Phase Behavior of Ternary Polymer Mixtures in a Common Solvent

The Edmond-Ogston model for phase separation isextendedto ternary polymer mixtures in a common solvent (de facto a quaternarymixture). The model assumes a truncated virial expansion of the Helmholtzfree energy up to the second-order terms in the concentration of thepolymers, and the second virial coefficients (B (11), B (22), B (33), B (12), B (13), B (23)) are the six parametersof the model. New results from this model are presented in relationto earlier work on binary mixtures: a necessary condition for thevirial coefficients for the occurrence of phase separation in twoor three phases, an analysis of the different regions of (local) thermodynamicinstability using the Descartes sign rule, an expression for the criticalcurves, a relation between the tangents in points along the criticalcurve, a relation between the concentration of components in the differentphases according to the so-called Lambert-W function, and a consistencycheck for the composition of coexisting phases in ternary mixtures.The obtained results are evaluated in the maximally symmetric versionof the model, where (B (11), B (22), B (33)) are equal and (B (12), B (13), B (23)) are equal, which leads to two remarkableobservations: the concentration range over which two phases are formedis relatively narrow; not all phase separation occurs within a Gibbstriangle, but also, out-of-Gibbs-triangle binodalsare observed. These results lead to a deeper insight into the phasebehavior of ternary mixtures and show promise as a stepping stonetoward modeling phase separation in mixtures with many components.

Automated summary

This article extends the Edmond-Ogston model to ternary polymer mixtures and presents new research results, including the necessary conditions for phase separation, analysis of thermodynamic instability, expression of critical curves, relationship between tangents on the critical curve, relationship between concentration of components in different phases, and consistency check of coexisting phase composition in ternary mixtures. The results show that the concentration range for two-phase formation is relatively narrow and phase separation can occur both within and outside the Gibbs triangle. These findings deepen our understanding of the phase behavior of ternary mixtures and have potential applications in modeling phase separation in mixtures with multiple components.