Disordered Hyperuniform Heterogeneous Materials

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Disordered hyperuniform heterogeneous materials

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Journal of Physics: Condensed Matter

J. Phys.: Condens. Matter J. Phys.: Condens. Matter 28 (2016) 414012 (16pp) doi:10.1088/0953-8984/28/41/414012

28 Disordered hyperuniform heterogeneous

2016 materials

JPCM Department of Chemistry, Princeton University, Princeton, NJ 08544, USA Department of Physics, Princeton University, Princeton, NJ 08544, USA Princeton Institute for the Science and Technology of Materials, Princeton, NJ 08544, USA 414012 Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA

E-mail: [email protected] S Torquato Received 15 February 2016 Accepted for publication 20 April 2016 Disordered hyperuniform heterogeneous materials Published 22 August 2016

Printed in the UK Abstract Disordered hyperuniform many-body systems are distinguishable states of matter that lie between a crystal and liquid: they are like perfect crystals in the way they suppress large- CM scale density fluctuations and yet are like liquids or glasses in that they are statistically isotropic with no Bragg peaks. These systems play a vital role in a number of fundamental and applied problems: glass formation, jamming, rigidity, photonic and electronic band structure, 10.1088/0953-8984/28/41/414012 localization of waves and excitations, self-organization, fluid dynamics, quantum systems, and pure mathematics. Much of what we know theoretically about disordered hyperuniform Paper states of matter involves many-particle systems. In this paper, we derive new rigorous criteria that disordered hyperuniform two-phase heterogeneous materials must obey and explore their consequences. Two-phase heterogeneous media are ubiquitous; examples include composites 0953-8984 and porous media, biological media, foams, polymer blends, granular media, cellular solids, and colloids. We begin by obtaining some results that apply to hyperuniform two-phase media d in which one phase is a sphere packing in d-dimensional Euclidean space R . Among other results, we rigorously establish the requirements for packings of spheres of different sizes to 41 be ‘multihyperuniform’. We then consider hyperuniformity for general two-phase media in d R . Here we apply realizability conditions for an autocovariance function and its associated spectral density of a two-phase medium, and then incorporate hyperuniformity as a constraint in order to derive new conditions. We show that some functional forms can immediately be eliminated from consideration and identify other forms that are allowable. Specific examples and counterexamples are described. Contact is made with well-known microstructural models (e.g. overlapping spheres and checkerboards) as well as irregular phase-separation and Turing- type patterns. We also ascertain a family of autocovariance functions (or spectral densities) that are realizable by disordered hyperuniform two-phase media in any space dimension, and present select explicit constructions of realizations. These studies provide insight into the nature of disordered hyperuniformity in the context of heterogeneous materials and have implications for the design of such novel amorphous materials. Keywords: hyperuniformity, fluctuations, heterogeneous media, disordered materials (Some figures may appear in colour only in the online journal)

1. Introduction was brought to the fore only about a decade ago in a study that focused on fundamental theoretical aspects, including how it The unusual suppression of density fluctuations at large length provides a unified means to classify and categorize crystals, scales is central to the hyperuniformity concept, whose broad quasicrystals and special disordered point configurations [1]. importance for condensed matter physics and materials science Moreover, it was shown that the hyperuniform many-particle

Figure 1. A disordered non-hyperuniform many-particle configuration (left) and a disordered hyperuniform many-particle configuration (right) [9]. The latter is arrived at by very tiny collective displacements of the particles on the left. These two examples show that it can be very difficult to detect hyperuniformity by eye, and yet their large-scale structural properties are dramatically different. systems are poised at a unique type of critical point in which density fluctuations and yet are like liquids or glasses in that (normalized) large-scale density fluctuations vanish such that they are statistically isotropic with no Bragg peaks. In this the direct correlation function of the Ornstein–Zernike rela- sense, they can have a hidden order (see figure 1 for a vivid tion is long-ranged [1]. This is to be contrasted with a standard example) and appear to be endowed with novel physical prop- thermal critical point in which large-scale density fluctuations erties, as described below. are infinitely large and the total correlation function (not the We knew of only a few examples of disordered hype- direct correlation function) is long-ranged [2–5]. runiform systems about a decade ago [1, 6, 10, 11]. The Roughly speaking, a hyperuniform (or superhomoege- importance of the hyperuniformity concept in the context of neous [6]) many-particle system in d-dimensional Euclidean condensed matter started to become apparent when it was d space R is one in which (normalized) density fluctuations are shown that classical many-particle systems with certain long- completely suppressed at very large length scales, implying ranged pair potentials could counterintuitively freeze into that the structure factor S()k tends to zero as the wavenumber disordered hyperuniform states at absolute zero with singular k ≡|k| tends to zero, i.e. scattering patterns, such as the one shown in the right panel of figure 2 [12, 13]. This exotic situation runs counter to our lim S()k = 0. (1) ||k →0 everyday experience where we expect liquids to freeze into crystal structures (like ice). Mapping such configurations of Equivalently, it is one in which the number variance of particles particles to network structures, what was previously thought within a spherical observation window of radius R, denoted by 2 d to be impossible became possible, namely, the first disordered σN()R , grows more slowly than the window volume (R ) in the dielectric networks to have large isotropic photonic band gaps large-R limit. Typical disordered systems, such as liquids and comparable in size to photonic crystals [14]. We now know structural glasses, have the standard volume scaling, that is, that these exotic states of matter can exist as both equilibrium 2 d σN()RR∼ . By contrast, all perfect crystals and quasicrystals and nonequilibrium phases across space dimensions, including 2 d−1 are hyperuniform with the surface-area scaling σN()RR∼ . maximally random jammed particle packings [15–17], jammed Surprisingly, there are a special class of disordered particle athermal granular media [18], jammed thermal colloidal configurations, such as the one shown in the right panel of packings [19, 20], dynamical processes in ultracold atoms figure 1, that have the same asymptotic behavior as crystals. [21], driven nonequilibrium systems [22–27], avian photore- There are scalings for the number variance other than surface- ceptor patterns [28], geometry of neuronal tracts [29], certain area growth. When the structure factor goes to zero in the limit quantum ground states (both fermionic and bosonic) [30, 31], ||k → 0 with the power-law form classical disordered (noncrystalline) ground states [9, 12, 13, 32, 33]. A variety of groups have recently fabricated disordered α S()kk∼| | , (2) hyperuniform materials at the micro- and nano-scales for var- where α > 0, the number variance has the following large-R ious photonic applications [34–36], surface-enhanced Raman asymptotic scaling [1, 7, 8]: spectroscopy [37], the realization of a terahertz quantum cascade laser [38] and self-assembly of diblock copolymers [39]. ⎧Rd−1,1α > , Moreover, a computational study revealed that the electronic ⎪ 2 d−1 bandgap of amorphous silicon widens as it tends toward a σN()R ∼ ⎨RRln ,1α =∞(→R ). (3) ⎪ hyperuniform state [40]. Recent x-ray scattering measure- ⎩Rd−α,0<<α 1 ments indicate that amorphous-silicon samples can be made Disordered hyperuniform systems can be regarded to be to be nearly hyperuniform [41]. Finally, we note that the hype- exotic states of matter that lie between a crystal and liquid: they runiformity concept has suggested new correlation functions are like perfect crystals in the way they suppress large-scale from which one can extract relevant growing length scales as

