Rouse model

The Rouse model is one of the simplest coarse-grained descriptions of the dynamics of polymer chains. It treats a single polymer as an ideal chain of N point-like beads connected by harmonic springs and neglects both excluded volume and long-range hydrodynamic interactions. Each bead experiences random thermal forces and a Stokes drag, so the chain undergoes overdamped Brownian motion described by Langevin dynamics. Although first proposed for dilute solutions, the model also describes polymer melts whose chain length is below the entanglement threshold.

Description

A flexible polymer is represented by an ideal freely jointed chain of beads with mean bond length l. Neglecting inertia, the overdamped equation of motion for the position of bead n is
where k is the spring constant, the one-bead friction coefficient and random force a zero-mean Gaussian noise that fulfills the fluctuation–dissipation theorem. At either chain end the missing neighbor term is omitted.

Key dynamical properties

Solving the coupled stochastic equations yields several characteristic quantities:

Centre-of-mass diffusion coefficient:, where is the Boltzmann constant, is the absolute temperature, and denotes the total number of beads that make up the ideal freely-jointed chain.
Longest Rouse relaxation time:, where is the mean bond length.
Single-segment mean-squared displacement for lag time τ

This subdiffusive behavior with time dependence is characteristic of Rouse dynamics and distinguishes polymer motion from simple Brownian diffusion.

Extension: The Zimm model

A significant extension was published in 1956 by Bruno Zimm: His model also takes into account *hydrodynamic interactions* between the beads of the chain.
These interactions are forces mediated by the surrounding solvent molecules: when a bead moves, it drags solvent molecules along, which in turn exert a force on adjacent beads. Because of this additional coupling, the Zimm model gives a more realistic description of polymers in dilute solution than the Rouse model and agrees with experimental data for certain dilute-solution polymers.
The Langevin equation of the Rouse model is extended by a tensor, which represents the hydrodynamic force between the -th and -th segments:
Here the tensor depends on the positions of all segments. Consequently, the equation is nonlinear and cannot be solved analytically. Zimm therefore replaced by its equilibrium average, which can be evaluated. From this approximation the following properties of a Zimm polymer are obtained:

Diffusion coefficient of the centre of mass:, where is the solvent viscosity.
Rotational relaxation time:.
Mean-square displacement of a single segment:, where represents the Gamma function.