PHYSICAL REVIEW E 83, 051308 (2011)

Hyperuniformity, quasi-long-range correlations, and void-space constraints in maximallyrandom jammed particle packings. I. Polydisperse spheres

Chase E. Zachary*

Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA

Yang Jiao†

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA

Salvatore Torquato‡

Department of Chemistry, Department of Physics, Princeton Center for Theoretical Science, Program in Applied and ComputationalMathematics, and Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA

(Received 20 January 2011; revised manuscript received 14 March 2011; published 31 May 2011)

Hyperuniform many-particle distributions possess a local number variance that grows more slowly thanthe volume of an observation window, implying that the local density is effectively homogeneous beyond afew characteristic length scales. Previous work on maximally random strictly jammed sphere packings in threedimensions has shown that these systems are hyperuniform and possess unusual quasi-long-range pair correlationsdecaying as r−4, resulting in anomalous logarithmic growth in the number variance. However, recent work onmaximally random jammed sphere packings with a size distribution has suggested that such quasi-long-rangecorrelations and hyperuniformity are not universal among jammed hard-particle systems. In this paper, weshow that such systems are indeed hyperuniform with signature quasi-long-range correlations by characterizingthe more general local-volume-fraction fluctuations. We argue that the regularity of the void space inducedby the constraints of saturation and strict jamming overcomes the local inhomogeneity of the disk centers toinduce hyperuniformity in the medium with a linear small-wave-number nonanalytic behavior in the spectraldensity, resulting in quasi-long-range spatial correlations scaling with r−(d+1) in d Euclidean space dimensions.A numerical and analytical analysis of the pore-size distribution for a binary maximally random jammed systemin addition to a local characterization of the n-particle loops governing the void space surrounding the inclusionsis presented in support of our argument. This paper is the first part of a series of two papers considering therelationships among hyperuniformity, jamming, and regularity of the void space in hard-particle packings.

DOI: 10.1103/PhysRevE.83.051308 PACS number(s): 45.70.−n, 05.20.Jj, 61.50.Ah

I. INTRODUCTION

Maximally random jammed (MRJ) packings of hard par-ticles are the most disordered structures, according to somewell-defined order metrics, that are rigorously incompress-ible and nonshearable [1]. These systems are prototypical“glassy” structures in the sense that they are structurallyrigid, yet lack Bragg peaks in their scattering spectra [2].In this sense, the idea of the MRJ state has replaced themathematically ill-defined notion of random close packing [1].Nearly half a century ago, these systems were thoughtto describe the disordered structure of liquids [3], but itis now known that three-dimensional MRJ monodispersesphere packings possess unusual quasi-long-range (QLR) paircorrelations decaying as r−4 [4]. This property is markedlydifferent from typical liquids, in which pair correlationsdecay exponentially fast [2,5,6]. Similar QLR behavior hasalso been observed in noninteracting spin-polarized fermionicground states [7,8], the ground state of liquid helium [9],and the Harrison-Zeldovich power spectrum of the densityfluctuations of the early Universe [10]. However, for each

*czachary@princeton.edu†yjiao@princeton.edu‡torquato@princeton.edu

of these examples and for MRJ hard-sphere packings, thestructural origins of these correlations have been heretoforeunknown, even for monodisperse systems. Furthermore, it isan open problem to generalize these QLR correlations forMRJ states to polydisperse packings, in which the jammingproperties are intimately related to the size distribution of theparticles [11].

Motivated by the observation that MRJ packings are struc-turally rigid with a well-defined contact network, Torquato andStillinger conjectured [5] that all strictly jammed (i.e., mechan-ically rigid), saturated [12] packings of monodisperse spheresin d-dimensional Euclidean space Rd are hyperuniform,meaning that infinite-wavelength local density fluctuationsvanish [5], a proposition for which no counterexample has beenfound to date [13]. This conjecture suggests that saturation andstrict jamming are sufficient to induce hyperuniformity, albeitnot necessary [5]. Hyperuniform systems play an integral rolein understanding the relationship between fluctuations in localmaterial properties and microstructural order [4–6,14–16].These systems have applications to the large-scale structure ofthe Universe [10], the structure and collective motion of grainsin vibrated granular media [17], the structure of living cells[18], transport through composites and porous media [19],the study of noise and granularity of photographic images[20,21], identifying properties of organic coatings [22], andthe fracture of composite materials [23]. For microstructures

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CHASE E. ZACHARY, YANG JIAO, AND SALVATORE TORQUATO PHYSICAL REVIEW E 83, 051308 (2011)

(a) (b)

FIG. 1. (Color online) (a) A binary packing of hard disks near the MRJ state. (b) A polydisperse packing of hard disks near the MRJ state.

consisting of “point” particles, one considers fluctuations inthe local number density within some observation window.Hyperuniform point patterns possess local density fluctuationsthat asymptotically grow more slowly than the volume ofthe window. Recently, the concept of hyperuniformity hasbeen extended to include systems composed of finite-volumeinclusions of arbitrary geometries [6]; in these cases, thequantity of interest is the fluctuation in the so-called localvolume fraction, defined as the fraction of the volume within anobservation window covered by a given phase. Hyperuniformheterogeneous media possess local-volume-fraction fluctua-tions that asymptotically decay faster than the reciprocal ofthe volume of the observation window, implying that the localvolume fraction approaches a global value beyond relativelyfew characteristic length scales.

Previous work on three-dimensional (3D) MRJ monodis-pere sphere packings [4] has supported the Torquato-Stillingerconjecture by showing that the structure factor S(k), propor-tional to the scattering intensity, approaches zero linearly asthe wave number k → 0, inducing a QLR power-law tail r−4

in the pair correlation function g2(r). This behavior impliesthat local-number-density fluctuations grow logarithmicallyfaster than the surface area of an observation window, but stillslower than the window volume. In this sense, the “degree”of hyperuniformity in MRJ packings is minimal among allstrictly jammed saturated packings. However, recent numer-ical [24] and experimental work [25] on polydisperse MRJpackings failed to detect vanishing infinite-wavelength local-number-density fluctuations, leading to the misconception thathyperuniform quasi-long-range correlations are not a universalsignature of the MRJ state. Unlike monodisperse systems,the distribution of particle sizes in a polydisperse packingintroduces locally inhomogeneous regions as the particlesdistribute themselves through space, as shown in Fig. 1. Onecan see in the binary packing that local clusters of small

particles are distributed near and around larger inclusions, andthe result is that the point pattern generated by the disk centerspossesses local inhomogeneities that are expected to (andindeed do) induce volume-order scaling within the numbervariance. The situation is apparently even more complex forthe polydisperse system in Fig. 1 since the size distributionresults in a highly inhomogeneous local structure with smallparticles trapped between larger ones with a high probability.

These observations have raised a number of quantitativelyand conceptually difficult questions. First, what is theappropriate extension of the Torquato-Stillinger conjecturefor monodisperse MRJ packings to systems with a size distri-bution? Clearly, one must explicitly account for the shapeinformation of the particles. Second, in the event that onecan generalize the Torquato-Stillinger conjecture, there isto date no satisfactory structural explanation for the linearsmall-wave-number scaling of the structure factor observed for3D MRJ monodisperse hard-sphere packings, which indicatesthe presence of QLR pair correlations. This extraordinarilydifficult problem is tantamount to providing an analyticalprediction of the MRJ state. Unfortunately, no such rigoroustheory currently exists, even for the considerably simplerproblem of predicting the scalar MRJ density [26]. Thepresence of QLR correlations makes this problem inherentlynonlocal, and, therefore, methods that attempt to predict theMRJ state based only on packing fraction and local criteria,such as nearest-neighbor and Voronoi statistics, are invariablyincomplete [26].

