The Hierarchy of Adsorption Models for Laterally ... - ACS Publications

Langmuir 1997, 13, 979-989

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The Hierarchy of Adsorption Models for Laterally Interacting Molecules on Heterogeneous Surfaces† Yu. K. Tovbin Karpov Institute of Physical Chemistry, Vorontsovo Pole Str., 10, Moscow, 103 064 Russia Received November 8, 1995X Real adsorbents are heterogeneous, therefore, the computation of the adsorption characteristics for any concentration of adsorbate must take account of the joint effect of the lateral interaction between adsorbed molecules and surface heterogeneity. The local coverage of any adsorption center depends on the adsorption capacity of this center and upon the local coverage of the neighboring centers. As a result, the distribution of the different types of adsorption centers plays an essential role. A comparison of the different approaches based on the fragment, cluster, and pair description of the surface structure is given here. A technique for increasing the precision of calculating adsorption characteristics is found. It uses cluster distribution functions and takes into account the mutual disposition of the different types of sites. As an example of a distributed system, multilayer adsorption is considered. A lattice-gas model for the case of capillary condensation in slitlike micropores gives results in accord with those obtained by molecular dynamics and Monte Carlo techniques. The possibility of using a simplified equation for multilayer adsorption on a heterogeneous surface is analyzed. This equation gives a good description of the experimental data. However, the model parameters are outside the region of their definition in the lattice-gas model. For an exact calculation of the adsorption characteristics on a small fragment of surface, a new method is proposed which allows us to consider fragments that are twice as large as in the well-known matrix technique.

1. Introduction The concentration dependence of the thermodynamic properties of adsorption systems is determined by the combined effects of surface nonuniformity and lateral interactions between adsorbed molecules. In most cases the two effects have been studied separately.1-6 Very few theoretical studies of the combined effects of the two factors have been reported.7 Among the most noteworthy are studies of nearest-neighbor interactions at arbitrary coverages of surfaces with either a random8 or a patchwise9 distribution of centers of different types and also at low coverage on surfaces with an arbitrary structure.10 In a group of papers11-20 on the theory of the adsorption of interacting particles on nonuniform surface, we have obtained expressions for the isotherms11,12,16,17 and the heats13,18 of adsorption over the whole range of coverages † Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovakia, September 4-10, 1995. X Abstract published in Advance ACS Abstracts, February 15, 1997.

(1) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361. (2) Temkin, M. I. Zh. Fiz. Khim. 1941, 15, 296. (3) Roginskii, S. Z. Adsorption and Catalysis on Non-uniform Surfaces; Izd. Akad Nauk SSSR: Moscow, Leningrad, 1948. (4) Kiperman, S. L. Introduction to the Kinetics of Heterogeneous Catalytic Reactions; Izd. Nauka: Moscow, 1964. (5) Trapnell, B. M. W. Chemisorption (Transl. in Russian); Izd. Inostr. Lit.: Moscow, 1959. (6) Adamson, A. Physical Chemistry of Surfaces (Transl. in Russian); Izd. Mir: Moscow, 1979. (7) Steele, W. A. In The Gas-Solid Interface; Flood, E. A., Ed.; Transl. in Russian; Izd. Mir: Moscow, 1970; p 270. (8) Hill, T. L. J. Chem. Phys. 1949, 17, 762. (9) Champion, W. M.; Halsey, G. D. J. Am. Chem. Soc. 1954, 76, 974. (10) Steele, W. A. J. Phys. Chem. 1963, 67, 2016. (11) Tovbin, Yu. K. Dokl. Akad. Nauk SSSR 1977, 235, 641. (12) Tovbin, Yu. K. Zh. Fiz. Khim. 1982, 56, 686. (13) Tovbin, Yu. K. Zh. Fiz. Khim. 1982, 56, 691. (14) Tovbin, Yu. K. Dokl. Akad. Nauk SSSR 1981, 260, 679. (15) Tovbin, Yu. K. Zh. Fiz. Khim. 1982, 56, 1698. (16) Tovbin, Yu. K. Teor. Eksp. Khim. 1982, 18, 417. (17) Tovbin, Yu. K. Poverkhnost 1982, No. 9, 15. (18) Tovbin, Yu. K. Zh. Fiz. Khim. 1987, 61, 3380. (19) Tovbin, Yu. K. Dokl. Akad. Nauk SSSR 1982, 262, 929. (20) Tovbin, Yu. K. Kinet. Katal. 1983, 24, 308, 317.

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which allow for nearest-neighbor interactions on surfaces having arbitrary compositions and different types of distribution of various centers. The interactions were calculated in the approximations which have been widely used for uniform surfaces: the mean field approximation11-16 and the quasi-chemical approximation (QCA).17-20 Two-dimensional condensation problems were specifically examined14 and the extension of the theory to include multicomponent adsorption systems was formulated.16,19,20 The main aspect of the theory concerns allowing for the actual structure of the nonuniform surface. At present, we have books21-23 where the theory of the mutual influence of both these factors is considered as a whole. However, the important problem of the effect of surface topography on the local coverages of different sites types is not discussed. In other words, how do the details of the topography of a heterogeneous surface affect the calculation of isotherms and isosteric adsorption heats? The traditional description of the coverage of a nonuniform surface relies on the functions θiq, which define the coverage (mole fraction) on centers of type q by particles of kind i. In the absence of interaction (ideal model) this approach is adequate, because each center “works” independently. In the case of interacting species (nonideal model) the degree of occupancy of any center (number f, say, in Figure 1) depends on the type of the center and on the occupancies of its neighbors (g), which are themselves dependent on the type of this center and on the occupancy of their neighbor h, including the center f from which the calculation started, and so on. As result the probability of finding particles on given centers depends not only on the type of the center but also on the type of the neighboring center, and depending on whether the effect of the neighboring centers on the distribution of the particles is (21) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (22) Tovbin, Yu. K. Theory of Physical Chemistry Processes at the Gas-Solid Interface; CRC Press: Boca Raton, FL, 1991. (23) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992.

