Mixtures, I. Hard Sphere Mixtures*

Proceedings of the Natioruil Academy of Scienccs Vol. 67, No. 4, pp. 1818-1823, December 1970 One- and Two-Fluid van der Waals Theories of Liquid Mixtures, I. Hard Sphere Mixtures* Douglas Henderson and Peter J. Leonardt IBM RE-llSEARCH LABORATORY, SAN JOSE, CALIFORNIA 95114; AND DEPARTMENT OF APPLIED MATHEMATICS, UNIVERSITY OF WATERLOO, WATERLOO, ONTARIO, CANADA Communicated by Henry Eyring, September 23, 1970 Abstract. The equation of state of a mixture of hard spheres is calculated using the one- and two-fluid van der Waals theories and the three-fluid theory. The one-fluid theory is found to be in the best agreement with the machinesimulation results. In the course of writing a review article1 on the theory of liquid mixtures, we found that there has been virtually no systematic examinations of the many theories of mixtures that have been proposed. This has been because until recently the results of a theory could only be compared with experimental results; unfortunately, for these experimental systems, the interaction between the unlike molecules is not well known. As a result, in comparison with experiment the depth, E12, of the interaction between unlike molecules is taken to be an adjustable parameter. The excess thermodynamic properties are extremely sensitive to small changes in 812; hence, a number of the current theories of mixtures all fit experimental data with about equal success and with values of 812 that are both reasonable and very similar. Fortunately, recent quasi-experimental simulation studies on computers2-7 have provided data that can be used in a systematic study of the theory of liquid mixtures. In earlier papers8'9 we examined perturbation theory. In this note, we examine the so-called one- and two-fluid van der Waals (vdwl and vdw2) theories10-13 and the three-fluid theory14 for hard-sphere mixtures. Hard-sphere mixtures are extremely useful for testing theories of fluid mixtures because of their relative simplicity. However, this usefulness seems not to be fully appreciated. For example, Leland et al. have made extensive tests of the vdwl and vdw2 theories for experimental systems12'13 but made no test of these theories for hard-sphere mixtures even though the simulation results2-4 were available at the time. This is unfortunate because the comparison with experiment does not distinguish between these theories, whereas comparison with the simulation results for hard-sphere mixtures clearly shows the vdw1 theory to be the better. Some years ago Salsburg and Fickett15 examined hard-sphere mixtures and considered a theory which is equivalent, for hard-sphere mixtures, to the vdw1 theory. However, the results of the vdw2 theory and a comparison of the vdw1 and vdw2 theories have Ilot been reported elsewhere. 1818

VOL. 67, 1970 LIQUID-MIXTURE THEORIES 1819 One-fluid van der Waals Theory. The simplest and most widely used theories of liquid mixtures are those based on the principle of corresponding states The idea behind these theories is that if two substances separately obey this principle, their mixture may also be expected to obey it as if the mixture were a singlecomponent fluid (which we call the equivalent fluid) characterized by some suitable (composition-dependent) average potential parameter. In their further development, the corresponding states theories consider ideal mixtures of two (or even three) pseudo-components. The most successful of the corresponding states theories is that of Leland et al.9-1' which has come to be referred to as the van der Waals (vdw) theory although it is not tied to the vdw equation of state. To avoid confusion with theories based on the vdw equation of state we refer to the one- and two-fluid versions of this theory as the vdwl and vdw2 theories, respectively. We consider a binary mixture of N, hard spheres of diameter ala (species 1) and N2 hard spheres of diameter 0a22 (species 2) occupying a volume V. The potential energy of this system consists of a sum of intermolecular potentials, V(ri,.. rn) = Ej U(ai, aj; Rij), ii (1) where ai = 1 or 2 if molecule i is of species 1 or 2, Rij = Iri -rj, u(x,m;r) =uxy(r) = + o R<<dd =0 R>uf, (2) anid 012 = 1/2(ll + 022). (3) It is possible to consider nonadditive hard sphere mixtures where a12 is not given by (3). However, no simulation results are available for such systems. Thus, we consider only additive hard-sphere mixtures. Consider the well-known pressure equation for a hard-sphere mixture: pv 2 r NkT = 1 + 3 p (4) where N = N, + N2, xi = Ni/N, p = N/V, and the gij(o-s) are the pair distribution functions of the mixture when hard spheres of species i and j are in contact. It should be kept in mind that the UgA are functions of the xi, p, and 022/0,, as well as oij even though we have not explicitly shown this. We now look for an approximation which will enable us to express this in terms of the properties of one or more pure substances. The most immediate suggestion is 5tsj(oij) = g(u) (5) for all i and j. Thus, pv 2r NkT 1 + 3 Epy()ZXiXjaij'. (6) NkT =1+3~~~~~~~~i

