Water Debye relaxation

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).

Considerable progress has been made in going beyond the simple Debye continuum model. Non-Debye relaxation solvents have been considered. Solvents with nonuniform dielectric properties, and translational diffusion have been analyzed. This is discussed in Section II. Furthermore, models which mimic microscopic solute/solvent structure (such as the linearized mean spherical approximation), but still allow for analytical evaluation have been extensively explored [38, 41-43], Finally, detailed molecular dynamics calculations have been made on the solvation of water [57, 58, 71]. 

Agmon N. 1996. Tetrahedral displacement The molecular mechanism behind the Debye relaxation in water. JPhys Chem 100 1072-1080. 

In our early work33 [50] the constant field model was applied to liquid water, where the harmonic law of particles motion, corresponding to a parabolic potential, was actually employed in the final calculations of the complex permittivity. In this work, qualitative description of only the libration band was obtained, while neither the R-band nor the low-frequency (Debye) relaxation band was described. Moreover, the fitted mean lifetime x of the dipoles, moving in the potential well, is unreasonably short ( ().02 ps)—that is, about an order of magnitude less than in more accurate calculations, which will be made here. 

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. 

The calculated spectra are illustrated by Fig. 25. In Fig. 25a we see a quasiresonance FIR absorption band, which, unlike water, exhibits only one maximum. Figure 25b demonstrates the calculated and experimental Debye-relaxation loss band situated at microwaves. Our theory satisfactorily agrees with the recorded a(v) and e"(v) frequency dependencies. Although the fitted form factor/is very close to 1 (/ 0.96), the hat-curved model gives better agreement with the experiment than does a model based on the rectangular potential well, where / = 1 (see Section IV.G.3). 

Using this calculation scheme, we have found the frequency dependencies a(v) and s"(v) for ordinary (H20) water at two temperatures. Figure 24 demonstrates for water a qualitative agreement in the calculated and experimental spectra in a very wide frequency band, comprising the microwave Debye relaxation region and the FIR range. This advance in application of the... 

With respect to water we shall conditionally extend the SWR from 10 to 300 cm this frequency region falls between the Debye relaxation range and the librational band. 

A principal drawback of the hat-curved model revealed here and also in Section V is that we cannot exactly describe the submillimeter (v) spectrum of water (cf. solid and dashed lines in Figs. 32d-f). It appears that a plausible reason for such a difference is rather fundamental, since in Sections V and VI a dipole is assumed to move in one (hat-curved) potential well, to which only one Debye relaxation process corresponds. We remark that the decaying oscillations of a nonrigid dipole are considered in this section in such a way that the law of these oscillations is taken a priori—that is, without consideration of any dynamical process. 

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. 

We conclude The submillimeter spectra calculated in terms of the harmonic oscillator model substantially differ from the spectra typical for the low-frequency Debye relaxation region. Such a fundamental difference of the spectra, calculated for water in microwave and submillimeter wavelength ranges, evidently reveals itself in the case of the composite hat-curved-harmonic oscillator model. 

Returning to our problem, we remark that the temperature dependences of such parameters as, for example, the Debye relaxation time td(7 ) (which determines the low-frequency dielectric spectra), or the static permittivity s are fortunately known, at least for ordinary water [17], As for the reorientation time dependence t(T), it should probably correlate with in(T), since the following relation (based on the Debye relaxation theory) was suggested in GT, p. 360, and in VIG, p. 512 ... 

Characteristic frequencies may be found from dielectric permittivity data or, even better, from conductivity data. The earlier data by Herrick et al. (6) suggest that there is no apparent difference between the relaxation frequency of tissue water and that of the pure liquid (7). However, these data extend only to 8.5 GHz, one-third the relaxation frequency of pure water at 37°C (25 GHz), so small discrepancies might not have been uncovered. We have recently completed measurements on muscle at 37°C and 1°C (where the pure water relaxation frequency is 9 GHz), up to 17 GHz. The dielectric properties of the tissue above 1 GHz show a Debye relaxation at the expected frequency of 9 GHz (8 ) (Figure 3). The static dielectric constant of tissue water as determined at 100 MHz compares with that of free water if allowance is made for the fraction occupied by biological macromolecules and their small amount of bound water (1, 9). 

