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Designing Disordered Hyperuniform Two-Phase Materials with Novel Physical

Properties

D. Chena, S. Torquatoa,b,c,d,∗

aDepartment of Chemistry, Princeton University, Princeton, New Jersey 08544, USAbDepartment of Physics, Princeton University, Princeton, New Jersey 08544, USA

cPrinceton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USAdProgram in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA

Abstract

Heterogeneous materials consisting of different phases are ideally suited to achieve a broad spectrum of desirable bulkphysical properties by combining the best features of the constituents through the strategic spatial arrangement of thedifferent phases. Disordered hyperuniform heterogeneous materials are new, exotic amorphous matter that behave likecrystals in the manner in which they suppress volume-fraction fluctuations at large length scales, and yet are isotropic withno Bragg peaks. In this paper, we formulate for the first time a Fourier-space numerical construction procedure to designat will a wide class of disordered hyperuniform two-phase materials with prescribed spectral densities, which enables oneto tune the degree and length scales at which this suppression occurs. We demonstrate that the anomalous suppressionof volume-fraction fluctuations in such two-phase materials endow them with novel and often optimal transport andelectromagnetic properties. Specifically, we construct a family of phase-inversion-symmetric materials with variabletopological connectedness properties that remarkably achieves a well-known explicit formula for the effective electrical(thermal) conductivity. Moreover, we design disordered stealthy hyperuniform dispersion that possesses nearly optimaleffective conductivity while being statistically isotropic. Interestingly, all of our designed materials are transparent toelectromagnetic radiation for certain wavelengths, which is a common attribute of all hyperuniform materials. Ourconstructed materials can be readily realized by 3D printing and lithographic technologies. We expect that our designswill be potentially useful for energy-saving materials, batteries and aerospace applications.

Keywords: Disordered hyperuniformity, Heterogeneous materials, Effective properties

1. Introduction

Heterogeneous materials that consist of different phases(or constituent materials) abound in nature and syntheticproducts, such as composites, polymer blends, porous me-dia, and powders [1–4]. In many instances, the length scaleof the inhomogeneities is much smaller than the macro-scopic length scale of the material, and microscopicallythe material can be viewed as a homogeneous materialwith macroscopic or effective properties [1, 5–8]. It hasbeen shown that given the individual phases, the effectiveproperties of the materials are uniquely determined by mi-crostructure of the phases [1]. Consequently, the discov-ery of novel guiding principles to arrange the constituentspresents a promising path to design and realize materi-als with a broad spectrum of exotic and desirable prop-erties by combining the best features of the constituents.The concept of disordered hyperuniformity provides guid-ing design principles for the creation of materials with sin-

∗Corresponding authorEmail address: torquato@electron.princeton.edu (S.

Torquato)

gular performance characteristics, as we will demonstratein this work.

The notion of hyperuniformity was first introduced inthe context of many-particle systems over a decade ago[9]. Hyperuniform many-body systems are those charac-terized by an anomalous suppression of density fluctua-tions at long wavelengths relative to those in typical dis-ordered systems such as ideal gases, liquids and struc-tural glasses. All perfect crystals and perfect quasicrys-tals, and certain special disordered systems are hyperuni-form [9, 10]. Disordered hyperuniform many-particle sys-tems are exotic amorphous state of matter that lie be-tween crystal and liquid states: they behave like crystalsin the way that they suppress density fluctuations at verylarge length scales, and yet they are statistically isotropicwith no Bragg peaks. There is a special type of hyperuni-formity called disordered stealthy hyperuniformity, char-acterized by the absence of scattering within a range ofsmall wavenumbers around the origin in the Fourier space[11, 12].

The exotic structural features of disordered hyperuni-form systems appear to endow such systems with novelphysical properties. For example, disordered hyperuni-

Preprint submitted to Elsevier August 6, 2018

form dielectric networks were found to possess completephotonic band gaps comparable in size to photonic crys-tals [13, 14]. Interestingly, such networks are isotropic,i.e., electromagnetic radiation propagates through the net-works independent of the direction, which is an advantageover photonic crystals, and thus makes them suitable forapplications such as lasers, sensors, waveguides, and opti-cal microcircuits [14]. Moreover, disordered hyperuniformpatterns can have optimal color-sensing capabilities, as ev-idenced by avian photoreceptors [15]. Recently it was re-vealed that the electronic band gap of amorphous siliconwidens as it tends toward a hyperuniform state [16]. In thecontext of superconductors, it was shown that hyperuni-form pinning site geometries exhibit enhanced pinning [17],which is robust over a wide range of parameters. In ad-dition, there is evidence suggesting that disordered hype-runiform particulate media possess nearly optimal trans-port properties while maintaining isotropy [18].

These tantalizing findings have provided an impetus forresearchers to discover and/or synthesize new disorderedhyperuniform systems. We now know that disordered hy-peruniformity arises in both equilibrium and nonequilib-rium systems across space dimensions; e.g., maximally ran-dom jammed hard-particle packings [19–22], driven nonequi-librium granular and colloidal systems [23, 24], dynamicalprocesses in ultracold atoms [25], geometry of neuronaltracts [26], immune system receptors [27] and polymer-grafted nanoparticle systems [28]. The reader is referredto Refs. [29] and [30] for a comprehensive review of disor-dered hyperuniform systems that have been discovered sofar.

