Dipole systems dielectric relaxation

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

In the second place, we shall study rotational dynamics. Rotational processes are of fundamental importance for dielectric relaxation. To shed light on some controversial issues in dielectric relaxation, Brot and co-workers did a computer simulation of a system of disks interacting via both Lennard-Jones potentials and electric dipole-dipole couplings. This is pre-... 

Similar anomalous kinetic behavior in alcohols was observed for other systems as well [148, 173]. Weaver and coworkers [3] discussed this behavior, considering the influence of high-frequency dielectric relaxation associated with rotation of solvent monomers and with translational motion of solvent dipoles. 

Relaxation of Interacting Dipoles.— The elementary systems considered so far may, of course, be used directly as models for the dielectric relaxation of bulk material, with more or less success. It is much more interesting to treat them as realistic models of molecular behaviour, but then the relation between the relaxing orientation of one molecule and the relaxing polarization of a bulk sample requires further attention. 

With local information given by INM analysis in mind, we next see the character of rotational relaxation in liquid water. The most familiar way to see this, not only for numerical simulations [76-78] but for laboratory experiments, is to measure dielectric relaxation, by means of which total or individual dipole moments can be probed [79,80]. Figure 10 gives power spectra of the total dipole moment fluctuation of liquid water, together with the case of water cluster, (H20)io8- The spectral profile for liquid water is nearly fitted to the Lorentzian, which is consistent with a direct calculation of the correlation function of rotational motions. The exponential decaying behavior of dielectric relaxation was actually verified in laboratory experiments [79,80]. On the other hand, the profile for water cluster deviates from the Lorentzian function. As stated in Section III, the dynamics of finite systems may be more difficult to be understood. 

FRACTIONAL ROTATIONAL DIFFUSION AND ANOMALOUS DIELECTRIC RELAXATION IN DIPOLE SYSTEMS... 

In the fixed axis rotation model of dielectric relaxation of polar molecules a typical member of the assembly is a rigid dipole of moment p rotating about a fixed axis through its center. The dipole is specified by the angular coordinate < ) (the azimuth) so that it constitutes a system of 1 (rotational) degree of freedom. The fractional diffusion equation for the time evolution of the probability density function W(4>, t) in configuration space is given by Eq. (52) which we write here as... 

The principal result of our calculation is that the Debye theory (based on the Smoluchowski equation), when extended to fractional dynamics via a onedimensional noninertial fractional Fourier-Planck equation in configuration space, can explain the Cole-Cole anomalous dielectric relaxation that appears in some complex systems and disordered materials. A further result of our calculation is that the aftereffect solution [Eq. (66)] is, with slight modifications, the moment generating function of the configuration space distribution function. Hence the mean-square angular displacement of a dipole, and so on, may be easily calculated by differentiation. We must remark, however, that the fractional Debye theory can be used only at low frequencies (got < 1) just as... 

Dielectric relaxation (DR) experiments measure the collective polarization response of all the polar molecules present in a given system. The DR time provides a measure of the time taken by a system to reach the final (equilibrium) polarization after an external field is suddenly switched on (or off). DR measures the complex dielectric fimction, s(w), that can be decomposed into real and imaginary parts as efca) = s (o) — is" (o) where s (co) and s fo ) are the real (permittivity factor) and imaginary (dielectric loss) parts, respectively. The total dipole moment of the system, at any given time t, M(t) = fift) where N is the total number of dipolar molecules and /Af is the dipole moment vector of the ith molecule. The complex dielectric function e((w) is given by the following relation. 

Enthalpy relaxation studies have also been used to assess the aging of polyether ether ketone blends with polyetherimide, PEEK/PEI = 50/50 (Hay 1992). The preparation of the blend produced an amorphous system with Tg 215 °C, but crystallization of the PEEK occurred after raising the temperature above Tg. Data could only be collected in the temperature range Tg to Tg — 50) and no aging could be detected at temperatures below 150 °C (Table 13.7). The enthalpic relaxation data were analyzed using the C-E model and fits )delded values of p = 0.4, intermediate between those of the pure polymers. It was noted that values of p were higher than those obtained from dielectric relaxation measurements (0.1-0.22). This technique probes the dipole relaxation spectmm, and the lower... 

The approach presented here was first developed for the dielectric relaxation of regular mesh-like polymer networks built from macromolecules with longitudinal dipole moments [30], and was later applied to disordered polymer networks [31,32]. Its key assiunption, namely the absence of any correlations in the orientations of the dipole moments of the different GGS bonds is obviously rather simphfied. However, it leads, as shown above, to simple analytical expressions for the dielectric susceptibility, a very important dynamical quantity in experimental studies of polymers we can now analyze it in great detail for particular GGS systems of interest. Another advantage of this model arises from the fact that one has a straightforward correspondence between the mechanical and the dielectric relaxation forms. From the expressions for the storage and loss modulus, Eqs. 20 and 21, and from those for the dielectric susceptibility Ae, Eqs. 41 and Eq. 42, one sees readily that [31]... 

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