Distribution of relaxation times from dielectric spectroscopy using Monte Carlo simulated annealing: Application toα−PVDF

Abstract

The existence of a distribution of relaxation times has been widely used to describe the relaxation function versus frequency in glass-forming liquids. Several empirical distributions have been proposed and the usual method is to fit the experimental data to a model that assumes one of these functions. Another alternative is to extract from the experimental data the discrete profile of the distribution function that best fits the experimental curve without any a priori assumption. To test this approach a Monte Carlo algorithm using the simulated annealing is used to best fit simulated dielectric loss data, ${\ensuremath{\varepsilon}}^{\ensuremath{''}}(\ensuremath{\omega}),$ generated with Cole-Cole, Cole-Davidson, Havriliak-Negami, and Kohlrausch-Williams-Watts (KWW) functions. The relaxation times distribution, $G(\mathrm{ln}(\ensuremath{\tau})),$ is obtained as an histogram that follows very closely the analytical expression for the distributions that are known in these cases. Also, the temporal decay functions, $\ensuremath{\varphi}(t),$ are evaluated and compared to a stretched exponential. The method is then applied to experimental data for $\ensuremath{\alpha}$-polyvinylidene fluoride over a temperature range $233 \mathrm{K}l~Tl~278 \mathrm{K}$ and frequencies varying from 3 MHz to 0.001 Hz. These data show the existence of two relaxation processes: the fast segmental ${\ensuremath{\alpha}}_{a}$ process associated with the glass transition and a ${\ensuremath{\alpha}}_{c}$ mode, which is slower and due to changes in conformation that can occur in the crystalline regions. The experimental curves are fitted by the simulated annealing direct signal analysis procedure, and the relaxation times distributions are calculated and found to vary with temperature. The decay function is also evaluated and it shows clearly its bimodal character and a good agreement with a KWW function with a temperature dependent $\ensuremath{\beta}$ for each mode. The relaxation plots are drawn for each mode and the Vogel-Tammann-Fulcher and Arrhenius parameters are found. The fragility parameter for polyvinylidene flouride (PVDF) is found to be 87, which characterizes this polymer as a relatively structurally strong material.