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

Haina Wang  _Department of Chemistry, Princeton University_ , Princeton, New
Jersey, 08544, USA  Salvatore Torquato<EMAIL_ADDRESS>_Department of
Chemistry, Department of Physics, Princeton Center for Theoretical Science,
Princeton Institute for the Science and Technology of Materials, and Program
in Applied and Computational Mathematics, Princeton University_ , Princeton,
New Jersey, 08544 USA

###### Abstract

Time-dependent interphase diffusion processes in multiphase heterogeneous
media arise ubiquitously 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 capability 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 (URL),
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 $|\mathbf{k}|\rightarrow 0$, 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 wavenumbers. 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 $\mathcal{S}(\infty)-\mathcal{S}(t)$
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 applicability 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.

##  I Introduction

Time-dependent interphase diffusion processes in multiphase heterogeneous
media arise in a wide range of physical, chemical and biological contexts,
including magnetic resonance imaging Wedeen  _et al._ (2005), surface
catalysis Torquato (2002); Sosna  _et al._ (2020), material design Torquato
(2002); Sahimi (2003); Tahmasebi (2018) and cell-behavior modeling Brownstein
and Tarr (1979); Langer and Peppas (1981). Heterogeneous media are ubiquitous;
examples include composites, geological media, gels, foams, cell aggregates,
among other natural and synthetic media Rintoul and Torquato (1996); Torquato
(2002); Milton (2002); Klatt  _et al._ (2019); Huang  _et al._ (2021);
Torquato and Kim (2021). It is well known that the effective transport
properties of heterogeneous media generally depend on an infinite set of
correlation functions that characterize the microstructure Torquato (2002);
Sen and Torquato (1989); Torquato and Kim (2021).

The spreadability concept, as very recently developed and explored by Torquato
Torquato (2021a), makes a direct link between time-dependent diffusive
transport and the microstructure of a given heterogeneous material across
length scales. Consider the time-dependent problem of mass transfer of a
solute in a two-phase media, where phases 1 and 2 occupy volume fractions
$\phi_{1}$ and $\phi_{2}$, respectively. Assume that at time $t=0$, the solute
is uniformly distributed throughout phase 2, and completely absent from phase
1. Assume also that the solute has the same diffusion coefficient $D$ in each
phase at any $t$. We call the fraction of total solute present in phase 1 as a
function of time $\mathcal{S}(t)$ spreadability, since it is a measure of the
spreadability of diffusion information as a function of time. Generalizing a
formula due to Prager in three-dimensional physical (direct) space Prager
(1963), Torquato showed that $\mathcal{S}(t)$ in any Euclidean space dimension
$d$ is exactly related to the microstructure via the autocovariance function
$\chi_{{}_{V}}(\mathbf{r})$ in direct space, or equivalently, via the spectral
density $\tilde{\chi}_{{}_{V}}(\mathbf{k})$ in Fourier space Torquato (2021a):

$\displaystyle{\cal S}(\infty)-{\cal S}(t)=$ $\displaystyle\frac{1}{(4\pi
Dt)^{d/2}\,\phi_{2}}\int_{\mathbb{R}^{d}}\chi_{{}_{V}}({\bf
r})\exp[-r^{2}/(4Dt)]d{\bf r}$ (1a) $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{d}\phi_{2}}\int_{\mathbb{R}^{d}}\tilde{\chi}_{{}_{V}}({\bf
k})\exp[-k^{2}Dt]d{\bf k},$ (1b)

where $\mathcal{S}(\infty)=\phi_{1}$ is the infinite-time limit of
$\mathcal{S}(t)$, and $\mathcal{S}(\infty)-\mathcal{S}(t)$ is called the
excess spreadability, i.e., spreadability in excess to its infinite-time
value. Equation (1) is a singular result because it represents one of the rare
examples of transport in two-phase random media where an exact solution is
possible only in terms of the first two correlation functions, namely,
$\phi_{1}$ and two-point statistics via either the autocovariance function or
spectral density.

Torquato Torquato (2021a) has shown that the relation (1) implies that small-,
intermediate- and long-time behaviors of $\mathcal{S}(t)$ are directly
determined by the small-, intermediate- and large-scale structural
characteristics of the two-phase medium. Thus, the spreadability has the
potential to serve as a simple and powerful dynamic figure of merit to probe
and classify all translationally invariant two-phase microstructures that span
from hyperuniform to nonhyperuniform media across scales. Hyperuniform two-
phase media are characterized by an anomalous suppression of volume-fraction
fluctuations relative to garden-variety nonhyperuniform disordered media
Torquato and Stillinger (2003); Zachary and Torquato (2009); Torquato (2018a);
see Sec. II.2 for exact mathematical definitions. Hyperuniformity is an
emerging field, playing vital roles in a number of fundamental and applied
contexts, including glass formation Donev  _et al._ (2005); Marcotte  _et al._
(2013a), jamming Torquato and Stillinger (2007); Atkinson  _et al._ (2016);
Coniglio  _et al._ (2017); Rissone  _et al._ (2021); Torquato (2021b),
rigidity Zhang  _et al._ (2016a); Ghosh and Lebowitz (2018), band-gap
structures Florescu  _et al._ (2013); Man  _et al._ (2013); Marcotte  _et al._
(2013b), biology Jiao  _et al._ (2014); Mayer  _et al._ (2015), localization
of waves and excitations Florescu  _et al._ (2009a, b); Sgrignuoli and Dal
Negro (2021), self-organization Hexner and Levine (2015); Hexner  _et al._
(2017); Ma and Torquato (2019), fluid dynamics Lei and Ni (2019); Lei  _et
al._ (2019), quantum systems Reatto and Chester (1967); Feynman (1998);
Torquato  _et al._ (2008); Abreu  _et al._ (2017); Crowley  _et al._ (2019),
random matrices Dyson (1970); Torquato  _et al._ (2008); Lacroix-A-Chez-Toine
_et al._ (2019) and pure mathematics Sarnak and Strömbergsson (2006);
Brauchart  _et al._ (2019); Torquato  _et al._ (2018, 2019); Brauchart  _et
al._ (2020). Because disordered hyperuniform two-phase media are states of
matter that lie between a crystal and a typical liquid, they can be endowed
with novel properties Degl’Innocenti  _et al._ (2016); Zhou  _et al._ (2016);
Torquato (2018a); Bigourdan  _et al._ (2019); Gorsky  _et al._ (2019); Kim and
Torquato (2020); Sheremet  _et al._ (2020); Zheng  _et al._ (2020); Ghosh and
Saha (2020); Chen  _et al._ (2021); Nizam  _et al._ (2021); Torquato and Kim
(2021); Zhang  _et al._ (2021); Yu  _et al._ (2021).

Torquato Torquato (2021a) showed that in instances in which the spectral
density has the power-law form

${\tilde{\chi}}_{{}_{V}}({\bf k})\sim B|{\bf k}|^{\alpha}$ (2)

in the limit $|\bf k|\rightarrow 0$, the long-time excess spreadability for
two-phase media in $\mathbb{R}^{d}$ is given by the inverse power-law decay

${\cal S}(\infty)-{\cal
S}(t)\sim\frac{B\Gamma[(d+\alpha)/2]\phi_{2}}{2^{d}\pi^{d/2}\Gamma(d/2)(Dt/a^{2})^{(d+\alpha)/2}},$
(3)

where $a$ represents some characteristic heterogeneity length scale. The
power-law (3) holds except when $\alpha\rightarrow+\infty$, which corresponds
to stealthy hyperuniform media with a decay rate that is exponentially fast.
The spreadability was computed for certain idealized ordered and disordered
model microstructures across dimensions. Torquato Torquato (2021a) also showed
that $\mathcal{S}(t)$ has remarkable connections to covering problem of
discrete geometry Conway and Sloane (1998); Torquato (2010) and nuclear
magnetic resonance (NMR) or magnetic resonance imaging (MRI) measurements
Mitra  _et al._ (1992); Sen and Hürlimann (1994); Novikov  _et al._ (2014).
Specifically, Torquato Torquato (2021a) identified precise mappings between
the long-time formulas of the spreadability and the NMR pulsed field gradient
spin-echo (PFGSE) amplitude Mitra  _et al._ (1992); Sen and Hürlimann (1994)
as well as diffusion MRI measurements Novikov  _et al._ (2014). Thus, any
analysis of the large-time behaviors of $\mathcal{S}(t)$ translates
immediately into corresponding analyses of NMR or MRI data, which enables one
to extract large-scale microstructural information of real heterogeneous media
from diffusion measurements alone.

Figure 1 schematically shows the “spectrum” of spreadability regimes in terms
of the exponent $\alpha$. The long-time excess spreadability for
antihyperuniform media ($-d<\alpha<0$) have the slowest decay among all
translationally invariant media, the slowest being when
$\mathcal{S}(\infty)-\mathcal{S}(t)$ approaches a constant. A long-time decay
rate of $\mathcal{S}(\infty)-\mathcal{S}(t)\sim t^{-d/2}$ corresponds to a
nonhyperuniform medium ($\alpha=0$) in which the spectral density is a bounded
positive number at the origin. The spreadability for nonstealthy hyperuniform
media ($0<\alpha<\infty$) has the aforementioned power-law decay
$\mathcal{S}(\infty)-\mathcal{S}(t)\sim t^{-(d+\alpha)/2}$. The limit
$\alpha\rightarrow+\infty$ corresponds to media in which the decay rate of
$\mathcal{S}$ is faster than any inverse power law, which is the case for
stealthy hyperuniform media. Figure 2 shows specific examples of model
microstructures corresponding to the generic ones indicated in Fig. 1. The
reader is referred to Sec. II for terminology.

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 $\alpha\rightarrow d$, the spreadability
decay rate at long times decays faster. i.e., the excess spreadability follows
the inverse power law $1/t^{(d+\alpha)/2}$, except when
$\alpha\rightarrow\infty$, which corresponds to stealthy hyperuniform media
with a decay rate that is exponentially fast. This figure is reproduced from
the one presented in Ref. Torquato (2021a).

