Past exam of the quantum condensed matter course of the University of Cambridge / 2006 / Section A / A3 / a / Solution

For smooth uncharged spheres, the basic attraction is the Van der Waals force, arising from correlated electric fluctuations. The Hamaker constant $A_H$ incorporates the particles and intervening medium. For equal spheres of radius $a$ separated by a surface gap $h\ll a$, the near-contact van der Waals attraction of spheres isHere positive force means repulsion, and the attractive case has $A_H>0$. At molecular separation, strong short-range repulsion prevents overlap and replaces the continuum divergence with a deep primary minimum. The energy approaches zero from below at large separation; an uncoated, uncharged attractive pair has no electrostatic barrier.

For illustration, $A_H=10^{-20}\mathrm{J}$, $a=100\mathrm{nm}$ and $h=1\mathrm{nm}$ give $|U|\simeq 8.3\times 10^{-20}\mathrm{J}\simeq 20k_{B}T$ at $300\mathrm{K}$. These are chosen scales, not universal material constants. Such a well can trap particles after a thermal encounter and cause particle aggregation, unless another mechanism such as steric stabilization intervenes. The neutral curve in the figure below shows the attraction regularized by an illustrative contact repulsion.