## JKR and Surface Adhesion Effects

### Quick Start

Unlike the Hertz case, when you put the rubber ball close to the surface with no force, it is pulled spontaneously into contact, giving a contact width. As with Herz, as you slide F to increase the force, the width increases. But to remove the ball you have to apply a negative force.

From the analysis of all this, the true surface energy of the contact can be calculated, though in the app we assume that the surface energy, γ, is 40mN/m, a very typical value. This is a much more fundamental way of measuring and understanding surface energies than contact angles etc.

There are variants of JKR (DMT, Tabor, Maugis-Dugdale, etc.) but for our purposes, JKR is good enough.

### JKR

Force mN
Width μm

Unlike the simple Hertz case, the actual case which includes surface energy γ has been analysed by Johnson, Kendall and Roberts and is called JKR theory.

Taking into account the surface energy γ, the JKR equivalent to the Hertzian formula is:

a^3=(3R)/(4E^**)(F+6γπR+sqrt(12γπRF+(6γπR)^2))

To see this in action, and assuming the radius and modulus of the Hertzian example plus a value of the surface energy γ =40dyne/cm (=0.04J/m2) try varying the Force as before:

When the Force=0 there is a significant contact width. As the force is increased the contact width increases in a manner similar to the Hertzian example. The big difference is that as the force goes below 0 there is still a finite contact width till suddenly the force is sufficient to pull the surfaces apart and the contact width becomes 0.

In a real experiment, the force needed to pull the surfaces apart can be used to measure the surface energy. This is a much better way of doing it than via contact angle measurements. Experiments on corona treated PE surfaces, for example, show a large scatter of values measured via contact angle, yet using the Surface Force Apparatus and the JKR formula it is possible to find reliable and subtle differences in surface energies depending on treatment level.