Adsorption of Fluids: Simple Theories for the Density Profile in a Fluid near an Adsorbing Surface BY GERHARD H. FINDENEGG* Lehrstuhl f. Physikalische Chemie 11, Ruhr-Universitat Bochum, D-463 Bochum, BRD, Postfach 2148 JOHANN FISCHER AND Dept. of Theoretical Physics, Ruhr-University, 463 Bochum, BRD Received 23rd December, 1974 Two simple theories of fluid adsorption are discussed. One starts from the first equation of the BBGKY-hierarchy whereas the other considers a reference system in which the fluid is bounded by a hard wall, and the attractive potential is added with a coupling parameter. For the latter theory numerical results are presented for the density profile and the integrated surface excess over a wide range of fluid densities. These results are compared with those of asymptotic theories.The problem of the density profile of the reference system is considered. Gas adsorption at or below the normal boiling point of an adsorptive can usually be considered as condensation of the vapour under the influence of the attractive potential of the surface. The density of such an adsorbed layer will be more or less uniform up to its outer surface where it falls off sharply to the density of the vapour phase. Such a model is less realistic at higher temperatures, and above the critical temperature there can be no discontinuity in the density of the fluid. Rather, the density will increase smoothly towards the surface. At low densities, when the fluid behaves as a perfect gas, the local density at a point Y near the surface, pcl)(r), is simply the bulk density po times a Boltzrnann-factor,l where u,(Y) is the interaction potential between an isolated molecule at Y and the solid.If u,(r) is only a function of the distance z from the surface, pcl'(r) = pcll(z), the amount of adsorption per unit area is given by Inserting eqn (1) into (2) leads to Henry's law, i.e., adsorption along an isotherm increases linearly with the pressure of the gas. At higher densities deviations from this law occur. It has been established by measurements at higher pressures that the adsorption isotherms pass through a maximum at pressures at which the fluid has about the critical (or a somewhat lower) density. Qualitatively, this behaviour is readily explained by the fact that the density near the surface cannot increase indefinitely ; hence, the surface excess decreases as the bulk density gradually approaches that near the surface.To obtain a quantitative isotherm equation it is necessary to have a theory for p'l'(z) which is applicable over an extended range of densities of the fluid. 38G . H. FINDENEGG AND J . FISCHER 39 Analytical expressions for the density profile in a liquid at relatively large distances from a wall have been derived by Steele and by Kuni and Ru~anov.~ If these are inserted into eqn (2) an isotherm equation r ( p ) is obtained which reproduces qualita- tively the main features of high pressure adsorption isotherms over a range of tem- perature~,~ even though it is based on an extrapolation of the asymptotic expression for pcll(z). It is of interest, therefore, to investigate the density profile close to the wall as a function of the bulk density and temperature.Below we discuss two simple theories for pcl’(z) which are derived from the same starting point along different ways. Numerical solutions are presented for one of them and the results are compared with the asymptotic expressions. THEORY We consider the following system: a fluid consisting of N spherical molecules is bounded by a structureless wall in the plane z = 0 that exerts attractive forces on the individual molecules. The potential energy of a molecule at a distance z from the wall is (cf. fig. 1) u,(z) = 00 for z < zo (34 u,(z) = u,(zo)(zo/z)3 for z 2 zo (3b) where zo denotes the collision diameter between the molecule and the wall, and the z - ~ dependence arises from the integration over the dispersion potential of the mole- cule with volume elements in the half-space of the wall.FIG. 1.