Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2022 / Paper 2 / Section B / 23 / i / Solution

$N_1$ and $N_2$ count the microscopic interacting units in bodies 1 and 2. Their positions are ${\mathbf{r}}_i$ and ${\mathbf{r}}_j$; the difference vector gives their pair separation. The positive coefficient $C$ specifies an attractive nonretarded pair energy $-C/r^6$ and has units $\mathrm{J}{\mathrm{m}}^6$.

Hamaker summation approximates a macroscopic interaction from Van der Waals forces by adding these microscopic pair energies. For homogeneous continua, sums are replaced by integrals over the two body volumes with number densities ${\rho }_1$ and ${\rho }_2$. Integrating the geometry changes the distance dependence and produces a Hamaker constant $A_H={\pi }^2{\rho }_1{\rho }_{2}C$ for the stated simple model.

The approximation assumes pairwise additivity, uniform material properties, a known geometry and a separation large enough to use a continuum but small enough for the chosen nonretarded interaction law. It neglects many-body electrodynamic screening, chemical bonding, atomic roughness and short-range repulsion. A medium changes the effective interaction strength; an effective measured $A_H$ across water need not equal a vacuum value inferred from isolated atom pairs.