Two Improvements in Adsorption Theory: Adsorption on an

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Langmuir 1999, 15, 6107-6116

6107

Two Improvements in Adsorption Theory: Adsorption on an Amorphous Surface and a New Estimation of Micropore Volumes† Yu. K. Tovbin Department of Matter Structure, Karpov Institute of Physical Chemistry, ul. Vorontsovo Pole 10, Moscow 103064, Russia Received November 3, 1998. In Final Form: April 5, 1999 Two improvements of the adsorption theory are offered. The first improvement expands the area of applicability of a lattice-gas model for description of adsorption of symmetric particles on amorphous surfaces. The second improvement raises the accuracy of determination of a microporous system volume. A new approach to calculating the equilibrium characteristics of the adsorption of spherically symmetric molecules on heterogeneous surfaces of adsorbents was developed. The new method was applied to adsorption of Ar atoms on an amorphous TiO2 (rutile) surface. On amorphous surfaces, adsorption centers are located chaotically and the distances between neighboring local minima of potential energy vary over a wide range. At adsorption of inert gases on amorphous surfaces, situations are possible where there is blocking by adspecies of several nearest adsorption centers. Therefore in general, the problem of the calculation of adspecies adsorption on amorphous surfaces represents a problem of adsorption of multisite particles on heterogeneous surfaces. Intermolecular adsorbate-adsorbate interactions are taken into account for nearest atoms in a quasichemical approximation. The energies of the lattice parameters of the all interactions of model are defined from the Lennard-Jones potential (12-6). The offered method gives a satisfactory quantitative description of the isosteric heat of adsorption and isotherm of adsorption in the submonolayer region. It is shown that the standard procedure of an estimation of micropore volume with help from the Dubinin-Radushkevich equation results in overestimated sizes. It follows that the filling of micropore volume occurs at pressures considerably smaller than vapor saturated pressure. The last fact is caused by joint influence of potentials of the adsorbate-wall and adsorbate-adsorbate. A more exact finding of mircopore volume with estimation of the pressure of filling of the micropore volume is suggested.

1. Introduction Real adsorbents are heterogeneous; therefore, the computation of the adsorption characteristics for any concentration of adsorbate must take into account the joint effect of lateral interactions between adsorbed molecules and surface heterogeneity. In most cases the two effects have been studied separately, although at present, there are instances where the theory of the mutual influence of both these factors is considered as a whole.1-4 As a rule, the modern adsorption theories for nonporous as well as for porous adsorbents are based on the latticegas model2,5-7 in explicit or implicit forms. According to this simple molecular model, the volume of the adsorption space (the multilayer region over the surface or the volume of the pores) is broken up into unit cells (sites) having the size of the adsorbate. The lattice-gas model takes into account the following main properties of the condensed phase: the proper volume of the molecules and interparticle interactions. The proper volume of the molecules is † Presented at the Third International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held inTorun, Poland, August 9-17, 1998.

(1) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1988. (2) Tovbin, Yu. K. Theory of Physical Chemistry Processes at a GasSolid Interface; CRC Press: Boca Raton, FL, 1991. (3) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (4) Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997. (5) Hill, T. L. Statistical Mechanics; McGraw-Hill: New York, 1956. (6) Shulepov, Yu. V.; Aksenenko, E. V. Lattice Gas; Naukova Dumka: Kiev, 1981. (7) Fisher, M. E. The Nature of Critical Points; Lecture in Theretical Physics, Vol. VII c, 1965.

taken into account by assuming that a lattice site can contain not more than one molecule of the adsorbate, and the interparcticle interactions are allowed by using the parameter of interaction (which is the average energy of interaction between the particles occupying neighboring sites of the lattice). In general, the modern lattice model makes it possible to reflect all properties of real adsorption systems, namely, intermolecular interactions and adspecies ordering, heterogeneity of open and pore surfaces, different characteristic pore sizes, the presence of monolayer (two-dimensional), multilayer, and capillary condensations, etc. The extension of the theory to multicomponent adsorption systems was formulated.1,2,8,9 The main aspect of the theory concerns allowing for the actual structure of the heterogeneous surfaces. Today within the framework of the lattice-gas model a question about describing adsorption of large molecules on heterogeneous surface is still open. This question has discussed in previous works.3,10,11 In this paper two improvements of the adsorption theory are offered. The first improvement expands the area of applicability of the lattice-gas model for description of adsorption of spherically symmetric molecules on heterogeneous surfaces. The second improvement raises the accuracy of determination of the microporous system volume. (8) Gerofolini, G. F.; Rudzinski, W. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997; p 1. (9) Tovbin, Yu. K. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997; p 105. (10) Nitta, T.; Kuro-Oka, M.; Katayma, T. J. Chem. Eng. Jpn. 1984, 17, 45. (11) Tovbin, Yu. K. Russ. Chem. Bull. 1997, 46, N3, 442.

