Polyelectrolyte Cylindrical Molecular Brushes

Molecular brushes are polymers composed of a long main chain (backbone), onto which side chains (grafts) are attached at regular intervals. See Fig. 4c for a graphic illustration. The graft-copolymers are classified as molecular brushes, provided the 

The box-like model considers a molecular PE brush with the backbone extended along the axis of an (infinitely long) cylindrical cell of radius D, wherein the end segments of the grafts are localized at a distance R D from the axis of the cell. The cell contains counterions who compensate the net charge of the brush. This model enables one to analyze the local conformational properties and the distribution of the counterions in a dilute salt-free solution of molecular PE brushes. The Coulomb interactions (calculated per graft) can be presented as  

The Gaussian elasticity of the grafts and spacers could be modified to account for the local swelling effects [102], Finally, the translational entropy of the counterions is accounted for as  

The characteristic branching parameter (grafting density), n/m = specifies the onset of counterion localization inside the molecular brush. Note that in the osmotic regime, the spacers get fully extended, /t m. It is therefore not surprising, that the counterion localization in a cylindrical molecular brush coincides (in scaling terms) with the Manning condensation threshold [25] for a charged cylinder, qh = 1. 

Molecular brushes composed of biopolymers, are typically more complex than their synthetic analogs. They may comprise different types of biomacromolecules, and involve self-assembly mechanisms. For example, in aggrecanes [108] and 

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