Johnson-Kendall-Roberts model

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. 

Mapping of the elastic modulus of the glassy and rubbery blocks and clay regions was probed by employing Hertzian and Johnson-Kendall-Roberts (JKR) models from both approaching and retracting parts of the force-distance curves. In order to determine the elastic properties of SEBS nanocomposites in its different constituting zones, the corrected force-distance curve was fitted to the Hertz model ... 

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 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). 

Adhesive Contact The Johnson-Kendall-Roberts (JKR) Model 386... 

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... 

In these equations, the repulsion of the sample became interrelated with the adhesion force via the contact area. Several models have been developed to include the effect of the adhesion forces [80-83]. Johnson, Kendall, and Roberts derived the following expression for the contact radius and surface indentation ... 

Another approach is the model proposed by Johnson, Kendall, and Roberts (JKR), which considers the elastic deformation and takes into account of the adhesion contribution in contact mechanics (Hugel and Seitz 2001 Janshoff et al. 2000). This model considers the influence of van der Waals forces within the contact zone, and the diminished force of elastic repulsion caused by the attraction. A general equation relating contact area and load is described as follows ... 

Plasma treatment of PDMS followed by adsorption of self-assembled silane monolayers has enabled us to controllably modify the surface energy of elastomer surfaces as described in the section on the Johnson, Kendall, and Roberts approach to deriving the surface free energy of solids. A similar treatment of silicon substrates has produced a useful, low—hysteresis model substrate for contact angle study. There are three types of PDMS contact angle substrates usually studied fluids baked or otherwise chemisorbed on solids such as glass or metals cross-linked coatings on flexible substrates, such as paper or plastic film PDMS elastomer surfaces. 

In Section 3.2.2 we treated Hertzian contact. We must consider that the case, where adhesive forces are negligible, is rather special. If adhesive effects are no longer negligible, one must switch to the adhesive contact model developed by Johnson, Kendall, and Roberts (the JKR model) [Johnson et al., 1971]. In the limit case of weak adhesive force, Fadh, one can also use the Derjaguin-Muller-Toporov... 

It was Johnson, Kendall and Roberts (JKR) who described the area of contact of two spheres including surface energy under the combined external load and the load of adhesion forces [119]. Figured shows the contacting geometry for an infinitely stiff (rigid) surface and an elastic sphere for the Hertzian (dashed line) and JKR (solid line) contacts, respectively. Near the contact the vertical arrows at the dashed contour represent the surface forces which cause an additional deformation of the elastic sphere thus increasing the contact radius from an (Hertz) to ajkr (JKR). The contact radius for the JKR model is a function of the external load, the work of adhesion, the radius... 

The elastic model was modified by taking into account the adhesion force acting between the tip and the surface using the Johnson, Kendall and Roberts theory (JKR model) 10). This leads to the following relations for the different tip geometries (spherical, paraboloid and conical) ... 

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