10.1021/la981556r CCC: $18.00 © 1999 American Chemical Society Published on Web 05/25/1999

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On amorphous surfaces, adsorption centers are located chaotically and the distances between neighboring local minima of potential energy vary over a wide range. On the adsorption of argon atoms at the amorphous surface of TiO2 (rutile), situations are possible where there is the blocking by adspecies of several of the nearest adsorption centers.12 Therefore in general, the problem of calculation of adspecies adsorption on amorphous surfaces represents a problem of adsorption of large (multicenters) molecules on heterogeneous surfaces (this case is discussed in section 2). A new approach to calculating the equilibrium characteristics of the adsorption of spherically symmetric molecules on heterogeneous surfaces of adsorbents was developed (section 3). The method was applied to the Ar/ TiO2 (rutile) system. Intermolecular adsorbate-adsorbate interactions are taken into account for nearest atoms in a quasichemical approximation. The energies of lattice parameters of all interactions of the model are defined from the Lennard-Jones potential (12-6). For estimation of microporous system volumes, the Dubinin-Radushkevich equation is actively used.13 Taking into account the results of research of microporous systems with the help of modern theoretical methods, it is shown (section 4), that the standard procedure of an estimation of micropore volume with help of the DubininRadushkevich equation results in overestimated sizes. It follows that the filling of micropore volume occurs at pressures considerably smaller than vapor saturated pressure. The last fact is caused by joint influence of potentials of the adsorbate-wall and adsorbate-adsorbate. A more exact finding of micropore volume with the help of an estimation of the pressure of filling of micropore volume has been suggested. 2. Basis of Adsorption Theory for Large Molecules on Heterogeneous Surfaces We shall describe the adsorption systems by means of a lattice model. When we consider a one-center adsorption, then each site of the lattice is an adsorption center and can be occupied by a particle of any kind i (including a vacancy i ) v), 1 e i e s, where s is the number of different kinds of molecules + 1 (for vacancy). The state of occupancy of each site f, 1 e f e N, where N is the number of sites in repeated fragment of system under consideration, will be characterized by the variable γfi, with γfi ) 1 if the site f contains a particle of kind i and γfi ) 0 if it contains a particle of any other kind. These variables obey the following equations γf1 + ... + γfs ) 1, γfiγfj ) ∆ijγfi,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 ηfq, which is assumed to be known and unvarying during the adsorption process (non-reconstructing surface), with 1 e q e t, where t is the number of types of sites. We have ηfq ) 1 if f is a site of type q and ηfq ) 0 in the opposite case. The complete set of {ηfq}, with 1 e f e N, uniquely defines the composition and topography of the surface, which can be of any kind. Large molecules can have different orientations in the surface plane. In the statistic description of large molecules, each orientation is considered as a kind of particle. Therefore, taking into account different orientations of the molecules is reduced to the adsorption theory of a mixture of molecules even for a one-component system. For multicenter adsorption, each molecule can block (12) Bakaev, V. A. Surf. Sci. 1988, 198, 571. (13) Dubunin, M. M.; Radushkevich, V. L. Comm. USSR Acad. Sci. 1947, 55, 331.

Tovbin

several sites, mi, where mi is the number of sites blocked by molecules with orientation i (let us designate the “molecule with orientation i” as the “particle i”, 1 e i e s (here s is the number of distinguishable orientations of one kind of molecule). To perform correct averaging over all configurations of large molecules, a rule of assignment of the molecule position to a specific site of the surface should be formulated. Let us choose one of the terminal parts of particle i (this part is located on site f), and let us count other sites occupied by the same particle from this site f. We shall say that particle i locates in site f and is designated by symbol {f}, the surface region occupied by particle i. This situation will be described by the new variable values Γ{f}i ) γfi∏hγhi, here h ∈ mi(f); i.e., h is the number of the blocking sites by the particle i located in site f. The local Henry constant a{f}i of component i on the local fragment {f} are defined by the set of the parameters ηfq for site f and for its neighboring sites h ∈ mi(f). We shall assume that lateral interaction is described by a pair potential and take into account the nearest interactions. Then the Hamiltonian of the heterogeneous lattice-gas model can be written as11 N

H)

s

1

υi{f}Γi{f} - ∑ ∑fgijΓi{f}Γj{g} ∑ ∑ 2 f,g i,j f)1 i)1

(1)

where υi{f} ) - β-1 ln(ai{f}Pi), ai{f} ) Fi{f}β exp(β Qi{f})/Fi0, and β ) (kT)-1, here Pi is the partial pressure of the i component (i * v); the index f denotes the site from which we begin to account for blocking sites; 1 e f e N; F{f}i and Fi0 are the partition functions of adsorbed molecule i on the local fragment {f} and of molecule i in the gas phase; Qfi is a energy binding of molecule i on the local fragment {f}; fgij is an interaction parameter for neighboring molecules i and j on the local fragments {f} and {g}; in the atom-atomic approximation we have fgij ) ij. The equations for the discrete model can be obtained with help of the so-called cluster approach.2 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 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,14 and therefore we need approximate methods of describing the systems. In the quasichemical approximation the closed system of equations can be presented as

a{f}iPθ{f}miv ) θ{f}iΛ{f}i Λ{f}i )

∑ ∏θ{f}{g}in exp[-β{f}{g}in]/θ{f}i

(2)

R(n) n

θ{f}{g}inθ{f}{g}miv,mnv ) θ{f}{g}i,mnvθ{f}{g}miv,n exp[-β{f}{g}in] (3) θ{f}i is the probability of filling of the local fragment {f} by particle i; θ{f}miv is the probability that the local fragment {f} of size mi is free; the symbol miv designates the free surface region of the size mi, in which the adsorption of (14) Baxter, R. Exactly Solved Models in Statistical Mechanics; Academic Press: London, 1982.

