Reptation, macromolecule

One can see that the approximation of the theory, based on the linear dynamics of a macromolecule, is not adequate for strongly entangled systems. One has to introduce local anisotropy in the model of the modified Cerf-Rouse modes or use the model of reptating macromolecule (Doi and Edwards 1986) to get the necessary corrections (as we do in Chapters 4 and 5, considering relaxation and diffusion of macromolecules in entangled systems). The more consequent theory can be formulated on the base of non-linear dynamic equations (3.31), (3.34) and (3.35). 

A very elegant linear model of reptating macromolecules was proposed by Doi and Edwards [63], This model can be considered as a limiting case of the above model of Curtiss and Bird [47] at a oo. It was proposed that, in this case of limiting anisotropy, the macromolecule moves inside the tube of radius... 

M. Antonietti, T. Pakula, and W. Bremser. Rheology of small spherical polystyrene mlcrogels A direct proof for a new transport mechanism in bulk polymers besides reptation. Macromolecules, 28 (1995), 4227-4233. 

M. Cates. Reptation of living polymers Dynamics of entangled polymers in the presence of reversible chain-scission reactions. Macromolecules 20 2289-2296, 1987. 

R. Granek. Stress relaxation in polymer melts and solutions Bridging between the breathing and reptation regimes. Macromolecules 2<5 5370-5371, 1995. 

Diffusion of flexible macromolecules in solutions and gel media has also been studied extensively [35,97]. The Zimm model for diffusion of flexible chains in polymer melts predicts that the diffusion coefficient of a flexible polymer in solution depends on polymer length to the 1/2 power, D N. This theoretical result has also been confirmed by experimental data [97,122]. The reptation theory for diffusion of flexible polymers in highly restricted environments predicts a dependence D [97,122,127]. Results of various... 

Obviously, the enthalpy gain can compensate for some unfavorable change in entropy. At a suitable value of adsorption energy, flexible macromolecules may reptate into narrow pores. Polystyrene of molar mass 173,000 g/mol and coil diameter 30 nm is reported to be capable of entering pores of silica gel with 10 nm average pore diameter8). This penetration will certainly be a slow process. It often has been observed that the amount of polymer adsorbed by porous adsorbents slowly increases over a protracted period of time. This may be due to similar effects. 

From this point of view the Doi-Edvards reptation theory can be regarded as the most perfect network theoryS2). In a molten polymer, macromolecules can not move notable in lateral direction since that is impeded by other polymer chains. This circumstance in the Doi-Edvards theory is taken into account by means of introduction of a... 

In description of effects observed in extension of molten polymers, the determinant is the phenomenon of anisotropy of the mobility of macromolecules. In the Doi-Ed-vards reptation theory the anisotropy of the mobility of macromolecules is taken into account topologically by means of placing a macromolecule into a certain hypothetical tube. In this case large-scale movements are allowed only along the macromolecule and are totally inhibited in the lateral direction. This, indeed, is a limiting case of mobility anisotropy. 

During gel electrophoresis, the migration of macromolecules is obstructed by the polymer matrix, and thus depends on the molecular weight as well as the frictional coefficient with the matrix. In the reptation model, DNA is assumed to move in a worm-like fashion through virtual tubes in the gel polymer matrix (Fig. 9.1A). The central result is that the electrophoretic mobility depends on the mean-square end-to-end distance of the macromolecule, and inversely on length (L) or molecular weight. The electrophoretic mobility is expressed as... 

Figure 9.1 Models for macromolecular electrophoresis. (A) Reptation of long DNA fragments through a polyacrylamide gel. Redrawn from Bloomfield et al. (2000). (B) Ogston sieve model, which applies when Rg of the macromolecule is smaller than the diameter of the pore. (C) Scanning electron micrograph of the interior of a 7.5% (w/v) polyacrylamide gel. Reprinted from Yuan et al. (2006) with permission.

The model described by equations (3.42)-(3.45) is valid for equilibrium situations. For chain in a flow, one ought to define displacements of the particles under flow and to consider the average values (3.44) to depend on the velocity gradient (Doi and Edwards 1986). McLeish and Milner (1999) considered mechanism of reptation motion of branched macromolecules of different architecture. 

These equations describe the reptation normal relaxation modes, which can be compared with the Rouse modes of the chain in a viscous liquid, described by equation (2.29). In contrast to equation (2.29) the stochastic forces (3.47) depend on the co-ordinates of particles, equation (3.48) describes anisotropic motion of beads along the contour of a macromolecule. 

There were different generalisations of the reptation-tube model, aimed to soften the borders of the tube and to take into account the underlying stochastic dynamics. It seems that the correct expansion of the Doi-Edwards model, including the underlying stochastic motion and specific movement of the chain along its contour - the reptation mobility as a particular mode of motion, is presented by equations (3.37), (3.39) and (3.41). In any case, the introduction of local anisotropy of mobility of a particle of chain, as described by these equations, allows one to get the same effects on the relaxation times and mobility of macromolecule, which are determined by the Doi-Edwards model. 

Derived from linear approximation of the equations (3.37), the equilibrium correlation function (4.29), defines two conformation relaxation times r+ and r for every mode. The largest relaxation times have appeared to be unrealistically large for strongly entangled systems, which is connected with absence of effect of local anisotropy of mobility. To improve the situation, one can use the complete set of equations (3.37) with local anisotropy of mobility. It is convenient, first, to obtain asymptotic (for the systems of long macromolecules) estimates of relaxation times, using the reptation-tube model. 

Each point is calculated as the asymptotic value of the rate of relaxation for large times (see examples of dependences in Fig. 6) for a macromolecule of length M = 25Me (x = 0.04, B = 429, ij) = 8.27) with the value of the coefficient of external local anisotropy ae = 0.3. The dashed lines reproduce the values of the relaxation times of the macromolecule due to the reptation-tube model. The labels of the modes are shown at the lines. Adapted from Pokrovskii (2006). 

A particular choice of the coefficients ae = 0.3 and a = 0.06 determines the value T = 417 r for the relaxation time of the first mode, which is close to the reptation relaxation time 370 r. The calculated relaxation times of the third mode 73 = 315 r is a few times as much as the corresponding reptation relaxation time 41.1 r, which indicates that the dependence of the relaxation times on the mode label is apparently different from the law (4.36). It is clearly seen in Fig. 7, where the dependence of the relaxation times of the first six modes of a macromolecule on the coefficient of internal anisotropy is shown. The relaxation times of different modes are getting closer to each other with increase of the coefficient of internal anisotropy. The values of the largest relaxation time of the first mode for different molecular weights are shown in Fig. 8. The results demonstrate a drastic decrease in values of the largest relaxation times for strongly entangled systems induced by introduction of local anisotropy. 

One can see that the diffusion coefficient of the macromolecule due to reptation does not depend on the length of the ambient macromolecules... 

The reptation diffusion is connected with the local anisotropy of mobility of particles, which can be confirmed by investigation of equations (3.37). As an example, Fig. 11 contains the results for displacement of a macromolecule of length M = 25Me (value of parameter x = 0.04) due to numerical integration... 

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