Spatial orientation and position distribution effect on the effective permeability of composites consisting of aligned flakes

In this work, the effective permeability of two-phase composite materials is studied based on a three-dimensional model. In this model, the magnetic inclusions embedded in a dielectric matrix are set with aligned orientation and calculated by finite-difference time-domain method in both lossless and lossy cases. First, the sensitivity of the effective permeability of composites to the spatial orientation and position distribution of inclusions is analyzed. Second, since it is difficult to give an accurate effective permeability with classic effective medium theories, we have modified the well-known Maxwell-Garnett mixing rule by introducing the equivalent demagnetization factors and two fitting parameters alpha and beta, indicating the extent of position distribution chaos. An excellent agreement is obtained between the theoretical and simulation results in the quasistatic limit. Finally, in the case of inclusions oriented at several directions in the matrix, an averaging method is proposed to calculate the effective permeability.