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molecule with orientation i is possible; θ{f}{g}in is the probability that particle i on the local fragment {f} and particle n on the local fragment {g} are adjacent, here 1 e n e s. The condition of normalization to the paired probabilities will be written as ∑nθ{f}{g}in ) θ{f}i; the summation over R(n) in eq 2 means the sum over all possible arrangements of all adjacent molecules n. The function Λ{f}i depends on the lateral interactions between neighboring molecules. If the lateral interactions can be neglected, Λ{f}i ) 1 and a{f}iP ) θ{f}i/(miθ{f}miv). The normalizing condition to the probability of coverage of the site with the number f has the form mi-1

s

Ffv +

∑ ∑ i)1 ξ)0

Ff-ξi ) 1

(4)

where Ffi ) θ{f}i/mi and index ξ denotes the shift of the molecule along the given orientation. Normalizing condition (4) reflects the whole totality of different manners of blocking the given site f; Ffv is the probability that site f is free.

thh+1vv ∏ h

θ{f}miv ) Ffv

(5)

In the case of macroscopic surface area, eqs 6 have the similar form if we replace indexes f and g on q and p, respectively: t

θ(P) )

∑

fqθq(P),

aqP )

q)1

θq 1 - θq

t

Λq ) (1 +

dqp xtqp)z , ∑ p)1

tqp )

Λq, 2θp

, bqp + δqp δqp ) 1 + x(1 - θq - θp), q

bqp ) (δqp2 + 4xθqθp)1/2

(7)

where fq is a discrete function of distribution of the sites of a heterogeneous surface that have local Henry constants aq, i.e., the fraction of sites of type q; dqp is the probability of finding the site p at the nearest distance from site q; 1 e q, p e t; Σp)1t dqp ) 1. Functions dqp describe a surface structure.2,15 For any size of adsorbed molecule the isosteric heat of adsorption can be computed by the thermodynamic definition:

Q(θ) ) - (d ln P/dβ)θ)const

(8)

L

thh+1vv(k) ) Fh+1v[Fh+1v +

∑i ∑φ Fh+1φQiφ/z]-1

where index h is the number of (mi - 1) sites blocked by particles i with size mi, thh+1vv is the conventional probability of the free site with number (h + 1) near the free site with number h, pair indexes h and h + 1 specify the orientation (k) of the pair of free sites, and Qiφ(k) is the number of contacts of class φ of adjacent particle i in the direction specified by index k. We must solve the system of eqs 2-5 numerically, and the final results for isotherm adsorption have been obtained by averaging of the local coverages over the all surface fragments of sizes mi and by summation over different orientations of adsorbed molecules (over different meanings of mi). For one-center adsorption eqs 2-5 reduced to the next system of equations allowing for lateral interacting molecules on heterogeneous surfaces2,15 N

θ(P) ) af P )

θf 1 - θf

θf(P)/N, ∑ f)1 zf

Λ f,

x ) exp(-β) - 1,

Λf )

(1 + xtfg), ∏ g)1

tfg )

2θg

bfg + δfg δfg ) 1 + x(1 - θf - θg), bfg ) (δfg2 + 4xθfθg)1/2

,

(6)

Here θf is the molar fraction of the particles in site f and index g numerates adjacent sites zf of site f. We shall restrict ourselves to the one-component sorption system s ) 2: i ) A is an adsorbed molecule and v is a vacancy (aqv ) 1); i.e., any site is occupied or vacant. This system of equations offers a detailed description of the distribution of particles upon different sites of the surface fragment (the distributed model2,15). (15) Tovbin, Yu. K. Langmuir 1997, 13, 979.

3. New Approach to Calculating Spherically Symmetric Molecules on a Heterogeneous Surface The main difficulty with using eqs 2-5 consists of the complexity of repeated calculation of functions Λ{f}i in eq 2, in which the sum of R(n) is the sum over all states of the nearest molecules. The account of specific properties of the adsorption system, which simplified these calculations, is useful. One such variant to simplify calculations of the right-hand side in eq 2 is adsorption of spherically symmetric molecules such as CCl4 or similar but greater size. By virtue of simple geometry of hard spheres for such molecules, eqs 2-5 can be reduced to the system of eqs 6 (subsection 3.1). This method was tested for an adsorption of argon atoms on an amorphous rutile surface. The experimental data16,17 serve as reference one. These data were considered also in previous papers.12,18 Amorphous surfaces are characteristic of a sufficiently wide range of sorbents widely used in adsorption processes. In general, the problem of calculating the adsorption of gases on amorphous surfaces consists of two parts: the first part is to simulate an amorphous surface, while the second one is to develop a procedure of calculating the adsorption on the generated surface. In this paper we have considered the second part of the problem. The first part has been discussed in ref 19, and here we brief present main results of the work (subsection 3.2). New method has been applied on such a constructed amorphous surface. To exclude the large number of lattice model parameters needed for describing any heterogeneous surface, these parameters were calculated using the atom-atom potential of the interaction of Ar atoms with each other and with oxygen and titanium ions. Calculations of the concentration dependence of the heat and isotherms of adsorption were carried out (subsection 3.3). 3.1. Quasi-One-Center Model for Calculation of Adsorption of Spherically Symmetric Molecules on Heterogeneous Surfaces. In the case of adsorption of (16) Drain, L. E.; Morrison, J. A. Trans. Faraday Soc. 1952, 48, 316. (17) Drain, L. E.; Morrison, J. A. Trans. Faraday Soc. 1952, 48, 840. (18) Bakaev, V. A.; Steele, W. A. Langmuir 1992, 8, 1372. (19) Gvozdev, V. V.; Tovbin, Yu. K. Russ. Chem. Bull. 1997, 46, 1060.

