Modelling of the grain size probability distribution in polycrystalline nanomaterials

Theoretical analyses have always resulted in nanomaterials’ grain size probability distribution being of varied form: approximately either lognormal, Rayleigh, normal, Weibull, etc. The isotropic Hillert’s model of grain growth which is more suitable for soap froth has been frequently used to establish these distributions with the hope of approximating experimental observations. Observed grain growth in nanomaterials shows departures from the Hillert’s model. In the present paper, the probability distribution of grain size in nanomaterials is dealt with. Use is made of a modified model of grain growth in polycrystalline nanomaterials developed recently by the authors. The modified model accounts for grain growth caused by curvature driven grain boundary migration and grain rotation-coalescence mechanisms. Since the grain size in the aggregate is random, the stochastic counterpart of the expression governing the incremental change in individual grain size is obtained by the addition of two fluctuation terms. The integro-differential equation governing the development of the probability density function of the grain size is obtained which is the generalised Fokker–Planck–Kolmogorov equation. Numerical solution to the integro-differential equation is obtained. Results from analytical modelling of grain size probability distribution in polycrystalline nanomaterials are different if the effect of grain rotation-coalescence mechanism on grain growth process is taken into account and, further, due to the addition of the fluctuation terms. Results also depend on the nature of the fluctuation term, which is a material property as the fluctuation in grain sizes varies from one material to another. It is shown that many of the major attributes of grain growth, such as self similarity (probability density approaching a stationary one), can be predicted by the solution of the Fokker–Planck–Kolmogorov equation