2 J. Phys.: Condens. Matter 28 (2016) 414012 S Torquato

Figure 2. Left: scattering pattern for a crystal. Right: scattering pattern for a disordered ‘stealthy’ hyperuniform material (defined in section 2.1). Notice that apart from forward scattering, there is a circle around the origin in which there is no scattering, a highly exotic situation for an amorphous state of matter. a function of temperature as a liquid is supercooled below its glass transition temperature [42], a problem of intense interest in the glass physics community [43–48]. The hyperuniformity concept was generalized to the case of two-phase heterogeneous materials [7], which are ubiquitous; examples include composites and porous media, biological media, foams, polymer blends, granular media, cellular solids and colloids [49, 50]. Here the phase volume fraction fluctu- ates within a finite-sized spherical window of radiusR (see figure 3) and hence can be characterized by the volume-fraction 2 variance σV()R . For typical disordered two-phase media, the 2 −d variance σV()R for large R goes to zero like R . However, for 2 hyperuniform disordered two-phase media, σV()R goes to zero asymptotically more rapidly than the inverse of the window volume, i.e. faster than R−d, which is equivalent to the following condition on the spectral density (defined in section 2):

lim χ˜V()k = 0. (4) ||k →0 As in the case of hyperuniform point configurations [1, 7, 8], Figure 3. A schematic indicating a circular observation window of radius R that is centered at position x0 in a disordered two-phase three different scaling regimes when the spectral density goes medium; one phase is depicted as a green region and the other α to zero with the power-law form χ˜V()kk∼| | : phase as a white region. The phase volume fractions within the window will fluctuate as the window positionx 0 is varied. ⎧R−+()d 1 ,1α > ⎪ 2 −+()d 1 conditions for a sphere packing to be stealthy and hyperuni- σV()R ∼ ⎨RRln ,1α =∞(→R ), (5) ⎪ form, and prove that when each subpacking associated with ⎩R−+()d α ,0<<α 1 each component is hyperuniform, the entire packing is hype- where the exponent α is a positive constant. runiform: a property called ‘multihyperuniformity’ [28]. In Much of our recent theoretical understanding of hyperuni- section 4, we consider hyperuniformity for general two-phase form states of matter is based on many-particle systems. The media that lie outside the special class that are derived from d purpose of this paper is to delve more deeply into theoretical sphere packings in d-dimensional Euclidean space R . Here foundations of disordered hyperuniform two-phase media by we apply realizability conditions for an autocovariance func- establishing new rigorous criteria that such systems must obey tion and its associated spectral density of a two-phase medium, and exploring their consequences. and then incorporate hyperuniformity as a constraint in order In section 2, we provide necessary mathematical defini- to derive new conditions. We demonstrate that some functional tions and background. In section 3, we derive some results forms can immediately be eliminated from consideration, but d concerning hyperuniformity of two-phase systems in R in also identify other forms that are allowable. Specific examples which one phase is a sphere packing and the spheres generally and counterexamples are described, including remarks about have different sizes. We determine the necessary and sufficient well-known microstructural models (e.g. overlapping spheres

3 J. Phys.: Condens. Matter 28 (2016) 414012 S Torquato and checkerboards) as well as irregular phase-separation and 2.2. Two-phase media

Turing-type patterns. We also ascertain a family of autocovari- d ance functions that are realizable by disordered hyperuniform A two-phase random medium is a domain of space V ⊆ R two-phase media in arbitrary space dimensions, In section 5, of volume V that is partitioned into two disjoint regions that we close with some concluding remarks. make up V: a phase 1 region V1 of volume fraction φ1 and a phase 2 region V2 of volume fraction φ2 [49].

2. Background 2.2.1. Two-point statistics. The phase indicator function i I ()()x for a given realization is defined as 2.1. Point configurations i ⎧1, x ∈ Vi, Consider statistically homogeneous point configurations in I ()()x = ⎨ (11) d ⎩0, x ∉ Vi, d-dimensional Euclidean space R . The standard pair correlation function g r is proportional to the probability density i 2() The one-point correlation function S()xx= I ()i (where associated with finding pairs of points separated by the dis- 1 () ⟨()⟩ angular brackets indicate an ensemble average) is generally placement vector r, and is normalized in such a way that it dependent on the position x, but is a constant for statistically tends to unity in the limit ||r ∞ in the absence of long-range → homogeneous media, namely, the phase volume fraction, i.e. order. The total correlation function h()r is defined as φ = I ()i x , (12) hg()rr=−2() 1. (6) i ⟨()⟩

The nonnegative structure factor S()k , which is proportional to such that φ12+=φ 1. The two-point correlation function is the scattering intensity, is trivially related to the Fourier trans- ()i ()ii() defined asS 2 ()xx12, = II()xx12(). This function is the form of h r : () probability of finding two points at positionsx 1 and x2 in phase i. For statistically homogeneous media, the two-point correla- S()kk=+1.ρh˜() (7) tion function will only depend on the relative displacement Appendix A provides definitions of the d-dimensional Fourier ()ii() transforms that we use in this paper. vector r ≡−xx21 and hence S2 ()xx12, = S2 ()r . The autocovariance function χ r associated with the random variable The local number variance σ2 R is determined entirely by V() N() I ()i x for phase 1 is equal to that for phase 2, i.e. pair correlations [1]: () χ rr≡−SS()1 φφ2 =−()2 r 2. (13) 2 ⎡ ⎤ V() 2 () 1 2 () 2 σρN()Rv=+1()Rh1;ρα()rr()rRd ⎢ ∫ d ⎥ ⎣ R ⎦ At the extreme limits of its argument, χ has the following V ⎡ 1 ⎤ asymptotic behavior = ρvR1() Sk()kkα();dR , (8) ⎢ d ∫ d ˜ ⎥ ⎣ ()2π R ⎦ χVV()rr==0,φφ12 lim χ ()= 0, ||r ∞ (14) dd/2 → where vR1()=Γπ Rd/(12+ /) is the d-dimensional volume of a spherical window, α()rR; is the intersection the latter limit applying when the medium possesses no long- volume of two identical hyperspheres of radius R (scaled by range order. If the medium is statistically homogeneous and the volume of a sphere) whose centers are separated by a dis- isotropic, then the autocovariance function χV()r depends tance r, which is known analytically in any space dimension only on the magnitude of its argument r =|r|, and hence is [1, 51], and α˜()kR; is its Fourier transform, which is nonnega- a radial function. In such instances, its slope at the origin is tive and explicitly given by directly related to the specific surface s (interface area per unit 2 volume); specifically, we have in any space dimension d, the dd2 [(Jkd 2 R)] απ˜()kR;2=Γ/ (/12+ d ) / . (9) asymptotic form [49], kd χ rr=−φφ β ds||+|O r|2 , (15) Here Jxν() is the Bessel function of order ν. V() 12 () () As mentioned earlier, the hyperuniformity property for where point configurations is specified by the structure-factor con- Γ(/d 2) dition (1). Stealthy configurations are those in which the β()d = . (16) structure factor is exactly zero for a subset of wave vectors, 21π Γ+((d )/2) meaning that they completely suppress single scattering of The nonnegative spectral density χ˜V()k , which can be incident radiation for these wave vectors [13]. Stealthy hype- obtained from scattering experiments [52, 53], is the Fourier runiform patterns [9, 12, 13] are a subclass of hyperuniform transform of χV()r , i.e. systems in which the structure factor is zero for a range of wave vectors around the origin, i.e. χ kr= χ ed−⋅ikr rk0, forall . ˜VV() d () ⩾ (17) ∫R S kk=|0for 0,| K (10) () ⩽⩽ For isotropic media, the spectral density only depends on where K is some positive number. An example of a stealthy dis k =|k| and, as a consequence of (15), its decay in the large-k ordered scattering pattern is shown in the right panel of figure 2. limit is controlled by the exact following power-law form:

