  The behavior of a polyelectrolyte adsorbed on a charged substrate of
high-dielectric constant is studied by both Monte-Carlo simulation and
analytical methods. It is found that in a low enough ionic strength medium, the
adsorption transition is first-order where the substrate surface charge still
keeps repulsive. The monomer density at the adsorbed surface is identified as
the order parameter. It follows a linear relation with substrate surface charge
density because of the electrostatic boundary condition at the charged surface.
During the transition, the adsorption layer thickness remains finite. A new
scaling law for the layer thickness is derived and verified by simulation.
