Complex permittivity of water

The theory of wideband complex permittivity of water described in the review drastically differs from the empirical double Debye representation [17, 54] of the complex permittivity given for water by formula (280b). Evolution of the employed potential profiles, in which a dipole moves, explored by a dynamic linear-response method can be illustrated as follows ... 

Table I. The complex permittivity of water at 70GHz. (The errors correspond to the 95% confidence intervals)...

Figure 2. Complex permittivity of water at frequencies above 70 GHz ((V) from the present 70-GHz study (G) i calculated from the Debye model using r = 9.3 ps, ts = 80.1, and — 5.5 data of Asfar and Hasted (2) (---------------) i (A) <")...

Figure 12 The complex permittivity of water and 2.8 molal aqueous glucose solution at 278 K O, t.d.s measurement of e water) , t.ds. measurement of e iyvater) ----------------------,., data interpolated from ref. 24 ...

Double Debye Approximation for Complex Permittivity of Heavy Water... 

Figure 30. Imaginary (a) and real (b) parts of the complex permittivity of liquid water H20 at 22.2°C. Ordinary water is represented by solid lines, heavy water is represented by dashed lines. To the left from vertical lines (for v < 20 cm 1). calculation is performed using approximation [17] modified as described in Appendix 3.2 in the rest region, it is performed using the data 51 given in Table XII.

Figs. 32a-c illustrate the absorption spectra, calculated, respectively, for water H20 at 27°C, water H20 at 22.2°C, and water D20 at 22.2°C dotted lines show the contribution to the absorption coefficient due to vibrations of nonrigid dipoles. The latter contribution is found from the expression which follows from Eqs. (242) and (255). The experimental data [42, 51] are shown by squares. The dash-and-dotted line in Fig. 32b represents the result of calculations from the empirical formula by Liebe et al. [17] (given also in Section IV.G.2) for the complex permittivity of H20 at 27°C comprising double Debye-double Lorentz frequency dependences. 

The solvent molecules form an oriented parallel, producing an electric potential that is added to the surface potential. This layer of solvent molecules can be protruded by the specifically adsorbed ions, or inner-sphere complexed ions. In this model, the solvent molecules together with the specifically adsorbed, inner-sphere complexed ions form the inner Helmholtz layer. Some authors divide the inner Helmholtz layer into two additional layers. For example, Grahame (1950) and Conway et al. (1951) assume that the relative permittivity of water varies along the double layer. In addition, the Stern variable surface charge-variable surface potential model (Bowden et al. 1977, 1980 Barrow et al. 1980, 1981) states that hydrogen and hydroxide ions, specifically adsorbed and inner-sphere... 

COMPLEX PERMITTIVITY OF ICE Ih AND OF LIQUID WATER IN FAR INFRARED UNIFIED ANALYTICAL THEORY... 

We use the relationships (4)—(7) to calculate the water spectra. In such a calculation for ice the static permittivity, s(ice) is not involved. For molecules reorienting in the hat well the high-frequency approximation is employed. The complex permittivity of the LIB state is represented, instead of Eq. (4), as... 

We do not consider the low-frequency spectra for ice, since the contribution to complex permittivity of rigid reorienting dipoles is calculated from the simplified expression (A29), which is applicable only in the high-frequency approximation. Indeed, the ice permittivity is found for v > 0.1 cm-1 (see Figs. 20a,b and 24a), while for liquid water Eq. (4) is used, applicable also in the relaxation region. 

Since the volume fractions of free, cpf, and bound, cp, water are both unknown, it is convenient to measure the dielectric permittivity in a frequency range where the dielectric loss of bound water may be safely neglected. The relaxation spectrum of free and bound water for our systems will safely satisfy this requirement at the measurement frequency of 75 GHz. In this case, die complex permittivity of the bound water is equal to its real part, i.e., = s -i-... 

One of the methods used to study emulsions has been the use of dielectric spectroscopy. The permittivity of the emulsion can be used to characterize an emulsion and assign a stability (1,42,48—54). The Sjoblom group has measured the dielectric spectra using time-domain spectroscopy (TDS) technique. A sample is placed at the end of a coaxial line to measure total reflection. Reflected pulses are observed in time windows of 20 ns, Fourier transformed in the frequency range from 50 MHz to 2 GHz, and the complex permittivity calculated. Water or air can be used as reference sample. The total complex permittivity at a frequency (co) is given by ... 

The dynamics of lubricant interaction with added water is even more complicated. In addition to high-frequency permittivity changes described by Eq. 7-7, the entire impedance spectrum undergoes a complicated pattern of time-dependent changes. The low-frequency impedance changes caused by water will be discussed in a later section of this chapter. In the bulk solution a complex kinetics of water-oil interactions occurs, combining several mutually dependent processes of emulsification of free nonbound water, formation of inverse micelles, and evaporation of free and micellated water [21]. 

