Gaussian distribution dielectric relaxation

Hermans and Van Beek626 have recently used the new model of polymer molecules suggested by Rouse.46 At high frequencies the whole molecule cannot follow the field so it is divided into a number of submolecules small enough to follow the field and yet sufficiently large to have a Gaussian distribution. Dielectric relaxation for the case of dipoles parallel to the chain has been calculated by Founder sum transforms. The distribution of relaxation modes appears though the multiplicity of the mathematical solution for the diffusion equation. 

Figure 14. Imaginary part of the dielectric permittivity e,(co) of (a) fluoroaniline (7 — 173 K) and (b) toluene (Tg = 117 K), both type B glass formers showing in addition to the main relaxation (a-process) a secondary relaxation peak (p-process) numbers indicate temperature in K. Unfilled symbols represent data obtained from a broad-band spectrometer [6,153]. Filled symbols represent data from a high-precision bridge [137] interpolations for fluoroaniline (solid lines) were done by applying the GGE distribution (a-process) and a Gaussian distribution (p-process) of relaxation times [142], and these for toluene (dashed lines) were done by the gamma distribution (a-process) and a Gaussian distribution (p-process) [6] (cf. Section IV.C.2).

Relevant dielectric results of type B glasses were already discussed in Section IV.C.2. The spectra below Tg exhibit a broad symmetric secondary relaxation peak that can be interpolated assuming a Gaussian distribution of activation enthalpies. Only recently it became clear that also NMR is able to identify secondary relaxation processes in glasses, moreover providing information on the mechanisms of molecular reorientation that is not easily accessible to most of the other methods. For detailed reports the reader is referred to the reviews by Bohmer et al. [11] and Vogel et al. [15]. Here, we summarize the major results. 

Similar heterogeneous model has been used to develop a relaxation function by Chamberlin and Kingsbury (1994), who consider the localized normal modes to be involved in the relaxation process. Localized (domains) regions are assumed to be present between Tg and T. They are described as dynamically correlated domains (DCD). A Gaussian distribution of the domain sizes has been assumed, with each domain characterized by a Debye relaxation time. Expressions for the dielectric susceptibility have been derived and used to fit the experimental susceptibilities of salol, glycerol and many other substances with remarkable agreement over 13 decades of frequency (even when only one adjustable parameter is employed). 

Figure 5 shows DISPA plots for composite line shapes consisting of a superposition of Lorentzian lines centered at a common resonant frequency, whose line widths vary as a log-Gauss distribution (i.e., a Gaussian distribution in log(T/To)). In this case the DISPA plots are displaced centrally inward from the reference circle, and the magnitude of the displacement is directly related to the width of the log-Gauss distribution of relaxation times. This situation is analogous to a distribution in dielectric relaxation times in the Cole-Cole plot, and a similar displacement is observed in that case.5 / 8... 

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