J . Chem. SOC., Faraday Trans. I , 1982, 78, 2361-2368 Adsorption from Solutions onto Solid Surfaces Effects of Topography of Heterogeneous Surfaces on Adsorption Isotherms and Heats of Adsorption BY WLADYSLAW RUDZINSKI* AND JOLANTA NARKIEWICZ-MICHALEK Department of Theoretical Chemistry, Institute of Chemistry, Maria Curie-Sklodowska University, 20-03 1 Lublin, Nowotki 12, Poland AND STANISLAW PARTYKA Centre de Recherches de Microcalorimetrie et de Thermochimie du C.N.R.S., 26 rue de 141e, R.I.A., F-13003 Marseille, France Received 7th August, 1981 A model investigation is presented, showing the effects of the topography of heterogeneous solid surfaces on adsorption from binary liquid mixtures onto solid surfaces. Equations are developed for excess adsorption isotherms and heats of immersion, for both patchwise and random surface topography, when the adsorbed solution is a quadratic mixture.Using the developed equations, numerical model calculations are carried out for some typical values of heterogeneity and interaction parameters. It has long been thought that the heterogeneity of a solid surface may play a dominant role in adsorption from liquid mixtures onto real surfaces.’ More than two dozen papers have been published in the last ten years providing either experimental evidence for this fact or a theoretical and numerical analysis of appropriate experimental data. We have referred to them in detail in a previous publication.2 We have shown that the contribution from the surface heterogeneity to the excess adsorption and the excess heat of immersion is comparable with the contribution due to the interactions between the adsorbed molecules.Only a few published papers correctly take into account both these physical factor^.^-^ The mutual interference of these factors raises new questions about the role of surface topography in adsorption from binary liquid mixtures onto heterogeneous surfaces of real solids. To our knowledge, no one has yet investigated this problem either experimentally or theoretically. It is the purpose of this paper to provide a model investigation showing the effect of surface topography on the adsorption isotherms and heats of immersion. In order to show this effect clearly we have confined ourselves to the simple model of adsorption of molecules of equal sizes forming a quadratic mixture in both the surface and the equilibrium bulk phase.THEORETICAL The starting point of our consideration will be the general equation describing adsorption from mixtures of molecules of equal sizes’ 236 12362 ADSORPTION FROM SOLUTION ONTO SOLID SURFACES where x denotes the mole fraction of components A and B in both the surface (s) and the equilibrium bulk phase (b) and y denotes appropriate activity coefficients. For the case of the regular bulk and surface solutions assumed here, the activity coefficients take the following form: = exp (2 [xg)]~) (2) where ap and a, are the mole fractions of the nearest-neighbour adsorption sites in directions parallel (p) and vertical (v) to the surface plane which fulfil the condition (ap + 2a,) = 1.The meaning of the parameter E, is known from the theory of regular bulk sol~tions.~ Further, it will be assumed in this paper that the structure of the solution adjacent to the surface is not perturbed by the presence of the solid. For the purpose of our illustrative numerical calculation, we will further assume that ap = f and a, = a. The constant K can be expressed in the following way? K = exp (s) (4) where Here and E~ are the energies of adsorption of molecules A and B from their gaseous state onto the solid surface, qAS and qss are the molecular partition furlctions of molecules A and B in the adsorbed state, and cAA and EBB are appropriate energies of the A-A and B-B pair interactions. In the case of adsorption from binary liquid mixtures onto heterogeneous solid surfaces, various adsorption sites will exhibit various adsorption energies, E,.It is usually assumed in theories of adsorption of single gases on heterogeneous solid surfaces that the adsorption energy eA or E~ may vary from some minimum (positive) value up to infinity.' If we accept these assumptions, then we have to assume that the adsorption energy E, may vary from - co to + co. Following the results at our previous publication, we assume that the differential distribution of adsorption sites among various values of the adsorption energy, E,, is described by the following quasi-gaussian function : where Y describes the width of the distribution function and E: is the most probable value of E, on the heterogeneous surface. When I + 0, the distribution function [eqn (S)] tends to the Dirac delta distribution B ( E , - E ~ ) , which means that the surface becomes homogeneous and is characterized by the adsorption energy E, = E:. Because of the dispersion of the adsorption energy, &,,the surface mole fraction of component A will be represented by the following average:w. R U D Z I N S K I , J.NARKIEWICZ-MICHALEK AND s. PARTYKA 2363 The form of the energy distribution function in eqn (8) makes