Characterizing the Hyperuniformity of Ordered and Disordered Two-Phase Media
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Characterizing the Hyperuniformity of Ordered and Disordered Two-Phase Media
Jaeuk Kim
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Salvatore Torquato ∗ Department of Physics, Princeton University, Princeton, New Jersey 08544, USADepartment of Chemistry, Princeton University, Princeton, New Jersey 08544, USAPrinceton Institute for the Science and Technology of Materials,Princeton University, Princeton, New Jersey 08544, USA andProgram in Applied and Computational Mathematics,Princeton University, Princeton, New Jersey 08544, USA (Dated: August 18, 2020)The hyperuniformity concept provides a unified means to classify all perfect crystals, perfectquasicrystals, and exotic amorphous states of matter according to their capacity to suppress large-scale density fluctuations. While the classification of hyperuniform point configurations has receivedconsiderable attention, much less is known about the classification of hyperuniform heterogeneoustwo-phase media, which include composites, porous media, foams, cellular solids, colloidal suspen-sions and polymer blends. The purpose of this article is to begin such a program for certain two-dimensional models of hyperuniform two-phase media by ascertaining their local volume-fractionvariances σ V ( R ) and the associated hyperuniformity order metrics B V . This is a highly challeng-ing task because the geometries and topologies of the phases are generally much richer and morecomplex than point-configuration arrangements and one must ascertain a broadly applicable lengthscale to make key quantities dimensionless. Therefore, we purposely restrict ourselves to a certainclass of two-dimensional periodic cellular networks as well as periodic and disordered/irregular pack-ings, some of which maximize their effective transport and elastic properties. Among the cellularnetworks considered, the honeycomb networks have the minimal value of the hyperuniformity ordermetrics B V across all volume fractions. On the other hand, among all packings considered, thetriangular-lattice packings have the smallest values of B V for the possible range of volume fractions.Among all structures studied here, the triangular-lattice packing has the minimal order metric foralmost all volume fractions. Our study provides a theoretical foundation for the establishment ofhyperuniformity order metrics for general two-phase media and a basis to discover new hyperuniformtwo-phase systems by inverse design procedures. I. INTRODUCTION
The hyperuniformity concept generalizes the tradi-tional notion of long-range order in many-particle sys-tems to include all perfect crystals, perfect quasicrystals,and exotic amorphous states of matter [1, 2]. A hyper-uniform point configuration in d -dimensional Euclideanspace R d is characterized by an anomalous suppression oflarge-scale density fluctuations relative to those in typ-ical disordered systems, such as liquids and structuralglasses. The hyperuniformity notion was generalized tothe case of heterogeneous (multiphase) materials [3–5],i.e., materials consisting of two or more phases [6, 7],such as composites, porous media, foams, cellular solids,colloidal suspensions and polymer blends. Subsequently,the concept was extended to quantify hyperuniformity ina variety of different systems, including random scalarfields, divergence-free random vector fields, and statisti-cally anisotropic many-particle systems [4]. Hyperuni-formity has been attracting great attention across manyfields, including physics [2, 8–21], materials science [22–26], mathematics [27–31] and biology [2, 32–34].In the case of point configurations, one can rank ordercrystals, quasicrystals, and special disordered systemswithin a hyperuniformity class according to the degree to which they suppress density fluctuations, as measuredby the hyperuniformity parameter B N [1, 3]. Much lessis known about the analogous rank ordering of hyper-uniform two-phase media via the appropriate hyperuni-formity parameter B V , as defined below. However, it ismuch more challenging to do so for two-phase media fortwo reasons. First, the geometries and topologies of thephases are generally much richer and more complex thanpoint-configuration arrangements. Second, one must de-termine length scales that are broadly applicable for themultitude of possible two-phase media microstructuresto make B V dimensionless. The purpose of this articleis to begin such a program for certain two-dimensionalperiodic and disordered models of two-phase media.For two-phase heterogeneous media in d -dimensionalEuclidean space R d , which include cellular solids, com-posites, and porous media, hyperuniformity is defined bythe following infinite-wavelength condition on the spectraldensity ˜ χ V ( k )[2, 3], i.e.,lim | k |→ ˜ χ V ( k ) = 0 , (1)where k is the wavevector. The spectral density ˜ χ V ( k )is the Fourier transform of the autocovariance