Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical approximation formulas. These property formulas include interfacial bounds as well as universal scaling laws for the mean survival time and the fluid permeability. We also estimate the principal relaxation time associated with Brownian motion among perfectly absorbing traps. For the propagation of electromagnetic waves in the long-wavelength limit, we show that a dispersion of dielectric MRJ spheres within a matrix of another dielectric material forms, to a very good approximation, a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio. We compare the effective properties of the MRJ sphere packings to those of overlapping spheres, equilibrium hard-sphere packings, and lattices of hard spheres. Moreover, we generalize results to micro- and macroscopically anisotropic packings of spheroids with tensorial effective properties. The analytic bounds predict the qualitative trend in the physical properties associated with these structures, which provides guidance to more time-consuming simulations and experiments. They especially provide impetus for experiments to design materials with unique bulk properties resulting from hyperuniformity, including structural-color and color-sensing applications.

Figure 1

Laminar flow through the void space of a two-dimensional hard-disk packing. The color (gray value) indicates the magnitude of the flow velocity. The image is reproduced from Ref. [9] by permission from Springer.

Figure 2

Diffusion of a particle through the void space of a three-dimensional packing of perfectly absorbing spheres. The blue trajectory shows an approximation of its Brownian motion. The particle is absorbed once it touches the surface of a sphere (indicated by a small white ball).

Figure 3

Interfacial bounds on the fluid permeability $k$ as a function of the packing fraction $\varphi$. The bounds for the MRJ sphere packings (at $\varphi =0.636$) are compared to those of an equilibrium hard-sphere liquid (at $\varphi =0.478$) and overlapping spheres, as well as cubic lattices of hard spheres: the simple cubic (gray lines), the body-centered-cubic (green lines), and the face-centered-cubic (dark-red lines) lattice. Because of the rescaling with $k_s=D^2/\left(18\varphi \right)$, the bounds converge for $\varphi \to 0$ to finite values.

Figure 4

Universal scaling of the fluid permeability $k$. The interfacial bounds for overlapping and equilibrium spheres as well as MRJ sphere packings are compared to the universal scaling law for hard and overlapping spheres (dark-red and black solid lines).

Figure 5

Mean survival time $\tau$ rescaled by the diffusion constant $\mathcal{D}$ and the diameter $D$ of a single sphere and plotted as a function of the packing fraction $\varphi$, compared for the MRJ sphere packings (red) to equilibrium hard-sphere liquids (blue) and overlapping spheres (black). The interfacial upper bound [see Eq. (9)] is indicated by closed triangles and a dash-dotted line; the pore-size lower bound [see Eq. (10)] by open triangles and dashed lines. The gray-shaded region indicates, for the overlapping spheres, the mean survival times between these bounds. The solid lines and the crosses indicate the prediction of the mean survival time using the universal scaling relation [see Eq. (12)].

Figure 6

Pore-size lower bounds on the principal diffusion relaxation time $T_1$ [see Eqs. (13) and (14)]. The bounds for the MRJ sphere packings is distinctly smaller than those for equilibrium or overlapping spheres and only slightly larger than dense lattice packings of hard spheres, which are perfectly ordered but anisotropic.

Figure 7

Schematic of a plane electromagnetic wave in the long-wavelength limit (the red and blue arrows depict its values along four lines) entering a heterogeneous dielectric medium that is formed by an MRJ hard-sphere packing. The spheres (steel gray) have a dielectric constant ${\varepsilon }_2$ and the intermediate space (green) has a dielectric constant ${\varepsilon }_1$.

Figure 8

Two-point approximation of the effective dielectric constant ${\varepsilon }_e$ in the strong-contrast expansion [see Eq. (20)] as a function of the ratio ${\varepsilon }_2/{\varepsilon }_1$ of the dielectric constants in the particle and void phase for various absolute values of the wave vectors $k_q$. Both the real part (top row) and imaginary part (bottom row) are plotted for the MRJ state (right column) and compared to the results for the equilibrium hard spheres (middle column) and overlapping spheres (left column).

Figure 9

Anisotropic packing of spheroids resulting from a linear transformation (stretching) of an MRJ sphere packing.

Figure 10

Upper bound on the mean survival time $\tau$ as a function of the aspect ratio $\epsilon =b/a$ of the spheroids. Depicted at the top are exemplary spheroids, from left to right: an oblate ellipsoid ($\epsilon <1$), a sphere ($\epsilon =1$), and a prolate ellipsoid ($\epsilon >1$).

Figure 11

Effective conductivity tensor ${\sigma }_{\mathbf{e}}$ as a function of the contrast between the conductivities ${\sigma }_1$ and ${\sigma }_2$ of the two phases. The three plots depict the results for different aspect ratios $\epsilon$ from disklike (top) to needlelike (bottom). The shaded areas indicate the upper and lower bounds on the diagonal elements of the tensors, distinguishing the $x$ and $z$ directions, where the latter is the direction of the dilation.

Figure 12

The integral $I\left(x\right)$ for each finite MRJ sphere packing oscillates but converges to the unbounded integral ${\int }_0^{\infty }r[S_2\left(r\right)-{\varphi }^2]dr$. Although the integrals over the finite packings still deviate significantly from this limit, the average over several packings appears to provide a robust estimate. However, depending on variations in the cutoff radius, an unknown systematic bias might appear. The unit of length is the diameter of a single sphere.

Figure 13

Extrapolation of the integral of the two-point correlation function for the void bound [see Eq. (A2)]. For $\alpha >1$, the limit ${lim}_{x\to \infty }I(x,\alpha )$ is well approximated by $I(maxx,\alpha )$. Thus, $I(maxx,\alpha )$ with $\alpha \in (1,2)$ allows for a robust extrapolation to $I$ (i.e., $\alpha \to 0$) for both equilibrium (top) and MRJ (bottom) hard spheres. Each plot shows the results for ten packings. The unit of length is the diameter of a single sphere.

Figure 14

Extrapolation of the integral of the correlation functions for the interfacial bound [see Eq. (A3)]. For $\alpha >1$, the limit ${lim}_{x\to \infty }J(x,\alpha )$ is well approximated by $J(maxx,\alpha )$. Thus, $J(maxx,\alpha )$ with $\alpha \in (0.12,0.30)$ allows for a robust extrapolation to $J$ (i.e., $\alpha \to 0$) for both equilibrium (top) and MRJ (bottom) hard spheres. Each plot shows the results for ten packings, where each contains 10 000 spheres. The unit of length is the diameter of a single sphere.

Figure 15

Wigner-Seitz (Voronoi) cells of the (a) sc, (b) fcc, and (c) bcc Bravais lattices.

Figure 16

For the fcc, bcc, and sc Bravais lattices, the analytic curves and simulation results for the mean pore size $\langle \delta \rangle$ and the second moment $\langle {\delta }^2\rangle$.