© 1997 American Chemical Society

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described with sufficient precision, the various models of the adsorption equilibrium will give different results.24,25 The exact description of the two-dimensional structure of an adsorbent’s surface layer implies a definition of the mutual dispositions of each of the adsorption centers. Such a detailed consideration of the heterogeneous surface is possible only for small fragments. In the case of macroscopic regions the averaged description is to be applied, namely, the distribution functions for the different sites types fq, 1 e q e t, where t is the number of types of sites, t with ∑q)1 fq ) 1, or the cluster distribution functions. The first does not take account of the connectivity between the sites while the second one does. In this paper the following topics are considered: (1) A comparison of the different approaches based on the fragment, cluster, and pair description of the surface structure is given. (2) A technique for increasing the precision of calculating adsorption characteristics is found which uses cluster distribution functions and takes account of the mutual disposition of the different types of sites. (3) As example of distributed systems, multilayer adsorption in pores and on flat substrate is considered. The results obtained are compared with those given by molecular dynamics and Monte Carlo techniques and with the experimental data on adsorption of N2 on silica as well as N2, O2, and Ar on KCl, and C6H6 on aerosil. (4) For exact calculations of the adsorption characteristics on a small fragment, a new method is proposed which allows us to consider fragments twice as large as in the well-known matrix technique.

where the subscript gr numbers the sites around of site f in the rth c.s., aiq is the adsorption coefficient of particle i on the site q, Fiq and F0i are the partition of the adsorbed particle i on site q and of the molecules i in the gas phase, β ) (kBT)-1, and Pi and mi are the partial pressure and the degree of dissociation of molecule i. If mi ) 1, we have Qiq ) iq, where iq the energy of the bond between the particle i and the site q, and if mi ) 2 we have Qiq ) iq Di, where Di is the dissociation energy of the molecule i in the gaseous phase. We shall restrict ourself to the onecomponent adsorption system s ) 2, i ) A, and ν; i.e., any site is occupied or vacant. The problem in theoretical statistics is to construct closed expressions describing the distribution of the adsorbed particles on the surface. The lateral interaction between these particles makes the state of occupancy of each site dependent on that of its neighbors. An exact solution of the “many-body” problem is possible only in special cases,26 and therefore we need approximate methods of describing the systems.

2. Lattice-Gas Model

3. The Basis of the Hierarchy of the Adsorption Models

We shall describe the adsorption systems by means of a lattice model. Each site of the lattice is an adsorption center and can be occupied by a particle of any kind i (including a vacancy), 1 e i e s, where s is the number of components. The state of occupancy of each site f (1 e f e N, N is the number of sites) will be characterized by the variable γif, with γif ) 1 if the site f contains a particle of kind i and γif ) 0 if it contains a particle of any other kind. These variables obey the equations γ1f + ... + γsf ) 1 and γifγjf ) ∆ijγif, where ∆ij is the Kronecker symbol, which states that any site can be occupied by any particle. The type of the site numbered f will be characterized by the parameter ηqf assumed to be known and unvarying during the adsorption process (nonreconstructing surface), with 1 e q e t, where t is the number of types of site. We have ηqf ) 1 if f is a site of type q and ηqf ) 0 in the opposite case. The complete set of {ηqf }, with 1 e f e N, uniquely defines the composition and topography of the surface, which can be of any kind. We shall assume that lateral interaction is described by a pair potential with an interaction radius R, where R has any given value. Distances will be expressed as the number of coordination shells (c.s.). The number of sites in the rth c.s. of a type-q site numbered f will be denoted by zf(r) [or by zq(r)], with 1 e r e R. The interaction parameter of the species i and j on site of types q and p numbered f and g at a distance ij r will be denoted by ijfg(r) [or by qp (r), since the number of the site and its type are uniquely correlated by the parameters ηqf ]. The total energy of the adsorption system in the grand canonical ensemble is described as follows22,25

3.1. Two-Dimensional Model. The initial lattice embodies at the atomic-molecular level a two-dimensional discrete set of lattice sites. It corresponds exactly to the two-dimensional discrete model, which can describe any surface fragment (of arbitrary shape). The use of twodimensional discrete models to describe the macroscopic characteristics of an adsorption system corresponds to a macroscopic lattice in which the defined fragment can fill the surface completely by translating in both directions. In many cases this model gives an exact description of the actual state of affairs. These include (1) stepped surfaces of low-index faces of crystals, with equally spaced steps, (2) the surfaces of highly ordered alloys, and (3) stepped surfaces of ordered alloys etc., on which the size of the fragment is not smaller than the self-repeating “pattern” formed by other surface imperfections, and also the case of isolated imperfections in uniform regions, if the number of sites in the fragment is large enough. This model operates through the occupancies of specified sites of the chosen surface fragment. The equations for the two-dimensional discrete model can be obtained with help of the so-called cluster approach.17,22 This method is based on using a local distribution function of particles on sites of the lattice. The very complex problem of calculating the statistical sum of the heterogeneous system is excluded by its help. Instead, a system of equations for local distribution functions have to be constructed. The following are essential features of the cluster approach: (1) The initial lattice is represented by a set of clusters. Each cluster consists of a group of central sites and their R c.s. In the case of a single central site the lattice is represented by N clusters (scheme A in Figure 1); if two

(24) Tovbin, Yu. K. Dokl. Akad. Nauk SSSR 1989, 306, 888. (25) Tovbin, Yu. K. Zh. Fiz. Khim. 1990, 64, 865.

N

H)

(

hif ) νiq -

1

s

hif, ∑ ∑ f)1 i)1

R

s

∑ ∑ ∑ ijqp(r) γgj 2 r)1 g j)1

r

r

νif ) -β-1 ln(Yiq),

)

ηgpr γif ηqf ,

Yiq ) (aiqPi)1/mi,

aiq ) (Fiq)mi β exp(βQiq)/F0i

(1)

(26) Baxter, R. J. Exactly Solved Models in Statistical Mechanics; Academic Press: London, 1982.

