Rouse modes normal coordinates

Two other possible segmental motions not depicted in Fig. 6 are in distinctly different time realms. At the slow end of the time scale (greater than milliseconds for reasonable amplitudes) are the Rouse-Zimm normal coordinate modes that result from the collective behavior of units of atoms (beads) along the chain pulling one another and acted on by Brownian forces, solvent frictional resistance, and other parts of the chain (Berne and Pecora, 1976). Librational motions, generally accorded to be of the order of 10 s , may also result from the thermally induced displacements of groups of atoms such wobbling motions have been proposed as important factors in the NMR relaxation of proteins (Howarth, 1979). 

There is some similarity between Ferry s treatment of concentrated systems (14), (123) [eq. (4.4)] and Cerf s just mentioned approach. In both cases the normal coordinate transformation is assumed to be possible along the lines given for infinitely dilute solutions of kinetically perfectly flexible chains (Rouse, Zimm). Only afterwards, different external (Ferry) or internal (Cerf) friction factors are ascribed to the various normal modes. 

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. 

To summarize, the relaxation times (or eigenvalues) of a rather complex system such as a 3-D topologically-regular network end-Unked from Rouse chains were determined analytically. In fact, one can do even better it is possible to construct all of the eigenfunctions of the network analytically (which amounts to the transformation from Cartesian coordinates to normal coordinates). Briefly, to construct the normal mode transformation, see Eqs. 84 and 85, one has to combine the Langevin equations of motion of a network jimction, Eq. 80, and the boundary conditions in the network junctions, Eqs. 87 to 92. After some algebra one finds [25,66] ... 

We finish this section with the schematic drawing displayed in Fig. 6.4, meant to indicate how the time dependent fluctuations of the amplitude of a Rouse-mode could look-like. The interaction of a chain with its surroundings leads to excitations of this mode at random times. In-between, the mode amplitude decreases exponentially with a characteristic relaxation time as described by the equation of motion. These are the only parts in the time dependent curve which show a well-defined specific behavior the excitations occur irregularly during much shorter times. We may therefore anticipate that the shape of the time correlation function is solely determined by the repeated periods of exponential decay. Regarding the results of this section, we thus may formulate directly the time correlation function for the normal coordinate... 

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 ... 

Analysis of the normal modes and transformation of the coordinates leads to another discrete spectrum which differs from that of the Rouse theory only in the values of the relaxation times. Thus, equations 15,16,21, and 26 are unchanged, but the relaxation times are obtained from the eigenvalues of a complicated matrix which includes the pairwise hydrodynamic interactions. - The strength of the latter is measured by either of two parameters ... 

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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