Theory of Adsorption from a Dimer–Monomer Solution on a Patchwise Heterogeneous Surface

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

182, 268–274 (1996)

0459

Theory of Adsorption from a Dimer–Monomer Solution on a Patchwise Heterogeneous Surface M. BORO´WKO 1

AND

W. RZg YSKO

Computer Laboratory of Chemistry, M. Curie-Skłodowska University, 20-031, Lublin, Poland Received January 5, 1996; accepted March 25, 1996

A theoretical model for adsorption from a dimer–monomer liquid mixture on a patchwise heterogeneous surfaces is proposed. A comparison with the results of canonical ensemble Monte Carlo simulation is discussed. The influence of patch sizes on the total adsorption and the structure of the adsorbed layers is analyzed. q 1996 Academic Press, Inc.

Key Words: adsorption from solution; adsorbent heterogeneity; dimer–monomer liquid mixture; Monte Carlo simulation.

INTRODUCTION

The history of modern colloid chemistry is closely connected with the problem of adsorption from solutions. In the past, a lot of theories dealing with this phenomenon were formulated. Their results have been summarized in several reviews (1–12). In the majority of the articles, adsorption on energetically homogeneous surfaces has been discussed. Although problems involving real adsorbents are much more frequently encountered in industrial processes, the behavior of a liquid mixture on a heterogeneous surface has been relatively less well studied. On the other hand, the theories including heterogeneity of the adsorbent surface refer usually to mixtures consisting of molecules of similar size. It is evident that the mathematical description of the mixture composed from molecules of different shapes in contact with a heterogeneous adsorbent is much more complicated. Therefore only few papers have been connected with this problem (12, 13). For this reason there is not a general theory of adsorption from a solution of molecules of different size on a heterogeneous surface. However, increasing experimental evidence shows that both these factors determine a situation on a liquid–solid interface (10, 11). Also, from a theoretical point of view it seems to be very interesting to study the simultaneous influence of these factors on the adsorption process. In this paper we present a theoretical approach to adsorption from a dimer–monomer liquid mixture in contact with 1

To whom correspondence should be addressed.

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0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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a patchwise heterogeneous solid surface. In the model, we introduce two important parameters determining the thermodynamic properties of the system, i.e., the difference in the molecular sizes of the components and adsorption heterogeneity. Our treatment is a generalization of the theory of adsorption from a solution consisting of equal-sized molecules on heterogeneous surfaces (3, 7–11) and the approaches of Ash–Everett–Findenegg (14) and Roe (15) to adsorption of dimers on homogeneous adsorbents. Let us discuss more closely the models we shall study in this paper. In spite of numerous attempts to found a theory of liquid solution the results are not yet satisfactory (16). One of the most popular theories is the ‘‘classical lattice model of solution,’’ the complete and rigorous development of which is due to Guggenheim (17). The major success of this model was the evaluation of the combinatory factor for a polymer solution by Flory (18) and Huggins (19). Unfortunately, the quantitative predictions of the model are in disagreement with experimental data. This model is not sufficient for the understanding of thermodynamic functions of the solution (16). In spite of all these limitations, the lattice model of the solution remains of great interest because it describes quite directly some difficult problems such as order–disorder effects and the combinatory factors in polymer solutions. Moreover, this model has been used in the fundamental papers underlying the theory of adsorption from solution on homogeneous surfaces (14, 15, 20–22). It seems to be quite natural to extend this approach to adsorption on a surface consisting of different active centers. The theory of adsorption on an energetically heterogeneous surface was primarily extended to a gas–solid interface (23). However, the influence of these ideas on the development of the theory of adsorption from solution was large and quite apparent (3, 7). In all these studies each adsorption site is unambiguously characterized by values of energies of interactions between molecules of all components and the solid. When these energies are the same for the whole surface the adsorbent is defined as homogeneous and the other surfaces are heterogeneous. The sites of various types can be distributed differently on the surface. Two models of the surface topography are most frequently used. The

