Grand Potential Multi-Phase, Multi-Order Parameter Model

A phase-field model for an arbitrary number of phases, grains per phase, and chemical constituents has been implemented in MOOSE. This model is based on a functional of the grand potential density (rather than the Helmholtz free energy density as is more commonly used in phase-field modeling). The model was originally described in Aagesen et al. (2018).

The model has certain advantages and disadvantages relative to other phase-field models in the literature.

Advantages

  • Allows interfacial thickness and energy to be set independently, enabling coarser mesh, improved computational performance

    • Chemical free energy contribution is removed from interfacial energy

    • Similar to Kim-Kim-Suzuki (KKS) phase-field model in this respect, but do not need separate phase concentration variables, so performance is improved

  • Prevents spurious formation of any additional phase ("ghost phase") at two-phase interfaces

  • Works well with ideal solution model/low equilibrium concentration situations compared with KKS

Disadvantages

  • Currently limited to parabolic, ideal solution, dilute solution chemical free energies

    • Need free energy to be convex, and the relationship between composition and chemical potential needs to be analytically invertible

  • Evolution equation for chemical potential less intuitive compared to evolution equation for composition

  • When using the evolution equation for chemical potential, discretization error results in small variations in mass (typically < 0.1 %). To prevent this from occurring, an alternative version of the model that strictly enforces mass conservation is available. However, this requires an additional concentration variable, increasing computational cost.

What is the grand potential?

The grand potential is the thermodynamic potential that is minimized at constant temperature, volume, and chemical potential. For phase , the grand potential density per unit volume is given by where is the Helmholtz free energy density of phase , is the number density of atoms of solute species , and is the chemical potential of solute species .

Grand potential functional

The model is derived from a functional of the total grand potential : where is a multi-well contribution to the grand potential density, is the gradient energy contribution, and is the chemical energy contribution. is a function of the grand potential densities of each of the phases in the system: where is a switching function that interpolates the chemical contribution based on what phase the microstructure is in at each point. is given by where is a constant with dimensions of energy per unit volume and is given by where is an order parameter representing grain of phase ( index phases and index grains). is a set of dimensionless parameters that adjust grain boundary energies and interfacial energies. This function has minima at the equilibrium values of the order parameters. is given by where is the gradient energy coefficient, which has dimensions of energy per unit length.

Evolution equations

The order parameters representing the grains of each phase each evolve by an Allen-Cahn equation, derived from the grand potential functional: where is the Allen-Cahn interfacial mobility.

In the original model formulation, the evolution of composition is transformed into an evolution equation for chemical potential. The general multi-species evolution equation is given in Aagesen et al. (2018). The simplest form of the evolution equation for solute species is given by where is the self-diffusivity of species and is the susceptibility, defined as depends on the form of the chemical free energy density used.

MOOSE implementation of the evolution equations

The kernels used to implement the terms in the evolution equations are shown below with underbraces. For the Allen-Cahn equation:

For the chemical potential evolution equation,

Creating an input file for this model can be simplified through the use of the MOOSE action GrandPotentialKernelAction, which automates the process of adding the required kernels.

Parameterization: Interface thickness and energy

For an interface between grain of phase and grain of phase , the interfacial thickness and interfacial energy are set by the combination of parameters , , and . For , analytical relationships exist: A convenient strategy for parameterization is to pick for one of the interfaces, preferably the one with the median interfacial energy of all the types of interface. The analytical relationships above can be used to calculate and . Once calculated, , , and can be set, normally using a GenericConstantMaterial. Once and are set, the interfacial energy for other types of interface can be set using the other parameters: where is a dimensionless function of for the other types of interfaces and can be determined based on the known , and for the other interfaces. The following polynomial approximation can be used to determine as a function of : The values for for the other interfaces can be specified using a GenericConstantMaterial. Alternatively, rather than calculating and specifying these values by hand, the material GrandPotentialInterface can be used.

Example input file

An example input file can be found here: (moose/modules/phase_field/test/tests/GrandPotentialPFM/GrandPotentialMultiphase.i)

Version with strict mass conservation

In the original formulation of the model, small variations in total solute concentration may occur due to discretization error of the chemical potential evolution equation. Typically such variation is less than 0.1% of total solute in the system, and decreases with decreasing time step. The careful choice of time step can be used to lower solute variation to a level low enough that it does not affect microstructural evolution. To eliminate this variation entirely, an alternative formulation has been developed that maintains the use of an evolution equation for composition as a field variable. However, the use of the additional field variable results in increased computational cost. In this formulation, the following equation i used:

where is the atomic volume of species and is related to using . The relationship between composition and chemical potential must also be specified, such as Eq. 22 in Aagesen et al. (2018) for parabolic free energy forms:

An example input file can be found here: (moose/modules/phase_field/examples/multiphase/GrandPotential3Phase_masscons.i)

References

  1. Larry K. Aagesen, Yipeng Gao, Daniel Schwen, and Karim Ahmed. Grand-potential-based phase-field model for multiple phases, grains, and chemical components. Phys. Rev. E, 98:023309, Aug 2018. URL: https://link.aps.org/doi/10.1103/PhysRevE.98.023309, doi:10.1103/PhysRevE.98.023309.[BibTeX]