function χ V ( r ) ≡ S ( i )2 ( r ) − φ i , where φ i is the volume fraction ofphase i , and S ( i )2 ( r ) gives the probability of finding twopoints separated by r in phase i at the same time [6, 7]. Itcan be easily obtained for general microstructures eithertheoretically, computationally, or via scattering experi-ments [35]. Hyperuniformity can equivalently be definedin terms of the local volume-fraction variance σ V ( R ) asso-ciated with a spherical window of radius R . Specifically,a hyperuniform two-phase system is one in which σ V ( R )decays faster than R − d in the large- R regime [2, 3], i.e.,lim R →∞ R d σ V ( R ) = 0 . (2)The local variance σ V ( R ) is directly determined by ˜ χ V ( k )via the following integral [2, 3]: σ V ( R ) = 1 v ( R ) (2 π ) d Z R d ˜ χ V ( k ) ˜ α ( | k | ; R ) d k , (3)where v ( R ) = π d/ [Γ( d/ − R d is the volume ofa d -dimensional sphere of radius R , Γ( x ) is the gammafunction, and˜ α ( | k | ; R ) ≡ d π d/ Γ( d/ (cid:2) J d/ ( | k | R ) (cid:3) | k | d , (4)is the Fourier transform of the scaled intersection volumeof two spheres of radius R that are separated by r .As in the case of hyperuniform point configurations [1–3, 36], there are three different scaling regimes (classes)that describe the associated large- R behaviors of thevolume-fraction variance when the spectral density goesto zero as a power-law scaling ˜ χ V ( Q ) ∼ | Q | α as Q tendsto zero: σ V ( R ) ∼ R − ( d +1) , α > R − ( d +1) ln R, α = 1 (Class II) ,R − ( d + α ) , < α < α is a positive constant. Classes Iand III are the strongest and weakest forms of hyperuni-formity, respectively. One aim of this paper is to computethe implied coefficient hyperuniformity order metric B V (defined in Sec. II A) multiplying R − ( d +1) for certainclass I structures, which is a measure of the degree towhich large-scale volume-fraction fluctuations are sup-pressed within that class.An overarching goal of this paper is to characterizethe hyperuniformity of models of two-phase media thatbelong to class I. Due to the infinite variety of possi-ble two-phase microstructures (geometries and topolo-gies), we purposely restrict ourselves to a certain classof periodic cellular networks as well as periodic and dis-ordered/irregular packings. Even this restrictive set ofmodels of two-phase media presents challenges, since onemust ascertain relevant length scales that are broadlyapplicable to make key quantities dimensionless, as dis-cussed in Sec. VI. In particular, we evaluate the volume-fraction variance as a function of the window radius R for all models. We also compute the aforementioned hy-peruniformity order metric B V for each model to rankorder them.In Sec. IV, we present relevant theoretical back-ground to characterize hyperuniform two-phase mediaand describe the computational methods employed inthis study. In Sec. III, we provide exact closed-formformulas of the form factors of general polyhedra in R and R , which are important to characterize periodicnetworks. We then describe the two-dimensional mod-els of two-phase media of class I hyperuniformity consid-ered in this investigation: periodic cellular networks (Sec.IV), periodic disk packings, and disordered/irregular diskpackings (Sec. V). In Sec. VI, we provide the ra-tionale for choosing the inverse of the specific surfaceas the characteristic length scale D in the hyperunifor-mity order metric B V . In Sec. VII, we investigate themicrostructure-dependence of the volume-fraction vari-ance and rank order all of our class I models accordingto B V . Finally, we present concluding remarks and out-looks for future research in Sec. VIII. II. BACKGROUND AND METHODSA. Asymptotic Analysis of HyperuniformTwo-Phase Media
For a statistically homogeneous and isotropic mediumin R d , the local volume-fraction variance σ V ( R ) can bewritten as the following large- R asymptotic expansion[2, 3]: σ V ( R ) = A V ( R ) Å DR ã d + B V ( R ) Å DR ã d +1 + o Å DR ã d +1 , (6)where A V ( R ) and B V ( R ) are dimensionless asymptoticcoefficients of powers R − d and R − ( d +1) , respectively, andthey are defined by A V ( R ) = 1 v ( D ) Z | r |≤ R χ V ( r ) d r (7) B V ( R ) = − c ( d )2 D v ( D ) Z | r |≤ R χ V ( r ) | r | d r , (8)where c ( d ) ≡ d/ / (cid:2) π / Γ(( d + 1) / (cid:3) , and D is a characteristic length scale of the medium. In thelarge- R limit, the coefficient A V ( R ) is proportional tothe spectral density at the origin, i.e., A V ≡ lim R →∞ A V ( R ) ∝ lim | k |→ ˜ χ V ( k ) , (9)and thus for any hyperuniform medium, A V = 0, andhence the expansion (6) reduces to σ V ( R ) = B V ( R ) Å DR ã d +1 + o Å DR ã d +1 . (10)It is noteworthy that, unlike σ V ( R ), the coefficient B V ( R ) depends on the choice of the length scale D .In the case of class I hyperuniform systems, σ V ( R )decays like R − ( d +1) for large R , as specified by σ V ( R ) ∼ B V Å DR ã d +1 , R → ∞ . (11)As R increases, the coefficient B V ( R ) converges to thehyperuniformity order metric B V for typical disorderedsystems. For some infinite media, such as periodic andaperiodic structures,the associated coefficient B V ( R ) os-cillates around some running average value. In suchcases, it is advantageous to estimate B V by using thecumulative moving average, as defined by [2] B V ≡ lim L →∞ L Z L B V ( R ) d R . (12)
