Precise Determination of the Long-Time Asymptotics of the Diffusion Spreadability of Two-Phase Media

Abstract

The time-dependent diffusion spreadability $\mathcal{S}(t)$ is a powerful dynamical probe of the microstructure of two-phase heterogeneous media across length scales [Torquato, S., \emph{Phys. Rev. E.}, 104 054102 (2021)]. It has been shown that when the spectral density takes the power-law form $\tilde{\chi}_{_V}(\mathbf{k})\sim |\mathbf{k}|^\alpha$ as the wavenumber $|\mathbf{k}|$ tends to zero, the normalized excess spreadability $\mathscr{s}^{ex}(t)$ [proportional to $\mathcal{S}(\infty)-\mathcal{S}(t)$] scales as $\mathscr{s}^{ex}(t)\sim t^{-\frac{d+\alpha}{2}}$ in the long-time limit $t\to\infty$, enabling one to determine the infinite-wavelength scaling exponent $\alpha$. An algorithm that allows one to reliably extract the exponent $\alpha$ from long-time spreadability data was previously devised [Wang, H., Torquato, S., \emph{Phys. Rev. Appl.}, 17 034022 (2022)]. In this paper, we further improve this procedure to obtain $\alpha$ even more accurately by incorporating higher-order correction terms to the long-time asymptotics and by utilizing analyticity properties of $\tilde{\chi}_{_V}(k)$ at the origin. We illustrate our procedure by analyzing hyperuniform ($\alpha> 0$), typical nonhyperuniform ($\alpha=0$), and antihyperuniform ($-d < \alpha <0$) models of two-phase media. In addition, by combining the large-$t$ asymptotic expansion of $\mathscr{s}^{ex}(t)$ with the small-$t$ expansion, we have devised a two-point Pad\'e approximant to approximate $\mathscr{s}^{ex}(t)$ for all $t$ with just a few parameters. Our findings facilitate the characterization of the microstructure of two-phase media across length scales as obtained from numerical spreadability data or experimental data obtained from NMR relaxation measurements. Our work can also be applied in the inverse design of two-phase microstructures with targeted spreadability behaviors.