Circles, Spheres and Drops Packings

TL;DR

This paper investigates the geometric and topological principles of polydisperse circle and sphere packings, revealing power-law size distributions and their dependence on packing strategies across dimensions, with applications to natural phenomena like vapor-deposited drops.

Contribution

It introduces a theoretical framework linking size distribution exponents to packing strategies and extends these results to higher dimensions and real-world systems like vapor-deposited drops.

Findings

Size distribution follows a power law with exponent related to packing strategy.

Disordered packings have an upper bound exponent of 2 in 2D and D in D dimensions.

Simulations agree with analytical predictions and describe natural drop formations.

Abstract

We studied the geometrical and topological rules underlying the dispositions and the size distribution of non-overlapping, polydisperse circle-packings. We found that the size distribution of circles that densely cover a plane follows the power law: $N(R) \propto R^{-\alpha}$. We obtained an approximate expression which relates the exponent $\alpha$ to the average coordination number and to the packing strategy. In the case of disordered packings (where the circles have random sizes and positions) we found the upper bound $\alpha_{Max} = 2$. The results obtained for circles-packing was extended to packing of spheres and hyper-spheres in spaces of arbitrary dimension D. We found that the size distribution of dense packed polydisperse $D$-spheres, follows -as in the two dimensional case- a power law, where the exponent $\alpha$ depends on the packing strategy. In particular, in the case of disordered packing, we obtained the upper bound $\alpha_{Max}=D$. Circle-covering generated by computer simulations, gives size distributions that are in agreement with these analytical predictions. Tin drops generated by vapour deposition on a hot substrate form breath figures where the drop-size distributions are power laws with exponent $\alpha \simeq 2$. We pointed out the similarity between these structures and the circle-packings. Despite the complicated mechanism of formation of these structures, we showed that it is possible to describe the drops arrangements, the size distribution and the evolution at constant coverage, in term of maximum packing of circles regulated by coalescence.