eqp(r)/%p(r) q,p = 1,2,3). The curves presented in Fig. 57 show reasonable agreement. This model describes all the pecularities of the experimental TDC: splittings of the peaks, correlations between the intensities and the areas under the low and high temperature maxima, the nature of the peak shifts along the temperature axis due to increases of the initial coverage, the broadening of the low temperature maxima, etc. The features related to the splittings of the peaks are determined by the heterogeneity of the (ll0)Ir 2xl edge and the features of the self peak are determined by the lateral interactions. Among these features one should emphasize a typical broadening of the low temperature peak as well as the fact that the position of the low temperature maximum is shifted to lower temperature (second order desorption) whereas the position of the maximum of the high temperature peak does not change (first order desorption). This suggests the existence of strong repulsions between adsorbed atoms locahzed on "weak" sites and a strong attraction between atoms on "strong" sites. Because the "strong" sites are isolated, the attraction between adsorbed atoms localized on these sites involves only the third coordination shell. This also follows from the value used for the interaction parameter e11(3). (For this reason the contributions Q1(3)and e~1(3)) are separated.)It should be noted that the authors reporting the experimental TDC (Ibbotson, Wittring and Weinberg [213]) have commented upon these curiosities of peak evolution and have drawn some conclusions about the the lateral interactions. However, they have not arrived at any quantitative estimates.
5.7. Continuous distribution functions. With many types of sites localized on a heterogeneous surface, macroscopic rates of adsorption processes can be studied using continuous distribution fonctions instead of
272 discrete ones. In this case, when estimating the lateral interactions it is reasonable to limit consideration to models of the third level. Fig.58 shows the effect of lateral interactions on the form of the TDC for non-dissociating adsorbed molecules on surfaces with different distribution functions f ( x ) . Increasng repulsion causes an increase of the spectral width and a decrease of the maximum height. At the same time, the maximum shifts toward lower temperatures. The degree of such changes depends on the ratio of the number of the sites characterized by higher (~ x2) and lower (.-- zl) adsorption energies. In the case of an exponential distribution function, the shift of position of the maximum for 7 > 0 is significantly less than for 7 < 0. Lateral interactions also can change the form of the TDC. Uniform and Gaussian distribution functions give, in the absence of lateral interactions, different forms for the TDC, but including repulsive interactions between the adsorbed molecules leads to TDC curves which become similar (curve 3). 0.29
0.33
dO
dO
dT
T 220 0.50 -
3 89
4 89
5 89
T
C
420 0.70
520
620
d
de
dO dT
dT
220
320
420
520
T
2:20
320
420
520
Figure 58. The influence of repulsion between nearest neighbor admolecules on the shape of TDS for heterogeneous surfaces with chaotic distribution of sites; f ( z ) - uniform (a), symmetric Gaussian (b), exponential 7 = 1/3 (c) and 7 = -1/3 (d); e = 0 (1),-0.5 (2) and-1.0 (3) kcal/mol [214]. Continuous distribution functions were used to describe the desorption of C O molecules from the surface of polycrystalline Rh [215]. The authors of [215] concluded that their TDC curves are similar to those determind for (111) and (11) edges [216,217]. The analysis of these experimental data has shown [218] that the process of desorption from the polycrystalline surface is not an exact superposition of desorption processes from the above-mentioned edges. The results of the description of the TDC curves for individual edges and for the whole polycrystalline surface are presented in Fig. 59. Among the most important factors taken into account in that study was the complete saturation of all
273 surfaces considered in the experimental work. The questionable point is that traditional TDC analysis deals usually with the position of peak maxima and the magnitude of the activation energy of the process connected with Tp, as well as with the estimation of the desorption order. In a given experimental system, the spectra cannot be described by assuming an ideally homogenous surface (either with or without consideration of lateral interactions) because of the broadening of both ends of the peak, especially at low temperature. For this reason, it was even necessary to use continuous distribution functions to describe the desorption from the (111) edge. DTC curves determined for both edges are characterized by commensurable intervals of different desorption activation energies. Continous Gaussian distribution functions were employed also for the C O / M g O system in a Monte Carlo study. The resulting TDC for this system is shown in Fig. 60. The spectra [219] consist of two peaks. The differences in the surface composition are due to different temperatures of sample treatment (500 and 1000~ The data deduced from isotope e-
d_o
a
dO dT
dT 1 II
300
!
!
400
500
i
T,K
300
400
500T, K
dO c
, ,/ , , / , ,/, 300
400
500
600 T,K
Figure 59. TDS curves for CO/Rh [218], solid lines - calculation, broken lines - experiment. (Part a) (110) face with z = 4, b = 50 K/s, Q = 31.2 kcal/mol, g ~ = 1013 s -1, c o o - c o = -2.5 kcal/mol, ebo_c o = O.14eco-co; 80 = 0.50 (curve 1), 0.24 (2), 0.15 (3), 0.07 (4), 0.02 (5); characteristics of the heterogeneous face with f ( x ) = exp(0.bx), Ax = 8 kcal/mol. (Part b) (111) face: z = 6, b = 5.7 K/s, Q = 32.5 kcal/mol, K~ = 4.1013 s -1, e c o - c o = -2.5 kcal/mol, ebo_co = O.14eco-co; 80 = 0.46 (curve 1), 0.36 (2), 0.18 (3), 0.09 (4), 0.046 (5); f ( z ) = exp(0.1z), Ax = 3 kcal/mol. (Part c) A polycrystal described as a superposition of (110) (fraction 0.2) and (111) face (fraction 0.77), and an additional peak with energy Q = 38 kcal/mol with the same values of the parameters: K~9 = 4- 1013 s -1, e c o - c o = -1.8 kcal/mol, e~o_co = 0.25eco-co; 8o = 0.47 (curve 1), 0.36 (2), 0.22 (3), 0.15 (4), 0.09 (5), 0.055 (6).
274 xchange of C 13 have shown that adsorbed molecules localized on strong and weak sites do not exchange one for other (essentially no migration on the surface). The Monte Carlo calculations reproduce totally the experimental TDC curves. For the samples treated at 500~ the amount of strong sites is 95% and for those treated at 1000~ - 33%. a
Q 350
450
550
T, K I
350
450
550
T,K
Figure 60. Experimental data (dots) and model predictions (lines) for the desorption of CO from MgO outgassed at 500~ (part a) and 1000~ (part b) [207]. The mean adsorption energies for a gaussian site distribution are E1 = 42, E2 = 25.5 kcal/mol, g D~ = 9-109 s-1, g D~ = 1014 s -1 for both surfaces. The energy dispersion for each kind of site is (part a) al = 2.19 and a2 = 1.48 kcal/mol (500~ and (part b) a~ = 1.1 and a2 = 1 kcal/mol (1000~ While considering the application of continuous distribution functions we should mention the simplified description of the kinetics of adsorption taking lateral interactions into account. If one introduces certain assumptions about the molecular properties, it becomes possible to arrive at relatively simple analytical expressions for the rates of elementary processes containing both evaluated factors. We will limit here ourselves to the application of the molecular field approximation and will consider disordered surfaces, assuming simultanously that the lateral interactions do not depend on the type of sites on which adsorbed molecules are localized. As a result of such simplifications, expressions are obtained [43] assuming that the contributions are additive. For example, the formula which is used for the rate of adsorption of non-dissociating molecules is written in the form UA = UA(r = O) exp[~ze*Q], where UA(e 0) is the adsorption rate for an ideal adsorption system described by equation (17) in which al,~ coefficients must be replaced by a12(0) = aoezp[~(xl,2 + z E/9]. In the same way, the form of the adsorption isotherms (18) changes [220]. The expressions for the adsorption rates are also simplified. Neverthless, discussion is given in ref. [43,40] of the combined effect of surface heterogeneity and of lateral interactions. =
6. C O N C L U S I O N S . An advanced theory taking into account both the heterogeneity of the surface and the lateral interactions between adsorbed molecules allows one to define the basic physical factors characterizing adsorption and creates possibilities of either a quantitative description of the kinetics of adsorption or of a molecular interpretation of some selected parameters.
275 Estimation of thermal desorption spectra on the basis of molecular models allows one to explain numerous rules of TDC i.e the nature of the peak splitting, the characteristics of peak shifts due to change of surface properties, the change of the desorption order with change of starting coverage etc. Computer studies increase the possibilities of modelling real processes in comparison to traditional theoretical methods and make it possible to clarify the basic physical factors of the system under study. Cluster methods have permitted the development of theory that reflects numerous factors of adsorption systems. Complex investigations of molecular models taking into account molecular properties of real adsorption systems are still at an initial stage. This review concerns only the kinetics of adsorption of individual components. Adsorption kinetics of a mixture of molecules on a heterogeneous surface taking account of lateral interactions will also be elaborated and presented in a future paper. Reconstruction of the surface and the effect of mobility of adsorbed molecules on the kinetics of surface processes will also be considered. Detailed studies of the role of all of these factors will allow a better understanding of the nature of these processes as well as eventually producing their quantitative description.
Acknowledgement Athour wishes to thank Profs. W.Rudzinski, W.A.Steele and V.A.Bakaev for preparing the English version of this manuscript.
276 References
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W. Rudzifiski, W.A. Steele and G. Zgrablich (Eds.) Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces Studies in Surface Science and Catalysis, Vol. 104 9 1997 Elsevier Science B.V. All rights reserved.
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S T A T I S T I C A L R A T E T H E O R Y AND THE M A T E R I A L P R O P E R T I E S C O N T R O L L I N G ADSORPTION KINETICS ON W E L L DEFINED SURFACES J. A. W. Elliott and C. A. Ward Thermodynamics and Kinetics Laboratory, Department of Mechanical Engineering, University of Toronto, Toronto, Canada M5S 1A4 The Statistical Rate Theory approach for predicting the rate of kinetic processes is developed from a simple quantum mechanical model of an isolated, thermodynamic system. The approach leads to an interpretation of irreversibility and to an expression for the rate in terms of molecular or material properties of the system. The approach is evaluated by examining the predictions that follow when it is used to examine two types of reactions: those occurring homogeneously within one phase and those occurring when a non-dissociating, diatomic molecule is exposed to a single crystal solid surface with one type of binding site. In the former case, the material properties are known. In the latter, it is found that they can be evaluated from the equilibrium adsorption isotherms. As evidenced by the agreement between the predictions and the measurements for both types of reactions, the expression for the rates obtained from Statistical Rate Theory appears to contain explicitly the concentration dependence. The adsorption kinetics of CO-Ni(111) was the adsorption system considered. When examined with Absolute Rate Theory, it was found that, as with many other systems, agreement between the measurements and the predictions can not be obtained without introducing coverage dependence into the rate expression through the introduction of the concepts of precursor states and of a coverage dependent sticking probability. Since the Statistical Rate Theory expression for the adsorption kinetics does not require the concepts of precursor states or of sticking probability, but nonetheless leads to the correct prediction of the adsorption kinetics for this system, it is not clear that these concepts need to be introduced. Other systems need to be examined before this possibility could be established.
1.0. I N T R O D U C T I O N When a molecule coming from a gas phase adsorbs on a solid surface, it must both cross a gas phase boundary and form an adsorptive bond as it encounters the solid. Similarly, when a molecule desorbs it must break the adsorptive bond and enter the gas phase. Although the collision of the molecule with the solid surface can be treated using either classical or quantum mechanics, the prediction of the rate of bond formation or breaking requires a quantum mechanical description because this process fundamentally involves the probability of the event occurring. The theoretical approach with which we are primarily concerned, Statistical Rate Theory, is based on quantum mechanics and thermodynamics. This theory was originally proposed in 1977 [1] and was published in its more current form in 1982 [2] and 1983 [3]. It has been used to predict the rate of gas absorption at liquid-gas interfaces. Its application to that rate process essentially involved predicting the net rate at which gas molecules cross the liquid-gas interface - a process similar to the first step in the adsorption process. The expression obtained for this rate from Statistical Rate Theory allowed certain anomalies in the reported values of diffusion coefficients in liquids ("surface resistance") to be resolved [4-6]. In the case of gas
286 absorption at a liquid-gas interface, it was found that the complete rate expression could be predicted in terms of molecular and material properties. The formation or breaking of a bond between an adsorbed molecule and the solid surface is the more complex of the processes involved in adsorption-resorption kinetics. It is basically a chemical reaction, and the Statistical Rate Theory approach leads to the definition of a new material property, the cross-section for adsorptive bond formation, that describes both the formation and breaking of bonds. Herein, we concentrate on the application of Statistical Rate Theory to the bond breaking or formation process, and examine the predictions that follow from this theory by using it to examine two types of "chemical reactions". The first chemical reaction considered is an electron transfer reaction between ionic isotopes dissolved in electrolytes. This case was chosen because it allows us to examine the theory in a form that requires a minimum of assumptions. The derivation of the rate equations makes use of the concept of an isolated system. Since there is no heat of reaction associated with electron transfer reactions between isotopes in solution, such intensive properties as temperature and pressure are constant throughout the reaction period, and since the reaction is between isotopes in solution, the activity coefficients do not appear in the expression for the reaction rate. These characteristics of electron transfer reactions between isotopes also lead to the value of the equilibrium constant being unity. As a result of these simplifications, the rate equation does not contain any unknown material properties; thus when the theory is compared with the data, a very "pure" form of the theory can be examined because questions associated with evaluating the material properties do not arise. Further, the structure of the equations is simple enough that some of the fundamental assumptions of Statistical Rate Theory can be examined quantitatively. In the next case that we consider, Statistical Rate Theory is applied to predict the net rate of molecular adsorption on single crystal solid surfaces. The evaluation of the material properties appearing in the rate equations is a major consideration. A second consideration is that the adsorption and desorption processes are assumed to take place isothermally. This requires that the interaction of the system with a reservoir be taken into account so that the total system considered is again an isolated system. A new approach for taking the interaction with the reservoir into account is presented. The final equations that are obtained from the Statistical Rate Theory approach for predicting the net rate of adsorption are the same as those that have been used in the past [7-10], but one major assumption is removed from their derivation. The objective of the Statistical Rate Theory approach is to obtain the explicit coverage dependence of the rate expression so that the other properties appearing in the rate expression are molecular and material properties. If successful, this approach would lead to a very different conceptual circumstance than has been obtained from the Absolute Rate Theory approach [11, 12]. It has been found by comparing the predictions from Absolute Rate Theory with experiments that the explicit coverage dependence is not obtained from the theory. This has led to the introduction of the concept of precursor states [ 13]. Precursor states are then used with either the Absolute Rate Theory approach [ 14] or with the sticking probability concept [ 15, 16] to try to derive the correct coverage dependence. As a first step toward determining whether it is necessary to introduce the concept of precursor states, we apply Statistical Rate Theory to a system that others [17] have reported not to be described by Absolute Rate Theory without empirically introducing coverage dependence. The system chosen for this purpose was CO-Ni(111). In addition to it having been studied using Absolute Rate Theory, the system has been subjected to other experimental studies and these can be used to establish the material properties for this system.
287
1.1. Absolute rate theory approach to adsorption kinetics The traditional theory used to describe adsorption kinetics is Absolute Rate Theory which is based on a particular model of reactions. This model involves an activated state [11, 12] and is based on the premise that in order for a molecule to take part in a reaction, it must first form an "activated complex". This requires the input of energy, the activation energy. It is assumed that molecules in the activated state are in equilibrium with the reactants and the rate of formation of products is calculated as the rate at which molecules pass through the activated state. The assumption of equilibrium between the reactants and molecules in the activated state and is one of the fundamental assumptions of Absolute Rate Theory, and its validity has never been established. In the case of the adsorption process, the reactant is a gas phase molecule which is approaching a vacant site on the surface. In order to become adsorbed, the molecule passes through a transition state which will be located approximately at the position of maximum potential energy on the path to adsorption. The activation energy for adsorption, E a, is thus defined as the energy difference between a free gas molecule and the molecule in the transition or activated state. Clark [12] has clearly described the development of the Absolute Rate Theory expression for the adsorption rate: (1.1,1) where k is the Boltzmann constant and T is the temperature; v is the rate of molecular collision with the surface; f(O) defines the fraction of surface that is available for adsorption and is, as yet, an unspecified function of the relative coverage, O" exp]-Ea/kT[ is the fraction of molecules striking the surface that have an energy greater J the product v f (.-0 ) e x p[- E a / k T ] is. the .rate at which molecules with thanL.E a . Thus sufficient energy to adsorb strike vacant adsorptxon sites, and a , the "condensation coefficient" [ 12], is the fraction of those meeting this condition that actually adsorb. The collision frequency, v, is given by v
P = ~ 42mnkT
(1.1,2)
where P is the pressure in the gas and m is the mass of an individual gas molecule. The relative coverage, 0, is expressed as a fraction of the coverage at saturation, i.e., ~= N a
Mr
(1.1,3)
where N ais the number of adsorbed molecules and M T is the total number of adsorption sites. For desorption, it is assumed that the desorbing molecule must pass through an activated state before desorbing. The activation energy for desorption, E d, is defined as the difference in energy between the adsorbed molecules and the activated state for desorption (see Fig. 1.1, 1), and it is assumed that the adsorbed molecules are in equilibrium with the molecules in the activated state. The expression for the desorption rate that is provided by Absolute Rate Theory is [12] (1.1,4)
288 where K d is called the desorption rate coefficient and g(0)is another unspecified function of coverage.
m
--
transition or activated state
I--4
r O
initial state (adsorbed molecules)
Figure 1.1, 1. Schematic diagram showing the activation energy for desorption. The allowable energy levels are shown for the initial and transition states. The activation energy is shown as the difference between the ground state energy level for the adsorbed molecule and the ground state energy level for a molecule in the transition state. A similar diagram can be drawn to represent E a. It should be noted that Absolute Rate Theory does not lead to an expression in terms of coverage for the condensation coefficient a , for the activation energy E,,, for the desorption energy E a, or for the desorption rate constant Kd . Nor is there any reason from Absolute Rate Theory to think that these are material properties and thus independent of coverage. Many experimental studies have indicated that additional coverage dependence must be added to the Absolute Rate Theory equations. This has affected the conceptual understanding of adsorption kinetics and has led to the introduction of concepts such as precursor states that may not be necessary.
1.2. Incorporation of the coverage dependence in the absolute rate theory equations through the assumed existence of precursor states Several approaches have been used to formulate the additional coverage dependence that is required by the Absolute Rate Theory equations. One approach has been empirically based. Coverage dependence is simply introduced into the activation energies and into the pre-exponential factors as required to obtain agreement between the measurements and the predictions. Another has been to introduce the concept of precursor states. For example, Gorte and Schmidt [ 14] used the idea of precursor states to derive an additional coverage dependence in the desorption term. They assumed that adsorption was a two step process in which the molecules first formed a weakly bound precursor phase before proceeding to the adsorbed phase. By evaluating desorption rates in a regime where the number of molecules in the precursor phase was constant (valid when the number of molecules in the precursor phase is small), and assuming desorption to the precursor phase to be the rate determining step (i.e., the rate from the precursor state to the gas is rapid), Gorte and Schmidt found that the desorption rate could be written
289
Jdes = k*kd ka
0 _
(1.2, 1)
1-0
where k*, k a and k a are the rate coefficients for the transition from the activated state to the gas, the adsorbed state to the activated state and the activated state to the adsorbed state respectively. The rate coefficients are made up of exponentials of activation energies and rate constants for the different processes. Gorte and Schmidt note that the only evidence for the existence of the precursor is that the above equation better describes experimental desorption results. It will be shown below that Statistical Rate Theory leads to the factor 0/(1 - 0) in the expression for the desorption rate, but it is obtained from the theoretical approach without introducing the concept of precursor states.
1.3. Precursor states and the sticking probability Kisliuk [15] also applied the idea of precursor states to derive a coverage dependent equation for the sticking probability which he defines as the number of molecules being adsorbed divided by the number of molecules striking the surface. Kisliuk considers the probabilities for a molecule to arrive at the surface above either an empty or filled adsorption site. There is then some probability for the molecule to be adsorbed on an empty site, evaporate from a filled site or move on to a second site. Kisliuk assumes that the probability of the second site being filled is independent of the state of the first site. The sticking probability is then found as the sum of the probabilities of adsorption on the first, second, third, etc. sites that are visited by a molecule. The equation Kisliuk found
was _ s so
1+
/ -1 K
(1.3, 1)
1- 0
where s is the sticking probability, s o is the sticking probability at the initial time, and K is yet a new parameter. Note that if K equals unity, the sticking probability reduces to the Langrnuirian form s=s0(1-0 )
(1.3,2)
King [ 16] considered two types of precursor states: one type was proposed to exist over a filled adsorption site and the other type over an unfilled site. He obtained a more complicated expression for the sticking probability that contains parameters that can depend on coverage. There is no way to derive the coverage dependence of these parameters from the theory he proposed [13, 16].
2.0. P R O B A B I L I T Y O F A T R A N S I T I O N IN " P A R T I C L E " C O N F I G U R A T I O N IN AN I S O L A T E D S Y S T E M In the Statistical Rate Theory approach, the expression for the rate of a particular process is obtained from the transition probability concept as defined in quantum mechanics. It has been previously shown that the expression for the transition probability of interest can be obtained from a first-order perturbation analysis of the Schr6dinger equation [2, 3]. In this section, we show that the same expression for the transition probability can be obtained from a simple physical argument.
290 To implement the Statistical Rate Theory approach, our first objective is to predict the rate of "particle" (molecule, atom or electron) transitions in an isolated system. The problem can be stated in terms of the particle configuration, A,j, that exists in the system at a particular instant. The particle configuration of the system defines the number of particles of each species that are present in each phase of the system and their spatial distribution within each phase. The phases considered may be either bulk or surface. Herein, we shall restrict our attention to cases where the spatial gradients within each phase are negligible. Consider an isolated system at an instant in time during its evolution toward equilibrium. Suppose that the particle distribution is known at that time and the system energy at that instant is known to within the energy uncertainty. Although the possible quantum mechanical energy levels for the system are degenerate, they are discrete in an isolated system. The restriction that the system energy lies within the energy uncertainty effectively limits the range of quantum mechanical states to which the system has access. This is indicated schematically in Fig. 2.0, 1, where the number of quantum mechanical states available when the system is in a particular particle distribution A,j is denoted as ~(~,j).
Quantum States of Z~
Quantum States of Z i
Quantum States of 2k
E
K[au,a,] J_ ~ f~( Z~)
] K[au,av] f~( Z j)
f~( Zk )
Figure 2.0, 1. Schematic of quantum mechanical states within a thermodynamically isolated system. The particle distribution within the system at an instant is indicated by ~,j. The virtual distributions that would result from the transfer of a single particle are ini:licated by ~'k and A,i, the total number of states corresponding to a given distribution by f~(&), the rates of transfer from one quantum mechanical state to another by K [ a u, a v ] and K[ a u, a t ], and the energy uncertainty by AE. We now introduce the probability, K[a(A,i)u,a(Ak)v], of a transition at any instant between the quantum mechanical state a ( 2 i ) u of the distribution A,i and the quantum mechanical state a(A k)v of the distribution ~,k where both quantum ~echanical states lie within the energy uncertainty. The numbers of states corresponding to these two particle distributions are denoted as f~(A,j) and s k); thus 1 < u < f~(~,j)
(2.0, 1)
and 1 < v < ~(&k)
(2.0, 2)
291 We now make two assumptions. The first is that if at any instant the quantum mechanical state of an isolated system is measured and its energy is found to be within the uncertainty AE, the system is as likely to be found in any one state as in any other. Except for its context, this assumption is not very different from one of the standard assumptions of statistical mechanics that was originally introduced by Gibbs in his study of equilibrium statistical mechanics [18]. When the assumption introduced by Gibbs was proposed, it was based on a system description that used classical mechanics. However, its equivalent has become a standard part of statistical mechanics that is based on quantum mechanics. It should be noted that we make this assumption for a system that is not in thermodynamic equilibrium. The second assumption, and one that is central to Statistical Rate Theory, is based on the probability of a transition between any two quantum mechanical states that lie within the energy uncertainty of the isolated system, but are associated with different particle distributions. We assume this rate, denoted as K[oc(Xj)u,O~(A,k)v],has the same value for all combinations of quantum mechanical states within the energy uncertainty range of the isolated system. We take its value to be Ke:
K[ot(A,r)u, Ot(Zs)vl = Ke for 1 _
(2.0, 3)
and for 1 < v < f2(& k)
We now wish to use these two assumptions to determine the expression for the probability of a transition from one particle configuration to another, independently of the system being in any particular quantum state initially. Suppose it is known that at any particular instant the particle distribution in an isolated system is ~,j and that the number of states within the energy uncertainty is f~(A,j), then the probability of measuring the state of the system and finding that it is in any of these states is (f2(A,j)) -1 . The product of this probability with the probability of the system making a transition from one quantum mechanical state to another would be the probability, P(&j; Otu), of a system making a transition from the particle distribution ~j tO any quantum mechanical state a u that lies within the energy uncertainty but corresponds to a different particle distribution. =
(2.0, 4)
However, of more interest to us is the probability of a transition from particle distribution ~,j to particle distribution A,k at any instant, z(2j, Ak), and since there are s k) quantum mechanical states corresponding to Ak, this latter transition probability would be given by
292 At the same instant that there is a probability of a transition from ~, to ~k there is a probability of a particle transition in the opposite sense, say to &i, (see Fxg. 2.0, 1) and the probability for a transition to this latter configuration from /],j would be given by
~(/]'i)
"c(/],j,/],i)= K e ~(/]'J)
(2.0,6)
Before examining these expressions for the transition probability, we note that if the Boltzmann definition of entropy is adopted:
S(l],r)
= k ln(f~(~,r))
(2.0, 7)
then the transition probability may be written in terms of entropy. For example, Eq. (2.0, 5) gives
Z()~j,2k ) = Ke exp[ S(,~,k) -k S()~j) ]
(2.0, 8)
We would emphasize that the expression for the transition probability given in Eq. (2.0, 8) is the same expression for this transition probability that was obtained from a first-order perturbation analysis of the Schr6dinger equation [2, 3]. Thus, in summary, the two assumptions listed above appear to be consistent with the more fundamental approach through the Schr6dinger equation. The expression for the net rate of particle transitions may now be obtained. We suppose that the transition from A,j to/t, k involves the transfer of one particle and that the transition probability z(A,i,A, k) is approximately constant as the result of the transfer of a single particle. The validity of the latter part of this assumption will be investigated below. The number of particles transferred is equal to the number of times this transition and if z(A,j,A, k) is the probability of this transition taking place at a takes place, 8Nik, J particular instant, then the nurhber of times the transition takes place in the interval d~t is given by
~Njk = ~(A,j, &k )St
(2.0, 9)
and the uni-directional reaction rate is given by
J(~j,/]'k ) = qd(Aj, ~k)
(2.0, 10)
When the system is initially in particle distributionAj, the probability of a particle transition occurring that leads to the system making a transition to A,k is given by Eq. (2.0, 8); however at the same instant, there is a probability of a particle making a transition in the opposite sense. This would cause the particle distribution in the system to be, say, ~,i- Considerations similar to those above lead to
T(i~j,t~i) = Ke exp[S(Xi)-k S(A,j) ]
(2.0, 11)
Since at any instant, there is a probability for the system to move in either the positive sense ( Aj to A,k) or in the negative sense ( A,j to Zi), the most probable transition when the system is in particle distribution Zj is to the particle distribution corresponding to a
293 large number of states._If this rate is denoted as J, then it would be the difference between J(~i,Ak) and J(~i,,~i). From Eqs. (2.0, 5) and (2.0, 6), the most probable, net rate at which'the system pro6eeds in the forward direction from A,j may be written
J(Zj) = n(K~~i[n(,~k)- n(Z~)]
(2.0, 12)
Or
J(Aj) = Ke(exp[ S(Ak) -k S(J~j) ]-exp[S(Ai ) -k S(Aj) ])
(2.0, 13)
To apply this relation to a particular kinetic process, it is necessary to obtain the expression for the change in entropy associated with that particular process. We shall consider two kinetic processes in subsequent sections.
2.0.1. The increase in entropy of an isolated, nonequilibrium system From Statistical Rate Theory, one obtains a simple interpretation of the "irreversible" increase in entropy that occurs in an isolated system as it evolves toward thermodynamic equilibrium. The Boltzmann definition of entropy has been adopted. This algebraic relation alone does not lead to the prediction of an increase in the entropy of an isolated system as the system evolves to equilibrium. It does, however, define thermodynamic equilibrium as corresponding to the condition existing in the system when the number of states is maximized. From Eq. (2.0, 12), it follows that if the system is in particle distribution 2i and if f~(~,k) > f~(A,i), then the most probable evolution is to particle distribution 2k .~At any instant then, the most probable transition is to the particle distribution corresponding to the larger number of states. Since this direction is also the direction of increasing entropy, the most probable evolution of the system is one corresponding to an increasing entropy. This increase would continue until the system reached a particle distribution where the number of states in either direction from the existing distribution was the same. This condition would correspond to a maximum in the number of states. In other words, if the equilibrium particle distribution is denoted as ~eq and those on either side as ~t and ~u, then the fact that ~"~(,)t,eq ) was a maximum would mean:
~(Zt) = ~'2(~eq ) = ~"~(Zu)
(2.0.1, 1)
Thus, from Eq. (2.0, 12) it may be seen that the most probable, net rate at equilibrium, J(/~eq), vanishes, but from Eqs. (2.0, 5), (2.0, 6) and (2.0, 10), it may also be seen that the uni-dxrectional rates reduce to Ke:
J(,~,eq,Zt) = J(Zeq,,~u)= K e
(2.0.1, 2)
This latter relation provides an interpretation of the rate of exchange between quantum mechanical states. Under conditions of thermodynamic equilibrium, it is the exchange rate between particle distributions. For example, for a reaction occurring homogeneously within one phase, as will be seen, it is the equilibrium exchange rate for the reaction. For the case of adsorption kinetics (i.e., particle exchange between two phases), it is the equilibrium exchange rate between the adsorbed phase and the gas phase. It should be noted that the expression for the rate given in Eq. (2.0, 13) is conceptually different than that obtained from either Absolute Rate Theory or the sticking probability approach. For example, in Statistical Rate Theory the view is taken that under equilibrium conditions in an isolated system a kinetic process takes place in either the
294 forward or negative direction at the same rate, K e. Then if that system is displaced from equilibrium in either direction and released, at any instant, there is a probability of the system moving in the equilibrium direction, and at the same instant, a probability of the system moving in the direction away from equilibrium. The rate in either direction is the equilibrium exchange rate multiplied by an amplification factor. As indicated in Eq. (2.0, 12), if ~(&i) is the number of states available to the system when it is in the displaced particle cofifiguration and f2(/~k) is the number of virtual states in the particle configuration closer to equilibrium, then the amplification factor for the rate in the direction of equilibrium is ~(&k)/f~(A,i). Similarly, one finds the amplification for the rate away from equilibrium is given by" f ~ ( & i ) / ~ ( s Thus if f2(~ k) > f2(~,i), then the rate in the direction of equilibrium is augmented relative to the equilibrium exchange rate, whereas the rate away from equilibrium is reduced relative to the equilibrium exchange rate. The most probable, net rate is then the difference between these two rates. Thus there is only one equation for predicting the net rate of the process. For example, when this procedure is applied to adsorption kinetics, the same equation is used to predict the net rate at which particles are transferred from the gas phase to the surface (adsorption) or the net rate at which particles are transferred from the surface to the gas phase (desorption). Only the values of the physical properties appearing in the rate equation will be needed to predict the net rate in either direction. Both Absolute Rate Theory and the sticking probability approach treat these process as two separate processes. There are, for example, different values of the pre-exponential factor and of the activation energy for adsorption and for desorption in the Absolute Rate Theory approach (see below). 2.1. Expression for the rate of an electron transfer reaction We now propose to obtain the expression for the rate of the electron transfer reaction between ionic isotopes in solution. Such a reaction may be expressed:
(2.1, 1)
a+b--+c+d
where components a and b are isotopes with charges of n + and (n + 1)+, respectively; c is an isotope of b, and has the same charge as b and d is an isotope of a and has the same charge as a. Physically, the reaction represents the transfer of a particle (an electron) from one component to another in a homogeneous, single phase, isolated system. There is no heat of reaction or expansion associated with the production of the products of this reaction; thus the temperature and pressure would be constant. The other intensive properties, the chemical potentials, are also approximately constant as a result of transferring a single electron, except in the limit where there are no products present and the transfer of one electron gives rise to the presence of the first product particles. This limit will be considered as a special case. We first consider the case where there is a sufficient amount of the product present so that the creation of an additional single particle of each product type is negligible compared to the amount of product already present. In this limit then, all of the intensive properties are unchanged as a result of one electron transfer. However, the transfer of one electron does give rise to a change in total entropy as may be seen from calculating the entropy change. The Euler relation for the entropy may be written l (2.1,2) where T and P are the temperature and pressure and #n is the chemical potential of component-n; U and V are the internal energy and the system volume, and N n is the
295 total number of particles of component-n. There are a total of l components present 9The particle distribution %j may now be stated explicitly.
