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 (ε*r = ε′r − iε′r, where ε′r and ε″r are the real and imaginary parts of ε*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 ε*r(iω,T) (where ω is the angular frequency and T is the temperature). The isothermal and isochronal diagrams obtained from the proposed model of ε′r and ε″r clearly show three dielectric relaxation phenomena; in the isochronal case, each relaxation mode manifests by an increase in ε′r with increasing temperature, and this behavior is associated with a peak of ε″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: β, β*, and α.