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value θ ) θk(P), 1 e k e n2, which corresponds to the minimum value of the free energy Fk of the adsorbed molecules, is chosen for each specific value of the pressure P. The expression for Fk can be written as21

Fk )

∑f ∑l Mflθfl

1 Mfl ) vfl + β-1 ln θfl - β-1 2

∑g ln[FfgllFfglD/(θfl)2 FfgDl]

Figure 1. Scheme of partitioning at n ) 2 for using quasione-center model. On an initial lattice structure (full line) the site with center 0 is divided on four small parts. The centers of small parts are denoted by 1, 2, 3, and 4. Full lattice structure with the nearest neighbor z ) 4 is reproduced around each new center. A new lattice with dash lines corresponds to the center 1. A new lattice with dash-dot lines corresponds to the center 3.

spherically symmetric molecules (which are approximated by a sphere or square), a simpler approximated calculation method can be used. The physical basis of the new method is the following. For small densities of the adsorbate, the contributions of the lateral interaction are low, and the distribution of the molecules is determined only by the local adsorbate-adsorbent potential (i.e., by a local Henry constant). In the region of great coverages of the surface, the adsorption occurs mainly on the weakest sites, and the contribution of the lateral interactions is maximum. In the intermediate region of coverages, the heterogeneity of the surface and lateral interactions appear simultaneously. This results in the nonequivalent coverage of different sites, leading to the formation of a certain compact arrangement of the molecules at first local and then larger (domain), on the most favorable subset of sites with the maximum binding energy and to blocking the adjacent sites. This phenomenon is analogous to the effect of ordered arrangement of the molecules in the case of strong repulsion of the nearest neighbors.20 The proposed procedure for the calculation of the adsorption is the following. The surface is divided by a lattice structure, as for one-center molecules, with the number of the nearest neughbors z and the lattice constant equal to the diameter of the molecule. Then the surface region in site each site is divided into smaller surface areas (for example, the square side is divided into n parts, n ) 2, 3, ..., and the site itself is divided into n2 parts). The center of any small surface area can be accepted as the center of the “whole” site, which is then propagated over the whole surface (Figure 1). The division of the surface into a set of sites with their sets of local Henry constants for each site corresponds to each position of the center of the site. For the chosen division of the surface, the adsorption θk(P) is calculated in the same manner as for the one-center system2,15 by solving the system of eq 6. Such a calculation is repeated n2 times (1 e k e n2): each small surface area is the center of the site. Then the (20) Roberts, M. W.; McKee, C. S. Chemistry of the Metall-Gas Interface; Claredon Press: Oxford, 1978.

(9)

where Ffgij ) θfgij exp(-βij), l ) A and v, and index D denote one of the fixed indexes A or v. The total dependence θ(P) thus constructed corresponds to the adsorption isotherms. Thus, the system of equations for one-center particles2,15 is solved n2 times instead of solving the system of equations for multicenter particles. This calculation procedure is sufficiently adequate to the physical picture. The results of the calculation θ f 0 and θ f 1 are independent of the method of division of the surface into cells. Because θ f 0, the Henry constant is the additive value independent of the value of summands, and at θ f 1 the total value of the contributions of the lateral interactions is independent (or hardly depends) on the character of surface heterogeneity. In the region of intermediate coverages, all possible modes of the ordered arrangement of the molecules is considered in the explicit form. This procedure was used below for calculating the isotherms and heat of the argon atom adsorption on the amorphous rutile surface in the whole range of the monolayer coverage. A gas-solid interface can be conceived as a lattice with a constant number of neighbors (z ) 4) whose sites are adsorption centers. A site occupies a surface area of about the size of an adsorbed atom d2 (d is the diameter of the molecule), and the volume of each site is of order d3. Each site f, 1 e f e N, on the surface fragment under consideration at each chosen division of surface fragment is characterized by the energy of bonding of the atom with the amorphous surface of the adsorbent (Qf) and the local Henry constants (af). For each chosen division of surface fragment the sets of parameters , Qf, and af are obtained and then the systems of eqs 6 are solved for local coverages of sites of different types. 3.2. Procedure for Formatting of the Amorphous Adsorbent Structure.19 The amorphous structure of an adsorbent is assumed to be rigid and unchanged at all degrees of surface coverage. Simulation of an amorphous structure was carried out taking into account the conventional geometric concept of an amorphous threedimensional structure as a completely random arrangement of atoms in a solid. We simulated the surface structure of the adsorbent by building up a subsurface range with the help of randomly generated positions of the atoms in the three-dimensional structure, beginning from the basis crystalline cell of the TiO2 located at a depth of R ) 1.77 nm. Here R is the maximum of two radii of the potential for Ar-O2- and Ar-Tu4+ interactions. These potentials are nearly equal to zero at R > 1.77 nm. With a random number generator, the random vectors of positions of new atoms relative to those of previously “arranged” atoms were obtained. Since the build-up is carried from the depth toward the surface, random displacements are repeatedly accumulated from layer to layer, and the degree of amorphization of the surface layer reaches its maximum value. This procedure is character(21) Tovbin, Yu. K. Russ. J. Phys. Chem. 1992, 66, 1395.

