Parallel charged surfaces

A high-frequency limit for the applied potential is encountered above several kilohertz where the impedance of the conductance cell again begins to deviate from the resistance R. Since the solution medium itself is a dielectric situated between two parallel charged surfaces, it can assume the characteristics of a capacitor placed in parallel across the solution resistance as shown in Figure 8.9a. The magnitude of this capacitance is given by... 

It has been noted that the electrostatic forces between two parallel flat surfaces uniformly charged with density cr are distance independent if they are interacting in the air or vacuum. When the two surfaces are charged oppositely (positive and negative, respectively), the intensity of the electrical field between them is cr/eg, and there would be no electrical field in the gap between similarly charged surfaces. 

We shall explain this by means of the following model. Imagine a plane-parallel slab of semiconductor of thickness L, both surfaces of which contain chemisorbed particles. The energy band scheme of such a semiconductor in the case of negatively charged surfaces is shown on Fig. 26. Suppose first that Ly>l (Fig. 26a). Then the inner region of the semiconductor is electrically neutral, and the energy bands inside it are horizontal, as shown in Fig. 26a. From this condition for electrical neutrality, one determines the position of the Fermi level ,+ inside the crystal is thus insensitive... 

Besides magnetic perturbations and electron-lattice interactions, there are other instabilities in solids which have to be considered. For example, one-dimensional solids cannot be metallic since a periodic lattice distortion (Peierls distortion) destroys the Fermi surface in such a system. The perturbation of the electron states results in charge-density waves (CDW), involving a periodicity in electron density in phase with the lattice distortion. Blue molybdenum bronzes, K0.3M0O3, show such features (see Section 4.9 for details). In two- or three-dimensional solids, however, one observes Fermi surface nesting due to the presence of parallel Fermi surface planes perturbed by periodic lattice distortions. Certain molybdenum bronzes exhibit this behaviour. 

In the past decade, many new techniques have been developed and applied to the study of interfaces. While earlier measurements involved only macroscopic characteristics of the interface (e.g., surface charge, surface tension, and overall potential drop), new spectroscopic techniques have opened a window to the microstructure of the interface, and insight at the atomic level in this important region is now possible. Parallel to these discoveries and supported by them, more realistic theoretical models of the interface have been developed that combine quantum mechanical theories of metal surfaces and the statistical mechanics of solutions. 

In the preceding section we discussed the problem of the variation of potential with distance from an interface from the highly artificial perspective of a parallel plate capacitor. The variation of potential with distance from a charged surface of arbitrary shape is a classical electrostatic problem. The general problem is described by the Poisson equation,... 

In two or three dimensions the answer is not entirely obvious. On the one hand, according to Proposition 2.1 the field induced by each particle at the surface of its counterpart is expected to saturate as the charge increases. In parallel, the surface charge density increases indefinitely, so that the electric surface force density on the particle (proportional to the field intensity multiplied by the surface charge density) may seemingly increase indefinitely, too. 

Let us start by considering a liquid on a planar, charged surface. If we apply an electric field parallel to the surface the liquid begins to move (Fig. 5.12). This phenomenon is called electro-osmosis. Why does the liquid start to move The charged surface causes an increase in the concentration of counterions in the liquid close to the surface. This surplus of counterions is moved by the electric field towards the corresponding electrode. The counterions drag the surrounding liquid with them and the liquid starts to flow. 

The dramatic increase of water density at a charged surface was observed by Toney et al. in their in situ X-ray scattering experiments, which has not yet been confirmed by simulation results.58,70 In another MD simulation work, Kiselev et al. found that selfdiffusion coefficient strongly decreases with increasing electric field.27 However, no difference between the self-diffusion coefficients for motion parallel and perpendicular to the external field was observed. 

The one-absorption example used for illustration here is appropriate to most van der Waals charge-fluctuation forces because of the important, usually dominant, UV-frequency range of fluctuations. Temperature is not usually a consideration the summation over sampling frequencies can often be smoothed into a continuous integration. However, retardation screening acts in any situation in which separations are more than a mere 20-30 A. Only for distances less than 20 A and, sometimes, for distances greater than 10,000 A, can the van der Waals interaction between parallel-planar surfaces be said to vary by the 1/Z2 power law that it demurely reveals in its simplest representation. 

The accuracy of these approximations, when neither the surface potential nor the surfece charge density are constants, will be investigated below, for two identical, parallel planar surfaces. 

Poly(amidoamine) dendrimers such as 91, which differ only in the presence of an ethylenediamine (instead of ammonia) core, were used to host the electron transfer reaetion between photoexcited polycyelic aromatic hydrocarbons (PAHs) and ni-tromethane [169]. Quenching studies in aqueous solution indicate that the PAHs associate with the dendrimer, residing in the interior region, far from the charged surface. It was also found that within the dendrimer structure, nitromethane quenches alternant PAHs, while nonalternant PAHs are not quenched. This selective behavior of nitromethane parallels that commonly observed in solution, while in the presence of traditional micelles both classes of PAHs are quenched by nitromethane. 

Consider now two parallel flat surfaces, each with equal charge density a, separated by a distance D. Although charged, there is no net electrostatic repulsion of the surfaces, since the counterions in the gap between the plates cancel the charges on the plates. However, because the counterions must be drawn into the gap, there is an entropy cost that becomes greater as... 

Electro-osmosis refers to the movement of the liquid adjacent to a charged surface, in contact with a polar liquid, under the influence of an electric field applied parallel to the solid-liquid interface. The bulk fluid of liquid originated by this electrokinetic process is termed electro-osmotic flow (EOF). It may be produced both in open and in packed capillary tubes, as well as in planar electrophoretic systems employing a variety of supports, such as paper or hydrophilic polymers. 

Figure 10. Electrical double layer models. Top right (a) typical type of potential vs. composition plot for a charged surface compared to (b) constant capacitance model. Top left Two double-layer models, (a) diffuse double layer, (b) part parallel plate capacitor and part diffuse layer.. Bottom left Stem layer model. Incorporation of adsorbed ions to surface. From Hiemenz and Rajagopalan (1997) Bottom right Comparison of Gouy-Chapman and Stem-Grahame models of the electrical double layer. From Davis and Kent (1990).

A common feature of electrokinetic phenomena is a relative motion of the charged surface and the volumetric phase of the solution. The charged surface is affected by the electric field forces, and the movement of such surfaces toward each other induces the electrical field. That is a question of slip plane between the double layer and a medium. The layer bounded by the plane at the distance d from surface (OHP) can be treated as immobile in the direction perpendicular to the surface, because the time of ion residence in the layer is relatively long. Mobilty of ions in the parallel direction to the interfacial surface should also be taken into account. However, it seems that the ions in the double layer and in the medium surrounding it constitute a rigid whole and that the layer from x = 0 to X = d is immobile also in the sense of resistance to the tangent force action. There is no reason why the boundary plane of the solution immobile layer should overlap accurately with the OHP plane. It can be as well placed deeply in the solution. The potential in the boundary plane of the solution immobile layer is called potential (. Strictly speaking it is not a potential of interface because it is created in the liquid phase. It can be considered as the difference of potentials between a point far from the surface (in the bulk solution) and that in the slip plane. 

Contrary to the case of two identically charged surfaces, which always repel each other (see Equation 5.178), the electrostahc interaction between two plane-parallel surfaces of different potentials, /j, and /j2, can be either repulsive or attractive. - Here, we will restrict our considerations to the case of low surface potenhals, when the Poisson-Boltzmann equation can be linearized. Despite that it is not too general quantitatively, this case exhibits qualitatively all features of the electrostatic interaction between different surfaces. 

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