Research Article

Multifunctional composites for elastic and electromagnetic wave propagation

Jaeuk Kim and Salvatore Torquato
PNAS April 21, 2020 117 (16) 8764-8774; first published April 9, 2020; https://doi.org/10.1073/pnas.1914086117

  1. Edited by Paul M. Chaikin, New York University, New York, NY, and approved March 4, 2020 (received for review August 13, 2019)

Significance

We establish accurate microstructure-dependent cross-property relations for composite materials that link effective elastic and electromagnetic wave characteristics to one another, including effective wave speeds and attenuation coefficients. Our microstructure-dependent formulas enable us to explore the multifunctional wave characteristics of a broad class of disordered microstructures, including exotic disordered “hyperuniform” varieties, that can have advantages over crystalline ones, such as nearly optimal, direction-independent properties and robustness against defects. Applications include filters that transmit or absorb elastic or electromagnetic waves “isotropically” for a range of wavelengths. Our findings enable one to design multifunctional composites via inverse techniques, including the exterior components of spacecraft or building materials, heat sinks for CPUs, sound-absorbing housings for motors, and nondestructive evaluation of materials.

Abstract

Composites are ideally suited to achieve desirable multifunctional effective properties since the best properties of different materials can be judiciously combined with designed microstructures. Here, we establish cross-property relations for two-phase composite media that link effective elastic and electromagnetic wave characteristics to one another, including the respective effective wave speeds and attenuation coefficients, which facilitate multifunctional material design. This is achieved by deriving accurate formulas for the effective electromagnetic and elastodynamic properties that depend on the wavelengths of the incident waves and the microstructure via the spectral density. Our formulas enable us to explore the wave characteristics of a broad class of disordered microstructures because they apply, unlike conventional formulas, to a wide range of incident wavelengths (i.e., well beyond the long-wavelength regime). This capability enables us to study the dynamic properties of exotic disordered “hyperuniform” composites that can have advantages over crystalline ones, such as nearly optimal, direction-independent properties and robustness against defects. We specifically show that disordered “stealthy” hyperuniform microstructures exhibit novel wave characteristics (e.g., low-pass filters that transmit waves “isotropically” up to a finite wavenumber). Our cross-property relations for the effective wave characteristics can be applied to design multifunctional composites via inverse techniques. Design examples include structural components that require high stiffness and electromagnetic absorption; heat sinks for central processing units and sound-absorbing housings for motors that have to efficiently emit thermal radiation and suppress mechanical vibrations; and nondestructive evaluation of the elastic moduli of materials from the effective dielectric response.

A heterogeneous material (medium) consists of domains of multiple distinct materials (phases). Such materials are ubiquitous; examples include sedimentary rocks, particulate composites, colloidal suspensions, polymer blends, and concrete (1⇓⇓⇓⇓–6). When domain (inhomogeneity) length scales ℓ are much smaller than the system size, a heterogeneous material can be regarded as a homogeneous material with certain effective physical properties, such as thermal (electric) conductivity σe, dielectric tensor ϵe, or stiffness tensor Ce (1, 2, 5). Such effective properties depend on the phase properties, phase volume fractions ϕi, and higher-order microstructural information (1, 2, 5, 7⇓⇓⇓⇓–12). Heterogeneous materials are ideally suited to achieve multifunctionality since the best features of different materials can be combined to form a new material that has a broad spectrum of desired properties (1, 13⇓⇓⇓⇓⇓–19). Because the effective properties of a heterogeneous material reflect common morphological information, knowledge of one effective property can provide useful information about a different effective property (1, 13⇓⇓⇓⇓⇓–19). Such cross-property relations can aid in the rational design of multifunctional heterogeneous materials that possess multiple desirable effective properties (1, 13⇓⇓⇓⇓⇓–19) via inverse techniques (20).

All of the previous applications of cross-property relations for multifunctional design have focused on the static transport and elastic properties. Remarkably, however, nothing is known about analogous cross-property relations for various effective dynamic properties, each of which is of great interest in its own right. For example, in the case of propagation of electromagnetic waves in two-phase media, the key property of interest is the frequency-dependent dielectric constant, which is essential for a wide range of applications, including remote sensing of terrain (21), investigation of the microstructures of biological tissues (22), probing artificial materials (23), studying wave propagation through turbulent atmospheres (24), investigation of electrostatic resonances (25), and design of materials with desired optical properties (22, 26). An equally important dynamic situation occurs when elastic waves propagate through a heterogeneous medium, which is of great importance in geophysics (27, 28), exploration seismology (29), diagnostic sonography (30), crack diagnosis (31), architectural acoustics (32), and acoustic metamaterials (33).

Our study is motivated by the increasing demand for multifunctional composites with desirable wave characteristics for a specific bandwidth (i.e., a range of frequencies). Possible applications include sensors that detect changes in moisture content and water temperature (34), thin and flexible antennas (35), materials that efficiently convert acoustic waves into electrical energy (36), materials that can attenuate low-frequency sound waves and exhibit excellent mechanical strength (37), and materials with negative modulus in the presence of magnetic fields (38) (ref. 39 and references therein).

However, systematic design of multifunctional materials with desirable elastodynamic and electromagnetic properties has yet to be established. In this paper, we derive accurate microstructure-dependent formulas for the effective dynamic dielectric constant ϵe and effective dynamic bulk Ke and shear Ge moduli, each of which depends on the appropriate wavelength (or wavenumber) of the incident waves. We show that these formulas can accurately capture the dynamic responses of composites that are valid for a wide range of wavelengths (i.e., beyond the long-wavelength limitation of conventional approximations). From these effective properties, one can obtain the effective wave speed ce and attenuation coefficient γe for the electromagnetic waves and the analogous quantities ceL,T and γeL,T for the longitudinal and transverse elastic waves. From these formulas, we are able to derive cross-property relations that link electromagnetic and elastodynamic properties to one another. Such cross-property relations facilitate multifunctional design. Two striking multifunctional design applications are schematically illustrated in Fig. 1.

Fig. 1.
Fig. 1.

