J . Chem. SOC., Faraday Trans. 1, 1988, 84(3), 739-749 Monolayer Adsorption of Non-spherical Molecules on Solid Surfaces Part 2.-The Application of First-order RAM Theory to Nitrogen on Graphite Jaros4aw Penar and Stefan Sokolowski" Department of Theoretical Chemistry, institute of Chemistry UMCS, 20031 Lublin, Nowotki 12, Poland First-order RAM (reference average Mayer) perturbational theory is used to calculate the adsorption characteristics of monolayer nitrogen on graphite. The model used for this purpose considers the adsorbed layer to be strictly two-dimensional and neglects the periodic structure of the underlying solid. However, considering the interactions in the adsorbed layer, we also discuss the effects of screening by the substrate on the interaction between a pair of admolecules.The results of theoretical calculations are compared with computer simulations and with the experimental data. In the preceding paper of this series' (henceforth referred to as Part 1) we advocated the use of a two-dimensional counterpart of the RAM2' (reference average Mayer) theory to describe monolayer adsorption of linear molecules on flat solid surfaces. According to this approach, the submonolayer adsorbed film is treated as a strictly two- dimensional phase with molecules oriented parallel to the surface. The theory was tested in the case of adsorption of diatomic molecules, and excellent results were obtained for structural and thermodynamic properties of this system. In this paper we apply the theory reported in Part 1 to a description of fluid submonolayer phases of nitrogen adsorbed on a model, structureless surface of graphite.Among other things we will consider here the effect of screening by the substrate of interactions between adsorbed particle^.^ RAM Theory for Two-dimensional Systems As in Part 1, we consider here a strictly two-dimensional system of diatomic molecule^^*^^^ interacting via a pair potential u(r, 8,, 8,) which is a function of the distance r between two selected points located within both molecules and molecular orientations 8, and 8,. Although other options can also be taken into account [cf. ref (3) and (7)], the angles describing orientations of a pair of molecules will be in this work measured with respect to the line joining centres of mass of both molecules.According to the RAM theory,'v3 the structural and thermodynamic properties of the system of interest are computed from perturbational expansions about spherically symmetric reference fluid interacting uia the potential (1) uref(r) = - kT In (exp [ - u(r, O,, S,)/kTl),l,p where (. . .) denotes an unweighted average over orientations. The first-order perturbational expansions for the two-particle background correlation function z(r, 8,, 8,) and for the Helmholtz free energy I; are given by 739 25-2740 Monolayer Adsorption of Non-spherical Molecules Afir, el, 0,) = exp [ - u(r, O,, O,)/kT-exp [ - d e P ( r ) / k ~ ] r is the two-dimensional density, N = rS,, Sa is the surface area, g(r,O1,8,) = exp[ - u(r, el, B,)/kT] z(r, el, O,),Af,,. ( r ) denotes the circular harmonic coefficient of A .0 - 9 61, 02) and the superscript ref denotes the reference system. There are several theoretical routes to evaluating the two-dimensional pressure $. In the case of bulk fluids composed of Lennard-Jones diatomic molecules, the best estimation of the pressure is obtained by differentiating the free energy.2* This approach has also been applied in our calculations. Thus The adsorption isotherm is then evaluated by using the Gibbs equation. We have where n is the number of molecules in unit volume of the gas phase and KH is the Henry constant. The average potential energy resulting from interparticle interactions can be evaluated as U=-==rc Jom ezz49 r d r ) rdr (8) where zlz’ and e,,, are the circular harmonic coefficients of the two-particle background correlation function and the function e(r, el, 0,) = u(r, O,, 0,) exp [ - u(r, O,, 8,)/kT].We note that according to the first-order RAM theory, only zl0 and z,, coefficients appear in the circular expansion of z(r, el, 0,). A knowledge of U allows one to calculate the dependence of the