Self-consistent field theory study of the effect of grafting density on the height of a weak polyelectrolyte brush

The height of weakly basic polyelectrolyte brushes in the osmotic brush regime is studied as a function of the grafting density using a numerical self-consistent field theory derived from the (semi)grand canonical partition function. The theory is shown to properly account for the local nature of the charge equilibrium and to capture the basic behaviors of polyelectrolyte brushes. On one hand, we find, in agreement with recent experiments, that the scaling of brush height with grafting density can be qualitatively different at intermediate chain lengths than that predicted by basic scaling arguments. This difference is attributed to the relative strength of electrostatic type interactions compared to finite segment size packing constraints. On the other hand, the trend of decreasing brush height with increasing grafting density predicted by the classic scaling analysis is recovered for large molecular weight polymers immersed in a solution of very weak ionic strength.