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patchwise model of the adsorbent assumes that the surface is composed of a number of homotatic domains which are considered independent each of other. According to the second model, the geometrical configuration of active centers is completely random (3). The general statistical-mechanical formulation of monolayer adsorption from mixtures has been presented in (7). This treatment is based on a concept of ‘‘local adsorption isotherm’’ which describes adsorption on any given type of site. The total adsorption isotherm is defined as the weighted average of the local isotherms. The method can be used only when adsorption on any given type of active center is not influenced by adsorption on other sites. Although this assumption has usually been made, it is not very realistic. An attempt to overcome this difficulty has been made by using the patchwise model of an adsorbent surface. When the homotatic domains are macroscopically large, equilibrium is established independently on the different patches. In this case interactions between molecules adsorbed on various patches are neglected. Moreover, this theoretical approach does not involve domain size as a parameter characterizing the adsorbent surface. However, in real systems the finite domains have been observed (24–27). There are few references in the literature that demonstrate the role of edge effects in adsorption from the gaseous phase (3, 23– 29). These effects can be more important for the liquid/ solid interface because all active centers are covered by the solution molecules and molecular interactions play a dominant role in the condensed phases. The latter problem has not been studied so far. In spite of all the limitations discussed above we have decided to extend these approaches to the adsorption of a dimer–monomer solution on heterogeneous surfaces because even the application of the elementary models can give an interesting picture of the physical situation in the complex system. The usefulness of the theoretical approach presented in the next section can be judged only in light of extensive and precise experimental data. Unfortunately, the results of such measurements are not accessible in the literature. Therefore, we have compared the theoretical values of adsorption with pseudoexperimental data obtained from Monte Carlo simulation. The aim of our studies was to state whether the deviations between these results arise from the mathematical approximations or the deficiences of the model connected with neglect of the edge effect and the assumption of independent adsorption on different active centers. Moreover, we wanted to discuss the influence of homotatic domain size on adsorption and the structure of the fluid near the surface. MODEL

A liquid mixture consisting of monomer and dimer molecules in contact with a solid surface is considered. We adopt the quasicrystalline model of the solution. Each lattice site

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may be occupied by the monomer molecule or by one segment of the dimer. It has z nearest neighbors, of which lz are in the same layer and mz are in each of the adjacent ones. The total interaction energy between molecules is the sum of contributions from near-neighbor pairs of segments. The interactions between the liquid and the solid surface are confined only to those segments in the first layer of the lattice. The solid surface is assumed to be patchwise heterogeneous one (3). According to this model, the surface is composed from homotatic patches which are considered totally independent of each other. In this case interactions between molecules adsorbed on different patches are excluded from the theory. The grand canonical partition function of the system may be expressed in the form

F S t

JÅ

∑ Q( f )exp N ∑

i Å1

f

D

1 (i) f 1 m1 / f (2i ) m2 /(kBT ) 2

G

, [1]

where f Å ( f (11 ) , f (12 ) , . . . , f t1 ) and f (j i ) is the volume fraction of the jth component ( j Å 1, 2; subscript ‘‘1’’ refers to the dimer whereas ‘‘2’’ denotes the monomer) in the ith layer, mj is the chemical potential of the jth liquid, and N is the number of sites in a given layer. The summation in Eq. [1] is over all possible concentration distributions f. The canonical partition function for the system with a patchwise heterogeneous surface is given by r*

QÅ

∏ Qr ( fr ),

[2]

rÅ1

where Qr is the canonical partition function for the rth homogeneous patch, fr Å ( f (1,r1 ) , f (1,r2 ) , . . . , f (1,rt ) ) and f (j,ri ) is the volume fraction of the jth component in the ith layer for the rth patch; r* is the number of types of homotatic patches. The equilibrium values of fr associated with the maximum term of the grand canonical partition function may be obtained from the (t 1 r*) equations

S

D

Ì ln Qr N 1 Å r m1 0 m2 (i) Ìf 1,r kBT 2

i Å 1, 2, . . . , t,

r Å 1, 2, rrr r*,

[3]

where Nr is the number of adsorption sites of the rth type. According to the Roe theory (15) the canonical partition function for a given homogeneous patch is given by Qr Å Q1pQ2p Vr ( fr )exp[N(EO 1,rf (1,r1 ) / EO 2,rf (2,r1 ) 0 vV N12,r )/(kBT )],

