This is an advanced 2D Cellular Automata (CA) model for the simulation of microstructural evolution during grain growth in polycrystalline materials. The model is built on physical principles and it can directly work with experimental EBSD data. It is written in Dyalog APL.

This model incorporates a sophisticated set of features to achieve a balance between physical realism and computational efficiency:

• Crystallography: Crystallographic orientations are represented as Euler angles (Bunge convention) to the user, and internally as quaternions. Misorientations are calculated considering cubic crystal symmetry

• User configurable grid: The neighbourhood of a cell can be defined as von Newmann (square grid, 4 neighbours), Moore (square grid, 8 neighbours) or hexagonal (6 neighbours). The total boundary energy of each cell $\gamma_o$ is calculated as:

  • Von Newmann: $\gamma_o=\gamma_N+\gamma_E+\gamma_S+\gamma_W$

  • Moore: $\gamma_o=\gamma_{NW}+\gamma_N+\gamma_{NE}+\gamma_E+\gamma_{SE}+\gamma_S+\gamma_{SW}+\gamma_W$

  • Hexagonal: $\gamma_o=\gamma_{NW}+\gamma_{NE}+\gamma_E+\gamma_{SE}+\gamma_{SW}+\gamma_W$

• Robust Simulation Mechanics: The model calculates the transformation rate ($r_n$) for a cell to flip to the orientation of neighbour $p$ (the parent cell) based on the equation

  $$r_n = \left(\frac{k_r}{A_c N}\right) M_n K |\Delta\gamma_n|$$

  The rate is scaled by cell area $A_c$ and number of neighbours $N$ to ensure physical kinetics are independent of grid resolution, and by an heuristic factor $K$ (expained below). The $k_r$ parameter, configurable by the user, works as an overall scaling factor for the model.

  Each time step, an accumulator of transformation into each neighbor is updated such that $a_n{t+\Delta t}=a_n(t) + r_n \Delta t$. The cell is then transformed or not interpreting this accumulator as a probability (scaled by the user configurable factor $\xi_f$, such that a higher $\xi_f$ implies higher randomness). Therefore, the condition to transform into neighbor $n$ is

  $$\xi(0,1)<a_n(t)^{\frac{1}{\xi_f}}$$

• Physically-Based Kinetics:

  • Grain boundary energy is calculated using the Read-Shockley equation, dependent on the misorientation angle with the neighbor $n$, given by $\theta_n$

  $$\gamma_n=\gamma_0 \frac{\theta_{on}}{\theta_H}\left(1 - \log\left(\frac{\theta_{on}}{\theta_H}\right)\right)$$

  • Grain boundary mobility is a function of both temperature and misorientation, according to Humphrey's equation

  $$M_n=M_0 \exp\left(\frac{-Q_g}{R T}\right) \left(1-\exp\left(-A_h\left(\frac{\theta_{on}}{\theta_H}\right)^n_h\right)\right)$$

• Flexible Local Factors: The driving force can be modified by any number of user-provided local property maps. The following factors are already predefined:

  • Thermal Energy: Adds a deterministic driving force for recrystallization

  $$\Delta\gamma_T=w_T\frac{k T}{A_c}\xi(-1,1)$$

  • Image Quality: Higher energy push at cells with more deffects (more image quality, also related with stored energy)

  $$\Delta\gamma_{IQ}=w_{IQ}\gamma_0 \left(1 - \frac{IQ_o}{IQ_{max}}\right)$$

  • Confidence Index: Higher energy fluctuations at cells with a more uncertain orientation (a lower confidence index)

  $$\Delta\gamma_{CI}=w_{CI}(1 - CI_o) \xi(-1,1)$$

  The total transformation energy for each neighbor $n$ is then calculated as:

  $$\Delta\gamma_n=(\gamma_n - \gamma_0)+\Delta\gamma_T+\Delta\gamma_{IQ}+\Delta\gamma_{CI}$$

• Advanced Heuristics: The factor $K$ is obtained as the product of three components

  $$K=K_c K_m K_g$$

  • Cooperative factor: For each neighbours for which there is a misorientation with respect to one of its neighbours of the same angle as the misorientation between the parent and transforming cell ($\delta_{np}$ is 1), it is considered that the boundary energy after transition will be decreased by a factor $k_c$

  $$K_c=1-\delta_n\left(1 - k_c\right)$$

  • Geometric factor: This factor accounts for the influence of the boundary topology in the neighborhood, considering both coherent front advancement and the "squeeze" pressure on shrinking grains, adding the contribution of all moving boundaries

  $$K_g=\sqrt{\left(\sum_n{|u_{nx}|}\right)^2+\left(\sum_n{|u_{ny}|}\right)^2}$$

  • Multiple-joint factor: The factor $K_m$ takes different values depending if the cell is part of a simple boundary, triple point or quadriple point (the quadruple point factor is also used when more than four grains meet)

• Boundary Conditions: Supports both periodic and "mirror" (no-flux) boundary conditions for realistic simulation of finite samples

• EBSD Integration: Can directly read .ctf (HKL) and .ang (TSL) EBSD files, clean non-indexed points, and crop regions of interest

To run the model, use the provided .apls script. To run it, specify the .json5 configuration file(s) as argument(s). If no file name is given, the default file gg.json5 is used. See this file for further details about configuration options.