Rouse model

The Rouse model is one of the simplest, used in polymer physics model for the dynamics of polymers.

Description

Rouse the model describes the polymer chain of the ideal mass points (often referred to as English beads ), which are connected by springs. The temporal change of the conformation is then realized by Brownian motion of the individual mass points by attacking a random (thermal ) force at each. In this case, no volume exclusion effects are observed, the polymer can thus intersect with itself. The model was proposed in 1953 by Prince E. Rouse.

The Rouse model leads to the following stochastic differential equation ( Langevin equation) for the position of the n- th material point:

Where k is the spring constant of the springs within the model, the coefficient of friction of a mass point ζ in the solvent and N is the number of chain links. The term is in each case the linear restoring force n by the spring to the previous n -1 and the successor 1 ( at the two free ends of the polymer eliminates one of the two terms ). The term describes a random thermal force ( Brownian Molekuardynamik ), which has no preferred direction and is spatially and temporally uncorrelated. For this approach, the following properties of the polymer obtained:

• Rotational relaxation:
• Mean square displacement of a single segment:

Extension: The Zimm model

An important extension was published in 1956 by Bruno Zimm: His model (often referred to simply as " Zimm model " ) also takes into account hydrodynamic interactions between the mass points of the chain. These are interactions (forces ), which are mediated by the solvent molecules to the polymer: the mass points draw the solvent molecules as they move with it, which also leads to a force on adjacent chain links ( see figure at right ). The Zimm model leads to a better description of real polymers as the Rouse model, which agrees well with experimental data for dilute solutions of certain polymers.

The above equation for the Langevin Rouse model is extended by a tensor (matrix ) which is the hydrodynamic force between the n-th and m- th segment:

It should be noted that the tensor depends on the positions of all the segments. Thus, the above Langevin equation is nonlinear and can not be easily solved. Bruno Zimm therefore replaced by its equilibrium middle value can be calculated. This yields the following properties of a Zimm polymer derive:

• Diffusion coefficient of the center of mass: ( ηs: viscosity of the solvent)
• Rotational relaxation:
• Mean square displacement of a single segment:

Experimental observation

In this section, it is worth stressing some real polymers which can best be described as one of the top models:

• Single-stranded DNA is a relatively flexible polymer and shows on short time scales in dilute solution Zimm -like dynamics for individual segments, as demonstrated by fluorescence correlation spectroscopy.
• Double-stranded DNA is significantly stiffer than single-stranded DNA, so the hydrodynamic interactions play a much smaller role, so their Monomerdynamik can be described in dilute solution well with the Rouse model