ABSTRACT:The mechanical and dielectric relaxation phenomena in PEN-films have been studied using fractional models. A mechanical fractional model for the description of dynamic modulus, E* ¼ E 0 þ iE 00 , and a dielectric fractional model for the dynamic relative permittivity, e à r ¼ e 0 r À ie 00 r . These models takes into account three relaxation phenomena and the corresponding differential equations have derivatives of fractional order between 0 and 1. In applying the Fourier transform to fractional differential equations and in considering that each relaxation mode is associated to cooperative or noncooperative molecular movements, we calculated E*(io,T) and e à r (io,T). The isochronal diagrams of the real and imaginary parts of either E* and e à r obtained from fractional models have been used to study the three relaxation phenomena (a, b*, and b) of poly (ethylene-2,6-napthalene dicarboxylate). An agreement between experiments and fractional models has been achieved for both mechanical and dielectric relaxation phenomena, and the effect of morphology samples on the fractional order parameters of the Fractional Models are related to molecular motions associated to a, b*, and b relaxations.

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ABSTRACT:A model based on the concept of fractional calculus is proposed for the description of the relative complex permittivity (* r ϭ Ј r Ϫ iЈ r , where Ј r and Љ r are the real and imaginary parts of * 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 * r (i,T) (where is the angular frequency and T is the temperature). The isothermal and isochronal diagrams obtained from the proposed model of Ј r and Љ r clearly show three dielectric relaxation phenomena; in the isochronal case, each relaxation mode manifests by an increase in Ј r with increasing temperature, and this behavior is associated with a peak of Љ 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: ␤, ␤*, and ␣.

ABSTRACT:A model based on the concept of fractional calculus is proposed for the description of the dynamic elastic modulus, E* ϭ EЈ ϩ iEЉ, of polymer materials. This model takes into account three relaxation phenomena (␣, ␤, and ␥) under isochronal conditions. The differential equations obtained for this model have derivatives of fractional order between 0 and 1. Applying the Fourier transform to the fractional differential equations and associating each relaxation mode to cooperative or noncooperative movements, E*(i,T) was evaluated. The isochronal diagrams of EЈ and EЉ clearly show three relaxation phenomena, each of them is manifested by a decrease of EЈ when temperature increases. This decrease is associated with a maximum in EЉ(T) diagram for each relaxation mode. The shape of the three peaks (three maxima in EЉ(T) diagrams) depends of the fractional orders of this new fractional model. The mathematical description obtained of E* corresponds to a nonexponential relaxation behavior often encountered in the dynamics of polymer systems having three relaxation phenomena. This model will enable us to analyze the viscoelastic behavior of polymers.