1820 PHYSICS: HENDERSON AND LEONARD PROC. N. A. S. Eq. (6) is in one-fluid form if we interpret g(o) as the distribution function at contact corresponding to a fluid in which cr3 = E XjXjaij. (7) Once a has been chosen, the thermodynamic properties may be calculated from: A(p)/NkT = 0(pau) - 3 in af (8) or G(p)/NkT = 41(pa3/kT) - 3 in a, (9) where the functions 4 and V/ are determined from the equation of a single-component hard-sphere fluid. It is often asserted that (8) and (9) are equivalent. They are in the sense that they produce the same isotherm. That is, if we solve (8) for p by iterating until it gives the experimental p we obtain the same p that comes directly from substituting the experimental p into (9) and solving for p. In this note we use (8) and (9) in this manner and so we do obtain the same results from either (8) or (9). However, it is common to use (8) and (9) in another manner. We can fix p and calculate p from (8) or we can fix p and calculate p. When used in this manner, (8) and (9) may not give equivalent results. This can be seen in Fig. 1 p FIG. 1. Typical isotherm of a fluid A mixture at high densities. The solid line Pexpt - -is the experimental isotherm at constant concentration and the dotted line is the vdwl prediction. Pexpt P where we see that using (8) and fixing p = Pexpt we calculate point B with error in pressure AB while fixing p = pexpt and using (9) we calculate C with error in density AC. For hard spheres bp/bp > 1 and for most experimental situations bp/?p >> 1. Thus AB > AC for most applications and (9) is to be preferred to (8) even though there is only one unambiguous isotherm. In Fig. 1 we have compared the equations of state of a hard-sphere mixture for which U22/U11 = 5/3 computed from (7) and (8) and the Ree-Hoover'6 singlecomponent hard-sphere isotherms with the simulation results.2 The agreement is quite good.

VOL. 67, 1970 LIQUID-MIXTURE THEORIES 1821 Other possible mixing rules that can be used in place of (7) are: 3= E xoii3 (10) and a = Exixjeyij = E xffii, (11) where we have assumed that (3) is satisfied. The results of (10) and (11) when used with (8) and the Ree-Hoover isotherm are plotted in Fig. 1. Neither (10) nor (11) yields results as good as those obtained from (7). Eqs. (7) and (8) give the vdwl prescription for calculating the properties of a hard-sphere mixture. We shall see that the vdwl theory provides the best agreement with the simulation results. Two-Fluid van der Waals Theory. A two-fluid theory"3 can be obtained from (4) by means of the approximation 912(al2) = '/2{gn1(all) + 922(a22)}. (12) Thus, (4) becomes pv 27r NkT = 1 + fpe xici3'gii(yjj). (13) The gej are not the same in the mixture as in the pure fluid. In order that each term in (13) be of pure-fluid form we must assume that gin is the distribution function corresponding to a pure hard-sphere fluid with diameter Ad given by at3 = Ex.a3 (14) Thus, we have the vdw2 approximation: the equation of state is that of an ideal mixture of two pure-hard-sphere fluids whose diameter is given by (14). The question arises as to whether the mixing is at constant density, i.e., A (p)/nkt = xi in xi + E x,(pa3) - 3 x it in ot, (15) or at constant pressure, i.e., G(p)/NkT = xi in xi + E x4(pa3/kt) - 3 E xi in as. (16) In the vdwl approximation, we saw that mixing at constant density or at constant pressure gave the same isotherm. This is not the case in the vdw2 approximation. In practice we find that (16) gives better results than (15). Also p,t are the more natural variables for a theory of mixtures. Eqs. (14) and (15) [or (16)] give the vdw2 prescription for calculating the properties of a hard-sphere mixture. In Fig. 2, the results of the vdw1 and the vdw2 theories for a hard-sphere mixture for which a22/all = 5/3 are compared with the simulation results.2 The constant p version of the vdw2 theory is better than the constant p version, but neither is as good as the vdw1 theory.

