Probability distribution dielectric relaxation

It was soon realized that a distribution of exponential correlation times is required to characterize backbone motion for a successful Interpretation of both carbon-13 Ti and NOE values in many polymers (, lO). A correlation function corresponding to a distribution of exponential correlation times can be generated in two ways. First, a convenient mathematical form can serve as the basis for generating and adjusting a distribution of correlation times. Functions used earlier for the analysis of dielectric relaxation such as the Cole-Cole (U.) and Fuoss-Kirkwood (l2) descriptions can be applied to the interpretation of carbon-13 relaxation. Probably the most proficient of the mathematical form models is the log-X distribution introduced by Schaefer (lO). These models are able to account for carbon-13 Ti and NOE data although some authors have questioned the physical insight provided by the fitting parameters (], 13) ... 

Relaxations tend to divide into two types those that obey a simple Arrhenius temperature dependence and those that do not. For simple thermally activated processes Arrhenius behaviour is observed. The probability of the dipole reorientating depends directly on the thermal energy distribution. The relaxation time is related to the frequency of maximum dielectric loss ... 

The a- and ajS-processes are characterized by a broad asymmetric dielectric relaxation spectrum, which can be well represented by the Kohlrausch Williams-Watts (KWW) decay function (cf. eqn. (4.17)). The major factor leading to the broad DR spectra for a- and ajS-relaxations is that chain segments relax in cooperation with their environment. In order to explain the mechanism of this relaxation, the concepts of defect diffusion and free-volume fluctuation are used. For example, Bendler has proposed a model in which the KWW function is interpreted as the survival probability of a frozen segment in a swarm of hopping defects with a stable waiting-time distribution At for defect motion. 

These relations allow the majority of the curves (e, e") = /(w) to be described for dielectrics. In these formulas, the deviation from the ideal curve is explained by relaxation time or probability jump distribution. In fact, the a and P values only describe the curves and cannot easily be connected to the material s physical features. So, the results analysis is often limited to the determination of a distribution function or to an equivalent electrical circuit. 

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