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# Multifunctional composites for elastic and electromagnetic wave propagation

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

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

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 *n* and field quantities as well as a judicious choice of the expansion parameter (1, 9⇓⇓–12). Here, the quantity *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 **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 *Materials and Methods* has the formal expansion) accurate approximate formulas that also depend on the spectral density *Results*, we employ these modified formulas to investigate the effective elastic wave characteristics, including effective wave speeds

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 that

In *Conclusions and Discussion*, we describe how our microstructure-dependent estimates enable one to design materials that have the targeted attenuation coefficients

## Preliminaries

We consider two-phase heterogeneous materials in d-dimensional Euclidean space

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 (

When these assumptions are met, inside each domain of phase

Formulas for the effective dielectric constant

## 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

For macroscopically isotropic media, this formula depends on a functional **17**). Physically, the attenuation function *Materials and Methods* and *SI Appendix*, section IV for a derivation of *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. S2*B*).

For statistically isotropic media in three dimensions, the attenuation function can be rewritten as*SI Appendix*, section IV.

*Effective elastic moduli*.

We extract the approximations for **19** and **20**) at the two-point level:**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

Note that the approximations (Eqs. **8** and **9**) are conveniently written in terms of the wavenumber **2**) and more suitable to describe microstructural information rather than the temporal quantity ω. For these reasons, we henceforth use the longitudinal wavenumber

*Static limit*.

In the long-wavelength limit (*SI Appendix*, section V).

*Long-wavelength regime*.

The effective dynamic properties

In the long-wavelength regime (**6**. When the spectral density follows the power law form *Transparency Conditions* has details.

*Transparency conditions*.

Our formulas (Eqs. **8** and **9**) predict that heterogeneous media can be transparent for elastic waves [**12**] are simply given as**12**] is simplified as *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

Overlapping spheres are systems composed of spheres with centers that are spatially uncorrelated (1). At *SI Appendix*, section II.

Equilibrium hard spheres are systems of nonoverlapping spheres in the canonical ensemble (76). To evaluate its **25** and the Percus–Yevick approximation (77) (*SI Appendix*, section II).

Stealthy hyperuniform dispersions are defined by *Materials and Methods* has details. Each of these obtained systems consists of **25** (*Materials and Methods* and *SI Appendix*, section III).

Stealthy nonhyperuniform dispersions are defined by *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 **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 *A*).

### 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 *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

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, *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 **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

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 *Upper* and *Lower* are contour lines of *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 *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

Similarly, we can obtain cross-property relations that links **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.,

### 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 (**13** (Fig. 5*A*). Fig. 5*B* shows the cross-property relation [**13**] with the chosen polarizabilities (i.e., *C* and *D* that the resulting materials are indeed transparent to electromagnetic waves at long wavelengths but exhibit resonance-like attenuation of sound at

## Conclusions and Discussion

We have obtained accurate approximations for the effective dynamic dielectric constant **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., *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

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

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

While we primarily focused on 3D two-phase media, our microstructure-dependent formulas (Eqs. **4**, **8**, and **9**) are valid for **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 *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 **5**). The attenuation function **5** is defined as**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 **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:*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 **15**. In order to obtain better estimates of **23** and **24** with **18**, which leads to Eqs. **10** and **11**. The justification for such replacements is based on two observations: 1) **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)*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):*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

↵

^{1}J.K. and S.T. contributed equally to this work.- ↵
^{2}To 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.

