Dielectric relaxation and the Debye model

Assuming the validity of standard linear dielectric response theory, the electrostatic displacement and the electrostatic field 5 in a dielectric medium are related to each other by 

In what follows we assume that the response is local, that is, e(r — r, t — / ) = e(r, t — — I ) This assumption is not really valid for dielectric response 

To account for the fast and slow components of the dielectric response we take e(Z) in the form 

The Debye model iskQS for the slow part of the dielectric response the form 

The dielectric response in this model is thus characterized by three parameters the electronic Sg and static Sg response constants, and the Debye relaxation time ro. 

In what follows we assume that the response is local, that is, e(r — r, / — / ) = e(r,t — t )S r — r ). This assumption is not really vaUd for dielectric response on molecular lengthscales, but the errors that result from it appear to be small in many cases while the mathematical simplification is considerable. Also, while in general the dielectric response is a tensor, we take it for simplicity to be a scalar, that is, we consider only isotropic systems. In this case it is sufficient to consider the magnitudes V and of T and . Thus, our starting point is the local scalar relationship 

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As a second example, we consider liquid fluoromethane CH3F, which is a typical strongly absorbing nonassociated liquid. For our study we choose the temperature T 133 K near the triple point, which is equal to 131 K. The relevant experimental data [43] were summarized in Table IV. As we see in Table VIII, which presents the fitted parameters of the model, the angle p is rather small. At this temperature the density p of the liquid, the maximum dielectric loss and the Debye relaxation time rD are substantially larger than they would be, for example, near the critical temperature (at 293 K). At such small (5 the theory given here for the hat-curved model holds. For calculation of the complex permittivity s (v) and absorption a(v), we use the same formulas as for water. 

Even if we consider a single solvent, e g., water, at a single temperature, say 298K, depends on the solute and in fact on the coordinate of the solute which is under consideration, and we cannot take xF as a constant. Nevertheless, in the absence of a molecular dynamics simulation for the solute motion of interest, XF for polar solvents like water is often approximated by the Debye model. In this model, the dielectric polarization of the solvent relaxes as a single exponential with a relaxation time equal to the rotational (i.e., reorientational) relaxation time of a single molecule, which is called Tp) or the Debye time [32, 347], The Debye time may be associated with the relaxation of the transverse component of the polarization field. However the solvent fluctuations and frictional relaxation occur on a faster scale given by [348,349]... 

This approximation requires that cos. This behavior in fact follows from a Debye dielectric continuum model of the solvent when it is coupled to the solute nuclear motion [21,22] and then xs would be proportional to the longitudinal dielectric relaxation time of the solvent indeed, in the context of time dependent fluorescence (TDF), the Debye model leads to such an exponential dependence of the analogue... 

In the simplest model investigated, including a single Debye mode (X(f) -exp(-t/ r, ), xL being the longitudinal dielectric relaxation time), the spectral effect was found to be small and negative -0.2 <[Pg.332]

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... 

Relaxations observed in polymers show broader dispersion curves and lower loss maxima than those predicted by the Debye model, and the (s" s ) curve falls inside the semicircle. This led Cole and Cole (1941) to suggest the following semi-empirical equation for dielectric relaxations in polymers ... 

The dielectric constant is a natural choice of order parameter to study freezing of dipolar liquids, because of the large change in the orientational polarizability between the liquid and solid phases. The dielectric relaxation time was calculated by fitting the dispersion spectrum of the complex permittivity near resonance to the Debye model of orientational relaxation. In the Debye dispersion relation (equation (3)), ij is the frequency of the applied potential and t is the orientational (rotational) relaxation time of a dipolar molecule. The subscript s refers to static permittivity (low frequency limit, when the dipoles have sufficient time to be in phase with the applied field). The subscript oo refers to the optical permittivity (high frequency limit) and is a measure of the induced component of the permittivity. 

The dielectric a-relaxation of the DGEBA oligomer is not fitted to the Debye model that is applicable to simple liquids. The correlation between r and x of the... 

Other relationships which have been used to describe dielectric relaxation data include the Cole-Cole and Cole-Davidson equations [29]. These are preferred when a distribution of relaxation times rather than a single relaxation time is more appropriate to describe the data in a given frequency range. Nevertheless, the Debye model in its simple version or multiple relaxation versions works quite well for most of the solvents considered here. 