Recently the concept of disordered hyperuniformity hasbeen generalized to two-phase heterogeneous materials [10,29, 30]. These materials possess suppressed volume-fractionfluctuations at large length scales, and yet are isotropicwith no Bragg peaks. This can sometimes offer advantagesover periodic structures with high crystallographic symme-tries in which the physical properties can have undesirabledirectional dependence [13, 14]. Specifically, the spectraldensity χ

V(k) of such system goes to zero as the wavenum-

ber k goes to zero with a power-law scaling [10, 19–21, 31],i.e.,

χV(k) ∼ |k|α, (1)

where α is the scaling exponent. Equivalently, the lo-cal volume-fraction variance σ2

V(R) associated with a d-

dimensional spherical observation window of radius R pos-sesses the following scaling at large R [10, 19–21, 31]:

σ2V(R) ∼

R−(d+1), α > 1,

R−(d+1) lnR, α = 1,R−(d+α), 0 < α < 1.

(R → ∞) (2)

where d is the dimension. Note that in all three casesσ2

V(R) decays more rapidly than the inverse of the win-

dow volume, i.e., faster than R−d, which is different fromtypical disordered two-phase materials.

Our ability to design disordered hyperuniform two-phasematerials in a systematic fashion is currently lacking andhence their potential for applications has yet to be ex-plored. In this work, we develop for the first time a Fourier-space based numerical construction procedure to design atwill a wide range of disordered hyperuniform two-phasematerials by tuning the shape of the spectral density func-tion across phase volume fractions. This is equivalentto tuning the degree and length scales at which there isanomalous suppression of volume-fraction fluctuations inthese materials. We note that the Fourier-space settingis the most natural one, since hyperuniformity is definedin the Fourier space. This setting is crucial for captur-ing accurately the long-wavelength, or equivalently, small-wavenumber k behavior. Our designed disordered hyper-uniform microstructures include ones with phase-inversionsymmetry as well as a stealthy dispersion. We computethe two-point cluster function, which incorporates nontriv-ial topological connectedness information and is known toprovide a discriminating signature of different microstruc-tures [32].

Subsequently, we investigate the effective transport prop-erties and wave-propagation characteristics of these ma-terials. We demonstrate that the anomalous suppressionof volume-fraction fluctuations in hyperuniform two-phasematerials endow them with novel and often optimal trans-port and electromagnetic properties. In the case of phase-inversion-symmetric materials, we show that they indeedachieve a well-known explicit formula for the effective elec-trical (thermal) conductivity. On the other hand, thestealthy dispersion possesses nearly optimal effective con-ductivity while being statistically istropic. It is noteworthythat the frequency-dependent effective dielectric constantof any two-phase hyperuniform material cannot have imag-inary part, implying that any such material is dissipation-less (i.e., transparent) to electromagnetic radiation in thelong-wavelength limit. Hence, all of our designed hyper-uniform materials possess such characteristics. Moreover,our constructed dispersion is transparent for a range ofwavelengths as well.

It is noteworthy that our tailored composite microstruc-tures can be readily realized by 3D printing and litho-graphic technologies. We expect that our designs will bepotentially useful for energy-saving materials [33], batter-ies [34] and aerospace applications [35].

In Sec. 2, we describe the Fourier-space based con-struction technique to design disordered hyperuniform two-phase materials. In Sec. 3, we employ our constructiontechnique to generate disordered hyperunform two-phasemicrostructures with prescribed spectral densities. In Sec.4. we compute the corresponding effective transport prop-erties and wave-propagation characteristics of the designedtwo-phase materials. In Sec. 5, we offer concluding marks,and discuss potential application and extension of our re-sults.

2

2. Fourier-Space Construction Procedure

2.1. Algorithm Description

The microstructure of a random multi-phase materialis uniquely determined by the indicator functions I(i)(x)associated with all of the individual phases defined as

I(i)(x) =

{1, x in phase i0, otherwise

(3)

where i = 1, ..., q and q is the total number of phases [1].For statistically homogeneous two-phase materials wherethere are no preferred centers, the two-point probability

function S(i)2 (r) measures the probability of finding two

points separated by vector displacement r in phase i [1].The autocovariance function χ

V(r) is trivially related to

S(i)2 (r) via

χV(r) ≡ S

(i)2 (r)− φ2

i , (4)

where φi is the volume fraction of phase i [1]. The spectraldensity χ

V(k) is the Fourier transform of the autocovari-

ance function χV(r), where k is the wavevector [10, 19–

21, 31]. In practice we generally deal with finite-sized dig-itized materials, i.e., materials consisting of pixels (squareunits) in two dimensions or voxels (cubic units) in threedimensions with each pixel (or voxel) entirely occupied byone phase. We apply periodic boundary conditions to ma-terials as approximation of the infinite system that we areinterested in.

The Yeong-Torquato stochastic reconstruction proce-dure [36, 37] is a popular algorithm to (re)construct digi-tized multi-phase media from correlation functions in phys-ical (or direct) space. Liu and Shapiro have further em-ployed advanced structure synthesis techniques that utilizea variety of microstructure descriptors in physical spaceto design functionally graded materials [38]. We note thatthere is a variety of other methods that have been devel-oped to generate or reconstruct microstructures from lim-ited structural information in the direct space; see Refs.[39–47] and references therein.

