Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces

Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces by W. Rudzinski, William A. Steele, G. Zgrablich (Editor)

• ISBN: 0444822437 • Publisher: Elsevier Science & Technology Books • Pub. Date: January 1997

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

xiii

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

Table of Contents

Preface Author Index Theoretical Principles of Single- and Mixed-Gas Adsorption Ch. 1

1 Equilibria on Heterogeneous Solid Surfaces Application of Lattice-Gas Models to Describe Mixed-Gas

Ch. 2

105 Adsorption Equilibria on Heterogeneous Solid Surfaces Theories of the Adsorption-Desorption Kinetics on

Ch. 3

153 Homogeneous Surfaces Theory of Adsorption-Desorption Kinetics on Flat

Ch. 4

201 Heterogeneous Surfaces Statistical Rate Theory and the Material Properties

Ch. 5

285 Controlling Adsorption Kinetics on Well Defined Surfaces A New Theoretical Approach to Adsorption-Desorption

Ch. 6

Kinetics on Energetically Heterogeneous Flat Solid Surfaces

335

Based on Statistical Rate Theory of Interfacial Transport Surface Diffusion of Adsorbates on Heterogeneous Ch. 7

373 Substrates

Computer Simulation of Surface Diffusion in Adsorbed Ch. 8

451 Phases Multicomponent Diffusion in Zeolites and Multicomponent

Ch. 9

487 Surface Diffusion Energy and Structure Heterogeneities for the Adsorption in

Ch. 10

519 Zeolites Static and Dynamic Studies of the Energetic Surface

Ch. 11

573 Heterogeneity of Clay Minerals Multilayer Adsorption as a Tool to Investigate the Fractal

Ch. 12

625 Nature of Porous Adsorbents

Ch. 13

Heterogeneous Surface Structures of Adsorbents

679

Characterization of Geometrical and Energetic Ch. 14

Heterogeneities of Active Carbons by Using Sorption

715

Measurements Structure of Porous Adsorbents: Analysis Using Density Ch. 15

745 Functional Theory and Molecular Simulation

Ch. 16

Dynamics of Adsorption in Heterogeneous Solids

777

Ch. 17

Sorption Rate Processes in Carbon Molecular Sieves

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 9 9 9 9

the gas is perfect; adsorbed molecules are classical objects localized on their adsorption sites; the surface is characterized by Afs identical sites; adsorbed molecules do not interact laterally; and 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 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 _ 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)

where O'(p, e) is the local disotherm O'(p,r O0(p,e)/Op. 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 If O[(p) has not this form (for instance, if O~(p) = r with j > l or 0't E LI\L2), eq. (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)

@


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 _~ 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 0,

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 -I

-Xi(eei)= [l+ exp ( eci-

(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 ) =

E Kjpj exp kT ]

exp (~--~)

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

eo, ~ Kip~exp (~,kT ]

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 in II* and the pressure Pm~ at which attains its maximum value and below 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 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

- f

0

p

PM

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 (f,i'~) k,Z]

NtiVis = F

(224)

'

where k~ is the coalescing factor for adsorbate i relative to a standard adsorbate. Rudzifiski a/.[269] have shown that their theoretical development of the PT approach 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 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 In 71 = In 0"1 -- /~1 /~I 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 a x 1030 erg cm2/molecule 2 123.9 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

9fAS -- 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, In XifliO 2 k(X.B" 3 \.~,,i~.~3 (y2Riii ' 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, one9may 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 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

fl --i 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)= H f ( X m )

,

/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 2fin , 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 , values from Qi2"(QT) to [')fin ~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 ~ > 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:

( ( a ~ p e f ) k ) ~/k 0 = 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/

_( 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~ ~'/ 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 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

]

ij (kn [ 1)Oip(n) --flZq(1) E E dqp(kn [ 1) eqp n=k-1 p=l j=l

(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 >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 _ >-

i0 Isz,,, o rY

101~.ug

T

z,2

o

~

t

1012

3.6

>

101]~ c/1

~u J3 . t .

~.0

.

o 0o

1011

u..J

,~ .'"

38

>.L.~

c~

!

"

0:3.2 t~ z z I--n

36

9

30

CX:

C~

m2.8 "'

0

I 1 1 02 0t. 06 COVERAGE 8 (ML)

3z'~ 7~

I 08

10 =-

0

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

f

E

f

..J

o

,....

mO.O

I

0.2

II

i

0.4

0.6

i

0.8

Coverage (ML) 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

0

~

o

-.I

-.1 -.2

-.2

-j

-.3

I

!

-.4

j \.

-.4

i

-.s

-4

-.6

!

~

-.5

~, o .-~ 0

-.6

,

-.7

~

-.~

-~ 0 (5

-,7 -.8 -.9 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

. . . .

"

"

"

'

'

. . . .

'

a

9

b o

3

0 0 •

x

3

c

2 o

O 1

I

.

0

.15

.20

Reduced

.25

.30

.35

Temperature

~

9 9

.15

.20

T/V 2

.

.

.

25

.30

Reduced Temperature

.35

T/V 2

.

,

.

.

,

,

,

.

.

.

.

,

.

.

.

.

~

.

.

.

.

c

oo 3

\ \

~2

I

0

.15

.

.20 25 JO 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 ) 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) 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

i

-l

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 [ 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 present9 The 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)]

(2.1,6)

kT 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 (r + No -1 )2 ( N a ( O ) _ r)(Nb(O) _

+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 of Ag at a different temperature (11.4~ Statistical Rate Theory has been used to 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

( h(,oi "~

a': expr 9]176 ~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 ~

(3.1, 15)

exPt,,--k-~-J-

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

:> 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)

on __ Oe (Psamp) 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.

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

~###3

.

S

m

0.60 --

/,"

.=

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, 0~0~

6 m - - El

s

ot =

exp{-

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


.,..,

60.00 l I l 1 l

40.00

l l I I f

1

I

20.00

I

I

i

~

i

I

I I~~

i

I

t

t

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.

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

375

,(a)

!

. (b) ,

, ,

o

~a,~

,

X

..

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

=

(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

Ed=-[

[a~nr(o)] O~ o-

[~J

r alnOO~

0~] -10H~

- E-

La-~ejr

alno -

~]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

V(x,yo)

Y

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:

(2.50)

((--.))----/---/(...)~n(Sl,...,En)dSl...dEn 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]

P.~,~ = PAo,~ / 0

(2.59)

i

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

" N 2 9 30~

"

v

-'

h_ O

25

-~Oo~ 0

O

0 o

Ov

v

o

%,

0

OV

~

v

0

o

0

0v

0

0

0 0

0

0

m

E

E lS 111-

A

~ ~ % 9

•

_

A

~oO~o @

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 _N89 and 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-

IIIII

I

I

I I IIII

I

I

I

I III

Key Data Key Data 9 6 [] 1 7 2 o 3 9 8

I 0

~D~ i 0 -2

9

4

v

9

5

e

10

CO

r~

~

. m

'~

10 -3

.,m

Key Data Key Data A 11 9

,'

13

v

16

[]

17

14 i l II

10

I

-3

I

I

i i IIII

I

10

-2

i

I

I I III

10

F[-] 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]

[52]

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

397

e

'i I

B

i Ca)

(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

(a)

0.1

I

I

I

0 0.2 0.4 0.6 0.8

I

(b) 0.1

I

i

i

I

I

0 0.2 0.4 0.6 0.8

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

}

5[" o

~'4 o'=1

~s 3 2

a=0.5

.

1 i

I

9

I

,

I

0 0.2 0.4 0.6

I

I

I

0.8

1

e 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

tl

t

0

it

t!

0.2

. L

0,~

I-

I~"~'=--"P ~,

"

O.B

0.8

"

[

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 278K 9 248K 9 225K

-

~8

!: : 0

.

-

206K

~

9 194K

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

9 283K 9 298K = 323K 9 348K *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.

9 413K ~8

-

i7

_

E ~6

9 358K 9 323K 9 298K 9 283K

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) ] AE s

l/hE

=

$ 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 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

kw

E~

--,,,~',,. ~ ~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

---lal--o

9

9

9

,

,

,

,

9

,

9

.

.

.

.

9

.

,

,

,

,

,

,

,

,

,

9

,,

,

o

,

.

,~

,

,,

o

~

9

.

.

o

,

9

,

,

~

,

o

,

.

,

,

AN

~

9

9

9

9

9

~

,

,

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

O

1

I

I

I

I

0.2

0.4

0.6

0.8

1

0 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 whole, the coverage dependence of D depends m

m

426 e=o

(a) ioo

arI'--2

O'/T=I

ZsI J T = ~

=100

g 1

/~

o

|

0.1 0

0.5

0.5

1

0,5

a=.I/3

c~/T---I

a/T=2

/ 1

e

1

o'/T--4

m

~o

0.1

,

0

0.5

1

1~176

I

!

I

o.[r=l

,~

0.5 w=lO. crfI'=2

1

0.5 8

J9 a/T--4

I

i

zE1/Z'=2

1

0,5

C3 1

0.1

0

0.5

1

0.5

1

8

Figure 35: Diffusion coefficient as a function of coverage for the Gaussian distribution of sites.

427

10 0 o~=I/3

Cb

/

0

0.2

0.4

0.6

0.8

1

0 Figure 36: Diffusion coefficient as a function of coverage for the log-normal energy distribution of sites with EmIT = 4 and r/T = 0.7. on the relative weight of heterogeneity and lateral interactions. If heterogeneity is rather considerable and lateral interactions are repulsive, the coverage dependence of D is very strong because both factors lead to an increase in diffusivity with increasing coverage (Fig. 34(a) for z~W = 3 or Fig. 35(a) f o r / 3 a = 4 and z/3W = 2). If heterogeneity is rather strong but lateral interactions are attractive, the coverage dependence of D can be weaker because the two effects compensate each other (Fig. 34 for z13W = - 3 and Figs. 35(a) and 35(5) for 3~ = 4 and zl3W = - 2 ) . Finally, if heterogeneity is weak and lateral interactions are attractive, D decreases with increasing coverage (Fig. 35 for/3a = 1 and z~W

5.1.2.

=

-2). C o m b i n e d effect of c~ a n d r0

We now study the most general case within the GGM, where the combined effects of the structure parameter ~ and the correlation length r0 are present. From Eqs. (2.61), (2.62), and (5.1) to (5.7), the chemical diffusion coefficient can be expressed as

- = u~/]a2 -0/3# D(O) ~ R(0)

(5.19)

where R(O) is given by

,

+

-

}

(1 - 0z)(1 - 02) dEs1 des2 Here, as before, we have taken E s = 0. Calculations representing D(O)/Do(O) where D0(0) is the Value of D(~) for ~ = c~ = r0 = 0, corresponding to T = 400K, 13a = 1

428

10 2

~,~ 0

(a)

(c)

(b)

/ / . .. /-" J / s d

.

7

1 0.5

!

!

0.5

0.5

0

Figure 37: Surface diffusion coefficient for a noninteracting adsorbate, W=0;-,ro=0;---,ro=l;--.,r0=2. (a) a = 0 ; (b) a = l / 4 ; (c) a= 1/2. and W* = 0 are shown in Figs. 37 and 38 for noninteracting and interacting adsorbates, respectively. In order to discuss properly the influence of the correlation length r0 on the diffusion coefficient it is convenient to use some results obtained by Monte Carlo simulation on a square lattice in the framework of the GGM [70]. Fig. 39 shows the behavior of the vacancy factor V, defined as the mean number of nearest-neighbor empty sites per occupied site, as a function of 8. As we can see, at intermediate coverages the vacancy factor decreases strongly for surfaces with a larger correlation length and for attractive lateral interactions. In Fig. 40 snapshots of the adsorbate taken during Monte Carlo simulation are shown for different correlation lengths, coverages and lateral interactions. W'e start by discussing the behavior of D(~) for a traps surface, a = 0, and for a noninteracting adsorbate, Fig. 37(a). As already pointed out in the previous section, for a random traps surface D(~) increases strongly with ~ due to the fact that the activation energy decreases as adsorbed molecules cover less energetic sites (shallower traps). However curves for r0 = 1,2 show a much faster increment for ~ > 0.2 reaching much higher values at ~ ~ 1. The cause of this behavior for correlated traps surfaces must be found in the "texture" of the adsorbate, or, more precisely, in the topology of the regions of empty sites, Fig. 40. In fact, it can be seen that at high coverages, for r0 = 0 migrating molecules must move in a very intricate region (white areas), while for r0 > 0 white areas form larger islands so that the net flux through a given line is clearly enhanced. The contrary effect due to the behavior of the vacancy factor is not so important in this case due to the fact that it is weaker for non-interacting adsorbates and that it vanishes for m

As a increases, Fig. 37(b), (c), the barrier to be jumped over depends not only on the energy of the starting site but also on the energy of the destination site in such a way

429

10

i

i

(a)

(c)

C3 i .-

/

"-.

I

I

I

0.5

0.5

0.5

1if'

\

o

Figure 38: Same as Figure 37 for an interacting adsorbate, 13W = - 0 . 5 .

I

V 0.8

nO 0 17

0,6 0.4

0 0

0

9

0 n

0

9

17

0,2

0 0

0

0 []

DO

0,5

0

Figure 39" Monte Carlo results for the vacancy factor (mean number of nearest-neighbor empty sites per occupiedsite) as a function of mean coverage. Symbols" for ro = 0, o/3W = 0.5, [3flW = 1; for ro = 2, o/3W = 0.5, DflW = 1. The solid line represents the well-known behavior of a noninteracting_ adsorbate on a homogeneous surface, V = 1 - 0 .

430 that barriers are lowered for strongly adsorptive starting sites (low coverage) and raised for weaker starting sites (high coverage) giving rise to an increase (decrease) of D at low (high) coverages in addition to the general behavior described for a = 0.5. The symmetry of curves for a = 0.5, showing a maximum at 8 = 0.5, is a consequence of the symmetrical role played by the two sites participating in a jump in determining the activation energy, Eq. (5.7), and the assumed symmetry of the site energy distribution. A more complex situation arises in the case of an interacting adsorbate. W'e have represented the more interesting case of attractive interactions in Fig. 38. Here a qualitatively different behavior is produced by the correlation length for a given value of the structure parameter. The minimum clearly displayed by the curves in (a) and (b) for r0 > 0 at low 0 is due to the competing effects of the decreasing activation barriers as 0 increases and the strong decrease in the vacancy factor as r0 increases at low 0. At intermediate and high coverages D is higher for higher ro due to the effect of the adsorbate texture, which is more pronounced for an attractive adsorbate. At high coverage lateral interactions take over by strongly decreasing the diffusion coefficient, producing the maximum observed near the monolayer in (a) and (b) for r0 > 0. In (b), for r0 = 0, and in (c), for all values of r0, the influence of attractive lateral interactions competes with the decreasing of activation barriers even at low 0 producing a monotonous decrease of D with 0. Finally, it is interesting to study the behavior of the factor R(0), Eq. (5.20), which is closely related to the effective jump frequency F(8). Analytical calculations and comparison with Monte Carlo simulations were performed in the framework of the GGM for noninteracting particles, [73], and are shown in Fig. 41. The behavior of R(O) can be analyzed in a similar way as for D(0) by taking into account that two main factors contribute to R, i.e. the vacancy factor, through Po0,~, which decreases with coverage, and the activation energy factor, exp(-flE~J), which increases with coverage. We have seen through the GGM how surface diffusion is strongly affected by the adsorptive energy correlations (a way of controlling the energetic topography) through induced correlations on the activation energies for diffusion as well as through the adsorbate texture, or morphology of adsorbate clusters. 5.2.

Site-Bond Model

The SBM was first introduced in the context of describing the structure of porous media [74-76] and later developed for the description of adsorptive energy topography of heterogeneous surfaces [77-79]. The basic idea is to describe statistically both site and bond energies and give a correlation function between a site and its connected bonds through which different energy topographies can be generated. In what follows we change our sign convention for the adsorptive energy surface in such a way that site and bond energies Es and EB, are positive numbers measuring the depth of the gas-solid potential energy at those points, see Fig. 42. Statistically, site and bond energies are described by the probability density functions Fs(Es) and F s ( E s ) and the distributions functions

S(Es) = 0E s Fs(E) dE

B(Es) = /0E s Fs(E) dE

(5.20)

o

r~

o

o

r~

o

o

oo o

~,.~o

k

0=-0.65, W = 0 . 5 K c a l / m o l - ~,,,-I,

J

e=o.65, w = o

iJ

,.,~-*.

I ~ . ~ # l l _ ' ( ~

, ...~ :.~:

~":-~'.:.~.r-,::~'~; i ~ ' ~~, ' ~ -..-,,;:..:-.~ ~2 !..........

0=036, W = O

m

u ~

,-,

8O.3, W = O

m

.,-"

i

"

;"

='~;:'

-

~

''C,

"),.~,

-_ ~ . . -

,.,~'. ,~.~,

;~,'a

..

~,.'~.

~

~.

"'"

~"~'ld~ ~

' " " "~ "

...'

~-

::' " , " . i

~,.~

.'.~

:

"~

~',

...

,~ ~ . ,

,,~

~- :~

..:~, :, .~..,,,~:, .', ~, ;p,,/, .~.
0, but it is not possible to generate a correlated barriers one, so that the two models are in some sense complementary to each other in order to describe a bigger variety of energy topographies. The behavior of the tracer diffusion D* on these surfaces can be easily simulated through the Monte Carlo method. Figs. 45 and 46 show the results for uniform distributions with different correlation lengths. The first result we observe is that D* for traps surfaces is not affected by the correlation length r0, behaving exactly as the analytical solution found for the random traps surface (broken line), the same is true for the jump correlation factor f. This can be understood on the basis of the fact that for any trap the jumping frequency and jump direction are independent of the depth of surrounding traps (this will not be true when other adsorbed particles are present). In the case of barriers, on the other hand, both D* and f are influenced by the correlation length. It is

436

/

",.-...,~,.~.~,

. t . . r~

t

.......... .................. ~

................................'

(b)

Figure 44: Adsorption energy surfaces obtained through the Site-Bond model for different overlapping degrees: (a), f / = 0.3; (b), f / = 0.5.

437

(c)

(o) Figure 44: Adsorption energy surfaces obtained through the Site-Bond model for different overlapping degrees: (c), ft = 0.7; (d), f / = 0.95.

438 TRAP MODEL ,,

,

,

,

,

-4

-8 13 " -12 ,.J

v

ro=O ro=l to=2 Theory

-16

Homogeneous

-20

i

I

0

i

I

2

!

i

6

4

I

i

lO

8

12

Eact/RT

TRAP M O D E L 1.5

,

,

,

1.0

0.5

+ • [] 0 , 0

ro=O ro=l ro=2 i

0

I

2

i

I

i

4

I

6

i

I

8

i

10

Eact/RT

Figure 45: Tracer diffusion coefficient D* and jump correlation factor f for correlated traps surfaces.

439 BARRIER

MODEL

,

[]

-2

•

[]

1:3

v

r" ..J

~0

-6

~2 Homogeneous

-8

I

i

I

1

i

I

i

I

3

2

4 Eact/RT

BARRIER

1.0

MODEL

i

i

+o x + []

X +

0.5

[] XEI -+

0.0

+

to=0

x

ro=l

[]

ro=2

,

0

I

2

E] X +

[]

X +

,

,

4

'

[]

[] []

'

6

"

17

'

P

8

10

12

Eact/RT

Figure 46: Tracer diffusion coefficient D* and jump correlation factor f for correlated barriers surfaces.

440 interesting to notice that if r0 is sufficiently large, D* can change from the characteristic barrier feature (D* is larger than that corresponding to an homogeneous surface) to a trap feature (D* smaller than that of an homogeneous surface). The explanation of this is that for large r0 barriers will group together according to their heights, forming patches in such a way that a group of low barriers may be surrounded by bigher barriers, and this region of the surface will actually behave like a super-site showing the trap feature. This particularity explains both the decrease of D* and the increase of f as r0 increases, since the flip-flop effect observed for a barriers surface, which is responsible for the strong fall in f as temperature decreases, is clearly reduced as r0 increases. Finally, it is also clear that both D* and f need to be analyzed in order to disclose the kind of energy topography characterizing the heterogeneous surface. 5.2.2.

Tracer diffusion on a general site-bond surface

When neither Fs nor FB are g-functions and their overlap fl is not null, we have a general site-bond surface characterized by a correlation length r0 corresponding to the given value of ft. Figure 47 shows results obtained by Monte Carlo simulation for the case of uniform distributions for Fs and FB. The behavior of D* and f can be analyzed by taking into account that both the trap feature and the barrier feature are now acting simultaneously. However, it is by now clear that the trap feature will be predominant in the behavior of D* while the barrier feature will determine that of f. Thus, there is a hope that in the future it will be possible to determine how much of each feature is present for a given heterogeneous surface by analyzing the behavior of D* and f obtained by independent experiments. This hope can be made possible by the rapid development of surface analysis techniques at the atomic scale. We end by discussing the hypothesis advanced by Limoge and Bocquet [80] on the basis of Monte Carlo simulations, according to which for a general heterogeneous surface there could be a compensation between the statistical distributions of sites and bonds in such a way that D* could behave just like for an homogeneous surface. We see, from Fig. 47(a) that such a compensation can exist only for noncorrelated surfaces, r0 = 0, and for low values of E~. This discrepancy can be due to the fact that Limoge and Bocquet model of overlapping sites and bonds distributions does not take into account the construction principle. 5.2.3.

Chemical diffusion coefficient for a general s i t e - b o n d surface

The chemical diffusion coefficient behavior for a general site-bond surface is similar to that predicted in the framework of the GGM. However, it is worth to discuss it in the framework of the SBM, at least for the simple case of noninteracting particles, because it will help to understand how the model works [78]. The general equations for the adsorption isotherm, Eqs. (5.2), (5.4) and (5.5), are still valid now but r be obtained in terms of Fs(Es), FB(EB)and r EB). By using the joint distribution of Es and EB, Eq. (5.23), it is easy to obtain

r

E& ) = Fs(Esl )Fs(E& )r

, E& ),

(5.30)

441

SITE-BOND

[]

~..

A

[]

+

A

.

X x []

X-~

A

-5

O

X

b~

[]

X

§ §

[] X

a v

MODEL

-t-

r

+

--I

x

ro--0

-10 x

ro= 1

[]

ro-2

ro=10 99 ,

-1 5

Homogeneous , , , 2 4

0

,

, 6

,

h

I

i

,.,

lO

12

Eact/RT

SITE-BOND 1.0

,

,

OX

,

,

,

,

,

,

MODEL ,

,

,

,

,

,

~-

[] •

+

_ Ox

,

,

§

r =0

•

[]

(=1 rU=2

~

"Or=10

O

+

0.5

,

O

_A A

[]

A

[]

x

+ X

[]

+

A

x []

-t-

X

A []

+

A

X []

A

A ,

0.0 0

[]

~

2

,

-I-

,

4

9

Bx

6

,

~x

8

,

h,

10

,

, +,

12

,

14

,

~

16

t,

18

Eact/RT

Figure 47: Tracer diffusion coefficient D* and jump correlation factor f for general correlated surfaces.

442 where

~(Es~, Es~)

=/min(Es~ ,Es2) r .,o

(5.3~)

des

With this, and remembering that now r0 = ~ / ( 1 - ~ ) , the isotherm equation and then the function # = #(8) and its derivative can be calculated. Now, as before, D(8)

~ z~,Ofl~ = a ~,e -o-~R(O)

(5.32)

where now R(0) = / / / [ 1 -

81(Esl)] [ 1 - 82(E&)] eZEBF3(Esl,EB, E&)des1 dEsdE&

(5.33)

and

F3(Esl, EB, E&) = Fs(Esl)Fs(E&)FB(EB)r

, Es)r

EB)

(5.34)

Fig 48 shows the behavior of D(8), normalized to D(0), for noninteracting particles on site-bond surfaces characterized by uniform distributions Fs and FB with different overlapping degrees. We see that a maximum accurs for 0 < 8 < 1 and that this maximum shifts to lower 8 values as the overlapping degree increases. This behavior can be understood by analyzing the effect of the correlation function r which is schematically represented in Fig. 49 for fl = 0.5. For any given Es, each of the bonds connected to it is distributed according to the function FBr EB). This function is represented in Fig. 49 for three particular values of Es, namely" 1) Es = s; 2) Es = s + A/4; 3) Es = b. As coverage increases, sites with lower binding energies are being occupied by adsorbed molecules. For a fixed fl, the mean energy of bonds connected to sites of a given energy is closer to the site energy when it is high (low coverage) and farther when it is low (high coverage), as long as Es > b + A. Therefore, for fl = 0, while the mean site energy is always decreasing with coverage, the mean bond energy is constant, and, as a result, the activation energy for migration continuously decreases with 8 and D('8)/D(O) presents no maximum. For fl > 0, on the contrary, the mean bond energy is constant as long as Es > b + A and then starts to decrease, compensating the decrease in Es. This produces a minimum in the activation energy for migration and then a maximum in D(-8)/D(O) which shifts to lower values of 8 as fl increases. Modifications in the behavior of D(8) due to lateral interactions follow the same pattern as discussed for the GGM. 6.

CONCLUSIONS AND OPEN PROBLEMS

Surface diffusion presents many still not well understood aspects and open problems, even for homogeneous surfaces, which have been widely reviewed [4,9,81]. Here we concentrate on those aspects which are strictly related to the surface energetic heterogeneity.

443

C~

....... I = 0 . 1 .

.

.

.

I=0.5 I= 0.7 I-->l

.

~2

"

(3

.

-

s

s

"v

i

0.5

0

1

0

Figure 48: Normalized surface diffusion coefficient for T = 300K, A = 2Kcal/mol and W = 0.

1

2

/

3

__/____ .

.

.

.

! I ! ! ! 0 ! ! ! !

____.I_ Figure 49: Schematic representation of the effect of the correlation function ~b for uniform energy distribution with ~t = 0 . 5 .

444 The general diffusion equations, developed in Section 2, are quite well established, though the validity of the Reed- Ehrlich equation for heterogeneous surfaces is not clear and should be further investigated through Monte Carlo simulation. This is an important aspect since the great majority of diffusion models are based on the Reed-Ehrlich picture. Surface diffusion in porous media is influenced by so many effects, and experimental determinations are so indirect, that an attempt to use refined models taking into account the energetic topography would be totally worthless. The simple phenomenological models discussed in Section 3 are representative of a great amount of effort dedicated to this problem and many more semiempirical equations have been developed trying to modify the Arrhenius law or by considering mechanistic models for the jumping molecules. In this respect, the percolation model is quite unique and it is based on clear and deep physical arguments leading to a comparatively simple final expression for D(~). The fitting of some experimental data are satisfactory and reasonable values of the parameters are obtained. The model seems to be promising, but a much wider comparison with experimental data would be necessary to make it reliable. It would also be useful to perform Monte Carlo simulations folowing closely the hypothesis of the model in order to determine its selfconsistency, and in particular to check the approximations made in obtaining the critical conductivity. The study of simple ideal topographies in Section 4 is instructive, the principal goal being that of understanding the two principal features present in any further analysis involving energetic topography, the "trap feature" and the "barrier feature". Traps and barriers surfaces are sufficiently simple limit topographies to encourage theoretical efforts in the field of statistical mechanics to obtain exact results. Two models, the GGM and the SBM, presented in Section 5, are first approximations toward a general statistical description of surface heterogeneity taking into account the energetic topography. It is not easy to say which of these models is more convenient. In the GGM three important parameters must be determined: or, a and r0. The SBM, on the other hand, depends on Fs, FB and the overlap f~. If uniform distributions are assumed for Fs and FB, then the model depends on two parameters: A and ft. On the other hand we have already pointed out that some kinds of topographies cannot be generated by one of the models and the other is necessary for it. In this sense both models are complementary. Principal questions to be answered in this field are: a) What is the relation between a given physical surface and the adsorptive energy surface? b) Which kinds of defects, impurities, etc., produce a given energetic topography? c) How to determine topography parameters (characterization) from the analysis of gas-solid experimental data? Monte Carlo or Molecular Dynamics, simulations will be of essential help to answer a) and b). With respect to c), tracer surface diffusion is a potentially useful tool for the characterization of the adsorptive energy topography since D* is quite sensitive to it and does not present the inconvenience of D(0). Here, heterogeneity and lateral interaction effects compete. However, it is still necessary that experimental techniques for surface

445 analysis at the atomic scale be perfected in such a way that both D* and the jump correlation factor f be measured. In fact we have learned that, for a general site-bond heterogeneous surface, the trap feature dominates in the behavior of D* versus ~E~, while the barrier feature dominates in the behavior of f versus 1/T. Finally, a still open problem of great practical importance is the development of a theory for surface diffusion of mixtures of gases on heterogeneous surfaces for both non porous and porous adsorbents.

ACKNOWLEDGEMENTS The present monograph is the result of many years of continuous efforts of my students and collaborators at the Surface Physical-Chemistry Group of the Universidad Nacional de San Luis, San Luis, Argentina. I am indebted to all of them for their enthusiasm, creativity, dedication and friendship. Special thanks are due to W. A. Steele, V. P. Zhdanov, J. L. Riccardo, R. J. Faccio and F. Bulnes for critically reading the manuscript.

REFERENCES

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448 59. R. Zallen, "The Physics of Amorphous Solids", Wiley, New York (1983). 60. D. Stauffer, "Introduction to Percolation Theory", Taylor and Francis. London (1985). 61. V. P. Zhdanov, Surface Sci., 149 (1985) L13. 62. J. Riccardo, V. Pereyra, J. L. Rezzano, D. A. Rodrigez Sas and G. Zgrablich, Surface. Sci., 204 (1988) 289. 63. J. Marchese, "Influencia de la heterogeneidad del s61ido sobre la difusi6n superficial de gases adsorbidos", Ph. D. Thesis, Universidad Nacional de San Luis, San Luis, Argentina (1979). 64. J. P. Hobson, in "The Solid-Gas Interface", Ed. E. Alison Flood, Vol 1, Dekker, New York (1967). 65. J. W. Haus, K. W. Kehr and J. W. Lyklema, Phys. Rev., B25 (1982) 2905. 66. Y. Sinai, Theor. Prob. Appl., 27 (1982) 256. 67. K. Binder and D. W. Hermann, "Monte Carlo Simulation in Statistical Physics", Springer, Berlin (1988). 68. K. Sapag, "Factor de correlaci6n de salto para la difusidn traza en medios amorfos", Tesis de Licenciatura, Universidad Nacional de San Luis, San Luis, Argentina (1991). 69. P. Ripa and G. Zgrablich, J. Phys. Chem., 79 (1975) 2118. 70. J. L. Riccardo, M. Chade, V. Pereyra and G. Zgrablich, Langmuir, 8 (1992) 1518. 71. V. Mayagoitia, F. Rojas, V. Pereyra and G. Zgrablich, Surface. Sci., 221 (1989) 394. 72. V. Mayagoitia, F. Rojas, J. L. Riccardo, V. D. Pereyra and G. Zgrablich, Phys. Rev., B41 (1990) 7150. 73. F. Bulnes, J. L. Riccardo, G. Zgrablich and V. Pereyra, Surface Sci., 260 (1992) 304. 74. V. Mayagoitia and I. Kornhauser, in "Principles and Applications of Pore Structural Characterization", Ed. J. M. Hynes and P. Rossi-Doria, pp. 15-26, Arrowsmith, Bristol (1985). 75. V. Mayagoitia, F. Rojas and I. Kornhauser, J. Chem. Soc. Faraday Trans., I84 (1988) 785. 76. V. Mayagoitia, B. Gilot, F. Rojas and I. Kornhauser, J. Chem. Soc. Faraday Trans., I84 (1988) 801. 77. V. Mayagoitia, F. Rojas, V. Pereyra and G. Zgrablich, Surface Sci., 221 (1989) 394.

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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.

451

C o m p u t e r S i m u l a t i o n of Surface Diffusion in A d s o r b e d P h a s e s William Steele Department of Chemistry, The Pennsylvania State University, 152 Davey Laboratory, University Park, PA 16802, U.S.A.

1

Introduction

Computer simulation is a now well-known technique for the evaluation of the thermodynamic properties of gases adsorbed on solid surfaces or in pores. The basic idea is to use the power of the computer to bypass the simplifying assumptions made in the past to pass from a molecular model of the adsorption system to the evaluation of its observable, macroscopic properties. In essence, one defines the solid adsorbent in terms of its interaction energies with gas molecules that come in contact with its surface. This model is completed by defining the interactions of the adsorbate molecules with each other. The simulation which can then be performed relies on one of two general algorithms: Monte Carlo or molecular dynamics [1, 2]. For both, tricks are needed to make such systems mimic the macroscopic adsorption of a gas on (or in) a solid: the potential energies of interaction must be modeled accurately, but not so accurately as to prevent the computer from evaluating them in a reasonable time; and periodic boundary and minimum image conditions must be applied to allow one to associate the properties of a few hundred molecules with those of a macroscopic fluid. These well-known ideas have been utilized in a very large number of problems explored using this technique. Features that are perhaps unique to adsorption are that the solid adsorbent is taken to be rigid so that one has a mechanical system of adsorbate particles with translational and rotational degrees of freedom moving in the external force field of the solid. Vibrational degrees of freedom for the adsorbed molecules are usually neglected, at least when the molecule is small- obviously, intramolecular motions are an important aspect of the simulations of long-chain adsorbed molecules. The molecular dynamics algorithm produces molecular trajectories for interacting molecules over periods of time ranging up to a nanosecond. This is accomplished by numerically integrating the classical equations of motion for each molecule of adsorbate to determine the time-dependent trajectories followed. Time-averages of quantities such as the average potential energy of the entire sample of adsorbate molecules can then, by the postulates of statistical mechanics, be equated to ensemble averages. Of more importance to the present discussion is the fact that the information about the time-dependent positions of the molecules in the system allow one to evaluate dynamical properties such as mean square displacements. It is well-known that these displacements are related to self-diffusion constants (or tracer diffusion constants, as they are sometimes called) in a straightforward but exact manner. An alternative calculation of the diffusion constant relies on the fact that molecular velocities are also known at each instant in time. This allows one to evaluate velocity autocorr~elation functions and, by integration of this function over time, to evaluate the self-diffusion coefficient in a calculation which is completely equivalent to the mettiod based on the long time limit of the slope of a plot of mean square displacement versus time. The formalism and the explicit equations for the self-diffusion constant

452

are written out in the chapter in this book by Zgrablich. In fact, other dynamical properties of a fluid such as the shear viscosity and/or the thermal conductivity can also be evaluated using suitable analogues to the mean square displacements or the velocity autocorrelation functions [2]. An important point is that all these quantities can be evaluated for fluids in geometries where the confining dimensions are of the order of nanometers. Furthermore, for inhomogeneous fluids, the dependence of the transport coefficients upon position within the fluid and upon the direction of the flux (of mass, momentum or energy, for diffusion, shear viscosity and thermal conduction, respectively) relative to the confining geometry can also be evaluated. It should also be emphasized here that these very detailed results can be generated by either molecular dynamics or by a suitably modified form of the Monte Carlo algorithm. Here, we will discuss some of the results that have been obtained over the past decade or so for diffusional motion in fluids on surfaces. In the application of the Monte Carlo algorithm to transport, dynamics are not followed per se, but the passage of the system through phase space is evaluated by making a lengthy series of small random displacements of the molecules. These trim displacements are accepted or rejected by a probability rule (hence, Monte Carlo) that has been shown rigorously to generate the statistical ensemble of interest. Many different types of ensemble can be produced (Grand Canonical, isobaric-isothermal, canonical, etc.) by appropriate modification of the probability rules that govern the random process. The tricks of periodic boundary conditions, rigid solid, etc. mentioned above for molecular dynamics are also utilized in Monte Carlo simulations. Since there is no time variable in this algorithm, one might think that evaluation of transport properties via Monte Carlo simulation is not possible. However, many workers have argued that this is unnecessarily restrictive. The "kinetic Monte Carlo" algorithm is based on a master equation with phenomenological rate constants [3]. In this way, molecules trace out trajectories in space and time that allow one to evaluate mean square displacements (in particular) and thus, a kind of diffusion coefficient. The problem is that the units of time remain undefined; nevertheless, many useful results have been obtained in this way [4]. In fact the kinetic Monte Carlo algorithm is indispensable for adsorbed phases that have been modeled as lattice gases. In this case, motion is due to instantaneous jumps between lattice sites and molecular dynamics is not applicable. This is a particularly appropriate model for the diffusive motion of strongly adsorbed (i.e., chemisorbed) species, especially on clean metallic surfaces - see for example, [5]. However, the lattice gas model is widely used for systems other than chemisorption - specifically, nearly all current models of heterogeneous surfaces involve the concept of adsorption on distributions of sites with varying adsorption energy. In this case, jump models plus kinetic Monte Carlo are employed to evaluate surface diffusion - the only added assumption needed is the specification of the energy barriers to jumps between neighboring sites. Simulation studies of such model systems are rather wide-spread [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Over the past decade or two, molecular dynamics simulations of transport in adsorbed fluids have been reported for a large number of model systems. We give here a brief and not exhaustive review of this work and subsequently discuss in detail simulations of self- (or tracer-) diffusion in thin films (one or two atomic layers at most) of gases adsorbed on model surfaces. Many of the molecular dynamics simulations of transport in confined fluids have been concerned with gases sorbed in pores. Initially, model pores with a simple geometry were studied. Two of the most popular are slit pores with planar parallel walls and straight-walled cylinders.

453

For such models, one must then specify the roughness of the pore walls. Perfectly smooth walls can be studied, but in more realistic systems, one attempts to include the atomic structure of the wall in a way that is simple enough to allow one to evaluate fluid-wall interactions reasonably efficiently. Within the category of transport of pore fluids, one should distinguish two rather different cases: 9 Pores that are essentially filled with adsorbed fluid [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] 9 Pores that have submonolayer or at most, one or two strongly held layers of fluid on the pore walls (see below). In the first case, one is interested in the behavior of either diffusive or shear flow in the small-size limit where the nature of the boundary conditions has a significant effect on the hydrodynamics of flow, i.e., on the solutions to Fick's equation for diffusion or to the Navier-Stokes equation for viscous flow down a microscopic tube (Poiseuille flow) or between two planar walls (Couette flow). In the second case, the molecular motion in a thin layer on the wall tends to be rather similar to that for a fluid adsorbed on a free surface - that is, the presence of another wall at some distance from a very thin adsorbed film has a minimal influence on the dynamics of the molecules in the film except in the case of extreme constriction where the second wall is at a separation distance of a couple of atomic diameters. Well known examples of transport through constrictions are fluids in the pore systems of zeolites, [33, 34, 35, 36, 37, 38, 39] where the sizes of the openings to the cages inside the solid are of the order of a molecular diameter, thus producing the molecular sieve effect. As an example of the simulation of molecular motions of rare gas atoms, Figure 1 shows the trajectories followed by xenon atoms at 300 K in the pore system of a zeolite known as zeolite rho [33]. This crystal contains two interlaced non-connecting cubic lattices of cages with diameters _~ 12 ~t. The cages are connected by short tubes with diameters of "~ 3.6 )1. When xenon is brought into contact with this system, the atoms first are adsorbed quite strongly in the connecting tubes and then begin to fill the cages. Figure 1 shows three cages as the volumes within the hexagonal rings that actually define one end of the connecting tubes to the six neighbors of each cage. The small dots show the limited trajectories followed by atoms trapped in the tubes and vibrating on their sites at this point. All these sites are filled, but the periodic b o u n d a r y conditions cause some of them to be invisible (each atom of this kind is half on one face and half on the face on the opposite side, but is shown here as a single atom). If we focus our attention on the pair of cages vertically aligned in Fig. 1, one of which is a periodic image, we see t h a t the 6 atoms that are in each cage are translating across the inner wall of their cage in a random walk that is produced by collisions with neighboring xenon atoms. During the duration of this simulation, one atom leaves the tube that connects the central cage and its image above. It moves into the upper image cage, where it collides with the image atoms (not shown) and diffuses within the second cage. As it makes the change of cages, an atom from the cage below moves into the vacated tube. In this way, one sees that inter-cage diffusion controls the overall diffusive motion of the xenon sorbed in this pore system. Since only one such event has occurred in 37 pic0seconds, it is evident that a very long simulation will be needed to observe a sufficient number of inter-cage jumps to give a statistically satisfactory estimate of the jump rate in this system. A problem that frequently arises in this and other zeolite studies is that the rigid lattice model assumed in this work may give results for transport that are significantly

454

different from those expected if the atoms making up the solid lattice are allowed to undergo vibrational displacements. This problem is magnified in importance in porous systems that are "tight" - i.e., where the sorbate and the pore dimensions are sufficiently close to be at the edge of the molecular sieving that occurs when the gas atoms are too large to pass through the cage

F i g u r e 1 Computer generated trajectories for 16 xenon atoms in cages taken from the two independent networks of connected cages in zeolite rho. Also shown is a periodic image of one of the cages. Six of the atoms are tightly held in the openings to the cages (not all are shown) and the remaining 10 are evenly divided between the two cages. During the time of this simulation, one atom jumps from its position in the window into a neighboring image cage and is replaced in the window by a different atom from the cage. openings. If we now focus on the molecular dynamics simulations of self-diffusion of molecules confined to few-layer films on flat surfaces, one can again divide the problem into homogeneous and heterogeneous model adsorbents. In the homogeneous case, we begin with the fact that a very large effort has been devoted to experimental, theoretical and simulation studies of gases adsorbed on graphite. This work has included a number of simulation studies of diffusion on this surface together with a few experimental reports [40]. As is well-known, the surface of this solid can readily prepared so that it is nearly entirely made up of large-area perfect basal planes. Furthermore, the basal plane is believed to be remarkably smooth compared to ionic solids such as MgO or oxides such as Ti02. Thus, one expects very small barriers to translation across such a surface and diffusion that is very different from the standard descriptions of activated jumps over energy barriers with height equal to an activation energy Ea. It would appear that the best description of diffusive motion in monolayers on such surfaces might be based on

455 a two-dimensional gas approach, slightly modified to take account of the effects of the small variations in gas-solid energy as a molecule translates across the surface [41]. In fact, even for the rougher surfaces presented by adsorbents with atomic structure such as those presented by model amorphous or crystalline oxides, the barriers to translation of a molecule across the surface tend to be small enough that a description of diffusive motion in such systems in terms of jump motion should be carried out with caution. For the highly irregular gas-solid potential functions presented by heterogeneous surfaces, one has a wide range of "activation energy" in a complicated surface geometry that leads to trajectories that exhibit frequent changes in direction and which do not necessarily pass near the bottoms of the saddle points in the energy surfaces [42]. Many of these points will be illustrated by the simulation results that will be presented here. Another group of dynamical simulations involves the diffusive motion of isolated molecules over an adsorbing surface [43, 44]. Such systems, which can be studied experimentally by techniques such as field emission microscopy, are of course primarily probes of the gas-solid potential function, which for molecules with rotational degree of freedom and/or internal flexibility (such as short hydrocarbon chains) can be quite complex. It should be noted that molecular trajectories can be used to determine a variety of dynamical quantities, some of which are essentially

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8

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losr ,o41_ ,o31_ I01~) lit

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u_,,

" ' 5 I0 timeslel~/lO00

-

F i g u r e 2 Number of atoms in a dense N2 monolayer on the graphite basal plane at 73.6 K as a function of time. Note that 1000 timesteps = 2.2 picosecond. The coverages, relative to commensurate, are 1.04, 1.10, 1.25, 1.35, 1.50, 1.75, 2.00 and 2.50 for the indexes on the panels increasing from 2 to 9. (Panel 1 corresponds to the commensurate layer - no interlayer transfers were observed for this case during the time of observation.

