The Debye Relaxation Model

The return to equilibrium of a polarized region is quite different in the Debye and Lorentz models. Suppose that a material composed of Lorentz oscillators is electrically polarized and the static electric field is suddenly removed. The charges equilibrate by executing damped harmonic motion about their equilibrium positions. This can be seen by setting the right side of (9.3) equal to zero and solving the homogeneous differential equation with the initial conditions x = x0 and x = 0 at t = 0 the result is the damped harmonic oscillator equation  

There is no oscillation the polarization merely relaxes toward zero with a time constant t. In the following paragraphs, we shall use (9.35), the basic assumption of the Debye theory, to derive an expression for the dielectric function of a collection of permanent dipoles. 

If at time t0 a constant electric field E0 (i.e., a step function) is suddenly applied to a sample of polarizable matter, the polarization will follow the time evolution illustrated schematically in Fig. 9.13. There are contributions to the polarization from electrons, lattice ions, and permanent dipoles. We shall assume that the response of the permanent dipoles is much slower than that of the electrons and ions that is, the response of the latter may be considered to occur instantaneously compared with the time required for the matter to reach 

With a bit of rearranging (9.36) can be put in a form which is easier to interpret and which will enable us to generalize it to a series of step functions  

The first term on the right side of (9.37) is the contribution to the total 

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Figure 9.15 Dielectric function of water at room temperature calculated from the Debye relaxation model with r = 0.8 X 10 11 sec, eQcl = 77.5, and e0l, = 5.27. Data were obtained from three sources Grant et al. (1957), Cook (1952), and Lane and Saxton (1952).

The fact that estimates of Xq are the correct order of magnitude suggests that there is an approximate relationship between xp and p, at least for liquids which obey the Debye relaxation model (see section 4.5). This point is illustrated in fig. 6.2 where Xj) is plotted against with logarithmic scales for the aprotic solvents listed in table 6.1. A very good correlation is found, confirming that equation (6.3.9) is approximately correct. On the basis of a least-squares fit for 14 solvents, the relationship is... 

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... 

An Evaluation of the Debye-Onsager Model. The simplest treatment for solvation dynamics is the Debye-Onsager model which we reviewed in Section II.A. It assumes that the solvent (i) is well modeled as a uniform dielectric continuum and (ii) has a single relaxation time (i.e., the solvent is a Debye solvent ) td (Eq. (18)). The model predicts that C(t) should be a single... 

For a reasonable set of the parameters the calculated far-infrared absorption frequency dependence presents a two-humped curve. The absorption peaks due to the librators and the rotators are situated at higher and lower frequencies with respect to each other. The absorption dependences obtained rigorously and in the above-mentioned approximations agree reasonably. An important result concerns the low-frequency (Debye) relaxation spectrum. The hat-flat model gives, unlike the protomodel, a reasonable estimation of the Debye relaxation time td. The negative result for xD obtained in the protomodel is explained as follows. The subensemble of the rotators vanishes, if u —> oo. 

In the low-frequency range (with x spectral function L(z) depends weakly on frequency x. Then Eq. (32) comes to the Debye-relaxation spectrum given by Eq. (33). Its main characteristics, such as the dielectric-loss maximum Xd and its frequency xD, are given by Eq. (34). A connection between these quantities and the model parameters becomes clear in an example of a very small collision frequency y. In this case, relations (34) come to... 

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. 

Thus, evolution of semiphenomenological molecular models mentioned in Section V.A (items 1-6) have led to the hat-curved model as a model with a rounded potential well. This model combines useful properties of the rectangular potential well and those peculiar to the field models based on application of the parabolic, cosine, or cosine-squared potentials. Namely, the hat-curved model retains the main advantage of the rectangular-well model—its possibility to describe both the librational and the Debye-relaxation bands. 

In Section V the reorientation mechanism (A) was investigated in terms of the only (hat curved) potential well. Correspondingly, the only stochastic process characterized by the Debye relaxation time rD was discussed there. This restriction has led to a poor description of the submillimeter (10-100 cm-1) spectrum of water, since it is the second stochastic process which determines the frequency dependence (v) in this frequency range. The specific vibration mechanism (B) is applied for investigation of the submillimetre and the far-infrared spectrum in water. Here we shall demonstrate that if the harmonic oscillator model is applied, the small isotope shift of the R-band could be interpreted as a result of a small difference of the masses of the water isotopes. 

The reflectivity of human skin is expected to decrease with increasing frequency. This is due to the reduction in the real part of the refractive index as well as an increase in absorption with frequency. In vivo studies have been conducted and used to derive parameters fitting a double Debye relaxation model for the impedance of human skin [58],... 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

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. 

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. 

Non-ideality has been shown to be due to ionic interactions between the ions and consideration of these led to the concept of the ionic atmosphere (see Sections 10.3 and 10.5). These interactions must be taken into account in any theory of conductance. Most of the theories of electrolyte conduction use the Debye-Hiickel model, but this model has to be modified to take into account extra features resulting from the movement of the ions in the solvent under the applied field. This has proved to be a very difficult task and most of the modern work has attempted many refinements all of which are mathematically very complex. Most of this work has focused on two effects which the existence of the ionic atmosphere imposes on the movement and velocity of the ions in an electrolyte solution. These are the relaxation and electrophoretic effects. 

There are very good reasons why there was the long delay of 30 years between formulation of the Debye-Hiickel model, recognition of the necessity to consider the effect of relaxation and... 

The hat model is parameterized to obtain an agreement between the calculated and experimental spectra in the region 400—1000 cm-1 and at low frequencies. To parameterize the hat model with respect to the relaxation band, in which the Debye relaxation time td is regarded to be known, we set x0Y = 0 in the SF argument and vary y0r in order to satisfy the equation... 

The frequency effect of K mainly originates from Ac because the remaining parameters are all independent of frequency in the low frequency region. Based on Debye relaxation model, Ac has following form ... 

Debye s original rokational diffusion model corresponds to ki = 2Dy where is the rotational diffusion coefficient which for nis model of the molecule as a sphere of volume V reorienting in a medium of shear viscosity Q ia s k T/6qV giving the Debye relaxation time... 

In contrast to the stretched exponential relaxation time, the Debye relaxation time T2 does not vary with hydration and stays around t2 = 4.7 0.4 ps and 2.5 0.6 ps for Fi and F2, respectively (Fig. 120). These values are noticeably larger than the bulk values for the same water model (2.5 and 0.9 ps [648]). The parameter a in equation (31) reflects the fraction of these weakly bound water molecules with Debye rotational dynamics. The amplitude a increases with hydration level, as it is shown in Fig. 121. At low hydrations, a is negligibly small and therefore cannot be estimated from the fits with a reasonable accuracy. At the surface of a rigid lysozyme, we have detected the appearance of the water molecules with Debye-like rotational dynamics only when Ny, > 300. On... 

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