# Phase Field Outline

- Method Introduction
- Equation Summary
- Solving with FEM
- Free-Energy Based System
- Multi-Phase Free Energies
- Grain-Boundary Migration
- Coupling to Mechanics and Heat Conduction
- 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 variablesâ€¦

### Non-Conserved Order Parameters

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