456

inaccessible to experiment. The rate of transfer of molecules in and out of a monolayer film can be monitored- for example, see Figure 2, which shows the time dependence of the number of molecules in several monolayer and submonolayer films of nitrogen adsorbed on the graphite basal plane at 73.6 K [45]. The fluctuations in the number of molecules in the first layer can be due either to transfer from (or to) the gas phase or from (or to) the second layer. Figure 2 indicates that the jumps occur rather infrequently on a molecular time-scale with a frequency on the order of a few per picosecond, depending upon coverage. The jumps themselves are completed very quickly on this time scale. Information of this kind is of interest to the determination of rates of adsorption and desorption, but there appears to be little use of simulations in this connection to date. Another type of dynamics that can be simulated is the rotational motion of polyatomic molecules in an adsorbed film. In this case, angular velocity and reorientational time correlation functions can be evaluated (and compared with experiment- see [40]). In this case, one must

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F i g u r e 3 Time-correlation functions for N2 on the graphite basal plane at low temperature. Part (a) shows the in-plane tcfs < cos(m$r > for m = 1 - 4 (top curve to bottom) for a commensurate monolayer at 18.8 K; part (b) shows the same functions for 41.4 K; and part (c) shows angular velocity time correlation functions for 41.4 K - the solid line is for out-of-plane tilting motion and the dashed line is for the in-plane reorientation.

457

realize that in-plane and out-of-plane molecular reorientation can be very different from each other due to the relatively large torque which will, for linear molecules, usually have the effect of constraining them to lie mostly parallel to the surface. In the case of in-plane rotation, the torques are due to the variations in gas-solid potential energy as a molecule moves parallel to the surface and to the torques arising from interactions with other adsorbate molecules in the same layer as the rotating species. Simulations of motion in the dense nitrogen monolayer [46, 47, 48, 49] gave the curves in Figure 3 which show in-plane libration at very low temperature (usually, a damped more-or-less harmonic oscillation as in the solid curve of Fig. 3c). At high temperatures, the in-plane molecular reorientation may show an approximation to rotational diffusion. In this case, theory gives exponential time-decays of the angular correlation functions gin(t)=< cos(m~r >, where ~ r the change in the in-plane angle of the molecular axis in time t and m is an integer. In fact, the solution of the two-dimensional rotational diffusion equation yields gin(l) = exp [-m2~t], where ~ is a rotational diffusion constant. In this limit, the angular velocity time-correlation function has been assumed to have a negligibly short life-time compared to the decay time of the orientational correlation function. Note that the in-plane rotational motion can be primarily diffusive even when the out-of-plane reorientation is a damped harmonic torsional oscillation. Part (a) of Figure 3 illustrates the in-plane orientational behavior for a fairly dense (15.7 .~l2 per molecule, or v/3 x v/3 commensurate) monolayer of N2 adsorbed on the graphite basal plane at a temperature below the in-plane orientational ordering transition which occurs for this system at ~_ 27 K and part (b) shows the behavior at a temperature above this transition. The almost constant time correlation functions (tcfs) of part (a) are indicative of small in-plane torsional oscillations, while the tcfs in part (b) correspond to in-plane motion which is more or less diffusional. Figures 4 and 5 show simulated orientational tcfs at 40.3 K for a N2 layer on graphite that is 5% denser than commensurate. In Fig. 4, rotational diffusion

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0.8 9

.

4..*

E -0.4

o C

-0.8

F i g u r e 4 In Cm(t)/m 2 for in-plane motion of N2 at 40.3 K on the graphite basal plane in a layer that is slightly compressed relative to commensurate.

458

theory is tested by plotting the in-plane orientational tcfs for for m = l to m = 4 on a scale that should give a single straight line for all m if the theory is obeyed [50]. Deviations from linearity at short time are due to the persistence of angular velocity correlations. It is probable that the

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:

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-0.12

F i g u r e 5 In 7>~(t)/l(l + 1) for out-of-plane reorientational motion in the system of Fig. 4. l = l corresponds to the lowest curve with the other curves in ascending order for l=2 - 4 and ~t(t) denotes a tcf of t h e / ' t h spherical harmonic function of the out-of-plane angle.

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2 TIME (PICOSEC)

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F i g u r e 6 Reorientational tcfs for monolayer N2 on the basal plane of graphite. Temperatures are shown on the Figure. The nearly constant curve is for out-of-plane (tilting) motion, and the decaying curves are for in-plane reorientation.

459

differences shown in Fig. 4 between the observed and the predicted slopes at long time are due to the fact that reorientation is no longer strictly in-plane at this temperature and surface density, but has become three-dimensional. This idea is supported by the out-of-plane reorientational tcfs that are shown in Figure 5. The decays of these functions, which should be linear and identical for all l if the out-of-plane reorientation is diffusional, indicate torsional oscillation at short times and non-diffusional but significant exponential decays at long time. Figure 6 shows the in-plane orientational tcfs for molecules in the commensurate N2 monolayer on graphite as a function of temperature for the case m = l . Also shown is the out-of-plane reorientational tcf. Clearly, out-of-plane motion is minimal for all the temperatures considered, which range up to the bulk N2 boiling point, even though the in-plane motion is becoming ever more rapid as temperature increases. It should be noted here that .,N2 on graphite exhibits two-dimensional melting at a temperature that depends strongly upon coverage, but is close to 48 K for an isolated commensurate patch on the surface [51]. The conclusions to be drawn from the simulation studies of rotational motion in dense N2 monolayers on the graphite basal plane are that orientational ordering is almost complete at sufficiently low temperature, and that the in-plane reorientation is considerably less hindered than the out-of-plane at higher temperatures. In fact, the out-of-plane reorientational motion is essentially torsional oscillation that is induced by the strong gas-solid forces on a N~ molecule on the graphite surface (or more accurately, strong torques). The torques are much weaker in the in-plane direction because of the flatness of the basal plane. Thus, hindrance in that degree of freedom is due primarily to interactions with neighboring N2 molecules in the adsorbed film. Although simulations of orientational dynamics for other adsorbates on other surfaces are quite limited, it is not unreasonable to assume that this behavior of the N2/graphite system might be characteristic of small non-spherical molecules on relatively fiat surfaces. Furthermore, persistent velocity correlations can have a significant effect upon the relevant tcfs in the translation problem as well the rotational, as we will see in the following.

2

D i f f u s i o n on n e a r l y flat s u r f a c e s

An obvious but important feature of diffusional motion in an adsorbed film is that it is extremely anisotropic, with motion perpendicular to the surface tending to be mostly harmonic oscillation, interrupted by occasional desorption events a n d / o r collisional changes in the oscillatory motion. Thus, the three dimensional molecular motion in real physisorption systems must be treated as two distinctly different types. One expects that the mean square displacements (msds) < ~z2(t) > and < 5p2(t) > = < ~x2(t) + ~y~(t) > as well as the corresponding velocity tcfs, will behave quite differently. Here, z is the coordinate perpendicular to the surface and x, y (and p) are parallel to it. The brackets here denote an average over all (or over a specific subset - see below) of particles and time-origins and 5z2(t) is (z(t) - z(0)) 2, the square of the displacement in the z direction in time t. The tcf and the displacement are related by the equation [52]: < t~z2(t) > - 2

( t - r) < v'._(O)vz(,) > dr

(1)

Analogous equation hold for the x and y msds and their velocity tcfs. If the molecules that give rise to the statistical quantities in eq. 1 are unbounded in their z-displacements, one has the

460

the well known long-time limiting form: ~ m < ~z2(t) > = 2t t----* OO

jo

< ~z(0)~(~)>

d,

(2)

and Dz, the diffusion constant for motion in the z-direction is thus Dz =

< v~(O)vz(,)

(3)

> d~

We define:

Cz(t) = < vz(O)vz(r) >

(4)

At short times, a power series expansion for the velocity tcf gives [52]:

Cz(t) = k T

< F~ > t 2

m

m2

2 + .....

(5)

where m is the molecular mass and Fz is the z-component of the force on a molecule. When this expression is substituted in eq. 2, one finds that the short-time msd is given by: < ~z2(t)

>=

kTt2

< F~ > t 4

~

m

m2

12 +

...

""

(6)

This argument shows that simulated msds must be quadratic functions of time initially, with a curvature that depends only upon the mean square velocity of a particle at temperature T with mass m, and not upon the density or forces acting in the adsorbed film. At the other limit of time, diffusive motion yields msds that are linear in time with slopes that give the diffusion 6

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I

I

1.5

2

2.5

3

3.5

4

time (picosecond) F i g u r e 7 The solid curve shows the mean square displacement for a system with an exponentially decaying velocity tcf with the decay time r = 1 picosecond. The straight dashed line is for strictly diffusional motion with diffusion constant ~ = 1 picosecond-:. The value of (kT/m) has been set equal to (1 A/picosecond) 2 in this case.

461

constants I)z, T~z and I)y. The approach to linearity in the msds is governed by the decay time of the velocity tcf, as can be seen explicitly by supposing for the moment that Cz(t) = (kT/m)exp[-t/Tx], where 7"z is the correlation time for the tcf Cx(t) and is equal to ~-1 where ~z is the diffusion constant. An integration of eq. 1 for this case gives:

kT < &~(t) >= --~-~[t + T~(~xp(-t/T~)- 1)]

(7)

?72

This msd has the correct initial (quadratic) time dependence but becomes linear in t only for times much greater than the decay time of Cx(t). The approach to linearity given by eq. (7) is shown explicitly in Figure 7. Simulated translational tcfs for dense physisorbed films are far from exponential function of time, as will be shown shortly for mono- and bi-layer films of oxygen on the graphite basal plane. Nevertheless, there is a class of systems that might be expected to conform to this simple expression. The physical basis goes back to the kinetic theory of dilute three-dimensional gases. A molecule in such a gas undergoes a series of binary collisions with, on average, a frequency vcou. It is reasonable to assume that a molecular velocity is constant until a collision occurs and that the velocity vector of the molecule after a collision is completely uncorrelated with its previous value. Since the fraction of molecules which have not yet collided in a time interval t is just exp[-~,coU t], the decay of the velocity tcf is given by this expression. If one asks whether there is an analogous situation in adsorbed films, the answer is a qualified "yes" if one considers a dilute film at high temperature on a very flat surface. Although the out-of-plane motion can be essentially vibrational, the in-plane motion could be well approximated by a freely translating two-dimensional gas. In this case, one might begin with the simple two-dimensional version of the collisional expression for the in-plane diffusion coefficient T~ll which is: 1 v,I = ~

< c>

(8)

where < c > is the mean speed = (8kT/rcm) 1/2 and )~ is the mean free path in two-dimensions and approximately equals 1/2v/2ap, with a equal to a hard disk diameter and p equal to the density in mole/cm 2. Simulations of a system that might fit this simple picture have been reported by two groups [53, 54] who have studied methane at room temperature in slit pores with parallel walls made up of the graphite basal plane. However, the point of view taken was rather different in the two studies. In [54], the total flux of molecules down the pore and the associated msds were evaluated after considering carefully the boundary conditions that controlled the collisions of the gas phase molecules with the pore walls. (This was a non-trivial problem because the gas-wall interaction had been assumed to be that for a perfectly fiat wall, thus yielding specular reflection and a physically unlikely flow mechanism in the absence of extra assumptions for the boundary conditions.) In [53], the msds were evaluated separately for the molecules in the monolayer on the wall and for those molecules in the interior of the pore. The molecules that jumped between the two regions and those in the central volume were excluded from the calculation. In fact, the local density within the pore shown in Figure 8 indicates that nearly all the gas in these systems was in fact in the monolayer. In [62], the gas-wall interaction was taken to have the periodic variation with position along the wall that has been used in the most accurate representations of the energy and that had the

462

advantage of spoiling the specular nature of a gas-wall collisions. In addition, two other systems were considered in which the wall had been roughened by adding "sulfur" atoms amounting to 1/13 (in a v/7 x ~/7 lattice) and 1/7 (in a 2x2 lattice) of the wall atoms. The variation in minimum methane-pore wall energy as an atom translates parallel to the wall is small for the non-sulfided surface, amounting to -'~ 400 joule/mole, but is "~ 2100 joule/mole for the dilute sulfided case and _ 2900 joule/mole for the more concentrated example. Apparently, these variations were insufficient to affect the surface diffusion process very much. In spite of the differences in outlook of the studies in [53] and [54], the self-diffusion coefficients obtained were also numerically similar over most of the range of methane loading. Since only the motions of the molecules in the monolayer films were considered in [53], the displacements in this case are "surface diffusion". The variation of the in-plane T~ with surface coverage ranging up to the approximate monolayer of 8 x 10 -1~ mole/cm 2 is shown in Figure 9 and is compared with the curve calculated from the crude two-dimensional coUisional expression given in eq. 8. The agreement is as good as one could expect from such a simple treatment. Note that the rapid variation of T~ with coverage is ascribed to the decrease in the mean free path of the adsorbate 60

9

,

1.65 1 0 - 1 0 m o l e / e r a 2 ...... . . . . . ......

3.30. I 0 - 1 0 m o l c / c m 2 4.95.10- l O m o l c / c m 2 6.6010- lOmole/cm2 . 5.

-

i o

[~ []

.

cr

. 0 and L22 > 0,

Lx2 = L21 = w(Lx,L22) 1/2

(39)

where

1

(40)

Following the derivation procedure discussed above, for w = 0, the diffusivity formula Eqs.37 and 38 are obtained for the binary diffusion case when a binary Langmuir isotherm equation is applied to express the gas phase partial pressures. However, when the absolute value of w is between 0-1, one obtains

Dij = Dioqi 0 Oqj In Pi +w(qiqJ)~(Di~176 ,

Oqj

i,j = 1,2

(41)

where Pi is a function of ql and q2, and can be correlated by using binary equilibrium isotherm data. w is the interaction parameter for diffusing molecules. It can be positive or negative depending on whether attractive or repulsive molecular interactions are operative. The value of w must be pre-determined based on information on single component diffusion, and it should satisfy the following conditions 0~(0j) .... >0

Eij

>0

~

>0

w

)0

Following the constraints above, one possible relationship between w and Eij is: w = (1

-

e -Ei'/'T) OiOj

(42)

where Ei i is the interaction energy between two different diffusing species. It can be calculated from E/i and Ejj where the energies E/i and Ejj can, in principle, be obtained from single component diffusion data. Here the geometric-mean rule or mixing rule is needed to calculate the cross-term energy

Eij= ~/Eii Ejj

(43)

Equation 42 is an empirical correlation that has the correct limits. More recently, Chen, et a1.[34] have taken an entirely different thermodynamic approach by considering the rate processes involved. For the diffusion of a multicomponent system, the phenomenological flux equation for component i is

Ji = - L i V ~ i .

(44)

497 where i* denotes the activated molecule i. The underlying assumption for this equation is that all migration steps must involve the activated species. The interactions between the unlike molecules will be accounted for in the calculation of V#i., so no cross-term is needed in Eq.44. Assuming ideal behavior where the activity coefficient is equal to unity (it needs to be stressed here, however, that this model is not limited to this condition; an activity coefficient may be added to take into account the lateral interactions between molecules), the chemical potential gradient is expressed as V # i = R T V In Oi

(45)

where 0 is the surface coverage. The relationship between the phenomenological coefficient Li and the Fickian diffusion coefficient at zero surface concentration Di0 is already expressed by Eq.34. Therefore, the Hux equation can be recast to Ji = - D i o qiV in 0i.

(46)

where qi = qisOi, and qis is the saturated amount adsorbed. In order to relate 0i and 0i., one needs to understand the rate process. For diffusion in zeolite, the rate processes are[33]" Activation:

Mi k,)

(47)

Mi. + V

Deactivation (to a vacant site)" (48)

mi. + V k'r ~ Mi

Deactivation (to an occupied site)" (49)

Mi. + Mj k,,> Mj. + Mi

In the step indicated by Eq.49, the exchange of the activated molecule i* with the adsorbed molecule j causes the adsorbed molecule to become activated due to the exchange of energy while i becomes adsorbed. The rate of formation for the activated i* is given by OOi. '~ Ot = k~O~ - k~O~.O~ - y ~ k~jO~.Oj

(50)

j----1

where the first term on the RHS in Eq.50 is the rate of activation, the other terms are the rate of deactivation on a vacant site (ki~Oi.O~) and sites already occupied by j; here j could be equal to i. The deactivation terms include both forward and backward movements, Furthermore, the steady-state condition stipulates that 00i. =0 Ot

498 It follows then kiOi

kiv 1 - ~ (1 - Aij) Oj j=l where O~ = 1 - ~in__l Oi, and ki...,i _ sticking probability of molecule i on adsorbed molecule j -ei,,-eij)/RT (53) Aij = kiv sticking probability of molecule i on vacant site = e

In Eq.53. kij and ki,, are, respectively, the rate constants for an activated i" to land on and stick to an adsorbed molecule j or a vacant site. A further underlying assumption for this equation is that the transit time between sites is negligible relative to the residence time at either site (whether vacant or occupied). Substituting Eq.52 into Eq.46, one gets J = -[D]V~

(54)

where J and V~ are two vectors of n components for, respectively, flux and concentration gradient. The diffusivity matrix [D] is given by

Dii = D,o 1 +

(ln-A,,)O, 1 - ~(1

-

(55)

A,j)Oj

j=l

(i - Aij )Si

Dij = Dio

n D

~(1 j:l

j #{

(56)

- )~,j )oj

Procedures for calculating the values of A and application of the above equations will be shown in the next section. Based on Eq.35, which was derived from the phenomenological flux equation, Hu and Do[41,42] have modified it by using different equilibrium isotherm formula. In their models, they used single component Langmuir or Toth isotherms to formulate the single component diffusion equation and applied IAS (ideal adsorption solution) model to compute the multicomponent adsorption equilibrium. Since the IAS model can be used to predict multicomponent equilibrium adsorption data from single Langmuir or Toth models without requiring any additional information, this model can also be used to predict multicomponent diffusion behaviors of the adsorbed species from the information of single component diffusion. More recently, Hu and Do[45] further modified Eq.35 by using the heterogeneous extended Langmuir model proposed by Kapoor et a1.[46]. This model computes the gas mixture equilibrium by using an extended Langmuir isotherm on a patch of surface and then integrates it over a uniform energy distribution, From the information obtained by fitting

499 single component dynamic data, they were able to predict the multicomponent diffusivities.

5. K I N E T I C ) k P P R O A C H 5.1. Model

formulation

Diffusion in zeolite and on surface are activated processes in which the adsorbed molecule must be activated in order to migrate to an adjacent site. Migration is then the result of a hopping process of the activated molecule. Before the derivation for binary diffusion, some assumptions must be made. First, the pore spaces are small but an adsorbed molecule will not cause pore blocking. Second, unlimited multilayer adsorption is not allowed but the adsorbed molecules may grow as a cluster[50]. The second assumption is equivalent to stating the desorption of the adsorbed molecules A or B requires the same activation energy regardless whether they are activated directly from the sorbent surface or from the occupied site. Following the above assumptions, a binary mixture of A and B may undergo the following rate processes for molecule A located at lattice site at (x - ~), where x is the distance coordinate and 5 is the inter-site distance (each hop covers a distance of 5). A potential energy diagram for the diffusion process is given in Figure 1.

A*

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

x'--~2

x

x+~

Figure 1: Potential energy diagram for activated diffusion along distance coordinate x. 1. Activation (A)x_ ~

'

ka

) (A')x_~6

(57)

rl = ka~A,~_~

2. Deactivation (to vacant site) (A*)=_~ + (Y)=_~ r2 --

km)

(A)=_~

km~A',x-~ ,x-~

(58)

500 3. Forward Migration (to vacant site) (a*) x - 6 ~ + (V)x+~ ,k,%(a)~+,~r3 -- k m O A , , x _ _ ~o~, ~+~

(59)

4. Forward Migration (to site occupied by A) (A*)~_~ + (A)~+~ koo>(A 9A)~+~ r4 -- kaaOA.,x_~OA, x+ ~

(60)

5. Backward Migration (to site occupied by A) (a*)~_~ + (a)~_~ % ( a . Ak_, r5 -- kaaOA.,x_ ~ 0 A

(61)

6. Forward Migration (to site occupied by B)

(a*)~_~ + (B)~+~ k~ (B. A)~+~ r6 = kobOA.,~_~Os,~+ ~

(62)

7. Backward Migration (to site occupied by B)

(a*)x_~ + (B)~_~ k.b>(B. A)~_~ r7 = kobOa.,~_~Os,~_ ~

(63)

Similarly, for molecule B at lattice (x - ~) one can have the following steps: 1. Activation (B)~_~ kb> (B*)~_~

r~ = kbOs,~_~

(64)

2. Deactivation (to vacant site) (B*)~_~ + (V)~_~ --~ (B)~_~ r2 = k~OB.,~_~O~,~_ ~

(65)

3. Forward Migration (to vacant site)

(B')~_~ + (V)~+~ kn >(B)~+~ r3 = knOB.,~_~O,,,~+~

(66)

4. Forward Migration (to site occupied by B) (B')x_~ + (B)~_~ kbb,, (B 9B)~+~ r4 -- kbbOB,,x_~OB,x+ ~

(67)

501 5. Backward Migration (to site occupied by B)

(B'),~_~ + (B)x_~ % (B. B),~_~ rs = kbbOs.,~_~Os,~

~_~

(68)

6. Forward Migration (to site occupied by A) (B')~_~

+ (A).+,

~ kbo, , (A 9 B)~+,

r6

kba 0 B,, x- ~,0A z+ ,

=

(69)

7. Backward Migration (to site occupied by A) (B')~_~ + ( A ) . _ ~ ~

=

k~o,, (A. B)~_~

kbo0s.,~_~0A,~_~

(70)

One advantage of using these rate equations is that one does not need to have the knowledge of the actual events occurring during the molecule migration on the surface or pores. In the RHS of Eqs.60-63, for example, molecule A can either stick on sites occupied by molecule A or B, or switch between A or B (exchange energies); the rate equations are the same. As mentioned earlier, for diffusion in narrow channels and pores, blockage or partial blockage (by adsorbed A or B) may occur, then two additional rate processes (blocking by A and B) should be included. Such a blockage process has been attributed to be the cause for the decrease of diffusivity with increasing concentration, and has been treated previously[51] for the case of single-component diffusion. For simplicity, the blockage processes are not included in this treatment. In steps 60-63, the mobile molecule sticks to the adsorbed A or B by forming a bond. There is a possibility, however, that it does not stick. In that event, two possibilities arise (for one-dimensional diffusion): it continues the movement either forward or backward (to make multiple jumps). For simplicity, we assume that these two probabilities are equal and therefore the subsequent events do not contribute to the net flux. 5 The net rates of forward migration for molecules A and B located at lattice site ( x - 5) are. respectively: MA, x-~ -- kmOA" ,z-~'Ov, x+~' + kaaOA',z-~OA, x+~, + kabOa. ,x- ~OB,x+~'

(71)

' z - ~Os,~+ ~, + kb~Os ., ~_ ~0a,~+~ B , ~_ 5~ = k,O s , ,z-~~O~+~+kbbOs. ,

(72)

Following the same procedure, one may write for the net rates of backward migration for molecules A and B located at lattice site (z + MA ~+~) MA, z+ ~ = kmSa.,x+~Ov, z_ ~ + ka~SA.,z+~OA, z_ ~ + kabSA.,z+~SB,z_ ~

(73)

Ms~+ ~ = k , Os.,~+~O~ ~_~ + kbbOs . ~+~0s ~_~ + kb~Os. ~+~OA~_ ~

(74)

502 The net rates for formation of activated molecules, A* and B* at two lattice sites are:

Ot

'

~-

'

~-~

-~o~o~.~ (o~,~_~ + o~,~+~) (rs) oo~. ~

=

Ot

~o~ ~

-

'

~oo~. ~ ( = N~=o

E

Na=O

(3.11)

K~7~,.{K'P}N~'[z'({:T,V,)F"

For the adsorption of non-rotating rigid molecules, the expression for the configuration integral Zs can be replaced by a simpler form N~

Zr(Na, T, Vs) = / exp{-~[ E

CZ(ri) + E E r i

i < j

rj)]} H dri

(3.12)

i=l

The adsorption of non-rotating rigid molecules often is taken as a first approximation in many theoretical treatments on adsorption. The more general situation can then be

524 built on the simple case by separating the configuration integral into that of the mass center (as a rigid non-rotating molecule) and the integral over other degrees of freedom. Z~(N~, T, Vs) = / exp[-fl Z

(I)intra(r i

"

Zr(Na' T, V~; @i, r

1 I d@idr

(3.13)

i

Let us assume that the Eulerian angles @i and internal degrees of freedom r can all be lumped by its effect to the host-guest potential. We then write ZS(N~, T, V~) = f Zr(N~, T, V~; e)-f(e)de

(3.14)

and (N,.) = f (Na) r (e)-g(e)de

(3.15)

where Zr(Na, T, Vs; e) is the configuration integral of a rigid molecule having an average host-guest potential of e, and (N~)r (e) is the average adsorption amount for the same rigid molecule. The f(e) and g(e) are two related but unknown distribution functions. In many discussions of heterogeneous surfaces, the energy heterogeneity has been stressed as a property of the solid. In the case of zeolite, the possibility of crystalline faults and foreign atoms may indeed introduce a distribution of adsorption energy. However, as we have demonstrated, energy heterogeneity may also come from the averaging over internal degrees of freedom, and therefore would be adsorbate dependent. This part of energy heterogeneity persists even in the case of a perfect zeolite crystal, and should not be disregarded when discussing adsorption data.

4. F I N I T E L A T T I C E M O D E L F O R C A G E T Y P E

ZEOLITES

The adsorption of gases in microporous zeolite has been applied in many industrial separation processes [23, 24]. Zeolites A, X and Y are among the most widely employed materials for such purposes. For example, both A and X zeolites have been used in the separation of air [25-27]. The Isosieve process [28] that separates small normal and branched paraffines in zeolite A is one of the most important adsorptive processes in petrochemical industry. The X and Y zeolites are also employed in the PARAX and SORBEX [29, 30] types of processes to separate the isomers of xylene and other akylbenzenes [31, 32]. The micropore voids in the above mentioned A, X and Y zeolites consist of three dimensionally arranged cages connected by windows. An upper limit can be assumed for the number of molecules that can be packed into each cage. The interaction of framework atoms and small admolecules may create distinct potential minima. Adsorbed molecules are confined in these local potential minima during the course of adsorption. Each local potential minimum can, thus, be considered as a discrete adsorption site. Van Tassel et al. [33, 34] proposed to consider the adsorption of small molecules in cage type zeolites as localized at nodes of a polyhedron. This picture was supported by their own molecular simulation studies and by many others [35-39].

525 Presumably, the maximum number of admolecules m, that a cage can accommodate, is the same as the number of local potential minima, or adsorption sites in the cage. If these sites are located at the nodes of a polyhedron, they may be considered as energetically identical. However, if the cage size is very large compared to that of the admolecule, one may have to include a second polyhedron of sites inside the first shell. For example, it has been observed in the simulation of Razmus and Hall [37] that oxygen and argon are initially adsorbed at the sites near the cage wall of A zeolite. At higher loadings, when most of the near wall sites are filled, the admolecules are forced to adsorb on the sites which are closer to the cage center. With the above picture in mind, a grand partition function can be written for a single zeolite cage. The zeolite-adsorbate interaction @Z(r) is assumed to concentrate in the vicinity of m identical local potential minima. Since these m local potential minima (sites) are defined by the zeolite-sorbate interaction @Z(r) alone, and are therefore independent of the inter-molecular force, we can write [33]

ZS(1, T) =/ca e-ZCz(r)dr~ m -

Vsite.

e -z ,'r T)

(4.7.1)

Then from the variational principle stating that 8r p, T) should approximate best Langmuir isotherm, one arrives at the definition of de = er defined in Eqn (4.6.9). The overall adsorption isotherm/gt is then given by the condition oo

St(p, T) = ] x(,)d, =

(4.7.2)

I v

~r

To be correct one has to replace +oo by a finite maximum value of e, era- In the case of strongly heterogeneous surfaces em is usually large, so replacing em by +c~ does not affect much 0t(p, T). And then for physically reasonable functions X(e), like these in Eqn (4.6.10) or Eqn (4.6.13), A'(+oo) = 0, i.e., we arrive again at Eqn (4.6.8). Thus, when the adsorption process is perfectly gradual (stepwise), we have the one-to--one dependence between ~ and p, and a one-to-one dependence between e~ and 0t, the particular form of which depends on the form of X(e). Let us assume for the moment, that

544 w = 0 i.e. neglect the interactions between the adsorbed molecules. Then, for the function X(e) defined in Eqn (4.6.10), the corresponding Eqn (4.6.11) yields

1 -0t

er - eo = c In 0----~

(4.7.3)

As we are neglecting for a moment attractive interactions between the adsorbed molecules, we compare Eqn (4.7.3) with the Eqn (4.6.22) for the heat of adsorption in which w = 0. It means, that in the process of the gradual filling Qst changes with the surface coverage in the same way as er Thus, at every step of adsorption process, (at every surface coverage), the observed heat of adsorption will be equal to the adsorption energy of the sites which are being filled at this coverage, within a constant independent of surface coverage. Of course, the process of filling will never be perfectly gradual, and at high temperatures, in particular. Nevertheless, accepting this approximation makes it possible to draw interesting conclusions about the change of the energy of the gradually induced sites, from the behaviour of the heat of adsorption. As our adsorption system is composed of many subsystems, (cavities), being only in thermal and material contact, the one-to-one dependence between er and 0t must exist also at a level of one cavity. The experimentally monitored one-to-one dependence (4.7.3) represents a certain statistical interrelation averaged over all the cavities in the system. To a certain approximation we may assume this statistical interrelation to be close to the local one, existing at a one cavity level. So, let ei denote the adsorption energy of the adsorption site induced when the ( j - 1) molecules are already adsorbed. Then er should be the function ec (~), averaged over all cavities. Thus, to a good approximation, we can accept the following relation cj - c o =

cln

1-• • m

(4.7.4)

m

Assuming a perfectly gradual filling of the induced adsorption sites, we assume implicitly that their number is equal to the number of the adsorbed molecules. So, the configurational degeneracy factor in Zs will be equal to unity, and we can write

(4.7.5)

ZS(s,m,T) = exp { ~ (ei - e~ } j=~ kT Thus ASQ~(s, v, T) = p*(K~ s exp

{s

~

In

1

}

(4.7.6)

j=l where QS(s, v, T) is the canonical partition function for s molecules in one cage, and where K ~ = K' exp

~

(4.7.7)

545 We replace now symbol n by s to denote the number of molecules in one cavity in order to emphasize that instead of mobile adsorption we consider now localized adsorption on s sites. Now it is the time to realize that the relation (4.7.5) corresponds to the assumption that ~ E ( - o c , +co), because X(C)in Eqn (4.6.10)is defined in this interval. However, for obvious physical reasons there must exist a maximum value of ~ on a heterogeneous surface and inside zeolite cavity (channel), Cm, and a minimum value cl. The assumption that ~ varies in the interval ( - o c , +oc) is accepted frequently in the theories of adsorption on heterogeneous surfaces for the purpose of mathematical convenience. Assuming such non-physical energy limits does not have a serious negative impact on the behaviour of the calculated theoretical isotherm until we do not consider the adsorption at very small and very high surface coverages, i.e. when p ~ 0 o r Ot - + 1. The most striking consequence of assuming that Cm = +OC is that the Langmuir-Freundlich or the Dubinin-Astakhov isotherm do not reduce to Henry's isotherm when p ---, 0. In the case of the corresponding theoretical heats of adsorption (4.6.22), (4.6.24) the assumption that ~,~ = +oc results into infinite values of Qst, when ~t --"> 0 (and to - o c when Ot --'+ ])" Meanwhile, the experimental adsorption isotherms in zeolites have the Henry region as a rule, and experimentally measured heats of adsorption become constant in that region. This suggests the following modification to be made in Eqn (4.7.6) ,kSQS(s,m, T) = p~KH(K~ ~-1 exp

~

in ~

j=2

j/m

for s < m -

1

(4.7.8)

where K H is the Henry's constant KS=

K'exp ~

(4.7.9)

which can be easily determined experimentally. In the case of the adsorption energy distribution (4.6.10) some further modification is to be made, to eliminate the situation that Zs = 0 for s = j = m. That situation is due to the fact that el = - o c is assumed in the function (4.6.10). For s = m, AmQ~ should be written as follows )~mQm(m,m, T) = pmKH(K~

~-~ In ~1 -j/m}j/m

exp

(4.7.10)

k j=2

where K TM is another temperature-dependent constant, which should be found by fitting the isotherm equation to the experimental data. However, as it can be deduced from Eqn (4.7.10), the values of ~SQS will decrease quickly with s, so, one can neglect safely the difference between zero and the true value )~mQm. Thus, Eqn (4.7.8) can formally be written in the form c_s kT

j/m /

(4.7.11)

546 Now, let us consider the case when, in the region of mediate surface coverages, the experimental adsorption isotherm can be correlated better by the Dubinin-Astakhov equation _ C0 r }

0t = exp { - ( e r

E

)

(4.7.12)

or 1

so, we assume now that 1_

ej = e~ + E In

(4.7.14)

Then ASQS(s' m' T) = p*KH(K~

exp

E k-Tj=2

In

;

(4.7.15)

So, finally, we arrive at the following equations for the adsorption isotherm. For the gaussian-like dispersion (4.6.10) of the adsorption energy of the induced adsorption sites we have m

E psKH(K~ s-is s=l

?(

1--j/m~

=2

j/m ]

r

n=

r

(4.7.16)

(1--j/m

1 + E psKH(K~ ~-1 fi s=1 =2 \ j/~

whereas for the adsorption energy dispersion (4.6.13) we obtain E p~KS(K0)~-ls exp n=

Ej:~

s--1

1 + s=lEpsKH(K~ ~-x exp

(4.7.17) k~j=~

The applicability of this new approach is demonstrated by considering again nitrogen adsorption in the zeolite 10X, studied by Nolan et al. [51] In view of the good correlation of these data, obtained by using the linear form of the Langmuir-Freundlich equation (4.6.20), we decided to verify that new theoretical approach using now Eqn (4.7.16). As the volume of this zeolite cages is comparable with the volume of 5A zeolite cages, we assumed that m = 10, according to our discussion in the section 4.5 of this review. The n value in Eqns (4.7.16) and (4.7.17) can, thus, be expressed as Nt/(M/10).

547 Theoretically, while fitting Eqn (4.7.16) to the experimental adsorption isotherms, M should be treated as a best-fit parameter, together with K H, K ~ and kT/c. It is, however, well known that the "adsorption capacity" is estimated best from the adsorption data measured at higher surface coverages. But there, Langmuir-Freundlich isotherm may be applicable, because the existence of Henry's law is not the main point here. So, we took M values listed in Table 4. The other parameters K H, K ~ and kT/c were found by computer by accepting the following ERROR function

1 1 (theor_i-expi) 2 ERROR = ~

]- k

expi

i=l

The parameters K s, K ~ and kT/c found in that way are collected in Table 5. Figure 8 shows the agreement between experimental adsorption isotherms of nitrogen adsorbed in the zeolite 10X with the theoretical ones, calculated from Eqn (4.7.16) by using the parameters collected in Table 5. Table 5

The parameters found by computer when fitting Eqn (4.7.16) to the experimental isotherms of nitrogen adsorption in the zeolite 10X, reported by Nolan et al. [51]. T

KH

K~

kT/c

ERROR

172.04 227.60 273.15

3.023-10 -1 1.176.10-2 7.219.10 -3

1.194-10-3 2.084.10 -4 4.877.10 -4

0.478 0.850 1.000

6.125-10 -s 8.289.10 -4 1.015-10 -4

10.0 .

8.o_

6.0~

T=172.04 K T 2276 K 73. 5K

4.0 2.0 0.0

0

400

860

1200' 1600 2000 p [ramHg]

Figure 8: Comparison of the theoretical isotherms (--) calculated from Eqn (4.7.16) by using the parameters collected in Table 5, with the experimental isotherms of nitrogen adsorption in 10X, measured by Nolan et al. [51] at the three temperatures: 172.04 K, 227.6 K and 273.15 K.

548

5.0 ~4.4 ~'~ .~. 3.8 3"2t N 2.6 2.0] , ~ { x ~ tga=0.38 I

0.0

I

1.6

I

I

l

I

3.2

I

I

4.8

I

8.0

6.4

ln(p) [ram Hgl Figure 9: The ln Nt vs. In p plot for the data on nitrogen adsorption in 10X at 172.04 K, reported by Nolan et al. [51].

3.0 r~

2.0 1.0 0.0-1.0-2.0

a~ tg a = 1.0 0.0

I

1.6

3.2

I

I

4.8

I

I

I

6.4

8.0

ln(p) [ramHgl Figure 10: The In Nt vs. In p plot for the data on nitrogen adsorption in 10X at 273.15 K, reported by Nolan et al. [51]. Eqns (4.7.16) and (4.7.17) predict existence of Henry's limit at sufficiently low surface coverages (zeolite loadings). It means that the tangent of the double logarithmic plot In Nt vs. In p should approach unity at a sufficiently low adsorbed amount. Figures 9 and 10 show that plot for the two temperatures: 172.04 K and 273.15 K, respectively. Looking at Figure 9 one can see a typical Freundlich linear plot at the lowest surface coverages, the tangent of which is equal to 0.38. Then, the lowest surface coverages (zeolite loadings) studied at the temperature 273.15 K are lower by two magnitude orders. At such low loadings the double logarithmic plot has a slope equal to unity, indicating thus the existence of Henry's region.

549

5. H E T E R O G E N E I T Y ZEOLITES

EFFECTS

IN ADSORPTION

IN MFI

MFI type zeolite is one of the most versatile and valuable zeolites in modern hydrocarbon processing. A wide range of applications including catalytic and adsorptive processes has been proposed based on the size and configuration of its pore structure. In particular, the adsorption of aromatics in MFI zeolite aroused tremendous research interest in the recent years. The general understanding is that there are several energetically non-equivalent locations in this zeolite for the adsorption of aromatics. Due to the comparable size of the pore channels and of the adsorbate molecules, a strong and position sensitive potential field is experienced by the aromatic molecules. This adsorption system may be successfully modelled by a regular lattice of energetically different sites. Such a model is the simplest case where both the energy heterogeneity and the structure heterogeneity (packing arrangement) are involved. The adsorption of aromatic compounds in ZSM-5, and its aluminum-deficient structural analog silicalite, has been extensively studied in the recent years. Various experimental methods including isotherm measurements [53-58], isosteric heat [59, 60] or calorimetry [7, 61, 62], gas chromatography [63, 64], detailed X-ray or neutron powder diffraction [4, 65-72], or NMR [73, 74] have been used. The results reveal some unusual adsorption characteristics of the MFI/aromatic systems, associated with the site heterogeneity and the possibility of a combined zeolite-adsorbate phase transition. This unusual behavior is the consequence of both the peculiar framework structure of MFI zeolites, and of the strong interactions induced by the tight fit situation. At least three different locations for adsorption have been suggested by considering the topology of the MFI framework. Site heterogeneity in MFI (silicalite) was first reported in the calorimetric studies of Thamm [7, 61]. The adsorption heat exhibited a strange jump at intermediate loading for benzene, ethylbenzene, toluene as well as for some alkanes. This could only be explained in terms of a rapid change in the state of the admolecules. A phase transition in the ZSM-5/p-xylene system was confirmed for sure [6, 75]. Talu et al. [5, 76] outlined the phase boundaries of ZSM-5/benzene and ZSM-5/p-xylene systems with isotherms measured at different temperatures, but the appearance of a phase transition in the benzene/silicalite system is still an open question. Using a deuterium NMR, Portsmouth and Gladden [74] proved that the adsorbed benzene molecules were indeed under different environments before and after the step on the adsorption isotherm. More recently, the powder diffraction studies of Mentzen et al. [65-68, 71] of the MFI/aromatics systems revealed the presence of several types of zeolite/admolecule complexes. In many cases [4, 68, 71, 72], the location of the admolecule was identified by comparing the model simulation with the refined diffraction data. According to Mentzen [71], there exist at least three types of polymorphic modifications of the MFI framework, depending on the nature and the amount of admolecules; a monoclinic (with P21/n. 1-1 symmetry), a first orthorhombic (Pnma) and a second orthorhombic phase (P21212a). They have assigned the name MONO, ORTHO and PARA for these structures respectively. ZSM-5 loaded with fewer than 4 benzene molecules/u.c. is in the MONO form. The case of toluene and ethylbenzene is the same. On the other hand, ZSM-5/n. p-xylene, (the short hand for ZSM-5 zeolite loaded with n p-xylene molecules per unit cell), is in the ORTHO form if n < 4 and ZSM-5/8-p-xylene is in the

550 PARA form. Apart from the difference in symmetry, there is also a slight variation of the lattice constants among the three polymorphic structures. A variety of theoretical approaches, based either on the computer simulation [10, 11, 77-82] or using the methods of statistical thermodynamics [83-86], have been proposed to model these adsorption systems. Thus, Thamm [7] assumed that this might be adsorption of dimers, or a cooperative redistribution of adsorbed molecules, whereas Stach et al. [58] saw it as a pore filling with the energy and entropy of different sites determined by the Monte Carlo simulations. Talu and co-workers [5, 76] launched a two-patch model of collective mobile adsorption with the surface phase transition. They argue that in spite of the channel structure of zeolite, the adsorption of aromatics in ZSM-5 cannot be treated in terms of one-dimensional systems. The idea to treat the whole adsorption system as a three-dimensional cooperative system has also been accepted by Chiang and co-workers [86]. They used a lattice gas model, with three kinds of adsorption sites and showed equivalence of their lattice gas model to an Ising model. The authors postulated next that the sharp increase in the adsorption isotherms is due to the phase transition predicted by this model. While considering the p-xylene adsorption in ZSM-5 Pan and Mersmann [85] launched a model of localized adsorption on two independent subsystems of sites, corresponding to the channel intersection and zigzag channels. Adsorption in the channel intersections was described by the Langmuir isotherm, while the adsorption in the zigzag channels was modelled by the isotherm corresponding to the quasi-chemical approximation for cooperative adsorption. The interactions between the molecules in the neighbouring zigzag channels were assumed to be mediated by structural relaxations in the zeolite itself, that occur when p-xylene is adsorbed at high loadings in ZSM-5 type zeolites. Pan and Mersmann proposed also a construction of the hysteresis loop for p-xylene on the basis of their model. That construction of the hysteresis loop of p-xylene has, next, been questioned by Snurr et al. [82] who launched another model of adsorption in ZSM-5 zeolites. They proposed that the phase change from the ORTHO to PARA form, which occurs when more than four molecules of p-xylene per unit cell are adsorbed, may also take place in the case of benzene adsorption. Snurr et al. accepted the three-site model launched by Chiang and co-workers, but assumed that the phase change in the zeolite structure causes the free energies of adsorption to change, on the three kinds of sites. In their model the sharp increase in the adsorption isotherms is essentially due to the change of the zeolite structure. Thus, in spite of the cooperative character of their model, the sharp increase in the adsorption isotherm is due to a cooperative redistribution of adsorbed molecules and not to a phase transition in the phase of the adsorbed molecules. The theoretical isotherms calculated by means of their lattice model were compared with GCMC molecular simulations but were not fitted to the experimental adsorption isotherms. Although calculations were performed for four temperatures, the corresponding heats of adsorption were not calculated. Heats of adsorption were estimated by these authors only by using the GCMC simulations for the changing zeolite structure. As well known, the behavior of isosteric heats of adsorption is a more stronger test for a theory than the behavior of adsorption isotherms. It should, however, be emphasized that none of the above discussed models can predict appearance of two steps which have been observed experimentally on the isotherm of benzene adsorption in silicalite at higher temperatures.

