Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 355 / 3 / a

At large separation, the finite-thickness van der Waals combination has the expansionThus the attraction governed by the Hamaker constant decays asand the screened electrostatic term decays exponentially on the Debye–Hückel screening length. The Helfrich repulsion, however, decays only asConsequently 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 $d_{\ast }$ to infinity, making this a discontinuous, first-order unbinding transition.