Statistical mechanics of the adsorption and collapse transitions in polymers

Two different models of a polymer molecule in a dilute solution near an adsorbing interface are studied. The first one consists of a self-interacting self-avoiding walk on the cubic lattice terminally attached to and interacting with the impenetrable z = 0 plane. The second one employs a self-interacting self-avoiding walk on the cubic lattice terminally attached to and interacting with the penetrable z = 0 plane. A number of results about the form of the free energy and the form of the phase diagram are derived for such walks, and for similarly defined polygons, using rigorous mathematical arguments. The second part of the thesis consists of a numerical treatment of this problem using a highly efficient Monte Carlo algorithm. Conjectured phase diagrams for the penetrable plane case and the impenetrable plane case are given and compared with each other.