Localization, electrical double layer

Solving the Poisson-Boltzmann equation with proper boundary conditions will determine the local electrical double layer potential field y/ and hence, via Eq.(3), the local net charge density distribution. 

IHP) (the Helmholtz condenser formula is used in connection with it), located at the surface of the layer of Stem adsorbed ions, and an outer Helmholtz plane (OHP), located on the plane of centers of the next layer of ions marking the beginning of the diffuse layer. These planes, marked IHP and OHP in Fig. V-3 are merely planes of average electrical property the actual local potentials, if they could be measured, must vary wildly between locations where there is an adsorbed ion and places where only water resides on the surface. For liquid surfaces, discussed in Section V-7C, the interface will not be smooth due to thermal waves (Section IV-3). Sweeney and co-workers applied gradient theory (see Chapter III) to model the electric double layer and interfacial tension of a hydrocarbon-aqueous electrolyte interface [27]. 

For an ideally polarizable electrode, q has a unique value for a given set of conditions.1 For a nonpolarizable electrode, q does not have a unique value. It depends on the choice of the set of chemical potentials as independent variables1 and does not coincide with the physical charge residing at the interface. This can be easily understood if one considers that q measures the electric charge that must be supplied to the electrode as its surface area is increased by a unit at a constant potential." Clearly, with a nonpolarizable interface, only part of the charge exchanged between the phases remains localized at the interface to form the electrical double layer. 

A theoretical approach based on the electrical double layer correction has been proposed to explain the observed enhancement of the rate of ion transfer across zwitter-ionic phospholipid monolayers at ITIES [17]. If the orientation of the headgroups is such that the phosphonic group remains closer to the ITIES than the ammonium groups, the local concentration of cations is increased at the ITIES and hence the current observed due to cation transfer is larger than in the absence of phospholipids at the interface. This enhancement is evaluated from the solution of the PB equation, and calculations have been carried out for the conditions of the experiments presented in the literature. The theoretical results turn out to be in good agreement with those experimental studies, thus showing the importance of the electrostatic correction on the rate of ion transfer across an ITIES with adsorbed phospholipids. 

As suggested before, the role of the interphasial double layer is insignificant in many transport processes that are involved with the supply of components from the bulk of the medium towards the biosurface. The thickness of the electric double layer is so small compared with that of the diffusion layer 8 that the very local deformation of the concentration profiles does not really alter the flux. Hence, in most analyses of diffusive mass transport one does not find any electric double layer terms. For the kinetics of the interphasial processes, this is completely different. Rate constants for chemical reactions or permeation steps are usually heavily dependent on the local conditions. Like in electrochemical processes, two elements are of great importance the local electric field which affects rates of transfer of charged species (the actual potential comes into play in the case of redox reactions), and the local activities... 

Thus summarizing, we note that at the leading order the asymptotic solution constructed is merely a combination of the locally electro-neutral solution for the bulk of the domain and of the equilibrium solution for the boundary layer, the latter being identical with that given by the equilibrium electric double layer theory (recall (1.32b)). We stress here the equilibrium structure of the boundary layer. The equilibrium within the boundary layer implies constancy of the electrochemical potential pp = lnp + ip across the boundary layer. We shall see in a moment that this feature is preserved at least up to order 0(e2) of present asymptotics as well. This clarifies the contents of the assumption of local equilibrium as applied in the locally electro-neutral descriptions. Recall that by this assumption the electrochemical potential is continuous at the surfaces of discontinuity of the electric potential and ionic concentrations, present in the locally electro-neutral formulations (see the Introduction and Chapters 3, 4). An implication of the relation between the LEN and the local equilibrium assumptions is that the breakdown of the former parallel to that of the corresponding asymptotic procedure, to be described in the following paragraphs, implies the breakdown of the local equilibrium. 

At the electrical double layer in Fig. 4.2(a), the charges on the electrolyte side are mostly localized at the (outer) Helmholtz layer if the electrolyte concentration... 

This electrokinetically driven micro mixer uses localized capacitance effects to induce zeta potential variations along the surface of silica-based micro channels [92], The zeta potential variations are given near the electrical double layer region of the electroosmotic flow utilized for species transport. Shielded ( buried ) electrodes are placed underneath the channel structures for the fluid flow in separate channels, i.e. they are not exposed to the liquid. The potential variations induce flow velocity changes in the fluid and thus promote mixing [92],... 

Due to the existence of the electrical double layer, the concentration of an ion near the interface varies, which should influence the ion transfer rate. Typically, the double layer effect is accounted for by assuming an equilibrium ion distribution up to a point located close to the interface, from which the ion becomes driven by the local potential gradient to cross the interface. The assumption of a three-step (four-position) mechanism has been introduced by Gavach and coworkers [109], but the idea can be traced back to a review by Buck [110]. Following this assumption, Eq. (20) can be expanded into... 

The ion and electrical potential distributions in the electrical double layer can be determined by solving the Poisson-Boltzmann equation [2,3]. According to the theory of electrostatics, the relationship between the eleetrieal potential ij/ and the local net charge density per unit volume at any point in the solution is deseribed by the Poisson equation ... 

Composition. While the average environment of a molecule in solution is well represented by the bulk concentrations of the various constituents of a reaction mixture, the environment of a molecule bound to a heterogeneous interface may be strongly perturbed. The properties of the motionally restricted phase may dictate. rather large deviations in the local concentrations of ions, reactants, and other mobile species from their respective bulk concentrations. The most important examples of this have been demonstrated for electrode-solution interfaces where, for example, the pH in the electrical double layer may differ significantly from its value in the bulk solution and can change with applied potential (1). A similar, though less extreme,example involves the interface between aqueous solutions and hydrophobic polymers. 

A film can only break up into droplets after a disturbance the film locally thins to less than t)q)ically 1000 nm (see Fig. 6.40). In this region the interaction force (van der Waals, electrical double layer, for example) between the liquid-solid and liquid-air surface of the film becomes important. Attraction forces can rupture the thin film and a dry patch is nucleated. Such a film is called a non-wetting film. When the interaction between the two film interfaces is repulsive the so-called disjoining pressure (see also p. 162) of the film, i.e. the pressure difference between the film and bulk liquid, is negative. In the other case of negative disjoining pressures, it may also be called conjoining pressure. 

Eqs. (41) and (42) neglect shares of free ions H" " in the plane of charge well as of ions C and A in the plane 0. Eqs. (39) and (40) point to the presence of such ions at the surface in the compact area of the electric double layer. Appropriate calculations show, that the charge coming from the free ions is so small compared with that coming from the localized charges, that it can be neglected. 

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