Conformal Solution Theory : Hard - Sphere Mixtures * ( computer simulation / three - fluid theory / one - fluid van der Waals theory )

Conformal solution theory is examined. It is suggested that most difficulties associated with previous applications of this theory arise from the use of concentration-independent reference fluids. For the particular case of a hard-sphere mixture, it is shown that if the reference fluid is chosen so as to make the first-order term in the theory vanish, good results are obtained. When two species of molecules that are not very different are mixed, and when full information is presumed to be available on the pure components, it is natural to look for a perturbation treatment using one of the components or some similar fluid as the reference system. One such theory was first proposed by LonguetHiggins (1) and was named by him conformal solution theory (CST). The definition of a conformal solution is a formalization of what is meant by "not very different species" and may be rephrased by saying that the interactions must be of the form that suffices to ensure the validity of the corresponding-states principle. That is, the total potential energy consists of a sum of pair potentials: U(ri, ... ., r) = E u(aj, aj; Rij), i<j (1) where a1 = 1 or 2 if molecule i is of species 1 or 2 and u(X,Iu, ?R) = ux,(R) = E;F(R/0)x,. (2) If we write Ejj = e(l + Ifsj) and rjj3 = a3(l + Shij1) (3) where e and a characterize the arbitrarily chosen reference fluid, we have to a first order of approximation: 16G\ ~~.,bGi1 G = Go + E (4af)6 + E G) bu + = Go +GtAf+G/h~+ ......... (4) where the ideal entropy of mixing term has been dropped, Go, Gf, and Gh are properties of the reference fluid, and (5) Af = E xixb~f1 ii Ah=E=xxjbhj, ij where xi = N,/N, N, is the number of molecules in the mixture of species i, andN = N1 + N2. The quantities Gf and Gh can be expressed in terms of the thermodynamic functions of the reference fluid. Thus, G = Go + (Eo + 3/2NkT)Af + (pVo NkT)Ah + (7) This expansion has been taken to second order (2, 3). We do not write down the second-order term as it is complex. However, it is to be noted that the secondorder term cannot be written in terms of the thermodynamic functions of the reference fluid. It involves integrals over the 2-, 3-, and 4-body distribution functions of the reference fluid. The formal result (7) does not depend on the particular choice of the reference fluid. However, the convergence of the series is affected by this choice. Usually, one of the components of the mixture is chosen as the reference fluid. Occasionally, the choices E = (el + E22)/2 (8) a = (all + om)/2 (9) have been made. However, all these choices are independent of concentration. If such a concentrationindependent choice is made, the perturbation cannot be small at all concentrations. Thus, it is not surprising that, in the past, CST has not given good results. It is much more promising to use a reference fluid that becomes the pure components when xi or x2 is zero and that is as close as possible to the properties of the mixture. One such choice would be that given by Af = Ah = 0 (10) or equivalently (11) C = E XjXjEjj ii a3 = E XfXj0j3. ii (12) This choice makes the first-order term zero and also simplifies the second-order term by eliminating the most complex term (4). Recently, Tan and Luks (5) have made calculations for 6:12 mixtures and have concluded that CST is inherently poor. We believe that the cause of their poor 2354 Abbreviation: CST, conformal solution theory. * This is the first paper in a series, "Conformal Solution Theory". Conformal Solution Theory 2355 results was the use of concentration-independent e and o, rather than any inherent problems with CST. HARD-SPHERE MIXTURES We may get some feeling for the effect of the choice for e and o by considering a hard-sphere mixture. For such a mixture A = Ao + (poV-NkT)Ah+... (13) or G = Go + (pVo-NkT)Ah+... (14) For simplicity, we will use (13) rather than (14). If we use the Percus-Yevick (PY) result poV/NkT -1 = 2rq 2 + ) (15)