Density profile of a polymer in a slab

We consider the problem of a polymer, modelled as a self-avoiding walk on a square lattice on the (x,y greater than or equal to 0) semi-plane, which is confined between walls located at x = -m and x = m. For each monomer incorporated into the walk and located at one of the walls the partition function is multiplied by a Boltzmann factor w = exp[-epsilon/k(B)T], so that the walls may be attractive (epsilon < 0) or repulsive (epsilon > 0). The activity of a monomer will be denoted by z. Using a recursive procedure which allows us to obtain the partition function of the problem for values of m up to 4, we calculated the fraction of monomers in each column x of the slab, at the critical value of the activity z(c), where the mean value of the number of monomers diverges. As expected, this density profile is convex for sufficiently attracting walls and concave for repulsive walls. For m > 1, there exists an interval of values for w in which the profile is neither convex nor concave.