Figure 41. Typical dielectric spectra of 20 mol% of glycerol—water mixtures at (a) 185 K (supercooled state) and (b) 218 K (frozen state), where solid and dashed curves show the real and imaginary parts of complex dielectric permittivity. Each relaxation process in the frozen state was fitted by (114) and by Cole-Cole and Debye relaxation functions, respectively, in order to separate the main process, the process due to interfacial water, and the process due to ice. (Reproduced with permission from Ref. 244. Copyright 2005, American Chemical Society.)...

Shablakh et al. (1984) investigated the dielectric properties of bovine serum albumin and lysozyme at different hydration levels, at low frequency. Besides a relaxation attributed to the electrode—sample interface, they detected a further bulk relaxation that can be confused with a d.c. conduction effect. The latter relaxation was explained by a model of nonconductive long-range charge displacement within a partially connected water structure adsorbed on the protein surface. This model has nonconventional features that differ from the assumptions of other more widely accepted models based on Debye relaxations. 

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... 

One system of particular importance is trans-Cr(NH3)2(NCS)4 in an alcohol/H20 solution at —65° where 50% of the reaction is unquenchable [46]. Although the viscosity of the solvent was not measured it was greater than 300 cP [53]. is reduced a bit more than 50% at high viscosities in polyalcohol/water mixtures of trans-Cr(NH3)2(NCS)4 at 20°C [54], Most of this decrease occurs below 3cP. The Debye relaxation time is less than lOps in water (r] = 1 cP) and 0.5ns in 1-propanol (r = 2cP). Viscosity is not an ideal measure of solvent mobility but it is unlikely that solvent motion is involved in the fast reaction when the viscosity is 300 cP. 

All protic solvents undergo multiple relaxation processes due to the presence of hydrogen bonding. In the case of water and formamide (F), the data can be described in terms of two Debye relaxations. For the alcohols and A-methyl-formamide (NMF), three Debye relaxations are required for the description. In all of these solvents, the low-frequency process involves the cooperative motion of hydrogen-bonded clusters. In the case of water and the alcohols the high-frequency process involves the formation and breaking of hydrogen bonds. The intermediate process in the alcohols is ascribed to rotational diffusion of monomers. Studies of dielectric relaxation in these systems have been carried out for the -alkyl alcohols up to dodecanol [8]. Values of the relaxation parameters for water and the lower alcohols are summarized in table 4.5. 

For water, in the low-frequency limit, the mechanism aformally (after a proper parameterization) describes the nonresonance Debye relaxation band, whose loss peak is located at microwaves. A useful molecular interpretation of this band was given, for example, by Agmon [18]. Liebe et al. [19] obtained a convenient empirical formula for g(v). 

We should remark that no one of the mechanisms previously elaborated for water enables analytical description of the above fall of e"(v) for the case of ice. This phenomenon probably characterizes the transition from collective motions in a fluid typical for Debye relaxation, characterized by the relaxation time td, to vibration of individual molecules, characterized by much shorter lifetimes Tor, rq, Tjj. The latter times have the same order of magnitude in water and ice, but td drastically differs in these fluids. That is why the theory of far-IR spectra is rather similar in ice and water. On the contrary, it appears that the molecular theory pertaining to the low-frequency ice spectra (which still is not elaborated)... 

The LIB fraction (the fraction of monomers) containing about 55-70% of all water molecules comprises a permanent dipole, namely, a neutral molecule librating in a tight surrounding of other molecules in a condensed medium. This motion, governed by mechanism a, is responsible for the librational band, placed near the boundary of the IR region, and for the low-frequency Debye relaxation band, placed at microwaves. The Debye spectrum is not considered in this work. 

Pure liquids - water and alcohols Water and peroxides (HO-OH) represent a limiting state of such interactions. In the liquids state, water molecules associate by hydrogen-bond formation. Despite its apparently complex molecular structure, because of its strong association, water closely followed simple Debye relaxation (at 25 °C sr = 78.2, ooj = 5.5, t = 6.8 x 10 s, and h = 0.02, the Cole-Cole term). 

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. 

For the dielectric re-orientational times of the water molecules, the Debye relaxation expression holds ... 

---