Double-Debye formula

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 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 ... 

In Figs. 66 and 68 the calculated absorption and loss spectra are depicted for ordinary water at the temperatures 22.2°C and 27°C and for heavy water at 27°C. The solid curves refer to the composite model, and the dashed curves refer to the experimental spectra [42, 51]. For comparison of our theory with experiment at low frequencies, in the case of H20 we use the empirical formula [17] comprising double Debye-double Lorentz frequency dependences. In the case of D20 we use empirical relationship [54] aided by approximate formulae given in Appendix 3 of Section V. The employed molecular constants were presented in previous sections, and the fitted/estimated parameters are given in Table XXIV. The parameters of the composite model are chosen so that the calculated absorption-peak frequencies ilb and vR come close to the... 

In applying the Debye formulae to the calculation of heat capacities we require values of the double integral... 

Studies of dielectric spectra of water in a range of temperatures present a fundamental physical problem that has also important practical applications. Experimental investigation of these spectra has a rich history. We refer here only to a few works. In Downing and Williams [22] and Zelsmann [21] tables for optical constants of water were presented for the temperature T = 300 K and for a wide T-range, respectively. In recent publications by Vij et al. [32] and Zasetsky et al. [33] in addition to original investigations the results of many other works are also discussed. In work by Liebe et. al. [19] a useful empirical double Debye-double Lorentz formula for the complex permittivity e(v, T) is suggested. 

To compare our theory with low-frequency experimental data, we estimate the static permittivity ss and the Debye relaxation time td using for s(v) the empirical double Debye-double Lorentz formula by Liebe et al. [19], where the temperature T is involved in terms of 6 T) = 1 — 300T-1 ... 

Smoluchowski,1 treating the problem from a much more general point of view, obtained the same equation (28), and all subsequent workers have found an equation similar in form, notw ithstanding that it is universally agreed now that the double layer is not of the simple, plane parallel type used as an illustration by Helmholtz. All the formulae agree except as to the exact value of the constant. Smoluchowski found Debye and Huckel2 6 for spherical particles ... 

If the solution to the problem of the equilibrium double layer is known, then the velocity is determined by the above formula. Thus, the solutions to the respective flow problems for the equilibrium situations considered in the previous subsection are readily written down. Solution to the electrokinetic equations is facilitated if the Debye layer thickness may be assumed small compared to the characteristic channel width Wq. This however is not usually the case in nanochannels since wq and are both on the order of nanometers. Exact analytical solutions... 

Guoy and Chapman developed a theory of the charge distribution in the double layer about 10 years before Debye and Hiickel developed their theory of ionic solutions, which is quite similar to it. If one neglects nonelectrostatic contributions to the potential energy of an ion of type i with valence Zi, the concentration of ions of type i in a region of electric potential cp is given by the Boltzmann probability formula, Eq. (9.3-41) ... 

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