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Introduction to the Rouse Model
The Rouse model is frequently used in polymer physics.

The Rouse model describes the conformational dynamics of an ideal chain. In this model, the single chain diffusion is represented by Brownian motion of beads connected by harmonic springs. There are no excluded volume interactions between the beads and each bead is subjected to a random thermal force and a drag force as in Langevin dynamics. This model was proposed by Prince E. Rouse in 1953. The mathematical formalism of the dynamics of Rouse model is described here. In particular, the looping time of the Rouse model has the mixed scaling law, $$\tau= \frac{N\sqrt{N}}{\epsilon}(d_1+(d_2+d_3\sqrt{N})\epsilon+\mathcal{O}(\epsilon^2))$$, where $$N$$ is the number of beads and $$\epsilon$$ is the capture radius.

An important extension to include hydrodynamic interactions mediated by the solvent between different parts of the chain was worked out by Bruno Zimm in 1956. Whilst the Rouse model overestimates the decrease of the diffusion coefficient D with the number of beads N as 1/N, the Zimm model predicts D~1/Nν which is consistent with the experimental data for dilute polymer solutions (where $$\nu$$ is the Flory exponent).

In a polymer melt, the Rouse model correctly predicts long-time diffusion only for chains shorter than the entanglement length. For long chains with noticeable entanglement, the Rouse model holds only up to a crossover time τe. For longer times the chain can only move within a tube formed by the surrounding chains. This slow motion is usually approximated by the reptation model.

Scaling and Relaxation of the Rouse Model
The Rouse model describes the dynamics of $$N$$ monomers connected by ideal springs. These monomers also undergo random jostling in the form of Brownian motion. We can analyze this model in a continuum limit using a method first introduced by Sam Edwards and Masao Doi extract scaling laws about its dynamics. If we let $$\vec{r}_n $$ be the position of particle $$1\leq n\leq N $$, the equation of motion governing $$\vec{r}_n $$ is $$m \ddot{\vec{r}}_n(t) = k(\vec{r}_{n+1}(t)+\vec{r}_{n-1}(t) - 2\vec{r}_n(t)) - \gamma \dot{\vec{r}}_n(t) + \gamma \vec{\zeta}_n(t) $$

where $$m $$ is the mass of the bead, $$k $$ is the spring constant of each spring, $$\gamma $$ is the drag coefficient for each of the monomers, and $$\vec{\zeta}_n(t)$$ is the noise term on each bead, typically called a white noise. We handle the motion of the ends of the polymer with the special boundary condition $$\vec{r}_{-1} = \vec{r}_0, \vec{r}_n=\vec{r}_{n+1} $$. In the Brownian limit, the inertial term $$m\ddot{\vec{r}}_n$$ can be dropped as the drag terms dominate. The final simplification we will make is to notice that the interaction term due to spring forces can be approximated as a second derivative

$$\vec{r}_{n+1}(t)+\vec{r}_{n-1}(t) - 2\vec{r}_n(t) \approx \partial^2_n \vec{r}(n,t) $$

Where $$\vec{r}(n,t) $$ is the continuum limit of the discretized $$\vec{r}_n(t) $$. We make a similar simplification for the noise term. All of these simplifications give our equation of motion a simple form.

$$\dot{\vec{r}}(n,t) = \frac{k}{\gamma}\partial^2_n\vec{r}(n,t) + \vec{\zeta}(n,t) $$

In real space, solving this equation is very difficult. This difficulty arises from the $$\partial^2_n $$ term, which represents local interactions between adjacent monomers. However, if we work in Fourier space we can actually remove a lot of this complexity. We can expand any function $$f(n) $$ in Fourier space as

$$f(n) = \int \frac{dq}{2\pi} f(q) e^{iqn} $$ and its inverse $$f(q) = \int dn f(n) e^{-iqn}  $$

using the convention that we distinguish the function $$f(n) $$ from its Fourier transform $$f(q)  $$ by their arguments. When we change into Fourier space, our equations of motion decouple and become

$$\dot{\vec{r}}(q,t) = -\frac{k}{\gamma}q^2\vec{r}(q,t) + \vec{\zeta}(q,t) $$

This equation tells us that each mode (i.e. at each fixed value of $$q $$) evolves independently of all the others. It will be interesting and useful to work out some of the statistics of our transformed noise terms. These are well established from the properties of Brownian Motion. In real space, the noise term for a white noise has correlations

$$\langle \zeta_i(n,t)\rangle = 0 $$

$$\langle \zeta_i(n,t) \zeta_j(n',t') \rangle = 2D \delta_{ij} \delta(t-t') \delta (n-n') $$

where $$i,j $$ label the components of the noise vector $$\vec{\zeta}  $$. In Fourier space, we get very similar relations.

$$\langle \zeta_i(q,t) \rangle = \int dn\ \langle \zeta(n,t)\rangle e^{-iqn} = 0 $$

$$\langle \zeta_i(q,t)\zeta_j(q',t') \rangle = \int dn dn'\ \langle \zeta_i(n,t) \zeta_j(n',t') \rangle e^{-i(qn+q'n')} = 4\pi D\delta_{ij}\delta(t-t')\delta(q+q') $$

Using the first of these correlation relations for the noise, we can see that the equation of motion for the average of any particular mode (where $$q \neq 0 $$) is

$$\langle\dot{\vec{r}}(q,t)\rangle = -\frac{k}{\gamma}q^2\langle\vec{r}(q,t)\rangle $$ which we can solve by integrating to find that $$\langle \vec{r} (q,t)\rangle = \langle \vec{r}(q,0)\rangle e^{-\frac{kq^2}{\gamma}t}  $$.

This tells us that each average mode of the polymer relaxes independently over a timescale $$\tau_q = \frac{\gamma}{kq^2} $$. We also know that the wavenumber should be proportional to one over the length of the polymer $$q \sim N^{-1} $$ which tells us that the relaxation time scales as

$$\tau_q \sim N^2 $$, i.e. longer polymers take longer to relax.

Diffusion in the Rouse Model
Of particular interest is the $$q = 0 $$ mode which we omitted from the previous section. The reason for this becomes clear when we look more closely at our Fourier expansion

$$\vec{r}(q=0,t) = \int dn\ \vec{r}(n,t) = N\ \vec{r}_{cm}(t) $$

where $$\vec{r}_{cm}(t) $$ is the coordinate of the center of mass of all the monomers. Setting $$q = 0 $$, our equation of motion is

$$\frac{d}{dt}\vec{r}_{cm}(t) = \frac{1}{N}\vec{\zeta}(0,t) $$ where $$\vec{\zeta}(0,t) = \int_0^N dn\ \vec{\zeta}(n,t)  $$

Integrating our equation of motion gives

$$\vec{r}_{cm}(t) = \int_0^t d\tau \int_0^n\frac{dn}{N}\vec{\zeta}(n,t) $$

The statistics for the center of mass are actually of great interest. We can use our statistics for our white noise to find

$$\langle \vec{r}_{cm}(t) \rangle = 0 $$

$$\langle \vec{r}_{cm}^2(t) \rangle = \frac{6Dt}{N} $$

The center of mass of the beads diffuses with a diffusion constant $$D/N $$. It is also apparent that the diffusion of the center of mass is independent of the strength of the springs!