Johnson-Kendall-Roberts force measurement

In the limits of established contact mechanics models, including those developed by Johnson-Kendall-Roberts (JKR) [5] or by Derjaguin, Muller, and Toporov (DMT) [6], the measured forces are a function of the chemical identity of the contacting surfaces (via the work of adhesion W12 that depends on the surface and interfacial free energies involved). In addition, we need to consider the nature of the medium, the radius of the AFM tip, and also temperature and loading rate. 

The pull-off forces obtained in f-d measurements can be related to the work of adhesion and the respective surface free energies utilizing, eg, (continuum) contact mechanics theories, such as the Johnson-Kendall-Roberts (JKR) theory (80). In particular for monomolecular model systems (65), but also polymers, this approach has yielded a satisfactory description of the experimental data, despite the fact that the contacting bodies are treated as purely elastic (81). 

In recent years it has been demonstrated that also adhesion (or adhesion hysteresis) plays an important role in friction. Israelachvili and coworkers could show that friction and adhesion hystereses are, in general, directly correlated if certain assumptions are fulfilled. These authors have proposed models based on data obtained by surface forces apparatus (SFA) experiments, e. g. the cobblestone model of interfacial friction (4). In addition, several groups described the application of continuum contact mechanics (e.g. Johnson-Kendall-Roberts (JKR) theory (5)) to describe friction data measured between flat surfaces and nanometer sized contacts (d). 

Johnson, Kendall and Roberts [47] measured d for some natural rubber spheres and found deviations from the Hertz equation at low loads, but conformity at high loads. Data are shown in Fig. 10. At low loads the zones of contact were greater than predicted by Hertz. This was due to the forces of attraction between the surfaces of the two spheres, and it was shown that the diameter of the zone of contact was now given by Eq. (12), where W is the work of adhesion. 

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