(a)

(b)

(c)

(d)

(e)

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 ($0<\alpha<\infty$). (d) Disordered stealthy
hyperuniform media ($\alpha=\infty$). (e) Ordered stealthy hyperuniform media
($\alpha=\infty$).

In this work, we further study the applicability of the spreadability concept
as a dynamic measure to probe a wide range of hyperuniform and nonhyperuniform
two-phase media. We explicitly compute spreadabilities of 2D and 3D models
heretofore not previously considered that represent diverse hyperuniform and
nonhyperuniform classes (see Sec. II.2), including fully penetrable spheres
Torquato and Stell (1983); Torquato (2002), equilibrium hard spheres Wertheim
(1963); Torquato (2002, 2018b), as well as sphere packings derived from
uniformly randomized lattices (URL) Klatt  _et al._ (2020), perfect glasses
Zhang  _et al._ (2016a) and disordered stealthy point processes Zhang  _et
al._ (2016b); Torquato (2018a). These are compared and contrasted with
spreadabilities for hyperuniform and nonhyperuniform media previously obtained
by Torquato Torquato (2021a), including those for Debye random media Debye
_et al._ (1957); Yeong and Torquato (1998) and sphere packings derived from
Bravais lattices. We find that the small-time behavior of $\mathcal{S}(t)$ is
determined by the derivatives of the autocovariance function
$\chi_{{}_{V}}({\mathbf{r}})$ at the origin, the leading term of order
$t^{1/2}$ being proportional to the specific surface $s$. The corresponding
long-time behavior is determined by the form of the spectral density
$\tilde{\chi}_{{}_{V}}(\mathbf{k})$ at small wavenumbers, which enables one to
easily and robustly ascertain the class of hyperuniform or nonhyperuniform
media. We also find that for URL packings, the excess spreadability has
exponential decay (characteristic of the unperturbed lattice packing) at small
to intermediate times, but transforms to power-law decay for $t$ larger than a
well-defined transition time. Our study of the spreadability of the
aforementioned models enables us to formulate an algorithm that efficiently
and accurately extracts large-scale characteristics from spreadability data
(or equivalently, NMR and MRI data) alone. Lessons learned from such analyses
of our models is used to characterize the large-scale behaviors of a sample
Fontainebleau sandstone Coker  _et al._ (1996), which we show is
nonhyperuniform. Our study demonstrates the practical capability of the
diffusion spreadability to extract crucial microstructural information of a
class of heterogeneous media across length scales. We note that using
reconstruction techniques that target autocovariance functions Yeong and
Torquato (1998); Zachary and Torquato (2011); Ma and Torquato (2020), one
could potentially construct two-phase media that realize prescribed functional
forms of $\mathcal{S}(t)$, thereby designing materials with desired diffusion
properties.

In Sec. II, we provide basic definitions and preliminaries. We then describe
the models that we consider in our study for $d=2,3$ and methods to compute
their spreadabilities (Sec. III). Section IV presents and discusses the
spreadabilities of the aforementioned two-phase media models. Section V
describes the algorithm for extracting large-scale structural characteristics
from time-dependent diffusion behaviors, as measured by the spreadability, or
equivalently, by NMR or MRI measurements. Section VI applies the lessons
learned from the spreadabilities of the idealized models to analyze the
structural characteristics of a sample Fontainebleau sandstone. Finally, Sec.
VII provides some conclusions and proposes potential future work.

##  II Definitions and preliminaries

In this section, we introduce some fundamental definitions and concepts that
describe microstructures of two-phase media, including correlation functions,
hyperuniformity, as well as the classification of hyperuniform and
nonhyperuniform media.

###  II.1 Two-point statistics

Consider a two-phase medium, i.e., a partition of space into two disjoint
regions of phase 1 with volume fraction $\phi_{1}$ and of phase 2 of volume
fraction $\phi_{2}$. For a statistically homogeneous medium, the
autocovariance function $\chi_{{}_{V}}({\bf r})$ is directly related to the
two-point correlation function $S_{2}^{(i)}({\bf r})$ of phase $i$ Torquato
(2002):

$\chi_{{}_{V}}({\bf r})\equiv S_{2}^{(1)}({\bf
r})-\phi_{1}^{2}=S_{2}^{(2)}({\bf r})-\phi_{2}^{2}.$ (4)

The two-point function $S_{2}^{(i)}({\bf r})$ gives the probability that the
two end points with displacement vector $\mathbf{r}$ are in phase $i$. At the
extreme limits of its argument, $\chi_{{}_{V}}({\bf r})$ has the following
asymptotic behavior: $\chi_{{}_{V}}({\bf r}=0)=\phi_{1}\phi_{2}$ and
$\lim_{|{\bf r}|\rightarrow\infty}\chi_{{}_{V}}({\bf r})=0$ if the medium
possesses no long-range order. If the medium is statistically homogeneous and
isotropic, then the autocovariance function ${\chi_{{}_{V}}}({\bf r})$ depends
only on the magnitude of its argument $r=|\bf r|$, and hence is a radial
function. In such instances, its slope at the origin is directly related to
the specific surface $s$, which is the interface area per unit volume. In
particular, the well-known three-dimensional asymptotic result Debye  _et al._
(1957) is easily obtained in any space dimension $d$:

$\chi_{{}_{V}}({\bf r})=\phi_{1}\phi_{2}-\kappa(d)s\;|{\bf r}|+{\cal O}(|{\bf
r}|^{2}),$ (5)

where

$\kappa(d)=\frac{\Gamma(d/2)}{2\sqrt{\pi}\Gamma((d+1)/2)}.$ (6)

The nonnegative spectral density ${\tilde{\chi}}_{{}_{V}}({\bf k})$, which can
be obtained from scattering experiments Debye  _et al._ (1957), is the Fourier
transform of $\chi_{{}_{V}}({\bf r})$ at wave vector $\bf k$, i.e.,

${\tilde{\chi}}_{{}_{V}}({\bf k})=\int_{\mathbb{R}^{d}}\chi_{{}_{V}}({\bf
r})e^{-i{\bf k\cdot r}}{\rm d}{\bf r}\geq 0,\qquad\mbox{for all}\;{\bf k}.$
(7)

A particular class of two-phase media is sphere packings, i.e. a collection of
spheres in $d$-dimensional Euclidean space $\mathbb{R}^{d}$ in which no two
spheres overlap. The packing fraction of a packing of identical spheres of
radius $a$ is $\phi_{2}=\rho v_{1}(a)$, where $\rho$ is the number density and
$v_{1}(a)$ is the volume of a sphere:

$v_{1}(R)=\frac{\pi^{d/2}R^{d}}{\Gamma(1+d/2)}.$ (8)

The spectral density of such a packing can be expressed as

${\tilde{\chi}}_{{}_{V}}({\bf k})=\phi_{2}\,{\tilde{\alpha}}_{2}(k;a)S({\bf
k}),$ (9)

where $\tilde{\alpha}_{2}(k;a)$ is the Fourier transform of the scaled
intersection volume of two spherical windows, which is given by Torquato and
Stillinger (2003)

${\tilde{\alpha}}_{2}(k;a)=\Gamma(d/2+1)\frac{J_{d/2}^{2}(ka)}{k^{d}},$ (10)

and the structure factor $S(\mathbf{k})$ is defined as

$S(\mathbf{k})=1+\rho\tilde{h}(\mathbf{k}),$ (11)

where $h(\mathbf{r})$ is the total correlation function, and
$\tilde{h}(\mathbf{k})$ is the Fourier transform of $h(\mathbf{r})$.

For a single periodic configuration containing number $N$ point particles at
positions ${\bf r}_{1},{\bf r}_{2},\ldots,{\bf r}_{N}$ within a fundamental
cell $F$ of a lattice $\Lambda$, the scattering intensity $I(\mathbf{k})$ is
defined as

$I(\mathbf{k})=\frac{\left|\sum_{i=1}^{N}e^{-i\mathbf{k}\cdot\mathbf{r}_{i}}\right|^{2}}{N}.$
(12)

For an ensemble of periodic configurations of $N$ particles within the
fundamental cell $F$, the ensemble average of the scattering intensity in the
infinite-volume limit is directly related to structure factor $S(\mathbf{k})$
by

$\lim_{N,V_{F}\rightarrow\infty}\langle
I(\mathbf{k})\rangle=(2\pi)^{d}\rho\delta(\mathbf{k})+S(\mathbf{k}),$ (13)

where $V_{F}$ is the volume of the fundamental cell and $\delta(\mathbf{x})$
is the Dirac delta function Torquato (2018a). In simulations of many-body
systems with finite $N$ under periodic boundary conditions, Eq. (12) is used
to compute $S(\mathbf{k})$ directly by averaging over configurations.

###  II.2 Hyperuniformity

The hyperuniformity concept generalizes the traditional notion of long-range
order in many-particle systems to not only include all perfect crystals and
perfect quasicrystals, but also exotic amorphous states of matter Torquato and
Stillinger (2003); Torquato (2018a). For two-phase heterogeneous media in
$d$-dimensional Euclidean space $\mathbb{R}^{d}$, which include cellular
solids, composites, and porous media, hyperuniformity is defined by the
following infinite-wavelength condition on the spectral density
${\tilde{\chi}}_{{}_{V}}({\bf k})$ Zachary and Torquato (2009); Torquato
(2018a), i.e.,

$\lim_{|{\bf k}|\to 0}{\tilde{\chi}}_{{}_{V}}({\bf k})=0.$ (14)

An equivalent definition of hyperuniformity is based on the local volume-
fraction variance $\sigma^{2}_{{}_{V}}(R)$ associated with a $d$-dimensional
spherical observation window of radius $R$. A two-phase medium in
$\mathbb{R}^{d}$ is hyperuniform if its variance grows in the large-$R$ limit
faster than $R^{d}$. This behavior is to be contrasted with those of typical
disordered two-phase media for which the variance decays like the inverse of
the volume $v_{1}(R)$ of the spherical observation window given by (8). The
hyperuniformity condition (14) dictates that the direct-space autocovariance
function $\chi_{{}_{V}}({\bf r})$ exhibits both positive and negative
correlations such that its volume integral over all space is exactly zero
Torquato (2016, 2018a), i.e.,

$\int_{\mathbb{R}^{d}}\chi_{{}_{V}}({\bf r})d{\bf r}=0,$ (15)

which is a direct-space sum rule for hyperuniformity.