4 J. Phys.: Condens. Matter 28 (2016) 414012 S Torquato

γ()ds of ||r ); for example, it prohibits autocovariance functions of χ˜V()k ∼∞,,k → (18) 2 k d+1 a Gaussian form (e.g. exp((− ra/))). For statistically homogeneous media, another condition is the so-called ‘triangular where inequality’: dd21dd()−12/ 2 γπ()=Γ(( + )/ ) (19) ff()rs⩾()(+−f t) 1, (25) is a d-dimensional constant. where r =−ts. If the autocovariance function of a statis- The higher-order correlation functions S34,,S … [49, 54, 55] tically homogeneous and isotropic medium is monotoni- will not be considered here, but we note that they arise in rig- cally decreasing, nonnegative and convex at the origin (i.e. orous bounds and exact expressions for effective transport 22 ddfr/⩾0), then it satisfies the triangular inequality (25). The [49, 56–63], elastic [49, 58, 60, 61, 64] and electromagnetic triangular inequality implies several pointwise conditions on [65] properties of two-phase media. f ()r . For example, for statistically homogeneous and isotropic media, it implies the condition (24) and convexity at the origin: 2.2.2. Realizability conditions on autocovariance functions of two-phase media. A necessary and sufficient condition for d2f 0. the existence of a scalar autocovariance function of a stochas- 2 ⩾ (26) dr r=0 tically continuous homogeneous process is that its spectral function must be a nonnegative bounded measure [49, 66]. The triangular inequality is actually a special case of the However, it is known that for a two-phase system character- following more general condition: ized by the phase indicator function (11), the nonnegativity m m property of the spectral function (see (17)) is a necessary ∑∑εεijf ()rrij− ⩾ 1, (27) but generally not sufficient condition for the existence of an i==11j autocovariance function χV()r corresponding to a two-phase where εi =±1 (i = 1, ..., m and m is odd). Note that by medium [49, 51, 67–70]. The autocovariance function must choosing m = 3; ε12ε = 1, ε13εε==23ε −1, equation (25) can also satisfy other conditions, which are most conveniently be rediscovered. If m = 3; ε12εε==13εε23ε = 1 are chosen stated in terms of the scaled autocovariance function f ()r , instead, another ‘triangular inequality’ can be obtained, i.e. which is defined by ff()rs⩾(−−)(f t) − 1, (28) χV()r f ()r ≡ . (20) where r =−ts. Equation (28) was first derived by Quintanilla φφ12 [69]. Equation (27) is a much stronger necessary condition Comparing this to relation (14), we see that that implies that there are other necessary conditions beyond those identified thus far. However, equation (27) is difficult to ff()rr==01, lim ()= 0. (21) check in practice, because it does not have a simple spectral ||r →∞ analog. We let f˜ ()k denote the Fourier transform of f ()r , implying that 2.2.3. Local volume-fraction variance and spectral density. It is known that the volume-fraction variance σ2 R within a χ˜V()k V() f˜ ()k = ⩾ 0for all.k (22) d-dimensional spherical window of radius R can be expressed φφ12 in terms of the autocovariance function χV()r [71]: Among other conditions, the scaled autocovariance function must satisfy the following bounds for all r: 2 1 σχV()R = V()rrα()rR;d, (29) ∫ d vR1() R ⎡ φ1 φ2 ⎤ − min,⎢ ⎥ ⩽(f r)⩽1. (23) where ⎣ φ2 φ1 ⎦ πdd/2R Another necessary condition on f ()r in the case of statistically vR1()= (30) Γ+(/12d ) homogeneous and isotropic media, i.e. when f ()r is dependent only on the distance r ≡|r|, is that its derivative at r = 0 is the volume of a d-dimensional sphere of radius R, and α()rR; 1 is strictly negative for all 0 <<φi 1: is the scaled intersection volume, as defined in equation (8) . The alternative Fourier representation of the volume-fraction df < 0, (24) variance that is dual to the direct-space representation (29) dr r=0 is trivially obtained by applying Parseval’s theorem to (29) which is consistent with the fact that slope at r = 0 is propor- under the assumption that the spectral density χ˜V()k (Fourier tional to the negative of the specific surface s (see (15)). Since transform of χV()r ) exists: f ||r is an even function (i.e. ffrr=−) that is linear in () () () 1 Note that we have changed the earlier notation for the volume-fraction ||r at the origin, it is nonanalytic at the origin. This is rather 2 2 variance used in [7] from στ()R to σV()R to distinguish it from other variance a strong restriction because it eliminates any function that is functions that have been introduced elsewhere [72] to describe generaliza- analytic at the origin (which necessarily implies even powers tions of the hyperuniformity concept.