B. Spectral Density and the Local Volume-FractionVariance
Here we present explicit formulas for the spectral den-sity ˜ χ V ( k ) of general packings in R d , ordered or not[2, 37, 38]. We also describe the formula for the localvolume-fraction variance σ V ( R ) for class I hyperuniformtwo-phase media and the associated hyperuniformity or-der metric B V . Importantly, these formulas also can beapplied to characterize periodic cellular networks, as wewill discuss later.In the case of packings of identical particles P of arbi-trary shape, it is known that˜ χ V ( k ) = ρ | ˜ m ( k ; P ) | S ( k ) , (13)where ρ is the number density of particle centers, ˜ m ( k ; P )is the Fourier transform of (also called form factor ) of theparticle indicator function m ( x ; P ) defined by m ( x ; P ) = ® , x is inside P , otherwise , (14)where x is the position vector with respect to the cen-troid of P , and S ( k ) is the structure factor of particlecenters; see Refs. [2, 37, 38] for the definition of S ( k )and its computation. One can immediately obtain from(13) the specific formulas for a d -dimensional (Bravais)lattice packing in which a single particle P is placed ina fundamental cell F of the Bravais lattice L as follows:˜ χ V ( k ) = | V F | − | ˜ m ( k ; P ) | S L ( k ) , (15)where | V F | is the volume of the fundamental cell F of L , S L ( k ) is the structure factor of L given by [2] S L ( k ) = (2 π ) d | V F | X q ∈L ∗ \{ } δ ( k − q ) , (16) L ∗ denotes the reciprocal lattice of L , and δ ( x ) isthe Dirac delta function. For a periodic packing inwhich a fundamental cell contains M distinct particles( P , · · · , P M ) whose centroids are at r , · · · , r M , for-mula (15) can be easily extended as˜ χ V ( k ) = | V F | − | ˜ m ( k ; { P j } ) | S L ( k ) , (17)where ˜ m ( k ; { P j } ) ≡ M X j =1 ˜ m ( k ; P j ) e − i k · r j . (18)Equation (17) is a special case of the multicomponentpacking formula given in Ref. [38]. Thus, given the formfactors and structure factors for a particulate two-phasestructure, one can immediately compute the correspond-ing spectral density via either (13), (15), or (17). It iscrucial to note that these formulas also can be applied toany periodic cellular network by treating it as a periodicpacking of polygons (polyhedra for d = 3) defined by thevoid regions (shown in white in Fig. 3). In such cases,the set { P j } represents the regions of void phase (shownin white regions in Fig. 3).Given the spectral density of a general packing, one cancompute the local volume-fraction variance by comput-ing Eq. (3) either numerically or analytically. The asso-ciated hyperuniformity order metric B V is obtained fromthe running average associated with (12). In the case ofperiodic packings or periodic networks, it immediatelyfollows from Eqs. (3) and (17) that the associated lo-cal volume-fraction variance σ V ( R ) and the surface-areacoefficient B V ( R ) are written as σ V ( R ) = 2 d Γ( d/ R d | V F | × X k ∈L ∗ \{ } | ˜ m ( k ; { P j } ) | [ J d/ ( kR )] k d (19) ∼ B V ( R ) Å DR ã d +1 , ( R → ∞ ) . (20)Thus, we see that periodic packings fall in class I. Thehyperuniformity order metric B V is obtained by substi-tuting (20) into (12): B V = 2 d Γ( d/ πD d +1 | V F | X k ∈L ∗ \{ } | ˜ m ( k ; { P j } ) | q d +1 , (21)where we have used the identitylim x →∞ x − R x dx ′ x ′ [ J d/ ( x ′ )] = 1 /π . C. Computation of σ V ( R ) and B V Here we describe two methods that we employ to es-timate the local volume-fraction variance σ V ( R ) and theassociated asymptotic value B V : numerical integrationof Eq. (3) and the Monte Carlo (MC) method. For pe-riodic media, we mainly use the former method becauseof its accuracy and efficiency for such structures [cf. Eq.(20)]. The key step of this method is to compute thespectral density ˜ χ V ( k ) of a periodic structure via either(15) and (17). For periodic packings of identical circulardisks,we use the exact formula for the form factor of a d -dimensional sphere of radius a given by [6]˜ m ( k ; a ) = Å πak ã d/ J d/ ( ka ) . (22)In the case of periodic networks, we employ the formulasfor general polyhedra in two and three dimensions givenin Sec. III. Provided that ˜ χ V ( k ) given in Eq. (17) canbe computed, it is in practice sufficient to perform thesummations in Eqs. (20) and (21) up to | k || V F | /d < d = 2 , σ V ( R ) for disordered disk packings. Specifically, σ V ( R ) is estimated by uniformly sampling the local vol-ume fraction with a d -dimensional spherical observationwindow of radius R a single packing or an ensemble ofpackings. Since this method involves computing the vol-ume of domains in one phase intersected by a window,it is highly nontrivial and computationally expensive forgeneral packings. In the case of disk packings (spherepackings for d = 3), however, such a calculation can beefficiently carried out by using an exact closed-form for-mula for the intersection volume of two spheres of radii R and R whose centers are separated by r , given inRef. [6]. III. FORM FACTORS OF POLYGONS ANDPOLYHEDRA
