Diffusion coefficient 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... 

But p decreases with salt concentration with an apparent exponent of k which changes from 0 at low salt concentration to — at high salt concentrations. The N-independence of p arises from a cancellation between hydrodynamic interaction and electrostatic coupling between the polyelectrolyte and other ions in the solution. It is to be noted that the self-translational diffusion coefficient D is proportional to as in the Zimm model with full... 

From the Einstein relation [Eq. (8.4)] the diffusion coefficient of a chain in the Zimm model is reciprocally proportional to its size R ... 

In contrast, the Zimm model considers the motion of beads (or monomers) to be hydrodynamically coupled with other monomers. Both the polymer and the solvent molecules within the pervaded volume of the chain move together in dilute solutions. The diffusion coefficient of a chain in the Zimm model is of the same form as the Stokes-Einstein relation [Eq. (8.9)] for diffusion of a colloidal particle in a liquid ... 

During their relaxation time r, polymers diffuse a distance of order their own size (r R /D). The relaxation times of the Rouse and Zimm models are then easily obtained from the diffusion coefficients ... 

Compare this Zimm diffusion coefficient Dz with the Rouse diffusion coefficient Dr of part (ii). Hint. The viscosity of an unentangled melt of shorter /Vg-chains is predicted by the Rouse model [Eq. (8.53)]. 

The zero mode is the self-diffusion of the center of mass whose diffusion coefficient is given by the Stokes-Einstein relation D = k TIN. The time Tj will be proportional to the time required for a chain to diffuse an end-to-end distance, that is, R )/D = t N b lk T. This means that for time scales longer than Tj the motion of the chain will be purely diffusive. On timescales shorter than Tj, it will exhibit viscoelastic modes. However, the dynamics of a single chain in a dilute solution is more complex due to long-range forces hydrodynamic interactions between distant monomers through the solvent are present and, in good solvents, excluded volume interactions also have to be taken into account. The correction of the Rouse model for hydrodynamic interaction was done by Zimm [79]. Erom a mathematical point of view, the problem becomes harder and requires approximations to arrive at some useful results. In this case, the translational diffusion coefficient obtained is... 

In the Zimm model (see Fig. 2A) the hydrodynamic interactions are included by employing the Oseen tensor Him the tensor describes how the mth bead affects the motion of the /th bead. This leads to equations of motion that are not Unear anymore and that require numerical methods for their solution. In order to simplify the picture, the Oseen tensor is often used in its preaveraged form, in which one replaces the operator by its equiUb-rium average value [5]. For chains in -solvents, this leads for the normal modes to equations similar to the Rouse ones, the only difference residing in the values of the relaxation times. An important change in behavior concerns the maximum relaxation time Tchain> which in the Zimm model depends on N as and implies a speed-up in relaxation compared to the Rouse model. Accordingly, the zero shear viscosity decreases in the Zimm model and scales as Also, in the Zimm model the diffusion coefficient... 

Zimm molecular mass dependencies of the maximal relaxation time, of the diffusion coefficient, and of the zero shear viscosity are all consistent with the experimental findings [3]. The Zimm model also agrees with the experiment with respect to the frequency dependence of the storage and loss moduli of dilute polymer solutions under 0-conditions both G (co) and G" co) show... 

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. 

The self-diffusion coefficient is used to describe the center-of-mass motion for a simple liquid. It can also be used in connection with the Rouse-Zimm model to describe the behavior of a long chain. The determination of diffusion coefficient D... 

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... 

The above results for the Rouse model are applicable to the experimental conditions where the hydrodynamic and excluded volume interactions and the entanglement effects can be completely ignored. We shall identify such an experimental regime later on. Now, we attempt to incorporate the hydrodynamic interaction in describing the chain dynamics in infinitely dilute solutions. The Rouse chain model incorporating the effect of hydrodynamic interaction is called the Kirkwood-Riseman modeF or Zimm model. These models differ from each other in certain subtle features and the numerical prefactors only the predicted molecular weight dependence of the longest relaxation time, viscosity of the solution, diffusion coefficient, etc. are the same. 

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. 

The Kirkwood formula neglects hydrodynamic flucmations and is thus identical with the preaveraging result of the Zimm approach. When only the hydrodynamic part is considered, the Zimm model yields the diffusion coefficient... 

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