Reptation model diffusion constant

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... 

Figure 5.14 shows high-pressure isobars for the self-diffusion data of the n-alkanes in a plot of log D against log where is the molecular weight. These are the first data obtained with the titanium autoclave described in Section 1.4.2. Such results are commonly described by the Rouse model or by the reptation model, which both predict a linear correlation in this type of plot at constant pressure and temperature this linear correlation is clearly established in Fig. 5.14. Judging from the chain length of the polymethylenes the Rouse model should apply. This model predicts, that D should be proportional to M while the experiments give a D correlation, which is...

This result shows that the diffusion constant of a long polymer chain in a concentrated system, because of the constraint effect of entanglement, is inversely proportional to the square of the molecular weight. This molecular-weight dependence is distinctively different from the result, Dg oc M, given by the Rouse model (Chapter 3) and its observation is often regarded as the indication of the reptational motion. As shown in... 

It must also be realized that for very large Ni, reptation itself is not the dominant mobility process for the long chain. For Nj - < we can think of the V chains as forming a solvent of comparatively small molecules, with a certain viscosity tjat- This viscosity is discussed in the next section, and, in the reptation model, it scales like tik = Then the diffusion constant is given by the Stokes-Einstein equation... 

The reptation model, like the Rouse model, supposes that the friction involved in dragging the chain through its tube is proportional to the chain length, 5 = N i, Equation (33.33). The diffusion constant Dtubo for the chain moving through the tube is given by the Einstein-Smoluchowski relation, Equa-... 

The reptation model also predicts a second type of diffusion constant. The diffusion constant Dtube cannot be readily measured because experiments cannot track how the chain moves along its tube axis. But you can measure a diffusion constant D that describes how the center of mass of the polymer chain moves in space over time. In this case, the average distance moved is (x-) oc Rj, where = Nb . The reptation model Equation (33.38) predicts that chain diffusion slows as the square of the chain length. 

Experiments give D oc N - , which differs somewhat from this reptation model prediction of D oc N (see Figure 33.12). In contrast, the self-diffusion constant for a Rouse chain, which is based on t oc (Equation (33.34)), is... 

Fig.4 In the biased reptation model, the biased walk of the chain in its tube, which creates new tube sections, is similar to the motion of a point-like particle between two absorbing walls. A biased Jump ends when the molecule has migrated over a distance a along the tube axis, i.e., when it has reached the next point defining the end of the next pore. This process is similar to the absorbtion of the particle by one of the walls, each at distance a from the starting position. The particle and the chain both have a one-dimensional velocity and diffusion constant D.

We have already mentioned that, for most purposes, a semi-dilute solution can be viewed as a melt of blobs of size (, each containing g = c monomers. It is thus reasonable to assume that the reptation model that we just described can be applied to a semi-dilute solution. The tube diameter is proportional to the blob size in this model. The overall contour length of the tube is (N )/g. The local mobility Pq is the blob mobility Pq = l/ 6nrjQ ) and the tube diffusion constant is — (Tg)/ (6nrjQ N). This gives a reptation time for a polymer chain in a semi-dilute solution... 

Ito s model [68] bears resemblances to the model of Ref. [35], but is different by two aspects. Firstly, it assumes that the constant rate of the chain termination depends on the number of monomeric units (so-called polymerization degree) of tn and n radical chains taking part in the termination reaction and represents the sum of the independent contributions of m and n. Secondly, the dependence of the chain termination constant on the length of chains under two types of conditions is described the first condition is < n, controlled by segmental diffusion, and the second one is m > controlled by the reptation diffusion. In the reptation chemical mechanism of diffusion in the deep states of conversion the macroradicals move snake-like between the network joints. De Gennes connected a reptative moving of macroradicals with the dynamic properties of the medium with the use of scaling ratios [37-40] as applied in Refs. [41-46] for the description of constant chain termination in the late conversion state. 

In any case, whatever the model, since tube-renewal is more important (compared to reptation) and accelerates the motion more efficiently for short chains than for long chains, it introduces additional molecular weight dependences of the dynamical quantities, and certainly contributes to the experimental deviation of the viscosity/molecular weight exponent from the reptation value 3. All treatments, including tube renewal, exhibit such deviations which vanish for asymptotically long chains. Detailed quantitative tests are, however, very difficult to perform when tube-renewal is taken into account, polydispersity becomes an essential parameter (the shortest mechanism, reptation and tube-renewal, dominates the relaxation process). No complete set of experiments, either for diffusion or for viscoelasticity, with constant polydispersity at all molecular weights, are presently available. 

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