Conduction in heterogeneous systems in the low-frequency regime: variational principles and boundary integral equations.

The response of a homogeneous material to the presence of an external low-frequency oscillating electric field can be described by means of an effective complex conductivity. Low frequencies are characterized by negligible magnetic and radiative effects. The properties of heterogeneous systems, composed of multiple homogeneous regions, can be determined from those of the individual components and their geometric arrangement. Examples of such heterogeneous systems include soft materials such as colloidal suspensions, electrolyte systems, and biological tissues. The difference in the intrinsic conductivities between the homogeneous regions leads to the creation of an oscillating charge density localized at the interfaces between these regions. We show how to express key properties of these systems using this dynamic charge as a fundamental variable. We derive a boundary integral equation for the charges and reconstruct potentials and fields from its solution. We present a variational principle that recovers the fundamental equations for the system in terms of the oscillating charge and show that, in some formulations, the associated functional can be interpreted in terms of the power dissipated in the system. The boundary integral equations are numerically solved using a finite element method for a few illustrative cases.