Communications in Mathematical Sciences

Volume 21 (2023)

Number 3

Thermodynamically consistent dynamic boundary conditions of phase field models

Pages: 859 – 883



Xiaobo Jing (Beijing Computational Science Research Center, Beijing, China; and Department of Mathematics, University of South Carolina, Columbia, S.C., U.S.A.)

Qi Wang (Department of Mathematics, University of South Carolina, Columbia, S.C., U.S.A.)


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 steps 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 elaborated 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 the surface dynamics at the boundary. We then show that many existing thermodynamically consistent, binary phase field models together with their dynamic or static boundary conditions are derivable from this approach. As an illustration, we show numerically how dynamic boundary conditions affect crystal growth in the bulk using a binary phase field model.


thermodynamically consistent model, phase field model, dynamic boundary conditions, binary materials, energy dissipation

2010 Mathematics Subject Classification

35G30, 35G60, 35Q79, 35Q82

Xiaobo Jing’s research is partially supported by NSFC awards #11971051, #12147165 and NSAF-U1930402 to CSRC.

Qi Wang’s research is partially supported by NSF awards OIA-1655740 and a GEAR award from SC EPSCoR/IDeA Program.

Received 12 April 2022

Received revised 6 August 2022

Accepted 15 August 2022

Published 27 February 2023