In this paper, we generalize the Yeong-Torquato proce-dure to construct disordered hyperuniform materials withdesirable effective macroscopic properties but from struc-tural information in Fourier (reciprocal) space, i.e., thespectral density χ

V(k). Specifically, we define a fictitious

“energy” E of the system as the squared differences be-tween the target and (re)constructed spectral densities,i.e.,

E =∑

k

[χV(k)/ld − χ

V,0(k)/ld]2, (5)

where the sum is over discrete wave vectors k, χV,0

(k)and χ

V(k) are the spectral densities of the target and

(re)constructed microstructures, d is the dimension, and lis certain characteristic length of the system used to scalethe spectral densities such that they are dimensionless.We employ simulated annealing method [36] to minimizethe energy of the system. We start with random initial

configurations with prescribed volume fractions of bothphases. At each time step we randomly select one pixel (orvoxel) from each of the two phases and attempt to swapthem [45, 46]. In the later stages of the construction pro-cedure, we apply the different-phase-neighbor-based pixelswap rule, an advanced rule developed previously [47] toimprove efficiency of the algorithm and remove random“noise” (isolated pixel or small clusters of pixels of thephase of interest) in the media. We update the spectraldensity of the trial configuration χ

V(k) and compute the

system energy. We accept the trial pixel swap accordingto the probability

pacc(old → new) = min{1, exp(−Enew − Eold

T)}, (6)

where T is the fictitious ”temperature” of the system thatis set initially high and gradually decreases according toa cooling schedule [36, 37], and Eold and Enew are theenergies of the system before and after the pixel swap.Trial pixel swaps are repeated and system energy is trackeduntil it drops below a specified stringent threshold value,which we choose as 10−3 in this work.

2.2. Efficient algorithmic implementation of the construc-

tion technique

In this work, we consider digitized two-phase materi-als in a square domain in two dimensions subject to peri-odic boundary conditions and denote the side length of thesquare domain by L. Here we set L to be 300 pixels. Forsuch materials, the wave vector k can only take discretevalues k = 2π × (n1x + n2y)/L (n1, n2 ∈ Z), where x, yare two orthogonal unit vectors aligned with the bound-aries of the square domain. It can be easily shown thatthe spectral density of such materials can be computed as

χV(k) =

1

Am2(k)|J (k)|2, (7)

where A = L2 is the area of the system, m(k) is the Fouriertransform of the indicator function m(r) of a pixel andgiven by

m(k) =

sin(kx/2)(kx/2)

sin(ky/2)(ky/2)

, kx 6= 0, ky 6= 0sin(kx/2)(kx/2)

, kx 6= 0, ky = 0sin(ky/2)(ky/2)

, kx = 0, ky 6= 0

1, kx = 0, ky = 0

, (8)

and the generalized collective coordinate [48] J (k) is de-fined as

J (k) =∑

r

[exp(ik · r)(I(r) − φ)], (9)

where r sums over all the pixel centers, and I(r) [as definedin Eq. (3)] and φ are the indicator function and volumefraction of the phase of interest, respectively. Henceforth,when referring to the properties of the phase of interest,

3

we will drop the subscripts and superscripts for simplicity.In this work we focus on isotropic materials and employthe angular-averaged version χ

V(k) of χ

V(k) in the energy

functional E as defined in Eq. (5).A central issue in the construction procedure is to com-

pute the spectral density of the trial configurations effi-ciently. Here instead of computing χ

V(k) from scratch for

every new configuration, we have devised a method thatenables one to quickly compute χ

V(k) of the new config-

uration based on the old ones. Specifically we track thegeneralized collective coordinate J (k) at each k. At the

beginning of the simulation, J (k) of the initial configu-ration is explicitly computed. Then for every new trialconfiguration, the change of J (k) only comes from thepixel swap and thus can be updated as follows:

J (k) + dJ new(k) − dJ old(k) → J (k), (10)

wheredJ new(k) = exp(ik · rnew), (11)

dJ old(k) = exp(ik · rold), (12)

and rnew and rold are the new and old positions of themoved pixel that belongs to the phase of interest. Thenχ

V(k) of the trial configuration is computed using Eq. (7)

and subsequently binned according to k = |k| in orderto obtain the angular-averaged χ

V(k). If the trial swap

is rejected, J (k) of the old configuration can be easilyrestored by

J (k)− dJ new(k) + dJ old(k) → J (k). (13)

Note that the complexity of our algorithm is O(L2),where L is the linear size of the microstructure. The simu-lations were performed on an Intel(R) Xeon(R) CPU (E5-2665) with a clock speed of 2.40 GHz, and it took roughlyone day to generate a typical microstructure.

3. Designing disordered hyperuniform two-phasematerials with prescribed spectral densities

Previously certain necessary conditions that autoco-variance functions χ

V(r) have to satisfy so that they can

be realized by two-phase materials have been determined[49, 50]. It is noteworthy that these necessary conditionsare not sufficient to guarantee realizablity of χ

V(r) by two-

phase materials, which should be ultimately verified by thesuccessful construction of the targeted spectral densities.Also, certain parameterized autocovariance functions ex-pressible in terms of a set of chosen realizable basis func-tions have been identified [49, 50]. Here we utilize thisknowledge, but for a completely different purpose, i.e., todesign various disordered hyperuniform two-phase materi-als. Specifically, we first design realizable χ

V(r) with an

additional hyperuniform constraint [29, 30]:

∫

Rd

χV(r)dr = 0, χ

V(0) = 0. (14)

We then compute the Fourier transform χV(k) of χ

V(r)

and employ χV(k) as the target spectral density in the

aforementioned Fourier-space construction technique. Sub-sequently, we carry out this construction technique to con-struct two-phase materials corresponding to χ

V(k). The

procedure is schematically shown in Fig. 1.

Figure 1: Illustration of the numerical construction procedure to de-sign and generate disordered hyperuniform two-phase materials. Thehyperuniformity condition places constraints on χ

V(r) and χ

V(k),

as given Eq. (14).

As a proof-of-concept and for simplicity, we first de-sign a family of disordered hyperuniform materials withphase-inversion symmetry [1], i.e., the corresponding mi-crostructures at volume fractions φ can be generated byinverting the two phases of the microstructures at volumefractions 1−φ. Another reason to design these microstruc-tures is that they achieve a well-known explicit formula foreffective conductivity, as we show in detail in next section.The scaled autocovariance function χ

V(r)/[φ(1 − φ)] of

such microstructures is independent of φ [50]. Here we ex-plicitly consider constructions at volume fractions φ ≤ 0.5,and generate microstructures at φ > 0.5 by inverting thetwo phases of the microstructures at 1− φ.

