Exclusion zone of convex brushes in the strong-stretching limit

Abstract

We investigate asymptotic properties of long polymers grafted to convex cylindrical and spherical surfaces, and, in particular, distribution of chain free ends. The parabolic potential profile, predicted for flat and concave brushes, fails in convex brushes, and chain free ends span only a finite fraction of the brush thickness. In this paper, we extend the self-consistent model developed by Ball, Marko, Milner and Witten to determine the size of the exclusion zone, i.e. size of the region of the brush free from chain ends. We show that in the limit of strong stretching, the brush can be described by an alternative system of integral equations. This system can be solved exactly in the limit of weakly curved brushes, and numerically for the intermediate to strong curvatures. We find that going from melt state to theta solvent and then to marginal solvent decreases relative size of the exclusion zone. These relative differences grow exponentially as the curvature decreases to zero.

Dispersing hairy nanoparticles in polymer melts

“…Both the variational analysis by Li and Witten [51] and the numerical calculations by Belyi [52] show that near the open surface the density profile still maintains the same parabolic structure as that found in flat and concave brushes. At present, our best understanding of this convex problem is that for most cases when the exclusion zone is a small fraction of the grafted layer thickness, the width of the exclusion zone and the effects that arise from it vary as the exponential of minus the radius of curvature over brush thickness.…”

“…Although this exponential correction is only formally derived based on an argument of weak curvature limit, numerical calculations [50,52] show that this correction is applicable over a wide range from the weak curvature when R 0 [ L up to the intermediate curvature for R 0 z L/3. Thus, near the extremity of the interpenetrated layer as it is shown in Fig.…”

“…Also, theoretical studies discussed the interactions between particles mostly based on systems of flat brushes [41][42][43][44][45][46][47][48][49]. The equilibrium statistics of polymers grafted on convex surfaces has only been considered in recent years [50][51][52], where a proper treatment of the exclusion zone in spherical brushes is still lacking [52]. As a result, the effects of interactions between the convex curved brushes are less understood than those between flat brushes.…”

Electrostatic origin of the genome packing in viruses

“…This condition, often referred to as the ''equal travel time constraint'' for its classical particle analogy (50,51), can only be satisfied by a parabolic potential:…”

“…We will self-consistently justify this assumption in Discussion. The free energy of a single peptide arm in the brush may thus be written as (50,51):…”