Thermodynamically consistent dynamic boundary conditions of phase field models

Abstract

We present a general, constructive method to derive thermodynamically consistent models and consistent dynamic boundary conditions hierarchically following the generalized Onsager principle. The method consists of two steps in tandem: the dynamical equation is determined by the generalized Onsager principle in the bulk firstly, and then the surface chemical potential and the thermodynamically consistent boundary conditions are formulated subsequently by applying the generalized Onsager principle at the boundary. The application strategy of the generalized Onsager principle in two-step yields thermodynamically consistent models together with the consistent boundary conditions that warrant a non-negative entropy production rate (or equivalently non-positive energy dissipation rate in isothermal cases) in the bulk as well as at the boundary. We illustrate the method using phase field models of binary materials elaborate on two sets of thermodynamically consistent dynamic boundary conditions. These two types of boundary conditions differ in how the across boundary mass flux participates in boundary surface dynamics. We then show that many existing thermodynamically consistent, binary phase field models together with their dynamic or static boundary conditions are derivable from this method. As an illustration, we show numerically how dynamic boundary conditions affect crystal growth in the bulk using a binary phase field model.

Thermodynamically Consistent Models for Coupled Bulk and Surface Dynamics

“…We note that when M is a differential operator, such as in the Cahn–Hilliard equation system, the symmetric property of M is also determined by the boundary conditions of the system as well. For a system where inertia is non-negligible and there coexist irreversible and reversible dynamics in the non-equilibrium process, we extend the force balance equation to a generalized Onsager principle [ 1 , 24 ] where is the general chemical potential, represents the inertia force, and is a measure of mass. We next use the generalized Onsager principle to derive the general phase field model along with its consistent boundary conditions for a binary material system.…”

“…This gives one a great deal of flexibility to fine-tune the volume fraction flux at the boundary to control the bulk dynamics. So, there is no surprise that this model includes many existing thermodynamically consistent phase field models with DBCs in the literature [ 23 , 24 , 27 , 28 , 29 ].…”

“…The governing equation system together with the boundary conditions in this model is given as follows When , and the governing equation system reduces to The corresponding energy dissipation rate is given by Setting , this system reduces to the general model with the first type dynamic boundary condition derived in [ 24 ] by the authors. Similarly, we deduce the general model with the second type dynamic boundary condition in [ 24 ] by selecting the mobility operator corresponding to the second type boundary condition alluded to in the previous section.…”

“…In particular, when one considers surface dynamics, an issue emerges right away that is what is the relation between the phase field variable in the bulk and the one on the boundary? By assuming these two are the same at the boundary, a series of studies have examined the issue of thermodynamically consistency [ 23 , 24 , 25 , 26 , 27 ]. By not assuming that they coincide on the boundary, Knopf et al derived a set of boundary conditions for the Cahn–Hilliard model (known as the Knopf–Lam model) in 2020 [ 28 ] and for non-local models (known as the Knopf–Signori model) in 2021 [ 29 ].…”