Schematics illustrating multifunctional applications of heterogeneous materials. Elastic and electromagnetic waves at two different wavenumbers (A) kI and (B) kII incident to, inside of, and transmitted from a composite material (a large ellipse) consisting of a matrix phase (shown in yellow) and a dispersed phase (shown in cyan). Parallel lines and sinusoidal curves represent elastic and electromagnetic waves, respectively. (A) For an elastic wave with a wavenumber kI, while the wave fronts inside this material experience microscopic disturbances, they effectively behave like a plane wave inside a homogeneous material with an effective wavenumber (ke)I=ωI[ceL+iγeL]−1 and effective elastic moduli Ke and Ge (definition [3]). Analogously, for an electromagnetic wave, this material behaves like a homogeneous material with an effective dielectric constant ϵe. For instance, both elastic and electromagnetic waves are attenuated due to scattering if this composite has a nonzero scattering intensity at kI (i.e., γeL<0 and Im[ϵe]>0). (B) For both types of waves (red) of a wavenumber kII, this composite can be effectively transparent if it has a zero scattering intensity at kII (i.e., γeL=0 and Im[ϵe]=0).

The challenge in deriving cross-property relations is that the effective properties depend on an infinite set of correlation functions (1, 2, 5, 7⇓⇓⇓⇓–12, 40). To derive the pertinent cross-property relations for the aforementioned effective wave characteristics, we rely on strong-contrast expansions (1, 9⇓⇓–12, 40). The strong-contrast formalism represents a very powerful theoretical approach to predict the effective properties of composites for any phase contrast and volume fraction (1, 9⇓⇓–12, 40). They are formally exact expansions with terms that involve functionals of the n-point correlation function Sn(i)x1,…,xn for all n and field quantities as well as a judicious choice of the expansion parameter (1, 9⇓⇓–12). Here, the quantity Sn(i)x1,…,xn gives the probability of finding n points at positions x1,…,xn simultaneously in phase i. Remarkably, the rapid convergence of strong-contrast expansions has enabled one to extract accurate estimates of the effective properties of a wide class of composites (dispersions of particles in a matrix) by approximately accounting for complete microstructural information. Specifically, higher-order functionals are approximated in terms of lower-order functionals (SI Appendix, section VI). Such microstructure-dependent approximations have been obtained for the effective static dielectric constant (9, 10), the effective static stiffness tensor (11), and the effective dynamic dielectric constant (12).

In the latter instance involving electromagnetic waves, the wavenumber-dependent effective dielectric constant ϵek1 for macroscopically isotropic two-phase composites depends on a functional involving the spectral density χ∼VQ (12). The quantity χ∼VQ is the Fourier transform of the autocovariance function χVr≡S2(i)r−ϕi2, where r≡x2−x1, and can be measured from scattering experiments (43). In principle, this approximation is valid only in the long-wavelength regime (i.e., k1ℓ≪1, where k1 is the wavenumber of the electromagnetic waves in the reference phase [phase 1]). However, we modify this formula in order to extend it to provide a better estimate down to the intermediate-wavelength regime (i.e., k1ℓ≲1) by accounting for spatial correlations of the incident plane waves (Eq. 4). This modified formula is superior to the commonly employed Maxwell–Garnett approximation (22, 44) that, unlike formula [4], fails to capture salient physics in correlated disordered systems (SI Appendix, section V). A capacity to accurately predict the effective dielectric constant is essential for the aforementioned applications (21⇓⇓⇓⇓–26).

To obtain analogous microstructure-dependent formulas for the effective dynamic elastic moduli KekL1 and GekL1, we utilize strong-contrast expansions for them that we have derived elsewhere that also apply in the long-wavelength regime. Here, kL1 is the wavenumber of longitudinal elastic waves. This dynamic formulation is considerably much more challenging mathematically than its dielectric counterpart (12) because of the complexity and nature of the fourth-rank tensor Green’s functions that are involved. In the present work, one primary objective is to extract from these expansions (Materials and Methods has the formal expansion) accurate approximate formulas that also depend on the spectral density χ∼VQ. As we did in the case of the dielectric formula, we modify these strong-contrast approximations for the effective dynamic moduli so that they are valid at the extended wavelengths (kL1ℓ≲1). In Results, we employ these modified formulas to investigate the effective elastic wave characteristics, including effective wave speeds ceL,T and attenuation coefficients γeL,T, for four models of disordered dispersions. Knowledge of the effective elastodynamic properties is of importance in the aforementioned disciplines and applications (27⇓⇓⇓⇓⇓–33).

We establish accurate cross-property relations linking the effective elastic and electromagnetic wave characteristics by using the aforementioned microstructure-dependent formulas and by eliminating the common microstructural parameter among them. Thus, these results enable one to determine the response of a composite to electromagnetic waves from the corresponding response to acoustic/elastic waves and vice versa. The resulting cross-property relations will have practical implications as discussed in Sound-Absorbing and Light-Transparent Materials and Conclusions and Discussion.

The primary applications that we have in mind are disordered microstructures, both exotic and “garden” varieties, because they can provide advantages over periodic ones with high crystallographic symmetries, which include perfect isotropy and robustness against defects. Such disordered media have recently been exploited for applications involving photonic bandgap materials (45, 46), gradient index photonic metamaterials (26), compact spectrometers (47), random lasers (48, 49), bone replacement (50, 51), and impact absorbers (52, 53).

We are particularly interested in studying the wave characteristics of exotic disordered two-phase media, such as disordered hyperuniform and/or stealthy ones, and their potential applications. Hyperuniform two-phase systems are characterized by anomalously suppressed volume-fraction fluctuations at long wavelengths (54⇓–56) such thatlimQ→0χ∼VQ=0,[1]where Q≡|Q| refers to a wavenumber. Such two-phase media encompass all periodic systems as well as exotic disordered ones (ref. 56 and references therein). These exotic disordered structures lie between liquids and crystals; they are like liquids in that they are statistically isotropic without any Bragg peaks and yet, behave like crystals in the manner in which they suppress the large-scale density fluctuations (54⇓–56). One special class of such structures is disordered stealthy hyperuniform media that are defined by zero scattering intensity for a set of wavevectors around the origin (57⇓⇓–60). Such materials are endowed with novel physical properties (26, 49, 60⇓⇓⇓–64), including that they are transparent (dissipationless) to electromagnetic waves down to a finite wavelength (12, 61). We also explore the wave characteristics of disordered stealthy nonhyperuniform media that possess zero scattering intensity for a set of wavevectors that do not include the origin (65).

