Adsorption in Pores with Heterogeneous Surfaces - American

---

Langmuir 1993,9, 2652-2660

2652

Adsorption in Pores with Heterogeneous Surfaces Yurii K. Tovbin' and Evgenii V. Votyakov Karpov Institute of Physical Chemistry, Moscow, Russia Received September 3,1992. In Final Form: February 1 6 , 1 9 9 9 Adsorption in porous sorbentsis described by a lattice-gasmodel consideringlateral interactionsbetween adsorbed molecules and using a quasichemical approximation. Chemical and structural imperfections of pore walls are taken into account. Surface composition and the occurrence of different adsorption centers are included by introducing certain distribution functions. Homogeneous, stepped, and rough walla of pores are considered. The self-consistentdescriptionof equilibrium(the local occupations of the different adsorptioncenters, isotherms,and isosteric heata of adsorption) and kinetic characteristics(the diffusion coefficientsof the adsorbed molecules) can be performed with the present model. Isotherms as well as a dependence of the isosteric heat of adsorption and the diffusion coefficient on concentration are investigated,and a relationbetween these generatedtheoreticalcurves and the correspondingexperimental data is briefly discussed. In addition, qualitative description of SFe isotherms on controlled-poreglasses is presented.

Introduction Adsorption processesin porous solids are widely applied to the separation and purification of gas mixtures. Problems in the theoretical description of the adsorption processes are related to the real properties of a sorptive sorbent Consideration of the real atomicmolecular structure of pore walls and ita effect on the equilibrium and kinetic characteristics of adsorption are among this problems. Most theoretical investigations concerning the equilibrium processes in pores have been carried out in pores with homogenous walls. Such studies are exemplified by the Brunauer-Emmett-Teller and Frenkel-Halsey-Hill modelsFJ0 lattice-gas mode1s,l1J3as well as Monte Carlo and molecular dynamics numerical methods.1k16 It is well-known that the walls of the existing adsorbents are heterogeneous; therefore, the description of the adsorption processes, using the above homogenous models, is ineffective. The usual molecular parameters of the adsorption systems under consideration (moleculemolecule and molecule-wall interaction potentials, Henry constants, monolayer capacity, etc.) are not sufficient to calculate the remaining equilibrium (e.g., isosteric heat of

* Abstract published in Advance ACS Abstracts, August 15,1993.

(1)Barrer, R. M. Diffueion in and through Solids; University Press: Cambridge, 1941. (2)Carman, P. C. Flow of Gases through Porous Media; Butterworths: London, 1956. (3)Sattertield, C. N. Mass Transfer in Heterogeneous Catalysis; M.I.T. Press: Cambridge, 1970. (4)Chizmadjev,Yu. A,; Markin, V. S.; Tarasevich, M. P.; Chirkov, Yu. G. Macrokinetics of Processes in Porous Media; Nauka: Moscow, 1971. (5)Kheyfeta, L. I.; Neimark, A. V. Multiphase Processes in Porous Media; Chimia: Moscow, 1982. (6)Mason, E. A.; Malinauskas, A. P. Gas Transport in porous media: The Dusty-Gas Model; Elsevier: Amsterdam-Oxford-New York, 1983. (7)Ruthven, D.M. Principle of Adsorptionand AdsorptionProcesses; Butterworths: London, 1984. (8)Yang, R. T. Gas Separation by Adsorption Processes; Butterworths: Boston, MA, 1987. (9)Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974. (10)Gregg,S.J.;Sing, K. S. W. Adsorption, Surfoce Area and Porosity, 2nd ed.; Academic Press, Inc.: London and New York, 1982. (11)Lane, J. E.Aust. J. Chem. 1968,21,827. (12)Ono, S.;Kondo, 5. Molecular Theory of Surface Tension in Liquids; Hanbuch der Physik, Springer-Verlag: Berlin-GottingenHeildelberg, 1960; Band X. (13)Tovbin, Yu. K. Kolloid. Zh. 1983,45,707. (14)Nicholson, D.;Parsonage, N. G. Computer Simulations and the StatisticalMechanics ofddsorption; AcademicPreas,Inc.: London, 1982. (15)Brodskaya, E.N.; Piotrovskaya, E. M. Rasplauy 1988,2,29. (16)Bojan, M. J.; Steele, W. A. Surf. Sci. 1988,199,L395.

the adsorption) and kinetic characteristics. Consideration of the surface heterogeneity and interactions between the adsorbed molecules (lateral interactions) is essential for properly estimating the molecular properties of the systems under study. Such a model based on the lattice-gas theory has been developed elsewhere17-lgwhere multilayer adsorption equations are presented. These equations take into account a surface heterogeneity by introducing distribution functions for the centers with differing adsorption capacities. A quasichemical approximation for lateral interactions is applied to the description of the equilibrium molecule distribution. Moreover, multilayer adsorption on heterogeneous surfaces can be studied by numerical methods. Adsorption of argon molecules on stepped graphite surfaces was the only subjected studied by a molecular dynamics technique.16 Computation of the molecule diffusion coefficient in the region above the substrate is another problem in the theory of adsorption processes. The attraction of pore walls carries the molecules to the adsorbed state, where the jump mechanism of molecular diffusion takes place. This mechanism depends on the substrate properties and potentials of the interaction between the molecules themselves and with the substrate. As the centers with a higher adsorption capacity are occupied, the molecules will be located on the other centers with a lower adsorption capacity. Therefore, the diffusion coefficient for the multilayer adsorption depends on concentration of molecules inside the pore. The description of the diffusion process is especially intricate when the pore walls are heterogeneous and lateral interactions take place. To the present time, this problem has been solved for the case of low concentrations,6and some attempts have been made to describe the diffusion on heterogeneous surfaces.2G22 For example, Sladecet alSm take into account the energetic heterogeneity of the solid using the dependence of the isosteric heat of adsorption on surface coverage, and (17)Tovbin, Yu. K. Dokl. Akad. Nauk SSSR 1990,312,918. (18)Tobvin, Yu. K. Theory of Physical Chemistry Proceeaes a t GasSolid Interface; Mir Publiihers: Moscow, and CRC Press, Inc.: Boca Raton, FL, 1991. (19)Tovbin, Yu. K. Dokl. Akad. Nauk SSSR 1990,312,1423. (20)Sladeck, K. J.;Gilliland, E. R.; Baddour, R. F. Ind. Eng. Chem. Fundam. 1974,13,100. (21)Zgrablich,C.; Pereyra, V.; Ponzi, M.; Marchese, J. AIChE J. 1986, 32,1158. (22)Pereyra, V.; Zgrablich, C. Surf. Sei. 1989,209,512.