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Figure 1. The g, h, and l are the first, second, and third neighbors of the site f. An image of a lattice fragment as a set of clusters consisting of a central site (scheme A) or two central sites (scheme B) and their nearest neighbors. For the average model (c) a lattice “separates” into pairs of sites.

central sites are used, and the distance between them is F, the lattice is represented by zq(F)N/2 clusters (scheme B in Figure 1). The order in which the clusters are distributed is “memorized”. (2) An exact system of equations for the relative probability of different possible configurations of particles in the cluster is written for each cluster. For example, if R ) 1, then the relative probability of possible configuraij1j2...jz . This tions of particles in the cluster is denoted by θfg 1g2...gz local distribution function characterizes the probability of realizing the following configuration of the particles: the particle i is at site f, the particle j1 is at site g1, and so on up to the particle jz. For these local distribution functions we have the following systems of equations (here R ) 1):

numbered f), and θijfg(r) (the probability of finding i and j particles in sites f and g at a distance r) becomes22,25

ij1j2...jz kj1j2...jz θfg ) exp[β(hkf - hif)] θfg 1g2...gz 1g2...gz

This system of equations offers a very detailed description of the distribution of particles. The possibility of practical application of the system (2), (3) for large N is determined by the power of the available computer. Obviously, it is of interest to establish whether alternative (and simpler) models could be constructed, allowing on the one hand a geometrically larger region of the surface to be described and on the other hand the use of the molecular parameters of the adsorption systems given in eq 2. The change can be made by adopting distributions of the lattice sites averaged over separate regions of the surface. 3.3. Averaged Models. Averaged (point) models are obtained if averaging of particle distribution is performed over the whole surface. In other words, the averaged models are obtained if the clusters, instead of being “returned” to their positions in the initial lattice (as done in point 3 of subsection 3.1), are regrouped so that all the clusters of the same type are placed in the same group irrespective of the number of central sites. The clusters having central sites of the same type as well as sites of the same type in their R coordination shells are treated as identical. The number of identical clusters gives the statistical weight of a given type of cluster. The full set of normalized values of various clusters with a fixed number of central sites and their types forms a cluster

1 e i, j1j2...jz e s These systems are finite because the number of sites in the cluster is finite, but they are not closed. (3) A single method of closing the system of equations which allows for the actual sequence of arrangements of the clusters in the initial lattice must be introduced for all the systems relating to different clusters. After the approximate closure of the system of equations, all the clusters are “returned” to their original positions, and the initial lattice is completely “reestablished”. The result is a single closed system of equations for the lattice as a whole. For example, if we use the random approximation with R ) 1 (in which correlation between interacting ij1j2...jz z ) θif∏k)1 θgjkk, then for any site particles is absent), θfg 1g2...gz of the lattice we obtain the similar equations, which couple to each other. (But this approximation is very crude and we shall use a more precise approximation.) 3.2. The Equation for the Two-Dimensional Discrete Model. We limit the discussion to direct correlations between the interacting particles (if R ) 1, then ij1j2...jz ijk z θfg ) θif∏k)1 θfg /(θif)z), the closed system of equations 1g2...gz k i for θf (the probability of finding i particles in a site

R

(1 - θAf )YAq ) θAf

∏ ∏ r)1

A [1 + θAA fg (r)xqp(r)/θf ] (2)

g∈zf(r)

xqp(r) ) exp[-βAA qp (r)] - 1, A A θAA fg (r) ) 2θf θgr/[δfg(r) + bfg(r)],

δfg(r) ) 1 + xqp(r)(1 - θAf - θAg ), 4xqp(r)θAf θAg }1/2,

(3)

bfg(r) ) {[δfg(r)]2 +

Aν A θAA fg (r) + θfg (r) ) θf ,

νν ν θνA fg (r) + θfg (r) ) θg,

θAf + θνf ) 1

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distribution function. Such functions characterize the probabilities that the clusters of sites of a specific type are present on the surface. When the same clusters are collected in one group, the connection between disposition of different sites in the initial lattice is broken. As result, if we know the distribution functions of sites and their clusters, we cannot reproduce the topography of the initial lattice. (In this case we must specify the type of the site but not its number, as in the distributed (two-dimensional) model, where the type and number of the site are unambiguously related by the parameters ηqf .) Nevertheless, cluster distribution functions characterize local surface structure around central sites. The crudest information about surface structure gives a pair distribution function fqp(r), which characterizes the probability of find two sites of some types qp at a separation r: fqp(r) ) Nqp(r)/[zq(r)N], where Nqp(r) is number of pairs of sites of t type qp at a distance r; ∑p)1 fqp(r) ) fq. The function fq ) Nq/N, characterizes the composition of the surface (mole fraction of sites of type q), and Nq is the number of sites of type q. The problem in representing the initial lattice in terms of clusters is to ensure that the clusters give a true description of the real structure of the initial nonuniform surface. In order to preserve the local structure of the surface, we shall describe it in terms of two groups of clusters. The first group includes clusters with one central site; the second group consists of clusters with two central sites, at distances F e R, and all the adjacent sites at distances r e R from at least one of the central sites (for R ) 1 see Figure 1). These clusters give a symmetric allowance for the effect of the neighboring sites around each of the central sites f and g, on which we find the particles i and j characterized by the function θijfg(r). For convenience in numbering the neighbors we introduce the concept of a single c.s. of radius r for a dimeric “molecule” occupying the central sites, consisting of a group of sites at a fixed distance r from the nearest of the central sites. For such clusters we must determine the distribution functions f(q{λ}R) and f(qp{λ}R|F). The first defines the probability of finding on the surface a cluster with a central site of type q and its neighbors specified by the set of numbers λqp(r), with 1 e p e t and 1 e r e R, and λqp(r) is the number of sites of type p at a distance r from the site q, {λ}R is the compact form of the description of this set of numbers. The number of different types of c.s. for clusters with one central site q will be denoted by σq. The second defines the probability of finding on the surface clusters with two central sites q and p at a distance (and all their neighbors at distance r e R). For these clusters {λ}R denotes the λqpξ(ωr|F) numbers; λqpξ(ωr|F) is the number of sites of type ξ in the rth c.s. with the orientation ωr(F) relative to two central sites q and p at a distance F. The number of different types of c.s. with two central sites q and p at a distance F will be denoted by σqp(F). We also introduce the conditional distribution functions for the sites of different types:

