7.4. A repulsive interlude[[Student version, January 17, 2003]] 233
a
- σq
bc
xx2 DFigure 7.8:(Schematics.) (a)Counterion cloud outside a charged surface with surface charge density−σq.Positive
values ofxrepresent locations outside the surface. The pointx 1 lies inside the surface, where there are no free charges;
x 2 lies just outside. (b)When two similarly charged surfaces approach with separation 2D,their counterion clouds
begin to get squeezed. (c)When two oppositely charged surfaces approach, their counterion clouds are liberated,
gaining entropy.
7.4.3 Charged surfaces are surrounded by neutralizing ion clouds
The mean field Now we can return to the problem of ions in solution. A typical problem might
beto consider a thin, flat, negatively charged surface with surface charge density− 2 σqand water
on both sides. We can think of the surface concretely as a cell membrane, since these are in fact
negatively charged.^6 Forexample, you might want to coax DNA to enter a cell, for gene therapy.
Since both DNA and cell membranes are negatively charged, you might want to know how much
they repel.
An equivalent, and slightly simpler, problem is that of asolidsurface carrying charge density
−σq,with water on just one side (Figure 7.8a). Also for simplicity, suppose that the loose positive
counterions aremonovalent(for example sodium, Na+). That is, each carries a single charge:
q+=e=1. 6 · 10 −^19 coul.Inareal cell, there will be additional ions ofbothcharges from the
surrounding salt solution. The negatively charged ones are calledcoions,because they have the
same charge as the surface. We will neglect the coions for now (see Section 7.4.3′on page 250).
As soon as we try to find the electric field in the presence of mobile ions, an obstacle arises: We
are not given the distribution of the ions, as in first-year physics, but rather mustfindit. Moreover,
electric forces are long-range. The unknown distribution of ions will thus depend not only on each
ion’s interactions with its nearest neighbors, but also with many other ions! How can we hope to
model such a complex system?
Let us try to turn adversity to our advantage. If each ion interacts with many others, perhaps we
can approach the problem by thinking of each ion as moving independently of the others’detailed
locations, but under the influence of an electric potential created by theaveragecharge density
of the others, or〈ρq〉.Wecall this approximate electric potentialV(x)themean field,and this
approach themean-field approximation.The approach is reasonable if each ion feels many others;
then the relative fluctuations inV(x)about its average will be small (see Figure 4.3 on page 103).
(^6) More generally, any curved surface will be effectively flat if the thickness of the ion cloud found below is much
smaller than the surface’s radius of curvature.