In order to compute the spectral density of a periodiccellular network using relation (17), one needs to com-pute the form factors of the relevant polyhedra. Here, weprovide the exact closed-form formulas for general poly-hedra in three dimensions and two dimensions (polygons)that were derived in Ref. [39].We first consider a planar polygon Γ that is placed inan arbitrary orientation in R [see Fig. 1(a)] and consistsof J vertices V , · · · , V J in a cyclic order, implying thatthe adjacent vertices V i − and V i are connected by asegment, and V J +1 = V . It is convenient to considera planar polygon in three dimensions since such planarpolygons will be employed to define a polyhedron in R later. For two adjacent vertices V i − and V i , we define R i ≡
12 ( V i + V i − ) , (23) E i ≡
12 ( V i − V i − ) , (24) FIG. 1. Illustrations of parameters used to compute the formfactors of polygonal figures in R . (a) A pentagon has a faceΓ surrounded by five vertices V , · · · , V . The vertex indicesincrease counter-clockwise when the normal vector ˆ n is to-wards the reader. (b) A polyhedron with five faces, for eachof which the ordering of vertices fulfills the right-hand-rulewith the normal vector ˆ n j . Note that all normal vectors pointtowards the outside of the polyhedron. where R i is the center of the two vertices, and E i standsfor the segment from V i − to R i . The form factor of Γ at a wavevector k is˜ m ( k ; Γ ) = 2 − ik k k × · J X j =1 E j sinc( k · E j ) e − i k · R j , (25)where ˆ n is the unit normal vector of the face Γ , k k ≡ k − ( k · ˆ n )ˆ n , k × = ˆ n × k , andsinc( x ) ≡ ® , x = 0 , sin xx , otherwsie . (26)Importantly, the ordering of vertices should fulfill the right-hand-rule with respect to the normal vector ˆ n [seeFig. 1(a)], implying that the vertex index increasescounterclockwise when ˆ n is towards the reader. In two-dimensional applications, one should take k = k k .We now consider a polyhedron P with K faces( Γ , · · · , Γ K ) in which a face Γ j is a polygon with J j vertices. For each face Γ j ( j = 1 , · · · , K ), its unit nor-mal vector ˆ n j points towards the outside of P , and theorder of vertices fulfills the right-hand-rule with ˆ n j ; seeFig. 1(b). Then, the form factor of P is˜ m ( k ; P ) = − ik k · K X j =1 ˆ n j ˜ m ( k ; Γ j ) , (27)where k ≡ | k | . The reader is referred to Ref. [39] forderivations of Eqs. (25) and (27) [40]. IV. PERIODIC NETWORK MODELS
In this paper, we consider six different periodic net-works with volume fractions that span the entire inter-val [0,1]: square, rhombic, honeycomb, square-octagon,triangular, and kagom´e networks. Figure 2 shows eachof these networks at small, intermediate, and large solidvolume fractions ( φ = 0 . , . , and 0 .
95, respectively).Figure 3 provides the dimensional parameters for theunit cells, which determine the corresponding solid-phasevolume fractions. Except for the kagom´e and square-octagon networks, all “wall” thicknesses are uniformacross all volume fractions. In the cases of the formertwo structures, the wall thicknesses are uniform for eachdifferent polygon but are proportional to their area ratiosin order to span all volume fractions in the interval [0,1].Note that in the limit of φ →
1, these six network mod-els can be regarded as periodic point patterns. For ex-ample, the square, honeycomb, and triangular networksbecome the square-lattice, triangular-lattice,and honey-comb crystal, respectively.It is noteworthy that these cellular solids can opti-mize certain effective physical properties. In the limit φ →
0, these network structures maximize certain ef-fective transport and elastic properties. Specifically, allnetworks maximize the effective conductivity σ e and ef-fective bulk modulus K e [41]. The effective shear mod-ulus G e is maximized for the triangular network [41, 42]as well as the kagom´e network [43]. The triangular andkagom´e networks are nearly optimal for σ e , K e and G e over the possible range of volume fractions [44]. Dueto the well-known mechanisms that lead to optimality inthe aforementioned networks, we can report here that therhombic and square-octagon networks maximize the ef-fective conductivity and effective bulk moduli in the limitof φ → V. PERIODIC AND DISORDERED PACKINGMODELS
Here we consider four different two-dimensional dis-persions of identical nonoverlapping circular disks on thesites of the triangular and squares lattices as well as thesites of honeycomb and kagom´e crystals (see Fig. 4 forillustrations of each of the periodic packings). We alsoinvestigate red different disordered/irregular packings ofcircular disks: stealthy hyperuniform packings ( χ = 0 . χ = 0 .
