Mechanical and dielectric relaxation phenomena of poly(ethylene‐2,6‐napthalene dicarboxylate) by fractional calculus approach

Abstract

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.

The effect of iron oxide nanoparticles on the mechanical relaxation of magnetic polymer hybrid films composed of a polystyrene matrix, a fractional calculus approach

“…The molecular mobility associated to experimental measurements of E * was analyzed by a fractional calculus approach, using a fractional Zener model (FZM). Fractional calculus is the branch of mathematics that deals with the generalization of integrals and derivatives of arbitrary (real and complex) orders . The application of fractional calculus to the study of structural relaxation of polymers is done by replacing two spring‐pot elements instead of the dashpot element on the classical Zener model .…”

Analysis of free vibrations of a viscoelastic oscillator via the models involving several fractional parameters and relaxation/retardation times

“…However, as is shown in [1][2][3][4][5], such models give good fit with the experimental data on the creep and relaxation of some polymeric materials.…”

“…This phenomenon is explained by the fact that the considered models (13) and (14) with the kernels of the form of the fractional exponents are Boltzmann-Volterra relationships with fading memory, that is why they are physically meaningful for any combination of the rheological parameters. The presence of variety of the rheological parameters, in its turn, allows one to describe adequately the behaviour of the advanced polymeric materials possessing the complex relaxation and creep functions involving several transition zones [1][2][3][4][5], in doing so utilizing only one rheological model with several relaxation (retardation) times and several fractional parameters. Moreover, the analytical procedure developed for the numerical calculation of the characteristic equation roots enables one to analyze the dynamic response of the mechanical systems, the hereditary elastic features of which are described by such perspective for the rheology models.…”

Application of fractional calculus to the modeling of the complex magnetic susceptibility for polymeric‐magnetic nanocomposites dispersed into a liquid media

“…(1), V is the circuit applied voltage, R and L are the electric resistance and inductance magnitudes, respectively, and τ = L / R is the characteristic response time, called relaxation time, which can be associated with the time required to the motion for a complete reorientation of a given particle (with a dipolar magnetic moment) to a new equilibrium state. Finally, D I ( t ) is the fractional derivate of the a th order of the electrical current in the FRI with respect to time,16, 17 which can be defined by the Riemann‐Liouville derivative: where Γ is the Gamma function: and “ y” is a mathematical variable used in Riemann‐Lioville derivative. It is important to mention that eq.…”

“…By introducing fractional calculus tool into the FRI, it is possible take into account a distribution of relaxation times, associated with the system magnetic response. This approach has been successfully used over the past few years in the case of the dielectric manifestation of the viscoelasticity of polymer‐dielectric materials 16, 17…”