Stretch-collapse transition of polyelectrolyte brushes in a poor solvent

This paper describes the behavior of charged, polymer brushes in electrolyte solutions of varying solvent quality. The brush height, d, dependence on the chain length, L (=Nl, where l is the Kuhn length), the grafting density σ, and solvent conditions is determined. We consider a monomer–monomer potential consisting of three components: (1) a long-ranged, screened Coulombic component of strength v̄/l (l is the Kuhn length) and range κ−1; (2) a short-ranged, two-body component of strength w̄l; and (3) a short-ranged, three-body component of strength ūl3. In particular, we examine the transition from a stretched state to a collapsed state in a poor solvent (w̄<0) as the solvent quality is decreased. Using dimensional analysis, Monte Carlo methods, and a variational technique, a first order transition is observed as predicted by the scaling arguments of Ross et al. and Borisov et al. for high charge/grafting densities. Using a variational procedure, we derive an analytical expression for the brush size and determine, quantitatively, the critical conditions for a first order transition in terms of key dimensionless variables, vN5/2, κlN1/2, wN3/2, and uN2 (where v=2πσl2v̄, w=σl2w̄, and u=σ2l4ū).