Double layer, electric relaxation

Overbeek and Booth [284] have extended the Henry model to include the effects of double-layer distortion by the relaxation effect. Since the double-layer charge is opposite to the particle charge, the fluid in the layer tends to move in the direction opposite to the particle. This distorts the symmetry of the flow and concentration profiles around the particle. Diffusion and electrical conductance tend to restore this symmetry however, it takes time for this to occur. This is known as the relaxation effect. The relaxation effect is not significant for zeta-potentials of less than 25 mV i.e., the Overbeek and Booth equations reduce to the Henry equation for zeta-potentials less than 25 mV [284]. For an electrophoretic mobility of approximately 10 X 10 " cm A -sec, the corresponding zeta potential is 20 mV at 25°C. Mobilities of up to 20 X 10 " cmW-s, i.e., zeta-potentials of 40 mV, are not uncommon for proteins at temperatures of 20-30°C, and thus relaxation may be important for some proteins. 

A persistent question regarding carbon capacitance is related to the relative contributions of Faradaic ( pseudocapacitance ) and non-Faradaic (i.e., double-layer) processes [85,87,95,187], A practical issue that may help resolve the uncertainties regarding DL- and pseudo-capacitance is the relationship between the PZC (or the point of zero potential) [150] and the point of zero charge (or isoelectric point) of carbons [4], The former corresponds to the electrode potential at which the surface charge density is zero. The latter is the pH value for which the zeta potential (or electrophoretic mobility) and the net surface charge is zero. At a more fundamental level (see Figure 5.6), the discussion here focuses on the coupling of an externally imposed double layer (an electrically polarized interface) and a double layer formed spontaneously by preferential adsorp-tion/desorption of ions (an electrically relaxed interface). This issue has been discussed extensively (and authoritatively ) by Lyklema and coworkers [188-191] for amphifunctionally electrified... 

Mobility is also reduced slightly by another phenomenon termed the relaxation effect. Here the charged species, as it is displaced by the electric field from the center of the double layer, is acted on by the opposite charge of the double layer to pull it back [38]. 

Forces due to the action of the electric field on ions of the opposite charge to that of the particle within the double layer relaxation effect)... 

For a cylindrical particle oriented at an arbitrary angle between its axis and the applied electric field, its electrophoretic mobility averaged over a random distribution of orientation is given by pav = /i///3 + 2fjLj3 [9]. The above expressions for the electrophoretic mobility are correct to the first order of ( so that these equations are applicable only when C is low. The readers should be referred to Ref. [4-8, 10-13, 22, 27, 29] for the case of particles with arbitrary zeta potential, in which case the relaxation effects (i.e., the effects of the deformation of the electrical double layer around particles) become appreciable. 

We derive approximate mobility formulas for the simple but important case where the potential is arbitrary but the double-layer potential still remains spherically symmetrical in the presence of the applied electric field (the relaxation effect is neglected). Further we treat the case where the following conditions hold... 

With this in mind, the impossibility of forming a double layer by electric forces only is obvious. Any ion that may attach to a particle will create a potential that keeps out all other ions of the same sign. Accumulation of a number of identical charges on a surface can take place only if the adsorbing ions experience a non-electric affinity for the surface so that they can move against the adverse potential. The extent to which this occurs depends on the balance between the attractive non-electrostatlc and the repulsive electric forces. In summary the reason for the formation of relaxed double layers is the nonelectric afflnlty of charge-determining ions for a surface the extent to which the double layer develops is determined by the non-electrostatlc electrostatic interaction balance. 

Here a° and yr° are the surface charge (density) and surface potential, respectively. The primes Indicate the variable values when the double layer is reversibly charged from cr° =0 to its final value, cr°. As y/° and a° have the same signs, AG°(el) > 0, so a non-electric contribution is needed to make the overall Gibbs energy change negative, as required for a spontaneous process. For relaxed... 

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 very simplified situation that the sphere behaves electrically as a pure capacitor, and the solution as a pure resistance, the relaxation can be described by a Maxwell-Wagner mechanism, with T = e e/K, see (1.6.6.321. Although some success has been claimed by Watillon s group J to apply this mechanism for a model, consisting of shells with different values of e and K, generally a more detailed double layer picture is needed. In fact, this Implies stealing from the transport equations of secs. 4.6a and b. generedizing these to the case of a.c. fields. 

Calculation of the Zeta Potential. The conversion of electrophoretic mobility to zeta potential is complicated somewhat by the existence of the electrophoretic relaxation effect. Figure 5 shows a schematic diagram of this effect. As we expose an emulsion droplet and its surrounding double layer to an electric field, the double layer distorts to the shape shown in the figure. This distorted double layer now creates its own electric field that... 

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