Hyperuniformity disorder length spectroscopy for extended particles

The concept of a hyperuniformity disorder length $h$ was recently introduced for analyzing volume fraction fluctuations for a set of measuring windows [Chieco et al., Phys. Rev. E 96, 032909 (2017).]. This length permits a direct connection to the nature of disorder in the spatial configuration of the particles and provides a way to diagnose the degree of hyperuniformity in terms of the scaling of $h$ and its value in comparison with established bounds. Here, this approach is generalized for extended particles, which are larger than the image resolution and can lie partially inside and partially outside the measuring windows. The starting point is an expression for the relative volume fraction variance in terms of four distinct volumes: that of the particle, the measuring window, the mean-squared overlap between particle and region, and the region over which particles have nonzero overlap with the measuring window. After establishing limiting behaviors for the relative variance, computational methods are developed for both continuum and pixelated particles. Exact results are presented for particles of special shape and for measuring windows of special shape, for which the equations are tractable. Comparison is made for other particle shapes, using simulated Poisson patterns. And the effects of polydispersity and image errors are discussed. For small measuring windows, both particle shape and spatial arrangement affect the form of the variance. For large regions, the variance scaling depends only on arrangement but particle shape sets the numerical proportionality. The combined understanding permit the measured variance to be translated to the spectrum of hyperuniformity lengths versus region size, as the quantifier of spatial arrangement. This program is demonstrated for a system of nonoverlapping particles at a series of increasing packing fractions as well as for an Einstein pattern of particles with several different extended shapes.

Figure 1

Poisson arrangement of various extended particles: (a) $25\times 25$ square, (b) circular, (c) diamond, (d) $5\times 125$ rectangular, (e) sine-squared, (f) Gaussian. For each the image size is $200\times 200$ square pixels, the total area fraction is $\varphi =1$, and the particle area is approximately $25\times 25$ square pixels (exactly for square, rectangular, and sine-squared particles). The insets in panels (e, f) show isolated particles.

Figure 2

Relative variance vs. measuring window size for simulated Poisson patterns of rectangular particles, with dimensions as labeled in pixel units $p_o$. For each particle width there are four volume fractions, $\varphi =\{0.02,0.15,1.0,5\}$, indicated by increasing symbol size except for $\times$ for $\varphi =5$ (500%). To within statistical uncertainty these collapse together and agree with the prediction of Eqs. (21, 22, 23), shown by the solid curves. The simulation system size is $3000\times 3000$ square pixels, and has periodic boundary conditions.

Figure 3

Relative variance vs. measuring window size for simulated Poisson patterns of various particles, as labeled. As per Fig. 1 the particle areas are approximately $V_P={\left(25p_o\right)}^2$ (exactly for square, rectangular, and sine-squared cases), the total volume fraction is $\varphi =1$, and the simulation system size is $3000\times 3000$ square pixels. The corresponding solid curves represent the predictions of Eqs. (21, 22, 23) for rectangular particles, Eqs. (26) and (27) for sine-squared particles, and Eq. (24) for Gaussian particles. There are two data sets for the rectangular particles, one with all horizontal alignment and one with a 50:50 mixture of horizontal and vertical. The $+$ symbols represent the relative variance results for the “center-pixel” pattern associated with the squares, where the entire weight of the particle is given to the central pixel.

Figure 4

Relative variance at $L=p_o$ vs. particle width $p$ for various $d=2$-dimensional particles, as labeled. For pixelated particles this represents the intercept and is shown by horizontal lines for square and sine-squared particles, and by symbols for simulated pixelated Gaussian particles. For continuum particles, $\mathcal{V}\left(p_o\right)$ values are given by Eqs. (21), (23), and (26) with pixel width set to zero and by Eqs. (24) and (23) as-is; they are plotted as dashed curves and are smaller than the $L=0$ intercept when the particle is not large. The difference is a measure of how well pixelated particles may be approximated by continuum predictions.

Figure 5

Relative variance vs. measuring window size for simulated Poisson patterns of bidisperse mixtures of $3\times 3$ and $30\times 30$ square particles, labeled by the area-fraction weight $W_{30}$ of the larger particles. The total volume fraction is $\varphi =1$, and the simulation system size is $3000\times 3000$ square pixels. The corresponding solid curves represent the predictions of Eqs. (21, 22, 23), computed separately for each particle size and averaged together with area-fraction weighting. The inset shows a $200\times 200$ sample with equal area fractions for the two species.

Figure 6

Relative variance versus window size for nonoverlapping square particles of area ${\left(15p_o\right)}^2$ placed at random into a pixelated image of area ${\left(6000p_o\right)}^2$, with various total area fractions as labeled. The dotted curves are results for the corresponding central-pixel representation. The red-dashed curves are the expectations for a totally random arrangement of the same particles. This is given by ${(15p_o/L)}^2$ for the central-pixel representation, and by Eqs. (21, 22, 23) for the actual extended particles. The inset shows an example pattern for a ${\left(600p_o\right)}^2$ sample with area fraction of 30%. For clarity, error bars are plotted on only two of the data sets.

Figure 7

(a) Variance ratio and (b) hyperuniformity disorder length based on the results in Fig. 6 for nonoverlapping square particles of area ${\left(15p_o\right)}^2$ and labeled area fractions. For clarity, error bars are plotted on only two of the data sets. As in Fig. 6, the solid curves are for the actual extended-particle patterns, and the dotted curves are for the corresponding central-pixel representations. The curves in (a) are literally the ratio of the Fig. 6 data to the corresponding random-arrangement expectation shown there by the red dashed curves. The curves in (b) are from solving Eq. (38) for $h$. In (a), the $\varphi =0.20$ data at $L>45p_o$ are fit to $R\left(\infty \right)[1+ap/L]$, as shown by the blue dashed curves. Extrapolation results for $\mathcal{R}\left(\infty \right)$ are plotted by solid gray circles on the right $y$ axis for all five data sets. Note that the error bars reflect the expected scatter for different patterns of the same size [6], and that this grows at large $L$.

Figure 8

Large-$L$ asymptotic limit of the variance ratio versus area fraction. Values are found by fitting data to the form $\mathcal{R}\left(L\right)=\mathcal{R}\left(\infty \right)[1+ap/L]$ as illustrated in Fig. 7. Squares are nonoverlapping ${\left(15p_o\right)}^2$ square particles. Circles are for nonoverlapping pixelated circles with the same area. The solid curves represent a cumulant expansion, $\mathcal{R}\left(\infty \right)=exp(-3.7\varphi -b{\varphi }^2)$, with $b$ equal to 3.3 for squares and 5.1 for circles; the initial behavior is $1-3.7\varphi$ as shown by the dashed line.

Figure 9

(a) Relative variance, (b) variance ratio, and (c) hyperuniformity disorder length versus measuring window size for four Einstein patterns with triangular lattice spacing $b=30p_o$ and root-mean-square displacement $3b$. The list of particle centers, the area $a={\left(15p_o\right)}^2$ of each particle, and the global area fraction $\varphi =0.29$, are the same for all four patterns. The only difference is particle shape: open squares for square particles, solid squares for pixel particles, circles for circular particles, and $\times \mathrm{s}$ for sine-squared particles. The solid curves in (a) that match the data at small $L$ are predictions from the text for a random arrangement of particles of the labeled shape; these three curves asymptote to ${(15p_o/L)}^2$ at large $L$, while the relative variance data all asymptote to ${(35p_o/L)}^3$. A small example pattern, for square particles, is shown in (b).