Predictive value of unstable modes for the number of equilibrium phases

Establish the relationship between the number of negative eigenvalues U of the Hessian of the free-energy density (linear stability of the homogeneous state) and the number of coexisting equilibrium phases M in the extended Flory–Huggins model with quadratic and cubic (binary and ternary) interactions for incompressible multicomponent mixtures, and determine the conditions under which the hypothesis M = U + 1 holds or fails to predict fully phase-separated states.

Background

The paper analyzes multicomponent mixtures with quadratic and cubic interactions using both linear stability of homogeneous states and full equilibrium computations of phase coexistence. The number of unstable modes U (negative eigenvalues of the Hessian) is often used as a proxy for the number of phases M, leading to the heuristic M = U + 1. However, the authors note that linear stability probes only infinitesimal deviations from homogeneity and may not reflect fully phase-separated equilibria.

Clarifying the quantitative link between U and M would guide when linear stability is a reliable predictor and when full equilibrium calculations are required, particularly in systems with higher-order interactions.