Repton model

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... 

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. 

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,... 

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 ... 

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. 

The numerically calculated diffusion coefficients of the repton model for 2 = 2. 6 and 12 are compared with the asymptotic expression D N,z) = (2-I)a /(z iV t ) in Fig. 2. This asymptotic behavior is reached relatively quickly (between iV = 30 and 50 reptons). Equation 11 suggests a molecular weight dependence stronger than for the diffusion coefficient... 

The results of the repton model qualitatively agree with the conclusions of Doi, > but represent a more accurate treatment of tube-length fluctuations and are in better agreement with experiments at intermediate molecular weights. The two models probably coincide in the high-molecular-weight region where expansion used by Doi is... 

Fig. 4. Repton model predictions of the viscosity hq as a function of cluster size N. Straight lines are best fits of the higher JV parts of the plot. Their slopes are 3.22 for coordination number 2=2, 3.36 for 2=6 and 3.52 for 2= 12.

M. Rubinstein Yes, it does. In fact, this is one of the main features of the repton model that it includes this type of fluctuation of tube length neglected in the conventional reptation model. 

Fig. 4.4 Illustration of the repton idea, from Ref. 28. (a) A polymer in an entangled net is confined to a tube. Filled circles divide the chain into segments of stored length, (b) Repton model representation of conformation in part (a). Cells of the entanglement net along the confining tube are represented by a one-dimensional lattice. Sections of chain length stored in these cells are modeled by reptons on the lattice sites, (c) Directed walk representation of the same conformation.

Fig. 4.5 Repton model prediction for the viscosity r] as function of the nmnber of reptons per chain N for different values of z, as indicated. The slope varies from 3.22 (z = 2) to 3.52 (z = 12), (from Ref 28).

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