Arbitrarily curved polymer brushes and their end-exclusion zones

Surface-grafted polymer brushes are seen in a wide verity of scenarios, ranging from coated nanoparticles to complex block copolymer phases. Their substrates can take on just about any shape, from highly curved spheres to gyrating saddles. In many of these situations, chain ends can exist anywhere within the brush and the thermodynamics are well-modeled by the parabolic brush theory (PBT). However, when a convex substrate curvature forces brush ends to splay outward, chain ends are depleted away from the substrate, resulting in an "end-exclusion zone" (EEZ) that is inconsistent with the PBT. Here, we generalize previous descriptions of the EEZ that were limited to spherical and cylindrical surfaces to surfaces of arbitrary curvature. We find that the combination of surface curvatures contribute to non-local adjustments to chain packing constraints, either promoting EEZ growth or depressing it, which in turn affects the chain end distribution and the local polar order of chains. Finally, we explore the consequences of EEZ corrections on the stability of complex block copolymer phases.