Electromagnetics - Modelling, Simulation, Control and Industrial Applications - Abstract
Giovangigli, Laure
The electric behavior of a biological tissue under the influence of an electric field at frequency $omega$ can be characterized by its effective admittivity $k_ef:= sigma_ef + i omega varepsilon_ef $, where $sigma_ef$ and $varepsilon_ef$ are respectively its effective conductivity and permittivity. Electrical Impedance Spectroscopy measure the admittivity across a a range of frequencies producing a spectrum showing the change of admittivity with frequency. The aim of our work is to prove that this spectrum carries information on the microscopic structure of the medium. We study the case of a periodic and dilute repartition of cells in the medium. We consider a periodic suspension of identical cells of arbitrary shape. We apply at the boundary of our medium an electric field of frequency $omega$. The medium outside the cells have an admittivity $k_0:=sigma_0+ iomega varepsilon_0$. Cells are composed of an isotropic homogeneous core of admittivity $k_0$ and a thin membrane of constant thickness $delta$ and admittivity $k_m := sigma_m + i omega varepsilon_m $. $delta$ is considered to be very small compared to the typical size of the cell and the membrane very resistive, i.e. $sigma_m ll sigma_0$. We set : $beta : = k_0 ,delta / k_m $. . In this context, the potential in our medium undergoes a jump across each cell boundary, which is proportional to its normal derivative with a coefficient $beta$. Its normal derivative is besides constant across the cell boundaries. We use homogenization techniques with asymptotic expansions to derive an homogenized problem and define an effective admittivity of our medium. We prove a rigorous convergence of our initial problem to the homogenized one with the two-scale convergence method. Using layer potential techniques, we expand our effective admittivity in terms of $rho$, i.e. we consider that the suspension of cells is dilute, that the fraction volume occupied by the cells goes to zero. Through $beta$, the first order, $M$, also called polarization tensor, depends on the frequency $omega$ of the source. For circular cells, we retrieve Maxwell-Wagner-Fricke formula. The imaginary part of $M$ is maximal for a frequency near $1/tau$, where $tau$ is the first Debye relaxation time. We can distinguish, only with $tau$, between different microscopic organizations of the medium. We computed numerically with realistic parameters, $tau$ for different initial configurations of our medium : one circular or elliptic cell, two or three nearly touching cells and obtained significant results. We proved in our work that one can classify with the first Debye relaxation time the different microscopic structure of a medium.