Diffusion coefficient Rouse

The viscosity relates to the longest relaxation time in a system. If we consider Rouse diffusion along the tube with a Rouse diffusion coefficient DJ l/ NQ) then an initial tube configuration is completely forgotten when the mean-square displacement along the tube fulfils (r (t))tube=(contour length ly. Thus, for the longest relaxation time, we obtain ... 

At low momentum transfer A2 describes the translational Rouse diffusion coefficient of the whole diblock, considering N l-f) segments exerting the friction Q and A[fsegments exerting the friction In the high Q hmit, RPA predicts... 

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)]. 

Body force of component i Polymer mass concentration Macroscopic diffusion coefficient Cooperative diffusion coefficient Stokes-Einstein diffusion coefficient Rouse diffusion coefficient Self diffusion coefficient... 

The above consideration suggests that dynamic entanglements per chain are few even in concentrated solutions. Hence, in timescales comparable to the motion of individual chains in such a system may be hardly affected by topological constraint and essentially Rouse-like. Even so, it should differ from the motion of a Rouse chain in dilute solution, because each chain drags another through dynamic entanglements. Thus, Dg for polymer concentrates may be expressed by the Rouse diffusion coefficient Dg — k T/ N if C is corrected for the dragging effect. Skolnick et al. assumed to be proportional... 

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 Doi-Edwards theory assumes that reptation is the dominant mechanism for conformational relaxation of highly entangled linear chains. Each molecule has the dynamics of a Rouse chain, but its motions are now restricted spatially by a tube of uncrossable constraints, illustrated by the sketch in Fig. 3.38. The tube has a diameter corresponding to the mesh size, and each chain diffuses along its own tube at a rate that is governed by the Rouse diffusion coefficient (Eq. (3.37)). If the liquid is deformed, the tubes are distorted as in Fig. 3.39, and the resulting distortion of chain conformations produces a stress. The subsequent relaxation of stress with time corresponds precisely to the progressive movement of chains out of the distorted tubes and into random conformations by reptation. The theory contains two experimental parameters, the unattached mer diffusion coefficient T>o... 

Typical dynamic properties like the scaling of relaxation times, e.g., ri, or diffusion coefficient with N are found in the simulation to change systematically from typical Rouse-like behavior Dj< ocN, t oc to... 

MC simulations and semianalytical theories for diffusion of flexible polymers in random porous media, which have been summarized [35], indicate that the diffusion coefficient in random three-dimensional media follows the Rouse behavior (D N dependence) at short times, and approaches the reptation limit (D dependence) for long times. By contrast, the diffusion coefficient follows the reptation limit for a highly ordered media made from infinitely long rectangular rods connected at right angles in three-dimensional space (Uke a 3D grid). 

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-... 

Therefore, remarkably, the coupled diffusion coefficient becomes independent of N and c in the Rouse regime of salt-free polyelectrolyte solutions. This is to be... 

Rouse rate at 470 K is 8100 AVns. The corresponding mode-dependent characteristic times are represented by the dashed-dotted line in Fig. 5.1. Taking into account the relationship between Dr and the corresponding diffusion coefficient Dr becomes Dr=1.22 0.17x10" mVs. 

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). 

In 2% agarose gel, the diffusion coefficient of linear DNA is proportional to power functions of N0 (Figure 20.4), as predicted by Eqs. 7 and 8. However, the experimental data are slightly different from theoretical predictions. The exponents of power functions are -0.52 and -1.55, rather than -0.5 and -2, in Rouse and reptation regimes, respectively (Pluen et al., 1999). 

We are in the process of examining time dependent properties such as the diffusion coefficient and the relaxation times of various correlation functions as a function of both chain length and density. Following the work of Kirkwood (24) and Rouse (24) we assume that the velocity of the polymer is proportional to the forces acting on it at any time this is the high-viscosity limit in which inertial terms are neglected. Neglecting also hydrodynamic forces, we then have for the velocity of the jth bead at time t... 

---