Computational study of interfacial polarization in the cluster of dielectrics
Keywords: Materials Science and Engineering.

Surface charges are induced when charge distributions in the dielectrics to both sides of the interface are polarized differently by the externally applied field. Resolving these surface charges is prerequisite to accurately evaluate the electrostatic interactions in systems with spatially varying dielectric permittivity, such as electrolytes near polarizable interfaces, dielectric composites, and electrostatic self-assembly of particles. The standard numerical method for this boundary value problem is limited by two intrinsic challenges: the polarization is a many-body effect, and induced surface charges may apparently diverge for dielectrics in close proximity. First, the electric field generated by induced surface charges on one particle polarizes all nearby dielectrics, making the resolution of surface charges a many-body problem. Second, the areal density of surface charges is high when the permittivity ratio across the interface is large, increases rapidly as the inter-particle separation shrinks, and may diverge apparently in certain cases. In this thesis, I introduce an expansion to the free energy of a particulate composite under an external static field, that accounts for the many-body effect systematically, and develop a framework to resolve the strong induced surface charges for dielectrics in close proximity. The free energy expansion is used to study the effective static dielectric permittivity of, e.g., polymer-ceramic composite, which depends not only on the permittivity but also the volume fraction of each constituent. The widely used Maxwell-Garnett mixing rule or its variants---valid only at dilute limit--- neglects the contribution from inter-particle polarization and estimates the composite permittivity using a linear average of the component's polarizability. I develop a parameter-free density expansion through a virial-type expansion for the effective dielectric permittivity to take inter-particle polarization into account. The Maxwell-Garnett mixing rule is shown to be the simplest form of this density expansion; the contributions from inter-particle polarizations can be included systematically through the evaluations of virial coefficients for the expansion. The virial coefficients depend on the ratio of particle's and medium's permittivity and are efficiently evaluated by reducing the surface charge integrals into those over image charge lines. The predictions of the theory, with the lowest order polarization corrections, are shown to agree with experimental data for up to 60% of inclusion's volume fraction, beyond which inclusions form close contacts with their neighbors. For clusters of closely separated dielectric particles, the conventional numerical approach for resolving polarization charges is computationally prohibitive in practice, because the electrical field in the gap region intensifies as the gap distance decreases. In the limit of high permittivity, this effect leads to a logarithmic singularity in surface charges. For the general dielectric cases, I show that the singular contribution to surface charge depends on both gap distance and permittivity ratio logarithmically. Analytically isolating this singular behavior from numerically calculated surface charges allows for the calculation of the electrostatic energy for clusters of dielectric particles with arbitrarily small gap distances. The exact cohesive energies for clusters of different configurations are examined, which reveals an unexpected crossover: the extended configurations are more stable when the permittivity ratio is high, whereas the compact ones become more stable when the permittivity ratio is low