Zimm Dynamics

This velocity at the ith segment is converted to the local force by multiplying by Adding this additional hydrodynamic force to the Rouse equation (Equation 7.24), we get 

The translational friction coefficient ft of the polymer chain is calculated by defining the net force acting on the chain - as the net frictional force 

Substituting the preaveraged result of the Oseen tensor and performing the normal mode analysis of the Zimm equation for the various Rouse modes, we can calculate the mean square displacement of the center-of-mass of the chain, mean square displacement of a labeled monomer, translational friction coefficient of the chain, and the relaxation times of the various Rouse modes with the Zimm dynamics (Doi and Edwards 1986). The main results of these calculations are the following. 

The mean square displacement of the center-of-mass of the chain obeying Zimm dynamics is the same as the Einsteinian dynamics for sufficiently long times as in Equations 7.9 and 7.10 with the translational friction coefficient given by 

This result follows from Equations 2.3,7.11, and7.42. When the hydrodynamic interaction among segments dominates over the frictional contribution from the segments, the second term in the denominator is much greater than unity so that ft for the Zimm dynamics is 

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B. Zimm. Dynamics of polymer molecules in dilute solutions viscoelasticity, low birefringence and dielectric loss. J Chem Phys 24 269-278, 1956. 

Table 1. Dynamic structure factors for Rouse and Zimm dynamics... 

Fig. 57. Relaxation spectra of the fully labelled star ( , + ) and the star core (, ) at two different Q-values. The solid lines represent the result of a fit for the Zimm dynamic structure factor to the initial relaxation of the fully labelled star. The dashed lines are visual aids showing the retardation of the relaxation for the star core. (Reprinted with permission from [154]. Copyright 1990 American Chemical Society, Washington)...

A feature of theories for tree-like polymers is the disentanglement transition , which occurs when the tube dilation becomes faster than the arm-retraction within it. In fact this will happen even for simple star polymers, but very close to the terminal time itself when very little orientation remains in the polymers. In tree-like polymers, it is possible that several levels of molecule near the core are not effectively entangled, and instead relax via renormalised Rouse dynamics (in other words the criterion for dynamic dilution of Sect. 3.2.5 occurs before the topology of the tree becomes trivial). In extreme cases the cores may relax by Zimm dynamics, when the surroundings fail to screen even the hydro-dynamic interactions between the slowest sections of the molecules. 

This formulation is used in the literature however, see also Eq. 6.48 which yields a better approximation of the Zimm dynamics. 

For the Zimm model the mean-square displacement of monomers is faster [Eq. (8.70)] leading to the logarithm of the Zimm dynamic structure factor scaling as the 2/3 power of time for tq < r < zz-... 

Only a small number of studies have addressed the problem of direct measurements of the dynamics of the surfactant layer in a bicontinuous microemulsion [16, 73, 75, 76]. The Zilman-Granek (ZG) model, which assumes membrane Zimm dynamics on an ensemble of free membrane patches [75] is expected to be applicable to the problem. In the framework of this model the intermediate scattering functions should then be describable by a stretched exponential function of the type... 

For flexible polymers the structural change due to intramolecular motions must be large enough for the light wave to detect the difference between the various molecular shapes. Only under these circumstances will intramolecular interference affect the lightscattering spectral distributions. An extreme example of this case, the Rouse-Zimm dynamic model of the Gaussian coil, is discussed in detail in Section 8.8. 

The depolarized scattering for the Rouse-Zimm dynamical model of flexible polymer chains (cf. Section 8.8) may also be calculated. Ono and Okano (1971) have performed this calculation for q = 0 (zero scattering angle) and find that the scattered light spectral density is a series of Lorentzians each with a relaxation time characteristic of one of the Rouse-Zimm model modes. However the contribution of each mode to the spectrum is equal. This behavior should be contrasted with that of the isotropic spectrum where the scattering spectrum is dominated by contributions from the longest wavelength modes. 

Some possible approximations have been considered by Cates [56], who concentrated attention on macromolecular entanglements, which play an important role in the description of the behaviour of block polymers [86-89]. Cates believes that the fact that the concept of polymer fractal neglects the effects of macromolecular entanglements is the main drawback of this theory. Nevertheless, Cates [56] introduced several simplifications that make it possible to ignore these effects for dilute solutions and relatively low molecular masses. However, in the opinion of Cates, even in the case of predominant influence of entanglements, theoretical interpretation of this phenomenon is impossible without preliminary investigation of the properties of the system in terms of Rouse-Zimm dynamics, which can serve as the basis for a more complex theory. It was assumed [56] that the effects of entanglement can be due to the substantially enhanced local friction of macromolecules. 

M. Troll, K.A. Dill, and B.H. Zimm, Dynamics of poiymer soiutions. 3. An instrument for stress reiaxations on dilute solutions of iarge poiymer moiecuies, Macromoiecuies. 13 436 (1980). 

Rouse-Zimm dynamics in a good solvent. Derive the dependence of the relaxation time T on the chain length N, for a dilute polymer solution in a good solvent. 

The above results are valid only if the time of measurement is longer than the characteristic time for the relaxation of the various Rouse modes of the Zimm chain. The longest relaxation time for a chain with the Zimm dynamics (corresponding to the Rouse mode p = 1), where the hydrodynamic interaction dominates, is called the Zimm time given by... 

The reduced viscosity of a dilute solution of chains with the Zimm dynamics is... 

The results of Equations 7.46, 7.48, and 7.50 for the Zimm dynamics are entirely consistent with the universal laws expected in Section 7.2.1 and are fully supported by experimental data in dilute solutions. If the hydrodynamic interaction among segments is suppressed in the Kirkwood-Riseman-Zimm equation, then the problem reduces to the Rouse dynamics and all results of Section 7.2.2 are recovered. 

Figure 7.4 In dilute solutions, intrachain hydrodynamic interaction dominates (Zimm dynamics). At higher polymer concentrations, chain interpenetration screens the hydrodynamic interaction, resulting in an apparent Rouse dynamics.

By repeating the same calculation as in the derivation of Zimm dynamics, it can be shown (Muthukumar 1996b) that the electrophoretic mobility follows as... 

To simulate polymer solutions, the monomers have to be coupled to the SRD fluid. Different strategies to achieve this aim have been proposed [173,174,176-179,181). Typically, the EOM of the polymers themselves are integrated with the standard MD method. The monomers are included into the collision step, thereby exchanging momentum with the SRD fluid. This is carried out under conservation of the total momentum. Often it is suffldent to treat the monomers as hydrodynamic point sources and to neglect the excluded volume interaction between solvent and monomers [173], This method reproduces hydrodynamic behavior on large length scales and therefore reveals Zimm dynamics for dilute linear polymers in static equilibrium [173, 176, 177]. 

Goupling a stochastic version of LB with standard M D has allowed the reproduction of Zimm dynamics for dilute polymers in static equilibrium [183]. It has been... 

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