551 Next, theoretical studies of aromatics adsorption in ZSM-5 focus, almost exclusively, on the comparison between the calculated and experimental adsorption isotherms. Although the experimental data on the coverage (loading) dependence of the isosteric heats of aromatics adsorption in ZSM-5 were reported as long as a decade ago, they were largely ignored in the theoretical studies. A reliable theory of adsorption should lead to a simultaneous good fit of the experimental adsorption isotherms, and of the accompanying heats of adsorption.

5.1. T h r e e

site lattice gas model

To include the true geometric arrangement of the adsorption sites, we have proposed a lattice model with three different kinds of sites; namely the I (intersection), Z (zigzag channel) and S (straight channel) sites. They are shown in Figure 11.

lO-ring

Rlliptical

.[0.5x*o.57 .m) ,

2.006

b

nm

_ j

"

Z-L_; t

Circular lO-ring

(0.54 nm)

•

,i.T

~

S

S ,I

1"2

T

T

g.J

11

r7

TI

9

S 9 I s

g.J

7

T!

9

S r7 L.J

T] 9

S

S

S

S _

9

q

"1"i_

_

S Figure 11: (A) The channel structure of silicalite. (B) The schematic model of the three-site lattice structure accepted by Lee and Chiang in their theoretical considerations. Channel dimensions shown in this figure are those reported by Flanigen et al. [87]. In literature some different dimensions of channels in ZSM-5 can be also found [88]. Namely, the straight channels are nearly circular with dimensions 0.54 x 0.56 nm, while the sinusoidal channels are elliptical with dimensions 0.51 x 0.55 nm.

552 None of the phenomenological models mentioned above took the observed structure change of the zeolite framework into account. By the atom-atom grand Canonical Monte Carlo (GCMC) or molecular dynamics (MD) simulations, such a framework change may be reflected in the model. The atom-atom simulations are, however, time consuming and require access to a super computer or a dedicated workstation. The situation is even worse for the tight-fit system such as aromatics in ZSM-5 (Li and Talu [11], Snurr et al. [81]) The lattice model, on the other hand, is less computationally intensive but requires a large number of energy parameters to be determined by data fitting. To circumvent this difficulty, Snurr et al. [82] proposed recently a hierarchical atom/lattice strategy, in which the detailed atom-atom simulations are used to obtain the required parameters for the lattice model. They tested this approach by considering the benzene/silicalite system. Two sets of parameters were obtained corresponding to the ORTHO and the PARA symmetries respectively. An additional empirical parameter, which still had to be fitted, was included to account for the free energy change upon the transformation of a framework. In our previous study [86], we proposed to model the adsorption of aromatics in silicalite as a lattice gas with three types of sites, located at the straight channel, the zig-zag channel and the channel intersection respectively. In order to obtain an exact solution for that model, we considered only an attractive interaction J~ for the occupied nearest neighbor Z-I pairs. The adsorption on neighboring S and I sites was assumed to be mutually exclusive due to size restriction. One could have included more interactions between the adsorbed molecules. However, in such a case, only a mean field solution could be obtained. For example, we could include attractive interactions J~z, Jzi, J~s, Jii for the occupied S-Z, Z-I, S-S and I-I pairs, and a very large repulsive interaction Jsi as a way to account for the mutual exclusion assumption of the occupied S-I pairs.

5.2 M e a n field s o l u t i o n s For a system with N sites of each type, the canonical partition function of the three site lattice gas can be written as E(#,N,T; J~,J~,Jsz, Jzi, Ji~,e~,ez, ei)=

E

E

E

exp[-~H]

(5.2.1)

ns--0,1 nz--0,1 hi--0,1

where - ~ H = J~ ~ n s2 + J~ ~] n~ + Jsz ~] nsn~ + Jzi ~ nzni + Ji~ ~ nin~ +(es + In A) E ns + (ez § In A) E

nz + (el + In A) E

ni

(5.2.2)

Above es, Q, ei are the adsorption energies on sites S, Z, I, and the activity expk-~T is written as A. The summation is over all possible occupancies, with ni -- 1(0) denoting site I being occupied (unoccupied). This canonical partition function can be evaluated by a mean-field approximation. First we write the Helmholtz free energy as a function of H F = E - TS = Tr(pH) + kW-Tr(p In p)

(5.2.3)

553 where the entropy S = - k . Tr (p In p) and p is the density matrix. By the mean-field approximation we assume

P = H Pi

(5.2.4)

i

where the product is over all sites. We choose the pi's that minimize F subject to Wr(p,) = 1.

Following this procedure, we obtain from equation (5.2.3) BF

N = J~0~2 + Jn0~ + 4Js~0s0~ + 2Jzi0z0i + 2Jsi0s0i

+(e, + In A)Os+ (~ + In A)0, + (ei + In A)0~- [Tr(p~ In p~) + Tr(p, In p,) + Tr(p~ in pi)](5.2.5) where 0j is the fraction of j sites that are occupied, defined as 1 8j = ~ < E

nj > = < nj > = Tr(pjnj).

(5.2.6)

One then finds the ps, pz, pi that minimize F subject to Tr(p~) = Tr(p~) the Lagrange's multiplier method. The final result reads as 0s = Tr(p~n~)=

e, = T r ( p , ~ , ) =

Oi = Tr(pini)=

1 1 + -~exp [-(2J~,0s + 4J~0~ + 2J,i0i + Ca)] 1

1 + }exp [-(4J,,0, + 1 + 89

2JziOi

@s

1 [-(2Jii0i + 2J~i0z + 2Jsi0~ + ei)]

= Wr(pi) =

1, with

(5.2.7)

(5.2.8) (5.2.9)

For the special case with J~ = J~ = Jss = 0, a complete equivalence of the lattice gas with an Ising model exists. Having all energy parameters o's and J's, one can solve these equations for any A value and obtain the ffs. The pressure of the system is related to the chemical potential by the relation p = ~ - e x p ( - # ~ where #o is a reference chemical potential. Lee et al. [86] used their three site lattice gas model to fit the experimental isotherms of benzene and p-xylene adsorbed in silicalite. To arrive at the phase transition in the adsorbed phase they were forced to assume certain values of the parameters es, Q, ei and some interrelations of the gas-solid and gas-gas parameters to be fulfilled. However, the calculated isosteric heat of adsorption could not be seen as reproducing well the qualitative features of the experimental heats of adsorption reported by Thamm [61]. The estimated energy of adsorption es was larger than el, which would mean that the straight channel sites S are filled first. Meanwhile, the reported atomistic simulations [11, 77, 78, 80, 81] suggest that channel intersections I are filled first. There are some experimental findings [89, 90] that seem to support such a view. As the reported simulations already provide a solid support for the three-site model, we have decided to investigate further possibilities of arriving at a consistent theoretical

554 description explaining all the experimental findings. Looking for possible revisions and improvements of the three-site model, we rejected the concept that this is only the phase change in the zeolite structure which might be the source of the step on adsorption isotherms. Firstly, because in the case of benzene it is only a hypothesis at present. Secondly, because that hypothesis itself does not lead to predict the two steps on the adsorption isotherm of benzene which have been observed at higher loadings and temperatures. Further, looking to the computer simulations reported by Snurr et al. [82], one can see, that the zeolite phase change should not affect much the adsorbate-adsorbate interactions. As far as the solid-adsorbate interactions are concerned these simulations suggest that this zeolite phase change could affect strongly the adsorption on Z sites, but should not affect the adsorption on I and S sites much. Thus, we decided to see first, which agreement between theory and experiment could be obtained by neglecting the changes in the adsorption parameters which might be caused by the possible phase transition in the zeolite structure. Finally, we focused our attention on the recent discovery reported by Mentzen et al. [69]. They have found, that two equally probable orientations of benzene molecules adsorbed in the channel intersection are possible, and that these orientations are characterized by different values of the adsorption energy e. Thus, we can assume, that these forms denoted herefrom by I1 and I2 compete for occupying channel intersections. In such a case the equation system (5.2.7)-(5.2.9) takes the following form Aexp{(E~+ j~. a,jSj)/kT} 8, =

(5.2.10)

9 =

l + A e x p { ( E z + j~. wzj)/~j)/kT)

=

(5.2.12)

and

0i2 =

(5.2.13)

555 where Ej = ej + kT In fj, fj is the molecular partition function of the admolecules occupying the sites of type j, and wij is the interaction energy of a molecule adsorbed on the site i with the molecules adsorbed on the nearest sites j. Let us remark that the parameters Ej's in the above equations are related to the nonconfigurational part of the free energy of adsorption per molecule. The overall adsorption isotherm 0 obtained by solving the equation system (5.2.10)(5.2.13) is then given by 0 = g1 Z 0j J

" i2,s,z j = 11,

(5.214)

Theoretically, such an isotherm equation with the appropriate interaction parameters wij, might show a critical point at a temperature Tr such that

( )

:0

and a non-physical loop at still lower temperatures. Then, one would have to perform the Maxwell construction to arrive at the physical isotherm, represented by a straight line fulfilling the conditions f0~ # (0)d0 /~M --~

0G _ 0L

and

#M = # (0G) = # (0L)

(5.2.16)

where 0G and 0L are the surface coverages corresponding to the beginning and the end of the phase transition, respectively. The theoretical isotherms based on the concept of a phase transition in the adsorbed phase will predict a vertical increase in the experimental adsorption isotherm at a certain pressure defined in Eqn (5.2.16). Looking more carefully to the behavior of the reported experimental isotherms, one can see indeed a sharp increase which, however, is never perfectly vertical. It seems, that this fact has not received enough attention. Then, within the range of the surface concentrations corresponding to the phase transition region, the predicted heat of adsorption is independent of the coverage (loading), and is given by Q = 0G - 0L

f01~ Q (0)dO

(5.2.17)

Let us remark that such constant heats of adsorption have not been observed at higher loadings by the adsorbed aromatics, where the phase changes were believed to occur. This must rise serious doubts as to whether the sharp increase(s) in the adsorption isotherms of aromatics are due to phase changes in the adsorbed phase. Now, let us investigate the behavior of the heat of adsorption of aromatics predicted by our equations (5.2.10)-(5.2.13). For that purpose, we rewrite Eqns (5.2.10)-(5.2.13) as follows

E~ + Ew~jOj F~= #~- ~ =ln~0~ _ kT 1 - 0~

j kT

-ln~=0

(5.2.18)

556 E~ + E ~zjOj J - In A - 0 kT

F~ = #~ - # = In ~ 0 z _ kT 1 - 0, Fia

=

#ix

-- #

__

kT

In

1 --

0i~ 0il

_

- - 0i2

(5 9 19) '

Ei, + ~ wi,jOj j - In A = 0 kT

(5.2.90)

Ei~ + ~ COi2j0j Fi2

-

#i: -- ~ __

kT

In

0i~

1 - 0i~ - 0i2

--

J

kT

--

In A = 0

(5.2.21)

Let Qj ({0m}) denote the molar differential heat of adsorption on the site of type j, at a certain set of the average surface coverages {0m} (m = s,z, il,i2). It is given by Qj ({0m})=-k~0-~ (#JkT#){0,,) Thus Qj's take the following explicit forms

Q, ({0m})_. QO + ~ a~,j0j

(5.2.23)

J

Qz ({Ore})= Q~ + )-" ~'~jOj

(5.2.24)

J Qil ( { 0 m }) --

Qi~ + ~

~'ilj0il

(,5.9.25)

o.,,i2j 0j

(5.2.26)

J Qi~ ( { 0m} ) = Qi~ + ~ J where d ( Ej +tz~ o

Qi = k

)

dT1

(5.2.27)

and #~ in Eqn (5.2.27) is the standard chemical potential of adsorbate molecules in the gaseous phase. An incremental increase in ~, dp, will result in an incremental increase of 0, dO, represented by ( 00j '~ dO = Z . \~-~,] d#

(5.2.28)

J

That incremental increase will be accompanied by a heat effect dQ dQ-

Z

QJ (00J

(5.2.29)

557 Thus, the overall (measured) differential heat of adsorption Q will be given by

Q=

~Q5

\0u)

(5.2.30)

The derivatives \ 0u ] can be evaluated from the equation system (5.2.18)-(5.2.21). It can be done as follows. Let Gj denote Fj multiplied by kT. Thus 0Gj = _1 + Z

0Gj 00m = 0

0~

00 m 0~

m

(5.2.31)

The derivatives (oo \ 0u ) are found by solving this system of four linear equations. Let G~ denote the derivative

Gj = \OOm] The solution of the equation system (5.2.31) reads 00 m

=Dm

0~

D

(5.2.33)

where D=

G:

G:

G~ ~ a~ ~

(5.2.34)

and Dm is obtained from D by replacing the m-th column of the determinant D by the column os constants. From Eqns (5.2.30) and (5.2.33) one will be able to evaluate the heat of adsorption as a function of coverage. The isosteric heat of adsorption of benzene in silicalite, reported by Thamm [61] (Germany) was taken for analysis, along with the adsorption isotherm measured in Chiang's Laboratory (Taiwan). This fact has to be taken into account while considering the obtained agreement between theory and experiment. The experimental isotherms of benzene in ZSM-5 measured by Chiang and co-workers [8] are shown in Figure 12. In the case of benzene adsorption two steps are observed: the first is at around 4 M/u.c. and the second one at around 6 M/u.c. The sharp steps in the isotherms of aromatics adsorbed in ZSM-5 silicalite at the loadings above 4 M/u.c. have been observed by many authors [59, 60, 62, 75, 76]. It seems, however, that we were first to report on the two steps on the experimental adsorption isotherm of benzene at higher temperatures. At the same time no hysteresis was found in our benzene adsorption isotherms. Such hysteresis was reported earlier by Thamm [7], using much smaller crystals than these

558 used in Chiang's experiment. It seems, therefore, that the observed hysteresis was due to desorption from the intraparticle pores. While considering the interaction parameters win, (m,n = il,i2,s,z), we have carried out numerical best-fit calculations, seeking for a minimum number of these parameters that would allow for a reasonably good (simultaneous) fit of experimental adsorption isotherms and heats of adsorption.

.....

8.0 e"

::3

,..

6.0

---r :3

4.0

(D Q.

0

(D 0

E

2.0 0.0 I 0.0

'

i 1.0

'

I

'

2.0

I

II

4.0

3.0

log p r e s s u r e

(Pa)

Figure 12: Experimental isotherms of benzene adsorption in ZSM-5 at 273 K (, 9 e), 283 S (ooo), 293 S ([::]DO) and 303 K (AAA), measured by Chiang and co-workers [8]. The solid lines were drawn by hand to emphasize the trends in the experimental data.

Table 6 The values of parameters found by fitting simulataneously the experimental isotherms and heats of adsorption of benzene in silicahte, measured at 303 K, by our equations (5.2.10)-(5.2.13) and (5.2.30). The Henry constant Kj is defined as exp{(Ej + p~ Kil (Pa -a ) 8.54 7-10 -2

Ki2 (Pa -1 ) 3.873-10 -3..

Ks (Pa -1 ) 2.605.10 .3

K= (Pa -z ) 3.643.10 -2

w=z (kJ / mol) 7.60

~sz

~si2

~zi2

~sil

~ziz

(kJ/mol) -2.75

(kJ/mol) 21.00

(kJ/mol) -30.30

(kJ/mol) -I1.00

(kJ/mol) -I4.00 ,,

~iziz

qi~

Q~

qo

QO

(kJ/mo~)

(kJ/mol)

(kJ/mol)

(kO/mol)

(kJ/mol)

0.00

56.00

105.00

69.00

70.00

559 Table 6 collects the values of the parameters found by computer, while fitting simultaneously an experimental adsorption isotherm and the corresponding isosteric heat of benzene adsorption. Before we comment on the values of the parameters found by computer, we will analyze first the behavior of our adsorption system predicted by our equations for that particular set of parameters. Figure 13 shows graphically the agreement between the experimental adsorption isotherm of benzene in ZSM-5 and the theoretical one calculated by using the parameters collected in Table 6. In Figure 14 the comparison between the experimental and theoretical heats of adsorption is presented. Apparently, the number of the parameters is large. If, however, one looks to the complicated shape of the benzene isotherm and of the related heat of adsorption curve, and realizes that we fit them simultaneously by using the same set of parameters, one can see a thin margin for an arbitrary choice of these parameters. As a matter of fact, we have performed numerous model calculations which showed, that the calculated data are very sensitive to a particular choice of these parameters. In Figure 15, in addition to the overall theoretical adsorption isotherm, also the contributions are shown from the adsorption on various adsorption sites. One can see that neither the total theoretical isotherm, nor its composite isotherms on a particular kind of sites, (in a particular configuration), show loops which could be associated with phase transitions. The sharp, but not exactly vertical, changes in the adsorption isotherm of benzene are due to rapid changes in the occupation of various adsorption sites. We called it already - the cooperative redistribution of adsorbed molecules. Our theoretical calculations confirm, thus, the long-shared feeling by many scientists, that the adsorption of aromatics in silicalite is governed by a delicate balance between the adsorbate-solid, and adsorbate-adsorbate interactions. The reason why in Figure 15 we used z(s) and s(z) to denote the occupancy of Z and S sites is following: From a purely theoretical point of view, it is impossible to judge which of the calculated solid lines means the occupancy of Z sites, or S sites alternatively. This is because of the symmetry of Eqns (5.2.10)and (5.2.11). Eqn (5.2.11) can be obtained from Eqn (5.2.10) by replacing the index s by z, and vice versa. The discrimination between the calculated contributions must be made on a rational physical basis. For us that basis are the theoretical and experimental findings reported by Mentzen et al. [69]. They argue that at the highest loadings of silicalite, the adsorbed benzene molecules form one-dimensional polymer-like structures in the straight channels. The formation of such one-dimensional polymers means full occupation of both I and S sites at the highest possible loading. Looking to Figure 15 we can deduce that such an occupation could really exist, provided that the solid line s(z) means the occupancy of sites S rather than of the Z sites. The symbol z(s) has a similar meaning. If, however, we interchange the interpretation of these theoretical isotherms, our conclusion that I and Z sites are fully covered at the highest loadings, will agree with the conclusion drawn by Snurr et al. [81, 82]. Looking to Figure 15 one can see, that at a loading of about 4 molecules/u.c, a sudden increase of the z(s) and s(z) forms takes place, and a sudden disappearance of the form il. This takes place even in the absence of the phase transition in the zeolite structure assumed so far. We can imagine, however, a situation when this sudden redistribution induces the silicalite phase transition which, in turn, promotes further this sudden rearrangement.

560

_

/"

8.0-

r

=

6.0-

ll...,

| cL

_.r

4.0-

0 .,...

0

E 2.0"

0.0

I

'

I

0.0

'

I

2.0 4.0 log pressure (Pa)

Figure 13" The comparison of the experimental isotherm of benzene adsorption in silicalite (9 9 .) measured by Chiang at 303 K, with the theoretical one, ( ~ ) , calculated from Eqns (5.2.10)-(5.2.13) by using the parameters collected in Table 6.

0

80.0

8 60.0 -~

.'/

-

o 'v /

'B t~

.

I.,

"0

Lo

/'.

v

o e-

.2 ~- 40.0 0 t~

"-

'

0.0

I

'

I

'

I

'

I

2.0 4.0 6.0 8.0 molecules per unit cell

Figure 14: The comparison of the experimental heat of adsorption of benzene molecules adsorbed in silicalite (o 9 o), reported by Thamm [61], and the theoretical ones ( ~ ) , calculated by us, using the parameters collected in Table 6.

561

8.0

6.0 a._.

(D

r

t

4.0

(D 0

"~

E

2.0

0.0

I

0

~

I

1000 2000

~

I

3 0 0 0 4000

I

5OO0 60O0

pressure (Pa)

Figure 15: The occupation by benzene molecules of various adsorption sites in silicalite, calculated from Eqns (5.2.10)-(5.2.13), by using the parameters collected in Table 6. The line rising sharply at small adsorbate pressures is the occupancy of I sites by benzene molecules being in a certain state denoted here by il, whereas the solid line rising sharply at the highest adsorbate pressures is the occupancy of I sites by benzene molecules being in a state denoted here by i2. The lines denoted by z(s) and s(z) are the occupancies of the sites Z and S, or vice versa. The black circles (. 9 o) are the experimental data measured by Chiang.

~__, 8C).0-

~o

|

9

o9

On

0

O0

9

Oo O

9

o40"0-

J

0.0

w I

"-

0.0

/

n

i

l

2.0

i

l

/

i

~

i l l l i i l l i l l

4.0 6.0 8.0 molecules per unit cell

Figure 16: The contributions Qcj'S to the total isosteric heat of adsorption of benzene in silicalite, calculated by us using the parameters collected in Table 6. These contributions are denoted in the same way as the contributions to the total adsorption isotherm from various sites, shown in Figure 15. The black circles (o 9 o) are the experimental data reported by Thamm.

562 Such a view could be supported by the values of the parameters found by computer, and collected in Table 6. One striking property observed there is the high value of the interaction (attraction) parameter w,z for benzene adsorption. This would suggest a high positive cooperativity of adsorption of molecules adsorbed on Z sites. That means, the total energy of the molecules adsorbed on Z sites grows rapidly with the number of molecules adsorbed on these sites. However, the calculations show, that adsorption on Z sites starts rapidly only when the surface loading exceeds 4 M/u.c, i.e., when the phase transition in the zeolite structure can take place. One may, therefore, assume, that this unusual positive cooperativity simulates, in fact, another factor leading to such a sudden increase of adsorption on Z sites. This may well be a sudden increase of the adsorption energy Ez, induced by the zeolite phase transition. In case of the argument that for benzene the phase change in the silicalite phase is only a hypothesis at present, we might offer another explanation following the arguments by Pan and Mersmann [85]. They believe that a strong attraction exists between two molecules on the nearest Z sites, transmitted through the solid phase. Finally, a certain compromise between the views expressed by Pan and Mersmann, and those launched by Snurr et al. [82] also seems to be possible. As well known, the behavior of theoretically predicted isosteric heats of adsorption is a much stronger test for the theory than the behaviour of theoretical adsorption isotherms. This is because the behavior of the experimental heats of adsorption is much more sensitive to the nature of an experimental adsorption system. Thus, we believe, that a special attention should be given to the agreement between theoretical and experimental heats of adsorption. Figure 16 shows the contributions to the total isosteric heat of adsorption of benzene, from the heats of adsorption Qcj generated by the adsorption on various adsorption sites.

QJ [o_h \0~)

Qcj - ~ f~,~

(5.2.35)

The additional index "c" in Qcj means that this is the "contribution" to the total heat of adsorption, from the molecules occupying sites j. The relatively less successful fit of the experimental data by the theoretical heats of adsorption is probably due to neglecting another physical factor. These are the chemical and geometrical defects in the silicalite structure. Zeolites are usually viewed as very regular crystallographic structures. The common existence of various structure defects has been known for a long time but has not received enough attention so far. Indeed, such defects should not affect strongly adsorption of small molecules in large cavities and channels. If, however, the dimensions of cavities and channels become comparable to the values of Van der Waals interaction parameters, even small changes in the zeolite local dimensions may result into considerable changes in the gas-solid interactions. This will cause the appearance of a new level of surface heterogeneity. Not only the features of the sites S, Z, I are different. The local imperfections in the silicalite structure will introduce an additional dispersion of the nonconfigurational free energies, even for the same kind of adsorption sites.

563 This was Thamm [7] himself who first emphasized that the strongly decreasing heats of aromatics adsorption at small surface coverages must be due to structure imperfections. Similar views were expressed by Talu et al. [76]. This can be seen in Figure 14. Our model assuming that all the sites of the same type have identical adsorption features is not able to reproduce w~ll the strong decrease in the initial part of the heat of adsorption curve. From the existing literature on the adsorption on heterogeneous surfaces [1], it is known, that this decrease could be reproduced by assuming a certain dispersion of Ej values on various sites j. Because of the local distortions of the zeolite structure, Ej may vary when going form one to another site of the same type j. Then, it seems reasonable to assume that these local distortions have a random nature. In such a case, the external force field acting on an adsorbed molecule, and created by its interactions with other molecules adsorbed nearby, should be a function of the average occupancy of sites of all types. Let xj(Ej) denote the differential distribution of the number of sites j having adsorption energy Ej, among the j-type sites, normalized to unity. Traditionally, the surface energetic heterogeneity was viewed as the dispersion of e-values, (adsorption potential values at the local gas-solid minima), not affecting fj. Here we must consider the dispersion of Ej rather, because the local distortions may affect also the local movement of the adsorbed molecules, i.e., the molecular partition function of these molecules. This is because the dimensions of the channels of ZSM-5 and of the adsorbed aromatics are comparable. Then, it seems natural to assume that the local distortions of the zeolite structure will result in a gaussian-like dispersion of adsorption energies for all types of sites. If we assume that the dispersion of E i is represented by the gaussian-like function (4.6.10), the expression for the average surface coverage of sites of j - th type given by the integral (4.6.2) will take the following form

o~t =

(5.2.36)

[ {( )~exp

Ozt

--"

E~ + E a,jOjt

)

/kT (5.2.37)

J

~ 1+ [exp { (Eo + ~

jkT}l

~i 1

+

[ {( A exp

) }1"

E~12 + y~ wi2jOit / k T J

(5.2.38)

564

~i2t --

l+[Aexp{(E~ wiljOjt)/kT}]'r"+[Aexp{(E~

"},i2

(5.2.39) where, to a first approximation, -yj < 1 m a y be identified with kT/cj, and E ~ is t h e most probable value of Ej. Now, let us investigate the related behavior of the heat of adsorption. To t h a t purpose, we rewrite Eqns (5.2.36)-(5.2.39) as follows

E~ + E ~,joj~ Fs=#s-#=--I kT

ln~0St _ % 1 - 0st

J kT

-lnA=0

(5.2.40)

-InA-0

(5.2.41)

E~ + E ~jej, F~=tt~-# =--i I n ~ & t _ kT

%

J

1 - 0~t

kT

Fil - #il # _ 1 In 0ilt E~ '~ Et'dilJOJt - -J kT % 1 -- Oil t -- Oi2t kT

- lnA = 0

(5.2.42)

E9 + ~ ~)i2jOjt ~2 J - In A = 0 kT

(5.2.43)

and F i ~ -- # i ~ - t t kT

0i~t _ - ~1 In % 1 - 0i~t - 0i~t

T h e molar differential heat of adsorption Qj ({0=t }) on the site of t y p e j, at a certain set of the average surface coverages {Omt }, takes then the following explicit form

Qs ({e=~}) = QO+ r

~ 0st

+ y~ ~jej~

(5.2.44)

J

0zt Qz ({emt}) -- Qz~ "+- ~'z In 1 - ezt

+ ~ ~jej,

(.5.2.45)

J

Qi,, ({Omt}) = Qi~ + (i, In

Qi2 ({o~t}) - Qi~ + ffi2In

•ilt 1 - Oilt -

Oi2t-~- E j

~i2t 1 - 0il t - 0i2 t

Wilj0jt

+ Z ~ej, J

(5.2.46)

(5.2.47)

565 where

(~.~+.~ o

Qj = k

d

kT

(5.2.48)

"J

(5.2.49)

dT1

and Cj = - k

Inserting Eqns (5.2.44)-(5.2.47) into Eqn (5.2.30) allows to evaluate the heat of adsorption as a function of the total surface coverage (loading of zeolite channels). Table 7 collects the values of the parameters found by computer when fitting simultaneously the experimental isotherm and heat of adsorption of benzene by our equations (5.2.36)-(5.2.39) and (5.2.44)-(5.2.47). Table 7 The values of the parameters found by fitting simulataneously the experimental isotherms and heats of adsorption of benzene in silicalite, measured at 303 K, by our equations (5.2.36)-(5.2.39) and (5.2.44)-(5.2.47). The Henry constant Kj is defined as exp{(E ~ + #~ Kil Pa -1 4.53.10 -2 Wsz (kJ/mol) -19.80

Q~: kJ/mol 57.5

Ki2 Ks Pa -1 Pa -1 2.95.10 -3 6.23.10 -3 wsi2 Wzi2 (kJ/mol)(kJ/mol) 24.90 -5.41

QO kJ/mol 97.0

qo kJ/mol 97.0

Kz Pa -~ 1.33.10 -2

wilil kJ/mol 3.00

Wzz kJ/mol 10.00

"Yil

~'i2

%

wsi~ kJ/mol -17.13 7z

0.99

0.86

0.93

0.90

qo kJ/mol 55.0

C~ Ci: kJ/mol kJ/mol kJ/mol -2.72 -0.60 -2.92

w~l kJ/mol -12.40

kJ/mol -2.80

Figures 17 and 18 show the comparison of the experimental adsorption isotherm and heat of adsorption with the theoretical ones, calculated by using the parameters collected in Table 7. While comparing Figures 13 and 17, one can see, that taking into account that additional level of heterogeneity due to the dispersion of Ej's values, improves the agreement between the theoretical and experimental adsorption isotherms, and especially in the region of low adsorbate pressures. Much better agreement is also observed between the experimental and theoretical heat of benzene adsorption. The rapid decrease in the heat of adsorption at small surface coverages is better reproduced, as well as the two local minima observed at the surface coverages of about 4 and 6.5 molecules/u.c. Trying to understand which is the adsorption mechanism behind that improvement, we have displayed in Figures 19 and 20 the contributions to the adsorption isotherm and heat of adsorption from the benzene molecules adsorbed on various sites and in various configurations.

566

8.0r :3

,..

6.0"

et u~ 0

4.0-

=,=,=

0

E

2.0-

0.0

TM

~

,

0.0

I

'

I

2.0 4.0 log pressure (Pa)

Figure 17" The comparison of the experimental isotherm of benzene adsorption in silicalite (e 9 .), measured by Chiang at 303 K, with the theoretical one, ( ~ ) , calculated from Eqns (5.2.36)-(5.2.39) by using the parameters collected in Table 7.

A

80.0

7o.o\ ~

6o.o I

=o

5o.0

.=_

40.0"~' 0.0

. - . - ..... -,,~../

'

-'I -"I"~-

, , ', 2.0 4.0 6.0 8.0 molecules per unit cell ,

,

,

,'

Figure 18: The comparison of the experimental heat of adsorption of benzene molecules adsorbed in silicalite (eoe), reported by Thamm, and the theoretical ones ( ~ ) , calculated by us, using the parameters collected in Table 7.

567

8.0

9

8.0

f" ~

o

~

9 9

2o! 0.0-

111~0

0

2OOO 3OOO pressure (Pa)

Figure 19: The occupation by benzene molecules of various adsorption sites in silicalite, calculated from Eqns (5.2.36)-(5.2.39), by using the parameters collected in Table 7. The solid line rising sharply at small adsorbate pressures is the occupancy of I sites by the benzene molecules being in the state il, whereas the solid line rising sharply at the highest adsorbate pressures is the occupancy of I sites by the benzene molecules being in the state i2. The solid fines denoted by z(s) and s(z) are the occupancies of the sites Z and S, or vice versa. The black circles (. 9 e) are the experimental data measured by Chiang.

..•

J "1

IPdP ee

~n

',d

oo

qb'%~e

9

9 o

9 ,

9

g 4o.o ~

"-~

0.0

0.0

2.0

4.0 6.0 8.0 molecules per unit cell

Figure 20: The contributions Qcj to the total isosteric heat of adsorption of benzene in silicalite, calculated by us using the parameters collected in Table 7. The black circles (o 9 9 are the experimental data reported by Thamm.

568 Looking at Figure 19 one can see that at the loadings smaller than 4 molecules/u.c., benzene molecules fill mainly the channel intersections. At the coverages above 4 molecules/u.c, we observe a sharp increase of adsorption in the sinusoidal channels, accompanied by a decrease and reorientation of molecules occupying the channel intersections. At still higher loadings (above 6.5 molecules/u.c.) a second redistribution of adsorbed molecules takes place. At the maximum loading of about 8 molecules/u.c, all straight channels and channel intersections are filled by benzene molecules. These redistributions of adsorbed molecules are responsible for the two steps observed on the adsorption isotherm of benzene. From the comparison of Figures 15 and 19, it follows, that both our models predict the same location of adsorbed molecules at low and at the highest adsorbate loadings. The difference occurs at the loadings between 4 and 8 molecules/u.c.. The model accounting for the energetic heterogeneity of the adsorption sites of the same type predicts, that in this coverage region all zig-zag ( or stright) channels and most of the channel intersections are occupied, whereas the model neglecting this additional level of heterogeneity predicts the occupation of zig-zag and straight channels (see Figure 15). From Figure 20, one can deduce that the contributions to the total isostric heat of adsorption coming from the forms Z, S and I in the orientation denoted by i2 decrease at low loadings. Such a decrease of the heat of adsorption curve is usually attributed to the adsorption on an energetically heterogeneous surface. The heterogeneity parameters (kT/cj) for these forms listed in Table 7 are less than unity and the temperature derivatives r defined in Eqn (5.2.49) are equal to -cj, indicating thus that c5 are temperature independent. So, this is the dispersion of the free energy of adsorption of benzene molecules on these sites that is responsible for the rapid decrease of the heat of adsorption at very low loadings. On the other hand, the heterogeneity parameter obtained for the form il is very close to unity and the derivative (il is not equal to -cil. This would indicate that the adsorption of benzene molecules in the channel intersections at low loadings, i.e. in the state of benzene molecules denoted here by il, is not sensitive to imperfections in the zeolite structure.

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[82] R.Q. Snurr, A.T. Bell, D.N. Theodorou, J. Phys. Chem., 98 (1994) 5111. [83] H. Stach, R. Wendt, K. Fiedler, B. Grauert, J. Janchen, H. Spindler, in Characterization of Porous Solids (K.K. Unger et al., Eds.) Elsevier, Amsterdam, 1988, p. 109. [84] P.T. Reischman, K.D. Schmitt, D.H. Olson, J. Phys. Chem., 92 (1988) 5165. [85] D. Pan, A. Mersmann, in Characterization of Porous Solids II (F. Rodriguez-Reinoso et al., Eds.) Elsevier, Amsterdam, 1991, p. 519. [86] C.K. Lee, A.S.T. Chiang, F.Y. Wu, AIChE J., 38 (1992) 128. [87] E.M. Flanigen, J.M. Bennett, R.W. Grose, J.P. Cohen, R.L. Patten, R.M. Kirchner, J.V. Smith, Nature, 271 (1978) 512. [88] W.M. Meier, D.H. Olson, Atlas of Zeolite Structure, Juris Druck und Verlag, Zurich, 1978. [89] H. van Koningsveld, F. Tuinstra, H. van Bekkum, J.C. Jansen, Acta Cryst., B45 (1989) 423. [90] C.A. Fyfe, Y. Feng, H. Grondey, G.T. Kokotailo, J. Chem. Soc., Chem. Commun., (1990) 1224.

W. Rudzifi.ski,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.

573

Static a n d d y n a m i c s t u d i e s of the energetic surface h e t e r o g e n e i t y of clay m i n e r a l s F. Villi4ras, L. J. Michot, J. M. Cases, I. Berend, F. Bardot, M. Francois, G. G4rard and J. Yvon Laboratoire Environnement et Min6ralurgie. ENSG et URA 235 du CNRS. BP 40.54 501 Vandoeuvre les Nancy cedex. France

INTRODUCTION Clays are important constituents of soils. They are finely divided materials: the size of elementary crystallites does not exceed some micrometers. Because of these low dimensions, clay minerals develop large specific surface areas. This is why, clays generally carry, together with iron and aluminium oxy-hydroxides, the largest part of the specific surface area of soils and, as a consequence, they largely control the reactivity of the solid soil fraction. Furthemore, because of their high specific surface areas, clay minerals are reservoirs of water in soils as, in contrary to water trapped in the soil porosity, water adsorbed at the surface of clays is slowly released from the surface. This keeps acceptable moisture levels for plants in hot and dry conditions. Clay minerals feature variable exchange capacity for organic and inorganic cations. These properties play an important role in heavy metal cations abstraction. The exchange capacity can be limited to external surfaces for minerals such as kaolinite and illite. It can be also extend to the interlamellar space when interlamellar cations can be hydrated; it is the case of swelling clay minerals such as montmorillonite. Finally, other mineral species such as sepiolite and palygorskite present large adsorption capacities because of their structural microporosity. These natural properties of clay minerals render them attractive for industrial applications. They have long been used as fillers in paper, paint, polyphased materials, cosmetic industries... Clays are also used to control waste deposits. They play an important role in the design of geochemical barriers: clay formations ensure low permeabilities preventing pollutants migration and can trap heavy metals as well. On the contrary, the presence of swelling clays can be very detrimental in civil engineering applications as their swelling behavior can affect the stability of buildings. Swelling and microporous clays are also used in the pharmaceutical industry where their water retention properties make them suitable as antacids. Microporous clay species are also used for their retention properties, for instance in cat litter, or as catalytic supports.

574 Therefore, it is extremely important to carefully study the surface properties of clays in order to understand the different mechanisms involved in natural conditions (hydration, pollutants adsorption and release...) or to improve the surface properties of the minerals used in industrial processes. As it is the case for all crystalline materials, structural defects can be observed at the surface of clay minerals such as dislocations and steps. Because of their natural origin, clay minerals are also chemically very heterogeneous solids. The cristallochemical composition can feature local variations inside a same particle. It is then necessary to study precisely the nature of such variations and their effect on hydration, dehydration and swelling properties as well as to determine the energetic heterogeneity of surfaces in order to obtain reliable informations about textural properties of clay minerals. For the reasons mentioned above, the present review will start with a summary of clay minerals cristallochemical properties in order to provide the reader with the major keys governing their surface heterogeneity. Then, specific techniques developed in our laboratory for analyzing surface heterogeneity of clay minerals will be described. Finally, different examples will be examined including non swelling minerals such as kaolinite, muscovite and talc, fibrous and microporous minerals such as sepiolite and palygorskite. Finally, the particular case of water adsorption on minerals such as montmorillonite will be discussed. In this case, the mecanism of adsorption is driven by the access of water inside the interlamellar space; then these minerals swell in presence of water yielding structural heterogeneities depending on the nature of the interlamellar cations.

1. ORIGIN OF SURFACE HETEROGENEITY OF CLAY MINERALS Clay minerals are hydrous layered silicates and belong to the phyllosilicates family. Elementary crystallites are generally platy and sometime fibrous. This specific habitus is due to the crystallochemical properties of clay minerals (comprehensive reviews on crystal chemistry of phyllosilicates can be found elsewhere, for instance in ref. I and 2). 1.1 Structure of phyllosilicates Phylosilicates are made of the arrangement of successive sheets of octahedra and tetrahedra which are named tetrahedral and octahedral sheets. The composition of the tetrahedral sheet is generally relatively simple. The basic pattern is the SiO4 4- tetrahedra. Interconnection of silicium by oxygens defines the tetrahedral network. In the case of silica, this network is tridimensional as every oxygen anion is connected to two silicium cations. In the case of clay minerals, the network is two dimensional as only three of the four corner (basal oxygens) of one tetrahedron are linked to other tetrahedra while all the free corners (apical oxygens) are pointing in the same direction perpendicular to the tetrahedral sheet. A plan view of the tetrahedral network shows that tetrahedra connections feature hexagonal rings (Figure 1). Silicium cations can be

575

_

_

_

y

X ~,,

l

Figure 1" Idealized plan view of the tetrahedral sheet of phyllosilicates. The filled circles represents the silicium cations located at the center of each tetrahedron. substituted by trivalent cations. The most important one is A13+ as its size is very close to that of Si4+. The charge difference between A13+ and Si4+ generates, in the tetrahedral sheet, a localized charge defect which must be compensated by other cations. The substitution level for four tetrahedra ranges between 0 and 2. Small quantities of other cations such as Fe 3+ can be found in substitution for silicium. The apical corners form part of the adjacent octahedral sheet. Individual octahedra are linked together by sharing octahedral edges and to the tetrahedral sheet by sharing apical oxygens and unshared hydroxyls. These hydroxyls lies at the center of the tetrahedral hexagonal ring and can be substituted by fluor. The octahedral cations are mainly Fe 2+, Fe 3+, Mg 2§ and A13+. Other elements such as Li, Ti, V, Cr, Mn, Co, Ni, Cu and Zn can occur in some species. If all octahedra contain a cation at their center, the sheet is said to be trioctahedral. If two centers are occupied and one is vacant, the sheet is said to be dioctahedral. In trioctahedral minerals, OH groups vibrates in a direction a p p r o x i m a t e l y perpendiculary to the sheet whereas, in dioctahedral minerals, the OH is tilted in the direction of vacant octahedral site [3-4]. The organization of octahedral and tetrahedral sheets defines different types of layers (Figure 2). The classification of phyllosilicates including clay minerals and micas is presented table 1. 1:1 layered silicates are made of one octahedral sheet and one tetrahedral sheet. In this structure, the unshared plan of the octahedral sheet consists of hydroxyls unstead of oxygens. 2:1 layers links one octahedral sheet between two tetrahedral ones. If, because of cation substitutions, the layer is not electrostatically neutral, the excess layer charge is neutralized by various interlayer materials such as individual or hydrated cations or hydroxide octahedral sheets. Smectites and vermiculites are swelling minerals. Sepiolite and palygorskite are 2:1 minerals made of an arrangement of small 2:1 units generating small structural micropores.

576

e, ~ : ~ : : : : : : . :

.....