###  II.3 Classification of hyperuniform and nonhyperuniform media

The hyperuniformity concept has led to a unified means to classify equilibrium
and nonequilibrium states of matter, whether hyperuniform or not, according to
their large-scale fluctuation characteristics. In the case of hyperuniform
two-phase media Zachary and Torquato (2009); Torquato (2018a), there are three
different scaling regimes (classes) that describe the associated large-$R$
behaviors of the volume-fraction variance when the spectral density goes to
zero as a power-law scaling

${\tilde{\chi}}_{{}_{V}}({\bf k})\sim|{\bf k}|^{\alpha}$ (16)

in the limit $|\bf k|\rightarrow 0$:

$\displaystyle\sigma^{2}_{{}_{V}}(R)\sim\begin{cases}R^{-(d+1)},\quad\quad\quad\alpha>1\qquad&\text{(Class
I)}\\\ R^{-(d+1)}\ln R,\quad\alpha=1\qquad&\text{(Class II)}\\\
R^{-(d+\alpha)},\quad 0<\alpha<1\qquad&\text{(Class III).}\end{cases}$ (17)

Classes I and III are the strongest and weakest forms of hyperuniformity,
respectively. Class I media include all crystal structures, many quasicrystal
structures and exotic disordered media Zachary and Torquato (2009); Torquato
(2018a). Stealthy hyperuniform media are also of class I and are defined to be
those that possess zero-scattering intensity for a set of wavevectors around
the origin Torquato (2016), i.e.,

${\tilde{\chi}}_{{}_{V}}({\bf k})=0\qquad\mbox{for}\;0\leq|{\bf k}|\leq K.$
(18)

Examples of such media are periodic packings of spheres as well as unusual
disordered sphere packings derived from stealthy point patterns Torquato
(2016); Zhang  _et al._ (2016b).

By contrast, for any nonhyperuniform two-phase system, the local variance has
the following large-$R$ scaling behaviors Torquato (2021b):

$\displaystyle\begin{split}\sigma^{2}_{{}_{V}}(R&)\\\
&\sim\begin{cases}R^{-d},&\alpha=0\quad\text{(typical nonhyperuniform)}\\\
R^{-(d+\alpha)},&-d<\alpha<0\quad\text{(antihyperuniform)}.\\\
\end{cases}\end{split}$ (19)

For a “typical” nonhyperuniform system, ${\tilde{\chi}}_{{}_{V}}(0)$ is
bounded Torquato (2018a). In antihyperuniform systems,
${\tilde{\chi}}_{{}_{V}}(0)$ is unbounded Torquato (2021b), i.e.,

$\lim_{|{\bf k}|\to 0}{\tilde{\chi}}_{{}_{V}}({\bf k})=+\infty,$ (20)

and hence are diametrically opposite to hyperuniform systems. Antihyperuniform
systems include systems at thermal critical points (e.g., liquid-vapor and
magnetic critical points) Stanley (1987); Binney  _et al._ (1992), fractals
Mandelbrot (1982), disordered non-fractals Torquato  _et al._ (2021), and
certain substitution tilings Oğuz  _et al._ (2019).

##  III Two-phase media models

To further explore the potential of the spreadability as a dynamic-based
figure of merit to probe microstructures of two-phase media, we compute it for
a variety of models in two and three space dimensions that represent diverse
hyperuniform and nonhyperuniform classes, as described in Sec. II.3. The
nonhyperuniform models that we study include fully penetrable spheres Torquato
and Stell (1983); Torquato (2002), equilibrium hard spheres Torquato (2002);
Weeks  _et al._ (1971) and Debye random media Debye  _et al._ (1957); Yeong
and Torquato (1998). The hyperuniform models that we study include
monodisperse sphere packings derived from perfect glasses Zhang  _et al._
(2016a), uniformly randomized lattices (URL) Klatt  _et al._ (2020),
disordered stealthy hyperuniform point processes Zhang  _et al._ (2016b) and
Bravais lattices. We describe each model below and the methods to compute
their spreadabilities. We choose the volume fraction $\phi_{2}$ of phase 2,
(i.e., the phase initially containing the solute,) to be 0.25 for 2D models
and 0.20 for 3D models. These relatively small values of $\phi_{2}$ were
chosen so that we could derive sphere packings from the aforementioned point
processes by decorating each point with a circular disk for $d=2$ or a sphere
for $d=3$. We also note that for our chosen $\phi_{2}$’s, the values of
$\phi_{2}/\phi_{m,d}$ are similar for $d=2,3$, where $\phi_{m,2}=0.9069$ and
$\phi_{m,3}=0.7405$ are the maximum packing fractions for sphere packings in
$\mathbb{R}^{2}$ and $\mathbb{R}^{3}$, respectively. This facilitates
comparison of spreadabilities across dimensions. For all packings, phase 2 is
chosen to be the particle phase. Figures 3 and 4 show snapshots of all the
models considered in this study.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

Figure 3: Two-phase media in two dimensions with $\phi_{2}=0.25$ 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 URL with $b=0.3$. (g) Disk packing corresponding to URL with
$b=0.2$. (h) Sphere packing corresponding to 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.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 4: Two-phase media in three dimensions with $\phi_{2}=0.20$ 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 hard
spheres. (d) Perfect glass sphere packing with $\alpha=1$. (e) Sphere packing
corresponding to URL with $b=0.3$. (f) Sphere packing corresponding to URL
with $b=0.2$. (g) Sphere packing corresponding to URL with $b=0.1$. (h)
Disordered stealthy hyperuniform sphere packing with $Ka=1.5$. (i) Cubic-
lattice sphere packing. (j) BCC-lattice sphere packing.

###  III.1 Nonhyperuniform fully penetrable spheres

We consider the well-known model of “fully penetrable spheres” (FPS) Torquato
and Stell (1983); Torquato (2002), which is nonhyperuniform. It is constructed
by decorating each point of a Poisson point process by a $d$-dimensional
sphere of radius $a$ that generally may overlap with one another. The
simplicity of the $n$-body statistics of the Poisson process enables us to
exactly express $\chi_{{}_{V}}(r)$ of the matrix phase, i.e., the space
unoccupied by the particles, which is designated phase 2 in this work Torquato
(2002):

$\chi_{{}_{V}}(r)=\exp\left[-\eta\frac{v_{2}(r;a)}{v_{1}(a)}\right]-\phi_{2}^{2},$
(21)

where $\eta=-\ln\left(\phi_{2}\right)$ and $v_{2}(r;a)$ is the union volume of
two $d$-dimensional spheres of radius $a$. Specifically,

$\frac{v_{2}(r;a)}{v_{1}(a)}=2\Theta(r-2a)+\frac{2}{\pi}\left[\pi+\frac{r}{2a}\left(1-\frac{r^{2}}{4a^{2}}\right)^{\frac{1}{2}}-\cos^{-1}\left(\frac{r}{2a}\right)\right]\Theta\left(2a-r\right),\qquad
d=2,$ (22)
$\frac{v_{2}(r;a)}{v_{1}(a)}=2\Theta(r-2a)+\left[1+\frac{3r}{4a}-\frac{r^{3}}{16a^{3}}\right]\Theta\left(2a-r\right),\qquad
d=3.$ (23)

The specific surface for fully penetrable spheres is given by Torquato and
Stell (1984)

$as=-d\phi_{2}\ln(\phi_{2}).$ (24)

###  III.2 Nonhyperuniform equilibrium hard spheres

Equilibrium hard sphere packings provide good models of condensed matter in
both liquid and crystal states when the interactions are dominated by strong
short-range repulsions Weeks  _et al._ (1971); Cheng  _et al._ (1999). We
study such systems along the stable fluid branch Torquato (2002, 2018a). We
generated ensembles of 1,000 equilibrium hard-sphere configurations using
Monte Carlo simulations in 2D and 3D at the aforementioned values of
$\phi_{2}$. The simulations used square or cubic simulation boxes under
periodic boundary conditions. The ensemble-averaged structure factors were
then computed from Eq. (12). The specific surface for all sphere packings is
given by Torquato and Stell (1984)

$as=d\phi_{2}.$ (25)

###  III.3 Nonhyperuniform Debye random media

Useful models of nonhyperuniform two-phase media are Debye random media Debye
_et al._ (1957); Yeong and Torquato (1998), defined entirely by the following
monotonic radial autocovariance function:

$\chi_{{}_{V}}(r)=\phi_{1}\phi_{2}\exp(-r/a),$ (26)

where $a>0$ is a length-scale parameter. Such media can never be hyperuniform
because the sum rule (15) requires both positive and negative correlations
Torquato (2016). Debye et al. Debye  _et al._ (1957) hypothesized the simple
exponential form (26) to model three-dimensional media with phases of “fully
random shape, size, and distribution.” Such autocovariance functions were
shown to be realizable in two Yeong and Torquato (1998); Chiu  _et al._
(2013); Ma and Torquato (2020) and three Jiao  _et al._ (2007); Torquato
(2020) dimensions. Torquato obtained explicit expressions of $\mathcal{S}(t)$
for Debye random media in any space dimension Torquato (2021a). We consider
them in this study as they are good approximation of certain realistic
heterogeneous materials Debye  _et al._ (1957), including Fontainebleau
sandstones Coker  _et al._ (1996). They also serve as “benchmark” models to
test our algorithm that extracts large-scale microstructural information from
time-dependent diffusion data, as their spreadabilities are exactly known. The
specific surface for $d$-dimensional Debye random media is given by Torquato
(2021a)

$as=\frac{\phi_{1}\phi_{2}}{\kappa(d)},$ (27)

where $\kappa(d)$ is given by (6).

