Multifunctionality of Particulate Composites via Cross-Property Maps

There exist closed-form analytical expressions for the effective electrical (thermal) conductivity and elastic moduli of a wide class of 3D isotropic particulate composites for all phase contrasts and volume fractions. In Ref. [1], we employ these expressions to derive and apply “cross-property maps” that connect combinations of pairs of the aforementioned effective transport and elastic properties to one another. Our cross-property maps are valid for a wide class of dispersions in which the inclusions may exhibit polydispersivity in shape and size and are spatially distributed in a matrix with varying degrees of order/disorder. We demonstrate maps for a variety of cases such as incompressible or compressible composites with conducting/insulating inclusions that may be rigid, compliant, or auxetic. We also discuss an example that enables us to design a disordered dispersion with desired values of the effective dielectric constant and effective Young’s modulus.

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1. S. Torquato and D. Chen, Multifunctionality of Particulate Composites via Cross-property Maps, Physical Review Materials, 2 095603 (2018); bibtex

Multifunctional Composites for Elastic and Electromagnetic Wave Propagation

Composites are ideally suited to achieve desirable multifunctional effective properties since the best properties of different materials can be judiciously combined with designed microstructures. In Ref. [1], we establish the first 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 (see Figure 1). We are able to establish these novel cross-property relations 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.

Figure 1: Schematic depicting a multifunctional composite material that has been engineered to absorb sound waves of a certain frequency while simultaneously transmitting light waves of a certain frequency. Figure adapted from Ref. [1].

Our formulas enable us to explore the wave characteristics of a broad class of disordered microstructures because they apply, unlike conventional formulas, for 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 CPUs 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.

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1. J. Kim and S. Torquato, Multifunctional Composites for Elastic and Electromagnetic Wave Propagation, Proceedings of the National Academy of Science and the United States of America, 117(16) 8764-8774 (2020).

Optimal Multifunctional Materials and Minimal Surfaces

Composite 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. There are presently no simple examples that rigorously demonstrate the effect of competing property demands on composite microstructures. To illustrate the fascinating types of microstructures that can arise in multifunctional optimization, we maximize the simultaneous transport of heat and electricity in three-dimensional, two-phase composites using rigorous optimization techniques in Ref. [1]. Interestingly, we have discovered that the optimal three-dimensional structures are bicontinuous triply periodic minimal surfaces (see Figures 1 and 2).

Figure 1: A bicontinuous optimal structure (for which volume fractions of both phases are identical) corresponding to maximization of the sum of the effective electrical and thermal conductivities for ill-ordered phases. Left panel: A 2 x 2 x 2 unit cell of the composite. Right panel: Corresponding morphology of phase 2 which has low thermal conductivity but high electrical conductivity. Figure taken from Ref. [2].

Significantly, the aforementioned optimal bicontinuous composites can be made using sol-gel processing techniques. Interestingly, cell membranes resembling periodic minimal surfaces have been observed in cytoplasmic organelles, such as mitochondria and chloroplasts, in which a variety of different transport processes occur. Our work tantalizingly suggests that it may be fruitful to explore the origins of such structures under a new light, namely, whether the optimization of competing functionalities within organelles can explain their resulting structures. Finally, note that our methodology is quite general and can be employed to discover the novel optimal microstructures that are expected to emerge when any combination of functionalities (e.g., mechanical, optical, chemical, electrical, thermal, and fluid flow properties) compete against one another. Such analyses will lead to insights into the genesis of optimal microstructures and will be pursued in future work.

Figure 2: Unit cells of two different minimal surfaces with a resolution of 64 x 64 x 64. Left panel: Schwartz simple cubic surface. Right panel: Schwartz diamond surface. Figure taken from Ref [2].

In Ref. [3], we further established the multifunctionality of the aforementioned bicontinuous two-phase systems by showing that they are also extremal when a competition is set up between the effective bulk modulus and the electrical (or thermal) conductivity of the composite. The implications of our findings for materials science and biology, which provides the ultimate multifunctional materials, are discussed.

