Rouse limit

Fig. 5.3. Log-log plot of the self-diffusion constant D of polymer melts vs. chain length N. D is normalized by the diffusion constant of the Rouse limit, DRoUse> which is reached for short chain lengths. N is normalized by Ne, which is estimated from the kink in the log-log plot of the mean-square displacement of inner monomers vs. time [gi (t) vs. t]. Molecular dynamics results [177] and experimental data on PE [178] are compared with the MC results [40] for the athermal bond fluctuation model. From [40]...

Chain-Rigidity and Rotationai Transitions as Rouse Limiting Processes... 

Therefore, in the free-draining. Rouse limit, the molecular weight dependence of [ij] is the same as that of . 

The Rouse limit is encountered when the chain is sufficiently short that v q) 1 in Eqs. (3.1.6), so that we have l q) C In this case we get for the open chain from (3.1.11)... 

It is noteworthy that in the Rouse limit the diffusion constant is independent of the coil s size.) From the preceding, in the long-time limit, the characteristic time ti/2 of the incoherent dynamic structure factor exp[- Q B(0, t)] is linked to Q by the power law... 

Just as in the Rouse limit the dominant term contributing to v q) in Eqn. (3.1.9) is unity, in the Zimm limit the sum over k is much larger than unity. Going to the integral for g 1 and substituting with its unperturbed value kf, we have [82]... 

The correlation function B(k, t) cannot be evaluated in closed form in the present case. However, for t comprised between T ,j and numerical calculations [54] show that B(0, t) cc whereas, for t x q), B(0, t) oc t due to globule diffusion. Consequently, the exponent of the relationship hiiQ = const has values 3 and 2 in the two respective regimes, as with the unperturbed chain [see Eqs. (3.2.15 ) and (3.2.16)]. We do not obtain any plateau of B 0,t) in this case, unlike the Rouse limit. Figure 7 shows the coherent dynamic structure factor S Q, t) as a function of t for two different Q values both for the unperturbed and for the collapsed chain. The two Q s correspond to observation distances /Q [ref. 15, note 6] just below and... 

The Rouse limit applies to unentangled polymer melts because hydrodynamic interactions are screened in melts (just as excluded volume interactions are screened in melts). Polymer dynamics in the melt state (with no solvent) are described by the Rouse model, for short chains that are not entangled. 

The diffusion constant is, in the Rouse limit, inversely proportional to the molecular mass, this being true even when excluded volume effects are incorporated. 

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