# 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

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# Atomistic vs Mesoscale Modeling

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

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### 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)
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### 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
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* 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.

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# Microstructure Evolution Approaches

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    Frost et al., 1988

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    Anderson et al., 1984

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    Fan and Chen, 1997

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* 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

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# The Phase Field Method

* Microstructure described by a set of continuous variables…

### Non-Conserved Order Parameters

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