PhasePot integrates Phase-Field and Monte-Carlo Potts models for microstructure simulation that can be both efficient and quantitative, particularly for polycrystalline materials.

Microstructure Simulation

Simulation can enhance our understanding of how best control and design microstructures for superior material performance. Most engineering materials are polycrystalline; they consist of many small micron or nano-sized crystals or grains. 

Phase-Field Model

The phase-field approach is used extensively to simulate various processes on a microstructural level. Practical applications of the approach range from solidification and solid-state phase transformation, to sintering and electrochemical processes.

Microstructural changes in the phase-field approach are handled by considering temporal evolution of one or more phase-field variables. These variables represent structural or chemical order at any location within the material. This approach can be used to characterise and simulate movement of diffuse interfaces that form between different phases or domains.

A key assumption in the phase-field approach is that thermodynamic properties depend on the gradient of certain field variables. For example, the free-energy of an ordered intermetallic phase can be formulated as a function of both local value and the gradient of chemical order parameter. Through minimization of this function, using Ginzburg-Landau formalism, the build-up and movement of a diffuse interface between two anti-phase domains can be simulated.

There are several different forms of the phase-field approach. While the numerical implementation of a quantitative phase-field model tends to be very complex, phase-field simulation can be very simple. To get a glimpse of how the phase-field approach works, download a simple Excel file (snapshot shown below) for the simulation of one-dimensional solidification/melting.

In PhasePot, the evolution of structural and chemical order parameters (i.e. phase-field variables) is simulated based on similar (Allen-Cahn) formulations as used in the above simple example. Both geometric and thermodynamic formulations are available in the standard module of the software.

Potts Model

Potts model is used extensively in combination with the Monte Carlo algorithm, for simulating changes in polycrystalline materials, such as recrystallization and grain coarsening. Despite versatility and wide-spread applications of the phase-field approach, it is still not as efficient and robust as is the Potts model in terms of being able to simulate changes in grain structure.

The combination of the phase-field and the Monte Carlo Potts models allows for quantitative simulation of diffusion and interface kinetics during phase transformation in polycrystalline materials. In addition to this, the combination enables computationally efficient calculation of orientation field and realistic description of the grain boundary characteristics. By combining these models, the simulated grain boundaries become sharp in the orientation field, whilst remaining diffuse in the concentration, structural order and orientational order fields.

In PhasePot, the evolution of the crystal orientation and the orientational order parameter is simulated based on the Monte-Carlo Potts model, with temperature-dependent kinetics. The orientational order parameter and the respective mismatch energy are also considered in the free-energy functional for the evolution of phase-field variables.

Heat and Mass Diffusion

Simulating microstructural changes often involves considering changes in temperature and concentration fields, in addition to changes in the phase and orientation fields.

In PhasePot, the evolution of the concentration variables is simulated based on a modified Cahn-Hilliard formulation, whereas the temperature field evolves through heat equation with heat generation. The mass diffusion equation contains a special anti-trapping term, which allows for quantitative simulation of phase transformation, over a wide range of interface thicknesses and interface velocities. The heat equation contains a heat generation term, which is linked directly to the thermodynamic properties of the system.