dqp(r) ) fqp(r)/fq, d(q{λ}R) ) f(q{λ}R)/fq, d(qp{λ}R|F) ) f(qp{λ}R|F)/fqp(F), d(u(qp{λ}R|F)) ) f(qp{λ}R|F)/f(q{λ}R), t

∑ dqp(r) ) 1,

∑

p)1

σqp(F)

d(qp{λ}R) ) 1,

d(q{λ}R) ) 1, ∑ σ q

∑ d(u(qp{λ}R|F)) ) 1

(4)

σup

where u(qp{λ}R|F) denotes the subset of the sites remain-

ing after removing from cluster with two central sites q and p the sites of a cluster with one central site q, i.e., the sites of the truncated cluster p{λ}R. The number of this clusters p{λ}R will be denoted by σup. So, to construct the averaged models we can use various distribution functions of the sites, pairs of sites, and clusters of the different sites. Each of such functions gives the own equations for local occupancies. 3.4. Equations for Averaged Models. Equations for the local occupancies of the sites of different types for the averaged models are obtained as follows: The equations of subsection 3.2 are written down for a cluster of a specific type and are averaged over different types of coordination shells of clusters with the distribution functions discussed in subsection 3.3. (a) A model which allows for the effects of the types of neighboring site in the cluster upon the occupancy of the central site

YAq (1 - θˆ Aq ) ) R

θˆ Aq

zq(r)

AA d(u(qp{λ}R|r))[1 + θˆ qp (r)xqp(r)/θˆ Aq ] ∑ ∏ ∏ r)1 k)1

(5)

k

σpu

The subscript k in p gives the number of the site in the rth c.s. of the central site gives the type of site. The equation for the functions θˆ AA fg (r) has a form similar to eq 3, in which they appear in place of the functions θAA ˆ Af , we can fg (r). After solving the systems of eq 5 for θ calculate the average degree of coverage by A particles of the site of type q and the proportion of the pairs of particles ij on sites of the qp type at a distance r

θAf )

d(q{λ}R)θˆ Aq ∑ σ q

∑

θAA qp (r) )

d(qp{λ}R|r)θˆ AA qp (r),

(6)

σqp(r)

(b) A model which does not allow for the effect of the types of neighboring site in the cluster on the occupancy of the central site

YAq (1 - θAq ) ) θAq

∑ σ

R

d(q{λ}R)

q

t

A λ [1 + θAA ∏ ∏ qp (r) xqp(r)/θq ] r)1 p)1

qp(r)

(7)

In spite of this limitation the model can reflect the local structure of surface regions of size comparable with zq(R). (c) A model which uses the distribution function of the pairs of sites of different types as an approximation to the distribution function of the clusters: R

YAq (1 - θAq ) ) θAq

∏ r)1

t

[1 +

A z (r) dqp(r)θAA ∑ qp (r)xqp(r)/θq ] p)1 q

(8)

This model gives the crudest description of the structure of the surface. This approximation transforms the initial lattice into the separate pairs of the sites (see Figure 1). It coincides with model 7 in the case of equilibrium distribution of sites of different types17,20 and also for special cases of surface with patchy or chaotic distribution of sites or for some types of ordered distribution. In models AA (r) are given by formulas (3) in 7 and 8 the functions θqp AA which functions θfg (r) and θAf should be replaced by AA θqp (r) and θAq .


Langmuir, Vol. 13, No. 5, 1997 983

Figure 2. Comparison of the local and average coverages of the first (solid) and the second (dashed lines) types of sites for a random surface f1 ) f2 ) 0.5) with β ) 1.2 and z ) 4: 1 ) the local coverage θˆ q of the first (second) type sites on the cluster with four nearest neighbors of the first type; 2 ) the same for the cluster with four nearest neighbors of the second type; 3 ) the average coverage θq of the first (second) type site for the model (a) (scheme B); and 4 ) the same for the model (b) (scheme A).

The models with the fragment, cluster, and pair distribution functions form the hierarchy of the adsorption models for interacting molecules on heterogeneous solid surfaces. Obviously, the more precise descriptions give equations containing the distribution functions with the larger dimensionality. All these models use the same set of molecular parameters. 4. Comparison of the Averaged Models The models discussed differ in their description of the character of the distribution of sites on a nonuniform surface. In order to establish the method of dealing with the surface structure, we shall compare the distributions of particles of one kind over the sites of a surface consisting of two types of sites given by models 5 and 7 and allowing for nearest-neighbor interactions. The results of calculations for random distributions of sites of the first and second type are shown in Figure 2. The occupancies of the central sites of clusters with different c.s. and the average value of θq (q ) 1 and 2) obtained by eq 6 are shown as a function of the surface coverage θ. (For θ > 0.7, all curves converge almost linearly toward the point (1.0; 1.0).) In models 5 and 7 the average surface coverage θq is given by curves 3 and 4. The differences between these curves are not greater than about 10% (adopting the conditions R ) 1, qp ) , zq ) z ) 4). However, an analysis of the local coverage of the central sites of the first type in cluster 1[4,0] with four nearest neighbors of the first type (curve 1) and in the cluster 1[0,4] with four nearest neighbors of the second type (curve 2) shows a larger difference θˆ Aq both between these clusters and with respect to the averages θAq . (The curves for the local coverages of clusters 1[3,1], 1[2,2], and 1[1,3] lie between the curves for 1[4,0] and 1[0,4] and are omitted from Figure 2.) The same is true of the local coverages of sites of the second type for the clusters 2[4,0] and 2[0,4], corresponding to curves 1 and 2, in comparison with the average value of 3 and 4 (dashed lines). Thus in the case of relatively insensitive properties such as the adsorption isotherms, it is often sufficient to use eqs 7. When more “exact” characteristics are being examined (for example, heats, specific heats, or adsorption kinetics), the use of eqs 7 can often lead to large deviations from the results given by eqs 5. It follows from the definition (6) that the functions θiq and θˆ iq are equal only for d(q{λ}R) ) 1. Systems to which this condition applies include those with ordered particles, with ordered sites, with transition regions at phase boundaries, among others.