Xj. Na,Nb,Nc,N d
(2.1,3)
There may be other species present but their numbers do not change when the reaction takes place and are thus not listed. The particle distribution that results from the transfer of an electron so that the reaction proceeds in the positive direction, i.e., from the distribution ~,j to the distribution Zk is Zk:N a -
1,N b - 1,N c + 1,N d + 1
(2.1, 4)
The change in entropy may be calculated from Eq. (2.1, 2). One finds 1
S(Zk ) - S(Zj ) = T(Zj ) [#a (Zj ) + #b (J~j ) - #c (J~j ) -- ~d(J~j )]
(2.1, 5)
It should be noted that the change in entropy between the particle distributions %, and Xk has been expressed only in terms of the chemical potentials evaluated whex4 the particle distribution is Xj. That is, the change in entropy is a function of the instantaneous particle distribution /~,. This result is obtained because of our assumption that the intensive propemes do not change as a result of one particle making a transition. It will have important consequences when the expression for the rate of the reaction is obtained. From Eqs. (2.0, 8) and (2.1, 5), one finds that the probability of a transition from ~,j to A,k at any instant is given by 9
,
"((Xj,Zk)
,
= K e
.]
exp[ ~ a ( z j ) + ldb(Xj)- l l c ( Z j ) - /2d(~,j)]
kT
(2.1,6)
Note then that the transition probability depends only on the intensive properties and the exchange rate between quantum mechanical states, K e. Thus, the assumption that the intensive properties do not change as a result of one electron transfer is consistent with the assumption of the transition probability being constant over an interval of time St. These assumptions will be discussed further after explicit expressions for the chemical potentials are available. From Eq. (2.0, 10), one finds that the uni-directional rate, J(2,j,)~ k), is given by J ( A j , X k ) = K e exp[#a(Aj) + ~2b (J~j ) -- # c ( Z j ) -- #d(J~j ) ]
(2.1, 7)
kT
At the same instant that there is a probability of the reaction proceeding from ~j. tO ~'k, i.e., in the positive direction, there is a probability of an electron transfer occumng that would cause the reaction to move in the negative direction. This would mean that the particle distribution would move from A,j to ~,i where
Xi:Na + l, Nb + l, Nc - l, Nd - 1
(2.1,8)
The corresponding change in the entropy may be obtained from Eq. (2.1, 2):
S(Zi) - S(Zj )
1
- -
T(Xj'j [-12a(Aj ) - #b(Xj ) + l.tc(Xj ) + btd(Xj )]
(2.1,9)
296 The probability of a transition from /~j to A,i is given by
v(~,j,X i) = K e exp[
--/aa(~j ) --/ab(~j)-F/ac(~j ) +/ad (~j) ] kT
(2.1, 10)
and the uni-directional rate in the negative sense is given by
J(~j,~i)=Keexp[ -/aa(')t'J)-/ab(ZJ)T/ac(ZJ)W/ad('~l'J)]
(2.1, 11)
kT
From Eqs. (2.1, 7) and (2.1, 11), one finds as the expression for the net, most probable rate
J(~j) = 2Kesinh[/aa(&j) + /ab('~'J)- /ac(~J)- /ad(~J)] kT
(2.1
12)
Note that the expression for the net rate at an instant only depends on the intensive properties evaluated when the instantaneous particle distribution is A,j and the constant K e. This is a reflection of the fact that the entropy change between A,j and Xk or ~i only depended on the value of the intensive properties evaluated in this condition and the constant K e. 2.2. The expression for the rate of an electron transfer reaction To examine the Statistical Rate Theory expression for the rate of a chemical reaction, the expression for the rate will be used to predict the concentration as a function of time for electron transfer reactions between isotopes dissolved in electrolytes and then these predictions will be compared with experimental results [3]. It is necessary to treat the solutions as nonideal. The chemical potentials must be written in terms of activity coefficients, rather than the concentrations, and we shall take the chemical potentials to be of the following form [19]:
/aa = /a~ (T,P) + kTln(r,~x~)
(2.2, 1)
where /a~ (T,P) is the reference chemical potential of component a , x a is the mole fraction of this component, and Ya is its activity coefficient. When the system is in thermodynamic equilibrium, the intensive properties must satisfy certain relations. These relations may be obtained by requiring that the entropy of the system be a maximum. As a necessary condition for equilibrium, one finds
#a ( ~eq ) +/ab(l'~eq)= llc ( ~eq ) + ~ld( ~,eq)
(2.2, 2)
and if the equilibrium constant, E c, is defined as
E c = exp(/aa~ + lab~ -/ac~ --/ad~ kT
(2.2, 3)
then as may be seen from Eq. (2.2, 1), it is only a function of temperature and pressure. If the reaction defined in Eq. (2.1, 1) has reached equilibrium, then Eq. (2.2, 2) must be valid and after inserting the expressions for the chemical potential of each of the
297 reacting components into Eq. (2.2, 2), one finds that the result may be written in terms of the equilibrium constant Ec: (2.2, 4)
Xa~'aXb~b eq where the subscript eq indicates that the term in the brackets is to be evaluated at equilibrium. After introducing the expression for the chemical potential given in Eq. (2.2, 1) and the expression for E c, one finds from Eq. (2.1, 5) that the change in entropy as a result of a single particle transition is given by S ( ~ k) - S(&j) = k In[ Ecxayaxb 7b ] x~7r
(2.2, 5)
and from Eq. (2.1, 12) that the net rate may be expressed
j = ge[Ecxa~taXb'Yb XcTcXd~td
Xc~tcXd~td ] Ecxa~'aXb ~'b
(2.2, 6)
Since components a and d are ions of the same charge, but are isotopes, the ratio of their activity coefficients is unity. The same would be true of the activity coefficients of components b and c, and hence 7a 7b = 1
(2.2, 7)
?'cTd
Also, the ratio of mole fractions may be written in terms of numbers of particles, since (2.2, 8)
XaX~b = NaNb
X~Xd N~Nd The number of reactant particles present in the system, after the reaction defined in Eq. (2.1, 1) is initiated, may be expressed in terms of the reaction variable r N a = N a (0)- r
o~ = a,b
(2.2, 9)
where N a (0) is the initial number of particles present. For the products, one may write N o = NI3(O ) + r
fl = c,d
(2.2, 10)
To establish the equilibrium constant for the reaction, we again take advantage of the fact that components a and d are isotopes of the same charge, as are b and c. If equal quantifies of components a and b were added initially, then under equilibrium conditions it would be expected that there would be equal concentrations of component a and its isotope, component d. Since the system we are considering is spatially homogeneous,
(Na)eq "- (Nd)eq
(2.2, 11)
298 A similar argument could be applied to components b and c:
(2.2, 12)
(Nb)eq = (Nc)eq
As may be seen from Eqs. (2.2, 4), (2.2, 7) and (2.2, 8), if these conditions are to be met, then Ec = 1
(2.2, 13)
Since the value of the equilibrium constant has been found in this special case to be unity and since its value is independent of the initial concentration, as may be seen from Eq. (2.2, 4), its value may be taken as unity for all initial concentrations. After making use of Eqs. (2.2, 7) to (2.2, 10) and Eq. (2.2, 13), one finds from Eq. (2.2, 5) that the expression for the change in entropy between particle distributions Xj and A.k is given by
S(~, k) - S ( Z j ) = k
r)(Nb(O)-~]
ln[(Na (0) -(~cc(O) + r ) ( N d ( O ) +
(2.2, 14)
where the initial concentration of each species is indicated by Na(0). From Eq. (2.2, 6), one finds that the expression for the net rate of the reaction may be expressed j = K e [ ( N a ( O ) - r)(Nb(O ) - r) (0) + r)(Nd(O ) + r)
(Nc(O)+ r)(Na(O)+ r) ] (Na(O) - r)(Nb(O ) - r)
(2.2, 15)
The latter equation is the one we shall use to make predictions of the concentration for particular electron transfer reactions that can be compared with a set of measurements. We would emphasize that two important simplifications have resulted from the fact that the reaction being considered is an electron transfer reaction between ionic isotopes in solution: 1) Since the reactants and products are isotopes, the activity coefficients do not appear in the final expression for the rate. No assumption was made regarding the value of the activity coefficients. Their elimination occurred because of the system being considered. 2) The expression for the rate has been written in terms of the number of particles of each component that would be present in the system at any instant in time and the equilibrium exchange rate for the reaction, K e. For the isolated system that we consider, once the initial concentrations of the components are given, K e is fixed, although its value is unknown. The predicted expression for the net rate of the reaction can be examined by determining if there is one value of K e that leads to a prediction of the concentration that is in agreement with the measured concentration as a function of time. Equation (2.2, 15) gives the expression for the rate in terms of the reaction variable r. As may be seen from this relation, if the initial concentrations of the products are both zero, the initial rate of the reaction is predicted to be infinite. This infinity results from the breakdown under these conditions of the assumption that there is no change in the chemical potentials of the products as a result of the reaction going one step in the forward direction. Under these conditions, the creation of the first products gives rise to a significant change in the value of the chemical potential of the products.
2.2.1. Expression for the net rate at very short times In preparation for calculating the rate in the very initial period, we first develop the expression for the initial rate without assuming that the chemical potentials are unchanged
299 as a result of one electron being transferred. The particle distribution, &(r), existing in the system at any time may be represented in terms of the reaction variable r and the initial amount of the reactants: (2.2.1, 1)
~(r):(Na(O ) - r),(Nb(O ) - r),r,r
The distribution that would exist after the reaction had proceeded one step in the forward direction would be (2.2.1, 2)
X ( r + 1):(Na ( 0 ) - r - 1),(Nb (0) - r - 1),r + 1,r + 1
The change in entropy may be calculated from Eq. (2.1, 2). If it is assumed that the reaction variable r and the initial concentrations are measured in moles, then one finds that S(r + No-l) - S(r) = Na (O)ln[
( N a ( O ) - r) (ga(O) - r - No-l)
(Nb(O) - r)
]+ gb(0)ln[ ( N b ( O ) - r - N o
+ r l n [ ( N a ( O ) - r - No - 1 ) ( N b (0) - r - No -1) r2 ( N a ( O ) _ r)(Nb(O) _ (r + No -1 )2
+No_ 1ln[(ga (0) - r - No-1)(Nb(O) - r - No-l)] (r + No-1) 2
-1)
]
r) ] (2.2.1,3)
where N O is Avogadro's number. For macroscopic systems, one may suppose that N a , N b >> No -1
(2.2.1, 4)
The change in entropy that results from the first electron transfer is obtained from Eq. (2.2.1, 3) by imposing the condition given in Eq. (2.2.1, 4) and then taking the limit of r going to zero: S(No -1) - S(0) = kln[ Na(O)Nb(O)) ] No -2
(2.2.1, 5)
In this limit then, there would be no product present and the initial rate may then be calculated by combining Eqs. (2.0, 8) and (2.0, 10) and inserting Eq. (2.2.1, 5). One finds J(O) = Ke(No2Na(O)Nb(O))
(2.2.1, 6)
In the latter relation, the term in parentheses is the product of the initial number of particles of the reactants. The rate J(0) is measured in the same units as the equilibrium exchange rate. Unless the equilibrium exchange rate is very small or the initial number of particles of the reactants is very small, the initial rate may be very large, but not infinite. The initial rate will be considered in specific cases in the following section.
2.3. Experimental examination of the expression for a reaction rate The electron exchange reaction between isotopes of A g has been examined experimentally by Gordon and Wahl [20]. This reaction may be written
300 *Ag + + A g 2+ --> *Ag 2+ + A g +
(2.3, 1)
When it took place in a 5.87 f H C I O 4 solution, measurements of *Ag2+/(*Ag2+)ea at different times were reported [20] and are shown in Fig. 2.3, 1. There wei'e no prod[acts initially present. The theoretical prediction of these measurements may be made from Eq. (2.2, 15) after it has been integrated. We first write the equation in terms of the reaction variable, r. dr = K e [ ( N a ( O ) d--7
r)(Nb(O ) - r) r2
r2 - ( N a ( O ) - r ) ( N b ( O ) - r) ]
(2.3, 2)
After separating the variables in Eq. (2.3, 2), integrating and applying as the initial condition that r(0) is equal to zero, one finds
get =
cl r
7.2
4C~
4C 2
C~ arc tanh[ ~ -~~l + C~ arc tanh[ C2~4 ] - 4r
8~4
8~4
Ca log(-Cs) _ G l o g ( G ) _ G 2 l o g ( - G + G r ) 2C23 16 2C23
-t C2 l~
- C2r + 2r2 ) 16
(2.3, 3)
where the values of C i are determined from the initial concentrations of the reactants: C 1 = (Na(0) 2 + Nb(0) 2) C2 = (Na(O)+ Nb(O)) C3 = (Na(O)- Nb(O))
(2.3, 4)
C4 = Na(0) 2 -6Na(O)Nb(O)+ Nb(0) 2 C5 = N a (O)Nb (0)
If the theoretical expression for the rate is correct, then there should be one value of K e that gives a prediction of the concentration of each of the products throughout the period of the reaction. To determine the value of K e that gives the best description of the
data, we shall take as the definition of the error, E, between the n-measurements and the corresponding calculations: n
E 2 = ~ [ t c ( r i ) - tm(ri)] 2 i=1
(2.3, 5)
where tc(ri) is the calculated time at which the reaction variable has the value r i and t m (r i) is the time at which the reaction variable is measured to have the same value of r i. Equation (2.3, 3) may be viewed as being of the following form: Ket = F(r)
(2.3, 6)
301 The time at which the concentration of the products is predicted to reach a value of r i may be obtained from this latter relation. After this expression is substituted into the expression for the error, one finds
F(ri) E 2 = ~[ - tm(ri)] 2 i=1 ge
(2.3, 7)
After differentiating the latter relation with respect to K e and requiring the result to vanish, one finds that the value of K e that minimizes the error, t is given by
~[F(ri)] 2 ge = i=1
(2.3, 8)
~ F ( r i ) t i ( r i) i=1 It should be noted that the minimization of the error has been based on minimizing the difference between the predicted time and the measured time at which a particular concentration is reached. By proceeding in this fashion, we avoid the necessity of inverting the function F(ri).
0.8 0
@ 0
0.6 = @ .m I.
=
0
0.4
=
o
0.2 .
.
.
.
i
0.2
.
.
.
.
i
9
0.4
I
0.6
.
9
0.8
9
i
1
Time, s Figure 2.3, 1. Comparison of the predicted and measured concentration ratio *Ag2+/(*Ag2+)eq for the reaction indicated in Eq. (2.3, 1) taking place in a 5.87 f HCIO 4 solution at 0.2~ with initial concentrations of *Ag § and Ag 2+ of 0.01062 f, and 0.00179409 f and no product initially present. The solid dots indicate the reported measurements of Gordon and Wahl [20] and the solid line the predicted concentration ratio made from Eq. (2.3, 3) for the stated initial concentrations. t In Ref. [3] the expression for K e was incorrectly given, but the calculations were made on the basis of the correct equations.
302 Equation (2.3, 8) and the reported experimental results may be used to calculate the value of K e. One finds
K e =0.00017044 f
(2.3,9)
S
A necessary condition for the theory to be valid is that when this value of K e is used in Eq. (2.3, 3), the prediction of the product concentrations be in agreement with the data. The predictions for one set of experiments are shown as the solid line in Fig. 2.3, 1. The data are said to be accurate to approximately 10%. Thus, as may be seen by comparing the experimental data and the predictions, there is no measured disagreement between the theory and experimental results for this case. In Ref. [20], data are given for the electron exchange reaction between isotopic ions Statistical Rate Theory has been used to of Ag at a different temperature (11.4~ predict the results at this second temperature. The close agreement between the theory and experiment was found at the second temperature as well. Further, Statistical Rate Theory has been applied to examine the electron transfer reactions:
*Mn042- + MnO4- .__>*Mn04- + Mn042-
(2.3, 10)
and *V 2+ -!-V 3+ --->* V 3+ + V 2+
(2.3, 11)
using data reported by Sheppard and Wahl [21] and by Krishnamurty and Wahl [22].
~"*0"8 o
0.6
.~
0.4
'~
0.2
r." .
.
.
.
.
5
-
-
i
i
i
10
.
.
.
.
9
i
i
15
20
25
30
Time, s
Figure 2.3, 2. Comparison of the predicted and measured concentration ratio Mn04-/( MnO4-)ea for the reacuon indicated m Eq. (2.3, 10) taking place m a 0.16 f NaOH solution at 0.1~ with initial concentrations of the reactants of 0.00004 f, and 0.000096 f. The solid dots indicate the reported measurements of Sheppard and Wahl [21 ] and the solid line the predicted concentration ratio made from Eq. (2.3, 3) for the stated initial concentrations.
303 Only in the case of the manganate-permanganate reaction is there any indication of disagreement between the theoretical predictions and the experimental results. A comparison for this case is shown in Fig. 2.3, 2. It is not clear that this difference is significant. For the other five systems for which there is data available, there is no indication of any disagreement between the experiments and the predictions made from Statistical Rate Theory [3]. 2.3.1. Examination of the predicted initial rate If the approximation is made that the chemical potentials are unchanged as the result of one electron transfer taking place, one obtains the expression for the rate shown in Eq. (2.2, 15). As may be seen from this equation if the initial concentration of the products is zero, Eq. (2.2, 15) indicates that the initial rate will be infinite. However, when the initial rate was calculated without introducing this approximation on the chemical potentials, the expression for the rate is that given in (Eq. 2.2.1, 6) and as may be seen from this latter equation, it indicates a finite rate initially when no product particles are present. From the information obtained in the previous section, we may calculate the initial rate from Eqs. (2.2.1, 6) and (2.3, 9). One finds J(0) = 1.7806 x 1038f
(2.3.1, 1)
S
Thus when the initial concentration of the products is zero the initial rate is finite, but very large. Assuming the rate stays constant over the initial period, the period of time required for the reaction variable, r, to become large compared to one t~article can be estimated. For r to be a thousand particles would require only 10-`59s. Thus the approximation that the chemical potentials do not change as the result of the reaction going one step in the forward direction appears to be valid except at the very shortest times. Using the prediction of the number of product and reactant particles as a function of time that is obtained from Statistical Rate Theory, the entropy as a function of time may be predicted. From Eq. (2.1, 2), one finds that the difference between the entropy at the time t and its initial value is given by
Na(O) ) + N b ( 0 ) l n ( N b ( O ) ) S(t)- S(O)= k[Na(O) ln( Na(O)-------~r Nb(O)-------~r +rln((Na(O)- r)(Nb(O ) - r) r2
)]
(2.3.1, 2)
If one uses Eq. (2.3.1, 2) and the predicted expression for the reaction variable given in Eq. (2.3, 3), then for the electron transfer reaction between Ag ions, one finds the entropy as a function of time that is shown in Fig. 2.3.1, 1. It should be noted that although there is a rapid initial rate of entropy increase, the entropy is predicted to remain finite at all times, unlike the reaction rate. The predicted and measured rates of the reaction are shown in Fig. 2.3, 1. In this case then, the prediction is that the most probable path from the initial nonequilibrium particle distribution to the equilibrium particle distribution corresponds to an increase in entropy at every instant of time for the isolated system.
304
0.005 o r~
i
0.004
r~
0.003 0.002 -
0.001
0.25
0.5
0.75
1 ]]me, s
1.25
1.5
1.75
Figure 2.3.1, 1. Predicted entropy change as a function of time resulting from the reaction *Ag + + Ag 2+ --->*Ag 2+ + Ag + taking.~lace in a 5.87 f HCIO 4 solution at 0.2~ with initial concentrations of *Ag + and A g z of 0.01062 f and 0.00179409 f and no product initially present.
2.4. Predicted rate at the initial time from absolute rate theory and statistical rate theory As may be seen in Fig. (2.4, 1), for the electron transfer reaction between isotopic ions of Ag in solution, Statistical Rate Theory leads to the prediction of a very large rate initially and also to a rate that approaches zero as the system approaches equilibrium. Although there is an approximation at short times, the approximation appears valid except for the very shortest times. The prediction of both of these rates is one of the characteristics of Statistical Rate Theory, and one that distinguishes it from previous theories. When Absolute Rate Theory is applied to electron transfer reactions, the concentration dependence of the rate expression is obtained from an interpretation of the relation for the equilibrium constant. As seen in the previous section, for the type of system that we consider, the equilibrium constant may be expressed
c~ca
Ec = [ CaCb ]eq
(2.4, 1)
where C i is the concentration of component-i. The Absolute Rate Theory interpretation of this latter relation strongly involves the assumption of equilibrium between the activated complexes and the reactants and leads to the denominator being interpreted as the concentration factor driving the reaction forward and the numerator being the concentration factor driving the system in the reverse direction [11, 12]. Thus, the expression for the forward rate of the reaction, according to Absolute Rate Theory, is
305
&(AbRr) =
(2.4, 2)
klc c
and similarly the reverse rate is assumed to be of the following form
Jr(abRT) = krCcCd
(2.4, 3)
Under equilibrium conditions, Jf.(abRT) and Jr(AbRT) must be equal and since the equilibrium constant, E c , is unity, it follows that the rate constants must be equal. Thus, the net rate, according to Absolute Rate Theory [ 11 ], is
J(AbRT) = kAbRT(CaCb -CcCd)
~
0.8
(2.4, 4)
S tatis ti cal Rate T heor y
0.6
g
o i
0.4
;olute Rate Theory
o.2
0.2
0.4
0.6
0.8
1
Time, s
Figure 2.4, 1. Comparison of the predictions from Absolute Rate Theory and the measured concentration ratio *Ag2+~(*Ag2+)eq for the reaction indicated in Eq. (2.3, 1) that took place in a 5.87 f HCI04 solution at 0.2~ with initial concentrations of *Ag+and Ag2+of 0.01062 f and 0.00179409 f. The solid dots indicate the reported measurements of Gordon and Wahl [20] and the solid line the predicted concentration ratio made from Absolute Rate Theory for the stated initial concentrations. Note the poor agreement during the initial period. We note that the interpretation of the equilibrium constant (i.e., Eq. (2.4, 1)) used by Absolute Rate Theory is arbitrary [23]. For example, if K e is the equilibrium exchange rate, then
XetqCd Ec Cafb ]eq
(2.4,5)
306 would also be the equilibrium exchange rate. Further, if (1/Ec)[CcCd/CaC b ] were greater than unity, then it would mean that there was an excess of products compared to reactants and it would be reasonable to expect that the rate at which the reaction would proceed in the reverse direction, Jrv, would be
Ke [C~Ca Jrv--Ec CaCb]
(2.4,6)
If (1/Ec)[CcCd/CaCb] were less than unity, then there would be an excess of reactants compared to products and one would expect the reaction to be driven in the forward direction. As the driving force, one might reasonably assume that it would be the inverse of the less-than-unity (1/Ec)[CcCd/CaCb] or
CaCb
Jfd = KeEc[ CcCd ]
(2.4, 7)
Thus the net rate of the reaction would be
e~CaCb_ Ccq
Jnet = Ke[ "Ccc-~d EcCaCb]
(2.4, 8)
To compare this latter expression with the Statistical Rate Theory expression, we may rewrite Eq. (2.2, 6) that was obtained from Statistical Rate Theory. If Eqs. (2.2, 7) and (2.2, 8) are used to simplify Eq. (2.2, 6), then it is found that the latter equation reduces to
j= K~tE~UaUb_ UaU~
EcNaNb ]
(2.4,9)
which is the equivalent of Eq. (2.4, 8). From the point of view of interpreting the equilibrium constant, the Absolute Rate Theory approach does not appear to offer any advantage compared to the latter interpretation and the second one leads to the same expression as the Statistical Rate Theory approach. To see that the expression for the rate that is obtained from the Absolute Rate Theory approach leads to poor agreement between theory and experiment during the initial period, it has been applied to electron transfer reactions [3]. An example of the type of results obtained is shown in Fig. 2.4, 1 where the results predicted from Absolute Rate Theory may be compared with the measurements reported by Gordon and Wahl for an electron transfer reaction between isotopes of Ag and with the results obtained from Statistical Rate Theory. The value of kabRr was determined from an equation that was obtained in the same manner as Eq. (2.3, 8). When the results shown in Fig. 2.4, 1 are compared, one notes a much better agreement between the Statistical Rate Theory approach and the experimental results in the initial period than between the Absolute Rate Theory and the experimental results. Both theories give an adequate prediction of the experimental results when the concentration ratio is near its equilibrium value. For the electron transfer reactions, the Absolute Rate Theory approach was found to lead to poor agreement in the initial period for each of the reactions examined [3]. It can be made to agree with the measurements in the initial period, but it does not then agree with the measurements near equilibrium. The disagreement in the initial period was originally thought to arise from experimental error; however Absolute Rate Theory also gives a poor prediction of adsorption kinetics in the initial period (as will be seen in sub-
307 sequent sections) and the experimental techniques used to examine adsorption kinetics are of a different nature than those used to study electron exchange reactions. 3.0. A D S O R F H O N K I N E T I C S We shall now consider the problem of non-dissociative adsorption in a constant volume, isothermal system. This represents a change in constraints as compared to the case of the reaction occurring in a homogeneous, isolated system. Since adsorption is an exothermal process, if the system is to be isothermal, then it is necessary to include a reservoir so that the system and reservoir define the isolated system to which Statistical Rate Theory may be applied. To obtain the expression for the net rate of adsorption, we again turn to Eq. (2.0, 13), this time determining the change in entropy associated with the adsorption of one molecule within the system. The entropy changes of the reservoir, the solid, the gas phase and the adsorbed phase must be taken into account in order to derive the total entropy change of the isolated system. We assume that there are no spatial gradients in intensive properties within each phase and that the molecular transport rate between the gas and the adsorbed phase is the rate determining step in the evolution of the system to equilibrium. We will assume that the reservoir maintains the entire system at a constant temperature, T. The Euler relation for the entropy of the gas phase may be written
S g "-
U g -b
Vg -
N g
(3.0, 1)
where the superscript g on a quantity refers it to the gas phase. In the model that we will consider, the solid substrate will be approximated as having uniform properties up to a dividing surface which is located at a position such that no solid atoms are in the interphase. Thus, the Euler relation for the adsorbed phase may be written Sa = ( T ) U a -
(-~)A a - ( @ 1 N t r
(3.0, 2)
where U Cr is the internal energy of the adsorbed molecules, ?' is the solid-gas surface tension, A Cr is the area of the interphase, N a is the number of adsorbed molecules and /2 a their chemical potential. For the solid phase, we shall suppose that there is no absorption of the gas phase molecules and neglect any effects due to shearing stresses. Then the approximate Euler relation may be written Ss =
(11
Us +
Vs _
N s
(3.0, 3)
where the superscript s on a quantity refers it to the solid phase. When one molecule is transferred from the gas phase to the adsorbed phase, the molecular configuration is changed from &j to A,k where ~,j:Na(j),Ng(j)
(3.0, 4)
and )ck:N(r(j)+
1,Ng (j) - 1
(3.0, 5)
308 We assume that during this process the reservoir, denoted with the superscript R, is heated quasi-statically so that
AS R
=
Aug AUa AUS T T T
(3.0, 6)
where A is an operator that defines the following operation Atp -= tp(;l,k ) - tp(~j)
(3.0, 7)
We shall suppose that the intensive properties of each of the phases are unchanged as a result of a single molecule being adsorbed. If initially, there are no adsorbed molecules on the surface, this approximation would be invalid at the initial instant; however, it was illustrated for the reactions considered in earlier sections that the period of time for which it would be invalid is so short as to be negligible. We shall assume the same to be true for adsorption kinetics. By conSidering the volumes of the gas and of the solid phase and the area of the adsorbed phase to be constant, and by using Eqs. (3.0, 1), (3.0, 2), (3.0, 3) and (3.0, 6), one may then write the change in entropy associated with the adsorption of one molecule as
+As,, +As, + As,, -
,u g
/2 a
(3.o, 8)
And hence, using Eqs. (2.0, 8) and (2.0, 10) along with Eq. (3.0, 8), one may write the uni-directional adsorption rate in terms of the chemical potentials
](A'J)=Keexp
kT (3.0, 9)
Following similar arguments, one may derive the change in entropy of the system as the result of a single molecule desorbing, and then using Eq. (2.0, 11) the uni-directional desorption rate may be written (3.0, 10)
And thus one gets for the net rate of adsorption J(Aj) =
2Ke sinh[l'tg(zj ) - l'ttr(~'j)
(3.0, 11)
By requiring that the differential of entropy vanish for virtual displacements about the equilibrium state, one obtains the condition for equilibrium
(3.0.12)
309 If this equilibrium condition is substituted into Eqs. (3.0, 9) and (3.0, 10) then, at equilibrium
J(~eq)'-Y(~eq)=ge
(3.0, 13)
so that K e, the exchange rate between quantum mechanical states of different particle distributions, is seen to correspond physically to the equilibrium exchange rate between the gas and adsorbed phases. Note that in the derivation of the net adsorption rate above, the only assumptions made about the solid substrate were that the shear stresses could be neglected, that no gas phase molecules were absorbed and that its volume did not change as a result of the adsorption. Equation (3.0, 11) is, therefore, applicable to a number of different systems and is consistent with a change in both the internal energy and the entropy of the solid as a result of the adsorption taking place. In order to calculate the surface coverage as a function of time from Eq. (3.0, 11), explicit relations are required for the equilibrium exchange rate, K e, and the chemical potentials of both the gas and adsorbed phases,/~g and/1 ~. The equilibrium exchange rate has been formulated as [ 1, 9] (3.0, 14)
K e = VeAeff~
where v e is the equilibrium collision frequency of molecules in the gas with the solid surface, Aeff is the area available for adsorption to occur per unit surface area and ~ is the probab~ity that a molecule striking an available adsorption site will adsorb. As defined, the equilibrium exchange rate has units of molecules per time per unit surface area. The equilibrium collision frequency may be found from the Maxwellian velocity distribution to be
Pe
(3.0, 15)
v e = ~itcmkT e
where m is the molecular mass and T e and Pe are the equilibrium temperature and pressure respectively. We shall assume the solid surface is a single crystal surface that is uniform in composition and consists of only one type of bonding site, that the total number of adsorption sites per unit area is M, and that at equilibrium nee of these sites per unit area have molecules adsorbed on them. The fractional area available for adsorption under equilibrium conditions would then be given by Aeff = ( M -
(3.0, 16)
n~e ) a a
where act is the area of an individual adsorption site. The equilibrium exchange rate may now be written
"e
g e = 42'~kZe
4)Oe
(3.0, 17)
where the area of an adsorption site and the probability that a molecule striking an available site adsorbs have been combined into a new quantity, the equilibrium adsorption cross-section
310 cre = aa~
(3.0, 18)
The quantity tr e is particularly important to our considerations. The subscript e has been included to emphasize that this cross-section is an equilibrium parameter. As such, its value does not change as the kinetic process proceeds. Since the adsorption cross-section is to be evaluated only at equilibrium, it appears to be a material property. If this is true, it may be tabulated for a given, well defined (single crystal) interface that has one type of bonding site? and used to predict the adsorption rate in a completely independent circumstance. We shall investigate this possibility in subsequent sections. An approximate expression for the chemical potential of ideal diatomic, nonsymmetric molecules in the gas phase can be obtained using the Born-Oppenheimer approximation and Boltzmann statistics [24]
l.tg = kTln(P(~)
(3.0, 19)
where P is the pressure in the gas phase and ~(T) is given by
[1
l]ex l /h
=
(3.0, 20)
(2~rkT)~ 2mlm2r2 (ml + m 2)1//2 where h is Planck's constant, COg is the characteristic vibration frequency of the gas molecules, D O is the dissociation energy, r e is the separation distance of the two atoms and m 1 and m 2 are the masses of the two atoms. These molecular properties of CO have all been previously established [24] and they are listed in Table 3.0, 1.
Table 3.0, 1 Gas Phase Prop.erties of Carbon Monoxide
Ogg(Hz)
re(nm )
DO(J / molecule)
6.394 x 1013
11.28
1.464 x 10 -18
3.1. Thermostatistical formulation for the chemical potential of the adsorbed molecules Following Ref. [ 10], we shall approximate the adsorbed CO molecule as a bound, double harmonic oscillator and obtain the expression for the chemical potential of the adsorbed molecules from statistical thermodynamics. The possible energy levels of such a molecule may then be represented in terms of the six characteristic frequencies as
fIf the single crystal surface has more than one type of bonding site, then there is one value of O"e for each type of bonding site.
311
eijklmn= E~ +(i +2)h(01+(j +l)h(02+...+(n+l)h(06 i,j,k,l,m,n
(3.1, 1)
=0,1,2 ....
where E~) is the minimum potential energy of the adsorbed molecule and co1, (02 . . . . . ( 0 6 are the six characteristic vibration frequencies. The potential seen by the adsorbed molecule can be thought of as consisting of two parts; the first arising from the interaction of the adsorbed molecule with the substrate atoms, and the second arising from the interaction of the adsorbed molecule with the rest of the adsorbed molecules collectively. Since changing the number of adsorbed molecules will affect this interaction, both because newly adsorbed molecules will add to the adsorbate-adsorbate interaction and because the addition of the adsorbate may affect the relative positions of the substrate atoms, E 6 will be allowed to depend on surface coverage 0 0=~
n ~
(3.1,2)
M0
where na is the number of adsorbed molecules and M 0 is the number of surface substrate atoms, each per unit surface area. Note that although interactions with neighbouring molecules are taken into account through E 6 , the vibration frequencies are assumed not to be affected by neighbouring adsorbed molecules and the individual harmonic oscillator degrees of freedom are assumed to be independent of one another. The allowed energy levels of the collectively adsorbed molecules are then
E(II,I2 ..... /6) = NaEo +
Na ~ ( h(0i ~ /=1~ , T j +
6
i=lEIih(0i
(3.1,3)
where Ii is the number of phonons with the characteristic frequency (0i" The canonical ensemble has been chosen to treat the problem of determining the thermodynamic properties of the adsorbed gas. The ensemble consists of a collection of surfaces, each with the same number of adsorbed molecules, the same surface area and the same temperature. The canonical partition function may be written
where Ea is the energy of the system when it is in state a . To change from a summation over states to a summation over energy levels, degeneracies arising from the ways of distributing the phonons of each type over No. oscillators and the ways of distributing N a adsorbed molecules over a total of M r adsorption sites must be included. After changing the summation to include these degeneracies and performing the required summations, one finds as the expression for the partition function of the adsorbed molecules
Qa= where
MT! (qlq2...q6)NCr Na ?(Mr - Na ) ?
