J. CHEM. SOC. FARADAY TRANS., 1994, 90(8), 1153-1156 Adsorption in Energetically Heterogeneous Slit-like Pores : Comparison of Density Functional Theory and Computer Simulations G. Chrniel, L. Lajtar and S. Sok&owski*t Computer Laboratory, Faculty of Chemistry, MSC University,20031 Lublin, Poland A. Patrykiejew Department of Chemical Physics, Faculty of Chemistry,MCS University,20031 Lublin, Poland We present a comparison of density functional theory results with Monte Carlo calculations of adsorption in narrow, slit-like pores with energetically heterogeneous walls. The calculations have been carried out assuming Gaussian distribution of the adsorption energy and random topography of adsorbing sites. We have found a reasonable agreement between theoretical predictions and computer simulations.A system of spherical particles inside a slit formed by a pair of parallel walls in equilibrium with a bulk fluid is one of the simplest of the models of confined fluids that have been inten- sively studied in recent years. lL1z The investigated models have included slits with simple hard walls,'*2*'0.' ' slits with attractive, but energetically uniform walls, interacting via a potential, which is a function of the wall-particle distan~e~-~,'and slits with crystalline walls built of regularly arranged atom^.^.^,^.'^ All these three types of systems have been investigated by means of computer simula-tions,2,3,5-11,14 and the first two models have also been studied by using theoretical approaches, based either on inte- gral equation^'*^^'^*'^ or on application of density functional theory.3*4,93 ' It is well known that the energetic heterogeneity of adsorb- ing surfaces plays a significant role in adsorption.Classical studies concerning that problem have been usually based on the so-called 'integral adsorption eq~ation'.'~ Unfortunately, this equation did not always provide reliable information concerning the real influence of energetic heterogeneity on the thermodynamics of adsorbed films and for this reason development of new approaches is de~irable.l~-~ Recently, an increasing number of papers have reported computer modelling of gas adsorption on non-uniform adsorb-ents.9.16.19-24 In our previous works "*' we have initiated investigations of adsorption on energetically non-uniform surfaces by using a density functional method.The theoretical results obtained, however, have not been tested against computer simulations. Therefore, the primary goal of this paper is to compare the results of computer simulations with the theoretical predic- tions of density functional theory. Calculations We considered a simple fluid, i.e. the interaction energy between the molecules is pairwise additive and depends on the scalar distance between the molecules. The cut Lennard- Jones (12,6) function has been assumed as the pair potential u(r)= {;[(u/r)l2 -(~/r)~];for r < 2.5~ (1); for r < 2.50 We used o as a unit of length and the calculations were carried out for ~T/E0.8 and 1.5.= Each pore wall of size XL = 11 and YL= 6J3 has been built of hexagonally arranged atoms placed at a distance c one from another.Periodic boundary conditions in the X and Y directions have been applied. t Also at: HLRZ-KFA Julich, 5160 Julich, Germany. The interaction of adatoms with the adsorbent has been calculated by summing the Lennard-Jones (12-6) energies for gas atom-solid atom interactions The energy parameter, characterizing a single surface atom, ,~,E~~ has been drawn according to a Gaussian distribution X(E,s)* (4) In each case we used kT/E,, = 0.8, sg = 50 and kT/A = 0.7. Random topography (cf ref. 13), i.e. quite random assignment of the values of cgs to the wall atoms, has been assumed and the final adsorption results have been evaluated by taking averages over six random configurations. We used the GCEMC implementation due to Adams." Each step of this method consists of an attempted movement of a randomly chosen particle followed by either an at-tempted deletion of a randomly chosen particle or an attempted creation of a particle in a randomly chosen posi- tion inside the pore. We report the results us.relative activity, a, defined as a = exp[,u/kT -3 in A/a] (5) where A is the DeBroglie wavelength. Usually, more than 2 x lo6 initial steps were discarded and the final results have been obtained taking averages over at least 3 x lo6 sub-sequent steps. The local densities evaluated from Monte Carlo simula- tions have been compared with the results of density func- tional theory.This theory is based on a variational principle for the Helmholtz energy, F, and its practical implementation requires the construction of an approximate expression for F[p(r)], being a functional of the local density. Among various schemes proposed in