Parallel plate interfaces

It is interesting to note that, in the absence of parasitics, the sensitivity of the parallel-plate interface is independent of capacitor size... 

Since, in most cases, the nominal overlap Xo is much larger than the displacement x, parallel-plate interfaces have considerably larger sensitivity. Lateral combs are most often used for actuation where large range is required or in electromechanical oscillators, where the position-dependence of the negative spring of parallel-plate interfaces introduces nonlinearity [13]. Parallel-plate structures are usually preferred for maximum displacement resolution. 

A number of more or less equivalent derivations of the electrocapillary Eq. V-49 have been given, and these have been reviewed by Grahame [113]. Lippmann based his derivation on the supposition that the interface was analogous to a parallel-plate condenser, so that the reversible work dG, associated with changes in area and in charge, was given by... 

Eutectoid structures are like eutectic structures, but much finer in scale. The original solid decomposes into two others, both with compositions which differ from the original, and in the form (usually) of fine, parallel plates. To allow this, atoms of B must diffuse away from the A-rich plates and A atoms must diffuse in the opposite direction, as shown in Fig. A1.40. Taking the eutectoid decomposition of iron as an example, carbon must diffuse to the carbon-rich FejC plates, and away from the (carbon-poor) a-plates, just ahead of the interface. The colony of plates then grows to the right, consuming the austenite (y). The eutectoid structure in iron has a special name it is called pearlite (because it has a pearly look). The micrograph (Fig. A1.41) shows pearlite. 

Any real sample of a colloidal suspension has boundaries. These may stem from the walls of the container holding the suspension or from a free interface towards the surroundings. One is faced with surface effects that are small compared to volume effects. But there are also situations where surface effects are comparable to bulk effects because of strong confinement of the suspension. Examples are cylindrical pores (Fig. 8), porous media filled with suspension (Fig. 9), and thin colloidal films squeezed between parallel plates (Fig. 10). Confined systems show physical effects absent in the bulk behavior of the system and absent in the limit of extreme confinement, e.g., a onedimensional system is built up by shrinking the size of a cylindrical pore to the particle diameter. 

The dimensionahty of a system is one of its major features. Despite the fact that our surrounding space is three-dimensional, one can prepare situations that lead to an effective lowered dimension. A typical example regarding colloids is the surface between the solvent and air. One can prepare the particles to be trapped at that interface, so that they float on top of the solvent, building up a two-dimensional (2d) system. Another realization is strong confinement between parallel plates that leads to an effective 2d system. Concerning simulations, it is very convenient to simulate 2d systems, as one has fewer degrees of freedom to deal with e.g., plotting snapshots is easier in 2d than it is in 3d. So, besides their experimental realizations, 2d systems are also important from a conceptual point of view. 

A simple model of the e.d.l. was first suggested by Helmholz in which the charges at the interface were regarded as the two plates constituting a parallel plate capacitor, e.g. a plate of metal with excess electrons (the inner Helmholz plane I.H.P.) and a plate of excess positively charged ions (the outer Helmholz plane O.H.P.) in the solution adjacent to the metal the... 

The electrified interface is generally referred to as the electric double layer (EDL). This name originates from the simple parallel plate capacitor model of the interface attributed to Helmholtz.1,9 In this model, the charge on the surface of the electrode is balanced by a plane of charge (in the form of nonspecifically adsorbed ions) equal in magnitude, but opposite in sign, in the solution. These ions have only a coulombic interaction with the electrode surface, and the plane they form is called the outer Helmholtz plane (OHP). Helmholtz s model assumes a linear variation of potential from the electrode to the OHP. The bulk solution begins immediately beyond the OHP and is constant in potential (see Fig. 1). 

Helmholtz had proposed such a parallel plate capacitance model for the entire interface in 1853. 

The Ravenfield model BS viscometer is a wide shear rate range instrument with several possible measurement systems cone—plate, parallel plates, concentric cylinders, and taper plug. The last gives shear rates of up to 106 s-1, and the cone—plate of up to 8 x 104 -1. The viscosity range is 102 108 mPa-s. Measurements can normally be made up to 170°C, but with special modifications even higher temperatures can be achieved. A computer interface permits two-way communication with a computer. 

Going a step further, what does the parallel-plate model of the double layer have to say regarding the capacity of the interface Rearranging Eq. (6.119) in the form of the definition of differential capacity [Eq. (6.97)],... 

It appears that an electrified interface does not behave like a simple double layer. The parallel-plate condenser model is too naive an approach. Evidently some crucial secrets about electrified interfaces are contained in those asymmetric electrocapillaiy curves and the differential capacities that vary with potential. One has to think again. 

What happens when the concentration c0 of ions in solution is very large Equations (6.124) and (6.130) indicate that while CG increases with increasing c0, CH remains constant. Thus, as c0 increases, (1/CG) (1/CH), and for all practical purposes, C CH. That is, in sufficiently concentrated solutions, the capacity of the interface is effectively equal to the capacity of the Helmholtz region, Le., of the parallel-plate model. What this means is that most of the solution charge is squeezed onto the Helmholtz plane, or confined in a region vety near this plane. In other words, little charge is scattered diffusely into the solution in the Gouy-Chapman disarray. 

Starting from the Lippmann equation, derive an expression for the variation of the radius of a mercury drop as a function of potential in a solution where there is no specific adsorption. Assume that the double layer at the mercury solution interface can be treated as a parallel-plate capacitor. (Contractor)... 

Fig. 7.7. (a) A double layer, a simple hypothetical type of electrified interface in which a layer of ions on the outer Helmholtz plane constitutes the entire excess charge in the solution. The solvation sheaths of these ions and the first row of water molecules on the electrode are not shown in the diagram, (b) The electrical equivalent of such a double layer is a parallel-plate condenser. 

FIG. 11.2 The variation of electrochemical potential in the vicinity of the interface between two phases, a and / (a) according to a schematic profile and (b) according to the parallel plate capacitor model. 

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,... 

Parallel plates arc corrugated (like roofing material) with the axis of corrugations parallel to the direction of flow. The plate pack is inclined at 45° and bulk water flow is forced downward. The oil sheet rises upward, counter to water flow, and is concentrated in the (op of each corrugation. When oil reaches the end of the plate pack, it is collected in a channel and brought to (he oil/water interface. 

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