###  III.4 Hyperuniform sphere packings derived from “perfect glasses”

Perfect glasses proposed by Zhang et al. Zhang  _et al._ (2016a) are
hyperuniform many-body systems with positive bulk and shear moduli that banish
any crystalline or quasicrystalline phases and are unique disordered states up
to trivial symmetries Zhang  _et al._ (2017). Thus, they form soft-interaction
analogs of the maximally random jammed (MRJ) packings Torquato  _et al._
(2000) of hard particles. These latter states can be regarded as prototypical
glasses since they are out of equilibrium, maximally disordered, hyperuniform,
mechanically rigid with infinite bulk and shear moduli, and can never
crystallize due to configuration-space trapping. A perfect glass is created by
cooling many-body systems with a potential that involves certain two-, three-,
and four-body soft interactions upon cooling from high temperature to zero
temperature Zhang  _et al._ (2016a). Using this procedure, Zhang et al. Zhang
_et al._ (2016a) generated perfect-glass point configurations with various
values of $\alpha>0$ in both 2D and 3D. Specifically, their 2D and 3D perfect
glasses with $\alpha=1$ and $\alpha=2$ can be transformed into packings with
our prescribed packing fractions by decorating each point with a sphere of
radius $a$ Zhang  _et al._ (2016b). We obtained numerical structure factors
for these perfect glasses, which are available as supplementary material of
Ref. 31.

###  III.5 Hyperuniform sphere packings derived from uniformly randomized
lattices (URL)

Perturbed lattices serve as important models in cosmology, crystallography and
probability theory Welberry  _et al._ (1980); Gabrielli  _et al._ (2002).
Consider a perturbed lattice in which each point $\mathbf{x}$ of a lattice
$\mathcal{L}$ is perturbed independently by a displacement vector
$\mathbf{u}_{\mathbf{x}}$ that follows the probability density function
$f(\mathbf{u_{x}})$. The structure factor $S(\mathbf{k})$ of this point
process is given by Gabrielli and Torquato (2004); Kim and Torquato (2018)

$S(\mathbf{k})=1-|\tilde{f}(\mathbf{k})|^{2}+|\tilde{f}(\mathbf{k})|^{2}S_{\mathcal{L}}(\mathbf{k}),$
(28)

where $S_{\mathcal{L}}(\mathbf{k})$ is the structure factor of the unperturbed
lattice $\mathcal{L}$ and $\tilde{f}(\mathbf{k})$ is the characteristic
function, i.e., the Fourier transform of $f$.

Here, we study a special class of hyperuniform sphere packings associated with
uniformly randomized lattice (URL) models derived from cubic lattices,
introduced by Klatt et al. Klatt  _et al._ (2020). In these models, each point
in a $d$-dimensional simple cubic lattice $\mathcal{L}=\mathbb{Z}^{d}$ is
displaced by a random vector that is uniformly distributed on a rescaled unit
cell $bC=[-b/2,b/2)^{d}$, where $b>0$ is a scalar factor and $C$ is a unit
cell of the lattice. Equation (28) implies that URL point processes are Class
I hyperuniform with $\alpha=2$. The Appendix gives explicit formulas for the
characteristic function $\tilde{f}(\mathbf{k})$ of the uniform perturbation on
$[-b/2,b/2)^{d}$, as well as plots of the angular averaged function
$1-|\tilde{f}(k)|^{2}$, which is equal to the diffuse part of the structure
factor $S(k)$ and can be numerically evaluated with arbitrary precision. It
can be shown that in the limit $b\rightarrow 0$ and $k\rightarrow 0$, one has
$1-|\tilde{f}(k)|^{2}\propto b^{2}k^{2}$ Klatt  _et al._ (2020). Remarkably,
Klatt et al. showed that the Bragg peaks of URL models disappear when $b$
takes integer values Klatt  _et al._ (2020). We derived monodisperse 2D and 3D
sphere packings from URL models by decorating each point with a sphere of
radius $a$, which is determined by the prescribed $\phi_{2}$ values. We
verified that the spheres do not overlap for $b\leq 0.3$ in both 2D and 3D.

###  III.6 Disordered stealthy hyperuniform sphere packings

Disordered stealthy hyperuniform materials are exotic amorphous states of
matter that have unusual structural characteristics (hidden order at large
length scales) and physical properties, including desirable photonic and
transport properties Torquato (2016); Zhang  _et al._ (2016b). Disordered
stealthy hyperuniform packings have been generated using the collective-
coordinate optimization procedure Torquato  _et al._ (2015); DiStasio  _et
al._ (2018) by decorating the resulting ground-state point configurations by
nonoverlapping spheres of radius $a$ Zhang  _et al._ (2016b); Kim and Torquato
(2020). The degree of order of such ground states depends on a tuning
parameter $\chi$, which measures the extent to which the ground states are
constrained by the size of the cut-off value $K$ relative to the number of
degrees of freedom. For $\chi<1/2$, the ground states are typically disordered
and uncountably infinitely degenerate in the infinite-volume limit Torquato
_et al._ (2015). Using the fact that $\rho\chi=v_{1}(K)/[2d(2\pi)^{d}]$
Torquato  _et al._ (2015), it immediately follows that for identical
nonoverlapping spheres of radius $a$ that the dimensionless stealthy cut-off
value $Ka$ in terms of the packing fraction $\phi_{2}$ for any space dimension
$d$ is given by

$(Ka)^{d}=d2^{d+1}\Gamma^{2}(1+d/2)\phi_{2}\chi.$ (29)

We obtained numerical structure factors for disordered stealthy hyperuniform
many-body systems with $N=1000,\rho=1$ in 2D and 3D described in Ref. Kim and
Torquato (2021). The stealthy systems have $Ka=1.3$ for 2D and $Ka=1.5$ for
3D.

###  III.7 Ordered hyperuniform sphere packings

For models of ordered hyperuniform media, we consider sphere packings derived
from Bravais lattices in 2D and 3D, including the triangle lattice, the square
lattice, the body-centered cubic (BCC) lattice and the cubic lattice. For each
medium, the lattice points are decorated with identical spheres with radius
$a$. The structure factor of the sites of a Bravais lattice in
$\mathbb{R}^{d}$ is given by Torquato (2021a)

$S({\bf k})=\frac{(2\pi)^{d}}{v_{c}}\sum_{{\bf Q}\neq{\bf 0}}\delta({\bf
k}-{\bf Q}).$ (30)

where $v_{c}$ is the volume of a fundamental cell in direct space and
$\mathbf{Q}$ denotes a reciprocal lattice (Bragg) vector.

##  IV Results for the spreadabilities

In this section, we present the results for the spreadabilities of the models
in Sec. III. We use the angular-averaged versions of Eq. (1) derived in Ref.
Torquato (2021a), which were shown to be exact for translationally invariant
two-phase media. To compare the spreadabilities of different models, we scale
the microstructures such that all models possess unit specific surface, i.e,
$s=1$. This simple microscopic length scaling choice has been previously
applied by Kim and Torquato Kim and Torquato (2021) because it has the
advantage that $s$ is easily computable for many well-known models and $1/s$
is directly proportional to the arithmetic mean of the mean chord length
$\ell^{(i)}_{C}$ of both phases Torquato (2002). Specifically, given the
chosen values of $\phi_{2}$ in this study, Table 1 lists the values of the
characteristic dimensionless length scale $a^{*}=as$ for the models that we
consider, obtained from (27), (24) and (25).

Table 1: Values of the characteristic dimensionless length scale $a^{*}=as$ for the models in this study. Model | $a^{*}=as$
---|---
2D Debye | 0.589
2D FPS | 0.693
2D disk packings | 0.5
3D Debye | 0.64
3D FPS | 0.966
3D sphere packings | 0.6

It is instructive to compare our results for the spreadabilities for the
aforementioned models to that for nonhyperuniform Debye random media Debye
_et al._ (1957); Yeong and Torquato (1998); Ma and Torquato (2020); Skolnick
and Torquato (2021), as by Eq. (26). Torquato Torquato (2021a) obtained
closed-form expressions of the spreadability for a Debye random media in any
space dimension $d$:

${\cal S}(\infty)-{\cal S}(t)=\frac{d\omega_{d}\phi_{1}}{(4\pi
Dt/a^{2})^{d/2}}I_{d}(t),$ (31)

where

$I_{d}(t)=\frac{1}{a^{d}}\int_{0}^{\infty}r^{d-1}\exp(-r/a)\exp[-r^{2}/(4Dt)]dr.$
(32)

Specifically, in 2D and 3D,

$I_{2}(t)=\frac{2Dt}{a^{2}}\left\\{1-\exp(Dt/a^{2})\,\sqrt{\pi
Dt/a^{2}}\,\left[1-\mbox{erf}(\sqrt{Dt/a^{2}})\right]\right\\},$ (33)
$I_{3}(t)=\frac{2Dt}{a^{2}}\Big{\\{}\exp(Dt/a^{2})\,\sqrt{\pi
Dt/a^{2}}\,\left[1-\mbox{erf}(\sqrt{Dt/a^{2}})\right]\left[1+2Dt/a^{2}\right]-2Dt/a^{2}\Big{\\}}.$
(34)

The spreadabilities associated with fully penetrable spheres were computed
from (1a) and (21). For this particular model, the direct-space representation
(1a) is the preferred method to compute spreadability than the Fourier-space
representation, as $\chi_{{}_{V}}(r)$ is known exactly and vanishes
identically for $r\geq 2a$. Therefore, the integration in (1a) can be
evaluated to arbitrary precision. On the other hand, the spreadabilities for
all the sphere-packing models were computed by substituting (9) into the
Fourier-space representation (1b), because structure-factor data for sphere
packings are readily available. For periodic sphere packings, Torquato
Torquato (2021a) obtained

${\cal S}(\infty)-{\cal S}(t)=\phi_{2}\sum_{{\bf Q}\neq{\bf
0}}\frac{{\tilde{\alpha}}_{2}(|{\bf Q}|;a)}{v_{1}(a)}\exp[-|{\bf Q}|^{2}Dt].$
(35)

In practice, we truncated the numerical integration in (1b) at
$|\mathbf{k}|a=100$. Similarly, the summation in (35) was truncated at
$|\mathbf{Q}|a=100$. We verified that errors due to this truncation are
negligible.

