Non-interacting lattice random walks for calculating diffusion controlled growth in solid state for dilute concentrations

To connect the molecular length scale phenomena to the macroscopic length scale in diffusion controlled growth in solid state, there is need to consider the movement of individual atoms in the crystal lattice and examine the length scale effect where the average density of the atoms approaches to the continuum macro scale. For this purpose a lattice random walk model has been constructed to represent the diffusion of atoms to form a precipitate. Once the atom is in contact with the precipitate surface, the precipitate grows and the atom is not anymore contributing to the random walk. Through the model, it is possible to evaluate the concentration fluctuations at different length scales in diffusion controlled growth and connect the continuum description of diffusion to the atomic level description. We connect the different length scales in theoretical description from atomistic scale through random atom movements to macroscale. In the current study, two-dimensional lattice random walks and growth are considered. The study contributes to the modelling efforts of understanding diffusion controlled precipitate growth in steels.