Cone-Paraboloid Transition of the Johnson-Kendall-Roberts-Type Hyperboloidal Contact

Atomic force microscopy (AFM)-based nanoindentation technique has been widely used to investigate the mechanical properties of compliant specimens. When a sharp probe is indented into a soft and adhesive specimen, not only the rounded end of the probe but also the pyramidal base may be in contact. However, even in such a case, a contact model that assumes a paraboloidal tip geometry (the Hertz model or one of its expansions) is mainly employed to derive the mechanical properties; the error on the mechanical properties induced by the inaccurate tip geometry assumption has not been systematically clarified. Therefore, the focus of this work was put on quantifying this error with the assumption that the actual contact occurs between a hyperboloidal indenter and an elastic and adhesive sample surface. We demonstrated that the cone-paraboloid transition of the indentation curve is governed by a single parameter, A̅ = [4RE/3πw(1 - ν²)]^1/3cotα, where E and ν are Young's modulus and Poisson's ratio of the specimen, respectively, R and α are the curvature radius and half-angle of the indenter, respectively, and w is the work of adhesion. Employing the general two-point method, we quantified the errors on elasticity and surface energy caused by the assumption of the paraboloidal and conical Johnson-Kendall-Roberts (JKR) models as functions of A̅ and the normalized load. AFM force measurements with cantilevers of different radii supported this A̅ dependency. These results showed unsuitable geometry assumption can give a large error, which is generally more serious than those caused by inappropriate choice of the adhesive interaction from the JKR and DMT (Derjaguin-Muller-Toporov). It can be said that the conical model gives a good approximation to a hyperboloidal contact when A̅<1 and so does the paraboloidal model when A̅>4. A̅ is expected to be an important index that validates the paraboloidal and conical approximation in a soft and adhesive contact.