Do triply periodic minimal surfaces optimize any other properties? In Ref. [3], we also compute the fluid permeabilities of the Schwartz P and D surfaces and other triply periodic bicontinuous at a porosity \( \phi >= 1/2 \) using the immersed-boundary finite-volume method. The other triply periodic porous media that we study include the Schoen gyroid (G) minimal surface, two different pore-channel models, and an array of spherical obstacles arranged on the sites of a simple cubic lattice. We find that the Schwartz P porous medium has the largest fluid permeability among all of the six triply periodic porous media considered in this paper. The fluid permeabilities are shown to be inversely proportional to the corresponding specific surfaces for these structures. This leads to the conjecture that the maximal fluid permeability for a triply periodic porous medium with a simply connected pore space at a porosity \( \phi >= 1/2 \) is achieved by the structure that globally minimizes the specific surface.

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1. S. Torquato, S. Hyun, and A. Donev, Multifunctional Composites: Optimizing Microstructures for Simultaneous Transport of Heat and Electricity, Physical Review Letters, 89, 266601 (2002).
2. S. Torquato and A. Donev, Minimal Surfaces and Multifunctionality, Proceedings of the Royal Society of London A, 460, 1849 (2004).
3. Y. Jung and S. Torquato, Fluid Permeabilities of Triply Periodic Minimal Surfaces, Physical Review E,92, 255505 (2005).

Predicting the Transport Properties of Porous Media

Establishing quantitative structure-property relationships for random heterogeneous media is critical to the understanding and prediction of their physical properties. When examining fluid flow in porous materials, the geometry of the pore microstructure is a powerful predictor of the fluid permeability \(k\), which is a key property that has been studied extensively in material science, geophysics, and chemical engineering (see Figure 1). In Ref. [1], we present an approximation of the fluid permeability for porous media with a well-connected pore space which depends on the second moment of the pore-size probability distribution function, \(\langle\delta^2\rangle\). This approximation is particularly useful since \(\langle\delta^2\rangle\) is straightforward to compute. Furthermore, in Ref. [2], we confirm that this approximation is accurate for various hyperuniform and nonhyperuniform systems with well-connected pore spaces.

Figure 1: An example of a simulated steady state flow through a Gaussian random field microstructure with porosity φ = 0.7. Regions with slow and fast flow are indicated by blue and red flow lines, respectively. Figure and caption taken from Ref. [3].

In Ref. [3], we use statistical learning methods to predict \(k\) for various porous microstructures. We find that providing the neural network with void-void, surface-void, and surface-surface two-point correlation functions as well as the geodesic tortuosity enable it to make the most accurate predictions of the permeability. Furthermore, we find that this combination of descriptors leads to a significant improvement in predictive performance over conventional Kozeny-Carman regression which utilizes only one-point statistical descriptors (porosity and specific surface). Overall, our results suggest that artificial neural networks are superior to conventional regression techniques for establishing quantitative structure-property relationships.

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1. S. Torquato, Predicting Transport Characteristics of Hyperuniform Porous Media via Rigorous Microstructure-Property Relations, Advances in Water Resources, 140 103565 (2020).
2. M. A. Klatt, R. M. Ziff, and S. Torquato, Critical pore radius and transport properties of disordered hard- and overlapping-sphere models, Physical Review E, 104 014127 (2021).
3. M. Röding, Z. Ma, and S. Torquato, Predicting Permeability via Statistical Learning on Higher-Order Microstructural Information, Scientific Reports, 10 15239 (2020).

Debye random media

So-called Debye random media [1] are unique models of statistically isotropic and homogeneous two-phase media in that they are defined entirely by the two-point correlation function \(S_2(r)\), namely,

\( S_2^{(i)}(r) = \phi_i(1-\phi_i)e^{-r/a} + \phi_i^2 \)

where \(r=|\bf{r}|\) is a radial distance, \(a\) is a characteristic length scale of the medium, and \(\phi_i\) is the volume fraction of phase \(i=1,2\). Debye et al. [2] proposed this two-point correlation function as a model of media with phases of “fully random shape, size, and distribution.” In Ref. [1], we demonstrate that Debye random media are realizable by using the Yeong-Torquato stochastic reconstruction procedure [1,3] to construct two-phase media that exhibit the exponentially decaying two-point correlation function above.