(a)

(b)

Figure 5: (a) Autocovariance functions $\chi_{{}_{V}}(r)$ for 3D models with
$\phi_{2}=0.20$ and unit specific surface. (b) Spectral densities
$\tilde{\chi}_{{}_{V}}(k)$ of the 3D models. The positions of the Bragg peaks
for the cubic lattice packing agree exactly with those for the URL packing and
are therefore not shown.

###  IV.1 Two-point statistics for the models

To get a sense of how two-point statistics vary across the models considered
here, we plot in Fig. 5 the angular-averaged autocovariance functions
$\chi_{{}_{V}}(r)$ and spectral densities $\tilde{\chi}_{{}_{V}}(k)$ for the
3D models, except for the BCC packings, for which the spectral density just
consists of Bragg peaks. Figure 5(a) shows that all autocovariance functions
have the same the slope at the origin due to our choice that each model
possesses the same specific surface $s$. It is also obvious that all the
sphere packings possess very similar forms of $\chi_{{}_{V}}(r)$ for $r\leq
a=0.6/s$, because at low packing fractions, $S_{2}(\mathbf{r})$ at small $r$
is predominated by the probability that the two end points of $\mathbf{r}$ are
in the same sphere. It has been shown that
$s_{2}=d^{2}\chi_{{}_{V}}(r)/dr^{2}|_{r=0}$ vanishes for unjammed sphere
packings Torquato (2002), which is the case for all the sphere packings in
this study. On the other hand, it is clear from (26) and (21) that $s_{2}>0$
for the Debye random media and FPS. We also note that all the hyperuniform
media possess both positive and negative correlations, whereas the
nonhyperuniform Debye random media and FPS possess only positive correlations.

Figure 5(b) shows the small-$k$ behaviors of the 3D models that we study.
Among these models, the Debye random media has the largest value of
$\lim_{k\rightarrow 0}\tilde{\chi}_{{}_{V}}(k)$, followed by FPS and
equilibrium hard spheres. For the hyperuniform models, the perfect glass
packing, the URL packing and the disordered stealthy sphere packing represent
two-phase media with $\alpha=1,2$ and $\infty$, respectively. As expected from
Eq. (28), $\tilde{\chi}_{{}_{V}}(k)$ for the URL packing is composed of a
smooth function and Bragg peaks characteristic of the unperturbed cubic
lattice packing.

###  IV.2 Small-, intermediate- and large-time behaviors of the
spreadabilities

Spreadabilities for the 3D models with $\phi_{2}=0.20$ are shown in Fig. 6.
Figure 6(a) gives the spreadabilities associated with the three
nonhyperuniform media in 3D at small dimensionless times ($0\leq Dts^{2}\leq
0.01$). Torquato Torquato (2021a) derived short-time asymptotic expansion for
$\mathcal{S}(t)$:

${\cal
S}(t)=\frac{s}{\phi_{2}}\left(\frac{Dt}{\pi}\right)^{1/2}-\frac{2\,d\,s_{2}}{\phi_{2}}\,(Dt)+{\cal
O}(Dt/a^{2})^{3/2}.$ (36)

The initial decay rates are very similar for all models due to their identical
specific surfaces. However, as soon as $Dts^{2}>0.002$, the second term in Eq.
(36) plays an important role in the distinguishing the spreadabilities. The
positive values of $s_{2}$ for the Debye random media and FPS contribute to
their slower spreadabilities compared to sphere packings, for which $s_{2}=0$.

(a)

(b)

(c)

Figure 6: Spreadabilities for the 3D models considered with $\phi_{2}=0.20$
and unit specific surface. (a) Spreadabilities for the nonhyperuniform media
at small dimensionless times ($0<Dts^{2}<0.01$). (b) Spreadabilities for the
3D models at intermediate dimensionless times ($0.01<Dts^{2}<1$). (c)
Spreadabilities for the 3D models at intermediate and large dimensionless
times ($0.01<Dts^{2}<200$).

Figure 6(b) shows the spreadabilities for the 3D models at intermediate
dimensionless times ($0.01\leq Dts^{2}\leq 1$). Table 2 lists the values of
$\mathcal{S}(t)$ at dimensionless time $Dts^{2}=1$ for all the models. Note
that the intermediate-time behavior of $\mathcal{S}(t)$ clearly captures
intermediate-scale correlations of the two-phase media. Among the
nonhyperuniform media, the intermediate-time spreadabilities are highly
dependent on the coefficient $B=\tilde{\chi}_{{}_{V}}(0)$. For example, Debye
random media has a much larger value of $B$ than the other models, and its
intermediate-time excess spreadability is also much larger. To explain this
observation, we note that the spectral densities are nonnegative by definition
and are continuous functions of $k$ for the nonhyperuniform models. Therefore,
a larger value of $B$ implies that $\tilde{\chi}_{{}_{V}}(k)$ must be larger
over a finite range $[0,k^{*})$ of wavenumbers, where $k^{*}$ is some positive
number. Integrating larger values of the integrand
$\tilde{\chi}_{{}_{V}}(k)\exp[-k^{2}Dt]$ over the range $[0,k^{*})$ in Eq.
(1b) then results in slower spreadabilities at intermediate and large times.
We also find similar correspondence between $B$ and intermediate-time
spreadabilities for the URL packings in which $\alpha=2$.

Table 2: Spreadability at $Dts^{2}=1$ for the 3D models. Model | $\mathcal{S}(\infty)-\mathcal{S}(1/Ds^{2})$
---|---
Debye | 0.0560
FPS | 0.0289
HS equil. | 0.00437
$\alpha=1$ perfect glass | 0.00285
URL $b=0.3$ | $5.05\times 10^{-4}$
URL $b=0.2$ | $2.27\times 10^{-4}$
URL $b=0.1$ | $5.73\times 10^{-5}$
Disord. stealthy | $8.27\times 10^{-5}$
Cubic | $1.78\times 10^{-7}$
BCC | $5.72\times 10^{-9}$

Interestingly, Fig. 6(b) shows that the spreadability curve for the disordered
stealthy packing crosses those for the URL packings at intermediate
dimensionless times. At small times, the latter spreadabilities are faster due
to the presence of Bragg peaks in their associated spectral densities. On the
other hand, at large times, the spreadability for the disordered stealthy
packing is faster because the associated spectral density vanishes for a
finite range of wavenumbers around the origin. The aforementioned crossing
points represent the times beyond which the contribution from the stealthiness
behavior of the disordered stealthy packing outcompetes the contribution from
the Bragg peaks for the URL packings. The crossing time increases with
decreasing $b$, since smaller $b$ values result in larger Bragg peaks for the
URL packing [cf. (28)], and thus faster spreadabilities at small to
intermediate times. Therefore, we have shown that in addition to probing
microstructures on the very-small and very-large length scales,
$\mathcal{S}(t)$ provides a powerful tool to study important intermediate-
scale phenomena in heterogeneous media.

To highlight large-time behaviors of the spreadabilities, Fig. 6(c) plots
$\mathcal{S}(\infty)-\mathcal{S}(t)$ against $Dt$ on a log-log scale in the
dimensionless time range $0.001<Dts^{2}<10$. It is clear that the large-$t$
behaviors of the spreadabilities is determined by the form of the spectral
densities at small wavenumbers. Specifically, the excess spreadabilities for
typical nonhyperuniform media (for which $\alpha=0$), which include Debye
random media, fully penetrable spheres and equilibrium hard spheres, decay
asymptotically as $t^{-3/2}$ Torquato (2021a). The spreadabilities for
nonstealthy hyperuniform media (for which $0<\alpha<\infty$), including sphere
packings derived from perfect glasses and URL, decay as $t^{-(3+\alpha)/2}$.
The fastest decay rates are achieved by stealthy hyperuniform media (for which
$\alpha=\infty$), including sphere packings derived from disordered stealthy
point processes and Bravais lattices. The values of $\alpha$ can be very
easily and accurately extracted from the slopes of the curves on the log-log
plot at large times.

For nonstealthy models with the same value of $\alpha$, Eq. (3) predicts that
the asymptotic decay rate of $\mathcal{S}(\infty)-\mathcal{S}(t)$ is directly
proportional to the coefficient $B$. Table 3 shows values of $B$ and $C$,
where $C$ is the numerically determined coefficient for the long-time excess
spreadability, i.e., $\mathcal{S}(\infty)-\mathcal{S}(t)\sim C/t^{(d+3)/2}$.
The results confirm the predicted proportionality between $B$ and $C$. Among
the three typical nonhyperuniform models ($\alpha=0$), the Debye random media
possesses the largest value of the coefficient $B=\tilde{\chi}_{{}_{V}}(0)$,
and therefore has the slowest large-time spreadability among the models that
we have considered. For the URL packings ($\alpha=2$), Table 3 suggests that
$B\propto C\propto b^{2}$. Indeed, it can be shown from (28) that
$\tilde{\chi}_{{}_{V}}(\mathbf{k})$ for URL packings is proportional to
$b^{2}k^{2}$ at small $k$ Klatt  _et al._ (2020). For stealthy hyperuniform
media, our calculations confirm the trend predicted in Ref. Torquato (2021a)
that the large-time spreadability is faster for models with larger values of
$K$, where in the ordered case $K=Q_{1}$ is the first (smallest positive)
Bragg wavenumber. Torquato Torquato (2021a) showed that the long-time behavior
of spreadabilities for both ordered and disordered stealthy media involve
exponential decay rate $\exp(-K^{2}Dt)$, and hence possess faster decay rates
than any hyperuniform medium governed by a power-law decay. Torquato Torquato
(2021a) has shown that the BCC lattice gives the optimal packing for the
spreadability among all packings of identical spheres in three dimensions, as
it possesses the largest value of $Q_{1}/s$. Our findings are consistent with
this optimal property of the BCC packing.

