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ID: past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2022/6/b/solution

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Past exam of the biological physics course of the University of Oxford / 2022 / 6 / b / Solution by

Codex  0 2026-10-25

An ionic atmosphere. A negatively charged molecule attracts positive counterions and repels negative co-ions; signs reverse for a positive molecule. The resulting electric double layer partially screens the bare electric field. Far from an isolated molecule, the electrolyte recovers its neutral bulk composition. At thermodynamic equilibrium, spatial uniformity of each species' electrochemical potential requires

μ_{i} = μ_{i}^{0} + k_{B} T ln n_{i} (r) + z_{i} e ψ (r), n_{i} (r) = n_{i}^{\infty} e^{- z_{i} e ψ (r) / (k_{B} T)} .

(1)

The logarithm implicitly uses a common reference density. This result for Boltzmann ion densities in a symmetric electrolyte neglects ion correlations, finite size and nonideal activities, but shows immediately why equal concentrations of the two signs do not persist close to a charged surface.

Self-consistent screening. Insert these densities into Poisson's equation, with solvent permittivity

ϵ

and fixed charge density

ρ_{f}

- ϵ \nabla^{2} ψ = ρ_{f} + \sum_{i} z_{i} e n_{i}^{\infty} e^{- z_{i} e ψ / (k_{B} T)} .

(2)

This is the Poisson-Boltzmann equation. For a symmetric monovalent salt with bulk density

n_{s}

of each sign, outside the fixed charge it becomes

\nabla^{2} ψ = \frac{2 e n _{s}}{ϵ} sinh (\frac{e ψ}{k _{B} T}) .

(3)

For

∣ z_{i} e ψ ∣ ≪ k_{B} T

, linearizing and using bulk neutrality gives the Debye–Hückel approximation

\nabla^{2} ψ = κ^{2} ψ, κ^{2} = \frac{e ^{2}}{ϵ k _{B} T} i \sum z_{i}^{2} n_{i}^{\infty}, λ_{D} = κ^{- 1} .

(4)

Thus added salt shortens the Debye length: electrostatic screening becomes stronger over a shorter distance. The factor of two in a

1 : 1

salt comes from two ionic species, not from double-counting the density of salt pairs.

For example, a charged sphere of radius

a

and total charge

Q

has the exterior linearized solution

ψ (r) = \frac{Q}{4 π ϵ ( 1 + κa )} \frac{e ^{- κ (r - a)}}{r}, r \geq a,

(5)

assuming fixed uniform surface charge and no ions inside. It follows from the decaying spherical solution and

- ϵ ψ^{'} (a) = Q / (4 π a^{2})

. At zero salt it reduces to the Coulomb potential; at finite salt it decays exponentially as well as geometrically. The excess mobile charge integrates to

- Q

in an infinite neutralizing reservoir, although a finite sphere surrounding the molecule generally contains only part of that ionic atmosphere.

Biophysical scales and limitations. In water at room temperature, the monovalent formula is approximately

λ_{D} = 0.304 nm / I / (mol L^{- 1})

, with

I

the ionic strength. It gives about

0.79 nm

0.15 mol L^{- 1}

and

9.6 nm

1 mmol L^{- 1}

. Consequently the same charged protein can exert very different long-range interactions mediated by electrostatics at different salt concentrations. Close to strongly charged DNA, the linear approximation fails; nonlinear screening, counterion condensation, ion-specific adsorption and changes of surface ionization through charge regulation may matter. Mean-field theory also misses strong multivalent-ion correlations and finite-size saturation: the exponential law cannot be extrapolated to arbitrarily large near-surface concentrations.

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