The interaction between double-layers

So far, we have used the Maxwell equations of electrostatics to determine the distribution of ions in solution around an isolated, charged, flat surface. This distribution must be the equilibrium one. Hence, when a second snrface, also similarly charged, is brought close, the two surfaces will see each other as soon as their diffuse double-layers overlap. The ion densities aronnd each surface will then be altered from their equilibrinm valne and this will lead to an increase in energy and a repulsive force between the snrfaces. This situation is illustrated schematically in Fignre 6.12 for non-interacting and interacting flat snrfaces. 

For ideal solutions the osmotic pressure is simply given by CkT. Hence, the osmotic pressure difference between the mid-plane region and the 

If we can now determine V /ni as a function of the separation distance d between the surfaces, we can calculate the total double-layer (pressure) interaction between the planar surfaces. Unfortunately, the PB equation cannot be solved analytically to give this result and instead numerical methods have to be used. Several approximate analytical equations can, however, be derived and these can be quite useful when the particular limitations chosen can be applied to the real situation. 

Using the Debye-Hiickel (DH) approximation the potential decay away from each flat surface is given by 

This result shows us that the repulsive double-layer pressure (for the case of low potentials) decays exponentially with a decay length eqnal to the Debye length and has a magnitnde which depends strongly on the surface potential. 

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The total interaction between the two metal spheres can therefore be classified into two parts (1) the surface, or double-layer, interaction determined by the Gouy-Chapman potential t f0e"Krand (2) the volume, or bulk, interaction —Ar-6 + Br 12. The interaction between double layers ranges from indifference at large distances to increasing repulsion as the particles approach. The bulk interaction leads to an attraction unless the spheres get too close, when there is a sharp repulsion (Fig. 6.131). The total interaction energy depends on the interplay of the surface (double layer) and volume (bulk) effects and may be represented thus... 

As two particles approach one another in the liquid, the diffuse double layers will start to overlap. It is the interaction between the double layers that gives rise to the repulsion between the particles. If the repulsion is strong enough, it can overcome the van der Waals attractive force, thereby producing a stable colloidal suspension. As a prelude to examining the interactions between double layers, we start with an isolated double layer associated with a single particle. 

Interaction between double layers, one of the building bricks of colloid stability, is an important theme planned for Volume IV. It has a large number of spin-offs, in, for instance ion exchange, thin wetting films, free films, membranes, association colloids, vesicles, polyelectroljdes, emulsions and rheology. The dramatic influence of electroljrtes on these phenomena finds its origin in the changes in the double layer, discussed in this chapter. 

In view of the distance over which electrical double layers extend into the solution, interaction between double layers become effective at a separation between the surfaces of at least a few nanometers. Hence, electrical double layer forces can be considered as long-range forces they operate over distances comparable to those over which dispersion forces between particles of colloidal dimensions reach significant magnitudes. 

More sophisticated approaches to describe double layer interactions have been developed more recently. Using cell models, the full Poisson-Boltzmann equation can be solved for ordered stmctures. The approach by Alexander et al shows how the effective colloidal particle charge saturates when the bare particle charge is increased [4o]. Using integral equation methods, the behaviour of the primitive model has been studied, in which all the interactions between the colloidal macro-ions and the small ions are addressed (see, for instance, [44, 45]). 

In some cases, e.g., the Hg/NaF q interface, Q is charge dependent but concentration independent. Then it is said that there is no specific ionic adsorption. In order to interpret the charge dependence of Q a standard explanation consists in assuming that Q is related to the existence of a solvent monolayer in contact with the wall [16]. From a theoretical point of view this monolayer is postulated as a subsystem coupled with the metal and the solution via electrostatic and non-electrostatic interactions. The specific shape of Q versus a results from the competition between these interactions and the interactions between solvent molecules in the mono-layer. This description of the electrical double layer has been revisited by... 

The interaction between two double layers was first considered by Voropaeva et a/.145 These concepts were used to measure the friction between two solids in solution. Friction is proportional to the downward thrust of the upper body upon the lower. However, if their contact is mediated by the electrical double layer associated with each interface, an electric repulsion term diminishes the downward thrust and therefore the net friction. The latter will thus depend on the charge in the diffuse layer. Since this effect is minimum at Eam0, friction will be maximum, and the potential at which this occurs marks the minimum charge on the electrode. 

In Section 15.6, the retention of proteins in ion exchange chromatography is discussed. The ions in the surrounding electrolyte form an electrical double layer around a charged macromolecule, e.g., a protein. The interaction between a protein and an oppositely charged surface can, therefore, be described as taking place between two overlapping double-layer systems. 

The generation of colloidal charges in water.The theory of the diffuse electrical double-layer. The zeta potential. The flocculation of charged colloids. The interaction between two charged surfaces in water. Laboratory project on the use of microelectrophoresis to measure the zeta potential of a colloid. 

Not only do double layers interact with double layers, the metal of one sphere also interacts with the metal of the second sphere. There is what is called the van der Waals attraction, which is essentially a dispersion interaction that depends on r-6, and the electron overlap repulsion, which varies as r-12. These interactions between the bulk... 

The secondary electroviscous effect refers to the change in the rheological behavior of a charged dispersion arising from interparticle interactions, i.e., the interactions between the electrical double layers around the particles. 

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