Normal coordinates and the spectrum

If we introduce the mobility tensors  Eq. (5.56) into Eq. (7.7), we are left with a completely intractable set of equations. One way out of this is by noting that in equilibrium, on average, the mobility tensors will be proportional to the unit tensor. We therefore assume that $\Psi$ will never differ much from $\Psi _{\mathrm{eq}}$, and that we may replace the mobility tensor in Eq. (7.7) by

$$\langle \bar{\mu}_{jk}\rangle _{\mathrm{eq}}=\int d^{3}R_{0}... ...c{R}_{k})\Psi _{\mathrm{eq}}(\vec{R} _{0},\ldots ,\vec{R}_{N})$$ (7.10)

where $\Psi _{\mathrm{eq}}$ is the Gaussian equilibrium distribution. A simple calculation yields

$$\langle \bar{\mu}_{jk}\rangle _{\mathrm{eq}} \frac{1}{8\pi \eta }\langle \frac{1}{R_{jk}}\rangle _{\mathrm{eq}}(\mathbf{1}+\langle \hat{R}_{jk}\hat{R} _{jk}\rangle _{\mathrm{eq}})$$ =

$$\frac{1}{6\pi \eta }\langle \frac{1}{R_{jk}}\rangle _{\mathrm{eq}}\mathbf{ 1}$$ =

$$\frac{1}{6\pi \eta b}\left\{ \frac{6}{\pi \vert j-k\vert}\right\} ^{1/2}\mathbf{1}$$ = (7.11)

The next step is to write down the equations of motion of the normal coordinates Eq. (6.26):

  

$$\frac{d\vec{X}_{k}}{dt} -\sum_{p=0}^{N}\mu _{kp}\frac{3k_{B}T}{b^{2}} 4\sin ^{2}\left( \frac{p\pi }{2(N+1)}\right) \vec{X}_{p}+\vec{F}_{k}$$ = (7.12)

$$\langle \vec{F}_{k}(t)\cdot \vec{F}_{k^{\prime }}(t^{\prime })\rangle 3k_{B}T\frac{\mu _{kk^{\prime }}}{N+1}\delta (t-t^{\prime }),$$ = (7.13)

where

 

$${\mu _{kp}=\frac{2}{N+1}\sum_{n=0}^{N}\sum_{m=0}^{N}\frac{1}{6\pi \eta b}\left\{ \frac{6}{\pi \vert n-m\vert}\right\} ^{\frac{1}{2}}}$$

$$\times \cos \left( \frac{k\pi }{N+1}(n+\frac{1}{2})\right) \cos \left( \frac{p\pi }{N+1}(m+\frac{1}{2})\right)$$ (7.14)

Equation (7.12) still is not tractable. It turns out however (see Appendix), that for large N approximately

$$\mu _{kp}=\left\{ \frac{N+1}{3\pi ^{3}p}\right\} ^{\frac{1}{2}}\frac{1}{\eta b}\delta _{kp}$$ (7.15)

Introducing this result in Eq. (7.12), we see that the normal modes , just like with the Rouse chain, constitute a set of decoupled coordinates.

For small values of k we find

  

$$\frac{d\vec{X}_{k}}{dt} -\frac{1}{\tau_k} \vec{X}_k + \vec{F}_k$$ = (7.16)

$$\langle \vec{F}_k(t) \cdot \vec{F}_{k^{\prime}}(t) \rangle 3k_BT \frac{ \mu_{kk}}{N+1} \delta_{kk^{\prime}} \delta (t-t^{\prime})$$ = (7.17)

where the first term on the right hand side of Eq. (7.16) equals zero when k=0, and otherwise

$$\tau_k = \frac{3\pi\eta b^3}{k_BT} \left( \frac{N+1}{3\pi k} \right)^{\frac{3 }{2}}$$ (7.18)

We are now in a position to calculate the diffusion coefficient D_G of a Zimm chain, and the viscosity of a dilute solution of Zimm chains.