Simple model of a polymer

Question

Let's describe a polymer as a linear chain of $N$ links which all have a bendable part. The energies of one link are $\epsilon_-$ (unbended) and $\epsilon_\wedge$ (bended). Additionally, the lengthsof a single link are given by $l_-$ (unbended) and $l_\wedge < l_-$ (bended).

With this information, I want to compute the canonical partition function for a given length $L$ of the polymer:

$$Z(T,L,N) = \sum_\text{states $n$} e^{-\beta E_n}$$

with $N$ the total number of links in the chain and $\beta = \frac{1}{kT}$. $L$, $N$ and the number of bended links $N_\wedge$ are not independent, so the sum runs over all states which satisfy the relation which connects $L$, $N$ and $N_\wedge$.

My problem is that I have no idea how to tackle this. I tried to express the total energy and length by

$$E = N\epsilon_- + N_\wedge(\epsilon_\wedge-\epsilon_-)\qquad\text{and}\qquad L = Nl_- + N_\wedge(l_\wedge-l_-)$$

but this still leaves me with the question how to evaluate the sum for the partition function. Can anybody help me with this?

Answer 1

You can connect the equations for $E$ and $L$ by using $$N=n_{\wedge}+n_-$$

Write $L$ as: $$L=n_-(l_--l_{\wedge})+Nl_{\wedge}=d n_-+Nl_\wedge$$ and $E$ as: $$E=n_-(\epsilon_--\epsilon_\wedge)+N\epsilon_\wedge=f n_-+N\epsilon_\wedge$$

Where $d=l_--l_\wedge$ and $f=\epsilon_--\epsilon_\wedge$

Now use the equation for $L$ to write $n_-$ in terms of $L$ and $N$: $$n_-=\frac{L-Nl_\wedge}{d}$$ Insert this into the equation for the energy of a certain configuration: $$E=\frac{f(L-Nl_\wedge)}{d}+N\epsilon_\wedge$$

And we can now write the equation for the partition function in terms independent of the sum over states: $$Z=\sum_{\mathrm{states} \ i}exp\Bigl\{-\beta\Bigl(\frac{(L-Nl_\wedge)f}{d}+N\epsilon_\wedge\Bigr)\Bigr\}=exp\Bigl\{-\beta\Bigl(\frac{(L-Nl_\wedge)f}{d}+N\epsilon_\wedge\Bigr)\Bigr\}\sum_{\mathrm{states} \ i}1$$ We just need to count the ways that the kinks and straights can be arranged. Imagine the chain completely straightened out. We want to distribute $n_\wedge$ kinks over $N$ positions. There are: $$\frac{N!}{n_\wedge!(N-n_\wedge)!}=\frac{N!}{n_\wedge!n_-!}=\frac{N!}{\frac{L-Nl_-}{d}!\frac{L-Nl_\wedge}{d}!}$$ ways to do this. So in full, you have the partition function: $$Z(T, N, L)=\frac{N!}{\frac{L-Nl_-}{d}!\frac{L-Nl_\wedge}{d}!}exp\Bigl\{-\beta\Bigl(\frac{(L-Nl_\wedge)f}{d}+N\epsilon_\wedge\Bigr)\Bigr\}$$

This is easier to write if you change from a function of $L$ to a function of $n_-$ (They contain the same information): $$Z(T, N, n_-)=\frac{N!}{n_-!(N-n_-)!}exp\Bigl\{-\beta\Bigl(n_-f+N\epsilon_\wedge\Bigr)\Bigr\}$$