Zimm regime

NSE results [40,117], which will be presented in the next section, showed that the situation is more complex. With increasing concentration, instead of the expected single crossover from Zimm to Rouse, indications for an intermediate Zimm regime were observed. 

Fig. 6.25 Relaxation rates as function of Q in semidilute PDMS/toluene solutions. A collective concentration fluctuation seen in normal contrast, C single chain motion as seen in zero average contrast. B Zimm regime of local chain relaxations, equal in both contrasts. (Reprinted with permission from [325]. Copyright 1991 EDP Sciences)...

The transition from the Zimm regime [Eq. (17), [f]] N ] at dilute concentrations to Rouse-like behavior [Eq. (16), [rf N] at the intermediate concentrations is related to the hydrodynamic screening at high concentrations the hydrodynamic interactions between segments become negligible because the local velocity around the beads falls quickly. 

The theoretical approach described before dealt with the short-time dynamic response of the star molecules. However, in the case of completely labelled stars [148] it was found that the line shape of the Zimm model provides a good description of the NSE spectra not only in the short-time regime (t < 5 ns), but also on longer time scales. 

Fig. 59. Incomplete screening of hydrodynamic interactions in semi-dilute polymer solutions. Presentation of different regimes which are passed with increasing concentration. A,C Unscreened and screened Zimm relaxation, respectively, B enhanced Rouse relaxation. (Reprinted with permission from [12]. Copyright 1987 Vieweg and Sohn Verlagsgemeinschaft, Wiesbaden)...

If a residual hydrodynamic interaction over large distances does not exist (l/r H(c) = 0), the regime of screened Zimm relaxation vanishes, and only the crossover from unscreened Zimm to enhanced Rouse relaxation remains. 

Dynamic light scattering, coupled with modern computer programs and auxiliary equipment, automatically graphs Zimm plots and computes Mw, Rg, , Cl, and flow rates. New techniques have expanded the method to the semidilute and concentrated regimes (Barth and Sun, 1991). 

Interestingly, the three-dimensional (3D) network of interconnected cavities described in Reference 22 is a perfeet practical reahzation of the lakes-straits model proposed by Zimm [39] as an early model of gel electrophoresis. To be in this regime, a molecule must span several pores, so DNA molecules significantly larger than those used in the experiments described above would be necessary to reach this regime. It would be interesting to check the validity of the lakes-straits model with such molecules. 

A similar expansion can be obtained for scattering intensity S q,C) oc 1/(1 + q R +...). For very low q, in the Guinier regime qR<, this expansion is the basis for the so-called Zimm plots that are commonly used to determine the radius of a chain and the second virial coefficient of a solution. 

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. 

As the hydrodynamic interaction is screened in semidilute solutions, the molecular weight dependencies of the diffusion coefficient, the longest relaxation time, and the viscosity change in the semidilute solutions are exactly the same as in the Rouse model. However, since the Rouse model was originally designed for an isolated chain, the concentration dependencies of these quantities are not captured by the Rouse model. Nevertheless, we shall refer to the correct description of polymer dynamics in semidilute solutions as the Rouse regime. A summary of the main results for the Zimm model in dilute solutions... 

There are two theoretical models to determine the relaxation time, -zp, and selfdiffusion time of the PE in the solution. One is the Rouse model, which is based on the assumption that the PEs are comprised of mass centered points connected with springs these points only interact with these springs. The other is the Zimm model which is an extension of the Rouse model and takes hydrodynamic effects into account [24, 30]. The Rouse model was used to establish the scaling theory of PEs in solution, which allows an estimation of the PE structure from the viscosity in the different solution regimes. 

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