Associative Phase Behavior and Negative Second-Order Virial Coefficients in Analytical Calculations of the Spinodal for Complex Mixtures Composed of Many Components

The calculation of spinodals for mixtures composed of many macromolecular components ( N ≥ 2, not counting the solvent explicitly as a component) for a model including interactions up to quadratic terms in concentration is extended to the case of negative (second-order) virial coefficients, typically indicating attractive interactions between the components. Surprisingly, it is found that different permutations of the signs of the virial coefficients do not always lead to different spinodals. This observation is explained by means of an analysis of the effects of the signs of the virial coefficients in calculations in a parameter space introduced previously as “linear programming space”, in which calculations for the spinodal can be done efficiently. The analysis also gives rise to an algorithm that indicates which parts of the parameter space can be ignored when calculating the physically relevant coordinates of the spinodal. After completing the formal analysis, examples of the role of negative virial coefficients for binary, ternary, and quaternary mixtures are discussed. The results impact fields like polymer and food science (e.g., predicting effects of polydispersity on phase behavior) and biophysics (e.g., liquid–liquid phase separation phenomena in cells).