We start with realizable basis functions for χV(r) that

were identified previously [50]. One such example is

χV(r)/[φ(1 − φ)] = e−r/a cos(qr), (15)

where q is the wavenumber associated with the oscillationof χ

V(r), and a can be considered as the correlation length

of the system. In previous work [30] it was found thatwhen qa = 1.0, the corresponding autocovariance functionsatisfies all the known necessary realizable conditions andthe hyperuniformity constraint. Here we set as q = 0.1to ensure high enough resolution for the microstructure.Then we set a = 10.0 such that qa = 1.0 is satisfied. Theresulting χ

V(r) is shown in Fig. 2(c).

We compute the corresponding spectral density χV(k)

and employ it to construct two-phase materials. We con-struct microstructures at volume fractions φ =0.1, 0.2,0.3, 0.4, 0.45, and 0.5. The constructed microstructuresare shown in the third colume of Fig. 3. Representativetarget and constructed spectral densities at φ = 0.5 areshown in Fig. 5(c). It is noteworthy that spectral densi-ties at other values of φ only differ by certain constants.Note that χ

V(k) goes to zero quadratically as k goes to

zero, i.e., the scaling exponent α = 2. In addition, in the

4

0 20 40 60 80 100r

-0.2

0

0.2

0.4

0.6

0.8

1

χ V(r

)/[φ

(1−φ

)]

0 20 40 60 80 100r

-0.2

0

0.2

0.4

0.6

0.8

1

χ V(r

)/[φ

(1−φ

)]

(a) (b)

0 20 40 60 80 100r

-0.2

0

0.2

0.4

0.6

0.8

1

χ V(r

)/[φ

(1−φ

)]

0 20 40 60 80 100r

-0.2

0

0.2

0.4

0.6

0.8

1

χ V(r

)/[φ

(1−φ

)]

(c) (d)

0 20 40 60 80 100r

-0.2

0

0.2

0.4

0.6

0.8

1

χ V(r

)/[φ

(1−φ

)]

0 20 40 60 80 100r

-0.2

0

0.2

0.4

0.6

0.8

1

χ V(r

)/[φ

(1−φ

)]

(e) (f)

Figure 2: Realizable autocovariance functions χV(r)/[φ(1− φ)] that

correspond to hyperuniform two-phase materials, where φ is the vol-ume fraction of the phase of interest. Functions in (a), (b), (d), (f)are given by Eq. (17) with the parameters (q, a, b, c) = (5/2, 5,5√15/2, 1/4), (3, 4, 4

√6, 1/2), (5, 4, 24, 1/2), and (8, 15, 15

√14,

1/2), respectively. Function in (c) is given by Eq. (15) with q = 0.1and a = 10.0. Function in (e) is given by Eq. (16) with q = 0.2,a = [5(1 +

√3)3/2]/[

√2(3 +

√3)], and b = 5(

√2 +

√6)/2.

opposite asymptotic large-k limit, χV(k) decays like 1/k3,

which is consistent with the fact that χV(r) is linear in r

for small r.Another basis function investigated previously [50] is

χV(r)/[φ(1 − φ)] =

1

2[e−r/a + e−r/b cos(qr)], (16)

where q, a, and b are parameters (here we choose a asa characteristic length of the system). In previous work[30] we find that when a = [((qb)2 − 1)1/2]/[(qb)2 + 1]and 1 < qb ≤ (

√2 +

√6)/2, the autocovariance func-

tion (16) satisfies all the known necessary realizable con-ditions and the hyperuniformity constraint. Here, sim-ilar to the previous case, we choose q = 0.2. To ob-tain quartic behavior of χ

V(k) near the origin, we set

b = 5(√2 +

√6)/2 such that qb = (

√2 +

√6)/2 is sat-

isfied, and then set a = [5(1 +√3)3/2]/[

√2(3 +

√3)] to

satisfy a = [((qb)2−1)1/2]/[(qb)2+1]. The resulting χV(r)

is shown in Fig. 2(e). The constructed microstructures areshown in the fifth column of Fig. 3. Representative targetand constructed spectral densities at φ = 0.5 are shownin Fig. 5(e). It is noteworthy that spectral densities atother values of φ only differ by certain constants. Theconstructed spectral density indeed demonstrates hyper-uniformity; moreover, χ

V(k) is indeed quartic in k around

the origin. In addition, in the opposite asymptotic large-klimit, χ

V(k) decays like 1/k3, which is consistent with the

asymptotic behavior of χV(r) for small r.

In order to obtain materials with other types of hy-peruniformity, i.e., other power laws of χ

V(k) around the

origin, we introduce a new class of autocovariance function

χV(r)/[φ(1 − φ)] = (c+ 1)e−r/a − c

bq

(r + b)q, (17)

where the parameters q, a, b and c are positive. For thisχ

V(r) to be realizable and hyperuniform, the parameters

have to satisfy the following conditions:

(v − 2)(v − 1)(1 + c)a2 = cb2, (18)

− 1 + c

a+

cv

b< 0, (19)

and1 + c

a2− cv(1 + v)

b2≥ 0. (20)

The relation (18) corresponds to the hyperuniformityconstraint, Eq. (19) corresponds to the realizability con-dition that the first derivative of χ

V(r) should be negative

at r = 0 [50], and Eq. (20) corresponds to the realizabil-ity condition that the second derivative of χ

V(r) should

be nonnegative at r = 0 [50]. It is noteworthy that thisχ

V(r) scales like −1/rq for large r, which translates into

a scaling of kq−d for χV(k) at small k, where d = 2 is the

dimension. Thus by tuning the value of q, we can manipu-late the type of hyperuniformity that results. Specifically,we aim to obtaining materials with their χ

