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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]

For two equal spheres of radius $a$ separated by a surface gap $h\ll a$, a positive Hamaker constant $A_H$ gives the nonretarded Van der Waals force approximation $U=-A_{H}a/(12h)$ and $F=-dU/dh=-A_{H}a/(12h^2)$. The Derjaguin approximation derives the geometry factor from the planar interaction energy per area $W=-A_H/(12\pi h^2)$ and effective radius $a/2$. A sphere and plane instead have effective radius $a$, doubling the force at the same gap. Molecular repulsion, roughness and retardation limit these continuum formulas.

For two equal spheres of radius $R$, the effective radius in the Derjaguin approximation is $R_{\mathrm{eff}}=R/2$. The nonretarded attractive van der Waals potential isDifferentiating with respect to the surface gap givesThus the requested approximation gives an attraction of about $8.3$ pN. Here $h/R=2$, whereas the Derjaguin approximation requires $h\ll R$. This is the formal estimate requested by the hint, not a quantitatively reliable near-contact approximation for the printed dimensions.

For two equal spheres, the Derjaguin approximation uses $R_{\mathrm{eff}}=R/2$. A planar nonretarded Van der Waals force interaction has energy per area $W(h)=-A_H/(12\pi h^2)$, soThe force is attractive, with magnitude $13.0\mu \mathrm{N}$, taking separation increase as positive. Using the sphere–plane value of $R_{\mathrm{eff}}$ would overestimate it by a factor of two. The gap is much smaller than either radius, as required. Treat the supplied Hamaker constant as the effective one for the intervening medium; roughness, solvation and molecular-scale contact can modify a literal $1\mathrm{nm}$ experiment.

For nonretarded attraction between planar half-spaces, the interaction free energy per area is $W(h)=-A_H/(12\pi h^2)$. The Derjaguin approximation gives $F=2\pi R_{\mathrm{eff}}W(h)$, with $R_{\mathrm{eff}}=R_1{R}_2/(R_1+R_2)=a/2$ for equal spheres. ThusThe force is $13.0\mu \mathrm{N}$, attractive. Scaling both radii and the gap by two gives $F^{\prime }=F/2$, or $6.50\mu \mathrm{N}$ attractive. This agrees with the invariant energy $V\simeq -A_{H}a/(12h)$: differentiating with respect to a gap that doubles makes the force halve. The approximation requires $h\ll a$ and neglects roughness and retardation.

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]

The force is determined by the total normal stress of the ionic liquid. The ions contribute osmotic pressure because confining an excess number density costs translational entropy. The osmotic stress in a symmetric electric double layer, relative to the bulk reservoir, isThe second term is the normal electrostatic stress. The Poisson-Boltzmann equation makes this combination independent of position between planar surfaces. At the midpoint of identical equally charged surfaces, symmetry gives $E=0$, soCalculating only the repulsion of bare surface charges would omit both the screening ion cloud and its entropic stress. Evaluating the total stress at the midpoint packages their combined force in a particularly simple form.

For equal spheres, assume $h\ll a$, $\kappa a\gg 1$, weak surface potential and weak overlap, with prescribed ${\psi }_0$. Then ${\psi }_{\mathrm{mid}}\simeq 2{\psi }_0{e}^{-\kappa h/2}$ gives $\Pi \simeq 2\epsilon {\kappa }^2{\psi }_0^2{e}^{-\kappa h}$. Integrating once gives the planar interaction energy per area, $W={\int }_h^{\infty }\Pi (s)ds\simeq 2\epsilon \kappa {\psi }_0^2{e}^{-\kappa h}$, and the Derjaguin approximation gives the repulsive sphere force. Combining it with the near-contact van der Waals attraction of spheres yields the DLVO theory approximationPositive force is repulsive. The exponential term can produce a repulsive barrier in the corresponding pair energy; the short-range attraction leads to a primary minimum, while a shallower secondary minimum may occur farther out. Adding salt shortens the repulsion and can destroy the barrier, allowing particle aggregation. Molecular contact requires a repulsive-core correction, and changing the charge-regulation boundary condition changes the electrostatic prefactor.