5 J. Phys.: Condens. Matter 28 (2016) 414012 S Torquato

2 1 is the packing fraction, defined to be the fraction of space cov- σV()R = χαV()kk()kR;d. (31) d ∫ d ˜ ˜ ered by the nonoverlapping spheres, and v1(a) is the volume of vR1()()2π R a sphere of radius a defined by (30). Note that the hyperuniformity condition (4) dictates that the 2 We can bound the volume-fraction variance σV()R from direct-space autocovariance function χV()r exhibits both posi- 2 above in terms of the number variance σN()R for some fixed tive and negative correlations such that its volume integral R. This is accomplished by substituting the second line of over all space is exactly zero, i.e. (36) into the integral expression (31), employing the number- variance relation (8) and using the fact that mk; a achieves its χ rrd0= , ˜ () d V() (32) ∫R maximum value of v1(a) at k = 0. This leads to the following which can be thought of as a sum rule. The generalization upper bound: of the hyperuniformity concept to two-phase systems has 2d 2 ⎛ a ⎞ 2 been fruitfully applied to characterize a variety of disordered σV()R ⩽(⎜⎟σN RR) forall , (39) ⎝ R ⎠ sphere packings [15, 20, 73–75]. In [7], the same bound was given, but was derived for the large-R asymptotic limit. Bound (39) is in fact valid for any R. 3. Hyperuniform sphere packings We now show that the hyperuniformity of a sphere packing in terms of volume-fraction fluctuations can only arise if the Here we collect in one place various known results scattered underlying point configuration (determined by the sphere throughout the literature concerning the autocovariance func- centers) is itself hyperuniform. Since α˜()ka; is analytic at tion χV()r and spectral density χ˜V()k for two-phase media in d k = 0, we have that in the limit k → 0, R in which one phase is a sphere packing in order to make dd/2 2 some remarks about hyperuniformity and stealthiness. A par- π R ⎡ ()ka 4 ⎤ α˜()ka; = ⎢1 − + O()k ⎥, (40) ticle packing is a configuration of nonoverlapping (i.e. hard) Γ+1/d 2 ⎣ d + 2 ⎦ d () particles in R . For statistically homogeneous packings of d Because α ka; is a positive well-behaved function in the congruent spheres of radius a in R at number density ρ, the ˜() vicinity of the origin, it immediately follows from expression two-point probability function S2()r of the particle (sphere) phase is known exactly in terms of the pair correlation func- (36) that if the underlying point process is hyperuniform, as tion [49, 76], yielding the autocovariance function as per the structure-factor condition (1), then the spectral density χ˜V()k inherits the hyperuniformity property (4) only through 2 χρV()rr=⊗mr();;am()ra+⊗ρ mr();;am()ra⊗ h() the structure factor, not α˜()ka; . The stealthiness property (no int 2 int scattering at some finite subset of wave vectors) is a bit more =+ρρvr2 ();;av2 ()ra⊗ h()r , (33) subtle. We see from relation (36) that χ˜V()k is zero at those where wave vectors where S()k is zero as well as at the zeros of the function α˜()ka; , which is determined by the zeros of the ⎧1, ra⩽ , Bessel function Jd/2(ka). mr(); aa=Θ()−=r ⎨ (34) ⎩0, ra> , To illustrate the utility of these results, we now consider an example where the spectral density as well as the is the sphere indicator function, and vrint ;;av= ara is 2 () 1()α() volume-fraction variance can be calculated exactly for a the intersection volume of two spherical windows of radius a sphere-packing model as density increases up to a maximal whose centers are separated by a distance r, where v1(a) and value corresponding to hyperuniform state. Specifically, α()ra; are defined as in (29), and ⊗ denotes the convolution of we compute these quantities for sphere packings corre two functions F r and G r : () () sponding to a g2-invariant process introduced by Torquato and Stillinger [1]. A g -invariant process is one in which F rr⊗=GFxrG − xxd. 2 () () d () () (35) ∫R a chosen nonnegative form for the pair correlation func- Fourier transformation of (33) gives the corresponding spec- tion g2 remains invariant over a nonvanishing density tral density in terms of the structure factor [7, 49, 76]: range while keeping all other relevant macroscopic variables fixed [77]. The upper limiting ‘terminal’ density is 222 χρ˜V()kk=+mk˜ ();;amρ ˜ ()kah˜() the point above which the nonnegativity condition on the 2 structure factor (see (7)) would be violated. Thus, when- = ρ mk˜ (); aS()k ever the structure factor attains its minimum value of zero at = φα ka; S k (36) ˜()() k = 0 at the terminal or critical density, the system, if real- where izable, is hyperuniform. In [1], a variety of hyperuniform 2 d g2-invariant processes in which the number variance σN()R 1 2 12⎛ πa ⎞ 2 d−1 α ˜()ka; ==mk˜ (); a ⎜⎟Jkd 2()a , (37) grows like the window surface area (i.e. R ) were exactly va va⎝ k ⎠ / 1() 1() studied in arbitrary space dimensions. For our purposes, we employ the ‘step-function’ g2-invariant process, namely, φ = ρva, 1() (38) a g2(r) that is defined by the unit step functionΘ ()rD− , where

6 J. Phys.: Condens. Matter 28 (2016) 414012 S Torquato 0.03 The top panel of figure 4 shows the spectral function χ˜V()k for the aforementioned g2-invariant packing process ρ=ρc/2 in three dimensions at two different densities: one at a non- ρ=ρ c hyperuniform density ρ = ρc/2 and the other at the hyperuni- 0.02 form terminal density ρc, where ρc ==ρπ34/( ), as obtained from (42). As noted above, the degree of hyperuniformity ~ (k) χV reflected inχ ˜V()k is inherited from the properties of the structure factor. Note that value of the spectral density at the origin 0.01 for ρ = ρc/2 would monotonically decrease as the density increases up to the terminal density at which point it is exactly zero. The bottom panel of this figures depicts the associated local volume-fraction variance 2 R multiplied by R3 for 0 σV() 024 6 8101214 16 18 these two packings, as obtained from relation (31). Observe k 2 that because σV()R for the non-hyperuniform curve decays −3 2 3 0.006 like R for large R, the product σV()RR asymptotes to a con- 2 3 stant value. By contrast, the product σV()RR for ρ = ρc decays −1 ρ=ρc/2 like R for large R, as it should for this three-dimensional hyperuniform two-phase system. ρ=ρc The aforementioned results for the pair statistics in both 0.004 direct and Fourier spaces for identical spheres have been gen-

(R) eralized to the case of impenetrable spheres with a continuous V

2 or discrete size distribution at overall number density ρ [49, 78]. σ

3 We collect these results in appendix B in order and prove there

R 0.002 that when each subpacking associated with each component is hyperuniform, the entire packing is hyperuniform, what has been termed multihyperuniformity [28]. It is important to note that examining the structure factor S()k of the point configurations derived from the centers of spheres with a polydispersity 0.000 246 in size could lead one to incorrectly conclude that the pack- R ings were not hyperuniform. It has been demonstrated [15, 73, 74] that the proper means of investigating hyperuniformity in this case is through a packing s spectral density k . This has Figure 4. Top panel: a hyperuniform spectral density χ˜V()k versus ’ χ˜V() wavenumber k for sphere packings corresponding to the step- also been confirmed in experimental studies of maximally function g2-invariant process in three dimensions at two different random jammed packings of colloidal spherical particles with densities: one at a non-hyperuniform density ρ = ρc/2 and the other a size distribution [20]. at the hyperuniform terminal density ρc, where ρc ==ρπ34/( ) [1]. Bottom panel: the corresponding volume-fraction variance 2 σV()R versus window sphere radius R for the non-hyperuniform and hyperuniform cases. The diameter of a hard sphere is the 4. Hyperuniformity conditions for a general class of unit distance that makes all relevant dimensional variables two-phase media dimensionless. Our interest here is to elucidate our understanding of hype- D = 2a is the sphere diameter. The corresponding structure runiformity in general two-phase media that lie outside the special class that are derived from sphere packings, as per the factor in the density range 0 ⩽⩽ρρc is given by previous section. This is accomplished by applying the realizability conditions for an autocovariance function of a two- ⎛ 2 ⎞d/2⎛ ρ ⎞ S()kd=−11Γ+(/2)(⎜⎟⎜ ⎟ Jkd/2 D), (41) phase medium that is also hyperuniform. We show that some ⎝ kD ⎠ ⎝ ρc ⎠ functional forms can immediately be eliminated from consideration and that other forms are allowable. Specific examples d −1 where ρc = [(22vD1 /)] is the terminal density at which the and counterexamples are described. We note that it trivially packing is hyperuniform [1]. For , the packing is not ρ < ρc follows from (32) that the scaled autocovariance f ()r obeys hyperuniform. Substitution of (41) into relation (36) yields the the sum rule associated spectral density for this model in d dimensions:

f ()rrd0= . (43) d ⎡ d/2 ⎤ ∫ d ⎛ πD ⎞ 2 ⎛ 2 ⎞ ⎛ ρ ⎞ R ⎜⎟ ⎜⎟ χ˜V()k =−ρ Jkd/2(/Dd21)(⎢ Γ+12/) ⎜ ⎟Jkd/2()D ⎥. ⎝ k ⎠ ⎣⎢ ⎝ kD ⎠ ⎝ ρc ⎠ ⎦⎥ When f ()r is a function of the modulus r =|r|, this sum rule (42) reduces to the following one-dimensional integral condition:

7 J. Phys.: Condens. Matter 28 (2016) 414012 S Torquato

Figure 5. Realizations of overlapping circular disks at φ2 = 0.885 (left) and of a random checkerboard at φ2 = 0.5.

∞ d 1 between different pattern instabilities with selected wave- rf− ()rrd0= . (44) ∫0 lengths. Such phenomena have been theoretically described by, for example, Cahn–Hilliard equations [82] and Swift– Hohenberg equations [83], whose solutions can lead to irreg- 4.1. Monotonic autocovariance functions ular phase-separation and Turing patterns with a well-defined characteristic wavelength. Thus, it is plausible that binarized To begin, it is instructive to illustrate the capacity of the sum (two-phase) patterns obtained by thresholding such scalar fields rule (43) to eliminate an enormous set of two-phase struc- might be hyperuniform or even stealthy and hyperuniform. An tures from the hyperuniform class. First, we make the simple example of a Turing pattern with an irregular labyrinth-like observation that any two-phase medium with a scaled auto- structure [84] is shown in figure 6. The distance between adja- covariance function f r that monotonically decreases from () cent channels of the labyrinth-type pattern is a physical dis- its maximum value of unity at the origin to its long-range ‘ ’ play of the wavelength (or wavenumber) that has been selected, value, such as the well-known overlapping-sphere and sym- which is roughly equal to the mean chord length [49]. Also, metric-cell models [49, 79], cannot be hyperuniform at any C depicted in this figure is the autocovariance functionχ r positive volume fraction, since the sum rule (43) requires that V() associated with the thresholded binarized (two-phase) version the autocovariance function r possess both positive and χV() of the Turing image. This function exhibits strong positive as negative values such that its volume integral over all space well as negative correlations at short distances. The top panel be zero. The overlapping-sphere model in d consists of the R of figure 7 shows the spectral density χ k obtained from the union of spheres that circumscribe the points generated from ˜V() thresholded image. Note that it exhibits a well-defined annulus a Poisson distribution. The symmetric-cell model is derived in which scattering intensity is enhanced relative to that in from a tessellation of space into cells with cells being ran- ‘ ’ the region outside this annulus, which is radially centered at domly designated as phase 1 and phase 2 with probability φ 1 k ≈ 7. The bottom panel of figure 7 shows the angular-aver- and , respectively. Figure 5 shows two-dimensional realiza- C φ2 aged spectral density from which we conclude that the thresh- tions of each of these models. We note that while these are olded Turing pattern is neither stealthy nor hyperuniform. idealized models, there are many real two-phase systems (e.g. While this outcome does not mean that thresholded Turing- sandstones and ceramic-metal composites) that have similar type patterns can never be hyperuniform, it does lead to the monotonic autocovariance functions [49, 80] and hence can be following question: are there disordered stealthy and hype- immediately ruled out as hyperuniform structures. Moreover, runiform two-phase systems with spectral densities in which it is noteworthy that there is a huge class of two-phase sys- scattering is concentrated within some relatively thin annulus tems that exhibit strong positive and negative pair correlations defined by a small range of wavenumbers away from the at small pair distances (e.g. equilibrium and nonequilibrium origin? To answer this question, we consider the following distributions of nonoverlapping particles) that nonetheless are hypothetical, idealized scaled spectral functions to see if they not hyperuniform by virtue of the fact that their autocovari- can fall within this possible stealthy and hyperuniform class: ance functions violate the sum rule (43) [49, 76, 81].

fc˜A ()k =−A()dkδ()K (45) 4.2. Remarks about phase-separation and turing patterns and ⎧cdB(),,Kk12⩽⩽K There are a variety of interesting spatial patterns that arise in f˜B ()k = ⎨ (46) biological and chemical systems that result from a competition ⎩ 0, otherwise,

8 J. Phys.: Condens. Matter 28 (2016) 414012 S Torquato

0.25 0.2 0.2 0.15 (r) 0.15 0.009 χV 0.1 0.006 0.05 0.003 ~χ (k) V 0.1 0 0 0 0.3 0.6

-0.05 0.05 0 246 r 0 Figure 6. Top left panel: image of a Turing pattern with a labyrinth- 024 6 8 like structure [84]. Bottom panel: the autocovariance function k χV()r associated with the thresholded version of the Turing image Figure 7. Top panel: scattering pattern as obtained from the (with φ12≈=φ 12/ ), showing strong short-range order, including anti-correlations (negative values). The unit of distance is the mean spectral density χ˜V()k associated with the thresholded version of the Turing image shown in figure 6. Bottom panel: angular-averaged chord length of the ‘yellow’ phase [49], which is roughly equal to the characteristic width of the ‘channels’. spectral density, χ˜V()k , obtained from the 2D scattering pattern shown in the top panel. The unit of distance used in both spectral plots is the mean chord length of the ‘yellow’ phase [49]. where δ()k is a radial Dirac delta function is d-dimensional Fourier space, K21> K , where