4) and perturbed-lattice packings (see Fig. 5for illustrations of each of these packings).Stealthy hyperuniform packings of identical particles,which are also class I, are defined by the spectral densityvanishing around the origin, i.e.,˜ χ V ( k ) = 0 , for 0 ≤ | k | ≤ K. (28)Specifically, we first generate stealthy hyperuniformpoint configurations that include N particles in a pe-riodic fundamental cell F via the collective-coordinateoptimization technique [45–47]. We then circumscribethe points by identical nonoverlapping disks [48]. Forstealthy hyperuniform packings (or point patterns), it isuseful to define the χ parameter, which the ratio of con-strained degrees of freedom to total number of degrees offreedom [8, 47], i.e., χ ≡ M d ( N − . (29)For 0 < χ < /
2, the stealthy hyperuniform point pat-terns are highly degenerate and disordered, whereas for1 / < χ < χ below 1/2) in a liquidalso have nearly maximal effective diffusion coefficientsas well as maximal effective thermal/electrical conduc-tivities for perfectly insulating inclusions [48].In this work, we numerically generate 30 different pointpatterns of 10 particles with χ = 0 . χ = 0 . φ min , equalto about 0 .
153 (i.e., φ ≥ φ min = 0 .
85) and 0 .
377 (i.e., φ ≥ φ min = 0 . − a/ , a/ [49]. We then circumscribe the re-sulting points by identical nonoverlapping disks. Theresulting point pattern (or packing) is class I hyperuni-form; see Refs. [19–21] for details. In this work, we FIG. 2. Illustrations of the six different periodic cellular networks considered in this paper: from top to bottom, the square,rhombic, honeycomb, square-octagon, triangular, and kagom´e networks. We show each of them at three solid-phase volumefractions: φ = 0 . φ = 0 .
50, and φ = 0 .
95. Note that these networks can be regarded as periodic point patterns in the limitof φ → numerically generate 50 configurations of 10 particlesand a = 0 .
48. We find that their largest possible frac-tion of space covered by the disks, which is equivalent tothe smallest possible solid-phase volume fraction φ min , isaround 0 .
213 (i.e., φ ≥ φ min = 0 . VI. CHARACTERISTIC LENGTH SCALES
When ranking class I hyperuniform systems accord-ing to the hyperuniformity order metric B V ( B N for thepoint-configuration counterparts [1, 3]), it is critical tochoose an appropriate characteristic length scale D be-cause these order metrics depend on D , as we noted in Sec. II A. In the case of hyperuniform point patternsin R d , it is natural to choose D = ρ − /d , where ρ is thenumber density of points. However, the choice of a lengthscale in the case of two-phase media is highly nontrivialbecause the geometries and topologies of the phases aregenerally much richer and more complex than point con-figurations. Indeed, there are an infinite number of waysof decorating a point configuration to produce two-phasemedia, all of which cannot be universally characterized.In this paper, we consider and evaluate several possiblechoices for the length scale D according to the followingthree criteria: (i) D must be defined for general two-phase media, (ii) D must be independent of the choice ofphase, and (iii) the associated order metric B V must be (e)(a) /2 (c) (b)(f)(d) FIG. 3. Unit cells of two-dimensional periodic networks: (a)square, (b) rhombic, (c) honeycomb, (d) square-octagon, (e)triangular, and (f) kagom´e networks. While (a) and (d) havesquare fundamental cells, the rest of networks have rhombicfundamental cells. The length parameters L and L (shownin black and blue arrows, respectively) determine the volumefraction of the solid phase (red regions) φ = 1 − ( L /L ) .These periodic networks can be treated as packings of poly-gons defined by the white regions: (a)-(c) can be expressedby a single polygon in the fundamental cells, whereas (d)-(f)needs multiple polygons. a finite number for any volume fraction. Seemingly ob-vious choices for D , including the size of a fundamentalcell for periodic systems or the mean nearest-distance fordisordered or irregular packings, fail to meet the crite-ria (i) and (ii). There are several candidates that satisfythe criterion (i), such as the mean chord length of onephase (i.e., the expected length of line segments in thephase between the intersections of an infinitely long linewith the two-phase interface [6, 50, 51]). However, cri-teria (ii) and (iii) immediately eliminate the mean chordlength of an individual phase. The resulting B V divergesat either φ = 0 or φ = 1. Averages based on the meanchord length for each phase, such as the arithmetic andgeometric means, satisfy all criteria. One such exampleis the inverse of the specific surface (i.e., the mean inter-face area per volume) 1 /s , which turns out to be directlyproportional to the arithmetic mean of the mean chordlength ℓ ( i ) C of both phases, i.e., 1 /s = ( ℓ (1) C + ℓ (2) C ) /π [6].Explicit formulas for the specific surface s of all modelsconsidered in the paper are provided in in Appendix A.Henceforth, we employ D = s − . FIG. 4. Illustrations of the four different models of the two-dimensional periodic dispersions of nonoverlapping identicaldisks considered in this paper with different solid-phase vol-ume fractions: φ = 0 .
50 and φ = 0 .
95. From top to bottom,we present dispersions associated with the square and trian-gular lattices and honeycomb and kagom´e crystals.
VII. RESULTS
We consider two-dimensional ordered and disorderedtwo-phase media, shown in Figs. 3, 4, and 5 with tak-ing the inverse of the specific surface as the character-istic length scale, i.e., D = 1 /s = 1. Figure 6 showslog-log plots of local volume-fraction variances σ V ( R )as a function of window radius R at a solid-phase vol-ume fraction φ = 0 .