We have presented arguments in a recent Letter to suggeststrongly that hyperuniformity and quasi-long-range paircorrelations are signatures of saturated MRJ packings of hardparticles, including binary disks, ellipses, and superdisks [27].In this paper, we provide detailed evidence to show thatpolydisperse MRJ packings of hard spheres, althoughinhomogeneous with respect to the number variance, possess

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local-volume-fraction fluctuations decaying faster than thereciprocal of the volume of an observation window. Thisobservation is consistent with the generalized Torquato-Stillinger conjecture that all strictly jammed saturatedpackings of spheres are hyperuniform with respect to fluctua-tions in the local volume fraction. Our major results include thefollowing:

(1) Infinite-wavelength local-number-density fluctuationsdo not vanish for MRJ packings of polydisperse hard spheres.Local-volume-fraction fluctuations provide the appropriatestructural description of these packings because they accountcorrectly for the size distribution of the particles. Importantly,our work studying local-volume-fraction fluctuations containspreviously published results for 3D monodisperse MRJ hard-sphere packings as a special case.

(2) Signature QLR pair correlations scaling asymptoticallyas r−(d+1) in d Euclidean dimensions are observed for allsystems that we study, including binary disks with varyingsize ratios and compositions and polydisperse disk packings.Our results suggest that these special correlations may be auniversal feature of the MRJ state.

(3) Strict jamming places a strong constraint on the dis-tribution of the available void space external to the particlessuch that hyperuniformity is observed when considering local-volume-fraction fluctuations, even when the point pattern ofthe sphere centers is locally inhomogeneous.

(4) The competition between maximal randomness andstrict jamming of the packings ensures that the void-spacedistribution is sufficiently broad to induce QLR correlationsbetween particles, thereby providing a direct qualitative struc-tural explanation for the linear small-wave-number region ofthe generalized scattering intensity.

In a companion paper [28], we investigate hyperuniformityand quasi-long-range pair correlations in MRJ packings ofnonspherical hard particles. The results of both papers suggestthat these features are structural signatures of the MRJ state.

II. BACKGROUND AND DEFINITIONS

A. Point patterns

We consider point patterns to be realizations of stochasticpoint processes. Formally, a stochastic point process is amethod of placing points in some space (such asRd ) accordingto an underlying probability distribution. This random settingis quite general, incorporating cases in which the locations ofthe points are deterministically known, such as in a Bravaislattice. A Bravais lattice is defined by the integer linearcombinations of a set of d basis vectors in Rd . By takingthe union of a Bravais lattice with its translate(s) by one ormore vectors, one can also form periodic non-Bravais lattices,also known as lattices with a basis in the physics literature.Unless otherwise stated, in this paper we use the term latticeto refer exclusively to a Bravais lattice.

A statistically homogeneous point process is completelydetermined by the number density ρ and the countably infiniteset (in the thermodynamic limit) of n-particle correlationfunctions. The n-particle correlation function gn(r1, . . . ,rn)is proportional to the probability density of finding n particlecenters in volume elements around the positions r1, . . . ,rn,regardless of the positions of the remaining particles in the

system. For an arbitrary point process, deviations of gn fromunity provide a measure of the correlations among pointsin the system. Note that specifying only a finite number M

of the n-particle correlation functions defines a class ofmicrostructures with degenerate M-particle statistics [29].Of particular interest is the pair correlation function g2,which defines the average number of particles surroundinga reference particle of the point process. Closely related tothe pair correlation function is the total correlation functionh(r) = g2(r) − 1. Since g2(r) → 1 as r → +∞ (r = ‖r‖) forisotropic, translationally invariant systems without long-rangeorder, it follows that h(r) → 0 in this limit, meaning that h isgenerally integrable with a well-defined Fourier transform.

It is common in statistical mechanics when passing to re-ciprocal space to consider the associated structure factor S(k),which, for a translationally invariant system, is defined by

S(k) = 1 + ρh(k), (1)

where h is the Fourier transform of the total correlationfunction, ρ is the number density, and k = ‖k‖ is themagnitude of the reciprocal variable to r. We utilize thefollowing definition of the Fourier transform:

f (k) =∫Rd

f (r) exp (−ik · r) dr, (2)

where k · r = ∑di=1 kiri is the conventional Euclidean inner

product of two real-valued vectors. For radially symmetricfunctions [i.e., f (r) = f (‖r‖) = f (r)], the Fourier transformmay be written as

f (k) = (2π )d/2∫ ∞

0rd−1f (r)

J(d/2)−1(kr)

(kr)(d/2)−1dr. (3)

B. Two-phase random heterogeneous media

Closely related to the notion of a stochastic point process isthat of a two-phase random heterogeneous medium (or randomset), which we define to be a domain of space V ⊆ Rd ofvolume V � +∞ that is composed of two regions: the phase-1region V1 of volume fraction φ1 and the phase-2 region V2

of volume fraction φ2 [30]. The statistical properties of eachphase i of the system are specified by the countably infiniteset of n-point probability functions S(i)

n , which are definedby [11,31–33]

S(i)n (r1, . . . ,rn) =

⟨n∏

j=1

I (i)(rj )

⟩, (4)

where I (i) is the indicator function for phase i:

I (i)(x) ={

1, x ∈ Vi

0, else.(5)

The function Sn defines the probability of finding n points atpositions r1, . . . ,rn all within the same phase.

Upon subtracting the long-range behavior from S2, oneobtains the autocovariance function χ (r) = S2(r) − φ2, whichis generally integrable. It is important to recognize thatthe autocovariance function is independent of the choice ofreference phase, meaning that it is a global descriptor of

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correlations within the system. This property will play aparticularly important role in this paper when we consider therelationship between local-volume-fraction fluctuations andthe void space between inclusions in the microstructure. Theanalog of the structure factor in this context is the so-calledspectral density, which is the Fourier transform χ of theautocovariance function [34]. The autocovariance functionobeys the bounds [11]

−min{(1 − φ)2,φ2} � χ (r) � (1 − φ)φ, (6)

where φ is the volume fraction of an arbitrary reference phase.We remark that it is an open problem to identify additionalnecessary and sufficient conditions that the autocovariancefunction must satisfy in order to correspond to a binarystochastic process [11,35–38].

C. Sphere packings and a categorization of jamming

A sphere packing is a collection of nonoverlapping spheresin d-dimensional Euclidean space Rd . The packing densityφ (equivalent to the volume fraction of the particle phase) isdefined as the fraction of space covered by the spheres, whichmay be polydisperse. An important characteristic of a packingis its degree of randomness (or the antithesis, order), whichreflects nontrivial information of the packing structure. Thedegree of randomness (order) can be quantified by a set oforder metrics ψ [1]. It is an open and challenging problem toidentify good order metrics, but it has recently been proposedthat hyperuniformity is itself a measure of order on large lengthscales [5,6].

One method of classifying sphere packings involves char-acterizing the extent to which particles are jammed. Torquatoand Stillinger [39,40] have provided a precise definition of theterm jamming and have proposed a hierarchical classificationscheme for sphere packings by invoking the notions of local,collective, and strict jamming. A packing is locally jammedif no particle in the system can be translated while fixingthe positions of all other particles. A collectively jammedpacking is locally jammed such that no subset of spheres cansimultaneously be continuously displaced without moving itsmembers out of contact both with one another and with theremainder set. A packing is strictly jammed if it is collectivelyjammed and if all globally uniform volume nonincreasingdeformations of the system boundary are disallowed by theimpenetrability constraints. The reader is referred to Ref. [39]for further details.

As previously mentioned, the maximally random jammedstate is defined as the most disordered jammed packingin a given jamming category (i.e., locally, collectively, orstrictly jammed) [1]. The MRJ state is well defined for agiven jamming category and choice of order metric, and ithas recently supplanted the ill-defined random close-packed(RCP) state [1]. In this paper, we focus on maximally randomstrictly jammed polydisperse sphere packings in Rd .

D. Hyperuniformity in point processes: Local-number-densityfluctuations

A hyperuniform point process has the property that thevariance in the number of points in an observation window �

grows more slowly than the volume of that window. In the caseof a spherical observation window, this definition implies thatthe local number variance σ 2

N (R) grows more slowly than Rd ind dimensions, where R is the radius of the observation window.Torquato and Stillinger [5] have provided an exact expressionfor the local number variance of a statistically homogeneouspoint process in a spherical observation window

σ 2N (R) = ρv(R)

[1 + ρ

∫Rd

h(r)α(r; R)dr]

, (7)

where R is the radius of the observation window, v(R) isthe volume of the window, and α(r; R) is the so-calledscaled intersection volume. The latter quantity is geometricallydefined as the volume of space occupied by the intersectionof two spheres of radius R separated by a distance r andnormalized by the volume of a sphere v(R). Exact expressionsfor α(r; R) in arbitrary dimensions have been given byTorquato and Stillinger [41].