According to the Debye model there are three parameters associated with dielectric relaxation in a simple solvent, namely, the static permittivity s, the Debye relaxation time td, and the high-frequency permittivity Eoq. The static permittivity has already been discussed in detail in sections 4.3 and 4.4. In this section attention is especially focused on the Debye relaxation time td and the related quantity, the longitudinal relaxation time Tl. The significance of these parameters for solvents with multiple relaxation processes is considered. The high-frequency permittivity and its relationship to the optical permittivity Eop is also discussed. 

This estimate of 19 ps is much larger than the experimental estimate of 4 ps, which is obtained by applying the Debye model to dielectric relaxation data for the pure solvent (see table 4.2). However, an approximate relationship between td and t r is found when data for more solvents are considered, as shown... 

The role of vibrational relaxation and solvation dynamics can be probed most effectively by fluorescence experiments, which are both time- and frequency-resolved,66-68 as indicated at the end of Sec. V. We have recently developed a theory for fluorescence of polar molecules in polar solvents.68 The solvaion dynamics is related to the solvent dielectric function e(co) by introducing a solvation coordinate. When (ai) has a Lorentzian dependence on frequency (the Debye model), the broadening is described by the stochastic model [Eqs. (113)], where the parameters A and A may be related to molecular... 

The width of the dielectric loss peak given by equation (9.28b) can be shown to be 1.14 decades (see problem 9.3). Experimentally, loss peaks are often much wider than this. A simple test of how well the Debye model fits in a particular case is to make a so-called Cole-Cole plot, in which s" is plotted against e. It is easy to show from equations (9.28) that the Debye model predicts that the points should lie on a semi-circle with centre at [(fis + oo)/2, 0] and radius (e — Soo)/2- Figure 9.3 shows an example of such a plot. The experimental points lie within the semi-circle, corresponding to a lower maximum loss than predicted by the Debye model and also to a wider loss peak. A simple explanation for this would be that, in an amorphous polymer, the various dipoles are constrained in a wide range of different ways, each leading to a different relaxation time r, so that the observed values of s and e" would be the averages of the values for each value of r (see problem 9.4). 

But Jonscher was apparently not aware that Cole already in 1928 used the circle segment analysis in the Wessel plane and found many circular arcs with suppressed circle centers. The concept of CPE was introduced, and in Cole and Cole (1941), the idea was introduced that a dielectric could have a distribution of many relaxation time constants. The Debye model with ideal components presupposed one single relaxation time constant and therefore a complete semicircle. However, the Cole—Cole model implied that the distributed time constants do not correspond to one exponential, but a fractional power law. It seems that Jonscher (1996) did not accept the flieory of distributed... 

In the case of DiMarzio and Bishop, they solved the hydrodynamic equations for the Debye model and the non-Newtonian case exactly. The important result of their analysis is that the dielectric response is no longer a Debye type but depends explicitly on how the local viscosity depends on time. In other words the nature of the viscoelastic properties surrounding the sphere determines the shape of the dielectric relaxation process. This result is in marked contrast to the results of the model... 

The advantage of the H-H model is that the starting point is a more general formulation of the dielectric relaxation problem, that is, it is less specific than is the Debye model. Another advantage of this approach is that the relationship between strain and electrostatic energies is clearly incorporated into the model. This incorporation has the effect of approximating real molecules as point dipoles situated on bodies that have an arbitrary shape. Furthermore it is reasonable to assume that the relationship between dipole moment and shape factor is given by a tensor. In any case there is no reason to assume that the moment of the sphere and its distortion are collinear when the electric field is applied. 

In the study of dielectric relaxation, temperature is an important variable, and it is observed that relaxation times decrease as the temperature increases. In Debye s model for the rotational diffusion of dipoles, the temperature dependence of the relaxation is determined by the diffusion constant or microscopic viscosity. For liquid crystals the nematic ordering potential contributes to rotational relaxation, and the temperature dependence of the order parameter influences the retardation factors. If rotational diffusion is an activated process, then it is appropriate to use an Arrhenius equation for the relaxation times ... 

The DFN material chosen to verify Eq. (10-22) was the mixture MLC2048 (EM Industries, NY), as its dielectric properties are well described by the Debye model, see Refs. [11, 12] and Figure 10-3. The relaxation time has been determined to be T = 13.4 xs at room temperature [11]. 

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