JKR equation

The so-called JKR equation relates the adhesion-induced contact radius a to the particle radius R and the applied load P by... 

There are several distinctive features worth noting about the JKR equation. The first is in the limit of no adhesion (or, equivalently, large applied loads, as commonly occurs with macroscopic particles), Eq. 24 reduces to the Hertz equation... 

Johnson [111] addressed the problem of viscoelastic flow by attempting to modify the JKR equation. In his approach, he postulated a creep compliance function... 

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 ... 

Israelachvili and his colleagues have used the SEA to study the interactions between surface layers of surfactant and of other molecules representing functionalised polymer chains, adhesion promoters or additives. Typically a monolayer of the molecule concerned is deposited onto cleaved mica sheets. The values of surface energies obtained from the JKR equation (Eq. 18) throw some interesting light on the nature and roughness of surface layers in contact. 

The JKR equation enables values for surface energy to be obtained both when the surfaces are advancing into closer contact, and when they are receding further apart, These two values would be expected to be the same, as indeed they some times are. In many cases, however, there is hysteresis with > > a-Israelachvili and his colleagues have studied this phenomenon in some detail [23,24,86,87]. 

In a typical experiment, Israelachvili deposited monolayers of surfactants onto cleaved mica sheets, and evaluated the surface energies using the JKR equation. Fig. 11 contrasts results for mica coated with monolayers of (a) L-a-dipalmitoyl-phosphatidylethanolamine (DMPE) where j/a = = 27 mJ/m and (b) hexa-decyltrimethylammonium bromide (CTAB) where = 20 mJ/m and = 50 mJ/m. ... 

In the limit X—>oo we get the JKR equations, whereas in the opposite limit A—>0 the DMT equations are obtained. In practice, the JKR regime may be defined by the condition X>5 and the DMT regime by /L<0.1 [105]. In the intermediate range, the transition region, the M-D model has to be applied. The quasi-analytic M-D model, however, does not provide closed formulas and its application for curve-fitting procedures is rather cumbersome due to the fact that a system of equations has to be solved. This is why the development of a direct relation between force and contact deformation is a matter of current efforts [105-107]. 

The JKR explanation of latex coalescence was proposed in 1982. Padget had observed the hexagonal structure of coalesced rubber latex (Fig. 9.21 (a)) and Kendall had measured the contact spot sizes between latex particles using electron microscopy (Fig. 9.21(b)). When the results were plotted in Fig. 9.21(c), they fitted the JKR equation and macroscopic observations, taking the elastic modulus to be 5.64MPa and the work of adhesion to be 26.5 Jm ... 

Fig. 11. Use of the JKR equation (Eq. 18) relating applied force, F, to radius of contact, a, to analyse results from the surface forces apparatus. Surface energy, y, of surfactant layers (DMPE), cf. (CTAB) (after Chen et al. [24]).

But infinite stress is always present in cracking problems and poses no difficulty if an energy-balance theory of crack equilibrium is used (see Fracture Mechanics). Applying this method to the above stress distribution gave the following equation, the so-called JKR equation, for the elastic contact spot diameter d of equal spheres, diameter D and... 

The JKR equation is usually employed to interpret the results of the Surface forces apparatus and of Atomic force microscopy, which may be employed to study adhesion between two surfaces. 

Eq. (12) has received much attention in the literature and is generally called the JKR equation. 

FIGURE 16. Variation in the work of adhesion as a function of dwell time. Open circles values obtained by fitting the JKR equation for equilibrium contacts. Full triangles values obtained by shifting the G(v) curves of Figure 15. (From Reference 62.)... 

From the stability condition that dQ/dA > 0, it can he shown that only the positive determinant in Eq. (8.66) has to be taken into account. By using again Eq. (8.52), to replace Pi with a, we obtain the JKR equation for the contact radius... 

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