Dynamic Measure of Hyperuniformity and Nonhyperuniformity in Heterogeneous Media via the Diffusion Spreadability

Time-dependent interphase diffusion processes in multiphase heterogeneous media are ubiquitous phenomena in physics, chemistry and biology. Examples of heterogeneous media include composites, geological media, gels, foams, and cell aggregates. The recently developed concept of spreadability, $\mathcal{S}(t)$, provides a direct link between time-dependent diffusive transport and the microstructure of two-phase media across length scales [Torquato, S., Phys. Rev. E., 104 054102 (2021)]. To investigate the capacity of $\mathcal{S}(t)$ to probe microstructures of real heterogeneous media, we explicitly compute $\mathcal{S}(t)$ for well-known two-dimensional and three-dimensional idealized model structures that span across nonhyperuniform and hyperuniform classes. Among the former class, we study fully penetrable spheres and equilibrium hard spheres, and in the latter class, we examine sphere packings derived from “perfect glasses,” uniformly randomized lattices (URLs), disordered stealthy hyperuniform point processes, and Bravais lattices. Hyperuniform media are characterized by an anomalous suppression of volume fraction fluctuations at large length scales compared to that of any nonhyperuniform medium. We further confirm that the small-, intermediate-, and long-time behaviors of $\mathcal{S}(t)$ sensitively capture the small-, intermediate-, and large-scale characteristics of the models. In instances in which the spectral density ${\tilde{\chi }}_{{}_V}(\mathbf{k})$ has a power-law form $B|\mathbf{k}{|}^{\alpha }$ in the limit $|\mathbf{k}|\to 0$, the long-time spreadability provides a simple means to extract the value of the coefficients $\alpha$ and $B$ that is robust against noise in ${\tilde{\chi }}_{{}_V}(\mathbf{k})$ at small wave numbers. For typical nonhyperuniform media, the intermediate-time spreadability is slower for models with larger values of the coefficient $B={\tilde{\chi }}_{{}_V}(0)$. Interestingly, the excess spreadability $\mathcal{S}(\mathrm{\infty })-\mathcal{S}(t)$ for URL packings has nearly exponential decay at small to intermediate $t$, but transforms to a power-law decay at large $t$, and the time for this transition has a logarithmic divergence in the limit of vanishing lattice perturbation. Our study of the aforementioned models enables us to devise an algorithm that efficiently and accurately extracts large-scale behaviors from diffusion data alone. Lessons learned from such analyses of our models are used to determine accurately the large-scale structural characteristics of a sample Fontainebleau sandstone, which we show is nonhyperuniform. Our study demonstrates the practical utility of the diffusion spreadability to extract crucial microstructural information from real data across length scales and provides a basis for the inverse design of materials with desirable time-dependent diffusion properties.

Figure 1

“Phase diagram” that schematically shows the spectrum of long-time spreadability regimes in terms of the exponent $\alpha$, which depends on the large-scale properties of the microstructure. As $\alpha$ increases from the extreme antihyperuniform limit of $\alpha \to d$, the spreadability decay rate at long times decays faster, that is, the excess spreadability follows the inverse power law $1/t^{(d+\alpha )/2}$, except when $\alpha \to \mathrm{\infty }$, which corresponds to stealthy hyperuniform media with a decay rate that is exponentially fast. This figure is reproduced from the one presented in Ref. [14].

Figure 2

Specific examples of microstructures corresponding to the generic ones indicated in Fig. 1. (a) Antihyperuniform media ($\alpha =-d$). (b) Typical disordered nonhyperuniform media ($\alpha =0$). (c) Disordered nonstealthy hyperuniform media ($0<\alpha <\mathrm{\infty }$). (d) Disordered stealthy hyperuniform media ($\alpha =\mathrm{\infty }$). (e) Ordered stealthy hyperuniform media ($\alpha =\mathrm{\infty }$).

Figure 3

Two-phase media in two dimensions with ${\varphi }_2=0.25$ studied in this work. A square portion of side length $70.90a$ is shown for each medium. Phase 2 is colored red. (a) Debye random media. (b) Fully penetrable circular disks. (c) Equilibrium hard disks. (d) Perfect glass disk packing with $\alpha =1$. (e) Perfect glass disk packing with $\alpha =2$. (f) Disk packing corresponding to an URL with $b=0.3$. (g) Disk packing corresponding to an URL with $b=0.2$. (h) Sphere packing corresponding to an URL with $b=0.1$. (i) Disordered stealthy hyperuniform disk packing with $Ka=1.3$. (j) Square lattice disk packing. (k) Triangle lattice disk packing.

