Relaxation, Debye longitudinal

In Debye solvents, x is tire longitudinal relaxation time. The prediction tliat solvent polarization dynamics would limit intramolecular electron transfer rates was stated tlieoretically [40] and observed experimentally [41]. 

Mozumder (1969b) pointed out that in the presence of freshly created charges due to ionization, the dielectric relaxes faster—with the longitudinal relaxation time tl, rather than with the usual Debye relaxation time T applicable for weak external fields. The evolution of the medium dielectric constant is then given by... 

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... 

The correlation function C(t) is purely phenomenological. Interpretation of its time evolution is often based on theory in which the longitudinal relaxation time, tl, is introduced. This time is a fraction of the Debye relaxation time ... 

Experiments in the picosecond time range show that C(t) is non-exponential in most solvents with an average spectral relaxation time greater than the longitudinal relaxation time tl and smaller than the Debye time td-... 

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]

Debye-Onsager model for C(t) and the longitudinal relaxation time Tj... 

The pre-exponential factor in this case includes the solvent longitudinal relaxation time tl, which will be discussed further on when the recent works concerned with the role of the solvent will be considered. This longitudinal relaxation time is related to the usual Debye relaxation time according to... 

This longitudinal relaxation time differs from the usual Debye relaxation time by a factor which depends on the static and optical dielectric constants of the solvent this is based on the fact that the first solvent shell is subjected to the unscreened electric field of the ionic or dipolar solute molecule, whereas in a macroscopic measurement the external field is reduced by the screening effect of the dielectric [73]. 

Recent theoretical treatments, however, suggest instead that the dynamics of solvent reorganization can play an important and even dominant role in determining vn, at least when the inner-shell barrier is relatively small [43-45]. The effective value of vos can often be determined by the so-called longitudinal (or "constant charge ) solvent relaxation time, rL [43, 44]. This quantity is related to the experimental Debye relaxation time, rD, obtained from dielectric loss measurements using [43]... 

In the calculation of the rate constants for comparison with experimental results one uses, in addition to other parameters, the longitudinal relaxation time Tl (see Eqs. (39) and (40)). This parameter, discussed by Friedman [194], is related to the Debye relaxation time by Eq. (37). 

In the analysis of experimental kinetic data, more attention should be paid to a careful determination of the longitudinal relaxation times. In the literature there are discrepancies between permittivities used for calculation of that parameter from the Debye relaxation time. Static dielectric permittivities and, to some extent, the Debye relaxation times exhibit a dependence on the electrolyte concentration. Therefore, in any analysis of the kinetic data, carefully measured and selected values fo the above parameters should be used. 

The Co longitudinal relaxation rates of tris(acetylacetonato)cobalt(lIl) in dichrolo-methane and benzene were measured over the concentration range between 20 and 110 mol m 3 at several temperatures. The dependence of the relaxation rate on the temperature and the complex concentration is primarily attributable to the change in the viscosity of solutioa The values of eQqlh were calculated from the concentration dependence of the Co relaxation rate by using the Debye-Stokes-Einstein equation and the Einstein relationship between molar volume and viscosity B coefflcienL... 

The plot shows a distribution closely around a slope of unity indicated by the solid line in Figure 2 except for the alcohols and nitrobenzene. Such anomaly in alcohols is also reported for other chemical processes and time-dependent fluorescence stokes shifts and is attributed to their non-Debye multiple relaxation behavior " the shorter relaxation components, which are assigned to local motions such as the OH group reorientation, contribute the friction for the barrier crossing rather than the slower main relaxation component, which corresponds to the longitudinal dielectric relaxation time, tl, when one regards the solvent as a Debye dielectric medium. If one takes account of the multiple relaxation of the alcohols, the theoretical ket (or v,i) values inaease and approach to the trend of the other solvents. (See open circles in Figure 2.)... 

The picture in terms of the decoupled Eangevin equations (98) and (99) (omitting the inertial term in Eq. (98) is that the orientational correlation functions of the longitudinal and transverse components of the magnetization in the axially symmetric potential, Kvm sin" . are simply multiplied by the liquid state factor, exp(—t/T ), of the Brownian (Debye) relaxation of the ferrofluid stemming from Eq. (99). As far as the fenomagnetic resonance is concerned, we shall presently demonstrate that this factor is irrelevant. 