-Two particles at different distances z from the wall. The spheres with radius r, indicate the range of the total correlation function ho. The potential of the wall us(z) and the collision diameter zo are also indicated. The first way to obtain the density profile in the fluid is by considering the first equation of the BBGKY-hierarchy for the above system. Let u(r,J be the pair potential between two fiuid molecules, then the total potential energy of the system is Hence the configurational partition function is ZN = I exp[ - E/kT] drl . . . dr,,40 ADSORPTION OF FLUIDS and the density profile (All configurational integrations are to be taken over the half-space z 3 zo.) Differ- entiation of this expression leads to where the usual definition of the pair correlation function g(rl, r,) has been used.To close the BBGKY-hierarchy one may assume that g(rl, r2) can be replaced everywhere by the pair correlation function go(rl, r,) in the bulk fluid. To proceed it seems indispensable for higher densities to take go(rl, r2) from some theoretical work and to evaluate the resulting integrodifferential equation numerically. For low densities we may use the approximation which gives % h 3 y2) = exPC - u(r1 ,)/kT3, Q O h , r,) -v1- = hgo(r19 r2) = VlhO(b r2), Nr1, r2> = dr1, r2) - 1 (5) ( u:3 where the total correlation function is used. We then obtain or h~[p(~)(r,)/p~] = -'%I+ p(')(r2)hO(rl, r2) dr,-p, h,(oo, r,) dr,.s s (6) In the second integral (rl = co) the volume of integration is effectively determined by the range of ho (rm in fig. 1) whereas in the first integral the wall causes a cut-off. A second method is to construct a reference system where the wall exerts no attractive forces on the molecules. The attractive potential is then added with a coupling parameter 5. The density profile can then be written as and again differentiation gives Formal integration then yields Here, pcl)(r, = 1) is the density at r when the attractive potential u,(r) is acliug. When the wall exerts no attractive forces on the particles of the fluid the densityG. H. FINDENEGG AND J . FISCHER 41 p"'(r, 5 = 0) will still deviate from po, as a consequence of packing effects near the wall, and also because of the unbalanced attractive potential of the fluid (as in the surface region of a liquid).who circumvented the last-mentioned difficulties by choosing a reference system (5 = 0) such that an imaginary boundary plane is drawn through a large sample of the fluid and the coupling parameter transforms the fluid in one half-space into the wall. With this choice of reference state us@) in eqn (7) is to be replaced by a perturbation potential where ufl(r) is the potential for a molecule at r (z>zo) with the reference fluid at z < 0. To proceed from eqn (7) one assumes that h(rl, r2, <) can be replaced by the total correlation function of the bulk fluid ho(rl, r2). In order to obtain a simple approximate equation for p<l'(r) Steele also replaces p"'(rY <) by p o and up(r2) by up(rl) and takes the integral over the fluid volume (z > z,) as an integral over an infinite volume.Thus Nearly the same method as outlined above has been proposed by Steele ulsr) = USW - ufl(r)Y = -'&I[ kT 1 + p o Jrn dr, h0(r1, r2)] where the well-known compressibility equation Po j h(l.12) dh2 = PokT- 1 (9) has been used (K is the isothermal compressibility of the bulk fluid). The resulting density profile differs from eqn (1) only by the thermodynamic factor pokTI. From the assumptions that have been made it seems clear that eqn (8) should hold far away from the wall. There up(z) will be small compared to ( p O ~ ) - l and eqn (8) can be approximated by This represents the leading term of another asymptotic theory which was derived in a different way by Kuni and Ru~anov.~ If this expression is used (in spite of its asymptotic character) to calculate the total surface excess [eqn (2)] the adsorption isotherms are in qualitative agreement with experimental res~lts.~ The asymptotic density profile can be checked by calculating p("(z) by less restrictive assumptions.In principle, eqn (4) would be preferred for this purpose and some calculations for low densities were made with eqn (6). The starting point for most of our calculations is, however, eqn (7) with the following assumptions: (i) h(rl, p i , c) is replaced by ho(rl 2) ; (ii) p(')(r, 5) depends linearly on <, i.e. (10) The density profile ~ ' ~ ' ( r , < = 0) could be obtained from eqn (4) with us@) = 0 but this involves solving an integro-differential equation.For the present work we assume (iii) that p"'(z) - p* = - p$KUp(Z). pc1+, <) = (1 - &P(v, = 0) + {p{l)(r, 5 = 1). pc1)(r, = 0) = po. (1 1)42 ADSORPTION OF FLUIDS With these approximations we obtain Eqn (12) refers to a reference state in which the fluid is bounded by a hard wall and the attractive potential of eqn (3b) is added by the coupling parameter c. Eqn (8) is based