# Phase Field Modeling

#### Introduction

Phase 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 solid-state transformation. The phase field model starts with description of microstructure using a set of conserved and/or non-conserved 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 Cahn-Hilliard nonlinear diffusion equation [1] and the time dependent Ginzburg-Landau (Allen-Cahn) relaxation equation[2].

Images

Thermally induced tetragonal to monoclinic martensitic phase transformatio in 2D zirconia single crystal. The picture on left shows the result from phase field simulation versus picture on the right from TEM micrograph [3]
Phase field simulation of 3D single crystal Zirconia. The model was able to predict the properties like surface uplift and variation impingement[4]

#### Governing Equations

The temporal and spatial evolution of a non-conserved order parameter(s) can be founded by the time dependent Ginzburg-Landau equation:

$\frac {\part\eta_p(r,t)}{\part t} = - L \times \frac {\delta F_{total}}{\delta \eta_p(r,t)}$

where, η represents the order parameter, L is the kinetic coefficient, F is the total energy of the system, $\frac {\delta F_{total}}{\delta \eta_p(r,t)}$ is the thermodynamic driving force for spatial and temporal evolution of η.

Evolution of conserved variables is given by Cahn-Hilliard equation:

$\frac {1}{V_m} \frac{\part x_B(r,t)}{\part t} = \nabla . M\nabla \frac {\delta F}{\delta x_B(r,t)}$

For an martensitic phase transformation, the total free energy can be written as:

$F = F_{ch} + F_{el}$

where, Fch is chemical free energy and Fel is elastic strain energy.

The total chemical free energy can be written as:

$F_{ch} = \int_V [f(\eta_1,\eta_2,\eta_3,...,\eta_n) + \frac {1}{2} \sum_{p=1}^{n} \beta_{ij}(p) \nabla_i \eta_p \nabla_{,j} \eta_p]dV$

$n=1,...,p$

where βij(p) is a positive gradient energy coefficient and ∇ is the gradient operator. f(η123,...,ηn) is the local specific free energy, which defines the basic bulk thermodynamic properties of the system.

The local specific energy can be approximated by Landau polynomial in terms of long-range order parameters ηp. For an example, the equation below shows the simplest sixth-order polynomial for for the local specific free energy;

$f(\eta_1,\eta_2,...,\eta_n) = \Delta G[\frac {a}{2} (\eta_1^2 + \eta_2^2 + ... + \eta_n^2) - \frac {b}{4} (\eta_1^4 + \eta_2^4 + ... + \eta_n^2) + \frac {c}{6} (\eta_1^2 + \eta_2^2 + \eta_3^2 + ... + \eta_n^2)^3]$

where ΔG is the chemical driving force representing the difference in the specific chemical free energy between the parent and the product phases.

Similarly, as shown by Khachaturyan[5]the elastic strain energy can be expressed as a function of order parameters as shown below:

$F_{el} = \frac {1}{2} \int_V \sigma_{ij} \epsilon_{ij}^{el} dV = \frac {1}{2} \int_V C_{ijkl} \epsilon_{kl}^{el} \epsilon_{ij}^{el} dV$

where the elastic strain εijel(r) is the difference between the total strain, εijtot(r), and the stress-free strain, εij0(r):

$\epsilon_{ij}^{el} (r) = \epsilon_{ij}^{tot} (r) - \epsilon_{ij}^{0} (r) = \epsilon_{ij}^{tot} (r) - \sum \epsilon_{ij}^{00} (p) \eta_p^2(r)$

where, εij00(p) is the transformation strain and is defined by.

$\epsilon_{ij}^{00} (p) = U_{ij}(p)-\delta_{ij}$

where Uij(p) is the symmetric right stretch tensor of the deformation gradient.

## References

1. J.W. Cahn, J.E. Hilliard, The Journal of Chemical Physics 28 (1958) 258.
2. S.M. Allen, J.W. Cahn. Acta Matellurgica 27 (1979) 1085-1095.
3. Mamivand M, Zaeem MA, El Kadiri H, Chen LQ. Acta Materialia 2013;61:5223.
4. Mamivand M, Asle Zaeem M, El Kadiri H. Effect of variant strain accommodation on the three-dimensional microstructure formation during martensitic transformation: Application to zirconia. Acta Mater. 2015;87:45-55.
5. Khachaturyan A. Theory of structural transformations in solids. New York: Wiley;1983.