I Serpentine kaolin

Talc Pyrophyllite

Mica and Brittle Mica

d=7.1 -7.3A

d = 9.1 - 9.4 A

d = 9.6- 10.6 A

0

Oxygen

@ hydroxyl water molecule ~t Smectite Vermiculite d = 14.4- 15.6 A

Interlayer cation

Chlorite d = 14.0- 14.4 A

Figure 2: View of major non modullated hydrous phyllosilicates groups

1.2 Textural and energetic superficial dependence From the above description, the reader can immediately perceive the heterogeneous nature of clay minerals. In addition to natural substitutions and defects, the different faces of clay minerals have different surface composition and therefore, different energetic properties. For instance, kaolinite with its 1:1 organization presents two distinct basal compositions. The tetrahedral surface is made of oxygens while the octahedral surface is made of hydroxyls. Lateral surfaces are made of A1-OH and Si-OH groups. In the case of 2:1 minerals, both basal surfaces have the same composition. However, talc and pyrophyllite which are not substituted present oxygen surfaces while smectites, vermiculites and micas present in addition to oxygens, monovalent or divalent cations at their basal surfaces in order to neutralize the charge defect of the 2:1 layer. Chlorite is a special case as the interlayer is made of an octahedral hydroxide layer. Then, as for 1:1 minerals, the 2:1 surface is made of oxygens while the other, the hydroxide surface, is made of hydroxyls. In fact, energetic distribution is more complicated as basal and lateral surfaces have their own energy distribution because of crystal defects, distribution of compensating cations, heterogeneity of cationic substitution and charge defects

577 Table 1. : Classification of phyllosilicates (From ref. [2]). Only few examples are given, x refers to an O10(OH)2 formula unit for smectite, vermiculite, mica and brittle mica. (t) : trioctahedral, (d): dioctahedral. group and exemples Layer interlayer charge (x) per sub-groups formula unit type Chrysotile, antigorite AntigoriteSerpentines (t) empty !I kaolin Kaolinite, dickite 1:1 Kaolins (d) or H20 ,x=0 Talc, wiliemseite empty Talcs (t) i TalcPyrophyllite pyrophyllite Pyrophyllites (d) i x=0 Saponite, hectorite Trioctahedral hydrated ! Smectite Beidellite, Dioctahedral exchangeable x = 0,2 - 0,6 montmorillonite cations Di and trioctahedral 2:1 hydrated Vermiculite Di and trioctahedral exchangeable vermiculites vermiculites cations Phlogopite, Biotite anhydrous Micas s.s. Trioctahedral micas Muscovite, illite cations x = 0,5 - 1,0 Dioctahedral micas Clinotite Brittle micas anhydrous Trioctahedral micas Margarite x = 2,0 Dioctahedral micas cations Hydroxide Chlorites Trioctahedral chlorites Clinochlore,chamoisite (2:1:1) layer x variable Dioctahedral chlorites Donbassite Di-trioctahedral Cookeite, sudoite chlorites 2:1 Sepiolite, loughlinite SepioliteSepiolites inverted Palygorskite H20 palygorskite Palygorskite ribbons .

.

.

.

.

i

r e p a r t i t i o n . These h e t e r o g e n e i t i e s result f r o m the c o m p l e x m e c h a n i s m s of crystallization and e v o l u t i o n of clay minerals. The distribution and location of heterogeneities are c o m m o n l y studied using different spectroscopic m e t h o d s such as X ray diffraction and absorption, UV-visible-infrared spectroscopy, solid state NMR ... [5-16]. The heterogeneity of clay minerals can also be studied by using the so-called "molecular probes" m e t h o d s based on the detailed study of the adsorption of one or more adsorbate at the surface. In that case the surface of clay minerals m u s t be considered as being formed of patchwise-like domains (basal and lateral surfaces), each d o m a i n bearing a r a n d o m energetic distribution of adsorption sites: i--n e t = ~ xi.eit i=l

(1)

with

eit = J ei (e).%i (e).de f~

(2)

578 where 0t is the total adsorption isotherm, 0it the adsorption isotherms on the different faces of the mineral, Xi is its contribution to 0t, 0i(a) a "local" theoretical adsorption isotherm and Xi(a) the dispersion of the adsorption energy a on the i th domain.

2. EXPERIMENTAL M E T H O D S FOR HETEROGENEITY OF PHYLLOSILICATES

ANALYZING

THE

SURFACE

Clay minerals are naturally heterogenous materials as they are generally associated to other mineral species. Then, separation processes must be used in order to get products as pure as possible. For instance, in the case of swelling materials, the swelling properties, and then the water retention properties, depend on the size and charge of the interlayer exchangeable cation [2]. In order to control the swelling behavior of these minerals, it is necessary to understand the mechanisms of water sorption and swelling for homoionic minerals. In order to obtain a good description of the geometric properties of clay minerals, a set of suitable experimental methods must be used. If the nature of the different species are generally well known, their respective extension must be determined accurately. Such results can be obtained by coupling different complementary experimental techniques with modern modelisation methods. In this chapter a short description of the experimental techniques and modelisation methods employed in our laboratory will be given. Most of the experimental techniques described here have been developed by Prof. Rouquerol, Grillet and Davy in the Centre de Thermochimie et de Microcalorim6trie (Marseille, France). They were further adapted for studying the energetic and geometric heterogeneities of clay minerals. A special X-ray diffraction device has also been designed with the collaboration of Mr. Lhote and Uriot (Centre de Recherche en P6trographie et G6ochimie, Vandoeuvre les Nancy, France) in order to study the structural heterogeneity induced during the swelling of smectites. 2.1 Constant Rate Thermal Analysis Before any adsorption study, the reference state of the surface must be properly characterized. This means that the outgassing conditions, i.e. the v a c u u m - t e m p e r a t u r e couple, must be defined precisely, especially for microporous and swelling minerals. These fundamental informations are obtained by using the "Constant Rate Thermal Analysis" (CRTA) also called reciprocal evolved-gas-detection thermal analysis [17-20]. In conventional thermal analysis, a physical property of a substance is measured as a function of temperature that is controlled by a constant heating rate. For clay minerals, this procedure usaUy results in a partial overlap of successive dehydration steps. This problem is overcome by the use of CRTA. In this method, the heating rate of the sample is controlled by the sample as the rate of desorbed gas is kept constant (or controlled) over the entire temperature range of the experiment. This system operates in the reverse way of conventional thermal analysis because the

579 measured temperatures are dependent of the properties of the sample. The rate of desorption, or gas flow, is controlled by keeping constant (at a chosen value) the pressure drop through a restriction leading to the pump (Figure 3). This rate can be controlled at any value low enough to ensure a satisfactory elimination of the t e m p e r a t u r e and pressure gradients within the sample. When a family of molecules desorbs from the surface, the system remains at constant temperature until the desorption is finished. Then, temperature increases again until another desorption or decomposition is found. For a constant rate of water vapor loss, the temperature vs time data may be immediately converted into temperature vs mass loss data if the total weight loss is measured at the end of the experiment. R o u q u e r o l and coworkers have extensively d i s c u s s e d the multiple advantages of this method : the technique allows to detect the desorption of very small quantities; the determination of equilibrium temperature is more accurate than in conventional thermal analysis. Furthermore, if needed, the experiments can be directly carried out with the sample cell used for gas adsorption experiments. The most recent apparatus are now e q u i p p e d with a mass spectrometer which yields information about the nature of the desorbed gases. A typical curve is displayed figure 4. It was obtained for a microporous mineral, sepiolite, under a residual pressure of 4 Pa [21]. This curve shows four dehydration and dehydroxylation steps. The first one, region I, occurs at 25~ and corresponds to the desorption of zeolitic water adsorbed in the micropores. Regions II and III correspond to the expulsion in two steps (154 and 428~ of water bound to Mg atoms on the edge of the sheet. The last region IV is due to the dehydroxylation of the mineral which occurs at 725~

I. . . . . . . .

Mass Spectrometer

--

Pressure Gauge

]

P

,=

Leak ,

(~

,

Vacuum

Furnace .

.

.

Figure 3: Schematic representation of CRTA

.

.

.

.

.

.

=

constant

580

_

r~

.:~"

10-

15

I ,-u

'

t,,}/,,~

0

II I '

III ! '

200

'

'

IIV ,

400

,

,

,,

~ 1 ,

,

600

'

800

I

1000

Temperature ~ Figure 4: Controlled rate thermal anlysis of sepiolite [21]. In the case of Wyoming sodium montmorillonite CRTA experiment has shown that after dehydration at 25~ under a residual pressure of 4 Pa, the water content is 0.97 molecule per exchangeable cation (Na +) while at 100~ there is still 0.87 H20 per Na [22].

2.2 Methods based on the evaluation of energetic heterogeneity In order to get information about microporosity distribution of microporous minerals and lateral to basal surface areas ratios of platy minerals, methods taking into consideration the energetic differences of these surfaces are used. The first one is the low temperature adsorption microcalorimetry (LTAM) that allows to measure the isosteric heats of adsorption as a function of surface coverage. The second one is the low pressure quasi equilibrium volumetry (LPQEV) that allows to m e a s u r e continuously and precisely a d s o r p t i o n isotherms in the submonolayer region.

2.2.1 Low temperature adsorption microcalorimetry In this method, adsorption isotherms are performed in a microcalorimeter [23]. It associates quasi equilibrium adsorption volumetry [24] and isothermal low temperature microcalorimetry. The heat flow and quasi equilibrium pressure are then recorded simultaneously as a function of the amount of gas introduced into the system. Adsorption isotherm and the derivative enthalpy of adsorption versus adsorbed quantities are easily derived. Low temperature adsorption experiments are performed at CTM (Marseille) in collaboration with Y. Gillet. A typical curve obtained by this method is shown figure 5 in the case of argon adsorption on a kaolirute [25]. The derivative heat of adsorption decreases rapidly

581 15

14113 -12

1 ~0

,

0

,

,

I

0.2

,

,

,

I

0.4

,

0.6

0.8

1

1.2

1.4

Surface Coverage Figure 5: Isosteric heat of adsorption vs surface coverage obtained by Ar LTAM on FU7 kaolinite [25]. in the low coverage region and then remains nearly constant up to the monolayer region. The first part was assigned to lateral faces, which are very heterogeneous and the most energetic faces, and the constant part to basal faces of the mineral, which are more homogeneous and less energetic. In the case of microporous samples, this method is preferred to conventional calculations derived from experiments performed at different temperatures as the structure of the gas adsorbed in the micropores might be different at different temperatures.

2.2.2 Low pressure quasi equilibrium volumetry It was showed by Michot et al. [26-27] that using low pressure quasi equilibrium volumetry proposed by Grillet et al. [28] and Rouquerol et al. [24] the resolution of adsorption isotherms can be enhanced in the low relative pressure domain, i. e. when the first layer of gas is adsorbed. Then, this method equipped with pressure sensors that work at low pressures allow to study in satisfactory conditions the surface heterogeneity of solids.

2.2.2.1 Experimental procedure The experimental procedure has been discussed by Rouquerol et al. [24] and Michot et al. [26-27]. The sheme of the most recent apparatus of our laboratory [29] is presented figure 6. A slow, constant and continuous flow of the adsorbate is introduced into the adsorption system through a microleak. The flowrate is constant, at least up to the BET domain, and can be adjusted by the pressure imposed before the leak. If the introduction rate is low enough, the measured pressures can be considered as quasi equilibrium pressures. Then, from the

582 recording of the quasi equilibrium pressure (in the range of 10-3, 3.104 Pa)as a function of time, the adsorption isotherm is derived. The quasi equilibrium state can be tested by comparing adsorption isotherms performed at different flow rates with a same sample. Two high accuracy Datametric differential pressure gauges are used for pressure measurements: 1) 0-1.3x102 Pa and 2) 0-1.3x105 Pa (590 type Barocel Pressure Sensors). For each pressure gauge, four ranges of reading: xl, xl0, xl00, xl000 are available, providing 0-10 volts signals. The minimal sensitivities are 1.3x10 -3 and 1.3x10 -1 Pa for gauges 1) and 2), respectively. Pressure accuracy is 0.05% P + 0.01% of reading range. A dynamic vacuum of 10-7 Pa is ensured on the reference side by the use of a turbomolecular vacuum pump. An accurate constant level of liquid nitrogen is ensured using a home made electronic controlled device [29-31]. The frequency of pressure recording is adjusted after each measurement to ensure between 100 and 200 experimental values per unit of ln(P/Po). Thus, 2000 to 3000 experimental points are collected for relative pressures lower than 0.15. The potentiality of this experimental device is illustrated in figure 7 in the case of an activated carbon. The expansion of the isotherm using a logarithmic scale reveals inflection points. Due to the large quantity of experimental data points, the derivative of the adsorbed quantity as a function of the logarithm of relative pressure can be calculated accurately. As shown figure 7b, the derivative of the adsorption isotherm is much more sensitive as it features different peaks that can be simulated using theoretical derivative adsorption isotherms. Then, using a quasi equilibrium procedure, one can consider the adsorbate as a probe scanning directely the full spectrum of energetic heterogeneity of the surface.

r ...... ! ! ! 9

'c~ ,..om~,uLer '~

! ! ! I

By pass Vacuum control gauge PrimaR vacuum pump

]

~

Gas input Microleak

Turbomolecular vacuum pump

Sample Figure 6: Schematic representation of the quasi-equilibrium gas adsorption apparatus.

583 100

500

a)

b)

375

'~' 75

~t~ 250

-' 50

~

125

0 -16

i

-12

-8 -4 ln(P/Po)

I

0

25

0 -16

|

-12

-8 -4 ln(P/Po)

|

0

Figure 7: Argon a d s o r p t i o n on CX activated carbon (CECA). a) isotherm vs ln(P/Po), b) derivative of adsorption vs ln(P/Po).

2.2.2.2 the Derivative Isotherm Simulation method The example presented Figure 7 shows that different adsorption domains are present on the surface of the solid. In the case of non microporous clay minerals, at least two d o m a i n s corresponding to basal and lateral surfaces should be observed. The total adsorption isotherm is then defined as the sum of a limited n u m b e r of a d s o r p t i o n isotherms (eq. 1). The next step is to determine the different 0i i s o t h e r m s and their contribution to the total isotherm. These isotherms are different if adsorption occurs in micropores or on external surface areas; they have to take into account the energetic distribution of each domain (eq. 2). In the first method proposed to simulate experimental derivative isotherms, each domain, i.e. lateral and basal surfaces, was considered homogeneous [29]. The local isotherms 0i are the Langmuir/Temkin-Bragg-Williams and BET/Hill adsorption isotherms on h o m o g e n e o u s surfaces for one layer and multilayer adsorption respectively. The derivatives of L a n g m u i r / T e m k i n - B r a g g - W i l l i a m s are s h o w n figure 8 for different values of lateral interactions. Analytical expressions of the derivatives are:

e=

C'eae'P/P~ 1+ C.e ae.P / Po

de = e.(1- e) din(P/Po) 1- a@.(1- e)

(3)

(4)

584 '

'

'

'

'

I

' '

' ' '

I '

' ' '

l

1

'

'

'

'

'

I '

'

'

'

'

I

t

co=0kT co = 1.5 kT co= 3kT o

m

0.8

0.6

0.4

0.2

0

t

i

-15

i

I

|

,

-12

-9

~

,

-6

"" ~'f --I

-3

=

I

0

In(PIPe) Figure 8: Theoretical derivative of Temkin adsorption isotherms vs l n ( P / P o ) w i t h C=1000 and various values of co calculated from eq. 4 (co = 0: L a n g m u i r isotherm).

d2e = e.(1-e).(1-2e) [dln(P / Po)] 2 [1- ae.(1- e)] 3

(5)

w h e r e P is the m e a s u r e d pressure, Po the condensation pressure of the bulk phase, C is the energetic constant describing the n o r m a l adsorbent-adsorbate interaction and a a constant describing the adsorbate-adsorbate interaction : a = c0/kT (6) w h e r e coo is the average force field exerted on one adsorbed molecule interacting with nearest neighbor adsorbed molecules at surface coverage 0, k the Boltzmann constant and T the absolute temperature. The condition of a m a x i m u m is derived from equation 5: 0*=0.5. Then, from equation 4, one finds:

dO 1

dln(P / Po) p=p.

=

1

(7)

4- a

and from equation 3: P~

c = p"~

-a/2

(8)

585 The same method can be applied in the case of multilayer adsorption (BET isotherm) taking into account the possibility of lateral interactions (extension of Hill [30]) :

0[1 -

C'eae'p / Po P / Po ].[1 + (C.e ae - 1).P / Po]

(9)

In this case, the derivatives are more complicated and computed methods can be used to determine C and a from the position of a maximum [29]. In this method, lateral interactions are introduced as best fit parameters : their role is to adjust the shape of the theoretical derivative to the experimental one. The simulation are performed without any automatic calculation as the operator chooses directely the adsorption models (one layer isotherms for space limited adsorption and multilayer isotherms for external surface areas) and adjusts the position of the maxima and the intensity of lateral interactions by try and errors until the total simulated derivative isotherm matches the experimental one. Then, the derivative isosteric heat of adsorption can be calculated using the energetic parameters and the monolayer capacity of each domain. The comparison of the calculated curve with experimental ones measured by LTAM shows that the agreement is not perfect but that the general features are conserved [29]. The observed discrepancies could be attributed to the fact that patches are not homogeneous and a random (or patchwise) distribution of energy must be taken into account on basal and lateral surfaces. Experimental results show that lateral interactions are rarely nil and values ranging between 1 and 3 k T , and 0 and 3kT are often found for argon and nitrogen, respectively. Recent tests show that these lateral parameters can be used as estimates of the energetic heterogeneity on the different domains. Indeed, there is a close relationship between lateral interaction and the width of a gaussian distribution of normal adsorption energies: O(P / P0)= yOl(C,c0i,P / P0).Gauss(ln(Ci),Ai / kT).dln(C) f~

(10)

where In(Ci) is the first order m o m e n t u m and Ai/kT the standard deviation of the gaussian distribution. Computer simulations demonstrate that the isotherm defined eq. 10 is an isotherm which can be modelled by the same equation as for Oi but with Cf> Ci and a)f < c0i and this for monolayer or multilayer isotherms and for patchwise or random distribution (Figure 9). Figure 10 gives the relationship between r and Ai/kT for roi = 2 and for random and patchwise distributions. This property suggests that, in the DIS method, lateral interactions with negative intensities can be used and correspond to very heterogeneous surfaces. Another consequence is that the higher c0i, the more homogeneous the surface.

586 0.6 Patch

a ) 0.5

_

~kT=l

~

Rand~kT=l

- ........

~/kT

b)

= 0

~ 9 , ,

, , ,

, i

~0.4

~

,

0

_

=0.3

ii

0.2

/ I

I I

,'/:

0.1

0.0

,/; ,

i

]

I

i

i

[

,

,

":',k, / ,

,

i

-4 0 ln(P/Po) In(P/Po) Figure 9: Calculated theoretical derivative isotherms using eq. 10 for patchwise and random energy distributions, a) adsorption limited to one layer; b) rnultilayer adsorption. -13

2.C-

-9

9

-13

-4

-9

0 9

0

0

9

9

Patch

o

Random

0 o 9

0

1 . 5 -

0 0

1.0-

0 0

0.5O O

0.s 0.0

,

,

,,

,

I

0.5

s

,,

~

,

I

,

1.0

,

,

I

1.5

I

,

I

|

|

2.0

Ai / k T Figure 10: relationship between (of and Ai/kT for patchwise and random distribution. The calculated values are the same for multilayer and monolayer initial isotherms.

587 However, the local isotherms discussed above correspond to symmetrical distributions of the normal interactions. In some cases, especially for microporous solids, non symmetrical distributions are needed to obtain a satisfactory representation of the experimental derivative isotherms. In a recent improvement, the generalization of the Dubinin-Asthakov isotherm was used [33]. This isotherm which can be derived from the Rudzinski-Jagiello approach [34-35] writes:

I i/ ll

_ kTln 0it(P / P0) = e

(11)

where Ei is the variance of Zi(e)and ri a parameter governing the shape of the distribution function. It is a gaussian like function widened on the low energy side for ri3. Its features are presented figure 11. For ri=2, equation 11 becomes the Dubinin-Raduskhevich isotherm. In practice, in application of the Dubinin-Asthakov equation, Pi 0 is commonly identified with saturated vapour pressure P0- If this approximation may be justified in some cases, Pi 0 can generaly not be identified with P0 and must be treated as a best fit parameter which represents the pressure at which the largest micropores of the family are filled. The mathematical relations between El, ri and Pi0 can be easily derived from the expression of the first and second derivatives of 0.25

"~'~~, 0.15

I,'"

--

\',,

0.1

"~ 0.05

//

ll _

/

-

~

4

J-

/'

I

0 -15

-12

-9

-6 ln(P/Po)

-3

0

Figure 11: Derivatives of Dubinin-Asthakov isotherms using Pi 0 = P0, Ei = 5 kT, co - 0 and different values of ri.

588 equation 11. In practice, ri and Pi 0 are fixed and from the position of a maximum, the adsorption capacity and Ei are calculated, ri and Pi 0 are adjusted until the simulated curve matches the experimental curve. A multilayer extension of Equation 11 has been also proposed in order to simulate peaks which correspond to adsorption on external surfaces [33]. In conclusion, the concept of derivative of adsorption can be used to get quick and reliable results on energetic heterogeneity which can then be used to derive geometric properties of phyllosilicates and other finely divided solids [7, 29-30, 33, 33-40].

2.3 Water vapor adsorption In the case of clay minerals, water adsorption is particularly important. Water vapor adsorption is studied using quasi equilibrium gravimetry of adsorption and i m m e r s i o n microcalorimetry measurements. In the case of swelling minerals, additional informations are obtained by using a special X Ray Diffraction apparatus which allows to measure the size increase of the interlayer upon water adsorption.

2.3.1 Quasi equilibrium adsorption/desorption gravimetry As water vapor is not an ideal gas, its adsorption cannot be studied by the classical volumetric techniques. It is therefore necessary to measure directly the adsorbed quantities by the use of gravimetric methods. The experimental apparatus is based on that described by Rouquerol and Davy [41] and was presented by Poirier et al. [42] (Figure 12). It is built around a Setaram MTB 10-8 symmetrical microbalance sensitive down to the ~tg. Pressures are measured with a Pirani gauge for the 0.01-10 Pa range and a Texas fused silica Bourdon tube automatic gauge for the 1-65 000 Pa range. The adsorption isotherms (i.e mass of H 2 0 adsorbed at 303K vs quasi equilibrium pressure) are recorded on an X-Y recorder connected to the mass signal (Y-axis) and to the pressure signal (X-axis). Prior to each experiment, the samples are outgassed in situ for 16h under a residual pressure of 0.1Pa. Water vapor is supplied to the adsorption cell from a source kept at 41~ at a slow flow rate through a Granville-Phillips leak valve to ensure quasi equilibrium conditions at all time. The vapor supply is immersed in a air thermostat in which the balance and the pressure gauges are encased. The sample cell is immersed in a liquid thermostat and the temperature difference between the air thermostat and the liquid thermostat is always higher than 10~ in order to ensure that the sample is the "cold point" of the system. Desorption isotherms are obtained by connecting the balance and the vacuum pumps to the leak valve. Adsorption-desorption isotherms are obtained for 0.005 < P / P o < 0.98. The sensitivity is limited to samples with total surface areas higher than 0.5 m 2. The vapor flow rate can be adjusted during the experiment by changing the setting of the leak valve.

589

Vacuum ~ (outgassing) Pressure Gauge

1~11 j ///AN~ ,,,

Balance I

~

Vacuum (desorption) Watervapor supply

Reference Analyzed sample sample Figure 12: S c h e m a t i c representation adsorption/desorption apparatus.

of

the

water

vapor

2.3.2 Water immersion microcalorimetry The adsorption of water onto a solid can be studied on an energetical point of view by measuring the heat signal obtained when a solid is immersed in water. Hydration can be studied step by step by immersing solids previously covered with a known quantity of adsorbate. Depending on the relative pressure of precoverage, water will adsorb on the surface according to the energy distribution of surface sites. Thus, the heat of immersion will decrease when precoverage pressure increases as most energetic sites are first screened by adsorption (Figure 13). For a given value of precoverage relative pressure which is generally greater than 0.75, the organization of the outermost adsorbed layers is not influenced anymore by the surface energy of the solid. Then, these layers have the same organization as bulk liquid water and the measured enthalpies of immersion are constant. This value is proportional to the surface area of the solid as shown by equation 12: OTIv. Qw = S (?lv - T-~--). Cos 0

(12)

where Qw is the measured heat of immersion at the asymptote, S the specific surface area of the sample, Tlv the liquid-vapour surface tension of water and 0 the contact angle at the solid-liquid-vapour interface. This method was first developed by Harkins and Jura [44]. In the case of hydrophilic solids (cos 0 = 1), the specific surface a r e a s a r e easily determined as Qw is measured and ?Iv and 0?lv/OT known. The determined surfaces are called absolute surfaces as the method does not require any assumption on the cross sectional area of the adsorbate molecule. In the case of non porous solids, the obtained values are close to the surfaces determined by gas adsorption. In the case of micro and mesoporous solids, the specific surface areas derived from

590 500

E

400

o

o~

E E

300

om

o

200 J

r,z,,1

A

100

. . . .

~

I

I

~

~

~

~

I

2

t

~

~

~

I

3

I

A

I

I

J

4

I

!

i

|

5

Surface coverage

Figure 13: Enthalpy of immersion of a kaolinite as a function of the statistical number of preadsorbed water layers [43]. immersion experiments are the external surface areas as the porosity is already filled at these relative pressures. Using these concepts, it was possible to show that the surface field of most solids does not influence the structure of water further than three layers i.e., roughly 10A [43, 44-46]. In practice, a glass bulb (with brittle end) containing a known amount of solid is outgassed and put under a certain pressure of adsorbate [47]. The pressure is controlled by using a thermostated bath. After equilibrium, the bulb is sealed and introduced into the experimental cell (half-filled with the immersion liquid) of a differential scanning microcalorimeter. The brittle end is then broken by depressing a glass rod to which the bulb is attached, the liquid adsorbate enters the cell and wets the solid. The resulting heat flow is then recorded as a function of time. The integration of this curve is proportional to the total heat exchanged. The standardization is carried out by joule effect and the parasite effects (breaking of the brittle end, heat of vaporization) are determined by immersing empty bulbs. Some difficulties are encountered if the solid is partly soluble in the immersion liquid. In order to avoid thermal effects or surface modifications due to dissolution, it is necessary to immerse it in a liquid which has been preequilibrated with an aliquot of the solid. In the case of ionic exchangers, the solution must be equilibrated with the ions exchanged by the solid.

591 2.3.3 X ray diffraction

In the case of swelling clay minerals, the size of the structural unit changes significantly with the hydration conditions. As these compounds are crystalline, their microscopic swelling can be studied by isothermal X-ray diffraction under controlled pressure. This can be achieved using an experimental setup, such as in figure 14, which results from a collaboration with Mr. Uriot and Lhote (CRPG). Oriented clay films containing kaolinite (non swelling) used as a reference are placed in the diffraction chamber (F). They can be outgassed at the desired temperature using the vacuum group (A+B). A given water relative pressure can then be applied to the sample by changing the temperature of the water source (D), the temperature of the diffraction chamber being regulated at 30~ by water circulation (C) in the double wall of the chamber. The X-ray diffraction data are then obtained from the Co Ks radiation on art Intel CPS 120 curved detector (G). Data are collected simultaneously over 60 degrees (20) during 20 to 40 minutes and processed using a multichannel Varrox analyzer (2048 channels for 60 degrees). The kinetics of the swelling phenomenon can be studied by recording Xray diffraction patterns at different times. XRD powder patterns can then be compared with simulated ones. The software used was developed in the Laboratory of Crystallography of Orl6arts, France, using the approach proposed by Drits and Sakharov [48], Drits and Tchoubar [49] and adapted by Besson and Kerm [50] for the study of the intensity diffracted along the 00 rod in reciprocal space from disordered phyllosihcates. The input data file includes (1) the abundances and positions of the atoms in the different clay sheets (2:1 layer structure) and of water molecules in the irtterlamellar space (Data proposed by Pezerat and Mering and by Drits and adapted from Ben Brahim et al. have been used [51-56]); (2) The basal spacings corresponding to the different hydration states (zero, one, two or three layers of water). (3) The proportions of the different hydrated states and the probabilities of succession of two kinds of layer. (4) The distribution of the number of clay layers per quasi-crystals (tactoids). The detailed simulation procedure can be found elsewhere [22, 57-59].

3. ENERGETIC HETEROGENEITY OF NON POROUS MINERALS 3.1 Determination of basal and lateral surfaces of clay minerals

Previous LTAM studies have shown that the most reliable results can be obtained using argon as an adsorbate as it does not have any permanent or inducible polarization [25]. As previously claimed (w 2.2.1., [25]), basal surfaces are assumed to be relatively homogeneous whereas lateral surfaces are assumed to be strongly heterogeneous. In addition, lateral surfaces are expected to share more energetic interaction with gases than basal surfaces. Then, the sharp decrease of differential enthalpy of adsorption can be assigned to lateral faces and the quasi horizontal branch to basal faces of phyUosilicates (Figure 5).

592

T

C A

I

V

1

Figure 14: Upper part: Water pressure and temperature controlled X-ray diffraction device. A: primary vacuum pump. B: turbomolecular pump. C: thermal regulator. D: spring of water vapour. E: diffractometer. F: Sample. Lower part: In situ X-ray diffraction device. G: diffraction chamber. H: bent detector. I: X-ray generator. J: Water pressure input. K: thin window (15~'n aluminum film). L: micrometric slits. M: Thermocouple.

593 By applying the DIS method to low pressure derivative adsorption isotherms these two types of surfaces can be distinguished [29]. Figure 15 displays the adsorption derivative obtained in the case of a kaolinite sample. This derivative can be simulated using four theoretical multilayer derivative isotherms. As with LTAM, the less energetic domain is assigned to adsorption on the basal faces and the three most energetic isotherms are assigned to adsorption on the lateral faces. This result was further validated through the comparison of calculated and experimental derivative heats of adsorption [29]. In the case of non porous phyllosilicates, argon derivative isotherms generally exhibit the same features. Some differences can be observed at high energy due to the presence of high energy adsorption sites. For instance, it could be demonstrated that high energetic adsorption sites observed in the case of talc correspond to the adsorption of argon onto octahedral OH located at the center of the hexagonal cavities of the basal faces [38]. A special section will be devoted to the particular behavior of talc. In the case of chlorite minerals, high energetic sites are also observed (Figure 16). The shape in the high energy part of the derivative isotherm is very similar to that of the derivative when adsorption occurs on basal surfaces. This suggests that the observed high energetic sites are located on the basal surfaces. Their exact physical meaning remains unknown. The DIS procedure yields more quantitative information about basal surfaces. First, the position of the maximum is quite constant in the case of clay minerals; around - 4kT. This result is relatively surprising as basal surfaces have different

1.0 0.8

Experimental Recalculated _- ........ Local derivative isotherms

f~ ./,",~ J' '~

O a.

f;

\

i! .t

a. 0.6 r"

V

m

"O

~0.4 "D m

0.2

0.0 -15

-10

-5

0

In(P/Po) Figure 15: Experimental and simulated Ar adsorption derivative isotherms at 77K on GB3 well crystallized kaolinite.

594

0.20

t

.........

Experimental Local derivative isotherms

0

,,-0.15 n_ c

"~

m

> 0.05

....... :::ii ............ ""-.-..

.."'

"0

0.00 -15

T

r

t

-10

L

-5

'

'

'

0

I n ( P / P o )

Figure 16: Derivative of argon adsorption on chlorite (Talc de Luzenac Sa) at 77K. Magnification of the high energy part shows the similitude between high energy domains and adsorption on basal surfaces. chemical compositions (Table 2). The isotherms obtained on kaolinite raise some questions. Indeed, its two basal surfaces are chemically different (Si-O-Si and A1OH) [7, 29]; however, only one theoretical basal isotherm is involved in the decomposition as in the case of talc that exhibits only one type of basal surface (SiO-Si) [38]. If the position of the maximum is relatively constant, the shapes of the derivatives are highly variable as evidenced by the wide distribution of lateral interactions (Table 2). As discussed in section 2.2.2.2, this reveals large variation in the variance of the energetic distributions. This property is illustrated in the case of kaolinites coming from Charentes orebodies, France [7]. The plot of lateral

Table 2: Some characteristics values obtained on phyllosilicates from high resolution quasi equilibrium volumetry of argon adsorption. Mineral Basal surfaqes Lateral surfaces -ln(P/Po) w/kT -ln(P/Po)l -ln(P/Po)2 Kaolinite 4.1 -4.5 0-1.5 7.6- 8.4 9.9- 12,6 Talc 4.1 1.5 6.8 - 7.1 Chlorite Mg 3.15, 4.3 2.7, 0.7 7.1 Muscovite 4.8 1.5 6.9 10, 11 Biotite 4.9 1.8 6.6 9, 11

595 interactions as a function of basal surface areas (Figure 17) shows linear inverse relationships between these two parameters. The samples can be discriminated into two families: the second one corresponds to a special orebody of Charentes province where the samples exhibit anomalously high surface areas for kaolinite minerals. The "decrease of c0/kT with respect to basal surface area is lower than the decrease observed for the other samples (family one). It can be noticed that in both families, lateral surface areas are relatively constant, around 2.5 and 7 m 2 / g for family one and two, respectively. These results can be related to crystallographical properties of kaolinites. Indeed, due to petrogenetic considerations, it is well known that the extend of basal faces is lower for well crystallized kaolinites than for poorly crystallized ones. This can be related to iron content in kaolinite which poisons the crystal growth. Then, the relationship between apparent lateral interactions and the extend of basal faces show that these crystallographical properties are also observed at the surface of the mineral. Lateral surface areas are generally decomposed using two local derivative isotherms. The maximum of the less energetic one is located b e t w e e n - 7 . 6 and -8 kT for kaolinites [7], at -7 kT for talc [38], chlorite and muscovite and at-9 kT for biotite. In all cases, the lateral interaction parameter is equal to zero, indicating the highly heterogeneous nature of lateral faces. Due to experimental limitations

1.80 0

1.60

(1) ,,~ ,..~ 1.40 0

"~

1.20

.oo 0.80 .~.,-

0.60 0.4(

,

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. . . . .

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I

20

,

,

,

,

I

,

,

,

30

,

I

40

,

,

,

,

50

Basal Surface Area (m2/g) Figure 17: Relationship between basal surface areas and their energetic heterogeneity expressed as lateral interactions between adsorbed molecules. (1) Charentes orebody except of (2) which corresponds to a special orebody of Charente Bassin.

596 at very low relative pressures, the behavior of the most energetic isotherm can not be interpreted safely. In the case of kaolinite of Charentes, the amount of lateral surface areas derived from the DIS procedure was used to calculate the theoretical cationic exchange capacity (CEC) which is borne by the lateral faces for this mineral [7, 25]. A linear relationship is observed between calculated and experimental CEC values (Figure 18). The slope of 0.92 obtained in the regression indicates that lateral surfaces could be underestimated by 8%. In conclusion, at the moment the DIS method seems to yield the most accurate determination of the flakiness ratio of platy materials. The lateral interactions used to simulate adsorption on basal surfaces can be used as an index of surface homogeneity. Such treatment can not be applied to nitrogen adsorption as previously evidenced by LTAM experiments on kaolinite minerals [25] due to the polarizability (quadrupolar momentum) of nitrogen molecules in the presence of protons of OH groups at mineral surfaces. Experimental evidences of such interactions were given by infrared experiments of Frohmsdorff et al. [60]. In the case of kaolinite minerals (Figure 19), the energetic distribution derived from nitrogen adsorption onto basal and lateral surfaces is not fully understood at the present time. However, the polarizability of nitrogen can be advantageously used as it can give information about electropositive groups of the surface as discussed in the next section.

12

m

9

9

10 r.,j r,j

9 m

-

9

O

9

r m

r

r,j 0 0

2

4

6

8

10

12

14

Experimental CEC (meq/lOOg) Figure 18" Relationship between experimental and corrected cationic exchange capacities of Charentes kaolinites. Experimental values have been corrected of exchange capacity due to octahedral isomorphic substitution of A13+ by Fe2+ [7].

597

0.8 0.7

Experimental Recalculated ......... Local derivative isotherms

i(~

[!

/

~

l

/

I

~//

A0.6 O n_

a. 0.5 c "O

0.4 0.3

> 0.2 "O

0.1 0.0

........

-16

.---~_:t-~-~--~.-_-.:-~:--r":_'~'_:::~..... " : : :

-12

-8 In(P/Po)

. . . . . . J .... ~:::',

-4

Figure 19" Experimental and simulated N2 adsorption derivative isotherms at 77K on GB3 well crystallized kaolinite.

3.2 Evidence of basal high energy sites of muscovite Muscovite is a mica mineral. It is a 2:1 phyllosilicate with a tetrahedral charge defect due to the substitution of aluminium for silicium. This charge defect is compensated by potassium cations located between 2:1 layers. K + ions are located above half of the hexagonal cavities. On the basis of crystallographic data the surface area of a hexagonal cavity is 24 ~2, then K + density on basal faces is one per 48 ~2 on basal faces. As mentioned above, basal and lateral surface areas can be d e t e r m i n e d from the DIS method applied to argon adsorption isotherm (Figure 20a). Nitrogen adsorption isotherms (Figure 20b) are clearly different as the derivative exhibits two high energy sites and lateral and basal surfaces can not be distinguished (manuscript in preparation). The high energy site distribution depends on the outgassing temperature. This means that water adsorbed on high energy sites is not totally desorbed at classical outgassing temperatures used for non porous samples. By applying the DIS method to the nitrogen isotherm, the adsorbed quantities on both energetic sites can be measured. The total adsorbed amount can then be compared to the number of surface hexagonal cavities deduced from the basal surface area determined by argon derivative isotherms. The results (Table 3) show that the amount of nitrogen adsorbed on high energy sites corresponds exactly to half the number of hexagonal cavities. It can then be concluded that these high energetic sites are located on the basal surfaces either on exchangeable

598 0.6

........

I .........

I ........

0.6

a) 0.5

0.5

0.4

0.4

e,,

-

c

-"

0.3

-"

0.3

"0

1).1

"~

0.1

0

'

-15

-10

-5

ln(P/Po)

,,

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#

K

,-i ~ r~_,.,~

-15

,,

~ EL,--,"i't_

-10

"-

_

Jr,,

I-_r-.~"=F r , ,

-5

,,

,

0

In(P/Po)

Figure 20: Muscovite, experimental derivative of adsorption (straight line) and decomposition results (doted lines), a) argon at 77K and b) nitrogen at 77K. Table 3: Description of surface properties of muscovite and biotite derived from simulation of ex ~erimental derivative of adsorption of argon and nitrogen at 77K Argon Nitrogen Adsorbed quantities on high basal surface area: 6.8 m 2/g Muscovite energy sites: 0.405 cm3/g STP Lateral surface area: 1.9 m2/g number of sites: 1.36 1019/g hexagonal basal sites: 2.83 1019/g Adsorbed quantities on high basal surface area: 3.9 m 2/g Biotite energy sites: 0.209 crn3/g STP Lateral surface area: 1.0 m2/g number of sites: 5.6 1018/ g hexagonal basal sites: 1.63 1019/g cations or above hexagonal cavities non occupied by K +. Experiments are in progress to clarify this assignment. Such high energy sites have already been observed on several samples such as apatite (calcium phosphate, in this case, the high energy sites correspond to superficial P-OH groups [39-40]), synthetic saponite (2:1 clay mineral with tetrahedral charge defect, in this case high energy sites are not assigned at the moment), synthetic silica (Si-OH groups) and chlorites (in this case, the high energetic sites are located on basal surface but their exact location is unclear at present time). The case of talc will be discussed in the next section. However, it seems that this phenomenon is not always observed. For instance, biotite, a trioctahedral mica with the same tetrahedral substitution as muscovite, features some high energy adsorption sites wich can not be easily assigned to the same basal sites as muscovite (Figure 21, Table 3).

599 0.35

0.3

. . . . .

I ....

J"

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'

"

'

'

I

. . . . .

'i

. . . . .

.

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0.2

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.

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.

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i

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,

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,

,

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-6

!

I

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'

!

-3

,

,

,

=

,

0

In(P/Po) Figure 21- Biotite, derivative of argon (straight line) and nitrogen adsorption (doted line) at 77K.

3.3 The microscopic hydrophilicity of talc Talc is a trioctahedral phyllosilicate with no layer charge which appears hydrophobic with experimental water contact angle around 80 ~ [61-64]. However, immersion microcalorimetry measurements show that outgassed talc can not be considered as truly hydrophobic [27, 38, 65-67]. The immersion enthalpy value is more than tripled for outgassing temperature of 25~ and 400~ indicating that the wettability of talc increases with outgassing temperature (Figure 22). Concomittantly, adsorption-desorption isotherms of water vapor are influenced by outgassing temperature (Figure 23). Their shape also suggests a hydrophilic behaviour. Furthermore, for high outgassing temperature all the adsorbed water cannot be removed even by pumping the samples under a vacuum of 0.1 Pa at 30~ As the crystallochemical properties are not affected at temperature below 800~ [30], it can be stated that the changes of water affinity toward the talc surface must be due to surface modifications. These surface modifications were studied using argon and nitrogen as probes (Figure 24) [38]. Derivative adsorption isotherms show that high energy sites appear when the outgassing temperature is increased. This is especially the case with nitrogen. The use of the DIS method shows that basal (=12 m2/g) and lateral (---4 m2/g) surface areas can be evaluated by argon and nitrogen. The adsorbed quantity on high energy sites measured with nitrogen is maximal (1,7 cm3/g) for outgassing at 250~ Close values are obtained with argon.

600 16

12

"E 8 4

0 0

0.2

0.4

0.6

0.8

1

P/Po Figure 22: Immersion entha]py of talc in water as a function of water v a p o r

precoverage and outgassing temperature.

1.5

1.5 / ' ' , 1 , , , i , , , i , , , i , , ,

: 30~ o

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m "0 m

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0.5

0

,

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,

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. . . .

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x

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0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 P/Po P/Po Figure 23: Water vapor adsorption isotherms at 303K on talc outgassed at 30 and 250~

601 1.5

'

'

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'

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'

'

I

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N 2 25~ A

o o. 13.

,

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9

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o .,

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-8

In(P/Po)

-4

0

.::.. -16

.... .:...: -12

....

