Counterion distribution between charged plates

Q The Counterion Distribution between Charged Plates in Solution... 

Equation (48) is identical in form to Equation (13) for a parallel plate capacitor, with pQ replacing A p and k 1 replacing 8. This result shows that a diffuse double layer at low potentials behaves like a parallel plate capacitor in which the separation between the plates is given by k . This explains why k 1 is called the double-layer thickness. It is important to remember, however, that the actual distribution of counterions in the vicinity of a charged wall is diffuse and approaches the unperturbed bulk value only at large distances from the surface. 

As is well known, the counterions distributed close to the plates work to shield their surface charges. Figure 6.3 shows, however, that there exist counterions at the middle of the inner region that do not belong to either of the plates. It is those counterions that result in an attraction, a counterion-mediated attraction, between... 

The fact that the origin of the repulsion is different for plates and for rods means that the overall phase behavior can be different, because the interaction depends sensitively on the competition between the correlation-attraction and the repulsion. However, the mechanisms of attraction that have been proposed for rods are the same as for plates. Thus the ionic crystal picture for plates has been applied to rods [20-23], as has the thermal fluctuation picture (which was developed for rods more than 30 years ago by Oosawa [24-26]). In the case of rods, there are several versions of the ionic crystal model, which differ in microscopic details [21-23,27]. In the thermal fluctuation picture, fluctuations in the condensed counterion density along the rods lead to nonuniformities in the charge distribution, which can become correlated from one rod to another [26,28], leading to an attraction similar to the van der Waals interaction. We have introduced a third approach, called the charge fluctuation approach [29,30], which is an extension of the thermal fluctuation approach to ions of nonzero size, and which captures aspects of both the thermal fluctuation picture and the ionic crystal pictures. 

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