In Conclusions and Discussion, we describe how our microstructure-dependent estimates enable one to design materials that have the targeted attenuation coefficients γeL,T for a range of wavenumbers (or equivalently, frequencies) via inverse-problem approaches (20). Using the stealthy hyperuniform materials, we explicitly demonstrate that such engineered materials can serve as filters for elastic waves, which selectively absorb (66, 67) or transmit (68, 69) waves “isotropically” for a prescribed range of wavenumbers. Furthermore, we show that we can engineer composites that exhibit anomalous dispersion (70), yielding resonance-like attenuation in γeL,T.

Preliminaries

We consider two-phase heterogeneous materials in d-dimensional Euclidean space Rd. For simplicity, the results reported here mainly focus on the case of d=3. We also make three assumptions on the phase dielectric properties (12): 1) the dielectric tensors of both phases are isotropic, 2) their dielectric constants are real-valued and independent of frequency, and 3) their magnetic permeabilities are identical.

The three analogous assumptions for the elastodynamic problem are 1) both phases are elastically isotropic, 2) their elastic moduli are real numbers independent of frequency, and 3) they have identical mass densities (ρ1=ρ2). The last assumption is achievable for many pairs of solid materials [e.g., nickel, copper, and cobalt have mass densities about 8.9 g/cm3, and tin and manganese have mass densities about 7.2 g/cm3 (71), but they have considerably different elastic moduli].

When these assumptions are met, inside each domain of phase p(=1,2), the elastic wave equation is given as (72)ω2ui+cLp2−cTp2∂2ul∂xi∂xl+cTp2∂2ui∂xl∂xl=0,where a displacement field oscillates sinusoidally with a frequency ω [i.e., uix,t=uixe−iωt], indices span integers between 1 and d, and the Einstein summation is implied. Here, cLp and cTp represent the longitudinal and transverse wave speeds* in phase p, respectively, and they are given ascLp2≡[Kp+2(1−1/d)Gp]/ρp,cTp2≡Gp/ρp,where Kp and Gp are the bulk modulus and the shear modulus of phase p, respectively. For a frequency ω, the corresponding longitudinal and transverse wavenumbers for elastic waves in phase p (=1, 2) are denoted bykLp≡ω/cLp and kTp≡ω/cTp,[2]respectively. Henceforth, we take “reference” and “polarized” phases to be phases 1 and 2, respectively (1, 11).

Formulas for the effective dielectric constant ϵek1 and the effective elastic moduli KekL1 and GekL1 also lead to estimates of the effective wave characteristics, including effective wave speeds ce and attenuation coefficient γe. For electromagnetic and elastic waves, the analogous quantities are given byce/c1+iγe/c1=ϵ1/ϵe,ceL+iγeL=[Ke+2(1−1/d)Ge]/ρe,ceT+iγeT=Ge/ρe,[3]where c1 is the wave speed of electromagnetic waves in the reference phase and ρe is the effective mass density (ρe=ρ1=ρ2). Note that, for the scaled attenuation coefficients γe/ce, γeL/ceL, and γeT/ceT, a quantity exp[−2πγe/ce] represents the factor by which the amplitude of the incident wave is attenuated for a period 2π/ω.

Results

We first derive the microstructure-dependent formulas for the effective dynamic dielectric constant, bulk modulus, and shear modulus that apply from the infinite wavelengths down to the intermediate wavelengths. Then, we use these estimates to establish cross-property relations between them by eliminating a common microstructural parameter among them. Using these formulas, we estimate the effective elastic wave characteristics and the cross-property relations for the four different three-dimensional (3D) models of disordered two-phase dispersions, including two typical nonhyperuniform ones. Finally, we discuss how to employ the established cross-property relations in designing multifunctional materials.

Microstructure-Dependent Approximation Formulas.

Effective dielectric constant.

We begin with the strong-contrast approximation formula for ϵek1 in the long-wavelength regime (k1ℓ≪1) derived by Rechtsman and Torquato (12). Here, we modify this approximation so that it becomes valid down to intermediate wavelengths (k1ℓ≲1).

For macroscopically isotropic media, this formula depends on a functional A2Q involving χ∼VQ (12):ϵek1ϵ1=1−dβϕ22A2k1β+ϕ2(βϕ2−1),[4]where k1 is the wavenumber of the electromagnetic waves in the reference phase (phase 1), d is the dimension, β≡(ϵ2−ϵ1)/[ϵ2+(d−1)ϵ1] is the dielectric “polarizability,” and ϵ1 and ϵ2 are the dielectric constants of phases 1 and 2, respectively. Here, we modify the functional A2Q stated in ref. 12 by including contribution from spatial distribution of incident electric fields [e.g., exp(ik1x)]:A2Q=−(d−1)π2d/2Γd/2FQ,[5]where FQ is what we call the attenuation function (Eq. 17). Physically, the attenuation function FQ incorporates the contributions from all diffracted waves due to single, elastic scattering events when the wavenumber of the incident waves is Q. The reader is referred to Materials and Methods and SI Appendix, section IV for a derivation of FQ. Numerical simulations of ϵek1 reported in SI Appendix, section V validate the high predictive power of Eqs. 4 and 5 for a wide range of incident wavelengths, which popular approximation schemes (22, 44) cannot predict (SI Appendix, Fig. S2B).

For statistically isotropic media in three dimensions, the attenuation function can be rewritten asIm[FQ]=−Q2(2π)3/2∫02Qqχ∼Vqⅆq,[6]Re[FQ]=−2Q2πp.v.∫0∞ⅆq1q(Q2−q2)Im[Fq],[7]where p.v. stands for the Cauchy principal value. We elaborate on how to compute the attenuation function in SI Appendix, section IV.