(3.1,5)
312
q/
--
oxp( ) i exp(h~
(3.1, 6)
and where 9
,
eoi = E~
CO
oi
hr i 2
+
(3.1, 7)
~, ooj j=l
The Helmholtz function for the system may be written (3.1, 8)
F = Fg + Fa + Fs
where F g, Fcr and F s are the Helmholtz functions of the gas phase, interphase and solid respectively. The Helmholtz function of the adsorbed molecules may be found from statistical thermodynamics [24] to be (3.1, 9)
F a = - k T l n Qcr
According to thermodynamics, the chemical potential of the adsorbed molecules is given by (3.1, 10)
O N cr A tr , T
Thus, the chemical potential of the adsorbed molecules may be written (3.1,11)
where
9 6
( h(,oi "~
a': expr ]17 ~xpt'2-ff)
t, kT )i=1e x p(ho~i ~ - - ~"~ )- 1
(3.1, 12)
and where fl' is a function of coverage that is defined as
oNcr
r
(3.1,13)
313 Note that 8'(T, 0) contains as parameters the six fundamental vibration frequencies. All of the vibration frequencies have not been resolved for most systems. The method that we describe below can be applied when any number of frequencies have been resolved. We shall suppose that only two frequencies have been resolved by electron energ3, loss spectroscopy (EELS), that they are denoted coj, (j = 1, 2) and that the unresolved frequencies are denoted toi, (i = 3 to 6). Then S' may be factored so that all of the unknown frequencies appear in one factor. From Eq. (3.1, 12), one may write ~' = ~'V
(3.1,14)
where
r
~'(T, 0) -= exp
(~-T)/I~3 =
expt 2--~) ('ho)i ~
exPt,,--k-~-J-
(3.1, 15)
and exp h ~ J / zKs) j=l
Now .u a (T, N ~
(3.1, 16)
-- 1
can be written
/Ia (T,N a) = kTln (M r _ N a ) ~ ,
Note that in order to have the expression for the chemical potential of the adsorbed molecule, all that remains is to determine the unknown function ~'(T, 0). This function plays a fundamental role in the determination of the material properties.
3.2. Determination of the material properties necessary to predict the adsorption kinetics of CO adsorbing on Ni(111) For a given gas-surface combination for which there are unresolved vibration frequencies, we now introduce a method by which ~'(T, 0) may be determined from the empirical isotherms. The system of CO-Ni(111) is one such system and we shall assume that one type of bonding dominates in this system. The vibration frequencies for CO adsorbed on Ni(111) that have been resolved using EELS [25] are thought to correspond to the stretching of the adsorptive bond and to the stretching of the C-O bond when the CO molecule is adsorbed. The values of the fundamental frequencies for these two bonds correspond to (1.23 x 1013Hz) and ( 5.57 • 10 '~ Hz) respectively. The C-O stretching frequency is known to vary slightly with coverage [25]. This will be neglected herein. By the nature of the isotherm fitting procedure, any error resulting from this assumption will be included in ~'(T, 0). The number of surface atoms per area for Ni(111), M 0, may be found from geometry to be 1.86 x 1019a t o m s / m 2 . This is for a uniform, single crystal and does not include any surface roughness. The number of adsorption sites available in
314
this system, M, is known only approximately from the maximum coverage observed. The value we will use is 0.53M 0 [26]. Christmann et al. [26] measured the work function change of the Ni surface as a function of temperature and CO pressure. The measured data can be converted to isotherms if a relationship between work function and coverage is assumed. Christmann et al. state that the work function is a linear function of coverage up to a work function value of 1.1 eV, and that above 1.1 eV, the work function may be correlated with coverage using LEED patterns. They also state that the maximum value of the work function (1.31 eV) corresponds to a coverage of 0.53. Using a linear relationship* between work function and coverage, we have converted the reported data [26] to the isotherm data shown in Fig. 3.2, 3. Now we propose to use this information to determine the function ~'(T, 0). The first step is to determine the values of this function from the measured isotherm data. This requires an analytical expression for the isotherm. Under equilibrium conditions, the chemical potential of the molecules in the gas phase must be equal to the chemical potential of the adsorbed molecules. This condition allows us to obtain an expression for the equilibrium isotherms. We first introduce n~, M 0, and 0e: the equilibrium number of adsorbed molecules, the number of substrate atoms (each per unit area) and their ratio. 0e = n~
(3.2, 1)
M0 The chemical potential for the gas phase molecules is given in Eq. (3.0, 19) and that for the adsorbed molecules in Eq. (3.1, 17). After equating the chemical potentials and using the definitions of 0 e and M 0, one finds
0e Pe = (0M --
(3.2,
where we have also introduced the number of adsorption sites per substrate atom, 0 M M
oM = . .
(3.2, 3)
/no
The analytical expression for the isotherm, Eq. (3.2, 2), plays a fundamental role in what follows. It may be inverted to obtain an expression for ~'(T, 0) =
0e
(3.2, 4)
Pe ( OM - O, ) I[I"* To obtain an empirical isotherm, values of 0e are measured for each of several different pressures. Thus from each empirical isotherm, several values of ~'(T, 0) can be calculated from Eq. (3.2, 4) and the values of the parameters listed in the upper portion of Table 3.2, 1. Next, note that without loss of generality, Eq. (3.1, 15) may be written in the form
?Another possibility for determininga relationship between work function and coverageis to make use of the shape of the work function versus coverage plot measured by Wedler et al. [27], even though the measurementwas done on a polycrystalline Ni surface. This was the method used in a previously published paper [9].
315
/ where b is a function of temperature only and /3' is a function of coverage only. Then, from Eq. (3.2, 5) one finds kTln(~') = b - / 3 '
(3.2, 6)
so that the coverage and temperature dependence have been separated and thus to obtain the function ~(T,O), it is necessary to obtain the functions b(T) and fl'(O). To obtain the values of b(T) from the empirical isotherm data, one isotherm was selected as a reference (we chose T = 370 K). The reference isotherm was arbitrarily assigned a value of b (we chose b = 0). There is no loss of generality in the choice of the reference value of b, since b wilJ only be determined to within a constant. Once these ~rej were calculated at each of 7/points on the reference choices were made, values of isotherm from Eq. (3.2, 4). To select the value of b(T i) for the i th isotherm, the following equation was used
10 ~
5
~
0
~
-5 -10
I
,~ -15 -20 .
.
.
9
300
.
.
.
.
9
325
.
.
.
.
|
350
.
.
.
.
I
.
.
375
.
.
,
400
.
.
.
.
,
425
.
.
.
.
9
450
Temperature, K
Figure 3.2, 1. The function b as determined from the isotherms obtained from the measurements of Christmann et al. [26]. The data points were obtained from the empirical isotherms by the procedure described herein. The solid line is a fourth order polynomial fit of the data points.
316 7"/i
7"/i
Z bj(zi) Z b(Zi)=j=l ~i = j=l
(3.2, 7)
17i
where ~i is the number of coverage values selected on the i th isotherm at which to compare the values of krln(() with the reference isotherm. The superscript ref on a quantity refers it to the reference isotherm and the subscript j on a quantity refers it to the jth measured point. As may be noted from Eq. (3.2, 7), the value of b(T i) selected by this procedure is an average of the values measured at the different coverages. Since the empirical isotherms determined from the measurements of Christmann et al. [26] do not all span the same coverage range, isotherms that did not have enough points with coverages in common with the reference isotherm had to be referred to an intermediate isotherm. The value of b(Tn), already determined for.the intermediate isotherm, could then be added to determine the required value of b(T ~) with respect to the reference isotherm. Equation (3.2, 7) was used for each measured isotherm to obtain a value of b(T i) at the temperature of the i th isotherm. In Fig. 3.2, 1, the values of b(T ~) obtained by this procedure are shown as data points. Since each isotherm has several measured points (at different values of coverage), there will be several values of ~'(T, 0) and hence several values of (kTln(~')-b) or fl'(O) are obtained from each isotherm. In Fig. 3.2, 2, the
1.75 [ l
'~
1.74 1.73
1.72
"r
~e, I
1.71 .7
t .
0.1
-
.
0.2
l
l
.
0.3
,
-
9
0.4
l
,
0.5
Coverage (na / Mo) Figure 3.2, 2. The function fl' as determined from the isotherms obtained from the measurements of Christmann et al. [26]. The data points were obtained from the empirical isotherms by the procedure described herein. The solid line is a fifth order polynomial fit of the data points.
317 values of fl'(0) obtained in this manner are shown as data points. All of the measured points on an isotherm yield values for fl' even though only selected points (those with coverages in common with the reference or intermediate isotherms) are used to determine the single value of b for a given isotherm. Table 3.2, 1 Properties o f C arb.on Monoxide Adsorbed on Ni(111) Property
Value
Source
Mo(atoms / m2)
1.86 • 1019
geometry
M(sites / m 2)
0.53M0
Christmann et al. [26]
tOl(HZ)
1.23 x 1013
EELS [25]
o92(Hz)
5.57•
EELS [25]
bo - Co(J / molecule)
-2.0980 • 10 -19
b1(J / moleculeK)
2.0996 x 10 -20
b2 (J / moleculeK 2 )
-8.6513 x 10 -23
b3 (J / moleculeK 3 )
1.6017 x 10 -25
t)4 ( J / moleculeK 4 )
-1.1128x10 -28
c1(J / molecule)
1.5423•
c 2 (J / molecule)
-1.3471 x 10 -18
c3( J / molecule)
6.2692 x 10-18
c4 (J / molecule)
-1.3580 x 10-17
Cs(J / molecule)
1.1572x10 -17
-19
present work, obtained by fitting the measured data of Christmann et al. [26]
318
296 K --.------------- 315 K ~ _ ~ ~ 332 K
0.5
~
3
5 _~
4
347 K K 362 K
0.4 370K 377 K 384 K 0.3 391 K O
r~
399 K 0.2 405 K
412K 0.1
418K 425 K 431 K 437 K 2
4
6
8
10
Pressure, 10 -9 ton"
Figure 3.2, 3. Comparison of the measured and recalculated isotherms. The data points represent the values measured by Christmann et al. [26]. The solid lines are calculated from the properties listed in Table 3.2, 1 at the temperatures (in kelvins) listed for each isotherm. Note that the points shown in Figs. 3.2, 1 and 3.2, 2 are a translation of the isotherms into the functions fl'(O) and b(T). The double harmonic oscillator model of the adsorbed molecules dictates that b should be a function of temperature only. If there were any coverage dependence in b (i.e., different values were obtained for the bj(T ~) for a given
319 isotherm), it would result in the functions (kTln(~')-b) not lying exactly on top of each other for the same value of coverage. This would result in a temperature dependence in fl'(O). Indeed, in Fig. 3.2, 2 a slight temperature dependence can be seen in the data points; however the scatter in the data in Fig. 3.2, 2 is very small (note the expanded scale of the ordinate). To obtain the function ~'(T, 0) corresponding to a set of measured isotherms, we shall introduce the approximation b - fl' = b0 + bIT + b2T 2 + b 3 T 3+...-c o - q 0 - c 2 02 - c 3 0 3 . . .
(3.2, 8)
where the coefficients bi and ci will be found from the method of least-squares applied to the data obtained from the isotherms and shown in Figs. 3.2, 1 and 3.2, 2. Polynomial fits of the functions b(T) and fl'(O) are shown in Figs. 3.2, 1 and 3.2, 2 along with the data points. A fourth order fit was used for b(T) and a fifth order fit was used for fl'(0).t Note that the functions b(T) and fl (0) obtained in this way are only obtained to within an additive constant because the choice of the reference isotherm is arbitrary. This does not cause difficulty because in the equations in which these parameters will be used, they appear as b - f l ' and the arbitrary constant is eliminated when the difference is taken. Once the coefficients b i and c i have been determined from the empirical isotherms, they may be tabulated along with other material properties (see Table 3.2, 1) and used to obtain an expression for the chemical potential. The fitting technique that has been used to obtain the values of these coefficients has a number of advantages over techniques that fit individual isotherms. By introducing a reference isotherm and then including all of the available data points, one is able to use higher order polynomials in the fitting procedure than would be possible if only a single isotherm were used. Often on a single isotherm, there are a limited number of points available and the order of the polynomials used in the fitting procedure would, therefore, be restricted by the number of points on the isotherm. In addition, separating the coverage and temperature dependence allows one to introduce only one function, fl'(O), that applies to all isotherms so that one need not have the complete coverage range for a particular isotherm in order to use this procedure. The coverage range for which the function ((T,O) may be used is the entire coverage range spanned by the set of isotherms. An example of these aspects of the fitting procedure is illustrated in Fig. 3.2, 3 by the isotherm at 296 K where there are only two data points spanning a very small coverage range. This isotherm was included in the fitting and indeed the two points are reproduced from the empirically determined function ((T, 0). The function ((T, 0) was also obtained over the complete temperature range of the data in Ref. [26]. This is important to this study because we wish to obtain isotherm information over a temperature range that overlaps the kinetic data of Ref. [ 17]. This will then eliminate the necessity for extrapolation of the chemical potential function so that any error in the predicted kinetics can be attributed to the Statistical Rate Theory approach. In order to determine if the chemical potential expression that is based on the function ~'(T, 0) leads to a good description of the isotherms, it may be used to reproduce the isotherms from which it was determined. In Fig. 3.2, 3, the measured isotherms are given along with the isotherms that were calculated from the values of the coefficients listed in Table 3.2, 1. The agreement is seen to be remarkable. This result provides support for the double harmonic oscillator model of the adsorbed CO molecule on Ni(111). It was this model of the adsorbed molecule that led to the form of the isotherm equation, and it was this equation that allowed the coverage and temperature dependence of ~(T,O) to be separated. tThe fitting was done using the routine "Fit" in the softwarepackageMathematica. |
320 3.3. Comparison of statistical rate theory predictions with kinetic data We are now in a position to apply Statistical Rate Theory to the CO-Ni(111) system. The relations for Ke,/z g and /zCrgiven in Eqs. (3.0, 17), (3.0, 19) and (3.1, 17) may now be substituted into Eq. (3.0, 11) to obtain the expression for the net rate of adsorption
I(M-na)(llrP~p na ] j=dn.....~a= Pe e (M_nae)tYe dt ~/2~'mkT na - (M-n~) (~eO
(3.3, 1)
Note that in the above theoretically derived equation, the term ncr / ( M - n a), introduced by Gorte and Schmidt [14] for purely empirical reasons (see Section 1.2), appears in the desorption term in the rate expression. This coverage dependence in the desorption term has not been previously predicted from a theoretical formulation other than Statistical Rate Theory. The coverage dependence predicted by Statistical Rate Theory for the adsorption rate, particularly the ( M - n ~) / nCr factor, is different than that predicted in
1.1 1.o
0.9 0.8
0.7 ~]
0.6 ~
'0.5
.•••___,
0.4
r~
0.3
__ .~
0.2 0 .lu
0.1 r,r.,
0.0 -0.1 0
2
4
6
8
10
12
Time, s Figure 3.3, 1. Fractional coverage of CO on Ni(111) during isothermal (395 K) isobaric (5 x 10 -3 torr) adsorption (left branch) and isothermal (395 K), isobaric (5 x lb -6 torr) desorption (fight branch). The data points were reported in Ref. [17]. The solid line was calculated from Statistical Rate Theory using an equilibrium adsorption cross-section, o"e, of 0.083/~ 2. For the adsorption curve, the starting time used was 0.19 seconds and for the desorption curve, the starting time was 0.45 seconds.
321 the other theories. The coverage dependence in both of these terms will be examined by comparing the Statistical Rate Theory predictions with the experimental results reported by Rubloff [ 17]. In his experimental procedure, a fast-acting valve and nozzle were used to suddenly expose the surface to a high pressure / ) o n . This allowed adsorption to take place isothermally and lsobarlcally. After a period of ume, the valv~ was suddenly closed; thereby rapidly reducing the pressure to a lower value, P~ p. After the pressure reduction, isothermal, isobaric desorption took place. The coverage as a function of time was measured using ultra-violet photoelectron spectroscopy. Adsorption-desorption cycles were recorded at several different temperatures. Four of the experiments reported by Rubloff are shown in Figs. 3.3, 1-4. These are all of the experiments reported by Rubloff that are within the temperature range of the isotherm data that we have examined herein. After making use of the function definitions given in Eqs. (3.0, 20), (3.1, 16), (3.2, 5), and (3.2, 8) and the parameters listed in Tables 3.0, 1 and 3.2, 1, the only unknown quantity in the kinetic expression for the net rate of adsorption, Eq. (3.3, 1), is the equilibrium adsorption cross-section, cre . We shall assume it is a material property that is 9
o
.
'
9
,y
~
1.1 1.0 0.9 ~
D DO D 9 D 9
0.8 0.7 0.6
~ ~
P
0.5
P D
0.4
P
t,.
~.
0.3
O
--
0.2
.~
0.1
L.
0.0 -0.1 0
2
4
6
8
10
12
Time, s
Figure 3~3, 2. Fractional coverage of CO on Ni(111) during isothermal (408 K), i~obaric ( 5 x 10 -~ torr) adsorption (left branch) and isothermal (408 K), isobaric ( 5 x 10-" torr) desorption (right branch). The data points were reported in Ref. [ 17]. The solid line was calculated from Statistical Rate Theory using an equilibrium adsorption cross-section, cre, of 0.12~t 2. For the adsorption curve, the starting time used was 0.19 seconds and for the desorption curve, the starting time was 0.45 seconds.
322 only dependent on temperature. From Statistical Rate Theory, it is predicted to be an equilibrium property so it could also be pressure dependent. To examine this aspect of the theory, the value of a e will be inferred from the desorption portion of an adsorptiondesorption curve. If the coverage dependence of the predicted net rate of adsorption has been obtained explicitly from Statistical Rate Theory, then one value of (re should provide a prediction of the coverage as a function of time that is in agreement with the measurements for all values of the coverage during the desorption process. And if a e is, in fact, a material property that is only temperature dependent, then since the adsorption measurements were performed at the same temperature, the value of (re that was determined from the corresponding desorption measurements should also lead to a good prediction of the adsorption measurements [9]. Thus, the type of data shown in Figs. 3.3, 1-4 potentially would allow us to examine two aspects of the Statistical Rate Theory approach. As will be seen, there is a difficulty with the data that prevents them from serving this dual purpose. To infer the value of (re from the desorption portion of the experimental curves, we first integrate Eq. (3.3, 1). This operation gives an equation of the form (3.3, 2)
C a e ( t - t o ) = r i n g)
where
c= q2
Pe
kr,
(3.3, 3>
(M_nee)
and f ( n a) is given by
:<.~
n-f E
n~
ha(to)
na
-(M-na)~ll/P~p
],
dna
(3.3, 4)
The latter equation may be integrated numerically once a value of na has been specified. Thus, the time at which the system is predicted to reach the value na is given by (3.3, 5)
t = t o + f(na-----2) Cae
Next, it is necessary to determine the experimental time, tm, at which the system was observed to reach the coverage n a. The square of the difference between the predicted time to reach a particular coverage and the measured time summed over several data points is a measure of the error. To determine the time tm requires that the ordinate of the figures reported by Rubloff be converted to coverage. We take this ordinate, Y ( t ) , to be N (:r(t)- g e (P'samp)
a
off
(3.3, 6)
" "e \" samp )
Once the pressure and temperature are known, the isotherm equation, Eq. (3.2, 2), may be used to calculate the necessary equilibrium coverages indicated in Eq. (3.3, 6). The
323 experimental procedure adopted by Rubloff was to equilibrate the system at a low CO pressure with the valve off and then to suddenly expose the surface to CO at a pressure that was three orders of magnitude higher by suddenly opening the valve to the nozzle. It is indicated that the higher pressure is less than 5 x 10 -3 torr. We investigated several different possible pressures that span the range of pressure that could be reasonably expected to occur in the experiment For the results that follow we take p~ to be off 9 "~" 5 • 10- 3 torr and Psamv to be 5 • 10" - 6 torr. (The effect of errors' an these pressures on determining the adsorption cross-section will be discussed below.) Note that the fractional coverage is equal to unity when the surface has equilibrated at the higher pressure and is equal to zero when the surface has equilibrated at the lower pressure. The figures published by Rubloff were enlarged so that n values of the fractional coverage corresponding to n experimental times could be read approximately. For the i th data point, we then have a measured time, tmi, required for the system to reach the
1.1 1.0 0.9 0.8
~
0.7
~
0.6
~:~
0.5 0.4
@ --
@
0.3 0.2 0.1 U
I..,
r.r.,
0.0 -0.1 0
2
4
6
8
10
12
Time, s
Figure 3~3, 3. Fractional coverage of CO on Ni(111) during isothermal (422 K), isobaric ( 5 x 10-" tort) adsorption (left branch) and isothermal (422 K), isobaric (5 x 10 torr) desorption (fight branch). The data points were reported in Ref. [ 17]. The solid line was calculated from Statistical Rate Theory using an equilibrium adsorption cross-section, o"e, of 0.16,~ 2 . For the adsorption curve, the starting time used was 0.19 seconds and for the desorption curve, the starting time was 0.45 seconds. TM
324 particular coverage and a calculated time, tci, found from Eqs. (3.3, 3), (3.3, 4) and (3.3, 5). The square-error, E 2 , is then defined by E 2-
~(tmi--tci) 2
(3.3, 7)
i=1
If Eq. (3.3, 5) is substituted into the above equation, then requiring that the derivative of Eq. (3.3, 7) with respect to cre vanish, yields an equation for the value of cre that minimizes the square-error
1.1 1.0 09
,~ ~
0.8
0.6 0.5 0.4 0.3
0.2 .~,
0.1 0.0 m
-0.1 0.0 I
9 |
i
0.5
|
i
|
1.0
i
1.5
|
l
2.0
|
,
i
2.5
Time, s
Figure 3.3, 4. Fractional coverage of CO on Ni(111) during isothermal (435 K), isobaric ( 5 x 10-3 torr) adsorption (left branch) and isothermal (435 K), isobaric (5 x 10-" torr) desorption (fight branch). The data points were reported in Ref. [ 17]. The solid line was calculated from Statistical Rate Theory using an equilibrium adsorption cross-section, o"e, of 0.37~t 2. For the adsorption curve, the starting time used was 0.15 seconds and for the desorption curve, the starting time was 0.36 seconds.
325
o.e =
n
i=1
(3.3, 8)
C~.(tmi -to)Yi i-1 The equilibrium adsorption cross-section will be taken to be the one calculated from Eq. (3.3, 8) using the desorption portion of the measurements seen in Figs. 3.3, 1-4. The inferred values of o"e at the four temperatures that we have investigated are shown in Fig. 3.3, 5. The results shown in Fig. 3.3, 4 were measured at 435 K. At this temperature, Rubloff [ 17] reports approximately 4.5 seconds of data for the adsorption-desorption cycle. In another figure of Ref. [17], however, a blow-up of the data is shown in which the adsorption portion can be seen more easily. All of the desorption data were used to determine the equilibrium adsorption cross-section. The predictions of both the adsorption and desorption portion of each experiment are shown in Figs. 3.3, 1-4 as solid lines. In assessing the agreement obtained by this procedure, it should be noted that the predicted adsorption and desorption portions of the curves are slightly dependent on the choice of the location of zero fractional coverage and the choice of the location of a fractional coverage of unity. Also, starting times must be selected for both the adsorption curve and the desorption curve. The initial time for the start of the adsorption portion of a curve was taken as the time at which the coverage began to rise. Likewise, the initial time for the desorption portion of a curve was taken as the time at which the coverage began to decrease. Even with these uncertainties in fitting the desorption curve to obtain the equilibrium adsorption cross-section, the maximum deviation in the value of the crosssection so obtained was 10%. It is expected that this is within the error to which the crosssection is known when one considers the experimental scatter in the data. Further, we note that Rubloff does not explicitly state the pressure at which the experiments were performed, but only the ratio of the adsorption pressure to the desorption pressure. The effect of changing the nominal pressure at which his experiments were performed results in a change in the inferred values of the adsorption cross-section, o"e, but not the degree of agreement between the Statistical Rate Theory predictions and the experimental data. For example, if the pressures used in the experiment were an order of magnitude lower, the calculated cross-sections would be approximately an order of magnitude higher. To assess the explicit coverage dependence that is predicted from the Statistical Rate Theory approach, one may examine the desorption portion of each of the curves shown in Figs. 3.3, 1-4. There does not appear to be any measured disagreement between the predictions and the measurements for this portion of the adsorption-desorption cycles. The results in each of these figures would support the contention that the coverage dependence of the net adsorption rate is given explicitly by the Statistical Rate Theory approach. And since a single value of the adsorption cross-section in each of these experiments allows one to predict the coverage dependence, it appears that o"e is a material property. It is clear from Fig. 3.3, 5 that a e is temperature dependent. Whether it can be approximated as being only temperature dependent as opposed to being both temperature and pressure dependent is not entirely clear at present. Although the same value of a e that was inferred from the desorption portion of each experiment was used to predict the adsorption portion of each of the four experiments, and although the pressure for the adsorption portion of each experiment was three orders of magnitude higher than the existing pressure during the desorption portion of each experiment, and although there is no measured disagreement between the theoretical predictions of the net adsorption rate in any of the four experiments, we are not able to argue that cre is independent of pressure because the time for the adsorption process to be completed is so short that a range of
326
values of tr e will give an equally good fit of the data in each case. Certainly, the results are consistent with tre being only temperature dependent, but this possibility must be viewed as unestablished. It should be noted that although it would be useful if tr e were only temperature dependent, there is no theoretical reason from Statistical Rate Theory to think that it must be the case.
o,
0.5
0.4 r~
r,.)
0.3
-N
~k 0.2
0.1
.
.
.
.
.
390
.
.
.
.
.
400
.
.
.
.
|
410
.
.
.
.
.
.
420
.
.
.
.
430
.
.
.
.
.
440
.
.
.
.
!
450
T~n per at ur e, K Figure 3.3, 5. Values of the equilibrium adsorption cross-section inferred from R.obloff' s data [17]. In order to make the calculation, experimental pressures P~ and P~ were taken to be 5 x 10 -3 and 5 x 10-" torr respectively. -
With more detailed experimental results, tr e could be examined more carefully. According to Eq. (3.0, 18), its maximum value is the physical cross-sectional area of an adsorption site. Otherwise, the probability that a molecule striking an available adsorption site will adsorb, ~, would have to be greater than unity. The maximum possible crosssection would be the surface area per site (or l/M) which is approximately 10 .~2 for CO on Ni(111). As indicated in Fig. 3.3, 5, the values of cre inferred from these experiments ranged from 0.1 to 0.4 .~2.
3.4. Comparison of absolute rate theory predictions with kinetic data Rubloff [17] pointed out that his experimental results could be used to assess the coverage dependence that is derived from Absolute Rate Theory. He examined the desorption portion of the measurements performed at 435 K that are shown in Fig. 3.4.1, 1. He assumed "first-order" desorption kinetics and examined the possibility that K d and E d are independent of coverage. However, he also made an assumption
327 regarding the equilibrium coverages at the two pressures that, in view of the isotherms, is not justified. We first present his argument. on ) >> O. - (e,~, ) O.(~.m,
3.4.1. P r e d i c t e d d e s o r p t i o n r a t e u n d e r t h e a s s u m p i o n o f -
In the Absolute Rate Theory expression for the desorption rate, Eq. (1.1, 4), RuNoffs assumption of first-order desorption means that the function g(0) is taken to be g(0)= 0
(3.4.1,1)
After equating the rate of desorption to the negative rate of change of coverage, Rubloff found
d__~Od-t -oKa exp(- kff-~>
(3.4.1, 2)
And if both K a and E d are independent of coverage, the latter relation may be integrated to give 0(t) = Bexp(-t / z)
(3.4.1, 3)
where B is the constant of integration and 1/z is related to the pre-exponential factor Kd and the desorption activation energy E d
1/z = K d exp(-ff-~)
(3.4.1, 4)
Since in the experiments, the pressure on the sample when the valve was open was three orders of magnitude larger than the pressure when the valve was closed, Rubloff assumed
o~ >> Oe(P'f/~mp)
be (PJamp) of:
(3.4.1,5)
The validity of this assumption is discussed below; however, if this assumption is made and the ordinate Y(t), defined in Eq. (3.3, 6), is required to be unity initially, then according to Absolute Rate Theory, it is given by
Y(t) = exp(-t / "r)
(3.4.1, 6)
Rubloff then compared the values of Y(t) that could be predicted from Eq. (3.4.1, 6) for various values of z with the experimental results. These calculations are shown in Fig. 3.4.1, 1 as plain lines. As may be seen there, no single value of z gives agreement between the theory and experiment throughout the experimental period. Rubloff was led to conclude that both the pre-exponential factor and the desorption activation energy must change "markedly" with coverage if one is to obtain agreement between theory and experiment.
328
1.0
0.8
0.6
o,
0.4
u
02 .im
00
0.0 0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
4.0
4.4
4.8
Time, s Figure 3.4.1, 1. Fractional coverage of CO on Ni(111) during isothermal (435 K), isobaric ( 5 x 10 -6 torr) desorption. The data points were reported in Ref. [ 17]. The bold line was calculated from Statistical Rate Theory using an equilibrium adsorption crosssection, o"e, of 0.37/~ 2. The starting time was 0.36 seconds. Curves found from Absolute Rate Theory in which no coverage dependence was included in either Ke or E d and for o~ > > Oe(esamp off ) are shown are shown as plain lines. which it was assumed that Oe(Psamp)
3.4.2. Predicted desorption rate under the assumption of 0, (Ps, mp) = Oe(P,~,) On
"
Off
The assumption listed in Eq. (3.4.1, 5) allowed Rubloff to simplify his equation for the ordinate. However, since the analytical expression for the isotherm is now available, on we may calculate the values of Oe(Psam-) and Oe(P' soHm'.p ) 9 We note that this calculation p . . could not be made previously because an expression ~or the isotherm was not available. There is only one data point available on the isotherm at 437 K and only two data points available on the 431 K isotherm. As may be noted from Fig. 3.2, 3, the fitting procedure outlined in Section 3.2 reproduces the points on these isotherms when the properties in Tables 3.0, 1 and 3.2, 1 are used to obtain the expressions for ~'(T, 0), ~(T) and q~(T) and they are used with Eq. (3.2, 2) tO calc~alate the isotherms. Equation (3.2, 2) may also be used to calculate Oe(P~ and Oe(P~ at 435 K. One finds on Oe(Psamp) = 0.977
(3.4.2, 1)
and off ) Oe(Psamp
=
0.752
(3.4.2, 2)
329 Since these values of the relative coverage differ by less than 25%, neglecting one relative to the other does not appear justified. If the assumption in Eq. (3.4.1, 5) is not made, but one otherwise proceeds in the same fashion as outlined in the previous section, one finds that the ordinate may be expressed ""
on
off Oe(Psamp)
Y(t) = Oe(Psamp)exp(-t / "C)- ~
(3.4.2, 3)
__ Oe (Psamp) on Oe(Psamp) off
1.0
0.8
0.6
r
0.4
0 m I= omal
0.2
f,,,
00
0.0 L. 0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
4.0
4.4
4.8
Time, s Figure 3.4.2, 1. F r a c t i o n a l coverage of CO on, Ni(111) during isothermal (435 K), isobaric ( 5 x 10--" torr) desorption. The data points were reported in Ref. [ 17]. The bold line was calculated from Statistical Rate Theory using an equilibrium adsorption crosssection, o"e, of 0.37~ 2. The starting time was 0.36 seconds. Curves found from Absolute Rate Theory in which no coverage dependence was included in either K d or E d are shown as plain lines. In this latter expression, the parameter ~: still bears the same relation to the preexponential factor and desorption activation energy as that indicated in Eq. (3.4.1, 4). Thus, we may determine if there is any constant value of 1: for which there is agreement between the measurements and the predictions. The predictions obtained from
330 Eq. (3.4.2, 3) for four values of z are shown in Fig 3.4.2, 1 along with the measurements and the Statistical Rate Theory predictions. As may be seen there, no value of z leads to agreement between the predictions and the measurements throughout the experimental period. Thus, although Rubloff concluded, on the basis of an unjustified assumption, that the pre-exponential factor and activation energy of adsorption would have to depend on coverage in order to obtain agreement between the Absolute Rate Theory predictions and the measurements, his conclusion nonetheless appears valid. This circumstance is similar to the situation found from Absolute Rate Theory when it was applied to examine electron transfer reactions between isotopes. 3.4.3. Motivation for the introduction of precursor states Results of the type shown in Figs. 3.4.1, 1 and 3.4.2, 1 have provided the motivation for introducing the concept of precursor states into the Absolute Rate Theory equation for desorption [28, 29], as such states had been introduced earlier to account for a lack of agreement between the observed and predicted adsorption rates [ 13]. However, since the Statistical Rate Theory correctly predicts the coverage as a function of time, there does not appear to be any need for introducing this concept, at least for the CO-Ni(111) system.
