Approximate Analytical Expressions for the Electrical Potential between Two Planar, Cylindrical, and Spherical Surfaces

Electrical potentials of two identical planar, cylindrical, and spherical colloidal particles in a salt-free medium

A perturbation solution based on the curvature of the surface of a particle is derived for the electrical potential of two planar, cylindrical, and spherical colloidal particles immersed in a salt-free medium. Compared with the available results in the literature, the present approach improves both the accuracy and the applicable range in the level of the surface potential, the valence of counterions, and the thickness of the double-layer surrounding a particle. The zero-order perturbation solution is exact for the case of two planar particles. Under typical conditions, the averaged percentage deviation of the first-order perturbation solution is on the order of ca. 5% for both two cylindrical and two spherical particles.

Stability of soft colloidal particles in a salt-free medium

The stability of a salt-free dispersion containing soft spherical colloidal particles is investigated theoretically. Here, a particle comprises a rigid core and an ion-penetrable membrane layer; the ionic species in the liquid phase come solely from those dissociated from the functional groups in the membrane layer. We show that, similar to the case of a salt-free rigid dispersion, the total energy, which comprises the electrical energy and the van der Waals energy, is always positive far away from the surface of a particle and does not have a secondary minimum. Both the Derjaguin approximation for the estimation the electrical energy of two spheres and the criteria for the critical coagulation concentration in the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory are inapplicable. If the molar concentration of the fixed charge in the membrane layer exceeds ca. 0.1 M, the stability of a dispersion remains roughly the same. The maximum allowable particle concentration for a stable dispersion for the case of soft particles is lower than that for the case of the corresponding rigid particles.

The stability of a salt-free colloidal dispersion

The electrical potential for the case of two identical, planar parallel particles immersed in a salt-free medium, where the ionic species in the counterions come solely from those that dissociated from the surfaces, is evaluated. Analytical expressions for the electrical potential, the concentration of counterions, and the electrical energy are derived. We show that in a salt-free dispersion, if the separation distance between two particles is sufficiently far, the electrical repulsive force dominates, that is, the total energy is positive and does not have a secondary minimum, which is not the case for a dispersion where both coions and counterions are present. Also, the conditions used to calculate the critical coagulation concentration in the classic Derjaguin-Landau-Verwey-Overbeek theory become inappropriate and the Derjaguin approximation is inapplicable. We show that if the surface charge density exceeds approximately 0.04 Cm(2), the stability of a salt-free dispersion remains essentially the same. If the surface charge density is sufficiently high, the maximum separation distance between two particles below which coagulation occurs is in the ranges of [0,1 nm] and [1,7 nm] for the cases where the Hamaker constant is 10(-20) and 10(-19) J, respectively.

Approximate analytical expressions for the electrical potential in a cavity containing salt-free medium

The electrical potential in a closed surface such as a cavity containing counterions only is derived for the cases of constant surface potential and constant surface charge density. The results obtained have applications in, for example, microemulsion-related systems in which ionic surfactants are introduced to maintain the stability of a dispersion and electroosmotic flow-related analysis. An analytical expression for the electrical potential is derived for a planar slit, and the methodology used is modified to derive approximate analytical expressions for spherical and cylindrical cavities. The higher the surface potential, the better the performance of these expressions. For the case where the surface potential is above ca. 50 mV, the performance of the approximate analytical expressions can further be improved by multiplying a correction function.