0743-7463/93/2409-2652$04.00/0 0 1993 American Chemical Society

Adsorption in Pores with Heterogeneous Surfaces Zgrablich et al.21*22 suggest a nonlocal model of adspecies diffusion based on the percolation the0ry.~8 In order to obtain the equilibrium and kinetic characteristics of adsorption, it is necessary to find out the equilibrium distribution of molecules over different adsorption centers. This means that computations of these characteristics should be self-consistent and based on common molecular parameters. Such an approach was also presented in refs 17-19 using the transition state the0ry.2~ The purpose of this work is to illustrate a wide variety of practical possibilities of the lattice-gas model.l7-l9 We computed the diffusion coefficients,isotherms, and isosteric heats of adsorption taking into account a width of a slitlike homogeneous pore, topography of flat heterogeneous walls (two kinds of atoms are considered), and surface structural imperfections (monoatomic steps and columns). Moreover, relation between the generated theoretical curves and the corresponding experimental data is discussed. Also, effects of pore size and geometry on a critical-pointshift in narrow porePZ8areconsidered. Finally, experimental isotherms of SFs on controlled-pore glasses29 are described qualitatively. Model17-19 Just as in the usual lattice-gas model,12we assume that the region above the substrate accessible to gas molecules is divided into a set of sites (adsorption centers), which compose a lattice. No more than one gas molecule is permitted in a site. Sites are classified in the groups and the total number of groups is t . Sites belonging to group q (q-typesites, 1Iq It ) have the same adsorption capacity (molecule-sitebond energy Q,) and the same number of neighboring sites located in the rth coordinationshell (z,(r), coordinationnumber), 1Ir IR , where R is the maximum radius of lateral interactions. There are two kinds of species in the lattice: moleculesA and vacancies V. Their equilibrium distribution over the sites is described by the followingfunctions: 62 (i = A, V) is the fraction of q-type sites occupied by species i , and OqPij(r)(i,j = A, V) is the fraction of the pairs consisting of q- and p-type sites at a distance r from each other and occupied by species i (on a q-type site) and species j (in a p-type site). Then, BqA e", = 1,e,ij(r) = eppBi(r), )+,e + BPpVj(r)= 88, eqPiA(r) 8,"(r) = Bpi for any q, p , and r; e,d'(r) = 8;BpJ at r > R. In order to calculate eqpij(r)the following "reactionwas considered: AA,&) + VV,(r) = AV,(r) VA,&), where ijqP(r)denotes species i on a q-type site and species j on a p-type site. This is an extended quasichemical approachla applied to the pairwise distribution of species. Then, assuming the existence of equilibrium in the "reaction", we have

+ +

+

where 0 = (k~!l')-' and cqp(r)is the parameter of lateral interactions between the molecules located on the q- and p-type sites at a distance r from each other. The plus sign of Q, and CqP(r)denotes attraction. (23) Ambegaokar, V.; Halperin, B. I.; Langer, J. S . Phys. Rev. E 1971, 4,612. (24) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rates Processes; McGraw-Hill. New York, 1941. (25) Fisher, M. E.; Nakanishi, H. J. Chem. Phys. 1981, 75,5857. (26) Nakanishi, H.; Fisher, M. E. J. Chem. Phys. 1983, 78,3279. (27) Taraeona, P.; Marconi, U. M. B.; Evans, R. Mol. Phys. 1987,60, 573. (28) Bruno, E.; Marconi, U. M. B.; Evans, R. Physica 1987,14lA, 187. (29) Keizer, de, A.; Michalski, T.;Findenegg, G. H. Pure Appl. Chem. 1991,63, 1495.

Langmuir, Vol. 9, No. 10,1993 2653 Distribution functions need to be introduced in order to describe the lattice composition and structure. Let f, = N,/N, where N , is the number of q-type sites and N is the totalnumber of sites (C,=ltf,, = l),then the distribution function f,, 1 I q It , describes the lattice composition. The surface structure is defined by the cluster distribution function d(q{m}R),which is the conditional probability of finding a q(m}Rcluster consisting of a central q-type site and R coordination shells around the central site, where the state of each coordination shell is described by a set mqp(r),1Ip It , 1Ir IR , m,(r) is the number of p-type sites in the rth coordination shell of the central q-type site, Cp&n,(r) = z,(r). The sign q(m}Rdenotesthe m,(r) set at 1I p It , 1Ir S R. Clusters with the same type of the central sites but different surroundingsq(mjRbelong to different types, and the number of such types with a central q-type site is u,(R). The normalization relationship Xu ( ~ ) d ( q ( m J=R )1. %he averaged pair distribution function d,(r) is also used to describethe heterogeneouslattice structure. This function is a conditional probability of fiiding a p-type site located at a distance r from a q-type site. The values of d,(r) are calculated as follows:

aq(R)

