Single Phase Field Model with Isotropic Surface Energy

Single Phase Field Model with Isotropic Surface Energy

Seunghwa Ryu and Wei Cai

This tutorial describes the theory and numerical implementation of a basic 3D phase field model. The model has been implemented in both matlab and in MD++. We describe the algorithm of these codes and how to use them in detail. A brief introduction of the phase field model can e found in Chapter 11 of

Computer Simulations of Dislocations, V. V. Bulatov and W. Cai, Oxford University Press (2006).

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Download files

The matlab program can be downloaded here. You can download and extract the matlab scripts into a directory of your choice and run the test cases by executing, e.g. pf3d_single_iso.m.

To run the MD++ test cases, first you need to install MD++ on your computer by following the instructions on MD++ Manuals.

Second, copy files ??? and ??? to your input file directory of MD++. You can do so by the following commands (assuming you have installed MD++ in ~/Codes/MD++).

 export MDPP=~/Codes/MD++
 mkdir -p ${MDPP}/scripts/work/phasefield
 cd ${MDPP}/scripts/work/phasefield
 wget http://micro.stanford.edu/mediawiki-1.11.0/images/phasefield.tcl.txt -O phasefield.tcl

Compile executable file

This step is only necessary for running MD++ test cases. In this tutorial, we describe how to run these examples on a linux computer (using mc2.stanford.edu as an example).

You can can compile the executable phasefield by typing

 cd ${MDPP}
 make phasefield build=R SYS=mc2

If compilation is successful, this will create binary file sworig_mc2 file in your ${MDPP}/bin directory. To compile MD++ on a different system, you should set variable SYS to a different value.

Theory

In this simple phase field model, the fundamental degree of freedom is a phase field, $\phi (\mathbf {x} )$. The range of the value for $\phi$ is between 0 and 1. For example, we can think of the region where $\phi =1$ as the solid phase and the region where $\phi =0$ as the liquid phase. The region where $\phi$ transitions (rapidly) from 0 to 1 is then the liquid-solid interface.

A free energy functional $F[\phi (\mathbf {x} )]$ is defined as the integral of the local free energy density $f(\phi ,\nabla \phi )$,

$F[\phi (\mathbf {x} )]=\int f(\phi (\mathbf {x} ),\nabla \phi (\mathbf {x} ))d^{3}\mathbf {x}$
$f(\phi ,\nabla \phi )=\varepsilon (\nabla \phi )^{2}+U\phi ^{2}(1-\phi )^{2}$