Reptons

Dependence of viscosity, reduced by the cube of molar mass, on the number of entanglements per chain. Filled circles are data from the Repton model of Section 9.6.2, courtesy of... 

D. Shirvanyants. Open symbols are experimental data for the three polymers in Fig. 9.5, shifted parallel to the rjolM axis to coincide with the Repton model data. The curve is the Doi fluctuation model [Eq. (9.84)] with... 

Repton model of chain motion in a confining tube... 

Figure 9.34 displays the mapping of a chain in an array of fixed obstacles to the repton model. Topological obstacles form a lattice with cell size equal to the tube diameter. Roughly Ar monomers are in cell I, between the end of the chain A and the point B where the chain finally leaves cell I for... 

The dynamics of a single chain in a tube is mapped onto the dynamics of the cluster of reptons along the one-dimensional lattice. To properly describe chain motion along the tube, the motion of reptons must obey a specific set of rules. The reptons can never vacate a site in the middle of the... 

The chain segments corresponding to repton 3 could be in cell 11 or in any cell adjacent to cell IT, but the mapping to reptons always places repton 3 in cell II. 

Thus, each repton contains slightly fewer than Nq monomers ... 

These simple rules for connectivity, order, and motion allow the repton model to be analysed analytically and easily solved numerically. The time dependence of such motion is shown in Fig. 9.35(a) for the extremities of the repton chain. The repton model allows direct visualization of tube length fluctuations. 

The ensemble average utilizes only positive values of. tr(0 — xi t), and averages over many different chains. All negative values of. tr(/) XL(t) are replaced by zeroes. The stress relaxation modulus for a chain of 80 reptons is shown in Fig. 9.35(b). 

Viscosity is determined by integration of the stress relaxation modulus [Eq. (7.117)] Tj = jG(t)dt. The numerically calculated dependence of viscosity on the number of reptons (for the coordination number z — 6) is... 

Carlo simulations of the Evans-Edwards model. The fact that the results of the Evans-Edwards model that does not assume the existence of the tube agree with the results of the repton model validates the concept of the confining tube. 

Repton model results for a chain of 80 reptons, courtesy of D. Shirvanyants. 

Viscosity of linear polymer calculated from Monte Carlo simulations. Filled circles are repton model data from M. Rubinstein, Phys. Rev. Lett. 59,1946 (1987), with data range extended by D. Shirvanyants and open squares are Evans-Edwards model data from J. M. Deutsch and T. L. Madden,... 

Determine the assignment of reptons by starting at the opposite end of the chain in Fig. 9.34. Explain the differences between these assignments and the original assignments. Will they affect the diffusion coefficient ... 

Botrytls cinerea fabae paeoniae reptons spectabilis species... 

An alternative to the continuous Rouse motion inside the tube is the repton model,which represents the chain dynamics by free diffusion of stored length in discrete space. Namely, at each moment of time, the chain is represented by a set of integer variables S, (one for each tube segment i = 1... Z), which records the number of chain beads in the i-th tube segment. At each time step, the segment is chosen at random. If it contains more than one monomer S,>1, one of the monomers can move either to the left or to the right ... 

In order to analyze the tube survival function /r(t), we use end-to-end relaxation function d>(t) which is proportional to fi(t) in mbe and repton models. Figure 24(c) shows the negative derivative of (t) multiplied by we see that this... 

Figure 24 Basic results of repton model mean-square displacement of the middle monomer (filled symbols) and end monomers (open symbols) are shown in panel (a) divided by and in panel (b) divided by Panel (c) shows normalized derivative rf the end-to-end vector, demonstrating the same scaling regimes as the tube model. Panel (d) shows normalized center-of-mass displacement (Q with both axis normalized by If.

In conclusions, the repton model seems to be very similar to the tube model for times much larger than and thus suffers from the same problems as the tube model. Therefore, the criticism of the tube model in this chapter is equally applicable to the repton model. 

A discretized version of the reptation model is proposed. The tube is modeled by a one-dimensional lattice, and the polymer is modeled by a cluster of walkers, called reptons, on this lattice. Each repton represents a part of the chain. Reptons are allowed to hop between neighboring sites in such a way that the cluster always remains connected. The fluctuations in tube length correspond to cluster length fluctuations in the repton model and are not pre-averaged. In the experimentally accessible range of molecular weights M, the repton model predicts the diffusion coefficient and viscosity 0.1 ... 

The subject of this paper is limited to the dynamics of a single entangled polymer chain, just as in the original de Gennes paper on reptation. In a melt or concentrated solution, the dynamics of any chain would be affected by the motion of surrounding polymers (by constraint release), and this effect has to be self-consistently taken into account. In order to do that, one has to start from a reliable model of the single-chain dynamics, such the reptation model, Doi s fluctuation theory,or the repton modeldescribed in the present paper. 

In the following section we review the concepts of the confining tube, the primitive path, and unentangled loops. The repton model is introduced and discussed in detail in the third section. Relations between the repton model and other models are outlined in the fourth section. In the last section we mention some generalizations and possible applications of the repton model to various problems. 

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