It is convenient to introduce a dimensionless density φ,which need not correspond to the volume fraction, according to

φ = ρv(D/2) = ρπd/2Dd

2d(1 + d/2), (8)

where D is a characteristic length scale of the system (e.g., themean nearest-neighbor distance between points). The numbervariance admits the following asymptotic scaling [5]:

σ 2N (R) = 2dφ

{AN

(R

D

)d

+BN

(R

D

)d−1

+O

[(R

D

)d−1]}

,

(9)

where O(x) denotes all terms of order less than x. This resultis valid for all periodic point patterns (including lattices),quasicrystals that possess Bragg peaks, and disorderedsystems in which the pair correlation function g2 decays tounity exponentially fast [5]. Explicit forms for the asymptoticcoefficients AN and BN are given by [5]

AN = 1 + ρ

∫Rd

h(r)dr = lim‖k‖→0

S(k), (10)

BN = −ρκ(d)

D

∫Rd

h(r)‖r‖dr, (11)

where κ(d) = (1 + d/2)/{π1/2[(d + 1)/2]}.Any system with AN = 0 satisfies the requirements for

hyperuniformity. Although the expansion (9) will hold for allperiodic and quasiperiodic point patterns with Bragg peaks,this behavior is not generally true for disordered hyperuniformsystems. For example, it is known that if the total correlationfunction h ∼ r−(d+1) for large r [S(k) ∼ k for small k], thenσ 2

N (R) ∼ (a0 ln R + a1)Rd−1 [7]. Such behavior occurs inmaximally random jammed monodisperse sphere packingsin three dimensions [4] and noninteracting spin-polarizedfermion ground states [7,8]. Other examples of “anomalous”local density fluctuations have been characterized by Zacharyand Torquato [6].

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E. Hyperuniformity in two-phase random heterogeneousmedia: Local-volume-fraction fluctuations

In order to define hyperuniformity for heterogeneousmedia, we introduce the local volume fraction τi(x) of phase i

according to

τi(x; R) = 1

v(R)

∫I (i)(z)w(z − x; R)dz, (12)

where v(R) is the volume of the observation window and w

is the corresponding indicator function. Using this definition,the variance σ 2

τ (R) in the local volume fraction is given by

σ 2τ (R) = 1

v(R)

∫Rd

χ (r)α(r; R)dr, (13)

which is independent of the choice of reference phase. Thevariance in the local volume fraction admits the asymptoticexpansion [6]

σ 2τ = ρ

2dφ

{Aτ

(D

R

)d

+ Bτ

(D

R

)d+1

+ O

[(D

R

)d+1]}

,

(14)

Aτ =∫Rd

χ (r)dr = lim‖k‖→0

χ(k), (15)

Bτ = −κ(d)

D

∫Rd

‖r‖χ (r)dr. (16)

The coefficients Aτ and Bτ in (15) and (16) control theasymptotic scaling of the fluctuations in the local volumefraction. It then follows that σ 2

τ decays faster than R−d asR → +∞ for those systems such that

lim‖k‖→0

χ(k) = 0, (17)

which generalizes hyperuniformity for a two-phase randomheterogeneous medium [6].

F. The effect of polydispersity on local fluctuations

Here we consider how the presence of polydispersityin a sphere packing affects the fluctuations in the localnumber density and local volume fraction. For a packingof polydisperse spheres, the distribution of sphere radii isdetermined by a probability density f (R), and the fundamentalstatistical descriptors of the medium therefore involve averages〈·〉R [42] over the distribution of sphere sizes [11]. For apacking of polydisperse spheres with M distinct radii, thedensity function f (R) takes the form

f (R) =M∑i=1

γiδ(R − Ri), (18)

where γi = Ni/N is the mole fraction of species i. As anexample, we will continually refer in this paper to the averageparticle diameter 〈D〉R of a polydisperse hard-disk packing:

〈D〉R =M∑i=1

γiDi, (19)

where Di is the diameter of species i.

The autocovariance function χ (r) for a heterogeneousmedium consisting of impenetrable polydisperse spheres isgiven by [11,43,44]

χ (r) = ρ〈vint(r; R)〉R + ρ2∫

〈h(x; R1,R2)

× vint(r − x; R1,R2)〉R1,R2dx, (20)

which implies that

Aτ =∫

χ (r)dr (21)

= ρ〈v2(R)〉R + ρ2〈v(R1)v(R2)∫

h(r; R1,R2)dr〉R1,R2 .

(22)

Unlike for monodisperse sphere packings [6], the result (22)shows that it is generally not possible to separate the shapeinformation of the inclusions from the details of the point pat-tern generated by the sphere centers. This observation suggeststhat, for polydisperse microstructures, hyperuniformity of theunderlying point pattern does not induce hyperuniformity withrespect to local-volume-fraction fluctuations; conversely, it isalso possible to find heterogeneous media for which Aτ = 0but AN > 0.

We note that the first term contributing to χ (r) in (20) can beinterpreted as the probability of finding two points, separatedby a displacement r, in a single particle of the packing. Thesecond term is therefore related to the probability of finding thepoints in two different particles. It is clear that only this latterterm, containing the pair correlation function, can contributeto the linear small-wave-number region of the spectral density,and it is therefore responsible, albeit in a highly nontrivial way,for the onset of QLR correlations in MRJ packings.

III. LOCAL DENSITY AND VOLUME FRACTIONFLUCTUATIONS IN POLYDISPERSE MRJ PACKINGS

A. Generation of MRJ polydisperse hard-disk packings

Motivated by the generalized conjecture that all strictlyjammed packings of d-dimensional spheres are hyperuniformwith respect to local-volume-fraction fluctuations, we havegenerated several packings of binary disks (2D) near the MRJstate using a modified Lubachevsky-Stillinger (LS) packingalgorithm [45–49], wherein particles with a fixed size ratio un-dergo event-driven molecular dynamics while simultaneouslyincreasing in size according to some prescribed growth rate.The initial growth rate used in our simulations is γ = 0.01,

but near the jamming point, a much smaller growth rate γ =10−6 is used to establish well-defined interparticle contacts.The Lubachevsky-Stillinger algorithm has been shown toproduce MRJ packings with these parameters consistent withthe positively correlated translational and orientational ordermetrics [50,51]. Our statistics for binary packings are averagedover 50 configurations of 10 000 particles; for polydispersepackings, we have averaged over 10 configurations of 10 000particles. Our results have been compared to systems of up to106 particles to verify invariance of the statistics with respectto system size.

For this study, we have chosen a particle size ratioβ = Rlarge/Rsmall = 1.4 and mole fractions γsmall = 0.75 and

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(a)

(c) (d)

(b)

FIG. 2. (a) Structure factor S(k) fora binary packing of disks near theMRJ state. (b) The corresponding spec-tral density χ(k). Note that infinite-wavelength local density fluctuationsare not suppressed, unlike local-volume-fraction fluctuations on the equivalentlength scale. (c) Spectral density forparticle concentrations γsmall = 2/3 andγlarge = 1/3 with size ratio β = 2.5.Hyperuniformity of the packing is un-affected by changes in these parametersas expected by the Torquato-Stillingerconjecture. (d) Spectral density for apolydisperse MRJ packing of disks witha uniform distribution of radii in theinterval [Rmin,Rmax].

γlarge = 0.25; however, the focus of this paper is to elucidatea universal property of polydisperse MRJ packings, and ourresults are expected to apply for a range of size ratios, sizedistributions, and mole fractions (see Fig. 2) [52]. For theremainder of this paper, we therefore focus on the binary casewith the disclaimer that our results are expected to apply forgeneral polydisperse MRJ packings. This algorithm has beenused in previous studies of 3D monodisperse sphere packingsnear the MRJ state [47]. The resulting packings in this studyhave a final volume fraction φ ≈ 0.8475, which is below theclose-packed density φcp = √

3π/6. Figure 1 (Sec. I) providesa typical realization of a binary packing; note that the particledistribution is saturated and nonperiodic.

