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1.Electrochemical equilibrium. Consider an electrolyte with positive and negative
ions(charged particles in solution) of charges, ±e, diffusion constants, D±, and
concentrations, c±(x), respectively
. Instead of treating the electrical force between each ion pair, consider the “ mean field
approximation ”that each ion feels only an average electric force, �edφ/dx, with a
mobility given by the Einstein relation. The electrostatic potential, φ(x), satisfies Poisson’s
equation,
where c is the permittivity, of the solvent (e.g. water). In this approximation, each ion
moves independently in the self-consistent electric field, so c+(x) and c_ satisfy steady
Fokker-Planck equations'. Consider ions in solution near a charged surface, x > 0, with
c±(oo) = c,, and 0(oo) = 0 and 0(0) = —c (the °zeta potential").
(a) Show that potential satisfies the dimensionless “Poisson-Boltzmann equation" (PBE),
where 11)(y) = —e0(x)1111' and y = x/A. What is A?
(b) Solve the linearized PBE. and explain why A is called the "screening length".
(c) Integrate the nonlinear PBE once to obtain the total interfacial charge,
Problems

2. First passage of a set of random walkers. Consider N independent random walkers
released from x = xo > 0, and let T1 be the first passage time to the origin (1 < i < N) with PDF
Mt). Approximate the PDF of the position of each walker by the solution to the diffusion
equation,
(a) Solve with P(0, t) = 0 to obtain the survival probability, Si(t) (that the ith walker has not
yet reached the origin). [Hint: introduce an “image" source.] Show that Mt) = —SW) is the
Smirnov density, derived by other means in lecture.
(b) Find the PDF f(t) of the first passage time for the set of N walkers, 7' = mini<i‹N Z, when
one of them reaches the origin for the first time.
(c) Show that (T") coo if and only if N > 2m. In particular, the expected first passage time is
finite if and only if N > 3, as mentioned in class.
3. Escape from a symmetric trap.. Consider a diffusing particle which feels a
conservative force, f (x) = —0(x), in a smooth, symmetric potential, 0(x) = 0(—x), causing
a drift velocity, v(x) = bf(x), where b = D/kT is the mobility and D is the diffusion constant.
If the particle starts at the origin, then PDF of the position, P(x,t), satisfies the Fokker-
Planck equation,

with P(x,0) = b(x). Suppose that the potential has a minimum = 0 at x = 0 with 9(0) = Ko n 0
and two equal maxima 0 = E > 0 at x = ±x1 with eV]) = —K1 < O. Let r be the mean first
passage time to reach one of the barriers at x = tx1 (and then escape from the well with
probability 1/2).
(a) Derive the general formula
(b) In the low temperature limit, kT/E y 0, calculate the leading-order asymptotics of the
escape rate, R= 1/2r N Ro(T), using the saddle-point method. Verify the classical result
of Kramers: Ro(T) cc c—E/kr.
(c) Calculate the first correction to the Kramers escape rate:

Solutions
I. Electrochemical Equilibrium.
(a) The Fokker-Planck (or Nemst-Pluck) equations for diffusion and drift of ions in the 'mean-
electrostatic potential. ᶲ are
where ill = DIIkT are the ionic mobilities, given by the Einstein relation. Assuming steady
state and constant D±, we obtain ODES for equilibrium:
where Ab = —0IkT. Integrating twice, using the boundary conditions, 11)(oo) = 0 and
c±(co) = co, we obtain the expected concentration profiles of Boltzmann equilibrium:
The diffuse charge density of ions is then
which combines with Poisson's equation of electrostatics :

to produce the Poisson-Boltzmann equation. Changing variables, we obtain the required
dimensionless form:
where y = x/A with a characteristic length scale,
EkT A - 2Wo. (7)
(b) Putting the units back, the linearized problem is
which is easily solved:
It is clear that the influence of the surface potential C decays exponentially with a character-istic
length scale, A. Equivalently, the (linearized) charge density
is "screened" in the bulk solution beyond a distance, A, usually called the "Debye screening
length" (even though it was derived earlier by Gouy).

(c) The region near the charged surface is commonly called a "double layer" since it looks like
a capacitor, with the charge q on the surface, equal and opposite to the diffuse charge in
solution which screens it:
To obtain the charge-voltage relation q((), and thus the differential rnpqritance, dq/d(, we
integrate the dimensionless PBE, using the trick of multiplying by 4?:
Integrating and requiring Coo) = = 0, we obtain
We choose the "-" square root
because the surface charge q in Eq. (11) has the opposite sign of the diffuse charge density,
p(0) a 0(0) oc zeta. Putting units back and using Eq. (11), we obtain the desired result:
(Note that in the limit of small voltage, c kTle, the interface behaves like a parallel-plate
rapiritor of dielectric constant E and width A, since the capacitance is dqld( elA.)

subject to an absorbing boundary condition P = 0 at the origin, the "target". By linearity, we
can satisfy the boundary condition by introducing an image source of negative sign at -xo,
which represents "anti-walkers" which annihilate with the true walkers whenever they meet,
at the origin (e.g. see Redner's book):
In terms of this solution, the survival probability is
in terms of the error function
The PDF of the first pmage time is then
which is the Lkvy-Smirnov density, derived by different means in lecture.

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