Phase Field Lectures

Regular solution model for phase separating and ordering systems; Failure of classical diffusion equation for systems that undergo spinodal decomposition.

Mathematical preliminaries: Taylor expansion of free energy, Gauss' theorem, group theory, and variational calculus.

Modification of classical diffusion equation for systems that undergo spinodal decomposition; Cahn-Hilliard equation and the linear stability analysis.

Analytical solution of Cahn-Hilliard equation in 1-D; Interface width and interfacial energy and their scaling with simulation parameters.

Order-disorder transformation and Allen-Cahn equation; Analytical solution of Allen-Cahn in 1-D; Characterization of the tanh solution for the Allen-Cahn equation.

Coherent misfitting precipitates; Eshelby problem; Khachaturyan's generalization.

Introduction to asymptotic analysis; Derivation of motion by mean curvature from phase-field models of grain growth.

Finite difference and spectral implementation of Allen-Cahn equation (order-disorder transformation, grain growth).

Incorporating elastic stress effects into phase-field models.