JKR and Other Adhesion Models

Figure 1 shows the schematic of a particle of diameter d attached to a flat surface. Here P is the external force exerted on the particle, a is the contact radius and is the adhesion force. The classical Hertz contact theory provides for the elastic deformation of bodies in contact, but neglects the adhesion force. Several models for particle adhesion to flat surfaces were developed in the past that improves the Hertz model by including the effect of adhesion (van der Waals) force.

Figure 1: Spherical particle of diameter d attached to a flat, horizontal surface.

JKR Model

 Johnson-Kandall-Roberts (1971) developed a model (The JKR Model) that included the effect of adhesion force on the deformation of an elastic sphere in contact to an elastic half space. Accordingly, the contact radius is given as

(1)

Here  is the thermodynamic work of adhesion, and K is the composite Young's modulus given as

(2)

In Equation (2), E is the elastic modulus,  is the Poisson ratio, and subscript 1 and 2 refer to the materials of the sphere and substrate.

In the absence of surface forces,  and Equation (1) reduced to the classical Hertz model. That is

(3)