Dielectric relaxation characteristic time

For transport in amorphous systems, the temperature dependence of a number of relaxation and transport processes in the vicinity of the glass transition temperature can be described by the Williams-Landel-Ferry (WLF) equation (Williams, Landel and Ferry, 1955). This relationship was originally derived by fitting observed data for a number of different liquid systems. It expresses a characteristic property, e.g. reciprocal dielectric relaxation time, magnetic resonance relaxation rate, in terms of shift factors, aj, which are the ratios of any mechanical relaxation process at temperature T, to its value at a reference temperature 7, and is defined by... 

Fig. 4.8 Temperature dependence of the dielectric characteristic times obtained for PB for the a-relaxation (empty triangle) for the r -relaxation (empty diamond), and for the contribution of the -relaxation modified by the presence of the a-relaxation (filled diamond). They have been obtained assuming the a- and -processes as statistically independent. The Arrhenius law shows the extrapolation of the temperature behaviour of the -relaxation. The solid line through points shows the temperature behaviour of the time-scale associated to the viscosity. The dotted line corresponds to the temperature dependence of the characteristic timescale for the main peak. (Reprinted with permission from [133]. Copyright 1996 The American Physical Society)...

Fig. 4.9 Temperature dependence of the characteristic time of the a-relaxation in PIB as measured by dielectric spectroscopy (defined as (2nf ) ) (empty diamond) and of the shift factor obtained from the NSE spectra at Qmax=l-0 (filled square). The different lines show the temperature laws proposed by Tormala [135] from spectroscopic data (dashed-dotted), by Ferry [34] from compliance data (solid) and by Dejean de la Batie et al. from NMR data (dotted) [136]. (Reprinted with permission from [125]. Copyright 1998 American Chemical Society)...

Finally, recently depolarized light scattering spectra [191] display an additional process that shows a much faster characteristic time and a much weaker temperature dependence than the dielectric j0-relaxation (more than three orders of magnitude faster time at -200 K and an activation energy of 0.16 eV, about half of the dielectric value). Also atomistic simulations on PB have indicated hopping processes of the frans-double bond [192,193] with an associated activation energy of -0.15 eV. Whether these observations may be related with the discrepancy in the apparent time scale of the NSE and dielectric experiments remains to be seen. 

The first thought experiment corresponds to dielectric measurements. It involves applying a voltage to a capacitor containing a dielectric medium at t = 0, and then holding the voltage constant at t > 0. The dependent variable is the time-dependent current which decays as dielectric relaxation of the medium occurs. From the current, the characteristic relaxation time of the time-dependent displacement ( >(r))) field can be calculated. The time is td. This is essentially a time domain analog of e(cu) dielectric measurements. 

The second thought experiment resembles transient solvation. At t = 0, a certain amount of charge is put on the capacitor plates. This charge jump (D field jump) is analogous to the photon induced change of the dipole moment in the fluorescence solvation experiment. Subsequently (t > 0), the decay of the voltage on the capacitor due to dielectric relaxation of the medium is measured. Note the capacitor in this experiment is not connected to an external power supply for t > 0. The characteristic relaxation time for the decay of the voltage (and electric field E) is t,. 

Two more difficulties require comment. The first is that in most of the early literature, authors did not recognize the importance of electrode polarization, and, hence, failed to make quantitative allowance for the presence of blocking and/or release layers. Thus, in most cases, it is not possible to reconstruct quantitative bulk properties from the data presented. (The present authors were not immune. They reported a correlation between a dielectric relaxation time and viscosity54), failing at that time to realize that the relaxation time being studied was actually the characteristic time for electrode polarization, and, hence was dominated by conductivity.)... 

Here r is a characteristic time constant, usually called the dielectric relaxation time. To conform to analogous theories of visco-elastic behaviour we should really use the term dielectric retardation time, because it refers to a gradual change in a strain (the polarisation or resulting electric displacement) following an abrupt change in stress (the applied field). Dielectric relaxation time is, however, still most commonly used in spite of this inconsistency. By integration of Equation (3.12) ... 

Measurements of the dynamic properties of the surface water, particularly NMR measurements, have shown that the characteristic time of the water motion is slower than the bulk water value by a factor of less than 100. The motion is anisotropic. There is litde or no irrotadonally bound water. Study of a protein labeled covalently with a nitroxide spin probe (Polnaszek and Bryant, 1984a,b) has shown that the diffusion constant of the surface water is about 5-fold below the bulk water value. The NMR results are in agreement with measurements of dielectric relaxation of water in protein powders (Harvey and Hoekstra, 1972). 

As reasonably expected, the agreement between Xq and I/t, is especially good in the low-temperature region. Furthermore, the other eigenvalues are proven to correspond to frequencies much higher than the characteristic frequency of the first band. Therefore, this theoretical result reproduces the experimental data which, at microwave frequencies, can be fitted by a single dielectric relaxation time. 

I) The H-bond mean lifetime, thb> from mf-18- (2) The probability that an hydrogen bond is randomly intact, pg, from Eq. (S.2). (3) The relaxation time calculated from Eq. (6.1) with conditions (6.2). (4) Measured relaxation time r, from refs. 52 and S3 (+) tg, values foreseen by our model using the parameters in columns 5-7, which list the rotational diffusion parameters utilized for a more accurate calculation of dielectric behavior. (8) The principal relaxation band amplitude Ag. (9) and (10) The characteristic time l/Xj and the amplitude A of the only other significant band, as calculated from Eqs. (6.1) and (6.4), respectively. 

The diffusion equations involved in our theoretical analysis, which are derived via the AEP from the fluctuating reduced model of Section IV, can be regarded as a five-state version of the Anderson two-state model supplemented by a quantitative description of the secondary process. We would especially stress that, in accordance with the point of view of other authors, the characteristic time of the principal dielectric relaxation band roughly coincides with the residence time in the structured part of the liquid. ... 

Now, we shall demonstrate that the characteristic times of the normal diffusion process, namely, the inverse of the smallest nonvanishing eigenvalue 1 //.], the integral and effective relaxation times xint and xef obtained in [8,62,63], also allow us to evaluate the dielectric response of the system for anomalous diffusion using the two-mode approximation just as normal diffusion (Ref. 8, Section 2.13). Here, we can use known equations for xint, x,f, and X for the normal diffusion in the potential Eq. (163) [8,62,63] these equations are... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

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