Multi Phase Field Model with Isotropic Interface Energy

Multi Phase Field Model with Isotropic Interface Energy

Seunghwa Ryu, Yanming Wang and Wei Cai

This tutorial describes the theory and numerical implementation of a 3D multi phase phase field model with isotropic interface energy. 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 be 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 from this link: Pf3d multi iso.m. For the test case you need only one matlab file. You can download it into a directory of your choice, using the following command

wget http://micro.stanford.edu/mediawiki/images/b/b7/Pf3d_multi_iso.m.txt -O Pf3d_multi_iso.m

and run it within Matlab.

Theory

The fundamental degrees of freedom in multi phase field model are a series of phase fields 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_i(x)} ,where i=1:N and N is the number of different phases in the system. Thereby, Similar to the previous wiki page but considering the interfaces between different phases, the free energy functional and the local free energy density are given by

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 F[\phi_1(\mathbf{x}),...,\phi_N(\mathbf{x})] = \int \sum_{k\neq{i}}{f_{ik}(\phi_i(\mathbf{x}), \nabla\phi_i(\mathbf{x}),\phi_k(\mathbf{x}), \nabla\phi_k(\mathbf{x})) d^3 \mathbf{x}} }

$f_{ik}(\phi _{i},\nabla \phi _{i},\phi _{k},\nabla \phi _{k})=\varepsilon _{ik}(\phi _{i}\nabla \phi _{k}-\phi _{k}\nabla \phi _{i})^{2}+U_{ik}\phi _{i}^{2}\phi _{k}^{2}$

Intuitively, if we let 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_k=1-\phi_i} , then 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 f_{ik}} will be reduced to a formalism similar to the single phase model.

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 f_{ik}(\phi_i, \nabla\phi_i,\phi_k, \nabla\phi_k) = \varepsilon_{ik}|\nabla\phi_i|^2 + U_{ik}\phi_i^2(1-\phi_i)^2 }

In the Ginzburg-Landau model, The evolution of the ith phase without constraint can be expressed with the following equation of motion,

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 \frac{\partial \phi_i}{\partial t} = - M_{ik} \sum_{k\neq i}\frac{\delta F_{ik}}{\delta \phi_i} }
  (GL)

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 M_{ik}}
 is the mobility of i-k interface, 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  F_{ik}=\int f_{ik}(\phi_i, \nabla\phi_i,\phi_k, \nabla\phi_k) d^3 \mathbf{x}}
 

For the free energy density function given above, the explicit expression of the variational derivative 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 \frac{\delta F_{ik}}{\delta \phi_i} = 2 U_{ik} \phi_i \phi_k (\phi_i - \phi_k) - 2 \varepsilon_{ik} (\phi_k\nabla\phi_i^2-\phi_i\nabla\phi_k^2)}

In the Cahn-Hillard model, the system evolves with the following equation of motion that conserve the total amount of each phase,

 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 \frac{\partial \phi_i}{\partial t} = \nabla \cdot \left[ M_{ik} \nabla \left( \frac{\delta F_{ik}}{\delta \phi_i} \right) \right] }
 (CH)

Algorithm

In matlab, We present liquid 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_1} , solid 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_2} , and vapor 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_3} , three phase fields by a 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 1\times3} cell. Each element phi{i} of the cell is a three-dimensional array,