Figure 3. Q(θ) for a surface with impurities: f1 ) c, z ) 4, Q1 ) 113, Q2 ) 88 kJ/mol, β(1) ) -4.

When d(q{λ}R) * 1, we must allow for the whole spectrum of states of the c.s. of the clusters, which greatly increases the dimensionality of the system of eqs 5 as compared with system 7, and the applicability of the θiq functions must be verified in each case. 5. The Method of Describing the Type of Surface Nonuniformity The use of distribution functions for the sites of a nonuniform surface does not adequately depict the real situation and thus can produce artificial effects. Let us consider, as an example, a surface consisting of two types of sites (t ) 2), the site with the stronger bond being in the minority (f1 , 1). Assume that the conditions qp ) and zq ) z ) 4 are obeyed, with R ) 1. We shall use model 8. If the interactions are not strong [β(1) ≈ -1] the distribution function f1 ) c, where c is concentration of the impurity; then f2 ) 1 - c, d12 ) 1 correctly reflects the thermodynamic properties of the adsorption. However, in the case of strong repulsion between the particles, the heat of adsorption Q(θ) has the form of curve 1 in Figure 3. In order to explain its unusual shape, we shall examine the source of the actual process. Since Q1 > Q2, the impurity sites are filled first at low values of θ, but as θ increases, the particles landing on sites of the second type will tend to settle as far as possible from the filled sites of the first type owing to the repulsion between the particles. But a model including two types of sites cannot deal with this situation: all sites of the second type treated as equivalent. As a result of the repulsion between the particles on sites of the first type and on the second type, we find a strong decrease in the heat and a redistribution of the particles from sites of the final type into those of the second type. Further adsorption takes place on the sites of first type which have been set free, which produces a maximum. Thus in order to depict the real state of affairs we must define around any impurity site the group of neighbors whose occupancy probability differs from the average population of the other sites of the second type; in other words, we must change to a model with a greater number of types of site. Curve 2 in Figure 3 was calculated for t ) 4: f1 ) c, f2 ) f3 ) zc (we are examining the first and

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the second neighbors of the impurity site), f4 ) 1 - (2z + 1)c, Q2 ) Q3 ) Q4, d12 ) 1, dqq ) 0, q ) 1, 2, 3; d21 ) d23 ) 1/4; dqp ) 1/2, qp ) 23, 32, 34; d13 ) d14 ) 0. Such a model accounts of the local distribution of adsorption centers near the impurity. Its trend is consistent with the above conclusions: no minima or maxima are formed. However, even this description requires some adjustments. It is well-known that particles chemisorbed on homogeneous surfaces become ordered when the temperature is lowered,27,28 and their distribution becomes nonuniform (with large differences between the occupancies of adjacent sites). For R ) 1 a single c(2×2) superstructure is formed on the (100) face: each occupied site is surrounded by free γ sites and vice versa. In light of these considerations we propose the following distribution function to describe the surface: t ) 4, f1 ) c, f2 ) zc, f3 ) f4 ) [1 - (z + 1)c]/2 [q ) 2 identifies the nearest neighbors of the impurity, and the remainder of the surface is divided into sites of the sublattice R (q ) 3) and γ (q ) 4)], d12 ) 1, dqp ) 0, qp ) 11, 22, 33, 44, 13, 14, 24, 31, 41, 42; d21 ) 1/4; d23 ) 3/4; d34 ) (1 - 11c)/(1 - 5c), d43 ) 1; d32 ) 6c/(1 - 5c). This model accounts for the distribution of adsorption centers over the surface. Curve 3 of Figure 3 shows the actual Q(θ) dependence. The steps in the heat curve during the filling of sites with constant bonding energy Qq (q ) 2, 3, 4) are due to forward or reverse order-disorder phase transitions. So, allowing for the spacial distribution of different site types allows us to correctly calculate isosteric adsorption heats. Without taking account of the “intermediate” sites or the formation of ordered superstructures, the results disagree. As well as showing the possibility of obtaining misleading effects, our example shows how to avoid them. By using a detailed description of the distribution of different types of lattice site (whose molecular characteristics may be identical), we magnify the connectedness of the lattice sites and produce a description as close as possible to the real surface. If the distribution of the species is correctly described, satisfactory results can be obtained even for a small number of types of sites and clusters. A high level of detail in the distributions of sites and clusters can also provide a test of the stability of the particle distributions which have been obtained. Of course, the detailed description should reflect the main physical features of the real system, because otherwise any formulation of the distribution of particles would be useless. This manner is most advantageous for describing multilayer adsorption and when sharp borders between the different kinds of fragments on the surface are present. 6. Lattice-Gas Model of Multilayer Adsorption 6.1. Porous Adsorbent. The space of a pore can be divided into monoatomic layers parallel to slitlike pore walls. A cubic lattice of adsorption sites with z ) 6 was used. The isotherm of multilayer adsorption determining t the filling of pore volume has the form θ(P) ) ∑q)1 fqθq (P), for pore walls with homogeneous surfaces and with t ) H/2, H ) the width of the pore. The equations describing the filling of the qth layer θq(P) are given by formula 7. The interaction potential between an adsorbate molecule and a homogeneous wall, represented by a lattice of atoms with lattice parameter ∆, is found by summing (27) Ohtoni, H.; Kao, C.-T.; v.Hove, M. A.; Somorjai, G. A. Prog. Surf. Sci. 1987, 22, 155. (28) Roberts, M.; McKay, C. Chemistry of the Metal - Gas Interface (Transl. in Russian); Izd. Mir: Moscow, 1981.