85 for honeycomb network, triangular-lattice packing, and disordered stealthy hyperuniformpackings (with χ = 0 . σ V ( R ) globally decays as fast as R − in the large- R regime, which are of class I hyper-uniformity [cf. Eq. (5)], and fluctuates on “microscopic”length scales, which in the case of periodic structures,are associated with the spacing of the underlying Bravaislattice.We plot the surface-area coefficients B V ( R ) for themodels considered in Fig. 6 to more closely investigatesuch local fluctuations of σ V ( R ); see Fig. 7. As pointedearlier, B V ( R ) oscillates around an average value B V . (a) (b) (c) FIG. 5. Representative images of the three different models of the two-dimensional disordered/irregular packings of identicalcircular disks considered in this paper at different volume fractions: (a) stealthy hyperuniform packing of χ = 0 .
49 and φ = 0 . χ = 0 .
40 and φ = 0 .
85, and (c) perturbed-lattice packing of φ = 0 . R/D -8 -6 -4 -2 σ V ( R ) Honeycomb network φ =0.85
D=1/s=1 (a)
R/D -8 -6 -4 -2 σ V ( R ) Triangular-lattice packing φ =0.85
D=1/s=1 (b)
R/D -8 -6 -4 -2 σ V ( R ) Stealthy hyperuniformpacking ( χ = 0.4) φ =0.85
D=1/s=1 (c)
FIG. 6. Log-log plots of the local volume-fraction variances σ V ( R ) of two-dimensional ordered and disordered cellular solids ata selected solid-phase volume fraction φ = 0 .
85: (a) honeycomb network, (b) triangular-lattice disk packing, and (c) stealthyhyperuniform packings of χ = 0 .
4. The first two models are periodic structures, whereas the last is a disordered one. Here wetake the inverse of the specific surface 1 /s to be unity, i.e., D = 1 /s = 1. For disordered systems [shown in Fig. 7(c)], such oscilla-tions typically decay as R increases, whereas for periodicnetworks, the amplitude of the oscillations does not de-crease, even in the limit of R → ∞ .Figure 8 shows how the hyperuniformity order metric B V for two-dimensional two-phase media varies with thevolume fraction φ . For periodic networks or disk pack-ings, B V is evaluated from Eq. (21), whereas the disor-dered/irregular counterparts are evaluated by applyingthe running average associated with (12) to the MC re-sults. For periodic networks, where φ can span from 0 to1, as shown in Fig. 8(a), B V exhibits the following threecommon characteristics: (i) it vanishes trivially at φ = 0and φ = 1, (ii) it is proportional to φ for small φ , and(iii) it has a maximum at around φ = 0 .
4. By contrast,for periodic or disordered disk packings, B V trivially van-ishes at φ = 1, but the other characteristics are not ob-served; see Fig. 8(b). We first investigate the rankings of B V for periodic networks shown in Fig. 8(a) and thosefor disk packings in Fig. 8(b) separately and then dis-cuss the rankings for all models. Among the consideredperiodic networks, honeycomb and kagom´e ones achievethe minimum and maximum values of B V , respectively, at a given value of volume fraction φ . For the six net-work models, the values B V increases from honeycomb,square, rhombic, square-octagon, triangular, to kagom´eones. Note that the ranking for the honeycomb, square,and triangular networks are consistent with the rankingof the corresponding metrics for the point counterparts ofthese three network models (triangular lattice, square lat-tice, and honeycomb crystals, respectively, as discussedin Sec. IV) given in Ref. [1, 3]. Moreover, we note thatperiodic networks with a single void region in the funda-mental cell (honeycomb, square, and rhombic) tend to bemore ordered (i.e., smaller B V ) than those with multi-ple void regions in the fundamental cell (square-octagon,triangular, and kagom´e).Figure 8(b) shows that among the periodic disk pack-ings, the triangular and kagom´e packings achieve theminimum and maximum values of B V , respectively. Con-sidering all models of disk packings, the values of B V at a given volume fraction φ increases from triangu-lar, square, disordered stealthy ( χ = 0 . χ = 0 .
22 24 26
R/D B V ( R ) × Triangular-lattice packing φ =0.85
D=1/s=1 (a)
22 24 26
R/D B V ( R ) × Honeycomb network φ =0.85
D=1/s=1 (b)
22 24 26
R/D B V ( R ) × Stealthy hyperuniformpacking ( χ = 0.4) φ =0.85
D=1/s=1 (c)
FIG. 7. Surface-area coefficient B V ( R ) as a function of window radius R of two-dimensional ordered and disordered cellularsolids at a selected solid-phase volume fraction φ = 0 .
