Mean electrostatic potential

The total energy of this adsorption reaction can be found experimentally from the microscopic activity quotient, and separated theoretically into the following components (1) transfer of the ion to be adsorbed from the bulk of solution to the oxide surface plane, at which the mean electrostatic potential is t/>q with respect to the bulk of solution (2) reaction of the adsorbate in the surface plane with a functional group at the surface (3) transfer of a fraction of the counter charge from solution into the solution part of the double layer by attraction of counter ions and (4) transfer of the remainder of the counter charge by expulsion of co-ions from the solution part of the double layer to the solution. 

For two planar charged surfaces separated by a distance, h, and by a symmetric v v electrolyte, the mean electrostatic potential is given by [4,13],... 

An improvement is to permit Eq. (4.19) to account for a nonzero mean electrostatic potential exerted by the solution on the distinguished reference solute, writing... 

The inner layer is a concept within the framework of the classical Gouy-Chap-man-Stern model of the double layer [57]. Recent statistical-mechanical treatments of electrical double layers taking account of solvent dipoles has revealed a microscopic structure of inner layer" and other intriguing features, including pronounced oscillation of the mean electrostatic potential in the vicinity of the interface and its insensitivity at the interface to changes in the salt concentration [65-69]. 

In sec. 1.4.3c we have already introduced the mean (average) electrostatic potential. The mean electrostatic potential yr x ) is related to the work necessary to bring an Infinitely small probe charge from Infinity to Xj without disturbing the environment. This potential can be defined In terms of the one- and two-body interactions, mentioned above, according to... 

The mean electrostatic potential, experienced by a test charge at a distance z from the surface, is obtained from... 

Thus the surface charge density is proportional to the negative slope of the mean electrostatic potential at z = 0. In fact, the initial shape of the mean electrostatic potential is necessarily linear with slope -47rir/e. 

It should be appreciated that in contrast to the simple free electron models used in much of our discussion of metals and semiconductors, a treatment of screening necessarily involves taking into account, on some level, the interaction between charge carriers. In the Thomas-Fermi theory this is done by combining a semiclassical approximation for the response of the electron density to an external potential with a mean field approximation on the Hartree level—assuming that each electron is moving in the mean electrostatic potential of the other electrons. 

We use the Gouy-Chapman theory for the diffuse layer which is based on the Poisson-Boltzmann (P.B.) equation for the potential distribution. Although the different corrections to the P.B. equation in double-layer theory have been investigated (20, 21, 22, 23), it is difficult to state precisely the range of validity of this equation. In the present problem the P.B. equation seems a reasonable approximation at 0.1M of a 1-1 electrolyte to 50mV for the mean electrostatic potential pd at the ohp (24) this upper limit for pd increases with a decrease in electrolyte concentration. All the values for pd calculated in Tables I-IV are less than 50 mV— most of them are well below. If n is the volume density of each ion type of the 1-1 electrolyte in the substrate, c the dielectric constant of the electrolyte medium, and... 

FIG. 16 Ion distribution function P(r) (left) and mean electrostatic potential if/j) (right) for DNA-like systems (see Table 1) with 0.5 mol/L added 2 2 salt. The six curves differ in the line charge density of the rod, producing Manning parameters between 1.05 and 10.5 as indicated in the key. The value 4.2 corresponds to DNA. Notice that the radial distance is only plotted up to one third of the cell radius. 

Another important phenomenon is indicated by the behavior of the mean electrostatic potential in the right frame of Figure 16. It concerns the value of this potential at the distance of closest approach between ions,... 

FIG. 19 Ion distribution function P(r) (left) and mean electrostatic potential i/4r) (right) for systems with Manning parameter = 4.2. The nine curves correspond to different numbers Ns of 2 2 salt molecules added to the box Ns 8, 17, 34, 68, 135, 270, 380, 540, 760. In the distribution function the salt content increases from bottom to top in the mean electrostatic potential it increases from top to bottom. 

The popular Poisson-Boltzmann equation considers the mean electrostatic potential in a continuous dielectric with point charges and is therefore, an approximation of the actual potential. An improved model and mathematical solution resulted in the MPB equation (26). This equation is based on a restricted primitive electrolyte model that considers ions as charged hard spheres with diameter d in a continuous uniform structureless dielectric medium of constant dielectric permittivity s. The sphere representing an ion has the same permittivity e. The model initially was developed for an electrolyte at a hard wall with dielectric permittivity and surface charge density a. The charge is distributed over the surface evenly and continuously. 

This theory takes into account the finite size of ions, the fluctuation potential, and image forces in the electrolyte solution next to a rigid electrode, but it still an approximation. The MPB theory begins with the Poisson equation for the mean electrostatic potential ir in solution ... 

FIGURE 3.15 Dimensionless mean electrostatic potential (a) and surface-ion distribution function (b) as predicted by the Gouy-Chapman-Stern (GCS) and modified Poisson-Boltzmann (MPB) theories for a 1 1 electrolyte with a = 0.425 nm and c = 0.197 M. (Outhwaite, Bhuiyan, and Levine, 1980, Theory of the electric double layer using a modified Poisson-Boltzmann equation. Journal of the Chemical Society, Faraday Transactions 2 Molecular and Chemical Physics, 76, 1388-1408. Reproduced by permission of The Royal Society of Chemistry.)... 

A second nonlinear problem to consider is the Poisson-Boltzmann (PB) mean-field theory of ionic solution electrostatics. " " This approximate theory has found wide application in biophysics involving problems such as protein-protein and protein-membrane interactions. At a qualitative level, the PB equation arises from replacement of the exact mobile-ion charge distribution by its average, assuming that the average is given by the Boltzmann factor for the ions interacting with the mean electrostatic potential ... 

Fig. 8. The variation of the mean electrostatic potential, Y, with distance, z, from a negatively charged surface. The surfaces are non-adsorbing, with two different choices of a-u, as in Figure 7.

The replacement of the potential of mean force with the mean electrostatic potential by Debye and Hiickel (and implicit in the Gouy-Chapman approach) has caused the greatest amount of concern for those applying the PB equation. Fowler severely criticized use of the PB equation on this basis, but his investigation was soon shown to be overly restrictive.Still, the effect of neglecting ion-ion correlation, which this mean-field approximation implies, is a continual source of study. Hence there have been published numerous comparisons between PB theory and more detailed statistical-mechanical theories or calculations that do include correlation. While the size of the effect depends on the particular system studied, calculations on the cylindrical and all-atom models of DNA show that PB calculations tend to underestimate ion concentrations at the surface by 15-25% for mono- or divalent ions, respec-tively. " "- ... 

In addition to neglecting ion correlation, using the mean electrostatic potential has the undesirable consequence that the (nonlinear) PB equation no longer satisfies a reciprocity condition that use of the potential of mean force would obey. Linearization of the equation by Debye and Hiickel regained this condition. These considerations led Outhwaite and others to propose modifications of the PB equation to treat these problems. Within this modified Poisson-Boltzmaim (MPB) theory, the effect of ion correlation is expressed in terms of a fluctuation potential for which a first-order (local) expression, written as an activity coefficient, can be derived. Their result for bulk hard-sphere electrolyte ions of valence z, and common radius a gives the formula ... 

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