B. Hyperuniformity and local-volume-fraction fluctuations

We have calculated both the structure factor S(k) andthe spectral density χ (k) for the binary MRJ disk packingsusing discrete Fourier transforms of the local density ρ(r) andindicator function I (r) of the particle phase of the packing.Specifically,

S(k) =

∣∣∣∑Nj=1 exp(−ik · rj )

∣∣∣2

N(k �= 0), (23)

χ(k) =

∣∣∣∑Nj=1 exp(−ik · rj )m(k; Rj )

∣∣∣2

V(k �= 0), (24)

where

m(k; R) ≡∫Rd

exp(−ik · r)�(R − ‖r‖)dr (25)

is the Fourier transform of the indicator function for ad-dimensional sphere of radius R [53]. Note that the shape ofthe enclosure (defined by a set of basis vectors {ei}) restrictsthe wave vectors such that k · ei = 2πn for all i, where n ∈ Z.Since the zero wave vector is removed from the spectrum inthe expressions above, one must define

χ(0) ≡ limN,V →+∞

χ(‖k‖ = kmin), (26)

with a similar expression for S(0). The limit here is taken atconstant number density ρ = N/V , and kmin is the smallestcomputable wave vector as determined by the shape ofthe boundary. To obtain radially symmetric forms of thestructure factor and spectral density, we angularly average overall wave vectors with equal magnitude. Our results for thestructure factors and spectral densities of the binary andpolydisperse MRJ hard disk packings are shown in Fig. 2.The structure factors for these systems lack Bragg peaks,reflecting the absence of long-range order. Of particularimportance is the behavior of the structure factor near theorigin. A fit of the small-k region (k � 0.5) with a third-order polynomial suggests S(0) ≈ 0.104 > 0, meaning thatthe point pattern generated by the disk centers does notpossess vanishing infinite-wavelength local-number-densityfluctuations, an observation verified by direct calculationof the number variance in Fig. 3. Note that the local

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HYPERUNIFORMITY, . . . . I. POLYDISPERSE SPHERES PHYSICAL REVIEW E 83, 051308 (2011)

(a) (b)

FIG. 3. (Color online) (a) Asymptotic fluctuations in the local number density for the MRJ binary disk packing. (b) Asymptotic fluctuationsin the local volume fraction for the binary MRJ packing, demonstrating that the system is indeed hyperuniform.

number variance asymptotically scales with the volume ofthe observation window. This behavior differs from MRJpackings of monodisperse spheres in three dimensions [47],where the structure factor decays linearly to zero for smallwave numbers. Bidisperse distributions of disks are inherentlyinhomogeneous with respect to the locations of the particlecenters. For the systems studied here, the large concentrationof small particles results in local clusters that are essentiallyclose packed. However, the introduction of large particles intothe system generates effective grain boundaries between theselocal clusters, breaking the uniformity of the underlying pointpattern.

We note that the structure factor for our binary packing canbe decomposed as

S(k) = SS(k) + SL(k) + SC(k), (27)

where

SS(k) =

∣∣∣∑NS

j=1 exp(−ik · rj )∣∣∣2

N, (28)

SL(k) =

∣∣∣∑NL

j=1 exp(−ik · xj )∣∣∣2

N, (29)

SC(k) = 2 Re

{[ ∑NS

j=1 exp(−ik · rj )][ ∑NL

�=1 exp(ik · x�)]

N

}

(30)

are the particle structure factors incorporating small-small,large-large, and small-large correlations, respectively. Notethat NS is the number of small particles with positions {ri},and NL is the number of large particles with positions {xi}.These partial contributions to the structure factor are shown inFig. 4. None of the partial contributions to the structure factorpossess a vanishing small-wave-number region, implying thatthe local density fluctuations of the small and large particleseach scale with the volume of an observation window as does

the covariance. These observations clearly demonstrate thatinformation contained in the structure factor is not sufficientto characterize the packings because it neglects the details ofthe particle shapes.

A direct calculation of the spectral density for the binaryMRJ system shows markedly different behavior from thestructure factor as seen in Fig. 2. One notices that infinite-wavelength local-volume-fraction fluctuations are suppressedby the system. We have fit the small-k region of the spectraldensity (in units of 〈D〉2

R) using a third-order polynomialof the form a0 + a1K + a2K

2 + a3K3 and have found a0 =

(1.0 ± 0.2) × 10−5, strongly suggesting that this system ishyperuniform with respect to local-volume-fraction fluctu-ations. We have verified this claim by directly calculatingthe variance in the local volume fraction, shown in Fig. 3.The local-volume-fraction fluctuations decay faster than thereciprocal of the volume of an observation window and loga-rithmically slower than R−(d+1) consistent with the presenceof QLR correlations.

0 1 2 3 4K = k<D>/(2π)

-0.5

0

0.5

1

1.5

2SS(K)SL(K) SC(K)

FIG. 4. (Color online) Partial contributions SS(k), SL(k), andSC(k) to the structure factor of the binary MRJ disk packings.

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0 1 2 3 4

K = k<D>/(2π)

0

0.2

0.4

0.6

0.8

<|m

(K; R

)|>

/<D

>2

^

FIG. 5. The average magnitude 〈|m(k; R)|〉R of the spatial contri-bution to the spectral density.

C. Properties of the spectral density of binary MRJhard-disk packings

The spectral density is dominated by a peak with wave-length just below 〈D〉R , approximately corresponding to thesmall-particle diameter. This observation is consistent with themicrostructure of the medium, which contains several regionsof almost close-packed clusters of small particles. In orderto further understand the behavior of the spectral density, itis important to recognize that both the local distribution ofinclusions and the shape information of the particles contributeto χ . However, the relative influence of each componenton the spectral density varies throughout the spectrum. Tocharacterize this effect, we recall the representation (24) of thespectral density, which admits the following upper bound:

χ (k) � Nρ (〈|m(k; R)|〉R)2 . (31)

Figure 5 plots the average 〈|m(k; R)|〉R , which, according toEq. (31), controls the upper bound on the spectral density.Note that, for small wave numbers [k〈D〉R/(2π ) � 2], the

bound (31) places only weak constraints on the spectraldensity, implying that this region of the spectrum is controlledby information in the local structure of the heterogeneousmedium. We emphasize that the shape information of the parti-cles must still be included at small wave numbers to account forvanishing infinite-wavelength local-volume-fraction fluctua-tions, but, as we show below, such geometric information onlyprovides the appropriate weights for the partial contributions tothe structure factor to induce hyperuniformity. In contrast, thelarge-wave-number region of the spectrum closely follows thebehavior of the upper bound (31), meaning that the length scaleimposed by the decoration of the particle centers, here chosento be disks, controls the spectral density. We emphasize thatthis portion of the spectrum arises from the shape informationof the inclusions themselves and is almost entirely independentof the distribution of particles in the system. Specifically, theparticle indicator function m(r; R), which has compact support[0,R], possesses a Fourier representation m(k; R) that is bothlong ranged and has an intrinsic period, which, for a binarypacking, leads to interference effects in the large-wave-numberregion.

Similar behavior arises in monodisperse systems, wherethe spectral density is exactly given χ(k) = ρm2(k; R)S(k),

0 0.5 1 1.5 2 2.5

K = kD/(2π)

0

0.05

0.1

0.15

0.2

χ(K)/D2

4φ m2(k; D)/(πD

4)

^^

FIG. 6. (Color online) Spectral density and corresponding shapecontribution for a microstructure generated by the step-function g2

process. Note that the shape contribution controls the spectral densityfor large wave numbers.

and one can rigorously separate information contained in thepoint pattern generated by the sphere centers from the shapeinformation of the inclusions. Figure 6 shows the spectraldensity and shape contribution for a system of impenetrabledisks with pair correlation function g2(r) = �(D − r). Notethat the bound (31) only applies when S(k) is at a maximum;for clarity, we have omitted the associated scaling factor onthe shape contribution. As with the binary MRJ packing, this“step-function process” possesses a spectral density that, forsmall wave numbers, depends signficantly on the distributionof sphere centers; the large-wave-number region is almostexactly equal to the shape contribution since the structurefactor approaches an asymptotic value of unity. Although, forbinary systems, polydispersity precludes the direct separationof the shape information from the underlying point pattern,these examples show that qualitatively these componentscontinue to affect only specific portions of the spectral density.