85, as per Fig. 6. φ B V × SquareRhombicHoneycombSquare-octagonTriangularKagome
D=1/s=1 (a) φ B V × SquareTriangularHoneycombKagomeSHU ( χ=0.49)
SHU ( χ=0.40)
PLP
D=1/s=1 (b)
FIG. 8. Asymptotic values of the surface-area coefficient B V as a function of the solid-phase volume fraction φ for (a) two-dimensional periodic networks and (b) ordered and disordered disk packings. We take the length scale as D = 1 /s = 1, where s is the specific surface. In (b), SHU and PLP stand for the stealthy hyperuniform packing and perturbed-lattice packing,respectively. The inset in (b) is a magnification of the larger panel. ordered than non-Bravais-lattice packings (honeycomband kagom´e). Importantly, such a ranking of disk pack-ings is identical to the rankings of the point counter-parts that were reported in Ref. [2, 3], i.e., triangu-lar, square, disordered stealthy ( χ = 0 . χ = 0 . φ is varied. However, when rank-ings all models in both periodic networks and periodicand disordered disk packings, the resulting rankings canchange with φ , and hence the volume fraction φ shouldbe specified. For this purpose, we tabulate B V for theperiodic networks, periodic disk packings, and disordereddisk packings at selected values of the solid-phase volumefraction φ in Tables I, II, and III, respectively. FromTables I and II, we immediately see that while the tri- angular disk packing is generally more ordered than thehoneycomb network, their rankings change for φ . . φ = 0 .
85, the triangular-lattice packing and perturbed-lattice packing have the smallest and highest values of B V , respectively. We also note that at φ = 0 .
85, thetriangular- and square-lattice packingshave lower ordermetrics than their network counterparts (i.e., honeycomband square networks, respectively). This implies that thelength scale D = 1 /s = 1 penalizes the order metric B V of a packing of nonspherical particles compared to thecorresponding sphere packing. VIII. CONCLUSIONS AND DISCUSSION
In this work, we took initial steps to characterize arestricted subset of class I hyperuniform two-phase me-dia in two dimensions by ascertaining their local volume-0
TABLE I. Hyperuniformity order metric B V of the six models of two-dimensional periodic cellular networks at various valuesof volume fraction φ ; see Fig. 3. The quantities are computed by Eq. (21) and taking the characteristic length scale to be theinverse of the specific surface, i.e., D = 1 /s = 1. Note that for a given value of φ , B V increases from top to bottom. φ × − × − × − × − × − × − Square 3.1288 × − × − × − × − × − × − Rhombic 3.1320 × − × − × − × − × − × − Square-octagon 3.7286 × − × − × − × − × − × − Triangular 6.7514 × − × − × − × − × − × − Kagom´e 6.9411 × − × − × − × − × − × − TABLE II. Hyperuniformity order metric B V of the four models of two-dimensional periodic disk packings at various valuesof volume fraction φ ; see Fig. 4. The quantities are computed by Eq. (21) and taking the characteristic length scale to be theinverse of the specific surface, i.e., D = 1 /s = 1. Note that for a given value of φ , B V increases from top to bottom. φ × − × − × − × − × − × − Square − × − × − × − × − × − Honeycomb − − − × − × − × − Kagom´e − − × − × − × − × − fraction variances σ V ( R ) and the associated hyperunifor-mity order metrics B V . These models include a varietyof different periodic cellular networks, periodic packings,and disordered/irregular packings, some of which maxi-mize their effective transport and elastic properties [41–44, 48]. Using the estimated B V and a judicious choicefor a length scale to make it dimensionless (as discussedbelow), we ranked these class I models of two-phase me-dia according to the degree to which they suppress large-scale volume-fraction fluctuations. Among the periodicnetworks, the honeycomb and kagom´e networks alwaysachieve the lowest and highest B V , respectively, and therankings do not change as the solid-phase volume fraction φ varies. Similarly, the rankings for disk packings alsodo not change with φ . The triangular-lattice packings(whose Voronoi tessellations are honeycomb networks)and the perturbed-lattice packings have the minimumand maximum values of B V , respectively. Not surpris-ingly, however, the overall rankings for both network andpacking models with their distinctly different geometriesand topologies are difficult to unscramble because theychange with φ . Nonetheless, we summarize these rank-ings by making two general observations. First, the rank-ings for packings of identical disks are consistent withthose of the point-configuration order metric B N corre-sponding to their underlying point patterns [1–3] at anyconsidered volume fraction φ . Second, for both periodicnetworks and periodic packings with the same underlyingBravais lattice, the structures with smaller specific sur-faces have lower values of B V . We note that the secondobservation is generally true, with a few notable excep-tions in which the volume fraction of the solid phase be-comes so low that the disks are nearly in contact with oneanother. Specifically, among all models considered in thiswork, triangular-lattice packing has the minimal B V forall solid-phase volume fraction greater 0.1. Otherwise, the honeycomb networks record the smallest valueWhen establishing these rankings according to B V , itis crucial to determine a characteristic length scale D to make B V dimensionless, which is highly nontrivialdue to the need to account for a wide spectrum of two-phase structures. Among various possibilities, we chosethe inverse of the specific surface 1 /s as the length scale D by considering the three criteria of generality, phase-independence, and boundedness of the associated B V .This choice is also reasonable in that s is easy to com-pute. Furthermore, it is one of the Minkowski function-als (i.e., volume, surface area, integrated mean curva-ture, and Euler number), which are fundamental shapedescriptors that have been widely used in various appli-cations [52, 53]. The integrated mean curvature mightalso serve as a choice of the length scale D . Although wehave made a specific choice D = s − , we note that onecan easily convert our results for B V tothe correspondingquantity for any another length scale D = ℓ by use of therelation B V | D = ℓ = B V / ( sℓ ) d +1 , (30)where d is the space dimension.Our study lays the theoretical foundation to establishthe hyperuniformity order metrics of more general two-phase systems. Towards this end, one needs to developmethods to estimate σ V ( R ) for a wider class of hyper-uniform two-phase media than what can be handled bythe methods used in this work, such as labyrinth-like pat-terns associated with spinodal decomposition [13]. Sucha development will also be beneficial in detecting (ef-fective) hyperuniformity of relatively small systems, inwhich the asymptotic analysis of σ V ( R ) [cf. (2)] is morereliable than the spectral-density condition (1) [54]. Fur-ther studies in three and higher dimensions will be helpfulin determining whether B V scaled by D = 1 /s is a ro-1 TABLE III. Hyperuniformity order metric B V of the three models of two-dimensional disordered disk packings at a selectedvolume fraction φ = 0 .