4.0. SUMMARY AND CONCLUSIONS To examine the Statistical Rate Theory approach, it has been applied to predict the concentration dependence as a function of time in two types of reactions. The objective of this approach is to derive explicitly, the complete concentration dependence of the reaction rate expression and to have only molecular or material properties appearing in the expression for predicting the rate of a process. The basis for the derivation is a simple quantum mechanical model of an isolated system and the Boltzmann definition of entropy. The instantaneous state of the system is described in terms of the particle (electron, atom, or molecule) distributions within a phase or phases. As a special case, the particle distributions within each phase are assumed to be spatially homogeneous. This type of system is general enough to consider either chemical reactions which take place homogeneously within one phase or adsorption kinetics where the rate of adsorption is controlled by the kinetics at the gas-solid interphase. Two assumptions are introduced that allow the expression for the quantum mechanical transition probability for a change from one particle distribution to another at an instant to be obtained in terms of the number of microscopic states in the original distribution and in the virtual distribution, and the rate of exchange between quantum mechanical states of the two distributions, K e. These assumptions lead to a simple interpretation of the reason for the increase in entropy in an isolated system. Also, they lead to the prediction of the rate of change of the particle distribution in terms of molecular and material properties and in terms of a non-equilibrium variable. In the case of the reaction occurring homogeneously within one phase, this variable is the bulk concentration and in the case of adsorption kinetics, it is the surface coverage. It is found that K e is equal to the equilibrium exchange rate of the reaction occurring homogeneously within one phase and, in the case of adsorption kinetics, it is the equilibrium exchange rate between the gas phase and the adsorbed phase. Thus, if the theory is correct, one value of K e should allow the concentration to be predicted from an initial nonequilibrium value throughout the period of system evolution. This is the basis for testing the theory in two circumstances. In the first circumstance, electron transfer reactions between isotopes dissolved in electrolytes are considered. The special properties of this type of reaction lead to an expression for the reaction rate that does not contain any material properties and the concentration dependence of the reaction rate is predicted explicitly [3]. The predictions
331 that follow are compared with concentration measurements in reactions between silver ions, between vanadium ions, and between manganate and permanganate ions. The measurements had been previously reported by Wahl and co-workers [20-22]. It is found that for one value of K e in each case, the Statistical Rate Theory predictions are in reasonable agreement with the measurements throughout the period of system evolution [e.g., see Fig. 2.3, 1 and 2.3, 2]. In contrast, Absolute Rate Theory does not give an explicit expression for the rate of a chemical reaction. The concentration dependence of the reaction is obtained from an interpretation of the equilibrium constant. This interpretation depends strongly on the concept of equilibrium between the activated complex and the reactants, but it is arbitrary. The Absolute Rate Theory prediction is such that it can not be in agreement with the measurements both initially and finally. An equally valid interpretation of the expression for the equilibrium constant leads to the Statistical Rate Theory expression for the rate of the reaction [23] and it is in agreement with the data both initially and as the system approaches equilibrium. In the second process considered (adsorption kinetics), material and molecular properties play a major role in the Statistical Rate Theory expression for the rate [9]. To examine the Statistical Rate Theory approach, it is applied to the isobaric, isothermal desorption of CO from Ni(111). The molecular properties of CO appearing in the rate expression have been previously tabulated by others (see Table 3.0, 1). To determine the material properties of the C O - N i ( l l l ) interface, the adsorbed CO molecule is approximated as a double harmonic oscillator. Based on this model, it is found that the information required to predict the adsorption rate necessitates knowledge of the number of adsorption sites and a set of equilibrium isotherms. If the fundamental vibration frequencies of the adsorbed molecule are known, they can be incorporated, but it is not necessary that they be available. This empirical information is found to be sufficient to determine all of the material properties appearing in the rate expression except one. The one remaining property arises from the equilibrium exchange rate K e. The equilibrium exchange rate is expressed in terms of the adsorption cross-section, cre (T,P). According to the theory, the equilibrium adsorption cross-section is a material property of the CONi(111) interface and as such depends only on the temperature and pressure at which the process occurs. Thus for isothermal, isobaric kinetics (adsorption or desorption), there should be one value of ae(T,P) that allows the surface concentration to be predicted throughout the period of the kinetic process. This is the basis on which the theory was tested [9]. Rubloff has reported measurements of the isothermal, isobaric adsorption of CO on Ni(111) and then at a pressure three orders of magnitude less, but at the same temperature, isothermal, isobaric desorption in the same system. The value of Cre(T,P) was inferred from the desorption portion of a cycle. In each of four experiments that were performed at different temperatures, it was found that one value of Ge(T,P) led to predictions that were in close agreement with the measurements (see Figs. 3.3, 1-4). Thus it would appear that for this system, the Statistical Rate Theory approach successfully led to the explicit coverage dependence of the rate expression. If it is assumed that O'e(T,P) is independent of pressure, then the value of Cre(T) inferred from the desorption portion of a cycle may be used to predict the surface concentration during the adsorption portion of a cycle. This calculation is performed using the same equation as that used to predict the desorption portion of a cycle. Only the initial surface concentration and the pressure are different than for the desorption portion. Although it is found that the predicted surface concentration for the adsorption portion of the cycle is in agreement with the measurements for all four temperatures (see Figs. 3.3, 1-4), we are not able to conclude that cre is independent of pressure. It is found that a range of values of cre also gives reasonable agreement for the adsorption portion. Only one equation is used in the Statistical Rate Theory approach to predict either the rate of desorption or of adsorption. It is only necessary to take into account the physical
332 conditions under which the process takes place. This is very different than the Absolute Rate Theory approach. In the latter approach, there are two different equations used, one for desorption and one for adsorption. When the Absolute Rate Theory equation is used to predict the desorption data reported by Rubloff, it has been previously reported [17] that agreement can not be found both initially and finally unless coverage dependence is introduced into the pre-exponential factor K,/ and the desorption energy E d. This observation is similar to the results found for the electron transfer reactions where Absolute Rate Theory was also found to be unable to predict results that were in agreement with the data both initially and near equilibrium. It is common to introduce the concept of precursor states with Absolute Rate Theory equations as a means of introducing the necessary coverage dependence or to use the concept of precursor states with sticking probability [13]. In the latter case, the expressions for the rate then contains parameters that can depend on coverage [ 16]. The coverage dependence that is obtained from the Statistical Rate Theory approach is obtained from the expression for the chemical potential. The coverage dependence of the latter is obtained by modeling the adsorbed diatomic molecule as a double harmonic oscillator. Then, the coverage dependence of the energy of the adsorbed molecule is required to be such that the total coverage dependence of the chemical potential expression is in agreement with the measured equilibrium adsorption isotherms. It might be argued that this is only the equilibrium expression for the chemical potential and that the expression would not be valid when the system is far from equilibrium; however no evidence is seen in the predictions that would suggest that the expression is strongly limited in this regard. Since the Statistical Rate Theory approach appears to predict the coverage dependence of the rate equation, it is not clear that there is any need to introduce the concept of precursor states. Before this conclusion can be strongly drawn, Statistical Rate Theory needs to be applied to other well defined systems besides CO-Ni(111). In its present state of development, this requires the availability of the equilibrium adsorption isotherms. The Absolute Rate Theory approach, even with the precursor states included, does not require any relation between adsorption kinetics and the adsorption isotherms of the system.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
C.A. Ward, J. Chem. Phys., 67 (1977) 229. C.A. Ward, R. D. Findlay and M. Rizk, J. Chem. Phys., 76 (1982) 5599. C.A. Ward, J. Chem. Phys., 79 (1983) 5605. P. Tikuisis and C. A. Ward, In: Chhabra, R. and DeKee, D. (eds.), Transport Processes in Bubbles Drops and Particles, Hemisphere Publishing Co., New York (1992) 114-132. C.A. Ward, M. Rizk and A. S. Tucker, J. Chem. Phys., 76 (1982) 5606. C.A. Ward, P. Tikuisis and A. S. Tucker, J. of Colloid and Interface Science, 113 (1985) 388. C.A. Ward and R. D. Findlay, J. Chem. Phys., 76 (1982) 5615. R.D. Findlay and C. A. Ward, J. Chem. Phys., 76 (1982) 5625. C.A. Ward and M. Elmoselhi, Surf. Sci., 176 (1986) 457. C.A. Ward and M. B. Elmoselhi, Surf. Sci., 203 (1988) 463. K.J. Laidler, Theories of Chemical Reaction Rates, McGraw-Hill Book Company, New York (1969) 45. A. Clark, The Theory of Adsorption and Catalysis, Academic, New York (1970), 210. M.A. Morris, M. Bowker, and D. A. King, "Kinetics of adsorption, desorption, and diffusion on metal surfaces" in Simple Processes At The Gas-Solid Interface, eds. C. H. Bamford, C. F. H. Tippler, and R. G. Compton, Elsevier, New York, 1984. R. Gorte and L. D. Schmidt, Surf. Sci., 76 (1978) 559.
333 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
P. Kisliuk, J. Phys. Chem. Solids, 3 (1957) 95. D.A. King, Surf. Sci., 64 (1977) 43. G.W. Rubloff, Surf. Sci., 89 (1979) 566. J. Willard Gibbs, Elementary Principles in Statistical Mechanics, Yale University Press, 1902; Dover Publications, N. Y. (1960) 16. I. Prigogine and R. Delay, Chemical Thermodynamics, Longmans, London, (1954) 357. B.M. Gordon and A. C. Wahl, J. Am. Chem. Soc., 80 (1957) 273. J.C. Sheppard and A. C. Wahl, J. Am. Chem. Soc., 79 (1957) 1020. K.V. Krishnamurty and A. C. Wahl, J. Am. Chem. Soc., 80 (1958) 5921-5924. F. K. Skinner, C. A. Ward and B. L. Bardakjian, Biophysical Journal, 65 (1993) 618-629. T.L. Hill, An Introduction to Statistical Thermodynamics, Dover Publications Inc., New York (1986) 147. W. Erley, H. Wagner and H. Ibach, Surf. Sci., 80 (1979) 612. K. Christmann, O. Schober and G. Ertl, J. Chem. Phys., 60 (1974) 4719. G. Wedler, H. Papp and G. Schroll, Surf. Sci., 44 (1974) 463. M.R. Shanabarger, Surf. Sci., 44 (1974) 297. C O. Steinbrtichel, Surf. Sci., 51 (1975) 539.
This Page Intentionally Left Blank
W. Rudzifiski, W.A. Steele and G. Zgrablich (Eds.)
Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces Studies in Surface Science and Catalysis, Vol. 104 9 1997 Elsevier Science B.V. All rights reserved.
335
A N e w T h e o r e t i c a l A p p r o a c h to A d s o r p t i o n - D e s o r p t i o n K i n e t i c s on E n e r g e t i c a l l y H e t e r o g e n e o u s F l a t Solid Surfaces B a s e d on S t a t i s t i c a l R a t e T h e o r y of I n t e r f a c i a l T r a n s p o r t
W. Rudzifiski Department of Theoretical Chemistry Maria Curie-Sktodowska University Lublin 20-031, POLAND
1. I N T R O D U C T I O N In the theories of adsorption-desorption kinetics, the mass balance of the adsorbate over the entire heterogeneous solid surface is usually written in the following form, = - M Z x_, _
(I.I)
(~0i
-
i
where M is the total number of sites on the solid surface; 8i means the fractional coverage of sites of i-th type; Xi is the fraction of these sites on the solid surface and t is the time. Most frequently, 00 is taken to be the expression offered by the Theory of Activated Adsorption-Desorption, (TAAD) hi- = K
p(Z -
e)
exp
-
where s is the number of adsorption sites involved in an elementary adsorption-desorption process, p is the pressure in the gas phase, e= and ed are the activation energies for adsorption and desorption respectively, K~, Kd are slightly temperature dependent parameters. Further, T and k are the absolute temperature and the Boltzmann constant respectively. Equation (1.2) represents the most commonly used, particular application of that approach thoroughly discussed in the previous chapter by Elliott and Ward. Now, let us consider for simplicity physisorption and the case s -- 1. So, at the equilibrium when ~0e = 0, eq.(1.2) yields the Langmuir isotherm equation 0(e)(p,W ) =
Kpexp {k-~} 1 + Kpexp {~}
(1.3)
where K = - ~ , e = ( e d - e~), and where the superscript (e) refers to equilibrium. At equilibrium, the assumption of a discrete distribution of the fraction X of adsorption sites among corresponding values of e, expressed in eq.(1.1), leads to the following expression, 0(e)(p,T)=~Xi Kipexp{ kW } i 1 + Kip exp { kW-a}-
(1.4)
336 where now 0}*) means the "total", (average) fractional occupancy of all adsorption sites. In the case of the actual (real) solid surfaces one usually deals with a dense spectrum of adsorption energies which should be represented rather by a continuous function X(e), so that, 0~e)(p, T) = f ~g
II
Kpexp {k-~} X(e)de 1 + Kpexp {k-Y}
(1.5)
where X(e) fulfills the normalization condition
/
=1
f~
and ft is the physical domain of e. For the mathematical convenience, fI is frequently assumed to be the interval (-co, § or (0, § It was shown that replacing the true physical domain 12 E (fl, em), (et and em meaning the lowest and the maximum values of e for a heterogeneous solid surface), by (-oo, +oo) or by (0, +c~) does not affect much the behaviour of the calculated isotherm provided that extremely low or high surface coverages are not considered[l]. Hundreds of papers have been published showing how much the behaviour of the actual gas/solid adsorption systems is influenced by the adsorption energy distribution in these systems. Finally, two monographs on these surface heterogeneity effects have been published[i,2]. At small surface coverages, the second term on the r.h.s, of eq. (1.2) can be neglected. Many papers on the experimental studies of adsorption kinetics[3] were published, but the reported data did not obey the Langmuirian kinetics represented by the first term on the r.h.s, of eq. (1.2). Thus, various empirical laws were formulated to correlate the experimental data for adsorption kinetics. The first attempts by Roginski and Zeldovich [4,5] to provide a theoretical explanation for these empirical laws employed the concept of adsorption on an energetically heterogeneous solid surface. Later on that concept was more thoroughly elaborated by Aharoni and coworkers[6-8], and more recently by Tovbin[9] and Cerofolini[10]. Studies of desorption kinetics were carried out even more extensively. They were stimulated by the wide application of the Temperature Programmed Desorption (TPD) experiments to study the energetic properties of catalysts and catalyst supports[l 1]. The lack of applicability of the Langmuirian desorption kinetics represented by the second term on the r.h.s, of eq. (1.2) was detected very soon. It was observed, that the activation energy for desorption ed changed with the surface coverage, so, the theoretical analyses of TPD desorption spectra started with the following equation,
O}e)(p,T),
0t = Kd0S exp
--
(1.7)
In most cases, the dependence of ed on 0 was explained as originating from in the energetic heterogeneity of the actual solid surfaces, characterized by the dispersion of the activation energies for desorption.
337 Although every adsorption process is accompanied by a simultaneous desorption process, and vice versa, the studies of adsorption-desorption kinetics proceeded historically along two separate routes. One group of scientists studied kinetics of adsorption at low surface coverages, and neglected desorption phenomena in their studies. This group of scientists treated the surface energetic heterogeneity as a dispersion of the activation energies for adsorption, across a solid surface. The second group of scientists investigating desorption at high surface coverages, mostly in TPD experiments, treated the surface heterogeneity as the dispersion of activation energies for desorption. Provided that we accept TAAD, we should consider the energetic surface heterogeneity as a simultaneous variation of e~ and ed values, from one adsorption site to another. So, let us consider the generalization of eq.(1.2) for a heterogeneous solid surface, characterized by continuous spectra of the adsorption and desorption energies. Provided that every adsorption site is characterized by a certain pair of values (e~ and ed) for the whole heterogeneous surface, we have OOt
--s
--ed
t~ ~d
where X(e=, ed) is a two-dimensional differentialdistribution of the fraction of the surface sites among corresponding pairs of the values {c=, ca}. Following the theoretical results obtained for the case of adsorption equilibria,one may assume that K= and Kd are practicaUy the same for all adsorption sites. Apart from the mathematical problems posed by the solution of the integral equation (1.8) with respect to X(e=, ca), one faces problems of a fundamental nature. Namely, first of all, one must know the analytical form of 0 as the function of e= and ed. Statistical theories of adsorption equilibria provide us with the functions = 0(ed - e=). Thus, in order to make use of the statistical theories of adsorption, one must establish the relationship between c= and ed- Seeking for the relationship between ed and e= on different adsorption sites seems to be a difficultfundamental problem. Finally, the lack of applicability of Langmuirian adsorption kinetics was reported for typical physisorption systems in which the sense of the activation energy for adsorption seems to be difficultto interpret. All the above mentioned difficultiesdisappear when, as the starting point, one applies Statistical Rate Theory of Interracial Transport[12]. 2. A S I M U L T A N E O U S D E S C R I P T I O N OF A D S O R P T I O N E Q U I L I B R I A AND KINETICS
The starting point of our consideration is the equation developed by Ward and Findlay[12].
0o ,[ 0t = Kgs exp{
kT
]
} - exp{-(PgkT- #') }
(2.1)
where #g, #s are the chemical potentials of the gaseous and adsorbed (surface) molecules respectively, and K;s is a constant. They assumed next that the transient configurations of adsorbed molecules are close to the equilibrium ones.
338 Now, let us consider the Langmuir model of adsorption , i.e., one-site occupancy (localized) adsorption when no interactions exist between adsorbed molecules. Then, #s
0
k--T = In q~(1 - 0-----~
(2.2)
where qS is the molecular partition function of the adsorbed molecules. Accepting the ideal-gas approximation for #g, we have
~g
~0g
kT = k"-T+ In p
(2.3)
Now, let us consider the region of low surface coverages, i.e., neglect the second term within the square brackets in eq (2.1). As the adsorption kinetics is essentially a non-equilibrium process, we introduce the superscript (n) at the surface coverages appearing in the equations for adsorption kinetics. Then, 0 0 (n)
0t
/~g - #~
= K'g~exp
kT
,
~
#og ~ 1 - 0 (n)
= Kq~pq exp{ kT"
0(n)
(2.4)
The main effect of the energetic heterogeneity of the actual (really existing) solid surfaces is related to the dispersion of the minima in the gas-solid potential function across a solid surface. In the case of localized adsorption, these local minima are called "adsorption sites", and the value of the gas-solid potential at these local minima, taken with a reverse sign, is called - "the adsorption energy", and usually denoted by e. So, while considering the kinetics of adsorption on a heterogeneous solid surface we will write the molecular partition function qS as the following product, s qS= qos exp{ ~_..~}
(2.5)
where q~ is the same for all the adsorption sites, and e varies from one to another site. We will denote further by K the following product, K = q0 exp
~-~
and rewrite eq (1.3) to the following form,
[
011= 1 +
(00 /0t)
K~sp K exp{~-~}
]_1
(2.7)
The above equation describes the rate of adsorption on adsorption sites having adsorption energy equal to e. The experimentally measured surface coverage 0}n), and the mean rate
339 of adsorption (O0}")/Ot), are the values 0 ('0 and (O0('O/Ot) defined in eq (2.7), averaged over all kinds of adsorption sites taken with an appropriate statistical weight. With a dense spectrum of adsorption energies on a solid surface, that statistical weight becomes practically a continuous differential distribution of the number of adsorption sites among corresponding values of adsorption energy, X(e). Thus, in the case of a heterogeneous solid surface, the experimentally determined values O}~) and (O0}n)/Ot) are to be related to the following average;
/I 1 + /00K~spoj0t/ 0}n)= K exp{~--~} +oo
--1
(2.8)
)~(e)de
When equilibrium is attained, 0 = 0 (~) is given by the condition #~ = tt g, i.e., 0 (e)=
[l + ~ e x1p {
-e
(2.9)
}
and the experimentally measured surface coverage O~~) is given by, +co
0~e) = /
1 e x p { ~- e } ] -1 X(e)de 1 + ~pp
(2.10)
--OO
where the superscript (e) in 0~~) refers to equilibrium conditions. Both eq (2.8) and eq (2.10) can be written in the same form,
+?[
O~i) =
c~ i) _
1 +exp{
kT
]_1
}
X(e)de,
(2.11)
i = e, n
For the non-equilibrium conditions e!i) = e!'~), e~n) = kW in (00(n)/Ot) K~spK
(2.12)
whereas when equilibrium is attained, e!0 = e!'), e~~)
-
(2.13)
- k T ln(Kp)
When T ~ 0,
0 (i) tends to the step function
O~i), oo
l i m = 0 ~ i)= { 0 ' f o r e < e ! i) and then 0t(i)= f X(e)de T--.0 1, for e >_ e~i);. ' e( i )
(2.14)
340 for both equilibrium and non-equilibrium conditions. This is shown in Figure 1, where the function 8(e!0, T), written in the following form exp { ~ } 8(e~i), T) = 1 + exp { u }
(2.15)
is shown, as the function of the dimensionless variables er = ( e - e!O)/kTo, and r = T/To. For the reduced temperature r = 1, the function (2.15) is very close to the step function
(2.14). I:=1 1.00 -I f . =
.=
i
,
0.80 m
0.60 --
~###3
.
S
/,"
.=
m
"~II J
0.40 -.
.--'
0.20 --
0.00
-
-
"
,,
i
.=
jill
-20.00
I"llllllallllllllllllllllilllJlllll -10.00 0.00
10.00
II 20.00
Figure 1. The temperature dependence of the function 8(e, ec). The dimensionless temperatures are:r= 1)(--),r=5(---),andr= 10( . . . . ). In the theories of adsorption equilibria on heterogeneous solid surfaces, the step function defined in eq (2.14) is called usually the "condensation isotherm". The application of CA (Condensation Approximation) was studied thoroughly in the theories of equilibria of adsorption on heterogeneous solid surfaces[i]. It was shown there that in the case of adsorption on a heterogeneous solid surface, the essential condition for the applicability of the CA approach is that the variance of x(e) must be at least 10% larger than the variance of k--~-]. Of course, the same will be true also for
341 adsorption-desorption kinetics. Replacing the true kernel 0(0, i = e, n in eq. (19) by its corresponding step function (2.14) means assuming that the adsorption proceeds gradually on various adsorption sites in the sequence of the decreasing adsorption energy e. At a given temperature T, pressure p, and (0~___..2)),adsorption "front" is on the sites whose /
__
%
f oo(")
energy e is equal to ec given in eq. (2.12). But then, the overall adsorption rate ~ ~ )
is, in fact, governed by the local rate of adsorption, on the sites whose adsorption energy is equal to ec, through the obvious relation
( ) ~=~
00~n) = Const. x(ec) 0t ~, 0t
(2.16)
In other words, e~'~) in equation (2.15) can be considered as,
•
0t
e!~) = kT In
(2.17)
X(e!"))I~g~pK
where Kgs = Const. K'gs. One may argue that because eq. (2.1) does not apply at very small (0 --+ 0) or very high (0 --+ 1) surface coverages, one cannot accept the step isotherm in our consideration. However, it is obvious that the picture of a sharp "adsorption front" will be a very good representation for the true isotherm (2.15) at not too high temperatures, and the true isotherm (2.15) does not violate the condition that 0 cannot be very small or very high at e = e!i), i.e. on the adsorption sites where the kinetics of the "local" adsorption governs the kinetics of the "total" adsorption through the relation (2.16). For some distributions X(e), the integration in eq. (2.14) can be performed in an exact analytical way. This, for instance, is the case of the rectangular adsorption energy distribution,
X(e)
r 0,
=
1
.
forez<e<em" . . . elsewhere
(2.18)
leading to the generalized Temkin adsorption isotherm, in the case of adsorption equilibria. 0~r _
1 + exp{ ~=-r kT } ] k____~TIn -4"' era-el [l+exp{ q ]kZ}
(2.19)
The corresponding equation for adsorption kinetics is obtained from eq (2.19) by replacing e~r by e~'~)n. Then
oe~n) = 0t
Kg, pK exp{ el exp{e~-q#=)} _ exp{ e=-q } kT v t RT } 1{ ~ }
w h e r e / ( g s - Gs(em --r
kT
-1.
(2.20)
342 Eqs (2.19) and (2.20) are not subjected to any temperature limits, and are valid at high temperatures too. (Provided that the adsorption can be considered in terms of localized adsorption). Eq. (2.20) is convenient to demonstrate that it is not essentially the condition T ---, 0 for the Condensation Approximation to be applicable. The essential condition is that the variance of the adsorption energy distribution must be larger than kT. In the limit T ~ 0 or more generally when .kT is small, eq. (2.20) yields the equation, r Ote _._ ~m - er em--el
(2.21)
obtained for the adsorption energy distribution (2.18), by adopting the Condensation Approximation formulated in eq. (2.14). The corresponding equation for adsorption kinetics reads, s
0~0~
ot =
exp{-
6 m - - El
kT
(2.22)
Eq (2.22) is just the Elovich equation which is probably the most popular one to correlate experimental data for adsorption kinetics. Originally, it was launched as an empirical equation, but later on, it was associated intuitively with a constant, (rectangular), of adsorption (desorption) energies. The attempts to derive it theoretically always started with the Temkin isotherm for adsorption equilibria. But next, several additional assumptions had to be made, to arrive at the Elovich kinetic equation. This made all these deviations obscure and the nature of Elovich equation remained still a half-empirical one. The general procedure proposed here by us leads to the kinetics equations, in which the parameters have precise thermodynamic interpretation. That procedure allows one to predict adsorption kinetics from the adsorption equilibria and vice versa. The integration of eq (2.22) leads to the following expression, 8~=) =
k_.....T_TIn [apt + 11
(2.23)
6m - - 61
where c~ =
6m-
kT
61
s
K~K exp{~--~}
(2.24)
The comparison of eqs (2.22) and (2.23) must bring one to the conclusion, that investigating linearity of the plot ln(OO~)/Ot) vs. 0~~) In -ae~ ~ = const,
kT -
~
(2.25)
is to be recommended to check whether the adsorption kinetics is Elovichian. The linearity of the plot 0['~) VS. ln(apt + 1), predicted in eq (2.23) involves the necessity of making an always suspicious assumption concerning the value of a-parameter.
343 As eqs (2.19) and (2.20) represent the exact results of the integrations in eqs (2.10) and (2.8) when X(e) is the function defined in eq (2.18), they reduce correctly to appropriate equations for a homogeneous solid surface when ~ ~ 0. (To check, one must apply de l'Hospital's rule.) Though the shape of X(e) for an actual solid surface is expected to be a complicated curve in general, its "smoothed" version can, to a first crude approximation, be represented by the gaussian-like curve, x-exp{ ~-=-~} r X(e) = [1 + exp{~-~}] 2
(2.26)
centered at e = e0, the dispersion of which is related to the heterogeneity parameter c. This function has a physical background related to some universal features of energetically heterogeneous solid surfaces[I]. When equilibrium is attained, from eqs (2.13), (2.14) and (2.26) we obtain, /9~r =
[
1 + exp{ c(r - eo }
]_1
(2.27)
C
or, in another form, 0~,)_ (Kop) kT/r - 1 + (Kop) kT/r
0 < kT/c < 1
(2.28)
where s
Ko = K exp{~-~}
(2.29)
Eq (2.28) is the well-known "generalized Freundlich', (Bradley's) isotherm which is so commonly used to correlate experimental adsorption isotherms. It reduces to Freundlich isotherm at low surface pressures (coverages), lim 0~~) = (Kop) kT/r
p--.,O
(2.30)
Now, let us consider the form of 0}'0, and (00~'~)/0t) corresponding to the gaussian-like adsorption energy distribution (2.26). For obvious physical reasons, there must be a certain minimum, and a maximum value of the adsorption energy e, on a heterogeneous solid surface, ez and era. Thus, the gaussian-like function (2.26) should, for various values of the parameter c, be viewed correctly in the way shown in Fig.2. All the functions shown in Fig. 2 are normalized to unity, and given by the equation, 1 ~1 exp{ ~--=-~r l X(e) = ~NN [1 + exp{~-~}] 2
(2.31)
344 0.60 O
r~
0.48
(D (D
0.36
O ...
~
O
<
0.24 /
."
/
,
9
/
0.12 -
-%
-
/
~f
~. 9
9
%
.
9
%
"
%
o
0.00
I
0
I
2
"" 3
I
I
4
5
I 6
9* 9 .
I 7
o
8
9
Energy of adsorption Figure 2. The effect of the heterogeneity parameter c on the shape of the adsorption energy distribution X(E), given in eq. (39). Here EU is a certain energy unit comparable to kT. As it can be deduced from this figure, el = 2EU, e0 = 5EU, and em= 8EU. The values of the parameter c are: 0.5EU (. . . . ), 1.0EU ( - - - ) , and 5EU (--). One can see that as the heterogeneity (parameter c) increases, X(e) tends to rectangular adsorption energy distribution. where the normalization factor
FN reads, -1
FN=
l+exp{e]
eo}
-- l + e x p {
kT
}
(2.32)
Figure 2 shows that when c ~ oo, the function in eq (2.31) becomes the rectangular (constant) energy distribution. It means, the Elovich equation is likely to be a limiting form of all kinetic equations when the surface is strongly heterogeneous. With finite integration limits, the applicability of the Condensation Approximation was discussed in the recent monograph by Rudzifiski and Everett[l]. As it was demonstrated above, in the case of the rectangular energy distribution, the condition for the CA approach to be applied is not solely T ~ 0, but ~------Lt ~T ~ c~. Practically, that condition means, that the ratio of the variance of (00(e)/Oe), or (00(n)/Oe) to the variance of X(e), should be smaller than 0.9. In the case of the function X(e) defined in eq. (2.26), that ratio is represented by kT/c. As the function (2.31) resembles to some extent the rectangular distribution (2.18), we will accept now the relation (2.17) as an approximate one in eq. (2.14). Namely we will treat X(e!n)) as a constant, while calculating the integral (2.14). For the function defined
345 in eq. (2.26), the expression for 0~'0 reads, ~ =
[
1 + exp{ d~ c
}
1
Considering eq (2.17) and solving eq (2.33) with respect to O~
"- KgspK~ (1 -r n) 0~n))
(2.33)
(O01")/Ot),we obtain, (2.34)
though the constant Kgs in eq. (3.34) is not identical to that in eq. (2.20). The comparison of eqs (2.34) and (2.4) shows, that the gaussian-like dispersion of adsorption energies causes the term ( 1 - 0~n))/0~'0 to be powered by c/kT. Now, let us investigate the simplified form of eq (2.34) valid at low surface coverages, 00~=) 0t = Kg~pK0(O~"))-r
(2.35)
In other words, let us investigate the kinetics of adsorption corresponding to the Freundlich adsorption isotherm. After integration we obtain, (2.36) In the case of a homogeneous solid surface, instead of Freundlich region we have Henry's region in the limit of low pressures (coverages). Then instead of eq (2.36), we have, 0 (') = [2Kg~Kptexp{~-~}] 1/2
(2.37)
Thus, in the case of a homogeneous solid surface, the double-logarithmic plot In 0('0 vs. 1 In t should, (at small pressures), be a straight line with the tangent (01n 0(")/0 In t) = ~. In the case of a heterogeneous solid surface that double-logarithmic plot should also be linear, but the tangent (OInSC'O/Olnt)= (1 + ~T) -1. Because the values of c/kT are larger than unity, the tangents (0 In 0('0/0 In t) smaller will indicate adsorption kinetics on a heterogeneous solid surface. As the values thane k-Y _- ~a would be very typical for many of the actual solid surfaces, so, the tangent (OlnO(=)/Olnt) = ~ should also be typical. One can see it in Fig.3 in the paper by Aharoni and Ungarish[13], which is redrawn in Figure 3. With rising temperature the ("condensation") approach will be less and less accurate. Rudzifiski and Jagietto (RJ) have shown, that for typical heterogeneous surfaces and typical physical regimes at which the gas-solid adsorption experiments are carried out, the CA approach can be safely applied. If a question arises whether CA approach is
346
-/
, /O
/
O /
u
9
~,,~,
e
I
!
D i /
-0.5'
r 'i !
