Debye relaxation mode

The frequency dependence of the linear susceptibility (4.102) is determined by a superposition of the Debye relaxation modes as... 

Implicit in (9.40) is the assumption that co is small compared with lattice vibrational frequencies. The susceptibility in the frequency region where Debye relaxation is the dominant mode of polarization is therefore... 

Since the reduced spectrum x"( ) clearly shows the low-ftequency Raman modes, we introduced a simple model to analyze the spectral profile of x"(.v) for obtaining the quantitative information. The model is composed of two damped harmonic oscillator modes and one Debye type relaxation mode (liquid water) or one Cole-Cole type relaxation mode (aqueous solution). Cole-Cole type relaxation is usually adopted in analyzing the dielectric relaxation. The formula of Cole-Cole type relaxation is represented as ... 

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). 

Since the frequency-dependent response of the solvent is included in the kernel of the integrals collected into X , the resulting hyperpolarizabilities will also depend on the frequency spectrum of the dielectric function e(u ) of the solvent. When e(w) is described by the Debye formula (i.e. in terms of a single relaxation mode), i.e. 

The time dependent solvation funetion S(t) is a directly observed quantity as well as a convenient tool for numerical simulation studies. The corresponding linear response approximation C(t) is also easily eomputed from numerical simulations, and can also be studied using suitable theoretical models. Computer simulations are very valuable both in exploring the validity of such theoretical calculations, as well as the validity of linear response theory itself (by comparing S(t) to C(t)). Furthermore they can be used for direct visualization of the solute and solvent motions that dominate the solvation process. Many such simulations were published in the past decade, using different models for solvents such as water, alcohols and acetonitrile. Two remarkable outcomes of these studies are first, the close qualitative similarity between the time evolution of solvation in different simple solvents, and second, the marked deviation from the simple exponential relaxation predicted by the Debye relaxation model (cf Eq. [4.3.18]). At least two distinct relaxation modes are... 

However, in polymer systems there can be many internal relaxation modes and it is unreasonable to assume that a single relaxation process is responsible for the complex TSC curves typically recorded. In order to deconvolute complex TSC spectra into individual relaxation processes, where the Debye and Arrhenius relations are more applicable, two approaches are used. The first approach is called the partial heating method or peak cleaning method (Figure 6.29(H)). Following quenching and extinction of the applied electric field, the TSC curve of the polarized sample is measured as the sample is heated at a controlled rate to. The sample is requenched from to Tq, and the TSC curve is subsequently... 

The root. i simply indicates that infinite distances are correlated with infinite time, S2 is the reciprocal of the Debye relaxation time, and 3 is the kinetic relaxation frequency of the system. Depending on the kinetic parameters of the chemical process, the kinetic relaxation frequency can be faster or slower than the Debye frequency of the system. If the kinetic relaxation frequency is much smaller than the Debye mode, it can be determined experimentally by conductance fluctuation analysis. 

Processes during a cell cycle are evidenced to be closely controlled cooperative events, including synchronisation within the ensemble. This caused us to describe relaxation within each cell by a Debye process, the relaxation time of which should increase with the size of the cell involved ( finite-size effect ). In that way ensemble structure and relaxation processes of cell ensembles are strictly interrelated. The universal energy density distribution and the universal relaxation mode distribution turn out to be copies of each other. Consequently, the spectrum depends only on the universal properties of the ensemble structure, i.e. on the value of p. Since all the cell populations studied here belong to the / = 3 class, the linear relaxation behaviour should show the same features. 

Let us now apply this Debye model to the orientational relaxation modes of a nematic with fixed director, n (f) = const, assuming that each corresponding to the orientational modes in Eq. (10-6) follows the Debye behavior (11). Using Eq. (10-5) and (10-6), we find... 

It is important from a practical viewpoint to predict the shear viscosity of mixtures from those of pure melts. For alkali nitrate melts, a linear dependence has been found between the reorientational line width obtained by Raman measurements and the ratio of temperature divided by shear viscosity.For NO3 ions, the depolarized Raman scattering from 1050cm" total stretching vibrational mode (Al) has a contribution to the line width L, which is caused by the reorientational relaxation time of the Csv axis of this ion. The Stokes-Einstein-Debye(SED) relation establishes a relation between the shear viscosity r of a melt and the relaxation time for the reorientation of a particle immersed in it ... 

Consider a mixture of acoustic-mode (rL) and ionized-impurity (r,) scattering. For tL t, we would expect r 0 = 1.18 and for r, tl, rn0 = 1.93. But for intermediate mixtures, r 0 goes through a minimum value, dropping to about 1.05 at 15% ionized-impurity scattering (Nam, 1980). For this special case (sL = i, s, = — f), the integrals can be evaluated in terms of tabulated functions (Bube, 1974). For optical-mode scattering the relaxation-time approach is not valid, at least below the Debye temperature, but rn may still be obtained by such theoretical methods as a variational calculation (Ehrenreich, 1960 Nag, 1980) or an iterative solution of the Boltzmann equation (Rode, 1970), and typically varies between 1.0 and 1.4 as a function of temperature (Stillman et al., 1970 Debney and Jay, 1980). 

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]

An alternative approach that was used in the past was to treat the photoelectrochemical cell as a single RC element and to interpret the frequency dispersion of the "capacitance" as indicative of a frequency dispersion of the dielectric constant. (5) In its simplest form the frequency dispersion obeys the Debye equation. (6) It can be shown that in this simple form the two approaches are formally equivalent (7) and the difference resides in the physical interpretation of modes of charge accumulation, their relaxation time, and the mechanism for dielectric relaxations. This ambiguity is not unique to liquid junction cells but extends to solid junctions where microscopic mechanisms for the dielectric relaxation such as the presence of deep traps were assumed. 

The concentration dependence of ionic mobility at high ion concentrations and also in the melt is still an unsolved problem. A mode coupling theory of ionic mobility has recently been derived which is applicable only to low concentrations [18]. In this latter theory, the solvent was replaced by a dielectric continuum and only the ions were explicitly considered. It was shown that one can describe ion atmosphere relaxation in terms of charge density relaxation and the elctrophoretic effect in terms of charge current density relaxation. This theory could explain not only the concentration dependence of ionic conductivity but also the frequency dependence of conductivity, such as the well-known Debye-Falkenhagen effect [18]. However, because the theory does not treat the solvent molecules explicitly, the detailed coupling between the ion and solvent molecules have not been taken into account. The limitation of this approach is most evident in the calculation of the viscosity. The MCT theory is found to be valid only to very low values of the concentration. 

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