the literature,26 one of most widely used is the so-called weighted density approximation proposed by Tarazona et ul.27,28 The approach starts from the definition of the grand poten- tial, Sl i2 = F + idrp(r)[v(r) -p] (6) The functional F is divided into two parts, representing the contributions due to repulsive, F, ,and attractive forces, FA, between the molecules. The former are modelled by hard spheres with suitably chosen diameter d and the latter are treated in the mean-field approximation.The Helmholtz- energy contribution, F,, is calculated by introducing a smoothed density function, F(r) P(r)= dr'p(r')wC I r -r' I, p(r)l (7)s where w is a weight function W(I, P) = wow + Wl(I)P + W2(I)P2 (8) and the coefficients wo, w1 and w2can be found in ref. 27. The Helmholtz energy takes the form F = s drp(r)(kT[ln p(r)A3 -11 +f[P(r)]} dr dr'p(r)p(r')u,( I r -r' I ) (9) where The Helmholtz-energy density of hard spheres,f, has been calculated from the Carnahan-Starling equation*' f(P)/kT= v(4 -3vMl -vI2 (11) where q = nd3p/6 and the hard-sphere diameter has been assumed to be equal to cr. The equilibrium density profile minimizes the grand poten- tial Q, thus the local density is evaluated from the condition We recall also that the excess adsorption isotherm, r, is defined as = dr[P(r) -Pbl (13)s where Pb is the density of a bulk fluid at a given temperature and chemical potential.The dependence of the bulk gas density on chemical potential (activity) was calculated by using the bulk counterpart of the theory. The minimum condition [eqn. (12)] can be rewritten in the following form 0 = In &)A3 +fC&)I + u(rl)-P (14) which is an integral equation for the density profile. This equation has been solved by using an iterational procedure with the grid size along each axis equal to 0.05. Because the local density is a function of three variables, the solution of the density profile equation is rather time-consuming. Fortu- nately, the form of eqn.(14) makes possible the application of a parallel algorithm for calculating subsequent iterational solutions on parallel computers. This method was used in our work. Results and Discussion Fig. 1 compares adsorption isotherms r, evaluated from density functional theory and from computer simulations. Density functional theory predicts the existence of a capillary condensation loop with the adsorption state at Pb = 0.001 95 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 0 0.000 0.002 0.004 Pb Fig. 1 Adsorption isotherms at slit-like pores. The points (diamonds) are the results of computer simulations and the lines have been obtained from density functional calculations.The calculations were carried out for H = 5, kT/&= kT/&,,= 0.8. and the desorption step at Pb = O.OOO91. The equilibrium capillary condensation point is located at Pb = 0.001 53. Except for the phase-transition region, the agreement between theory and computer simulation is rather satisfac- tory. In particular, we observe a very good coincidence between computer simulations and theory at higher bulk gas densities. The Monte Carlo simulations predict no first-order capil- lary condensation for this system. According to the per- formed simulations, the filling of the pore with a dense, condensed liquid is smooth and reversible. A sharp, but con- tinuous adsorption step is located at Pb % 0.014. Note, however, that in the case of the pore with walls built of iden- tical atoms, interacting via a potential [eqn.(3)] with ~T/E~,= 0.768, the phenomenon of capillary condensation is evident and the adsorption jump is at Pb = 0.00172.30 Instantaneously, for pore walls built of identical atoms, density functional theory predicts capillary condensation at Pb = 0.00208, the adsorption step at Pb = 0.00293 and the desorption step at Pb = 0.001 27.30 Thus, the heterogeneity causes the hysteresis loop to become narrower. The observed discrepancy between theory and computer simulation is a direct consequence of neglect of statistical fluctuations by mean-field theory. It is known3' that in systems with random impurities any phase transition is smeared and rounded.Therefore, the observed finite slope of the adsorption isotherm is a manifestation of this effect. The structure of the adsorbed fluid has been characterized by one-particle distribution functions, p(r). Representative examples of the local densities, evaluated from density func- tional theory and from computer simulations are displayed in Fig. 2 and 3. Fig. 2 shows some selected local densities obtained from density functional theory and averaged over the y coordinate, (p(x, y, z)),. We see that the density profiles inside the pore with H = 5, filled with a gas-like phase, exhibit only one single maximum at each of the walls, and the capillary