J . Chem. SOC., Furuduy Trans. 1, 1988, 84(1), 19-28 The Second Planar Virial Coefficient for Nitrogen, Oxygen and Carbon Monoxide adsorbed on Graphite Leszek tajtar" and Stefan Sokolowski Department of Theoretical Chemistry , Institute of Chemistry, M. Curie- Sklodo wska University, 20-031 Lublin, PI. M. Curie, Sklodowskiej 3, Poland We present the results of calculations of the second planar virial coefficients for N,, 0, and CO adsorbed on graphite. In particular, we discuss the effect connected with the influence of the solid substrate upon interaction energy of a pair of adsorbed molecules, the role of electrostatic interactions and the effects of deviations of adsorbed film from an ideal planar configuration. Numerous experimental and theoretical studies of physical adsorption of small linear molecules on well characterized solid surfaces have indi~atedl-~ that at coverages below one monolayer and at sufficiently low temperatures the adsorbed fluid forms an almost two-dimensional film with molecules oriented parallel to the surface.To a first approximation the thermodynamic description of such systems can be formulated6, ' in terms of a two-dimensional model, by neglecting out-of-plane motion of adsorbed molecules. Different theoretical techniques, such as thermodynamic perturbation theory, can be proposed to calculate the properties of adsorbed two-dimensional submonolayer molecular fluid.6 In the limit of very low surface coverages, the properties of such systems can be represented exactly by the virial expansion.*-'' In addition, a comparison of the theoretically predicted and experimentally evaluated values of the virial coefficients constitutes an unambiguous test for the model potentials used in calculation^.^^ l2 In this work we present the results of calculations of the second planar coefficient for nitrogen, oxygen and carbon oxide adsorbed on graphite.The van der Waals interactions between a pair of adsorbed molecules are represented by the sum of four site-site potentials ; the electrostatic interactions, however, are handled by using the well known analytical formula for an ideal quadrupoleequadrupole energy,13 as well as by modelling these interactions by a discrete molecular charge distribution. 2* 3, '* Moreover, we also consider the effects connected with the substrate screening of the interactions between adsorbed m01ecules,~~-~~ and in the case of N, and 0, the effects connected with fluctuations of adsorbed particles out of a common plane." The Second Virial Coefficient For a diatomic fluid interacting via the potential u(r, O,, O,), the second planar coefficient is given by8 B, = -I JOm r dr J l n f i r , O,, 8,) do, do, 4n where is the Mayer function, r is the separation of the centres of the molecules and 0, and 0, are the angles describing molecular orientations.If the temperature is not too low, some deviations from an ideal two-dimensional configuration of adsorbed fluid can be observed and in such cases one should rather use the so-called pseudo-planar second virial coefficient, which is defined by'' fir, 0',0,) = exp [ - u(r, @,,0,)/kTI - 1 B, = ( K ~ B;- w,) K;, x - W, K;, 1920 Second Planar Virial Coeficient where Bi is the bulk three-dimensional second virial coefficient" and KH is the Henry constant.ASKH = k(R)-l]dR (3) I,,, 2 4 wz = J=2"g(Rl)dRlJ a3 k(R2)- 11JIRl,R2)dR2. where A, is the surface area, g(R) = exp [ - u(R)/kT], u(R) is the gas-solid potential, R is an abbreviation for Cartesian r and orientational o coordinates, and the cluster integral W2 is given by'l (4) In the above the subscript 00 means that the integration is extended over all spac? and z, is the position of the Gibbs dividing surface." The difference between B, and B, can be treated as a measure of deviations of adsorbed layer from a strictly two-dimensional configuration.10* l1 The problem of numerical evaluation of the integrals (3) and (4) can be simplified by considering