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 $f(\phi ,\nabla \phi )$,

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

The system evolves in the direction that reduces the free energy. In the Ginzburg-Landau model, the phase field evolves with the following equation of motion that does not conserve the total amount of the solid phase (or liquid phase),

${\frac {\partial \phi }{\partial t}}=-M{\frac {\delta F}{\delta \phi }}$  (GL)
where ${\frac {\delta F}{\delta \phi }}={\frac {\partial f}{\partial \phi }}-\nabla \left({\frac {\partial f}{\partial \nabla \phi }}\right)$ is the variational derivative.  

In the Cahn-Hillard model, the phase field eolves with the following equation of motion that conserve the total amount of the solid phase (or liquid phae),

${\frac {\partial \phi }{\partial t}}=\nabla \cdot \left[M\nabla \left({\frac {\delta F}{\delta \phi }}\right)\right]$ (CH)

Note here that we have been treating $f$ as a function with two independent input variables, which take the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla\phi} , respectively.

A stationary, planar interface (between liquid and solid) is described by the following solution that satisfies the boundary conditions: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x=+\infty) = 1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x=-\infty) = 0} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x) = \frac{1}{2} [ 1 + \tanh(x/\xi) ] }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi = 2\sqrt{\varepsilon/U}}
 is the interface width.

Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{x} = (x,y,z)} but the phase field in this solution is independent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} , i.e. the interface is normal to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} axis.

The interface energy (per unit area) of this solution is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \int_{-\infty}^{\infty} f(\phi,\nabla\phi) dx = \frac{1}{3} \sqrt{\varepsilon U} }

Algorithm

Represent phi by a three dimensional grid of size Nx x Ny x Nz.

How to compute variational derivative. Centered numerical difference.

How to constrain the total amount of solid phase within the Ginzburg-Landau equation. Subtract off average forces, equivalent to using a Lagrangian multiplier, or adjusting the chemical potential of the solid phase.