Zimm modes

Similar scaling analysis of the mode structure can be applied to the Zimm model. The relaxation time of the pth mode is of the order of the Zimm relaxation time of the chain containing Njp monomers [Eq. (8.25)]  

The index p of the mode relaxing at time t = Xp after a step strain imposed at time t = 0 is obtained by solving the above equation for p  

The number of unrelaxed modes per chain at time t — Zpis p. The stress relaxation modulus is proportional to the number density of chain sections with N/p monomers  

In 0-solvents 1/2), the stress relaxation modulus decays as the - 2/3 power of time, while in good solvents (i/ 0.588) G(t) decays approximately as the - 0.57 power of time. Like the stress relaxation modulus of the Rouse model [Eq. (8.47)], Eq. (8.63) crosses over from kT per monomer at the monomer relaxation time tq to kT per chain at the relaxation time of the chain tz TqN [Eq. (8.25)]. Once again, an excellent approximation to the stress relaxation modulus predicted by the Zimm 

The polymer contribution to the solution viscosity is obtained by integrating [Eq. (7.117)]  

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In contrast to the Rouse modes, which have the dispersion xp 1 p2, the pure Zimm modes (B 0 and Bx/2N/p 1) lead to... 

Figure 64 presents the results of the line-shape analysis for c = 0.18 and c = 0.45. In the first case the polymer relaxation is still determined by the Zimm modes at larger Q-values, while at smaller Q the Rouse modes become dominant. Qualitatively, this behavior is expected for the crossover from unscreened Zimm to enhanced Rouse relaxation. At c = 0.45 the Q-dependence P is... 

The observation that branches A and B in Fig. 6.25 merge at large Q is consistent with the predictions for and T since 6ti and 18.84 deviate from 16 by less than 15% and statistical errors of the experiment and systematic uncertainties in methods to extract the cumulant exceed this difference. In [325] for both the collective concentration fluctuations and the local Zimm modes the observed rates are too slow by a factor of 2 if compared to the predictions with T (the solvent viscosity) and (the correlation length) as obtained from the SANS data. It is suggested that this discrepancy may be removed by the introduction of an effective viscosity qf that replaces the plain solvent viscosity Finally at very low Q, i.e. 1, branch C should level at the centre of mass... 

The depolarized spectrum is dominated by local fluctuations of the optical anisotropy owing to an overall orientation of stiff molecules, local motions of optically anisotropic segments or internal Ronse-Zimm modes for flexible chain molecnles. The depolarized spectrum thus reflects dynamics of collective molec-nlar orientation when the interaction between translation and rotation can be neglected. 

At finite frequencies o> the viscosity increase due to solute coils 8i) = T) — 1), becomes dependent on Mechanical data on this function are difficult to obtain in the dilute regime, but some results have been achieved. Their classical analysis was in terms of Zimm modes, allowing for backflow effects but ignoring any excluded volume effects. We present here a more general scaling picture. 

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