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where Vr is the combinatory factor, Qj p ( j Å 1, 2) is the partition function of the pure liquid ‘‘ j’’, ( 0 Eˆ2,r ) is the free energy change accompanying a transfer of a monomer molecule in the pure monomer from its interior to the first layer and ( 0 Eˆ1,r ) is defined likewise for a dimer segment, N12,r is the total number of nearest neighbour contact pairs between a monomer and a dimer segments, and vV is the excess energy per contact. When the Bragg–Williams approximation of random mixing is used we have

where fj ( j Å 1, 2) is the volume fraction of the jth component in the mixture f j Å lim ( f (j i ) )

[11]

i r`

and v is the well-known Flory–Huggins parameter given by

S

v Å 0z u12 0

D

1 (u11 / u22 ) /(kBT ). 2

[12]

t

N12,r Å ∑ zNrf (2,ri ) (mf (1,ri0 1 ) / l f (1,ri ) / mf (1,ri /1 ) )

[5]

iÅ1

The combinatory factor Vr is given by (15) t

ln Vr Å 0Nr ∑ iÅ1

S

The set [8] is a generalization of the equations derived for homogeneous surfaces (14, 15). From these equations we can obtain the ‘‘local adsorption isotherm’’ for the particular homogeneous patches. The volume fraction for the whole layer is defined as the average

f (1,ri ) ln f (1,ri ) / f (2,ri ) ln f (2,ri )

0

r*

1 (i) f 1,r r f 2

(i) r

D

,

[6]

1

(i) r

Å

∑ ckf (1i/k )

[7]

where pr Å Nr /N is the probability that a given adsorption site is of the rth type. The determination of the concentration profile f has two steps: (i) solving Eqs. [8] for all patches and (ii) calculating the average from Eq. [13]. The surface excess of the dimer is defined as

kÅ01

t

GÅ

for c0 Å l and c01 Å c1 Å m. Combination of Eqs. [3] – [7] leads to the set of simultaneous equations

1 ln f 2

1

(i) r

0

1

1 i0n/k )] ∑ [cnf (1,ri0n ) /( ∑ f 1,r 2 nÅ01 kÅ01

1

0v

∑ ck ( f (1,ri/k ) 0 f (2,ri/k ) ) 0 EO 12d (1 0 i) Å g

THE MONTE CARLO SIMULATION

i Å 1, 2, . . . , t

r Å 1, 2, . . . , r*

[8]

for

F

E1,r 0 E2,r 0

G

mr z (u11 0 u22 ) /(kBT ), [9] 2

where ( 0 Ej,r ) is the adsorption energy on the rth patch and ( 0uij ) denotes the energy of interaction between segments ‘‘i’’ and ‘‘ j’’, whereas d is the delta function of Kronecker. In the above equations g is the function defined for the bulk system g Å ln( f1 / f2 ) 0

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[14]

iÅ1

kÅ01

EO 12,r Å

∑ ( f (1i ) 0 f1 ).

For the monomer we have Gm Å 0 G. The model discussed above may be extended for a surface characterized by continuous distribution of adsorption energy by replacement the sum in the average [14] by a suitable integral (30). Similar considerations may be performed for a random model of the surface topography.

ln( f (1,ri ) / f (2,ri ) ) 0

[13]

r Å1

where f

∑ f (j,ri ) r pr j Å 1, 2,

f (j i ) Å » f (j,ri ) … Å

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The simulation was performed according to procedure used in the previous paper (13). The standard Metropolis sampling method in the canonical ensemble was applied (31). The simulation box was assumed to be a three-dimensional cubic lattice 24 1 24 1 20 or 30 1 30 1 20. Standard periodic boundary conditions were used in both directions parallel to the adsorbent surface. The reflecting wall was placed at the top of the system. For the solid surface the model with patchwise regular heterogeneity was used (29). We have assumed that the surface was composed of two different kinds of adsorption sites which have equal contributions (p1 Å p2 Å 0.5). Moreover, the chessboard type of surface topography (13, 29) with square patches of different sizes (s 1 s; s Å 1, 2, 5) was assumed.