expansions of angular-dependent functions into spherical harmonics Km(o).18 Introducing the reference frame with the z-axis perpendicular to the solid surface we can define the following expansions : (5) g(R) = ( 4 ~ ) ' C glrn(r) Km(m> Im and f(R1, R2) = 4~ C fil12mlrn2(r1, '2) K,m,(W1) K2m,(w& (6) 11 l2m1 m2 where and I 1 sin 81 sin 02 do1 do2 d41 d 4 2 ~ ~ 1 , ~ 2 ) mjU1) x GmJm2)- (8) Jl= KH = /2>2* kOO(Z)-- 11 dz wz = c n 1 gz10(z1) dz1 Jya3 kl*0(Z2) - 4201 dz2 A, I , m, m2(rl, '2) = Now we consider the simplest case of a flat solid surface. Substituting eqn ( 5 ) and (6) (9) into eqn (3) and (4) we obtain and 4 1 1 2 0 0 ( ~ , Zl, 22) dr.(10) I l l 2 z>z* 121-221 If, additionally, the interparticle potential u(R,, R2) does not depend upon the position of two interacting molecules with respect to the solid surface, the coeffi~ient~llpOO can be related to the spherical harmonic coefficient of the Mayer function calculated in the bulk reference frame, i.e.in the frame with the z-axis passing through molecular centres.'* We have : fi11,00(r12, 21 22) Al1,00(r12, 212) = C f;:,(,m Dkm(aBr) D?-rn(aBr) (1 1) O<m<min(l1,1,) where Dim are the elements of the three-dimensional rotation matrix,ls a = y = 0 and = arccos (z12/r12). In such case the one-dimensional integralsL. Lajtar and S. Sokoiowski 21 appearing in eqn (10) can be computed separately and stored, and consequently the integral ( 10) effectively becomes two-dimensional.In the presence of a surface, the molecular interactions become modified. Usually, the modification of the interparticle potential consists of adding to the original (free gas- phase) potential u an additional term Au. In this work we consider the case when Au does not depend upon molecular orientations. Thus f;;zA;0(r12, Zl,Z,) = exp [- Au(r12, Zl,Z,)l lfE(,zz00(~12, z12) + 4 , o SZ201 - 4 , o dZ2O (12) where the superscripts denote the interparticle potentials and the spherical harmonic coefficients fi,z200 can be again determined by using eqn (1 1). The Interaction Potentials In this section we give detailed information about the potentials used in numerical calculations. Because in the case of carbon monoxide our calculations are restricted to the two-dimensional model we will not present here the CO-graphite potentials.Information concerning the interactions of CO with graphite can be found in ref. (2) and (3). Nitrogen and Oxygen The interactions between two gaseous N, and 0, molecules are described by the sum of four site-site Lennard-Jones (1 2,6) terms to which the quadrupole-quadrupole potential is added (1 3) dR1, R,) = 4~ C [(0/rijY2 - (0/rJ61 + ue1 f, i 3Q2 ue, = - [ 1 - 5(cos2 0; + cos2 0;) + 2(sin 0; sin 6; cos & - 4 cos 0; cos e;), 4r5 - 15 C O S ~ e; C O S ~ e;] (14) where the primed angles are measured in the reference frame with the z axis along the line joining the centres of mass of the two molecules and Q is the quadrupole. The values of the parameters of the potentials (13) and (14) are 9s follows (the subscripts denote the interacting species) : E5N/k = 35 K, cr,, = 3.32 A, Qgz = (QN2/€” aiN)r = 1.06, eoo/k = 54.3 K, o,, = 3.05 A, Q& = -0.1965 and the elongations of both molecules are dN2 = 1.1 A and doz = 1.208 A.1292o The electrostatic interactions between two N, molecules can be also modelled by the sum of nine coulombic interactions between three partial charges.Charges of -8.49 x low2’ C are placed at the Lennard- Jones centres and a charge of 16.98 x In the presence of a solid surface, the free gas-phase potential (13) is modified. Following earlier author^^^-^' we allow for this effect using the molecule-molecule version of McLachlan theory [cf. ref. (3)]. In the case of the two-dimensional model we calculate Au according to the equation C at the bond centre.’, where p = 1 + 4L2/r2 and L is the height of the adsorbed layer above an effective image plane. For graphite” (17) L = (2) - dG/222 Second Planar Virial Coeflcient Fig.1. (a) The second planar B, and the second pseudo-planar l$ virial coefficients for different models of nitrogen adsorbed on graphite, Part (a) was calculated assuming the free gas-phase pair energy between two nitrogen molecules, whereas the curves presented in part (b) were obtained by using the surface-mediated interparticle potential. The meaning of all symbols is explained in the text. (a) (-) 2QG, (---) 20G, (...) 