P-1

The cluster pair distribution function d(qp(X}RIp)is introduced to describe the jumps of molecules. The quantity d(qp{X)Rlp)is a conditional probability of finding a cluster with two central sites of q- and p-types located at a distance p, 1 I p I p*, from each other; p* is the maximum length of a jump. The cluster qp(X)Rlp has R common coordination shells. The location of any site in the rth common coordination shell, 1Ir IR , is defined by the orientation parameter Wr, 1 IWr Ir r , where r r is the total number of orientations in the rth common coordination shell. The quantity Wr is characterized by rl and r2, rl being the distance to the central q- and r2 to the central p-type site, respectively. The state of the rth common coordination shell is defined by a set Xqp,,(w,./p), 1 I9 It , 1 IWr Ir r ; XqPtl(wdp) is the number of tptype sites with the orientation wr. The number of different cluster types with the same central pair qp(p) is denoted as uqp(RIp).The normalizationrelationshipC,,w(a)d(qp{X}Rlp) = 1.

Equations17-19 In order to determine the equilibrium distribution of molecules on the lattice sites, we have to solvethe following system of algebraic equations relative to the unknown quantities e,*, 1 Iq It , at either the fixed P (pressure of molecules in the bulk) or 0 (the total lattice occupation)17918 t

Toubin and Votyakov

2654 Langmuir, Vol. 9, No. 10,1993 exp(@Q,),bqis the preexponential factor, and Qq is defined above. The function S,(r) takes into account (in a quasichemical approximation) lateral interactions of molecules located on the q- and p-type sites at a distance r from each other. Solution of the system (2) at 0 < 0 < 1 gives the adsorption isotherm W). To determine the molecule concentration in the gas phase (e,), no more than one equation from system (2) should be used (the lattice is homogeneous,Q, 0, and all indices q andp are omitted): R

&,P(I- e,) = e,n](l+t,(r)x(r))z(r)

(3)

r=l

Once the equilibrium distribution of molecules is established the isosteric heat of adsorption can be computed by the thermodynamic definition

Q(e) = -(d h(P)/dB), (4) The diffusion coefficient was derivedl8 considering the direct and oppositejumps of a molecule from a q-type site to a p-type site at a distance p, 1 Ip Ip*. The sites of the lattice are places for molecule localization;therefore acertain amount of activation energy is necessary to jump. The transition state theory enables one to estimate the direct and opposite jump rates, and a difference between the values of these rates gives the diffusion coefficient. It is expressed as follows1g ”*

t

t

where Z,(p) is thenumber of paths in the diffusion direction for a moleculelocatedon a q-type site and jumping through a distance p and Vqp(p) is the rate of the elementary jumps from a q-type site to a p-type site at a distance p R

*I

t

under consideration. In order to find out this potential relief, a knowledge of both the adsorbent structure and all the interaction potentials is essential. We carried out all the computations for model systems to illustrate the main properties as a function of concentration. In these computations the following values were used the lattice symmetry is a simple cubic, R = p* = 1 (indices r and p are omitted below),6, = l?, = 1,tqp*= eqp/2,eqp = E = 2.10. The case of p* > 1was considered in ref 30. The parameter E, was defined using the following equation: E, = E11 + AE,, where E11 is the activation energy for the jumps over the homogeneous region (first-type sites) and AE,, is its change when the jumps take place over q- andp-type sites. Expanding AE, into one-site contributions AEqp = AE,‘ + AEp and assuming that AE,‘ = a(Q, - 81)and AEp = (a - l)(Qp - 811, where a is the parameter of Brensted-Polyany’s type, we have E, = En + a(Qq- 41) (a - l)(Qp - Q1). Such a representation gives selfconsistent changes of E,, Qq, and Q p The computations use the following values: E11 = 1.2Q1and a = 1.1. All the energetic parameters are given in kJ/mol. The distribution functions were constructed by meam of a specialcomputer program and will not be presented below (instead, specific lattices will be shown). Unless otherwisespecified,all the computations use a slitlike pore having a width x (in units of monolayer thickness) and periodic boundary conditions in other directions. The diffusion coefficients are in the same units as p2RqP(i.e. d2/t,d is the monolayer thickness). Used as temperature is the reduced parameter T = (Be)J (BE), where is the critical parameter of the bulk fluid, whose state is described by eq 3. The quasichemical approximation at R = 1gives (Be), = 2 h(z/(z - 2)), and the saturation pressure, when (Be) 1: $,’ = exp(-~Be/2).~l For T I 1, the van der Waals loops can be present on the adsorption isotherms. We established the of any loop section by Maxwell’s rule applied to the isotherm plotted in the 0 - ln(P) coordinates.

