Precise determination of the long-time asymptotics of the diffusion spreadability of two-phase media

The time-dependent diffusion spreadability S(t) is a powerful dynamical probe of the microstructure of two-phase heterogeneous media across length scales [S. Torquato Phys. Rev. E104, 054102 (2026) 2470-0045 10.1103/PhysRevE.104.054102]. The spreadability can be exactly represented as a certain functional of the spectral density χ̃_V(k), where k is the wave vector. Experimentally, it is intimately related to nuclear magnetic resonance (NMR) measurements in fluid-saturated media. The short-, intermediate-, and long-time behavior of the spreadability reflects structural features at small, intermediate, and large length scales, respectively. It has been shown that when the spectral density takes the power-law form χ̃_V(k)∼|k|^α as the wave number |k| tends to zero, the normalized excess spreadability s^ex(t) [proportional to S(∞)−S(t)] scales as s^ex(t)∼t−d+α/2 in the long-time limit t→∞, 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 [H. Wang and S. Torquato, Phys. Rev. Appl.17, 034022 (2026) 2331-7019 10.1103/PhysRevApplied.17.034022]. 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 χ̃_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 sex(t) with the small-t expansion, we have devised a two-point Padé approximant to accurately approximate 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.