-8 In(P/Po)

....iiii.i -4

" 0

Figure 24- Experimental derivative of argon and nitrogen adsorption at 77K (straight line) and decomposition results (doted lines).on talc outgassed at 25 and 250~ The octahedral layer of talc is trioctahedral and almost purely magnesian. Consequently, the octahedral hydroxyls point directly toward the surface inside the hexagonal cavity formed by the arrangement of the silica tetrahedra of the tetrahedral layer. These OH groups could then exhibit a particular reactivity toward polar molecules as evidenced by computer simulation of water adsorption [68]. From crystallographical parameters, it appears that each hexagon of the tetrahedral layer has an area of 25 ]k2. Then, the theoretical quantity of nitrogen adsorbed on OH groups can be calculated by taking into account the basal

602 surface area of 12 m2/g and an adsorption ratio of one nitrogen per hydroxyl group. The obtained value is 1.79 cm3/g which is very close to the experimental value (1.72). Therefore, the high reactivity of OH groups proposed by Skipper et al. [68] is experimentally confirmed in the case of nitrogen. The increased wettability with outgassing temperature can be assigned to outgassing of adsorbed molecules screening OH groups. Controlled rate thermal analysis coupled to mass spectrometric analysis confirmed that numerous surfaces species including water, carbon dioxide, nitrogen and organic molecules which seem to be long chain amines and nitriles are outgassed between 150 and 400~ (Figure 25). As a consequence, water adsorbs first on liberated OH groups and screens the high energy sites. The surface is then hydrophobic and water adsorption occurs only by cluster growth through hydrogen bonding with the first adsorbed molecules. If structural OH groups are replaced by fluor ions, talc is totally hydrophobic and no water adsorbs on its surface [38]. Another interesting consequence of the determination of basal and lateral surface areas is that they can be used in the Harkins and Jura formula to calculate water contact angle on the basal surface of talc. Because of talc hydrophobicity, the external Harkins and Jura surface can not be derived from enthalpy immersion after water precoverage at P/Po greater than 0.85 as this method assumes'a water contact angle equal to zero. Indeed, the obtained value is 4.2 m2/g and is very different from the values obtained by argon or nitrogen adsorption (16 m2/g). As 2 10 "11 '

1.5 10 "11

"'i

'

'

I

"'

.....

m/z=12

.....

m/z=28

'

""

'

I

'

'

'

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,

'

,

,

a....

m/z=30(xlO) ......... m/z=44

,,r ~

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m ~

1 10 "~ ~ i

B

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--

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9

~ 1 7 6

9

~

o

,

5 10 "~2

,=,.

100

200

.

.

.

.

300

Temperature

.

400

.

500

~

Figure 25: Evolution of gases (except of water) release as a function of temperature obtained using controlled rate thermal analysis coupled to mass spectrometry. The mass m/z = 12, 28, 30, 44 can be assigned to C, CO and N2, NO, CO2 respectively.

603 the basal and lateral surface areas have been measured, it is possible to calculate the water contact angle with basal surfaces if one assumes that lateral surfaces are totally wetted. This calculation leads to a contact angle of around 86 ~ which is close to the macroscopic experimental value [38]. 4. P O R O U S M I N E R A L S : SEPIOLITE

AND PALYGORSKITE Sepiolite and palygorskite are natural microporous clay minerals. Their microporosity originates from structural properties as they are made of ribbons of 2:1 talc-like layers organized in quincunxes. The linked ribbons form a 2:1 layer continuous along the x direction but of limited lateral extension along the y axis creating channels or structural micropores of 13.4A x 6.7A and 13.7A x 6.4A for sepiolite and palygorskite, respectively [69-70]. Sepiolite and palygorskite exhibit a fibrous habit and the association of fibers produces an interfiber microporosity [69, 71]. The external surfaces consists predominantly of {011} crystal corrugated faces [72-73]. Most studies on sepiolite and palygorskite textural properties have been based on adsorption isotherms obtained from conventional volumetric adsorption apparatus, typically using N2 [74-84]. Such techniques are unable to reveal information at relative pressure lower than 0.05. Hence, the usual method used to detect and study microporosity cannot be used to reveal the two types of microporosity. From the use of high resolution transmission electronic microscopy, well controlled thermal treatment and outgassing procedures, argon and nitrogen low temperature adsorption microcalorimetry coupled to quasi equilibrium volumetry, carbon dioxide quasi equilibrium volumetry, water vapor adsorption gravimetry and immersion microcalorimetry in water [21, 85], it has been possible to distinguish between the structural or intrafiber microscopy (c~ Figure 26), the interfiber microscopy (15 Figure 26) and the external surface of the fibers (7 Figure 26). The dehydration steps must first be characterized precisely. This was achieved using CRTA experiments (Figures 4 and 27). The case of sepiolite was described part 2.1. In the case of palygorskite, the dehydration behavior is close to that observed for sepiolite but, the dehydration of the first half of crystallized water occurs at lower temperature, around 100~ and the second dehydratation is immediately followed by the dehydroxylation of the 2:1 layers between 250 and 420~ For both samples, the structure folds after the dehydration of the first half of crystallized water i.e. above 350 and 120 ~ for sepiolite and palygorskite, respectively. After the folding the accessibility of structural micropores disappears. The study of the textural properties of these samples as a function of outgassing temperature can give valuable information about their geometric heterogeneity. The majors results can be obtained by Ar and N2 LTAM (Figure 28). Indeed, the derivative enthalpy of adsorption remains constant during the filling of the microporosity as it is the case for zeolites. When the samples are outgassed at temperatures higher than folding temperatures, this part of the isotherms disappears. The horizontal branch of the derivative enthalpy of adsorption can then be assigned to structural micropores.

604

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Figure 26: Schematic representation of the texture of sepiolite fibbers perpendicularly to their axe. c~" structural microporosity, 13" interfiber microporosity and ~. external surface area.

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Temperature

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Figure 27: Controlled rate thermal analysis of palygorskite [85].

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Figure 28: Low temperature adsorption of argon and nitrogen at 77K microcalorimetry on palygorskite outgassed at 25 and 150~ [85]. The direct distinction between filling of interfiber micropores and adsorption on external surfaces from LTAM experiments is less easy. However, using Harkins and Jura's method, the external surface area of the fibers can be measured without any assumption on the cross sectional area of water molecules. The obtained results were 125 and 63 m 2 / g for sepiolite and palygorskite, respectively. The adsorbed quantities in interfiber micropores can then be derived if the adsorbed volumes in intrafiber micropores and on external surfaces are subtracted from the monolayer capacity. More recently [29,31], it has

606 been shown that the derivative isotherm summation can be used to distinguish between structural micropores, interfiber micropores and external surface area (Figure 29). The evolution of the textural properties of sepiolite with outgassing temperature shows that the accessibility of structural micropores decreases between 200 and 250~ and disappears at 350~ The external surface area decreases from 125 to 48 m 2 / g when the structure collapses. In the case of palygorskite, it was observed that the folding was dependent on the couple vacuum-temperature. Furthermore, water vapor adsorption showed that the folding was reversible for outgassing temperatures below 225~ which is the temperature of folding of sepiolite. In contrary to sepiolite, the external surface area of palygorskite decreases only slightly from 65 to 54 m 2/g when the structure folds. The geometric microporosity detected by nitrogen and argon adsorption can be compared to the theoretical volume taking into account the density of the bulk phase, i.e. 0.808 and 1.427 for nitrogen and argon respectively [86]. The filling of the micropores was only partial as 20% of the total microporosity was available to nitrogen (= 15% for argon) for outgassing temperatures lower than 100~ However, adsorption of carbon dioxide at 293K using a quasi-equilibrium volumetric technique shows that the adsorbed volumes in micropores, as calculated by the Dubinin method, correspond to 100% of the theoretical volume

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607 for sepiolite [26] and palygorskite [85]. This feature of CO2 adsorption was recently confirmed in the case of All3 pillared saponites. Therefore, CO2 quasi-equilibrium adsorption volumetry appears as the most reliable method for total microporosity assessment [87].

5. THE SPECIAL BEHAVIOR OF SWELLING CLAY MINERALS The adsorption of water on swelling clay minerals is a very complicated phenomenon. Swelling clay minerals are 2:1 clay minerals with octahedral charge defects. In this case, the charge is not localized at the surface of the layer and the cohesion between layers, ensured by the interlayer cations, is weaker than in the case of clay minerals with tetrahedral charge defects. Due to the delocalization of the layer charge, the interlayer cations can share high interactions with water molecules which can enter between 2:1 layers. This interlayer cation hydration phenomenon is considered as being the driving force for swelling structure. The interlayer space increases with adsorbed water, creating structural heterogeneities. Then, the surface accessible to water changes constantly from 20-60 m2.g -1 in the dry state to = 800 m2.g -1 in the fully hydrated state. The swelling properties depend also on the nature of the interlayer cations:. mixtures of interlayer cations are generally observed in natural samples. This represent a source of heterogeneity in addition to crystal defects, faces distribution, and layer charge distribution. In the case of montmorillonite which is dioctahedral, the complexity of the phenomena involved makes it necessary to simplify the system by studying homoionic samples. The mechanisms of water adsorption and the influence of the nature of the interlayer cation can then be studied. Nine homoionic samples were prepared by ionic exchange starting from the homoionic sodium form [22, 57-59]. In the present review we have chosen to present the case of two monovalent cations, sodium and cesium and two divalent ones, calcium and barium. In order to approach such complicated phenomena various experimental techniques have to be used. The initial state or "dry" state, i.e. the outgassing conditions which were used in all adsorption experiments, was first carefully defined by CRTA (see paragraph 2.1.). The size of clay tactoids (i.e. the number of clay layers per quasicrystal) in the initial dry state and in the fully hydrated state was obtained from nitrogen adsorption-desorption volumetry at 77K and immersion microcalorimetry respectively. Once the initial and final states are well defined, the mechanisms of water adsorption can be approached by coupling water adsorption/desorption gravimetry and X-ray diffraction studies under controlled water vapor relative pressure. Subsequent modelling was used to characterize the various hydrated states obtained with increasing water vapor relative pressure.

608 5.1. Characterization of the dry state 5.1.1. CRTA Analyses. Figure 30 shows the CRTA curves corresponding to the four different homoionic montmorillonites. Three regions can be defined on these curves. Region I, between room temperature and 100~ corresponds to the desorption of physically adsorbed water. Region II (100~ 500~ corresponds mainly to the expulsion of water supposedly bound to exchangable cations whereas region III (above 500~ corresponds to the dehydroxylation of the structure. In the central region, the water content decreases in the order Ca2+> Na + > Ba2+> Cs + following the decreasing solvation energy of the cations. In the initial state chosen for the hydration studies (outgassing at 100~ there is still some water associated to the exchangable cations. This quantity varies between 4.6 and 1.7 molecules per cation in the case of Ca-montmorillonite and Cs-montmorillonite, respectively. 5.1.2. Analysis of the size of the tactoids. The "dry" state can be further characterized by nitrogen adsorption-desorption isotherms (Figure 31). Depending on the interlayer cation, the quantity of nitrogen adsorbed is very different as revealed by the BET surface areas displayed in Table 4. The high surface area of Cs-montmorillonite is due to the size of the interlayer cation which allows nitrogen molecules to enter the interlayer space. Each curve exhibits a H3 hysteresis loop [88] in desorption characterizing the

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100

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300

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400

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500

,

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600

l

I

700

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800

l

I

900

J

1000

T e m p e r a t u r e (~

Figure 30: Controlled montmorillonites.

rate

thermal

analysis

of Na,

Cs,

Ca

and

Ba

609 presence of slit-shaped pores. The size of Cs also explains the low pressure hysteresis observed in this case. The low pressure hysteresis observed in the case of Ca-montmorillonite and to a lesser degree in the case of Na-montmorillonite has a different origin. It results from the modification of the arrangement of the clay layers in quasi-crystals (or tactoids) brought by the adsorption of nitrogen at high relative pressure. This was confirmed by carrying out successive adsorptiondesorption cycles [59]. The average number of clay layers per quasicrystal was calculated from the nitrogen adsorption data assuming that the plates (square parallelipipeds = 3000A by side and a multiple of the d001 in height) are perfectly stacked as in a deck of cards (Table 4). In the case of Cs-montmorillonite the contribution of the microporosity was first subtracted. The results show strong differences depending on the nature of the cation as Ca-montmorillonite has an average of 69 layers per tactoid whereas Cs montmorillonite is formed of quasicrystals of around 11 layers. 5.2. Characterization of the hydrated state. As the surface area of the materials changes upon water adsorption, it is necessary to define precisely the state of the quasicrystals after immersion in water. This can be achieved through immersion microcalorimetry experiments. The results displayed in Figure 32 reveal that the immersion enthalpy at zero coverage increases with the hydration energy of the exchangable cation : Cs + < N a + < Ba 2+ < Ca 2+. In each case, the asymptotic behavior is observed for precoverage relative pressures higher than = 0.8. Using Harkins and Jura's treatment, the external surface area of the wet samples was then derived and the number of clay layers per hydrated tactoid was subsequentely obtained in the same way as in the case of nitrogen adsorption (Table 4). The differences are less pronounced than in the dry state with 8 layers per quasicrystal in the case of monovalent montmorillonites and 12-14 layers for the divalent ones. The breaking of the tactoids into smaller units must then be taken into account for studying water adsorption.

Table 4: Specific surface areas and number of layer per tactoid deduced from BET surface area,.(dry state) and Harkins and Jura surface area (wet state). .Surf.a.ce area m 2 / g Number of layer per tactoid N2 BET Harkins and Jura Dry state Wet state Na 42 84 33 8 Cs 130 86 11 8 Ca 17 63 69 12 Ba 25 56 32 14 ,

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F i g u r e 32: E n t h a l p y i m m e r s i o n i n w a t e r at 3 0 3 K of N a , Cs, Ca a n d Ba m o n t m o r i l l o n i t e s as a f u n c t i o n of r e l a t i v e p r e s s u r e of w a t e r v a p o u r p r e c o v e r a g e at 303K.

612 5.3 Study of water adsorption mechanisms. 5.3.1. Water adsorption-desorption isotherms. Water adsorption isotherms corresponding to the four homoionic montmorillonites are displayed in Figure 33. The shape of the isotherm is strongly influenced by the nature of the interlayer cation. The desorption branch closes in the case of monovalent cations in contrary to the case of divalent cations where all the adsorbed water is not desorbed at low relative pressure. The shape of the isotherms suggests that water molecules enter the interlayer space for very low relative pressures in the case of Ca- and Ba-montmorillonite whereas in Csor Na-montmorillonite it seems that water enters the interlayer space for a given value of the relative pressure. These curves can not be interpreted safely without any data concerning the texture of the quasicrystals after adsorption and the evolution of the interlayer distance with water relative pressure which can be obtained using the X-ray diffraction system described in Figure 14. 5.3.2. X-ray diffraction studies under controlled water vapor pressure. The evolution of the d(001) with water relative pressure is presented in Figure 34. All the points were taken after at least 4 hours equilibrium time as the swelling kinetics of monovalent montmorillonites are rather slow for high relative pressures [58]. The absence of defined steps in adsorption shows the presence of interstratified states, i.e. heterogeneous distribution of the hydrated states, in the whole range of water relative pressure. The desorption branches generally exhibit more pronounced steps revealing a tendency towards more homogeneous hydrated states. Figure 35 presents the proportions of the different hydrated states obtained for the four montmorillonites in adsorption. Basal spacings of the models were set taking into account the harmonics of the quasi-homogeneous states or, in the absence of such states, results from the litterature [55-57, 89-93]. The mean numbers (M) of layers per stack were determined in order to obtain a good superimposition of the positions of the experimental and calculated reflexions. Generally these values agree with those obtained using the width of the d(001) reflexion and Scherrer's formula except in the case of Csmontmorillonite which present symmetrical (hexagonal cavities face to face) and non symmetrical (hexagonal cavities shifted) stacking types noted 0L sym and 0L non Sym in figure 35. In order to explain the broadening of the peaks d(002) and d(004) and the high value of the position of the peak d(002), the different states (0L sym, 0L non sym and 1L) were assumed to be separated (demixed) for relative pressure values between 0.15 and 0.50 (Figure 35). In the case of Camontmorillonite at high relative pressure, homogeneous series of values of basal spacing are observed that do not correspond to one layer-two layer hydrate state interstratifications. The patterns were then simulated using a set of two values of basal spacing corresponding to sub-states 2a and 2b according to the study of Suquet and Pezerat [94].

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F i g u r e 34: X-ray basal spacing of Na, Cs, Ca a n d Ba m o n t m o r i U o n i t e s as a function of w a t e r v a p o u r relative p r e s s u r e .

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616 In adsorption, the one layer hydrate appears at very low relative pressure for both divalent cations and for Cesium. In contrary, in the case of sodium it starts developping only for a relative pressure of around 0.25. The two layers hydrate appears at relative pressure values of 0.2, 0.4, and 0.6 for Ca-, Ba- and Namontmorillonite, respectively. It never develops for Cs-montmorillonite. For m e d i u m relative p r e s s u r e s (around 0.5) all m o n t m o r i l l o n i t e s exhibit interstratified states. Quasi-homogeneous two layer hydrates are present from relative pressure values of 0.7 and 0.9 for Ca and Na-montmorillonite, respectively whereas Ba-montmorillonite exhibits a two layers-three layers interstratified state at high relative pressure. Cs-montmorillonite tends towards a quasi-homogeneous one layer hydrate. In any case real homogenous hydrates are never observed. All the results obtained give some insight about the way water adsorbs on these four montmorillonites. In the first hydration stage Na, Ca and Ba-montmorillonite exhibit a splitting of the initial dried quasicrystals into smaller ones. This is not the case of Csm o n t m o r i l l o n i t e where the size of the tactoids appears u n c h a n g e d upon hydration. The repartition of water between internal and external surfaces can be derived assuming that the adsorption on the external surfaces of the clay layers can be represented using the reference isotherms obtained on non porous solids by Hagymassy et al. [95]. The amount of water in the internal surfaces can then be obtained for each relative pressure by subtracting to the water adsorption isotherm presented in Figure 32 the amount of external water and by adding the amount of residual water in the dry state determined from CRTA experiments (Figure 36). These curves reveal that for Ca and Ba-montmorillonite water starts entering the interlayer space for the lowest relative pressures investigated. This is not the case for Cs-montmorillonite where water starts adsorbing in the interlayer region for relative pressures higher than 0.05. The case of Na-montmorillonite is peculiar as the water starts entering the interlayer space to form a predominant one layer hydrate, once a monolayer is formed on the external surface. The same situation appears for the formation of the two layer hydrate which starts once a bilayer is formed on the external surfaces of the tactoids. It then seems that in the case of N a - m o n t m o r i l l o n i t e the bidimensional pressure is a factor governing the hydration as well as the hydration energy of the interlayer cation. Once the distinction between water adsorbed on the external or internal surfaces of the clay is clarified, it is possible to study the degree of filling of the interlayer space by combining data obtained from x-ray experiments and water adsorption data. The maximal amount of water in the interlayer region can be calculated using the following assumptions. 9 The internal specific surface area is given by Sint = 801.3 -(Sext- Sext lat) where Sext is the total external specific surface area and Sext lat the lateral external specific surface area. 9 Models of the one-layer hydrate and two-layer hydrate proposed by Ben Brahim et al. [55] for Na-beidellite are valid for homoionic montmorillonites. Cross sectional areas of 7.8 ~2 (c~1) and 8.7A 2 (c~2) are assumed for the water

617 molecule in the one layer-hydrate and two or three-layer hydrate, respectively. The amount of water Qmi, in mmol.g -1, adsorbed in the interlayer space as a monolayer (i=1), bilayer (i=2) and third layer (i=3) is given by the relation:

i.Sint Qmi = 2c~i.Na 9 Then for given abundances of relative proportions Wi of each type of layer, the maximal adsorbed amount in the interlayer region is given by: Qint max = W0*Qinit + Wl*Qml + W2*Qm2 + W3*Qm3. The filling of the interlayer space can then be calculated from the ratio (Qint / Qint max)- It is presented in Figure 37 for the adsorption of water on the four homoionic montmorillonites as a function of the % of interlamellar swelling defined as the sum of Wi balanced by i. It corresponds to the solvation of the exchangeable cations and to the filling of the remaining interlamellar space. The filling lies between 40 and 70% for the monovalent cations and is higher around 80-90% for divalent cations. This latter value is close to what is observed in the case of beidellite. In the case of Na-montmorilonite, the filling decreases upon the transition from a dominant one layer hydrate to the two layer hydrate. The variation of the filling reveals the complexity of the structure of the interlamellar water that should be studied using spectroscopic techniques. This study on homoionic montmorillonites shows the complexity of the phenomena involved in the adsorption of water on such swelling materials that are strongly heterogenous. More work is needed before the chemical heterogeneity of the nature of the interlayer cations in natural soil clays can be taken into account.

GENERAL CONCLUSIONS Clay minerals are typical examples of heterogeneous adsorbents showing both surface geometric and energy distributions. The heterogeneity of clays is governed by the geochemical crystallisation conditions generating a strong relationship between the structure, shape and chemical surface properties of these solids. Such complexity obliges to develop m o d e r n experimental techniques and modelling methods for studying solid surfaces. In this way, the high resolution low pressure quasiequilibrium adsorption technique, first developped to characterize geometrical heterogeneity of clay minerals, looks very powerful and promising for studying their energetic heterogeneity. Furthermore, the possibility to detect very high energy surface sites opens new investigation fields as those sites are always involved in interfacial interactions.

618 20

20

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C$

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|

~0 O

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10

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0 .....

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External water I Internal water

0.8

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~ 0.8

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20

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15 A

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~0

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0 1

0

.... ,

"i-;, -i-i .~, 0.2 0.4 0.6 P/P 0

Figure 36: Water distribution on external surfaces and in internal surfaces of Na, Cs, Ca a n d Ba montmorillonites as a function of relative w a t e r v a p o r pressure.

619 120 9 9

o Na A Cs

Ca Ba

100

o

80 0

&

9

A

60 o

40

20

0

i

0

I

100 % of inteflamellar swelling

i

200

Figure 37: Percentage of water filling as a function of the degree of swelling of Na, Cs, Ca and Ba montmorillonites.

The case of swelling clay minerals is of great applied and fundamental interest as water adsorption generates structural heterogeneities which render the interactions mechanisms rather complex. Here again, the d e v e l o p m e n t of adapted techniques such as X-ray diffraction under controlled water vapor pressure is required for a clearer understanding of the behavior of these minerals. This review also shows that it is important to combine different experimental approaches to analyze properly the surface heterogenities of these solids in order to understand their behavior in natural or industrial conditions. The succesful results obtained in the case of talc, microporous and swelling clay species validate the developped methods which are currently applied to other solid surfaces.

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622 53. J. Ben Brahim, Contribution a l'~tude des syst~mes eau argile par diffraction des rayons-X. Structure des couches ins6r6es et mode d'empilement de feuillets dans les hydrates homog~nes ~ une et deux couches d'eau de la beidellite Na. Thesis Doctorates Sciences, Orl6ans University, France, 1985. 54. J. Ben Brahim, G. Besson, C. Tchoubar, Journ. Appl. Cryst., 17 (1984) 179. 55. J. Ben Brahim, G. Besson, C. Tchoubar, 5th Meeting of the European Clay Groups, Prague, Konta J. (ed.), Univerzita Karlova, Praha, (1985), 65. 56. J. Ben Brahim, N. Armagan, G. Besson, C. Tchoubar, Clay Min., 21 (1986) 111. 57. I. Berend, Les m6canismes d'hydratation de montmorillonites homoioniques pour des pressions relatives inf6rieures a 0,95. Th~se, Institut Polytechnique de Lorraine, Nancy, France, 1991. 58. I. Berend, J.M. Cases, M. Francois, J.P. Uriot, L.J. Michot, A. Masion, F. Thomas, Clays and Clay Minerals, (1995) In press. 59. I. Berend, J.M. Cases, M. Francois, J.P. Uriot, L.J. Michot, F. Thomas, Clays and Clay Minerals, (1996) In press. 60. C. G. C. Frohnsdorff and G. L. Kington, Trans. Faraday Soc., 55 (1959) 1173. 61. M. E. Schrader and S. J. Yariv, J. Colloid Interface Sci., 136 (1990) 85. 62. R. F. Giese, C. J. Van Oss, J. Norris, P. M. Constanzo, in Proc. Int. Clay Conf., Strasbourg 1989, V.C. Farmer and Y. Tardy, (eds.), Sci. Geol. Mem., 86 (II) (1990) 33. 63. R. F. Giese, P. M. Constanzo, C. J. Van Oss, Phys. Chem. Miner., 17 (1991) 611. 64. J. Norris, R. F. Giese, P. M. Costanzo, C. J. Van Oss, Clay Miner., 28 (1993) 1. 65. J. Yvon, E16ments sur les propri6t6s cristallochimiques, morphologiques et superficielles des min6raux constitutifs des gisements de talc, Th~se de Doctorates Sciences Physiques, INPL, Nancy, France, 1984. 66. L. Michot, J. Yvon, J.M. Cases, J.L. Zimmermann, R. Baeza, C. R. Acad. Sci. Paris, S~rie II, 310 (1990) 1063. 67. L.J. Michot, J. Yvon, J.M. Cases, In Advances in Measurement and Control of Colloidal Processes, N. De Jaeger and R.A. Williams (eds), ButterworthHeinemann, (1991) 233. 68. N. T. Skipper, K. Refson, J. D. C. McConnell, Clay Miner., 24 (1989) 411. 69. M. Rautureau and C. Tchoubar, Clays Clay Miner., 24 (1976) 43. 70. M. Rautureau, C. Clinard, A. Misfud, S. Caill6re, in Proc. 104~me Cong. Nat. des Soci6t6s Savantes, Bordeaux, S6ries Sciences, 3 (1979) 199. 71. M. Rautureau and A. Misfud, Clay Miner., 12 (1977) 309. 72. P. Fenoll Hach-ali and J. L. Martin Vivaldi, An. R. Soc. Esp. Fis. Quim., 64B (1969) 77. 3. J. L. Martin Vivaldi and P. FenoU Hach-ali, in Differrential Thermal Analysis, Vol. 1, Fundamental Aspects, R. C. Macenzie ed., Academic Press, London, (1969) 553. 4. R. M. Barrer and N. Mackenzie, J. Phys. Chem., 58 (1954) 560. 75. K. P. Moiler and M. Kolterman, Z. Anorg. Allg. Chem., 41 (1965) 36. 76. A. J. Dandy, J. Phys. Chem., 72 (1968) 334. 77. A. J. Dandy, J. Chem. Soc. A, (1971) 2383. 78. A. J. Dandy and M. S. Nadiye-Tabbiruka, Clays Clay Miner., 23 (1975) 428. 79. J. F. Delon and J. M. Cases, J. Chimie Physique, 4 (1970) 662.

623 80. T. Fernandez Alvarez, In Compte-rendu de la Reunion Hispano-Belga de minerales de la Arcilla, J. M. Serratosa (ed.), Consejo Superior de Investigaciones Cientificas, Madrid, (1970), 202. 81. T. Fernandez Alvarez, Clay Miner., 13 (1978) 325. 82. A. Jimenez-Lopez, D. de Lopez-Gonzales, A. Ramirez-S~ienz, F. RodriguezReinoso, C. Valenzuela-Colahorro, L. Zurita-Herrera, Clay Miner., 13 (1978) 375. in Proc. Int. Clay Conf. Oxford 1978, M. M. 83. C. Serna and G. E. Van Sr Mortland and V. C. Farmer (eds.), Elsevier, Amsterdam, 1979,197. 4, B. F. Jones and E. Galan, in Hydrous phyllosilicates, S.W. Bailey (ed.), Review in Mineralogy Mineral. Soc. Amer., Washington, D.C., 19 (1988) 628. 85. J. M. Cases, Y. Grillet, M. Francois, L. Michot, F. Villi6ras, J. Yvon, Clays Clay Miner., 39 (1991) 191. 86. L'Air Liquide (ed.), Gas encyclopedia, Elsevier, Amsterdam, 1976. 87. L. Bergaoui, J. F. Lambert, M. A. Vicente-Rodriguez, L. J Michot., F. ViUi6ras, Langmuir, 11 (1995) 2849. 88. K. S. W. Sing, Pure and Applied Chem., 54 (1982) 2201. 89. H. Suquet, C. De la CaUe, H. Pezerat, C.R. Acad. Sci. Paris, 284D (1977) 1489. 90. U. Del Pennino, E. Mazzega, S. Valeri, A. Alietti, M.F. Brigatti, L. Poppi, J. Colloid Interf. Sci., 84 (1981) 301. 91. M.W. Kamel, Etude de l'Imbibition, du Gonflement et du D6ss~chement de quelques Argiles, Th~se Universit6 Toulouse, France (1981). 92. E.C. Ormerod and A.C.D. Newman, Clay Miner., 18 (1983) 289. 93. T. Iwasaki and T. Watanabe, Clays and Clay Minerals, 36 (1988) 73. 94. H. Suquet and H. Pezerat, Clays Clay Miner., 35 (1987) 353. 95. J. Hagymassy, S. Brunauer, R.S. Mikhail, J. Colloid Interf. Sci., 29 (1969) 485.

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.

625

Multilayer Adsorption as a Tool to Investigate the Fractal Nature of Porous Adsorbents Peter Pfeifer and Kuang-Yu Liu Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211, U.S.A.*

This chapter surveys the use of physical adsorption, from a monolayer upward, as an experimental method to study the fractal surface structure found in many porous and irregular adsorbents. The fractal structure leads to power laws of the Frenkel-Halsey-Hill (FHH) type for the adsorption isotherm, with exponents depending on the fractal dimension of the surface and on whether the dominant force is the substrate potential (van der Waals wetting, low coverage) or the film-vapor surface tension (capillary wetting, high coverage). We derive the power laws from a unifying framework which treats the two forces as competing effects and automatically identifies well-defined coexistence lines in the pressure-dimension diagram between the submonolayer regime, the van der Waals wetting regime, and the capillary wetting regime. We compute the resulting phase diagram for several adsorbate/adsorbent pairs, predicting which of the two power laws will be observed in what pressure range for a given surface geometry and adsorbate. A detailed comparison of the adsorption isotherm on a fractal surface with that in a single pore exhibits many parallels and differences between the two, which we also discuss in terms of t-plots and comparison plots. The aim of the presentation is to provide a simple, but complete set of guidelines for the interpretation of experimental adsorption isotherms, with a minimum number of parameters, in a thermodynamically and geometrically consistent way. A variety of recent experimental studies using multilayer adsorption for fractal analysis are reviewed as illustrations. The examples include some important test cases and range from metal films to carbon blacks, activated carbons, carbon fibers, pyrogenic silicas, silica xerogels and aerogels, porous glasses, and cements.

1. I N T R O D U C T I O N Much of our understanding of how solid surfaces interact with their surroundings, physically or chemically, depends on quantitative models for their structure. Until quite recently, most of these models have been based on Euclidean geometry, such as planar surfaces, straight-line step edges, cylindrical or slit-shaped pores, etc. However, many systems of practical importance (colloidal aggregates, adsorbents, catalyst supports, electrodes, hightemperature superconductors, reinforced materials, and other micro-engineered or naturally porous solids) have a complex structure which cannot be adequately described in such terms. * Research supported in part by the Petroleum Research Fund, administered by the American Chemical Society, Grant No. 28052-ACS

626 Typical of these complex geometries is that they exhibit structural features (departures from a planar surface) over a whole range of length scales----often several decadesmrather than features of one characteristic size only. Often the coexistence of features at different length scales leads to surface geometries where iteratively small pores (or other features) are subpores of larger pores. Such nested pore hierarchies render the concept of individual pores, cylindrical or otherwise, as underlying the conventional notion of pore-size distribution, meaningless. Similarly, other Euclidean descriptors, such as terrace-width distribution or surface height fluctuation, are inapplicable. This creates an eminent need for models in which structural features span a whole range of length scales, i.e., that treat surface irregularities as recurrent and nested, rather than isolated, entities. The simplest such model is that of a fractal surface [ 1-11 ]. A fractal surface has the s a m e structural features (Fig. 1) at length scales between s (inner cutoff) and s (outer cutoff) and is characterized by the fractal dimension D, 2 < D < 3, describing the irregular surface geometry in terms of its space-filling ability in the interval [gmin, s ]- At the low end, D = 2, the surface is planar, at length scales between s and gmax. At the high end, D = 3, the surface is maximally convoluted and fills a volume, at length scales between s and gmaxAt intermediate values, 2 < D < 3, the surface interpolates in a natural way between a plane and a volume. These basic properties have made fractal geometry, since its first application to surface problems [12-18], a highly appealing and successful new tool to study a wide range of phenomena associated with complex and disordered surfaces [4, 5, 7-11, 19-35]. The appeal comes from the fact that (i) the complexity is captured by a single number, the fractal dimension (for phenomena involving connectivity properties of the surface, also the spectral dimension is important [36]); (ii) the fractal dimension identifies the recurrence of the same features at different levels of resolution as a hidden symmetry (self similarity or self affinity) in an otherwise irregular structure; (iii) the resolution analysis automatically focuses on a range of features and is indifferent to whether small features are subfeatures of larger features or not; (iv) the metric properties of a fractal surfacemsuch as the number of pixels required to digitize the surface at a prescribed level of resolution, the number of surface sites present within a prescribed radius from a given site, or the number of surface sites as a function of the surface diameter--are as simple as for a planar surface (power laws with D-dependent exponents); (v) the metric properties, combined with the irregular structure of the surface, generate a wealth of explicitly D-dependent properties (Table 1.A);

Figure 1. Two fractal surfaces (cross sections), with small features being replicas of large features and small pores being subpores of large pores. The two cross sections have the same fractal dimension by construction.

627 (vi) a given value of D can be realized in many different ways (Fig. 1), giving the properties in (iv, v) a high degree of universality and making the description in terms of D stripped of all redundancy. The practical success comes from the fact that (vii) many of the random processes that produce complex surfaces (polymerization, aggregation, vapor deposition, electrolytic deposition, phase separation, drying, leaching, decomposition, corrosion, fracture, laser ablation) give rise to fractal structures by fairly well-understood mechanisms [ 1-5, 7, 8, 11, 22, 31, 34, 35]; (viii) when a surface has features at length scales between s and s and the ratio s is not very large, one may always approximate the features of different size as being in lowest order the same (least-biased guess), which makes the fractal model a good approximation even if the surface is not genuinely fractal; (ix) the D-dependent properties in (iv, v) generate explicit relations, usually in the form of power laws, between a wide variety of experimentally measurable quantities, offering many experimental methods of fractal surface analysis (structural analysis, Table 1.B) and predictions of how the structure controls the performance of the surface in physical and chemical processes (structure-function relations, Table 1.C). Common to all methods of fractal analysis is that the surface is subjected to the interaction with probes ("yardsticks") of different size. Depending on the method, the probes may be molecules of different size, electromagnetic waves diffracted at different angles, energy transfer from a donor molecule to acceptor molecules at different distances, liquid menisci with variable radius of curvature, films of variable thickness, molecules diffusing to the surface starting from different locations in the pore space, etc. In multilayer adsorption, the subject of this chapter, the probes are films which vary both in thickness and radius of curvature. Multilayer adsorption plays a special role among the methods listed in Table 1. First of all, in many applications one is interested in the morphology of the surface starting at atomic length scales, i.e., one would like to know whether the surface is fractal in an interval [gmin, gmax] with ~min of the order of a few ,~,ngstroms and s of the order of hundreds of .&ngstroms (nanostructured surfaces). Films of adsorbed N 2, Ar, Kr, or other inert gases can easily span this range and offer a structural resolution down to atomic length scales by virtue of the atomic size of the adsorbate particles. Second, no matter how convoluted and porous the surface is, gas diffusion through pore space and complete wetting of the surface by the adsorbate guarantee that the film probes the entire surface. Multilayer adsorption is therefore an important tool whenever the surface is too tortuous for methods like scanning tunneling microscopy, atomic force microscopy, or reflectometry to be applicable. The only other methods which compete with multilayer adsorption, in terms of length scales and applicability, are molecular tiling, small-angle X-ray or neutron scattering, electronic energy transfer, and preadsorbed films (Table 1). Combined, these other methods have provided some of the most extensively investigated case studies of fractal surfaces [43, 80, 81 ]. But they are quite time-consuming or require instrumentation that is not readily available in most laboratories. Thus, an important rationale for investigating fractal surfaces by multilayer adsorption is the wide availability of gas adsorption instruments and the ease with which adsorption isotherms can be measured up to quite high relative pressures. The first experimental study of a fractal surface by multilayer adsorption was published in 1989 [44] and has been followed by numerous investigations since. The purpose of this

628 Table 1 Geometric quantities controlled by the fractal dimension, experimental methods of fractal analysis, and applications controlled by the fractal dimension. References A. Geometric quantities: Pore-size distribution Chord-length distribution Porosity Density-density correlation function Height-height correlation function Fourier transform of surface cross sections Dimension of cross sections and projections of the surface

[14,20,21] [19,32] [19,32] [8,23] [8,11,31] [17] [ 1, 6, 30]

B. Experimental methods: Molecular tiling and monolayer capacity Small-angle X-ray and neutron scattering Electronic energy transfer Kelvin porosimetry Thermoporosimetry Hg porosimetry Surface area of preadsorbed films Scanning tunneling microscopy X-ray reflectivity NMR spin relaxation Multilayer adsorption

[12-16,29] [21,29,37] [29,38] [29,39] [40, 41] [42] [43] [44-46] [47, 48] [29,49] this chapter

C. Applications: Surface-diffusion controlled reactions Pore-diffusion controlled reactions Catalysis Dissolution and combustion Chromatography Electrochemical impedance Debye-Hfickel screening Electrical conductivity Hydrodynamic flow Magnetic phase transitions 4He phase transitions Vibrations Thermal conductivity Scattering and absorption of light Compaction Sintering

[20,50] [18,20,51-55] [ 18, 20, 25, 51-55]

[55-57] [58] [53, 54, 59-61 ] [62-64] [19,28,32,33] [19,28,32,33] [65-67] [21, 68, 69] [70-73] [73] [74-76] [77] [78, 79]

629 chapter is to give an account of these developments and to describe the various aspects which have placed multilayers on fractal surfaces at the crossroads of several different areas of surface science. One of the distinguishing features is that an adsorbed multilayer is not a "premanufactured" probe which is brought to the surface and interacts with it like a rigid object, but is a "self-assembled" probe formed by complicated substrate-adsorbate and adsorbate-adsorbate interactions (repulsive at short range and attractive at long range). These interactions result in a layer that is controlled by a strong interplay between energetic and geometric factors when the surface is irregular. At temperatures above the triple point and below the critical point of the adsorptive, i.e., at which gas and bulk liquid of the adsorptive coexist, the adsorbed multilayer is a liquid. So our first task will be to find an adequate description of the energy and thermodynamics of a liquid film, in equilibrium with its own vapor, on an arbitrarily shaped solid (Sect. 2). The description we shall use models the film as homogeneous liquid with sharp liquid-gas interface. The substrate-adsorbate interaction enters through the substrate potential, dependent on the surface geometry, and the adsorbate-adsorbate interaction enters through the liquid-gas surface tension. This describes the two interactions with a minimum number of parameters and provides a framework to calculate adsorption isotherms in various approximations. The key is that equilibrium shape of the film-gas interface is determined by a variational principle (minimization of the grand potential). In Sect. 3, we apply this variational principle to derive the adsorption isotherm for multilayers on fractal, self-similar surfaces. The genetic form of the isotherm is N or [_In(p/p0)] -(3-D)/3 N 0r [_In(p/p0)] -(3-D)

for low p, for p ---) P0,

(1 a) (lb)

where N is the number of adsorbed particles adsorbed at gas pressure p, and P0 is the coexistence pressure for gas and bulk liquid. These are the power laws that most experimental studies have used to infer fractal surface properties from multilayer data. The power law (1 a), called van der Waals wetting regime, is characteristic of multilayers in which the substrate potential is the dominant interaction. The power law (lb), called capillary wetting regime, results when the liquid-gas surface tension is the dominant interaction. Together, the two regimes provide a unified description of the transition from substrate-controlled adsorption to capillary condensation. The description includes the form of the crossover from (la) to (lb), explicit expressions for the prefactors in (1), the connection between gas pressure and length scales probed by the film, and refinements of (1) when the length scales probed approach the inner or outer cutoff of the fractal regime. Section 4 analyzes the transition from van der Waals to capillary wetting as a function of the fractal dimension. For every D value, there is a pressure which defines this transition in a natural way. Similarly, there is a pressure which defines the transition from submonolayer to multilayer adsorption. The resulting transition lines divide the D-p plane into a phase diagram with three distinct regions: submonolayer adsorption, van der Waals wetting, and capillary wetting. For example, when D decreases, capillary wetting is restricted to progressively higher pressures and is completely absent at D = 2. By explicitly predicting in what pressure intervals the power laws (1) occur for a given solid and adsorbate, the phase diagram offers important consistency tests for the interpretation of experimental data of the form (1). Armed with these theoretical prerequisites, we review experimental results in Sect. 5. The examples illustrate the wide applicability of the power laws (1) and amplify earlier accounts of the pervasiveness of fractal surfaces at nanoscales [29]. Our main focus, however, is on

630 studies in which the surface structure inferred from (1) has been tested for internal and external consistency. Internal consistency includes meaningful values for D, groin, and grnax, consistency with the phase diagram, etc. External consistency means agreement with results from other experimental analyses (fractal or otherwise). These tests form an important experimental confirmation of the adequacy of the liquid-film framework used in Sect. 2 to describe multilayers adsorbed on arbitrarily shaped solids. Indeed, the ultimate goal is to develop a general framework from which it will be possible infer structural properties of solids with arbitrarily shaped surfaces (fractal or not), without geometric model assumptions. This is the inverse problem of the program that wishes to predict the isotherm for an arbitrary given surface geometry. The common view is that both the direct problem and the inverse problem are intractably difficult. We therefore would like to point out that, in Sect. 3.6, we will present a solution of both the direct and the inverse problem. The solution will, of course, not be free of approximations; but the approximation will be one in the variational determination of the equilibrium film, not in the surface geometry. In this sense, this chapter may be viewed as fractal illustration of a much more general framework.

2. LIQUID-GAS E Q U I L I B R I U M ON NONPLANAR SURFACES When analyzing the 3-phase equilibrium between a liquid, its vapor, and a solid, we first must determine whether the liquid completely wets the solid or not. If the surface of the solid is planar, the macroscopic condition for complete wetting is that the contact angle between the liquid and the solid, 0, be zero (Fig. 2). Partial wetting corresponds to 0 < 0 < r~. The contact angle is determined by Young's equation, (2)

COS 0 = (O'sg -- O'sl)/(Ylg,

where the o's are the solid-gas, solid-liquid, and liquid-gas surface tension. Thermodynamic equilibrium ensures that the right-hand side of (2) always lies between -1 and 1 [82]. Thus complete wetting is characterized by the condition Csg- Cysl= c]g. Microscopically, complete vs. incomplete wetting can be distinguished by considering the high-pressure behavior of the adsorption isotherm: If N --->o0 for p --->P0, one has complete wetting; if N approaches a finite limit for p --->P0 (i.e., if there is a discontinuous jump to N = oo at p = P0), wetting is incomplete [83-85]. From this criterion and a large body of adsorption data, one concludes that nitrogen and other inert adsorptives, at their normal boiling temperature, completely wet most solids. Consequently, we assume the liquid to be completely wetting in the sequel.