+

Adsorption on Flat Heterogeneous Surfaces It is known that application of a lattice gas model developed for homogeneous surfaces results in steplike isotherms at a temperature lower than critical?JO*32 Isotherms on real surfaces are not steplike due their heterogeneousnature. For example,this fact was observed e~perimentally3~ and the steplike isothermswere obtained exp[B(e,/*(r,) + ~,/*(r,) - E ~ ~ % ~ N I with increasingsurface h0mogeneity,~3Also, it was shown where Kqp(p)’sare the rate constants for the jumps from theoretically3*using the lattice-gas model that the mula q-type site to a p-type site at a distance p, related with tilayer adsorption isotherms on patchwise heterogeneous each other by Kqp(p)/Kqp(p)= ap/aq. A conventional surfaces with an exponential distribution function for representation is Kqp(p) = Rqp(p) exp(-BE,(p)), where adsorption centers are distinguished by depressed steps. Rqp(p) and EqP(p) are the preexponential factor and Let a surface consist of some kind of atoms interacting activation energy, respectively. eqPAi*(r)and eqpAi(r)are with adsorbate molecules by means of a Lennard-Jones the interaction parameters for molecules A located on type potential and t’ is the number of atoms kinds. Each q-type sites in the transition and the ground states, surface atom is an adsorption center for molecules in the respectively, with species i located on p-type sites at a first layer. Surface composition is described by the distance p from molecules A in the ground state; tqpa*(r) distribution function f d denoting the fraction of q-kind = eqp*(r),eqpAV*(r)= cqpAV(r)3 0,EqpM(r) cqp(r)for any atoms; Cq=lt’fql= 1. The region above the substrate is q, P, and r. divided into x layers, and the number of sites in each layer The function Tqp(p) is estimated by means of the is equal to that of surface atoms. Here, x is defined by the following equation: condition (8, - eg)/t9, < w, where 0, is calculated by eq 3, 8, is the concentration of molecules in the last layer x , and w is prefixed criterion. The computations showed that x equal to 10 is sufficient at w = 10s. A molecule in layer Differentiation of eqs 2 is necessary when we use eqs 4 and 6. The values of the parameters in eqs 2-6 (the distribution functions, bq,Qq, Rqp(r),Eqp(r),Eqp*(r),and Eqp(r))can be obtained from a potential relief of the adsorption system

(30) Tovbin, Yu. K.; Votyakov, E. V. Zh.Fiz. Chem., in press. (31) Hill, T. L. StatisticalMechaniccr;McGraw-Hill: New York,1956. (32) Tovbin, Yu. K.; Votyakov, E. V. Zh. Fiz. Khim. 1992,66, 1597. Smith,W.R.J. Phy.9. Chem. 1963, (33) Polley, M. H.; Shaeffer, W. D.; 57, 469. (34) Champion, W. M.; Halsey, G. D. J. Am. Chem. SOC.1964,76,974.

Adsorption in Pores with Heterogeneous Surfaces

Langmuir, Vol. 9, No.10, 1993 2656

k above a q-kind atom interacts with this atom and its neighbors. Hence, to describe the surface topography, a distribution function is essential. It is of the same type as the above function d(q{m]R)but related to the surface atoms only. The parameter d’(q{m’]R’)expresses the probability of finding a cluster q{m’)R’ consisting of a central q-kind atom and its mlqP(r)neighboring atoms of p-kind (1I p I t’) in each rth coordination shell (1I r IR’). Let uql be the number of the clusters with the same central q-kind atom but the different surrounds q{m’]R’, then C,;d’(q(m’]R’) = 1. Here R’ is the maximum radius of the interaction between the molecules and the surface atoms. The Lennard-Jones potential is short-ranged, therefore the computations used R‘ = 6. The molecule energy in a layer k (1I k Ix ) , above a q{m’JR’cluster is Qk(q{m’)R’)= E,(l(k,O))+ R‘

t‘

whereE,(l(k,r)) is the interaction potential of the molecule with a q-kind atom and Z(k,r) is the distance between the molecule and the atom, depending on k and r. In this equation, the interaction is considered for surface atoms only; therefore eq 7 represents an integrated potential of type 4-10.9In order to take into account the interaction with the atoms of the second,third, and other inner solid’s layers, it is necessary to know the composition and structure of these layers. All the energies Qk(q{m’]R’)are computed by eq 7 for each site [(k)of the lattice (4 = q{m’]R’,1I [ I Cp=lt’uql), and so a set Q&k) is formed. The lattice structure is simple: the sites [(k + 1)and [(k - 1)are over and under the site [(k),respectively, and the innermost cluster [ defines the surroundings of the site [(k) inside layer k. Taking into account these conditions, we constructed the distribution functions. It is possible to limit the total number of site types t = x Cq=lt’uqlby an arbitrary criterion A: the sites [(k)and M k ) are of the same type if l Q ~ ( k-) Qr(k)l I A. The computations used A = 0.056, with t defined above. Then t = C k l l X t k , where tk is the total number of site types in layer k, this quantity decreasing as k increases (clearly, t , = 1). This work considers the surfaces consisting of the two kinds of atoms (t’ = 2). The interaction potential was chosen as follows: Eq(z)= 4e:~[(u~/z)~ - (uq/z)l21,where z is the distance and eql is the depth of the potential characterizing the interaction between the molecules and a q-kind atom relative to that between the molecules themselves. (Note, that uqis a potential parameter here.) If z is counted off in the monolayer thickness d (I = z / d ) , and if all up = u and d = 2%, then E&) = eqle (224 - 1-12). The computations used this form of the potential with 1 defined by k and r (e.g., 1 = (k2+ 1 V 2 for any k and r = 1). Three surface structures were analyzed: patchwise, chaotic, and regular (Figure 1). The regular surface looks like a “chessboard”at f1’ = f2’ (Figure la), Le., each-first kind atom is surrounded by the second-kind atoms, and vice versa. When fi’ is not equal to fd, the regular surface consistsof two patterns, a “chessboard”and a homogeneous region arising from the other kind sites. The patchwise surface consists of two distinct homogeneous regions (Figure lb), and the chaotic surface can be represented as the ideal mixed “solution” of two kinds of atoms (Figure IC). In the latter case, the clusters with differing in their f i t coordination shells were only considered. Summation over r L 2 in eq 7 was made with z’(r)fp’instead of mqp‘-

a

L 8

C

+++-e--+

Figure 1. Clusters of different type on regular (a),patchwise (b), and chaotic (c) surfaces. Empty and fiied circles denote first- and second-kind atoms, respectively.

