Bispherical Coordinates

Under such condition, the steady-state density, n, is identical to the escape probability, p (see Chap. 7, Sect. 2.3). By using bispherical coordinates [499], Samson and Deutch [258] were able to show that the density n (r) is satisfied by... 

Electrophoretic interactions between spherical particles with infinitely thin double layers can also be examined using the boundary collocation technique [16,54]. This method enables one not only to calculate the interactions among more than two particles, but also to deal with the case of particles in contact, for which the bispherical coordinate solution becomes singular. Analogous to the result for a pair of spheres, no interaction arises among the particles in electrophoresis as long as all the particles have an equal zeta potential. This important result is also confirmed by a potential-flow reasoning [10,55]. 

When the gap width between two particles becomes very small, numerical calculations involved in both the bispherical coordinate method and the boundary collocation technique are computationally intensive because the number of terms in the series required to be retained to achieve a desired accuracy increases tremendously. To solve this near-contact motion more effectively and accurately, Loewenberg and Davis [43] developed a lubrication solution for the electrophoretic motion of two spherical particles in near contact along their line of centers with the assumption of infinitely thin ion cloud. The axisymmetric motion of the two particles in near contact can be approximated as the pairwise motion of the spheres in point contact plus a deviation stemming from their relative motion caused by the contact force. The lubrication results agree very well with those obtained from the collocation method. It is shown that near contact electrophoretic interparticle... 

Numerical Solution to Nonlinear Poisson-Boltzmann Equation for Bispherical Coordinates... 

The direct numerical solutions of PB equation for spheres have been reported by a number of researchers, including. In the numerical computation, PB Eq. (13) is conveniently expressed and solved in the bispherical coordinates. Due to the rotational symmetry of the interaction along the centerline, Eq. (13) simplifies into... 

To determine the bispherical coordinates of a point p, rj, O from specified values of Cartesian coordinates x, y, z, it is necessary to invoke formulae of reverse transformation [89] ... 

These confluent bispherical coordinates are related to Cartesian coordinates as follows ... 

Vm,n confluent bispherical coordinates Solution of Laplace equation in ... 

However, many numerical solutions of the PB have become available now which can be exploited for estimating the validity of the approximate models. In these calculations, pioneered by Hoskin and Levine [21,44], one uses the finite-difference method and the PB equation is formulated in the bispherical coordinate system. The advantage of this orthogonal coordinate system is that the boundary conditions at the sphere surfaces can be accurately expressed. This coordinate system (with more mesh points) was subsequently used by Camie et al. [45], who performed calculations of the interaction force for two spherical particles in a 1-1 electrolyte. The authors proved that the electrostatic fields distribution within the particles exerted a negligible effect on interaction force characterized by 8 < 5 (e.g., polystyrene latex particles). 

For smaller separations, the force on a dielectric sphere in a imiform field E can only be calculated numerically using the bispherical coordinate system. The results plotted in Fig. 6 indicate that at h /a < 1, the force decreases with the distance less abruptly than for larger distances and significantly depends on the relative dielechic constant Ep/E. In the limit of Ep/E 0, the force can be interpolated by the function... 

An analysis of particle motion parallel to a planar interface in a qmescent fluid is more complicated because the translational motion induces a coupled rotational motion, which produces a hydrodynamic torque on the particle. These effects have been calculated by Goren and O Neill [99] using bispherical coordinates. It was shown that particle velocity can be expressed as... 

---