238 Chapter 7. Entropic forces at work[[Student version, January 17, 2003]]
tan(Dβ), largeDsmallD2 πBσq
εβtan(Dβ),β 1 β 2 βFigure 7.10: (Mathematical functions.) Graphical solution of Equation 7.29. The sketch shows the function
2 πBσq/(εβ), as well as tanDβfor two values of the plate separation 2D.The value ofβat the intersection of the
rising and falling curves gives the desired solution. The figure shows that smaller plate separation gives a large solution
β 2 than does large separation (yieldingβ 1 ). Largerβin turn implies a larger ion concentrationc 0 =β^2 /(2πB)at
the midplane and larger repulsive pressure.
As expected, the charge density is greatest near the plates; the potential is maximum in the center.
Wewantaforce. Examining Figure 7.8b, we see that our situation is essentially the opposite
of the depletion interaction (Figure 7.3b on page 222): There, particles were forbidden in the gap,
whereas now they arerequiredto be there, by charge neutrality. In either case some force acts on
individual particles to constrain their Brownian motion; that force gets transmitted to the confining
surfaces by the fluid, creating a pressure drop ofkBTtimes the concentration difference between
the force-free regions (see Idea 7.14 on page 226). In our case, the force-free regions are the exterior
and the midplane (becauseE=−ddVx=0there). The corresponding concentrations are 0 andc 0
respectively, so the repulsive force per unit area on the surfaces is just
f/(area) =c 0 kBT. repulsion of like-charged surfaces, no added salt (7.31)In this formulac 0 =β^2 /(2π B), andβ(D, σq)isthe solution of Equation 7.29. You can solve
Equation 7.31 numerically (see Problem 7.10), but a graphical solution shows qualitatively thatβ,
and hence the osmotic pressure, increases as the plate separation decreases (Figure 7.10). This is
the expected behavior.
Note that the force just found is not simply proportional to the absolute temperature, sinceβ
has a complicated temperature dependence. This means that our pressure is not a purely entropic
effect (like the depletion interaction, Equation 7.10), but rather a mixed effect: The counterion
layer reflects abalancebetween entropic and energetic imperatives. As remarked at the end of
Section 7.4.3, the qualitative effect of adding salt to the solution is to tip this balanceawayfrom
entropy, shrinking the diffuse layers on the surfaces andshortening the rangeof the interaction.
This theory works (see Figure 7.11). You’ll make a detailed comparison to experiment in Prob-
lem 7.10, but for now a simple case is of interest:
Your Turn 7g
Show that at very low surface charge density,σq
1 /(D B), the density of counterions in the