V(k) going to

zero as k goes to zero with the following exponents α:1/2, 1, 3, and, 6, respectively. We find that by settingthe parameters (q, a, b, c) = (5/2, 5, 5

√15/2, 1/4), (3,

4, 4√6, 1/2), (5, 4, 24, 1/2), and (8, 15, 15

√14, 1/2), re-

spectively, these desired small-k asymptotic behaviors ofχ

V(k), i.e., α = 1/2, 1, 3, and, 6, are achieved. The re-

sulting χV(r) are shown in Fig. 2(a), (b), (d), and (f),

and the constructed microstructures in the first, second,fourth, and sixth columns of Fig. 3, respectively. Repre-sentative target and constructed spectral densities of thesemicrostructures at φ = 0.5 are shown in Fig. 5(a), (b), (d),and (f), respectively. It is noteworthy that spectral den-sities at other values of φ only differ by certain constants.In addition, we note that the microstructures in the firstand second columns of Fig. 3 possess the third and secondtypes of volume-fraction variance scaling described in Eq.(2), respectively, and all of the rest microstructures in Fig.3 possess the first type of scaling described in Eq. (2).

5

Figure 3: Realizations of disordered hyperuniform two-phase materials. From left to right, each column correspdonds to one autocovariancefunction in Fig. 2. The quantity φ is the volume fraction of the phase of interest, and α specifies the asymtotpic behavior of χ

V(k) as k

goes to zero, i.e., χV(k) ∼ kα. Note that since these microstructures possess phase-inversion symmetry, the corresponding microstructures at

volume fractions φ > 0.5 can be generated by inverting the two phases of the microstructures at volume fractions 1−φ. Note that dependingon the exponent α, the volume-fraction variance scaling will behave according to Eq. (2).

0 50 100 150r

0

0.1

0.2

0.3

0.4

0.5

C2(r

)

materialpore

0 50 100 150r

0

0.1

0.2

0.3

0.4

0.5

C2(r

)

materialpore

(a) (b)

Figure 4: (a) Two-point cluster function C2(r) of the microstructurewith α = 2.0 at φ = 0.5 in Fig. 3. (b) Two-point cluster functionC2(r) of the microstructure with α = 4.0 at φ = 0.5 in Fig. 3. Notethat C2(r) in (a) decays more slowly than C2(r) in (b), implyingbetter long-range connectedness of the microstructure with α = 2.0in Fig. 3.

The designed materials shown in Fig. 3 possess a va-riety of morphologies: as φ increases, the materials grad-

ually transition from particulate media consisting of iso-lated “particles” to labyrinth-like microstructures. Notethat both phases in the microstructures with α = 2.0 andα = 4.0 at φ = 0.5 in Fig. 3 percolate with nearest-neighbor and next-nearest-neighbor connections (along thepixel edges and corners), which is a singular topologicalfeature for a two-dimensional composite [1]. Normally,only one phase can percolate (with the other phase beingtopologically disconnected). It is known that d-dimensional(d ≥ 2) two-phase materials that possess phase-inversionsymmetry are bicontinuous (i.e., both phases percolate)for φc < φ < 1− φc, provided that the percolation thresh-old φc < 1/2 [1]. For example, two-dimensional randomcheckerboard systems are bicontinuous for 0.4073 < φ <0.5927 with nearest-neighbor and next-nearest-neighborconnections [1]. Also, to quantify the differences in long-range topological connectedness of the microstructures withα = 2.0 and α = 4.0 at φ = 0.5 in Fig. 3, we havecomputed their corresponding two-point cluster functionsC2(r), which measure the probability of finding two points

6

0 0.2 0.4 0.6 0.8 1ka/(2π)

0

0.5

1

1.5

χ V(k

)/a2

TargetConstruction

~

0 0.2 0.4 0.6 0.8 1ka/(2π)

0

0.5

1

1.5

χ V(k

)/a2

TargetConstruction

~

(a) (b)

0 0.2 0.4 0.6 0.8 1ka/(2π)

0

0.1

0.2

0.3

0.4

0.5

χ V(k

)/a2

TargetConstruction

~

0 0.1 0.2 0.3 0.4 0.5ka/(2π)

0

0.2

0.4

0.6

0.8

1

χ V(k

)/a2

TargetConstruction

~

(c) (d)

0 0.2 0.4 0.6 0.8 1ka/(2π)

0

0.5

1

1.5

2

χ V(k

)/a2

TargetConstruction

~

0 0.1 0.2 0.3 0.4 0.5ka/(2π)

0

0.2

0.4

0.6

0.8

1

χ V(k

)/a2

TargetConstruction

~

(e) (f)

Figure 5: Representative target and constructed dimensionless spec-tral densities χ

V(k)/a2 that correspond to realizations of hyperuni-

form microstructures at φ = 0.5 in Fig. 3, where a is certain char-acteristic length scale of the systems. It is noteworthy that spectraldensities at other values of φ only differ by certain constants.

separated by r in the same cluster of the phase of inter-est [1, 32], as shown in Fig. 4(a) and (b). A cluster isdefined as any topologically connected region of a phase.Clearly the microstructures with α = 2.0 is less connectedthan the one with α = 4.0 on large length scales, whichis consistent with the observation that the exponentiallydecaying term in Eq. (16) gives rise to clusters of randomsizes and shapes [45, 46].