## References

- ↵
- S. Torquato

- ↵
- G. W. Milton

- ↵
- T. I. Zohdi

- ↵
- A. M. Neville

- ↵
- M. Sahimi

- ↵
- ↵
- M. J. Beran

- ↵
- P. H. Dederichs,
- R. Zeller

- ↵
- ↵
- A. K. Sen,
- S. Torquato

- ↵
- S. Torquato

- ↵
- M. C. Rechtsman,
- S. Torquato

- ↵
- ↵
- ↵
- ↵
- S. Torquato,
- D. Chen

- ↵
- ↵
- ↵
- Y. Wang,
- Z. Luo,
- N. Zhang,
- Q. Qin

- ↵
- ↵
- L. K. Tsang

- ↵
- A. Sihvola

- ↵
- N. P. Zhuck

- ↵
- V. I. Tatarskii

- ↵
- R. C. McPhedran,
- D. R. McKenzie

- ↵
- B. Y. Wu,
- X. Q. Sheng,
- Y. Hao

- ↵
- ↵
- G. T. Kuster,
- M. N. Toksöz

- ↵
- R. E. Sheriff,
- L. P. Geldart

- ↵
- ↵
- A. Sutin,
- V. Nazarov

- ↵
- F. R. Watson

- ↵
- M. Yuan,
- Z. Cao,
- J. Luo,
- X. Chou

- ↵
- E. Ekmekci,
- G. Turhan-Sayan

- ↵
- U. Ali et al.

- ↵
- ↵
- Y. Tang et al.

*Sci. Rep.***7**, 44972 (2017). - ↵
- K. Yu,
- N. X. Fang,
- G. Huang,
- Q. Wang

- ↵
- R. Lincoln,
- F. Scarpa,
- V. Ting,
- R. S. Trask

- ↵
- G. W. Milton

- G. W. Milton

- ↵
- ↵
- V. A. Markel

- ↵
- M. Florescu,
- S. Torquato,
- P. J. Steinhardt

- ↵
- W. Man et al.

- ↵
- B. Redding,
- S. F. Liew,
- R. Sarma,
- H. Cao

- ↵
- ↵
- R. Degl’Innocenti et al.

- ↵
- A. Rabiei,
- L. J. Vendra

- ↵
- ↵
- M. Garcia-Avila,
- M. Portanova,
- A. Rabiei

- ↵
- J. Marx,
- M. Portanova,
- A. Rabiei

- ↵
- S. Torquato,
- F. H. Stillinger

- ↵
- C. E. Zachary,
- S. Torquato

- ↵
- S. Torquato

- ↵
- O. U. Uche,
- F. H. Stillinger,
- S. Torquato

- ↵
- S. Torquato,
- G. Zhang,
- F. Stillinger

- ↵
- G. Zhang,
- F. H. Stillinger,
- S. Torquato

- ↵
- D. Chen,
- S. Torquato

- ↵
- O. Leseur,
- R. Pierrat,
- R. Carminati

- ↵
- G. Zhang,
- F. Stillinger,
- S. Torquato

- ↵
- G. Gkantzounis,
- M. Florescu

- ↵
- F. Bigourdan,
- R. Pierrat,
- R. Carminati

- ↵
- ↵
- G. Ma,
- M. Yang,
- Z. Yang,
- P. Sheng

- ↵
- C. Chen,
- Z. Du,
- G. Hu,
- J. Yang

- ↵
- A. Khelif,
- P. A. Deymier,
- B. Djafari-Rouhani,
- J. O. Vasseur,
- L. Dobrzynski

- ↵
- V. Romero-Garíca,
- N. Lamothe,
- G. Theocharis,
- O. Richoux,
- L. M. García-Raffi

- ↵
- J. D. Jackson

- ↵
- N. W. Ashcroft,
- N. D. Mermin

- ↵
- L. Landau,
- E. Lifshitz

- ↵
- F. H. Kerr

- ↵
- ↵
- M. D. Rintoul,
- S. Torquato

- ↵
- J. P. Hansen,
- I. R. McDonald

- ↵
- S. Torquato,
- G. Stell

- ↵
- G. W. Milton

- ↵
- W. Jiang et al.

- ↵
- H. Guan,
- S. Liu,
- Y. Duan,
- J. Cheng

- ↵
- A. P. Surzhikov,
- T. V. Fursa

- ↵
- T. V. Fursa,
- B. A. Lyukshin,
- G. E. Utsyn

- ↵
- J. M. Carcione,
- B. Ursin,
- J. I. Nordskag

- ↵
- K. V. Wong,
- A. Hernandez

- ↵
- K. Zhao,
- T. G. Mason

- ↵
- O. V. Tereshchenko,
- F. J. K. Buesink,
- F. B. J. Leferink

- ↵
- L. V. Gibiansky,
- S. Torquato

- ↵
- ↵
- ↵
- G. Zhang,
- F. H. Stillinger,
- S. Torquato

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