0=0

0 Ag c,

(19a) (19b)

for R ---) oo,

(20)

for Ag = Ag c.

(21)

For low Ag, one may use (17) to evaluate (19a), which in the limit of large R leads back to the FHH isotherm, Eq. (14). So adsorption at low Al.t is dominated by the substrate potential and, in a large pore, is indistinguishable from adsorption on a planar surface. As Ag increases, however, the film thickness grows faster than on a planar surface because the decrease of the energy (5~I d2x' resulting from a small interface radius, outweighs the increase of the particlenumber t e r m - n A g If(S,I) d3x" At Ag = Ag c, the enhancement of adsorption due to surface tension leads to spontaneous pore filling, resulting in a jump of the isotherm (Fig. 5). The jump describes capillary condensation, corresponding to a first-order phase transition ("thin-f'dm/fullpore" coexistence), in a highly idealized setting. The film thickness jumps from z c [Eq. (21)] to R. During desorption, the pore remains full at all Ag because 13 never turns unstable. Thus neither the adsorption nor desorption isotherm jumps at the point Ag c' < Ag e at which the integral (18b) vanishes (coexistence of 11 and I 3 as equilibrium states, "Maxwell construction"). The example illustrates that capillary condensation is controlled by surface tension, but that a thermodynamically consistent construction of the equilibrium interface is not possible without taking the substrate potential into account. This point has already been emphasized by Broekhoff and Linsen [ 107], and the example here has in fact been analyzed along similar, somewhat less explicit, lines before [ 108]. The example will serve as a useful comparison frame to appreciate similarities and differences in adsorption on a fractal surface. On a fractal surface, the adsorption regime controlled by surface tension will no longer exhibit any jump because the surface will have a hierarchy of pores and voids of many sizes. Instead, above some characteristic value Agc of Ag, one will have a continuous sequence of first-order transitions in each of which the "next" larger void in the hierarchy fills by capillary condensation. This turns the first-order transition at Al.t = Ag e in a single pore into what resem-

638

A~t

Al.t A~tc Figure 5. Film shape and adsorption isotherm in Examples 1 and 2 (schematic).

bles a second-order transition at A~t = Al.tc on a fractal surface. The change from a single pore to a fractal surface spreads the event of capillary condensation from a single chemical potential, Al.tc, to a whole range of chemical potentials, A~t > A~te. Since capillary condensation is often equated to an isolated jump in the isotherm, we refer to the regime Al.t > A~tc on a fractal surface as capillary wetting instead. Accordingly, we refer to the transition at A~tc as transition from van der Waals wetting to capillary wetting. This transition is worked out in the next section.

3. ADSORPTION ISOTHERM ON FRACTAL SURFACES The variational principle for the grand potential provides a powerful tool to construct approximations of the isotherm (12) when the Euler equation (13) has no simple solution, as is the case for a general fractal surface. We note that such approximations may be more desirable than the exact isotherm: The two fractal surfaces shown in Fig. 1 give isotherms that certainly differ in their exact details, but in practice will be indistinguishable from each other. So, an approximation well may capture this indistinguishability, while the exact isotherm does not. One chooses a family of trial interfaces and minimizes the grand potential with respect to this family. If the interfaces are sufficiently flexible, the one with the lowest grand potential should be a good approximation of the exact equilibrium interface. A simple and natural choice are equidistant interfaces I z [36, 43-45, 81,101,109, 110], defined as the locus of all points x outside the solid whose distance from the surface, dist(x, S), equals z (Fig. 6). The distance z is the variational parameter (0 < z < oo), making the trial family a one-parameter family. In comparison, the exact equilibrium interface may be regarded as member of an inf'mite-parameter family. The equidistant interfaces form a natural trial family for two reasons. First, they are equipotential surfaces of the substrate potential (7) because the latter depends only on dist(x, S). This makes them automatically solutions of the Euler equation (13) in the absence of surface tension. Second, they offer a convenient characterization of fractal surfaces. There are several, essentially equivalent ways of measuring the space-filling ability of a fractal surface [36]. The most precise one, well-defined for any surface S, is the function V(z) := If(S, Iz) d3x'

(22)

639 which is the volume of all points outside the solid whose distance from the surface is less than or equal to z. The derivative dV/dz equals the area of the interface I z, by the equidistance property of I z. We shall refer to f(S, I z) as film of thickness z, and to dV/dz as the corresponding film area. Equivalenfly, dV/dz may be interpreted as the surface area of the solid measured by tiling the surface with spheres of diameter z [36]. A self-similar surface with fractal dimension D, inner cutoff gmin, and outer cutoff gmax then is characterized by the power law dV dz

(23)

oc z 2-D

for gmin < z < s It describes how the film area on a fractal surface decreases with increasing film thickness by filling progressively larger pores and voids (D > 2), and remains constant on a planar surface (D = 2). The film area dV/dz plays a special role among different ways of assessing fractality because it depends only on the structure of the surface at length scale z, whereas the film volume V(z) and other interrogator functions depend cumulatively on all structure below z, including the regime below the inner cutoff, which may lead to departures from a pure power law for V(z) as z approaches s [43]. If the fractal regime starts at the smallest resolvable scale, i.e., at z = ao where a0 is the thickness of a monolayer (diameter of an adsorbate particle) as in (11), then (23) can be integrated and expressed entirely in terms of a0 and the number of particles in a monolayer, N m. The result, in its simplest form, is V(z) = Nma~ (z/ao) 3-D

(2 < D < 3),

(24a)

V(z) = Nma~ [1 + ln(z/ao)]

(D = 3, nonuniform space filling),

(24b)

V(z) = Nma~

(D = 3, uniform space filling),

(24c)

for ao < z < s Here the unspecified prefactor in (23) and the integration constant have been calibrated by the condition V(z)/V(%) -- (z/ao) 3-D or V(z)/V(a o) -- ln(z/a o) for large z/a o (scale invariance), and by equating the monolayer volume V(a 0) to Nma~. Other calibrations yield similar results [43, 110].

89 liquid

Figure 6. Equidistant interface, I z, at distance z from a self-similar fractal surface.

640

The two ways in which a three-dimensional surface can fill space, nonuniformly or uniformly, are discussed in [36, 43, 81, 111 ]. In the nonuniform case, the surface visits only certain regions in a compact manner and therefore, by leaving other regions empty, contains pores of many sizes. Examples for such surfaces, satisfying (23) with D = 3 and a nonzero prefactor, are certain silica and alumina xerogels [12, 39, 80, 81, 111 ]. A uniformly space-filling surface, by contrast, contains pores of one size only. A good example is a zeolite with channel width w and comparable wall thickness, which is uniformly space-filling at length scales above w. In this case, the volume V(z) trivially remains constant for all z > ao, if we choose a0 = w. This illustrates (24b) and the mechanism by which the prefactor in (23) may be zero. The expressions (24) show that the film volume grows, with increasing film thickness, increasingly slowly as the fractal dimension increases from two to three. They quantify that on an increasingly convoluted surface there is less and less space for a film to grow. The slow growth on a high-dimensional surface may seem to contradict the intuition that a highly irregular surface should have a large surface area and hence should be able to support a large film volume. There is no contradiction, however: In (24), we compare surfaces with variable D and fixed number of surface sites (adsorption sites), N m. In the surface-area consideration, one compares surfaces with variable D and fixed diameter L (largest distance between two points on the surface). Thus, the two opposed conclusions are due to different comparison frames. To switch from one frame to the other, one simply uses the relation Nm =

(L/gmax)3(gmax/ao)D,

(25)

in which the first factor counts how many fractal, identical "pieces" the surface consists of and the second factor counts the number of surface sites on each piece. For example, if the surface is fractal over the entire range of length scales between a0 and L ( g m a x = L), then N m = (L/ao) D and substitution into (24) shows that the film volume indeed increases with increasing D, for fixed L and' z. Similar differences in the performance at constant N m and at constant L, as a function of fractal surface dimension, exist also for diffusion-controlled reactions [20, 52-54], electrochemical response [53, 54, 59, 61 ], and other applications. This completes our analysis of the equidistant interface I z and the associated film volume V(z), in which we have treated the thickness z as a variable that can take arbitrary values. The variational determination of the equilibrium value for z and the construction of the isotherm is now easy: Substitution of (7) and (22) into the grand potential, Eq. (6a), gives A.Q[I z, Al.t] = n rjz [_tx(z,)_3 _ Akt]dV(z') + ~dV(z)/dz, a0

(26)

where we have used that dV(z') is the volume of a layer of thickness dz' at distance z' from the surface (equidistance property of Iz,), and have set the lower integration limit equal to the monolayer thickness to ensure a finite integral. Minimization with respect to z yields the algebraic equation (replacing the Euler equation (13)) n[--txz - 3 - Al.t]dV(z)/dz + t~dV2(z)/dz 2 = 0

(general surface)

(27)

n[-o~z -3 - A~t] - ( D - 2)~z -1 = 0

(fractal surface )

(28)

for the equilibrium value of z. The use of (24) in arriving at (28) is limited to the cases (24a, b), of course. In the case (24c), there is no isotherm to construct and we shall disregard it in the

641 sequel. The left-hand side of (28) increases with increasing z, so that the equation has a unique positive solution z for every value of Ag. The equation is equivalent to a cubic equation for z, which can be solved exactly. But the basic features of the solution can be deduced directly from the low- and high-Ag regime in (28): For zig ---) ..oo, the surface-tension term can be neglected, while for zig ~ 0, the substrate-potential term can be neglected. This gives z - ( ~__~)1/3

(Ag --+ _._~),

(29a)

z ~ - (D - 2).._.____~ nag

(Ag ~ 0).

(29b)

-

The crossover between the two regimes occurs at the chemical potential and film thickness, Ag c and zc, at which the asymptotes (29a, b) intersect, i.e., at

A~I,c 1= - 0~-1/2 [(D zc :=

2)6/n] 3/2,

(30)

(D - 2)6 "

(31)

Substitution of the equilibrium film thickness z as a function of Ag into N = nV(z) [recall Eqs. (12, 22)] yields the adsorption isotherm. Thus, by extrapolating the asympotic expressions (29) to where they intersect, inserting them into (24), and using ao - n-u3 to eliminate %, we obtain the results displayed in Table 3.

Table 3 Multilayer adsorption on a fractal surface, with Ag and Agc given by (5) and (30). Van der Waals wetting

Capillary wetting

Range of Ag 9 D --~ 2 9 D --, 3

Ag < Ag c range increases range decreases

Ag >__Ag c range vanishes range increases

Isotherm: 9 2 Nm) and that fractality starts at the length scale of the thickness of a monolayer. The regimes A~t < A~tc and All, > A~tc are identified as van der Waals wetting and capillary wetting because the isotherm depends only on the substrate potential strength a and surface tension ~, respectively The chemical potential A~tc given by Eq. (30) is therefore the transition point anticipated in Sect. 2.4 for a fractal surface. It is the fractal analog of the capillary-condensation point A~tc, Eq. (20), in a single pore. We divide the discussion into several subsections. The transition from van der Waals wetring to capillary wetting, which for simplicity we treat as sharp transition in this section, will be analyzed further in Sect. 4. Experimental examples will be presented in Sect. 5. 3.1 Earlier Derivations The power law for van der Waals wetting was first obtained in [44] (see also [ 109]). It was derived by assuming that films are sufficiently thin that surface tension can be ignored. The whole set of isotherms (32-35), including the logarthmic dependence for D = 3 and the coexistence point (30, 31) of the two types of wetting, were obtained in [101] (see also [45]). In fact, our derivation here is a streamlined version of the one in [ 101]. A more refined analytic calculation, in which the interface was not equidistant and the substrate potential was not simply a function of dist(x, S), was performed in [112] and reproduces the results (30-33) within a factor of order one. Other analytic treatments give similar agreement [ 113, 114]. The power law for capillary wetting or equivalents thereof, on the other hand, has been discovered and rediscovered in several different contexts, such as pore filling under hydrostatic pressure [ 115-117], third sound in superfluid 4He films [ 118], adsorption in micropores [ 119], and capillary condensation in mesopores [120-124]. Even the fractal generalization of the BET isotherm [109, 110, 125-128], which is unphysical beyond a few layers, coincides with (33b, 35b) in the limit p ~ P0- Most of these contexts neglect the substrate potential or model it inadequately. As a result, they give no estimate of the pressure above which (33b) would be valid, give prefactors which differ from the one in (32b), or give no prefactor at all. We therefore will focus our discussion on the present framework. 3.2 Nonclassical FHH Isotherms The classical FHH isotherm, Eq. (14), is of the form

N o, [-ln(p/p0)]-Y,

(36)

with y = 1/3. It is included in the fractal isotherm as the special case in which the surface irregularity vanishes. Indeed, in the planar-surface limit D -~ 2 the transition point Al.tc approaches zero and the associated film thickness zc diverges, so that for D = 2 the capillary wetting regime is absent in the isotherm (32) and only the van der Waals regime with y = 1/3 exists. We call (36) with y ~ 1/3 a nonclassical FHH isotherm because a departure from the value 1/3 signals a departure from the classical planar surface geometry, a departure from the classical inversedistance-cubed law for the energy of a particle at some distance from the surface, or both. Experimentally, it is weN-known that many adsorption data can be fitted to (36), but rarely with an exponent equal to 1/3. Typical experimental exponents y range between 0.4 and 0.7 [85, 87, 129-131]. Following Halsey [132], the discrepancy has been rationalized for a long time by a model in which the surface is planar, but patchwise energetically heterogeneous.

643 Each patch supports a film obeying Eq. (14), and the substrate potential strength ot varies from patch to patch. This model, under suitable assumptions for the distribution of or, leads to isotherms that can be fitted to (36) with an exponent y = 0.4 [87, 131,132]. The nonclassical FHH isotherms in Table 3, covering the whole range 0 < y < 1 by virtue of 2 < D < 3, offer a very different interpretation of experimental isotherms with y ~: 1/3. They replace Halsey's hypothesis that the surface is geometrically homogeneous (planar) and energetically heterogeneous by the hypothesis that the surface is geometrically heterogeneous (fractal) and energetically homogeneous. The merits of the fractal interpretation, as a working hypothesis, are considerable: (i) Halsey's interpretation requires the specification of a whole function, namely the distribution of ~ values, which is difficult to test by independent experiments and therefore must be regarded as phenomenological. The fractal interpretation requires the specification of a single parameter, D, which can be tested by independent experiments (Table 1) and by internal thermodynamic consistency (Sect. 4). (ii) The variable o~ values in Halsey's model may be attributed to variable-index domains in a polycrystalline surface or other defects in a planar surface. But such local energy differences are rapidly averaged out as the adsorbed film grows beyond a few layers and cannot account for y ~ 1/3 for thick films. By the same argument, the local energy differences (departures from the substrate potential (7)) that may exist on a fractal surface as a result of the geometric heterogeneity, are averaged out in films exceeding a few layers. Exact calculations of the potential U(x) for a fractal surface confirm this [133]. Even at coverages less than a layer, energetic heterogeneities are irrelevant on a fractal surface at sufficiently high temperatures [ 125] (see also [29]). Thus, under a wide range of circumstances, energetic heterogeneities of Halsey's type can be disregarded, leaving geometric heterogeneity as the most likely source of experimental exponents y ~: 1/3. (iii) Halsey's model is designed to explain exponents y > 1/3. But experimentally, one finds also y < 1/3, although less frequently. The fractal model is capable of explaining both deviations and covers to the best of our knowledge all experimental exponents that have ever been observed. The correspondence between y and the fractal dimension is illustrated in Fig. 7. It shows that, for 1/3 < y < 1, the dimension can be uniquely reconstructed from y as D = 3 - y, that the resulting D value lies between 2 and 8/3, and that the value of y necessarily implies capillary wetting. This agrees remarkably well with the earlier suggestion, predating the fractal concept, that exponents y > 1/3 are "probably explicable by some reversible capillary condensation" [ 129], and converts that suggestion into a quantitative statement about surface irregularity. For 0 < y < 1/3, the correspondence between the isotherm exponent and the fractal dimension is no longer one-to-one. That is, the fractal dimension may be either D = 3 - y (capillary wetting), giving 8/3 < D < 3, or else D = 3 - 3y (van der Waals wetting), giving 2 < D < 3.

Zyy

= 3 - D [capillary wetting]

.

.

.

.

.

.

.

.

= (3 - D)/3 [van der Waals wetting] 1/3

-ilj~176 2

8/3

3

D

Figure 7. Isotherm exponent y as function of the fractal dimension D, as given by Eq. (33).

644 Which of the two possibilities is the appropriate assignment when only y is known, is a question that has been raised in several experimental studies using the isotherms in Table 3. We will answer that question systematically in Sect. 4. A partial answer is provided by the inequality 3-3y

< D < 3-y,

(37)

which translates the two possibilities into upper and lower bounds for the sought-after dimension (Fig. 7). These bounds, which are tight when y is small, may be taken as substitute for the missing one-to-one correspondence between y and D for 0 < y < 1/3. If the y value under consideration comes from the low-coverage regime in the isotherm, i.e., from the regime starting at N/N m = 1, one additionally has the role of thumb that the closer to 1/3 the y value lies, the more likely is the choice D = 3 - 3y (van der Waals wetting) the correct one. Indeed, the only way how a value y -- 1/3 can originate from capillary wetting at low coverage is if D -- 8/3 and simultaneously the film thickness z c at which capillary wetting sets in, Eq. (31), is of the order of a monolayermwhich would be an unlikely coincidence. Together, this rule and the above bounds give fairly strong guidelines for the interpretation of exponents in the range 0 < y < 1/3 in terms of fractal dimension [ 134].

3.3 Reduced Isotherm and Law of Corresponding States The isotherms (32, 34) offer a fully developed theory of multilayer adsorption on irregular surfaces with a minimum number of parameters: The number of particles in the monolayer, N m, describes the sample size; the fractal dimension D describes the surface geometry; the potential strength ~ describes the substrate-adsorbate interaction; the surface tension ~ describes the adsorbate-adsorbate interaction; and the number density n describes the adsorbate particle size (recall a0 = n-1/3). Interpreted in this way, the five parameters are clearly independent, none of them is dispensable, and none of them is adjustable. One gains additional insight into the analytic structure of the isotherms by expressing them in reduced variables. The reduced form is N/N m = [max{~, (D-2)T~3}] 3.D

(2 < D < 3),

(38a)

N/N m = 1 + In(max{ ~, y~3 })

(D = 3),

(38b)

/ an/l/3

; := ~-S-ff] y :=

'

t~ . t~n5/3

(39) (40)

The expression max {....... } selects the larger of the two arguments and automatically reproduces the van der Waals regime for ~ < ~c [N/Nrn < (N/Nm)e], and the capillary wetting regime for ~ ___~e [N/Nm > (N/Nm)c], where ~c := [(D-2)Y] -1/2,

(41)

(N/Nm) c := [(D-2)y] -(3-D)/2.

(42)

The variables ~, ~c, and T are all unitless and take the role of Ag, Ag e, and the triple (~, tj, n) in Table 3. The variable ~ is the film thickness z, measured in units of the thickness of a monolayer, one would have at chemical potential Al.t if the surface were planar. In agreement, the isotherm (38) for D = 2 reads N/N m = ~. The value ~c predicts the film thickness z c at which

645 capillary wetting sets in, Eq. (31), in units of the monolayer thickness; and the value (N/Nm) c predicts the associated coverage. By the remarks at the beginning of this subsection, the constant 3' measures the strength of the adsorbate-adsorbate interaction relative to the substrateadsorbate interaction. Since ao = n -1/3, it may be interpreted as the ratio of the liquid-gas free energy, ca~, to the potential energy, ~a~ 3, of a particle in a monolayer. We therefore call 3' the capillary-energy ratio. In wetting dynamics (spreading of liquids), a similar energy ratio, namely surface tension divided by the product of liquid velocity and viscosity, is known as the inverse of the capillary number [95, 133]. The reduced form of the isotherm displays the full D dependence of the adsorption process (note that ~ and 3' do not involve D), encapsulates the competition between the substrate potential and surface tension in the single number q,, and describes the dependence on the chemical potential through the single variable ~, the number of equivalent layers on a planar surface. As a result, it reduces the dependence on four parameters in Table 3 (D, ~, (~, n) to a dependence on only two parameters (D, 3'). The fact that 3' enters exclusively via the combination (D - 2)3' [Eqs. (38, 41, 42)] says that the liquid filling the hierarchy of small and large pores of the fractal surface has an "effective surface tension" equal to ( D - 2)~. Thus, compared to the liquid-gas equilibrium in a single pore, the fractal surface lowers the effective liquid-gas free energy by a factor of D - 2. A similar lowering of effective energy exists in chemical reactions on a fractal surface, for stationary diffusion of reactant molecules from an external source to the surface: There the effective activation energy is lowered, by a factor of 1/(D - 1), compared to the reaction on a planar surface [52, 136, 137]. The reduced isotherm leads to the following simple way of testing an experimental isotherm for fractality of the underlying surface. The idea is to employ the coverage N/N m, rather than the chemical potential A~t, as calibration scale, i.e., as variable to distinguish van der Waals wetting from capillary wetting and to determine absolute film thicknesses. Assuming that the monolayer value N m is known (say from a BET analysis of the low-pressure part of the isotherm), all one needs to do is to plot the coverage N/N m as a function of In(p/p0); test whether the data, starting at N/N m = 1, obeys one of the power laws (33a, b); andmin the event that the data obeys (33a) at low coverage and (33b) at high coverage (with the same D)---test whether the crossover from (33a) to (33b) occurs at the coverage predicted by Eq. (42). The start of the power law at N/N m = 1 reflects the start of the fractal regime at the molecular scale. The test for D = 3 proceeds similarly. Since the expression max {....... } in the reduced isotherm is the film thickness, in units of the monolayer thickness, on the fractal surface (both for van der Waals and capillary wetting), one can convert any experimental coverage into a corresponding film thickness z: / N ~l/(3-D) z = /N~-~) a~

(2 < D < 3),

(43a)

z = exp

(D = 3).

(43b)

- 1 ao

This gives the length scale of surface irregularities probed at coverage N/N m [ 134]. The smallest and largest film thickness computed from (43), as the coverage varies over the range in which the data obeys (33) or (35), gives the inner cutoff s (-- %) and outer cutoff s respectively, of the fractal regime of the surface. Representative values of the parameter 3' and related data are listed in Table 4. The film thickness ~/o~n/o is the smallest possible film thickness at which capillary wetting may set in. It

646 Table 4 Substrate potential strength or, film thickness ~/~n/c~ at which capillary wetting sets in for D = 3 [Eq. (31)], and capillary-energy ratio 3' [Eq. (40)], for nitrogen on different solids. The surface tension and number density of liquid nitrogen, in coexistence with its vapor at T = 77.347 K, are o = 8.85.10 -16 erg,~ -2 and n = 1.738-10 -2 ,~-3 [138]. The resulting monolayer thickness is a 0 = n -1/3 = 3.86 ,~. Solid

c~ (erg,~ 3)

SiO 2 (quartz) C (graphite) Si A1 Ag Au

6.19-10 -13 1.42.10 -12 1.59.10 -12 2.13-10 -12 2.35-10 -12 2.62.10 -12

[89] [89] [89] [89] [112] [89]

qo~n/o (.~)

7

3.49 5.28 5.58 6.46 6.80 7.18

1.23 0.534 0.478 0.357 0.323 0.289

measures the competition between the substrate potential and surface tension in terms of a length and is a characteristic length also for multilayer adsorption on other nonplanar surfaces (see Sect. 2.4 and [ 100-104]). When the substrate potential is so weak or surface tension is so strong that qom/c < ao, which is equivalent to ~/> 1, and if D > 2 + 1/3', then capillary wetting sets in at a film thickness that is nominally less than the thickness of a monolayer [Eqs. (31, 41)]. In this case, adsorption obeys the capillary-wetting isotherm for all N/N m > 1, i.e., the regime of van der Waals wetting is absent. This means that capillary forces may dominate already at the stage of a monolayer and conforms with the observation that the monolayer has an effective liquid-gas free energy exceeding the substrate-potential energy whenever (D - 2)7 > 1. On a planar surface, the situation 3' > 1 expresses that the lateral attraction among adparticles in a monolayer is stronger than their attraction to the solid. An instance of such a weak substrate potential is SiO 2 in Table 4. The weak substrate potential of SiO 2 can be attributed to its low polarizability (electric insulator). Accordingly, the increase of the substrate potential, as we go down in the table, reflects the increase in polarizability with increasing metallic character of the solid. If one views the capillary-energy ratios in Table 4 as significantly different, then the variety of ratios predicts a corresponding variety of different behaviors of the isotherm (38). But if one regards the capillary-energy ratio as only weakly varying, T-- 1, then the reduced isotherm has the status of a law of corresponding states for films on a fractal surface, quite analogous to the law of corresponding states for bulk fluids. Indeed, just as bulk fluids satisfy, at least semiquantitatively, a universal equation of state that makes no reference to material constants if expressed relative to the pressure Pc, temperature T e, and number density n c at the liquid-gas critical point [139], the films on a fractal surface satisfy a universal isotherm making no reference to material constants (other than the fractal dimension) if expressed in the reduced form (38). The independence of material constants manifests itself in the relation pe/(kTenc) = 0.27 + 0.04

(normal bulk fluids),

(44)

o/(otn 5/3) = 0.76 + 0.47

(adsorbed films),

(45)

respectively, where the estimate (45) is based on Table 4 and hence is preliminary.

647

3.4 Does the Roughness of the Substrate Enhance Wetting? In the study of general 3-phase equilibria between a liquid, its vapor, and a solid (Fig. 2), this question [ 140] has been investigated from many different angles (interfacial energies, contact angle, spreading kinetics, transition to complete wetting) and for several types of roughness [83, 85, 135, 140-152]. The genetic answer is yes, in the sense that the contact angle decreases or increases with increasing roughness (enhanced or diminished wetting), on a surface without overhangs, depending on whether the contact angle on the planar surface is less than or larger than rff2 [85,140]. Here, for completely wetting films on a fractal surface with Overhangs (Fig. 6), we address the question in terms of whether the coverage N/N m grows faster or slower than on a planar surface as A~t ~ 0 (enhanced or diminished wetting). We assume for simplicity that the surface is fractal at all length scales above a 0, i.e., that the outer cutoff is formally infinite. The surface then has voids of all sizes and, based on the picture that capillary condensation in voids of ever larger size systematically enhances adsorption [ 129], one might expect that the coverage always grows faster than on a planar surface. However, this is not the case. The isotherms (32, 34), or equivalently (38a, b), show that the coverage grows faster in the limit Akt ~ 0 if and only if 1/3 < 3 - D < 1. We therefore have enhanced wetting if 2 < D < 8/3, diminished wetting if 8/3 < D < 3

(46a) (46b)

[ 112]. In Fig. 7, the situation (46a) corresponds to 1/3 < y < 1 and identifies the isotherm exponents which give enhanced wetting as those from which D can be uniquely reconstructed. To explain how (46) arises, including the paradoxical consequence that the coverage grows fastest on a nearly planar surface (D ~ 2), we begin at low A~ and note that the film thickness at low A~t is the same as on a planar surface, Eq. (29a), so that the coverage depends on the fractal dimension only through the film volume, which decreases with increasing D. Hence, for van der Waals wetting the coverage always decreases with increasing D. When we enter the capillary wetting regime, the film thickness becomes D-dependent and grows faster than on a planar surface, Eq. (29b). The question then is, can this fast growth of the film thickness compensate for the slow growth of the film volume, so as to make the isotherm for D > 2 overtake the classical FHH isotherm at sufficiently high Al.t? The answer is yes if the film volume V(z) does not grow too slowly with increasing film thickness z, i.e., if the fractal dimension is not too high. Thus, a surface with 2 < D < 8/3 is sufficiently porous to induce capillary condensation and, at the same time, sufficiently "open" for the condensate to grow in excess of the growth on a planar surface. A surface with D _>8/3 is also sufficiently porous to induce capillary condensation, but too "closed" for the condensate to grow as fast as on a planar surface. On a nearly planar surface, with D very close to 2, the van der Waals regime will be very extended and the coverage will be close to that on a planar surface up to very high values of Alx; but no matter how close to zero A~tc may be, once Akt exceeds the value Al.tc, the nonplanarities of the surface at those large length scales will be sufficiently magnified to induce capillary condensation and make the coverage eventually grow as (-A[.t) -(3-o). Thus the paradox of fastest growth on a nearly planar surface is resolved by the observation that the limits Al.t -~ 0 and D --~ 2 are not interchangeable [ 112]. A convenient way to analyze the growth of the coverage in more detail and compare it with the growth on a planar surface is to plot the reduced isotherm (38) as a function of ~. Since ~ is the coverage one would have at chemical potential Akt if the surface were planar, such a plot is in fact equivalent to a t-plot or comparison plot [ 129], using the classical FHH isotherm as

648

4

3 -

~

N/N m 2

-

. ~

(b)

1

0 0

1

I ,, 2

I 3

I 4

J

5

Figure 8. Reduced isotherm (38) as a function of the unitless variable ~: (a) for D = 2; (b) for D = 2.40 and T = 0.323; (c) for D = 2.40 and y = 2.50. Depending on whether ~ is interpreted as film thickness (in units of the monolayer thickness) or amount adsorbed (in units of a monolayer) on the planar reference surface, the curves represent the t-plot or the comparison plot of the adsorption isotherm on the respective substrates.

standard isotherm. Such plots for selected D values and capillary-energy ratios y are shown in Fig. 8. The straight line (a) is the isotherm on the planar reference surface. Curves (b) and (c) are the isotherms on a 2.4-dimensional surface for two different values of y. They both lie above the planar-surface isotherm for sufficiently large ~, in agreement with (46a). In (b), the value of T, representing nitrogen on silver, is low enough for the van der Waals regime to extend up to ~ = 2.78 [Eq. (41), second dotted line] and for capillary wetting to make the isotherm cross the planar-surface isotherm only at ~ = 4.64 [third dotted line]. In (c), the value of y has been chosen high enough for the van der Waals regime to be absent, i.e., for capillary wetting to set in at ~ = 1 [Eq. (41), first dotted line]. This makes the isotherm (c) lie above the planarsurface isotherm for all ~ > 1. We note that the break points at ~ = 2.78 and ~ = 1 in (b) and (c) result from our treating the crossover from van der Waals wetting to capillary wetting as a sharp transition in this section. They are smoothed out if one solves Eq. (28) for the film thickness exactly and puts the resulting isotherm into reduced form. Likewise, the submonolayer regime in the isotherms in Fig. 8 (N/N m < 1) should not be taken at face value, because the underlying expression (32a) does not describe submonolayer adsorption properly. Figure 8 gives a fine-tuned picture of how a surface with 2 < D < 8/3 enhances wetting, by displaying the value ~e at which the fractal isotherm intersects and overtakes the planar-surface isotherm. We call this point the onset of enhanced wetting and define it generally as ~e := min{~'" (~---m-m)(D,~)>(~mm}(2,~)for all ~> ~'; ~'> 1} = max{ 1, [ ( D - 2)T]-'(3-D)/(8-3D)},

(47a) (47b)

where the condition ~'> 1 serves to exclude the submonolayer regime from consideration. The result (47b) is simply the solution of [ ( D - 2)]t~3}] 3-D = ~ if this solution is > 1, and equal to

649

20. 15. I0.

2 1.5 ~ ~

~

~

'

~

I

~

I

~ - ~

I 1

2

2 2

2 4

,

,

,

i

2.6

,

I,

~ ,

,

~

~

...._._.

_........

I

9

2.8

3

D

Figure 9. Onset of capillary wetting, ~c [Eq. (41), dashed curves], and of enhanced wetting, [Eq. (47), full curves], as function of the fractal dimension. The curves from top to bottom are for y = 0.323, y = 1.50, and 3t = 2.50. For fixed D and y, the regimes ~ < ~c, ~ > ~c, and ~ > ~e amount to van der Waals wetting, capillary wetting, and enhanced wetting. To the right of the vertical line at D = 8/3, enhanced wetting no longer exists.

one else. The significance of ~e is that it distinguishes whether in (46a) the onset of enhanced wetting occurs at low or high chemical potential (low or high ~e)" We illustrate this in Fig. 9. For D ~ 2, all curves ~ diverge in agreement with our discussion of Eq. (46a). For D ~ 8/3, the curves ~e diverge, too, if'f < 3/2. In this case capillary condensation is not strong enough to overcome, at some low chemical potential, the slow growth of the film volume as D ~ 8/3. Accordingly, the curve ~e for nitrogen on silver first drops with increasing D until it reaches a minimum of ~e = 4.41 at D = 2.32, and then rises again. If 3t > 3/2, however, capillary condensation is strong enough to make the onset of enhanced wetting drop all the way down to the chemical potential at which the planar reference surface supports a monolayer, as D increases. For example, the fact that the isotherm (c) in Fig. 8 lies above the planar-surface isotherm for all > 1 is mapped into the value ~e = 1 for D = 2.40 and y = 2.50 in Fig. 9. We note that Fig. 9 also clarifies the status of the borderline case D = 8/3 in (46): For D = 8/3, wetting is enhanced if 3t > 3/2 (~e < oo), and diminished if T < 3/2 (~e = oo). These results for the onset of enhanced wetting on a fractal surface lead to guidelines for the interpretation of t-plots and comparison plots which differ substantially from the traditional guidelines. The traditional wisdom [129] is that (i) capillary condensation, with or without hysteresis, manifests itself in an "upward swing" of the t-plot or comparison plot, i.e., in a plot that lies above the straight line representing the standard isotherm; and that (ii) the presence of micro- and/or mesopores manifests itself in an upward swing of the plot, with or without the adsorption equilibrium being controlled by capillary forces. The presence of an upward swing, also called enhanced adsorption in [ 129], amounts to a finite value of ~e in Fig. 9. Thus, Fig. 9 confirms the validity of statement (i) for 2 < D < 8/3, but shows that the statement is not valid in the range 8/3 < D < 3 because in that range capillary

650 condensation does not enhance adsorption. The validity of statement (ii) is even more restricted. Indeed, if we interpret the statement as saying that enhanced adsorption starts at ~ = dmin/a0 where dmi n is the smallest pore diameter, enhanced adsorption should start at ~ = 1 on a fractal surface with inner cutoff a 0. But since van der Waals wetting never enhances adsorption, the statement is valid only if ~e = 1. Hence statement (ii) is valid only if 2 + 1/3, < D < 8/3 [Eq. (47b)]. It is easy to see how the guidelines need to be modified to be valid without restrictions. All one has to do is to weaken them from "if-and-only-if" statements to "if" statements. That is, capillary condensation is operative and micro- or mesopores are present if there is an upward swing in the plot. The converse, however, is not true: As fractal surfaces show, pore filling may occur, by capillary condensation or otherwise, without an upward swing.

3.5 Comparison with Adsorption in a Single Pore Having obtained the transition from van der Waals wetting to capillary wetting in a single pore (Sect. 2.4) and on a fractal surface (Table 3), we now wish to highlight the parallels and differences between the two transitions. We compare the transitions by analyzing their dependence on the energy parameters and surface geometry. The chemical potential Agc, Eqs. (20) and (30), sets the energy scale for the transition. The film thickness z c, Eqs. (21) and (31), sets the associated length scale. As in Sect. 3.3, we express Agc in units of the potential energy per particle in a monolayer, czn, and z c in units of the monolayer thickness n -1/3. This reduces the dependence on o~ and ~ to a dependence on the capillary-energy ratio 3, and leads to the results in Table 5, the fractal entries being familiar from Sect. 3.3. We begin with the dependence on the energy parameters. Since ABc and z c predict the point at which capillary forces become stronger than the substrate potential, their value must decrease with increasing 3,. Lines 1 and 2 in Table 5 confirm this both for the fractal surface and the single pore. The specific dependence on 3, is quite different, however: The magnitude of the exponent of 7 is consistently larger in fractal case (by a factor of 3/2 for Agc, and 2 for Zc). Thus the onset of capillary wetting depends strongly on the competition between cz and o in the fractal case, and by comparison weakly in the single-pore case.

Table 5 Onset of capillary wetting on a fractal surface and in a single pore, with 3, given by (40). The first two entries in the last column are the leading terms of the asymptotic expansion. Fractal surface

Single pore

Agc/(om)

- [ ( D - 2)T] 3/2

-27/(Rn I/3) [R --4 oo]

Zcn 1/3

[(D - 2)3,]-1/2

(23'/3)-1/41/Rnl/3 [R --4 oo]

Dependence on energy

strong

weak

Dependence on geometry

D

R

Dependence on size

none

R

Planar-surface limit

z c ~ oo [D ~ 2]

z c --~ oo [R ~ oo]

Maximally nonplanar surface

z c = n-1/3T-1/2

Zc= n-1/3(l+ ~/l+V8T/3 ) -1

= 3-7 ~, [ D = 3 ]

= 1-2 ,~ [R = (1/2)n -1/3]

651 Next we consider the transition as function of surface geometry, restricting attention to the film thickness z c. Just as z c in the single pore depends on the surface geometry via the pore radius R, the value of z c on the fractal surface depends on the surface geometry via D. However, while in the single-pore case this dependence on geometry involves simultaneously a dependence on size, R, there is no size dependence in the fractal case because a fractal surface is scaleinvariant (D is a size-independent measure of surface irregularity [12]). Capillary effects should become weak and z c should increase, as the degree of nonplanarity of the surface decreases. This is indeed the case, as the planar-surface limits in Table 5 show; both for the fractal surface and the single pore, there is no capillary condensation at any finite film thickness in this limit. Conversely, capillary effects should be strongest and z c should be smallest when the surface is maximally nonplanar. The notion of maximally nonplanar is naturally defined as D = 3 in the fractal case, and as a pore that can hold exactly one particle in the single-pore case. The resulting values for z c in Table 5 (based on the values from Table 4 for n and 7) are of atomic size in both cases, reinforcing the conclusion that capillary forces may control the formation of already the first layer. It might be argued that the onset of capillary consensation at a film thickness of a few ,~mgstroms, on a maximally nonplanar surface, is not really surprising because, if pores with diameter comparable to the diameter of a single adsorbate particle are present, these pores are automatically full after the first or second layer. But capillary condensation at the level of one or two layers is not trivial because in the fractal case a surface with D = 3 has room for an unlimited number of layers [Eq. (34)]. In the single-pore case, it is not trivial because condensation occurs at z c < R (Table 5). In this light, we consider it as a signature of remarkable internal consistency that nonplanar surface geometries of such different types yield similarly low values for z c in the limit of maximum nonplanarity. This consistency suggests that the model of capillary condensation in a single pore, when treated including the effects of the substrate potential and applied to micropores, may in fact perform considerably better than what recent model studies of adsorption in micropores seem to suggest [ 153, 154]. A somewhat different picture of the transition from van der Waals wetting to capillary wetting emerges when we compare the fractal and single-pore case away from the transition point. On the fractal surface, the isotherm gives the full structural information about the surface (i.e., the dimension D) both below and above the transition point, as shown by the D dependence of Eqs. (32, 34) or (38). In principle, therefore, the D value can be obtained in three different ways, namely from the isotherm exponent in the van der Waals regime, from the isotherm exponent in the capillary regime, and from the transition point itself. The multiple manifestation of D is naturally a consequence of the recurrence of the same structure at all length scales. In a single pore, by contrast, the isotherm exhibits only little information about R below the transition point (the film thickness z 1 in Fig. 4 depends only weakly on R for Ag zc

(small pores), (large pores).

(48a) (48b)

The filling of small pores, taking place when Ag < Ag c, is governed by the substrate potential, in the sense that the isotherm jump in a single pore due to capillary condensation is small. In this sense capillary condensation is negligible in small pores. The filling of large pores, taking place when Ag > Ag c, is governed by capillary condensation, in the sense that most of the increase in adsorption comes from the jump in pores with radius R-- 2~/(-nAg) [Eq. (20)], while the contributions from van der Waals wetting in larger pores (where capillary condensation has not occurred yet) are negligible. The reason why van der Waals wetting is negligible in large pores is because the film thickness is at most R1/E(6om/tJ) TM prior to capillary condensation [Eq. (21)], which is small compared to R for large R. Thus small and large pores are indeed filled by van der Waals and capillary wetting, in agreement with Table 3. Explicit calculations of the isotherm on a fractal surface in terms of single-pore adsorption have been carried out in Refs. [44, 112].

3.6 Effective Potential and Extension to Arbitrary Surface Geometries Can the variational isotherm calculation, as underlying the results in Table 3, be extended to other irregular surface geometries? If so, can the calculation be inverted to extract information about the geometry from a general experimental isotherm? The answer is yes. To see that and gain additional insight into the nature of the adsorption equilibrium, we return to Eq. (28) which determines the equilibrium film thickness z on a fractal surface and can be written as Ueff(z) - Ag = 0 Ueff(z) := - ~ z 3 - (D - 2)~/(nz).