isosteric enthalpy of adsorption, qst, on the amount adsorbed. We have N ZZ’ where 4:‘ = qst(T = 0) is the isosteric heat of adsorption at zero coverage. The application of the equations listed above requires a detailed knowledge of the properties of reference system. As numerous calculations performed for three- dimensional fluids have indicated,,~~ the most serious errors in predictions of the RAM theory are caused by an inaccurate description of the reference system.Although the thermodynamic and local properties of adsorbed monolayers are now of great interest, only some scattered computer simulation results have been published for monomolecular fluid adsorbed phases composed of linear molecules.8* For this reason, an unequivocal choice of the best method of calculating the properties of the reference system is difficult.J. Penar and S. Sokolowski 74 1 In this work the reference-system two-particle correlation functions have been evaluated by using the two-dimensional Percus-Yevick (PY) equation, and the reference-system pressure has been calculated from the pressure equation, using gref (r) resulting from the PY approximation. Because the methods of solution of the PY equation are now quite standard,lO~ l1 they are not presented here.The Interaction Potentials In developing practical theories for the thermodynamic and structural properties of molecules adsorbed, it is crucial to know the potential energies that appear in the general statistical-mechanical equations for the problem. Thus in this section we give detailed information of the potentials used in our numerical calculations. The model which has been most frequently', used in theoretical studies of bulk nitrogen assumes that the energy of interaction between a pair of N, molecules is represented by the sum of four atom-atom Lennard-Jones (12-6) potentials : and the quadrupole-quadrupole energy uQQ(r, 8,, 6,). In two dimensions we have + 2(sin 8, sin 8, - 4 cos 8, cos 8,), - 15 cos 8, cos O,] (1 1) where Q is the quadrupole.Alternatively, the quadrupole interactions can be evaluated as the sum of nine coulombic potentials between three charges placed in each nitrogen molecule; two which equal q are located on the Lennard-Jones sites and one which is opposite in sign but equal to the sum of the other two is at the molecular centre. However, in contrast to eqn (1 l), the presence of three point charges on the molecular axis gives rise to multipoles of higher order thaa the leading quadrupole. According to Murthy et a1.l' for ~ / k = 36.4 K, 0 = 3.318 A, Q = 3.91 5 lo-*' C m2 (or q = 6.49 x 1 OP2' C ) and a molecular elongation given by I = 1.098 A, the three-dimensional analogue of the above model accurately reproduces the thermodynamic and structural properties of bulk solid and liquid nitrogen.We will refer to this model as model B. It is well known that in the presence of an adsorbing surface the intermolecular interactions between adsorbed particles become modified. Following earlier authors we allow for this effect by using the molecule-molecule version of MacLachlan theory,*! l4 according to which the free gas-phase pair potential is altered by adding the term where p = 1 +4L2/r2 and L is the height of the adsorbed layer above an effective image plane. In the Sase of nitroge! adsorbed on graphite15 C,/k = 231 288.26 K A6, C,/k = 11 726.84 K A6 and L = 5 A. We will label the model based upon these equations by the symbol S1. An attractive alternative16 to the approach based on the theory of MacLachlan is to use the potential model B with 'effective' parameters Q (or q), E and 0.The values of these parameters can be evaluated from temperature dependence of the second surface virial coefficient. The theoretical analysis of the N,-graphite experiments carried out by Bojan and Steele16 indicated that the- best agreement between computed and experimenotal values of the second surface virial coefficient is achieved for ~ / k = 28 K, 0 = 3.32 A and Q = 3.91 x loP4' C m2 (q = 6.49 C ) . The potential model outlined above will be denoted by S2.742 0.6 B2 0 - -0.6 Monolayer Adsorption of Non-spherical Molecules - - 100 80 60 c \ \ \. \. a '*\ 10 