Hyperuniformity scaling of maximally random jammed packings of two-dimensional binary disks

Jammed (mechanically rigid) polydisperse circular-disk packings in two dimensions (2D) are popular models for structural glass formers. Maximally random jammed (MRJ) states, which are the most disordered packings subject to strict jamming, have been shown to be hyperuniform. The characterization of the hyperuniformity of MRJ circular-disk packings has covered only a very small part of the possible parameter space for the disk-size distributions. Hyperuniform heterogeneous media are those that anomalously suppress large-scale volume-fraction fluctuations compared to those in typical disordered systems, i.e., their spectral densities ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{{}_{V}}(\mathbf{k})$ tend to zero as the wavenumber $k\ensuremath{\equiv}|\mathbf{k}|$ tends to zero and are often described by the power-law ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{{}_{V}}(\mathbf{k})\ensuremath{\sim}{k}^{\ensuremath{\alpha}}$ as $k\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow}\phantom{\rule{0.16em}{0ex}}0$ where $\ensuremath{\alpha}$ is the so-called hyperuniformity scaling exponent. In this work, we generate and characterize the structure of strictly jammed binary circular-disk packings with a size ratio $\ensuremath{\beta}={D}_{L}/{D}_{S}$, where ${D}_{L}$ and ${D}_{S}$ are the large and small disk diameters, respectively, and the molar ratio of the two disk sizes is 1:1. In particular, by characterizing the rattler fraction ${\ensuremath{\phi}}_{R}$, the fraction of configurations in an ensemble with fixed $\ensuremath{\beta}$ that are isostatic, and the $n$-fold orientational order metrics ${\ensuremath{\psi}}_{n}$ of ensembles of packings with a wide range of size ratios $\ensuremath{\beta}$, we show that size ratios $1.2\ensuremath{\lesssim}\ensuremath{\beta}\ensuremath{\lesssim}2.0$ produce maximally random jammed (MRJ)-like states, which we show are the most disordered strictly jammed packings according to several criteria. Using the large-$R$ scaling of the volume fraction variance ${\ensuremath{\sigma}}_{{}_{V}}^{2}(R)$ associated with a spherical sampling window of radius $R$, we extract the hyperuniformity scaling exponent $\ensuremath{\alpha}$ from these packings, and find the function $\ensuremath{\alpha}(\ensuremath{\beta})$ is maximized at $\ensuremath{\beta}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}1.4$ (with $\ensuremath{\alpha}=0.450\ifmmode\pm\else\textpm\fi{}0.002$) within the range $1.2\ensuremath{\le}\ensuremath{\beta}\ensuremath{\le}2.0$. Just outside of this range of $\ensuremath{\beta}$ values, $\ensuremath{\alpha}(\ensuremath{\beta})$ begins to decrease more quickly, and far outside of this range the packings are nonhyperuniform, i.e., $\ensuremath{\alpha}=0$. Moreover, we compute the spectral density ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}}_{{}_{V}}(\mathbf{k})$ and use it to characterize the structure of the binary circular-disk packings across length scales and then use it to determine the time-dependent diffusion spreadability of these MRJ-like packings. The results from this work can be used to inform the experimental design of disordered hyperuniform thin-film materials with tunable degrees of orientational and translational disorder.