In Ref. [4], we conduct an extensive survey of two-dimensional Debye random media realized with the Yeong-Torquato procedure. Using various analytical and numerical approaches, we ascertain several other microstructural descriptors of these microstructures, including their surface correlation functions, pore-size distributions, lineal-path function, and chord-length probability density function. Notably, we find that Debye random media possess a wide spectrum of hole sizes, including a substantial fraction of large holes.

Since Debye random media are defined only by an exponentially decaying two-point correlation function, they provide a singular opportunity to study the degeneracy of a two-point correlation function [5-7]. In Ref. [8], we consider three such “classes” of degenerate Debye random media. First, we generate the “most probable” class using the Yeong-Torquato construction algorithm. A second class of Debye random media is obtained by demonstrating that the corresponding two-point correlation functions are effectively realized in the first three space dimensions by certain models of overlapping, polydisperse spheres. A third class is obtained by using the Yeong-Torquato algorithm to construct Debye random media that are constrained to have an unusual prescribed pore-size probability density function (see Figure 1). We structurally discriminate these three classes of Debye random media from one another by comparing several of their microstructural descriptors, physical properties, and percolation thresholds. Notably, we find that these degenerate Debye random media are distinguished from one another to varying degrees by these measures; with their differences in percolation thresholds being the most profound. Our findings further confirm the well-known fact that scattering information is insufficient to determine the effective physical properties.

Figure 1: Three degenerate “classes” of Debye random media. (left) Debye random media realized with the Yeong-Torquato stochastic reconstruction procedure. (middle) Debye random media realized by overlapping disks with exponentially distributed radii. (right) Debye random media with an unusual pore-size probability density function. Figure adapted from Ref. [8].

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1. C. L. Y. Yeong and S. Torquato, Reconstructing Random Media, Physical Review E, 57, 495 (1998).
2. P. Debye, H.R. Anderson Jr., and H. Brumberger, Scattering by an Inhomogeneous Solid II. The Correlation Function and Its Application, Journal of Applied Physics, 28, 679 (1957).
3. C. L. Y. Yeong and S. Torquato, Reconstructing Random Media: II. Three-Dimensional from Two-Dimensional Cuts, Physical Review E, 58, 224 (1998).
4. Z. Ma and S. Torquato, Generation and structural characterization of Debye random media, Physical Review E, 102 043310 (2020).
5. C. J. Gommes, Y. Jiao, and S. Torquato, Microstructural Degeneracy Associated with a Two-Point Correlation Function and its Information Content, Physical Review E, 85, 051140 (2012).
6. C. J. Gommes, Y. Jiao, and S. Torquato, Density of States for a Specified Correlation Function and the Energy Landscape, Physical Review Letters, 108, 080601 (2012).
7. Y. Jiao, F. H. Stillinger, and S. Torquato, A Superior Descriptor of Random Textures and Its Predictive Capacity, Proceedings of the National Academy of Sciences, 106, 17634 (2009).
8. M. Skolnick, and S. Torquato, Understanding degeneracy of two-point correlation functions via Debye random media, Physical Review E, 104 045306 (2021).

Metrics for Quantifying Order and Phase-Mixing Across Length Scales

The capacity to devise order metrics to characterize and classify microstructures of multiphase heterogeneous media across length scales is an outstanding but highly challenging task, given the richness of the possible geometries and topologies of the phases that can arise (see Figure 1 for representative examples). In Ref. [1], we formulate several order metrics for characterizing the degree of order/disorder of two-phase microstructures in \(d\)-dimensional Euclidean space \(\mathbb{R}^d\) across length scales. We specifically use the local volume-fraction variance \(\sigma_{_V}^2(R)\) associated with a spherical window of radius \(R\) as our order metric. We determine \(\sigma_{_V}^2(R)\) as a function of \(R\) for 22 different models across the first three space dimensions, including both hyperuniform and non-hyperuniform systems with varying degrees of short- and long-range order. Overall, we find that the local volume-fraction variance as well as asymptotic coefficients and integral measures derived from it provide reasonably robust and sensitive order metrics to categorize disordered and ordered two-phase media across all length scales. Such order metrics could be employed to accelerate the discovery of novel heterogeneous materials by tailoring their degree of order/disorder.

