Permittivity water, frequency dependence

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.

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. 

Figure 35. Frequency dependence in the submillimeter wavelength region of the real (a, b) and imaginary (c, d) parts of the complex permittivity. Solid lines Calculation for the composite HC-HO model. Dashed lines Experimental data [51]. Dashed-and-dotted lines show the contributions to the calculated quantities due to stretching vibrations of an effective non-rigid dipole. The vertical lines are pertinent to the estimated frequency v b of the second stochastic process. Parts (a) and (c) refer to ordinary water, and parts (b) and (d) refer to heavy water. Temperature 22.2°C.

The conducting properties of a liquid in a porous medium can provide information on the pore geometry and the pore surface area [17]. Indeed, both the motion of free carriers and the polarization of the pore interfaces contribute to the total conductivity. Polymer foams are three-dimensional solids with an ultramacropore network, through which ionic species can migrate depending on the network structure. Based on previous works on water-saturated rocks and glasses, we have extracted information about the three-dimensional structure of the freeze-dried foams from the dielectric response. Let be d and the dielectric constant and the conductivity, respectively. Dielectric properties are usually expressed by the frequency-dependent real and imaginary components of the complex dielectric permittivity ... 

In the first part of this work (Sections II through V) we have combined the formula for x given there without derivation, with the formulas for xq, and Xor> accounting for dielectric response, arising, respectively, from elastic harmonic vibration of charged molecules along the H-bond (HB), from elastic reorientation of HB permanent dipoles about this bond, and from a rather free libration of a permanent dipole in a defect of water/ice structure modeled by the hat well. The set of four frequency dependences, namely of Xor(v)> (v), X (v), and X (v), allows us to describe the water/ice wideband spectra. For these dependences and those similar to them—namely 0r(v), Asq(v), Ae/1(v), and Ae (v) for the partial23 complex permittivity—we refer to mechanisms a, b, c, and d. 

The Hamaker constant can be evaluated accmately using the continuum theory, developed by Lifshitz and coworkers [40]. A key property in this theory is the frequency dependence of the dielectric permittivity, e( ). If this spectrum were the same for particles and solvent, then A=0. Since the refractive index n is also related to t ( ), the van der Waals forces tend to be very weak when the particles and solvent have similar refractive indices. A few examples of values for for interactions across vacuum and across water, obtained using the continuum theory, are given in table C2.6.3. 

Earth resistivity, as mentioned earlier, is weather/climate dependent. The resistivity after the rains is lower than that measured during dry days. Also, it may be frequency dependent. The frequency dependence of earth permittivity may be far more significant than that of earth resistivity. Furthermore, water (H2O), which is a dominant factor for earth permittivity, is extremely temperature dependent [18]. As a result, the error due to the uncertainty of earth resistivity and permittivity might be far greater than that due to the incompleteness of the earth-return impedance derived by Carson and Pollaczek. This should be remembered as a physical reality that is important in engineering practice. 

The first term, which contains the the static dielectric permittivities of the three media , 2, and 3, represents the Keesom plus the Debye energy. It plays an important role for forces in water since water molecules have a strong dipole moment. Usually, however, the second term dominates in Eq. (6.23). The dielectric permittivity is not a constant but it depends on the frequency of the electric field. The static dielectric permittivities are the values of this dielectric function at zero frequency. 1 iv), 2 iv), and 3(iv) are the dielectric permittivities at imaginary frequencies iv, and v = 2 KksT/h = 3.9 x 1013 Hz at 25°C. This corresponds to a wavelength of 760 nm, which is the optical regime of the spectrum. The energy is in the order of electronic states of the outer electrons. 

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 ... 

D. Bertolini and A. Tani, The frequency and wavelength dependent dielectric permittivity of water, Mol. Phys., 75 (1992) 1065-88. 

B. Protein Solutions. The dielectric properties of proteins and nucleic acids have been extensively reviewed (10, 11). Protein solutions exhibit three major dispersion ranges. One occurs at RF s and is believed to arise from molecular rotation in the applied electric field. Typical characteristic frequencies range from about 1 to 10 MHz, depending on the protein size. Dipole moments are of the order of 200-500 Debyes and low-frequency increments of dielectric permittivity vary between 1 and 10 units/g protein/100 ml of solution. The high-frequency dielectric permittivity of this dispersion is lower than that of water because of the low dielectric permittivity of the protein leading to a high-frequency decrement of the order of 1 unit/g protein/... 

Figure 8. Three-dimensional plots of the frequency and temperature dependence of the dielectric permittivity s (a) and dielectric losses e" (b) for AOT-water-decane microemulsion. (Reproduced with permission from Ref. 143. Copyright 1995, The American Physical Society.)...

Nanoscale confinement significandy affects water polarizability P = P(r), whose dependence on F may be obtained rigorously from first principles, starting from the Debye relation [2] V.(e0E + P)(r) = p(r) (e0= vacuum permittivity, p(r) = charge density, E= electrostatic field). In Fourier-conjugate frequency space (v-space) we get [2]... 

The full spectra of permittivities are required in the calculation of the van der Waals interaction energy using Eqs. (3) and (4). For water and a few materials, the dependence of the permittivity, on the sampling frequency, is available (see Fig. 2). For highly polar liquids such as water, the relaxation in the micro-wave and infrared appears signifieant and the oscillator model for the speetrum of water permittivity has a number of terms as deseribed by Eq. (5), and the model parameters are given in Table 1. 

The moment of inertia 7or (its value is given in Table IV) is estimated for a freely rotating symmetric top molecule (rotation about the principal axis normal to the symmetry axis). The moment 7or determines the frequency scale relevant to reorientation of dipoles in the hat well. Thus, 7or determines the temperature dependence of the fitted lifetime Tor and the strong-collision frequency yor estimated from Eqs. (2a) and (2b) with account of lifetime Tor. The dipole moment /ior of a librating dipole is expressed from Eq. (3) through that (/i0) of a free water molecule and through the fitted dipole-moment factor kiL, with n1 being the optical (measured near the frequency 1000 cm-1) permittivity (n 1.7). The factor kfl is rather close to unity. 

The factor (1 in Eq. (2) measures the tangential electric field at the particle siuface. It is this component which generates the electrophoretic or electroacoustic motion. For a fixed frequency, it can be seen from Eq. (4) that (1 +J) depends on the permittivity of the particles and on die function X - Kg/K a, where Ks is the surface conductance of the double layer X measures the enhanced conductivity due to the charge at the particle surface. It is usually small unless the zeta potential is very high, so for most emulsions with large ka, X has a negligible effect. The ratio fp/f is also small for oil-in-water emulsions. Equation (4) can then be reduced to/= 0.5 and hence the dynamic mobility becomes ... 

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