an analytical integration in eqn (9) possible:s where xgb is the value of xg) ate, = E:, and Bm is Bernouli's number. We have shown2 that for typical adsorption systems a sufficient accuracy of integration is achieved when only the first term of the sum in eqn (10) is retained. Thus, truncating expansion (10) after the first term we arrive at the following equation for x f i : where D&') is the second derivative of x f ) taken at the point eS = ez.Now let us consider the effect of the topography of a heterogeneous surface. As in the theories of adsorption of single gases on heterogeneous solid surfaces, we shall consider two extreme models of the surface topography. The first is the 'patchwise' model, which assumes that adsorption sites having the same adsorption energy are grouped into patches on the heterogeneous surface. These patches are sufficiently large that statistical thermodynamics can be applied to any of them, but interactions between admolecules on different patches are neglected in calculating the state of the adsorption system. Thus the adsorption system can be considered as a collection of independent subsystems, being in only thermal and material contact.The potential of the average force acting on a molecule adsorbed on a certain patch will depend only on the surface concentration of this patch. Consequently, the surface activity coefficient, yfb, referred to this patch will be given by the expression [ I - x2)(Es)]2 + a, 2 (xg))~) (12) where the subscript p in ygk refers to the patchwise topographical model. The other extreme model of surface topography considered so far is the 'random' model of heterogeneous surfaces. It is assumed in this model that no spatial correlations exist between adsorption sites having the same adsorption energy. In other words adsorption sites having various adsorption energies are distributed on the heterogeneous surface completely at random.Thus any local concentration on such a heterogeneous surface will be the same as the average surface concentration. Consequently the surface activity coefficient 72; should be written in the following form : yk; = exp ( a 5 (xf3~)2 + a, 5 (xg)>2 (13) kT kT where the subscript r in yg! refers to the random surface topography. For this case, the first and second derivatives of xg) take the following explicit form:2364 ADSORPTION FROM SOLUTION ONTO SOLID SURFACES For the patchwise surface topography, we obtain Now let us consider the heat of wetting of a heterogeneous surface by a solution of varying composition. As usual we start by writing an appropriate expression for the heat of wetting of a hypothetical homogeneous surface, Qw, G?, = 4) Q ~ A +xP Q ~ B +P (18) where QwA and QwB are the heats of wetting of the homogeneous surface by the pure liquids A and B.With certain approximations they can be written in the following (19) form2 Q ~ B = M(EB-~~EBB) (20) where M is the total number of the adsorption sites on the homogeneous surface. Further /? is an interaction term, which for the Bragg-Williams approximation takes the following form (21) Q ~ A = M(%-~v&AA) /? = Mcin [a, xg) xfjs) + a, (xg)xf3b) + xfjs) xff) - xff) xf3b))l where &in = E m - (3). The interaction term, /?, describes the change in energy of the pair interactions A-A, B-B and A-B owing to the formation of the solid/solution interface, relative to the situation in an infinite bulk solution.The surface heterogeneity will affect this interaction term, but in different ways, depending on the kind of surface topography present. In the case of surfaces characterized by random surface topography, the interaction term p will have the same form as in eqn (21), except that appropriate local surface concentrations have now to be replaced by their averaged values (23) Pr = P o +Ex (24) /?r = M&in [a, xg! xgi + a, (xfl xf3b) + xgi xff) - xp) @))I. We will now write Pr in the following form: where Po is the value of /3 at E, = E:. Thus, Ex is the contribution to the interaction term owing to the dispersion of the adsorption energy E,. Or, in other words, this is an excess of caused by the surface heterogeneity. In the case of patchwise surface topography, the interaction term Pp will be represented by the following average : Using the expansion (10) and neglecting terms of order higher than Co(r/kT)2, we obtain Pp = Mcin [ a, (xgb x& + (W [ Dg) (xfjsb - x f i ) - 2( DP))’]) + a, (xpi xf3b) + xgi xp) - xp) xg))] .(26)w. RUDZINSKI, J. NARKIEWICZ-MICHALEK AND s. PARTYKA 2365 As in the case of random surface topography, we can write Pp in the following form: Pp = Po+8”,”. (27) Now, let us consider the contribution to the heat of wetting, Q,, due to solid-solution interactions. This contribution is represented by the first two terms on the right-hand side of eqn (18). We can rearrange them in the following form:2 ~ 2 ’ QwA + xg’ QwB = Q,, + M x ~ ) E, - M x ~ ) t?, (28) where A = kT1n (qAs/qBS).