We now consider a construction of hyperuniform mate-rials that does not have phase-inversion symmetry. We em-ploy random disk packings as initial conditions and startfrom very low initial temperature T0 = 10−10. We em-ploy a pixel selection rule that favors the swap of pix-els of different phases at the two-phase interphase (seeAppendix A for detail), and only constrain χ

V(k) to be

zero for wavenumbers within a circular exclusion regionaround the origin with a radius K. We obtain disordered

stealthy hyperuniform dispersion at relatively high volumefraction φ = 0.388, which appear like the patterns of leop-ard spots, as shown in Fig. 6. Here K is chosen suchthat K/(2πρ1/2) ≤ 0.864, where ρ is the number densityof the “particles” (We note that for a microstructure witha linear size of 300 pixels, the distance between neigh-boring k points in the Fourier space is 2π/300, and if wechoose a bin size roughly twice as large as this distancefor k to compute χ

V(k), the exclusion region with a ra-

dius 0.864 includes k points within the first 5 bins). Thisexample serves to demonstrate that by varying the initialconditions and cooling schedule, there is a wide diversity ofmicrostructures that can be generated by our constructiontechnique. Note that this dispersion is transparent to elec-tromagnetic radiation with wavenumbers smaller than Kin the single-scattering regime. Such materials should alsobe transparent for a range of wavelengths in the multiple-scattering regime when the incident wavenumber of theradiation is less than about K/4 [51].

Figure 6: Designed disordered stealthy hyperuniform dispersion thatis transparent to long-wavelength electromagnetic radiation and itsassociated dimensionless spectral density ρχ

V(k) (scaled by the num-

ber density of the “particles” ρ). To generate such a pattern, weemploy random disk packings as initial conditions and a pixel se-lection rule that favors the swap of pixels of different phases at thetwo-phase interphase. We start from very low initial temperatureT0 = 10−10, and only constrain χ

V(k) to be zero for wavenumbers

within the exclusion region, which is shown in the spectral densityplot (the region on the left of the blue dash line). Interestingly, thisdispersion possesses nearly optimal effective electrical (or thermal)conductivity for a realization with the individual phase conductivitycontrast σ2/σ1 = 10.0, where σ1 and σ2 are the electrical (or ther-mal) conductivities of the “particle” and matrix phases, respectively.

4. Transport and wave-propagation properties ofthe designed materials

4.1. Effective conductivity of the designed materials

In this section, we first compute the effective electric(or thermal) conductivity σe of the aforementioned de-signed materials. According to the homogenization theory

7

[1], the effective conductivity σe is defined through theaveraged Ohm’s (or Fourier’s) law:

〈J(x)〉 = σe〈E(x)〉, (21)

where angular brackets denote an ensemble average, 〈J(x)〉is the average flux and 〈E(x)〉 is the average field.

We consider the case where the phase conductivity con-trast of the two individual phases σ2/σ1 is 10.0. To com-pute the effective conductivities σe of the constructed dig-itized materials, we employ the first-passage-time simu-lation techniques [18, 52, 53]. Specifically we release over105 random walkers to sample each material and record themean squared displacement 〈R2(t)〉 of the random walkersat sufficiently large t. Then σe is computed as

σe = limt→∞

〈R2(t)〉/(2dt), (22)

where d is the dimension.Torquato [1] has derived a “strong-contrast” expansion

of σe that perturbs around the microstructures that real-ize the well-known self-consistent (SC) approximation foreffective conductivity:

φ2σe + (d− 1)σ1

σe − σ1+ φ1

σe + (d− 1)σ2

σe − σ2

= 2− d−∞∑

n=3

[A

(2)n

φ2βn−221 +

A(1)n

φ1βn−212 ],

(23)

where A(p)n is the n-point parameter, and β12 = −β21 =

(σ1 − σ2)/(σ1 + σ2). If we truncate Eq. (23) after third-

order terms and set A(2)3 = φ1φ

22, Eq. (23) reduces to the

SC approximation, which in two dimensions is given by

φ2σe + σ1

σe − σ1+ φ1

σe + σ2

σe − σ2= 0. (24)

The reader is referred to Ref. [1] for more details about thisSC approximation. Milton showed that multiscale hierar-chical self-similar microstructures realize the SC approxi-mation [54]. Torquato and Hyun further found a class ofperiodic, single-scale dispersions that realize this approxi-mation [55]. Here we discover that a family of disorderedsingle-scale microstructures can also realize this approxi-mation, as shown in Fig. 3. The effective conductivityresults for these microstructures across phase volume frac-tions φ are shown in Fig. 7. The Hashin-Shtrikman (HS)two-point bounds on σe as well as the SC approximationas described by Eq. (24) [1] are also plotted alongside thesimulation results. The effective conductivities σe of thesemicrostructures indeed agree well with the SC approxima-tion. This agreement demonstrates our ability to constructmicrostructures with targeted transport properties.

Now we determine the dimensionless effective conduc-tivity σe/σ1 of the disordered stealthy hyperuniform dis-persion described in Fig. 6. Again, using first-passage timetechniques [18, 52, 53], we find that σe/σ1 = 4.92. The cor-responding HS upper and lower bounds on the dimension-less effective conductivity for any two-phase material with

such a phase volume fraction and phase conductivities aredetermined to be 5.18 and 3.01, respectively. Thus, we seethat the effective conductivity σe of the stealthy disper-sion is close to the upper bound, which means that thisdisordered stealthy dispersion possesses a nearly optimaleffective conductivity.

