Johnson, Kendall, and Roberts

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... 

The results are most commonly analysed by the Johnson, Kendall and Roberts (JKR) equation [22]. For two elastic spheres, of radii R and Rj, in contact this takes the form ... 

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 ... 

For the interactions between a sphere with radius R and a planar surface, a general equation relating contact area A and load L has been derived by Johnson, Kendall, and Roberts (4.3). 

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... 

A third experiment which shows unequivocally that molecules leap into adhesive contact was performed by Johnson, Kendall and Roberts in 1970. This experiment was similar to Newton s original test on glass telescope lenses (Fig. 3.1) but used rubber surfaces because they adhere much more reliably than glass. Roberts had developed a way of moulding rubber in concave glass lenses to produce remarkably smooth elastomeric spherical surfaces as shown in Fig. 3.12. The rubber composition was mixed and then pressed hot into the glass lens. After... 

The correct solution to the problem of contact between elastic spheres with surface adhesion was obtained by Johnson, Kendall and Roberts 37 years later. This came about because Roberts and Kendall had both been supervised by Tabor while studying for doctorates in Cambridge, while Johnson had collaborated over many years with Tabor on the contact problems associated with friction and lubrication. ... 

The importance of viscoelastic effects in adhesion as measured by methods such as peeling tests means that it is quite difficult to measure the limiting fracture energy Gq. One method that has been used successfully for elastomers was developed by Johnson, Kendall and Roberts (Johnson et al. 1971) and is commonly known as the JKR experiment. The experimental arrangement is... 

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... 

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. 

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... 

Johnson, Kendall, and Roberts (JKR) calculated the Hertzian contact area between two spherical surfaces when the adhesion energy could not be disregarded [15]. They verified their theory by their own experiments using the material combination of rubber and glass. In the JKR theory, it is assumed that adhesion energy is proportional to the contact area and that the attractive force acting on the outside of the contact area can be ignored. The JKR theory is outlined below. 

Modem theories of adhesion mechanics of two contacting solid surfaces are based on the Johnson, Kendall, and Roberts (JKR theory), or on the Derjaguin-Muller-Toporov (DMT) theory. The JKR theory is applicable to easily deformable, large bodies with high surface energy, whereas the DMT theory better describes very small and hard bodies with low surface energy. The JKR theory gives an important result about the adhesion or pull-off force. That is, the adhesion force is related to the work of adhesion, and the reduced radius, R, of the AFM tip-surface contact as... 

In the early 1990s, Chaudhury and Whitesides proposed using a standard adhesion test developed in 1971 by Johnson, Kendall, and Roberts (known as the JKR test) to determine interfacial tensions between a solid and a liquid or a solid and a vapor. The procedure consists of bringing a hemispherically shaped solid, of radius typically 1 mm, in contact with a planar surface made of the same material (Figure 2.25). 

The case of a sphere of radius R is more subtle, since the contact is no longer conformal and elastic energy is stored under the action of molecular forces. To evaluate it, Johnson, Kendall, and Roberts( ) (JKR theory) first apply the Hertzian load... 

Table IV shows contact angle data for this PDMS film grafted onto silicon wafer. Note that these contact angles were measured in the conventional manner to allow a Zisman critical surface tension of wetting determination (aj to be made from the advancing -alkane and paraffin oil data. This is shown in Figure 1, a value of 22.7 mN/m being obtained. This surface exhibited small hysteresis and the value is gratifyingly close to the contact mechanics (Johnson, Kendall and Roberts approach) value of 22.6 mN/m (13). Note also that paraffin wax has a very similar value of 23 mN/m (3),...

By far the most commonly used approach is that due to Johnson, Kendall and Roberts - the JKR theory (63). In this approach, the sphere deforms subject to an adhesive force, without accounting for surface forces. Thus, an attractive neck forms at zero load when the surfaces are in contact. The contact radius is given by the following ... 

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) ... 

The same arguments apply to adhesion of two spheres where the contact spot is much smaller than the sphere diameter. In this case the stress inside the contact is the sum of the Hertzian hemispherical distribution and the flat punch Boussinesq distribution. This problem was solved by Johnson, Kendall and Roberts in 1971 [22] to give the diameter d of the equilibrium contact spot for equal spheres diameter D... 

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