9
i
/
q !
tg c z -
/ /
1
2.5
-1.0' -3
-2
-1
0
0.5
log ((A/N o )t) Figure 3. The black circles (o 9 .) are the data for the kinetics of hydrogen adsorption on chromia drawn by Aharoni and Ungarish[13] in log(q/No) vs. log[(A/No )t] coordinates, where q is the adsorbed amount, and No, A axe some constants. The dotted line (- - -) drawn here by us shows that at the smallest surface coverages (c9log q/c9 log t) = 2-~" sufficiently accurate in some cases, one may apply the more accurate RJ approach[i], to represent the result of the integration in eq. (2.11). As we focus our attention on the most fundamental problems, we will accept the CA approach for the sake of simplicity. Now let us consider, why the model of localized adsorption along with the CA approach (assumption of a strong surface heterogeneity) leads to these generally observed laws for adsorption kinetics. (The Bangham's power, and the Elovich equations). It is now widely realized that this is the geometric nonuniformity of the real solid surfaces which is the main source of their energetic heterogeneity for adsorption. The surface geometrical distortions lead to a dispersion of the value of the local minima in the gas-solid potential function described by X(e) function. But if these local minima are the consequence of geometrical distortions, the creation of various energy barriers separating these local minima will be an unavoidable accompanying effect. These energy barriers will hinder the translational movement of adsorbed molecules across a solid surface. In other words, the stronger the surface heterogeneity, the stronger tendencies to localized adsorption. This is why the assumption of localized adsorption along with the CA appro-
347 ach work together so well. The observations of the actual adsorption systems provide an impressive support for this intuitive hypothesis. By taking into account the dispersion of the local minima in the gas-solid potential functions, (e.g. the dispersion of ~), we have arrived at a more accurate result that the rate of adsorption is proportional to t[!-0(n) e~-) ]c/kT. For low surface coverages, the above statement leads to the Bangham's empirical power law. The proportionality r~,e~")]c/kT is likely to L e(n) be valid in the case of moderately heterogeneous surfaces. In the case of strongly heterogeneous surfaces, the adsorption rate will be proportional to exp{--~T~----x'0~n)},because in this limit every (one-modal) adsorption distribution degenerates into the rectangular adsorption energy distribution. However, as demonstrated in Fig.2, this degeneration proceeds gradually, so the rate of adsorption will always be a hybrid between that described by the term rl-e<")]c/kT and the L e(n) .... rate of adsorption described by the term exp{ - ' =~-~' ~t a(n) }. And, this is why the experimental data can be correlated by both the Power Law and the Elovich equations, but always with a limited success. One can only say, that this hybrid behavior resembles more the e(,,) , than that predicted by the term exp{ - ~ Akr( ' ~"t) }, one predicted by the term trl--O(")]~lkT or vice versa. The purpose of our consideration in the present section was to provide a standard routine for a simultaneous description of equilibria and kinetics of adsorption on a non-porous, but energetically heterogeneous solid surface, characterized by a certain distribution of adsorption energy. To illustrate our new procedure we have chosen some expressions for the adsorption energy distribution, which lead to compact well-known equations for adsorption kinetics. In general, the functions O~)(p,T) and o0~") _- - f(O~) p,T, t) will have to be evaluated at numerically. Then, in the case of adsorption kinetics, numerical calculations will involve solving, for an assumed function X(~), the systems of the two equations (2.14) and (2.17) to eliminate e!'~) and to obtain ~ Ot as the function of 0~") p, T. It was Ward and coworkers [12] who proposed to express #" in eq. (2.1) by appropriate functions of coverage and temperature developed for adsortpion equilibria. Their proposal has been fully supported by the computer simulations of adsorption kinetics, published recently by Talbot et. a1114]. While assuming that adsorption is accompanied by desorption, they found, that the transient configurations of adsorbed particles can be approximated well by the corresponding equilibrium configurations at the same coverage.
3. M U L T I - S I T E O C C U P A N C Y A D S O R P T I O N : E F F E C T S O F S U R F A C E T O P O G R A P H Y ON A D S O R P T I O N K I N E T I C S . The generally accepted quantitative measure of the energetic heterogeneity of the actual solid surfaces is the differential distribution of a number of adsorption sites among the corresponding values of adsorption energy. This function, called usually - - " the adsorption energy distribution", is sufficient for a complete thermodynamic description of adsorption equilibria in the systems where one adsorbed molecule occupies one adsorption site and no interactions exist between the adsorbed molecules (Langmuir model).
348 However, in the systems where the interactions between adsorbed molecules cannot be ignored, or when one adsorbed molecule occupies more than one site, another important physical factor comes into play. This is the way in which adsorption sites characterized by different adsorption energies are distributed on a heterogeneous solid surface. In other words, this is the topography of a heterogeneous solid surface. So far, two extreme models of surface topography have, almost exclusively, been considered in theoretical works on adsorption. The first one was the "random" model, introduced to literature by Hill[15]. It assumes that adsorption sites characterized by different adsorption energies are distributed on a solid surface completely at random. The other extreme model of surface topography was the "patchwise" model, introduced to literature by Ross and Olivier[16]. It assumes, that adsorption sites having the same adsorption energy are grouped on a heterogeneous surface into "patches". These patches are large enough so that the states of an adsorption system in which two interacting molecules are adsorbed on different patches, could be neglected. Thus, the adsorption system can be considered as a collection of independent subsystems, being only in a material and thermal contact. Yang and coworkers[17-19], and Zgrablich and coworkers[20-24] have shown that even in the simplest case of Langmuirian adsorption the surface topography affects strongly the surface diffusion of adsorbed molecules. Although different expressions are obtained for adsorption equilibria or surface diffusion on surfaces characterized by different topographies, a practical discrimination between various topographies is difficult, at the present standard accuracy of adsorption measurements. Thus, experimental studies of one-site-occupancy adsorption do not practically allow at present to determine the surface topography by a suitable computer analysis of experimental data. On the contrary, the experimental and numerical studies of multi-site-occupancy adsorption seem to create such a chance[i,25]. It was demonstrated first by Rudzifiski and Everett in their monograph: "Adsorption of Gases on Heterogeneous Surfaces"[1]. That idea has been developed and elaborated further in the recent works by the author and coworkers [26 ,2 7] . Similarly to the case of adsorption equilibria, the starting point of our consideration was the Nitta's approach to multi-site--occupancy adsorption[25]. When Nt molecules are adsorbed at equilibrium on a heterogeneous surface, the system partition function Q(Nt, M,T), may be written as follows, q -
g(N (Nij}
,M,(N
j}/exp
, i
"
where r is the energy of adsorption of jth group (mer) on ith kind of sites. Nij is the number of adsorption-pairs of sites i and a group j; {N~j} is a distribution of these adsorption pairs, the subscript i0 being used for the empty site, g(Nt, M, {N~j}) is a combinatorial factor, expressing the number of the distinguishable ways of distributing the Nt molecules on M sites under the condition of a special distribution {Nij }; q~ is the product of internal and vibrational partition functions of an adsorbed molecule. As emphasized by Nitta, it is difficult to find a rigorous expression for g(Nt, M, {N~j}) by taking into consideration the mutual correlations between neighbouring sites and ne-
349 ighbouring groups in a molecule. A simple expression is obtained by assuming that all pairs of site-mer {ij ) are independent, under the constraints imposed by the distribution
{N,j}. To evaluate 0t, three kinds of surface coverages, 0t, {0tj } and
= s'iNt M , (j = 1 , . . . , s ) ,
0ij --- ~M i ~
(i =
...,
to be defined;
the fraction of sites occupied; (s is the total number of sites occupied by the adsorbed molecule), the fraction of sites occupied by the segment of type j; (sj is the number of sites occupied by the segments of type j), s), the fraction of sites of type i occupied by the segments of type j.
-, Ot -__- - ~sYt
Ot j
{O~j} are
W; j = 1 , . . . ,
The problem of evaluating Ot is not trivial, and Nitta proposed a complicated numerical procedure, in which only a discrete distribution of adsorption energy can be applied. However, that problem becomes easy when all the segments of an admolecule are of the same kind. The Nitta's approach leads then to the following expression for the chemical potential #" of the adsorbed (surface) molecules[l], Oi
#~ = -e,i - kT ln(q,(s) - (s - 1)kT In Ot + skT In 1 - Oi '
(3.2)
where Ni ei =
(3.3)
esi = seli
(3.4)
and ( is a constant related to the flexibility and symmetry number of a molecule. Thus, when all the segments are identical, the Nitta's assumption that all the pairs {i,j} are independent is true only if ith values are independent. This means a lack of spatial correlations between ith values on various sites which is true only in the case of random topography. It means, Nitta's expression (3.2) is valid for surfaces having random topography. When the surface becomes homogeneous, i.e., 8i ~ 8t, equation (3.2) reduces to the well-known Flory's isotherm for multi-site occupancy adsorption on homogeneous solid surfaces, ~ = -e~ - kT ln(q,(s) + skT In
(1
-0)
~ '
(3.5)
In the case of patchwise topography es will contain the additional index i referring to i-th patch, esi. Thus in the case of patchwise topography the isotherm equation reads, ~s = - e s i - kT ln(qs(s) + skT In
(1
(3.6) -
Oi) ~
350 The averaging of 0i in equation (3.6) with respect to the dispersion of e,i defined in (3.4) can be carried out easily by applying the Rudzifiski-Jagietto[1] method. Then, 0t = - X ( q r
Corr.
(3.7)
where
x(~) = / x(~)d~
(3.8)
X(e) is the differential adsorption energy distribution normalized to unity, and e,~ is found from the condition
( 020i~
--0
(3.9)
0'2i / ,,fi-'-r162
and Corr. is a correction term which can be safely neglected in the case of strongly heterogeneous surfaces[i]. Then, the Rudzinski-Jagiello approach becomes essentially equivalent to the Condensation Approximation. After performing the differentiation (3.9) in equation (3.2) one arrives at the following expression for e,~c, e,c = e~:) = - / ~ ' - kT l n ( q , r
(s - 1)kTln0t ,
(a.10)
where the superscript "r" refers to random topography. After replacing/z ~ by/~g of an ideal gas, /zg [pA3 l kW = In
(3.11)
[%kTJ
eq. (3.10) takes the following form, e~) = - k T ln(sK'p) - (s - 1)kW In 0t,
(3.12)
where K ' = q~(Aa %kT "
(3.13)
A is the thermal Broglie wavelength, and qg is the internal molecular partition function of the molecules in the bulk gas phase. In the case of patchwise topography the equation for e~ is obtained by applying eq. (3.9) to eq. (3.6), and then takes the following form,
e,r = e!rp) = - k T In
(s89+ ~)
,_~ K'p
(3.14)
Now let us consider the rectangular adsorption energy distribution, which is used so frequently to represent the energy dispersion on strongly heterogeneous solid surfaces,
x(~,)
' e--/-~-,,'
ror'<~,<e', ~ - -
0 , elsewhere
(3.15)
351 where e's and e~ are the lowest (l) and the maximum (m) values of the adsorption energies found on a heterogeneous solid surface. Then, according to eq. (3.7) for the random surface topography, at takes the following form O~e)=
~r kT In -~ kT In O~s-I) kT ln(sK')+ ~~ e~.(r) - ,~ - A--~ p ~ _
(3.16)
where A, = (e~ -ezs). For patchwise topography we obtain, 0~e ) -
--
e~)
kWln I / ~s(l+~)K' - - ~ ~ - - 1 1 + kT In ) ~ p ~Sg + 1)
e~-e~ = A--~
(3.17)
To emphasize that eqs. (3.16) and (3.17) describe the adsorption equilibria, we introduced again the superscript (e) in 01e). Now let us remark that for one-site-occupancy adsorption, (Langmuir model), i.e. when s = 1, eq. (3.16) developed for random topography reduces to the well-known Temkin's isotherm. For patchwise surface topography, 0t will always be the Temkin's isotherm for both one--site and multi-site--occupancy adsorption. Thus, we can see, that even in the absence of interactions between the adsorbed molecules, surface topography will affect the adsorption equilibria in the case of multi-site-occupancy adsorption on a heterogeneous solid surface. In the case of one-site-occupancy adsorption on heterogeneous solid surfaces, surface topography does not affect the kinetics of adsorption (desorption), until the interactions between adsorbed molecules can be neglected. It was shown in the series of papers published by Aharoni and coworkers[28-30]. Now we are going to show that surface topography must affect also the kinetics of multi-site-occupancy adsorption (desorption) on energetically heterogeneous solid surface. Then, for random surface topography and the rectangular adsorption energy distribution, for non equilibrium conditions we have, ~g - ~ = -A~0~ n) + kW ln(sK') + kW In p + (s - 1)kW In 0~n) = 0
(3.18)
whereas for patchwise topography, we obtain, ~ g - ~s = -A~0t + kT In
s(l+~)K , s89+ 1)(s-l)
+ kT In p = 0
(3.19)
After inserting eqs. (3.18) and (3.19) into eq. (2.1) one can express the experimentally monitored rate of adsorption \ 0t ] in terms of the actual (non-equilibrium) coverage 0~~) which is also monitored experimentally. Here, we will limit our interest to the very initial rate of adsorption, (very small surface coverages), when the second term within the square bracket in eq. (18) representing the
352 readsorption rate can be neglected. Then, for patchwise surface topography we arrive at the following equation for the rate of adsorption, -)
cOt = K'gs
(S89-[-1)(s-I)
p e x p { - }As k Ta(")V t
(3.20)
which is essentially the well-known Elovich equation. For random surface topography, we obtain,
i)O~n)=OtK'gssK'p [0~n)]s-1 exp { - ~ - ~ 0 t }
(3.21)
Comparing eqs. (3.20) and (3.21) one can see, that the surface topography will affect also kinetics of multi-site--occupancy adsorption on energetically heterogeneous solid surfaces, even in the absence of interactions between the adsorbed molecules. Now let us consider whether there might be a chance to elucidate the information about the nature of the surface topography of an investigated adsorbent sample, from the experimental data for adsorption kinetics. Our answer is yes, and to prove it we will take as an illustration the kinetics of chemisorption on scandia. There is a large body of experimental data in the literature on adsorption kinetics, but this particular example provides a very impressive proof. It is shown in Figure 4.
C02
There, in Figure 4 we show the initial rate of adsorption \ 0t ],,a~, measured at very small surface coverages, and reported by Pajares et. all31]. One has'to realize that the smaller is the pressure p at which the kinetics of adsorption is studied, the smaller is also the region of the studied surface coverages. In the case of patchwise topography, going down to smaller and smaller pressures (surface coverages studied) should lead to a better and better linear dependence of since exp { - ~-~0(' } k rt~)
tends to unity, and the derivative of exp { - ~-"-a(' } k~'t r~)
, on p, with respect to
0}'~) tends to the constant value -~--'Such a behaviour predicted for patchwise surface kT" topography is not observed in Figure 4. Let us remark at this moment that the surface of the scandia sample was very heterogeneous, as it can be deduced from the log-log plots of equilibrium adsorption isotherms measured by Pajares and owo k [a21. (Figu~ 11 in the paper by Fiero and Pajares[32].) Now let us assume that the scandia sample has random topography and adsorbs in a two-site-occupancy way by its two oxygens. Then, though the derivative of exp { - a-A(n)}kT't with respect to 0~n) tends to the constant value a_~kT when 0~n)~ 0, the
6"02
observed rate of adsorption will decrease with p, because p is multiplied by 0~") decreasing with p. (Smaller experimental pressure means smaller surface coverages studied). Our conclusion that the scandium oxide must have random topography is in agreement with the results published recently by Bakaev and Steele, carrying out computer simulations of adsorption on metal oxide surfaces[33]. They have shown that the best agreement
353 I
"7
I
I'
I
'I
f
I
I
I
~0 "7
1.5 -
~~'''~~
-,.-4
E
1.0-
~ 0
(,9 E
-
0.50-
~
C
-
1.0 0.5
m
4=
"
1.0-
[] o
0.50
I
0
!
2
I
I
4
I
I
6
I
I
I
8 p (Tort)
Figure 4. The initial rate of adsorption of C02 on scandia oxide, measured by Pajares et. al.[31] at low surface coverages. (After Pajares et. al.[31]). between experimental and theoretical adsorption isotherms is obtained for the surface model, in which surface oxygens are viewed as randomly packed hard balls.
4. D E S C R I P T I O N OF D E S O R P T I O N K I N E T I C S " A P P L I C A T I O N T O T P D SPECTRA The Temperature Programmed Desorption (TPD) is nowadays one of the most popular and frequently used methods to characterize the energetic properties of adsorbent and catalyst surfaces. The principles of that method were published by Amenomiya and Cvetanovic in 1963134], and 5 years later the first theoretical paper on the application of that experiment to study energetic surface heterogeneity was published by Cvetanovic and Amenomiya[35]. The number of the papers reporting on the application of this technique started to grow rapidly, and one can quote dozens of relevant papers published in this field. As our paper proposes an essentially new theoretical approach to the problem, we refer only to the most exhaustive reviews by Falconer and Schwarz[36], Kreuzer[37], and by Bhatia et al.[ll] presenting the approaches which have been used so far in the
354 theoretical interpretation of the TPD spectra. Tovbin[38] has reviewed the achievements of Russian scientists in this field. The theories of TPD were based commonly on the theory of activated desorption (adsorption). In fact, the TPD technique was used first of all to study the surface energetic heterogeneity of catalysts and adsorbents. The various peaks observed on TPD diagrams were ascribed to various kinds of surface adsorption sites characterized by different activation energies for desorption. In most cases, researchers were happy with such qualitative information about the surface energetic heterogeneity. Only few papers reported on more attempts to draw quantitative information about surface heterogeneity from the experimental TPD spectra. Czanderna et. a1.[39], Cvetanovic and Amenomiya[35], Carter[40], Dawson et. al.[41] and Tokoro et. a1.[42] assumed that the surface energetic heterogeneity manifests itself as a linear variation (decrease) of the activation energy for desorption with the decreasing surface coverage. King[43] was the first to assume that the activation energy for desorption may be a more complex function of the surface coverage. Also Tokaro et. al. followed that assumption in their second work on that problem[44]. An interesting paper in this field was published by KnSzinger and Ratnasamy[45]. After 1980, the papers concerning that problem were published by Davydov et. a1.[46], Unger et. a1.[47], Malet et. a1.[48], Leary et. a1.[49], Ma et. al.[50] and by Salvador and Merchan[51]. Very interesting reports concerning that problem have been presented during the "Second International Symposium on Surface Heterogeneity Effects in Adsorption and Catalysis on Solids" held in Zakopane- Levoca ( P o l a n d - Slovakia) in the autumn 1995152]. At low (partial) pressures p, and high coverages 0}"), (the condition usually found in the TPD experiments), the first term in eq. (1.2) is expected to be small compared to the second one. In this section we will limit our selves to one-site-occupancy adsorption, i.e., when s = 1. Then, to a good approximation, the experimentally observed desorption rate should be given by,
oNlo> 0t
= MKdO exp {-ed/kT}
(4.1)
where Nt(~) = M6} n). Experimentally, the desorption rate is monitored as the concentration of the desorbed species in the carrier gas, c, so, -ONt = F c 0t
(4.2)
where F is the volumetric flow rate of that carrier gas. Of course, eq.(4.1) is valid for a hypothetical homogeneous solid surface with the same constant desorption energy ed across the surface. Generally, it was observed that while applying eq. (4.1), the estimated ed values change with the surface coverage #~n) so, ed in eq. (4.1) was considered to be rather a function ed(O). The authors trying to generalize
355 eq.(4.1) for the case of the actual, energetically heterogeneous solid surfaces, wrote it in the following form[50], cgt = Kd
0t~)(ed)exp {-ed/kT} X(ed)ded
(4.3)
fl
where X(ed) is the distribution of the number of adsorption sites among various values of ed, normalized to unity, and the superscript "t" in O0~=)/Otmeans the average of O0(=)/Ot over the "total" heterogeneous surface. Then, f
X(ed)tied = 1
(4.4)
f~
where fl is the physical domain of ed. Eq.(4.3) could be treated as an integral equation for X(ed), provided that the kernel O('O(ed)is known, and provided that there exists a functional relationship between ed and ea on a heterogeneous solid surface. The existence of such a relationship is a mystery which, to our knowledge, has been considered in only one theoretical paper by Salvador and Marechan[51]. One can even ask the question whether ea and ed on various adsorption sites are correlated at all. What is only sure, is the fact, that the Langmuirian kinetics of adsorption, (represented by the first term on the r.h.s of eq.(1.2)), is not found in real adsorption systems. Instead, Elovich, Power Law, or other equations for adsorption kinetics must be applied. In terms of TAAD, it means, that there must exist a dispersion of e~ values on different adsorption sites.The concept of the dispersion of the activation energies for adsorption was accepted as fundamental by all the authors trying to explain the theoretical origin of these empirical equations. The TPD experiments, on the other hand, reveal a dispersion of ed values on the actual solid surfaces. So, generally, the correct generalization of eq.(4.1) should read, 0t
= Kd
o(n)(ea, ed) exp {-ed/kW} X(ea, ed) deaded
(4.5)
fl
where X(e~, ed) is a two-dimensional adsorption energy distribution. (The differential distribution function of the number of adsorption sites among the corresponding pairs of values of e~ and ed.) The common interpretation of the TPD spectra was based on the assumption that, at every moment, (at every adsorbed amount), the observed desorption rate can still be described by eq. (4.1). The value of ed corresponding to that adsorbed amount was still called "activation energy for desorption" and was given the same physical meaning as for a homogeneous solid surface characterized by this ed value. Looking to eq.(4.5) one can deduce easily that such an interpretation could be accepted only if the following two conditions were fulfilled:
356 1. There exists a functional relationship between
ed
and e= on various adsorption sites,
2. The desorption proceeds in an ideally stepwise fashion, in the sequence of increasing activation energies for desorption. While accepting the Statistical Rate Theory, along with the Langmuir model of adsorption, we have, 00(``) 1 - 0(") 0t = Kap 0(.) exp{e/kT} -
1 0 (~) Kd exp{-e/kT} p 1 - 0(``)
(4.6)
where !
$
(4.7)
Ka = Kgs qo exp{#~,/kT} Kd =
K
!
~ exp{-/z~/kT}
(4.8)
q~,
As it has already been mentioned, in typical TPD experiments the (partial) pressure of the desorbed species, (concentration c in the carrier gas), is usually small, so, the desorption rate can be well represented by the second term on the r.h.s, of eq.(4.6). Although we already have at hand a more general treatment taking into account the possibility of readsorption, we will neglect it in the present publication for the purpose of clarity of presenting the main idea of using the Ward's theory as the starting point. So, we assume that in the case of a homogeneous solid surface having adsorption energy e, 00(``) 1 0(``) (9t = -- Kd--p 1 -- 0(n) e x p { - e / k T }
(4.9)
Eq. (4.9) can be rewritten to the following form,
exp {(e - ~"))/kT} o(n) (s
,,
"--
1+
exv
(
-
~
(4.10)
)
where ~ p
e~``) = kT In
Kd
00(n) [l 0t J
(4.11)
Figure 1 shows the temperature dependence of the function 0('0(ec). At not too high temperatures, the desorption from a heterogeneous solid surface will proceed in a stepwise-like fashion. This assumption was intuitively made in the hitherto interpretation of TPD spectra, but this is the Statistical Theory of Interfacial Transport which provides now, for the first time, a rigorous proof for that intuitive common assumption. The step on 0(e, e!~)) will appear at e - e!~). Then the observed "total" desorption rate from the whole heterogeneous surface will be governed by the "local" rate of desorption from the sites the adsorption energy of which is equal to e!~).
357 The observed "total" rate of desorption (O0~")/cgt) will be proportional to that local desorption rate on the sites where e = e!'~), and to the fraction of these sites on the surface. Thus, lim
00~n) =
m-->o cot
Const
(O0~(e))
~=<{,,)
X(e! "))
(4.12)
where
a
1 + exp t ( e - e~ ))/kmj~
From now on, till the end of this chapter, we will consider only the desorption kinetics. (,~) Therefore, we will drop the superscript (n) in the expressions 0 (~), 0}n) and ec 9 While expressing (O0/Ot) in eq. (4.11) by (OOt/Ot) defined in eq. (4.12), we obtain, ec= _kT l n [ - p ~d where/T;'d =
0t =
1
(00t'~ ]
x(~) k-a-T)<<
(4.14)
Const Kd. Alternatively, eq.(4.14) can be rewritten to the following form,
p
The above equation is to be compared with the equation 00t0 t = --Kd0t exp { - skT
}
(4.16)
which is commonly used to analyze the TPD spectra from heterogeneous solid surfaces. (We consider here the simplest case of one site-occupancy adsorption). Although it is usually said not explicitly, or perhaps even not realized sometimes, the interpretation of the TPD spectra based on eq. (4.16) involves the assumption of an ideally stepwise character of the desorption process. For our further purposes, we will rewrite eq.(4.15) to the following form, X(ec) = exp {ec/kT} [--~Pd p ~__t.]
(4.17)
The function xc(ec) calculated from the relation,
00t _ ___1 0Nt(er
Xr162
8ec-
M
aec
(4.18)
is usually called the "Condensation Approximation" for the actual adsorption energy distribution X(e). It is defined as follows: x<(~<) =
oo /(oo) ~
0
x(~)a<
358 In the limit T ~ 0 the derivative ( ~ ) becomes the Dirac delta function 6 ( e - ec), and Xc(ec) becomes the exact function X(e). One of the common approximations used in the theoretical description of adsorption equilibria on heterogeneous solid surfaces is to replace the true kernel in eq.(4.13) by the step function 0c(e, ec)[53-55]. The essential condition for this approximation to be applicable is not solely the condition T ~ 0 but the condition (6e/6• < 1, where 6e is the variance of the derivative (00/0e), and 6• is the variance of X(e). Thus, to a certain degree of accuracy, the function Xc(ec) calculated in eq.(4.18) can be compared to the function X(ec) calculated in eq.(4.17). In the limit T ~ 0 or in the limit of a very heterogeneous solid surface, they will become equal. Then, from the comparison of eqs (4.18) and (4.17) we obtain, (4.20)
Oec
At every temperature T, the still existing adsorbed amount Nt(T), is found from the relation, T
(4.21)
Nt(T) - V o - ~F f c(T)dT To
where ~ is the heating rate, ~ = dT/dt. In typical TPD experiments ~ is a constant. Also, at every temperature T, the corresponding value of ec can be calculated from the desorption rate, if the value Kd and ec(T) are known. The preparatory steps to calculate Xc(er are the following: 1. To eliminate the noise in the experimental TPD curve, which would be multiplied in the subsequent differentiations, the TPD spectrum is approximated by a sum of gaussian-like functions: c(T) = E aici(T)
(4.22)
where ci(W) = ri
E[
exp
{
-
Ei
(4.23)
In eq.(4.23), E~ is the variance of c~(T), and r~ governs the shape of c~(T). This is a fairly symmetrical gaussian-like function for r - 3, right-hand widened for r < 3, or a left-hand widened gaussian-like function for r > 3. 2. We assume that at To the surface coverage is equal to 1, and that at the final temperature, Tk, it is equal to zero. Then, Wk
M= ~
c(T)dT To
(4.24)
359 3. The partial pressure p of this adsorbate is given by the relation p = cP (where P is the atmospheric pressure). 4. The derivative (ONt/Ot) in eq.(4.20)is replaced by, 0Nt = - F c (t(W)) 0t
(4.25)
5. For physical reasons ~c must be a one--to-one increasing function of T, we write the derivative (ONt/Oec) in eq.(4.20) as follows:
( ONt'~
fONt'~(OT)
Fc(t) Oer
(4.26)
Remembering that c = p/P, after certain rearrangements eq.(4.26) takes the following form, oqec I~d 1 [--er "l aT = fl-P c(T) exp t ~f
(4.27)
This is a differential equation, the solution of which will give the function ec(T), provided that the constant/~d is known. Let us assume for the moment that it is known. The boundary condition is the assumption that at T = To, desorption starts at the adsorption sites having adsorption energy e(To) = Co. 6. Having calculated the function ec(T), the temperature scale is converted into energy scale, and the function X(ec) is calculated from eq.(4.26), written in the following form,
x(cr =
[Oer
Fc(W(er /3vm
(4.28)
taT
or from eq.(4.17) considered as the following function,
While considering the limit T ~ 0 we had the purpose in mind to illustrate in an easy way the basic features of our new theoretical approach. In the actual TPD experiments c(T) is measured at finite temperatures. So, let us consider now this real physical situation. Eq.(4.26) does not change, as it is valid for any temperature. Changes will appear in eq.(4.14) obtained by inserting eq.(4.12), into eq.(4.11). To find the relation between the "local" rate of desorption (O0/Ot) and the "total" rate (00t/0t) at finite temperatures, we consider the following obvious relation
0r
or
f ( oe)
(4.30)
o
Then, we replace (i)O/i)T) under the integral by the following product,
oo (oo) OT-
~
(oo) OWl = -
~e
\OWJ
(4.31)
360 where, =
1
~-y exp { ~-:~ } kT [1 + exp {~-=m}]2 = -
( 00 )
(4.32)
kT
(Or
Because following form
is not a function of e, the r.h.s, of eq.(4.30) can be rewritten to the
jfo~ (i)O) ~ X(e)de =
(Oer 9~o~ (00)X(e)de = N
- f l \ 0T J
- f l (0er \ 0T,] Xr162
(4.33)
where the function X,(ec) is the same "condensation" function as that on the 1.h.s. of eq.(4.18). Now let us consider the "local" rate of desorption from the adsorption sites for which the energy is equal ec,
(O0/Ot)~=~r
(90
= fl
0T
s
= fl
~
s
\0T)
(4.34)
~:c
Because according to eq.(4.32), ~o = 4 k T ' we replace now (4.33).
and
~
~r = 4kT
(Oec/OT)ineq.(4.35),
0T)
(4.35)
by its value calculated from equations (4.30) and
0ec (o_~o ~ \Ot] 0---T = - -flXr162
(4.36)
Eq.(4.35) takes then the following form, 00
)
(oe,)
1 ~ 0t ~c = 4kW Xr162
(4.37)
In the limit T ~ 0 the fully accurate expression (4.37) reduces to the approximate expression (4.12) previously used by us. It means that the value of in eq. (4.12) is simply We should remember that for e = ec, the expression (4.9) reduces to the expre~sio. (4.11)because 0(~ = ~o)= ~. a It means, at finite temperatures eq.(4.28) is still valid, but the function X(e) calculated next from eqs (4.17), (4.18) and from (4.28) has the fully rigorous meaning of the "Condensation Approximation" Xc(ec) for the true adsorption energy distribution X(e). Having calculated Xc(e,) one can calculate X(e,) using the methods developed by Rudzinski and Jagiello[56-59]. In many cases sufficiently accurate values of X(e,) are obtained by using the Rudzinski-Jagiello approach[i],
4kT.
x(~) = Xc(~r - -g-(kT) ~ [-~2 j
Const
(4.38)
361 Eq.(4.38) is valid for the Langmuiric model of the "local" adsorption, assumed here by us. Rudzinski et.al, showed that including interactions between adsorbed molecules results mainly in using more complicated expressions to calculate X(ec) from Xc(ec), the latter function being always the key information about surface energetic heterogeneity[I,57]. Their considerations were based on the assumption, that lateral interaction parameters for a molecule adsorbed on a certain adsorption site does not depend on the adsorption energy of that site. Such an assumption seems to be fairly reasonable in the case of one-modal adsorption energy distributions. However, the actual TPD spectra from many solid surfaces are multimodal. They look like a number of more or less overlapping peaks, corresponding to various groups of sites existing on a studied solid surface. Thus, in many cases a certain generalization of the method developed by Rudzinski et. al. will be necessary, to account properly for the interactions between adsorbed molecules. A first necessary step toward such a generalization, will be "decomposition" of the calculated condensation function X~(ec) into the overlapping one-modal distributions corresponding to a certain kind of adsorption sites. In the present treatment we neglect diffusional effects, which may be responsible for a certain broadening of these overlapping peaks. Diffusional effects, however, are usually neglected in the published analyses of the TPD spectra from heterogeneous surfaces. It reflects the general belief that this is the energetic surface heterogeneity which predominantly governs the shape of a TPD diagram. A decomposition of a condensation function X~(e~), was done by Cases and coworkers[60] for a number of adsorption systems. It appeared then that such decomposition can be successfully made only for some systems, and failed as a general decomposition method. Then, these authors came to the conclusion that the general physical situation will be fol!owing. If even one may distinguish on a given surface, a number of distinct kinds of adsorption sites, (or surface domains), there will still exist a certain distribution of surface properties (adsorption energies) within each group. Therefore, one should consider the following representation for an actual adsorption energy distribution X(e), n
n
~(s -" ~ ~fiXi(s , ~ ")'i = 1 (4.39) i=l i=l where Xi(e) is the adsorption energy distribution for ith kind of adsorption sites. The condensation function X~(e) calculated from a TPD diagram will then have the form, n
Xc(s
E~ i ~-~6 ] X i ( , ) d , -- ~"[iXci(s i=1 0 i=l
(4.40)
where 0i(e) is the isotherm equation for adsorption on an ith kind of adsorption sites. Even, if it is still the Langmuir-like function, it may have different values of the constant Kd, for different kinds of adsorption sites. In their improved decomposition (DIS) method[61] Villieras et. al., used the following DA (Dubinin-Astakhov) functions to represent X~(e), (See the chapter contributed by these authors to this book), ~ci(s -- ri(s -- i ) Er
exp -
s __ 6o] Ei Ji
(4.41)
362 Although diffusional effects are commonly neglected, they can also be taken into account. Huang and Schwarz[62] showed that in the absence of intraparticle diffusion the more accurate description of mass transfer effect on the TPD spectra requires considering a nonsteady mass balance which includes both axial dispersion and convective transport. The equations that describe the concentration in the bulk fluid, c, and on the surface St, are given by, 0c 02c 0c 0-'-t = De ~ - U0z
1 -p
(4.42)
p 0Nt
0t
where D~ is an effective diffusion coefficient (m2/s), u is a temperature dependent constant, p is the porosity (cm3/cm3), and Y = Vm0t. It is known, that in addition to diffusional effects outside the solid phase, TPD spectra may be affected by intraparticle diffusion and lateral interactions between adsorbed molecules[49]. It is suspected that the two effects mentioned above may be a source of some extra peaks on the TPD diagrams. To illustrate the application of our new approach to analyze the TPD spectra, we take the experimental data obtained in the Department of Chemical Technology of Maria Curie--Sklodowska University in Lublin[63]. The Temperature Programmed Desorption measurements were conducted on AMI1 equipment (Altamira Instruments). Thermal Conductivity Detector was used as a detector. Experiments were conducted on the silica supported nickel catalyst (15%Ni) obtained by impregnation of silica Merck Kieselgel 100 by nickel nitrate solution. The catalyst made in this way was calcinated at 4000 and reduced right before the experiment. Hydrogen purified on the OXICLEAR deoxidant made by PIERCE CHEMICAL Co., and passed through the activated carbon absorber was used as a reducing agent. The carrier gas was argon (99.99%pure) additionally purified over the same OXICLEAR deoxidant and then by the molecular sieves 4A and 5A. Figure 5 shows the decomposition of the TPD diagram of hydrogen desorbed from the silica supported nickel catalyst, done by using the linear combination (4.22) of five functions Ci(T) defined in eq. (4.23). The parameters found by computer in the course of that decomposition are collected in Table 1. Table 1. The parameters found by computer in the course of the decomposition defined in eqs. (4.22) and (4.23) of the TPD diagram of hydrogen desorbed from the silica supported nickel catalyst.
r,
i=l i=2 i=3 i=4 i=5
10604 1405 2139 2004 1154
2.22 2.32 2.71 2.71 1.90
T~ (~
35.56 278.67 141.00 218.50 384.51
Ei (de9 ) 127.15 107.49 88.04 84.47 213.93
363
so oo -~
;> .,..,
60.00 l I l 1 l
40.00
l l
I I
I
20.00
f
1 I
I
i
~
i
I i
I
t
t
I I~~
t
4,
4
\
0.00 0
200
400
600
800 T (~
Figure 5. The decomposition of the experimental TPD diagram (--) by the linear combination (4.22) of the function (4.23), (. . . . ), done by using the parameters collected in Table 1. The values 1, 2, 3, 4, 5 at the maxima of the composite functions ci(T) denote the value of the index "i", in eq. (4.22) and in Table 1. Figure 6 shows the function ec(T) calculated by solving the differential equation (4.28) for three values of the parameter Ka, the temperature dependence of which was neglected. The choice of the parameters /~'d in our model calculation was dictated by the obvious requirement that the ec values should lie in a physically reasonable range. We are facing here a situation, which is similar to that in the papers reporting on calculation of X(e) from the equilibrium adsorption isotherms. There the calculation of X(e) requires the knowledge of the Langmuir constant K. Changing the assumed values of K resulted there primarily in shifting the calculated function X(e) on the energy scale toward higher or smaller values of e. In the case of X(e) calculated here from the TPD spectra, changing the assumed value of/(a results both in shifting X(e) on the energy scale, and in a simultaneous broadening (or vice versa) of X(e), as it is shown in Figure 7. The choice of the parameter e0 = e(T0) has a similar effect on the calculated function X(e). Figure 8 shows the solution of the differential equation (4.28) for three values of e0, and Figure 9 shows the corresponding calculated functions X(e). While thinking about a possibility of determining the parameters /(d and e0 in an in-
364 60.0o
E
50.0
40.02
30.0
20.0
1
10.0 .=
2
0.0-~
. . . . . . . . .