4.2. Frequency-dependent effective dielectric constant of

the designed materials

Here we consider the determination of the frequency-dependent effective dielectric constant εe(k1) of the con-structed microstructures, which we treat as two-phase di-electric random media with real phase dielctric constantsε1 and ε2. Here k1 is the wavenumber of the wave prop-agation through phase 1. In this case, the attenuationof the waves propagating through the effective mediumis due purely to scattering, not absorption [56]. We as-sume that the wavelength of the propagation wave is muchlarger than the scale of inhomogeneities in the medium.We are interested in the effective dielectric constant εe(k1)associated with the homogenized dynamic dielectric prob-lem [1, 56]. Rechtsman and Torquato [56] have deriveda two-point approximation based on the strong-contrastexpansion to estimate εe for two- and three-dimensionalmicrostructures with a percolating phase 2 and ε2 ≥ ε1:

ε1 − ε2ε1 + ε2

φ21[εe − ε2εe + ε2

]−1 = φ1 −A(1)2 × [

ε1 − ε2ε1 + ε2

], (25)

where A(1)2 is the two-point parameter that is an integral

over the autocovariance function weighted with gradientsof the relevant Green’s functions. The latter was explicitlyrepresented in three dimensions, but not in two dimen-sions. It can be shown (see Appendix B for details) that

A(1)2 in two dimensions is given by

A(1)2 = [−γk21

∫∞

0

χV(r)rdr − k21 ln k1

∫∞

0

χV(r)rdr

− k21

∫∞

0

χV(r)r ln(r/2)dr] + i

π

2k21

∫∞

0

χV(r)rdr +O(k41 ln k1),

(26)

where γ ≈ 0.577216 is the Euler’s constant.The imaginary part of εe(k1) accounts for attenuation

(losses) due to incoherent multiple scattering in a typicaldisordered two-phase material [56]. Because of the sumrule Eq. (14) on any χ

V(r) that corresponds to disordered

hyperuniform two-phase materials, it immediately follows

that the imaginary part of A(1)2 in Eq. (26), and hence

εe(k1) in Eq. (25) vanish in the long-wavelength limit forany such materials. As a result, any hyperuniform mate-rial is transparent to electromagnetic radiation, i.e., dis-sipationless in the long-wavelength limit according to theapproximation (26). This is because the attenuation ofpropagating waves in such composite materials with realphase dielectric constants can only come from scattering,as mentioned above. Also, the first two terms of the real

8

part of A(1)2 in Eq. (26) vanish for such materials, while

the remaining lowest-order term −k21∫∞

0χ

V(r)r ln(r/2)dr

generally does not vanish and depend on the exact form ofχ

V(r). We compute the real part of εe for two hyperuni-

form materials: a phase-inversion-symmetric case of a re-alization of Eq. (15) and a non-phase-inversion-symmetricone in Fig. 6 at φ2 = 0.612, where phase 2 is the percolat-ing matrix phase. We take k1 = 2π/(10a), where a = 10.0is the characteristic length in Eq. (15). We first compute

A(1)2 from Eq. (26), and then compute εe from Eq. (25)

using A(1)2 . The results are shown in Fig. 8. Clearly the

real part of εe differ for these two systems across valuesof ε2/ε1. At last, we note that the frequency-dependentdielectric constant problem is demonstrated to be equiv-alent to the static effective conductivity problems as thewavenumber of the propagation wave goes to zero [1].

0.2 0.4 0.6 0.8φ

0

2

4

6

8

10

σ e/σ1

α = 0.5α = 1.0α = 2.0α = 3.0α = 4.0α = 6.0HS lower boundHS upper boundSC approx.

Figure 7: The effective conductivities of the designed microstruc-tures across different values of α and φ in Fig. 3, as computed fromthe first-passage-simulation techniques. Here we consider the casewhere the contrast of phase conductivities σ2/σ1 is 10.0, and φ isthe volume fraction of phase 2, which is targeted in the construc-tion technique. The HS two-point bounds on σe as well as the SCapproximation as described by Eq. (24) are also plotted alongsidethe simulation results. The effective conductivities σe of these mi-crostructures indeed agree well with the SC approximation.

5. Conclusions and discussion

In this work, we developed for the first time a Fourier-space numerical construction procedure to design at willa wide class of disordered hyperuniform two-phase mate-rials. These materials possess anomalous suppression ofvolume-fraction fluctuations at large length scales, whichendow them with novel and often optimal transport and

2 4 6 8 10ε

2/ε

1

1

2

3

4

5

6

7

Re[

ε e]/ε 1

Non-phase-inversion-symmetricphase-inversion-symmetric

φ2 = 0.612

Figure 8: Real part of the effective dielectric constant Re[εe] of twohyperuniform materials: a phase-inversion-symmetric case of a real-ization of Eq. (15) and a non-phase-inversion-symmetric one in Fig.6 as a function of dielectric-contrast ratio ε2/ε1 at volume fractionφ2 = 0.612 and wave number k1 = 2π/(10a). In order to calculate

A(1)2 for each microstructure, Eq. (26) is used. Clearly Re[εe] differ

for these two systems across different values of ε2/ε1.

electromagnetic properties as we demonstrated. Our de-signed phase-inversion-symmetric materials possess vari-ous morphologies and different levels of topological con-nectedness, as revealed by the two-point cluster function.Moreover, they indeed achieve a well-known explicit for-mula for the effective electrical (thermal) conductivity. Onthe other hand, our designed disordered stealthy hyper-uniform dispersion possesses nearly optimal effective con-ductivity, while being fully isotropic. Such materials cansometimes offer advantages over periodic structures withhigh crystallographic symmetries where the physical prop-erties can be anisotropic, such as has been shown in thecase of photonic materials [13, 14]. All of our designedhyperuniform materials are dissipationless (i.e., transpar-ent to electromagnetic radiation) in the long-wavelengthlimit, which is a common characteristic of hyperuniformmaterials. Moreover, our dispersion is also transparent toelectromagnetic radiation for a range of wavelengths.