(29) Thus we can see that the total heat of immersion of a heterogeneous surface, Q,,, can be represented by the following expression: +a3 Q,, = QwB+Pt+MJ -a3 X & ~ ~ ~ E ~ - M X ~ ! A (30) where Pt is P,. or Pp depending on the topography of the heterogeneous surface. Using the expansion (lo), and neglecting the terms of order higher than O ( T - / ~ T ) ~ , we arrive at the following expression for Qwt: Finally, we rewrite eqn (31) in the following condensed form: where Q,, is the value of Q, at E , = E: Qwo = Qw~+Po+Mxfb($-A) and Q$$, is the excess in Qwt due to the heterogeneity of surface (33) NUMERICAL RESULTS AND DISCUSSION Simultaneous measurements of the excess adsorption isotherms and heats of immersion have rarely been reported in the literature. Thus in order to illustrate our theoretical consideration from the previous section we have performed appropriate model calculations.Their results are shown in fig. 1-4. Fig. 1 shows the excess isotherms, evaluated using various interaction and heterogeneity parameters, for both patchwise and random surface topographies. The values of the heterogeneity parameter, r, assumed here, are comparable to, or even smaller than, the value found by us for the system benzene-cyclohexane on silica geL2 Also the values of the interaction parameter, E,/kT, accepted here represent only moderate deviations from the model of an ideal solution. The general conclusion which can be drawn from fig. 1 is as follows. The increasing degree of surface heterogeneity causes the maximum in the excess isotherm of the preferentially adsorbed solvent to increase, and to be shifted to smaller solvent concentrations.In the case of negative deviations from Raoult’s law, this effect is stronger for random surface topography. On the other hand, this effect is stronger for patchwise surface topography in the case of positive deviations from Raoult’s law. In this case the surface heterogeneity causes azeotropes to appear.2366 ADSORPTION FROM SOLUTION ONTO SOLID SURFACES bulk mole fraction, x t ) FIG. 1.-Excess adsorption isotherms nzt = ( x ~ ~ - x ~ ) ) , evaluated for the models of random (-) and patchwise (---) surface topography. The assumed heterogeneity parameters are r/kT = 1 .O (circles), r/kT = 0.5 (crosses) and r/kT = 0.0 (free solid line).The most probable value of the adsorption energy cz assumed here was 2542 J mol-l. (A) shows the case of negative deviation from Raoult’s law, characterized by the parameter &,/kT = - 1.0. (B) shows the situation for a positive deviation from Raoult’s law, characterized by c,/kT = 1 .O. 0.3 1 -o,21 \ \ d / bulk mole fraction, xf“ FIG. 2.-Contribution to the excess adsorption isotherm due to surface heterogeneity : (-) surface excess for hypothetical homogeneous surface, characterized by the adsorption energy cz; (--), (---) adsorption excess nxt when the heterogeneity parameter r/kT = 1.0, for random and patchwise topography, respectively; (-@---@-), (-0- - - @-) contribution (nr)2 Db2)/6 due to surface heterogeneity, for random and patchwise topography, respectively.Other notation as in fig. 1.w . RUDZINSKI, J. NARKIEWICZ-MICHALEK AND s. PARTYKA 2367 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 bulk mole fraction, xf“ FIG. 3.-Behaviour of the heat of immersion, Qwt. The notation is the same as in fig. 1. Other parameters are: QwB = 0.0, T = 25 OC and 1 = 418.5 J mol-l. bulk mole fraction, x f ) FIG. 4.-Contribution to the heat of immersion due to surface heterogeneity. The notation is the same as in fig. 2, but (-@--.--) and (@---a) now denote @,“h for random and patchwise topography, respectively. Fig. 2 shows separately the contribution to the excess adsorption isotherm ( ~ r ) ~ Di2)’/6 caused by the surface heterogeneity. Fig. 3 shows the behaviour of the heats of immersion found for the same interaction and heterogeneity parameters.As in our previous paper,2 we have assumed that the term T(aem/i3T) is negligible when compared with E,. Further, we have accepted that 2 is equal to 418.5 J mol-l. This value is close to the A value found by us for the system benzene-cyclohexane on silica gel.9 Fig. 4 shows separately the contribution to the heat of immersion (3;; caused by surface heterogeneity. The general conclusion which can be drawn from fig. 3 and 4 is as follows: Surface2368 ADSORPTION FROM SOLUTION ONTO SOLID SURFACES heterogeneity causes large positive contributions to the heat of immersion, no matter what type of surface topography is present or in which direction deviation from Raoult’s law occurs. We hope that our present model calculations may be helpful in understanding the complicated interference of the basic physical factors governing the behaviour of solid-solution systems, i.e. the non-ideality of the bulk and adsorbed phase, the energetic heterogeneity of real solid surfaces, and the topography of these surfaces. We also believe that a simultaneous numerical analysis of the excess isotherms and the heats of immersion should provide much more reliable information on the nature of solid-solution systems. D. H. Everett, Trans. Faraday Soc., 1965, 61, 2478. A. Dqbrowski and M. Jaroniec, J. Colloid Interface Sci., 1980, 77, 571. E. A. 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