Phase segregation and interface dynamics in kinetic systems

TL;DR

This paper analyzes a kinetic model of two interacting particle species, deriving macroscopic interface dynamics in the diffusive and sharp interface limits, linking microscopic interactions to classical interface motions.

Contribution

It introduces a detailed derivation of interface motion equations from a kinetic model, connecting microscopic particle interactions to macroscopic interface dynamics.

Findings

Interface motion combines Mullins-Sekerka and Hele-Shaw dynamics.

Derived macroscopic equations match those in polymer blend modeling.

Formal expansions reveal the interface evolution in the sharp limit.

Abstract

We consider a kinetic model of two species of particles interacting with a reservoir at fixed temperature, described by two coupled Vlasov-Fokker-Plank equations. We prove that in the diffusive limit the evolution is described by a macroscopic equation in the form of the gradient flux of the macroscopic free energy functional. Moreover, we study the sharp interface limit and find by formal Hilbert expansions that the interface motion is given in terms of a quasi stationary problem for the chemical potentials. The velocity of the interface is the sum of two contributions: the velocity of the Mullins-Sekerka motion for the difference of the chemical potentials and the velocity of a Hele-Shaw motion for a linear combination of the two potentials. These equations are identical to the ones in Otto-E modelling the motion of a sharp interface for a polymer blend.