The dielectric behavior of nonionized PAAm network and a ionized P(AAm/MNa) network with xMNa = 0.03 in deionized-water-acetone mixtures was also studied [33]. High values of complex permittivity e were found for both networks (Fig. 19). For the PAAm network, the dependence of both component e and e" on acetone a is continuous. On the other hand, for the ionized network the jumpwise decrease in swelling at the transition is accompanied by a jumpwise increase in the values of both components of e at all... 

We do not know theoretical descriptions other than ours of the dielectric/FIR spectra applicable for water in the range from 0 to 1000 cm-1, which were made on a molecular basis in terms of complex permittivity s(m). 

It should, however, be noted that there exist rather complex and nontransparent descriptions made [15] in terms of the absorption vibration spectroscopy of water. This approach takes into account a multitude of the vibration lines calculated for a few water molecules. However, within the frames of this method for the wavenumber1 v < 1000 cm-1, it is difficult to get information about the time/spatial scales of molecular motions and to calculate the spectra of complex-permittivity or of the complex refraction index—in particular, the low-frequency dielectric spectra of liquid water. 

Employing the additivity approximation, we find dielectric response of a reorienting single dipole (of a water molecule) in an intermolecular potential well. The corresponding complex permittivity jip is found in terms of the hybrid model described in Section IV. The ionic complex permittivity A on is calculated for the above-mentioned types of one-dimensional and spatial motions of the charged particles. The effect of ions is found for low concentrated NaCl and KC1 aqueous solutions in terms of the resulting complex permittivity e p + Ae on. The calculations are made for long (Tjon x) and rather short (xion = x) ionic lifetimes. 

It would be important to find analogous mechanism also for description of the main (librational) absorption band in water. After that it would be interesting to calculate for such molecular structures the spectral junction complex dielectric permittivity in terms of the ACF method. If this attempt will be successful, a new level of a nonheuristic molecular modeling of water and, generally, of aqueous media could be accomplished. We hope to convincingly demonstrate in the future that even a drastically simplified local-order structure of water could constitute a basis for a satisfactory description of the wideband spectra of water in terms of an analytical theory. 

In this section we calculate the complex permittivity (v) and the absorption coefficient ot(v) of ordinary (H20) water and of fluoromethane CH3F over a wide range of frequencies. We shall first write down the list of the formulas useful for further calculations. 

In this section we have to calculate the complex permittivity s (v) and the absorption coefficient a(v) of ordinary (H2O) water over a wide range of frequencies. It is rather difficult to apply rigorous formulas because the fluctuations of the calculated characteristics occur at a small reduced collision frequency y typical for water (in this work we employ for calculations the standard MathCAD program). Such fluctuations are seen in Fig. 13b (solid curve). Therefore the calculations will be undertaken for two simplified variants of the hat model. Namely, we shall employ the planar libration-regular precession (PL-RP) approximation and the hybrid model.26... 

There appears some disagreement of the calculated complex permittivity e (v) with the experimental data [17, 42] recorded in the submillimeter wavelength range—that is, from 10 to 100 cm-1. It is evident from Fig. 15 and more clearly from Fig. 16 that a theoretical loss is less in this spectral interval than the experimental one. The reason of such a discrepancy can be explained as follows. Some additional mechanism of dielectric loss possibly exists in water. Such a mechanism will be studied in Sections VII, IX, and X, where we shall propose composite molecular models of water. 

In Table V the fitted free and estimated statistical parameters are presented. For calculation of the spectral function we use rigorous formulas (130) and Eqs. (132) for the hybrid model. For calculation of the susceptibility %, complex permittivity , and absorption coefficient a we use the same formulas as those employed in Section IV.G.2 for water.29... 

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. 

Starting with the important example of ordinary water, we choose temperatures 22.2°C and 27°C. We compare our theory with the recorded FIR spectra [42, 56] of the complex permittivity/absorption. At low frequencies we use for this purpose an empirical formula [17] by Liebe et al. these formulas were given also in Section IV.G.2.a. The values of the employed molecular constants are presented in Table VI and the fitted parameters in Table VII. The Reader may find more information about experimental data of liquid H2O and D2O in Appendix 3. 

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. 

Figure 28. Experimental frequency dependences of dielectric parameters recorded for liquid water (a) Real (curve 1) and imaginary (curve 2) parts of the complex permittivity at 27°C. The data are from Refs. 42 (solid lines) and 17 (circles), (b) Absorption coefficient. Solid line and crosses 1 refer to 1°C filled circles 2 refer to 27°C dashed line and squares 3 refer to 50°C. For lines the data from Ref. 17 were employed, for circles the data are from Ref. 42, for crosses and squares the data are from Ref. 53.

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