Rouse model normal coordinates

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

The long-range coupling via the flow field which only decreases with 1/r leads to a qualitatively different behavior from that of the Rouse model. Equation (75) is approximately solved by transformation to Rouse normal coordinates. Its solution [6,91] leads to the spectrum of relaxation rates... 

There is an alternative and very direct way to generalize the Rouse-Zimm model for non-Gaussian chains. This approach takes advantage of the expression given by the original theory for the chain elastic potential energy in terms of normal coordinates ... 

Equation (3.C.1) represents the Brownian motions of coupled oscillators. Similar to the discrete case (Eqs. (3.31)-(3.33)), the standard method to solve the differential equation of the continuous Rouse model is to find the normal coordinates, each with its own independent motion. Considering the boundary conditions given by Eq. (3.C.3), we may write the Fomier expansion for Rn(t) in terms of the normal coordinates X, as... 

The elastic dumbbell model studied iu Chapter 6 is both structurally and djmamicaUy too simple for a poljmier. However, the derivation of its constitutive equation illustrates the main theoretical steps involved. In this chapter we shall apply these theoretical results to a Gaussian chain (or Rouse chain) containing many bead-spring segments (Rouse segments). First we obtain the Smoluchowski equation for the bond vectors. After transforming to the normal coordinates, the Smoluchowski equation for each normal mode is equivalent in form to the equation for the elastic dumbbell. 

Let us now study the consequence of the Rouse model. Equation (4.9) represents a Brownian motion of coupled oscillators. A standard way of treating such a system is to find the normal coordinates, each capable of independent motion. It is shown in Appendix 4.II, that in terms of the coordinates Xp defined by... 

First we calculate the stress relaxation using the Rouse model. The stress tensor (eqn 7.4) is expressed by the normal coordinates Xp of the Rouse model (see Section 4,5.2, eqn (4.137))... 

Equation of Motion for the Normal Coordinates in the Rouse Model... 

For g we use the same definition as Eq. 3.133 with I given by Eq. 3.130. It does not mean that h is the same in the Rouse model and the Zimm model. We just use the same formula in the Zimm model to express the random force in the normal coordinates. As Eq. 3.125 is an inverse transform of Eq. 3.118, the following gives an inverse transform of Eq. 3.133 (Problem 3.21) ... 

The Rouse model can be solved by transforming to the normal coordinates Xp t) of the chain. For a discrete monomer chain these are given... 

The 3N -3 normal mode vectors are also called internal mode vectors, because they describe changes in the internal coordinates, the relative positions of the polymer beads. A set of internal coordinates - not to be confused with internal modes - that decompose changes in atomic coordinates into translations, rotations, and internal motions is given by Wilson, et a/. (42) and enhanced by McIntosh, et a/.(43). The Wilson, et al. coordinate decomposition is substantially the same as the Kirkwood-Riseman model, except that Kirkwood and Riseman focus on translation and rotation, while aggregating the internal modes into a fluctua-tional term(44). The Wilson decomposition differs from the Rouse and Zimm decompositions, which identify translations but focus on the internal modes. 

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