Potential distribution, charged spheres

O Brien s method was extended to study the electrophoresis of a nonuniformly charged sphere with thin but polarized ion cloud in a symmetric electrolyte [32]. The electrophoretic mobility depends on the charge distribution at the particle surface. It is found that the polarization effect of the ion could leads to different electrophoretic mobilities for particles with different zeta potential distributions but having an identical velocity for the limit of infinite Ka. This intriguing result is due to the fact that the theory for undistorted ion cloud is linear in the distribution of zeta potential, whereas the polarization effects are nonlinear. 

The potential distribution, and hence the extent of the band bending, within the space charge layer of a planar macroscopic electrode may be obtained by solution of the one-dimensional Poisson-Boltzmann equation [95]. However, since the particles may be assumed to have spherical geometry, the Poisson-Boltzmann for a sphere must be solved. This has been done by Albery and Bartlett [131] in a treatment that was recently extended by Liver and Nitzan [125]. For an n-type semiconductor particle of radius r0, the Poisson-Boltzmann equation for the case of spherical symmetry takes the form ... 

Figure 10.7 shows the potential distribution surrounding two spheres of the same size with the same surface potentials, one with constant potential and the other with constant chaise boundary conditions as they are moved together. At small separations, the potential distribution between the particles is again greatly effected by the other particles. The surface of the sphere with the constant charge boundary... 

Figure 1.11 gives the scaled potential distribution y(r) around a positively charged spherical particle of radius a with yo = 2 in a symmetrical electrolyte solution of valence z for several values of xa. Solid lines are the exact solutions to Eq. (1.110) and dashed lines are the Debye-Hiickel linearized results (Eq. (1.72)). Note that Eq. (1.122) is in excellent agreement with the exact results. Figure 1.12 shows the plot of the equipotential lines around a sphere with jo = 2 at ka = 1 calculated from Eq. (1.121). Figures 1.13 and 1.14, respectively, are the density plots of counterions (anions) (n (r) = exp(+y(r))) and coions (cations) ( (r) = MCxp(—y(r))) around the sphere calculated from Eq. (1.121). 

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... 

Bo is the potential inside the sphere caused by charge distribution induced the by the central charge. The potential energy of this charge is therefore qBo. This potential energy separates into a contribution due to the ion-ion interactions that is determined by n and a term due to the dipole-dipole interactions, which is governed by e — eo-... 

For a homogeneous solid dielectric sphere (a particle with a radius of R) in a homogeneous dielectric medium, charges will accumulate at the interface between the particle and the medium. This results in an effective or induced dipole moment across the particle. The potential of the effective dipole moment Pg can be considered as an increment to the potential distribution of the applied field, which is given by ... 

Many model potentials (pnuc f) have been used [131] but two have become most important in electronic structure calculations. These are the homogeneous and the Gaussian charge distributions. The homogeneously or uniformly charged sphere is a simple model for the finite size of the nucleus. It is piecewise defined, because the positive charge distribution is confined in a sphere of radius R. The total nuclear charge -f-Ze is uniformly distributed over the nuclear volume 4 rR /3,... 

Unfortunately this is still too severe a requirement in many cases. It is often possible for molecules to approach to distances at which such spheres would overlap, without encountering the repulsive part of the potential and even when they do not overlap, they may approach so closely that convergence becomes very slow. For this reason, it has become common to adopt a distributed multipole description, in which each molecule is divided into a number of regions, each described by its own multipole moments. There are many ways of determining these distributed multipole moments [6-11] many authors have used distributed charges alone, but it is now widely... 

Garcia etal. [41] developed a two-dimensional porous electrode model and accounted for potential and charge distributions in the electrolyte. They employed transport equations derived from dilute solution theory, which is generally not adequate for LIB systems. The stress generation effect is built into the 2D DNS modeling framework with a simplified, sphere-packed electrode microstmcture description. 

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

The spheres represent (roughly) the 2p atomic orbitals on C and N, and half an electron resides in each sphere. The mutual potential energy of this charge distribution can be easily calculated from elementary electrostatics. For small distances, a polynomial fit was used instead. 

In view of this equation the effect of the ionic atmosphere on the potential of the central ion is equivalent to the effect of a charge of the same magnitude (that is — zke) distributed over the surface of a sphere with a radius of a + LD around the central ion. In very dilute solutions, LD a in more concentrated solutions, the Debye length LD is comparable to or even smaller than a. The radius of the ionic atmosphere calculated from the centre of the central ion is then LD + a. 

---