22dd−12π / Γ d d/2 c d = (/) (47) ⎛ 1 ⎞ d A() d−1 F()rK; = cdB()⎜⎟KJd 2()Kr . (51) K ⎝ 2πKr ⎠ / and Using the results of appendix A, we can expand the afore- KKdd− mentioned putative autocovariance functions about r = 0 to c dd22d/2 1.()12 B()=Γ()π (/+ ) dd (48) yield KK12 fCr =−1 dr24+ O r (52) Using the results of appendix A, the corresponding hypothet- A () A() () ical scaled autocovariance function, which obeys the exact and limiting conditions (21), are given by 24 fCB ()r =−1,B()dr + O()r (53) ⎛ 2 ⎞d/21− ⎜⎟ where CA(d) and CB(d) are positive d-dimensional constants. fA ()r =Γ(/dJ2,)(d/21− Kr) (49) ⎝ Kr ⎠ It immediately follows the autocovariance functions (49) and and (50) cannot be realizable by two-phase media since such systems would have a vanishing specific surfaces , i.e. the fFB ()r =−()rK;;21Fr()K , (50) small-r expansion of a valid autocovariance function must be

9 J. Phys.: Condens. Matter 28 (2016) 414012 S Torquato

nonanalytic at the origin such that the slope is strictly negative for d = 2 and d = 3, hyperuniformity requires that (qb)2 = 1 (see section 2.2.2). This strongly suggests that scattering pat- and 3(qb)2 = 1, respectively, implying that the autocovariance terns in which power is concentrated within some concentric function for a hyperuniform system in a particular dimen- ring of the origin cannot be derived from a two-phase medium. sion generally does not correspond to a hyperuniform system Indeed, any function that is analytic at the origin cannot be in another dimension. Moreover, these are only necessary an autocovariance function that corresponds to a two-phase conditions on the parameters b and q for the existence of a medium. hyperuniform two-phase medium and one must still check whether the known realizability conditions for two-phase media (described section 2.2.2) are satisfied. As it turns out, 4.3. General considerations all of these realizability conditions are satisfied, including the A general formalism has been proposed that enables the nonnegativity of the spectral density (see (17)). Under these functional form of a realizable autocovariance function to be hyperuniform restrictions, the small-k behavior of the spectral expressed by a set of chosen realizable basis functions [68]. density f˜ ()kk≡ χφ˜V()/( 12φ ) associated with (58) for d = 2 For our limited purposes in this paper, we will make use of and d = 3 are given respectively by only some of these results. It is known that convex combi-

nations of a set of realizable scaled autocovariance functions f˜ ()k 3 2635 693 10 14 2 =−()kb ()kb ++()kb O()kq,1[( b)]= ffrr,, , f r is itself a realizable autocovariance func- b2 4ππ128 8192π 12() () m () tion f ()r [68], i.e. (59) m and ff()rr= ∑ αi i (), (54) i=1 f˜ ()k 27 24243 3645 816561 0 m 3 =−()kb ()kb +−()kb ()kb where 0 ⩽⩽αi 1 (im=…1, 2, , ) such that ∑i=1 αi = 1. b 4ππ32 512ππ1024 In what follows, we focus on basis functions that could cor- 14 2 +=O()kq,3[(b)]1, respond to statistically homogeneous and isotropic two-phase (60) media. A simple choice is the radial exponential function: where we have made use of the small-argument asymptotic fr1()r =−exp,(/a) (55) expansion of the Bessel function Jxν() given in appendix A. which is itself a realizable autocovariance function for all Notice also that the hyperuniformity constraint prohibits mul- positive and finite a [68]. For reasons discussed at the begin- tiple powers of four in the two-dimensional expansion (59) ning of this section, the monotonicity of f1 precludes it from and multiple powers of six in the three-dimensional expansion ever corresponding to a hyperuniform two-phase system. It (60). In the opposite asymptotic large-k limit, we respectively has been shown that a linear combination of f1 and the basis have for d = 2 and d = 3 function f˜ k 2π () , k fr2 ()r =−expc(/bq)(os r + θ) (56) 23∼∞→ (61) bk()b may be realizable for some parameters, but whether such a and linear combination can ever correspond to a disordered hyperuniform two-phase system has heretofore not been studied. f˜ k 8π () ,.k 34∼∞→ (62) Here b can be thought of as a characteristic correlation length bk()b and q determines the characteristic wavelength associated with the oscillations. These results are consistent with the general asymptotic result (18). Therefore, we explore here whether a disordered hyperuni- Figure 8 shows the autocovariance function and spectral form two-phase can have an autocovariance function of the density for a selected set of hyperuniform parameters (b, q) form in both two and three dimensions. Not surprisingly, the spectral densities associated with autocovariance functional form frr =−ααexpear+−xp bqcos,r + θ () 12(/)(/) () (57) (58) differ across dimensions. To verify that there are indeed where αα12+=1. For simplicity, we examine two special disordered hyperuniform two-phase media that correspond to cases. First, we consider the instance in which α1 = 0, α2 = 1 these autocovariance functions, well-established ‘construc- and θ = 0, i.e. tion’ (reconstruction) optimization techniques devised by Yeong and Torquato [49, 68, 85] are employed. Such proce- frr =−expcbqos r . () (/)() (58) dures utilize simulated-annealing methods that begin with a Notice that the specific surface corresponding to (58) is given random initial guess for a digitized two-phase system (hyper- by sd= βφ()/(b 12φ ), where β()d is the d-dimensional con- cubic fundamental simulation box that is tessellated into finer stant specified in (16). The hyperuniformity sum rule (44) pro- hypercubic cells) satisfying a prescribed volume fraction. The vides conditions on the parameters b and q, which will depend fictitious energy is a sum of squared differences between a on the dimension. For example, for d = 1, we immediately target correlation function (or corresponding spectral function) conclude that (58) can never correspond to a hyperuniform and the correlation function (or corresponding spectral func- medium because (44) cannot be satisfied. On the other hand, tion) of the simulated structure at any point along the evolution

10 J. Phys.: Condens. Matter 28 (2016) 414012 S Torquato

0.25 1

0.8 0.2 4 0.6 0.15 (R) R V 2 0.4 χ (r) σ V 0.1 d=3 0.2 0.05 0 0 5 10 15 20 25 30 d=2 R 0 2 Figure 9. Volume-fraction variance σV()R (multiplied by 01234 5 6 4 r R ) as a function of the window radius using (58) with φ12==φ 12/ in three dimensions where b = 1, q = 13/ . The fact that this scaled variance asymptotes to a constant value 1.4 d=3 (243/256 =…0.949 218 75 ) for large R implies that this three- dimensional hyperuniform system has a variance that decays like 1.2 R−4, which is consistent with the analytical formula (63).