Tovbin

Figure 4. Capillary condensation pressures (1-7) and capillary critical temperatures as functions of the width H of the slitlike pore at Tcc/Tc ) 0.7 (1, 2, 4-6) and 0.5 (3) and Q1/ ) 10 (2-6, 9) and 2 (1, 8); L is the width of the interface between the dense and dilute pore fluids.

the Lennard-Jones interaction potential for each surface atom and has the form29

U(r) ) 2πsfσsfFs∆(0.4(σsf/r)10 - 0.4(σsf/r)4 0.4(σsf/(3∆(r + 0.61∆)))3) where Fs is the surface density of wall atoms and σsf and sf are the Lennard-Jones interaction potential parameters. The calculation were performed for parameters corresponding to the interaction of nitrogen molecules with the basal plane of graphite: sf/kB ) 52 K, σsf ) 3.57 Å, ∆ ) 3.35 Å, and Fs ) 0.114. The interaction energy between adsorbate molecules in lattice sites nearest to each other was taken to be ff/k ) 95 K. In the approximation of the homogeneous surface, all adsorption sites in a monolayer parallel to the pore wall are of the same type q determined by distance rq from this layer to the wall. The binding energy for a site of type q was calculated as Qq ) U(rq), where r1 ) σsf, rq+1 + σff and σff ) 3.75 Å (q takes on values from 1 to H/2). For the parameters specified above, Q1/ ≈ 10. Adsorption isotherms were calculated for slitlike pores from 3d0 to 28d0 wide. The dependence of the pressure of capillary condensation Pcc the width of pores is given on Figure 4. Decreasing the temperature and increasing the potential of the wall increases the difference between Pcc and the saturated vapor pressure of the fluid in the bulk phase. Although the potential of interaction with the wall is close to zero already for the fourth adsorbent layer, the finite volume effect results in a decrease of Pcc in pores up to H ) 25d0 (75-100 Å). The symbols and the dot-and-dash line correspond to Monte Carlo calculations29 for the same parameter values as for curve 2. Although qualitative agreement is good, quantitative differences are observed for micropores with 4 e H/d0 e 10. The dashed lines were obtained using the Kelvin equation (T ) 0.7Tc, Tc is the critical temperature): ln(P/P0) ) -2σVLβ/ rm, where σ is the surface tension of the liquid, VL is the (29) Walton, J. P. R. B.; Quirke, N. Mol. Simul. 1989, 2, 361.


Langmuir, Vol. 13, No. 5, 1997 985

Figure 5. Density profiles across the interface between the dense and dilute pore fluids in a slitlike pore: H/d0 ) 8 (a,b,c) and 12 (d); T/Tc ) 0.7(a,c,d) and 0.5 (b); and Q1/ ) 10 (a,b,d) and 2 (c). The indices along the axes correspond to the division into lattice sites according to scheme given in Figure 4; h varies from 1 to H and l varies from 1 to L.

molar volume of the liquid, and rm is the mean radius of curvature of the meniscus. Curve 4 is obtained without taking the adsorbed film into account (rm ) H/2), curves 5 and 6 correspond to the formation of an adsorbed layer (rm ) H/2 - σfs), and curve 6 in addition takes account of a decrease in the surface tension of the liquid in the pore (see ref 30). For calculating the fluid density profile across the interface in the pore, a distributed lattice model is used. In the boundary region, all sites are numbered (the number runs over sites along pore walls and in the pore cross section, see the scheme in Figure 4). This region separates the equilibrium liquid and gaseous phase coexisting in the pore at pressure Pcc. The density profiles in the coexisting phases (i.e., the local occupancies of sites θq) are known from the preceding calculation of adsorption isotherms. It follows that, at given boundary conditions (in this scheme, θ1, θ2 and θ3 correspond to the liquid phase; θ19, θ20 and θ21 correspond to vapor), we can calculate the local occupancies of all transition region sites by solving eqs 7 with respect to θq. Typical calculated density-distribution profiles for adsorbed molecules in the transition region between the liquid phase filling the pore volume and the gas phase in the core are shown in Figure 5. The lines in the figure are constant density lines, and the scales on the axes refer (30) Tovbin, Yu. K.; Petrova, T. V. Zh. Fiz. Khim. 1995, 69, 127.

to positions and size of the lattice sites (the size of sites is equal that of the adsorbate molecules). The lattice model is used to calculate the concentrations of molecules in discrete lattice sites Quadratic interpolation is then applied to produce continuous fluid density profiles in the pores. The spherical boundaries obtained with and without taking into consideration the adsorbed layer (the dashed lines in Figure 5) illustrate the applicability of the Kelvin equation (see curves 4 and 5 in Figure 4). The data presented in the figure illustrate the influence of pore width (a, d), temperature (a, b), and the moleculewall interaction potential (a, c). At low temperatures (T e 0.5Tc), the boundary region tends to monolayer thickness (b). An increase in temperature increases the width of the transition zone between the liquid and vapor phases in the pore (contrary to what is implied in the macroscopic Kelvin equation, this zone is not a single layer). Decreasing the width of the pores (compare a and d) also increases the transition-zone width because of an increase in the importance of surface forces compared to interactions between molecules. Note that if interactions with the wall are weak and temperatures high (c), the densities of the liquid and gas phases in the adsorption layer are substantially different. That is, there is no unique liquid adsorbed film along the length of the pore; this film also includes a narrow transition region, which distorts the spherical shape of the pore fluid-pore gas interface.

