Assessing effective medium theories for conduction through lamellar composites

Abstract

In this study, we developed a numerical finite-difference model for 2D transient diffusion in lamellar structures. The explicit application is heat transfer, but it could equally be applied to dielectric constants, magnetic susceptibilities, electron/ion conduction, and mass diffusion as well. The control volume contains two phases A and B. The phases have different transport parameters. The modeling aims to evaluate the effect of grain size, grain boundaries, and phase contrast on apparent transport properties of composite materials, such as laminates, polycrystalline materials, and block copolymers, by examining a progression of increasingly complex structures. To validate the model, effective transport parameters of parallel and perpendicular structures from the numerical model are compared to analytical expressions. Effective Medium Theory provides an analytical expression in the limit of many, small, randomly oriented grains. The impact of coarse grains on transport is investigated. Specifically, the model is used to examine how the apparent transport parameters trend from the limit of a homogeneous material to small randomly oriented grains containing two different phases. The effective thermal conductivity (averaged over many random structures) was found not to be a function of grain size. However, the standard deviation decreased exponentially with decreasing grain size, reaching less than 2% variation for transport through 15 grains. Thus, the appropriate Effective Medium prediction is reliable for a surprisingly few number of grains, and connectivity of the more conducting phase is important only in coarse grains with significant contrast.