1 process to the global energy minimum (ground state) as the fictitious temperature tends to zero. Here we target hyperuni- ~ 0.8 χV(k) form spectral densities associated with (58). The bottom panel 0.6 of figure 8 shows a final construction in the case of two dimensions that corresponds to (58) with extremely high numerical 0.4 accuracy for a selected set of parameters. Apparently, the known realizability conditions on the function (58) are suf- 0.2 d=2 ficient to ensure that it corresponds to a two-phase medium in two dimensions. It is noteworthy that it becomes easier to 0 024 6 ensure realizability of a hypothesized autocovariance function k of specific functional form as the space dimension increases for exactly the same reasons identified for point-configuration realizability [51]. Figure 9 shows the volume-fraction vari- 2 ance σV()R as a function of the window radius R, as obtained analytically from (29), in the case of three dimensions for a selected set of parameters. We can analytically show that this specific three-dimensional volume-fraction variance has the following asymptotic scaling:

2 243 1 σV()R ∼∞(→R ). (63) 256 R4 As a second example, we consider the function (57) in which α1 = 12/ , α2 = 12/ and θ = 0, i.e. 1 1 fr()r =−exp(/ar)(+−expc/)bqos()r , (64) 2 2 which provides greater degrees of freedom to achieve hype- Figure 8. Top panel: hyperuniform autocovariance function χV()r given by (58) with φ12==φ 12/ in two dimensions where b = 1, runiformity relative to the form (58). Here a and b are q = 1 and in three dimensions where b = 1 and q = 13/ . Middle taken to be positive and thus characteristic length scales. panel: corresponding hyperuniform spectral density χ˜V()k in two The specific surface corresponding to (64) is given by and three dimensions. Bottom panel: a realization of a disordered sa=+()bdβφ()/(2ab 12φ ), where β()d is the d-dimen- hyperuniform two-phase system that corresponds to (58) in two sional constant specified in (16). For d = 1, we find that (64) dimensions with these parameters, as obtained using reconstruction −9 can never correspond to a hyperuniform medium because techniques [49, 68, 85]. The final ‘energy’ is smaller than 10 , indicating that the targeted function is achieved to very high (44) cannot be satisfied, which also was the case for the func- accuracy. tion (58). This indicates that the hyperuniformity condition

11 J. Phys.: Condens. Matter 28 (2016) 414012 S Torquato is more difficult to achieve in one dimension than in higher 0.25 dimensions. For d = 2 and d = 3, the hyperuniformity sum rule requires that 0.2 bqb 21− 1 /2 a (( )) = 2 (65) ()qb + 1 0.15 d=2 and χ (r) 213 V 0.1 bq((31b))− / a = 2 , (66) ()qb + 1 0.05 respectively. Even though these conditions ensure hyperuniformity in these dimensions, they are not sufficient to guar- antee the nonnegativity of the spectral density (see (17)) for 0 all k because the leading term in the series expansion of f˜ ()k about k = 0 is generally quadratic in k but may have a nega- 0246 tive coefficient. For example, for d = 2, to ensure positivity of r the quadratic term, qb must satisfy the following inequalities: 0.8 1 1 <+qb ⩽(62). (67) 2 0.6 If qb is equal to the upper bound in (67), the quadratic term d=2 vanishes identically such that the leading term in the expan- ~ sion of f˜ ()k about k = 0 is now quartic in k, which is to be χ (k) 0.4 contrasted with the hyperuniform spectral density associated V with (58) that goes to zero quadratically in k in the limit k → 0. Under the aforementioned restrictions on the parameters a, b and q, all of the known realizability conditions described 0.2 in section 2.2.2 are satisfied. Figure 10 shows both the autocovariance function and spectral density for a set of hyperuniform parameters in two dimensions. The bottom panel of 0 024 6 810 figure 10 shows a realization obtained by the construction pro- k cedure [49, 68, 85] that corresponds to (64) in two dimensions with extremely high numerical accuracy for a selected set of parameters.

5. Conclusions and discussion

d For two-phase media in d-dimensional Euclidean space R in which one of the phases is a packing of spheres, we presented explicit exact expressions for the autocovariance function and associated spectral density as well as upper bounds on the volume-fraction variance in terms of the number variance for any window radius R. We used these results to determine the necessary and sufficient conditions for a sphere packing to be stealthy and hyperuniform as well as to establish rigorously the requirements for a packing comprised of spheres of different sizes to be multihyperuniform. We then considered hyperuniformity for general two-phase media in d Figure 10. Top panel: hyperuniform autocovariance R function r given by (64) in two dimensions with outside the class consisting of sphere packings. We applied χV() 32/ realizability conditions for an autocovariance function and its a =+()13/( 23()+=3 ) 0.674 790 19 … associated spectral density of a two-phase medium and incor- b =+()62/2 =…1.931 8516 , q = 1 and φ12==φ 12/ . porated hyperuniformity as a constraint in order to derive new Middle panel: corresponding hyperuniform spectral density χ˜V()k conditions. We showed that some functional forms can imme- in two dimensions. Bottom panel: a realization of a disordered diately be eliminated from consideration and identified other hyperuniform two-phase system that corresponds to (64) in two dimensions with these parameters, as obtained using reconstruction forms that are allowable. Contact was made with well-known −9 techniques [49, 68, 85]. The final ‘energy’ is smaller than 10 , two-phase microstructural models (e.g. overlapping spheres indicating that the targeted function is achieved to very high and checkerboards) as well as irregular phase-separation and accuracy.

12 J. Phys.: Condens. Matter 28 (2016) 414012 S Torquato Turing-type patterns. We ascertained a family of autocovari- Appendix A. Fourier transformation in ance functions that are realizable by disordered hyperuniform d dimensions two-phase media in arbitrary space dimensions. Realizations of disordered hyperuniform two-phase media with targeted We employ the following definition of the Fourier trans- spectral densities were explicitly constructed. These studies form of some scalar function f ()r that depends on the d elucidate the nature of hyperuniformity in the context of het- vector r in R : erogeneous materials. In a subsequent work, we will explore more fully the ff˜ ()kr=−()expi[(kr⋅ )] d,r (A.1) ∫ d explicit construction of disordered hyperuniform two-phase R media and characterize their higher-order statistics (beyond where k is a wave vector. When it is well-defined, the corresp the two-point autocovariance function) as well as host of other onding inverse Fourier transform is given by microstructural descriptors that are well-known in homog- d enization theory [49]. A particularly important goal of such ⎛ 1 ⎞ ffrk=⋅⎜⎟ ˜ expikr d.k (A.2) () d () [( )] studies will be to develop a deeper understanding of the effect ⎝ 2π ⎠ ∫R of space dimensionality on the microstructural descriptors, If f is a radial function, i.e. f depends only on the modulus including the relevance of the decorrelation principle as the ‘ ’ r =|r| of the vector r, then its Fourier transform is given by space dimension is increased [51]. A fruitful direction for future research would be the study d ∞ d 1 Jk(/d 21)− ()r fk˜ = 2d2 rf− r r, and determination of the effective physical properties of dis () ()π () d 21− (A.3) ∫0 ()kr (/) ordered hyperuniform two-phase systems. There is already evidence demonstrating that disordered hyperuniform cel- where k =|k| is wavenumber or modulus of the wave vector k lular network structures possess novel photonic properties and Jxν() is the Bessel function of order ν. The inverse trans- [34–36]. However, the investigation of the bulk properties form of fk˜ () is given by of general disordered hyperuniform two-phase materials and ∞ 1 d−1 Jk(/d 21)− ()r their technological relevance is essentially uncharted terri- fr()= kf˜ ()k d.k (A.4) d ∫0 kr (/d 21)− tory, and its exploration may offer great promise for novel ()2π 2 () materials by design. Very recently, the hyperuniformity concept was gener- We recall the first several terms in the series expansion of Jxν() alized to spin systems and shown to exist as disordered about x = 0: spin ground states [86]. The implications and significance νν++24νν+6 (/xx2) (/2) (/xx2) (/2) ν+8 of the existence of such disordered spin systems warrants Jxν()= − + − + O()x , further study, including whether their bulk physical prop- Γ+()νν1 Γ+()2 23Γ+()νν64Γ+() erties, like their many-particle system counterparts, are (A.5) singularly remarkable, and can be experimentally real- which we apply in section 4. ized. Finally, we note that the notion of hyperuniformity has recently been generalized to include surface-area fluctuations in two-phase media as well as fluctuations associ- Appendix B. Packings of spheres with a size ated with random scalar and vector fields [72]. Now that distribution and multihyperuniformity we know what to look for, different varieties of disordered hyperuniform systems seem to be arising in surprising Both the autocovariance and associated spectral density places and contexts, and hence offer both intriguing fun- for packings of hard spheres with a continuous or discrete damental and applied research challenges and questions size distribution at overall number density ρ have been for the future. derived [49, 78]. We collect these results here and apply them to establish rigorously the requirements for multihyperuniformity [28]. In the case of a continuous distribution in radius R charac- Acknowledgments terized by a probability density function f ()R that normalizes to unity, The author dedicates this article to the honor the memory of Professor George Stell, mentor and colleague, for his ∞ f ()RRd1= , (B.1) seminal research contributions to statistical mechanics. The ∫0 author thanks Duyu Chen, Jaeuk Kim and Zheng Ma for the packing fraction and the autocovariance function are given their careful reading of the manuscript. He is especially respectively by [49, 78] grateful to Duyu Chen for his assistance in creating the ∞ microstructure-construction figure. This work was sup- φ = ρ ∫ fv()RR1()R d (B.2) ported by the National Science Foundation under Grant 0 No. DMS-1211087. and

