Quantifying Structure in Quasi 2D-Foams at All Lengths by Uncovering Long-Range Hidden Order with Hyperuniformity Disorder Length Spectroscopy and Relating Local Bubble Shape to Dynamics
Amorphous materials have constituent particles without translational order and appear to have no long range structure; one way to search for hidden order in these materials is to measure their long range density fluctuations and if the fluctuations are suppressed to the same extent as in crystals then the system is called hyperuniform. Using the tools developed to measure hyperuniformity we introduce a new length scale to quanitfy disorder in amorphous systems at all distances. This quantity is called the hyperuniformity disorder length h and it is a distance that controls the variance of volume fraction fluctuations for randomly placed windows of fixed size; only the particles that land within a distance h from the boundary of each window determine fluctuations. For cubic windows where the window volume scales like Ld, we propose to quantify disorder in particle configurations by the real-space spectrum of h(L) versus L. We call this approach Hyperuniformity Disorder Length Spectroscopy (HUDLS). Completely random patterns have long-ranged density fluctuations and h = L/2. Hyperuniform patterns have h = he is constant at large L. We investigate the short, medium, and long-range structure of soft disk configurations for a wide range of area fractions and simulation protocols by using HUDLS with square windows of growing side length L. Rapidly quenched unjammed configurations exhibit size-dependent super-Poissonian long-range features that, surprisingly, approach the totally-random limit even close to jamming. Above and just below jamming, the spectra exhibit a plateau, h(L) = he, for L larger than particle size and smaller than a cutoff Lc beyond which there are long-range fluctuations. The value of he is independent of protocol and characterizes the putative hyperuniform limit. Additionally, we use HUDLS to study the structure of foams by analyzing bubble centroids that are either unweighted or given a weight equal to the area of the bubble they occupy. The h\left( L \right) data for the unweighted foam centroids collapse for different ages of a coarsening foam and have random fluctuations at long distances. The area weighted patterns have h\left(L\right)= he for large L and its value shows the foams are most ordered when compared to other cellular structures. Finally, we quantify bubble shape with a quantity called “circularity” that is based on the curvature of the bubble edges and show that it significantly affects individual bubble coarsening. For wet foams we find 6-sided bubbles whose areas change in time; this is in direct violation of the usual von Neumann law but in agreement with a generalized coarsening equation that we describe. The data for these bubbles is in good agreement with predictions made by our generalized equation and the coarsening we observe is entirely due to the bubble shape as defined by its circularity