986


Tovbin

Figure 6. Isotherms of nitrogen adsorption on nonporous adsorbents at T ) 77 K. Experimental points34 and calculations by the BET equation (1) and the simplified equation (9) for σ ) 0 (2) and 0.6 (3).

So, the lattice-gas model gives the same results as molecular dynamics:31 the radius of the meniscus for interface between dense and rare phases in pores is smaller than the pore radius, and the density of the above-surface layer for dense phase is larger than that for the dilute phase. 6.2. Simplified Equations for Multilayer Adsorption on Flat Adsorbents. The lattice-gas model gives simplified equations for multilayer adsorption32 t

θ(P) ) θ1(P)γ(P),

θ1(P) )

∑ fqCqx/(1 + Cqx),

q)1

γ(P) ) [(1 - σx)/(1 - x)]1/k,

x ) P/P0 (9)

where k and σ are the parameters. The lattice-gas model gives k ) 2 and σ ) 0.5-0.6. The possibility of using the simplified equation of multilayer adsorption on heterogeneous surfaces (at small lateral interactions in relation to the contribution of surface heterogeneity) for the adsorption of nitrogen on silica as well as nitrogen, oxygen, and argon on potassium chloride and benzene on modified Aerosil have been analyzed.33 Equation 9 was used for the two first adsorbents on the assumption that the surfaces of the adsorbent are homogeneous. This simplest assumption will allow us to answer the question of what molecular information can be extracted from these equations. The experimental data34 shown in Figure 6 are θ ) n/nm, where nm is the capacity of the surface monolayer, for various surfaces and various specific surface values. Points I refer to silica specimens with specific area ranging from 2.6 to 11.5 m2/g, and points II, to aluminum oxide specimens with specific area ranging from 58 to 153 m2/g. The fitting procedure gives δ ) 7% with σ ) 0 (C ) 61, Xm ) 0.17) and δ ) 12% with σ ) 0.6 (C ) 27, Xm ) 0.35); here the simbol δ% denotes percent of the relative (31) Heffelfinger, G. S.; v. Swol, F.; Gubbins, K. E. Mol. Phys. 1987, 16, 1381. (32) Tovbin, Yu. K. Zh. Fiz. Khim. 1992, 66, 2162; Dokl. Akad. Nauk SSSR 1990, 302, 917. (33) Tovbin, Yu. K.; Petrova, T. V. Zh. Fiz. Khim. 1994, 68, 1459. (34) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, New York, 1982.

Figure 7. Isotherms of N2 (I), Ar (II), and O2 (III) adsorption on KCl crystals, T ) 85 K. Experimental points35 and calculations by the simplified eq 9 for σ ) 0 (dashed lines) and σ ) 0.6 (solid lines). The curve for O2 is shifted by 0.2 along the x axis.

deviation between experimental and calculated values per experimental point. Similar results were obtained for the multilayer adsorption isotherms of N2, O2, and Ar on KCl35 (see Figure 7). Real surfaces of KCl crystals are nonuniform, as is confirmed by differences in the heats of adsorption, which change linearly with increasing coverage.36 When the model used does not account for possible surface nonuniformity, the fitting parameters depend on surface nonuniformity. The same is true of nitrogen adsorption on nonporous adsorbents.34 In both cases, the isotherms of the second type can be described satisfactorily by the simplified isotherms for uniform surfaces, but the parameters obtained may not have physical meaning. Also, we consider adsorption of benzene on nonmodified (I) and on 60% modified (II) surfaces of silica (Aerosil).37,38 The modified surface contains sites of at least two types, with heats of adsorption higher and lower, respectively, than the heat of condensation. The choice of this system was dictated by the availability of experimental data not only for adsorption isotherms but for differential heats of adsorption measured independently by the calorimetric method. A comparison of the concentration dependences of the heats of adsorption calculated and measured experimentally sheds light on the molecular meaning of the model parameters determined from the adsorption isotherms and vice versa. Our analysis shows that simplified isotherms give a good description of isotherms I and II (Figure 8, solid lines). The parameters determined cannot, however, be used to predict of the concentration dependences of the heats of adsorption. At the same time, the isotherms calculated using the parameters obtained from the heat of adsorption yield values that are far lower than those obtained by experimental means. Thus the simplified equations cannot be used for prediction purposes, although they satisfactorily describe the separate adsorption characteristics (isotherms and heats). Some compromise is reached by simultaneously using the isotherms and heats of adsorption to determine a unified set of parameters (dashed lines). (35) Keenan, A. G.; Holmes, J. M. J. Phys. Colloid Chem. 1949, 53, 1309. (36) Young, D. M. Trans. Faraday Soc. 1952, 48, 548. (37) Kiselev, A. V. Q. Rev. Chem. Soc. 1961, 15, 116. (38) Babkin, I. Yu.; Kiselev, A. V.; Korolev, A. Ya. Dokl. Akad. Nauk SSSR 1961, 136, 373.


Langmuir, Vol. 13, No. 5, 1997 987

fragments studied, an alternative method has been proposed42sthe so-called fragment method. The fragment method deals with the exact calculation of the statistical sum Z for some fragment of the surface. A fragment with periodic boundaries is taken. The number of the sites in the fragment under consideration can be 20-30, which is twice larger than in matrix method. B

Z)

∑ exp[-βEk], k)1

1 e k e B ) 2Nfr, B

θk ) exp[-βEk]/Z,

∑ θk ) 1

(10)

k)1 Nfr

Ek )

Figure 8. Isotherms (a) and heats (b) of benzene adsorption on Aerosil. The experimental points37,38 and calculations by the simplified eq 9 are shown with parameters adjusted using the n(x) (solid lines), Q(n) (dash-and-dot lines), and both dependences (dashed lines).

