The expectation-maximization algorithm has been applied to the analysis of several impedance data measurements of the Ca-montmorillonite clays. This analysis models the imaginary component of the complex impedance as a series of Debye-like functions. The result of the analysis is a distribution of of these functions, each with a different relaxation time. The number-density of the Debye-like functions are the a(t). Plots of these the a(t) versus the relaxation time, t, produce the relaxation-time spectra shown below.
All data sets tried exhibit a shoulder at the long relaxation time constant side that is probably due to a mass-diffusion limited process. They also exhibit a peak, the maximum of which is the mean relaxation time. The maximum relaxation time and the relaxation time peak half width are listed next top the peaks in the relaxation time spectra illustrated below.
Temperature-Dependent Effects
An interesting trend may be observed in the temperature-dependent relaxation-time for the small Ca montmorillonite clay. The most probable relaxation time determined by EM decreases with increasing temperature. In general, this type of temperature dependent effect could indicate either a temperature-dependent rate process with an activation energy, Ea, or an equilibrium process with an enthalpy change between the two (or perhaps more) states, DH. The former is more likely and consistent with these data since the later equilibrium process would probably result in more than one relaxation time distribution peak.
Illustrated below is the "Arrhenius plot" used to determine the activation energy for the rate process. The effective first-order rate constant, k, is simply the inverse of the most probable relaxation time, tmax. The plot of ln(k) versus (1/T) is reasonably linear. Linear regression yields a line with a slope of -2.58x103 Kelvin. This corresponds to an activation energy barrier (to relaxation of the polarized state) 21.45 kJ/mole or 5.13 kcal/mole. This activation energy is about that of water. The increased peak width exhibited by the relaxation-time spectra (with decreasing temperature) indicates that there is a distribution of activation energies for the process. This is not surprising given the heterogeneity of the water-saturated clay sample. At higher temperatures, lower activation energy barriers do not limit conduction as long at the activation energy is less than kT.
The pre-exponential indicated by the above line is e23.2, or 1.2x1010 s-1. This is the high temperature limited relaxation rate constant. It represents a relaxation time of 84 psec. This limiting relaxation time is about the same as the dielectric relxation time of room temperature liquid water, ~25 psec, estimated by Debye in 1928.
The activation energy and high temperature limiting rate both support the notion that dielectric relaxation of water controls the rate processes limiting conduction in small particle Ca montmorillonite clay.
Last Updated on
Tuesday, August 03, 2004
By Stephen Bialkowski