Quantifying the Long-Range Structure of Foams and Other Cellular Patterns with Hyperuniformity Disorder Length Spectroscopy

Abstract: We investigate the local- and long-range structure of four different space-filling cellular patterns: bubbles in a quasi-2d foam plus Voronoi constructions made around points that are uncorrelated (Poisson patterns), low discrepancy (Halton patterns), and displaced from a lattice by Gaussian noise (Einstein patterns). We study distributions of local quantities including cell areas and topological features; the former is the widest for bubbles in a foam making them locally the most disordered but the latter show no significant differences between the cellular patterns. Long-range structure is probed by the spectral density and also by converting the real-space spectrum of number density or volume fraction fluctuations for windows of diameter $D$ to the effective distance $h(D)$ from the window boundary where these fluctuations occur. This real-space hyperuniformity disorder length spectroscopy is performed on various point patterns which are determined by the centroids of the bubbles in the foam, by the points patterns around which the Voronoi cells are created and by the centroids of the Voronoi cells. These patterns are either unweighted or weighted by the area of the cells they occupy. The unweighted bubble centroids have $h(D)$ that collapses for the different ages of of the foam with random Poissonian fluctuations at long distances. All patterns of area-weighted points have constant $h(D)=h_e$ for large $D$; $h_e=0.084 \sqrt{\left< a\right>}$ for the bubble centroids is the smallest value, meaning they are most uniform. All the weighted centroids collapse to the same constant $h_e=0.084 \sqrt{\left< a\right>}$ as for the foams. A similar analysis is performed on the edges of the cells where the spectra of $h(D)$ for the foam edges show $h(D) \sim D^{1-\epsilon}$ where $\epsilon=0.3$