30G, (--) 3QG, 0, 2DG; (b) (-) 2QS, (----) 20S, (--) 3QS, .,2DS. where dG = 3.37 A is the interlayer spacing of graphite and the distance (z) of the adsorbed layer from the top carbon layer can be evaluated from experiments [cf.ref. (16)] or from computer sir nu la ti on^.^ In many cases (2) can be approximated by the value of z at which the molecule-solid potential reaches its minimum value.lS In general (2) depends upon temperature, but in our calculations we assume that (2) is independent of temperature. A suitable generalization of eqn (16) to the three- dimensional case was developed in ref. (3). We have (18) where r;, is the distance between one of the adatoms and the image of the others in the substrate; and r12 and 4, subtend the angles a, and a, with respect to the surface: ri2 = (r;2 + 42, z,>+ a, = arcsin (12, -z21/r12) a, = arcsin [(z, + z2)/ri2]. Au(r,,, z,, 2,) = C,(2 + 3 cos 2a, + 3 cos 2a2)/6(r12 r'12)3 - C2/(r;2)6 In our calculations we seto Cl, N,/k = 23 1 288.26 K kJ C,, N,/k = 1 17 267.804 K ( z ) ~ ~ = 3.1 A.21 l7 Cl,02/k = 225953.8446 K A6, C2,02/k = 97775.9 K A6,17 ( z ) , ~ = 3-32 A5*12 andL.tajtar and S. Sokoiowski potentials. In our calculations we assume that the surface is completely flat and that 23 The molecule-substrate potential was assumed to be the sum of two atom-surface U(Z, 8) = U,(Z + 0.5d cos 8) + V,(Z - 0.5d cos 8) where a, = 2.46 A, oAG = (o+o,)/F, o, = 3.4 A, E,, = ( E , E ~ ) ~ and Ep/k = 15 K.22*23 Thus ENG/k = 31.3 K, oNG = 3.36 A, E,,/k = 39 K and o, = 3.225 A. The periodic terms neglected in eqn (19) are and we believe that their influence on the value of the second pseudo-planar virial coefficient is negligible [cf.ref. (24)]. Carbon Monoxide The van der Waals interactions between two CO molecules are described by the sum of four (exp-6) atom-atom potentials3 (20) with AC9/k = 3.81271 x lo7 K, A,./k = 3.60824 x lo7 K, A,, = 3.90893 K, a,, = 0.239 23 A, a,, = 0.255 75 A, a,, = 0.271 94 A, B,,/k = 21 1.8038 K A6, B,,/k = 170.79001 K A6, B,,/k = 130.4981 K A6 and the molecular elongation was d,, = 1.1282 A.3 The electrostatic interactions are handled by employing a three-site mod$ - 1.018935 x C , 1.231 345 x C and -0.31241 x 1O-l’ C located at - 1.0!2 A, -0.6446 A and 0.3256 A, respectively, with the atomic sites located at -0.6446 A (C) and 0.4836 A (0) with respect to the centre of mass. The effect of substrate screening on the pair intezaction energy is again described by the potential 116) with3 C,,,,/k = 272 181.66 K A6, C2,,,/k = 143 367.45 K A6 and (z),, = 3.35 A.uij(rij) = A exp ( - rij/a) - B/rfj Results and Discussion We begin our characterization by defining abbreviations for the system studied. The numbers 2 and 3 will refer to the planar and pseudo-planar virial coefficient. The symbols Q and D will denote the ideal quadrupole potential (14) and the discrete charge distribution model [eqn (1 5)] used in calculations of electrostatic interactions, whereas the label 0 means that the electrostatic interactions are completely neglected. The symbols G and S denote the free gas-phase potential and the substrate-mediated interactions between a pair of molecules. For example, the symbol N,-2QG means that the calculations for N, are performed according to the two-dimensional model assuming that the quadrupole-quadrupole interactions are described by eqn (14) and that the influence of the underlying solid on the interparticle potential is neglected. The numerical integrations were performed by using standard multidimensional Gaussian