  Theories of strongly stretched polymer brushes, particularly the parabolic
brush theory, are valuable for providing analytically tractable predictions for
the thermodynamic behavior of surface-grafted polymers in a wide range of
settings. However, the parabolic brush limit fails to describe polymers grafted
to convex, curved substrates, such as the surfaces of spherical nanoparticles
or the interfaces of strongly segregated block copolymers. It has been
previously shown that strongly-stretched, curved brushes require a boundary
layer devoid of free chain ends, requiring modifications of the theoretical
analysis. While this "end exclusion zone" has been successfully incorporated
into descriptions of brushes grafted onto the outer surfaces of cylinders and
spheres, the behavior of brushes on surfaces of arbitrary curvature has not yet
been studied. We present a formulation of the strong-stretching theory for
molten brushes on surfaces of arbitrary curvature and identify four distinct
regimes of interest for which brushes are predicted to possess end exclusion
zones, notably including regimes of positive mean curvature but negative
Gaussian curvature. Through numerical solutions of the strong-stretching brush
equations, we report predicted scaling of the size of the end exclusion zone,
the chain end distribution, the chain polarization, and the free energy of
stretching with mean and Gaussian surface curvatures. Through these results, we
present a comprehensive picture of how brush geometry influences the end
exclusions zones and exact strong-stretching free energies, which can be
applied, for example, to model the full spectrum of brush geometries
encountered in block copolymer melt assembly.