Figure 1: Representative images of each of the eight 2D two-phase models examined in Ref. [1] that illustrate the great diversity of phase geometries and topologies displayed by two-phase media: (a) Debye random media; (b) overlapping circular disks; (c) random checker-board; (d) equilibrium hard disks; (e) disordered hyperuniform; (f) stealthy hyperuniform disks; (g) hexagons on the triangular lattice; (h) circular disks on the triangular lattice. Figure and caption from Ref. [1].

Phase mixing and separation phenomena abound in the formation of natural and synthetic material systems, including alloys, composites, granular media, geological media, complex fluids, and biological media. While characterizing phase mixing is critical to understanding material microstructure formation and physical properties, previous metrics that measure the degree of mixing in condensed phases often rely on empirical observation, are applicable only to certain microstructures, and/or fail to detect nontrivial changes in mixing with respect to length scale. To overcome these shortcomings, in Ref. [2], we introduce a mixing metric that is applicable to a broad spectrum of complex microstructures and leverages the hyperuniformity concept to provide a unifying framework to rank and classify mixing in materials across both length and time scales. We specifically employ the local phase volume-fraction/field-intensity variance \(\sigma_{_{V/F}}^2(R,t)\) associated with a spherical sampling window of radius \(R\) at time \(t\) as our metric. We demonstrate that \(\sigma_{_{V/F}}^2(R,t)\) provides a sensitive measure of mixing across both length and time scales of a structurally diverse set of real and simulated material microstructures: exotic disordered multihyperuniform avian retina photoreceptor cell patterns, time-evolving spinodal decomposition patterns, and a model material mixed via the baker’s transformation (see Figure 2 for representative images).

Figure 2: Plots of the mixing metric for two-phase media \(\sigma_{_{V}}^2(R,t)\) across length and time scales for a model two-phase system mixed via the Baker’s transformation. Figure from Ref. [2].

1. S. Torquato, M. Skolnick, and J. Kim, Local order metrics for two-phase media across length scales, Journal of Physics A: Mathematical and Theoretical, 55 274003 (2022).
2. M. Skolnick and S. Torquato, Quantifying phase mixing and separation behaviors across length and time scales, Acta Materialia, 268 119774 (2024).

The Diffusion Spreadability

Time-dependent interphase diffusion processes in heterogeneous media are of great importance in various contexts and applications spanning biology, chemistry, materials science, and physics. In 1963, Prager [1] considered the time-dependent problem of mass transfer of solute between two phases in a heterogeneous medium whose solute diffusion coefficients are equal, i.e., \(D_1=D_2=D\), in three dimensions under the assumption that solute is initially uniformly distributed in one phase (phase 2) and absent from the other (phase 1). Recognizing that an exact solution was possible, he expressed the fraction of total solute present in phase 1 as a function of time, denoted \(\mathcal{S}(t)\), as a functional of the two-point probability function of phase 2. In Ref. [2], we expand on Prager’s remarkable result and derive exact direct-space formulas for \(\mathcal{S}(t)\), which we henceforth refer to as the spreadability, in any space dimension \(d\) in terms of the autocovariance \(\chi_{_V}(\mathbf{r})\) as well as corresponding Fourier representations of \(\mathcal{S}(t)\) in terms of the spectral density \(\tilde{\chi_{_V}(\mathbf{k})}\) (which is obtainable from scattering experiments), i.e.,

