Monomer motion

In this section we study the mean square displacements  of the individual monomers. Using Eq. (6.24) and the fact that different modes are not correlated, we get

$${\langle (\vec{R}_{n}(t)-\vec{R}_{n}(0))^{2}\rangle =\langle (\vec{X} _{0}(t)-\vec{X}_{0}(0))^{2}\rangle }$$

$$+4\sum_{k=1}^{N}\langle (\vec{X}_{k}(t)-\vec{X}_{k}(0))^{2}\rangle \cos ^{2}\left( \frac{k\pi }{N+1}(n+\frac{1}{2})\right)$$ (6.43)

Introducing Eq. (6.38) we get

 

$${\langle (\vec{R}_{n}(t)-\vec{R}_{n}(0))^{2}\rangle =6D_{G}t}$$

$$+\frac{4b^{2}}{\pi ^{2}}(N+1)\sum_{k=1}^{N}\frac{1}{k^{2}} (1-e^{-tk^{2}/\tau _{1}})\cos ^{2}\left( \frac{k\pi }{N+1}(n+\frac{1}{2} )\right)$$ (6.44)

where we have used Eqs. (6.30) and (6.31) to calculate the first term, and Eqs. (6.39) and (6.40 ) for the second term.

There are two different limits to Eq. (6.44). First, when tis very large, i.e. $t \gg \tau_1$, the first term will dominate, yielding

$$\langle (\vec{R}_n(t)-\vec{R}_n(0))^2 \rangle = 6D_G t t \gg \tau_1.$$ (6.45)

This is consistent with the fact that the polymer as a whole diffuses with diffusion constant D_G.

Secondly, suppose $t \ll \tau_1$. Then the sum in Eq. (6.44) will dominate. Averaging over all monomers, and replacing the sum over kby an integral we get

$$\frac{1}{N+1} \sum_{n=0}^N \langle (\vec{R}_n(t)-\vec{R}_n(0))^2 \rangle \frac{2b^2}{\pi^2} (N+1) \int_0^{\infty} dk \frac{1}{k^2} ( 1-e^{-tk^2/\tau_1} )$$ =

$$\frac{2b^2}{\pi^2} (N+1) \int_0^{\infty} dk \frac{1}{\tau_1} \int_0^t dt^{\prime} e^{-t^{\prime}k^2/\tau_1}$$ =

$$\frac{2b^2}{\pi^2} \frac{(N+1)}{\tau_1} \frac{1}{2}\sqrt{\pi\tau_1} \int_0^t dt^{\prime} \frac{1}{\sqrt{t^{\prime}}} .$$ = (6.46)

Performing the final integral we get

$$\langle ( \vec{R}_n(t)-\vec{R}_n(0) )^2 \rangle = \left( \fra... ...pi\gamma} \right)^{\frac{1}{2}} t^{\frac{1}{2}} t \ll \tau_1 .$$ (6.47)

So, at short times the mean square displacement of a typical monomer goes like the square root of t.