Exact Solutions to Lattice Models of Polymer Adsorption

Entropy calculations are important in determining the physical properties of a polymeric system. A classical method of modeling polymers is with self-avoiding walks, and entropy may be determined by counting the total number of weighted walks. If directed walks are used, recurrences may be formed and solved to study a variety of physical properties exactly. One solution method is to solve the recurrences with generating functions. Additionally, we may attempt to derive the partition function of the model, which explicitly provides the walk entropy per length. In this thesis, the solution to a variety of polymer models of adsorption are derived with generating functions. The partition functions of these models are then extracted when possible through combinatorial convolution identities or by solving their partial difference equations on the lattice.