End exclusion zones in strongly stretched, molten polymer brushes of arbitrary shape

TL;DR

This paper extends strong-stretching theory to molten polymer brushes on surfaces of arbitrary curvature, revealing how geometry influences end exclusion zones and free energy, with implications for block copolymer systems.

Contribution

It develops a generalized formulation of strong-stretching theory for curved surfaces, including regimes with positive mean curvature and negative Gaussian curvature, and predicts scaling behaviors.

Findings

End exclusion zones depend on surface curvature.

Scaling laws for end zone size and free energy are derived.

Brush behavior varies significantly with geometry.

Abstract

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.