Free Energy Function Materials

The free energy functional is the basis of the evolution of the phase field variables. However, the free energy itself is never used directly in the phase field equations, rather derivatives of the free energy are required. In the case of a free energies involving multiple coupled variables, different derivatives will be required for each residual equation.

In the free energy based model approach, a Function Material class defines these required free energy derivatives, which are used by the respective kernels. These derivatives can be defined in the code by the user or they can be generated using automatic differentiation. The derivatives can also come from CALPHAD type free energies. The required order of derivative that is required for the residual are shown on the Basic Phase Field Equations page for the Allen-Cahn equation and for the two solution approaches for the Cahn-Hilliard equations. One additional derivative is required to define the Jacobian term used in the solution of the nonlinear system of FEM equations. It is common to also define the full free energy, even when not required, for visualization and debugging purposes.

To demonstrate this approach, we focus on a simple system with a single conserved concentration that varies from -1 to 1. The free energy in this case is

and its derivatives are

The second and third derivatives would be required to use the direct solution method (via CahnHilliard) and the first and second derivatives would be required to use the split solution method (via SplitCHParsed). This model has been implemented in the MathFreeEnergy material found in the phase field module of MOOSE, inheriting from DerivativeFunctionMaterialBase. The code from MathFreeEnergy is shown below:

Real
MathFreeEnergy::computeF()
{
  return 1.0/4.0*(1.0 + _c[_qp])*(1.0 + _c[_qp])*(1.0 - _c[_qp])*(1.0 - _c[_qp]);
}

Real
MathFreeEnergy::computeDF(unsigned int j_var)
{
    return _c[_qp]*(_c[_qp]*_c[_qp] - 1.0);
}

Real
MathFreeEnergy::computeD2F(unsigned int j_var, unsigned int k_var)
{
    return 3*_c[_qp]*_c[_qp] - 1.0;
}

Real
MathFreeEnergy::computeD3F(unsigned int j_var, unsigned int k_var, unsigned int l_var)
{
    return 6*_c[_qp];
}

An alternative to writing your own free energy materials is to take advantage of the DerivativeParsedMaterial, where the free energy is entered in the input file and all required derivatives are taken automatically. This approach is highly encouraged, as it drastically simplifies model development.