 Solution

Use the colloidal probe technique: attach a sphere of known radius to the cantilever of an atomic force microscope and approach a chemically matched planar surface in the same liquid as the suspension. A laser reflected from the cantilever measures its deflection $d$; calibration of the spring constant $k_c$ converts this to force, $F=k_{c}d$. The surface separation is reconstructed from piezo motion after correcting for cantilever bending and defining the contact position. Slow approach reduces hydrodynamic drag; residual surface charge, roughness and hydration must be controlled when identifying a dispersion force.

The Derjaguin approximation accounts for the geometry. With interaction free energy per area $W(h)$ between flat plates, $F=2\pi R_{\mathrm{eff}}W(h)$, where $R_{\mathrm{eff}}=a$ for a sphere and plane but $a/2$ for equal spheres. Since $W_{\mathrm{vdW}}=-A_H/(12\pi h^2)$, the sphere-plane force is $-A_{H}a/(6h^2)$; divide it by two to infer the equal-sphere result in the same near-contact approximation. Attractive force gradients can produce a jump to contact, limiting the directly accessible approach curve.

An actual demonstration attached a silica sphere of radius $3.5\mu \mathrm{m}$ to an AFM force sensor and measured its interaction with a plane in sodium chloride solutions. Its double-layer contribution and short-distance deviations show why charge and surface conditions must be measured rather than simply presumed absent.^[ref]