on a continuous fluid as a reference state and the coupling parameter trans- forms one half space of the fluid into the solid, thereby adding the potential up. However, the assumptions in the derivation of these two equations can be made with either of these models and yield formally the same relations, the only difference being that up is replaced by us and vice versa.We prefer the former reference state since this coupling procedure seems to be better defined. Thus we compare eqn (12) with the asymptotic equation ln[p(l)(zl)/po] = - %(p,kTK). RESULTS A N D DISCUSSION Eqn (12) and (13) were evaluated numerically for a fluid whose particles interact by a Lennard-Jones potential u(r) = 4E[(;)12-(q)6]. 0.2. I , I.- I _ L IIL-.*___- 10 1.5 21) 2.5 11) 1.5 2.0 2.5 3.0 1.0 1.5 2.0 25 30 FIG. 2.-Density profiles p(')(z) in reduced units plotted against z/zo for different bulk densities p and temperatures T. The solid curves show the results from the numerical evaluation of eqn (12), the dashed curves correspond to the asymptotic theory, eqn (13).Tabulated go(r) data have been used at pa3 = 0.45 and 0.65; at pa3 = 0.1, go@) is obtained from the low density approximation, 2120 eqn (5).G . H. FINDENEGG AND J . FISCHER 43 For the solid-gas potential, eqn (3), zo = cr and u,(zo) = - 2 ~ were taken. These values correspond roughly to the interaction of argon with a graphite surface. The pair correlation function for low densities is given by eqn (5). For high densities tabulated values of go@) for a Lennard-Jones fluid obtained by molecular dynamics calculations were used. The thermodynamic quantity pokTic that appears in the asymptotic expression, eqn (13), is obtained from the total correlation function ho through eqn (9). The integrals in eqn (9) and (12) were computed for r < r,, where r, corresponds to the zero of ho(r) next to 5.00.0 0.2 0.4 0.6 0.8 PO3 FIG. 3.-(a) Adsorption l' in reduced units plotted against bulk density p at T = 2.84~lk. The solid line and the filled circles are obtained from eqn (12). The dashed line and the open circles show the results of the asymptotic theory, eqn (13). The adsorption isotherm for argon according to eqn (13) is also indicated (dotted curve). (b) The thermodynamic quantity pok?"x as obtained from the compressibility eqn (9) using tabulated go(r) data6 (full circles) and using the low density go(r) according to eqn (5) (full line). The dotted curve is calculated from tabulated PVTdata of argon.' In principle, a comparison of the density profiles following from the asymptotic expression (13) in combination with eqn (9), and from eqn (12), respectively, should be consistent even if the go(r) data employed are not entirely self-consistent.How- ever, eqn (9) is sensitive to minor variations in go(r) at large r and significant errors in pokTic (and, therefore, also in the asymptotic density profile) arise from an in- correct interpolation of the tabulated gO(r) data. Nevertheless, our procedure seems to be sufficiently accurate to be used to discover systematic deviations of the asymptotic expression (13) from eqn (12). This is also indicated by the fact, that the values of pokTK which are obtained from the tabulated go@) data with the help of eqn (9) are in reasonable agreement with the experimental values for argon (as one should indeed expect 6, if the parameters cr = 0.3405 nm, Elk = 119.8 K are used for the Lennard- Jones potential of argon.This is shown in the lower part of fig. 3 and 4. The curves44 ADSORPTION OF FLUIDS for argon were obtained from tabulated PYT data over an extended range of pres- sures.' These graphs also indicate the range of densities in which the low-density- approximation for go(r) [eqn (93 may be employed. \ \ 0 1 \ \ \ \ . \ \ \ \ \ \ \ \ '\. \ \ '. i .'\ 0.5 1 ;. . 0.2 0.4 0.6 0.8 i"' " 1 ' 0 PO3 FIG. 4.-A similar plot to that in fig. 3 for the isotherm at T = 1.584k. A comparison of density profiles according to eqn (12) and (13), respectively, is shown in fig. 2. The critical constants for a Lennard-Jones fluid are : Tc = 1.36~/k, pc = 0 . 3 6 r 3 (the experimental values for argon are somewhat lower: T .