Double Debye approximation

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

In accordance with Ref. 54 both for liquid H20 and D20 the double Debye approximation is applicable in the frequency range up to 2THz (i.e., up to 70cm-1) and in the temperature range from 273 K to 303 K. 

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... 

The high frequency limit of for this second process is therefore n. The result of the fit is shown in Table III where the mean values of the various parameters and their associated 95% confidence intervals are given. Considering the small amplitude of the second dispersion both in absolute t rms and in relation to the main dispersion the parameters 6m, n and Y are quite well defined, and therefore it may be concluded that the double Debye representation is an acceptable description of the dielectric behaviour of water up to around 2THz. Other alternative interpretations are clearly possible but no attempt has been made here to follow these up at this stage. What is clear is that a small subsidiary dispersion region in the far infrared is necessary to account for all the presently available permittivity data, and that such a dispersion is centred around 650GHz and has an amplitude of about 2.4 in comparison with that of the principal dispersion which is approximately 75. 

When reorganization of the medium is small, the rate constant in the Debye approximation is described (as shown in ref 161) by expression (62) at A = 0, but with doubled a , the contribution of zero-point vibrations is also doubled. 

One of the simplest equations is obtained using the Debye-Hiickel approximation (for low potentials) and the superposition principle. The latter assumes that the unperturbed potential near a charged surface can be simply added to that potential due to the other (unperturbed) surface. Thus, for the example shown in the Figure 6.12, it follows that /m = 2 /d/2- This is precisely valid for Coulomb-type interactions, where the potential at any point can be calculated from the potentials produced by each fixed charge, individually. However, the Poisson-Boltzmann equation is non-linear (this has to do with the fact that in the diffuse double-layer the ions are not fixed but move because of their kinetic energy) and so this is formally not correct although it still offers a useful approximation. 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

One of the most important quantities to emerge from the Debye-Huckel approximation is the parameter k. This quantity appears throughout double-layer discussions and not merely at this level of approximation. Since the exponent kx in Equation (37) is dimensionless, k must have units of reciprocal length. This means that k has units of length. This last quantity is often (imprecisely) called the thickness of the double layer. All distances within the double layer are judged large or small relative to this length. Note that the exponent kx may be written x/k a form that emphasizes the notion that distances are measured relative to k in the double layer. 

FIG. 11.5 Fraction of double-layer potential versus distance from a surface according to the Debye-Hiickel approximation, Equation (37) (a) curves drawn for 1 1 electrolyte at three concentrations and (b) curves drawn for 0.001 M symmetrical electrolytes of three different valence types. 

Even allowing for the fact that the Debye-Hiickel approximation applies only for low potentials, the above analysis reveals some features of the electrical double layer that are general and of great importance as far as stability with respect to coagulation of dispersions and electrokinetic phenomena are concerned. In summary, three specific items might be noted ... 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

Equation (62) describes the variation in potential with distance from the surface for a diffuse double layer without the simplifying assumption of low potentials. It is obviously far less easy to gain a feeling for this relationship than for the low-potential case. Anticipation of this fact is why so much attention was devoted to the Debye-Hiickel approximation in the first place. Note that Equation (62) may be written... 

The Derjaguin approximation illustrated in the above example is suitable when kR > 10, that is, when the radius of curvature of the surface, denoted by the radius R, is much larger than the thickness of the double layer, denoted by k 1. (Note that for a spherical particle R = Rs, the radius of the particle.) Other approaches are required for thick double layers, and Verwey and Overbeek (1948) have tabulated results for this case. The results can be approximated by the following expression when the Debye-Hiickel approximation holds ... 

Equations (6.4.43a-c) yield the central result of this section—the following expression for the electro-osmotic slip velocity ua under an applied potential and concentration gradient, in the Debye-Hiickel approximation for a thin double layer... 

AF4 is the change in the free energy of the electrical double layer accompanying the adsorption of charged trains on the charged surface, and if the Debye-Hiichel approximation is applied, it is given by... 

Here multiplier 2 approximately accounts for doubling of integrated absorption due to spatial motion of a dipole, which is more realistic than motion in a plane to which LCs(Z) corresponds. For representation (235), only one (Debye) relaxation region with the relaxation time rD is characteristic. At this stage of molecular modeling it was not clear (a) why the CS potential, which affects motion of a dipole in a separate potential well, is the right model of specific interactions and (b) what is physical picture corresponding to a solid-body-like dipole moment pcs. 

When the electrolyte concentration is increased, the range of the double layer decreases dramatically (the Debye-Hiickel length decreases) and the magnitude of the surface potential also decreases. In the linear approximation, ]f(x) = iJj(c E)cxp( — (x-dB)/X) (for x>dB) and the second right-hand-side term of Eq. (48) becomes ... 

For the sake of simplicity, in what follows it will be considered that the double layer potential is sufficiently small to allow the linearization of the Poisson—Boltzmann equation (the Debye—Hiickel approximation). The extension to the nonlinear cases is (relatively) straightforward however, it will turn out that the differences from the DLVO theory are particularly important at high electrolyte concentrations, when the potentials are small. In this approximation, the distribution of charge inside the double layer is given by... 

For relatively wide channels with negligible electrical double-layer overlap (r/8 > 10), a nearly flat flow profile is expected. It has often been stated that when the channel size and the Debye length are of similar dimensions (r 8), complete electrical double-layer overlap occurs and the EOF is negligible. However, when r 8, a significant EOF can still be created the EOF velocity in the central part of the channel is approximately 20% of that in an infinitely wide channel. Only at conditions where r/8 1 is the EOF fully inhibited by double-layer overlap [25], It should be noted here that the approximations made by using the Rice and Whitehead theory at r/8 < 10 may lead to significant errors in the calculation of the velocity distribution and magnitude of the EOF [17] compared to more sophisticated models. 

The quantity I /at has units of length and is called the Debye length it defines the extent of the double layer, i.e., the distance in which the potential decays to I je of its initial value k is called the Debye-Huckel parameter. Hence within validity of this approximation (low surface potentials < 25 mV) the potential decreases exponentially away from the surface. 

Equation (1.9) is the linearized Poisson-Boltzmann equation and k in Eq. (1.10) is the Debye-Htickel parameter. This linearization is called the Debye-Hiickel approximation and Eq. (1.9) is called the Debye-Hiickel equation. The reciprocal of k (i.e., 1/k), which is called the Debye length, corresponds to the thickness of the double layer. Note that nf in Eqs. (1.5) and (1.10) is given in units of m . If one uses the units of M (mol/L), then must be replaced by IQQQNAn, Na being Avogadro s number. 

Figure 1.4 shows y(x) for several values of yo calculated from Eq. (1.37) in comparison with the Debye-Hlickel linearized solution (Eq. (1.25)). It is seen that the Debye-Hiickel approximation is good for low potentials (lyol< 1). As seen from Eqs. (1.25) and (1.37), the potential i//(x) across the electrical double layer varies nearly... 

In Chapter 11, we derived the double-layer interaction energy between two parallel plates with arbitrary surface potentials at large separations compared with the Debye length 1/k with the help of the linear superposition approximation. These results, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and to the constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the linear superposition approximation can be applied not only to hard particles but also to soft particles. We now apply Derjaguin s approximation to these results to obtain the sphere-sphere interaction energy, as shown below. 

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