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.