and Gaussian-Chebyshev procedures [cf.ref. (25)]. The calculation of the sum (10) was truncated after the term with 1, = 1, = 6. Our previous investigations26 have indicated that for molecular elongations considered here, the higher-order terms in eqn (10) give quite negligible contributions. The dependence of the virial coefficients upon temperature can be described by using the formulae collected in table 1. The proposed approximations recover the values of the two-dimensional virial coefficients with an accuracy of 1 .O O/O, except in the vicinity of the Boyle temperatures (see table l), where larger deviations are observed.In the case of pseudo-planar virial coefficient this accuracy is worse and is 5 % . The variation in the second planar and pseudo-planar virial coefficients with temperature is also presented in fig. 1-3. The following observations can be made. (a) The effect of electrostatic interactions on the values of the virial coefficients computed for oxygen is very small. On24 Second Planar Virial Coeficient Table 1. Analytical approximations describing the temperature dependences of the second planar and pseudo-planar virial coefficients and the Boyle temperatures estimated for the investigated systemsa system Boyle temperature designation a b C d temperature/K range/K N2-20S 173.75 19.46 0.571 2.084 105 40-270 N,-2QS 168.04 22.01 0.7 13 2.140 102 40-280 N,-2DS 171.52 21.89 0.930 2.196 102 40-240 N2-20G 220.76 21.93 1.050 2.228 127 40-240 N,-2QG 215.74 24.12 1.212 2.297 125 40-260 N,-2DG 213.70 24.41 1.191 2.276 125 40-240 Ni-3QS 151.92 13.80 2.040 1.071 N2-30G 318.86 10.74 3.876 2.977 N,-3QG 234.55 23.89 1.836 2.184 O,?") 259.23 26.10 0.78 1 2.371 2QS 02JoG] 320.31 32.12 0.738 2.358 2QG 21 50-250 42 50-250 45 50-250 40 50-240 75 60-270 02-3QS 383.85 13.61 3.469 3.193 156 50-250 0,-3QG 434.63 23.42 3.494 3.101 205 50-250 c o - 2 0 s 254.90 39.92 0.351 2.234 155 70-240 CO-2DS 321.19 66.82 3.333 3.399 183 90-260 CO-2DG 302.13 76.23 0.197 2.298 210 110-240 a The general approximating formula is B, = -a/( T- b) - 0.001cT+ d.the other hand, the electrostatic interactions significantly effect the values of the virial coefficients determined for carbon monoxide. Over the whole range of investigated temperatures the curve evaluated for the CO-2DS model lies below the curve labelled by 20s. (b) In the case of nitrogen both the ideal quadrupole model and the discrete charge distribution model lead to very close values of the virial coefficients. (c) The differences between pseudo-planar and planar virial coefficients computed for nitrogen and oxygen are not negligible, even at low temperatures. Over almost the entire range of temperatures the values of the pseudo-planar virial coefficients are lower than the values of the planar virial coefficients.( d ) Modification of a pair potential by the presence of a solid surface causes a shift of the computed curves in the direction of lower temperatures. (e) We observe different effects of quadrupole-quadrupole interactions on second pseudo-planar and second planar virial coefficients of nitrogen adsorbed on graphite. These interactions reduce the values of the pseudo-planar virial coefficients, and at temperatures >80 K the values of the planar virial coefficients become higher than corresponding values of the virial coefficients evaluated for models which neglect the electrostatic interactions. To explore the reason for such different behaviour of pseudo-planar and planar virial coefficients of nitrogen we evaluated the two- and three-dimensional Boltzmann- averaged potentials (21) and (22) i&)(r) = -kT In (exp [ - u(r, el, B,)/kT])ff,i i&D(r) = - kT In Gfooo + 1) = - kT In (exp [ - u(R1, R,)/kq);D+ where (...)XI$: and (...>rAa denote unweighted averages over two- and three- dimensional rotations.Both potentials ziZD(r) and aaD(t-) depend upon temperature, andL. Lajtar and S. Sokoiowski 25 300 100 200 1 " " 1 " " 1 ' " ' 1 I ' TIK B Fig. 2. Fig. 2. As in fig. 1, but for oxygen. (- TIK 100 200 1 " " 1 " " 1 " " 1 " " 1 " B Fig. 3. ) 2QS, (---) 2QG, (--) 3QS, (....) 