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Attach a sphere to an atomic force microscope cantilever and approach a planar surface in a controlled fluid. Calibrated cantilever bending measures force as a function of corrected surface separation. The sphere's known curvature allows comparison with planar interaction models through the Derjaguin approximation. Approach speed, charge, roughness, hydration and jump-to-contact can limit interpretation.

A 1991 demonstration used a $3.5\mu \mathrm{m}$ radius silica sphere against a plane in sodium chloride solutions. The reported double-layer agreement and short-range deviations illustrate both the method and the need to distinguish electrostatic forces from a presumed purely dispersive interaction.^[ref]

The literal $u(r)=-C_4/r^4$ curve is monotone increasing from $-\infty$ at $r\to 0^{+}$ toward zero from below as $r\to \infty$:Taken literally at a zero gap, it predicts an unbounded collapse energy and force. That extrapolation is outside the far-field dipole approximation and cannot describe actual touching matter.

Finite atomic/ionic dimensions, electronic overlap and Pauli repulsion supply short-distance constraints. Solvent and hydration layers, steric coatings and surface chemistry can add further contributions. An illustrative regularized model is $u_{\mathrm{toy}}=A/r^{12}-C_4/r^4$ with $A>0$; it has a finite minimum at $r=(3A/C_4{)}^{1/8}$ and becomes repulsive at shorter distances. This is a schematic regularization, not a derived complete ion–sphere contact law.

Separately, correcting the center-distance convention to $R=a+r$ already removes this particular $r=0$ divergence from the far-field expression, but does not make a dipole approximation accurate at contact. Near-field multipoles and microscopic repulsion still require a better model. Calibrated colloidal probe technique measurements can test where a long-range force model ceases to agree with actual approach curves.^[ref]

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]