Slippage of polymer melts on grafted surfaces

We study the slippage of a highly viscous polymer melt (P monomers per chain) on a solid substrate grafted by a few smaller chains in the mushroom regime (N monomers per chain, grafting density v). The friction is provided by the sliding motion of the P chains of the ''skin'' (thickness R(p) = P(1/2)a) which are entangled with the tethered chains. At low grafting densities, only a fraction of the P chains in the skin are coupled to the N chains, and the friction on the mushrooms is additive. Above a threshold v(c), all P chains of the skin are trapped, and the low-velocity friction becomes independent of the grafting density. Above a certain threshold slippage velocity V*(v), the N chains are strongly stretched and reach a ''marginal state'', corresponding to a constant shear stress. We expect that for v > v(c), V*(v) increases linearly. Depending on N, P, v, and V, we predict a cascade of regimes, where the N chains may be ideal, stretched, or ''marginal'', while the trapped chains may be ideal or stretched and progressively disentangle from the N chains.