Effective elastic moduli.

We extract the approximations for KekL1 and GekL1 in the long-wavelength regime (kL1ℓ≪1) from the strong-contrast expansions (Eqs. 19 and 20) at the two-point level:KekL1K1=1−κϕ22[1+2(1−1/d)G1/K1]C2kL1+ϕ2κϕ2−1,[8]GekL1G1=1−(d+2)μϕ22[1+2(1−1/d)G1/K1]2(1+2G1/K1)D2kL1+ϕ2(μϕ2−1),[9]where C2kL1 and D2kL1 are microstructural parameters that depend on kL1 because kT1=(cL1/cT1)kL1. These strong-contrast approximations are also valid in the long-wavelength regime (kL1ℓ≪1). However, we modify them so that they are valid in the intermediate-wavelength regime (kL1ℓ≲1) by using the following modified forms of the microstructure-dependent parameters:C2kL1=π2d/2Γd/2FkL1κ,[10]D2kL1=π2d/2Γd/2dcL12FkT1+2cT12FkL1dcL12+2cT12μ,[11]where FQ is the attenuation function defined in Eq. 17. The reader is referred to Materials and Methods for a derivation of these relations. Computer simulations reported in SI Appendix, section V verify that these modified formulas accurately predict microstructure dependence of KekL1 and GekL1 down to intermediate wavelengths, where the conventional approximation schemes (73) are no longer valid.

Note that the approximations (Eqs. 8 and 9) are conveniently written in terms of the wavenumber kL1 associated with the reference phase. Furthermore, this wavenumber is directly proportional to the frequency ω (Eq. 2) and more suitable to describe microstructural information rather than the temporal quantity ω. For these reasons, we henceforth use the longitudinal wavenumber kL1 instead of ω or kT1 as an independent variable for the effective elastic properties.

Static limit.

In the long-wavelength limit (Q→0), the attenuation function vanishes as FQ∼O(Q2), which enables us to recover the previous corresponding static results (9, 11) in which the effective properties are real-valued and identical to the Hashin–Shtrikman bounds on ϵe, Ke, and Ge (1, 2, 74). Note that these static limits are also identical to the static Maxwell–Garnet approximations (SI Appendix, section V).

Long-wavelength regime.

The effective dynamic properties ϵek1 , KekL1, and GekL1 are generally complex-valued, implying that the associated waves propagating through this medium are attenuated. Such attenuation occurs due to scattering from the inhomogeneity even when both phases are dissipationless (i.e., real-valued as we assume throughout this study).

In the long-wavelength regime (Qℓ≪1), we now demonstrate that attenuation is stronger in a nonhyperuniform medium than in a hyperuniform one by comparing their leading-order terms of the effective attenuation coefficients. The imaginary parts of effective properties and importantly, the associated effective attenuation coefficients γe are approximately proportional to Im[FQ], which is easily evaluated from the spectral density by using Eq. 6. When the spectral density follows the power law form χ∼V(Q)≈a1+a2Qα, where for nonhyperuniform media a1>0 and hyperuniform media a1=0,a2>0 and α>0, γe in the limit Q→0 is given byγe∼Q3,nonhyperuniform,Q3+α,hyperuniform.Thus, hyperuniform media are less dissipative than nonhyperuniform media due to the complete suppression (in the former) of single scattering events in the long-wavelength limit. Note that, for stealthy hyperuniform media [χ∼VQ=0 for 0<Q<Qupper], which is the strongest form of hyperuniformity, γe=0 in the same limit; Transparency Conditions has details.

Transparency conditions.

Our formulas (Eqs. 8 and 9) predict that heterogeneous media can be transparent for elastic waves [γeL,TkL1=0] ifIm[FkL1]=Im[FkT1]=0.[12]Physically, these conditions imply that single scattering events of elastic waves at the corresponding frequency are completely suppressed. For stealthy hyperuniform media that satisfy χ∼VQ=0 in 0<Q<Qupper, the transparency conditions [12] are simply given as0<kL1a<(cT1/cL1)Quppera/2,where cT1/cL1=(1−2ν1)/[2(1−ν1)] and ν1 is the Poisson ratio of phase 1. For electromagnetic waves, the condition [12] is simplified as Im[Fk1]=0. Thus, the aforementioned stealthy hyperuniform media are transparent to the electromagnetic waves (γe=0) for 0<k1a<Quppera/2 (SI Appendix, section V). We will make use of these interesting properties in Effective Elastic Wave Characteristics and Conclusions and Discussion.

Models of Dispersions.

We investigate four different 3D models of disordered dispersions of identical spheres of radius a with ϕ2=0.25. These models include two typical disordered ones (overlapping spheres and equilibrium hard spheres) and two exotic disordered ones (stealthy hyperuniform dispersions and stealthy nonhyperuniform dispersions).

Overlapping spheres are systems composed of spheres with centers that are spatially uncorrelated (1). At ϕ2=0.25, this model will not form macroscopically large clusters since it is well below the percolation threshold ϕ2≈0.29 (75). To compute its attenuation function FQ , we use the closed form expression of its χVr given in ref. 1 and SI Appendix, section II.

Equilibrium hard spheres are systems of nonoverlapping spheres in the canonical ensemble (76). To evaluate its FQ , we use the spectral density χ∼VQ that is obtained from Eq. 25 and the Percus–Yevick approximation (77) (SI Appendix, section II).

Stealthy hyperuniform dispersions are defined by χ∼VQ=0 for 0<|Q|≤Qupper. We numerically generate them via the collective-coordinate optimization technique (57, 59, 65); Materials and Methods has details. Each of these obtained systems consists of N=103 spheres and satisfies χ∼VQ=0 for Qa<1.5. In order to compute FQ , we obtain χ∼VQ numerically from Eq. 25 (Materials and Methods and SI Appendix, section III).

Stealthy nonhyperuniform dispersions are defined by χ∼VQ=0 for Qlower<Q<Qupper. We numerically find realizations of these systems [N=103 and χ∼VQ=0 for 1.0<Qa<1.5] via the collective-coordinate optimization technique (57, 59, 65) (Materials and Methods). Its spectral density is obtained in the same manner as we did for the stealthy hyperuniform dispersions.

