7.4. A repulsive interlude[[Student version, January 17, 2003]] 239
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separation, nmpressure,PaFigure 7.11: (Experimental data with fits.) The repulsive pressure between two positively charged surfaces in
water. The surfaces were egg lecithin bilayers containing 5 mole% or 10 mole% phosphatidylglycerol (open and filled
circles respectively). The curves show one-parameter fits of these data to the numerical solution of Equations 7.29
and 7.31. The fit parameter is the surface charge densityσq. The dashed line shows the curve with one proton
charge per 24nm^2 ;the solid line corresponds to a higher charge density (see Problem 7.10). At separations below
2 nmthe surfaces begin to touch and other forces besides the electrostatic one appear. Beyond 2nmthe purely
electrostatic theory fits the data well, and the membrane with a larger density of charged lipids is found to have a
larger effective charge density, as expected. [Data from Cowley et al., 1978.]
gap is nearly uniform and equals the total charge on the plates divided by the volume of the gap
between them, as it must.Thus in this case the counterions act as an ideal solution, and the pressure they exert is that
predicted by the van ’t Hoff formula.
T 2 Section 7.4.4′on page 251 derives the electrostatic force directly as a derivative of the free
energy.
7.4.5 Oppositely charged surfaces attract by counterion release
Now consider an encounter between surfaces ofoppositecharge (Figure 7.8c). Without working
through the details, we can understand the attraction of such surfaces in solution qualitatively
using the ideas developed earlier. Again, as the surfaces approach from infinity, each presents a net
charge density ofzeroto the other; there is no long-range force, unlike the constant attractive force
between two such planar surfaces in air. Now, however, as the surfaces approach they can shed
counterion pairs while preserving the system’s neutrality. The released counterions leave the gap
altogether and so gain entropy, lowering the free energy and so driving the surfaces together. If the
charge densities are equal and opposite, the process proceeds until the surfaces are in tight contact
with no counterions left at all. In this case there is no separation of charge, and no counterions
remain in the system. Thus all the self-energy estimated in Equation 7.27 gets released. We have
already estimated that this energy is substantial: Electrostatic binding between surfaces of matching