7. Fragment Method An important theoretical question is related to the accuracy of the approximate equations (2) and (3). In order to resolve it, solution of eqs 2 and 3 should be compared with the exact one. The well-known matrix method26,39,40 allows us to consider patches of surface containing no more than 12 sites.41 The number of various topographies which can be studied by this method and which can be placed on the 2×4, 3×3, 3×4, and 2×6 fragments is not large. To increase the dimensions of the (39) Kramers, H.; Wannier, G. H. Phys. Rev. 1941, 60, 252. (40) Montroll, E. W. J. Chem. Phys. 1941, 9, 117. (41) Payne, S. H.; Zhang, J.; Kreuzer, H. J. Surf. Sci. 1992, 264, 185.

∑ f)1

β-1 ln(Yf)σ(k, f) +

1

Nfr

∑ fhσ(k, f)σ(k, h)

2 f,h)1

where Nfr is the number of sites on the lattice fragment, Ek is energy of the kth configuration of adspecies on the fragment, kth configuration is defined by the set of variables σ(k,f) (1 e f e Nfr), where σ(k,f) ) 1, if the site f contains adspecies in the configuration k and σ(k,f) ) 0 in the opposite case. The concentration dependences of the isosteric adsorption heat given in Figure 9 is a more sensitive characteristic of the system than the adsorption isotherm. As can be seen, all the curves Q(θ) are antisymmetric about point (θ ) 0.5; Q(θ ) 0.5)), therefore, the region 0.5 < θ e 1.0 is not shown. Pronounced differences in the results of the Q(θ) calculation by matrix and cluster methods already are seen at β g (1, while they are absent for the adsorption isotherms. Using these parameters on the fragment of 16 sites gives results that coincide with the curve from the matrix method. The larger bend of these curves in relation to the QCA curve is due to the allowance for indirect correlations in the fragment and matrix methods. At small values |β| < 1, the curves Q(θ), which have been calculated by the QCA approximation and by the fragment and matrix methods are almost coincident (4). If two-dimensional condensation occurs, the adsorption heat would be constant (curves 1 and 2). As matrix and fragment methods do not fix this phase transition, near the critical region (for the curve 2 β(1) ) 1, β(2) ) 0.33) smoother Q(θ) curves are observed that markedly differ from QCA results, but further from the critical region (β(1) ) 2.5 for curve 1), the bend of the curves does increase and results in practically total coincidence of curve 1 with the horizontal line of the QCA curve. The largest differences in the results of the methods under consideration are in the case of chemisorption near the critical region (β(1) ) -2, curve 5). In the lower part of Figure 9, Q(θ) curves with β(1) ) -2 are given on an increased scale for the following methods: the fragment (16 sites, d), the matrix method for the strip with M ) 4 (a), 6 (b), and 8 (c) sites, as well as the QCA (e) (i.e., eqs 2 and 3). Formation of the ordered c(2×2) phase is accompanied by an increase of the isosteric heat. Increasing the width of the strip in the matrix method results in an increase of the peak in Q(θ), moving it closer to the cluster curve. The last curve shows a discontinuity near the formation of the ordered phase due to the redistribution of adatoms (see also curves 6 and 7). Upon increasing the degree of repulsion, i.e., upon removal from the critical region, the isosteric heat practically does not change for (42) Tovbin, Yu. K. Russ. J. Chem. Phys. 1996, 15, (No. 2), 75.

988


Tovbin Table 1. Comparison of Adsorption Characteristics Obtained from the Fragment Method and the Quasi-Chemical Approximation for Homogeneous Surfaces (Percent of Difference Is Given, PT ) Phase Transition, E1 ) E(1)) β1

θ < 0.15

start PT

θ = 0.5

(1 2.5 -2 -5

0.1 0.1 0.1 0.1

Adsorption Isotherm 0.5 0.2 0.5 0.1 1.5 0.2 1.5 0.2

(1 2.5 -2 -5

0.1 0.1 0.1 0.1

Adsorption Heat 0.5 0.1 0.5 0.1 5.0 0.1 0.5 0.1

end PT

θ > 0.85

0.5 0.5 1.5 1.5

0.1 0.1 0.1 0.1

0.5 0.5 5.0 0.5

0.1 0.1 0.1 0.1

Table 2. Comparison of Adsorption Characteristics Made by the Fragment Method and Quasi-Chemical Approximation on Surface Fragment for Stepped Surface (Percent of Difference Is Given, the Length of the Terrace L ) 5) β1

Figure 9. Calculations of the isosteric heat adsorption on a homogeneous surface performed by matrix method (solid lines) with a strip of six-sites and the QCA (dashed line): Q0 ) 23 kJ/mol, z ) 4; β(1) ) 2.5 (1), 1.0 (2, 3), -1.0 (4), -2.0 (5, 6), -5.0 (7); β(2) ) 0 (1, 3-5, 7); and β(1)|/3 (2, 6). The curves with β(1) ) -2 and β(2) ) 0 are shifted down along the Q axis; they are on a scale larger by a factor of 2.5 relative to the others: the matrix method with M ) 4 (a), 6 (b), and 8 (c); the fragment method with Nfr ) 16 (d) (dash-and-dot line), and the QCA approximation (e).

0 < θ < 0.5. This means that the adatoms occupy only the sublattice in the structure c(2×2) (curve 7). Strictly speaking, all methods in the region near the phase transition (near the critical temperatures) give artificial effects because they are approximate. The cluster method results in a correct jump of the isosteric heat at the points of disorder-order and order-disorder phase transitions. This jump is due to difference in molecular distributions in the disordered and ordered phases at θ < θor and θ > θor, where θor is density at which the ordering appears. In the first case, repulsion of the molecules decreases Q(θ) at increasing θ (at θ < θor) in relation to Q0 ) Q(θ)0). In the ordered phase the molecules tend to occupy the sites at the distance of the second neighbors in order to free the nearest neighbors. Therefore, at the ordering θor < θ < 0.5, the molecules have no nearest

θ < 0.15

0.15 < θ < 0.85

(1 2.5 -2 -5

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