15 b X -Y E: 1 - 0.006 0.008 0.010 0.012 KIT Fig. 1. (A). The RAM reference system potentials for the models B1 (-), S1 (---) and S2 ( - -- -) of nitrogen adsorbed on graphite.T = 60 K (B). Dependence of the Henry constant and the second virial coefficient, B,* = B2u2, upon temperature. The labels S1 and S2 refer to appropriate N,-N, potentials, the points are the results of experimental determinationP and the curve marked S2-3D denotes the results of three-dimensional calculations of Bojan and Steele. l6 In fig. 1 (A) we show a comparison of the RAM reference system potentials [eqn. (l)] evaluated for the models B, S1 and S2 at T = 60 K. Both 'surface' models, S1 and S2, lead to similar results. Fig. 1(B) compares experimental and theoretical values of the second surface virial coefficient B, computed for the three models described above. According to the two-dimensional treatment B, = -nJoa (exp[-uref(r)/kT] -1)rdr.Moreover, the curve in fig. 1(B) corresponding to S2-3D denotes the results of three- dimensional calculations performed by Bojan and Steele.'' Model B does not correctlyJ . Penar and S. Sokolowski 743 reproduce the experimental quantities. We also note that models S1 and S2 lead to almost indistinguishable results. According to the two-dimensional treatment, information concerning molecule- surface interactions is contained in the Henry constant. We will identify this constant with that resulting from full, three-dimensional description. Thus KH = 1; dz exp [ - u(z, 8)/kT- 11 cos 8 do. (14) where v(z, 6) is the three-dimensional molecule-surface potential. The most widely used explicit model of molecule-solid interaction for graphite is based on a site-site approach in which each carbon atom interacts with each atom in the gas.If the site-site interactions are taken to be Lennard-Jones (12-6) functions, the potential v(z, 6) can be approximated by16 where E,, and ogs are the well-Cepth and size parameters for the nitrogenxarbo? atomic interaction and d, = 3.4 A and a, = 5.24 A2. For Egs/k = 33.4 and o,, = 3.36 A the values of the Henry constant resulting from the above model agree nicely with those evaluated experimentally16 [cf. fig. 1 (B)]. Results and Discussion In this section the theory reported above will be applied in numerical calculations of thermodynamic and structural properties of two-dimensional fluid nitrogen. Because the main purpose of this work is to propose a practical method of evaluating of properties of real monolayer adsorbed films, the theoretical predictions are confronted with the results of three-dimensional simulations,’ as well as with results of experimental determinations.”? l8 The first series of our numerical calculations has been performed for model B, investigated previously by Talbot et a1.8 using molecular-dynamics simulations.In fig. 2 we show representative examples of the circular harmonic coefficient g,1!2 (r). Fig. 2 (A) compares the results of theoretical predictions with computer simulations. Additionally, we also present the radial distribution function evaluated from the Percus-Yevick equation for spherically symmetric nitrogen molecules, lo! l1 according to which the pair potential is given by1’ u(r) = 4 & s [ ( 9 1 2 - ( y .with E,/k = 91.5 K and os = 3.68 A. Our calculations have confirmed previous findings’ concerning rapid convergence of the circular harmonic expansion of g(r, el, 02). The coefficients gllll(r) where ll and l2 are both 2 4 are very close to zero in the complete range. Remembering that the computer simulations were performed’ assuming the full, three-dimensional model, which also included the periodic variation of the adsorbing potential, the observed agreement between ‘ experimental ’ and theoretical coefficients gll ,2(r) is suprisingly good. We may expect that with an increase in the two-dimensional density r the errors introduced by the two-dimensional modelling of the system will increase, because under such conditions the adsorbed molecules are forced to tilt away from the surface.In fig. 3 (A) we present a comparison of the average potential energies of interparticle interactions computed from eqn. (8) and evaluated