Table 3: Values of the coefficients $B$ and $C$ for the nonhyperuniform models and the URL packings in three dimensions. Model | $B$ | $C$
---|---|---
Debye | 1.054 | 0.107
FPS | 0.3254 | 0.355
HS equil. | 0.0384 | 0.00389
URL $b=0.3$ | 0.00356 | $5.60\times 10^{-4}$
URL $b=0.2$ | 0.00158 | $2.49\times 10^{-4}$
URL $b=0.1$ | $3.95\times 10^{-4}$ | $6.25\times 10^{-5}$

###  IV.3 Spreadabilities for URL packings

Notably, we find that for the sphere packings derived from URL, the
spreadabilities at small to intermediate dimensionless times are very similar
to that of the unperturbed cubic-lattice packing, i.e. their initial decay
rates are nearly exponential. However, at large $t$, the spreadabilities tend
to a power-law decay, which is consistent with the small-wavenumber behavior
($\alpha=2$) of such packings. The nearly exponential decay up to intermediate
times is due to the fact that the small- to intermediate-scale two-point
correlations are mainly determined by the Bragg peaks in the URL packing Klatt
_et al._ (2020). On the other hand, it has been shown that the uncorrelated
stochastic displacements of the lattice points degrade the stealthy
hyperuniformity of the lattice into nonstealthy hyperuniformity described by
the power-law (2) with $\alpha=2$ for URL Kim and Torquato (2018); Klatt  _et
al._ (2020). We remark that for the URL packing with $b=0.1$, the snapshot in
direct space [Fig. 4(g)] is visually very similar to that of the unperturbed
lattice packings [Fig. 4(i)]. However, their spreadabilities are markedly
different at large $t$. Therefore, $\mathcal{S}(t)$ sensitively captures long-
range correlations that are not obvious in direct space.

(a)

(b)

Figure 7: (a) Spreadabilities for 3D URL packings and the cubic-lattice
packing with $\phi_{2}=0.20$ at intermediate dimensionless times
($0.01<Dts^{2}<10$). The transition time $t^{*}$ [see (38)] for each value of
$b$ are indicated by solid dots. (b) Plot of $t^{*}$ against $b$ for 3D URL
packings with $\phi_{2}=0.20$.

To quantitatively analyze the transition time $t^{*}$ of the spreadabilities
for a URL packing, we note that for any monodisperse sphere packing derived
from a perturbed lattice in which the perturbation characteristic function
$\tilde{f}(\mathbf{k})$ for the lattice points are independent and identical,
the excess spreadability can be obtained by substituting (9) into (1b), where
$S(k)$ in (9) is given by (28), i.e.,

$\begin{split}&\mathcal{S}(\infty)-\mathcal{S}(t)=\\\
&\phi_{2}\left[\int_{0}^{\infty}\tilde{\alpha}_{2}(k;a)(1-|\tilde{f}(\mathbf{k})|^{2})\exp(-k^{2}Dt)d\mathbf{k}+\sum_{\mathbf{Q}\neq\mathbf{0}}\frac{\tilde{\alpha}_{2}(Q;a)}{v_{1}(a)}|\tilde{f}(Q)|^{2}\exp(-Q^{2}Dt)\right].\end{split}$
(37)

From relation (37), one can extract the transition time $t^{*}$ that separates
small-time nearly exponential decay and large-time power-law decay of the
spreadability by equating the Bragg-peak part and the diffuse part of the
excess spreadability:

$\int_{0}^{\infty}\tilde{\alpha}_{2}(k;a)(1-|\tilde{f}(\mathbf{k})|^{2})\exp(-k^{2}Dt^{*})d\mathbf{k}=\sum_{\mathbf{Q}\neq\mathbf{0}}\frac{\tilde{\alpha}_{2}(Q;a)}{v_{1}(a)}|\tilde{f}(Q)|^{2}\exp(-Q^{2}Dt^{*}).$
(38)

The specific form of $\tilde{f}(\mathbf{k})$ for URL packings derived from the
hypercubic lattice for any space dimension $d$ and its angular average for
$d=2,3$ is given in the Appendix. Figure 7 shows the spreadabilities for the
URL packings with the corresponding transition times indicated by solid dots,
as well as the spreadability for the unperturbed simple-cubic lattice packing.
Note that on the log-log plots of $\mathcal{S}(t)$ for the URL packings,
gradual changes of slope occur in the vicinity of $t^{*}$. Figure 7(b) plots
the values of $t^{*}$ against $b$ for 3D URL packings with $\phi_{2}=0.20$, in
which we observe that $t^{*}$ is a decreasing function of $b$ and has a
logarithmic divergence in the limit $b\rightarrow 0$, i.e., $t^{*}\sim-\ln b$.
To derive this logarithmic divergence behavior, we first presume that $t^{*}$
is large in the limit $b\rightarrow 0$. The factors multiplying
$\exp(-k^{2}Dt^{*})$ on the left-hand side of (38) can then be replaced by
their small-$k$ behaviors, i.e., $1-|\tilde{f}(\mathbf{k})|^{2}\rightarrow
b^{2}k^{2}$ Klatt  _et al._ (2020), and
$\tilde{\alpha}_{2}(k;a)\rightarrow\exp(-ck^{2})$, where $c$ is a constant
Torquato (2018a). Therefore, the left-hand side of Eq. (38) has the small-$b$
limit

$\begin{split}\phi_{2}\int_{0}^{\infty}\exp(-ck^{2})b^{2}k^{2}\exp(-k^{2}Dt^{*})d\mathbf{k}\\\
=\frac{\phi_{2}b^{2}\exp(-ck^{2})\sqrt{\pi}}{4t^{3/2}}.\end{split}$ (39)

The small-$b$ behavior of the sum on right-hand side of (38) is dominated by
the Bragg peak with the smallest wavenumber $Q_{1}$. Therefore, the right-hand
side of (38) has the small-$b$ limit

$\frac{\phi_{2}\tilde{\alpha}_{2}(Q_{1};a)Z(Q_{1})}{v_{1}(a)}\exp(-Q_{1}^{2}Dt^{*}),$
(40)

where $Z(Q_{1})$ is the coordination number at radial distance $Q_{1}$.
Equating (39) and (40) yields

$2\ln b\sim\frac{3\ln t^{*}}{2}-Q_{1}^{2}Dt^{*},$ (41)

which corresponds to a logarithmic divergence of $t^{*}$ as $b\rightarrow 0$.
Our numerical solution of Eq. (38) at small $b$ confirms the asymptotic
relation (41).

###  IV.4 Effects of dimensionality on the spreadability

To compare spreadabilities across dimensions, we show spreadabilities of the
2D models in Fig. 8. We observe in Fig. 8(a) that for the same value of
$\alpha$, the excess spreadability decays asymptotically as
$t^{-(2+|\alpha|)/2}$, slower than that for 3D models. This confirms the
large-$t$ trend predicted in Ref. Torquato (2021a), i.e.,
$\mathcal{S}(\infty)-\mathcal{S}(t)\sim t^{-(d+\alpha)/2}$. Again, we notice
that the excess spreadabilities for the URL packings transform from
exponential decay at small and intermediate $t$ to power-law decay at large
$t$. Figure 8(b) compares the spreadabilities of equilibrium hard spheres at
small to intermediate $t$. The small-time decay of the excess spreadability is
faster for the 3D model. This is evident from Eq. (36), as $s/\phi_{2}$ is
larger for the 3D model.

(a)

(b)

(c)

Figure 8: (a) Spreadabilities of two-phase media in two dimensions with
$\phi_{2}=0.25$ and unit specific surface at intermediate dimensionless times
($0.01<Dts^{2}<1$). (b) Spreadabilities of two-phase media in two dimensions
with $\phi_{2}=0.25$ and unit specific surface at intermediate and large
dimensionless times ($0.01<Dts^{2}<200$). (c) Spreadabilities for equilibrium
hard spheres for $d=2,\phi_{2}=0.25$ and $d=3,\phi_{2}=0.20$ with unit
specific surface at small and intermediate dimensionless times
($0<Dts^{2}<0.5$).

##  V Algorithm for extracting long-time behaviors from Time-Dependent
Diffusion Measurements

Based on our study of spreadabilities of the idealized models, we introduce
here an algorithm that extracts large-scale behavior from time dependent
diffusion behaviors as measured by spreadability, or equivalently, by NMR or
MRI data. For simplicity, we first describe the algorithm using the
spreadability. Subsequently, we discuss the adaption of the algorithm to NMR
and MRI measurements using the precise mappings identified in Ref. Torquato
(2021a). For purposes of illustration, we treat the spreadability data as if
they were directly given by diffusion measurements when applying the
algorithm, despite the fact that we computed $\mathcal{S}(t)$ from two-point
correlations in this study.

To extract the asymptotic form of any spreadability data $\mathcal{S}(t)$, we
introduce a “set-in” time $t_{S}(\epsilon)$, defined as the time such that
$|\mathcal{S}(t)-\mathcal{S}_{l}(t)|=\epsilon$ for $t=t_{S}$ and
$|\mathcal{S}(t)-\mathcal{S}_{l}(t)|<\epsilon$ for all $t>t_{S}$. Here,
$\mathcal{S}_{l}(t)$ is the long-time excess spreadability given in Ref.
Torquato (2021a):

$\mathcal{S}(\infty)-\mathcal{S}_{l}(t)=Ct^{-\varphi}\qquad\text{Nonstealthy
media},$ (42) $\mathcal{S}(\infty)-\mathcal{S}_{l}(t)=C\exp(-\varphi
t)/t\qquad\text{Disordered stealthy media},$ (43)
$\mathcal{S}(\infty)-\mathcal{S}_{l}(t)=C\exp(-\varphi t)\qquad\text{Ordered
media},$ (44)

where $\varphi$ and $C$ are parameters to be determined from the spreadability
data. The self-consistent relations (42), (43) and (44) can be solved via the
following iterative procedure.

