Phase Field Modeling
IntroductionPhase field method is a power computational tool to model the temporal and spatial evolution of microstructure in mesoscale region. Most materials have a complex microstructure that arise due to grain, grain boundaries, different phases, compositions, orientations and crystallography. The material properties can be improved if we could understand the evolution of the microstructure. Phase field method has been successful to predict the microstructure evolution of the material using the phase field variable, which is driven by thermodynamic and kinetic properties. The phase field method is successful to model the microstructure evolution in wide variety of material processes, such as solidification, martensitic phase transformation, precipitate growth and coarsening, grain growth and solidstate transformation. The phase field model starts with description of microstructure using a set of conserved and/or nonconserved field variable that are continuous across the interfacial regions separating different phases. Then, the evolution of these variables is determined by the spatial and temporal evolution of these field variables. The evolution is governed by the CahnHilliard nonlinear diffusion equation ^{[1]} and the time dependent GinzburgLandau (AllenCahn) relaxation equation^{[2]}. 
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Governing EquationsThe temporal and spatial evolution of a nonconserved order parameter(s) can be founded by the time dependent GinzburgLandau equation:
where, η represents the order parameter, L is the kinetic coefficient, F is the total energy of the system, is the thermodynamic driving force for spatial and temporal evolution of η.
The local specific energy can be approximated by Landau polynomial in terms of longrange order parameters η_{p}. For an example, the equation below shows the simplest sixthorder polynomial for for the local specific free energy;

References
 ↑ J.W. Cahn, J.E. Hilliard, The Journal of Chemical Physics 28 (1958) 258.
 ↑ S.M. Allen, J.W. Cahn. Acta Matellurgica 27 (1979) 10851095.
 ↑ Mamivand M, Zaeem MA, El Kadiri H, Chen LQ. Acta Materialia 2013;61:5223.
 ↑ Mamivand M, Asle Zaeem M, El Kadiri H. Effect of variant strain accommodation on the threedimensional microstructure formation during martensitic transformation: Application to zirconia. Acta Mater. 2015;87:4555.
 ↑ Khachaturyan A. Theory of structural transformations in solids. New York: Wiley;1983.