- #1

CharlieCW

- 56

- 5

## Homework Statement

Consider a polymer formed by connecting N disc-shaped molecules into a onedimensional chain. Each molecule can align either its long axis (of length ##l_1## and energy ##E_1##) or short axis (of length ##l_2## and energy ##E_2##). Suppose that the chain is subject to tension ##\tau##.

a) Calculate the number of different ways of arranging the polymer such that there are n molecules aligned by its long axis.

b) Using the Gibbs canonical ensemble in 1-dimension find the average energy ##\langle E \rangle## and the average length of the chain ##\langle L \rangle##.

## Homework Equations

Gibbs canonical partition function

$$\Xi=\sum_{V_S}\sum_{i}exp[-\beta (E_i+pV_S)]$$

Hint: You can take the sum in the partition function as

$$\sum_{V_S}\sum_{i}=\sum_{\{ n \}}g(n)$$

where ##\{ n \}## denotes the states of the chain with ##n## molecules aligned by its long axis and ##g(n)## a degeneracy factor.

## The Attempt at a Solution

a) For the first problem, let's denote for simplicity a molecule in vertical position with ##0## and in horizontal position with ##1##. Since each molecule can only adopt two configurations ##(0,1)##, then ##N## molecules can adopt a total of ##2^N## possible configuration. Now if we want the number of ##n## molecules on state ##1##, it is the same to find the total number of ways of arranging n molecules in N spaces, that is:

$$g(n)={N \choose n}$$

b) I'm really not sure how to proceed on this one and how to even include the tension ##\tau##. I began by calculating the Gibbs canonical partition function as:

$$\Xi=\sum_{V_S}\sum_{i}exp[-\beta (E_i+pV_S)]$$

Using the hint, the sum can be simplified to:

$$\Xi=\sum_{\{ n \}}g(n)exp[-\beta (E_n+pV_S)]=\sum_{\{ n \}}{N \choose n}exp[-\beta (E_n+pV_S)]$$

The energy ##E_n## for the state with ##n## molecules aligned along it's axis is ##n(\epsilon_2-\epsilon_1)=n\Delta\epsilon##. On plus, it is subject to a tension which I think should be ##\tau (l_2-l_1)=\tau\Delta l##. I'm not sure if I should include this tension in the energy, or replace ##pV_S## by ##\tau \Delta l##.

After that, I have no idea on how to find the average energy and length.