Gradient of electrostatic potential

Fig. 4.2. The migration of ions resulting from a gradient of electrostatic potential (i.e., an electric field) in an electrolyte. The electric field is produced by the application of a potential difference between two electrodes immersed in the electrolyte. The directions of increasing electrostatic potentials and of ionic migration are shown below the diagram.

Figure 21.7 The cell membrane of a nerve acts as a capacitor. The membrane Is composed of lipids that, like oil, have a low dielectric constant. If the inside and outside solutions have different electrostatic potentials, there is a gradient of electrostatic potential across the membrane. 

Here s how you use Equation (22.4). Suppose that you are interested in the distribution of ions that are subject to a fixed spatial gradient of electrostatic potential. Consider two different locations in space, ri and r2, within a single solution. To eliminate some vector complexity, let s look at a one-dimensional problem. Suppose that ri = Xi and Tt = x->- At location Xi, some constellation of fixed charges or electrodes creates the electrostatic potential ip(xi), and at X2 you have ip(x>). Now consider a single species of mobile ions (so that you can drop the subscript i) that is free to distribute between locations Xi and x. The mobile ions will be at equilibrium when the electrochemical potentials are equal. 

Concentration gradients can cause particles to flow, whether the particles are neutral or charged. In addition, charged particles flow if they are subject to gradients of electrostatic potential. According to Equation (20.12), the force on a charged particle equals (charge ze) x (electrostatic field E),... 

If an ion is driven by both a gradient of its concentration and a gradient of electrostatic potential, then its flux Jp will be the sum of the component fluxes. 

However, in this case the EMF measured will be distorted by another effect [i.e., the variation of electrostatic potential within a given conductor, which is caused by a temperature gradient in the conductor (the Thomson effect, 1856)]. Potential gradients will arise even at zero current, in both the electrolyte (between points and Aj)... 

An important equation of electrostatics, which follows directly from Maxwell s equations (Jackson 1975) is Poisson s equation. It relates the divergence of the gradient of the potential charge density at that point ... 

Like the potential, other electrostatic functions can be expressed as Fourier summations over the structure factors (Stewart 1979). The electric field, being the (negative) gradient of the potential, is a Fourier series in which the power of the magnitude of H increases from —2 to —1, as expected from the reciprocal relationship between direct space and Fourier space. Starting with... 

A gradient in electrostatic potential can produce a driving force for the mass diffusion of a species, as discussed in Section 2.2.2. Two examples of this are the potential-gradient-induced diffusional transport of charged ions in ionic conductors such as those used in solid-electrolyte batteries and the electron-current-induced diffusion of interstitial atoms in metals. 

The relations developed here point up an interesting feature The streaming potential (V /VP)I+ 0 cannot readily be experimentally determined, since it forces imposition of a change in electrostatic potential in the absence of a net responding current. However, this quantity is also given by the ratio - (Jv/I+)yp o - /3, which can readily be determined experimentally. Here one measures the volume flux and current in response to the imposition of a gradient in electrostatic potential when the pressure in the two compartments are identical. Analogous remarks apply to the quotient in (6.8.7). 

In Eq. 12, FB is the boundary force at r0, F(r0 — rT) is the force of interaction between a particle at rT in RR and a particle at r0 in RZ, and dTTpTg( o rr) is the probability of the pair (0, T) having a separation r<> — tT. The boundary force may be written as the gradient of a potential, the boundary potential. The boundary potential for the oxygen atom of ST2 water12 in an 11 A reaction zone is plotted in Fig. 9. In the calculation of this potential only the van der Waals part of the ST2-ST2 interaction was included in Eq. 12. A methodology that consistently incorporates electrostatic forces into the boundary potential is under development.110 In its present simplified form, the model has proven successful in the simulation of localized regions of pure... 

For simplicity we discuss mainly solutions sufficiently dilute that the solute species and their gradients do not interact. The solution might be an un-ionized solvent containing ionized electrolytes. If a gradient in electrostatic potential is applied to the solution, there will be an electric force exerted on the ion species, which is proportional to the potential gradient. The electric field E is the negative gradient of the electrostatic potential ... 

Another difference between an electrochemical reaction and a catalytic reaction is that a so-called electrical double layer will form as the appearance of electrostatic potential gradient in the interface of electrolyte solution and electrode (conductor). Graham summarized in more detail the electrical double layer in 1947. He considered that this electrostatic potential, i.e. the double layer potential, is different from the electrode potential. He also discussed and observed in detail the double layer potential of Hg-electrode-water solution system. He found that it could not observe such potential when electrode reaction occurred while the ideal polarization happened in a wide range of electrode potential if there was no electrode reaction. Hg is a liquid and it is thus easy to observe its surface tension and calculate the relationship between surface tension and double layer potential. Therefore, its structure is clearer. The structure of electrical double layer is composed of Helmholz layer and diffusion layer. The Helmohloz face is located between Helmholz layer and diffusion layer. The external of Helmohloz face is diffusion double layer. The model of Helmholz electrical double layer corresponds to simple parallel-plate capacitor. According to its equation, it can quantitatively describe the structure of diffusion double layer. 

To display properties on molecular surfaces, two different approaches are applied. One method assigns color codes to each grid point of the surface. The grid points are connected to lines chicken-wire) or to surfaces (solid sphere) and then the color values are interpolated onto a color gradient [200]. The second method projects colored textures onto the surface [202, 203] and is mostly used to display such properties as electrostatic potentials, polarizability, hydrophobidty, and spin density. 

---