## Contact Mechanics and Adhesion

When two bodies come into contact the interacting forces across the contact boundary result in adhesion force which is the combination of different

Fig. 8.5 The schematic of a sphere of radius R attached to a rigid flat surface

contributions (Van der Waals force, electrostatic force, capillary force, and others). This adhesion force depends on different parameters such as the number of contributing forces, the strength of the contributing forces, area of contact between two bodies, the wettability of the two surfaces, and the ambient conditions.

Two bodies in contact deform as a result of either external or surface forces. The original work in contact mechanics dates back to 1882 when Hertz solved the problem involving contact between a sphere and a planar surface, and between two spheres [28]. The Hertzian model of contact does not consider the adhesion between two bodies. A schematic of a contact between a sphere and a flat surface is presented in Fig. 8.5. The radios of contact a and the indentation d based on Hertz model is given by

c a2

where F is the external force, E* the reduced Young's modulus, and R* is the reduced radius.

The Johnson-Kendall-Roberts (JKR) model [40] and the Derjaguin-Muller-Toporov (DMT) models [16, 63, 64] were the first attempts to extend the Hertz theory and include the contribution of adhesion forces. The JKR model has been employed to incorporate the effect of adhesion in the Hertzian model. This model considers the adhesion forces inside the area of contact and neglects the interactions outside the area of contact. The JKR model is more suitable for soft materials with high surface energy, and large radii of curvature. The radios of contact area a it and the JKR indentation d are obtained with the following equations:

where y is the surface energy.

The JKR model predicts that the force needed to remove the particle (the pulloff force) is given by

DMT model is another modified Hertzian contact theory taking into account the surface attraction forces. DMT model has been successfully applied to systems with small particles, small surface energy, and high elastic modulus. The pull-off force predicted by DMT is given by

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