(49) (50)

The function Ueff(z), which we call effective potential has a simple and important physical meaning. It is the difference between the energy of adding a particle to an equilibrium film of

653 thickness z on the fractal surface and the energy of adding a particle to bulk liquid, all effects included. It separates the adsorption isotherm into two regimes, van der Waals wetting and capillary wetting, by decomposing the energy into a sum of the substrate potential -ot/z 3, dominant at short distances, and the "capillary p o t e n t i a l " - ( D - 2)cy/(nz), dominant at large distances. The slow growth of the capillary potential at large distances induces the fast film growth indicated in Table 3. More generally, the lower the effective potential is as a function of distance z, the larger is the equilibrium film thickness at given chemical potential Ag. Remarkably, the capillary potential in (50) acts like a Coulomb potential, with "electric charge" ( D - 2)~/n. The electrostatic analogy makes clear that the capillary potential is overwhelmingly stronger than the substrate potential at large distances. In the spherical pore, the equilibrium (or metastable) interface is an equidistant interface, too, and the associated film thickness z is determined by Eq. (49), too, but now with the effective potential

Ueff(z) =

- ot/z 3 Al.tc

2(y/[n(R

- z)]

if z < z c

(51)

if z c < z < R

where z c and A~tc are defined as in Eq. (21) [recall Eqs. (15, 16) and Fig. 4]. The constancy of the effective potential (51) at distances larger than z c, which implies that the film can have any thickness between z c and R at A~t = A~tc, describes the vertical jump in the isotherm (Fig. 5). This extends the effective potential to include all physically realizable states on the isotherm, regardless of whether they are equilibrium states, metastable states, or even unstable states (vertical jump in the isotherm). Under this extension, constancy of the effective potential is the most extreme form of slow growth of the effective potential and entails the most rapid growth of the film thickness. For adsorption on a general surface, in terms of equidistant trial interfaces, the condition (27) for the equilibrium film thickness z leads again to Eq. (49), with effective potential Ueff(z )

=

o~ z3

~

(3"~VP'(z) . n V'(z)

(52)

The functions V'(z) and V"(z) denote the first and second derivative of the film volume V(z), Eq. (22), and the expression is subject to the condition that the right-hand side of (52) is monotone increasing with increasing z. Under this condition, Eq. (49) has a unique solution z for every A~ and the solution represents the equilibrium state. If the fight-hand side of (52) is not monotone increasing, Eq. (49) with (52) has more than one solution z and the physically realized film configuration, depending on whether one moves along the adsorption or desorption branch of the isotherm, is found by a stability analysis similar to Eqs. (18a, b). The stability analysis yields

Ueff(z) = nde+_. - ~ +

W(z)1

n V'(z)

(53+)

for the adsorption (+) and desorption (-) isotherm, respectively. Here nde_ denotes what we call the upper and lower nondecreasing envelope, defined as

654 nde+ qo(z) := max{ qo(t)" t ___z; qo'(t) _>0 }, nde_ q)(z) := min{qo(t): t _>z; ~o'(t) > O}

(54+)

(54-)

for a function q~(z) with derivative cp'(z). The construction of the two envelopes is illustrated in Fig. 11. As in the spherical-pore potential (51), which is a special case of (53+), the potentials (53+) lead to a vertical jump in the isotherm where they are constant. Clearly, they reduce to (52) if the function on which nde_+ acts is monotone increasing. We note that V'(z) is always positive, V"(z) is zero if and only if the surface is planar, V"(z) is negative for the spherical pore and every fractal surface, and V"(z) is positive for every convex solid. Thus, to compute the adsorption/desorption isotherm on an arbitrary surface S, one has the following simple steps: 9 Compute the function V(z), i.e., the volume of points outside the solid whose distance from the surface is less than or equal to z; this is the only surface-geometric input needed. If no analytic expression for V(z) is available, the computation requires only the sorting of distances dist(x, S) of points x close to the surface [ 156]. 9 Compute the effective potential Ueff(z) according to (53). If no analytic expression for V(z) is available, use V"(z)/V'(z) = (d/dz) [In V'(z)] for robust numerical differentiation. 9 Solve Eq. (49) for z as a function of Ag and substitute the result, z = z(Ag), into N = nV(z); this is the desired isotherm, N(Ag) = nV(z(Ag)).

(55)

The procedure is remarkable because, unlike the original problem (12, 13), it does not require to solve any differential equation. It operates with a minimum of geometric input, namely a function of one variable, which automatically maps two surfaces with "similar degree of irregularity" (Fig. 1) onto the same function V(z). The presence of the second derivative V"(z) in (53) corresponds to the Euler equation (13) being a second-order differential equation. The bifurcation generated by the operator nde+_in (53) automatically selects, from the several solutions of the Euler equation, the ones that are realized during adsorption and desorption, respectively. It may lead to hysteresis loops with any number of steps in either isotherm (Fig. 11). For the case in which there is no hysteresis, the procedure has been described before [43, 157].

Ueff

(+)

f f

S Figure 11. Construction of the effective potential Ueff(z) given by (53). The solid curve is the f u n c t i o n - ~ z 3 + a V"(z)/(nV'(z)) [schematic]. The dashed lines, together with the adjoining monotone parts of the solid curve, represent Ueff(z) for adsorption (+) and desorption (-).

655 When there is no hysteresis and therefore the form (52) of the effective potential applies, it is possible to invert the procedure and deduce the function V(z) from the adsorption isotherm N(A~t): By differentiation of (55) with respect to A~t one obtains V'(z(A~I,))

= n -1N'(A~I,)/z'(A].I,)

g"(z(A].l,)) = n-l[N"(Akl,)-

N'(A].t)z"(A~l,)[z'(Akt)]][z'(A~l,)] 2,

(56)

(57)

where N', N", z', z" are the derivatives of the functions N(A~t) and z(A~). Putting (56, 57) into (52) and using the equilibrium condition (49) yields

(58)

This is a second-order nonlinear differential equation for the film thickness z(Als), in terms of the given isotherm N(AI.t), subject to Akt > A~trn where Aktm is the chemical potential at monolayer coverage. The initial data for (58) is z(A~l,m) = n-l/3

z'(Aktm) = n-l/3 N'(A~m)/N(AILtm)'

(59a) (59b)

expressin~ that the monolayer thickness is n- 1/3 and that the area of th e monolayer, V'(n-l/3), equals n-2/3N(Alarn), which after substitution into (56) gives (59b). The solution of the initialvalue problem (58, 59) gives the film thickness z(A~t) at any chemical potential A~t > Atxm. Since z(A~t) is a monotone increasing function, it can be inverted to obtain the chemical potential as function of the film thickness, Akt(z), which yields the sought-after surface geometry as V(z) = n-lN(A~t(z)),

(60)

for any z > n-1/3. If desired, the volume V(z) can be converted into the pore-size distribution Vpore(Z), the cumulative volume of pores with radius less than or equal to z, which is defined-free of any model assumptions about the surface geometry--as the volume of space outside the solid that is inaccessible to spheres of radius z [21, 36, 43, 109, 128]. The connection between the two volumes is given by Vpor~(Z) = V(z) - z V ' ( z ) ,

(61)

for a surface that is neither convex nor concave [43]. This concludes our demonstration, by explicit construction, of the existence of a general one-to-one correspondence between surface geometry and experimental adsorption isotherms as announced in Sect. 2.3. More details will be published elsewhere. The inverse part of the correspondence, which transforms the isotherm N(AI.t) into the function V(z), is optimal in the sense that from a scalar function of one variable, N(Atx), one cannot expect to get more information about the surface than another scalar function of one variable. The correspondence shows that there is nothing special about fractal surfaces with regard to the computability and invertibility, as a matter of principle, of the isotherm. What does make fractal surfaces special is that their isotherms are power laws from which all geometric information (D, groin, gmax) carl be obtained without having to solve the differential equation (58).

656 4. W E T T I N G P H A S E D I A G R A M In our discussion in Sect. 3, we treated the transition from van der Waals wetting to capillary wetting on a fractal surface as a sharp transition. The sharp transition resulted from extrapolation of the asymptotic film thicknesses (29) to the point where they are equal to each other. But in reality, when Eq. (28) for the film thickness is solved exactly, the transition is smooth. This raises the question in what sense the notion of two distinct wetting regimes is well-defined independently of any approximation, how accurate the simplification of the transition between the two regimes as a sharp transition is, and in what sense the transition is a phase transition, similar to that for capillary condensation in a single pore. The same question arises for the transition from submonolayer adsorption to multilayer adsorption, which we treated as sharp, but which in fact is smooth, too. We address these questions in an operational way. Our goal is to map the qualitative regimes of submonolayer adsorption, van der Waals wetting, and capillary wetting onto quantitative intervals of chemical potential, (-oo, A~tm)' (Aktrn' A~tc)' and (A~tc, 0), so that the intervals, when used to analyze experimental isotherms, predict the pressure ranges in which one should, or should not, expect to find one of the power laws (33a, b). The challenge is to identify A~n (transition to multilayer adsorption) and Al.tc (transition to capillary wetting) in a way that is conceptually independent of whether the surface is fractal or not and independent of whether the isotherm computation ( 12, 13) is carried out exactly or approximately. With such an identification at hand, one may then compute A].I.m and A~c as function of D, which will be the desired wetting phase diagram for fractal surfaces. Depending on what approximations are used to compute the isotherm, the resulting phase diagram is approximate, too. We begin by examining how well the adsorption isotherm on a fractal surface, computed from Eq. (24) by solving Eq. (28) for z exactly, is approximated by the power laws (32a, b) [Eq. (24) with z from (29a, b)] which describe the transition as sharp. This is done in Fig. 12, for a choice of parameters that corresponds to one of the experimental examples in Sect. 5. The isotherm is plotted over a deliberately wide range of A~t values, including very low values where the multilayer framework no longer applies (recall Sect. 3.4), so as to exhibit the full asymptotic behavior for A~ --~ _.oo and A~t --~ 0. The figure shows that the isotherm rapidly approaches the power laws (32) as Akt moves in either direction from the point of intersection of the power laws. At the point of intersection, where the difference between the isotherm and the power laws is largest, the difference is about 20 %, which is small for a power-law crossover. The crossover interval, which we take as the range of N/N m values for which the isotherm differs by more than 10 % from the respective power law, spans a coverage ratio of 2.5, which is again small for a power-law crossover. In units of film thickness, Eq. (43a), the interval spans a ratio of 3.8; and in units of A~t, the interval spans a ratio of 10. Thus we conclude that the description of the transition from van der Waals wetting to capillary wetting as a sharp transition, as given by the power laws (32), is a good approximation for most practical purposes and can easily be replaced by an exact evaluation of Eqs. (24, 28) if need arises. The evaluation of Eqs. (24, 28) for other D values and substrates yields similar results for the rate of approach to the asymptotic power laws and the behavior in the crossover region. That is, there is no significant dependence of the crossover behavior, in relative units, on the fractal dimension and the material constantsmin agreement with the universal form of Eqs. (24, 28) when expressed in terms of the reduced variables ~ and ~t introduced in Eqs. (39, 40). The only thing that depends on D and the material constants is the position of the transition point on the Al.t and N/N m axis.

657

1000

100

10

zX~t/(kT)

[mL1tl I

[illl,tt

I

hillllll

-10000

],lltili~

-10

hiil|itl

'.htli~il i

Ir

-0.01

h,,t, ll t

]u,,~,~,

N/Nm

0.1

-0.00001

Figure 12. Adsorption isotherm for nitrogen on a silver surface with D = 2.30, computed from Eqs. (24, 28) and the constants in Table 4. The dashed straight lines are the asymptotic power laws (32a, b) for van der Waals wetting and capillary wetting. They intersect at Al.t = A~tc and N/N m = (N/Nm) c, given by Eqs. (30, 42), and have slopes (3 - D)/3 and 3 - D, respectively.

Figure 12 contains an important caveat, however: If one selects a sufficiently small portion of the isotherm from the crossover region and fits a power law to it, one may find a nonclassical FHH behavior, Eq. (36), but with an exponent y neither equal to (3 - D)/3 nor to 3 - D. Thus, an experimental exponent y obtained from a small pressure range must be interpreted with care. If the underlying surface is known to be fractal, one can use Fig. 12 to conclude that the exponent must satisfy (3-D)/3 < y < D-3,

(62)

which leads back to the bounds (37) for the fractal dimension. The exponent in this case, even though it is nonasymptotic, still provides valuable information. If nothing is known about the surface and the analyzed pressure range is small (e.g., in the sense that the range of film thicknesses calculated from Eqs.(43a) and (37) spans less than a factor of two [36, 158]), a fractal interpretation of the exponent is not meaningful, as a rule. Figure 12 depicts A~tc, the transition to capillary wetting, as the point of intersection of the asymptotic power laws (32a, b). More generally, we define h~tc as follows. We write the adsorption isotherm (12) on a surface with arbitrary geometry, computed from (13) exactly or approximately, in the form

N(AB) = NmO(A~t; n, o~, They expressed the observed chemical shift of Xe,

8,~ , by the sum of

the average

contribution of adsorbed Xe species, , and that of interporous Xe species, < ~ , ~ . Here, x ~ and xp,~ are the fraction of the adsorbed and interporous Xe species. Thus, the NMR behaviors of Xe atoms in mesoporous silicas are presumed to be caused by the adsorption of the Xe atom on very small pore site( the micropore size) and rapid exchange between Xe in the micropore and Xe adsorbed on the surface.

They also examined the

706 NMR signal change for mixing of two kinds of mesoporous silicas, then they showed two distinct signals due to different pore sizes. Consequently, NMR method is effective for direct checking the heterogeneity of pore structures.

The measurement of the chemical

shift of NMR will be widely used in the adsorption sicence field hereafter. 2D-echange spectroscopy (2D-EXSY) of 129XeNMR can provide information about the rate and the pathways of the Xe exchange between the different states of Xe in pores. ~29Xe 2D-exchange NMR should become another useful technique for elucidation of the relationship between surface heterogeneities and molecular adsorption, m If there is a macroscopic orientation in solid samples, another NMR principle is helpful to elucidate the pore structure and the molecular motional state of adsorbed molecules. electric quadrupole occurs only for spins with quantum number I > 1/2.

An

Deuterium has a

spin quantum number of 1, then the interaction between the nucleus and the local electric field shifts the energy level.

One observes two resonance lines by the following equation

for hydrocarbons whose principal axis z is given by the direction of the C-D bond so that the xx component of the electric field gradient tensor V~, equals to the yy component V.. to =

too

-4-

(toQ/2) (3 cos: 0m- 1 )

(26)

Here toQ is called quadrupole frequency, which is given by eq.(27). % = (3 ~2)e 2qQ/h

(27)

where Q is the quadrupole moment, eq

=

V= is the zz component of the electric field

gradient tensor, and 0mrepresents the orientation of the magnetic field in the principal axis of the electric field gradient. Different distributed

~\\\\N%

orientations and motional states of molecules having the C-D bond provide characteristic line shape. Deuterated benzene NMR provides

.~-~-~

E

ao~, oooo

~ ~

o

0 0

an important information, because benzene has been widely used as a probe adsorptive. Recently Fukazawa et al applied this D-NMR technique to analysis of adsorbed benzene states in slit-shaped micropores of boehmite microcrystalline aggregates. 24'129

The

microporous boehmite aggregate has two kinds

~

0.4

0

.................. @'V..,.

.

x

.x\\xx~

0.0

t

0.0

0.2

0.4

0.6

Or.8

1.0

PfPo

of effective micropores of 0.8 and 1.3 nm.

Figure 20.The benzene adsorption isotherm

Fig. 20 shows

the benzene adsorption

of boehnfite aggregate at 303 K. The solid

isotherm at 303 K. They measured D-NMR

symbol corresponds to the D-NMR meas-

707 at three adsorption stages denoted by arrows

ment and the filling state is shown.

and solid circles which correspond to 0.02, 0.2, and 0.4 of PIP0- Almost smaller micropores of 0.8 nm are filled with benzene at P/P0 = 0.02, while adsorption by larger micropores of 1.3 nm begins at P/P0= 0.2. Both types of micropores are filled with benzene at P/P0= 0.4. Fig. 21 shows the D-NMR spectra of benzene adsorbed in the micropores at P/Po = 0.02, 0.2, and 0.4.

The peak becomes narrower with the increase of P/P0. At P/P0 = 0.04 benzene

molecules have a considerably definite organized structure, but they are moving with the tilt angle of 44* against the pore wall. Under the conditions of P/P0 = 0.4, benzene molecules exchange more rapidly with the configurationof the tilt angle of 51-54 ~ The surface diffusion coefficient was 3 x 10s cmZs' (liquid benzene: 2.1 x 10 s cm2sI). Three spectra are different from each other and each spectrum is not the convoluted peak due to coexistence of different molecular motional states, suggesting that benzene molecular states are characteristic to the relative pressure and even at P/P0 = 0.04 benzene molecular states are uniform regardless of adsorption on the two kinds of sites. Hence, D-NMR should give important informations on the shape and size of micropores, the pore orientation, and also the adsorbed molecular states. There are other methods for characterization of pores and pore-walls. High resolution electron microscopy is one of powerful characterization method of the pore-wall and pore structures and the higher order structures. 13~

However, quantitative analysis is difficult.

The image analysis of high resolution electron transmission micrographs is a hopeful quantitatie technique. ~2 In future, scanning tunneling microscopy

(STM)

and

atomic

j••

force

microscopy (AFM) will become one of representative methods. 133'134 The mercury

0.02 P/Po

porosimetry is an established method for evaluation of mesopore analysis, although this method has problems such as deformation or fracture occurrence in porous specimen due to application

of

high

pressure

!

and

0.2

discrepancey from N 2 adsorption. 3'135 The mercury porosimetry is not described here, because there are many articles on it. The

0.4

heat of immersion has been associated with the pore structures. The depression of the melting point of solids in pores has been

'"|

20

[

i

II

I

10

0

- 0

-20

kHz

applied to the pore size determination by Quinson et al. t~ Yoshizawa et al showed that

Figure 21.D-NMR spectra of benzene

the EELS(electron energy-loss spectrometry)

adsorbed at different relative pressures

708 can determine the sp:- and spS-carbons at 303 K separately. ~3~ It is quite important to characterize ill-crystalline porous solids, and thereby new techniques and trials are desired. 5. HIGHER ORDER STRUCTURES

Activated carbon is quite important adsorbents which have been widely used in human activities. They are produced from natural products, coals, and pitches. The adsorption characteristics is influenced by the sample history. In catalysis field, Hedvall suggested "structural heredity effect" which is strongly associated with the sample history. 138 The scientific reason is not clearly elucidated yet. Even in the modem technology activated carbons the best fit to the industrial application have been selected from intuitive experimental examinations. Probably higher order structural difference lead to observed difference in adsorption properties. As the voids are pores, the examination of the particle agglomerates or pore walls provides the information of the higher order structures of pores.

We can

understand a clear feature of the higher order structures of an adsorbent through the micro graph. The high resolution transmission microscopy and AFM will contribute to understand the higher order structures. However, analysis with microscopy is not quantitative, but qualitative. Understanding of the higher order structures requires introduction of a qualitative analysis in microscopy.

The image analysis of micrographs is one of such possibilities, as

mentioned above. The stress in the image contrast enables to determine the size and shape of primary particles and their short-range order by a statistical averaging. The development of the image analysis of micrographs has been desired. A systematic measurement of classical properties such as the electrical conductivity or dielectric constant is often helpful to understand the higher order structures. The example of the electrical conductivity measurements is shown here. The porous solid can be regarded as a binary system of primary solid particles and pores. Both have own electrical conductivity's. Here, the pore has high resistance of an interparticle region. The pore resistance is not necessarily the same as that of vacuum, but it is determined by the atmosphere and the surface conductivity of primary particles. The total electrical conductivity R t of the porous solid samples between two electrodes depends on the higher order structure of solid particle resistors R, and the pore resistors Rp. There are two types of electrical contacts among component resistors according to the two layer model. ~a9 The two types are series and parallel contacts of R, and R r

The electrical conductivity change with the pore fraction is

completely different from each other for parallel and series connections. Then, we can determine the fundamental contact structure of the aggregates of primary particles. ~4o The electrical conductivity measurement was effective for determination of the higher order structure of the oxide mixture. If the pores and solids have capacitance character, the .

equivalent circuit is expressed is in terms of the resistor and condenser. Suck-an equivalent circuit shows a frequency dependence which is characteristic to parallel or series connection.

709 Thus, the ac electrical conductivity measurement is a hopeful method to determine the higher order structures. Especially, activated carbon is composed of conductive graphite crystals. The graphitic units are highly conductive, while the pore resistance is quite high. The electrical conductive measurement should be applied to characterization of activated carbon relating to the development of a superhigh capacitor. 14L142 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.

715

C h a r a c t e r i z a t i o n of g e o m e t r i c a l a n d e n e r g e t i c heterogeneities o f active c a r b o n s by u s i n g s o r p t i o n m e a s u r e m e n t s M. Jaroniec a and J. Chorea b aSeparation and Surface Science Center, Department of Chemistry, Kent State University, Kent, Ohio 44242, U.S.A. bInstitute of Chemistry, Military Technical Academy, 01489 Warsaw, Poland

A thermodynamic approach to the characterization of active carbons is presented including a brief review of selected methods and extensive experimental illustrations. This approach represents a comprehensive way to characterize the energetic and geometrical heterogeneities of active carbons, and is based on the differential adsorption potential distribution, which provides a quantitative measure of all changes in the Gibbs free energy of a given gas/solid sorption system. The distribution in question is associated with the differential enthalpy, differential entropy and immersion enthalpy via simple relationships. Also, it can be easily converted to the mesopore and micropore volume distributions.

1. INTRODUCTION Active carbons are most popular porous solids [1,2]. The presence of micropores in these solids increases substantially their sorption capacities over those for nonporous carbons [3,4]. According to the classification scheme of the International Union of Pure and Applied Chemistry (IUPAC), micropores have widths below 2 nm, macropores above 50 nm, and mesopores between 2 and 50 nm [5,6]. The macropores and large mesopores play an important role in the molecular transport process, whereas the remaining pores determine the sorption properties of a given carbon. By carbonization and activation of various raw materials (such as wood, coal, lignite, coconut shell, peat, polymeric resins, etc.) [1,2], it is possible to prepare carbons of differentiated surface and structural properties. In order to alter their properties, various chemical and physical modifications of active carbons can be performed [1]. One of the simplest modifications can be carry out through heat treatment in either an oxidizing gas or solution [1,2]. The types and concentration of surface functional groups are determined by the degree and extent of a given surface oxidation. Other modifications of the carbon surface include impregnation of carbons with various inorganic salts or organic compounds; coating with oils, waxes, and other high molecular weight liquid phases; chemical bonding of different ligands; and deposition of finely dispersed metals and their oxides [1,2]. A further progress in the preparation and applications of novel carbons of the controlled porosity and surface properties can be accelerated by creating novel

716 characterization methods. Among various methods used to study the surface and structural properties of these materials, various sorption-based techniques such as adsorption [7-10], chromatography [11] and thermal analysis [12] are still popular because they provide direct information about adsorbate-adsorbent interactions. For instance, the low-temperature nitrogen adsorption isotherm is recommended by the IUPAC to evaluate the BET specific surface area and the mesopore volume distribution function, which are essential for characterizing the sorption properties of mesoporous solids [5,6]. However, these classical quantifies are not sufficient to characterize active carbons, which usually are highly microporous solids of a complex surface functionality [1,2]. A more sophisticated characterization of these solids, which includes the evaluation of their energetic and geometrical (structural) heterogeneities, is desirable. In the current chapter a consistent thermodynamic approach to the characterization of energetic and geometrical heterogeneities of active carbons is presented with a brief discussion of selected methods. In this approach the adsorption potential distribution, which provides information about changes in the Gibbs free energy for a given system, is a key thermodynamic function. This distribution can be used to calculate the differential enthalpy, differential entropy and heat of immersion as well as it can be converted to the pore volume distribution. In addition, this approach demonstrates clearly the relationship between energetic and geometrical heterogeneities of active carbons.

2. ACTIVE CARBONS AS ENERGETICALLY AND GEOMETRICALLY HETEROGENEOUS SOLIDS The porous structure of active carbons is complex. These solids possess mostly slit-like pores of different sizes, which are contained between the twisted aromatic sheets forming the matrix [1,2]. Although the main feature of good active carbons is their high microporosity, they also contain large pores (i.e., mesopores and maeropores [1]). Thus, these materials are geometrically heterogeneous due to the existence of pores of different sizes and shapes. Their geometrical (structural) heterogeneity can be characterized partially by the mesopore volume distribution function. The IUPAC [5,6] has made some recommendations for evaluating this distribution, which can be evaluated from the high-pressure part of the equilibrium adsorption-desorption isotherm. Also, other methods (e.g., mercury porosimetry) can be applied for determining the porosity of solid materials [1]. While the mesopore distribution evaluated from the high-pressure part of the adsorption-desorption isotherm is useful for characterizing the structural heterogeneity of mesoporous and macroporous solids, it does not provide information about the micropore effects. This information can be obtained from the low-pressure part of the adsorption isotherm. An experimental illustration of the adsorptiondesorption isotherm from the gas phase is shown in Figure 1 for benzene vapor on Ambersorb 572 synthetic carbon (Rohm and Haas Co., Philadelphia, PA) at 293 K [13]. Of course, the shape of isotherm depends on the adsorbate-adsorbent interactions as well as on the structural and surface heterogeneities of the carbon. The classification of the gas-solid adsorption isotherms and the adsorption-desorption hysteresis loops are discussed elsewhere [7]. Here, it should be mentioned that adsorption occurs more or less gradually. At very low pressures, the most energetic adsorption sites are occupied

717 and adsorption takes place in the micropores, which are filled gradually. Adsorption data measured in the range of the micropore filling provide information about the structural heterogeneity of micropores and about the surface heterogeneity. At higher relative pressures, the layer-by-layer adsorption occurs on the surface of mesopores. Finally, capillary condensation takes place in mesopores and small macropores. The multilayer and condensation segments of the adsorption isotherm provide information about the mesopore and macropore volume distributions. 12.0 13) 10.0 0

E E

~

Amb572

Capillary Condensation

8.0

(D .Q

6.0

L_

0


r

(11) where r==2~/ran is the location of the minimum of the Lennard-Jones potential. The hard sphere term F hcan be written as the sum of two terms,

Fh[p(r);d] = kT~ drp(r)[ln(A3p(r)) - 1] + kT~ drp(r)f,,[-~(r);d]

(12)

where A = h~ (2zankT) u2 is the thermal de Broglie wavelength, m is the molecular mass, h and k are the Planck and Boltzmann constants, respectively, and f,~ is the excess (total minus ideal gas) Helrnholtz free energy per hard sphere molecule. The latter is calculated from the Camahan-Starling equation of state for hard spheres [22]. The fh'st term on the fight side of (12) is the ideal gas contribution, which is exactly local (i.e. its value at r depends only on p(r) at r), while the second term on the fight is the excess contribution, which is nonlocal.

751 The density ~(r) that appears in the last term of (12) is the smoothed or nonlocal density, and represents a suitable weighted average of the local density p(r),

(13)

~(r) = j'dr' p(r' )w[Ir- r'l;~(r)]

The choice of the weighting function w depends on the version of density functional theory used. For highly inhomogeneous confined fluids, a smoothed or nonlocal density approximation is introduced, in which the weighting function is chosen to give a good description of the hard sphere direct pair correlation function for the uniform fluid over a wide range of densities. In this work, Tarazona's model [23] is used for the weighting function. This model has been shown to give very good agreement with simulation results for the density profile and surface tension of LJ fluids near attractive walls. The Tarazona prescription for the weighting functions uses a power series expansion in the smoothed density. Truncating the expansion at second order yields 2

~(r)] = ~ w,(~'-

(14)

i=0

Expressions for the weighting coefficients w, are given by Tarazona et al. [23]. The equilibrium density profile is determined by minimizing the grand potential functional with respect to the local density, gff~[p(r)] =0 6p(r)

at p = p.~

(15)

A numerical iteration scheme is used to solve this minimization condition for p.q(r) for each set of values of (T,I.t,H); the hard sphere diameter is determined from the Barker-Henderson prescription [24] for each temperature. The chemical potential is related to the bulk pressure through the bulk fluid equation of state, (16)

P(P) = Ph (P) - ~Tc~ak_,..~ 2

where a is a van der Waals interaction parameter given by (17)

a = 4z~o~,~,(r)rZdr

For large pores, two minimum density profiles often arise; these are the liquid and vapor branches associated with thermodynamic hysteresis in individual slit pores. When more than one minimum exists, the density profile which has the lower grand potential energy is the stable branch. The chemical potential at which condensation occurs is the value for which the two minima have the same grand potential energy. In subsequent sections, the following dimensionless quantities are used: H * = H~ or#

T*= k T / e f l

p*= p ~ f

z* = z / a n

752 5.

Molecular Simulation

To check the accuracy of the density functional theory results, adsorption isotherms have been calculated using Gibbs ensemble simulation. The Gibbs ensemble method provides a direct route to the determination of the phase coexistence properties of a fluid, by Monte Carlo simulation of the fluid in two distinct physical regions that are in thermal, mechanical and material contact. Gibbs simulation was originally developed to determine phase equilibria in bulk fluids [25] and in fluids adsorbed in cylindrical pores [26]. In order to test the theory, Gibbs ensemble calculations are performed for fluid adsorption in the slit pore geometry of Figure 2. Two types of Gibbs simulation are performed: pore-pore and pore-fluid. The porepore calculations yield the coexisting liquid and vapor densities at the falling pressures of pores in which capillary condensation occurs. A schematic of the pore-pore simulation method is shown in Figure 3. The fluid molecules are confined in the slit pore geometry of each of the two simulation cells, designated regions I and II. At each simulation step, one of three perturbations is attempted: (a) particle displacement in each region; (b) particle interchange between the two regions; and (c) exchange of pore surface area between the two regions, such that the total surface area remains constant. The acceptance probability for each of these moves is chosen in a way that ensures that the system obeys the laws of equilibrium statistical mechanics [8,25,26]. Sampling configurations using these three perturbations ultimately brings the two regions into thermal, material and mechanical equilibrium. Such a state is phaseequilibrated, and thus the Gibbs method yields the equilibrium densities of the coexisting liquid and vapor phases. The potentials described in Section 3 are used to model the fluid-fluid and fluid-solid interactions. Starting from an initial facecentered cubic lattice configuration in each region, the system is equilibrated by successive perturbations over a sufficient number of configurations. Additional configurations are then sampled to measure the properties of the equilibrated phases. a

Io

! I

oO-

Ioko 101 0

L.-o

l9..--I Ii"I~ oo#o

L:-~

Ul.-"

o

0

0

"

0-:ol .--" !o

"'i

o

0

0

Ul.-"

"o

b

.--'I

0

0

II

I

0 0

0

I oi

0

o

L,

~ -

L.-~

o_.1 ~

Ul--"

I1'-'

I

l.-'~o

o

I

12

Figure 3. Schematic of Gibbs ensemble Monte Carlo simulation method for pore-pore equilibria calculations in slit-shaped pores. Solid lines denote pore walls; dashed lines denote periodic boundaries.

753 A typical calculation requires approximately 1.5 million configurations for equilibration and an additional 1.5 million configurations for collecting property data. For the inhomogeneous confined fluid, long range corrections are not computed because of the associated computational difficulties. However, the system size is chosen so that the minimum edge length of the pore surface is 1 0 ~ (i.e. all molecular interactions up to a distance of 5Grr are explicitly included in the calculations). This cutoff is presumed to be large enough so that long range corrections may be neglected. While the pore-pore calculation yields the equilibrium vapor and liquid states that coexist in a pore of specified width, it does not specify the pressure at which the equilibrium state occurs, nor does it indicate the amount adsorbed at pressures below and above the equilibrium pressure. Therefore, the f'flling pressure and the state points along the vapor and liquid branches of the adsorption isotherm are calculated using the Gibbs pore-fluid ensemble. In this variation of the Gibbs ensemble method, the fluid in region I is again confined in a slit pore, but region II is a homogeneous bulk fluid (all boundaries are periodic). In the pore-fluid calculation, no exchanges of pore volume are attempted, since the condition for mechanical equilibrium is automatically satisfied if the chemical potentials in regions I and II are equal (i.e. material equilibrium is achieved) [27]. Hence, only displacement and particle interchange steps are performed, with the same acceptance probabilities as stated for the aforementioned pore-pore simulation. In calculating the bulk fluid properties, a long range correction for the cubic simulation cell geometry is included [25,28]. At equilibrium, the adsorbed fluid density corresponding to the bulk fluid pressure is obtained. To facilitate comparison of theory and experiment, the bulk pressure is scaled with respect to the saturation pressure P0 of the bulk Lennard-Jones fluid, given by the modified Benedict-Webb-Rubin equation of state [29]. To determine'the pressure at which condensation occurs, the vapor coexistence density from the pore90 80

~'0 a.

I-

m6o E u

~

'u

E3o

_= o

>20 10 0

0

2

4

6 8 10 Film Thickness (A)

12

14

16

Figure 4. Surface area measurement of Vulcan nonporous carbon. The slope of the linear region of the plot is the specific surface area o f the sorbent.

754 pore calculation is interpolated so that it coincides with the vapor branch of the isotherm constructed from the pore-fluid results. 6.

P r e d i c t e d Nitrogen Adsorption Isotherms: Slit C a r b o n Pores

Calculations using several different values of the solid-fluid potential strength e J k showed that the pressure at which the monolayer forms, and the pore filling pressure for micropores, are both sensitive to this value, but the general form of the isotherm is not much affected. The solid-fluid potential strength was fitted to adsorption data for a nonporous material of the same chemical structure as the porous material. For carbons this was a Vulcan provided by BP Research. The surface area A of this material was estimated using the method of deBoer et al. [21], from the slope of the plot of volumetric uptake against the universal film thickness, or "tcurve" (see Figure 4); A is related to the slope St~ of the linear region as A = S ti,, / (P ,,~Vsrp)

(18)

where P.ds is the density of the adsorbed film, and Vsrp is the molar volume of nitrogen at standard conditions. Here P~ is usually taken to be the bulk saturated liquid nitrogen density. This procedure leads to A=78.6 m 2, as compared to A-71.76 m '~ from BET measurement. The ratio A/AB~T=I.10 is consistent with deBoer's results for Graphon and Spheron carbon blacks. The solid-fluid potential parameter Ej'k was then estimated by fitting the mean nitrogen density in a large mesopore, pDrr, as calculated from the DFT theory, to experimental data, using the relation

0.8 0.7

! ]

i

0.6 i

0.5

p'0.4 0.3 0.2 0.1

0

0.1

0.2

0.3

0.4

0.5 P/Po

0.6

0.7

0.8

0.9

1

Figure 5. Nonlocal theory isotherms for nitrogen adsorbed in carbon mesopores at 77 K. The pore widths, reading from left to right, are" H*=6, 7, 8, 9, 10, 12, 14, 20, 40, 60, and 100.

755

V,a, = lporrHA Vsre

(19)

where V ~ is the uptake of nitrogen (cm 3 at STP/g of carbon). This procedure gave a value of eJk=53.22 K as the one best fitting the pressure at which the monolayer forms. For oxide materials a similar procedure was followed, using adsorption data for a nonporous NBS alumina [8]. With these fitted potential parameters, individual pore isotherms were generated using nonlocal theory for arange of pore sizes. For carbons these ranged from H*=1.68 to 100 (6.0 A to 357 A) at T=77 K. Nitrogen adsorption isotherms in mesoporous carbon slits are shown in Figure 5. Pores in this size range exhibit type IV capillary condensation behavior. At low pressures, a monolayer is formed; as the pressure is increased, additional multilayers are adsorbed, until the condensation pressure is reached, whereupon a phase transition to the liquid state occurs. Above the condensation pressure, there is a gradual increase in the mean density due to compression of the nitrogen in the liquid-filled pore. As the pore width is reduced, the condensation pressure decreases, as shown in Figure 6. This follows from consideration of the solid-fluid slit potentials, shown for several pore widths in Figure 7. As the slit walls move closer together, the bulk-like region in the center of the pore, where V~,, is approximately zero, disappears, and adsorption is enhanced throughout the pore space. In addition, the proximity of the adsorbed fluid layers on opposing walls increases fluid-fluid interactions, and further promotes the adsorption of multilayers. A transition from capillary condensation to continuous filling occurs at a critical width of Hc~*=3.8, or 13.6 A (e.g. isotherm H*=3.75 in Figure 6). Interestingly, there is a second region of discontinuous pore filling in the nonlocal theory isotherms, separate from the condensation region, at pore widths below the critical width Hc~*. From Fig. 8 we see that for pore widths between Hc3"=2.55 (9.1 A) and H,2"=3.6 (12.8 A), a 0---~1 monolayer transition occurs wherein the 0.8 0.7 0.6 0.5

p'0.4 0.3 0.2 0.1 0 0.00001

0.0001

0.001

0.01

0.1

1

PlPo Figure 6. Nonlocal theory isotherms for nitrogen adsorbed in carbon supermicropores at 77 K. The pore widths, reading from left to right, are: H*=3.75, 4, 4.25, 4.5, 5, and 6.

756 5-~ .

-5 ,It

,,,,, X

~10 -

3

-15 -

-20 2 -25

-4

-3

-2

-1

0 Z*

1

2

3

4

Figure 7. Wall-fluid potentials for nitrogen on graphite for selected slit pore widths H* (indicated by the numerals). The wall-fluid potential V,=t* = V,=,/e:! is plotted as a function of position in the pore.

0.8

0.7,-[ 0.6 0.5

p'0.4 0.3

0.2 0.1

0 1E-07

1E-06

1E-05

IE-04

1E-03

1E-02

1E-01

1E+O0

P/Po

Figure 8. Nonlocal theory isotherms for nitrogen adsorbed in carbon supermicropores at 77 K. The pore widths, reading from left to right, are: H*=2.5, 2.6, 2.75, 3, 3.25, 3.5, and 3.75.

757 incomplete monolayer on each pore wall abruptly fills to completion. At "I"=77 K there is a narrow band of continuously fLlling pores between He2* and H:I*. Although the pores in the size range from He3* to H:2* fall within the IUPAC supermicropore classification (9.1 to 13.6 A), the phase transition in these slits is atypical of the continuous filling normally expected for such pores. However, Gibbs ensemble Monte Carlo simulations (Section 5) conf'u'm the presence of a 0--->1 monolayer transition in pores that can accommodate approximately two complete layers of adsorbate. For still smaller pore widths, a return to continuous pore f'dling is observed (Figure 9). Pores of this size, corresponding to the IUPAC ultramicropore range, arc too narrow to accommodate more than a single layer of adsorbate. The shape of the ulu'amicropore isotherms arc similar to the IUPAC Type I isotherm, characteristic of micropore adsorption, although the IUPAC representation uses a linear rather than a logarthmic pressure axis. As Figure 7 illustrates, the two minima of the solid-fluid potential coalesce into a single minimum at a pore width of H*=2.25 (8.0 A). This enhances the potential well strength, which is maximized to roughly double its original depth at a pore width of H*= 1.94, or 6.9 A. A corresponding reduction in the f'dling pressures is seen, with the minimum (at H*=1.94) occurring at approximately P/Po=10 "~~ a pressure on the order of 0.1 microtorr. As the pore width is reduced beyond this minimum, the repulsive portions of the opposing wall potentials begin to overlap. Hence, there is a rapid rise in f'dling pressure as pore width decreases below H*=1.94 (Fig. 9). For pores with physical width narrower than H*=1.69 (6.0 A), the entire solid-fluid slit potential is repulsive, and thus the pore space is inaccessible to nitrogen and no adsorption occurs. It is observed that 0.8

0.7 0.6

0.54: p'0.4 /t

0.3

/

,

0.1 p

1E-12

1E-10

1E-08

1E-06

I

I

/

i

!

I II

0

/

//

iI

I : i

0.2

/

i/

I I/

sI s

1E-04

I J ~*

1E-02

1E+O0

P/Po

Figure 9. Nonlocal theory isotherms for nitrogen adsorbed in carbon ultramicroPores at 77 K. The pore widths, reading the solid lines from left to right, are: H*=1.94, 2.25, and 2.5. For pore widths smaller than H*=1.94, the filling pressure increases with decreasing width, as shown by the dashed lines, reading from left to right: H*=l.8, 1.75, 1.72, 1.7, and 1.69.

758 the mean density of nitrogen in the ulwamicropores (H less than 9.1 ,1,) is considerably reduced, due to exclusion of the adsorbate from the region near the slit walls. Only in the largest mesopores (e.g. Fig. 5) does the mean fluid density in the pore approach the bulk saturated liquid density of p*=0.792. Although the IUPAC designation of pore sizes is a useful guide to anticipating pore filling behavior, it is evident from the results presented in this section that the nature of the adsorption depends as much upon the adsorbate characteristics as it does upon the structure of the adsorbent. For example, if the size of the adsorbate molecule was increased, and all other potential parameters were held constant, some of the slits that exhibited capillary condensation for the original value of o'~ would fill continuously were they to adsorb the larger fluid molecules. Therefore, it is more relevant to devise a classification scheme that uses a pore size scaled with respect to the adsorbatc molecular diameter, and that also accounts for the influence of temperature. A comparison of the filling behavior predicted by nonlocal theory and Gibbs ensemble Monte Carlo shows good agreement for both the density profiles in the pore and the adsorption isotherms, for the whole range of pore widths from the ultramicropore region to large mesopores. The Gibbs result also verifies the presence of the 0-->1 monolayer transition found in the theoretical isotherms for supermicropore-sized slits. Adsorption isotherms calculated from theory and simulation are compared in Figure 10, for a range of pore widths. In each case, the filling pressures predicted by nonlocal theory and Gibbs simulation are in excellent

U.8 ,

4

0.7 0.6 0.5

p'0.4 u

9

0.3 0.2 0.1 O

:

1E-10

|

| |

I

i

1E-08

1E-06

~

l

'

1E-04

'

' '''":

1E-02

I

'

' ''"~

1E+O0

P/Po

Figure 10. Comparison of nitrogen adsorption isotherms calculated from nonlocal theory and Gibbs simulation for adsorption in carbon slit pores at 77 K. Lines denote the nonlocal theory isotherms f o r pores of width, reading from left to right: H*=2, 2.5, 3, 3.75, 5, 8 and 12. Symbols show the corresponding Gibbs simulation isotherms. Open circles indz'cate equilibrium densities from pore-pore calculations; solid circles indicate pore-fluid equilibrium results.

759 agreement. Furthermore, the vapor and liquid branches of the theoretical and simulation isotherms agree quantitatively over the range of pressures sampled. There are some differences between theory and simulation for the adsorbed densities in smaller pores; this may arise because mean-field density functional theory predicts a higher bulk fluid critical temperature T~ than is obtained from simulation results for the Lennard-Jones fluid [30]. Hence, the nonlocal theory isotherms at T=77 K are at a lower reduced temperature T/To than the corresponding Gibbs simulation isotherms, and the phase splitting in the theoretical calculation is therefore more pronounced than in the Gibbs simulation. Overall, however, the nonlocal theory provides a quantitatively accurate description of pore filling. The variation of the adsorption isotherm with temperature has also been studied [6] for the range 70-85 K. The critical slit widths, Hc~*, Ha* and H~3* depend on temperature. At 70 K there are no continuously filling pores, but rather a range of pore widths which exhibit two discontinuous jumps in the isotherm. These "stepped" or IUPAC Type VI isotherms are typical of low temperature adsorption of simple gases on carbon surfaces [4,19]. At the higher temperature of 85 K (which is still subcritical) the range of continuous filling is broader than at 77 K. The results suggest that an upper critical temperature for the monolayer transition in nitrogen adsorption should occur slightly below T~*=I.0 (94 K). This critical temperature lies intermediate between the bulk Lennard-Jones fluid two-dimensional [31] and three-dimensional [32] critical temperatures of Tin*=0.515 and T3D*=l.316, respectively. The pores which exhibit the 0--->1 monolayer transition condense two layers of adsorbate, one on each surface. Thus, it follows that T~,,Tm, since the adsorbed molecules interact with molecules in the opposing film layer as well as those in their own layer (i.e. the fluid is not strictly two-dimensional). 7.

