Simulation of microstructure evolution enhances our capability to control the manufacturing process and to design alloys for superior material performance. For real engineering materials and processes, this often requires combining several models.

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 an Excel file for the simulation of one-dimensional solidification/melting.

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 transfer

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

In PhasePot the temperature field evolves through heat equation with heat generation. The heat equation contains a heat generation term, which is linked directly to the thermodynamic properties of the system.

Diffusion

Changes in concentration fields should often be considered in microstructure modelling of alloys.

In PhasePot, the evolution of the concentration variables is simulated based on a modified Cahn-Hilliard formulation. 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.

Fluid flow

Microstructure development in continuous or semi-continuous casting processes may involve material flow.

In PhasePot, fluid flow can be mimicked by considering a ‘moving frame’, where the material enters the domain from one side (inlet) and leaves from the other side (outlet). In addition to this, fluid flow can be simulated concurrently through a simple Lattice Boltzmann algorithm. This can include advection of the main field variables (phase, temperature and concentration) and free movement of the solid particles within the liquid.