270
I . . . . . . . . .
470
I . . . . . . . . .
670
I . . . . . .
870 T (K)
Figure 6. The functions r calculated from eq. (4.28) by assuming co(To) = 0, for three values of the parameter /fd; the curve No 1 is obtained by assuming that h'a = 105, the curve 2 is for R'd = 104, and the curve 3 corresponds to/x'a = 103.
dependent way, one might eventually consider independent calorimetric measurements. Calorimetric studies of adsorption equilibria show that the main contribution to the isosteric heat of adsortpion qst comes from the energies of adsorption. That possibility deserves obviously further theoretical studies. After determining the "condensation" function Xc(ec), we decompose it in the way outlined in eqs. (4.40) and (4.41), for its further theoretical analysis. Figure 10 shows an example of the decomposition of the condensation function Xc(ec). Table 2 collects the parameters; 7i, ri, q0 and Ei found by computer, while decomposing the function Xc(cc) calculated by assuming e(T0) = 0, and/(d = 104-
365 0.800
E
0.60-
0.403
0.20-
0.00
, ,
I
I
i
I
I
0.0
I
J
I
I
I
I
I'
I
I
I
10.0
I
I
'
'
'
'
'
'
20.0
,
i
,
I
'
i
['
'i
i
30.0 r (kJ/mole)
Figure 7. The condensation functions X~(E~) calculated from eq. (4.28) with the functions (Oec/OT) = f(ec), found by numerical differentiation of the functions ec(T) presented in Fiure 6. The numbers 1, 2, 3 correspond to the same values of the parameter /~d as in Figure
g
Table 2. The parameters: 7i, ri, ei, 0 Ei found by computer while decomposing the calculated function Xc(e~) by the linear combination (4.40) of the functions (4.41).
i=1 i=2 i--3 i-4
~/i
ri
0.4 0.105 0.34 0.1
3.5 2.4 2.1 2.2
s0
(kJ/mole)
8.3 10.05 8.58 10.8
Ei (kJ/mole)
1.48 0.6 2.9 4.0
The decomposition of the function Xc(ec) can be done by assuming at least four over-
366 ~, r
0
E ~
40.0-
-
30.0-
20.0 =
3
10.0-
0.00 5 27O l
I
I
I
l
I
l
I
I
I
I
470
I
I
I
I
a
l
I
I
i
l
670
l
I
I
I
I
I
I
I
i
I
l
I
I
I
l
870
T (K) Figure 8. The functions ec(T) calculated from eq. (4.28) by assuming that /fd = 104, for three values of the parameter co, 0 (curve No 1), 5 kJ/mole (curve 2) and 10 kJ/mole (curve 3).
laping peaks to exist. The estimated variances Ei's of the composite functions Xci(ec), are smaller than the values of kT corresponding to the related range of temperatures T(ec). Meanwhile, even if Xi(e) had been a Dirac delta function 6 ( e - e,), the variance of its corresponding condensation function should be close to kT value. It means, that while calculating Xc(er for this particular chemisorption system, from TPD spectra, one should consider values of e(T0), Kd leading to broader functions X~(e~). This can be done by assuming e(To) = 0, and/~d values smaller than 103. Such parameter values will result into broader calculated X~(er functions, but also shifted on energy scale toward higher chemisorption energies. In fact, Lee and Schwarz[64] argue that for the nickel-supported silica catalysts the activation energies for hydrogen desorption should vary from 50 to 90 kJ/mole. Thus, the decomposition of the calculated X~(e~) function, and then studies of its composites X~i(ec) seem to create a chance for arriving at reasonable range of the values of the parameter/(d-
367
o
60.
40.0
0.00 5.0
10.0
15.0
- - - - ~ 20.0 25.0 r
(U/mole)
Figure 9. The condensation functions Xc(Ec), calculated from eq. (4.28) by assuming that ~'d = 104 , for three values of the parameter r The numbers 1, 2, 3 correspond to the sazne values of the parameter E0 as those in Figure 8. CONCLUSIONS While presenting our new approach to the kinetics of adsorption (onto), and desortpion from flat but energetically heterogeneous surfaces, for the purpose of clarity, the consideration was limited to the model of a monolayer localized and ideal adsorbed phase. The key problem in this approach is to relate the "local" rate of adsorption OO/Ot to the "total" rate 0StlOt. This can also be done by considering other adsorption models - the models of mobile adsorption for instance. In the case of localized adsorption, the interactions between adsorbed molecules can easily be taken into account, just in the same way as in the case of adsorption equilibria. Similarly to the case of adsorption equilibria, including the lateral interactions there will be a much more difficult problem in the case of mobile adsorption on the surfaces posessing topography different from patchwise. The proposed method is based on the assumption that the transient states of the adsorbed phase are close to those corresponding to the adsorption equilibria. This also involves an implicit assumption that the surface diffusion proceeding via "hopping" or in another way is a fast process. The easy way in which the commonly observed laws of adsorption kinetics, are developed here, may suggest that the above assumption is true for the majority of adsorption systems and physical situations studied experimentally.
368
0"501 0
E
-
II
-
I I
0.10- _
/I..I.. 3,,"Jl I " ~
" --
0.008.0
I
/
rl
I
x~
z
"12.0
16.0
20.0 r162(kJ/mole)
Figure 10. The decomposition of the condensation function Xc(ec), calculated by assuming that e(To) = 0 and ~'d = 104. The broken lines are the composite functions "TcXc~(ec). REFERENCES 1. Rudzinski, W.; Everett, D.H., "Adsorption of Gases on Heterogeneous Surfaces", Academic Press, 1992. 2. Jaroniec, M.; Madey, E., "Physical Adsorption on Heterogeneous Solids", Elsevier, 1989. 3. See the review by Low, M.J.D., Chem. Rev., 60, (1960), 267. 4. Roginski, S.; Zeldovich, Ya. Acta Physicochim., URSS 1, (1934), 554. 5. Roginski, S.; Zeldovich, Ya. Acta Physicochim., URSS 1, (1934), 595. 6. Aharoni, C.; Ungarish, M., J.C.S. Faraday Trans. I, 73, (1977), 1943. 7. Aharoni, C.; Ungarish, M., J.C.S. Faraday Trans. I, 74, (1978), 1507.
369 8. Aharoni, C.; Suzin, Y., J.C.S. Faraday Trans. I, 78, (1982), 2329. 9. Tovbin, Yu., "Lattice-Gas Model in Kinetic Theory of Gas-Solid Interface Processes", Progress in Surface Science, 34, (1991), 1. 10. Cerofolini, G.F., in "Adsorption and Chemisorption on Inorganic Sorbents", Dabrowski, A.; Tertych, V.A., Editors, Elsevier, 1996. 11. For most recent and exhaustive review see, Bhatia, S.; Beltramini, J.; Do, D.D., Catal. Today, 7, (1990), 309. 12. Ward, C.A.; Findlay, R.D., J. Chem. Phys., 76, (1982), 5615. See also proceeding paper by Ward, A., Findlay, R.D., and Rizk, M., J. Chem. Phys., 76 (1982), 5599. 13. Aharoni, C., Ungarish, M., J.C.S. Faraday Trans. I, 73, (1976), 456. 14. Talbot J.; Jin, X.; Wang, N.-H., Langmuir, 10, (1994), 1663. 15. T.L.Hill, J.Chem.Phys., 17, (1949) 762. 16. S.Ross, and J.P.Olivier, "On Physical Adsorption", Interscience Publishers, Inc., N.Y. 1964. 17. Chen, Yu-D., and "fang, R.T., "Multicomponent Diffusion in Zeolites and Multicomponent Surface Diffusion", this monograph. 18. A. Kapoor, and R.T. Yang, A.I.Ch.E.J., 35, (1989) 1735. 19. A. Kapoor, and R.T. "fang, Chem. Engng. Sci., 45, (1990) 3261. 20. V. Pereyra, G. Zgrablich, and V.P. Zhdanov, Langmuir, 6, (1990) 691. 21. V. Pereyra, G. Zgrablich, Langmuir, 6, (1990) 118. 22. V. Mayagoitia, F. Rojas, V. Pereyra, and G. Zgrablich, Surface Sci., 221, (1989) 394. 23. V. Mayagoitia, F. Rojas, J.L. Riccardo, and G. Zgrablich, Phys. Rev., B41, (1990) 7150. 24. J.L. Riccardo, V. Pereyra, G. Zgrablich, F. Royas, V. Mayagoitia, and I. Kornhauser, Langmuir (in press). 25. T.Nitta, M.Kuro-Oka, and T.Katayama, J.Chem.Eng.Jpn., 17, (1984) 45. 26. W.Rudzifiski, K.Nieszporek, and A.D~browski, Adsorption Science and Technology, 10, (1993)35. 27. Rudzinski W., and Aharoni, C., Polish J. Chem., 69, (1995) 1066. 28. C.Aharoni, and M.Ungarish, J.C.S. Faraday Trans. I, 73, (1977) 1943. 29. C.Aharoni, and M.Ungarish, J.C.S. Faraday Trans. I, 74, (1978) 1507.
370 30. C.Aharoni, and Y.Suzin, J.C.S. Faraday Trans. I, 78, (1982) 2329. 31. I.A.Pajares, I.L.Garcia Fierro, and S.W.Weller, J.Catal. 52. (1978) 521. 32. I.L.Garcia Fierro, and I.A.Pajares, J.Catal. 66, (1980) 22. 33. V.A.Bakaev, and W.A.Steele, Langmuir 8, (1992) 1372, 8, (1992) 1379. 34. Amenomiya, Y.; Cvetanovic, R.J., J. Phys. Chem. 67, (1963) 144. 35. Cvetanovic, R.J.; and Amenomiya, Y., Catal. Rev. Sci. Eng.6_, (1972) 21. 36. Falconer, J. L.; Schwarz, J.A., Catal. Rev. Sci. Eng. 25, (1983) 141. 37. Kreuzer, H.J., Langmuir 8, (1992) 774. 38. Tovbin, Yu., in "Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces", (Rudzinski, W.; Steele, W.A.; and Zgrablich, G., Editors) Elsevier, 1996. 39. Czanderna, A.W.; Biegen, J.R., and Kollen, W., J. Colloid Interface Sci., 34, (1970) 406. 40. Carter, G., Vacuum, 1_.22,(1962) 245. 41. Dawson, D.T.; and Peng, Y.K., Surface Sci. 33, (1972) 565. 42. Tokoro, Y.; Misono, M.; Uchijima, T.; and Yoneda, Y., Bull. Chem. Soc. Japan, 51, (1978) 85. 43. King, D.A., Surface Sci. 47, (1975) 384. 44. Tokoro, Y.; Uchijima, T., and Yoneda, Y., J. Catal., 5..66,(1979) 110. 45. KnSzinger, H.; Ratnasamy, Catal. Rev. Sci. Eng. 1j_7,(1978) 31. 46. Davydov, V.Ya; Kiselev, A.V.; Kiselev, S.A; Polotryuk, V.O.V, J. Coll. Interface Sci. 74, (1980) 378. 47. Unger, K.K.; Kittelman,U.R.; Kreis, W.K., J. Chem. Techn. Biotechnol., 31, (1981) 435. 48. Malet, P.; and Munuera, G., in "Adsorption at the Gas-Solid and Liquid-Solid Interface", p.383, Rouquerol, J., and Sing, K.S.W., Eds., Elsevier 1982. 49. Leafy, K.J.; Michaels, J.N.; and Stacy, A.M., AIChE J., 34, (1988) 263. 50. Ma, M.C.; Brown, T.C.; and Haynes, B.S., Surf. Sci. 297, (1993) 312. 51. Salvador, F.; and Merchan, D., React. Kinet. Catal. Lett., 52, (1994) 211. 52. Proccedings of the "Second International Symposium on Surface Heterogeneity Effects in Adsorption and Catalysis on Solids", Zakopane-Levoca, (Poland-Slovakia), held in autumn 1995 (Brunovska, A.; Rudzinski, W.; and Wojciechowski, B.W., Editors).
371 53. Roginsky, S.; and Todes, O., Acta Phisicochim. USSR, 21, (1946) 519. 54. Harris, L.B., Surface Sci.,10, (1968) 129, 13, (1968) 377, 15, (1969) 182. 55. Cerofolini, G.F., Surface Sci., 24, (1971) 391. 56. Rudzinski, W.; and Jagiello, J., J. Low Temp. Phys. 45, (1981) 1. 57. Rudzinski, W.; Jagiello, J., and Grillet, Y., J.Colloid Interface Sci., 87, (1982) 478. 58. Jagiello, J.; Ligner, G.; and Papirer, E., J. Colloid Interface Sci., 137, (1989) 128. 59. Jagiello, J.; and Schwarz, J.A., J. Colloid Interface Sci., 146, (1991) 415. 60. Villi~ras, F., Cases, J.M., Francois, M., Michot, L.J., Thomas, F., Langmuir, 8_, (1992) 1789. 61. Villi~ras, F., Michot, L.J., Cases, J.M., Francois, M., Rudzinski, W., Langmuir (in press). 62. Huang, Y.J.; Schwarz, J.A., J. Catal., 99, (1986) 149. 63. Rudzinski W., Borowiecki T., Dominko A., Zientarska M., Chemia Analityczna, (in press). 64. Lee, P.I, Schwarz, J.A., J. Catal., 73, (1982) 272.
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w. Rudzifiski, W.A. Steele and G. Zgrablich (Eds.) Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces Studies in Surface Science and Catalysis, Vol. 104 9 1997 Elsevier Science B.V. All rights reserved.
373
Surface diffusion of adsorbates on heterogeneous substrates G. Zgrablich Centro Regional de Estudios Avanzados, Gobierno de la Provincia de San Luis, Av. del F u n d a d o r s/n, 5700 San Luis, Argentina and D e p a r t a m e n t o de Ffsica, Universidad Nacional de San Luis, C h a c a b u c o y Pedernera, 5700 San Luis, Argentina
Contents 1-
Introduction
2 - G e n e r a l theory of surface diffusion 2.1 - Tracer diffusion coefficient 2.1.1 - U n c o r r e l a t e d r a n d o m walks 2 . 1 . 2 - Correlated r a n d o m walks 2.2 - Chemical diffusion coefficient 2.3 - Relation between D and D* 2.4 - Phenomenological a p p r o a c h to D 2.5 - Arrhenius form for D 2.6 - Lattice-gas model for the diffusion coefficient on heterogeneous surfaces 3 - Surface diffusion in porous solids 3.1 - Surface diffusion coefficient from permeability e x p e r i m e n t s in porous solids 3.2 - Simple empirical and phenomenological models 3 . 2 . 1 - Variations on t h e Arrhenius equation. 3 . 2 . 2 - Effective m e d i u m approximation 3 . 2 . 3 - Percolation m o d e l 4 - Tracer diffusion on simple t o p o g r a p h y heterogeneous surfaces 4.1 - R a n d o m t r a p s
374 4 . 2 - Random barriers 5 - Diffusion on correlated heterogeneous surfaces: an approach for the description of general topography 5.1 - Generalized gaussian model 5.1.1 - Influence of a for random surfaces (r0 = 0) 5.1.2 - Combined effect of a and r0 5.2 - Site-bond model 5.2.1 - Tracer diffusion on correlated traps and barriers surfaces 5.2.2 - Tracer diffusion on a general site-bond surface 5.2.3 - Chemical diffusion coefficient for a general site-bond surface 6 - Conclusions and open problems 1.
INTRODUCTION
Diffusion of adsorbed particles on heterogeneous surfaces is a most important phenomenon ocurring in a variety of processes of great practical relevance such as gas separation and purification by adsorption, heterogeneous catalysis, crystal growth, wetting, sintering, etc. In spite of continuous theoretical and experimental efforts dedicated to its study for many years, surface diffusion on heterogeneous surfaces still presents many unsolved problems and doubts related to the mechanisms and physical nature of the migration of particles and to the characterization of surface energetic heterogeneity. In the last ten years, researchers have recognized the importance of including the energetic topography in the statistical description of heterogeneity. This, and the progresses in the experimental techniques for surface analysis at the atomic scale, have encouraged the development of more refined models of heterogeneous surfaces capable of including the topographic characteristics of the adsorptive energy surface. One of the important problems is then to study the effects of the energetic topography both on the tracer and chemical diffusion coefficients. Another important problem, specially due to its applications to heterogeneous catalysis, is that of surface diffusion in porous adsorbents. Here, many different effects contribute to the behavior of the chemical diffusion coefficient as a function of the surface coverage and other types of phenomenological models, or semiempirical equations incorporating apparent (or resultant from the superposition of several effects) quantities, are necessary. In the present monograph, a general basic framework is first given for describing surface diffusion on heterogeneous surfaces, and then particular models are presented and discussed for the two specific problems mentioned above. 2.
G E N E R A L T H E O R Y OF S U R F A C E D I F F U S I O N
This section is devoted to state the general kinetic equations which are independent of the particular model assumed to describe the energetic heterogeneity and are, in particular, valid for homogeneous surface. We end with an explicit description of surface
,(a)
!
. (b) ,
, ,
o
~a,~
,
X
375
..
X
Figure 1: Random walks in one (a) and two (b) dimensions. diffusion on heterogeneous surfaces based on the lattice-gas model, which is the most general description available at present. 2.1.
T r a c e r Diffusion Coefficient
Tracer diffusion Coefficient, D* , refers to the migration of a single particle (the "tracer") on a surface. It can be conveniently introduced by analyzing the particle movement as a "random walk" [1-3]. 2.1.1.
U n c o r r e l a t e d R a n d o m Walks
In an uncorrelated random walk each jump is completely independent from the previous one. For homogeneous surfaces this occurs when the particle transfers its energy rapidly to the phonon bath of the substrate after a jump, in such a way as to stay in the new site for a long time compared to the periods of typical vibrational modes involved in jumps [4]. If this happens, the particle loses memory of the previous jump and the next one will be made in a completely random direction. For heterogeneous surfaces, however, in addition to the above condition, the random walk will be uncorrelated only if at each site all jumping directions are equally probable. This new condition will be fulfilled obviously only by some kinds of heterogeneous substrates (for example a random traps lattice). For an uncorrelated random walk in one dimension, Fig. l(a), we have as well known results
[2]: (x(t)) = 0
(2.1)
(~(t)) = ~ ( t )
(2.2)
where n(t) is the number of jumps in a time t. The average (...) is intended to be taken over a large number of walks of duration t. In two dimensions we will have analogously
(R2(t)> = a2nz(t) + ayny
(2.3)
376 or, if a~ = ay = a and n(t) = n~(t)+ ny(t)
(R2(t)l = a2n(t)
(2.4)
Introducing now an "effective" jump frequency I/ef f "--
(2.5)
t
the mean square displacement for the tracer can be written as (R2(t))
=
a2Vefft
(2.6)
For homogeneous media vr
is a constant and the proportionality law
(R2(t)> c< t,
(2.7)
which is valid for a medium with any Euclidean dimension d, is fulfilled (normal diffusion). For heterogeneous media, v~ff can be in general time-dependent and (2.7) is replaced by the "anomalous diffusion" proportionality law [5] 2
(a2(t)) c< t ~ ,
d~ _ 2
(2.8)
where the value of d~ depends on the system considered. Anomalous diffusion occurs typically on fractal substrates and for some kinds of energetically heterogeneous surfaces, but also in a variety of other interesting physical problems [6]. The Tracer Diffusion Coefficient, D* , for a medium with Euclidean dimension d, is defined independently of the characteristics of the medium as:
(R~(t)) = 2dD*t
for t ~ ~
(2.9)
or equivalently, D * = tlim--,~[ (R2 (t))2dt
(2.10)
Combining (2.6) and (2.10) we have D * = tlim
a21/eff ] 2d ]
(2.11)
In the case of surface diffusion d = 2. However, it is useful to keep these equations in their most general form. Definition (2.10) is independent of the nature of the jumps (for example, these can be due to an activated process, i.e. jump over an energy barrier, or through a tunneling process). If the jumping process is an activated one, we can use the absolute rate theory [7,8] to write I / e f f "-
2dr e x p ( A S / k s ) e x p ( - E , / k s T ) ,
(2.12)
377 where v = k B T / h , A S is the activation entropy corresponding to the transition of a particle from a potential minimum to the saddle point of a barrier and E~ the activation energy (the energy height of the saddle point from the bottom of the potential minimum). With this, the tracer diffusion coefficient for an activated jumping process can be now written as (2.13)
D*= Do exp(-Ea/ksT),
where
(2.14)
Do=
Eq. (2.13) is known as the Arrhenius law and is strictly valid for an homogeneous surface. For heterogeneous surfaces AS and E~ generally vary from point to point. Some times, a satisfactory approximation is to replace these values by their mean values, AS and Ea, taken over the whole surface. However, as it will be shown later on, for some heterogeneous surfaces the Arrhenian behavior is not always appropriate. For an assembly of N classical (distinguishable) particles the tracer diffusion coefficient still has a perfectly good meaning through the generalized definition D * = t~oo lim 2 d1g t ~N( [ / ~ i ( t ) -/~i(0)[ 2) ,
(2.10')
i=1
although it cannot be measured experimentally due to the impossibility of labelling the N particles, but, of course, it can be determined in computer simulations. 2.1.2.
Correlated
Random
Walks
When the relaxation time of the particle after a jump is comparable to the vibrational period, or when energetic heterogeneity makes the jump probability depend on the direction, a given jump will not, in general, be independent from the previous one and we will have a correlated random walk. To illustrate how this can be due to heterogeneity, we refer to Fig. 2 where the adsorptive energy E along a direction x on the surface is represented. If the tracer particle located at site i jumps to the right, the next jump will probably be made in the reverse direction due to the lower potential barrier. Let us write the mean square displacement of the tracer particle as: n
n
i=l
j=l
= <(X:
zJ)>
(2.15)
where r'k is the displacement vector corresponding to the k th jump and the point (.) represents the scalar product between vectors. After developing the sums and grouping terms conveniently, and taking into account that r'k.r'k = a 2, we have = ha2(1 + 9
n
a2 k=l
2
+ --
a------y--- - t- .. .) ~-2~ ~'k.~'k+2
?'t k - 1
(2.16)
378
X Figure 2: Correlated random walk due to energetic heterogeneity. Introducing now the definition of the "jump correlation factor", f, as ~
/=(1+-~
? ~ - - T?Z
~
~
rk.rk+m
aS
)
(2.17)
?2 m--1 k--1
the mean square displacement can be expressed in the form:
(R2(t)) = n a ~ f
(2.18)
In this way, if the definition of tracer diffusion coefficient, Eq. (2.10), is maintained, then the most general expressions involving D* become: = 2dD*t = na2 f D*
=
lim
na2 f ] 2dt J
(2.19) (2.20)
The value of the jump correlation factor, f, ranges from 0 to n. For an uncorrelated random walk all jump directions are distributed completely at random in such a way that (~'~.~'j) = 0 for any pair (i, j) and f = 1. If jumps are so strongly correlated that given the first jump in a given direction all subsequent jumps will be taken in the same direction, then (~'~.~'j) = a 2 for all pairs (i, j) and f = n. Another limit situation arises when, given a jump in one direction, the probability of the next jump to be taken in the reversal direction is very high. In such a case f approaches zero. A very complete discussion of the properties of correlated random walks is given in [9].
379 2.2.
C h e m i c a l Diffusion Coefficient
The Chemical Diffusion Coefficient, D, is usually defined through the Fick's First Law expressing the proportionality between the diffusive flux of adsorbed particles, J, and the particle concentration gradient ~Tp(~'): 1st Fick's Law
J = -DV,.p(~')
(2.21)
Here the adsorbate concentration p is assumed to depend only on the spatial position K This definition is generally valid even for heterogeneous surfaces where D can be a function of the position. If the adsorbate concentration p were also a function of time, then taking into account the continuity equation -
0
V.J + -~p(~',t) = O
(2.22)
we have 0
O---~p(r, t) = Yr. [DVrp(r, t)]
(2.23)
which reduces to the well known 2=d Fick's Law
~tp(r t) DV~p(~',t)
(2.24)
=
only in the case that D is independent of the spatial coordinate K In the most general case, and specially for heterogeneous media, D not only will depend on the spatial coordinate but also on the direction (anisotropic medium) being then represented by a tensor [4]. 2.3.
Relation
Between
D a n d D*
A general relation between D and D* can be obtained by expressing the two diffusion coefficients in terms of velocity correlation functions. This can be easily achieved in the case of D*. In facts, the mean square displacement for a tracer particle can be written as {R2(t)) = {[ fotV(t ') dt'12)
= =
]o'dt'
dt"{g(t').g(t")>
(2.25)
380 where 5(t) is the instantaneous velocity of the particle at time t.The velocity correlation function K~ = (5(t').TY(t"))
(2.26)
in Eq. (2.25) depends only on the time difference ~- = t" - t', in such a way that K ~ ( t " - t ' ) = (g(t').g(t"))= (fi'(0).g(7-))= K . ( 7 ) = K ~ ( - r )
(2.27)
With this, Eq. (2.25) can be written as (R2(t)) = ~0t dt' /~,____.--Td~-(g(0).~'(T)) t'
= ~0t dT j~Ot-r dt' (g(O).~7(~-))+ drKv(T)(t - T)+
=fo' dTK.(v)(t
=2]o' d~'(t -
f
d~-
dt' (~'(0).77(r))
t
"7-
f t drKv(z)(t + T)
- T)+ j~Ot d r K . ( r ) ( t -
7)
~-) (g(O).~7(T))
(2.28)
Then, using the definition (2.10), we finally obtain (2.29)
D* = -~ 1 ~o,~ dt(5(O).g(t)) For a system of N tracer particles, this result can be easily generalized to D* = - f 1- j ~ ~=~ fo
(2.30)
~ dt(gi(O).g~(t))
It is interesting to see how the simple expression, Eq. (2.11), can be derived as a particular case of Eq. (2.29). If 5(0) and g(t) are correlated only over a "jump time" t ~ 2 , then
Veff
D*
1 [~
--- -d J O
115
dt(5(O).5(t)) ~ -~
(5(0/.5(
I/elf
)
(2.31)
and, given that g(0) = 5'(~-~) ~ a~'eff/2, this reduces to D* -- 2-~a 2lJef f
(2.32)
381 The analogous expression to Eq. (2.30) for D in terms of the velocity correlation function can be obtained by a simililar but much more lengthy procedure given, for example, in [10]. The result is the Kubo-Green formula: oo
N
N
1 fo dt(~ ~'~(0). ~ ~'j(t)) D = 2kBTp2xA i--1 j--1
(2.33)
where n is the two-dimensional compressibility of the adsorbed phase and A its area. It can be shown [4] that x is related to the mean square fluctuation of the number of molecules in A through ((6N)2> =
ksTp 2xA
(2.34)
so that Eq. (2.33) can be finally written as 1 r~ D = 2((5N) 2) Jo
N
N
dt(~ ff~(O).~ gj(t)) i--1
(2.35)
j--1
As a first important difference between D* and D, we notice that, while in D* only velocity correlations for each single tracer particle are involved, D depends also on cross correlations for the velocities of different particles. By separating in Eq. (2.35) the contributions of the direct and cross correlations, and combining with Eq. (2.30), we can write
D_D= D*
(N) ((SN) 2)
[l + f~ dt(~iCj gi(O).gj(t)) ] J dtf E~(g~(0).g~(t)) o
[
(2.36)
Thus a simple relation between D and D* exists only in the particular case in which there are no cross correlations between the velocity of different particles. In that case v~(0) and gj(t) are totally independent for i-r j and Eq. (2.36) reduces to:
D (N> D--7 = ((5N)2)
(2.37)
The relation between (N) and ((SN) 2) can be readily obtained through the partition function for the grand canonical ensemble of adsorbed particles at temperature T and chemical potential/~ as:
((/~> = o-~0(k-~) where 0 = D D*
N/A
(2.38)
is the surface coverage. Thus Eq. (2.37) becomes
0 ~]
~n-~ (k-~)
(2.39)
382 known as the Darken equation [11]. The right hand side is referred to as the "thermodynamic factor". The coverage dependence of the chemical potential, the isotherm equation, can be obtained through a great variety of adsorption models, depending on the kind of system to be considered. For a general heterogeneous surface we have already seen that single particle correlations could be present over several jumps, and it is then also probable that in some cases cross correlations for velocities would be important. The validity of Darken equation for heterogeneous surfaces should then be considered only as an approximation. The effects of adsorptive energy heterogeneity on the chemical diffusion coefficient, according to Eq. (2.39), will have two components, i,e. those acting through the tracer diffusion coefficient D* and those acting through the thermodynamic factor. Of them, the latter have been intensively studied through adsorption models for heterogeneous surfaces [12, 131.
2.4.
Phenomenological Approach to D
As discussed in the previous section, the calculation of D is a difficult problem even if D* and the coverage dependence of ~t were known, except in the most simple case in which the Darken equation is valid. For such reason, the development of a phenomenological approach to D is of great importance. One of the most transparent of such approaches is due to Reed and Ehrlich [14], who introduce a "phenomenological diffusion coefficient ", D, as
b = ~r(O)a 2
(2.40)
where F(0) is a coverage dependent effective jump frequency. The basic assumption of this model is that the Darken equation is still valid for the general case in which cross velocity correlations are not negligible if D" is replaced by D. In this way the chemical diffusion coefficient in the Reed-Ehrlich model can be written as
D = ~r(O)a2[O(g/ksT)/OlnO]T
(2.41)
The reasonability of this model can be further appreciated by combining the KuboGreen formula, Eq. (2.35), with Eq. (2.38) to obtain
D = [O(~/ksT [ OlnO
(2.42) =1
Then Eq. (2.41) has the same physical meaning as the Kubo-Green formula if the factor
lfo 2N
N g~(O).gj(t)) dt(~-~ i,j
is considered as an average b = ~F(O)a 2.
(2.43)
383 The validity of Eq. (2.41) has been tested by Monte Carlo simulation in the case of homogeneous surfaces [15] against the Bolzmann- Matano and the number fluctuation methods [8]. All three methods agree in general, except when repulsive nearest-neighbor and attractive next-nearest-neighbor interactions are present among adsorbates. In such case, predictions of eq.(2.41) are in concordance with those obtained by the transfer-matrix method [8,16]. Although the Reed-Ehrlich model was formulated for homogeneous surfaces, it is intuitively clear that it can be also used as an approximation for heterogeneous surfaces. In fact, Eq. (2.41) should be valid (to the same degree of approximation as for an homogeneous surface) for one realization (sample) of an heterogeneous one. Now, the chemical diffusion coefficient for an heterogeneous surface should be obtained as an average over a large number of surface samples (all prepared with the same statistical properties). If the effective jump frequency, F(0), and the thermodynamic factor are statistically independent quantities, the averaging process will preserve the factors separation and the form of Eq. (2.41). Thus the applicability of the Reed- Ehrlich model to the case of an heterogeneous surface amounts to statistical independence of the effective jump frequency and the thermodynamic factor. We shall see in Section 2.6 what this really means in a particular formulation of the diffusion coefficient for heterogeneous surfaces based on the lattice-gas model. 2.5.
