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 Incoming links: Second virial coefficient

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 355 / 3 / b / Solution Created 2026-09-24 Updated 2026-09-25

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For a dilute stack, the membrane number per unit normal length is

1/ d = ϕ / δ

. Dividing the fluctuation repulsion per area by the repeat distance gives its free-energy density

f_{rep} = \frac{c ( k _{B} T ) ^{2}}{k _{c} d ^{3}} = \frac{c ( k _{B} T ) ^{2}}{k _{c} δ ^{3}} ϕ^{3} .

(1)

The short-range electrostatic repulsion and long-range van der Waals attraction enter at second-virial order. In general, for an effective pair energy

U_{A_{H}} (Γ)

over relative configurations

Γ

, the second virial coefficient has the Mayer-integral form

B_{2} (A_{H}) = \frac{1}{2} \int [1 - e^{- U_{A_{H}} (Γ) / (k_{B} T)}] d Γ,

(2)

with a fixed normalization by the microscopic membrane thickness making

B_{2}

dimensionless here. The mean-field contribution is quadratic in membrane concentration. The required two-power free energy can therefore be written

f (ϕ) = \frac{k _{B} T}{δ ^{3}} B_{2} (A_{H}) ϕ^{2} + \frac{c ( k _{B} T ) ^{2}}{k _{c} δ ^{3}} ϕ^{3}, ϕ \geq 0 .

(3)

Changes of microscopic normalization merely rescale

B_{2}

by a positive constant and do not affect the transition.

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