In the present paper, we focused on the design of two-dimensional hyperuniform two-phase materials with pre-scribed spectral densities, but it is noteworthy that withslight modification our Fourier-space numerical construc-tion technique can be readily applied in three dimensionsto design disordered hyperuniform microstructures, whichare expected to be distinctly different from their two-dimesionalcounterparts. For example, bicontinuous microstructuresare much more common in three dimensions [1]. More-

9

over, all of our in-silico designed microstructures can bereadily realized by 3D printing and lithographic technolo-gies [57]. Also, in principle there is no constraint on thetypes of constituent materials that can be used in thesecomposite materials as long as the constituents (phases)are arranged in a hyperuniform fashion. In addition, wenote that a two-phase material can be viewed as a specialcase of random scalar fields, and our results can be furthergeneralized to design hyperuniform scalar fields, which hasreceived recent attention [29, 58].

Since all of our hyperuniform material designs are dissi-pationless in the long-wavelength limit, they will be poten-tially useful for energy-saving materials that prevent heataccumulation by allowing the free transmission of infraredradiation [33]. In addition, by employing a phase-changematerial as the particle phase and graphite as the matrixphase in our designed disordered stealthy hyperuniformdispersion, one could fabricate phase-change compositeswith high thermal conductivity [59]. Such composite ma-terials can absorb and distribute heat efficiently, which iscrucial for the normal operation of battery packs [34] andspacecrafts [35].

A natural extension of this work will be the statisticalcharacterization of our designed disordered hyperuniformtwo-phase systems by computing a host of different typesof statistical correlation functions. This not only includesvarious types of two-point correlation functions, e.g., pore-size functions, lineal-path functions, surface-surface corre-lation functions, to name a few, but their higher-order(three-point) generalizations as well [1, 60]. Moreover, togain further insight into the potential of these constructedmicrostructures, it will be beneficial to carry out a com-prehensive study to estimate other transport, thermal, me-chanical, photonic and phononic properties as well as ef-fective reaction rates of these composites.

Moreover, we note that identifying and utilizing process-structure-property relationships to design and manufac-ture novel materials with desirable properties is a holy grailof materials science. The emergence of Integrated Com-putational Materials Engineering (ICMG) has greatly ac-celerated this process by integrating materials science andautomated design [44]. Our present results demonstratethat by designing hyperuniform microstructures with tun-able spectral densities, which are then automatically gen-erated at the mesoscale, we can control the effective phys-ical properties of the materials. By combining our con-struction technique with existing material models and datainfrastructures [61], one may be able to create new power-ful ICMG platforms to efficiently design optimized mate-rials for various applications.

Acknowledgments

The authors are very grateful to Dr. Ge Zhang for hiscareful reading of the manuscript and Dr. Yang Jiao forhis helpful discussion. This work was supported by the

National Science Foundation under Award No. CBET-1701843.

Appendix A. Pixel-selection rule for constructionof disordered stealthy hyperuniformdispersion

To construct disordered stealthy hyperuniform disper-sion shown in Fig. 6, we modify the different-phase-neighbor-based (DPN-based) pixel selection rule proposed in Ref.[47]. In two dimensions, each pixel has 8 neighbors, andwe divide the pixels of each phase into different sets Si

based on the number of neighboring pixels i in a differentphase that they have. For example, if we consider a two-phase medium consisting of blue and red pixels, we dividethe blue pixels into different sets based on the number ofneighboring red pixels that they have.

In each pixel-swap iteration, for each phase in the medium,a set Si is first selected according to p(Si), which is givenby

p(Si) =

{0.6, i = M,wA(Si)(i + 1)4, 0 ≤ i < M,

(A.1)

whereM is the maximum number of different-phase neigh-bors of a pixel in the phase of interest, A(Si) is the num-ber of pixels in the phase of interest with i different-phaseneighbors, and w is the normalization factor given by

w = (1− 0.6)/[M−1∑

i=0

A(Si)(i + 1)4]. (A.2)

Then for each phase a pixel is randomly selected from thecorresponding chosen Si, and the two selected pixels be-longing to the two different phases are swapped, generatinga new trial microstructure.

Appendix B. Derivation of the two-point param-eter associated with the frequency-dependent dielectric constant in twodimensions

Here we apply the general formalism derived in Ref.[56] to two dimensions. Specifically, the dyadic Green’sfunction is given by

G(r, r′) = − I

2σ1δ(r − r

′) +G1(r, r′)I +G2(r, r

′)rr.

(B.1)where r is a unit vector directed from r

′ towards r, I is theunit tensor, δ(r − r

′) is the Dirac delta function, σ1 = k21(k1 is the wavenumber of the wave propagating throughphase 1), and

G1(r, r′) =

i

4[H

(1)0 (k1r)−

H(1)1 (k1r)

k1r], (B.2)

10

G2(r, r′) =

i

4[H

(1)1 (k1r)

k1r+

1

2H

(1)2 (k1r)−

1

2H

(1)0 (k1r)].

(B.3)

Here H(1)i (k1r) is the i-th order Hankel function of the

first kind. Note that the Green’s function G(r, r′) solvesthe following partial differential equation:

∇×∇×G(r, r′)− σ1G(r, r′) = Iδ(r − r′). (B.4)

The two-point parameterA(1)2 associated with the frequency-

dependent dielectric constant εe(k1) is given by

A(1)2 = k21

∫Tr[H(r)]χ

V(r)dr, (B.5)

where χV(r) is the autocovariance function, Tr[H(r)] is

the trace of H(r), and H(r) is the principle value of theGreen’s function given by

H(r, r′) = G1(r, r′)I +G2(r, r

′)rr. (B.6)

Substituting Eqs. (B.2) and (B.3) into Eq. (B.6), we

obtain the explicit expression for A(1)2 , which is Eq. (26).

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