A r r h e n i u s F o r m for D
It is usual, but not always useful, to force the chemical diffusion coefficient to obey an Arrhenius law of the type D = Doe -zE~
;
(2.44)
~ = 1/ksT
where the "apparent" Arrhenius parameters, i.e. the preexponential factor Do and the activation energy for diffusion Ed , are given by Do = lim D
(2.45)
t3~0
Assuming the validity of Eq. (2.41), Ed can be written as
[a~nr(o)] Ed=-[ O~ o- E-
La-~ejr
[~J
alno -
r alnOO~
0~] -10H~
~]r
(2.47)
0l,~o
where U is the internal energy per particle, H~ the heat of adsorption per particle and the effective activation energy of jumps given by -[aznr(e)/o~]e. Eq. (2.47)showsthat /~ is a function both of 8 and T, so that linear Arrhenius plots of InD against/3 can be expected only over narrow ranges of T. It is also shown that if the heat of adsorption
384 increases (decreases) with coverage then Ed is greater (smaller) than the effective activation energy of jumps, /). Lateral interactions and heterogeneity, however, will affect, in general, both terms in Eq. (2.47), in such a way that they are not really meaningful as separate contributions. Interpretation of experimental data in terms of the apparent Arrhenius parameters Do and Ed should be handled very carefully since the validity of Eq. (2.44) can be far from reality in the case of heterogeneous surfaces or when lateral interactions are strong enough to produce order-disorder phase transitions in the adsorbate. In the last case, values of D varying through several order of magnitude with coverage can be found [17-21]. 2.6.
L a t t i c e - G a s M o d e l for t h e Diffusion Coefficient on H e t e r o g e n e o u s Surfaces
Following [8,22,23] we present here a kinetic derivation of D for a general heterogeneous surface assuming that the adsorbate can be represented by a lattice-gas of interacting particles. More precisely, we assume that adsorbed particles are located in a two-dimensional array of adsorption sites S, Fig.3, each one being either vacant or occupied by a single particle. Diffusion occurs via activated jumps of particles to nearest-neighbor empty sites through the saddle points, or bonds, B. Jumps to the next-nearest-neighbor sites through the high energy barriers H are neglected (effects of long range jumps on diffusion, discussed in [24], will not be considered here). The adsorptive energy can be described by a three-dimensional potential energy surface E(x, y), where x and y are the cartesian coordinates on the substrate. Fig.3, left hand side, represents equipotential contour lines of such energy surface for an homogeneous (a) and an heterogeneous (b) surface. When one moves along, for example, the x direction, at a fixed y = y0, the adsorptive potential profile shown on the right hand side is found. We assume that the statistical properties of the adsorptive energy surface can be conveniently described by an n-point multivariate distribution function:
6,~( El, E2, ..., En)dE1 dE2...dEn
(2.48)
which is the probability of finding an energy Ea E (Ea,E1 + dE~) at the point (x~,y~), E2 E (E2, E2 + dE2)at (z2, y2),..., E,~ E (E,~,E,~ + dE,-,) at (x,~,yn). The n points (xl, yl), (x2, Y2),..., (z~, y~) can be chosen in some convenient way to describe the energetic topography. For interacting particles, the jump rate will be in general dependent upon the particular configuration "z" of adsorbed particles in the neighborhood of the jumping particle and on the potential energy topography around it. So, we can write the flux of particles from column 1 to column 2 (see the quare lattice of Fig. 3 (b)) as 1
J,,2 = ~<E PAo,iKAo,,)
(2.49)
where PAO,i is the probability that the site in the first column is occupied by a particle A and the nearest-neighbor site in the second column is empty, with an environment of
385
S
B
H
Y
V(x,yo) A
~
a---~
S
N 1
a-----~ 2
X
X
~
(a)
B
H
X
V(X,Yo)
(b)
X
Figure 3: Equipotential contour lines (left hand side) and adsorptive potential along x direction (right hand side) for homogeneous surface (a) and heterogeneous surface (b), showing adsorption sites S, saddle points B and barriers H.
386 adsorbed particles marked by the index i, and K is the rate constant for the transition
AO, i ---+OA, i. The summation on i is taken over all possible configurations of adsorbed particles surrounding the pair AO and the average (...) is taken over the ensemble of surfaces, i.e. should be calculated with the distribution function (2.48) as:
((--.))----/---/(...)~n(Sl,...,En)dSl...dEn
(2.50)
Using the grand canonical distribution, we have
(2.51)
PAO,i - Poo,i exp [/~(#1- E 1 - WAO,i)]
where Poo,i is the probability that a pair of nearest-neighbor sites is empty, with the environment marked by the index i, #1, is the chemical potential in column 1, E1 is the energy of the site occupied by the jumping particle in column 1 and WAO,i is the lateral interaction of a particle A with the neighborhood "{'. Substitution of (2.51) into (2.49) yields 1
J1,2 = ~(exp(~#l) ~ Poo,iKAo,~ exp[-13(E1 + WAO,~)])
(2.52)
i
In a similar way, the flux from column 2 to 1 is given by 1
J2,1 -" ~(eXP(fl~2) 2 Poo,iKoA, i eX,p [-/3(E2 + WOA,~)]) i
(2.53)
where #2 is the chemical potential in column 2. Application of the detailed balance principle to the present case gives KAO,i e x p [ - ~ ( E 1 + WAo,i)] =- KOA,i exp[-13(E2 + WOA,i)] so that the summations in Eqs. (2.52) and (2.53) are equal and the total flux J = can be expressed through a gradient in the chemical potential in the form O# J = -/3(exp(/3#)~-~z ~
Poo,~KAo,i exp[-3(E~ + WAO,,)])
Introducing now the adsorbate density p =
(9..54) J1,2-J2,1
(2.55)
O/a 2, we have
Olz Op J = -a213(-~ ~. Poo,iKAo,iexp [-/3(Ea + WAO,{)]-~x >
(2.56)
Comparing this equation with Fick's first law we finally obtain for the chemical diffusion coefficient:
D = -a2~(-~
~ Poo iKAo,iexp[-~(E1 + WAo,i)]> i
(2.57)
387 We now wish to discuss under what conditions Eq. (2.51) can be reduced to a generalized form of the Reed-Ehrlich representation. In the context of the present lattice-gas model, we can write for the effective jump frequency: F(0) - 4 ( ~ QAo,iKAo,i}
(2.58)
i
where QAO,i is the probability that a given A particle in the first column, has an empty nearest-neighbor site in the second column, with the environment marked by the subscript "z". Comparing the definitions of PAO,i and QAO,i, we see that they are related through: Q ~o,~ = PAo,~ /
Y] i
P.~,~ = PAo,~ / 0
(2.59)
where PA,i is the probability that a site with environment "i" is occupied. Using (2.51) and (2.59), Eq. (2.58) takes the form
s
4 ( ~ Poo,~KAo,i exp{/3 [#- (Ex 4- WAO,i)]}/O)
(2.60)
i
which can be considered as the generalized expression of the effective jump frequency for heterogeneous surfaces represented by a two-dimensional square lattice of sites. Assume now that the t h e r m o d y n a m i c a l factor O/~#/OlnO in Eq. (2.57), which is d e p e n d e n t u p o n a p a r t i c u l a r realization of a h e t e r o g e n e o u s surface a m o n g those belonging to the statistical ensemble, has a s m o o t h variation and can be replaced, t h r o u g h a m e a n field a p p r o x i m a t i o n , by O(/~#)/OlnO, i.e. t h e a p p a r e n t t h e r m o d y n a m i c a l factor calculated from the overall a d s o r p t i o n i s o t h e r m ~ = #(0). Then, combining Eq. (2.60)and (2.57), we obtain the generalized Reed-Ehrlich expression" _~
D-
0(9~) F(O) Oln-O
(2.61)
This equation is helpful for practical calculations and is currently used in the literature. However the validity of the main assumption stated above, on which it rests, is far from being evaluated. Future simulations to test its validity are encouraged. So far, the present analysis is applicable to any kind of diffusion (for example, activated and tunnel diffusion). For gas-solid phenomena, we are specifically interested in activated diffusion. For this process the rate constant can be written as
KAO,~ - u ezp{--/3 [E~(0)+ W~'- WAO,i]}
(2.62)
where u is the usual preexponential factor, E~(0) the activation energy for a jump at low coverage and W/* is the lateral interaction of the activated complex A* with the environment "z". Using (2.62) in (2.60), the effective jump factor becomes: -
s
-
//
= 4(~ ~ Poo,i exp{-~ [E~(0)+ Ea + W/* - #]}}
(2.63)
The lattice-gas formulation developed here will be used in Section 5 to discuss diffusion on well characterized simple and correlated heterogeneous surfaces. Before this, however, in the next Section, it is useful to discuss diffusion in porous adsorbents, to be analyzed through somesimple empirical and phenomenological models.
388 I--
zH I Ii
0
~"
D ii
C3
i
i
i
i
~-
1 2 3 4 COVERAGE ( m o n o b y e ~ ) Figure 4: Log of surface diffusion coefficient versus coverage for the system CF2C4 at 240K[29], showing a typical behavior. 3.
SURFACE
DIFFUSION
IN POROUS
SOLIDS
Porous solids constitute a wide variety of materials which are of fundamental importance in gas-solid processes of great practical interest, like catalysis and gas separation. Therefore the study of surface diffusion in porous solids has attracted the attention of researchers for a very long time [25-41,82-84], but even now a general theory explaining the main observed properties is far from being available. Two kinds of difficulties contribute to this situation: i) the experimental difficulties to measure accurately surface diffusivities, which must be obtained by subtracting the contribution due to the transport of molecules in the pore space in experiments where the total permeability through a cylindrical pellet is measured; and ii) the fact that the surface of a porous material is highly heterogeneous, an important component of such heterogeneity being due to surface rugosity. A typical behavior of the surface diffusion coefficient versus coverage is given in Fig. 4 for the system CF2C4 on silica at 240K [29]. Different mechanisms are usually assumed to explain the behavior of D in different coverage regions. Below the monolayer a diffusive mechanism based on activated jumps of individual molecules over energy barriers is considered. The very fast rise of D in this region (which often is greater than one order of magnitude) is attributed to the effect of strong heterogeneity. Around the monolayer, D stabilizes and even decreases for larger coverage. In the multilayer region two kinds of mechanisms are usually taken into account: i) a hopping mechanism where jumps are allowed both in the same layer or between different layers [37]; and ii) a mechanism known as hydrodynamic model, which regards the adsorbed gas phase as a laminar-flowing film of viscous liquid [26,30]. In a third region, identified as that where capillary condensation begins, D raises again. Here the hydrodynamic model is applied. We are mainly interested in the submonolayer region, where gas-solid interactions and
389 heterogeneity effects are most important. In this section we modify Fick's law in order to take into account the "tortuosity" of the surface of a pore, relating then the diffusion coefficient to the permeability which is the quantity obtained in experiments, and review some of the empirical equations and phenomenological models which have been used in the analysis of experimental data. 3.1.
Surface diffusion coefficient f r o m p e r m e a b i l i t y e x p e r i m e n t s in p o r o u s solids
The "permeability", I, of a gas through a cylindrical pellet of cross section A and length L is defined through: - -IA~z z
(3.1)
where N is flow rate through the pellet and dp/dx is the pressure gradient along the x axis of the cylinder. If the pellet is a porous solid capable af adsorbing the flowing gas, then the total permeability It will have two components
It = / g + Is
(3.2)
where Ig is the permeability in the gas phase, corresponding to transport of molecules in the pore space, and Is is the surface permeability, corresponding to surface transport of molecules in the adsorbed phase. If the mean free path of molecules of mass M in the gas phase is much greater than the mean pore diameter (say, ten times), then the flow in the pore space should obey the Knudsen equation C b = x/MT
(3.3)
where C is a constant characteristic of the pore space and is independent of pressure, temperature and the nature of the gas. The condition for Knudsen flow is frequently fullfilled for a great variety of gas-solid systems. Thus, by measuring the total permeability of a gas such as He which will not adsorb on the given solid at the temperature of interest, C is easily determined. Now, It is measured for the adsorbable gas whose surface diffusivity we want to study and this allows the determination of the surface permeability as
Is = It - C v / M T
(3.4)
Total permeabilities are easily measured through an ingenious permeation apparatus (see for example Ref [30]) which allows the measurement of the flux on the basis of the change in the pressures on both sites of the pellet under steady state conditions, or through time lag measurement. We must now relate Is to D. To do this we first immagine the total internal surface S of the porous pellet spanning a rectangle of length TL (along the x direction) and side
390
b = S/'rL. Now the distance ~ measured on the rectangle along the x direction is related to x by ~ = Tx. The parameter ~- is known as the "tortuosity" of the porous pellet. Writing now Fick's first law, Eq. (2.21), for this spread out rectangular surface we have N. = _bDdP(~) S dp(x) d~ = - r 2 L D dx
(3.5)
where Ns is the surface flow rate along the coordinate ~. Now dx d~ = dp dp dx dv then .~s= _ S--~--Ddp dp
7"2L dp dx and, taking into account by the definition of permeability, Eq. (3.1), that the surface permeability/~ is given by/~ = -IV~/(Adp/dx), we obtain the following relation between D and 5:
....
.]
D = Sdp/dpJ Is
(3.6)
or, in a more convenient form:
D __. [[ T2VsTP OP] ppV,~ -~ Is
(3.7)
where pp is the pellet density, VSTP is the gas molar volume at standard temperature and pressure, V,~ is the molar volume necessary to fill a monolayer and Op/O0 is obtained from the adsorption pressure-coverage isotherm. Thus the measurement of the adsorption isotherm and the monolayer capacity is necessary as an additional experiment to relate D to Is. For this reason, the pressure gradient applied to the pellet must be very small, in such a way that the equilibrium pressure of the adsorption isotherm at a given coverage may correspond to the mean pressure through the pellet. The only parameter which is still undetermined is the tortuosity T. This cannot be measured directly for the gas of interest, but can be estimated by comparing the measured flux of a non-adsorbable gas with the expected value predicted by the Knudsen equation for a straight cylindrical tube [42]. Typical values of ~- determined by this method range between 1.5 and 2.5. More recently, an alternative way to estimate 7 through the fractal measure of surface rugosity has been proposed by Avnir [43]. It is interesting to explore the relation between tortuosity 7 and the porosity r (the volume fraction accessible to the gas phase) in a porous material. This is a very difficult problem and only some few semi-empirical equations have been given. However, the subject is of great importance since r is a measurable quantity. Two frequently used equations are T = 1/r
(3.8)
due to Wakao and Smith [44], and = 45/r
(3.9)
391 40
I
I
I
i
o SO2, 1 5 ~ o S 0 2 . 30oc
35 30
i
= NH 3. ZS~
CO 2, -78 ~ 9 CO 2, -50 ~ * H e , 30~
= NH3, 40oc
" N2
"
930~
v
-'h_ 25 - ~ O o ~ O
0
O
0 o
Ov
v 0
0
%,
OV
~
o
v 0v
0
o
0
0
0 0
0
0
m
E
E lS 111-
~ ~ % 9
•
_
A
~oO~o @
A
A
~
~
4kA
@A
5
I
I
I
I
I
I
100 200 300 400 500 600 700
0
P, tort
Figure 5" Permeabilities of five gases in porous Vycor glass versus mean pellet pressure. due do Weisz and Schwartz [45]. Less frequently used is the relation 1
(3.10)
developed by Wissberg [46] who considered the solid as formed by an agglomerate of overlapping spheres. Some very high values of tortuosity reported in [47] follow instead the equation 7 - 1/r 2
(3.11)
Monte Carlo simulations performed by Evans and coworkers [48], who simulated the solid by placing spheres of different radii centered at random positions (allowing overlapping of spheres), follow closely Eq. (3.10) which gives the smallest tortuosity for all values of porosity. We can only conclude that the relation between ~- and r is strongly dependent upon the nature of the porous material. Figure 5, taken from Ref. [33], representing the measured total permeabilities for five gases in porous Vycor glass, gives a nice idea of the experimental errors involved in this kind of measurements (see the statistical fluctuations of experimental points). Notice, in particular, that for a weakly adsorbed gas, like NH3 at 25~ and 40~ statistical fluctuations are nearly of the order of the difference between the total permeabilities of
392 that gas and those of the reference non-adsorbable gases _N89and He. This means that for NH3 the statistical experimental error is almost of the order of the surface permeability which is being measured. In spite of these experimental dificulties, permeation experiments have provided for a long time valuable information which has been used to improve our understanding of surface diffusion on highly heterogeneous surfaces through simple phenomenological models. We review some of them in the following section. 3.2.
Simple empirical and phenomenological models
3.2.1.
V a r i a t i o n s on t h e A r r h e n i u s e q u a t i o n
Carman et al. [29] and Barrer et al. [49] made use of the Arrhenius equation D = Doe -zE'~
(3.12)
to analyse surface diffusion in a variety of systems. Carman realized that surface heterogeneity should be responsible for the increase in D at low coverage and that the activation energy E= should decrease with coverage by similarity with the behavior of the isosteric heat of adsorption qst for an heterogeneous surface. Barter found that the ratio E=/q,t was approximately in the range 0.2 to 0.5 for solids with smooth (but heterogeneous) surfaces and in the range 0.6 to 1.0 for solids presenting rough surface textures. These results were interpreted assuming that for rough surfaces there is a contribution of long molecular flight over "evaporation barriers" created by the crevices and blind pores. Higashi et al. [50] proposed a simple modification of Eq. (3.12) to take into account that a molecule not only can jump from the site it is occupying to a neighboring empty site, but to any neighboring site and, if it is occupied, the molecule will keep migrating in a random walk from site to site until an empty site is found and it is adsorbed again. The expected number of jumps until re-adsorption is given in such a model by
n(e) = ~ i ( 1
-
i'-1
e)e ' - ' =
1 1 -0
(3.13)
Thus, since D should be inversely proportional to the time of residence at a site, and this is now shorter by a factor (1 - 0 ) , we have D=
19'o e_ZE,, 1--0
(3.14)
Even though this modification of the Arrhenius law seems to work for nearly homogeneous surfaces and at 6 < 0.6, in many cases it predicts the wrong curvature for D(0) at low coverage [32]. Following Carman's thinkings about a relationship between the activation energy and the isosteric heat of adsorption, Gilliland et al. [33] assumed the linear relationship E~ = aqst
(3.15)
393 Table 1 Correlation of experimental data to Eq. (3.16) Gas Solid Do a Range of (cm2/s) Variaton of qs~ (Kcal/mol)
Mean deviation from correlation
Ref.
(%) C02 C02 NH3 C2H4 C3H6 i - C4H10
CF2CI2 SOz
Glass Glass Glass Glass Glass Glass Silica Carbon
0.037 0.018 0.20 1.15 1.20 0.025 0.27 0.22
0.48 0.47 0.60 0.81 0.75 0.46 0.63 0.43
4.1-6.3 5.5-7.7 6.5-8.8 5.3-7.0 6.3-7.5 5.9-7.1 6.5-7.8 6.7-8.8
6 4 9 15 17 10 4 8
33 33 33 51 51 51 52 53
and proposed the equation:
D = Doe -zaq''
(3.16)
With this, they could obtain satisfactory fitting of a variety of experimental data, as shown in Table 1. This partial success of the very simple Eq. (3.16) stimulated further research, by Sladek et al. [34], who proposed what is known as Sladek's correlation:
D~ = 0.016e -~
(3.17)
where m is a parameter whose value (1, 2 or 3) depends on the type of solid (conductor or dielectric) and the type of gas-solid interactions. With these three values of m, data from 30 gas-solid systems with D varying over 11 orders of magnitude were correlated to within 131 orders of magnitude. This success, on a coarse grained scale, reflects the fact that there is undoubtedly a strong relation between Ea and qst for a strong surface heterogeneity. However, equation (3.16) or (3.17) cannot be valid in general, since, for example, for a homogeneous surface it would predict a constant D value, contrary to what is generally observed. A more complete expression for D, was developed by Okazaki et al. [37] who included other elementary processes, like the hopping from one layer to another, as depicted in Fig. 6. The total residence time is t = (1 - O~)to + O~tx, where 0~ is an effective coverage for diffusion. Each elementary process is assumed to behave in an Arrhenius form. This leads to the equation D = D 0 ~ -,m,,
(1 - e - ~ i i l -2- 0~1 - ta/to)] f(q)dq
(3.18)
where t: =
(e - z ~ q -
to
(1
-
e-~'q)(1 - e -~'q') _
(3.19)
394
t1
LO
9
~
9
9
Figure 6: Hopping modes of adsorbed molecules and their respective residence times.
10-
I
IIIII
I
I I IIII
I
I
I
I III
Key Data Key Data 9 6 [] 1 7 2 o 3 9 8
0
~D~ i 0 -2
I
9
4
v
9
5
e
10
CO
r~
~
. m
'~
10 -3
Key Data Key Data A 11 9 13
,'
v
16
[]
17
.,m
14 i l II
10
I
-3
I
I
i i IIII
I
10
F[-]
-2
i
I
I I III
10
Figure 7: Correlation of surface diffusion coefficients data of Table 2 by Eq. (3.18).
-1
395 Here, f(q)dq is the number of molecules adsorbed with heat of adsorption between q and q + dq, q, is the heat of vaporization of the adsorbate and E'a is the activation energy for all layers above the first. For an adsorbate obeying the B.E.T. isotherm:
0
=
cx/(1
-
x)(1
-
x
+
cx)
(3.20)
where x = p/po is the relative pressure, 0e is estimated as 0~ = 0(1 - x). Fig. 7 shows the correlation of experimental data given in Table 2 by Eq. (3.18). Eq. (3.18) improves the correlation of diffusivity data near the monolayer and in the multilayer region. However, like other phenomenological equations based on Arrhenius behavior it does not allow a deeper understanding of the effects of heterogeneity and, on the other hand, still predicts a constant behavior for a homogeneous surface. 3.2.2.
Effective m e d i u m a p p r o x i m a t i o n
The effective medium approximation (EMA) is an old method which has been succesfully applied to a variety of problems related to conduction in disordered media [54-57]. The idea, in a sense similar to that of mean field theory, is the following: one assumes that the mean effective diffusion coefficient Dm is known. The random medium is then replaced everywhere but in a small region by the equivalent effective medium (which is then homogeneous). One then successively assignes to the small region elements the values Di of the initial random medium, each with its own weight f(Di). These values differ from Dr~ by amounts 5D which depend both on Dm and Di
Di = D~ + 5D(Dm, Di)
(3.2~)
Dm is then self-consistently determined by the condition
~f(Di)SD(D~,Di)
= 0
(3.22)
i
which states that, on the average, the correction induced by the disorder vanishes. We now discuss more deeply the EMA in the context of a network of random resistors, of conductances g, by closely following Kirkpatrick [57]. The distribution of electrical potentials in a random resistor network to which a voltage has been applied along one axis may be regarded as due to both an "external field" which produces an increase in voltages by a constant amount per row of nodes, and a fluctuating "local field", whose average over any sufficiently large region will vanish. We shall represent the average effects of the random resistors by an effective homogeneous medium which, for simplicity, we consider to be made up by a set of equal conductances gin. Consider now a different conductance gAB = g, oriented along the external field between nodes A and B, surrounded by the effective medium, Fig. 8(a). To the effect of the uniform field, represented by a constant voltage Vm, we add the effects of a fictious current, 5i, introduced at A and extracted at B, which is chosen in such a way as to satisfy current conservation at A and B:
Vm(gm - g) = 5i
(3.23)
396 Table 2 Correlation results for experimental data Gas Porous T 0 ~ Solid C2H4 Vycor 30.0 0.13~0.31 C3H6 Vycor 30.0 0.04~0.60 iC4H~o Vycor 30.0 0.04,-~0.73 S02 Vycor 30.0 0.29,~1.07 CF2Cl2 Linde -33.1 0.47~1.61 Silica -21.5 0.21~2.66 S02 Linde -10.0 0.46~2.13 Silica 0.0 0.36~1.71 CF2C12 Carbolac -33.1 0.85,-~2.30 -21.5 0.61~1.71 0.0 0.48~1.16 20.0 0.32,~1.10 C02 Carbolac -33.1 0.13~0.53 -21.5 0.14~0.41 0.0 0.04~0.26 20.0 0.03~0.17 S02 Vycor 15.0 0.37,~0.84 30.0 0.27-,~0.67 C02 Vycor -78.0 0.63~0.89 -50.0 0.38,~0.78 C3H6 Vycor 0.0 0.14-~0.99 25.0 0.07,-~0.71 40.0 0.07~0.61 iC4Hlo Vycor 0.0 0.16~0.48 C2H6 Vycor 0.0 0.02~0.25 25.0 < 0.15 50.0 < 0.12 C3H6 Vycor 0.0 0.12~0.81 25.0 0.06~0.69 50.0 0.03~0.50 CF2Cl2 Carbon -5.0 0.26~1.20 Regal 10.0 0.12~1.02 20.0 0.07,~0.86 Call6 Graphon 0.0 0.11,~1.14 25.0 0.05,-~0.97 50.0 0.02~0.73 nC4Hlo Graphon 30.0 0.15~1.12 41.7 0.25,-~1.14
using Eq. (3.18)
D xl0 s 4.4~9.1 2.2~7.5 2.2~6.7 0.8,~4.0 2.1~5.3 1.9~7.0 2.8~7.6 2.6~9.2 3.2~10 2.7~11 3.0~15 2.8~16 3.6,~13 5.8~14 5.1~16 7.5,~17 0.8,,~3.3 0.8~2.6 0.7,-~1.3 0.6,--2.5 0.5~7.7 0.9~6.2 0.5,~6.8 0.7~2.0 2.7,-~11 3.3~15 5.7,~8.8 1.1~15 1.1~11 0.9~10 4.0,~3.4 3.5~39 5.9~39 80~730 120~500 170~300 100--~1300 90~880
a
Do
Esl
Ref.
0.39 0.45 0.48 0.38 0.49
2.i3x10 -3 5.41x10 -3 ll.9x10 -3 2.16x10 -3 9.20x10 -3
4.98 7.79 10.4 8.75 7.49
[37]
0.41
8.14x10 -3
8.75
[52]
0.41
7.67x10 -3
7.49
[52]
0.54
3.41x10 -3
7.83
[52]
0.46
5.02x10 -3
8.75
[33]
0.51
9.95x10 -3
7.83
0.45
4.29x10 -3
7.79
[30]
0.56 0.54
1.06x10 -2 1.47x10 -2
10.4 6.95
[30] [32]
0.52
1.47x10 -2
7.79
[32]
0.66
5.75x10 -2
7.49
[35]
0.63
5.57x10 -1
7.79
[32]
0.48
6.39x10 -1
7.39
[52]
397
'i e
I
Ca)
B
i
(b)
Figure 8" Constructions used in calculating the voltage induced across a conductance g surrounded by a uniform medium. The extra voltage, 5V, induced between A and B, can be calculated if we know the conductance G~B of the nework between points A and B in absence of the conductance
(rig. S(b)): =
i/(g +
(3.24)
Now, G~B is related to the conductance GAB between A and B in the uniform effective medium through GAB = G~AS + gin, and this can be calculated by a simple symmetry argument: express the current distribution in Fig. 8(a) with gAS = g~ as the sum of two contributions, a current 5i introduced at A and extracted at a very large distance in all directions, and an equal current introduced at infinity and extracted at B. In each case, the current flowing through each of the z equivalent bonds at A and B is r so that a total current 25i/z flows through the A B bond. Then, it follows that GAS = (z/2)gm, or G'AS = ( z / 2 - 1)gin. Using (3.23) and (3.24) we obtain
5V = Vm(g~ - g ) / [g + (z/2 - 1)gm]
(3.25)
If the values of g are distributed according to a probability density function f(g), the requirement that the average value of 5V must vanish leads to the folowing condition for determining gm:
f dgf(g)(gm - g ) / [ g + (z/2 - 1)gin] = 0
(3.26)
Since in the equivalent diffusivity problem D is proportional to g, we may write for the EMA for the diffusion coefficient:
f d D f ( D ) ( D m - D ) / [ D + (z/2 - 1)Dm] - 0
(3.27)
We notice that Eq. (3.27) is valid both in 2 and 3 dimensions for a lattice of adsorptive sites with coordination .number z. Two nearest-neighbor sites are connected through a bond characterized by an energy barrier for molecular jumps.
398 100
100
50
5O
10
I0
5
Dm(e ) 5
Dm(e) Hom D O=O
IDe=o
0.5
0.5
0.1
I
I
I
0 0.2 0.4 0.6 0.8
(a) I
I
0.1
i
i
I
I
0 0.2 0.4 0.6 0.8
(b) I
0
Figure 9: Behavior of D~(O)/DH=~ ~ a function of 0 for different values of the heterogeneity parameter s. (a) E~ = e; (b) E~ = ~e. Through the derivation of Eq. (3.27) it is clear that the EMA is applicable to heterogeneous surfaces whose energetic topography is such that bond energies are distributed totally at random and such that the activation energy to jump from any site A to a nearestneighbor site B is the same as that for the reverse jump, i.e. a random barriers surface (see section 4.2). However, by considering that each resistor in Fig. 8(a) is really representing a homogeneous patch, it is also clear that Eq. (3.24) is applicable to a patchwise topography. Monte Carlo simulations performed by Kirkpatrick [57] show the shortcoming of EMA for a correlated bonds network corresponding to intermediate topographies. From a practical point of view EMA may provide an adequate basis (compared with other existing models) to describe diffusion on highly heterogeneous surfaces like those corresponding to porous solids. Kapoor and Yang [40] applied this approximation to the analysis of a variety of experimental data. They assumed for D the Higashi modification of Arrhenius law, Eq. (3.14), with a uniform distribution of activation energies given by
1
Emin_E~a z
f(E~) =
0
Eamin Emax for __~E~ < otherwise
(3.28)
They also assumed a Langmuir isotherm
o=
pb
(3.29)
b = boez'e
(3.30)
1 +pb
with
399
,r
i" (a)
(b)
Figure 10: Adsorptive energy contour lines: (a) thermal energy lower than critical energy; (b) thermal energy higher than critical energy. where ~ is the mean adsoption energy. Under these assumptions, integration of Eq. (3.27) over activation energies yields" eS+pb+(~-l)Dm(O)/DH=~ e -~ + pb + (~ - 1)D,-,.,(O)/DH~
[ = exp
e S - e -s ] ~D2 m (O)/DH=~ ~ '
(3.31)
where s is the "heterogeneity" parameter ( E y ~'x- Ey~'~)/2kT, and DH=~ = D'o e-~E~ Figure 9 shows the behavior of Dm(~)/DH=~ for different values of the heterogeneity m m 1parameter s in the case that E~ = ~ (a) and E~ = ~e (b). Diffusion coefficients for a variety of gases on vycor glass, Linde silica, Carbolac, Spheron 6 carbon black and activated carbon, were fitted with average relative errors ranging between 9% and 46%, the highest errors corresponding to activated carbon in which microporosity may introduce new transport mechanisms. 3.2.3.
Percolation model
Following the ideas introduced by Ambegaokar et al. [58] to treat the problem of the hopping conductivity of electrons in amorphous semiconductors, Zgrablich and co-workers [38,39] developed a quite general model considering the surface diffusion of adsorbed gases as a percolation process [59,60]. The underlying idea can be better understood by visualizing the adsorptive energy surface E ( x , y ) for an heterogeneous surface ( ( x , y ) defines a position on that surface) as a mountainous landscape where depths are adsorptive sites. If the thermal energy of adsorbed molecules is considered as the "height of waters" in the mountainous landscape, then we could describe the situation at two different temperatures as in Fig. 10, where the adsorption energy surface is represented by its "contour" lines, or equipotential curves. At low temperature, Fig. 10(a), molecules are confined to disconnected "lakes" (shadowed areas) through which they can migrate by a hopping mechanism (of course, some jumps
400
E 0
Gas Phase
Ec J
Figure 11: Adsorptive energy variation along a direction x on the surface. Due to lateral paths, in the true adsorptive energy surface the minimum energy level E~j connecting sites i and j may be at the lower position shown. outside the shadowed regions may occur but they are infrequent) and the overall surface diffusion coefficient is very low. As the temperature increases, the "water" level becomes higher and the lakes grow wider until a thermal energy is reached for which the "water" percolates to form an "ocean" where low diffusivity regions are disconnected islands, Fig. 10(b). This state defines the critical energy level, Ec, for surface diffusion of adsorbed molecules and its value is determined by the overall adsorptive energy topography and not by a "local" value of the activation energy. The formulation of the percolation model is based on the following assumptions: (i) For each pair of sites "i" and "j" separated by a distance Rij, there exists an energy level Ei3 which is the minimum energy level connecting both sites, such that E~j = Ej~ (see Fig. 11). In general, E~j will be lower than the critical energy level Ec defined by the percolation threshold. (ii) For a molecule to jump from an occupied site "i" to an empty site "j" a distance Rij apart (jumps to sites other than nearest neighbors are allowed) it must make a transition to an activated state determined by E~j and the interactions with other adsorbed molecules. (iii) These interactions are described by a mean field energy W, for adsorbate - adsorbate, and W*, for activated particle-adsorbate interactions. (iv) A molecule in the activated state moves from site to site and may either be readsorbed at an empty site (with probability 1) or, if the site is occupied, have a probability Pd of being desorbed to the gas phase and 1 - Pd of continuing its random walk. If the probability for the activated particle of either being readsorbed or desorbed in travelling a distance dR is defined as dR~A, then )~ is a "mean free path" and will depend on the surface coverage ~.
