Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 355 / 2 / b

Solution

In the high-tension, narrow-gap regime, the almost spherical membrane has area

4 π (R + δ)^{2}

. Its spherical bending energy

8 π k_{c}

is independent of radius, while Helfrich repulsion over area

4 π R^{2}

costs

4 π R^{2} \frac{( k _{B} T ) ^{2}}{64 k _{c} δ ^{2}} .

Up to terms independent of

δ

, a suitable free energy is consequently

F (δ) ≃ 8 π k_{c} + 4 πσ (R + δ)^{2} + 4 π R^{2} \frac{( k _{B} T ) ^{2}}{64 k _{c} δ ^{2}} .

8 πσ (R + δ) - 8 π R^{2} \frac{( k _{B} T ) ^{2}}{64 k _{c} δ ^{3}} = 0.

Using

δ / R ≪ 1

δ = (\frac{( k _{B} T ) ^{2} R}{64 k _{c} σ})^{1/3} .

The inverse-square entropic repulsion prevents contact, while membrane tension limits the area gained by opening the gap.

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