Recent advances in the evolution of interfaces: thermodynamics, upscaling, and universality

TL;DR

This paper reviews thermodynamic models of interface evolution in heterogeneous binary mixtures, introduces new upscaled equations, and reveals a universal coarsening rate independent of heterogeneity, with applications to lithium batteries.

Contribution

It presents two novel upscaled equations for binary mixtures in heterogeneous systems and uncovers a universal coarsening rate unaffected by system heterogeneity.

Findings

Upscaled equations accurately model binary mixtures in porous media.

Coarsening rate of ${ m O}(t^{1/3})$ is universal across heterogeneous and homogeneous systems.

Phase field models based on free energies are promising for practical applications.

Abstract

We consider the evolution of interfaces in binary mixtures permeating strongly heterogeneous systems such as porous media. To this end, we first review available thermodynamic formulations for binary mixtures based on \emph{general reversible-irreversible couplings} and the associated mathematical attempts to formulate a \emph{non-equilibrium variational principle} in which these non-equilibrium couplings can be identified as minimizers. Based on this, we investigate two microscopic binary mixture formulations fully resolving heterogeneous/perforated domains: (a) a flux-driven immiscible fluid formulation without fluid flow; (b) a momentum-driven formulation for quasi-static and incompressible velocity fields. In both cases we state two novel, reliably upscaled equations for binary mixtures/multiphase fluids in strongly heterogeneous systems by systematically taking thermodynamic features such as free energies into account as well as the system's heterogeneity defined on the microscale such as geometry and materials (e.g. wetting properties). In the context of (a), we unravel a \emph{universality} with respect to the coarsening rate due to its independence of the system's heterogeneity, i.e. the well-known ${\cal O}(t^{1/3})$-behaviour for homogeneous systems holds also for perforated domains. Finally, the versatility of phase field equations and their \emph{thermodynamic foundation} relying on free energies, make the collected recent developments here highly promising for scientific, engineering and industrial applications for which we provide an example for lithium batteries.