Values of the complex-valued attenuation function FQ (Eqs. 6 and 7) for the four aforementioned models of dispersions are presented in Fig. 2. Their imaginary parts are directly obtained from the spectral density based on Eq. 6. The associated real parts are then computed from an approximation of Eq. 7 (SI Appendix, section IV). For various types of dispersions, while the values of χ∼VQ up to intermediate wavenumbers (Qa≲3) are considerably different from one another, all of the curves approximately collapse onto a single curve for Qa≫5 (Fig. 2A).

Fig. 2.
Fig. 2.

Evaluation of χ∼V(Q) (A) and the attenuation function F(Q) (B and C) for four different models of 3D dispersions: stealthy hyperuniform dispersions, stealthy nonhyperuniform dispersions, overlapping spheres, and equilibrium hard spheres. B, Inset is a magnification of the larger panel. Since all of these dispersions are composed of spheres of radius a, their microstructures resemble one another at the small length scales r<a, indeed, all of the spectral densities effectively collapse onto a single curve for Qa≫5.

Effective Elastic Wave Characteristics.

We now investigate the aforementioned effective elastic wave characteristics of four different models of 3D dispersions using the strong-contrast approximations (Eqs. 8 and 9). In striking contrast to the other models, stealthy hyperuniform dispersions are transparent for both longitudinal and transverse elastic waves down to a finite wavelength. This result clearly demonstrates that it is possible to design disordered composites that exhibit nontrivial attenuation behaviors by appropriately manipulating their spatial correlations.

We first determine phase elastic moduli of the aforementioned four models of composites. Since this parameter space of phase moduli is infinite, we consider two extreme cases: a compressible matrix phase (phase 1) with a Poisson ratio ν1=1/3 that contains a rigid dispersed phase (phase 2; i.e., K2/K1=G2/G1=∞) (Fig. 3) and a compressible matrix with ν1=−0.1 that contains cavities (i.e., K2/K1=G2/G1=0) (SI Appendix, section VI). Investigating these two extreme cases will still provide useful insight into the wave characteristics in intermediate regimes of phase moduli. While the Poisson ratio of the compressible matrix phase can take any value in the allowable interval of −1≤ν1<1/2, we examine two different values of ν1=1/3 (typical of many materials) and −0.1. Negative Poisson ratio (“auxetic”) materials laterally dilate (contract) in response to axial elongation (contraction) (78). While we present the estimated effective elastic moduli up to kL1a=3, our approximations are, in principle, valid down to the intermediate-wavelength regime (kL1a<1.5).

Fig. 3.
Fig. 3.

Estimates of scaled effective elastic wave characteristics for 3D dispersions of rigid spheres of radius a in a compressible matrix phase with Poisson ratio ν1=1/3 (i.e., K2/K1=G2/G1=∞). Here, kL1 is the wavenumber of longitudinal waves in reference phase (phase 1), and cL1 and cT1 are elastic wave speeds of longitudinal and transverse waves, respectively, in phase 1. (A and C) Effective wave speeds and (B and D) effective attenuation coefficients are plotted in terms of kL1a. B, Inset and D, Inset are magnifications of B and D, respectively. In B, Inset and D, Inset, we see that stealthy hyperuniform dispersions have a zero γeL,S regime in kL1a<0.375, but others do not, which vividly demonstrates that γeL,S can be engineered by the spatial correlations of composites.

We estimate the scaled effective wave propagation properties of the models of 3D dispersions considered in Fig. 2. For each of the aforementioned cases of phase properties, four different models have similar effective wave speeds but significantly different attenuation coefficients. Instead, ceLkL1 and ceTkL1 depend largely on the phase properties. For rigid dispersed phase (Fig. 3), the effective wave speeds ceL,T are generally faster than those in phase 1 but tend to decrease with kL1 at most frequencies. By contrast, when the dispersed phase consists of cavities, the wave speeds are slower than those in phase 1 and increase with kL1a from kL1a≈1 (SI Appendix, section VI).

In both cases shown in Fig. 3 and SI Appendix, Fig. S5, stealthy hyperuniform dispersions are transparent to both longitudinal and transverse waves in 0<kL1a≲0.4 as predicted in Eq. 12. Such composites can be employed to design of low-pass filters for elastic as well as electromagnetic waves. By contrast, the stealthy nonhyperuniform dispersions do not attain zero attenuation at any finite wavelength because these systems can suppress scatterings at only specific directions.

Cross-Property Relations.

It is desired to design composites with prescribed elastic and electromagnetic wave characteristics as schematically illustrated in Fig. 1. The rational design of such multifunctional characteristics can be greatly facilitated via the use of cross-property relations for these different effective properties, which we derive here.

We first obtain a cross-property relation between the effective dynamic bulk modulus and effective dynamic dielectric constant from Eqs. 4 and 8 by eliminating FQ between them:KekL1K1=1+6βκν1−1ϕ2ν1+1×ϵekL1/ϵ1−12β+κ+3βκϕ2−2β−κϵekL1/ϵ1,[13]where the effective properties Ke and ϵe must be at the same wavenumber (i.e., kL1=k1) but possibly at different frequencies as illustrated in Fig. 1. Remarkably, this cross-property relation depends only on the phase properties, regardless of the microstructures of composites. Intuitively speaking, such cross-property relations can be established because the effective properties depend on the interference pattern of the associated waves, which are commonly determined by wavelengths and microstructures.