0.2

PIP.

1.0

Figure 2. Multilayer adsorption isotherms on flat regular (I), chaotic (2),and patchwise (3-7) surfaces. €1‘ = 3.26,at = 1.63; fi’ = 0.5 (1-3,6,7), 0.15 (4),0.86 (5);T 0.6 (1-5), 0.45 (6),0.8 (7).Curves are shifted along the PIPOaxis by 0.4 of this quantity.

(r)d(q{m’]R’),where z’(r) is the coordination number of the rth shell. The multilayer adsorption isotherms for different surface structures, compositions, and temperatures are shown in Figure 2. Curve 1for the regular surfaceis similar to isotherms for the homogeneous surfaces (one step is present in each layer). However, the isotherms for the patchwise surfaces (curves 3-7) have two steps in the second and third layers, the remaining parameters being the same. This is caused by the fact that two bond energies, which correspond to two different clusters on the regular surface, are close, while, for the patchwise surfaces, the difference in the bond energies is larger. A more complex shape of the isotherm at the same parameters is observed for the chaotic surface (curve2). In this case, ten different site types are possible in any layer and three values of Q&k) contribute to the isotherm for the second and third layers. Curves 3-5 illustrate the surfacecompositioneffect on the isotherm shape for the patchwise surfaces. When the fraction of the sites of any type increases, the corresponding steps on the isotherms become more pronounced. The same effect is produced by temperature (curves 3, 6, and 7). The present model enables one to take into account the surface structure and composition changes and to quantitatively reveal these changes in the isotherm shape. The larger number of different kinds of surface atoms is responsible for the appearance of depressed steps on the isotherms, and it is likely that two to four kinds of such atoms are sufficient for step elimination. This fact is in agreement with both e ~ p e r i m e n t a and l ~ ~ theoretical”

Toubin and Votyakou

2656 Langmuir, Vol. 9, No. 10,1993

a 0.6 -

C en(a"P)

-3

-4

-2

Figure 4. Occupations of different type sites in homogeneous pore aa a function of pressure and isotherm (dotted line): T = 1.1;x = 10;n = 3; CA 16.8. a

042

51

!

Figure 3. Latticesof pores with homogeneouswalls (a),stepped walls at x = 9 (b), and column walls at x = 10 (c). Each cell comprises lattice sites whose type is shown in the cell. studies. Also, it is shown that the surface topography produces a great effect on the isotherm shape.

Adsorption in Pores When a distance between the pore walls is large, a gas molecule is attracted by one wall only,and the contribution of the opposite wall is negligible. Decreasing a distance between the pore walls results in a larger contribution of the oppositewall. A microporous system is obtained when both walls effect on the molecules. These practically important systemslo are considered below. Firstly we present the results concerning adsorption in a slitlike pore with homogeneous walls. In this case, the lattice structure is simple because the layer's number correspond to the site type (Figure 3a). Let the second pore wall be the mirror image of the first one; therefore the total number of site types is x/2 at even x or ( x + 1)/2 at odd x. The bond energies for a molecule located in any layer are defined by interactions with the two pore walls

Qb = E ( k ) + E(x - k + 1)

(8) where E ( k )is the potential of interaction for the molecules with the homogeneous wall. The simplest case is E(k) = cA/kn, where EA is the arbitrary parameter, and the potentials of type 3-9,4-10, and 6-12 are modeled by n = 3, 4, and 6, respectively. This equation for E ( k ) corresponds only to the attraction term of the potential, and the hard core of the lattice sites takes into account a repulsive term. The occupations for different type sites in the homogeneous pore are shown in Figure 4 as a function of pressure. The bond energies decrease with increasingthe site type (layer number); therefore, the filling process goes from layer to layer. The adsorption isotherms, heats, and diffusion coefficients of this system at the different values of CA, n, and x are plotted in Figure 5 as a function of 8. It is clearly observed that the isotherm bends correspond to maxima on the Q(8) and D(8) curves. This is caused by the fact that the monolayer is completed. A novel finding is a qualitative dependence of the diffusion coefficient on concentration. When the pore fillingis small, most molecules, attracted by the pore wall,

/47

6

ETw 4 _.-

0.6

1

i

Dtei

c i /

I ;1" i

2

8z/2

5

Figure 5. Isotherms (a),heats of adsorption (b), and diffusion coefficients (c) for homogeneous pore walls: 7 = 1.1; x = 10 (1, 2, 5), 6 (31, 4 (4);n = 3 (1-4),4 (6);€A

16.8 (1,3-5), 8.4 (2).

are adsorbed in the first layer. Hence, the jumps of the molecules, by means of which the diffusion takes place, occur also in the first layer. With increasing 8, the attraction of the moleculesthemselves hinders the jumps, and, in addition, the concentration of the empty sites in the first layer decreases. These facta result in a decrease of D(8). As the first layer is completed, the moleculeswill be located in the second layer, whose bond and, hence, jump activation energies are lower. Then, D(8) sharply increases. Further growth of the concentration results in both an increased contribution of the lateral interactions and an decrease in the number of empty sites in the second layer, so that one can observe a fall in D(8) again. Therefore,a maximum on the concentrationplot is present. This picture is possible also in the subsequent layers, and such a behavior is controlled by the radius of the moleculewall potential. For example, the bond energies of the third and fourth layers are close, and they are filled almost simultaneously. Therefore, a drop of the D(8) curve in

