Electrical potential, distribution

Eig. 1. Current flow (—) and electrical potential distribution (—) between two planar electrodes separated by an iasulated channel. 

Figure 23. Electric potential distribution in electric double layer. HL, Helmholtz layer DL, diffuse layer.

The Electrical Potential Distribution at the ITIES in the Presence of Zwitterionic Phospholipids... 

The theory presented above accounts for the electrostatic effects on the apparent rate constant for ion transfer by relating the observed changes in to changes in c"(0), or equivalently to 0(0). In the following, we present the simulated electrical potential distributions and the corresponding enhancement factors for a cation transferring from the aqueous phase across the water-l,2-DCE interface (s" = 78.39, s° = 10.36). The rela-... 

Fig. 4.1 Structure of the electric double layer and electric potential distribution at (A) a metal-electrolyte solution interface, (B) a semiconductor-electrolyte solution interface and (C) an interface of two immiscible electrolyte solutions (ITIES) in the absence of specific adsorption. The region between the electrode and the outer Helmholtz plane (OHP, at the distance jc2 from the electrode) contains a layer of oriented solvent molecules while in the Verwey and Niessen model of ITIES (C) this layer is absent...

Fujino, M., Ogata, S., and Shinohara, H., The electric potential distribution profile in a naturally charged fluidized bed, Funt. Kog. Kaishi, 20 280-289 (1983) English translation inInt. Chem. Engrg., 25 149-159 (1985)... 

Let us show that no fluid equilibrium is compatible with the electrolyte concentration distribution given by (4.4.55d). Indeed, associated with this concentration distribution, there is an electric potential distribution, prescribed by (4.4.53). This latter is in turn inseparable from an electric space charge, whose density may be found from the Poisson equations as... 

Electric potential distribution change upon electric field turning-on... 

M 2] [P 2] A numerical simulation of the electrical potential distribution after applying the electric field was made at two times, 1 and 10 ms after the start of the turning-on of the field [94], The electric potential distribution changes considerably, as does the velocity field (see Velocity field change upon electric field tuming-on above). 

Piezoresponse force microscopy (pfm) [11] and Kelvin probe force microscopy (kpfm) [9] were applied to deduce the polarization and local electric potential distribution over the whole cross section of the pzt sample (see Figure 12.3 and Figure 12.4) under static conditions as well as after switching. The details of our setup are described elsewhere [9,11],... 

Figure 7 Three-dimensional ECVT sensor configuration and electrical potential distribution (Warsito and Fan, 2005) (see Plate 10 in Color Plate Section at the end of this book).

The electrical potential distribution can be calculated using the Poisson equation... 

Combined with the boundary conditions (eqs 8—11) and the charge density distributions (eqs 6 and 7), eq 5 can be solved numerically to obtain the electrical potential distribution. Introducing the value obtained for the electrical potential at the middle between the plates in eq 3 or 4, one can obtain the force that is acting between the two plates. 

Equation (2) and the boundary conditions (9) and (10) can be solved numerically to obtain the electrical potential distribution for selected overall concentrations of electrolyte and small particles. 

The electrical potential distribution between the two plates is given by the Poisson equations... 

Fig. 3. The electrical potential distributions between two plates for various bulk NaCl concentrations. The distance between the two plates is 100 A. The other parameters employed are a = —0.01 C/nr, A = 10, e11 = 80, 8 = 10 A. (1) 0.001 (2) 0.01 (3) 0.1 M NaCl. The solid lines are for the electrical potentials predicted by the new model and the dashed lines for the electrical potentials predicted by the Gouy-Chapman theory.

Fig. 10. The electrical potential distribution between plates at small distances for various stiffnesses of the polyelectrolyte chain. D=11 A. and the other parameter values used in calculation are as for Fig. 8. (1) a =4 A (2) a=5 A (3) a=8 A (4) a=10 A.

Figure 3.23. Schematic of electric potential distribution for two electrolyte concentrations as a function of distance from a constant charge surface (from Dixon and Weed, 1977, with permission).

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... 

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 ... 

In both cases the electric potential distribution recorded by the receiver electrodes is used to map the spatial resistivity distribution of the rock formation. The main limitation of the resistivity method is that direct current cannot penetrate through resistive formations. Electromagnetic induction methods, based on transient electromagnetic fields, overcome this difficulty because a transient field can easily propagate through resistors like a radiowave propagates in the air. At the same time,... 

Zaban A., Meier A. and Gregg B. A. (1997), Electric potential distribution and short-range screening in nanoporous Ti02 electrodes , J. Phys. Chem. B 101, 7985-7990. 

Figure 2 shows a simplistic model of a FGM thermoelectric power device. we proceed to calculate the temperature and electric potential distributions within each arm of the device. The internal physics is a coupled phenomena of heat flux and electric current, and the expressions of thermoelectric phenomenon in isotropic body are given by... 

Heat flux and temperature distributions are shown in Figvire 6, and current density and electrical potential distributions are... 

The electrical conductivity(<, carrier concentration(n) and Hall mobility(/i ) were measured at 300 Kfor the p and n-type sintered PbTe. The characterization of the p-n jimction was conducted at 300 K by measuring thermoelectromotive force within temperature difference of 5 K, and voltage (electric potential) distribution using 4-probe method and current(])-voltage(V) relationship in forward and reverse bias. 

Substitution of Ohm s law into Equation (26.59) (with k constant since there are no concentration gradients) yields Laplace s equation, which is the govenung relationship for the electric potential distribution in an electrolyte solution of uniform composition ... 

Some of the electric potential distributions in a two-dimensional set up can be expressed in infinite series terms, when other mathematical algorithms do not work. Special series terms of Bessel functions and Legendre polynomials are easily evaluated nowadays with the help of computer... 

When the primary distribution does not illustrate the current or electric potential distribution well, an additional resistance, that is, the charge transfer electrode resistance, has to be considered. In such cases, we need to account for the electrode kinetics, and the secondary current and potential distributions emerge from the models. For industrial purposes the porous or tortuous electrocatalyst has to be considered as a dynamic system. This means that its porosity shape and density besides the surface roughness and the real geometric area changes all the time. This point makes us think that it... 

In the numerical simulation of an electrochemical system, it is advantageous to perform the simulation on the smallest domain to better describe the behavior of the cell. Taking symmetry as a criterion often reduces the problem of electric potential distribution. The division of the entire domain into smaller sub-domains enables to obtain a simpler function of linear or parabolic expressions. Then, by successive iterations the values from the approximating functions can be improved. 

Figures 16.7 and 16.8 show the electric potential profile for the conditions explained in the legends. The influence of Ad>° on the electric potential distribution is not very critical, but the values of j0 and X strongly affect the absolute values of A< but not its linear dependence with y.

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