Modeling disordered hyperuniform heterogeneous materials: Microstructure representation, field fluctuations and effective properties

Disordered hyperuniform (DHU) materials are an emerging class of exotic heterogeneous material systems characterized by a unique combination of disordered local structures and a hidden long-range order, which endow them with unusual physical properties, including large isotropic photonic band gaps, superior resistance to fracture, and nearly optimal electrical and thermal transport properties, to name but a few. Here, we consider material systems possessing continuously varying local material properties $K\left(x\right)$ (e.g., thermal or electrical conductivity), modeled via a random field. We devise quantitative microstructure representation of the material systems based on a class of analytical spectral density function ${\tilde{\chi }}_{{}_K}\left(k\right)$ associated with $K\left(x\right)$, possessing a power-law small-$k$ scaling behavior ${\tilde{\chi }}_{{}_K}\left(k\right)\sim k^{\alpha }$. By controlling the exponent $\alpha$ and using a highly efficient forward generative model, we obtain realizations of a wide spectrum of distinct material microstructures spanning from hyperuniform ($\alpha >0$) to nonhyperuniform ($\alpha =0$) to antihyperuniform ($\alpha <0$) systems. Moreover, we perform a comprehensive perturbation analysis to quantitatively connect the fluctuations of the local material property to the fluctuations of the resulting physical fields. In the weak-contrast limit, i.e., when the fluctuations of the property are much smaller than the average value, our first-order perturbation theory reveals that the physical fields associated with Class-I hyperuniform materials (characterized by $\alpha \ge 2$) are also hyperuniform, albeit with a lower hyperuniformity exponent ($\alpha -2$). As one moves away from this weak-contrast limit, the fluctuations of the physical field develop a diverging spectral density at the origin, revealed by our higher-order analysis. We also establish an end-to-end mapping connecting the spectral density of the local material property to the overall effective conductivity of the material system via numerical homogenization. We show that the strong stealthy-like hyperuniform systems can achieve nearly optimal effective conductivity and observe a sharp decrease of the variance of effective properties across realizations as $\alpha$ increases from antihyperuniform values to hyperuniform values. Our results have significant implications for the design of novel DHU materials with targeted physical properties.