I am aware that the Potts Model can be used to simulate grain growth, and that Phase Field Models have also been very successful. What are the advantages and limitations of these models? What are the other modelling techniques that can be used to simulate microstructure evolution?
I could list the models that could be used for microstructural modeling as:
- Phase-Field: It is constructed based on non-equilibrium thermodynamics and Onsager reciprocal relations to derive a functional for Gibbs free energy and then find the order variables (i.e. phase-field variable to describe the fraction of phases, concentration, temperature, stresses, etc.). This method will give you a PDE that needs to be solved numerically with finite difference or finite element. The advantage with phase-field is that you have solid theory to capture phase transformations (e.g. grain growth) by using thermodynamics, but the disadvantage is that it is computationally expensive usually for real systems and you need a parallelized code with MPI or GPU to have a reasonable computational performance.
- Stochastic or Monte-Carlo: In this method you model the phase transformation in your material by calculating its Gibbs free energy and then calculate probability density function to decide if this transformation is going to happen or not. Note that stochastic is much more simpler than Monte-Carlo method. Basically in your random-walk trial you just try one time for comparing the generated random number with the probability drawn from your probability density function, but in Monte-Carlo you need to repeat this try even in your first move for certain times to make sure it's the most probable way for your system to go. The advantage is that it's much more simpler to code and it's much more faster than phase-field, but you would have lots of difficulties to capture complex phase transformation phenomena in your real systems.
- Molecular Dynamics: It's possible to simulate the grain growth or other phase-transformation problems with molecular dynamics. In fact, you need to define the interatomic potential (i.e. in case of metallic systems, it's simple: just choose EAM (Embedded Atom Method)) and the temperature of your system and then you need to track the time-evolution of your system and identify the interesting regions such as crystallized grains, etc. The advantage here is that you could approach complex systems with much more confidence because here molecular dynamics acts like a pseudo first-principle method for studying microstructures and phase transformations, but the challenge or disadvantage is that the molecular dynamics is usually computationally expensive and also the phase-transformation phenomena are considered rare-events which means you need to run your simulation for long-time at least based on diffusion time calculated from your diffusion coefficient to see a phase-transformation, which again adds more burden on computationally expensive molecular dynamics.
- Potts Models or Ising Models: This model is similar to stochastic and Monte-Carlo method but you need to define the Hamiltonian of the system (instead of defining the Gibbs free energy) based on mechanics of your system and then solve it semi-deterministic to find the state of your system and study the phase-transformation in your material. The advantage is that the coding is much simpler than phase-field for example and as a result, it's much faster than phase-field, but you would have lots of difficulties defining the Hamiltonian of your system for real case applications besides toy models available in books.
If you are looking for practical sides and want some suggestion for practical softwares or codes, feel free to open a new question to discuss the suitable codes or softwares for each of these methods mentioned above.