Adsorption in Pores with Heterogeneous Surfaces

Langmuir, Vol. 9, No. 10, 1993 2667

a

t

:

0.9 Die/

Figure 6. Isotherms (a),heats of adsorption (b), and diffusion coefficients (c) for pore walh with patchwise (1-3),chaotic (4), and regular (5)topography: x = 6; €11= 2.8, €2' = 1.4; T = 1.1;f1' = 0.0(6), 0.15 (l),0.5 (2, 4,5),0.85 (3),1.0 (7). these layers is not so fast as in the first or the second layers. When the filling of a pore is close to completion, an increase of D(8) curve is evident. This is caused by the fact that the number of empty sites at high concentrations is small, and further increase in 8 requires agreat pressure rise, Le., the value of dPld8 is large. Hence, the function T,,@),which contains dPld8 (see eq 61, increases to cause a D(8) growth. The following results illustrate the effect of flat heterogeneous surfaces on the concentration dependences under study (Figure 6). Here, the second pore wall is considered as a mirror image of the first wall; therefore bond energies can be calculated by means of eq 8, where each term is defied by eq 7. At these values of the parameters the main effect is observed in the first layer only, and the behavior of the concentration dependence in the other layers is similar to that of the above discussed homogeneous pores. The parameters of curves 6 and 7 correspond to two different homogeneous walls, and the surface compositions of the remaining curves lie in the intermediate region. A change in the surface composition at the surface topography under discussion (curves 1-3) gives different contributions from particular local surface regionsto the entire isotherm, isostericheat of adsorption, and diffusion coefficient. The centers with the highest bond energies are filled first, followedby the same process for the centers with lower bond energies. This gives rise to both isotherm bends and the appearance of maxima on Q(8) and D(8) (curves 1-31, depending on surface composition. Thus, the concentration behavior in the case of monolayerfilling can be also observed in each homogeneous surface region. Curves 2, 4, and 5 illustrate the effect of surface topography on the concentration dependence, the remaining parameters being the same. Curve 5 is of the same type as curves 6 and 7, because at f1' = fz' the regular surface (a "chessboard" pattern) can be represented as a homogeneoussurfacewitheffective(modified)parameters. The curves for the chaotic surfaces (curve 4) are averaged over the contributions from all the clustersshown in Figure ICand a relation between the fractions of different cluster

Figure 7. Occupationsof differenttypesitesin pore with stepped walls as a function of pressure and isotherm (dotted line): x = 9, L = 10; T = 1.1; Q1 = 12.6, Q2 = 4.2, Qa = Qd = (81+ Q2)/2, remaining 8;s = 0. 810 coincides with O1l. types defines the behavior of the curves under consideration. Therefore, a knowledge of the surface topography is important in the analysis of adsorption characteristics. The last two examples represent nonflat pore walls modeling a roughness of real surfaces. In these computations, bond energies were set directly for each site type. Figure 3b shows the lattice of a pore with monoatomic steps directed normal to the plane of this figure. The bond energies of the first-type (at the step base, QI),the second-type (on the terrace edge, 821, the third-type (betweenthe first-type sites, Q3), and the fourth-type sites (between the second-type sites, Q4) are nonzero and for the sake of simplicity it is taken that Q 3 = Q 4 = (Q1 Q2)/2. The occupations of the sites are plotted in Figure 7. In this case, the picture of pore filling differs from that of the homogeneous situation. For example, at the same > although Q3 = Q4, and OsA> OsA,while pressure QS = 8 6 . This is attributed to the lateral attractions of the molecules located on the neighboring sites: the moleculeson the first-type sites encouragethe occupation of the third- and fifth-type sites, while the molecules on the second-type sites do the same with respect to fourthand sixth-type sites. Due to the fact that Q 1 > Qz,we have elA > OzA and, hence, 83A> 84Aand OsA > OsA. The calculated isotherms, heats of adsorption, and diffusion coefficients for the stepped pore walls are shown in Figure 8. Curves 2,4, and 5 describe the effect of the bond energy differencesand curves 1-3 illustrate the effect of the step dimension L. The roughness of the pore walls is revealed in the first and in the second layers. Therefore, a shift of the features related to the monolayer filling (Q(8) and D(8) maxima, as well as the isotherm bend) can be observed along the concentration axis. The greatest effect of surface heterogeneity on the concentrationdependencestakes place with the adsorption occurring on the column pore walls. Figure 3c presents schematically such a slitlike pore with one monoatomic column on a (m X m) square. The second pore wall is a mirror image of the first one, and x is the distance between the homogeneous wall regions. In this case, the bond energies of the first- and second-type (on the column top), the third-type (on the column central part), the fourthtype (at the column base), and the seventh-type sites (on the homogeneous wall regions) are nonzero. The occupations of the sites are plotted in Figure 9. One can observe a complex picture of pore filling due to the joint effect of structure imperfections and lateral interactions. In the case of attraction of the homogeneous wall regions alone (Figure 9a), the fourth- and the seventh-type sites will be occupied first, being larger than at the same pressure, although Q, = Q4. This is caused by the fact that the number of the nearest neighbors of seventh-type sites is larger than that of the fourth-type sites, and, hence,

