Diffusion coefficient curvilinear

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... 

The primary mode of motion of a linear chain along its confining tube is reptation, first proposed by de Gennes. Reptation is a snake-like diffusion of a chain, as a whole, along the contour of its tube, with a Rouse curvilinear diffusion coefficient. The relaxation time of the melt is the time it takes the chain to reptate out of its original tube, called the reptation time Trep. The reptation time and the viscosity of entangled polymers are... 

Consider a segment 5 of the tube of an entangled A-mer at time t = 0. Assume that the chain moves along its tube by simple diffusion with curvilinear diffusion coefficient D. ... 

In the semidilute solution, the hydrodynamic interactions are shielded over the distance beyond the correlation length, just as the excluded volume is shielded. We can therefore approximate the dynamics of the test chain by a Rouse model, although the motion is constrained to the space within the tube. In the Rouse model, the chain as a whole receives the friction of N, where is the friction coefficient per bead. When the motion is limited to the curvilinear path of the primitive chain, the friction is the same. Because the test chain makes a Rouse motion within the tube, only the motion along the tube survives over time, leading to the translation of the primitive chain along its own contour. The one-dimensional diffusion coefficient for the motion of the primitive chain is called the curvilinear diffusion coefficient. It is therefore equal to Dq of the Rouse chain (Eq. 3.160) and given by... 

The process of disentangling, as it is envisaged in the reptation model, is sketched in Fig. 6.11. The motion of the primitive chain , the name given to the dynamic object associated with the primitive path, is described as a diffusion along its contour, that is to say, a reptation . The associated curvilinear diffusion coefficient can be derived from the Einstein relation, which holds generally, independent of the dimension or the topology. Denoting it D, we have... 

The Einstein relation, already employed in Eq. (6.121) to write down the curvilinear diffusion coefficient in the tube, gives us, when used as normally, also the diffusion coefficient of a polymer in a melt, provided there are no entanglements. We call it D and have... 

The mobility of the whole chain, free to move along the curvilinear axis of its tube is iV/iV jsmaller than that of one bead (the friction on the full necklace is the ft iction on one bead times the number of beads), leading to a curvilinear diffusion coefficient... 

Fig. 24. Illustration of the whip model of chain-end dynamics. A chain fold that happens to be near a chain end consists of a relatively immobile strand towards the middle of the chain left drawn in dark gray) and a chain section on the right (drawn in light gray) that can quickly migrate back and forth. Its curvilinear length between the chain end and the tip of the fold is decreased when it is displaced towards the tip of the fold and is increased when it moves in the chain-end direction. As a consequence, the curvilinear diffusion coefficient is increased or decreased, respectively, depending on the displacement direction. Therefore, motions towards the tip of the fold are favored, and the fold tends to unroll not unlike the cord of a cracking whip...

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