3QG. Fig. 3. As in fig. 1, but for carbon monoxide. (-) 2QS, (---) 20S, (--) 2QG. in general fi,,(r) # ~ ~ ~ ( r ) . We note that the definition of the second planar coefficients, eqn (l), can be rewritten as B = -0.5 {exp[-zi,,(r)/kT]- 1)dv (23) I and the leading term in the sum (10) involves the integral of (exp [ - u3,(r)/kT] - l}.In fig. 4 and 5 we give a comparison of the Boltzmann-averaged potentials computed for nitrogen and carbon monoxide. The addition of a quadrupole term to the intermolecular potential of nitrogen causes that the minima in the #3D(') curves to become deeper, and this effect does not depend upon temperature. A similar effect of electrostatic interactions is observed in the case of the potential UID(r) computed for carbon monoxide. The situation is different in the case of the two-dimensional potential #,,,(r) computed for nitrogen. At low temperatures the quadrupole-quadrupole interactions reduce the minimum in uzD(r), but at higher temperatures the addition of the quadrupole term to the nitrogen-nitrogen potential causes this minimum to become shallower. We can thus state that the observed different effects of quadrupole interactions upon the values of planar and pseudo-planar virial coefficients of nitrogen can be attributed to a different averaging of the pair potentials over angles in two and three dimensions.However, in all cases the modification of a pair potential by molecule-surface interactions causes the Boltzmann-averaged potential minima to be less pronounced. The calculations performed in this work clearly demonstrate that the application ofI ' I 3 L 5 1 I I I I 1 I 1 I L 5 6 rlA 3 L 5 I I I I I ( b ) I I I I I L 5 6 rlA I 1 1 I L 5 6 rl A Fig.4. Two- and three-dimensional Boltzmann-averaged potentials for nitrogen adsorbed on graphite. All symbols are explained in the text, (a) (-) 3QG, (---) 30G; (b) (-) 2QG, (---) 20G; (c) (-) 2QS, (---) 20s.L. Eajtar and S. Sokolowski 27 10 5 - i - 1: - 5 - 1c L 5 6 r i K Fig. 5. As in fig. 4, but for carbon monoxide (-) 2DS, (---) 20S, ( . . . ) 2DG at T = 200 K and (---) 20s at T = 100 K. a strictly two-dimensional model to the description of real adsorption systems may in some cases produce unexpected errors. The discrepancies between two- and three- dimensional virial coefficients would be at least partially removed by replacing the two- dimensional potential GzD(r) in eqn (23) by the three-dimensional potential UBD(r). The last hypothesis should be tested by comparing computed virial coefficients with those evaluated experimentally. References 1 R.D. Diehl and S. C. Fain Jr, Surf: Sci., 1983, 125, 116. 2 J. Piper, J. A. Morrison and C. Peters, Mol. Phys., 1984, 53, 1463. 3 C. Peters and M. L. Klein, Mol. Phys., 1985, 54, 895. 4 R. P. Pan, R. D. Etters, K. Kobashi and V. Chandrasekharan, J . Chem. Phys., 1982, 77, 1035. 5 J. Talbot, D. J. Tildesley and W. A. Steele, Faraday Discuss. Chem. Soc., 1985, 80, 119. 6 L. tajtar, J. Penar and S. Sokolowski, J . Chem. Soc., Faraday Trans. I , 1987, 83, 1405. 7 T. Boublik, 1985, Mol. Phys., 1985, 54, 1644. 8 J. S. Rowlinson, J. Talbot and D. J. Tildesley, Mof. Phys., 1985, 54, 1065. 9 W. A. Steele, The Interactions of Gases with Solid Surfaces (Pergamon Press, Oxford, 1974). 10 J. R. Sams, Prog. Surf. Membr. Sci., 1973, 8, 1. 11 S. Sokolowski, J . Chem. Soc., Faraday Trans. 2, 1981, 77, 405. 12 J. Talbot, D. J. Tildesley and W. A. Steele, Mol. Phys., 1984, 51, 1331. 13 J. D. Hirschfelder, C. F. Curtis and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New 14 P. A. Monson, W. A. Steele and W. B. Streett, J. Chem. Phys., 1983, 78, 4126. 15 A. D. McLachlan, Mof. Phys., 1964, 7 , 381. York, 1954).28 Second Planar Virial Coeficient 16 S. Rauber, J. R. Klein and M. W. Cole, Phys. Rev. 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