  A model of rod-like polyelectrolyte brushes in the presence of monovalent and
multivalent counterions but with no added-salt is studied using Monte Carlo
simulation. The average height of the brush, the histogram of rod
conformations, and the counterion density profile are obtained for different
values of the grafting density of the charge-neutral wall. For a domain of
grafting densities, the brush height is found to be relatively insensitive to
the density due to a competition between counterion condensation and inter-rod
repulsion. In this regime, multivalent counterions collapse the brush in the
form of linked clusters. Nematic order emerges at high grafting densities,
resulting is an abrupt increase of the brush height.
