Effective transport and mechanical properties of two-phase materials across the order-disorder spectrum

Two-phase heterogeneous materials arise in a plethora of natural and synthetic situations, such as alloys, composites, geological media, complex fluids, and biological media, exhibit a wide-variety of microstructures, and thus display a broad-spectrum of effective physical properties. Elucidating how microstructural details (e.g., specific surface, phase volume fractions, and phase geometries and topologies) influence two-phase materials’ effective transport and mechanical properties enhances our fundamental understanding of microstructure–property relationships, and is of great practical use in the design of structural and functional materials. Here, we compute the three-point microstructural parameters ζ₂ and η₂ for phase 2 to evaluate bounds and accurate approximation formulas on the effective thermal/electrical conductivities σ_e as well as the bulk K_e and shear G_e moduli of a wide class of disordered 2D and 3D material microstructures; including various dispersions of hard and penetrable particles, certain amorphous dispersions, and networks. These parameters are determined using a Monte Carlo integration scheme that incorporates certain algorithmic enhancements that improve upon the accuracy and performance of previous methods. Our results reveal that ζ₂ and η₂ are generally sensitive to the phase-connectedness properties of the diverse set of microstructures we consider here. Using numerical simulations and rigorous approximations that incorporate ζ₂ and η₂, we show that certain disordered microstructures nearly realize the Hashin–Shtrikman lower and upper bounds on σ_e, K_e, and G_e across volume fractions. We also describe how our algorithm can be used in inverse methodologies to realize ordered and disordered materials with desirable effective physical properties by targeting specific values of ζ₂ and η₂, and hence aid in materials by design.