The dominance of the shape information for large wavenumbers has the surprising effect of almost completelysuppressing fluctuations for K = k〈D〉R/(2π ) ≈ 2.5,thereby suggesting that local-volume-fraction fluctuationsessentially vanish on this length scale. In actuality, the

0 1 2 3 4

x = R/<D>

0

0.1

0.2

0.3

0.4

σ τ(x)

MRJsquare latticebinary square lattice

FIG. 7. (Color online) Standard deviation of the local volumefraction for the close-packed binary square and square lattices withthe corresponding result for the MRJ binary packing.

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(a) (b)

FIG. 8. (Color online) (a) Unit cell for the square lattice at close packing. (b) Unit cell for a close-packed binary variant of the square lattice.

nonuniformity of the microstructure precludes a completeextinction of the variance in the local volume fraction;nevertheless, the local-volume-fraction variance (Fig. 7)undergoes a sharp change in slope near this length scale,highlighting a transition to the “geometrically controlled”region of the spectral density. Physically, this behavior impliesthat observation windows with radii given by the appropriatewavelength capture the effective pore size surrounding theinclusions in such a way that the medium is essentiallyhomogeneous on this local scale. This observation suggeststhat the void space surrounding the particles plays a central rolein determining the spatial statistics of a heterogeneous system.

D. Probing the origin of QLR pair correlations

It is important to note that the small-k behavior ofthe spectral density for the binary MRJ packing results inan “anomalous” asymptotic scaling of the variance in thelocal volume fraction. Specifically, our results suggest thatthe spectral density is nonanalytic at the origin with anexpansion χ (k) ∼ a1k + O(k2) as k → 0, implying that theautocovariance function exhibits quasi-long-range behaviorand scales with r−3 [r−(d+1) in d dimensions]. Equivalentbehavior has been observed for the structure factor of MRJmonodisperse sphere packings in three dimensions [4], andthe number variance in that case has been shown to scaleaccording to

σ 2N (R) ∼ (b0 + b1 ln R)R2 + O(R). (32)

A related scaling must also hold for the variance in the localvolume fraction of our binary MRJ packings; specifically,

σ 2τ (R) ∼ (c0 + c1 ln R)

R3+ O(R−4). (33)

One can directly see the effect of this behavior in Fig. 7, wherethe variance in the local volume fraction for the MRJ packingsis compared to the square lattice packing and its binary variant,

the unit cells for which are shown in Fig. 8. Note that local-volume-fraction fluctuations in the MRJ packings decay moreslowly than in either of the periodic systems; indeed, as thesize distribution of the void space becomes smaller and moreuniform, local-volume-fraction fluctuations are more rapidlysuppressed on the global scale of the microstructure. Sincethe square lattice packing is not even strictly jammed, thiscomparison suggests that strict jamming is neither a necessarynor a strong determinant of hyperuniformity in heterogeneousmedia.

Identifying the origin of the linear small-wave-numberregion of the spectral density is an open problem that mustbe related to the structural features of the MRJ state. Thisproblem is particularly difficult for our polydisperse packingssince the underlying point pattern generated by the particlecentroids possesses nonvanishing infinite-wavelength local-number-density fluctuations. However, the spectral densityχ (k) can be expressed in terms of the partial structure factorsas

χ (k) = ρm2(k; RS)SS(k) + ρm2(k; RL)SL(k)

+ ρm(k; RS)m(k; RL)SC(k), (34)

where RS and RL are the radii of the small and large particles,respectively. This result follows directly from an expansion of(24). As k → 0, we therefore find

χ (0) = ρv2(RS)SS(0) + ρv2(RL)SL(0)

+ ρv(RS)v(RL)SC(0), (35)

implying that the particle volumes provide the appropriateweights to properly balance the small- and large-particle vari-ances with the covariance between the particles. Additionally,since m(k; R) possesses no linear term in its small-wave-number Taylor expansion, it follows that the appearance ofa linear small-wave-number region in the spectral density,and therefore QLR correlations, must involve an appropriate

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superposition of linear contributions in the partial structurefactors.

This observation suggests that quasi-long-range correla-tions in MRJ packings arise from the competing effects ofstrict jamming and maximal randomness. Incompressibilityof the structure from strict jamming implies that particlesshould be correlated over several characteristic length scales;indeed, along the equilibrium branch of the binary hard-diskphase diagram, one expects that correlations become fullylong ranged at the close-packed density, corresponding tophase-separated lattice structures [54]. However, maximalrandomness interferes with this long-range order, resulting inthe apparent r−(d+1) asymptotic scaling in the pair correlationfunction. For our binary packings, this scaling is encodedin the partial pair correlations of the structure, which, afterappropriate weighting with the shape information of theparticles, induces hyperuniformity.

We emphasize that a complete explanation for the ap-pearance of the linear small-wave-number region of thespectral density is intractable because the problem is inherentlynonlocal due to hyperuniformity and the presence of QLRcorrelations. Furthermore, it has recently been establishedthat decreasing the exponent α in the small-wavenumberscaling χ(k) ∼ kα of the structure factor in a many-particlesystem is associated with greater disorder, potentially inducingclustering among the particles [55]. Sublinear scaling [e.g.,χ (k) ∼ k1/2] is therefore inconsistent with strict jamming.More generally, hyperuniformity is associated with an effec-tive interparticle repulsion that attempts to evenly distributethe particles throughout space, and the length scale of thisrepulsion increases with increasing exponential scaling ofthe small-wave-number region of the spectral density [55].The conditions of saturation and impenetrability in an MRJpacking induce structural order that apparently competeswith the constraint of maximal randomness to minimizethe scaling of the small-wave-number region of the spectraldensity to its smallest integer value, an effect that weargue is physically tied to the void-space distribution of thesystem.

To motivate our discussion of the void space and its funda-mental importance to polydisperse MRJ packings, we mentiona peculiar property of hyperuniformity in these packings.Namely, one can maintain hyperuniformity in polydisperseMRJ hard-particle systems even upon shrinking the particlesat a fixed size ratio so long as one does not affect theunderlying statistical distribution of the point process; thiseffect is illustrated in Fig. 9. To understand this behavior,we note that the leading-order term governing the expansionof the Fourier transform of the particle indicator functionm(k; R) is the volume of the particle πR2, meaning that,for small wave numbers, we have (in the case of the binarypackings)

χ (k) ∼ 1

V

∣∣∣∣∣∣πR2small

Nsmall∑j=1

exp(ik · rj )

+πR2large

Nlarge∑�=1

exp(ik · r�)

∣∣∣∣∣∣2

, (36)

0 0.5 1 1.5 2 2.5

K = k<D>/(2π)

0

0.025

0.05

0.075

0.1

0.125

χ(K

)/<

D>

2

κ = 0.95κ = 0.85κ = 0.75

^

FIG. 9. (Color online) Spectral densities upon shrinking the par-ticles in the binary MRJ packings at fixed size ratio β = Rlarge/Rsmall.The parameter κ = Rshrink/RMRJ is the scaling factor.

which, upon rescaling Ri → κRi for κ < 1, suggests

χshrink(k) ∼ κ4χMRJ(k) (k → 0). (37)

Therefore, hyperuniformity of the MRJ binary packing is notlost when performing this scaling operation. Physically, thesmall-wave-number region of the spectral density effectivelyhomogenizes the medium due to the coupling between thewave number and the particle radius in the indicator functionm(k; R), meaning that this region is not affected by changesat the boundaries of the particles so long as the underlyingstatistics of the point pattern generating the medium remainconstant. This observation suggests that a saturated and strictlyjammed sphere packing extends naturally to an uncountablyinfinite family of hyperuniform heterogeneous media relatedby an appropriate scaling parameter. One can map this behaviordirectly using the result (35), which indicates that the spectraldensity at small wave numbers defines a quadratic form in theparticle radii when the underlying point pattern of the diskcenters is held fixed; the corresponding curve is plotted inFig. 10. Furthermore, this scaling has the effect of deformingthe void space surrounding the particles in a uniform mannerthat preserves the regularity of the pore distribution even uponbreaking both the jamming and saturation constraints. Themedium therefore is able to retain information from the strictlyjammed configuration even upon relaxing the sizes of theinclusions. Additionally, it follows from the quadratic form(35) that nonuniform changes in the particle radii inherentlybreak the hyperuniformity of the packing.