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ized by parameter δ (%), i.e., the maximum displacement of the atoms from the value of the lattice constant. Parameter δ allows one to prevent complete bond cleavage between the atoms since short-range order exists in an amorphous substance. Previously12 the model, which neglected the cationic sublattice, was used for amorphizing an adsorbent. Our model takes into account both cationic and anionic sublattices, which corresponds to a real amorphous substance containing cations and anions. The second peculiarity of surface amorphization consists of the removal of a certain portion, γ (%), of the oxygen ions after random displacement of the atoms in the surface layer as in ref 18. The procedure proposed takes into account two basic properties of an amorphous substance: the presence of short-range order and the absence of longrange order. The potential energy of the system was calculated in the atom-atom approximation using a set of potentials based on the Lennard-Jones (12-6) potentials

σ r

[( )

U(r) ) 4EAr-i

σ r Ar-i

12

Figure 2. Potential relief of an amorphous surface (fragment 20 × 20). Table 1. Distribution Function of the Number of Local Minima for Potential Energy F(m*) Corresponding to the One Lattice Sitea

6

() ] Ar-i

f(m*)

where indexes i are the O2- and Ti4+ ions and Ar atom, EAr-i is the depth of the potential well, r is the distance between the interacting atoms, and σAr-i values are the parameters of the potentials. Three sets of parameters for the O2- ions have been previously18 considered. The second set was proposed as providing the best qualitative agreement with the experiment. However, in the work19 the parameters of the Ar-Ti4+ and Ar-O2- potentials were varied, and here we used in the calculations parameters EAr-i/k ) 60 K (i ) Ti4+), 120 K (i ) O2-), 119.8 K (i ) Ar), and σAr-I ) 0.241 (i ) Ti4+), 0.325 (i ) O2-), 0.3405 (i ) Ar) nm, while those for the Ar-Ar potential were taken from ref 18. The parameters of the lattice model , Qf, and af are related to the interaction potentials in a complicated way.22 The minimum and maximum values of parameter  can be estimated using the following equations:23,24

(min) ) β-1 ln(Dfg) Dfg ) (max) )

∫V ∫V f

∫V ∫V f

g

g

exp[-βE(r)] dVf dVg/VfVg

(10)

∫V ∫V

E exp[-βE(r)] dVf dVg/

f

g

exp[-βE(r)] dVf dVg Qf )

∫V U(r) exp[-βU(r)] dVf/Af f

Af )

∫V exp[-βU(r)] dVf f

Expressions for the lattice parameters Qf and af are given in ref 24. In eqs 10 Vf is the volume (d3) of the site with index f, E(r) is the potential of the adsorbate-adsorbate interaction, U(r) is the potential of the adsorbateadsorbent interaction, which is obtained by summing over all paired contributions UAr-i(r) from the atom-atomic potential of the Ar atom interactions with titanium or oxygen ions (i ) Ti4+, O2-) that are located within a sphere of radius R. Since all atoms of the adsorbent are randomly (22) Tovbin, Yu. K. Russ. J. Phys. Chem. 1998, 72, 775. (23) Tovbin, Yu. K. Russ. J. Phys. Chem. 1995, 69, 118. (24) Gilyazov, M. F.; Kuznetsova, T. A.; Tovbin, Yu. K. Russ. J. Phys. Chem. 1992, 66, 305.

m* 0 1 2 3 4 5 6 7 8 9 Nm Mm

L × L ) 10 × 10 0.12 0.15 0.24 0.23 0.13 0.09 0.01 0.02 0.01 257 2.57

L × L ) 15 × 15

L × L ) 20 × 20

0.15 0.24 0.20 0.21 0.11 0.0.5 0.02 0.01 Tc(H), where Tc(H) is the critical temperature of the adsorbate in pore of width H. At temperatures T1, as pressure increase to the P1(H) value, line AHBH is crossed; therefore, condensation of the adsorbate in the pore occurs. If the system was closed, the stratification phase transition would occur in pores. However, a real experiment occurs in an open system; therefore, a pore is completely filled with the “liquid” adsorbate. The P1(H) value is an analogy of the saturated vapor pressure for a bulk liquid phase. At temperature T2 the adsorbate in a pore occurs above the critical state, and the volume filling of a pore is not accompanied by a phase

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transition. The potential of both walls attracts the adsorbate; hence, micropores are filled more efficiently than an open surface. Let us denote the pressure at which micropores are filled at temperature T2 by P2(H). In both cases, filling of micropores ends at a certain pressure P1,2(H), which is less than the saturated vapor pressure Ps at the same temperature. Below, we use a simplified variant of the system of eq 7 to estimate qualitatively the P1,2(H) values suitable for practical purposes. 4.2. Lattice-Gas Model for Slitlike Pores. Below we will restrict ourselves to the simplest variant of this theory according to which the walls in a slitlike pore with width H are considered to be homogeneous. We shall allow only interactions between the nearest neighbors.2,38 To calculate local coverages, the space in the slitlike pore is divided into monatomic layers parallel to the walls of the slitlike pore. The lattice of the sites is characterized by fixed numbers of the closest neighbor zq; 1 e q e H, where q is the number of layers. Let us denote by zqk the number of nearest neighbors, located in a layer k for a site in layer q; k ) q, q ( 1; zqq-1 + zqq + zqq+1 ) zq ) z. All sites in layer q possess identical properties, because the bonding energy Qq of the adsorbent-adsorbate interaction varies from layer to layer (1 e q e H) but remains constant within the same layer; Qq ) U(q) + U(H - q + 1), where U(q) is the potential of interaction of the adsorbate with pore wall: U(q) ) U/q3 corresponds to the 9-3 Mie potential in which the repulsive branch of the potential is taken into account by the lattice structure. An isotherm of multilayer adsorption for a slitlike pore system has the form of eq 7, where t is the number of different types of sites in the slitlike pore: t ) H/2 for even H, t ) (H + 1)/2 for odd H; fq is the fraction of type q sites in the lattice structure; fq ) 2/H, when 1 e q e t for even H and 1 e q e (t - 1) for odd H, fq ) 1/H for odd H and q ) t only; θq is the mole fraction of particles that occupy sites of type q; aq ) aq0 exp(βQq) is the local Henry constant for site of type q with the bonding energy Qq, aq0 is the preexponential factor; Λq is the term taking into account the lateral interactions between the particles located in the neighboring sites. In the quasi-chemical and the mean field approximations we have2,38 q-1