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We have considered two types of a change in system’s state: (i) displacement of the monomer by one segment of the dimer and (ii) displacement of two monomers by the whole dimer molecule. All possible translations and rotations of the dimer molecule were included. Our model assumed the competitive character of the adsorption process. The total energy was the sum of the energies of interaction between first-nearest neighbors. The liquid–solid interactions were confined to the first layer. The ‘‘bulk’’ values were calculated for layers: i § 7. We introduce the following notation: u* ij Å uij /u22

[15]

E* j Å Ej /u22

[16]

T* Å kBT/u22 .

[17]

The average values were calculated from the standard relations (32). The number of Monte Carlo steps used for the equilibration was usually 10 5 . The simulations runs consisted of 10 4 configurations over which the ensemble averages were obtained.

FIG. 1. Excess adsorption isotherms for the homogeneous surface. The points are the results of Monte Carlo simulation and the lines are obtained from theory. Parameters: E *1 Å 6, E *2 Å 4 (open symbols) and E *1 Å 1, E *2 Å 3 (filled symbols); u *12 Å 1.1 ( s ) ( v Å 00.6); u *12 Å 0.9 ( , ) ( v Å 0.6); u * 12 Å 0.8 ( h ) ( v Å 1.2).

RESULTS AND DISCUSSION

All calculations have been performed for the systems exhibiting complete mixing. We have assumed that the monomer–monomer and dimer–dimer interactions are the same, e.g., u * 11 Å u * 22 Å 1. The interaction between a pair of unlike segments u * 12 was equal to 0.8, 0.9, 1, or 1.1. These parameters correspond to the following values of the parameter v: 00.6, 0, 0.6, 1.2. The calculations were carried out for T* Å 1. For the cubic lattice we have z Å 6, m Å 1/6, l Å 2/3. The classical iteration procedure was applied to solve the set [8]. First, we discuss the results obtained for homogeneous surfaces. Examples of the excess adsorption isotherms obtained for different systems are presented in Fig. 1. The agreement among the results of Monte Carlo simulations is surprisingly good for values of the parameter v near zero. Relatively small deviations are obtained even for high positive values of v Å 1.2. One can conclude that for similar energies of molecular interactions the Bragg–Williams approximation used in the theoretical calculations does not contribute considerable error to the adsorption isotherm on the homogeneous surface. In this case the problems in the comparison with experiment arise probably from the other assumptions included in the model. The main results of this paper concern adsorption on heterogeneous surfaces. We have considered two surfaces: (ii) the first (denoted by code A) adsorbing dimers preferentially * Å 5, E * on both type of sites (E * 1,1 Å 6, E 2,1 1,2 Å 4, E * 2,2 Å 2), and (ii) the second (code B) composed from two kinds

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* Å 6, of active centers—one type adsorbed dimers (E 1,1 E* 2,1 Å 4) whereas the other strongly interacted with mono* Å 1, E 2,2 * Å 3). The average difference of adsorpmers (E 1,2 tion energies » E * 12,r … is equal to 1.5 for the surface A and it is zero for the surface B. The simulations were performed for different sizes of the homogeneous squares: s Å 1, 2 and 5. It should be pointed out once more that the theory discussed in the previous section does not involve the patch size as a parameter. Figures 2–4 show the results obtained for the surface A and liquid mixtures characterized by the three values of * . Comparison of the simulational data with parameter u 12 theoretical curves demonstrates clearly that Eqs. [8] and [13] lead to quite good approximation. It is noticeable in the figures that there are deviations of the theoretical curves from the simulation points in the region corresponding to the extreme values of the excess adsorption. These discordances are most visible for high positive deviations from Raoult’s law ( v Å 1.2). When f ú 0.5, Eqs. [8] and [13] give results consistent with the simulations. In the case of surface A an increase of the patch area * Å 0.9 and 1, when causes a decrease of adsorption. For u 12 homotatic domains increase we observe a gradual approximation of the simulations to the theoretical curves. This may be explained in the framework of the model. Namely, the condition of independent adsorption on different active sites may be satisfied for macroscopically large patches. On the contrary, when the simulations were performed * Å 0.8 ( v Å 1.2) different results have been obtained. for u 12

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FIG. 2. Comparison of theoretical excess adsorption isotherms for the patchwise surface A with the simulational data. Parameters: E *1,1 Å 6, E *2,1 Å 5, E *1,2 Å 4, E *2,2 Å 2, u *12 Å 1.1 ( v Å 00.6) and s Å 1 ( h ), 2 ( s ), 5 ( , ).

