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Variational Methods for Microstructural Evolution



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


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.
Journal of Metals


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


Carter, W. , Taylor, J. and Cahn, J. (2021), Variational Methods for Microstructural Evolution, Journal of Metals (Accessed May 29, 2024)


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Created October 12, 2021