Λq )

(1 + xtqp)z ∏ p)q+1

qp

q+1

Λq ) exp(- β

∑ zqpθp) p)q-1

(12)

Note that our main conclusions do not depend on the particular type of approximation used. 4.3. Estimations of Pressure, Adequate to Volume Filling of Micropores. To obtain qualitative estimates for the pressures P1(H) and P2(H), let us use an approach developed previously.45 The use of the condensation approximation and the equation for the spreading pressure in a multilayer system yielded an expression45 for the pressure at which two-dimensional condensation occurs within each layer during the layer-by-layer filling of an open surface. This expression can be easily extended to the model of slitlike pores considered here. Then we obtained a relation for the pressure at which the volume filling of the pores occurs (45) Tovbin, Yu. K.; Votyakov, E. V. Russ. J. Phys. Chem. 1992, 66, 1597.

a0P(H) )

∫01ln Λporedθpore - βQpore

(13)

where Qpore describes the interaction of the adsorbate with the wall of a pore. According to the results of the previous study37 for T1 < Tc(H), the capillary condensation always occurs when the first monolayer on each side of the pore is completely filled, i.e., θ1 ) θH ≈ 1; the remaining (H - 2) layers in a pore with H > 2 are considered to be a homogeneous phase with the density θpore. Then the complete filling of the pore can be represented, instead of eqs 7 and 12, as θ ) f1θ1 + fporeθpore, where f1 ) 2/(H - 2) and f2 ) 1 - f1. For simplicity, the equation for θpore can be written, for example, in the mean-field approximation: aporeP(1 - θpore) ) θporeΛpore, Λpore ) exp[-βEzporeθpore], here, the zpore and Qpore (apore ) a0 exp(βQpore)) values are found by averaging the corresponding zqk and Qq over the (H-2) layers of the pore. This gives zpore ) [zqq(H - 2) +2zqq+1(H - 3) + 2zqq+1θ1/θpore]/(H - 2) ≈ z + 2zqq+1/(H - 2), because θ1/θpore ≈ 2, and Qpore ) 2Q1φ(H)/(H - 2), where φ(H) ) ∑q)2H-1q-3 and Q1 is the adsorption heat of molecules at low coverages of the monolayer on an open surface. By substituting the above expressions into eq 13 and taking into account that Ps ) exp[-zβ/2], we obtain the target formula

P1(H) ) Ps exp(-b1) b1 ) β[z12 + 2Q1φ(H)]/(H - 2)

(14)

As the slit width H increase, the b1 value decreases, and the pressure at which the pore is filled rapidly (exponentially) approaches the bulk pressure Ps. This model adequately describes the above-mentioned variation of θc as a function of H. Then we obtain θc ) (H + 2)/2H; hence, θc ) 1/2 for H f ∞ and θc ) 0.75 and 0.83 for H ) 4 and 3. For the second case (T2 > Tc(H)), we will consider the last stage of layer-by-layer filling is a pore, i.e., the situation when the preceding layers have already been filled, except the last layer. Here we obtain two estimates (the upper and the lower) for the pressure corresponding to the filling of the last layer. To make the upper estimate, we will assume that in the last layer, two-dimensional condensation occurs. Then eqs 7 and 12 assume the form

a0 exp(βQlayer)P2+(H) ) θlayerΛlayer/(1 - θlayer)

(15)

in which the Λlayer ) exp[-βzlayerθlayer], zlayer ) z11, and Qlayer ) [2z12 + Q1D(H)] values are introduced; here D(H) is equal to [1/(H/2)3 + 1/(1 + H/2)3] for even H and 2/((H + 1)/2)3 for odd H. From an equation completely analogous to eq 13 we obtain (by replacing the index “pore” by “layer”)

P2+(H)/Ps ) exp{-b2+(H)} b2+(H) ) β[z12C(H) + Q1D(H)]

(16)

where C(H) is equal to 1 for odd H and 1/2 for even H. The lower estimate of P2-(H) can be obtained from eq 15 for θlayer f 1. Using the known relation46 and assuming that probability of finding a vacancy in the dense phase is (1 - θlayer) ≈ exp(-βz), we can write (46) Frenkel, Ya. I. Statistical Physics; Izd. Acad Nauk SSSR: Moscow, 1948.

