Interaction of two spherical double layers

Potential energy of interaction of two spherical double layers. 

The basic theory of gel formation from colloidal particles has been formulated by Thomas and McCorkle (228), who show that the Verwey-Overbeek theory for the interaction of two spherical double layers around adjacent spherical colloidal particles leads to isotropic flocculation. New particles can be attached more readily to the ends of a chainlike floe where the repulsion energy barrier is at a minimum. It is this type of aggregation that converts a sol to a gel at a certain point by forming an infinite network of chains of particles throughout the sol volume. (See also Chapter 3.)... 

THE INTERACTION OF TWO SPHERICAL DOUBLE LAYERS a. General considerations... 

Value of the stirface potential X a Method to be applied for the caiculation of the interaction of two spherical double layers ... 

In the second place the interaction of the double layers of two droplets of an emulsion will determine the stability of the emulsion A complete description of this effect offering still more mathematical difficulties than the interaction of two spherical double layers as treated in 4 will not be aimed at, as a superficial consideration will already be sufficient to demonstrate that the energy of interaction between two droplets of an emulsion is much smaller than that between solid particles This explains why emulsions of two pure liquids are never stable (c/ chapter VIII, 11, p 336) and why the addition of emulsifiers is necessary to prepare stable emulsions Let us first consider the meeting of... 

This equation can be transformed into a formula describing the interaction between curved surfaces, such as that between two spherical double-layers. This is carried out by using the Derjaguin equation. The latter connects the force between two spherical double-layers and the interaction energy per unit area, Vr, of two plane interacting double-layers. It is assumed that both the spherical and the plane double-layers carry the same surface charge density, which leads to the following ... 

Clearly, Eq. [338] represents an approximation to the actual interaction energy since (1) the rings become progressively less parallel as p and hence H increases, (2) this ignores contributions from that part of the larger sphere beyond the smaller sphere radius as well as those from the backsides of both spheres, and (3) there are pressure contributions on the dividing plane beyond min not considered at all. These concerns are minimized if the system meets two conditions (1) the closest spacing between spheres is much less than the smaller radius, and (2) the thickness of both spherical double layers is small ... 

There are, however, differences in the geometry of the two problems. These differences affect the mathematical development. Thus, the central ion puts out a spherically symmetrical field. In contrast, the electrode is like an infinite plane (infinite vis-a-vis the distances at which ion-electrode interactions are considered), and its field displays a planar symmetry. Otherwise, the technique of analysis of the diffuse double layer proceeds along the same lines as in the theoiy of long-range ion-ion interactions (Section 3.3).43... 

The calculation of the interaction energy, VR, which results from the overlapping of the diffuse parts of the electric double layers around two spherical particles (as described by Gouy-Chapman theory) is complex. No exact analytical expression can be given and recourse must be had to numerical solutions or to various approximations. 

For the case of two spherical particles of radii a and a2, Stern potentials, iftdi and i//d2, and a shortest distance, H, between their Stern layers, Healy and co-workers195 have derived the following expressions for constant-potential, V, and constant-charge, Fr, double-layer interactions. The low-potential form of the Poisson-Boltzmann distribution (equation 7.12) is assumed to hold and Kax and xa2 are assumed to be large compared with unity ... 

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 ... 

Ennis and White [56] employed the method of reflections to investigate the electrophoresis of two spherical particles with equilibrium double layers of arbitrary thickness. Their analysis assumes that the zeta potential of the particle is small and the double layers do not overlap significantly. One interesting finding from their study is that the particles with equal zeta potential do interact with each other when the double layer thickness is finite, unlike the no-interaction result for the case of infinitely thin ion cloud. The leading order interaction between two particles is still but the... 

In this chapter, we give approximate analytic expressions for the force and potential energy of the electrical double-layer interaction two soft particles. As shown in Fig. 15.1, a spherical soft particle becomes a hard sphere without surface structures, while a soft particle tends to a spherical polyelectrolyte when the particle core is absent. Expressions for the interaction force and energy between two soft particles thus cover various limiting cases that include hard particle/hard particle interaction, soft particle/hard particle interaction, soft particle/porous particle interaction, and porous particle/porous particle interaction. 

McCartney, L.N. Levine, S. An improvement on Derjaguin s expression at small potentials for the double layer interaction energy of two spherical colloidal particles. J. Colloid Interface Sci. 1969, 30, 345-362. 

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 ... 

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]. 

The expression for the electrostatic interaction energy between the double layers of two spherical bodies is given by ... 

Part I deals with the single double layer. Part II with the interaction of two flat plates, and here most of the fundamental results of the theory already come to the fore. Part III gives a treatment of the, interaction of spherical particles, which serves to clarify various details, especially the influence of particle dimensions and of the kinetics of flocculation. 

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... 

L. N. McCartney and S. Levine, /. Colloid Interface Sci, 30, 345 (1969). An Improvement on Derjaguin s Expression at Small Potentials for the Double Layer Interaction Energy of Two Spherical Colloidal Particles. 

In this section, the equations needed to calculate the interaction potential between two spherical particles will be presented. It will be assumed that the double-layer, hydration, and van der Waals interactions are independent of each other. 

When the radius of the spherical colloidal particles a is much larger than the shortest distance between the surfaces of two colloidal particles, the free energy between two identical spherical particles, due to double-layer, steric, and depletion interactions, can be calculated using the Derjaguin approximation 2... 

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]. 

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