Debye relaxation dielectric polarization, time-dependent

Relaxation processes are probably the most important of the interactions between electric fields and matter. Debye [6] extended the Langevin theory of dipole orientation in a constant field to the case of a varying field. He showed that the Boltzmann factor of the Langevin theory becomes a time-dependent weighting factor. When a steady electric field is applied to a dielectric the distortion polarization, PDisior, will be established very quickly - we can say instantaneously compared with time intervals of interest. But the remaining dipolar part of the polarization (orientation polarization, Porient) takes time to reach its equilibrium value. When the polarization becomes complex, the permittivity must also become complex, as shown by Eq. (5) ... 

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]... 

The first process prevails at relatively low frequencies. The electric component E of radiation orients dipole moments p along the field direction, while chaotic molecular motions hinder this orientation p and E are the vectors, and the field E is assumed to vary harmonically with time t. Due to inertia of reorienting molecules the time dependence of the polarization lags behind the time dependence E(f), so that heating of the medium occurs (the heating effect is not considered in this work). The dielectric spectrum obeys the Debye relaxation, for which the absorption monotonically increases with frequency. 

Let us first discuss estimates fi om DR measurements that provide several important pieces of information. These experiments measure the frequency-dependent dielectric constant and provide a measure of a liquid s polarization response at different frequencies. In bulk water, we have two dominant regions. The low-frequency dispersion gives us the well-known Debye relaxation time, Tq, which is equal to 8.3 ps. There is a second prominent dispersion in the high-frequency side with relaxation time constant less than Ips which contains combined contributions from low-frequency intermolecular vibrations and libra-tion. Aqueous protein solutions exhibit at least two more dispersions, (i) A new dispersion at intermediate frequencies, called, d dispersion, which appears at a timescale of about 50 ps in the dielectric spectrum, seems to be present in most protein solutions. This additional dispersion is attributed to water in the hydration layer, (ii) Another dispersion is present at very low frequencies and is attributed to the rotation of the protein. 

Maintaining polar order in a poled pol5uner is of great importance for second-order applications (88,89). The dielectric relaxation process leading to decay in the orientation of ordered polymers has been studied extensively and is the subject of another article (see Dielectric Relaxation). Several models that describe the chromophore reorientation for NLO materials have been proposed, including the Kohlrausch-Williams-Watts (KWW) model (90,91), biexponential and triexponential decay models (92), time-dependent Debye relaxation time models (93), and the Liu-Ramkrishna-Lackritz (LRL) model (94). For further information on... 

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 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. ... 

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... 

Such numerical simulations have played an important role in the development of our understanding of solvation dynamics. For example, they have provided the first indication that simple dielectric continuum models based on Debye and Debey-like dielectric relaxation theories are inadequate on the fast timescales that are experimentally accessible today. It is important to keep in mind that this failure of simple theories is not a failure of linear response theory. Once revised to describe reliably response on short time and length scales, e.g. by using the full k and (O dependent dielectric response function e(k,o , and sufficiently taking into account the solvent structure about the solute, linear response theory accounts for most observations of solvation dynamics in simple polar solvents. 

At sufficiently high frequencies the orientation polarization lags behind the measuring field, resulting in a decrease of the permittivity. Simultaneously, the system absorbs energy that appears as a dielectric loss. The frequency-dependent dielectric constant is usually expressed as a complex permittivity e = e - ie". In the simplest case of a single relaxation time t, the real and imaginary part of e can be described with the well-known Debye equations ... 

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