Critical Solution Behavior in a Binary Mixture of Gaussian Molecules

Abstract

We have examined in detail the cluster expansion for the mixing free energy of molecules whose repulsive interactions give rise to Ursell—Mayer cluster bonds of Gaussian form. By choosing the Gaussian width for unlike molecular pairs in a binary solution to exceed a common width assignment for like pairs, the resultant positive mixing free energy can produce phase separation at sufficiently high over‐all density. Using the exact evaluation technique for any Gaussian cluster integral of arbitrary complexity, the set of free‐energy contributions involving clusters of up to five particles were calculated and treated as input in a Padé approximant to estimate the width‐ratio‐dependent critical density. In the large unlike‐to‐like width‐ratio limit, only the relatively restricted class of clusters corresponding to ``bicolored'' graphs needs to be considered, and the free‐energy calculation may be easily extended to eight‐particle clusters. A Padé analysis of the second composition derivative of free energy (related to composition fluctuations and solution turbidity in the critical region) yields results at variance with simple solution theories, but of the same type that may by analogy be inferred from recent three‐dimensional Ising model susceptibility numerical analyses. A list is appended of the multiply connected mixture graphs through five points and bicolored graphs through eight points.