  1. 1.

Set $i=0$ and choose initial guesses of $t_{S,i},\varphi_{i}$ and $C_{i}$ as
well as the stopping criterion $\sigma$.

  2. 2.

Find $\varphi_{i+1}$, $C_{i+1}$ by fitting the spreadability for $t>t_{S,i}$
and obtain the large-time spreadability approximant $S_{l,i}(t)$ using (42),
(43) or (44).

  3. 3.

Find $t_{S,i+1}$, which is the time such that
$|\mathcal{S}(t_{S,i+1})-\mathcal{S}_{l,i+1}(t_{S,i+1})|=\epsilon$ and
$|\mathcal{S}(t)-\mathcal{S}_{l,i}(t)|<\epsilon$ for all $t>t_{S,i+1}$.

  4. 4.

If $|\varphi_{i+1}-\varphi_{i}|<\sigma$, stop and set
$t_{S}=t_{S,i+1},\varphi=\varphi_{i+1},C=C_{i+1}$. If not, set $i=i+1$ and go
to Step 2.

Because it is not known a priori whether a two-phase media is nonstealthy,
disordered stealthy or ordered, the aforementioned procedure must be executed
three different times, using Eq. (42), (43) and (44) alternatively each time.
The equation that yields the smallest $t_{S}$ is then accepted as the
functional form that best describes the asymptotic behavior of the
spreadability. Specifically, assuming nonstealthy media, $\varphi_{i+1}$ and
$C_{i+1}$ in Step 2 are obtained from a linear regression on the plot of
$\ln[\mathcal{S}(\infty)-\mathcal{S}(t)]$ against $\ln(t)$. Assuming
disordered stealthy media, the linear regression is performed on the plot of
$t\ln[\mathcal{S}(\infty)-\mathcal{S}(t)]$ against $t$. Assuming ordered
media, the linear regression is performed on the plot of
$\ln[\mathcal{S}(\infty)-\mathcal{S}(t)]$ against $t$. We chose the stopping
criterion to be $\sigma=10^{-6}$ for all executions of the algorithm.

Table 4 and 5 list the values of $t_{S}(10^{-6})$, $\varphi$ and $C$
determined from the algorithm for the 2D and 3D models studied in this work,
as well as the expected values of $\varphi$ and $C$. These parameters are
known exactly for Debye random media, FPS, URL packings and Bravais-lattices
packings, whereas those for the other models in Sec. III are inferred from the
corresponding spectral densities. As these idealized models span a wide range
of nonhyperuniform and hyperuniform classes, they serve as good benchmarks for
testing the general capability of the algorithm to accurately extract
$\varphi$ and $C$ from diffusion data of realistic heterogeneous media. Table
4 and 5 show that for all the models that we study, the values of $\varphi$
and $C$ extracted using our proposed algorithm agree closely with their
expected values. In cases in which these parameters are exactly known, the
extracted $\varphi$ values deviate less than 1% from the exact ones and the
extracted $C$ values deviate less than 5% from the exact ones. Indeed, we find
that the extracted $\varphi$ and $C$ converge to their exact values as
$\epsilon\rightarrow 0$, and that $\varphi$ can be estimated with reasonable
accuracy (less than $10\%$ error) as long as $\epsilon<10^{-4}$.

We also observe a general trend that $t_{S}$ decreases with increasing
nonhyperuniform or hyperuniform class. Among models with the same value of
$\alpha$, $t_{S}$ decreases with decreasing $B$. Among the stealthy models,
$t_{S}$ decreases with increasing $K$. This is because for models with a
higher degree of hyperuniformity, the corresponding spectral densities can
usually be approximated by the power law (2) over a larger range of
wavevectors. However, we point out that for URL packings with extremely small
$b$, the “set-in” time of the power-law decay is expected to increase due to
large Bragg peaks.

We remark that due to the connection between the spreadability and NMR
measurements, including the pulsed field gradient spin-echo (PFGSE) amplitude
$\mathcal{M}(k,t)$ Mitra  _et al._ (1992); Sen and Hürlimann (1994) and the
MRI-measured water diffusion in biological media Novikov  _et al._ (2014), our
algorithm can be directly applied on NMR or MRI data to infer large-scale
microstructural information of real materials. Specifically, consider a fluid-
saturated porous medium, which invariably contains paramagnetic impurities at
the interface, resulting generally partially absorbing boundary conditions.
Torquato Torquato (2021a) found that one can map the spreadability problem to
the PFGSE amplitude problem via the transformations
$\mathcal{S}(\infty)-\mathcal{S}(t)\rightarrow\mathcal{M}(\mathbf{q}=0,t)-\phi_{2}$
and $D\rightarrow\mathcal{D}(t)$, where $\phi_{2}$ is the porosity and
$\mathcal{D}(t)$ is the time-dependent diffusion coefficient for the porous
medium. For diffusion-weighted MRI of water-saturated biological media, one
has a similar mapping
$\mathcal{S}(\infty)\rightarrow\mathcal{D}_{e}=\mathcal{D}(\infty)$ and
$\mathcal{S}(t)\rightarrow\mathcal{D}(t)$ Torquato (2021a). Therefore, our
algorithm can be directly adapted to analyze experimental data for PFGSE or
MRI to extract the coefficients $\varphi$ and $C$.

Table 4: “Set-in” times $t_{S}$ and large-time parameters $\varphi$ and $C$ for the spreadabilities of the 2D models considered in this work. Model | $\phi_{2}$ | $Dt_{S}(10^{-6})s^{2}$ | $\varphi$ | $C$ | Expected $\varphi$ | Expected $C$
---|---|---|---|---|---|---
Debye | 0.25 | 309 | 1.003 | 0.133 | 1 | 0.130
FPS | 0.25 | 43.1 | 0.999 | 0.0592 | 1 | 0.0594
HS equil. | 0.25 | 17.2 | 1.002 | 0.0211 | 1 | 0.0209 sim
Perfect glass $\alpha=1$ | 0.25 | 13.06 | 1.499 | $6.12\times 10^{-3}$ | 3/2 |  $6.15\times 10^{-3}$ sim
Perfect glass $\alpha=2$ | 0.25 | 5.67 | 2.013 | $1.06\times 10^{-3}$ | 2 |  $9.87\times 10^{-4}$ sim ]
URL $b=0.3$ | 0.25 | 4.71 | 1.997 | $1.44\times 10^{-3}$ | 2 | $1.48\times 10^{-3}$
URL $b=0.2$ | 0.25 | 3.63 | 1.997 | $6.41\times 10^{-4}$ | 2 | $6.56\times 10^{-4}$
URL $b=0.1$ | 0.25 | 2.34 | 1.996 | $1.60\times 10^{-4}$ | 2 | $1.64\times 10^{-4}$
Disord. stealthy | 0.25 | 1.04 | 6.662 | 0.0329 | 6.648 sim | 0.0370 sim
Square | 0.25 | 0.710 | 12.55 | 0.418 | 12.57 | 0.430
Triangle | 0.25 | 0.600 | 14.450 | 0.548 | 14.50 | 0.558
Table 5: “Set-in” times $t_{S}$ and large-time parameters $\varphi$ and $C$
for the spreadabilities of the 3D models considered in this work.

Model | $\phi_{2}$ | $Dt_{S}(10^{-6})s^{2}$ | $\varphi$ | $C$ | Expected $\varphi$ | Expected $C$
---|---|---|---|---|---|---
Debye | 0.2 | 110 | 1.494 | 0.113 | 3/2 | 0.118
FPS | 0.2 | 28.1 | 1.499 | 0.0361 | 3/2 | 0.0365
HS equil. | 0.2 | 7.23 | 1.500 | $4.30\times 10^{-3}$ | 3/2 |  $4.31\times 10^{-3}$ sim
Perfect glass $\alpha=1$ | 0.2 | 4.02 | 2.001 | $3.14\times 10^{-3}$ | 2 |  $3.14\times 10^{-4}$ sim
URL $b=0.3$ | 0.2 | 3.39 | 2.495 | $6.06\times 10^{-4}$ | 5/2 | $6.25\times 10^{-4}$
URL $b=0.2$ | 0.2 | 2.66 | 2.494 | $2.68\times 10^{-4}$ | 5/2 | $2.78\times 10^{-4}$
URL $b=0.1$ | 0.2 | 1.80 | 2.494 | $6.69\times 10^{-5}$ | 5/2 | $6.94\times 10^{-5}$
Disord. stealthy | 0.2 | 1.06 | 5.377 | 0.0176 | 5.341 sim | 0.0168 sim
Cubic | 0.2 | 0.410 | 14.43 | 0.388 | 14.43 | 0.387
BCC | 0.2 | 0.380 | 18.18 | 0.554 | 18.18 | 0.556
Debye | 0.157 | 48.1 | 1.496 | 0.0686 | 3/2 | 0.0704
Fontainebleau | 0.157 | 24.9 | 1.501 | 0.0379 | - | -
FPS | 0.157 | 21.9 | 1.499 | 0.0253 | 3/2 | 0.0255

##  VI Application to Fontainebleau sandstone

In this section, we apply the lessons learned from the spreadabilities of the
idealized models to analyze the structural characteristics of a sample
Fontainebleau sandstone Coker  _et al._ (1996). Because Fontainebleau
sandstone is a porous medium with a pure mineral composition (almost 100%
quartz) and free of any clay, it has frequently been used to study the
relation between pore geometry and effective properties. Coker  _et al._
(1996); Bied  _et al._ (2002); Chen  _et al._ (2016); Hosa  _et al._ (2016);
Saadi  _et al._ (2017); Sun and Wong (2018). Understanding its microstructures
is of importance in many applications, including calibration of flow
experiments Saadi  _et al._ (2017); Hosa  _et al._ (2016), petroleum
engineering Bied  _et al._ (2002); Chen  _et al._ (2016) and geological dating
Missenard  _et al._ (2017). The morphology and physical properties (e.g. fluid
permeability) of Fontainebleau sandstone samples have been accurately studied
via X-ray tomographic imaging Coker  _et al._ (1996); Sun and Wong (2018).
Here, we further characterize their microstructures via diffusion
spreadability measurements using the analysis described in Sec. IV and V.

