Repulsive interactions between spherical double layers

For interacting identical spherical particles with spherical double layers, a similar calculation of the repulsive potential can be carried out (Overbeek 1972). Provided the thickness of the double layers is small compared with the particle size, the interaction between the double layers on the spherical particles can be assumed to be made up of contributions from infinitesimally small parallel rings, each of which can be considered as a flat plate (see Fig. 8.1.2). The energy of repulsion between the spherical double layers is then... 

A similar calculation can be carried out for the potential energy of repulsive forces between two identical spherical particles. In the case when the thickness of the electrical double layer around these particles is small compared to their radii, interaction of electrical double layers of these spheres may, according to Derjaguin, be considered as a superposition of interactions of infinitely narrow parallel rings (Fig. 10.2) [53]. 

A general expression for the repulsive interaction between the electrical double layers around two spherical particles is quite complex and does not warrant discussion here. A simple and relatively good approximate equation derived by Reerink and Overbeek is... 

A quantitative treatment of the effects of electrolytes on colloid stability has been independently developed by Deryagen and Landau and by Verwey and Over-beek (DLVO), who considered the additive of the interaction forces, mainly electrostatic repulsive and van der Waals attractive forces as the particles approach each other. Repulsive forces between particles arise from the overlapping of the diffuse layer in the electrical double layer of two approaching particles. No simple analytical expression can be given for these repulsive interaction forces. Under certain assumptions, the surface potential is small and remains constant the thickness of the double layer is large and the overlap of the electrical double layer is small. The repulsive energy (VR) between two spherical particles of equal size can be calculated by ... 

The potential half-way between the plates is no longer zero (as would be the case for isolated particles at x -mx>). For two spherical particles of radius R and surface potential and condition kR<, the expression for the electrical double layer repulsive interaction is given by. 

When charged colloidal particles in a dispersion approach each other such that the double layers begin to overlap (when particle separation becomes less than twice the double layer extension), then repulsion will occur. The individual double layers can no longer develop unrestrictedly, as the limited space does not allow complete potential decay [10, 11]. The potential v j2 half-way between the plates is no longer zero (as would be the case for isolated particles at 00). For two spherical particles of radius R and surface potential and condition x i <3 (where k is the reciprocal Debye length), the expression for the electrical double layer repulsive interaction is given by Deryaguin and Landau [10] and Verwey and Overbeek [11],... 

Figure 5. Cluster-network model for Nafion perfluorinated membranes. The polymeric ions and absorbed electrolyte phase separate from the fluorocarbon backbone into approximately spherical clusters connected by short, narrow channels. The polymeric charges are most likely embedded in the solution near the interface between the electrolyte and fluorocarbon backbone. This configuration minimizes both the hydrophobic interaction of water with the backbone and the electrostatic repulsion of proximate sulfonate groups. The dimensions shown were deduced from experiments. The shaded areas around the interface and inside a channel are the double layer regions from which the hydroxyl ions are excluded electrostatically.

The classical DLVO theory of interparticle forces considers the interaction between two charged particles in terms of the overlap of their electric double layers leading to a repulsive force which is combined with the attractive London-van der Waals term to give the total potential energy as a function of distance for the system. To calculate the potential energy of attraction Va between solid spherical particles we may use the Hamaker expression ... 

In an electrostatically-stabilized dispersion in which the particles do not approach within a distance of about 10 or 15 solvent diameters, the equilibrium behaviour depends only on the double-layer repulsion and van der Waals attraction. The interaction energy between pairs of spherical particles i and / a distance r apart is split into two parts, arising respectively from repulsive and... 

We now consider interactions between two spherical colloid particles of radius R in an electrolyte of bulk concentration cq. The expression for the repulsive potential can, to a good approximation, be derived from the electrical potential as a function of distance from a charged plane, if the radius of the particles is sufficiently large. When the electrical double layers are far... 

---