Nearest-neighbor distributions in thin films, sheets, and plates

Uniform random spatial distance distribution of point particles in thin films, sheets, and plates is modeled and an exact analytical expression for the nearest-neighbor distance distribution of such point particles is derived. The results can be applied to model spatial distributions of nano-clusters in thin films as well as for modeling spatial arrangements of brittle inclusions in sheets and plates. It is shown that the nearest-neighbor distance distribution is not only a function of the number density of point particles but also a function of film/sheet thickness and the distance of point particles of interest from the film/sheet surface. The overall mean nearest-neighbor distance (averaged over all distances from the surface of a film/sheet) is also a function of film/sheet thickness such that for films thinner than a certain critical value, the mean approaches the classic two-dimensional planar solution; and for films thicker than another critical value, the mean approaches the classic three-dimensional bulk solution. The critical thickness for approach to the three-dimensional bulk solution is very small, and consequently many thin films lie above this value and exhibit mean nearest-neighbor distances closer to those predicted by the classic three-dimensional bulk solution.