The real and imaginary parts of this cross-property relation (Eq. 13) are separately represented in Fig. 4 for the four models of 3D dispersions considered in Fig. 3. The surface plots in Fig. 4, Left depict the hypersurface on which any possible pairs of ϵek1 and KekL1=k1 of a composite must lie when its phase properties and ϕ2 are prescribed. The black dotted lines in Fig. 4, Upper and Lower are contour lines of Re[Ke/K1] and Im[Ke/K1] at level spacing 0.1, respectively. Fig. 4, Right represents the top views of the associated surface plots on Fig. 4, Left. We note that the resulting surface plots have a simple pole at ϵe=ϵpole with position that is determined by phase properties and packing fraction ϕ2 (SI Appendix, Figs. S6 and S7). In Fig. 4, the locus of points (shown with solid lines) depicts the effective dielectric constants and bulk moduli of the four different models of 3D dispersions as a dimensionless wavenumber k1a(=kL1a) varies from zero to five along with the arrows. Since these points should lie on the surfaces as depicted in Fig. 4, one can indirectly determine the wavenumber-dependent Ke by measuring ϵe at different wavenumbers (or frequencies) and vice versa.

Fig. 4.
Fig. 4.

Cross-property relation between the effective dielectric constant ϵe and the effective bulk modulus Ke for the four models of 3D dispersions with ϕ2=0.25, each of which consists of a compressible matrix with ν=1/3 and incompressible inclusions (Fig. 3), and the ratio of phase dielectric constants is ϵ2/ϵ1=10. (Left) Surface plots, evaluated from Eq. 13, represent the surface on which ϵe and the real (Upper) and imaginary (Lower) parts of Ke of any composites with the prescribe phase properties must lie. Contour lines (black dotted lines) are at level spacing 0.1. (Right) Contour plots are the top views of the surface plots in Left. We note that the surface plots for other values of the Poisson ratio ν1 and the contrast ratio ϵ2/ϵ1 are qualitatively similar except for the position of a simple pole (SI Appendix, Figs. S6 and S7). On these surfaces, the locus of points (solid lines with/without markers) represents effective properties of each dispersion model as the wavenumber k1 increases from k1a=0 to k1a=5 (in the directions of arrows). Note that all of the models attain the Hashin–Shtrikman bounds (blue stars) at k1a=0.

Similarly, we can obtain cross-property relations that links ϵe to Ge or Ge to Ke. The former case is explicitly given asGekL1G1=1−15(ν1−1)ϕ2μ×(5ν1−4)ϕ2(1+2μ)−1+2ββ+3ϕ23cL12+2cT12×  3cL12ϵecL1kL1/cT1/ϵ1−1+2cT12ϵekL1/ϵ1−1−1,[14]which depends on values of the effective dielectric constant ϵe at two different wavenumbers k1=kL1 and cL1kL1/cT1, making it difficult to graphically depict this cross-property relation. Instead, we list in Table 1 values of Ge that are computed from both Eqs. 9 and 14. Furthermore, by combining Eqs. 13 and 14, one can also establish cross-property relations that link the effective dielectric constant to the effective elastic wave characteristics (i.e., ceL,T and γeL,T).

View this table:
Table 1.

Evaluation of the effective shear moduli Ge/G1 for dispersions of spheres of radius a from the strong-contrast approximation (Eq. 9) and the cross-property relation (Eq. 14)

Sound-Absorbing and Light-Transparent Materials.

To illustrate how our results can be applied for novel multifunctional material design, we engineer composites that are transparent to electromagnetic waves at infrared wavelengths (long wavelengths) but absorb sound at certain frequencies. Importantly, designing such materials is not possible by using standard approximations (22, 44, 73) and quasistatic cross-property relations (1, 13⇓⇓⇓⇓⇓–19). Such engineered materials could be used as heat sinks for central processing units (CPUs) and other electrical devices subject to vibrations or sound-absorbing housings (39). A similar procedure can be applied to design composites for exterior components of spacecraft (79) and building materials (80). We will further discuss possible applications in Conclusions and Discussion.

We take advantage of the fact that stealthy hyperuniform dispersions are transparent down to a finite wavelength (k1≤Qupper/2) (Fig. 5). We then find polarizabilities β and κ that result in high attenuation coefficient γeL at kL1a=k′a≡1.3. This is achieved when ϵek′ is close to the simple pole ϵpole of Eq. 13 (Fig. 5A). Fig. 5B shows the cross-property relation [13] with the chosen polarizabilities (i.e., β=0.17 and κ=−12). Phase properties corresponding to these polarizabilities are degenerate, and we choose ϵ2/ϵ1≈1.63, ν1=23/49, and K2=G2=0. We see from Fig. 5 C and D that the resulting materials are indeed transparent to electromagnetic waves at long wavelengths but exhibit resonance-like attenuation of sound at kL1a≈1.0.

Fig. 5.
Fig. 5.

Multifunctional design of materials that are transparent at infrared wavelengths but absorb sound at certain acoustic frequencies. In order to attain such materials, we exploit exotic 3D stealthy hyperuniform dispersions. (A) Contour plot of the distance between the real part of the effective dielectric constant at a target frequency (the associated wavenumber is k′a=1.3) and the simple pole ϵpole as a function of the dielectric β and bulk modulus κ polarizabilities. We choose β=0.17 and κ=−12 (shown as a blue disk). (B) The cross-property relation between ϵe/ϵ1 and Im[Ke/K1] for the chosen parameters β=0.17 and κ=−12. The effective static dielectric constant is shown as a blue star. The effective dielectric constant at the target frequency and the pole are shown as the black and red circles, respectively. The associated composite consists of a nearly incompressible matrix phase with ν1=23/49 containing cavities (K2=G2=0), and its contrast ratio of the dielectric constants is ϵ2/ϵ1≈1.63. (C) The imaginary part of the resulting effective dynamic dielectric constant as a function of the wavenumber k1a. (D) The resulting scaled attenuation coefficient for the longitudinal elastic waves as a function of the longitudinal wavenumber kL1.

Conclusions and Discussion

We have obtained accurate approximations for the effective dynamic dielectric constant ϵe and the effective dynamic elastic moduli Ke and Ge of two-phase composites that depend on the microstructure via the spectral density χ∼VQ , which is easily computed theoretically/computationally or accessible via scattering experiments (Eqs. 4, 8, and 9). These formulas are superior in predicting these effective dynamic properties compared with commonly used approximations, such as Maxwell–Garnett and quasicrystalline approximations (22, 44, 73), as verified by computer simulations (SI Appendix, section V). Unlike these conventional approximations, our formulas are accurate for a wide range of incident wavelengths for a broad class of dispersions.