85 and their respective lowest volume fractions φ min ; see Fig. 5. The values of φ min for various modelsare provided in Fig. 5. The quantities are computed by the MC procedure and Eq. (12) and taking the characteristic lengthscale to be the inverse of the specific surface, i.e., D = 1 /s = 1. The uncertainties are estimated from the statistical errors inthe estimation of σ V ( R ). Model B V at φ min B V at φ = 0 . χ = 0 .
49) 8.9655(6) × − × − Stealthy hyperuniform packings ( χ = 0 .
4) 1.0313(1) × − × − Perturbed-lattice packing 3.6034(3) × − × − bust order metric. It would also be interesting to knowwhether the two-phase counterpart of the decorrelationprinciple [55, 56] for disordered two-phase media couldbe observed as the space dimension increases.It will be of interest to determine whether hyperuni-formity of fluctuations associated with the two-phase in-terface [4] leads to the same rank ordering as for volume-fraction fluctuations for the models considered in this in-vestigation. Another promising avenue for future studyis the construction of two-phase structures with a pre-scribed value of B V . This problem can be regarded asa type of Fourier-space based inverse design procedure[25], in which Eq. (21) is taken as the objective function.Such a procedure can be employed in discovering newtypes of periodic structures with a specified value of B V .The algorithm developed to solve this problem would alsoprovide a tool for determining whether the triangular-lattice disk packing, which we demonstrated minimizes B V among the considered models, is a global minimizerfor B V among a larger class of models. An interestingquestion is what physical properties are optimized by theglobal minimizer of B V under certain constraints. TABLE IV. Formulas for the specific surface s for all modelsconsidered in this work. For periodic networks and disk pack-ings, the specific surface can be expressed as s = C √ − φ/L ,where L is a length parameter of the unit cells; see Fig. 5.For any disordered/irregular disk packing of the number den-sity ρ , the specific surface is written as s = C √ − φρ / .Models C Periodic networks Honeycomb 4Square 4Rhombic 8 / √ / (1 + √ √ √ √ π/ / Square 2 √ π Honeycomb 4 √ π/ / Kagom´e 2 √ π/ / Disordered disk packing 2 √ π Appendix A: Specific Surface for Various Models ofTwo-Phase Media
Table IV provides the formulas for the specific surface s for all two-dimensional models of class I hyperuniformtwo-phase media considered in this work. We note thatall disk packings, ordered or not, have the same specificsurface if they are at the same number density ρ . ACKNOWLEDGMENTS
We thank M. Klatt, C. Maher, and T. M. Middlemasfor very helpful discussions. The authors gratefully ac-knowledge the support of the Air Force Office of Scien-tific Research Program on Mechanics of MultifunctionalMaterials and Microsystems under award No. FA9550-18-1-0514. ∗ [email protected]; http://torquato.princeton.edu [1] S. Torquato and F. H. Stillinger, “Local density fluctuations, hyperuniformity, and order metrics,”Phys. Rev. E , 041113 (2003).[2] S. Torquato, “Hyperuniform States of Matter,”Phys. Rep. , 1 – 95 (2018).[3] C. E. Zachary and S. Torquato, “Hyperuniformity inpoint patterns and two-phase random heterogeneous me-dia,” J. Stat. Mech: Theory Exp. , P12015 (2009).[4] S. Torquato, “Hyperuniformity and its generalizations,”Phys. Rev. E , 022122 (2016).[5] Z. Ma and S. Torquato, “Precise algorithms tocompute surface correlation functions of two-phase heterogeneous media and their applications,”Phys. Rev. E , 013307 (2018).[6] S. Torquato, Random Heterogeneous Materials: Mi-crostructure and Macroscopic Properties , Interdisci-plinary Applied Mathematics (Springer Science & Busi-ness Media, 2002).[7] M. Sahimi,
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