Quantifying the Structure of Space-Filling Disordered Cellular Patterns with Hyperuniformity Disorder Length Spectroscopy

A system is hyperuniform if the spectral density decays like χ(q)~q^-ε with ε >0 as q goes to zero. In real space the area fraction variance for randomly placed measuring LxL windows can be written σ²(L)=4<a>h(L)/L³ where <a> is the average particle area and h(L) is the hyperuniformity disorder length, which is defined by the distance from the window boundary where number density fluctuations occur [1]. The spectrum of h(L) versus L quantifies the degree of structural order; smaller h(L) indicates more order, and at large L hyperuniform patterns have constant h(L)=h_e. Here, we compare χ(q) and h(L) spectra for cellular patterns given by Voronoi construction around points that are (1) uncorrelated (Poisson), (2) low discrepancy (Halton), and (3) displaced from a lattice by Gaussian noise (Einstein), as well as (4) the centroids of bubbles in a quasi-2d foam. All four types are hyperuniform and have X(q)~q^-4 for small q. The h_e values indicate that Poisson, Halton, Einstein, and foams are ranked least to most ordered. The foam has h_e=0.082<a>^1/2 and the same value of h_e is found for the other cellular patterns if analyzed in terms of the Voronoi cell centroids; for comparison, h_e=0.084<a>^1/2 is found at jamming for bidisperse disks [1].

[1] Chieco et al. PRE 98, 042606 (2018)