Adsorption Theory Improvements

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P2-(H)/Ps ) exp{-b2-(H)}

(17)

b2-(H) ) β[z/2 + Q1D(H)]

(17)

This estimate gives P2-(H) < P2+(H). Note that if the Frenkel-Helsy Hill’s equation for multilayer adsorption47 is extended to porous systems and assumes the form ln(P(H)/Ps) ) -βQ1[q-3 + (H - q + 1)-3], 1 e q e H, then, due to the confined size of the pores, the right-hand side of the equation takes the minimum value at q ) H/2 (even H) or q ) (H + 1)/2 (odd H), this makes it possible to find the ratio of pressures P2+(H)/Ps ) exp{-βQ1D(H)}, at which the pores are completely filled. (In the case of multilayer adsorption, the right-hand side of the FrenkelHelsy Hill’s equation contains only q-3 and this gives P(H)f Ps at H f ∞.) All estimates made here do not depend on what particular approximation (quasi-chemical or mean-field) has been used. In any case, micropores are filled at pressures lower than the saturated vapor pressure corresponding to the same temperature. The contributions of the interaction of adsorbate molecules with the walls and with one another are manifested differently for temperatures below and above the critical temperature. As follows from the formulation of the model, for H ) 3 and 4, D(H) ) φ(H)/(H - 2), therefore b1 ) b2+; and for H > 4 we have D(H) < φ(H)/(H - 2). In this case, at H ) const the contribution from the walls for P2(H) is less than that for P1(H). The contribution of the lateral interactions to the coefficients b1,2(H) decreases with an increase in H in the presence of capillary condensation or remains equal to 1 or 1/2, depending on the evenness of H when there is no capillary condensation. Let us estimate the b1,2(H) for spherically symmetrical particles. For inert gases and molecules such as CH4, N2, and O2 adsorption experiments are carried out at T ∼ 0.5 - 0.7 Tc and for them Q1 ∼ 10E. The best description of the critical bulk data48 is provided by lattice structure with z ) 6. For this structure z12 ) 1 and in the quasichemical approximation, βcE ) 0.81. Assuming that β ) 1.4 we find that b1 varies 4.9 (at H ) 3) to 1.7 (at H ) 6); similar b2+ varies from 4.9 to 1.2 and b2- varies from 7.7 to 2.8. Equations 14, 16, and 17, like eq 12, relate to monodisperse systems of slitlike pores. Nevertheless, as the first approximation, these results reflect the most general conditions for filling of micropores, because the consideration was based on a characteristic value determining the limiting size of a micropore. Equations 14, 16, and 17 do not allow the run of an isotherm to be followed; they only distinguish the characteristic point on an isotherm corresponding to the filling of micropores. The heterogeneous surface of the walls or different sizes of the pores change the run of the isotherm but do not alter the relationship between the pressure at which narrow pores are filled and the width of the pores. Therefore, in determining the volume of micropores from an adsorption isotherm, it makes no difference in what coordinates the experimental data are plotted. The isotherms for microporous systems plotted in the θ-P/Ps coordinates, which contain an extended plateau, are well-known.47 These plots areas convenient for the determination of volumes of micropores as the curves draw in the coordinates of eq 11. It should be noted that the isotherm for systems consisting (47) Gregg, S. J.; Sing, K. G. W. Adsorption, Surface Area and Porosity, Academic Press: London, 1982. (48) Batalin, O. Yu.; Tovbin, Yu. K.; Fedyanin, V. K. Russ. J. Phys. Chem. 1980, 53, 3020.

Figure 6. Adsorption isotherms (1 and 2) in the coordinates of eqs 11 and 18 with the corresponding abscissa axes 1 and 2 at pressures lower than the P(H) pressure corresponding to the volume filling of micropores of width H; the dashed section of curve 1 corresponds to the standard procedure of determination of the micropore volume Wo.

of, e.g., CCl4, benzene, cyclohexane (all at 25 °C), or isopentane (0 °C) on ammonium phosphoromolybdate,49 benzene on carbon from anthracite,50 propane on 5A zeolite (T ) 273, 323, and 398 K),51 and argon on chabazite (from 138 to 195 K)52 flatten out at P/Ps from 0.1 to 0.3. These data correspond to b values from 2.3 to 1.1. Since experimental adsorption isotherms must be corrected to take into account filling of mezopores, the above b should be increased to ∼2.5-1.3. These results are qualitatively consistent with the obtained estimates. Note that in a previous study29 a value of the order 0.2P/Ps was introduced as convenient unit for measuring the number of adsorbed molecules of benzene, but nevertheless, a value extrapolated at P ) Ps was used as the volume of micropores Wo. 4.4. Estimation of Micropore Volume with Help of the Dubinin-Radushkevich Equation. To used real pressure corresponding to the filling of micropores with characteristic size H in the calculations in terms of eq 11, it is necessary to change the standard method of determination of Wo according to which the Wo is found by the linear extrapolation of the dependence of ln W on [a(x)]2 where x ) 1. In this situation the value of Wo(H) should be matched by the corresponding Ps(H). Therefore, instead of eq 11, the following equation should be used

ln(W/Wo(H)) ) -k(RT/γ)2(ln[P1,2(H)/P])2

(18)

The only difference between this equation and eq 11 is that Ps has been replaced by p1,2(H). The correlation with the traditional procedure for determination of the volume of micropores is shown in Figure 6. If the experimental curve is straightened in the coordinates of eq 18, its intersection with the ordinate axis determines the volume of micropores Wo(H) with characteristic size H. Note that dependence (18) cannot be continued to higher pressures; hence, the necessity to interpret the deviations of the experimental curve from linear dependence (18) at higher pressure is eliminated. These deviations have been traditionally explained by the presence of micropores (49) Gregg, S. J.; Stock, R. Trans. Faraday Soc. 1957, 53, 1355. (50) Cadenhead, D. A.; Everett, D. H. Conference on Industrial carbon and Graphite. Soc. Chem. Ind. 1958, 272. (51) Ruthven, D. M.; Loughlin, K. F. J. Chem. Soc., Faraday Trans. 1 1972, 68, 690. (52) Egerton, T. A.; Stone, F. S. Trans. Faraday Soc. 1970, 66, 2364.

