A geometric upper bound on the mean first nearest neighbour distance between particles in three-dimensional microstructures Original Research Article

Numerous mechanical and physical properties of microstructures depend on the spatial arrangement of features such as precipitates, voids, and inclusions. Mean first nearest neighbour distance is an important descriptor of spatial arrangement of microstructural features that is of significance in several microstructure-properties relationships, but it is very difficult to measure experimentally in three-dimensional microstructures of opaque materials. Therefore, it is of interest to determine the bounds on the mean first nearest neighbour distance to facilitate computation of the corresponding bounds on properties of interest. In this contribution, it is shown that for a given number density of particles in a microstructure of infinite extent, there exists a finite upper bound on the mean first nearest neighbour distance. The relationship between this upper bound and the number density of particles is derived. The result is applicable to point particles (zero-size) as well as to a microstructure containing particles of non-zero sizes; it is also equally applicable to penetrable, partially penetrable, or impenetrable ensembles of particles of any arbitrary shapes and sizes.