= Solution
At large separation, the finite-thickness van der Waals combination has the expansion
$$
\frac1{d^2}-\frac2{(d+\delta)^2}
+\frac1{(d+2\delta)^2}
=\frac{6\delta^2}{d^4}+O(d^{-5}).
$$
Thus the attraction governed by the <Hamaker constant> decays as
$$
V_{\rm vdW}\sim-\frac{A_H\delta^2}{2\pi d^4},
$$
and the screened electrostatic term decays exponentially on the <Debye–Hückel screening length>. The <Helfrich repulsion>, however, decays only as
$$
V_{\rm rep}\sim\frac{c(k_BT)^2}{k_cd^2}>0.
$$
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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