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characterized by a different parameter k or having a different characteristic size, although the deviations are observed at relative high coverages. When an experimental curve plotted in the coordinates of eq 18 deviates from linear dependence, this indicates that the microporous system is actually heterogeneous. The fact that deviations are observed at relative high pressure in the coordinates of eq 11 but are not observed in coordinates of eq 18 implies that there exist a function of volume distribution other than function f(a) ) 2ka exp(-ka2), where k is a parameter, which corresponds to eq 11.28 This bears no direct relation to polydispersity of the microporous systems. Figure 6 shows that the volume of micropores Wo(H) is smaller than Wo. As H increase, the difference between Wo(H) and Wo decreases. For practical purposes our estimations (14), (16), and (17) can be used, if the characteristic linear size of molecules at their dense packing in slitlike pores is known. The attitude of the pore width to this characteristic size of the adsorbate defines the number H of monatomic adsorbate layers at filling of the pore volume. At given H it is possible to use the obtained formulas for definition of sizes b1,2(H), which specify area of pressure P, at which it is possible to use eq 18: P < P1,2(H) (we should note, that in estimations (14), (16), and (17) the linear size of adsorbates is used instead of its volume). If the above-described estimates are applied to b1,2(H), the change in the volumes of micropores depends on the slope of the isotherm straightened in the coordinates of eq 11 or 18. For some experimental data27 the difference can range from 10 to 50%; this means that the real pressure corresponding to the filling of micropores should be taken into account. The similar amendments on real meanings of pressure, adequate to volume fillings of micropores systems, should be entered at using of all equations,30-36 which give the generalized Dubinin-Radushkevich equation for more complex cases of microporous adsorbents. In conclusion, we note that the estimates obtained here provide a correct description of the pressure dependence for the volume filling of micropores on the size of the adsorbate molecules: as the pressure increases, the number of monolayers H decreases. Therefore, the b1,2(H) values increase and, consequently, the pressure at which the pore is filled decreases. As a result, whereas in the case of eq 11 the curves should intersect at the same point Wo,30 the isotherms for various substances should end at the same Wo value but at different pressures P1,2(H). 5. Conclusion The proposed procedure for calculating the characteristics of spherically symmetrical molecules adsorbed on an amorphous surface and blocking several local minima of the potential energy of adsorbent-adsorbate within one local volume of the order d3 is efficient in the framework of the quasi-one-center lattice model, since a

Tovbin

small number of partitions of the site area into small parts is required. To perform reliable calculations of the adsorption characteristics, the fragment of the amorphous surface should contain no less than 15 × 15 sites of the lattice structure. A new approach gives the possibility of reproducing a picture of amorphous surface and allows analysis of its structure by using the experimental data on isotherms and/or heats of adsorption. An applicability area of the Dubinin-Radushkevich equation for determination of micropore volume of microporous systems from data of measurements of isotherm adsorption is analyzed. It is shown that the standard procedure of an estimation of micropore volume results in overestimated sizes. A more exact finding of micropore volume with the help of an estimation of the pressure of filling of micropore volume has been suggested. Nomenclature {f} ) adsorption coefficient of component i on the local fragment {f}; the index f denotes the site from which we begin account of blocking sites, 1 e f e N f(Q) ) distribution function for heat of adsorption F{f}i and Fi0 ) partition functions of a adsorbed molecule i on the local fragment {f} and of the molecule i in the gas phase H ) pore width measured in the number of monolayers mi(f) ) number of blocking sites by molecule with orientation i N ) number of lattice sites Pi ) partial pressure of the i component (i * v) Q ) heat of adsorption s ) number of components of lattice-gas model, including a vacancy (i ) v), for a one-component system: s ) number of different orientation of molecule + 1 (for vacancy) tfgij ) conventional probability that the site with number f is occupied by particle j near the particle i in site with number g W0 ) volume of the porous system z ) number of the nearest neighbors; zqp ) number of p type site around of the q type site a ) “adsorption potential” fgij ) interaction parameter for neighboring molecules i and j on the local fragments {f} and {g};  ) similar parameter for spherical particles γfi ) a random variable: γfi )1, if the site f contains the molecule i, and γfi ) 0 in opposite cases Λ ) function of nonideality, which takes into account lateral interaction θ{f}i ) probability of filling of the local fragment {f} by particle i; Ffi ) θ{f}i/mi θ{f}miv ) the probability that the local fragment {f} of size mi is free; symbol miv designates the free surface region of the size mi, in which the adsorption of molecule with orientation i is possible θ{f}{g}in ) probability that particle i on the local fragment {f} and particle n on the local fragment {g} are adjacent, here 1 e n e s ai

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