A stochastic tiling model of mean nearest-neighbor distances in three-dimensional uniform random distributions

The mean nearest-neighbor distance is an important clustering/ordering metric in quantitative spatial analysis of microstructures; especially for materials containing particles or voids. Nearest-neighbor distance (d_NN) distributions in three-dimensional (3D), random-sequential-addition, uniform, hard-sphere, computer-generated patterns have been studied for volume fractions from 0.0001 to 0.35. Normal distributions can well fit these d_NN distributions and they provide insight into the study of nearest-neighbor (NN) metrics such as means, medians, and modes. Furthermore, we report on the development of an estimator for the 3D mean d_NN (μ_NN) based upon stochastic tiling. Stochastic tiling models involve random rotations of polyhedral space-filling tiles which allows the calculation of the probabilities for d_NN distributions. Solutions are presented for volume fractions ≈0.10 to the cubic theoretical maximum of ≈0.52. Extrapolating solutions to lower volume fractions also provides reasonable estimates. There is good agreement between the μ_NN estimates compared to the computer-generated random patterns. In the volume fraction region where deviations from these estimates exist, there is an apparent transition in the relationship between the fitted normal distribution means and the NN distance means at a volume fraction ≈0.25. Suggestions for areas of future research are highlighted.

Graphical Abstract