Adhesion between rubbers and grafted solids

The adhesion energy, G (at low separation velocities), between a rubber and a solid surface, is expected to increase when flexible chains, chemically identical to the rubber, are attached to the surface. This can be set up with or without chemical binding of the chains to the rubber ("bound" chains or "free" chains). 1) For the "free" case, rupture is predicted to occur mainly in an adhesive mode. The calculated adhesion energy, G(a), has a maximum at a certain value of the grafting density, sigma = sigma (2) N-o(-1/2) (where N-o is the number of monomers between cross links in the rubber). 2) For the "bound" case, we expect a linear increase of G (sigma) up to a value (sigma = sigma (3)) where we switch from adhesive to cohesive rupture.