+

Toubin and Votyakou

2668 Langmuir, Vol. 9, No. 10,1993

Figure 8. Isotherms (a),heats of adsorption (b),and diffusion coefficients (c) for stepped pore walls: x = 9; T 1.1; L = 6 (l), 10 (2,4, 5), 20 (3);Qi 21.0 (1-4), 12.6 (5); Qz 4.2 (1-3,5), 12.6 (4), Qs = Q4 = (Q1 + Q d 2 , remaining 8;s = 0. 1.0

3,. e5.ei -4

-3

-2

-4 -3 -2 - 1 tn(@P) Figure 9. Occupations of differenttype sites in pore with column walls as a function of pressure and isotherm (dotted line): x = 10; m = 5; T = 1.1. For (a): Q4 = Q, = 8.4, remaining 8,'s = 0; Oe, Ole, and coincide with ea. For (b): Q1 = QZ= 8.4, remaining Q;s = 0; Ol0 and ell coincide with ee.

the larger number of the sites encourage the occupation of the seventh-type sites than that of the fourth-type site. As the surface is completed (e@,@ = 11,the occupations of the other type sites appear. A different picture in the mechanism of the pore fiiing is observed when the molecules are attracted by the column top only, and the bond energies of other type sites are zero (Figure 9b). In this case, the first- and the second-type sites are filled first. A difference between elAand e$ at the same pressure is small, because their bond energies and the number of

Figure 10. Isotherms (a),heats of admrption (b), and diffusion coefficients (c) for column pore: x = 10; m = 5; 7 = 1.1. All Q Is = 0 except for specified cases. For (1): 81- Qz = 21.0. For QI= 8 2 = 21.0, Qa = 8 4 = 8.4. For (3): 81 Q2 = QS = 8 4 = 8, = 21.0. For (4): Q1 = Qz = Qa = 8.4, Q 4 = Q, = 21.0. For (5): Q4 = Q, 21.0. Parmetere of curve 6 are same as those of m e 2, except for m = 9.

-

(4):

the nearest neighbors are equal. However, this difference is present, because the occupations of the third-, fifth-, and sixth-type sites are dissimilar. These sites are the nearest neighbors of the first- and the second-type sites. Due to the fact that ObA and e& is larger than ea*, el* > &A. Moreover, Os*, h O A , and ellA are similar, and e+, 0 8 < e#. This indicates that the central pore part is filled first, and "molecular slab" (Figure 9b) is formed. Thus, the contribution of both the surface hetarogeneity and lateral interactions to the mechanism of pore filling is evident. Figure 10 shows isotherms, heats of adsorption, and diffusion coefficients for the column pore walls. The computations were made for the following typical cases: (A) the columns attract molecules stronger than do the homogeneous wall regions (81and Q2 are higher than Q4 and Q,, curves 1 and 2); (B)the bond energies for the homogeneous wall regions are identical to those for the columns (curve 3); (C) the homogeneous wall regions attract molecules stronger than do the columns (curves 4 and 6). The parameters of curve 6 are equal to those of curve 2 except that the column concentration is lower. A change in the parameters affects a region, where pore ftakes place. Therefore, the various behavior of curves is observed in Figure 10. The important question is to what extent the generated theoretical curves resemble those observed in experiment. Firstly, we analyze the dependenceof the heat of adsorption on concentration. Curve 6 in Figure 6b represents qualitatively the experimental isosteric heat for nitrogen molecules adsorbed on graphitized carbon black at a total filling of 0.2 to 2 monolayers.% The fact that m e 6 for heterogeneous surfaces resembles curves 6 and 7 for homogeneous surfaces indicates that the shape of the (35) Dormant, L.M.;Adamson, A. 28,459.

W.J. Colloid Interface Sci. 1968,

Adsorption in Pores with Heterogeneous Surfaces concentrational dependences cannot always reflect the surfacestates. It is likely that many experimental systems consideredtraditionally as consistingof homogeneouspore walls do consist of heterogeneous walls. (Even for single crystals it is necessary to develop special methods for producing homogeneous surfaces,the problem for porous sorbents being especially intricate.) Qualitative agreement between the theoretical Q(0) and experimental isosteric heats can also be demonstrated on curves 2 and 3 (Figure 8b) for stepped walls, on the one hand, and experimental data for argon adsorbed on Spheron-6(T = lo00 and 1500OC)= as well as for n-hexane and n-heptane adsorbed on graphitized carbon b l a ~ k at 3~ a filling of 0.2 to 3 monolayers, on the other. The differences observed at a filling lower than 0.2 monolayer is likely to relate to alarger number of the active adsorption centers (we considered the monoatomic step only). The theoretical curves of diffusion coefficients are also in qualitative agreement with the corresponding experimental data: (1)D(0) decreases at 0 < 8 I0.4 (see, for example, the diffusion of Ar, Kr, and Xe molecules in 5A zeolites38);(2) D(0) increases at a high volume concentration of Xe molecules in 5A zeolites;3s(3) in the case of a monolayer filling one can observe a maximum for the diffusion of C3H6 (T = 0,25 and 50 0C)39and C4H10 (T = 30 and 41.7 oC)40in Graphen, as well as for CF2C12 and SO2 in Linde Silica.41 The presence of maximum is also confirmed by theoretical computations of Guilliland et al.42and Tamon et al.43 Critical Point Shift and Description of SFs Experimental Isotherms on Controlled-Pore Glasses Modern theories for the critical behavior of fluids in narrow lit like^^^^^ and cylindrical27t28 pores predict that a pore critical temperature (T,) is lower than the critical temperature of the bulk fluid (Tc).As shown in the above works a critical point shift is affected both by pore size and by molecule interaction with the pore walls. Briefly, this phenomenon can be rationalized by the argument that a fluid in narrow pores is intermediate between a three-dimensional state (when the distance between pore walls is infinite) and a state with the lower dimensionality (two-dimensionalstate for slitlike pores at x = 1or onedimensional state for cylindrical pores at 28 = 1,il is reduced radius). Therefore,the criticalpoint shift appears for fluids in narrow pores. In addition, the pore space is inhomogeneous due to the pore wall attraction, and as shown elsewhere4 using a mean-field approach,the critical temperature of inhomogeneous lattices is lower than that of their counterparts. We computed a critical point shift from the present model in a quasichemicalapproximation for both slitlike ores at different x's and for cylindricalpores at different 's, considering the pore walls as homogeneous. Lattice structures for the cylindrical pores were simple cubic as

