= 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, $F=k_cd$. 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, $F=2\pi R_{\rm eff}W(h)$, where $R_{\rm eff}=a$ for a sphere and plane but $a/2$ for equal spheres. Since $W_{\rm vdW}=-A_H/(12\pi h^2)$, the sphere-plane force is $-A_Ha/(6h^2)$; 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.<Direct measurement of colloidal forces using an atomic force microscope by William A. Ducker and others>[Nature 353, 239–241 (1991), experimental geometry and force-distance measurements summarized in the authors' institutional abstract]{ref}
Back to article page