Using the approximations [4], [8], and [9], we have shown that hyperuniform composites can have desirable attenuation properties for both electromagnetic and elastic waves. We analytically showed that hyperuniform media are less dissipative than nonhyperuniform ones in Long-wavelength regime. Remarkably, stealthy hyperuniform media are dissipationless (i.e., γe = 0) down to a finite wavelength as shown in Fig. 3 and SI Appendix, Figs. S2 and S5. Such composites can be employed to low-pass filters for elastic and electromagnetic waves.

Using Eqs. 4, 8, and 9, we also established cross-property relations [13] and [14] that link the effective dynamic dielectric constant ϵek1 to the effective dynamic bulk modulus KekL1 and shear modulus GekL1, respectively. Thus, when it is difficult to directly measure Ke or Ge, they can be indirectly evaluated from these cross-property relations by measuring the wavenumber-dependent dielectric constants (81, 82) as demonstrated in Fig. 4 and Table 1 and vice versa. For example, one can use them to indirectly determine physical/chemical properties for construction materials (81, 82) and oil exploration (83).

Our cross-property relations also have important practical implications for the rational design of multifunctional composites (1, 15⇓⇓⇓–19, 39) that have the desired dielectric properties for a particular range of electromagnetic wavelengths and elastic properties for a certain range of elastodynamic wavelengths. The validation of our formulas via computer simulations justifies their use for the design of novel multifunctional materials without having to perform full-blown simulations. In particular, we described how to engineer a sound-absorbing composite that is transparent to light via our cross-property relations, which again, could not be done using previous approximation formulas (1, 13⇓⇓⇓⇓⇓–19, 22, 44, 73). This is done by exploiting the exotic structural properties of stealthy hyperuniform dispersions (Fig. 5). Such engineered materials could be used as heat sinks for CPUs and other electrical devices subject to vibrations because they enable radiative cooling while suppressing prescribed mechanical vibrations. Another application is a sound-absorbing housing for an engine or a motor, which can efficiently convert cyclic noise into electric energy (39) and allow radiative cooling. It is natural to extend to the aerospace industry, where low-frequency engine noise is prevalent (39). A similar procedure can be applied to design composites with high stiffness that absorb electromagnetic waves at certain wavenumbers for use as exterior components of spacecraft (79) and building materials (80).

With the aid of our microstructure-dependent formulas (Eqs. 4, 8, and 9), one can employ inverse-design approaches (20) to design composites. We recall that inverse-design approaches enable one to prescribe the effective properties of composites and then, find the microstructures that achieve them. For example, one would first prescribe the material phases and then, compute the desired effective properties (say, attenuation coefficients γe for a given bandwidth) via the microstructure-dependent formulas. Then, one backs out the corresponding spectral density from the attenuation function, which would correspond to a particular microstructure if realizable. The latter can be constructed by using previously established Fourier-space construction techniques (57, 59, 60, 65). Finally, one can generate simulated microstructures via modern fabrication methods, such as 3D printing (84) or two-dimensional photolithographic technologies (85). The same inverse techniques also can be employed to design multifunctional composites using the cross-property relations. Remarkably, such inverse approaches were not possible with previously known approximations, such as Maxwell–Garnett and quasicrystalline formulas (22, 44, 73), because they are independent of microstructures.

It is instructive to briefly discuss how to measure the wavenumber-dependent effective properties in experiments. Here, for brevity, we focus on the dielectric constants because the same reasoning applies to the elastodynamic case (Eqs. 8 and 9). Clearly, the property ϵek1 is identical to the frequency-dependent one because a wavenumber k1 in the reference phase can be converted to a frequency ω via the dispersion relation of the reference phase [i.e., ω=ωk1]. The frequency-dependent effective dielectric constant ϵeω can be measured via conventional techniques, such as perturbation methods (measuring changes in a resonance frequency of a resonator due to a specimen) or transmission techniques (measuring the transmission/reflection by a specimen) (ref. 86 and references therein). However, when using cross-property relations, it is crucial to covert the independent variable of the effective properties from frequency ω to the associated wavenumbers (k1 and kL1) according to the dispersion relations of the reference phase.

While we primarily focused on 3D two-phase media, our microstructure-dependent formulas (Eqs. 4, 8, and 9) are valid for d≥2. Furthermore, the cross-property relations (Eqs. 13 and 14) can be extended to any dimension d with minor modifications.

Based on a previous study on the static case (87), it is relatively straightforward to generalize our microstructure-dependent formulas to composites whose dispersed phase is a piezoelectric (i.e., mechanical stress can induce an electric voltage in the solid material). Such extensions can be profitably utilized in the optimal design of materials for elastic wave energy harvesting to power small electrical devices (33).

Materials and Methods

Derivation of Eq. 4.

We begin with the original expression of the microstructure-dependent parameter A2Q in the long-wavelength regime derived in ref. 12:A2Q=(d−1)Q2∫i4Q2πrd/2−1Hd/2−1(1)QrχVrⅆr≡−(d−1)π2d/2Γd/2FQ,[15]where Q is a wavenumber, Γx is the gamma function, and Hν(1)x is the Hankel function of the first kind of order ν. Here, the function i/4(Q/2πr)d/2−1Hd/2−1(1)Qr in the integrand is the Green’s function of the Helmholtz equation characterized by a wavenumber Q in d-dimensional Euclidean space. Note that the imaginary part of FQ can be simplified asIm[FQ]=−Qdχ∼VQ/(2π)d/2.[16]The reader is referred to SI Appendix, section IV for discussion about physical interpretation of this quantity.