This behavior has important implications for MRJ packings.First, this effect is geometric in origin and depends on theexplicit inclusion of particle-shape information to appropri-ately balance the partial structure factors of the packing.This observation immediately implies that “point” informationcontained in the particle centers is not sufficient to describe thesystem. Second, the presence of interparticle contacts is not es-sential for the onset of hyperuniformity and QLR correlations.This subtle point, although easy to appreciate mathematically,is highly nontrivial from a physical perspective. It impliesthat, on the global scale of the microstructure, the space isregularized in such a way that hyperuniformity is preservedupon making uniform scaling deformations of the particlephase. Importantly, since the spectral density is a descriptor of

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00.1

0.20.3

0.40.5

0

0.2

0.4

0.6

0.80

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

RS

RL

FIG. 10. (Color online) The quadraticform (35), equal to the small-wave-numberlimit of the spectral density χ (0), as a functionof the radii of the particles in the packing. Auniform decrease in the particle radii definesa basin of hyperuniform systems.

both the particle and the void phases, this homogenization isinvariant to the choice of reference phase. Note, in particular,that upon breaking the contact network of the MRJ packing,the void space can become connected throughout the space,implying that the spatial statistics of the microstructure mustaccount for correlations over large length scales. It is for thesereasons that we turn our attention to the void-space distributionin the following sections.

Note that our scaling analysis does not apply if the particleradii are uniformly increased since breaking the impene-trability constraint implies that higher-order microstructuralinformation is necessary to characterize the medium, andEq. (36) is no longer valid. Additionally, if the particles areshrunk to the point that the small particle radius vanishes, thenthe packing will no longer be hyperuniform since importantshape information about the system has been lost. In termsof the void space, the distribution of pore sizes will beskewed toward higher values due to the sudden appearanceof “holes” in the microstructure, corresponding to the lostsmall particles, thereby deregularizing the microstructure andbreaking hyperuniformity.

IV. CHARACTERIZATION OF THE PORE-SIZEDISTRIBUTION

A. Numerical evaluation of pore-size statistics

It is clear from the discussion above that the void phase sur-rounding the disk inclusions of the binary MRJ packings playsa significant role in characterizing local fluctuations in themedium. Indeed, we argue that the conditions of strict jammingand saturation place strong constraints on the distribution of thepore sizes, effectively regularizing the local structure aroundeach disk such that the system is hyperuniform with respect to

local-volume-fraction fluctuations even though the disk cen-ters do not constitute a hyperuniform point pattern. To supportour arguments concerning the void space, we have quantifiedthe size of the available void space using the so-called comple-mentary pore-size cumulative distribution functionF (δ) [11],which represents the fraction of the void space external to theinclusion phase with a pore radius larger than δ. Equivalently,if we define P (δ) as the probability density that a randomlychosen point in the void space lies within δ and δ + dδ fromthe nearest point on the void-inclusion interface, then

F (δ) =∫ +∞

δ

P (�)d�. (38)

0 0.1 0.2 0.3 0.4 0.5

x = δ/<D>

0

0.2

0.4

0.6

0.8

1

F(x

)

MRJsquare latticebinary square latticeequil. HS; φ = 0.5

FIG. 11. (Color online) Cumulative pore-size distributions F forthe binary MRJ packing, the square lattice Z2, a saturated binaryvariant of Z2, and a system of equilibrium hard spheres with volumefraction φ = 0.5.

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TABLE I. First moments and standard deviations of the pore sizefor distributions of hard disks.

〈δ〉/〈D〉 σδ/〈D〉Binary MRJ 0.0438 0.0380Z2 0.0626 0.0480Binary Z2 0.0332 0.0258Equilibrium hard disks (φ = 0.5) 0.1612 0.1267

Figure 11 shows the cumulative pore-size distribution func-tion for the binary MRJ packings along with the correspondingresults for the square lattice and binary square lattice packingsand a system of equilibrium monodisperse hard disks atvolume fraction φ = 0.5. The equilibrium hard-disk packingis neither saturated nor jammed; it is also known that thissystem is not hyperuniform [5]. As a result, it possesses a broaddistribution of pore sizes, and the probability of finding largepores (i.e., on the order of a particle diameter) is nonvanishing.In contrast, both the square lattice packing and its binaryvariant possess narrow pore-size distributions with compactsupports, which result from the close-packed nature of thepackings. Interestingly, Fig. 11 indicates that the binary MRJpacking also possesses a narrow pore-size distribution thatessentially falls between the square lattice and binary squarelattice packings. This observation suggests that the void spaceis highly constrained by the condition of strict jamming andalmost regular in its distribution.

One can directly compute the moments 〈δn〉 of thepore-size density from the cumulative distribution F (δ) usingthe relation [11]

〈δn〉 = n

∫ +∞

0δn−1F (δ)dδ. (39)

Lower-order moments of the pore-size density arise in boundson the mean survival times and principal relaxation times ofheterogeneous materials and are thus important descriptorsof microstructures [56,57] (see Sec. V B below). Table Iprovides the first moments and standard deviations of the poresize for the systems mentioned in Fig. 11. The average poresizes for the hyperuniform systems are significantly smallerthan for the equilibrium hard-disk packing. In fact, the pore-size distribution of the binary MRJ packing appears to be morelocalized about its mean than even the square lattice packing.We conclude from this information that the available voidspace in the binary MRJ disk packing is sufficiently restrictedsuch that local-volume-fraction fluctuations decay fasterthan the reciprocal of the volume of an observation window,thereby reflecting the underlying regularity of the pore space.

B. Bounds on the pore-size distribution

It is important to note that the pore-size distribution for ahyperuniform hard-particle packing with volume fraction φ

must asymptotically decay faster than the correspondingdistribution for a nonhyperuniform medium with the same one-point statistics; this claim is a generalization of an argumentby Gabrielli and Torquato [15]. Since nonhyperuniform mediaare by definition “Poisson-like,” it suffices then to considerthe pore-size distribution for a polydisperse system of fully

penetrable disks. The cumulative pore-size distribution FP (δ)for such a system is known analytically [11,33] and is given by

FP (δ) = (1/φ1) exp [−ρ〈v(δ + R)〉R] , (40)

where the angular brackets denote an average over the diskradii and φ1 is the volume fraction of the matrix phase. We em-phasize in these calculations that we are considering the pore-size distribution for a 2D Poisson-distributed heterogeneousmedium with volume fraction φ that is the same as a referencehyperuniform system. However, equivalence of the one-pointstatistics does not imply that the number densities are the samebetween the systems. To account for this discrepancy, wewrite ρ = η/(π〈R2〉R) = ln(1/φ1)/(π〈R2〉R), which implies

FP (δ) = exp[−(s〈D〉R/φ1)δ2 − (s〈D〉R/φ1)δ], (41)

where s = ρφ1〈s(R)〉R is the specific surface of a Poisson-distributed medium and s(R) is the surface area of ad-dimensional sphere of radius R. Note that we have intro-duced the length scale 〈D〉R and reduced variable δ = δ/〈D〉Rin Eq. (41). Letting b = s〈D〉R/φ1, we find the following upperbounds for the first and second moments of the pore-sizedensity:

〈δ〉UL = 1

2

√π

bexp [b/4] erfc(

√b/2), (42)

〈δ2〉UL = 1

b(1 − b〈δ〉UL). (43)

