Modeling the permittivity of two-phase media containing monodisperse spheres: Effects of microstructure and multiple scattering

Abstract

A numerical modeling approach was developed to predict the dielectric properties of heterogeneous particulate materials with arbitrary microstructures. To test the method, simulation and experimental data were acquired for the effective permittivities of various glass sphere suspensions. Both ordered lattices and random microstructures of up to 3600 spheres were modeled for volume fractions of 0.025-0.60. The electric fields in the suspensions were computed using an iterative multipole method that included multiple-scattering effects. The effective permittivities were obtained by averaging the electric field and electric displacement over a representative volume. Frequency spectra, electric field images, and single-scattering results ͑i.e., no particleparticle interactions͒ were additionally generated. The results were compared to experimental data for random close-packed microstructures, to effective-medium approximations, to exact lattice models, and to a perturbation expansion model. The comparisons showed that the iterative models agreed with the exact lattice models to within 3.31% for crystalline suspensions. Results for random suspensions agreed with the perturbation expansion model to within 1.76% for volume fractions up to 0.50. Single-scattering models additionally predicted permittivities for the microstructures as well as or better than the Maxwell Garnett approximation ͓Philos. Trans. R. Soc. London, Ser. A 203, 385 ͑1904͔͒, suggesting that microstructural effects and multipole moments higher than the dipole are required for more accurate statistical prediction of effective permittivities. The effective permittivities, convergence behavior, and dispersion behavior of the simulations were sensitive to both microstructure and the extent of multiple scattering included in the models, illustrating how the macroscopic properties depend significantly on the microscopic details of the interactions. In contrast to other approaches, the iterative multipole method can model both the frequency and spatial dependencies of the electromagnetic properties of particulate materials, as well as a wide variety of microstructures, including polydisperse and hierarchical systems.

Effects of aggregation on the permittivity of random media containing monodisperse spheres

“…43 The rapid convergence of the computations can be attributed to the small (200 µm) diameter of the particles as compared to the wavelengths of the 0.01-10.0 GHz EM fields. These values correspond to size parameters of 2×10 -5 to 2×10 -2 .…”

“…43 Simulations were performed on particle packs containing up to 2440 particles to assess the effects of aggregation (clustering) and foaming (void formation) on random microstructures. The spherical particles were configured in a cylinder of fixed volume with the EM field propagating parallel to the cylinder's axis (z direction) and with the electric field vector perpendicular to the axis (x direction).…”

“…Although the size parameters are within the Rayleigh scattering approximation for single spheres and therefore suggest only small values (< 7) are required for n max , previous studies have demonstrated that higher n max values are necessary when multiple scattering is involved. 43 Computer algorithms for the model were written and compiled in FORTRAN 90…”

“…43,48 The EM fields of the spherical particles were modeled with vector multipole functions derived from the Maxwell equations for a macroscopic dielectric medium with no free charges. These functions are solutions to the vector Helmholtz equations in spherical coordinates, and are also known as vector spherical wave functions: 49 Rose, 50 Edmonds, 51 Greiner and Maruhn, 52 and Jackson.…”

“…42 Recently, an iterative multipole approach was developed that modeled the permittivity of up to 3600 monodisperse glass spheres in air. 43 The spheres were configured in either ordered lattices or random microstructures, and with particle volume fractions of 0.025-0. 60.…”