from molecular-dynamics simu-744 2 (a) 1 0 -1 Monolayer Adsorption of Non-spherical Molecules 1 1.5 2 2.5 rl3.332A 4 ( b ) 2 0 2 1 1.5 2 2.5 r/3.32A Fig. 2. (A) Comparison of theoretical (dashed lines) and resulting from simulationss (solid lines) circular-harmonic coefficients gZll $r). The numbers in parentheses denote the values of Zl and I, and the dotted line denotes the radi$ distribution function evaluated for spherically symmetric model of N,. T = 74.5 K, r = 0.0307 A-,. (B) The coefficients g, (r) obtained from the0 RAM theory for two state points. (a) (-) r = 0.0454 A-,, T = 745 K; (b) (---) r = 0.0272 A-2, T = 543 K.J.Penar and S . Sokolowski 745 0.02 0.03 0.04 rlK2 Fig. 3. (A) Comparison of average energies resulting from the RAM theory (the solid lines), computer simulations* (points) and the PY approximation for spherically symmetric model of N, molecules (the dashed lines). (a) T = 54.3 K, (b) T = 74.5 K. (B) Comparison of isosteric heats of adsorption at T = 74.5 K computed by using the RAM theory (a), computer simulations* (b) and the PY approximation for spherically symmetric model of N, (c). lations, whereas in fig. 3(B) we show the dependence of the isosteric heat of adsorption at T = 75.3 K on the two-dimensional density. The isosteric enthalpy of the simulation model is given by in kJ mol-1 when r is in We note that all simulation data have been treated as isothermal.The results presented here clearly demonstrate that the spherically symmetric approximation16 leads to worse results in comparison with RAM theory. qSt(r)-qit = 50.93 r746 Monolayer Adsorption of Non-spherical Molecules Fig. 4. Compressibility factors 2 = q5/TkT us. r resulting from the RAM theory for model B (temperatures in K). By way of illustration we present in fig. 4 the dependence of the two-dimensional compressibility factor 2 = $/r kT upon the density r. The critical temperature resulting from the RAM theory is close to 52 K. Since the convergence of our numerical PY procedure for the reference system two-particle background correlation function is difficult to achieve in the neighbourhood of the critical point, we have not yet completed our study in this region, and consequently cannot give accurate estimations of the critical temperature and density.Furthermore, the monolayer nitrogen adsorbed on graphite does not exhibit a gas-liquid phase transition, and its phase diagram is similar to that observed for monolayer krypton. 2o The method outlined above was next applied to analyse experimental data for nitrogen adsorbed on graphite. The interactions between adsorbed particles were described by using the model S2, and the results of numerical calculations were compared with experimental data measured at T = 79.3 K by Piper et a1.l' [system (a)] and with the data obtained at T = 7 K by Isirikyan and Kisielevl' [system (b)]. The value of the HeGry constant evaluated according to eqn (14) for both of these systems is 0.75 x 10' A.In fig. 5 we show the dependence of the isosteric heat of adsorption as a function of the amount adsorbed at T = 79.3 K. The dashed line denotes the experimental results of Piper et a1." The larger experimental values of qst at the lowest two-dimensional densities may be attributed to some heterogeneity in the energy of adsorption, but it is also likely that this effect is caused by adsorptioneon the inside wall of the calorimeter vessel. The sharp maximum in qst at r x 0.65 A-2 is connected with the transition from a two-dimensional fluid phase to the registered two-dimensional solid phase. Our theory fails completely in this region. A comparison of experimental and calculated adsorption isotherms for systems (a)J. Penar and S.Sokoiowski 747 0.0 2 OD 4 rJA-2 Fig. 5. Comparison of experimental’’ (---) and theoretical (-) values of isosteric heats of adsorption. The calculations were carried out for model S2. T = 79.3 K. and (b) is given in fig. 6(A). The error bars show the experimental error in r calculated assuming only 5% uncertainty in the surface area of both adsorbents [Sa z 21 m2 g-l for the system (a)16 and Sa z 12 m2 g-l for system (b)].’ Fig. 6(B), however, presents examples of the two-dimensional equation of state calculated for model S2. A comparison of the results presented in fig. 4 and 6(A) illustrates the screening of interactions between a pair of admolecules by the solid surface. The model considered here is very simple.In particular, we have neglected the effects of out-of-plane motion of adsorbed molecules and the role of periodicity of the gas-solid potential. The importance of both these effects depends on the temperature and surface coverage. At the lowest coverages and low temperatures, only a small fraction of the adsorbed molecules assumes non-planar configuration. Under such conditions the corrections to adsorption isotherms resulting from non-planar configuration can be evaluated by using a perturbational method, similar to that developed by Monson, Cale, Toigo and Steele for atomic adsorbed For this purpose we expand exp[ - u(RiD, R:D)/kT], where RfD denotes the three-dimensional positional and orientational coordinates of the ith molecule, into the following Taylor series about a planar configuration : exp [ - u ( R ; ~ , R,3”)/kT] = exp [ - u(ri, ri, Oi, 0,)/kT] x [ 1 - (zi - zj) u:, / k T - sli Un,/ k T - slj ud/k T + .. . ] . ( 1 7) In eqn (1 7) z denotes the coordinate perpendicular to the surface and Ri is the tilt angle of ith molecule, Moreover, for p = zi - z j , sZi and sli.748 0.06 N I 5 0.04 fi 0.02 Monolayer Adsorption of Non-spherical Molecules 1 2 n/ 10-7 A 0.02 0.04 r/K2 1.5 Z 1 Fig. 6. (A) Experimentall6, l7 (points) and theoretical (lines) adsorption isotherms. The labels (a) and (b) denote corresponding adsorption systems (see text). The solid lines were computed assuming a strictly two-dimensional model of adsorption and the dashed line was evaluated from eqn (24). (B) Compressibility factors 2 us.r for the model S2 of monolayer N, adsorbed on graphite. Temperatures in K. Substituting expansion (17) into the definition of the three-dimensional con- figurational integral for N molecules : V(RtD) + 2 u ( R , ~ ~ , Ria”)/kT). (19) i <j and retaining only the lowest-order terms, we obtain where and The symbol u ~ , ~ denotes the sites a-site p potential, &{(r) is the lowest-order circular harmonic coefficient of g(r, 0,, 0,) computed by using the site cc-site B reference frame g:{(r) = J dr, dr,(d4/24 (d0,/24&7 0,,0,) (23) (ri1,8- constant) and the sum in eqn (22) runs over all interacting sites. obtained from eqn (1). We have The lowest-order perturbational correction to the adsorption isotherm can be easily In nk, = Vplanar (r, T ) + A v ( ~ , T ) AV(r, T ) = (a/an[(r2/kn 11 4 1 (24) (25)J.Penar and S. Sokolowski 749 where v/ planar (r, T ) is the right-hand side of eqn (7). We stress that eqn (25) describes only small deviations from coplanarity and cannot be applied if the orientation of adsorbed particles differs significantly from the planar one. The dashed line in fig. 6(A) shows the adsorption isotherm (24) computed for system (a). The observed differences between adsorption isotherms (24) and (7) are relatively small, but we have not performed relevant calculations at high two-dimensional densities. Moreover, the correction term A@, T ) defined by eqn (25) is connected only with out-of-plane motion of adsorbed molecules and does not take into account the effects arising from lateral variation of the substrate potential.It is known20 that in the case of monolayers composed of spherically symmetric particles the periodic variation of the adsorbing potential may cause significant changes in the computed phase diagrams. We will return to these problems in future papers. This work was supported by the C.P.B.P. (Poland) under grant no. 01.08.E2. References 1 L. tajtar, J. Penar and S. Sokolowski, J . Chem. Soc., Faraday Trans. I , 1987, 83, 1405. 2 C. G. Gray and K. E. Gubbins, Theory of Molecular Fluids (Oxford University Press, Oxford, 1984), 3 W. R. Smith and I. Nezbeda, Adv. Chem. Ser., 1983, 209, 235. 4 S. Rauber, J. R. Klein and M. W. Cale, Phys. Rev. B, 1983, 27, 1314. 5 J. S. Rowlinson, J. Talbot and D. J. Tildesley, Mol. Phys., 1985, 54, 1065. 6 Y. P. Joshi and D. J. 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