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

TL;DR

This work advances the precise determination of the long-time diffusion spreadability exponent $\alpha$ in two-phase media by enriching the asymptotic theory with higher-order corrections and exploiting analyticity properties of the spectral density $\tilde{\chi}_V(k)$. It introduces three fitting types and two-point Padé approximants to accurately capture $\mathscr{s}^{ex}(t)$ across all times, enabling robust extraction of $\alpha$ for typical nonhyperuniform, hyperuniform, and antihyperuniform media from either simulations or experiments such as NMR/dMRI. The authors validate the approach on Debye random media, disordered hyperuniform, and antihyperuniform models, and demonstrate robustness to noise while providing insights into the underlying correlation functions, moments, and long-distance decay. The framework supports practical characterization of microstructures and offers avenues for inverse design of materials with targeted spreadability behavior.

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 $\tildeχ_{_V}(\mathbf{k})\sim |\mathbf{k}|^α$ 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 $\mathscr{s}^{ex}(t)\sim t^{-\frac{d+α}{2}}$ in the long-time limit $t\to\infty$, enabling one to determine the infinite-wavelength scaling exponent $α$. An algorithm that allows one to reliably extract the exponent $α$ 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 $α$ even more accurately by incorporating higher-order correction terms to the long-time asymptotics and by utilizing analyticity properties of $\tildeχ_{_V}(k)$ at the origin. We illustrate our procedure by analyzing hyperuniform ($α> 0$), typical nonhyperuniform ($α=0$), and antihyperuniform ($-d < α<0$) 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 Padé 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.

Figures (9)

• Figure 1: “Phase diagram” that schematically shows the spectrum of spreadability regimes in terms of the long-distance scaling exponent $\alpha$. As $\alpha$ increases from the extreme antihyperuniform limit of $\alpha\to-d$, the spreadability decay rate at long times decays faster, that is, the excess spreadability follows the inverse power law $1/t^{(d+\alpha)/2}$, except when $\alpha\to\infty$, which corresponds to stealthy hyperuniform media with a decay rate that is exponentially fast. This figure is reproduced from the one presented in Ref. torquatoDiffusionSpreadabilityProbe2021
• Figure 2: (a) The numerical results of the truncated series $P_0^{(n)}(t)$ and $P_\infty^{(n)}(t)$ [Eq. \ref{['eq:poly0']} and \ref{['eq:polyinf']}] against the exact $\mathscr{s}^{ex}(t)$. The small-, intermediate-, and large-$t$ regions distinguished by different shadows are settled semi-quantitatively based on the agreement between $P_0^{(n)}(t)$, $P_\infty^{(n)}(t)$, and $\mathscr{s}^{ex}(t)$. (b) Padé approximants $R_0^{(n)}(t)$ and $R_\infty^{(n)}(t)$ [Eq. \ref{['eq:pade0']} and \ref{['eq:padeinf']}] in contrast to the exact $\mathscr{s}^{ex}(t)$. (c) The two-point Padé approximant $R^{(1,1)}_{0;\infty}(t)$ [Eq. \ref{['eq:2pt-pade']}] that resumes the correct short- and large-$t$ behaviors up to the same order as $R_0^{(1)}(t)$ and $R_\infty^{(1)}(t)$. In all three plots (a), (b), and (c), both the diffusion coefficient $D$ and the characteristic heterogeneity length scale $a$ are set to be unity, i.e., $D=a=1$.
• Figure 3: The relative error of the truncated long-time asymptotic expansion $P_\infty^{(n)}(t)$ [Eq. \ref{['eq:polyinf']}] against the exact $\mathscr{s}^{ex}(t)$ of different order $n$ at different times $t$. Both the diffusion coefficient $D$ and the characteristic heterogeneity length scale $a$ are set to be unity, i.e., $D=a=1$.
• Figure 4: Results of the fitting functions (a)$\ln[\hat{\mathscr{s}}^{ex}_1(t)]$ [Eq. \ref{['eq:fit1']}] and (b)$\ln[\hat{\mathscr{s}}^{ex}_3(t)]$ [Eq. \ref{['eq:fit3']}] for DRM spreadability [Eq. \ref{['eq:large-t-Debye']}]. Positive and negative coefficients are indicated by filled circles and crosses, respectively. The underfitting region ($n<n_o$) and the overfitting region ($n>n_o$) are forward slash and backslash hatched, respectively, while the optimal fitting ($n=n_o$) is left blank.
• Figure 5: Relative fitting errors of (a)$\ln[\hat{\mathscr{s}}^{ex}_1(t)]$ [Eq. \ref{['eq:fit1']}] and (b)$\ln[\hat{\mathscr{s}}^{ex}_3(t)]$ [Eq. \ref{['eq:fit3']}] compared to the exact values for Debye random media [Eq. \ref{['eq:large-t-Debye']}]. Positive and negative relative errors are indicated by filled circles and crosses, respectively. The underfitting region ($n<n_o$) and the overfitting region ($n>n_o$) are forward slash and backslash hatched, respectively, while the optimal fitting ($n=n_o$) is left blank. The fit with $n=0$ in (a) is equivalent to Wang and Torquato's fitting method proposed in wangDynamicMeasureHyperuniformity2022.