Theoretical Prediction of the Effective Dynamic Dielectric Constant of Disordered Hyperuniform Anisotropic Composites Beyond the Long-Wavelength Regime

Torquato and Kim [Phys. Rev. X 11, 296 021002 (2021)] derived exact nonlocal strong-contrast expansions of the effective dynamic dielectric constant tensor that treat general three-dimensional (3D) two-phase composites, which are valid well beyond the long-wavelength regime. Here, we demonstrate that truncating this general rapidly converging series at the two- and three-point levels is a powerful theoretical tool for extracting accurate approximations suited for various microstructural symmetries. We derive such closed-form formulas applicable to transverse polarization in layered media and transverse magnetic polarization in transversely isotropic media, respectively. We use these formulas to estimate effective dielectric constant for models of 3D disordered hyperuniform layered and transversely isotropic media: nonstealthy hyperuniform and stealthy hyperuniform (SHU) media. In particular, we show that SHU media are perfectly transparent (trivially implying no Anderson localization, in principle) within finite wave number intervals through the third-order terms. For these two models, we validate that the second-order formulas, which depend on the spectral density, are already very accurate well beyond the long-wavelength regime by showing very good agreement with the finite-difference time-domain simulations. The high predictive power of the second-order formulas implies that higher-order contributions are negligibly small, and thus, it very accurately approximates multiple scattering effects. Therefore, there can be no Anderson localization in practice within the predicted perfect transparency interval in SHU media because the localization length should be very large compared to any practically large sample size. Our predictive theory provides a foundation for the inverse design of novel effective wave characteristics of disordered and statistically anisotropic structures

Extraordinary Optical and Transport Properties of Disordered Stealthy Hyperuniform Two-Phase Media

Disordered stealthy hyperuniform (SHU) two-phase media are a special subset of hyperuniform structures with novel physical properties due to their hybrid crystal-liquid nature. We have previously shown that the strong-contrast expansion of a linear fractional form of the effective dynamic dielectric constant leads to accurate approximations for disordered two-phase composites when truncated at the two-point level for distinctly different microstructural symmetries in three dimensions. Here, we further elucidate the extraordinary optical and transport properties of disordered SHU media. Among other results, we prove in detail that SHU layered and transversely isotropic media are perfectly transparent (i.e., no Anderson localization, in principle) within finite wavenumber intervals through the third-order terms. Remarkably, the results for these SHU media imply that there can be no Anderson localization within the predicted perfect transparency interval in practice because the localization length is much larger than any practically large sample size. We further contrast and compare the extraordinary physical properties of SHU layered, transversely isotropic, and fully isotropic media to other model nonstealthy microstructures, including their attenuation characteristics, as measured by the imaginary part of effective dielectric constant, and transport properties, as measured by the time-dependent diffusion spreadability. We demonstrate cross-property relations between them: they are positively correlated as the structures span from nonhyperuniform, nonstealthy hyperuniform, and SHU media. Establishing cross-property relations for SHU media for other wave phenomena (e.g., elastodynamics) and transport properties will also be useful. Cross-property relations are generally useful because they enable one to estimate one property, given a measurement of another

ECM Incorporation into Model.

<p>(A) Schematic representation of a cell and some of the associated ECM components. Note how the ECM forms in close proximity to the cell that produces it. Image is adapted from:<a href="http://courses.cm.utexas.edu/jrobertus/ch339k/overheads-2/figure-07-30.jpg" target="_blank">http://courses.cm.utexas.edu/jrobertus/ch339k/overheads-2/figure-07-30.jpg</a>. (B) Our representation of the ECM in the proposed three-phase model. The square represents a convex cell body and the “x-ed” network surrounding the cell represents the ECM.</p

Diffusivity of Two-Phase 3D Models.

<p>Effective diffusivities of our proposed two-phase 3D models are compared to the HS upper bound and to the experimentally observed effective diffusivity of brain tissue. Cubes have a fixed size (<i>L</i>×<i>L</i>×<i>L</i>), long cuboids are any permutation of a cuboid of size 2<i>L</i>×2<i>L</i>×<i>L</i>, and short cuboids are any permutation of a cuboid of size 2<i>L</i>×<i>L</i>×<i>L</i>.</p

First-Passage-Time Algorithm.

<p>Example of random walk in 2D using first-passage squares.</p

Proposed Two-Phase Model.

<p>Representative region of proposed microstructural model. (A) 2D two-phase model with ICS in red and ECS in blue. (B) 3D two-phase model (nonstaggered case) with ICS in red.</p

Properties of Three-Phase Model.

<p>(A) The left <i>y</i>-axis (dashed blue line with circles) gives the average gap width in the model and the right <i>y</i>-axis (solid green line with squares) gives the fraction of concave cells in the model. Both plots are given as a function of particle radius (in µm). B) Effective diffusivity of 3D three-phase media at ECS volume fraction <i>φ</i>₁ = 0.2 as a function of the particle radius. The results of the simulation are compared to the maximum two-point three-phase upper bound (Equation 3 with <i>a</i> = 1; solid red line) and the target diffusivity (dotted black line).</p

Fluorescence micrograph of a breast tumor stained to visualize carcinoma cells (phospho-p53, green) surrounded by macrophages (CD11b, red) (a).

<p>Nuclei appear blue (DAPI). Image courtesy of Michael Graham Espey, PhD, National Cancer Institute, NIH (private communication). (b) Representative pictures of dormant and fast-growing tumors and their vascular structure. Reprinted from Cancer Letters, 294, Almog N, Molecular mechanisms underlying tumor dormancy, 139–146, Copyright (2010), with permission from Elsevier.</p

Parameters characterizing the interactions between tumor suppression factors and tumor cells in the CA dormancy model.

<p>Note that the two “critical threshold” parameters themselves do not incorporate any additional CA rules.</p><p>Parameters characterizing the interactions between tumor suppression factors and tumor cells in the CA dormancy model.</p

Tumor area <i>A_T</i> normalized by the area <i>A</i>₀ of the growth permitting region of a simulated noninvasive tumor growing in the ECM under different killing rates by microenvironmental suppression factors.

<p>The parameter <i>k</i>₀ is the fraction that the suppression factors from the microenvironment kill the actively dividing proliferative cells when the suppression factors counteract these cells.</p