Biological Physics: Energy, Information, Life

234 Chapter 7. Entropic forces at work[[Student version, January 17, 2003]]

solve Poisson equation

Boltzmann distribution

charge

density

electric

potential

Figure 7.9: (Diagram.) Strategy to find the mean-field solution. Neither the Poisson equation nor the Boltz-
mann distribution alone can determine the charge distribution, but solving these two equations in two unknowns
simultaneously does the job.

Tomake the notation less cumbersome we will drop the averaging signs; from now onρqrefers to
the average density.

The Poisson–Boltzmann equation We wantc+(x), the concentration of counterions. We are
supposing our surface to be immersed in pure water, so far away from the surfacec+→0. The
electrostatic potential energy of a counterion atxiseV(x). Because we are treating the ions as
moving independently of each other in a fixed potentialV(x), the density of counterions,c+(x),
is given by the Boltzmann distribution. Thusc+(x)=c 0 e−eV(x)/kBT,wherec 0 is a constant.
Wecan add any constant we like to the potential, since this change doesn’t affect the electric field
E=−dV/dx.It’s convenient to choose the constant so thatV(0) = 0. This choice givesc+(0) =c 0 ,
so the unknown constantc 0 is the concentration of counterions at the surface.
Unfortunately we don’t yet knowV(x). To find it, apply the second form of the Gauss Law
(Equation 7.20), takingρqequal to the density of counterions timese. Remembering that the
electric field atxisE(x)=−dV/dxgives thePoisson equation: d

(^2) V
dx^2 =−ρq/ε.Given the charge
density, we can solve Poisson’s equation for the electric potential. The charge density in turn is
given by the Boltzmann distribution asec+(x)=ec 0 e−eV(x)/kBT.
It may seem as though we have a chicken-and-egg problem (Figure 7.9): We need the average
charge densityρqto get the potentialV. But we needVto findρq(from the Boltzmann distri-
bution)! Luckily, a little mathematics can get us out of predicaments like this one. Each of the
arrows in Figure 7.9 represents an equation in two unknowns, namelyρqandV.Wejust need to
solve these two equations simultaneously in order to find the two unknowns. (We encountered the
same problem when deriving the van ’t Hoff relation, at Equation 7.13 on page 225, and resolved
it in the same way.)
Before proceeding, let’s take a moment to tidy up our formulas. First, we combine the various
constants into a length scale:
(^) B≡
e^2
4 πεkBT

. Bjerrum length(in water) (7.21)

(^) Btells us how close we can push two like-charge ions together, if we have energykBTavailable.
Formonovalent ions in water at room temperature, (^) B=0. 71 nm.Next, define the dimensionless
rescaled potential
V(x)≡eV /kBT. (7.22)