Polymer in a strip: profile of the monomer density

We study a model of self-avoiding walks (SAWs) on a square lattice in the (x,y⩾0) semi-plane, which are confined between two impenetrable walls located at x=0 and x=m⩾0. Such a model may describe polymers inside a strip. The activity of a monomer (site of the lattice incorporated into the walks) is denoted by z=exp(−βμ), μ being the chemical potential, and monomers located at the walls interact with them, so that the energy of the system is increased by ε for each monomer located at x=0 or x=m. Using a transfer matrix formalism for the partition function of the model, we calculate the fraction of monomers in each column 0⩽x⩽m at the activity zc corresponding to the polymerization transition, where the number of monomers diverge, for 0⩽m⩽8. We find convex density profiles for sufficiently attracting walls and concave profiles when the walls are repulsive. The transition between these two regimes takes place in an interval of values for ε, in which the density profile is neither convex nor concave, a more complex behavior being observed.