Effect of the finite extensibility of polymer chains and of the finite size of added electrolyte ions on polyelectrolyte brushes

Abstract

Our earlier polyelectrolyte brush theory [Macromolecules 22, 4173 (1989)] was derived by equating the sum of the nonelectrostatic and electrostatic potentials of the polymer segment to an overall parabolic potential shown to exist in the brush by Milner, Witten, and Cates (MWC) [Macromolecules 21, 2610 (1988)]. Here we correct the theory to take into account the finite extensibility of the chains by adopting the non-Gaussian stretch (or entropy) term of Shim and Cates [J. Phys. France 50, 3535 (1989)]. This correction leads to brush heights which will never exceed the chain length; an aspect that was not accounted for in the earlier work of MWC. In addition, the ions of the added symmetric electrolyte are considered to be hard spheres, the size of the cation and anion assuming different values in general. The total electrochemical potential of the ions, therefore, includes a volume exclusion term as well. The ions are assumed to ‘‘view’’ the solvent as a continuum (characterized only by its dielectric constant) and the brush layer as an obstacle course of randomly distributed spherical hard segments. The treatment is made simple by assuming a regime where the volume fraction of ions is much lower than the segment or solvent volume fractions.