Biological Physics: Energy, Information, Life

7. Track 2[[Student version, January 17, 2003]] 251

Wecan now look at a more relevant limit for biology: This time hold the salt concentration fixed
and go out to large distances, where our earlier result (Equation 7.25) displayed its pathological
behavior. Forx λD,Equation 7.35 reduces to

V→−(4e−x∗/λD)e−x/λD. (7.37)

That is,

The electric fields far outside a charged surface in an electrolyte are exponen-

tially screened at distances greater than the Debye lengthλD.

(7.38)

Equation 7.34 shows the behavior we expected qualitatively (Idea 7.28): Increasingc∞decreases
the screening length, shortening the effective range of the electrostatic interaction.
In the special case of weakly charged surfaces (σq is small), Equation 7.36 gives e−x∗/λD =
πBλDσq/e,sothe potential simplifies to

V(x)=−

σqλD

ε

e−x/λD potential outside a weakly charged surface. (7.39)

The ratio of the actual prefactor in Equation 7.37 and the form appropriate for weakly charged
surfaces is sometimes calledcharge renormalization:Anysurface will at great distances look the
same as a weakly charged surface, but with the “renormalized” charge densityσq,R=λ^4 εDe−x∗/λD.
The true charge on the surface becomes apparent only when an incoming object penetrates into its
strong-field region.
In the presence of added salt, the layer thickness no longer grows without limit as the layer
charge gets smaller (as it did in the no-salt case, Equation 7.25), but rather stops growing when it
hits the Debye screening length. For weakly charged surfaces, then, the stored electrostatic energy
is roughly that of a capacitor with gap spacingλD,notx 0 .Repeating the argument at the end of
Section 7.4.3, we now find the stored energy per unit area to be

E/(area)≈kBT

(σq

e

) 2

2 πλD (^) B. electrostatic energy with added salt, weakly charged surface
(7.40)
7.4.4′ The crucial last step leading to Equation 7.31 may seem too slick. Can’t we work out
the force the same way we calculate any entropic force, by taking a derivative of the free energy?
Absolutely. Let us compute the free energy of the system of counterions+surfaces, holding fixed
the charge density−σqon each surface but varying the separation 2Dbetween the surfaces (see
Figure 7.8b on page 233). Then the force between the surfaces will bepA=−dF/d(2D), whereA
is the surface area, just as in Equation 6.17 on page 188.
First we notice an important property of the Poisson–Boltzmann equation (Equation 7.23).
Multiplying both sides by dV/dx,wecan rewrite the equation as
d
dx

[(

dV

dx

) 2 ]

=8πBdc+

dx

.

Integrating this equation gives a simpler, first-order equation:
(
dV
dx

) 2

=8πB(c+−c 0 ). (7.41)