Con?gurational entropy of square lattice models for polymers In this problem, we will consider, several models for a polymer on a 2D square lattice. In particular, we will focus on the con?gurational entropy that arises from the different conformations that a polymer can adopt. (Con?gurational entropy is distinct from translational entropy, which addresses polymer motion through space.) Any lattice polymer model requires that bonded monomers must occupy adjacent sites on the lattice. In class, we considered a 2D Self-AvoidingWalk (SAW) model for a polymer with the ?rst monomer tethered to a wall. The key aspect of the SAW model is that no two monomers may occupy the same lattice site, which provides a simple treatment of excluded volume.
Let us ?rst consider models of a polymer with n =3monomers.
i. The "ideal chain model" of a polymer does not account for excluded volume, but still treats polymer connectivity by requiring that successive monomers occupy adjacent lattice sites. Suppose that we ?xed the position of the ?rst monomer. How many con?gurations are accessible to an ideal chain model with n = 3 monomers. What is the entropy of this model?
ii. Now consider a SAW model for the same polymer. (This is the same model treated in class, except we are not yet considering the wall.) Determine the sample space of con?gurations accessible for this model. How many con?gurations are eliminated as a consequence of excluded volume and how does this change the con?gurational entropy of the polymer?
iii. Now consider a SAWmodel for the same polymer, but with the ?rst monomer bonded to the wall. (This is the model we treated in class.) How many con?gurations have now been eliminated as a consequence of the wall and what is the resulting entropy of this model? What had a bigger effect upon the polymer statistics, the excluded volume of the polymer or of the wall?