Figure 4 displays that an increase of domain size leads to slightly worse agreement with the theoretical curve in the region 0.1–0.5. It is probably caused by simultaneous contribution of the errors following from the random mixing approximation and the thermodynamic inconsistency connected with the concept of the ‘‘local isotherm.’’ Figures 5 and 6 present a comparison of the theoretical adsorption isotherms and those obtained from Monte Carlo

FIG. 3. The same as in Fig. 2 but for u * 12 Å 1 ( v Å 0).

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FIG. 4. The same as in Fig. 2 but for u * 12 Å 0.8 ( v Å 1.2).

simulations performed for the surface B and different sizes of the homotatic domains. Similarly, as for the surface A, the agreement of the theoretical curves and simulation points is quite good. However, the importance of the patch area is greater than in the previous case. For low values of f an increase of the domain size causes an increase of adsorption. This trend is reversed in the region of high dimer concentrations. The best agreement with simulational data is found

FIG. 5. Comparison of theoretical excess adsorption isotherms for the patchwise surface B with the simulational data. Parameters: E *1,1 Å 6, E *2,1 Å 4, E *1,2 Å 1, E *2,2 Å 3, u *12 Å 1.1 ( v Å 00.6) and s Å 1 ( h ), 2 ( s ), 5 ( , ).

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FIG. 6. The same as in Fig. 5 but for u * 12 Å 0.9 ( v Å 0.6). Moreover, the filled symbols refer to s Å 12.

for large patches (see Fig. 6 for s Å 12). In the previous paper (13) it has been shown that surface B with patch reduced to a single site (s Å 1) exhibits essentially different adsorption behavior than the surfaces consisting of greater homogeneous domains. An analysis of the results presented in Figs. 5 and 6 confirms the conclusion. For the crystalline surface the simulational data are considerably different from the theoretical values. Thus, for the specific surfaces the theoretical approach discussed in the previous section may be used only when the homogeneous patches are sufficiently large. The Monte Carlo simulations allow us to study the orientation effects in the liquid mixture near the surface (13). Let us define the excess of dimers oriented along the axis a,

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FIG. 7. The surface excesses of dimers oriented parallel (solid lines; a Å xy) and normal (dashed lines; a Å z) to the surface B. Parameters are the same as in Fig. 6.

Gxy Å Gx / Gy .

[21]

Figures 7 and 8 show the orientation excesses of dimers parallel ( Gxy ) and normal ( Gz ) to the surface for various sizes of the patches. The most striking conclusion arising from our simulation is that a change of the patch area from 1 to 2 leads to drastic differences in the orientational ordering near the surface B. In this case the normal orientation of

t

Ga Å

∑ ( f (ai ) 0 fa) a Å x, y, z

[18]

i Å1

for f (ai ) Å N (ai ) /N,

[19]

where N (ai ) denotes the number of lattice sites in the ith layer occupied by dimers oriented parallel to the x, y, or z axis and fa is the bulk value given by fa Å lim f (ai ) .

[20]

i r`

The excess of dimers oriented parallel to the adsorbent surface is equal to the following sum:

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FIG. 8. The same as in Fig. 7 but for the surface A. Parameters are the same as in Fig. 3.

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dimers is preferred. However, for all remaining systems the parallel orientation is predominant. When s ú 2 the patch areas affect the orientational excesses to a relatively small extent (Fig. 7). For the strong adsorbents (the surface A) the orientational excesses are practically topographically insensitive (Fig. 8). The results of the performed calculations indicate that the accuracy of the theory in predicting dimer adsorption is satisfactory for homogeneous surfaces and small deviations from Raoult’s law. The differences between theory and the simulational data are more visible for heterogeneous surfaces. Obviously, most essential deviations are observed when the energy of interaction between a pair of unlike molecules is low enough. The theory incorporates the requirement that adsorption on any given type of site is not influenced by adsorption on the other sites. This assumption is quite artificial and may be fulfilled for sufficiently large patches. It is confirmed by the results of our calculations. Moreover, it is displayed that geometrical distribution of adsorption sites may decisively affect the behavior of the system. For the surfaces composed from active centers of opposite adsorption affinities the topography may be a very important parameter determining adsorption and the structure of the adsorbed layer. One can suppose that the big dimer molecule is insensitive to local differences on the surface. The general conclusion which can be drawn is that Eqs. [8] and [13] seem to be useful to describe real adsorption systems. However, for the specific surfaces an additional parameter, e.g., the patch size, should be taken into account. ACKNOWLEDGEMENTS One of the authors (W.R.) thanks the Foundation for Polish Science for financial support.