The above Poisson bound will be rigorously true for the fullpore-size distribution of our polydisperse MRJ packings basedsolely on the presence of hyperuniformity. However, we cansignificantly tighten the bounds of the moments of the pore-size distribution for binary MRJ packings based on our analysisbelow of the local voids. Specifically, since the distribution ofvoids within the MRJ packings is dominated by three- andfour-particle loops, the average pore size must be less than thecorresponding average pore size of a saturated square lattice.This effect is explicitly shown in Fig. 11, where it is clear thatthe square lattice provides a good approximate upper boundfor the full pore-size distribution with the exception of the tail,due to the presence of higher-order loops in the MRJ packing.Since this tail must decrease faster than exponentially with acutoff at the small-particle radius (because of saturation), wetherefore have the following tighter upper bounds on the firstand second moments of the pore-size distribution:

〈δ〉 � 〈δ〉Z2 , (44)

〈δ2〉 � 〈δ2〉Z2 . (45)

It is also possible to find a simple lower bound on the pore-size distribution that is applicable for arbitrary heterogeneousmedia (hyperuniform or not). Specifically, we utilize thefollowing series representation for the cumulative pore-sizedistribution [11]:

F (δ) = 1

φ1

[1 +

+∞∑k=1

(−ρ)k(

1

[k + 1]

)

×∫ ⟨

gk(rk; Rk)k∏

j=1

m(‖x − rj‖; δ + Rj )

⟩Rk

drj

⎤⎦ .

(46)

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0 0.2 0.4 0.6 0.8 1φ

0

0.5

1

1.5

2

2.5

<δ>

/<D

>Poisson upper boundSeries lower bound

FIG. 12. (Color online) Upper and lower bounds on the meanpore size applicable for binary hyperuniform heterogeneous media atvolume fraction φ with the same composition as the MRJ binary diskpackings studied here.

It is well known that keeping terms up to order ρk in Eq. (46)places an upper bound on F (δ) for k even and enforces a lowerbound for k odd [11]. Therefore, by expanding the series toorder ρ, we obtain the following lower bound on F (δ):

F (δ) � max

{1

φ1[1 − ρ〈v(δ + R)〉R] ,0

}; (47)

the max operation in Eq. (47) enforces the trivial lowerbound F (δ) � 0. Note that Eq. (47) is equivalent to theO(ρ) expansion of the pore-size distribution for a Poisson-distributed medium; however, the number density (and there-fore the specific surface) of the system is not necessarilythe same as the Poisson medium. We will now assume thatour hyperuniform reference medium consists of impenetrablespheres with volume fraction φ = ρ〈v(R)〉R; we also remarkthat the definition of the specific surface for impenetrablespheres s = ρ〈s(R)〉R is different than for a fully penetrablesystem. It now follows that

F (δ) � max{1 − bδ2 − bδ,0}, (48)

where b = s〈D〉R/φ1 as in the Poisson case. The lower bound(48) first reaches zero at

δ∗ = −1 + √1 + 4/b

2. (49)

We thus obtain the following lower bounds on the first andsecond moments of the pore-size density:

〈δ〉LL = δ∗ − bδ∗2

2− bδ∗3

3, (50)

〈δ2〉LL = δ∗2 − 2bδ∗3

3− bδ∗4

2. (51)

Figure 12 plots the upper and lower bounds (42) and (50) onthe mean pore size for hyperuniform binary heterogeneous

TABLE II. Bounds on the moments of the pore-size distributionfor binary MRJ packings of hard disks.

〈δ〉 〈δ2〉Poisson upper bound 0.1135 0.0227Z2 upper bound 0.0626 6.219×10−3

Series lower bound 0.0224 6.635×10−4

MRJ binary packing 0.0438 3.362×10−3

media at volume fraction φ. We emphasize that these boundsaccount only for the hyperuniformity of the packings and thusplace constraints on the pore-size distributions of any binaryhyperuniform hard-particle packing.

For our binary MRJ packings, the specific surface is

sMRJ〈D〉R/φ1 = (φ/φ1)〈D〉2R/〈R2〉R

= (4φ/φ1)

[(γsmall + γlargeβ)2

γsmall + γlargeβ2

]≈ 21.6917.

(52)

The specific surface for the equivalent Poisson-distributedmedium is

sP 〈D〉R/φ1 = − ln(φ1)〈D〉2R

/〈R2〉R ≈ 7.340 37; (53)

as expected, this value is less than the corresponding result forthe impenetrable case. Using these parameters, we collect inTable II the bounds on the pore size of our binary MRJ diskpackings. We immediately notice that the numerical valuesare well within the bounds above; it follows that the one-pointstatistics of the medium do not place strong constraints onthe distribution of the void space. Furthermore, this evidencesuggests hyperuniformity strongly regularizes the pore spacecompared to the Poisson-distributed system.

It is interesting to note that the series lower bound providesa better estimate of the mean pore size than the Poissonupper bound. This observation is reasonable since the compactsupport of the lower bound more closely matches the behaviorof the actual pore-size distribution, which also has compactsupport due to the saturation of the packing. However, it isunclear if the constraint of saturation can be relaxed whilestill maintaining hyperuniformity in the medium. One canconjecture that so long as the probability of finding a large poredecays faster than the corresponding behavior for a Poissonpattern, then hyperuniformity holds.

Although our results provide a quantitative basis for under-standing the appearance of hyperuniformity in polydisperseMRJ packings, a complete explanation for the linear scaling ofthe small-wave-number region of the spectral density is still notapparent. Again, we emphasize that this problem is inherentlynonlocal, and its solution must account for the presence ofQLR correlations within the packing. Such correlations aredifficult to discern with the distribution F (δ) that we havepresented here, which is essentially a one-point descriptor ofheterogeneous media [11].

In the following section, we characterize the allowablen-particle loops within a binary MRJ packing. This analysissuggests that the variance in pore sizes is bounded by the strict

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FIG. 13. (Color online) The four distinct three-particle loops in a binary circular disk packing.

jamming of the packings, supporting our argument that theseconstraints are sufficient to induce hyperuniformity.

V. CHARACTERIZATION OF n-PARTICLE LOOPS IN MRJBINARY DISK PACKINGS

The preceding discussion shows how the void space is afundamental descriptor of a two-phase random heterogeneousmedium. As we have seen, fluctuations in the local volumefraction contain information about the void space and thereforeprovide a more complete picture of hyperuniformity inheterogeneous media. Here, we quantitatively characterizethe voids in MRJ binary disk packings and argue that strictjamming restricts the variance of pore sizes.

An n-particle void is associated with a loop of n contactingdisks (an n-particle loop) in which each disk only contacts twoneighbors. These loops are defined topologically, meaning thattheir identification is invariant to local shears of the particlecontacts. The smallest loops contain three mutually contactingparticles. The constraint of saturation places an upper boundon the number of particles in a loop, and the largest loop wefind in the MRJ binary disk packings contains six particles.The number of n-particle loops in a packing decreases rapidlyas n increases.

The area of the void associated with a loop can be rigorouslycomputed by substracting the area of particles falling intothe polygon constructed by connecting the centers of thedisks in the loop. We note that, except for the three-particleloop, the polygons associated with other n loops (n � 4)can be continuously deformed while maintaining the contactsbetween the particles. Such deformations change the area of thevoids associated with the loop. The rigidity of MRJ packingsalso strongly constrains the number of particles in a loop. Inthe MRJ packings of binary disks, the majority of the voidsinvolve three-particle loops, which form the “backbone” ofthe network. Certain four-particle loops can be observed atthe effective grain boundaries between disks with differentsizes. A large portion of these four-particle loops are verydistorted with a void area almost equal to that associated withtwo three-particle loops. Loops with more particles are rare inthe packings, and the observed ones are all strongly distortedwith void areas almost equal to those associated with three-or four-particle loops. Therefore, we only discuss the voidsassociated with three- and four-particle loops here, focusingon the maximum possible void areas associated with theseloops.

