The Spheroplastic Permittivity Estimates

Abstract

The composite materials with a matrix from polymer and silicone binders, reinforced by the hollow spherical inclusions (microspheres) are commonly referred to as spheroplastics. A widespred microsphere material is glass, but, it is also possible to use ceramic, carbon, and polymeric microspheres having a diameter within a millimeter and a wall thickness of several micrometers. Low density and thermal conductivity and sufficiently high mechanical properties of the spheroplastics decided their widespread use as the light and strong thermal insulation materials. But a selection of the microsphere and binder materials allows us to create the spheroplastics with appropriate properties to be used in variety of technology domains, including those with dielectric properties, which the structural materials of various wireless devices and objects should have. To create the spheroplastic with specified dielectric properties it is necessary to have a mathematical model allowing us to describe the mechanical and electrical interaction of the microspheres and binder and build the calculated relationships to estimate a spheroplastic permittivity, which can be obtained when using the various combinations of microsphere materials and binder. Such models can be built involving the essentials of mechanics and electrodynamics The paper offers a constructed mathematical model to represent the spheroplastic as a combination of four phases: a homogeneous material with the desired effective permittivity and a spheroplastic interacting with the representative structure element consisting, in turn, of a ball bonding layer, a hollow microsphere and and its filling. To ensure a two-sided estimate of the possible values of spheroplastic effective permittivity this model uses a dual variational formulation of electrostatic problems in nonhomogeneous continuum, including two alternative functionals (minimized and maximized), reaching the matching extreme values to true solution. The transition from the initial four-phase model of the spheroplastic structure to the simple three-phase one enabled us to derive the calculated relationships for the equivalent permittivity of microspheres, and then also the calculating formulas for effective permittivity of the spheroplastic as a whole. A quantitative analysis of these relationships proved the adequacy of the constructed mathematical models to describe the interaction between mechanical and electrical elements of the structure and the predicted accuracy of the effective spheroplastic permittivity determined owing to calculated dependences.