con- densation is connected with the filling of the two inner layers inside the pore.We can also realize that energetical hetero- geneity plays a more important role at lower bulk gas den- sities (activities). Indeed, the profile displayed in Fig. 2(a) indicates large variations in the density along the surface. The relative insensitivity of the liquid-like profile presented in Fig. 2(b) on energetical heterogeneity means that for a dense adsorbed film averaging along one direction is sufficient for smoothing the density distribution. Obviously, changes in the J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Fig. 2 Density profiles averaged over y coordinates, (p(x, y, z)),, resulting from density functional theory. The calculations were per- formed for H = 5, kT/c = kT/Egs= 0.8 and for pb = 0.001 (a) and 0.002(b).full density distribution n(x, y, z) for the liquid-like phase still remain correlated with changes in the adsorbing potential; in particular, in both Fig. 2(a) and (b) we can distinguish sub- sequent minima and maxima, connected with the crystalline arrangement of the solid wall atoms. Fig. 3(a)-(c)show a comparison of the Monte Carlo and density functional theory profiles p,(z) = (n(x, y, z)),,,, where (0 --),,,denotes an unweighted average over the coor- dinates x and y parallel to the pore walls. The calculations have been performed for two pore widths and for two strengths of fluid-fluid interaction. In each case the topo- graphical model of the surface is the same. In comparison with computer simulations, density functional theory leads to a slightly less pronounced oscillatory character of the density profiles.In all cases, however, the positions of the subsequent local density minima and maxima are well preserved. In general, we can state that even for a quite dense adsorbed film, the agreement between the density functional calcu- lations and Monte Carlo results is satisfactory. Fig. 4 compares the Monte Carlo density profiles p,(z) averaged over six different random topographical distribu- tions of energies, cgs, with the profiles obtained for a model pore with all wall atoms being identical. The differences between density profiles, obtained for the liquid like as well as for the gas-like phases, are really small. This indicates that adsorption on random heterogeneous surfaces can be described by considering a suitably defined model homoge- neous surface. This conclusion remains in agreement with the results of our previous workI5 on the application of the 1155 2 h 3c: 1 0 0 1 2 3 4 5 z .2i ;I c 0 2 4 6 a Fig. 3 Density profiles, p,(z). The points denote the results of Monte Carlo simulations and the lines have been evaluated from density functional calculations. The subsequent parts have been evaluated for: (a)H = 5, Tk/E = 0.8 and Pb = 0.002; (b) H = 5, Tk/ E = 1.5 and pb = 0.535; (c) H = 8, Tk/E = 0.8 and pb = 0.004. In each case kT/c, = 0.8. density functional method to description of monolayer gas adsorption on heterogeneous surfaces, in which we have emphasized that adsorption on random heterogeneous sur- faces can be modelled by an effective 'average' homogeneous surface.In this paper we have demonstrated that the classical density functional method can be successfully applied to the description of adsorption in pores with energetically non- uniform walls. Therefore, we hope that further studies on the application of density functional theory, including its lattice 1156 31 h-.!3 2' t'i 0 1 2345 L Fig. 4 Comparison of the Monte Carlo density profiles evaluated for the pore with heterogeneous walls (solid lines) and with walls built of identical vertices (dashed lines and points). In the latter case the energy parameter in eqn. (3) was constant and equal to Tk/&,= 0.8.The calculations were performed for Tk/c = 0.8, H = 5 and for pb = 0.0008(lower curves) and 0.004 (upper curves). counterpart, may be very useful in explaining the microscopic structure of adsorbed layers and for the description of surface phase transitions. The paper was supported by KBN under Grant No. 303.077.05. References 1 Y. Zhou and G. Stell, MoZ. Phys., 1989,66,767. 2 S. Sokolowski and J. Fischer, J. Chem. Phys., 1991,93,6787. 3 B. K. Peterson, K. E. Gubbins, G. S. Heffelfinger, U. M. B. Marconi and F. van Swol, J. Chem. Phys., 1988,88,6487. 4 P. C. Ball and R. Evans, Mol. Phys., 1988,63, 159. 5A. Papadopoulou, F. van Swol and U. M. B. Marconi, J. Chem. Phys., 1992, 97, 6942. J. CHEM. SOC. FARADAY TRANS., 1994, VOL.90 6 S. Sarman, J. Chem. Phys., 1990,92,4447. 7 M. Schoen, D. J. Diestler and J. H. Cushman, J. Chem. Phys., 1987,87,5464. 8 A. Delville and S. Sokolowski, J. Phys. Chem., 1993,97,6261. 9 S. Sokolowski, Phys. Rev. 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