Equilibrium double layer

Equation (1.35) is known as the Debye-Hiickel or Gui-Chapman equation for the equilibrium double layer potential. In terms of the original variable x (1.34), (1.35) suggest e1/2(r(j) is the correct scale of ip variation, that is, the correct scale for the thickness of the electric double layer. At the same time, it is observed from (1.32) that for N 1 the appropriate scale depends on N, shrinking to zero when N — oo (ipm — — oo). This illustrates the previously made statement concerning the meaningfulness of the presented interpretation of relectric potential

(—oo) — 0, < (oo) — —oo). 

A few remarks are due about this feature. The nonuniformity above is a formal expression of breakdown of the local electro-neutrality assumption in concentration polarization, described in the previous chapter. Essentially, this reflects the failure of a description based upon assuming the split of the physical region into a locally electro-neutral domain and an equilibrium double layer where all of the space charge is concentrated. The source of this failure, reflected in the nonuniformity of the corresponding matched asymptotic expansions, is that the local Debye length at the interface tends to infinity as the voltage increases. In parallel a whole new type of phenomena arises, which is not reflected in the simplistic picture above. The... 

The structure of the double layer can be altered if there is interaction of concentration gradients, due to chemical reactions or diffusion processes, and the diffuse ionic double layer. These effects may be important in very fast reactions where relaxation techniques are used and high current densities flow through the interface. From the work of Levich, only in very dilute solutions and at electrode potentials far from the pzc are superposition of concentration gradients due to diffuse double layer and diffusion expected [25]. It has been found that, even at high current densities, no difficulties arise in the use of the equilibrium double layer conditions in the analysis of electrode kinetics, as will be discussed in Sect. 3.5. 

Ennis and White [56] employed the method of reflections to investigate the electrophoresis of two spherical particles with equilibrium double layers of arbitrary thickness. Their analysis assumes that the zeta potential of the particle is small and the double layers do not overlap significantly. One interesting finding from their study is that the particles with equal zeta potential do interact with each other when the double layer thickness is finite, unlike the no-interaction result for the case of infinitely thin ion cloud. The leading order interaction between two particles is still but the... 

The exact numerical results for s at various ajlat were obtained by Keh and Yang [16] using the collocation results and are presented in Table 2. The corresponding approximate results from (93) are also shown in the same Table for comparison. It can be found that the results of s predicted by the method of reflections have significant errors when the ratio ajlat is small. For equilibrium double layer of finite thickness, an approximate analytic expression for atj was obtained by Ennis and White [56] using the reflection results, but no exact solution is available as yet. 

Returning to the Induced dipole moment, this vector is determined with respect to both sign and magnitude by the variety of fluxes in the nonequilibrium double layer. Because of this, and because it is measurable (by dielectric spectroscopy), p, j is the most basic characteristic of non-equilibrium double layers. 

Addition of isoconductlng particles to an electrolyte solution does not alter the conductivity. The isoconduction state plays a similar role in non-equilibrium double layers as the point of zero charge does for equilibrium double layers. For large ica Isoconduction is observed for Du = 1. In this case =-Er cos0. 

The presence of a near and far field in and around a non-equilibrium double layer leads to the distinction between (at least) two relaxation times. Relaxation to the static situation, after switching off the external field, can take place by conduction or by diffusion. Conduction means that ions relax to their equilibrium position by an electric field. Diffusion relaxation implies that a concentration gradient is the driving force. In double layers these two mechanisms cannot be separated because excess ion concentrations that give rise to diffusion, simultaneously produce an electric field, giving rise to conduction. For the same reason, if polarization has taken place under the Influence of an external field and this field is switched off, ions return to their equilibrium positions by a mixture of conduction and diffusion. 

For the diffuse part of the double layer can simply be obtained from the Gouy-Chapman theory CjCx) - c H =C H exp Z FVM/RT -1]. The Integrals refer to the equilibrium double layer. Their upper limits are for simplicity set at oo although the far field does not contribute. As in [4.3.101. is the slip velocity It Is the increase In the fluid velocity from the value at the particle surface to the same In the bulk. When the double layer Is not polarized, = E, V c = 0 and as for a symmetrical electrolyte... 

Dukhin. B.V. Deijaguin, Electrokinetic Phenomena, in Surface and Colloid Science. E. Matijevlc. Ed. Vol. 7 Wiley (1974). (Three chapters 1. S.S. Dukhin. Development of Notions as to the Mechanism of Electrokinetic Phenomena and the Structure of the Colloid Micelle, p. 1 2. S.S. Dukhin, B.V. Derjaguin. Equilibrium Double Layer and Electrokinetic Phenomena, p. 49 3. B.V. Derjaguin, S.S. Dukhin, Non-equilibrium Double Layer and Electrokinetic Phenomena, p. 273.)... 

Dukhin, V.N. Shilov. Kinetic Aspects of Electrochemistry of Disperse Systems. Part II. Induced Dipole Moment and the Non-Equilibrium Double Layer of a Colloid Particle. Adu. Colloid Interface Sci. 13 (1980) 153. (Review on the determination and interpretation of induced dipole moments of various colloidal particles.)... 

If ions of the metal are already present in the liquid when the metal electrode is immersed, far fewer metal ions will go into solution. If the concentration is higher than that of the equilibrium double layer, metal ions will deposit on the metal surface and return to the solid state, and the direction of the potential difference at the phase boundary is reversed. 

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... 

Dukhin, S.S. and Deijaguin B. V., Equilibrium double layer and electrokinetic phenomena, in Surface and Colloid Scierwe, Matijevic, E., Ed., WUey, New York, 1974, Vol. 7, chap. 2. 

With aqueous solutions in pressurised cells, the temperature can be varied in a very broad range. Many fundamental electrochemical data have been obtained in this medium. Thermodynamic quantities such as activity coefficients of ions [252], equilibrium double-layer capacity [261], zeta potential of metals [233], potential-pH diagrams [206] or properties of the palladium-hydrogen electrode were determined [262]. Exotic systems, e.g. the solvation of rare earth atoms in liquid gallium [234], have been studied. Main research interests in subcritical aqueous solution were focused on the kinetics, reaction mechanism and transport properties. 

Levich, B. Theory of the non-equilibrium double layer. Dokl AkadNauk SSSR 1949,67, 309-312. 

Dukhin, S. S. and B. V. Deijaguin. 1974. Equilibrium double layer and electrokinetic phenomena. In Surface and Colloid Science, E. Matijevic (Ed.). New York Wiley-Interscience. 

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