An Analysis of a Class of Variational Models for Heterogeneous Phase Separation

Suppose the free energy of the fluid mixture to have periodic heterogeneities of the scale $\delta$ and the phase separation to happen on the scale of $\varepsilon$. We will survey physical motivation and recent works, where variants of the famous Modica-Mortola functional are used to model this phenomenon. While $\Gamma-$limits depend on the rate of convergence between $\delta$ and $\varepsilon$ (and many of the regimes are still open—especially with thermally responsive fluid phases), we will specifically discuss recent results in the regimes $\delta\lless\varepsilon$ and $\varepsilon\lless \delta$.