Variational Methods for Microstructural Evolution

Published

October 12, 2021

Author(s)

W Carter, J E. Taylor, John W. Cahn

Abstract

The application of variational methods and gradient flows to principles of microstructural evolution are reviewed. For the cases considered in this paper, gradient flows specify a trajectory along with the free energy decreases most rapidly. The gradient depends on the choice of inner product and on kinetic coefficients for the system. The choice of inner product is a central theme for microstructural evolution. The L2 inner product gives microstructural evolution equations which follows from the gradient flow implied by the L2 inner product (e.g., the Allen-Cahn and grain boundary motion). The H-1 inner product gives microstructural evolution for conserved quantities. We tabulate how familiar microstructural evolution equations, such as Cahn-Hilliard the Mullins surface diffusion equation, follow from the choice of the H-1 inner product. The variational approach is not limited to continuous functions or free energies: we provide an example of L2 gradient flow by calculating the motion of a triple junction with completely faceted grain boundaries in two dimensions.

Citation

Journal of Metals

Pub Type

Journals

Keywords

Allen-Cahn, Cahn-Hilliard, gradient flows, Hilbert Spaces, interface motion, microstructural evolution, triple junctions

Citation

Carter, W. , Taylor, J. and Cahn, J. (2021), Variational Methods for Microstructural Evolution, Journal of Metals (Accessed February 15, 2026)