Analysis of two-dimensional energy and relaxation-time distributions from temperature-dependent broadband dielectric spectroscopy

We present a method to determine the two-dimensional 2D distribution $G(W,\ln {\tau }_o)$ of the relaxation time $\tau ={\tau }_o\exp {(W/k}_{B}T)$ using an inversion of temperature-dependent broadband dielectric data. In contrast to the usual evaluation of 1D distributions $g\left(\tau \right)$, the 2D analysis reveals whether the broadening of a relaxation peak has its physical origins in a variation of the activation energy $W$, or the preexponential factor ${\tau }_o,$ or both. A study of the $\beta$ relaxation of a polymer blend demonstrates the validity of the analysis. We compare the distribution obtained from dielectric ac data (5 Hz–300 MHz, 100 K–300 K) with the results of a thermally stimulated depolarization currents sampling technique. The contour line of $G(W,\ln {\tau }_o)$ is shown to follow the compensation law.