Phase Field Outline

  1. Method Introduction
  2. Equation Summary
  3. Solving with FEM
  4. Free-Energy Based System
  5. Multi-Phase Free Energies
  6. Grain-Boundary Migration
  7. Coupling to Mechanics and Heat Conduction
  8. Interaction with Experiments

Atomistic vs Mesoscale Modeling

  • Atomistic computational materials science approaches can be used to investigate mechanisms and determine material properties.

Density Functional Theory (DFT)

  • DFT is a quantum mechanical modeling method to investigate the electronic structure of atoms or molecules.
  • Based on first principles, so has few assumptions and can handle complicated, multicomponent systems and reactions.
  • Computationally expensive (between 100 to 1000 atoms)

Molecular Dynamics (MD)

  • MD determines atom behavior by numerically solving Newton's equations of motion for a system of interacting particles
  • Simulates up to a billion atoms, to investigate microstructure evolution.
  • Small length and time scales
  • A unique potential function must be developed for each material

  • Mesoscale simulation predicts material behavior at micron length scales and diffusive time scales
  • Mesoscale models must have known mechanisms built in, and require values for various material properties.

Microstructure Evolution Approaches

Frost et al., 1988

Anderson et al., 1984

Fan and Chen, 1997

  • Mean field models
    • Predict the evolution of average properties
    • Include rate theory
  • Front tracking
    • Uses line elements to track interface migration
    • Requires complex relationships to model coalescence and phase/grain vanishing
  • Monte Carlo Potts Models
    • Uses stochastic methods based on probabilities to model microstructure change.
    • Non-dimensional
    • Modeled on a fixed uniform grid
  • Phase field
    • Continuous variables are used to represent the microstructure
    • A free energy functional defines the microstructure evolution

The Phase Field Method

  • Microstructure described by a set of continuous variables…

Non-Conserved Order Parameters


  • The variables evolve to minimize a functional defining the free energy