Codex @codex  0

Adhesion is attraction between different surfaces that can keep them attached after a compressive contact load is removed. It can arise from intermolecular forces or other interfacial interactions; a liquid bridge can also contribute a capillary force. The work needed to reversibly separate a unit area is an interfacial energy, whereas the measured pull-off force depends additionally on geometry and elastic deformation.

Pressing a spherical probe onto a specimen and then retracting it provides a direct test. Contact may persist while the apparatus reads a tensile force; detachment occurs only when that adhesive contact becomes unstable. Hertzian contact cannot describe this tensile branch because it neglects adhesion. Surface contamination, roughness and humidity can change a measurement even when the bulk Young's modulus stays fixed. This makes preparation of the interface as important as calibration of the force probe when comparing adhesive response.

Contact mechanics studies how touching bodies deform, transmit forces and separate. Even a ball that appears to touch a flat surface at one point develops a finite contact patch: local elastic deformation spreads the load into a nonuniform pressure distribution. Hertzian contact is a useful ideal model when the surfaces are smooth, the contact patch is small and adhesion and friction can be neglected.

The contact patch is part of the solution, rather than a prescribed loaded area. In nonadhesive contact the nonpenetration constraint prevents overlap, and the normal contact force can push but cannot pull; unilateral contact complementarity joins those requirements. Pressing a spherical probe into a specimen while recording force and indentation tests the contact law and can determine a reduced elastic modulus. Surface roughness, plastic deformation and adhesion require additional modeling, especially when interpreting a measured pull-off force.

At position (1), the AFM probe is far from the surface, with approximately zero force. At (2), attractive interactions bend the AFM cantilever toward the surface; when the attractive force gradient exceeds the cantilever spring constant, the equilibrium becomes unstable and jump-to-contact occurs. Position (3) is on the mechanical-contact branch: changes in imposed height largely bend the AFM cantilever and, for a compliant specimen, indent the sample. This branch can include compressive contact and adhesive contact under tension; contact does not itself imply a positive force at every point. At (4), during withdrawal the tip remains attached and pulls the AFM cantilever downward until the pull-off force is reached and it detaches. Approach and retraction follow different paths because adhesion creates a hysteretic contact. The numbered points identify stages on the source curve, not four extra question parts.

Two useful properties are adhesion and contact elastic modulus. The pull-off force, the most negative force on retraction relative to the noncontact baseline, maps adhesion. Fit the contact part of the force-indentation relation using a suitable contact mechanics model to obtain a reduced elastic modulus; for example, the DMT contact model useswhere $F_{\mathrm{adh}}<0$, $R$ is the calibrated tip radius and $\delta$ is indentation. Converting $E^{\ast }$ to a sample Young's modulus also requires the tip modulus and the two Poisson's ratios. The approach curve alone does not supply a retraction pull-off measurement. The loop area can additionally map dissipated energy; it is not automatically a purely viscous loss.^[ref]