A continuum framework for phase field with bulk-surface dynamics

TL;DR

This paper develops a comprehensive continuum mechanical framework for phase field models with coupled bulk-surface dynamics, generalizing existing theories and deriving thermodynamically consistent equations and boundary conditions.

Contribution

It introduces a generalized principle of virtual powers and free-energy imbalance for bulk-surface phase field coupling, extending previous models and deriving explicit field equations and boundary conditions.

Findings

Derived explicit bulk-surface phase field equations resembling Cahn--Hilliard.

Generalized the principle of virtual powers for discontinuous normal fields.

Provided thermodynamically consistent boundary conditions.

Abstract

This continuum mechanical theory aims at detailing the underlying rational mechanics of dynamic boundary conditions proposed by Fischer, Maass, & Dieterich [1], Goldstein, Miranville, & Schimperna [2], and Knopf, Lam, Liu & Metzger, [3]. As a byproduct, we generalize these theories. These types of dynamic boundary conditions are described by the coupling between the bulk and surface partial differential equations for phase fields. Our point of departure within this continuum framework is the principle of virtual powers postulated on an arbitrary part $\mathcal{P}$ where the boundary $\partial\mathcal{P}$ may lose smoothness. That is, the normal field may be discontinuous along an edge $\partial^2\mathcal{P}$. However, the edges characterizing the discontinuity of the normal field are considered smooth. Our results may be summarized as follows. We provide a generalized version of the principle of virtual powers for the bulk-surface coupling along with a generalized version of the partwise free-energy imbalance. Next, we derive the explicit form of the surface and edge microtractions along with the field equations for the bulk and surface phase fields. The final set of field equations somewhat resembles the Cahn--Hilliard equation for both the bulk and surface. Lastly, we provide a suitable set of constitutive relations and thermodynamically consistent boundary conditions.