In order to extend the range of applicable wavelengths, we modify the microstructure-dependent parameter A2Q to account for the spatial variation of the (external) incident waves as in the Born approximation (70) (Eq. 5). The attenuation function FQ in Eq. 5 is defined asFQ=−iΓd/22πd/2Qd/2+1∫Hd/2−1(1)Qrrd/2−1e−iQk^⋅rχVrⅆr[17]=−Γd/22d/2πd+1Q2∫χ∼Vq|q+Qk^|2−Q2ⅆq,[18]where k^ is the unit wavevector in the direction of the incident waves. Eq. 18 is obtained by applying the Parseval theorem to Eq. 17. Importantly, comparison of the modified attenuation function (Eq. 17) to Eq. 15 reveals that the former has an additional factor exp(−iQk^⋅r) in its integrand, which accounts for the spatial variation of the incident waves. This change enables us to include the contributions from all scattered waves at wavevectors q+Qk^ (Eq. 18). We note that the modified approximation [4] with the attenuation function [18] shows excellent agreement with numerical simulations (SI Appendix, section V).

Dynamic Strong-Contrast Expansions for the Effective Elastic Moduli.

Elsewhere, we derived exact strong-contrast expansions for these moduli through all orders in the “polarizabilities.” These expansions are, in principle, valid in the long-wavelength regime. Here, it suffices to present their general functional forms when the effective stiffness tensor is isotropic:Ke−K1Ke+2(d−1)G1/d=ϕ22κϕ2−∑n=2∞Cn,[19]Ge−G1Ge+[dK1/2+(d+1)(d−2)G1/d]G1K1+2G1=ϕ22μϕ2−∑n=2∞Dn,[20]where Cn and Dn are functionals involving the n-point probability functions and double gradients of the appropriate Green’s functions andκ=K2−K1K2+2(d−1)G1/d,[21]μ=G2−G1G2+dK1/2+(d+1)(d−2)G1/dG1/(K1+2G1)[22]are the polarizabilities (strong-contrast parameters) for bulk and shear moduli, respectively. This expansion is the dynamic analog of the static strong-contrast expansion derived by Torquato (11), which can be viewed as one that perturbs around the Hashin–Shtrikman structures (74) that attain the best possible bounds on the effective elastic moduli of the isotropic composites for prescribed phase properties and volume fractions. In the family of such structures, domains of one phase are topologically disconnected, well separated from one another, and dispersed throughout a connected (continuous) matrix phase (88, 89). This means that strong-contrast expansions will converge rapidly, even for high-contrast ratios in phase moduli (SI Appendix, section VII.A), for dispersions that meet the aforementioned conditions (12), and hence, the resulting property estimates will be nearly optimal.

Derivation of Eqs. 8 and 9.

The original strong-contrast approximations [formally identical to Eqs. 8 and 9] depend on the following two-point parameters C2kL1 and D2kL1 are:C2kL1=π2d/2Γd/2FkL1κ,[23]D2kL1=π2d/2Γd/2dcL12FkT1+2cT12FkL1dcL12+2cT12μ,[24]where FQ is defined in Eq. 15. In order to obtain better estimates of KekL1 and GekL1 in the intermediate-wavelength regime, we modify the strong-contrast approximations in the same manner as we did for the electromagnetic problem. Specifically, we replace FQ in Eqs. 23 and 24 with FQ defined by Eq. 18, which leads to Eqs. 10 and 11. The justification for such replacements is based on two observations: 1) FQ involves the Green’s function of the Helmholtz equation at the wavenumber Q, and 2) the associated incident wave should have the wavenumber Q. We note that the modified formulas (Eqs. 8 and 9) show excellent agreement with numerical simulations (SI Appendix, section V).

Spectral Density.

For dispersions of nonoverlapping identical spheres of radius a, the spectral density can be expressed as (1, 9)χ∼VQ=ϕ2α∼2Q;aSQ,[25]where ϕ2 is the packing fraction, Q≡|Q|, α∼2Q;a≡2dπd/2Γ1+d/2[Jd/2Qa]2/Qd, and Jνx is the Bessel function of the first kind of order ν. Here, SQ is the structure factor for particle centers, which can be computed fromSQ=N−1|∑j=1Nexp−iQ⋅rj|2.Therefore, one can easily obtain stealthy hyperuniform and stealthy nonhyperuniform dispersions from the associated point patterns by decorating their point centers with nonoverlapping equal-sized spheres. More details are in SI Appendix, section I.

Stealthy Hyperuniform/Nonhyperuniform Hard Spheres.

We first generate stealthy point configurations in periodic simulation boxes via the collective-coordinate optimization technique (57, 59, 65) (i.e., numerical procedures that obtain ground-state configurations for the following potential energy):ΦrN=1vF∑Qv∼QSQ+∑i<jurij,[26]wherev∼Q=1, Qlower<|Q|≤Qupper0, otherwiseand a soft-core repulsive term (90)ur=(1−r/σ)2, r<σ0, otherwise.These ground-state configurations are stealthy [i.e., SQ=0 for Qlower<|Q|<Qupper] and due to the soft-core repulsions ur , and the interparticle distances are larger than σ. The resulting point configurations are stealthy hyperuniform if Qlower=0 but otherwise, stealthy nonhyperuniform. We then circumscribe the points by identical nonoverlapping spheres of radius a<σ/2. The reader is referred to SI Appendix, section III for details.

Data Availability.

There are no data associated with the manuscript.

Acknowledgments

We acknowledge the support of Air Force Office of Scientific Research Program on Mechanics of Multifunctional Materials and Microsystems Award FA9550-18-1-0514.

Footnotes

  • ↵1J.K. and S.T. contributed equally to this work.

  • ↵2To whom correspondence may be addressed. Email: torquato{at}princeton.edu.

  • Author contributions: S.T. designed research; J.K. and S.T. performed research; J.K. and S.T. analyzed data; and J.K. and S.T. wrote the paper.

  • The authors declare no competing interest.

  • This article is a PNAS Direct Submission.

  • ↵*Henceforth, “wave speeds” always refer to the phase speeds because the term “phase” is reserved for a constituent material in this paper.

  • This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1914086117/-/DCSupplemental.

Published under the PNAS license.

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Multifunctional composites for elastic and electromagnetic wave propagation
Jaeuk Kim, Salvatore Torquato
Proceedings of the National Academy of Sciences Apr 2020, 117 (16) 8764-8774; DOI: 10.1073/pnas.1914086117
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Proceedings of the National Academy of Sciences: 117 (16)
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