REFERENCES 1. Kipling, J. J., ‘‘Adsorption from Solutions of Non-Electrolytes.’’ Academic Press, London/New York, 1965.

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2. Os´cik, J., ‘‘Adsorption.’’ Horwood, Chichester, 1982. 3. Jaroniec, M., and Madey, R., ‘‘Physical Adsorption on Heterogeneous Solids.’’ Elsevier, Amsterdam, 1988. 4. Everett, D. H., Colloid Sci. 1, 49 (1973). 5. Brown, C. E., and Everett, D. H., Colloid Sci. 2, 52 (1975). 6. Everett, D. H., and Podoll, R. T., Colloid Sci. 3, (1979). 7. Jaroniec, M., Patrykiejew, A., and Boro´wko, M., Progress Surf. Membrane Sci. 14, 1 (1980). 8. Boro´wko, M., and Jaroniec, M., Adv. Colloid Interface Sci. 19, 137 (1983). 9. Jaroniec, M., Martire, D. E., and Boro´wko, M., Adv. Colloid Interface Sci. 22, 177 (1985). 10. Da– browski, A., and Jaroniec, M., Adv. Colloid Interface Sci. 27, 211 (1987). 11. Da– browski, A., and Jaroniec, M., Adv. Colloid Interface Sci. 31, 155 (1990). 12. Rudzin´ski, W., Narkiewicz-Michałek, J., and Pilorz, K., J. Chem. Soc. Faraday Trans. 81, 999 (1985). 13. Boro´wko, M., Patrykiejew, A., Rzg ysko, W., and Sokołowski, S., Langmain, in press. 14. Ash, S. G., Everett, D. H., and Findenegg, G. H., J. Chem. Soc. Faraday Trans. 60, 2645 (1968). 15. Roe, R. J., J. Chem. Phys. 60, 4192 (1974). 16. Prigogine, I., Bellemans, A., and Mathot, V., ‘‘The Molecular Theory of Solutions.’’ North-Holland, Amsterdam, 1957. 17. Guggenheim, E. A., ‘‘Mixtures.’’ Clarendon, Oxford, 1952. 18. Flory, P. J., J. Chem. Phys. 10, 51 (1942). 19. Huggins, M. L., J. Phys. Chem. 46, 151 (1942). 20. Everett, D. H., J. Chem. Soc. Faraday Trans. 61, 2478 (1965). 21. Lane, J. E., Aust. J. Chem. 21, 827 (1968). 22. Ash, S. G., Everett, D. H., and Findenegg, G. H., J. Chem. Soc. Faraday Trans. 66, 708 (1970). 23. Rudzin´ski, W., and Everett, D. H., ‘‘Adsorption of Gases on Heterogeneous Surface.’’ Academic Press, London/New York, 1992. 24. Larher, Y., Mol. Phys. 38, 789 (1979). 25. Bardi, U., Glachant, A., and Bienfait, M., Surf. Sci. 97, 137 (1980). 26. Bertz, M., Dash, J. G., Hickernell, D. C., McLean, E. O., and Vilches, O. E., Phys. Rev. A. 8, 1589 (1973). 27. Chung, T. T., and Dash, J. G., Surf. Sci. 66, 559 (1977). 28. Patrykiejew, A., Langmuir 9, 2562 (1993). 29. Patrykiejew, A., Thin Solid Films 223, 39 (1993). 30. Boro´wko, M., Jaroniec, M., and Rudzin´ski, W., Monatsh. Chem. 112, 59 (1980). 31. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E., J. Chem. Phys. 21, 1087 (1953). 32. Nicholson, D., and Personage, N. G., ‘‘Computer Simulation and the Statistical Mechanics of Adsorption.’’ Academic Press, London, 1982.

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