\(\mathcal{S}(\infty) – \mathcal{S}(t) = \frac{1}{(4\pi Dt)^{d/2}\phi_2} \int_{\mathbb{R}^d} \chi_{_V}(\mathbf{r})\exp[-r^2/(4Dt)] d\mathbf{r} =\frac{1}{(2\pi)^d\phi_2}\int_{\mathbb{R}^d} \tilde{\chi}_{_V}(\mathbf{k})\exp[-k^2Dt]d\mathbf{k}.\)
Here, \(\phi_2\) is the volume-fraction of phase 2 and \(\lim_{t\to\infty}\mathcal{S}(t)\equiv\mathcal{S}(\infty)=\phi_2\) is the infinite-time asymptotic value of the spreadability. We refer to the difference \(\mathcal{S}(\infty)-\mathcal{S}(t)\) as the excess spreadability. Illustrative snapshots of the solute concentration field for the spreadability process in a random two-phase medium are shown in Figure 1.

Figure 1: Concentration field for the diffusion spreadability process at different times for an amorphous two-phase material in two dimensions. (Left) The uniform concentration of the solute species within phase 2 (dark blue) at time \(t = 0\). (Middle) The spreading of solute at short times. (Right) The uniform concentration of solute species throughout both phases (pale blue) in the infinite-time limit. The phase interface is highlighted in black in all three panels. Figure and caption from Ref. [3].

These exact solutions are singular results because of their dependence on up to only the first two correlation functions, whereas the vast majority of effective properties of heterogeneous media rely on an infinite set of such functions. Using these relations we have shown that the short-, intermediate-, and long-time behavior of \(\mathcal{S}(t)\) directly maps to the short-, intermediate-, and large-scale microstructural characteristics of a heterogeneous medium. Thus, the spreadability serves as a powerful dynamic figure of merit for probing and characterizing heterogeneous microstructures across length scales.

For statistically homogeneous microstructures whose spectral densities exhibit the following power-law form around the origin:

\(\tilde{\chi}_{_V}(\mathbf{k}) \sim |\mathbf{k}|^{\alpha},\qquad \mathbf{k}\to0\)
where \( \alpha \in (-d, \infty) \), it is shown in Ref. [2] that the long-time excess spreadability for two-phase materials is given by the power-law
\(\mathcal{S}(\infty)-\mathcal{S}(t) \sim \frac{A}{(Dt)^{-(d+\alpha)/2}}.\)
This power-law holds except for instances where \(\alpha\to+\infty\), which, roughly speaking, may be regarded to correspond to stealthy hyperuniform materials whose excess spreadability decays exponentially fast, i.e.,
\(\mathcal{S}(\infty)-\mathcal{S}(t) \sim \begin{cases}
\exp(-BDt)/BDt \qquad\text{(disordered stealthy)} \\
\exp(-CDt) \qquad\text{(ordered stealthy)},
\end{cases}\)
in which \(B\) and \(C\) are microstructure-dependent parameters and \(C>>B\). From these scaling relations, we see that the more hyperuniform a microstructure (i.e., the greater the value of \(\alpha\)), the more rapidly the excess spreadability decays to \(0\) in the long-time asymptotic limit. The excess spreadability for various hyperuniform and nonhyperuniform two-phase media in dimensions 1 and 3 are plotted in Figure 2. A “phase-diagram” summarizing the spectrum of possible long-time excess spreadability behaviors for different hyperuniform and nonhyperuniform media is shown in Figure 3.

Figure 2: (Left) Log-log plots of the excess spreadability for an ordered hyperuniform (Integer lattice packing), a non-stealthy hyperuniform, and a nonhyperuniform microstructure in 1D. (Right) Log-log plots of the excess spreadability for an hyperuniform medium, a nonhyperuniform medium, and an antihyperuniform medium that highlight the different long-time asymptotic power-law scalings of these systems. Figures from Ref. [2].

Figure 3: “Phase diagram” that schematically shows the spectrum of spreadability regimes in terms of the exponent \(\alpha\). As \(\alpha\) increases from the extreme antihyperuniform limit of \(\alpha \to −d\), the spreadability decay rate gets faster, i.e., the excess spreadability follows the inverse power law \(t\sim−(d+α)/2\), except when \(\alpha\to+\infty\), which corresponds to stealthy hyperuniform media with a decay rate that is exponentially fast. Figure and caption from Ref. [2].

