Relaxation time Debye polarization

It has been noted that the conditions for observation of this type of absorption are much more favourable in solid polymers than in liquids. The point is that the relaxation times of polar liquids are of the order of magnitude of 10 s this means that the peak of Debye relaxation process will occur in the 1-10 cm" region and the broad absorption will extend through the region in which the Poley band is found. For polymers, however, the relaxation times are commonly much longer, typically ca 10" s, and the peak frequency of the Debye process is moved to a lower frequency therefore, the probability of resolving it from the Poley absorption is much higher. 

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]. 

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]... 

When a constant electric field is suddenly applied to an ensemble of polar molecules, the orientation polarization increases exponentially with a time constant td called the dielectric relaxation time or Debye relaxation time. The reciprocal of td characterizes the rate at which the dipole moments of molecules orient themselves with respect to the electric field. 

The remaining types of polarization are absorptive types with characteristic relaxation times corresponding to relaxation frequencies. Debye, in 1912, suggested that the high dielectric constants of water, ethanol, and other highly polar molecules were due to the presence of permanent dipoles within each individual molecule and that there is a tendency... 

Similar expressions can be generated for holes simply by letting coc - — relaxation time xB needs justification, which will not be attempted here. Suffice it to say that this assumption is not bad for elastic scattering processes, which include most of the important mechanisms. A well-known exception is polar optical-phonon scattering, at temperatures below the Debye temperature (Putley, 1968, p. 138). We have further assumed here that t is independent of energy, although this condition will be relaxed later. 

Complex dielectric susceptibility data such as those in Figure 15.6 provide a detailed view of the dynamics of polar nanodomains in rls. They define relaxation frequencies, /, corresponding to the e (T) peak temperatures Tm, characteristic relaxation times, r = 1/tu (where uj = 2nf is the angular frequency), and a measure of the interaction among nanodomains as represented by the deviation of the relaxation process from a Debye relaxation. Analysis of data on pmn and other rls clearly shows that their dipolar relaxations cannot be described by a single relaxation time represented by the Debye expression... 

Figure 15.8 A model of temperature-dependent relaxation time spectrum for PMN. G (r, T) is the number of polar regions having a relaxation time r, T/ is the freezing temperature, and td is the inverse Debye frequency (from [16]).

Debye obtained his result by solving a forced diffusion equation Ci.e., with torque of the applied field included) for the distribution of dipole coordinate p - pcosS, with 6 the polar angle between the dipole axis and tSe field, and the same result for the model follows very simply from equation (3) using the time dependent distribution function in the absence of the field (5). The relaxation time is given by td = 1/2D, which for a molecular sphere of volume v rotating in fluid of viscosity n becomes... 

P. J. W. Debye, Polar Molecules (Dover, New York, reprint of 1929 edition) presents the fundamental theory with stunning clarity. See also, e.g., H. Frohlich, "Theory of dielectrics Dielectric constant and dielectric loss," in Monographs on the Physics and Chemistry of Materials Series, 2nd ed. (Clarendon, Oxford University Press, Oxford, June 1987). Here I have taken the zero-frequency response and multiplied it by the frequency dependence of the simplest dipolar relaxation. I have also put a> = if and taken the sign to follow the convention for poles consistent with the form of derivation of the general Lifshitz formula. This last detail is of no practical importance because in the summation Jf over frequencies fn only the first, n = 0, term counts. The relaxation time r is such that permanent-dipole response is dead by fi anyway. The permanent-dipole response is derived in many standard texts. 

The change in the emission spectrum with time after pulsed excitation (TRES) is a method for assessing the overall response of the solvent to a change in solute geometry or polarity [22]. The precise values of the relaxation times depend upon the method of measurement. At room temperature the TRES solvent correlation times are subnanosecond and, in some cases subpicosecond. The Debye relaxation time in water is 8 ps, while the TRES correlation time is shorter [22]. Although there is not, in general, a... 

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 polarizability of the individnal molecules is also frequency dependent, but the characteristic values are of the order of lO Vs and lO Vs for the rotational and electronic polarization, respectively. " Therefore, in the typical frequency domain for investigation of dispersions (1/s < co < 10 /s) the polarizability, e, of the material building up the particles is frequency independent. On the other hand, the disperse medium (which is usually an electrolyte solution) has a dielectric permittivity, Ej, for which the freqnency dependence can be described by the Debye-Falkenhagen theory. Besides, the characteristic relaxation time of the bulk electrolyte solutions is also given by Eqnation 5.385. ... 

This behavior is analogous to that of a polar molecule in a fluid under the influence of an electric field. This was studied by Debye [14], who obtained the relaxation time... 

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. 

Debye established this model in the case of dilute solutions of polar molecules [122], A dipole with different possible orientations is considered. A dipole which is displaced from its equilibrium state, tends to return to it progressively, with a relaxation time r. 

Debye s model The Debye model could be built with these assumptions, and polarization and permittivity become complex as described by Eq. (18) where n is the refractive index and t the relaxation time ... 

Within the complex plane, two circles are obtained. The overlapping of these two circles depends on the vicinity of the relaxation time or relaxation frequency of the two polar groups. This assumption could be applied to more than two polar groups. Are there two isolated Debye s relaxations or a distribution of relaxation times for a single relaxation process If the latter, it is better to use the Cole and Cole or Davidson and Cole models. Results from permittivity measurements are often displayed in this type of diagram. The disadvantage of these methods is that the frequency is not explicitly shown. 

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