Application of fractional calculus to the modeling of dielectric relaxation phenomena in polymeric materials

A model based on the concept of fractional calculus is proposed for the description of the relative complex permittivity (epsilon(r)* = epsilon(r)' - i epsilon(r)', where epsilon(r)' and epsilon(r)'' are the real and imaginary parts of epsilon(r)*) in polymeric materials. This model takes into account three dielectric relaxation phenomena. The differential equations obtained for this model have derivatives of fractional order between 0 and 1. Applying the Fourier transform to fractional differential equations and considering that each relaxation mode is associated with cooperative or noncooperative movements, we have calculated epsilon(r)*(i omega,T) (where omega is the angular frequency and T is the temperature). The isothermal and isochronal diagrams obtained from the proposed model of epsilon(r)' and epsilon(r)'' clearly show three dielectric relaxation phenomena; in the isochronal case, each relaxation mode manifests by an increase in epsilon(r)' with increasing temperature, and this behavior is associated with a peak of epsilon(r)''(T) in each case. The model is matched with the experimental data on poly(ethylene naphthalene 2,6-dicarboxylate) (PEN) to justify its validity. Poly(ethylene 2, 6-naphthalene dicarboxylate) (PEN) is a semicrystalline polymer that displays three dielectric relaxation processes: beta, beta*, and alpha. (c) 2005 Wiley Periodicals, Inc.