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 Incoming links: Debye–Hückel screening length

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

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At large separation, the finite-thickness van der Waals combination has the expansion

\frac{1}{d ^{2}} - \frac{2}{( d + δ ) ^{2}} + \frac{1}{( d + 2 δ ) ^{2}} = \frac{6 δ ^{2}}{d ^{4}} + O (d^{- 5}) .

(1)

Thus the attraction governed by the Hamaker constant decays as

V_{vdW} \sim - \frac{A _{H} δ ^{2}}{2 π d ^{4}},

(2)

and the screened electrostatic term decays exponentially on the Debye–Hückel screening length. The Helfrich repulsion, however, decays only as

V_{rep} \sim \frac{c ( k _{B} T ) ^{2}}{k _{c} d ^{2}} > 0.

(3)

Consequently the total interaction approaches its unbound value zero from above as

d \to \infty

A finite bound minimum must have nonpositive energy to beat the state at infinity. Because the large-

d

interaction is positive, such a minimum cannot move continuously to infinity while remaining globally stable. At the transition it instead becomes degenerate with the

d = \infty

state at a finite spacing and then loses global stability. The equilibrium spacing therefore jumps from finite

d_{*}

to infinity, making this a discontinuous, first-order unbinding transition.

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