Johnson-Kendall-Roberts theory

These forces follow the trend predicted by the Johnson-Kendall-Roberts theory of contact mechanics [16,28]. 

The maximum contact area between the tip and the monolayer was estimated based on the Johnson-Kendall-Roberts theory of contact mechanics (Johnson, K. L. Kendall, K. Roberts, A. D. Proc. R. Soc. London A 1971, 324, 301) The number of ferrocene-terminated molecules in the estimated maximum contact area of the mixed SAM can be estimated to be 14 - 28 molecules (assuming a homogeneous dispersion of the 1% - 2% of ferrocene-terminated thiol in the SAM). This agrees reasonably well with the experimental observation of ca. 10 - 20 interactions per pull-off event. The load (L = 1.0 nN), surface energy per unit area (W = 10" N/m), and tip radius (R = 100 nm) were experimentally determined. The following parameters for the SAMs were assumed Young s modulus E = 0.5 GPa, Poisson s ratio v = 0.3. 

Perhaps the most significant complication in the interpretation of nanoscale adhesion and mechanical properties measurements is the fact that the contact sizes are below the optical limit ( 1 t,im). Macroscopic adhesion studies and mechanical property measurements often rely on optical observations of the contact, and many of the contact mechanics models are formulated around direct measurement of the contact area or radius as a function of experimentally controlled parameters, such as load or displacement. In studies of colloids, scanning electron microscopy (SEM) has been used to view particle/surface contact sizes from the side to measure contact radius [3]. However, such a configuration is not easily employed in AFM and nanoindentation studies, and undesirable surface interactions from charging or contamination may arise. For adhesion studies (e.g. Johnson-Kendall-Roberts (JKR) [4] and probe-tack tests [5,6]), the probe/sample contact area is monitored as a function of load or displacement. This allows evaluation of load/area or even stress/strain response [7] as well as comparison to and development of contact mechanics theories. Area measurements are also important in traditional indentation experiments, where hardness is determined by measuring the residual contact area of the deformation optically [8J. For micro- and nanoscale studies, the dimensions of both the contact and residual deformation (if any) are below the optical limit. 

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). 

It has recently become common to use the JKR theory (Johnson, Kendall Roberts, 1971) to extract the surface and inteifacial energies of polymeric materials from adhesion tests with micro-probe instruments such as the Surface Force Apparatus and the Atomic Force Microscope. However the JKR theory strictly applies only to perfectly elastic solids. The paper will review progress in extending the JKR theory to the contact mechanics and adhesion of linear viscoelastic spheres. The observed effects of adhesion hysteresis and rate-dependent adhesion are predicted by the extended eory. 

In the present communication, we report the results from studies of micromechanical properties on polymeric materials interpreted using classic theories of elastic contacts, Sneddon s, Hertzian, and Johnson-Kendall-Roberts (JKR). These models are tested for a set of polymeric materials with known Young s modulus, E, from 1 MPa to 3 GPa. Special attention is paid to the elucidation of applicability of different contact models and optimization of experimental probing procedures. 

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 used an energy-based contact mechanics approach to understand particle adhesion. In their theory, they deviated from the earlier Derjaguin and Krupp models by assuming that tensile stresses are present... 

Surface free energies can be obtained from direct work of adhesion measurements using the Johnson, Kendall, and Roberts (JKR) [7] approach. According to the JKR theory, the contact radius a between two elastic bodies of radii of curvature R and R2, Young s moduli of E and E2, and Poisson ratios of v and V2, under an applied load P is given by... 

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