Figure 9(a) shows a sample filtered slice of Fontainebleau sandstone Coker
_et al._ (1996). Figure 9(b) and (c) show the autocovariance function and
spectral density, respectively, for a sandstone sample with porosity
$\phi_{2}=0.157$ and specific surface $s=0.0154\mu\text{m}^{-1}$, which we
obtained from the corresponding two-point correlation function $S_{2}(r)$
published in Ref. Coker  _et al._ (1996). We observe that $\chi_{{}_{V}}(r)$
for Fontainebleau sandstone contains slightly negative correlations in the
range $2.2<rs<3.7$, and vanishes identically for $rs\geq 3.7$, which indicates
that the sample is not hyperuniform. The negative correlations in
$\chi_{{}_{V}}(r)$ corresponds to a maximum away from the origin in
$\tilde{\chi}_{{}_{V}}(k)$. Indeed, the plot of the spectral density, shown in
Fig. 9(c), definitively enables us to conclude that the sandstone is
nonhyperuniform of the typical kind; see Eq. (19). Therefore, it is
instructive to compare its two-point correlations to those for the idealized
typical nonhyperuniform models with $\phi_{2}=0.157$, including the Debye
random media and FPS. Figure 9(b) shows that $\chi_{{}_{V}}(r)$ for the
sandstone closely resemble that for the Debye random media at small to
intermediate values of $rs$. Figure 9(c) shows that the value of
$B=\tilde{\chi}_{{}_{V}}(0)$ for the sandstone lies in between those for the
Debye random media and FPS. As in the case of FPS, the fact that
$\chi_{{}_{V}}(r)$ for the sandstone has finite support enables us to
accurately compute the corresponding spreadability using Eq. (1a) in this
study. As we have stressed earlier, we can infer structural characteristics
across length scales by direct dynamic measurements via the time-dependent
spreadability or NMR/MRI techniques Mitra  _et al._ (1992); Sen and Hürlimann
(1994); Novikov  _et al._ (2014).

Figure 10 shows the spreadabilities for Debye random media, Fontainebleau
sandstone and FPS with $\phi_{2}=0.157$. The spreadability for the sandstone
closely resembles that for the Debye random media at small to intermediate
dimensionless times ($0\leq Dts^{2}\leq 1$), which reflects their similar
small-scale correlations. We applied the algorithm described in Sec. V to
extract large-time behaviors of these spreadabilities. As noted earlier, we
treat the spreadability data for the sandstone as directly given when applying
the algorithm described in Sec. V. The extracted parameters for the
Fontainebleau sandstone are $\varphi=(3+\alpha)/2=1.501,C=0.0253$, which
indicates clearly that the sandstone is typical nonhyperuniform (Table 5). The
trend of $C$ values for the three nonhyperuniform media confirms that the
spectral density for the sandstone at small to intermediate wavenumbers lies
in between those for Debye random media and FPS. The set-in time for the
spreadability of the sandstone sample $Dt_{S}(10^{-6})s^{2}=24.9$ is also on
the order of magnitude for typical nonhyperuniform media.

(a)

(b)

(c)

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 $\mu$m. This figures is reproduced from
the one in Ref. Coker  _et al._ (1996). (b) Autocovariance functions for
Fontainebleau sandstone Coker  _et al._ (1996), Debye random media and fully
penetrable spheres with $\phi_{2}=0.157,s=0.0154$ $\mu$m-1. (c) Spectral
densities for Fontainebleau sandstone, Debye random media and fully penetrable
spheres. Figure 10: Spreadabilities for Fontainebleau sandstone, Debye random
media and fully penetrable spheres with $\phi_{2}=0.157$ at small,
intermediate and large dimensionless times $0.001<Dts^{2}<200$.

##  VII Conclusions

In this work, we investigated the practical applicability of the spreadability
to probe microstructural information of real heterogeneous materials by
accurately computing $\mathcal{S}(t)$ for idealized 2D and 3D models that
represent a wide variety of hyperuniform and nonhyperuniform classes. These
models include Debye random media, fully penetrable spheres, equilibrium hard
spheres, which are nonhyperuniform, as well as sphere packings derived from
“perfect glasses”, uniformly randomized lattices, disordered stealthy
hyperuniform point processes and Bravais lattices, which are hyperuniform. Our
calculations confirm the previously predicted trends Torquato (2021a) that the
small-, intermediate and long-time behavior of $\mathcal{S}(t)$ sensitively
captures the small-, intermediate- and long-range characteristics of the
models. For typical nonhyperuniform models, the intermediate-time
spreadability is slower for models with larger values of the coefficient
$B=\tilde{\chi}_{{}_{V}}(0)$. Our study of the aforementioned models enabled
us to devise an algorithm that efficiently and accurately extract large-scale
structural characteristics from time-dependent diffusion behaviors, as
measured by the spreadability, or equivalently, by NMR or MRI data. The
algorithm also determines the “set-in” time of the large-time behavior of
time-dependent diffusion measurements. Such analysis was used to characterize
the large-scale behaviors of a sample Fontainebleau sandstone, which we show
is typical nonhyperuniform. Therefore, we have demonstrated that the
spreadability provides a simple and robust probe of crucial microstructural
information across length scales of realistic heterogeneous media. Our
proposed algorithm in Sec. V is shown to be capable of accurately extracting
large-scale behaviors of real samples from time-dependent diffusion data.

Remarkably, in the case of URL packings, we found that
$\mathcal{S}(\infty)-\mathcal{S}(t)$ has nearly exponential decay at small to
intermediate times, but transitions to power-law decay at large $t$, i.e.,
$\mathcal{S}(t)$ clearly captures both the lattice-like order on small to
intermediate length scales and the nonstealthy hyperuniform behavior on large
length scales. We showed that the transition time $t^{*}$ that separates
small- and large-time decay behaviors is well-defined and has a logarithmic
divergence as $b\rightarrow 0$. The behaviors of the spreadability for URL
packings stand in sharp contrast with those for disordered stealthy
hyperuniform packings, which are characterized by a relatively slow small-time
decay and an exponentially fast large-time decay.

We remark that the spreadability formula derived in Ref. Torquato (2021a) can
be regarded as a type of Gaussian smoothing of the spectral density, since it
is weighted by a Gaussian function Torquato (2002); see Eq. (1b). Indeed, we
observe that the numerically obtained spectral densities in this study contain
some degree of noise at small wavenumbers [Fig. 5(b)]. However, their
corresponding excess spreadabilities are very smooth functions of time and
exhibit unambiguous power-law or exponential decay at large $t$ [Fig. 6(a) and
8(a)], which can be analyzed using our proposed algorithm to extract large-
scale structural characteristics. Therefore, extracting the exponent $\alpha$
from spreadability offers a more robust and accurate means to compute it from
real data compared to the direct numerical fitting of the small-$k$ data of
the spectral density.

It is noteworthy that the microstructural characteristics extracted from time-
dependent diffusion measurements are capable of predicting other effective
properties that depend on two-point correlations. For example, Torquato
recently derived a formula for the fluid permeability of porous media that
depends on the spectral density ${\tilde{\chi}}_{{}_{V}}({\bf k})$, which has
been shown to provide reasonably accurate permeability predictions of certain
of hyperuniform and nonhyperuniform porous media Torquato (2020). He used this
formula to show that the dimensionless fluid permeability, i.e., (permeability
multiplied by $s^{2}$), for stealthy hyperuniform media is expected to be
lower relative to nonhyperuniform ones and that the BCC lattice packing
minimizes the fluid permeability among common crystal lattices (simple cubic,
FCC and BCC) at fixed porosity Torquato (2020). Thus, the BCC lattice packing
yields the fastest spreadability and relatively low values of the fluid
permeability for packings of identical spheres at a given porosity.

Our findings on the spreadabilities for the idealized models in this study
provide the basis for inverse design of materials with desirable time-
dependent diffusion properties. Specifically, one could realize desired
short-, intermediate- and large-time diffusion rates by constructing two-phase
media with the corresponding values of $s$, $\alpha$ and $B$ (or $K$ in the
case of stealthy media). Thus, a promising avenue for future study is the
determination of microstructures of two-phase media that realize prescribed
functional forms of $\mathcal{S}(t)$. This is an inverse problem that can be
potentially tackled by adapting methods employed to construct microstructures
with prescribed autocovariance functions and physical properties Yeong and
Torquato (1998); Zachary and Torquato (2011); Ma and Torquato (2020); Skolnick
and Torquato (2021).

## Acknowledgement

The authors thank Zheng Ma and Murray Skolnick for helpful discussions and the
code for reconstructing Debye random media. Acknowledgement is made to the
donors of the American Chemical Society Petroleum Research Fund under Grant
No. 61199-ND9 and to the Air Force Office of Scientific Research Program on
Mechanics of Multifunctional Materials and Microsystems under Grant No.
FA9550-18-1-0514 for support of this research.

*

##  Appendix A Characteristic functions for URL point processes

The characteristic function $\tilde{f}(\mathbf{k})$ of the uniform
distribution on $[-b/2,b/2)^{d}$ is given by

$\tilde{f}(\mathbf{k})=\left(\frac{2}{b}\right)^{d}\prod_{i=1}^{d}\frac{\sin(bk_{i}/2)}{k_{i}},$
(45)

where $k_{i}$ is the $i$-th component of the wavevector $\mathbf{k}$. Figure A
shows plots of $1-|\tilde{f}(k)|^{2}$, i.e., the diffuse part of $S(k)$, for
2D and 3D URL point processes with $b=0.1,0.2$ and $0.3$.

(a)

(b)

Figure A: (a) Plot of $1-|\tilde{f}(k)|^{2}$ for 2D URL point processes. (b)
Plot of $1-|\tilde{f}(k)|^{2}$ for 3D URL point processes.

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