The Fractional Model Of Dielectric Relaxation And Its Application

In the external electric field, the movement of each part in polymer is bound, themicroscopic structure of the polymer will be changed and its electrical properties willevolve over time, which results in the complex dielectric relaxation phenomenon.With the development of science and technology, the demands for the dielectricproperties of polymer material become higher and higher in electronic, electricaltechnology and cutting-edge science field.The dynamics process presented by the fractional calculus is in nature dissipative,and the fractional constitutive equations have been successfully applied in viscoelasticmechanical phenomena. A fractional viscoelastic element, called “spring-pot”, wasproposed by Koeller, the spring and dashpot in classical model were replaced.H.Schiessel et al. made use of “spring-pot” to establish the generalized fractal modelsand gave the corresponding constitutive equations. Some tentative works were alsoperformed to describe the dielectric relaxation with fractional calculus. The dielectricloss spectra of glycerol and propylene carbonate in a wide frequency was studied byR.Hilfer. The “cap-resistor”, a dielectric fractional element, was introduced anddielectric relaxation of poly (ethylene naphthalene2,6-dicarboxylate) was studied byReyes-Melo et al. But as far as we know, no more work with respect to the study ofthe dielectric relaxation phenomena applying the “cap-resistor” exist.Comparing with generalized fractional model of viscoelastic mechanicalprocess， fractional model of dielectric relaxation are established， the correspondingconstitutive equation and the relation between complex dielectric constant and thefrequency are deduced. The dielectric relaxation characters of fractional Maxwellmodel and fractional Poynting-Thomson model are analyzed, the common featuresare as following:(1) α not only influences relaxation behavior of both ε’ and ε" atthe high-frequency, but also determines the width of the loss peak,(2) the dielectricrelaxation time τ is the characteristic time corresponding to the frequency that thedielectric constants begin to decay or the position of loss peak, and the loss peakmoves to the lower frequency gradually with the increase of τ,(3) εiis determined byCi, which is proportional to the dielectric constant and the strength of dielectric loss.Through the effective combination of genetic algorithm and conjugate gradient method, the optimum parameters of parallel fractional Maxwell model is determined,and it describes the permittivity and dielectric loss of moist allophone and imogolitesuccessfully. The dielectric relaxations of polyvinyl alcohol and polyvinyl pyrrolidonemixtures are fitted by fractional Poynting-Thomson model, the result shows thatPoynting-Thomson model can perfectly describe the behavior of ε’ and ε"simultaneously.