Figure 4

Two-phase media in three dimensions with ${\varphi }_2=0.20$ studied in this work. Unless otherwise stated, a cubic portion of side length $8.21a$ is shown for each model. Phase 2 is colored red. (a) Debye random media, a cubic portion of side length $16a$ is shown here to better represent the distribution of the phases. (b) Fully penetrable spheres. (c) Equilibrium HS. (d) Perfect glass sphere packing with $\alpha =1$. (e) Sphere packing corresponding to an URL with $b=0.3$. (f) Sphere packing corresponding to an URL with $b=0.2$. (g) Sphere packing corresponding to an URL with $b=0.1$. (h) Disordered stealthy hyperuniform sphere packing with $Ka=1.5$. (i) simple cubic (SC) lattice sphere packing. (j) bcc lattice sphere packing.

Figure 5

(a) Autocovariance functions ${\chi }_{{}_V}(r)$ versus dimensionless distance $rs$ for 3D models with ${\varphi }_2=0.20$, where $s$ is the specific surface. (b) Dimensionless spectral densities ${\tilde{\chi }}_{{}_V}(k)s^3$ versus dimensionless wave number $k/s$ of the 3D models. The positions of the Bragg peaks for the sc lattice packing agree exactly with those for the URL packing and are therefore not shown.

Figure 6

Spreadabilities for the 3D models considered with ${\varphi }_2=0.20$ versus dimensionless time $Dt{s}^2$, where $s$ is the specific surface. (a) Spreadabilities for the nonhyperuniform media for small dimensionless times ($0<Dt{s}^2<0.01$). (b) Spreadabilities for the 3D models for intermediate dimensionless times ($0.01<Dt{s}^2<1$). (c) Spreadabilities for the 3D models for intermediate and large dimensionless times ($0.01<Dt{s}^2<200$).

Figure 7

(a) Spreadabilities for 3D URL packings and the cubic lattice packing with ${\varphi }_2=0.20$ for intermediate dimensionless times ($0.01<Dt{s}^2<10$), where $s$ is the specific surface. The dimensionless transition times $D{t}^{\ast }{s}^2$ [see (37)] for these values of $b$ are indicated by solid dots. (b) Plot of dimensionless transition time $D{t}^{\ast }{s}^2$ against $b$ for 3D URL packings with ${\varphi }_2=0.20$.

Figure 8

(a) Spreadabilities of two-phase media in two dimensions with ${\varphi }_2=0.25$ for intermediate dimensionless times ($0.01<Dt{s}^2<1$), where $s$ is the specific surface. (b) Spreadabilities of two-phase media in two dimensions with ${\varphi }_2=0.25$ for intermediate and large dimensionless times ($0.01<Dt{s}^2<200$). (c) Spreadabilities for equilibrium HS for $d=2,{\varphi }_2=0.25$ and $d=3,{\varphi }_2=0.20$ for small and intermediate dimensionless times ($0<Dt{s}^2<0.5$).

Figure 9

(a) Sample filtered slice of Fontainebleau sandstone. The black region corresponds to the grain phase. The diameter of the cylindrical core is 3 mm with a voxel resolution of 7.5 $\mu \mathrm{m}$. This figure is reproduced from the one in Ref. [80]. (b) Autocovariance functions ${\chi }_{{}_V}(r)$ versus dimensionless distance $rs$ for Fontainebleau sandstone [80], Debye random media, and fully penetrable spheres with ${\varphi }_2=0.157$, where $s$ is the specific surface and $s=0.0154\mu {\mathrm{m}}^{-1}$ for the sandstone. (c) Dimensionless spectral densities ${\tilde{\chi }}_{{}_V}(k)s^3$ versus dimensionless wave number $k/s$ for Fontainebleau sandstone, Debye random media, and fully penetrable spheres.

Figure 10

Spreadabilities for Fontainebleau sandstone, Debye random media, and fully penetrable spheres with ${\varphi }_2=0.157$ for small, intermediate, and large dimensionless times $0.001<Dt{s}^2<200$, where $s$ is the specific surface. The inset plots these spreadabilities on a linear scale at intermediate and large dimensionless times $1<Dt{s}^2<10$.

Figure 11

(a) Plot of $1-|\tilde{f}(k){|}^2$ versus dimensionless wave number $k$ (in units of the lattice constant) for 2D URL point processes. (b) Plot of $1-|\tilde{f}(k){|}^2$ versus dimensionless wave number $k$ for 3D URL point processes.