Length 54 Zimm model

Diffusion of flexible macromolecules in solutions and gel media has also been studied extensively [35,97]. The Zimm model for diffusion of flexible chains in polymer melts predicts that the diffusion coefficient of a flexible polymer in solution depends on polymer length to the 1/2 power, D N. This theoretical result has also been confirmed by experimental data [97,122]. The reptation theory for diffusion of flexible polymers in highly restricted environments predicts a dependence D [97,122,127]. Results of various... 

Consistent with the fact that the longest relaxation time of the Zimm model is shorter than the Rouse model, the subdiffusive monomer motion of the Zimm model [(Eq. (8.70)] is always faster than in the Rouse model [Eq. (8.58)] with the same monomer relaxation time tq. This is demonstrated in Fig. 8.8, where the mean-square monomer displacements predicted by the Rouse and Zimm models are compared. Each model exhibits subdiffusive motion on length scales smaller than the size of the chain, but motion becomes diffusive on larger scales, corresponding to times longer than the longest relaxation time. ... 

Explain the length scales over which the reptation, Rouse, and Zimm models describe dynamics in semidilute entangled solutions of linear polymers. 

Within this framework, one can draw a rather simplified picture about the dynamics of the polymers in the entangled state. If the characteristic length scale of a motion is smaller than a, the entanglement effect is not important, and the dynamics is well described by the Rouse model (or the Zimm model if the hydrodynamic interaction is not screened). On the other hand, if the length-scale of the motion becomes larger than a, the dynamics is governed by reptation. 

Note that this Dq is also the center-of-mass diffusion coefficient for the test chain. Dq decreases as N with an increasing chain length. The absolute value of the exponent is much greater compared with the center-of-mass diffusion coefficient of Unear chain polymer in dilute solutions in which Dq N for the Rouse chain and Af 2 for Zimm model in the theta condition. 

Figure 2. A depiction of the bead-and-spring (Rouse Zimm) model for a polymer chain. The chain is represented by n beads and (n-1) springs of length b.

In case of semidilute systems without entanglement (c > c ), in regions above the correlation length, the Rouse motion is faster than the Zimm motion because the Zimm motion is hindered by a coupling between the chains and Rouse dynamics apply [30]. For a detailed derivation of the Rouse and Zimm models, the scaling theory and some older viscosity reviews see references [24, 30, 90]. 

The dynamical behavior of macromolecules in solution is strongly affected or even dominated by hydrodynamic interactions [6,104,105]. Erom a theoretical point of view, scaling relations predicted by the Zimm model for, e.g., the dependencies of dynamical quantities on the length of the polymer are, in general, accepted and confirmed [106]. Recent advances in experimental single-molecule techniques provide insight into the dynamics of individual polymers, and raise the need for a quantitative theoretical description in order to determine molecular parameters such as diffusion coefficients and relaxation times. Mesoscale hydrodynamic simulations can be used to verify the validity of theoretical models. Even more, such simulations are especially valuable when analytical methods fail, as for more complicated molecules such as polymer brushes, stars, ultrasoft colloids, or semidilute solutions, where hydrodynamic interactions are screened to a certain degree. Here, mesoscale simulations still provide a full characterization of the polymer dynamics. 

For the Rouse model, the relaxation times Xp depend on chain length and mode number according to Tp 1/sin pn/2Nm), whereas for the Zimm model the dependence... 

The idea of Kirkwood (25) is combined with the Rouse model by Pyun and Fixman (14). The theory allows a uniform expansion of the bond length by a factor a such as introduced by Flory. The nondiagonal term of the Oseen tensor is considered but only to the first order by a perturbation method. Otherwise, their theory is identical to Zimm s theory in Hearst s version in the treatment of the integral equation (14). 

Theories of gel electrophoretic mobility are usually based on the reptation theory introduced for polymer melts by de Gennes, as well as Doi and Edwards [32,33,7]. Their approach succeeded in explaining the inverse relationship between mobility and chain length for short fragments of DNA [7d], By replacing the tube in the reptation model by open spaces and lakes connected by straights, Zimm explained the antiresonance phenomena observed in the field inversion experiments that occur when the time scale for the formation of the conformational change of the DNA coincides with the time scale for the field cycle [7c]. 

The estimation of g then enables all the quantities in equation (43) to be evaluated, and hence allows a relation for a hypothetical, linear dextran g = 1) to be obtained. By this method, Wales and coworkers found g values which could not be reconciled with theoretical values calculated from the randomly branched model of Zimm and Stockmayer. This led to dextran s being assigned a herring-bone type of structure, that is, a linear backbone having branches of uniform length distributed uniformly along the chain. 

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