Rouse model diffusion coefficient

Figure 14 Master curve generated from mean-square displacements at different temperatures, plotting them against the diffusion coefficient at that temperature times time. Shown are only the envelopes of this procedure for the monomer displacement in the bead-spring model and for the atom displacement in a binary Lennard-Jones mixture. Also indicated are the long-time Fickian diffusion limit, the Rouse-like subdiffusive regime for the bead-spring model ( ° 63), the MCT von Schweidler description of the plateau regime, and typical length scales R2 and R2e of the bead-spring model.

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

Thus the relaxation spectrum resulting from the average coordinates equation11 of our model has the same form as that of Rouse, of Kargin and Slonimiskii, or of Bueche. In order to relate the parameters of the model to those of the Rouse theory, the time scale factor a must somehow be connected to the frictional coefficient for a single subchain of a Rouse molecule. To achieve this comparison, we may23 study the translational diffusion coefficients as computed for the two models. 

When Rgdiffusion coefficient, Df. is described by the Zimin Model (Doi and Edwards, 1986). 

The primitive chain reptates along itself with a diffusion constant that can be identified as the diffusion coefficient of the Rouse model. Under the action of a force /, the velocity of the polymer in the tube is v =f /, where is the overall friction coefficient of the chain. It is expected that C is related to the friction coefficient of the individual segments, Q, by the expression... 

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,... 

Regardless of its complex architecture, any polymer relaxing with no topological constraints and no hydrodynamic interactions is well-represented by the Rouse model, with friction proportional to molar mass. To estimate the terminal response of randomly branched polymers, we apply this reasoning to the characteristic polymers, with size consisting of N monomers. The diffusion coefficient of these chains is given by the... 

The Rouse model is the simplest molecular model of polymer dynamics. The chain is mapped onto a system of beads connected by springs. There are no hydrodynamic interactions between beads. The surrounding medium only affects the motion of the chain through the friction coefficient of the beads. In polymer melts, hydrodynamic interactions are screened by the presence of other chains. Unentangled chains in a polymer melt relax by Rouse motion, with monomer friction coefficient C- The friction coefficient of the whole chain is NQ, making tha diffusion coefficient inversely proportional to chain length ... 

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)]. 

At times longer than the Rouse time tr, all monomers move coherently with the chain. The chain diffuses along the tube, with a curvilinear diffusion coefficient given by the Rouse model Dg... 

Constraint release has a limited effect on the diffusion coefficient it is important only for the diffusion of very long chains in a matrix of much shorter chains and can be neglected in monodisperse solutions and melts. The effect of constraint release on stress relaxation is much more important than on the diffusion and cannot be neglected even for monodisperse systems. Constraint release can be described by Rouse motion of the tube. The stress relaxation modulus for the Rouse model decays as the reciprocal square root of time [Eq. (8.47)] ... 

The diffusion properties in Region I are well described by the Rouse model, which predicts the self-diffusion coefficient will scale as 1/N, the number of monomeric units. Applied to the two lipopolymers of interest, the Rouse model predicts the ratio... 

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... 

Relaxation 67,70,96,99,111,155 Reptation model 1,24,42 Resolution 14 Resonance NSE 20 Rheology 35,55 Rotational isomeric state 118 Rotational transitions 117 Rouse diffusion coefficient 28,42, 175 Rouse model 24-26,30-35,38, 117, 119, 142, 193,200 —, generalized 47 Rouse time 27 RPA 162, 163, 199... 

The Rouse model was initially designed to treat the dynamics of polymers in very dilute solutions [1]. Ironically, however, it turned out that dilute solutions are not appropriate systems for it. Indeed, in the Rouse model the maximal relaxation time, Tchain> and the diffusion coefficient, I>chain> scale with the molecular weight, M, as and M respectively (see Eqs. 57 and 61). Furthermore, it is a straightforward matter to demonstrate that for the Rouse model at = 0 the zero shear viscosity [ ] (0)] is proportional to M, see Eq. 22. All these theoretical findings disagree with the experimental data... 

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... 

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