= 1.26 E/k; pc = 0.32 r3). The profiles in the upper row of fig. 2 therefore correspond to temperatures above 2Tc, those in the lower row to an isotherm relatively close to the critical temperature. At low temperatures and high densities the density profile of eqn (12) exhibits distinct maxima near z/z, = 2 and 3, indicating a tendency towards a layer-like structure near the wall. This mainly reflects the damped oscillations in the pair function g,(r) at higher densities, but a flat maximum in p{')(z) near z/z, = 2 is also obtained at low densities, where go(r) is approximated by the single-peaked function exp[ - u(r)/kT] of eqn (5). This is indicated in fig. 2 in the profile at pOo3 = 0.1 and The adsorption f was obtained from the density profiles by numerical integration between 1 < (z/z,) and (zlz,) < 5 plus a correction for the excess density at (z/z,) > 5.Some results along isotherms at 2.84 Elk and 1.58 E/k, covering a wide range of densities up to the triple point density of the liquid (0.84 r3 for argon), are plotted in fig. 3 and 4. The following conclusions emerge. (a) In comparison with the numerical computation of eqn (12) the asymptotic theory predicts a steeper increase of r at low densities and a steeper decrease of r at high densities. kT/& = 1.58.G . H. FINDENEGG AND J . FISCHER 45 (6) At high temperatures (T 3 2T, ; see fig. 3) the asymptotic theory represents a reasonable approximation to eqn (12) at low and moderate densities but seems to give too low a surface excess at high densities.The large scatter in the values of rasympl reflects the sensitivity of the coefficient pokTrc to errors in the pair function (c) The isotherm in fig. 4, corresponding to a temperature not far above T’, shows strong adsorption at around the critical density. However, the asymptotic expression (13) is not applicable in the critical region.8 Sufficiently far away from the critical density the results are qualitatively in agreement with those at higher temperatures. It remains to check to what extent these results are influenced by our assumptions. As an alternative to assuming eqn (10) we replaced pcl’(r, c) in the integral by p{’j(r, = 1), but this had no significant influence on the numerical results. A more serious error is introduced by replacing pcl’(r, ( = 0) on the left-hand side of eqn (7) by po [eqn (ll)] : for low densities pc”(r, < = 0) can be calculated by use of eqn (6) taking u,(z) = 0 (z 3 zo).It was found that close to the wall the correct reference density falls below po by up to 20 % for po = 0.1 r3 (at T = 1.58 e/k). Thus our results are probably based on too high values of the reference profile and should be too high; this should not affect the comparison of eqn (12) and (13) since both are based on the same reference profile. As was already mentioned some preliminary computations at low densities were also made for eqn (6). The resulting values of I‘ fall below those of eqn (12), probably as a consequence of the incorrect reference density profile used in eqn (12).go(r). CONCLUSION In this paper we have compared the density profile and adsorption r of the asymptotic theory [eqn (13)] with results obtained from eqn (12) which was derived using less restrictive assumptions. The main results are : (i) along an isotherm the asymptotic theory gives too great an increase in r at subcritical densities and too steep a decrease of r at higher densities ; (ii) even at low densities eqn (12) exhibits a tendency towards a layer-wise structured density profile, whereas the asymptotic theory yields monotonic behaviour if the potential u,(z) is monotonic. The most serious assumption in the derivation of eqn (1 2) seems to be introduced by replacing the density profile of the reference state (fluid bounded by hard wall without attractive potential) by the bulk density po. To clarify this point evaluation of eqn (4) is desirable. W. A. Steele, The Solid-Gas Interface, ed. E. A. Flood (Dekker, New York, 1966), vol. 1, chap. 10; see also W. A. Steele, The Interaction of Gases with Solid Surfaces (Pergamon, Oxford, 1 974). P. G. Menon, Chem. Rev., 1968, 68, 277; see also P. G. Menon, Aduunces in High Pressure Research, ed. R. S . Bradley (Academic Press, New York, 1969), vol. 3, p. 313. F. M. Kuni and A. I. Rusanov, Russ. J. Phys. Chem., 1968,42,443, 621. G . H. Findenegg, Ber. Bunsenges. phys. Chem., 1974, 78, 1281. F. Kohler, The Liquid Stare (Verlag Chemie, Weinheim, 1972), chap. 4 and 7. L. Verlet, Phys. Rev., 1968, 165, 201. F. Din, Thermodynamic Functions of Gases, ed. F. Din (Butterworth, London, 1962), voi. 2. V. L. Kuz’min, F. M. Kuni and A. I. Rusanov, Rum. J. Phys. Chem., 1972, 46, 1032. Note added in proof: In fig. 3 and 4 the scale of ra2 is to be reduced by dividing the displayed values by 20.