Potential distribution, parallel

The broadening of the surface state level up to 1 eV could reflect the time fluctuations of the water dipoles in the double layer. It could also be caused by the discreteness of the charge, leading to an inhomogeneous potential distribution parallel to the surface. In any case, a lineshape analysis should throw some light onto the microscopic structure of the interfacial region and its potential dependence. 

FIG. 4. Schematic representation (a) of a parallel-plate, capacitively coupled RF-discharge reactor, with unequal-size electrodes. The potential distribution (b) shows the positive plasma potential Vp and the negative dc self-bias voltage... 

Figure 3. Potential distribution in a parallel plate plasma etcher with grounded surface area larger than powered electrode area, (Reproduced with permission from Ref 12 J...

Here r is the magnitude of the projection of the vector r — f2 between oxygens onto a plane parallel to the metal walls, and z and zz are the distances of oxygens 1 and 2 from the wall. Values of the parameters appear in Ref. 69, where the resulting oxygen densities from the classical and direct dynamics simulations are compared. While the densities are rather similar, the potential distributions are not, as emphasized above. 

Figure 6.9 Change in the potential distribution when two parallel planar surfaces approach each other. The gap is filled with electrolyte solution.

Several cell configurations are common in electrochemical research and in industrial practice. The rotating disk electrode is frequently used in electrode kinetics and in mass-transport studies. A cell with plane parallel electrodes imbedded in insulating walls is a configuration used in research as well as in chemical synthesis. These are two examples of cells for which the current and potential distributions have been calculated over a wide range of operating parameters. Many of the principles governing current distribution are illustrated by these model systems. 

Fig. 33. Schematic representation of the potential distribution in the electrolyte as a result of an inhomogeneous distribution of the electrode potential, DL, and the effect of migration currents induced by the inhomogeneous potential distribution on the local temporal evolution of the potential (a) for the case that the length of the WE is much smaller than the distance between the WE and the CE and (b) for the case that the length of the WE is much larger than the distance between the WE and the CE. The length of the arrows in the representations below the potential distributions indicate how the contribution of the migration couplings to the temporal evolution of DL changes as a function of distance from the disturbance. (x, z spatial coordinates parallel and perpendicular to the WE, respectively. The electrode is assumed to be one-dimensional and the electrolyte two-dimensional.)...

C. Current and Potential Distribution for the Flow Channel with Parallel Plates... 

The proposed method is based upon the quantitative measurement of the contribution of differently charged nitroxide probes to the spin-lattice relaxation rate (1/T i) of protons in a particular molecule, followed by the calculation of local electrostatic potential using the classical Debye equation (Likhtenshtein et al., 1999 Glaser et al., 2000). In parallel, the theoretical calculation of potential distribution with the use of the MacSpartan Plus 1.0 program has been performed. 

This is the required expression for the interaction energy per unit area between two parallel similar plates with constant surface potential. Erom the nature of this linearization, the obtained potential distribution (9.177) is only accurate near the plate surface and thus the interaction energy expression (9.177) is also only accurate for small plate separations. Indeed, at h = 0, Eq. (9.177) gives the following correct expression, regardless of the value of yo,... 

FIGURE 10.1 Schematic representation of the potential distribution (soUd line) across two interacting parallel dissimilar plates 1 and 2 at separation h. Dotted line is the unperturbed potential distribution ath = oo. unperturbed surface potentials of plate 1 and 2, respectively. 

FIGURE 11.1 Schematic representation of the potential distribution i/ (x) across two parallel similar plates 1 and 2. 

FIGURE 13.2 Schematic representation of the potential distribution ij/ x) across two parallel interacting ion-penetrable semi-inhnite membranes (soft plates) 1 and 2 at separation h. The potentials in the region far inside the membrane interior is practically equal to the Donnan potential i/ doni or i/ don2-... 

In Fig. 13.3, we plot the potential distribution i/ (x) between two parallel similar ion-penetrable membranes with i/tdoni = iAdon2 = don (or >Aoi = o2 = >Ao) for Kh = 0, 1,2, and oo. In Fig. 13.3, we have introduced the following scaled potential y, scaled unperturbed surface potential y, and scaled Donnan potential yooN-... 

FIGURE 13.5 Scaled potential distribution y x) —e j/ x)/kT between two parallel oppositely charged dissimilar membranes 1 and 2 with Tdoni — 2 and Tdon2 —... 

Consider the double-layer interaction between two parallel porous cylinders 1 and 2 of radii and a2, respectively, separated by a distance R between their axes in an electrolyte solution (or, at separation H = R ai—a2 between their closest distances) [5]. Let the fixed-charge densities of cylinders 1 and 2 be and Pfix2. respectively. As in the case of ion-penetrable membranes and porous spheres, the potential distribution for the system of two interacting parallel porous cylinders is given by the sum of the two unperturbed potentials... 

Consider two parallel planar dissimilar ion-penetrable membranes 1 and 2 at separation h immersed in a solution containing a symmetrical electrolyte of valence z and bulk concentration n. We take an x-axis as shown in Fig. 13.2 [7-9]. We denote by Ni and Zi, respectively, the density and valence of charged groups in membrane 1 and by N2 and Z2 the corresponding quantities of membrane 2. Without loss of generality we may assume that Zj > 0 and Z2 may be either positive or negative and that Eq. (13.1) holds. The Poisson-Boltzmann equations (13.2)-(13.4) for the potential distribution j/(x) are rewritten in terms of the scaled potential y = zeif/IkT as... 

We start with the simplest problem of the plate-plate interaction. Consider two parallel plates 1 and 2 in an electrolyte solution, having constant surface potentials i/ oi and J/o2, separated at a distance H between their surfaces (Fig. 14.1). We take an x-axis perpendicular to the plates with its origin 0 at the surface of one plate so that the region 0solution phase. We derive the potential distribution for the region between the plates (0linearized Poisson-Boltzmann equation in the one-dimensional case is... 

FIGURE 15.2 Interaction between two parallel soft plates 1 and 2 at separation h and the potential distribution i/r(x) across plates 1 and 2, which are covered with surface charge layers of thicknesses d and d2, respectively. 

Consider two parallel identical plates coated with a charged polymer brush layer of intact thickness at separation h immersed in a symmetrical electrolyte solution of valence z and bulk concentration n as shown in Fig. 17. la [2]. We assume that dissociated groups of valence Z are uniformly distributed in the intact brush layer at a number density of No- We first obtain the potential distribution in the system when the two brushes are not in contact h > 2d. We take an x-axis perpendicular to the brushes with its origin 0 at the core surface of the left plate so that the region... 

FIGURE 17.1 Interaction between two identical parallel plates covered with a charged polymer brush layer and potential distribution across the brush layers, (a) Before the two brushes come into contact Qi > 2do). (b) After the two brushes come into contact (h < 2do). do is the thickness of the intact brush layer. From Ref. 2. 

Now consider two parallel identical ion-penetrable membranes 1 and 2 at separation h immersed in a salt-free medium containing only counterions. Each membrane is fixed on a planar uncharged substrates (Fig. 18.2). We obtain the electric potential distribution i/ (x). We assume that fixed charges of valence Z are distributed in the membrane of thickness d with a number density of A (m ) so that the fixed-charge density pgx within the membrane is given by... 

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