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quantitative nanomechanical mapping can image the flake without an electrical return path. Repeated AFM force curves provide surface topography, adhesion and deformation or effective elastic modulus channels. The step and differences in tip–surface interaction can distinguish graphene from glass. For an atomically thin supported sheet the contact response is strongly substrate dependent, so fitting a three-dimensional half-space contact model does not directly measure the intrinsic in-plane Young's modulus of free graphene. Calibrate the cantilever spring constant, deflection sensitivity and tip radius, use small loads, and compare bare substrate. The manufacturer's description specifies the force-curve origin of the mechanical channels.^[ref]

In quantitative nanomechanical mapping, the operational adhesion is the magnitude of the most attractive force immediately before detachment on a calibrated retraction curve:Here $k_c$ is the cantilever spring constant and ${\delta }_c$ its deflection after baseline subtraction. The area between approach and retraction relates to dissipated work, whereas the minimum force is a force, not an energy. Chemical attraction, Van der Waals forces and ambient-water capillary bridges can all contribute. Match loading rate and peak load, calibrate deflection sensitivity and $k_c$, compare clean reference surfaces, and control humidity; otherwise different maps may reflect different tips or capillary conditions rather than different specimens.

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.