401 Now, the number of molecules Fij migrating from i to j per unit time can be written in general as:
Fij = FoP+_P~exp [-Rij/)~(O)] P~j
(3.32)
where F0 is the characteristic vibration frequency of an adsorbed molecule, P+_ is the probability that site i is occupied (+) and site j empty ( - ) , P~ is the transition probability to the activated state for the molecule adsorbed at site i, exp [-R~j/$(O)] is the probability for the activated molecule to travel a distance R~j without being either readsorbed or desorbed and P~j is the probability for the activated molecule to be readsorbed at the empty site j (P~j is taken = 1, as usual). If a mean field approximation is used to take into account lateral interactions, then
P+_ - P__exp[fl(#i- E i - WO)] ~, (1 - O ) 2 e x p [ / 3 ( # i - E ~ - WO)I
(3.33)
where Ei is the adsorptive energy and #i the chemical potential at site i. Now, according to assumptions (i), (ii)and (iii),
P,o = ~xp{-Z [(E,j + W*O) - ( E , + WO)])
(3.34)
so that rij = F(1 - O)2exp[13(#i- W*0)] exp [-R~j//~(O)] exp(-flAEq)
(3.35)
where F = Foexp(-/3E---~), AEq = E~j- E---~and E~ is the mean energy of adsorptive sites. In a similar way: r~ = r(1 -
o)2ezp [~(m - w'o)] ezp [-R~j/,X(O)] ezp(-~AE~j)
(3.36)
At equilibrium, as #i = #j (and given that in all cases Eij = Eji) it turns out that Fij = F ji and there is no net current between i and j. However, when a surface concentration gradient is present, which can be expressed through a chemical potential gradient such that #j = # and tti = # + A#, then the net current Cq = Fij - F j i becomes:
r
- GijAO
(3.37)
where the "conductance" Gij is given by
a~j - rA(O)~xp{-[Rq/~(O) + /~XE~j]},
A(O) = (1 - O ) 2 e z p [ j 3 ( # - W*O)]
0(~)
0O
(3.38) (3.39)
The problem is now reduced to that of finding the conductance of a network of random resistors whose conductances Gij, given by Eq. (3.38), are composed by a slowly varying factor A(0), Eq. (3.39), depending on mean field values of coverage and chemical potential, and a stochastic factor exp [-(R~j/A + 3AE~j)]. An exact solution for such a problem is out of reach Until now, and some suitable approximation must be made. Of course, an
402
E
r
/ 4
v
R Figure 12" Representation of a "completely connected cornponent". approximation like EMA could be applied here, even though the greater generality of the present model (for example the distribution of the P~j is needed) may bring about some difficulties which were not present in the simple model discussed in the previous section. However, a very direct and powerful approximation leading to a very simple expression for D will be applied in this opportunity, say, to replace the total conductance of the medium by the critical value Gc given by the percolation threshold for conduction. Before the calculation of Gc is attempted, we give a brief discussion of why this is a suitable approximation. Different points of the three-dimensional space (R, E) are connected by resistors whose conductances G~j are distributed according to some probability distribution. If we choose resistors from that distribution in decreasing order of conductance values G1 > G2 > G3 > ..., and assign them randomly to pairs of points in the (R, E) space, then we will achieve a critical conductance Go, when a path crossing the entire region occupied by the system is formed, Fig. 12. At this moment the percolation threshold is reached and there will be conduction through the whole system. Now, if more resistors are added, new paths will be created but the total conductance will not increase appreciably because only small conductance resistors are available. Then the total conductance is approximately given by G~. Going now to the calculation of G~, we see that the condition for conduction, G~j > Go, can be expressed through Eq. (3.38) as: ..r
R~j/~(O) + ~AE~j < ln(Gc/FA) -1
(3.40) ..r
For a given site i, Eq. (3.40) defines a region in the three-dimensional space (R,E) within which conduction is possible. Then, for several sites we have a collection of partially overlapping conducting polyhedra in the (R, E) space, as shown in Fig. 13. The critical conductance will be reached when the conducting regions percolate in the continuum three-dimensional (R, E) space. The maximum spatial dimension for a typical conducting
403
-%
R Figure 13" Percolation three-dimensional space and conducting polyhedra. region is obtained by making AE~j = 0 (for example, a fiat adsorptive energy surface), in Eq. (3.37) and is
AR~
= ~l~(G~/rA) -~,
(3.41)
while, similarly, the maximum energetic dimension turns out to be the one corresponding to R U 0 -
-
1
AEm~: = ~In(Gc/FA) -1,
(3.42)
so that a typical volume of conducting polyhedra is given by ~ ~ (AR,~)2AEm~
(3.43)
Supposing a constant density of states po in the (/~, E) space, then the total conducting volume is p0~ and, as well known, the percolation threshold is reached when the total conducting volume is a fraction 6 (6 ~ 0.16 for 3 dimensions) of the total space, i.e.: p0~ = 6
(3.44)
From Eqs. (3.41) to (3.44) we obtain Gc as:
G~= rA(O)ezp{- [(/~(5/t00)1/3~-2/3]}
(3.45)
Now, the surface diffusion coefficient D (conductivity) will be related to the conductance through D = gG~, where g is a geometrical factor, so that we have
D = DoA(O)exp{-(136/po)l/3[A(O)] -2/3}
(3.46)
404 where Do = gF. We must still calculate the mean free path A(0). According to its definition in assumption (iv), and given that the main contribution to the diffusion coefficient comes from the critical conductivity, we may suppose that any activated molecule contributing to D is at the critical energy level Ec, so that the desorption probability at an occupied site is P~ - ~ z p
[-Z(AE~- W*0)]
(3.47)
where/kEg = E a - Ec, Ea being the boundary energy for the gas phase. Now, assuming that the activated molecule performs a random walk of step length A0 (notice that this is the first time that it is necessary to assume the existence of a regular array of sites), then
,~(t~) = )~o/P~(O)
(3.48)
where Pc(O) is the effective probability for the activated molecule to stop its walk (either by readsorbing at an empty site or by desorbing at an occupied site) and, according to (iv) and Eq. (3.47), is given by P~(O) = (~ - o) + O ~ p [ - Z ( ~ X E ~
-
w*0)]
(3.49)
With this, D can be finally expressed as-
D = DoA(O)exp{-(To/T)~/3[P~(O)] 2/3}
(3.50)
where we have introduced the characteristic temperature To as
To = ks poA~
(3.51)
It is important to investigate the significance of To. From the definition of po and for a constant adsorptive energy distribution between E - cr and E + a, where cr is the dispersion of adsorptive energies, we have po - 1/(2~r)~o2), so that To is proportional to cr through
To = (23/ks)a
(3.52)
hence, To is a measure of heterogeneity (ksTo ~ 0.32cr). In the limiting case To ~ 0 (a ~ 0), we obtain D = Do(1 - 0) 2ezp [/3(# - W'O)]
o(Z~) 00
(3.53)
which is the result obtained by Zhdanov [61] for an homogeneous surface. Actual calculations of D require that the isotherm equation be given so that the factor A(9) could be evaluated. Since a mean field approximation for the adsorbate has been made, the following isotherm equation for the mean coverage 0 can be used [62]:
0(#, T) = / f(E)OL(E, #, T)dE
(3.54)
405
o
}
5["
~'4
o'=1
~s 3 2
a=0.5
.
1 i
I
9
I
,
I
0 0.2 0.4 0.6
e
I
I
I
0.8
1
Figure 14: Effect of heterogeneity on surface diffusion coefficient for noninteracting molecules. where
OL(E,#, T) is the "local coverage" for a site with adsorptive energy E, given by
OL(E, #, T ) =
exp [ - # ( E + WO - #)] 1 + ~ p [ - # ( E + W O - ~,)]'
(3.55)
and f(E) is the adsorptive energy probability density (in our case a constant between Es - a and Es + cr and zero outside). Figure 14 shows the effect of heterogeneity on D(O)/D(O), while Figures 15 and 16 show the combined effects of heterogeneity and lateral interactions W and W*. All curves were calculated for E, = -2.5 kcal/mol and T = 300K. Heterogeneity values, a, are in kcal/mol. Finally, the model is tested by fitting experimental data from references [35,36]. Since experimental data are given for permeabilities/,, from Eqs. (3.7) and (3.50) we have: L = Io(1 -
O)2ezp(-nW*O)ezp{-(To/T)~/3[P,(O)] 2/3}
where Io =
gFppVmezp(##o)/r2VsTP and #o is given by the well known expression
#o=-kTln
\
h2 ]
]
kTq,,q,.
(3.56)
(3.57)
qv and qr being the internal vibrational and rotational partition functions. This is indeed a very simple equation for the permeability and has four fitting parameters: I0, To, AEd and W*. Two of them, i.e. To and W*, can be determined independently by fitting adsorption isotherms when these are available for the same gas-solid system in which permeabilities are measured. Figures 17 to 19 show the results for surface permeabilities of ethylene, 1,3-butadiene and n-butane on carbon Regal 660 [36], while Table 3 presents the parameter values.
406 [a]
L '
"
1.0 0"=1
0.8 0.6
o.,~ x,...~ ~ 00 0.2
lt
t
0
ti
!t
0.2
. L
0,~
I~"~'=--"P ~,
"
O.B
0.8
I-
"
[
I
I
. l
i
0.2
0
l
I
0.~
l
i
0.6
|
0.8
1
Figure 15" Effect of heterogeneity on surface diffusion coefficient for molecules with attractive interactions and activated complex with repulsive interactions (in kcal/mol)" ( a ) W = -1, W * = 0; (b) W = 0 , W * = 1.
2O
18 (O] 16 14
/
-
ff=l
(b)
/
"
0"=1
12-
c3 10 s
C
_
0
|
a
0.2
1
t
0,/,
L
l
0.6
|
I
0.8
l
t
E)
~
0.2
i
i
0./,
I
I
0.6
i
J
0.8
I
_
1
Figure 16: Effect of heterogeneity on surface diffusion coefficient for molecules with repulsive interactions and activated complex with attractive interactions (in kcal/mol); (a) W = 1, W* = 0; (b) w = 0, w ' = - ~ .
407
!: :
9
-
~8
0
.
-
278K 9248K 9225K
206K
~
9194K
I
I
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
Figure 17: Surface permeabilities for ethylene on carbon Regal 660. It can be seen that To is fairly constant for all data (which correspond to the same heterogeneous substrate) with a mean value of 168.4 and a standard deviation of 11.8(7%). This value corresponds to cr = 1.04kcal/mol for the dispersion of adsorptive energies, a very reasonable value given the experimentally observed [63] variation in isosteric heat of adsorption (Aqst ~ 1.13+0.3kcal/mol) from 0 ~ 0 to 0 = 0.5 calculated by estimating the slope near 0 = 0) at low coverage where the lateral interactions contribution is expected to be small. For A E d we have also a remarkably good behavior. In fact, A E d is practically constant for each gas-solid system and leads to values of the mean activation energy for diffusion, E~, = -qst - / k E g , which are between 0.56 and 0.59 of the desorption energy, a result widely supported by independent experimental evidence [49,64]. Finally, the values for W* are reasonable although their variation with temperature has no apparent explanation. In conclusion, the theoretical background on which the model is formulated and the encouraging results obtained by fitting experimental data makes it worth to be further investigated by analyzing a wider variety of well characterized gas-solid systems. We now turn over to the study of the effects of heterogeneity on surface diffusion for solids with simple and well characterized energy topography.
0
TRACER DIFFUSION ON SIMPLE TOPOGRAPHY OUS SURFACES
HETEROGENE-
In this section we study the tracer diffusion on model heterogeneous surfaces whose adsorptive energy topography is very simple. These surfaces probably have no close relation to real surfaces but their study is clarifying. We have chosen two opposite kinds
of
random
(RT)
408
9283K 9298K = 323K 9348K *403K
~16 ==
==
vl
~
e
~o 8
==
i
~
4 e..t.....-t---~-'~e
2
u~
0 0
i 0.1
i 0.2
i 0.3
9..... e ~
I I I 0.4 0.5 0.6
I 0.7
I 0.8
0
Figure 18" Surface permeabilities for 1,3-butadiene on carbon Regal 660.
~8
i7 E ~6
413K 9
-
9358K 9323K 9298K 9283K
_ i
,~4
i3
==
-- __T 1
..Ilium
0 0
.-
--
I
i
I
0.1
0.2
0.3
i
I
I
I
0.4 0.5 0.6 0.7
I 0.8
Figure 19: Surface permeabilities for nbutane on carbon Regal 660.
409 Table 3 Parameter values obtained by least-squares fit of Eq. (3.56) to experimental data of Ref. [36]. Here 1~ = 10sI0v/-M--T
[(molKg)l/2/s cm mmHg] G~ Ethylene
1,3-Butadiene
n-Butane
4.1.
Random
T
I~
-W*/kB
(K)
(cm2/s)
(K)
278 248 225 206 194 403 348 323 298 283 413 358 323 298 283
2.1 3.3 8.9 10.6 15.0 3.4 7.3 14.4 23.6 29.7 3.1 820 11.4 19.5 24.9
933 807 471 445 385 1160 785 491 371 420 904 820 684 594 556
/XEd/kB To
(K)
(K)
658 686 691 651 671 868 850 863 835 839 928 907 921 950 836
182 179 186 169 168 156 162 147 158 152 171 168 169 185 174
Traps
A random traps surface is characterized by the fact that all the saddle point (bond) energies for jumps from one site to another have the same value EB, while site adsorptive energies are randomly distributed according to some function Fs(Es), Fig. 20. This model surface presents the important symmetry property that the jumping probability is independent of the direction of the jump, just like in a non-correlated random walk and is the only one for which exact results have been obtained. One of the most important of these results [65] concerns the tracer diffusion exponent d~ of Eq. (2.8), which is found to be d~ = 2 with the only condition that the initial state must be one of thermal equilibirum (this means that the probability for the tracer to start its walk from a given site with adsorptive energy Esi must be proportional to e-~Esi). The mean square displacement of a tracer particle can also be obtained exactly by solving the master equation [9]. However we chose here to obtain it by using simple intuitive arguments. We start by assuming that the probability r for a tracer particle located at site "i" to jump over the energy barrier in a very short period of time dt is proportional to dt, i.e., r = dt/z~, where 1/z~ is the proportionality factor. Then, the probability 9~(t) for the particle to wait at site "i" during a time t satisfies the differential equation: ~i(t) - 9~(t - dt)(1 _ _dt) T~
(4.1)
410
K8
~-sj E.SI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 20" Adsorptive energy profile for a random traps surface. By integrating this equation, we get
~ ( t ) = e -t/~'
(4.2)
Then, the probability r of time (t, t + dt) is:
for a particle at site i to make its jump in the interval
= -dte - - tl~.' 7i
r
(4.3)
From this, the mean waiting time at a given site "i" can be calculated as
tr
< t >i=
= ~'i
(4.4)
Now, our tracer diffusion process on the random traps surface is just equivalent to the uncorrelated random walk described in Section 2.1.1, with the only difference that at each site "i" we have a waiting time ri. Thus, if we replace the number of steps n(t) of Eq. (2.4) by an effective mean number of steps t
=
(4.5)
we obtain for the mean square displacement (R2(t)> = a2t/(T~),
(4.6)
and for D* (through Eq. (2.10)) a 2
D~
__
"
(4.7)
411
Fs(Es) ] l/hE
AE s
=
$ i ..........
I
I
-E B
-Es
-~
-E
Eo
Figure 21" Uniform site distribution for the random traps surface. which is the exact solution for the random traps surface. We show now how (r~) can be calculated from a given distribution of adsorptive site energies. The simplest case is a uniform distribution of site energies, as shown in Fig. 21. If the jumpling process is an activated process, then the waiting time ~'i at a site i is
Ti
=
e3E~
eB(EB-E%)
=
(4.8)
where E~ is the activation energy for a jump from site i. Then 1 [Es+AEs/2 < Ti >-- A E s J-Es-AEs/2 m
e x p [ 3 ( E s - Es)] des
(4.9)
which gives D
e3E~
< Ti > = 2 3 A E s S i n h ( 3
/kE 2
s)
(4.10)
and, for D*, m
1 D* = -
_
3vE~
_
4 exp(3E~)sinh(37E~)
(4.11)
where we have introduced the dimensionless heterogeneity parameter "7 = AEs/2E~ (0 <_ 7 -< 1). The behavior of D* given by this equation is shown in Fig. 22. At very high temperatures, 3 -+ 0, D* reaches its maximum value 1/4. As temperature approaches zero, 3 ---, oc, D* vanishes asymptotically. If the surface is modified at fixed temperature T, in such a way as to increase E~ without changing 7, D* decreases. On the other hand, at fixed T and E~, D* decreases with increasing 7-
412
D
1/4
~_
/Y3 Y2 ~Ea
Figure 22: Tracer diffusion coefficient for a random traps surface. The values of the heterogeneity parameter are in increasing order, 71 < 3'2 < 3'3. In a similar way, D* can be obtained for a random traps surface with a truncated gaussian distribution for site energies (4.12)
Fs(Es) = 0;
otherwise
After some straightforward calculations, D* can be found to be given by
[Erf(b)- Evf(a)] exp(/3Es)ezp D* =
I_.-5-(/~ ~ + ~-)~
2d{Erf ["/7(b - ,8o"2 - Es)] - E r f t 2[ ~ c~,a
-/3~
l + ~ 2~2j -
~s)]
(4.13) }
where Erf(x) is the error function defined by
Erf(x)
2fo~exp(-u 2) du
= --~
(4.14)
It is instructing to discuss the Arrhenius behavior of D*. For simplicity we take the case of a uniform distribution of site energies, Eq. (4.11). In Fig. 23, InD* is represented versus ~Ea for different values of the heterogeneity parameter 3'- It can be immediately seen that the Arrhenius law is valid only for 3' = 0, i.e. an homogeneous surface. As 3' increases, the plot deviates more and more from a straight line until it reaches the boundary curve corresponding to 3' = 1. If we force D* to behave as exp(-~E~ff), where E~ff is an effective activation energy, then for the case of a uniform distribution for Es it can be easily obtained from Eq. (4.11) that 1
E~f] = E---~- -~ + 7E---~tgh-a(flT-E~)
(4.15)
413 n
13E a
.
In D
=0
t
!
7=1
Figure 23" Arrhenius plot of D* for a random traps surface. which is clearly temperature dependent. For 7 = 0 we have the expected result E,.:.:= Ea (Arrhenius behavior). The same limit is reached also for/3 ~ 0 (infinite temperature) and any 7- For 7 > 0 and low temperature,/3 ~ oo, the limit is m
XEs
(4.16)
2 i.e., as the temperature approaches zero the effective activation energy is the maximum available value. This result is reasonable since it is assumed that the tracer particle starts from the state of thermodynamical equilibrium which is coincident with the deepest trap at zero temperature. Another important fact emerging from Fig. 23 is that the Arrhenius plot for an heterogeneous traps surface is below that corresponding to the homogeneous surface. At first sight, this could be a little surprising because one is tempted to think that for the heterogeneous case there will be deeper traps but also shallower ones, as compared to the homogeneous case, and their effect could cancel. However, this is not what happens in reality, due to the fact that with the presence of very few deep traps the tracer particle will be retained by them for very long times, and this effect is stronger as the temperature decreases (/3 increases). This is a very important feature of this kind of energetic surface topography and we name it as the "trap feature". 4.2.
Random Barriers
A random barriers surface is characterized by the fact that all adsorptive site energies have the same value Es, while bond energies are randomly distributed according to some function FB (EB)~ Fig. 24. As we can immediately see, a fundamental property of the random traps surface is now lost, i.e. now the jumping probability will depend on the direction of the jump. In fact, for example in Fig. 24, the barrier to overcome to jump from site i to j is quite higher
414
Ell ..........
s
/
!
i
k .... X
Figure 24: Adsorptive energy profile for a random barriers surface. than that to jump from site i to k. In particular, this means that the jump correlation factor f of Eq. (2.18) will not be equal to 1 and will be at least temperature dependent. Another important difference with random traps is the possibility of having anomalous diffusion, i.e. d~ > 2. In fact anomalous diffusion for random barriers has been reported [5] in the case of barrier energy distributions of the power type, involving infinite values of activation energy, for example: r
- (SE06E~"(1+6)
(4.17)
with E0 _< E~ < oc and (5 > 0. Using this distribution, and assuming a one-dimensional geometry, the following temporal behavior for < R 2 > is found [66]"
< R2(t)>1/2or (ln t) ~
(4.18)
In general, the random barriers problem is more difficult to handle by analytical methods than the random traps problem, and approximate solutions exist only for the onedimensional case. For this reason if is more convenient to study the problem by means of Monte Carlo simulation [67,68]. For a fixed site energy Es, a truncated gaussian distribution was used for bond enegies, with mean value E--B,dispersion (r and limiting bond energy values E~ 'i'~ and E ~ ~x. Simulations were performed not only for a two-dimensional (2D) square lattice but also for three-dimensional simple cubic ( 3 D - SC) and face-centered cubic (30 - FCC) lattices. Results for In [D*(T)/D*(oo)] versus/gE~ are shown in Fig. (25), and compared to the homogeneous and traps cases. It can be observed that, at any fixed temperature,
D*(T) D'ic~)
traps
D*(T) D'(oo)
[ D*(T)] homo
(4.19)
< LD*(~176
and the ratio is higher for higher dimensions (or lattices with a higher coordination number). The reason for this is a very important feature of the random barriers topography, which we name "barrier feature", i.e. the tracer particle can "get around" a high barrier
415 o
5
lO
I
w /t
I~Ea
8
k-
E~ w
--,,,~',,. ~ ~mc
..J
\ HOMOGENEOUS~
,
-10
'k ~ " ~JPs
"~
Figure 25: Tracer diffusion coefficient for random barriers for a truncated gaussian distribution with E s - l O . 5 k c a l / m o l , E B = 7 k c a l / m o l , cr = 1 . 6 k c a l / m o l , E ~ ~'~ = 3 . 5 k c a l / m o l , E'~ ~x = lO.5kcal/mol.
Eeff
Ea
4 t
I
0.5
1200/1'
Figure 26: Effective activation energy for a random barriers surface with the same parameters as Fig. 25.
416
Figure 27: Trapping of tracer particle between high energy barriers at low temperature" flipflop effect. by choosing a jump in another direction with a lower activation energy. This evidently is more probable for lattices with a higher coordination number. The behavior of the effective activation energy of the Arrhenius equation is another difference w'ith respect to the traps case. Figure 26 shows E~ff versus 1/T as obtained by Monte Carlo simulation. It can be seen that the maximum activation energy corresponds now to the limit of high temperatures, while at low temperatures E~ff ~ 0. This can be understood by realizing that at low temperatures the tracer particle will be oscillating between sites connected by low energy barriers, see Fig. 27, while as the temperature increases it can jump over higher barriers. This "flip-flop" effect, which has been observed experimentally [4], is also responsible for the behavior of the jump correlation factor f for a random barriers surface. In the case of traps, f was equal to 1 for any temperature, due to the jump symmetry. Figure 28 shows f versus ~E~ obtained by Monte Carlo simulation for barrier surfaces with different degrees of heterogeneity. We see how f drops rapidly to zero (faster for surfaces with higher heterogeneity) as the temperature decreases. In fact, as T ~ 0 the tracer particle is trapped in sites like those of Fig. 27 in such a way that each jump is followed by the reversal jump, making the sum in Eq. (2.17) give -n/2 and then f ~ 0. This particular behavior is another important characteristic of the barrier feature. We proceed now to study more complex adsorptive energy topographies in the following Section, and we will see that the trap feature and the barrier feature are two components which will appear as contributions to the general behavior.
417
\
\
k
-,,
,
\\
x%
,
"..
c=0.5
0.5
\o=1
\ 9
\
\ o =1.6
0
"\
5
"\
"\
PEa
10
Figure 28: Behavior of the jump correlation factor as a function of ~E~ for random barrier surfaces with different degrees of heterogeneity. 5.
DIFFUSION APPROACH
ON CORRELATED HETEROGENEOUS SURFACES- AN TO THE DESCRIPTION OF GENERAL TOPOGRAPHY
The general description of energetic topography of an heterogeneous substrate is a very difficult and unsolved problem. As already stated in Section 2.6, the gas-solid interaction energy is in general a three- dimensional potential energy surface E ( z , y ) , and this, for an heterogeneous substrate, is a stochastic process depending on the two continuous variables, z and y. This means that the most complete mathematical description of such a surface would involve dealing with a doubly infinite continuous set of intercorrelated random variables, a formidable task! A more modest program would be the development of a formulation capable of describing the statistical behavior of a few "well chosen" points of the energy surface, in such a way that different energy topographies should result by changing the degree of statistical correlations between such points. Two models in the spirit of this picture have been developed, i.e. the Generalized Gaussian Model (GGM), [62,69,70], and the SiteBond Model (SBM), [71,72]. The GGM chooses as special points of the energy surface the deeper points, i.e. what are known as adsorptive sites, and establishes a correlation function between pairs of sites which decays with the distance between them. The SBM, instead, chooses as special points a site and its neighbor bonds (saddle points in the energy surface connecting two sites) establishing a correlation function between them. We shall use these models here to study the influence of energetic topography of a general heterogeneous surface on surface diffusion.
418 5.1.
Generalized Gaussian Model
The GGM was first introduced in 1975 [69] as an attempt to study the adsorption of gases on a generalized heterogeneous surface with correlations between pairs of adsorptive sites through a gas-solid virial expansion. The correlation function between adsorptive energies of two sites i and j separated a distance r was given as =<
-
-
>=
where a is a measure of the heterogeneity degree and ro is the "correlation length". According to this model, any two sites which are within a distance r0 are strongly correlated (the correlation decays with distance as a gaussian function of half width ro) in such a way that their adsorptive energies have a high probability of being nearly equal. For sites separated a distance greater than r0 the correlation is almost null. Two well known simple topographies, i.e. random sites and patchwise surfaces, are limiting cases of the GGM corresponding to ro = 0 and ro = cx~, respectively. Due to the difficulties in calculating higher gas-solid virial coefficients, the GGM was reformulated more recently in terms of a lattice-gas description for the adsorbed phase using mean field approximations to handle adsorbate-adsorbate interactions [62,70]. The resulting adsorption isotherm equation is expressed as:
(5.z) where r
r
is the bivariate gaussian probability distribution given by
= [(2r)2det(H)] -'/2
exp{--~l ~ [Es, - -Es)(H-1)ij(Es3-- ES)])
(5.3)
i,j-'l
H being the matrix whose elements are the Hij defined in Eq. (5.1), and 0 is the "local" coverage given by
----(01 + 02)/2
(5.4)
where 81 and 82 are the coverages of sites 1 and 2 and can be obtained through the system of coupled equations ~,[-n(E a +W0:+~00+~-,)] (5.5)
As indicated in Fig. 29, A0 is the mean field interaction with particles within the circle of radius r0, A is the mean field interaction with particles outside that circle and W is the interaction between particles adsorbed at sites 1 and 2. In the GGM only the statistical properties of pairs of adsorptive sites are considered. Bonds energies are determined by assuming that heterogeneity is really a random perturbation of a periodic potential, represented by a broken line in Fig. 30. This means that
419
o
9
9
9
AN
o
9
9
9
,
,
,
9
9
.
,
,
,
,
,
,,
,
,
,
,
,
.
.
.
.
,,
,
o
,
.
,~
~
9
.
.
o
,
9
9
9
---lal---
,
~
,
,
9
9
,
,
,
~
.
,
,
,
o
,
.
,
~
.
,
9
,
,
.
,
,
.
9
9
9
.
,
.
Figure 29: Different regions considered for adsorbate-adsorbate interactions in the GGM.
E
Eb E O
Q
Es X
Figure 30: Determination of bond energies in the GGM.
420 for the heterogeneous potential (full line) the perturbation of the bond energy should be proportional to the mean value of the perturbations of the energies of the two neighbor sites, [70], i.e. E--s - E~ = o~ [(Es - Es,)+ (-Es - Esj)]
(5.6)
This leads to the following expression for the activation energy for the jump from site i to site j:
E~j = E ~ + ( a - 1)(Es - Es,) + a(-Es- Esj)
(5.7)
The "structure" parameter a (0 < a < 1/2) determines partially the topography of the adsorptive energy surface, for example it is easy to see that a = 0 corresponds to a "traps" surface. The topography is more completely determined by a and the correlation length r0. Figures 31-33 show different topographies obtained by different combinations of these parameters. In Figs. 31(a), 32(a) and 33(a) we have random traps (to = 0, a = 0), correlated traps (r0 -~ 1, a = 0) and strongly correlated, or patchwise, traps (r0 ---* c~, a = 0), respectively. More general topographies arise in (b) and (c). We now go to the calculation of the chemical diffusion coefficient to discuss the effect of the adsorptive energy topography. 5.1.1.
I n f l u e n c e of a for r a n d o m surfaces (r0 = 0)
We first study the simple case of an uncorrelated, or random, surface characterized by null correlation length, r0 = 0, to observe the effect of the structure parameter a on D, [23]. We use here the general expression obtained for D for a lattice-gas description of the adsorbed phase, Eqs. (2.63), (2.61) and (2.50). In the limit ro ~ 0, [70], the isotherm equations (5.2) to (5.5) reduce to
=
/= 0
+
[Z('[Z(, -
Fs(Es)O(Es, O) dEs
(5.8) (5.9)
where z is the lattice coordination number and Es the adsorptive site energy distribution. From these equations the thermodynamic factor O3#/Oln-~ in Eq. (2.61) can be numerically calculated. Now, the activation energy at zero coverage, E~(0), of Eq. (2.63) is just E~j given by Eq. (5.7), so that F(~) = 4v~yf eZu (~-, P00,i exp [-/~(W~* + aEsl + aEs2 )]>
(5.10)
where u, H = uexp(-~E~ W~ is the interaction energy between the activated particle and the environment "i", and we have taken for simplicity the origin of the energy scale
e-~
~=
0
o
0
o~
o
o~
o
ee~.,~o
Oq
o
(1)
0~" O-
4~
o
tmL
~j
t~ oo
i-~o
o~ CT
v
423
Ca)
(b)
(c)
Figure 33: Same as Figure 31 for
ro
=
~.
424 such that E s = 0. Finally, through Eqs. (2.61) and (2.50), we get for the chemical diffusion coefficient"
D = Ueffa2ez"i)~#R;
(5.11)
where
R - exp [-/Y(2z - 2)W*0 ] i Fs(Es)exp(-l%~E8)[1 - 0(Es, 0)] des
(5.12)
where W* is the interaction energy between the activated particle and an adsorbed one. We immediately notice that the effect of W* on D(O) is given by the trivial factor exp [ - 3 ( 2 z - 2)W*0], so that in what follows we consider W * = 0. At very low coverage, 0 --+ 0 we can easily obtain the limit value D(0), which should also represent the tracer diffusion coefficient D*, as" L
.i
D(O) = u~fsa2[f Fs(Es)exp(-~aEs)dEs]2 f Fs(Es)exp(-~Es)dEs
(5.13)
It can be seen that since the adsorptive energy Es involves negative values, D(0) for a = 0 is smaller than that for a > 0 at any temperature (higher/3), in coincidence with the results obtained for the tracer diffusion coefficient for random traps surfaces. Several adsorptive site energy distributions were used in actual calculations of D(O). (i) For a uniform distribution 1
Fs(Es)=
7-~
0
; for - A < _ E s < _ A ;otherwise
(5.14)
the isotherm equation can be easily obtained as
ezp(~#) =
exp(zW-O) [exp(2/~O)- 1] exp(~A) - exp [/3(20- I)A]
(5.15)
The results for D(O) normalized to D(0) are shown in Fig. 34. (ii) For a gaussian energy distribution C
Fs(Es) = v~aezp(-E]/2a2),_
(5.16)
the energy values must be truncated at E,~i,~ and Ema~ in order to avoid unphysical situations and the normalization constant C determined such that
/EE'~~ Fs(Es)dEs = 1 rnin
The results for the Gaussian distribution are shown in Fig. 35.
(5.17)
425
/A -5
f
O
113 t o~= A/-r--5
'
rr-3
'
Zl~llT=3
'~10 2
0
o
-3
l 0
0.2
0.4
0.6
0.8
1
0
O
I
I
I
I
0.2
0.4
0.6
0.8
1
Figure 34: Diffusion coefficient for ro = 0 and a uniform distribution of site energies. (iii) For a log-normal energy distribution
(5 S)
Fs(Es)- TEsv/~exp
where Em and 7 are the median and dispersion, respectively. Results are given for the case of non-interacting molecules in Fig. 36. The coverage dependence of the chemical diffusion coefficient is seen to be much stronger for random traps surfaces (a = 0), Figs. 34(a), 35(a) and 36, than for surfaces with a > 0. This is explained as follows. At low coverage the deeper sites are preferentially occupied. For a = 0 these sites are also those corresponding to the highest activation energy for jumping. As coverage increases, particles from sites with lower activation energies contribute to migration and D increases. As soon as the surface differs from a traps surface, for example for a = 1/3, Figs. 34(5), 35(5) and 36, the energy of the destination site begins to influence the jump rate and the deepest sites do not necessarily correspond to those with the highest energy barriers for jumping. This effect leads to a much weaker increase of D with coverage for a > 0. It is interesting to observe that for a = 1/2 a maximum in D(8) occurs at 0 = 1/2 for the rectangular (not shown) and gaussian energy distributions, and D(8) is symmetric about that value. This is not true for a non-symmetric distribution like the log-normal. The absence of a maximum in D(0) seems to be another characteristics of the random traps surface. Let us now discuss briefly the effect of adsorbate-adsorbate interactions. Repulsive (attractive) interactions result in an increase (decrease) in the diffusion coefficient. This is explained by a decrease (increase) in the activation energy for diffusion due to a rise (fall) of the bottom of the potential energy of adsorbed particles. The effect is the same as in the case of homogeneous surfaces. On the w