POLYMERS GRAFTED TO A CONVEX SURFACE

We consider the equilibrium statistics of polymers grafted to the outside of a cylinder in the limit of high molecular weight for small, fixed grafting density. Under melt conditions, we find the exact self-consistent pressure profile, the chain configurations, and the free energy in closed form. These are qualitatively different from the case of concave or flat surfaces: the free chain ends are excluded from a zone near the surface, whose thickness grows from zero for a flat surface to a fraction 2/pi of the layer height for a strongly curved cylinder. Despite these differences, the free energy of the convex layer is very well described by the simpler law for the concave layer, analytically continued to convex curvature, for all but the most extreme curvatures. For a cylindrical layer in a marginal solvent, we reduce the conditions for equilibrium to a linear integral equation describing the exclusion zone. For spherical curvature the analogous equation has a quadratic nonlinearity: we discuss the conditions under which this equation is suitable for describing the spherical layer.