ii

(36) Beebe, R. A.; Young, D. M. J. Phys. Chem. 1954,58,93. (37) Kiwlev,A. V. InProceedingsof the SecondInternationalCongress on Surface Actiuity, II; Butterwortlw London, 1957; p 168. (38) Ruthven, D. M.; Derrah, R. 1. J. Chem. SOC.,Faraday Trans. 1 1975, 71, 2031. (39) Horiguchi, Y.; Hudgins, R. R.;Silveston,P. L. Can.J. Chem. Eng. 1971,49,76; (40) Ross, J. W.; Good, R. J. J . Phys. Chem. 1956,60, 1167. (41) Carman, P. C.; Rnd, F. A. Proc. R. SOC.London 1951,20lA, 38. (42) Gdilnnd, E. R.; Baddour, R. F.; Rueael, J. L. AIChE J. 1958,4,

"". WI

(43) Tamon, H.; Okazaki, M.; Toel, R. AIChE J . 1981,27,271. (4.4) Tovbin, Yu. K. Dokl. Akad. Nauk SSSR 1981,260,679.

Langmuir, Vol. 9, No.10, 1993 2669

Figure 11. Criticaltemperaturesfor fluids in homogeneous pores as a function of distance between homogeneous pore walls: slitlike (1, 2) and cylindrical (3, 4) pores. Curves 2 was computed by means of the mean-field approximation in ref 26. Fluid-wall interaction is zero (1-3) and corresponds to the SFdCPG-10 system (4, see text).

in ref 45. Distribution function f ,was determined using quanta of the sites in each cylindrical layer from ref 45, and all d(q{m)R)'swere equal to unity. The m&) at R = 1are as follows (index r is omitted): mkk = 4, mkk+l = 1and mkk-1 = 1. The total number of site types is equal to that of cylindrical layers. Unfortunately, such a description of the lattice structure at low R's is crude due to its simplification;thereforethe computationswere made at I? 1 4. Figure 11shows these findings. Curve 2 was computed in a mean-field approximation in ref 26 and is plotted for comparison (Tcis given in the same approximation). The potential of interaction between the pore walls and the fluid (F)for curve 4 corresponds to the interaction between SF6 molecules and a controlled-pore glass (CPG-10). The above system was studied by Keizer et al.,29who obtained the experimental and molecular dynamics simulation isotherms at = 8.19 (in units of UFF). In addition, we computed the SF$CPG-10 adsorption isotherms using the interaction potentials from ref 29 and the lattice structure at t = 7. The values of T,and POwere found in a quasichemicalapproximation, and the site size d is equal to om.Figure 12 demonstrates these results and experimental isotherms.29 The critical point shift for the system under consideration is presented in Figure 11 by the point A (TPJT, = 0.962 at R = 8.19); therefore no condensation takes place at TPJTc = 0.985. Thus, there exists a qualitative agreement between the experimental and our isotherms. Conclusions This paper illustrates the wide practical possibilities of the lattice-gas model17-lgas applied to the description of adsorption isotherms, isosteric heats, and diffusion coefficients. This model takes into account both a surface heterogeneity by constructing the distribution functions for different types of adsorption centers and includes lateral interactions of molecules in a quasichemical approximation. Also, it provides a self-consistentdescription of equilibrium and kinetic characteristics of adsorption, i.e., a set of input molecular parameters applies equally to computations of the characteristicsunder consideration. This was examplified by the consideration of adsorption both on flat heterogeneous surfaces and in slitlike pores with various heterogeneous walls. The existence of steps on the adsorption isothermsat low temperatures was found to depend on surface composition and topography. This (45) Nicholson, D. J. Chem. SOC.,Faraday Trans. 1 1976, 71, 238.

Tovbin and Votyakov

2660 Langmuir, Vol. 9, No. 10,1993

U

I

""i

Figure 12. Sorption isotherms of SFe molecules adsorbed on controlled-pore glasses at Tpc/T, = 0.985 (l), 0.920 (2), 0.857 (3); computation (a) and experimentaB(b).

is consistent with experiment.10133 In contrary to the theoretical results reported by Champion and Halseya who obtained depression by introducing a large number of adsorption centers described by the exponential distribution function, we found that two to four kinds of surface atoms are sufficient to depress the steps. Adsorption and diffusion in pores at temperatures above the critical values were studied to reveal the effects of the following physical factors: (1)width of a slitlike homogeneous pore; (2) topography of flat heterogeneous walls consisting of two kinds of atoms; (3) structural imper-

fections of pore walls (monoatomic steps and columns). The computations showed that the model is vary flexible and can be utilized to describe the adsorption processes in micropores, taking into account the atomic-molecular structure of the adsorbent used and varying the above factors. A critical point shift in narrow poressm was also computed for slitlike and cylindrical pores with different distances between the pore walls. Finally, the experimental isotherms of the SFe molecules adsorbed on controlled-pore glassesmwere qualitatively described by our model.