There are four distinct three-particle loops in the binarydisk packings as shown in Fig. 13. Defining the radii of smalland large disks as R1 and R2, respectively, the n-particle void

areas λnα , normalized by the volume of a small particle, areλ3a ≈ 0.0513, λ3b ≈ 0.0631, λ3c ≈ 0.0790 and λ3d ≈ 0.1060for a size ratio β = R2/R1 = 1.4. It can be seen clearly thatthe void areas constitute a small fraction of the small particlearea. Since these three-particle voids dominate the packing,it is reasonable to conclude that the regularity of these localclusters enforces hyperuniformity on the medium despite thenonuniform distribution of the sphere centers.

There are a total of six distinct types of four-particle loops(see Fig. 14) in the binary disk packings, each associatedwith a void that can possess a spectrum of shapes and sizesby distorting the quadrangle formed by the centers of thedisks. Although most of the void areas associated with thesehighly distorted four-particle loops are almost equal to thearea associated with two three-particle loops, there are stillrelatively large voids. The maximum normalized void areasare given by λ4a ≈ 0.2732, λ4b ≈ 0.3208, λ4c ≈ 0.3934,λ4d ≈ 0.3782, λ4e ≈ 0.4498, and λ4f ≈ 0.5355. These valuesare relatively large, but more tightly distributed around themean when compared to the normalized areas for three-particlevoids. As a result, they can be considered as an estimate ofthe upper bound on the degree of local inhomogeneity thatcan be consistent with hyperuniformity of the packings.

VI. CONCLUDING REMARKS

We have provided a detailed study of local-volume-fractionfluctuations in MRJ packings of polydisperse hard disksand have shown that these systems are hyperuniformwith quasi-long-range correlations. Our results stronglysuggest that these QLR correlations are a signature of MRJhard-particle packings, in contrast to previous misconceptionsin the literature [24]. Although it is true that the structurefactors for these systems do not vanish at small wavenumbers except in the monodisperse limit, we have shownthat the more appropriate structural descriptor of MRJpackings is the spectral density, which accounts appropriatelyfor the shape information of the particles. Our worktherefore generalizes the Torquato-Stillinger conjecture forhyperuniform point patterns by suggesting that all saturated,strictly jammed sphere packings are hyperuniform withrespect to local-volume-fraction fluctuations. Furthermore,MRJ sphere packings are expected to exhibit quasi-long-rangepair correlations scaling as r−(d+1) in d Euclidean dimensions.Importantly, this generalization contains previously publishedresults for monodisperse MRJ packings [4] as the specialcase where the particle-shape information can be rigorouslyseparated from the “point” information of the particle centers.

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FIG. 14. (Color online) The sixdistinct families of four-particle loopsin a binary circular disk packing.

Based on the observations that local-volume-fraction fluc-tuations are invariant to the choice of reference phase andthat one can maintain hyperuniformity in MRJ polydispersepackings under a uniform scaling deformation of the particles,we have argued that the onset of hyperuniformity and quasi-long-range correlations results from the homogenization ofthe void space external to the particles. The conditions ofsaturation and strict jamming limit the sizes and shapes ofthe local voids, which are completely determined by thecontact network of particles. By using a local analysis ofthe voids and rigorous bounds on the pore-size distribution,we have argued that the void space is highly constrained bystrict jamming, thereby suppressing infinite-wavelength local-volume-fraction fluctuations. Furthermore, we suggest that thepresence of quasi-long-range correlations reflects the inherentstructural correlations from the contact network between voidshapes. Specifically, saturation and strict jamming of thepackings compete with the maximal randomness of the particledistributions to drive the exponent of the small-wavenumberscaling of the spectral density to its smallest integervalue.

Although our work has addressed important problemsrelated to the structural properties of MRJ hard-particlepackings, a number of unanswered questions remain. First,a rigorous foundation for the observed linear scaling inthe small-wave-number region of the spectral density isstill missing. This problem is immensely difficult to handletheoretically because the problem is inherently long ranged,and any local analysis of the MRJ structure will therefore beunable to account correctly for this behavior [26]. Second,although we have considered only polydisperse MRJ packingsof d-dimensional spheres, our results are expected to holdmore generally for strictly jammed packings of hard particlesof arbitrary geometry. Indeed, our arguments concerning thevoid space of an MRJ packing are easily extended to includethese more general cases, and in a companion paper we willprovide direct evidence that MRJ packings of hard ellipses

and superdisks [28] are also hyperuniform with signaturequasi-long-range correlations indicated by linear scaling ofthe small-wave-number region of the spectral density. Theseresults will suggest a remarkably strong extension of theTorquato-Stillinger conjecture, namely, that all maximallyrandom strictly jammed saturated packings of hard particles,including those with size and shape distributions, are hyper-uniform with universal quasi-long-range correlations [via thetwo-point probability function S2(r)] scaling asymptotically

FIG. 15. (Color online) A jammed configuration of particles (upto rotations) surrounding a circular void. By replacing the wall with aclose-packed collection of particles, one can construct strictly jammedpackings in the plane with pores of arbitrarily large size.

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as r−(d+1). These observations support the important notionthat the topology and geometry of the void space are morefundamental than the particle space [11,58].

We mention that recent, independent work appearingin preprint form shortly after our own manuscript wasposted [59] has shown that hyperuniformity and vanishinginfinite-wavelength local-volume-fraction fluctuations in sat-urated MRJ packings of polydisperse disks are consistent withcertain fluctuation-response relations involving a generalized“compressibility.” However, this work does not address theappearance of quasi-long-range pair correlations in MRJhard-particle packings and does not consider MRJ packingsof nonspherical particles as in our companion paper [28].

Our arguments suggest, in particular, that certain quan-tum many-body systems and cosmological structures withthe same linear small-wave-number scaling in the structurefactor [7,9,10] are statistically “rigid” in the sense that theirmicrostructures are effectively homogeneous over large lengthscales with vanishing infinite-wavelength number-densityfluctuations. These unique features are inherently linked tothe structural properties of the system, independent of thephysical model itself.

Finally, we note that the systems examined in this paperare constrained to be both saturated and strictly jammed.Saturation of the packings is responsible for enforcing compactsupport in the pore-size distribution function and thereforeplays an important role in regularizing the void space surround-ing the jammed disks. However, it is not clear if saturation isa necessary condition to ensure hyperuniformity in strictlyjammed heterogeneous media. Namely, what conditions musta strictly jammed but unsaturated packing of hard spheres meet

in order to be hyperuniform? We note that the event-drivenmolecular-dynamics algorithm used here to generated thebinary MRJ disk packings inherently precludes the presence ofarbitrarily large “holes” within the packing. However, it doesproduce a small concentration (∼2.5%) of “rattler” particles,which are particles that are free to move within some smallcaged region of the packing. Removal of such particles isknown to break hyperuniformity of the medium even thoughthe strict jamming of the surrounding structure remains [4];therefore, strict jamming alone is not sufficient to inducehyperuniformity if large holes are “common enough” in thestatistics of the microstructure. This scenario corresponds toskewing the pore-size distribution and thereby deregularizingthe void space.

It is indeed possible to construct strictly jammed packingsof two-dimensional disks with a hole of arbitrarily large size bystarting with a ring of particles encompassing a large pore; thisring can be jammed by surrounding it with a close-packed col-lection of particles that approximates an impenetrable “wall”(see Fig. 15). One is then free to construct any strictly jammedsystem of particles outside the hole [60]. If such holes are suffi-ciently rare, then it is possible that the system may still be hype-runiform since the pores do not become very large on average.

ACKNOWLEDGMENTS

This work was supported by the Materials Research Scienceand Engineering Center (MRSEC) Program of the NationalScience Foundation under Grant No. DMR-0820341 and bythe Division of Mathematical Sciences at the National ScienceFoundation under Award Number DMS-0804431.

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[52] Although we only consider packings in 2D for this paper, ourresults are expected to apply in higher dimensions as well.We have already seen that our methodology contains knownresults for 3D monodisperse MRJ sphere packings as a specialcase, and the arguments we make in this section concerning theconstrained void space extend directly to higher-dimensionalpackings.

[53] One can generalize the expression (24) to include hard-particlepackings with inclusions of arbitrary geometry by appropriatemodification of m(k).

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