13 J. Phys.: Condens. Matter 28 (2016) 414012 S Torquato

∞∞ d d int 2 cell in R of side length L and volume V = L subjected to χρr =+fvRRr;dRRρ d 1 V() ∫∫0 ()2 () 0 periodic boundary conditions, which is given by [67]: ∞ 2 ×⊗d;RR21ff()()RR21mr()mr();;RR21⊗ h()r ,,R2 ||J˜ ()k ∫0 χ˜V()k = ,,k0≠ (B.11) V (B.3) where J˜ ()k is the discrete Fourier transform of phase indicator where h()r;,RR12 is the appropriate generalization of the total function minus the phase volume fraction, which generally correlation function for the centers of two spheres of radii R1 is a complex number for any k ≠ 0. Ultimately, we take the and R2 separated by a distance r. Fourier transformation of thermodynamic limit to make contact with (B.10). Consider (B.3) gives the corresponding spectral density the two-phase system to be a finite packing of N spheres con- ∞∞ ()i ()i 22 sisting of M components. Let r1 ,,… rN denote the positions χρk =+fmRRk;dRRρ d 1 i ˜V() ∫∫0 ()˜ () 0 of type-i spheres, where Ni is the total number of spheres of ∞ radius a and iM=…1, 2, , such that N = M N. The dis- × d;RR21ff()()RR21mk˜ ()mk˜ ();;RR21h˜()k ,.R2 i ∑i=1 i ∫0 crete Fourier representation of the ‘scattering amplitude’ J˜ ()k (B.4) for such a multicomponent packing was given in [73], which One can obtain corresponding results for spheres with M can be recast as follows: different radii aa12,,…, aM from the continuous case by letting M M J˜ ()kk= mk˜ ();;afii()a , (B.12) ρi ∑ fR()=−∑ δ()R ai , (B.5) i=1 ρ i=1 where where ρ is the number density of type-i particles, respectively, i Ni and ρ is the total number density. Substitution of (B.5) into ()i fa()kk;ei =⋅∑ xp()i,rn (B.13) (B.2)–(B.4) yields the corresponding packing fraction, auto- n=1 covariance function and spectral density, respectively, as and the product mk˜ ();;afii()k a represents the scattering M amplitude for the ith component (subpacking). Substitution of φ = ∑ ρiva1()i , (B.6) (B.12) into (B.11) yields i=1 M M 2 M M M χ˜V()kk=+∑∑ρρimk˜ ();;aaiiSH() ijρ mk˜ ();;amij˜ ()ka ()k;,aaij, int i=≠1 ij χV()rr=+∑∑ρρivr2 ();;ami ∑ ijρ ()raij⊗⊗mr();;ah()aaij, i==1 i 11j= (B.14) (B.7) where and fak; 2 M M ||()i 2 S()k; ai = (B.15) χ˜V()kk=+∑∑ρρimk˜ ();;aSii()amijρ ˜ ()ka;;ijmk˜ ()ah˜()k;,aaij, Ni i=≠1 ij (B.8) is the discrete structure factor for the ith subpacking and where fa()kk;;ijfa()V H()k;,aaij= . (B.16) S()kk;1ahi ≡+ρi ˜();,aaii (B.9) NNij is the structure factor for type-i particles. It immediately fol- The similarity between this discrete representation of the lows that the spectral density at the origin is given by spectral density of a multicomponent packing and the con-

M tinuous version (B.8) is readily apparent. 2 The chosen hypercubic fundamental cell restricts the χρ˜V()00= ∑ iva1()iiSa(); i=1 wave vectors to take discrete values, which are defined by M the vectors that span the reciprocal hypercubic lattice, i.e. + ρρ vavaha˜ 0; ,.a ∑ ij 11()ij()()ij (B.10) k = (/2,ππnL122,nL// ,2πnLd ), where ni (id=…1, 2, , ) ij≠ are the integers. Thus, the smallest positive wave vectors have We now prove that when each subpacking associated magnitude kmin = 2π/L. Now we constrain the scattering with each component is hyperuniform, i.e. the first term on amplitude for each component to be zero at ||k = kmin, the right side of (B.10) is zero, the second term must also be which from (B.12) implies that J˜ ()||k ==kmin 0 and hence identically zero (sum of cross terms vanish), leading to the χ˜V()||k ==kmin 0. This in turn means that the second sum hyperuniformity of the entire packing, i.e. χ˜V()00= . Such a (B.14) involving the cross terms must vanish at ||k = kmin. polydisperse packing has been called multihyperuniform [28]. To complete the proof, we must consider the thermodynamic We begin the proof by considering the spectral density χ˜V()k limit because the ensemble-average relation (B.10) applies for a general, very large but finite-sized two-phase heteroge- under this condition. Assuming ergodic media, the zero-wave- neous system that is contained within hypercubic fundamental vector behavior of the spectral density defined by (B.10) can be

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