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Manufacturer of scientific instruments, including atomic force microscopes.

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 atomic force microscope measures the cantilever deflection $d$, converted to normal force by $F=k_{c}d$ after calibration. On approach, the force is initially almost zero, then attractive; a sufficiently steep attractive force gradient causes jump-to-contact. After contact, further approach bends the cantilever and indents the sample, giving a rising repulsive branch. With tip-surface separation increasing to the right, approach proceeds from right to left. The instability occurs when the magnitude of the attractive force gradient exceeds the cantilever stiffness. The contact-branch slope depends on both cantilever and sample compliance.

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]

Use a force-tilted activation barrier for a generic one-coordinate escape model. A tensile force tilts a free-energy landscape to $G_F(x)=G_0(x)-Fx$. If $x_b$ is the initial minimum and $x_s$ the saddle, define $\Delta x=x_s-x_b>0$ and the zero-force barrier $\Delta G_0^{‡}=G_0(x_s)-G_0(x_b)$. The relevant activation barrier is measured relative to the initial minimum, not an arbitrary absolute transition-state energy. To first order in force,The attempt prefactor $\nu$ is assumed force independent, and the saddle displacement is approximated as constant. This single-barrier approximation need not survive large landscape deformations or a change of escape pathway. TODO: user-deferred biomolecular implementation; the protein-specific shear-force arrangement and attachment chemistry are not solved.

For a generic force-ramp measurement, a calibrated atomic force microscope cantilever pulls an attachment while a stage changes separation. cantilever deflection $d$ gives force $k_{c}d$, and the cantilever compliance correction subtracts $d$ from imposed separation to give tether extension, with further corrections for other apparatus compliance. A prescribed stage speed does not automatically produce constant force loading rate when tether stiffness varies; the assumed $F(t)=ft$ must be measured or controlled. The original schematic below illustrates a generic nonbiological bond/tether geometry. Protein-specific construction is deferred.

The no-rebinding approximation is a physical escape assumption: after the link breaks, separation removes the partners from the capture region before appreciable reassociation. It requires the integrated return/association probability during that separation to be small. If the separated state stays nearby or the force is held without separation, setting $k_{+}=0$ can fail. Under the stipulated approximation, with the link initially intact,This survival under a linear force ramp differs from the probability density for rupture. It starts at one, decreases to zero and shifts to larger force as the loading rate increases. For small $F/F_0$, $p\simeq e^{-k_{0}F/f}$; a slow ramp allows much escape before substantial force is reached.

The median rupture force under a linear ramp satisfies $p=1/2$, givingAt high loading rate, the slope of $F_{1/2}$ against $\ln f$ tends to $F_0$; it is not exactly $F_0$ for a slow ramp. Fit the full expression across loading rates to recover $F_0$ and $k_0$. The rupture-force probability density is $\rho (F)=k(F)p(F)/f$, whose most probable rupture force under a linear ramp is $F_{\ast }=F_0\ln [f/(k_0{F}_0)]$ if $f>k_0{F}_0$, and zero otherwise. median and mode are different observables and must not be interchanged when fitting.

At known temperature, $F_0$ yields the activation distance $\Delta x=k_{B}T/F_0$; $1/k_0$ gives the zero-force mean lifetime in this exponential model. Absolute barrier height additionally requires the attempt prefactor, and neither equilibrium binding free energy nor an association rate follows from dissociation-only data. A loading-rate-dependent rupture force is a kinetic readout, not an intrinsic equilibrium strength. TODO: user-deferred biological conclusions. The protein-specific stability interpretation and cell-adhesion strategy are left unsolved.

A surface-chemistry researcher whose work includes the colloidal probe technique for measuring forces with an atomic force microscope.

Homepage: surfaces.che.vt.edu/professor-william-ducker.html