Rouse modes amplitude

Fig. 11 Correlation functions of the Rouse-mode amplitudes for the modes p = 1 - 4 of Gaussian polymers. The chain lengths are = 20 right) and = 40 left). From [73]...

Fig. 13 Correlation functions of the Rouse-mode amplitudes for various modes as a function of the scaled time tp for polymers with excluded volume interactions. The chain lengths are Am = 20 left) and Am = 40 right). The calculated correlations were fitted by Ap exp(—r/vp) and have been divided by Ap. The scaling exponents of the mode numbers are a = 1.93 (Am = 20) and a = 1.85 (Am = 40), respectively. From [73]...

According to Rouse theory (eqn [27]), Ae average mode amplitudes are given by = ( 24(N-I-1)/... 

Figure 6 (a) intermonomer distance d(s) for all unentangled models (b) dimensionless derivative of d(s) from (a) (c) normalized amplitudes of the Rouse modes C . 

As we can see, the motion of a polymer chain in a non-entangled melt, as represented by the Rouse-model, can be described as a superposition of 3ATr linearly independent Rouse-modes, corresponding to Nr modes in x-, y- and 2-directions respectively. In a dynamic equilibrium state all these Rouse-modes become thermally excited and it is instructive to calculate their mean-squared amplitudes. The displacement pattern of mode m is given by... 

According to this result, the amplitudes of the Rouse-modes rapidly decrease with increasing mode order m. If we consider the contributions of the different Rouse-modes to the known total mean squared end-to-end distance, B ) = we find that a large part is already provided by the three lowest order Rouse-modes. Contributions to the end-to-end distance in the 2 -direction, R ) = i o/ 5 come from all z-polarized Rouse-modes with odd m s... 

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... 

Fig. 6.4. Time dependence of the amplitude Zm of a Rouse mode (schematic)...

Introducing the mean-squared amplitudes (X ) and (Z ) of the Rouse modes in thermal equilibrium, as given by Eq. (8.53), leads to... 

To farther characterize the internal dynamics of the molecular chain, a mode analysis in terms of the eigenfunctions of the discrete Rouse model [6,116] has been performed. The mode amplitudes %p are calculated according to... 

Because of hydrodynamic interactions. Rouse modes are no longer eigenfunctions of the chain molecule. However, within the Zimm theory, they are reasonable approximations and the autocorrelation functions of the mode amplitudes decay exponentially, i.e.,... 

Figure 12. (a) Plots of the Rouse amplitude correlation functions for several modes versus time... 

This Rouse stress relaxation time is half of the end-to-end vector correlation time because stress relaxation is determined from a quadratic function of the amplitudes of normal modes (see Problem 8.36). 

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). 

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