In Ref. [2], we also identify remarkable connections to the covering problem in discrete geometry, pulsed field gradient spin-echo (PFGSE) amplitude nuclear magnetic resonance (NMR) measurements, and diffusion magnetic resonance imaging (MRI) measurements. These previously unidentified connections between the spreadability and noninvasive NMR and MRI methodologies suggest ways that microstructural characteristics across length scales can be extracted from diffusion experiments alone.

In Ref. [3], we expand on the utility of the spreadability as a probe of microstructure across length scales by explicitly computing it for a broad-spectrum of two- and three-dimensional idealized model structures that span across the nonhyperuniform and hyperuniform classes. For non-stealthy media, we show that the long-time spreadability provides a simple means of extracting \(\alpha\) that is robust against noise. We also devise an algorithm for systematically extracting \(\alpha\) and diagnosing stealthy hyperuniformity from finite-size microstructural snapshots from the excess spreadability.

In Ref. [4] the utility of the excess spreadability as a diagnostic probe of microstructural information is further expanded by obtaining it directly from novel computationally efficient “lattice diffusion algorithm” (LDA) simulations. This algorithm simulates the underlying time-dependent concentration field associated with a diffusion process and provides a means of obtaining excess spreadability data in the absence of scattering information. Additionally, our method provides a means of calculating the rate of specific entropy production \(\dot{s}(t)\) for a heterogeneous microstructure, a measure of the rate of energy dissipation in a material for the diffusion process associated with the spreadability. Via our \(\dot{s}(t)\) simulations, we show that hyperuniform materials have smaller dissipation than any nonhyperuniform materials. We also use our algorithm to compute the more general case of the spreadability in which the phase diffusion coefficients are distinct and provide a method for extracting the effective diffusion coefficient of the two-phase material from such data.

In Ref. [5] we measure the long-time asymptotic behavior of \(\mathcal{S}(\infty) – \mathcal{S}(t)\) for maximally random jammed (MRJ) superball packings to determine the hyperuniformity classes of these materials and illustrate how the degree of hyperuniformity degrades as a superball deviates from perfectly spherical character. Similarly, in Ref. [6] we study MRJ packings of hyperspheres across \(d = 3, 4\) and \(5\) space dimensions and use the excess spreadability to assess hyperuniformity classes of sphere packings in each of these instances. Lastly, in Ref. [7] we utilize the excess spreadability as an accurate and robust means of extracting \(\alpha\) for quasiperiodic (1D Fibonacci chain and 2D Penrose tiling) and limit-periodic systems (period doubling chain). These particular applications demonstrate how the excess spreadability provides a route to ascertain the exponent \(\alpha\) from finite-sized microstructural snapshots that is much more robust to noise than directly extracting this value from noisy or discontinuous spectral density data.

1. S. Prager, Interphase transfer in stationary two-phase media, Chemical Engineering Science, 18 227-231 (1963).
2. S. Torquato, Diffusion spreadability as a probe of the microstructure of complex media across length scales, Physical Review E, 104 054102 (2021).
3. M. Skolnick and S. Torquato, Simulated diffusion spreadability for characterizing the structure and transport properties of two-phase materials, Acta Materialia, 250 118857 (2023).
4. H. Wang, and S. Torquato, Dynamic Measure of Hyperuniformity and Nonhyperuniformity in Heterogeneous Media via the Diffusion Spreadability, Physical Review Applied, 17 034022 (2022). (Editor’s Suggestion)
5. C. E. Maher, F. H. Stillinger, and S. Torquato, Characterization of void space, large-scale structure, and transport properties of maximally random jammed packings of superballs, Physical Review Materials, 6 025603 (2022).
6. C. E. Maher, Y. Jiao, and S. Torquato, Hyperuniformity of maximally random jammed packings of hyperspheres across spatial dimensions, Physical Review E, 108 064602 (2023).
7. A. Hitin-Bialus, C.E. Maher, P.J. Steinhardt, and S. Torquato, Hyperuniformity Classes of Quasiperiodic Tilings via Diffusion Spreadability, To appear in Physical Review E