Question

16. A polymer can be modelled as a path of N + 1 identical segments on a square lattice (see picture below). At each of the N joints (lattice sites), the polymer can either go straight ahead, or it can bend by 90° left or right. Different segments do not interact with each other, i.e., the path can intersect itself. A straight joint has zero energy, i.e., Est = 0, while a 90° turn has energy Etu = ? (? > 0). Assume that one end of the polymer is fixed at the origin of the coordinate system (dot in the picture). (a) Find the partition function of the polymer chain. (b) Find the Helmholtz free energy, internal energy and the specific heat. (c) Find the average number Nst of straight joints as a function of temperature T and the total number of joints N. Examine what happens in the limit T ? 0 and T ? ?

          16. A polymer can be modelled as a path of N + 1 identical segments on a square lattice (see picture below). At each of the N joints (lattice sites), the polymer can either go straight ahead, or it can bend by 90° left or right. Different segments do not interact with each other, i.e., the path can intersect itself. A straight joint has zero energy, i.e., Est = 0, while a 90° turn has energy Etu = ? (? > 0). Assume that one end of the polymer is fixed at the origin of the coordinate system (dot in the picture).

(a) Find the partition function of the polymer chain.
(b) Find the Helmholtz free energy, internal energy and the specific heat.
(c) Find the average number Nst of straight joints as a function of temperature T and the total number of joints N. Examine what happens in the limit T ? 0 and T ? ?

16. A polymer can be modelled as a path of N + 1 identical segments on a square lattice (see picture below). At each of the N joints (lattice sites), the polymer can either go straight ahead, or it can bend by 90° left or right. Different segments do not interact with each other, i.e., the path can intersect itself. A straight joint has zero energy, i.e., Est = 0, while a 90° turn has energy Etu = ? (? > 0). Assume that one end of the polymer is fixed at the origin of the coordinate system (dot in the picture).

(a) Find the partition function of the polymer chain.
(b) Find the Helmholtz free energy, internal energy and the specific heat.
(c) Find the average number Nst of straight joints as a function of temperature T and the total number of joints N. Examine what happens in the limit T ? 0 and T ? ?

Added by Jennifer W.

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