1822 PHYSICS: HENDERSON AND LEONARD PROC. N. A. S. 6 ~ ~~I ~ 022/01 5/=l / Eq.(7) 4/- Eq.(10),',/ /,' FIG. 2. Equation of state of a hardz //,/ Eq.( 11) sphere mixture for o022/0'i1 = 5/3 and xi = >_2 /, = 1/2. The points give the simulation // 7 results (ref. 2) and the curves give the // _ o results calculated from Eqs. (7), (10), 2 ~ ~,' ~and (1 1). 0.2 0.4 0.6 p Three-Fluid Theory. Scott'4 has observed that (4) is already in pure-fluid form if we assume that the gqi are equal in the mixture to their values in the respective pure fluids, and that g12(r) corresponds to that of a hard-sphere fluid with diameter 0'12. Once again we will find that the constant pressure version, G(p)NkT = xi in xi + E xixj,(pcj1/kt) - 3 E xixj In atj, (17) i ii ii is better than the constant density version, A(p)/NkT xxi in xi + E x-xj0(pa j3)-3 E xixj In aij (18) i ii ii In Fig. 3 the results of the three-fluid theory for hard-sphere mixtures for which 022/Tall = 5/3 are compared with the simulation results2 and with the vdwl and vdw2 results. The constant p version is better than the constant p version, but neither is as good as are the vdw1 and vdw2 results. Summary. We would like to emphasize that the simulation results are much more useful than the experimental results in assessing the relative merits of theories. For example, all three theories (Fig. 2) are about equally satisfactory in reproducing the experimental results"3" 7 (if E12 is adjusted), whereas comparison with the simulation results for hard-sphere mixtures shows the vdw1 results to be best. The vdw1 theory is clearly the best of the theories we have considered. In addition, the fact that the vdwl theory introduces no ambiguities between constant pressure and constant density makes it preferable to the other two theories. To our knowledge, this is the first paper to point out the superiority of the vdwl theory. However, the vdwl theory does not have a fully sound basis. The derivation of the vdw1 theory gives no insight into why the vdw1 theory yields better

VOL. 67, 1970 LIQUID-MIXTURE THEORIES 1823 3F /, vdw2 022/a, 1=5/3// / /vdwl 4 _ / ///_ 4-~~~~I FIG. 3. Equation of state of a hardsphere mixture for o-22/c = 5/3 andx=xi x2= 1/2. The points give the simulation / /W2&3F results (ref. 2), the solid curves give the > constant pressure vdwl, vdw2, and three-fluid-theory results, and the broken curves give the constant density 2- vdw2 and three-fluid-theoryresults. X 0.2 0.4 0.6 results than the vdw2 theory. Indeed, the derivation of the vdw2 theory would lead one to expect it to be an improvement over the vdw1 theory. In fact, it is still widely so regarded. But, even if the vdw1 theory cannot be put on a fully sound basis, it is a very useful approach. It is easy to use and gives very good results. Abbreviation: vdw1 and 2, van der Waals theories (one- and two-fluid, respectively). * Supported in part by grants from the Department of the Interior, Office of Saline Water, and the National Research Council of Canada. t National Research Council of Canada postdoctoral fellow. Present address: Department of Chemistry, University of Manchester, Manchester, England. 1 Henderson, D., and P. J. Leonard, in Physical Chemistry-An Advanced Treatise, eds. H. Eyring, D. Henderson, and W. Jost (New York: Academic Press, in press) vol. 8, chap. 7. 2 Smith, E. B., and K. R. Lea, Trans. Faraday Soc., 59, 1535 (1963). 3 Alder, B. J., J. Chem. Phys., 40, 2724 (1964). 4Rotenberg, A., J. Chem. Phys., 43, 4377 (1965). 5 Singer, K., Chem. Phys. Lett., 3, 165 (1969). 6 Singer, J. V. L., and K. Singer, Mol. Phys. 19, 279 (1970). 7McDonald, I. R., Chem. Phys. Lett., 3, 241 (1969). 8 Henderson, D., and J. A. Barker, J. Chem. Phys., 49, 3377 (1968). 9Leonard, P. J., D. Henderson, and J. A. Barker, Trans. Faraday Soc., 66,2439 (1970). 10 Leland, T. W., Jr., P. S. Chappelear, and B. W. Gamson, Amer. Inst. Chem. Eng. J., 8, 482 (1962). 11 Leland, T. W., Jr., and P. S. Chappelear, Ind. Eng. Chem., 60, no. 7, 15 (1968). 12 Leland, T. W., Jr., J. S. Rowlinson, and G. A. Sather, Trans. Faraday Soc., 64, 1447 (1968). 13 Leland, T. W., Jr., J. S. Rowlinson, G. A. Sather, and I. D. Watson, Trans. Faraday Soc., 65, 2034 (1969). 14 Scott, R. L., J. Chem. Phys., 25, 193 (1956). 15 Salsburg, Z. W., and W. Fickett, Los Alamos Scientific Laboratory Report La-2667, Los Alamos, N.M. (1962). 16 Ree, F. H., and W. G. Hoover, J. Chem. Phys., 40, 939 (1964). 17 Rowlinson, J. S., Liquids and Liquid Mixtures, 2nd ed. (London: Butterworths, 1969).