Pore Size Distribution: Slit Carbon Pores

The overall sorbent structure is envisioned as an array of noninteracting individual slit pores with a distribution of pore widths described by a function f(H). Clearly this distribution function must be non-negative for all pore widths H. For amorphous sorbents such as carbons, it is reasonable to assume also that f(H) is continuous. Two functions which satisfy these requirements are the gamma distribution and the lognormal distribution. The gamma distribution is

,~, lxi (Ti H) [J' f(H)=~=zT~ F(fl,)H exp(-7,H)

(20)

while the lognormal distribution is

f(H) = ~ a, -[ln H - 13i12 = 7iH(2z01/2 exp 27/2

(21)

where m is the number of modes of the distribution, and ai,/3i and 7i are adjustable parameters that define the amplitude, mean and variance of mode i. These equations are used to represent the PSD in the fitting of adsorption integral. The choice of the PSD function is discussed at greater length below. To determine the PSDs of porous carbons from experimental nitrogen adsorption data, the set of model isotherms presented in Section 6 are correlated as a function of pressure and pore width. The adsorption integral, equation (1), is then

760 solved numerically, inserting one of the model pore size distribution functions (equations (20) and (21)) and employing a simple minimization algorithm to optimize the parameters /xi, ]~iandTi of the PSD function. A least squares error minimization criterion is used to determine the optimum fit. The choice of the number of modes in the PSD function is arbitrary, provided that enough are used to give f(H) sufficient flexibility. In practice, the number of inflection points in the experimental isotherm can be used as an estimate of the number of modes required to yield an acceptable fit. This assertion is made on the basis of the one-to-one correspondence of filling pressure to pore width in the nonlocal theory results in Figure 1 (excluding the special case of the ultramicropores, with H100

...

0

_r ~-~--"

I E-06

i

i

i

i

1

1E-05

1 E-04

1E-03

1E-02

1E-01

"

1 E+00

PIP0

Figure 11. Nitrogen adsorption on microporous carbon CXV at 77 K. Symbols denote the experimental uptake measurement; the solid, shortdashed and long.dashed lines indicate the fitted isotherms from nonlocal theory, the HK equation, and the Kelvin equation.

0.2 0.18

0.,6

il

I ]i

. @Dp, then the apparent diffusivity is:

m~pp

(:- r

~ Ds (6) r +(1-r This case is applicable to solids having very high affinity, typically in the order of 1000 to 10000. The model is called the solid diffusion model, still commonly used for describing adsorption in solids such as resins. In contrast to the pore volume diffusion model where longer time is needed to equilibrate the particle of higher affinity, the solid diffusion model states that the same time is required to equilibrate the particle no matter how high the affinity K is. This is so because the higher is the affinity the larger is the driving force. The half time for the solid diffusion model is [3]: R2 to5 = 0.03055 (7) Ds =

779 The limitation of the linear models is that it can not be applied to most systems where solids are not ideal or the isotherms are not linear or the system is not isothermal. 2.1.2 Nonlinear models: Recognizing the nonlinear nature of many isotherms, researchers solved the diffusion equations for the case of nonlinear isotherm, for example the following Langmuir isotherm, one of the most widely used isotherm bC (8) q = f(C) = qs 1+ bC Weisz and co-workers [4-7] published a series of papers solving diffusion and adsorption in a particle with nonlinear isotherm. Nonlinear isotherm problems were generally solved by numerical techniques, but problems with rectangular isotherms were obtained analytically as in this case, the adsorption is so strong (irreversible) the adsorption occurs like a penetration process into the particle. By splitting the particle domain into two, by which one domain is saturated with the adsorbed species while the inner domain is free from any sorbates, the problems can be solved to yield analytical solution [8]. Using a rigorous perturbation method, Do [9] has proved that the penetration of the adsorption front in the case of rectangular isotherm can be derived from the full pore diffusion and adsorption equations under the condition of strong adsorption kinetics relative to diffusion, high sorption affinity and low capacity in the fluid phase compared to that in the adsorbed phase. Using the same technique, the fixed bed problem with rectangular isotherm can also be solved analytically [10]. The result is that the breakthrough curve is a almost linear function of time, an explicit form which is useful for design purposes. 2.1.3 Nonisothermal models: Adsorption m gas phase systems is an exothermic process, i.e. heat evolves during the adsorption. The adsorption heat is typically in the order of 10 to 100 kJoule/mol, with the high end usually associated with sorption of hydrocarbons m zeolite particle. The consequence of the heat release due to adsorption or heat uptake due to desorption will affect the rate of diffusion of molecules into the particle as well as the instant equilibrium capacity of the particle. For example, m the case of adsorption, heat released will increase the particle temperature ff the rate of energy dissipation to the environment is not fast enough (especially in systems under vacuum or stagnant environment). As a consequence the diffusion coefficients of bulk, Knudsen and surface diffusion increase and the adsorption affinity decreases in the early stage of adsorption. After this, the rate of adsorption will decrease and hence the rate of heat release will be more compensated by the heat of dissipation; the temperature will drop and finally the particle temperature will be equal to the environment temperature at the time when there is no further adsorption.

780 Early work of nonisothermal effect in adsorption system has been carried out [11, 12]. Detailed analysis of nonisothermal sorption in a single particle is due to Brunovska et al. [13]. They considered pore diffusion is the only mass transport mechanism into the particle, and allowed for heat conduction within the particle. No film heat resistance was considered by the authors. The case is applicable only to large particles in rigorously stirred environment, and surface diffusion is not important relative to pore diffusion. From the simulations they concluded that the temperature rise within the particle may be of order of 10 - 50 oc. This order of temeprature rise was confirmed in their experiment of adsorption of n-heptane in 5A zeolite particle [14]. Brunovska et al. [15] later proposed a lumped thermal model to describe adsorption in nonisothermal systems. They allowed for intraparticle mass transfer but heat transfer was limited to that through a stagnant film surrounding the particle. Pore diffusion and Langmuir isotherm were considered in their work. Later, Brunovska et al. [16] used this lumped thermal model to investigate the case when the adsorption affinity is very strong, i.e. rectangular isotherm. The maximum adsorptive capacity is assumed to be a linear function with respect to temperature: q ~" qs = qso[1 - c~(W-To) ] (9) where qso is the maximum capacity at temperature To. Haul and Stemming [17] allowed for heat conduction as well as heat transfer to the environment in their model. A better model was presented by Kanoldt and Mersmann [18] and they tested their model with water adsorption in molecular sieve 4A, carbon dioxide m molecular sieve 10A and water on silica gel. Surface diffusion in addition to the pore diffusion was considered in nonisothermal model [19]. Generally, numerical procedures are used to obtain the solution to the coupled heat and mass balance equations. Hills [20] obtained an analytical solution to a system where there are internal and external mass transfer and external heat transfer. However, linear isotherm was dealt with, limiting the applicability of his model. Further limitations of his models are (1) diffusion coefficient and slope of isotherm are independent of temperature and (2) solution is valid for short time, hence curvature of solid is lost. He compared his analytical solutions with the experimental results of water vapor sorption onto activated alumina obtained by Bowen and Rimmer [21]. The agreement was reasonably good. In the eighty and ninety, a lot more papers have appeared in the literature dealing with a variety of situations [22-26], such as linear driving force for the mass transfer [27], surface barrier for mass transfer [28], bidispersed particle [29], multicomponent systems [30], systems with macropore, surface and micropore diffusion [31]. 2.2 L D F M o d e l s

Solving diffusion and adsorption models by computer many decades ago requires a large amount of computing time. This is due to the fact that the model

781 equations are generally nonlinear partial differential equations and coupled, and that early computers were comparably slow. To overcome this, Glueckauf [32] developed a simple lumped model, in which the Laplacian operator was replaced by a constant; hence the driving force term is replaced by the difference between the surface concentration and the mean concentration in the particle. Glueckaufs model has the proportionality constant is ~2/R2. Jury [33] later used the frequency domain analysis to obtain a value of 15, instead of ~2. This model, he claimed, was derived from a sound mathematical foundation. This factor 15 was later proved by Liaw et al. [34] that it can be derived from the original Fick diffusion equation by assuming that the concentration profile at any time t is a parabolic function with respect to distance. They then applied this model to a fixed bed configuration where the external concentration varies with time. Film mass transfer was considered in their model. The parabolic profile assumption was later further utilized by Rice [35] to study the batch, packed tube and radial flow adsorber. Same as in the work of Liaw et al., Rice considered only linear adsorption isotherm, and the diffusion model is the homogeneous model, that is there is no distinction between the two phases existing reside the particle, i.e. the fluid and adsorbed phases. Hills and Pirzada [36] reanalysed the work of Liaw et al. and Rice and examined the approximate solution with the finite difference solutions. Hills [37] further studied the applicability of the linear driving force to a few cases and concluded that the LDF should be used with care when dealing with the batch adsorber. The parabolic profile approximation is not restricted to the homogeneous diffusion model, it is also applicable to the pore diffusion model. If the adsorption isotherm is linear the pore diffusion and homogeneous diffusion models reduce to the same nondimensional form [38]. The validity of the LDF model is justified when ~ (nondimensional time) is greater than 0.05, the LDF model can be used to approximate the system behaviour for both single particle as well as the batch adsorber. Do and Rice [38] also extended the parabolic assumption to a quartic profile assumption, and have found that the quaxtic approximate solution only improves slightly over the LDF model but the complexity involved with the quartic model does not justify its use in assessing system behaviour. Recognising the limitation of the LDF model especially at short times, Do and Mayfield [39] developed a so called power law model, and the new approximate solution, available in analytical form, improves signfficantly over the LDF model. This power law model was later improved [40]. Their solution is not only valid over the whole time domain but also exhibits the correct short time behaviour. For a solid diffusion model, they obtain m

qR which is very close to the exact solution at the short time: q _ 6 ~Dt

11 Dt

The following table lists the various LDF models used in the literature.

(11)

782 Table 1 Various LDF models used in the literature Authors Model equation Glueckauf [32] d_~ = ~2D(q R _q) dt

R "~

!

dq dt Glueskauf[32]; Jury [33]

m

R "~

2q

dq 15Dlq R[ -~- = R----V -

Do and Mayfield [39]

d:9 dt Do and Nguyen [40]

I o(qR

- 3 (3

a--0.59;

t21

~=1.9

The success of the power law model [39, 40] was applied to deal with more complicated problems, such as nomsothermal adsorption in a particle [41], and adsorption m a bimodal particle [42]. This model was later applied [43] to solve multicomponent diffusion problem, and they found that the power law model is the preferred choice over other approximations for systems having large differences in adsorption selectivity among components. Buzanowski and Yang [44] proposed a model by approximating the concentration profile as a cubic profle, and they have successfully applied it to single step changes as well as m cyclic adsorption - desorption processes. Applying the LDF concept was extended to nonlinear isotherm [45-52]. The popularity of the LDF model was later extended to reaction systems [53-55].

2.3 Bimodal Diffusion Models Since the wide spread use of molecular sieve adsorbents, researchers were concerned with ways how to describe adsorption kinetics in these particles. The particles normally have bimodal pore size distribution. The pellets consist of small microparticles (crystals in the case of zeolites), and the pores within the crystal provide space to store molecules while the space between the crystals provide space for the molecules to diffuse from the exterior into the particle interior. The pores in the crystals are called the micropores while the pores between the crystals are called the mesopore and macropores. The IUPAC classification is that pores having size smaller than 2nm are called micropores.

783 Pores larger than 50 nm are called macropores, and pores between these two are called the mesopores. With the bimodal structure, the mechanism of adsorption into particle is generally taken as follows. Molecules diffuse within the macropore and mesopore space by molecular and Knudsen diffusion mechanism. During their path of diffusion, they adsorb onto the pore mouth of the micropores. The adsorbed molecules at the pore mouth diffuse into the micropore via a process called micropore diffusion. This diffusion process is activated, i.e. molecules have to overcome certain energy barrier to diffuse into the particle. The mobility is far slower than that in the macropore space, but the length scale (typically 1~) over which the molecules have to diffuse is far smaller than the particle scale (typically 1 ram). Therefore, the time scale of diffusion in the micropore could be comparable to the time scale of diffusion in the macropore. For a pore diffusion mechanism in the macropore space, a parameter which characterizes the importance between the two time scales of diffusion is defined as [56] [~ + (1 - ~)K]DR 2 Time scale for macropore diffusion ~DpR~2 - Time scale for micropore diffusion

(12)

where ~ is the particle porosity (macropore and mesopore), K is the slope of the isotherm if the isotherm is linear or the ratio of the adsorbed concentration to the bulk concentration at equilibrium, D is the ~ s i v i t y in the micropore, Dp is the pore diffusivity, R is the particle radius and R~ is the radius of the microparticle. This parameter is basically the ratio of the time scale of macropore diffusion corrected by the adsorption capacity to the time scale of diffusion in the micropore. If this parameter is less than unity, the adsorption process is dominated by the micropore diffusion. The macropore diffusion will dominate the transport ff it is greater than unity. When it is of order umty both diffusion processes will control the uptake into the particle, and the following set of equations describe concentration distribution within a bimodal particle [57-58]. ~-~~ ) - +~ -(1/ ~ C 0q = ~DpV2C ," V2 = Laplacian operator in particle coordinate aq = D~V~q V, = Laplacian operator in microparticle coordinate 5t subject to the following boundary condition at the surface of the microparticle: r~ = R, ; q = f(C) (isotherm equation) (13) They have used the above set of equations for linear isotherm, i.e. f(C)=KC to review a number of systems in the literature and has found that many of them are controlled by both diffusion processes. The first paper published to deal with bimodal particle is due to Ruckestein et al. [59], which was later justified as the point sink approximation from a more exact analysis [60]. Since then a number of papers have appeared to investigate the various aspects of adsorption, for example in fixed bed chromatography [61-63], in fixed bed breakthrough curve [64-65], in adsorbent particle [66-70], and in nomsothermal adsorbent particle [29].

784 The bimodal diffusion model enjoys its application m zeolite particles or other particles that exhibit a bimodal structure. The carbon molecular sieve studied by Chihara et al. [71] shows that the bimodal diffusion model is applicable, and the microparticle (grain) was observed using the electron microscopy. 2.4 P o r e a n d S u r f a c e D i f f u s i o n Models: It is noted that the bimodal ~ s i o n model does not consider the transport of mass into the particle by surface diffusion. Surface diffusion was observed to occur in many systems, particularly those that exhibit large surface and with strongly adsorbing species. Surface diffusion was considered by Damkohler [2]. Experimentally, surface diffusion was observed [72-78]. The correlation between the activation energy of diffusion and the heat of adsorption was observed [79-80] to fall between one third and one. If the surface diffusion flux is expressed in terms of the adsorbed concentration gradient, the proportionality is called the surface diffusion coefficient, which is observed as a function of adsorbed concentration. This dependence does not follow any particular pattern, but generally it increases with the loading. To explain for this behaviour, a number of theories have been put forward [81-83]. Surface diffusion is no doubt the most difficult phenomenon to investigate. It always occurs in parallel with pore diffusion and it is affected by the equilibria between the fluid and adsorbed phase. Moreover, it is affected by the way solid surface is structured. If the surface were energetically homogeneous, the situation would be much simpler. Unfortunately, the surface of a real solid is complex and hence the study of surface diffusion relies on how we model the surface. Most work assume the surface as a homogeneous domain, and treat the diffusion flux in terms of the total adsorbed concentration gradient. This is no doubt the simplistic view at surface diffusion, but it serves a means for us to describe kinetics in systems revolving surface diffusion. The first experimental work to recognise the surface heterogeneity is probably due to Horiguchi et al. [84]. Review on surface diffusion was carried out by Kapoor et al. [85]. They pointed out the importance of surface diffusion, especially the role of heterogeneity of the surface on the phenomenon. This aspect of heterogeneity is studied in a systematic way by Zgrablich and co-workers [86-87]. Their approach is very general, but to develop a more analytical model for describing surface dii~sion on heterogeneous substrates Seidel and Carl [88] and Kapoor and Yang [89-90] have obtained expressions for the surface diffusivities for heterogeneous surface. Although the heterogeneity of the surface was recognised by researchers working in the area of sorption e q u ~ b r i a for quite some time, its utilization m dynamics studies was rather limited. This is due to the complexity os heterogeneity when sorption dynamics was considered. The first reason is our inability to describe the surface structure. The second reason is the description of surface diffusion rate in terms of surface structure, and the third reason is the complexity in mathematical modelling. Accepting the difficulties at hand, early researchers working with high surface area solids have to resort to the simple

785 Fickian type model for surface diffusion [2, 91-100]. Because of the importance of this model for high surface area, it has been used quite extensively in the literature. The surface diffusivity is normally found by subtracting the pore diffusion rate from the total observed rate, or it can be found by increasing the temperature to the extent that the contribution of the surface diffusion is negligible. This is due to the fact that the surface flux is a product between the surface diffusivity and the surface loading. As temperature increases, the surface diffusivity increases but not as fast as the decrease of the surface loading; therefore, the surface flux decreases [101-102]. This methodology has two limitations. One is that the temperature beyond which the surface diffusion is negligle may be too high for the solid, and the other limitation is that the surface diffusion rate is negligible compared to the pore diffusion rate as the temperature increases only when the adsorption isotherm is linear. Another possible way to overcome this problem is to use a method first proposed by Do [3,103] to extract the surface diffusivity from a simple linear plot. Starting from the set of equations: OC Oq ~DpVBC bC dp-~- + (1 - ~)-~+ (1- qb)DsV2q with q = qs 1 + bC (14) =

the following equation for the half time was obtained: to.5 =

oj

1 + Y b-Co)

(15)

where Co is the external bulk concentration and qo is the adsorbed concentration which is in equilibrium with Co and oc, 13, Y are defined in the following table. Table 2 Values of (z, 13, y for three .d~erent shapes of particle Particle shape a. ...... 13 . Slab 0.19674 0.25 Cylinder 0.06310 0.26 Sphere 0.03055 0.30

? 0.686 0.663 0.750

The experimental fractional uptakes were obtained at various bulk concentrations over which the isotherm curve is nonlinear [103]. Half times were collected for each bulk concentration. Rearranging the analytical haft time equation (eq. 15), we obtain the following equation which is useful for parameter determination.

oLR-~ , + (1- *)C7

1-

786

bCo 1( 0/ we This equation suggests that ff one plots the LHS versus I\l---------~--Jt,.------J~bC,0(-;ol+ would obtain a straight line, of which the slope is (1-r s and the intercept is ~Dp. This technique is only applicable when experiments are carried out over the nonJmear range of the isotherm. If experiments were to be carried out over the Henry's law range (i.e. bC 0 > h.

4.3. Results and Comparison with Experiment Figure 4.2 shows MC-REMA results for the network diffusion coefficients as a function of the coordination number of the network, Z, with the mean pore size as a parameter. For all these simulations, ~ = 0.5 and a lognormal pore size distribution function was employed in all cases. The standard deviation of the PSD is fixed at 0.01 nm and we present the results in terms of an "effective" pore width, > = - 3A. As for individual pores in this size range, oxygen diffuses more rapidly than nitrogen, and both diffusion coefficients are strong functions of Z. Using the best current value for the percolation threshold of the undiluted simple-cubic lattice, Pc = 0.2488 [Sahimi, 1994], and employing the dimensional invariance of the percolation threshold (Section 2.2), we obtain Z c ,~ 1.5. Above this value, the effective diffusion coefficients increase rapidly from zero. (There is an exception to this behaviour. The effective nitrogen diffusivity in a network with a mean pore size of 0.21 nm is non-zero only above Z = 3; this is because a pore of this size is the smallest pore to allow any measurable (in the MD simulations) diffusion of nitrogen. Thus, about half the pores are effectively closed to nitrogen and the "effective" coordination number for nitrogen at Z = 3 is only about 1.5, i.e. ~ Zc. ) Figure 4.3 shows the diffusivity ratio corresponding to the data of Figure 4.2. These are narrow PSDs so the networks would be expected to show a degree of kinetic selectivity similar to that of individual pores of the same dimension as the mean pore size, and this is confirmed by the data. Chihara et al. (1978) measured effective diffusion coefficients for nitrogen and oxygen in Takeda MSC-5A, over a range of temperatures. At 300 K (the temperature of our simulations), they report a nitrogen diffusivity equal to 2.1x10 -7 cm2/s and an oxygen diffusivity equal to 6.7x10 -7 cm2/s; the diffusivity ratio, R = 3.1. These diffusivities are about two orders of magnitude smaller than the MC-REMA results. However, the experimental diffusivity ratio is well matched by a network with a mean pore size of a little less than 0.25 nm (Figure 4.3). Chihara and Suzuki (1979) have measured diffusion coefficients in a different carbon, Takeda MSC 4A. For the unmodified carbon they report R = 4.5; after carbon deposition, they report R = 3.7. These values are matched by networks with mean pore sizes between 0.21 nm and 0.25 nm. Diffusion coefficients have been measured for oxygen and nitrogen in the BergbauForschung CMS, most recently by Chen et al. (1994) who report the following results (r is the characteristic microparticle size): DM,NJ eft' r2 = 9.5x10 -6 S"1 and DM,02/ eft r 2 = 3.5x10 "4 s "l, giving R = 36. This diffusivity ratio is matched by a network with a mean pore size in the region of 0.21 nm, provided Z > 3. Using the value r = 4.3 l~m, measured by Chen et al. using scanning electron microscopy, the effective diffusion coefficients are O(10 -10 - 10-12 cma/s), at least five orders of magnitude smaller than the MC-REMA results. Other results have been presented for the Bergbau-Forschung CMS [Kikkinides and Yang, 1993; Knoblauch, 1978; Ruthven et al., 1986; Farooq and Ruthven, 1991; Kuthven, 1992]. The reported diffusion coefficients vary widely (this is presumably at least partly due to the

869

,o > = 0.2 lnm [] j > = 0.25nm o , o > = 0.29nm o

eff

(10 -5)

yr.~.-m~''~r" .

0

4.0 4.5 5.0 5.5 6.0 Z Figure 4.2. The effective network diffusivity (in cm 2 Is) as a function of the network coordination number Z. The effective pore width is > = - 3A and the standard deviation of the effective pore width distribution is 0.01 nm. The open symbols and the f'dled symbols represent the data for nitrogen and oxygen, respectively. 1.5

2.0

2.5

3.0

3.5

100 > = 0.21 nm [] > = 0.25 nm > = 0.29 nm 0

eft DM'~ ,eft

10

./

l

El

I-I

A

A

v

v

i

2

I

..,

m

m

v

v

A

i

3

1 !

n

4

I

5

I

|

6

Z Figure 4.3.

The diffusivity ratio

eft eft DM, o2/DM,N2 as a function of the network coordination

number for the results provided in Figure 4.2. The horizontal arrows indicate the respective diffusivity ratios for the individual pores of width h = w + 3A in which w = 0.21, 0.25, and 0.29 nm.

870 differing pretreatment regimes used in these studies), and are all at least four orders of magnitude smaller than our simulation results. However, the diffusion coefficient ratios are broadly similar to the value obtained by Chen et al. (1994) (23 < R < 38) and are within the range of the simulation results. We return in Section 5 to the discrepancy between the absolute diffusion rates predicted by our simulations and measured experimentally. Figures 4.4 and 4.5 show diffusivity data for the same mean pore sizes and coordination numbers as in Figures 4.2 and 4.3, but for much wider PSDs, with a standard deviation of 0.25 nm. For these PSDs, a proportion of the pores are so small that they do not permit diffusion of nitrogen, and this proportion increases as the mean pore size decreases. The effective percolation threshold for nitrogen is thus larger than Z c ~ 1.5, and increases with decreasing mean pore size. Unlike in the case of narrow PSDs, the diffusivity ratio is a strong function of the coordination number. One of the most important conclusions implied by these results is that networks with a wide distribution of pore widths are able to display the same range of kinetic selectivities as individual pores, and are capable of reproducing the diffusivity ratios observed experimentally. Thus, the functioning of our model CMS does not depend on a narrow PSD; we see no reason why this observation should not apply also to real CMSs. The effect of the width of the PSD on the diffusion coefficients (for fixed mean pore width and porosity) is investigated further in Figure 4.6. Figure 4.7 shows the corresponding data for the ratio of diffusion coefficients. As expected, the increasing width of the PSD reduces the absolute values of the diffusion coefficients. The diffusivity ratio, on the other hand, goes through a minimum as the pore width is increased, so that a network with a very narrow PSD and a network with a wide PSD have similar selectivities. This behaviour is unexpected. The simulation results for single REG pores (see Figure 3.4) show that only pores in a very narrow size range offer a significant selectivity to oxygen and nitrogen. As the width of the PSD increases, the proportion of pores outside this narrow size range increases. These pores are either so large that they show little selectivity, or so small that nitrogen is unable to pass. So, one might expect the selectivity to be "washed out" by increasing the width of the PSD. An elegant analysis due to Ambegaokar et al. (1971) explains why this does not happen. Their analysis, which is not quantitative in three dimensions, involves the following thought experiment. Consider a network with an infinitely wide PSD. First, remove all the bonds from the network. Then, replace the bonds, in order of decreasing diffusional conductance, monitoring the diffusion coefficients. While the first few bonds are replaced, the diffusion coefficients remain zero. This is so until a percolating cluster is formed. As we are considering a network with an infinitely wide PSD, diffusion through the network is controlled by the last "critical" pore to be replaced. All the pores that were replaced earlier are much wider than the critical pore, and they are in series with it. Therefore these pores exert little additional resistance to diffusion. Now, add the remaining pores. These pores are all much smaller than the critical pore and, because they provide alternative diffusion paths, they are effectively in parallel with it. Therefore these smaller pores contribute little to diffusion. Thus, according to this "critical path" analysis, a single pore size controls diffusion in both the limits of zero and infinite width of the PSD. The isosteric heat of adsorption and the activation energy for diffusion offer another point of comparison between simulation and experiment and data reported in the literature for these properties are reproduced in Table 4.1. The theoretical results for qst provided in Figure 3. lb

871

o 9 > = 0.21 nm n , I > = 0.25nm

~/~

, 9 > = 0.29nm

(10 -5 )

2.0

1.5

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Z Figure 4.4. The effective network diffusivity (in cm 2/s) as a function of the network coordination number Z. The standard deviation of the pore width distribution is 0.25 nm and all symbols are as def-med in Figure 4.2.

10

3

10

2

Deft --M,02 1 eff 10

DM,N2

~.. 0

10

-1 10

Figure 4.5.

v

nm [] > = 0.25 nm o > = 0.29 nm

o

>

,

=

-

-

0.21

I

....

4

The diffusivity ratio

,

t

,

5 eft" eft DM, O2/DM, N2as a function of the network coordination

number Z for the results provided in Figure 4.4.

872 2.5 o Oxygen 2.0

eff

~

9 Nitrogen

1.5

-5 ( 1 0 ) 1.0

o.5 0.0 0.0

,

r

,

0.05

t

,

0.10

I

~

l

0.15

,

0.20

0.25

Standard Deviation of the Slit Width Distribution Figure 4.6. The effective network diffusivity (in cm 2/s) as a function of the standard deviation of the slit width distribution. The average slit width is > = - 3A = 0.25rim and the coordination number of the network is Z = 4.5.

D eft 3 -M,O 2 eft DM,N2 2

0

I

I

l

I

i

I

z

I

t

0.0 0.05 0.10 0.15 0.20 0.25 Standard Deviation of the Slit Width Distribution Figure 4.7.

The diffusivity ratio

eft"

eft

DM,o2/DM,N~ as

a function of the standard deviation of

the slit width distribution for the results provided in Figure 4.6.

873 Table 4.1.

Reported Isosteric Heats of Adsorption and Activation Energies for Diffusion within CMS Gas Oxygen

Nitrogen

CMS Takeda MSC 5A Unmodified Takeda MSC 4A Modified Takeda MSC 4A Bergbau-Forschung Unspecified Origin Takeda MSC 5A Unmodified Takeda MSC 4A Modified Takeda MSC 4A Bergbau-Forschung Bergbau-Forschung Unspecified Origin

qst (kJ/mol) 17.1 15.9

E a (kJ/mol) 10.04 5.86

11.7

27.6

17.8 - 19.5 18.8 15.9

23.4 16.3 10.9

11.7

30.9

20.0 - 21.1

27.2 28.7 10.0 - 11.1

Source Chihara et al. (1978a) Chihara and Suzuki (1979) Chihara and Suzuki (1979) Kikkinides et al. (1993) Fitch et al. (1994) Chihara et al. (1978a, b) Chihara and Suzuki (1979) Chihara and Suzuki (1979) Ruthven et al. (1986) Kikkinides et al. (1993) Fitch et al. (1994)

suggest that the isosteric heat of adsorption for CMS should lie between the asymptotic value 13.9 kJ/mol for the wide pore limit and maximum values of approximately 19 kJ/mol for nitrogen and 24 kJ/mol for oxygen. For Takeda MSC 5A [Chihara et al. (1978)] agreement between model and experimental values of qst is fair; the value for nitrogen is close to our largest value for a single REG pore for nitrogen, and is within the range of single-pore results for oxygen. This would appear to suggest that the controlling pore size for this CMS lies in the range 0.21 nm - 0.25 nm (h = 3.625A to 3.75A) as observed above for the diffusion coefficient ratio. For unmodified Takeda MSC 4A, Chihara and Suzuki (1979) report qst = 15.9 kJ/mol for both species which are also in good agreement with the model results. However, for the Takeda MSC 4A modified by carbon deposition, qst = 11.7 kJ/mol for both species which is significantly below the minimum value reported in Figure 3. lb. The activation energy for diffusion as defined by the Equation (1.1) may also be estimated from MD simulations over a range of temperatures (albeit expensively if the required statistical sampling of the particle trajectories is high) and the results shown in Figure 3.9 suggest that E a for oxygen and nitrogen diffusion within a REG pore of width h = 3.75 A at 300 K ( T* = 6.524 for 0 2 and T* = 7.296 for N2) are 3.9 kJ/mol (02) and 3.8 kJ/mol (N2). Both of these estimates are much smaller than the experimental values cited in the fourth column of Table 4.1 and this suggests that pore sizes of this magnitude (or larger) do not play a significant role in the experimentally observed kinetic selectivity of CMS. On the basis of our earlier observations for the diffusivity ratio and qst, we believe that the rate limiting paths for diffusion within CMS correspond to pores which are very close to the single pore percolation threshold for the given gas and to demonstrate this additional MD simulations were conducted over a range of thermal energies for nitrogen diffusing in pores of width

874 h = 3.625A and 4.0A. The diffusion coefficients determined using the MD simulation technique described in Section 3.2 are plotted in Figure 4.8 as a function of l/T* (-=5e~2c/2<X>) and the activation energies obtained from each of these sets of data are 13.0 kJ/mol (h = 3.625A), 3.8 kJ/mol (h = 3.75A), and 4.4 kJ/mol (h = 4.0A). These results suggest that an REG model subject to 50% etching may provide a reasonable description of the rate limiting pore structure in Takeda MSC 5A, unmodified Takeda MSC 4A, and in the carbon molecular sieves of unspecified origin which were investigated by Fitch et al. (1994). It is anticipated that by varying the extent of etching (and pore width) it should be possible to generate pore structures which accurately model the behaviour of the other CMS media cited in Table 4.1, particularly in view of the high sensitivity of the CMS kinetic selectivity to small changes in the degree of carbon deposition [see for example, Moore and Trimm (1977) and Chihara and Suzuki (1979)].

101

100

DIvI(T)10-1

10-2

9 h=3 9 h 3.75A

I I

1 03 - , -

,

0.0

l

0.05

0.10

0.15

0.20

,

0.25

1/T*

Figure 4.8. The diffusion coefficient for nitrogen in s~ngle REG pores as a function of the inverse reduced temperature. DM(T) is in units of g cc~/(eic/mi) and T* is in units of eic/kB.

5. SUMMARY AND CONCLUSIONS We summarise the comparison of our model with experimental data reported in the literature by the following observations. 1. The model reproduces the relative diffusion rates observed experimentally. 2. The absolute diffusivities predicted by the model are always larger than the experimental values, by at least two orders of magnitude. 3. The calculated activation energies are in agreement with experimental results for Takeda MSC 5A [Chihara et al., 1978], for the unmodified Takeda MSC 4A [Chihara and Suzuki, 1979], and for the carbons studied by Fitch et al. (1994). For

875 these samples, the model isosteric heats are a little too low. The activation energies measured on the other carbons are much higher than the model predictions. The inability of the model to give good absolute values for the effective diffusion coefficients is, on the face of it, a significant failing. However the calculation of diffusion coefficients from dynamic diffusion measurements requires the input of a typical linear dimension of the microporous regions in the adsorbent particle. In the analyses of experimental diffusion rates discussed here, this dimension is identified with the mean microparticle radius [Chihara et al., 1978; Ruthven, 1992; Chen et al., 1994], but this is not necessarily the most appropriate choice. This definition assumes that all the macropores surrounding the microparticles are part of a percolating network. It is at least plausible to suppose that when the microparticles are bound to form the adsorbent particle some of the macropores become disconnected from the percolating macropore network, so that not all microparticles are directly accessible from the macropore network. In this case, the molecules would have to diffuse through microporous regions larger than the microparticles themselves. If these regions were O(10) microparticles in diameter, our diffusion coefficient results would be brought into good agreement with the results of Chihara et al. (1978) for Takeda MSC 5A; as we noted above, there is good agreement between experimental and model results for the activation energy for nitrogen diffusion for this carbon. This is no more than a possible explanation for the discrepancy between our model predictions and some of the experimental results; we have no direct evidence that this occurs in practice. As we pointed out above, our model values for qst are a little lower than the experimental values, even where there is good agreement on activation energies. This suggests that the KEG model lacks a sufficient density of high energy sites; this could be remedied (without significantly changing the activation energy) by reducing the amount of etching. This hypothesis cannot explain the large discrepancy in diffusion rates between the model and the results for the Bergbau-Forschung carbon [Kikkinides and Yang, 1993; Knoblauch, 1978; Ruthven et al., 1986; Farooq and Ruthven, 1991; Ruthven, 1992], or for the modified Takeda MSC 4A [Chihara and Suzuki, 1979]. Our model fails to reproduce the low qst values (modified Takeda MSC 4A) and both the high activation energies and the low diffusion coefficients observed experimentally. This suggests that an acceptable model for these carbons should have fighter restrictions in the pore network and while it should be possible to achieve this with lower etching conditions and small pores it may still be difficult to reconcile the model and experimental values for qst- One possible avenue of investigation involves relaxation of the etched graphitic surface. This could result in locally fight restrictions with large cavities (low qst values on average) within the REG pores. Finally, we note that our model is consistent with the existence of a surface barrier to diffusion, as observed by Dominguez (1988) and LaCava et al. (1989). In our model, a surface barrier to diffusion would be observed experimentally if the portion of the pore network near the surface of the microparticles offered a greater resistance to diffusion than the pores in the interior, due for example to preferential carbon deposition. It is not necessary to assume that the surface resistance is due to pores actually at the surface of the microparticles.

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881

SUBJECT INDEX

Absolute Rate Theory approach 335 Activated carbon 679, 777 Activation energy for adsorption (desorption) 335 Active carbons 715 Adsorption 153, 715,745,837 Adsorption at low temperature 679 Adsorption/desorption isotherm 625 Adsorption-desorption processes 201 Adsorption energy distribution 1 Adsorption equilibria 777 Adsorption in pores 105 Adsorption isotherms 285 Adsorption kinetics 777 Adsorption potential distribution 715 Adsorption rate 285 Adsorptive energy surface 373 Allosteric surfaces 1 Argon 573 Aromatics adsorption in MFI zeolites 519 Arrhenius law 373 Asymptotically correct approximation 1

Barriers surface 373 BET isotherm 1 Bimodal diffusion model 777 Binary systems 777

Cage configurational integral 519 Calorimetric studies of adsorption 519 Capillary condensation 625,745 Capillary-energy ratio 625 Capillary wetting 625 Carbon dioxide adsorption 573 Carbon molecular sieves 837 Carbon monoxide 285 Carbons 745 Chemical diffusion coefficient 373

Chemical surface structure 679 Chemisorption 153 Chlorite 573 Clays 573 Cluster method 201 CO-Ni(111) 285 Collision model 201 Comparison plot 625 Complementary site-matching correlation Computer simulation 451 Condensation approximation I Construction principle 373 Controlled rate thermal analysis 573 Correlated heterogeneous surface 373 Cumulative adsorption energy distribution D Darken's equation 373 Density functional theory 1,745 Derivative isotherms 573 Desorption 153 Desorption kinetics 285,777 Differential adsorption bed 777 Diffusion 153,837 Diffusion in zeolites ---, concentration dependence of 487 ---, dependence on bond energies for 487 ---, energetics of 487 ---, hopping model for 487 ---, kinetic theory of 487 --, Molecular Dynamics simulation of 487 ---, Monte Carlo simulation of 487 ---, stochastic theory of 487 Displacement kinetics 777 Dual diffusion model 777 Dubinin-Astakhov isotherm 519 Dubinin-Radushkevich isotherm 1,625

Effective medium approximation 373 Effective potential 625 Electron exchange reactions 285

882 Electronic structure 679 Elovich equation 335 Empirical isotherm equations 1 Energetic/geometric heterogeneity 625 Energetic heterogeneity 715 Energetic topography 373 Energy correlations in mixed-gas adsorption Energy distribution 777 Enhanced wetting 625 Entropy increase 285 Entropy production 285 Equidistant interface 625 Equilibrium isotherms 285 Equilibrium properties 153 Equilibrium surfaces 1 Exact methods 1 Excluded-volume interactions 519 External surfaces 573

FHH isotherm I Fick's law 373 Fluctuations 625 Fowler-Guggenheim isotherm 1 Fractal dimensions 625, 715 Fractal surfaces 1,625 Frenkel-Halsey-Hill isotherm, FHH exponent 625 Frankel-Halsey-Hill equation 715 Freundlich isotherm 1 FT IR spectroscopy 679 Full isotherm 105

Gas-solid systems 105,201 Generalized gaussian model 373 Geometrical heterogeneity 715 Gibbs equation 1 Gibbs Monte Carlo 745 Grand potential 625 Graphite 451 H

Heat of adsorption 519 Heat of immersion 715 Heterogeneous ideal adsorbed solution 1 Heterogeneous models 777 Heterogeneous pore network model 837

Heterogeneous solid 777 Heterogeneous surfaces 105,201 High ordered structure of porous materials 679 High resolution alpha s-plot 679 Hill-deBoer isotherm 1 Homogeneous models 777 Horvath-Kawazoe method 715 Hydrophobicity 573 Hysteresis 625, 745

Ideal Adsorbed Solution theory 1 Immersion calorimetry 573 Induced heterogeneity approach 519 Inner/outer cutoff 625 Integral equation 1 Integral equation of adsorption 715 Inverse problem 1 Irreversibility 285 Isothermal desorption 285

Jagiello method 1 Jaroniec-Choma equation 715 Jump correlation factor 373

Kaolinite 573 Kelvin equation 625, 715, 745 Kinetic lattice gas 153 Kinetic selectivity 837 Kinetics 153 Kubo-Green's equation 373

Landman-Montroll method 1 Langrnuir-Freundlich isotherm I Langmuir isotherm 1 Langmuirian kinetics 335 Latent pore characterization 679 Lateral interactions 105,201 Lattice gas 153,373 Law of corresponding states 625 Layering transition 745 Light paraffins adsorption 777 Linearized plots for adsorption isotherms 519

883 Liquid-gas interface 625 Localized adsorption 1 Low-pressure adsorption, virial approach

1

M

Mean field approximation 105, 519 Medium approximation 837 Mesopores 745 Metals 153 MFI zeolites 519 Mica 573 Micropores 745 Micropores, diffusion in 487 Micropores, multicomponent diffusion in 487 Micropore filling 679 Micropore size induced heterogeneity 777 Micropore size distribution 679 Microporosity 5 73, 715 Mixed adsorption 105 Mixed-gas adsorption 1 Mobile adsorption 1 Mobile surfaces 1 Molecular dynamics 451 Molecular reorientation 451 Molecular sieves, multicomponent diffusion in 487 Molecular simulation 745,837 Monolayers 451 Monolayer coverages 105, 201 Monte Carlo renormalised effective 837 Montmorillonite 573 Multicomponent adsorption 777 Multicomponent diffusion 487 Multicomponent diffusion in zeolites 487 ---, comparison of models for 487 ---, models for 487 Multicomponent diffusivities, prediction from pure component diffusivities 487 Multicomponent surface diffusion 487 Multicomponent surface diffusion in zeolites 487 Multilayers 153,625 Multilayer adsorption I, 105 Multilayer adsorption, diffusion in 487 Multi-site-occupancy adsorption 1 N Networked pores 745 Nickel 285

Nitrogen 573 Nitrogen isotherm 745 Nonequilibrium thermodynamics 153 Nonideal competitive adsorption 1 Nuclear magnetic resonance method 679 O Order of adsorption-desorption kinetics 335 Oxides 745

Palygorskite 573 Partial isotherms 105 Patchwise topography 1,777 Peak splitting 201 Percolation model 373 Perfect positive correlations of adsorption energies 1 Permanganate electron exchange 285 Phase diagram 625 Phase transition 625 Phase transitions in lattice gas 519 Phase transitions in zeolites structure 519 Physical adsorption 679 Physisorption 153 Pore 625 Pore blocking 745 Pore characterization 679 Pore origin 679 Pore shape 745 Pore size 745 Pore structure 679 Porosity 715 Porous adsorbents 745 Porous solids 373 Potential Theory 1 Power Law equation 335 Precursors 153 Pre-exponential factor 285 Primary salt effect 285

Q Quantum mechanical pertubation theory 285 Quasichemical approximation 105, 201 Quasi-equilibrium 153,573 Queer surfaces 1

884

Random topography 1 Random walk 373 Reaction kinetics 285 Reconstructable surfaces 1 Redistribution of molecules adsorbed on different sites 519 Reduced isotherm 625 Reed-Ehrlich's equation 373 Rough surfaces 1 Rudzinski-Iagiello method 1 Ruthven's model for adsorption in zeolites 519

Self-diffusion 451 Self-similar 625 Separation coefficient 105 Sepiolite 573 Silica 745 Silver electron exchange 285 Sips' method I Site-bond model 373 Small angle X-ray scattering 679 Sorption in pores 451 Standard adsorption 715 Statistical mechanics 153 Statistical Rate Theory of interracial transport 335 Stepped surfaces 451 Sticking 153 Stoeckli's extension of DR equation 1 Submonolayer 625 Substrate potential 625 Sulfur dioxide adsorption 777 Surface diffusion 451,487, 777 Surface forces 745 Surface modification 679 Surface tension 625 Swelling 573

Talc 573 Temkin isotherm 1 Temperature dependence of adsorption 519 Temary systems 777 Thermodynamics of adsorption 715 Thermo-programmed desorption 201 Time-correlation-functions 451 Topography 625 Toth isotherm 1 t-plot 625 Tracer diffusion coefficient 373 Transfer matrix 153 Transition state model 201 Traps surface 373

Vacancy Solution Theory 1 Vanadium electron exchange 285 Van der Waals wetting 625 Volmer isotherm 1 W Water vapour adsorption 573 Wetting transition 625 X XPS-ratio method 679 X-ray diffraction 573,679 X-ray photoelectron spectroscopy 679

Zeolites structure and imperfections 519 Zero-time exchange 285