Interface dynamics in a degenerate Cahn-Hilliard model for viscoelastic phase separation

TL;DR

This paper derives non-local geometric evolution laws from a degenerate Cahn-Hilliard model for viscoelastic phase separation, revealing new third-order interface dynamics with gradient flow structure.

Contribution

It introduces a novel asymptotic analysis linking a degenerate Cahn-Hilliard model to new third-order interface evolution laws with gradient flow properties.

Findings

Zero level set evolution approximates intermediate surface diffusion flow.

Non-constant coupling leads to third-order evolution laws with square root of Laplace-Beltrami operator.

Constrained elliptic problem encodes the gradient structure of the interface dynamics.

Abstract

The formal sharp-interface asymptotics in a degenerate Cahn-Hilliard model for viscoelastic phase separation with cross-diffusive coupling to a bulk stress variable are shown to lead to non-local lower-order counterparts of the classical surface diffusion flow. The diffuse-interface model is a variant of the Zhou-Zhang-E model and has an Onsager gradient-flow structure with a rank-deficient mobility matrix reflecting the ODE character of stress relaxation. In the case of constant coupling, we find that the evolution of the zero level set of the order parameter approximates the so-called intermediate surface diffusion flow. For non-constant coupling functions monotonically connecting the two phases, our asymptotic analysis leads to a new family of third-order evolution laws with associated propagation operators behaving, at leading order, like the square root of the minus Laplace-Beltrami operator. In this case, the normal velocity of the moving sharp interface arises as the Lagrange multiplier in a constrained elliptic equation, which is at the core of our derivation. The constrained elliptic problem can be solved rigorously by a variational argument, and is shown to encode the gradient structure of the effective geometric evolution law. The asymptotics are presented for deep quench, an intermediate free boundary problem based on the double-obstacle potential.