The origin of the terms transverse and longitudinal dielectric relaxation times lies in the molecular theory of dielectric relaxation, where one finds that the decay of correlation functions involving transverse and longitudinal components of the induced polarization vector are characterized by different time constants. In a Debye fluid the relaxation times that characterize the transverse and longitudinal components of the polarization are T ) and rp = (ee/eslfD respectively. See, for example, P. Madden and D. Kivelson, J. Phys. Chem. 86, 4244 (1982). 

The well-known continuum models and also the microscopic theories of solvation dynamics suggest a close relation between solvation dynamics and DR. This is expressed as tl = (soo/so)td where tl is the longitudinal relaxation time and td is the Debye relaxation time. However, the solvation dynamics of an ion at the protein surface is difficult to understand because of the heterogeneous environment of the protein surface. Therefore, a straightforward application of the continuum model with a multiexponential description of DR is not possible. The continuum theory suggests that at short length scales, the relaxation time is essentially given by the DR time. Therefore, we certainly expect a slow component in the solvation dynamics. 

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. 

The temperature dependence of the longitudinal relaxation time tl is also an important quantity. For a Debye solvent, tl is given by the relationship... 

Estimation of the longitudinal relaxation time in solvents with multiple Debye relaxation processes is not straightforward. In fact, tl is a function of time in these systems [33, 34], and varies between two limiting values. For a solvent with two relaxation processes, the low-frequency limit for xl is... 

Table 7.9 Reactant System Characteristic Frequencies in Selected Debye Solvents Estimated Using Equation (7.10.12) Together with the Solvent Molecule s Effective Moment of Inertia and the Reciprocal of the Longitudinal Relaxation Time...

The longitudinal relaxation time for acetonitrile given a Debye relaxation time of 3.2ps is... 

In Section V we are concerned with Gilbert s equation as applied to the Debye relaxation of a ferrofluid particle with the inertia of the particle included. It is shown, by averaging Gilbert s equation for Debye relaxation corrected for inertia and proceeding to the noninertial limit, how analytic expressions for the transverse and longitudinal relaxation times for Debye relaxation may be obtained directly from that equation thus bypassing the Fokker-Planck equation entirely. These expressions coincide with the previous results of the group of Shliomis [16]. 

It is of interest to compare these results with those for the field dependencies of the relaxation times and for T for the longitudinal and for the transverse polarization components of a polar fluid in a constant electric field Eq. As shown in [52, 55] the relaxation times and T are also given by Eqs. (5.55) and (5.56), where = nEJkT, p. is the dipole moment of a polar molecule and is the Debye rotational diffusion time with = 0. Thus, Eqs. (5.55) and (5.56) predict the same field dependencies of the relaxation times Tj and T for both a ferrofluid and a polar fluid. This is not unexpected because from a physical point of view the behavior of a suspension of fine ferromagnetic particles in a constant magnetic field Hg is similar to that of a system of electric dipoles (polar molecules) in a constant electric field Eg. 

Figure 17. Time-resolved fluorescence spectra of a solute with one vibrational mode in ethanol at 247 K.68 The various frames show the fluorescence spectrum measured at successively later times after the application of a 1 ps excitation pulse. Each spectrum is labeled with the observation time. The steady-state fluorescence spectrum is given by the dashed curve in the bottom frame. In the electronic ground state, the solute vibrational frequency is400cm 1, and in the excited state, the frequency is 380 cm 1. The dimensionless displacement is 1.4. The permanent dipole moment changes by 10 Debye upon electronic excitation. The Onsager radius is 3A. The longitudinal dielectric relaxation time, xL, is 150 ps.

Working in dilute solution (< 80 mmolP ) is recommended whenever the Stokes-Einstein-Debye model is employed for data analysis, as illustrated in the study of Co longitudinal relaxation time of Co(acac)3 in CH3CN. ... 

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