Rouse chains

Muthukumar and Winter [42] investigated the behavior of monodisperse polymeric fractals following Rouse chain dynamics, i.e. Gaussian chains (excluded volume fully screened) with fully screened hydrodynamic interactions. They predicted that n and d (the fractal dimension of the polymer if the excluded volume effect is fully screened) are related by... 

We can consider the friction coefficient to be independent of the molecular weight. At times less than this or at a frequency greater than its reciprocal we expect the elasticity to have a frequency dependence similar to that of a Rouse chain in the high frequency limit. So for example for the storage modulus we get... 

The applied strain is affine and the whole of the tube is deformed along with the polymer. As the strain is infinitesimally small the contour length is unaltered. At very short times / after the strain is applied, t < re tr the stress is relaxed as a Rouse chain. At short times we can make an approximation and replace our sum by an integral ... 

The form of the time-dependence can be understood from the anomalous diffusion of a piece of Rouse chain, which displaces in time such that rather than t. As a result the exp(-k Dt) scattering from Fickian diffusers is replaced by The initial structure factor S(k,0) is... 

Early-time motion, for segments s such that UgM(s)activated exploration of the original tube by the free end. In the absence of topological constraints along the contour, the end monomer moves by the classical non-Fickian diffusion of a Rouse chain, with spatial displacement f, but confined to the single dimension of the chain contour variable s. We therefore expect the early-time result for r(s) to scale as s. When all prefactors are calculated from the Rouse model [2] for Gaussian chains with local friction we find the form... 

Compared to the available clear-cut comparisons between NSE results and simulations for short Rouse chains, the situation in the entangled regime is less satisfactory. The molecular dynamics (MD) simulation of ensembles of suffi-... 

A multicomponent dynamic RPA for incompressible diblock copolymer mixtures was developed by Akcasu [270]. The details of the approach may be found in the above reference. Here we recall some basic ideas dealing with the calculation of the dynamic structure factor S Q,t) for Rouse chains. 

The form of the spectrum is the same for Rouse chains fixed at one or both ends, although t, in these cases is no longer given by Eq. (4.32). This is also the spectrum at long times for various models with explicit forms for the local motions (91,92). The Rouse spectrum can also be represented by a continuous function for times which are small compared to xx. For N- 00 ... 

Chompff and Duiser (232) analyzed the viscoelastic properties of an entanglement network somewhat similar to that envisioned by Parry et al. Theirs is the only molecular theory which predicts a spectrum for the plateau as well as the transition and terminal regions. Earlier Duiser and Staverman (233) had examined a system of four identical Rouse chains, each fixed in space at one end and joined together at the other. They showed that the relaxation times of this system are the same as if two of the chains were fixed in space at both ends and the remaining two were joined to form a single chain with fixed ends of twice the original size. 

The failure of the Rouse theory was attributed to the pathological nature of medium motions in entangled systems, and not any special defect in the Rouse representation of the polymer chain itself. For Rouse chains in a deforming continuous medium, the frictional force depends on the systematic velocity of the bead relative to the medium. The frictional force on a bead is therefore a smootly... 

As for the multidimensional freely jointed chain, it is possible to relate a to the parameters which describe a Rouse chain by evaluating the translational diffusion constant D for the center of mass. In the stochastic model, we determine the square of the displacement per unit time of a single bead averaged over an equilibrium ensemble. For bead j,... 

The first two of the above equations represent Langevin equation for the Rouse chain in presence of extra random force 4>f. One can note that dynamics of a polymer chain in random fields was studied extensively (Baumgarter and Muthukumar 1996 Ebert et al. 1996), as a possible mode of motion of a macromolecule in entangled system. Note also that the two top equation... 

The short chains with Mrelaxation times on the order of the experimental time-scale. For example, the Rouse times of PSD3 and of PSDIO are approximately 9s and 19s seconds, respectively at 115°C [35,36]. Consequently, the residual orientation at long times for these chains can be attributed to orientational coupling interactions with the long chains of the polymer matrix. Similar orientational correlations have been observed on various systems by NMR spectroscopy studies of stretched elastomers where even dissolved solvent molecules and free chains were shown to possess a very short-length scale local orientation [37]. 

All the data confirm that the tube can be viewed as a Rouse chain (x e but with an elementary time which is not the reptation time of the passing... 

The three parameters necessary for the characterization of the primitive chain, L, D, and a can be expressed in terms of the Rouse model parameters N, b, and Thus D is given by Eq. (11.14), while La is equal to Nb, the mean square end-to-end distance of the Rouse chain. As a result, the length of the primitive chain can be written as... 

If constraint release were the only process for conformational rearrangement, the initial path motions would be the same as the chain motions of the N-element Rouse model. Equation 4 relates the longest relaxation time Ti to the diffiision coefficient for Rouse chains. The diffusion cxieffident from constraint release is gjven by Eq. 90 with Ns = N. With Eqs. 1, 2, 4 and 9,... 

The longest relaxation time for a Rouse chain of length Lo can be written 1... 

The total friction coefficient of the whole Rouse chain is the sum of the contributions of each of the N beads ... 

The viscous frictional force the chain experiences if it is pulled with velo-city V is —NCv. The diffusion coefficient of the Rouse chain is obtained from the Einstein relation [Eq. (8.4)]. 

For oscillatory shear, Eqs (7.149) and (7.150) allow calculation of the storage and loss moduli of a solution of linear Rouse chains (see Problem 8.14) ... 

On length scales larger than the correlation length but smaller than the tube diameter a, hydrodynamic interactions are screened, and topological interactions are unimportant. Polymer motion on these length scales is described by the Rouse model. The relaxation time Tg of an entanglement strand of monomers is that of a Rouse chain of N jg correlation volumes [Eg. (8.76)] ... 

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