Unentangled loops

Consider the dynamics of an entangled ring polymer in an array of fixed obstacles [Fig. 9.40(a)]. The ring is not permanently trapped by the obstacles, but is able to diffuse. The ring does not have free ends and, therefore, classical snake-like reptation is not expected for it. An ideal untrapped ring polymer in an array of fixed topological obstacles is an unentangled loop formed by double-folded strands of Ne monomers each, similar to an arm of a star at the moment of complete retraction. 

Thus, most of the monomers are in the side branches, corresponding to the unentangled loops. 

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. 

II. CONFINING TUBE, PRIMITIVE PATH, AND UNENTANGLED LOOPS... 

The physical significance of separating the unentangled loops and the primitive path is that the modes related to them have very different time scales. The relaxation times of unentangled loops are very short and are independent of the chain molecular weight M because the motion of these loops is not severely restricted by topology. The modes, related to the primitive paths, relax much slower and control the important long-time properties, such as viscosity and diffusion coefficient. 

In the following section we calculate these long-time properties from the simplest assumptions about the short-time dynamics of unentangled loops. These assumptions are based on the definition of unentangled loops given below, that provides a unique assignment of the loops to the steps of the primitive path. 

There are several possible ways of defining and counting unentangled loops. The one we choose in this paper is similar to the B- and C-loops defined and used in references 18 and 20. 

The general procedure for assigning unentangled loops to the cells is then as follows (see Fig. la) ... 

This definition divides the chain into a set of unentangled loops along the confining tube. In the next section we use this to analyze the moti<5n of the polymer along this tube. 

The motion of unentangled loops between the cells along the confining tube is represented in the discretized model by the hopping of reptons from site to site. In the repton bond representation foi, this motion corresponds to a random interchange of nearest neighbors... 

By modifying Axiom 2 we can discuss a wide class of problems. In the model described above, there are no interactions between reptons occupying the same site. The probability of an unentangled loop decreases exponentially with the size of this loop ° > (Eq. 3). As mentioned in the second section, the algebraic prefactor of this dependence introduces a weak interaction between reptons occupying the same site and could be included in a more elaborate version of the model. 

The fluctuations in the number of monomers in each unentangled loop also relax. For N i to go up by one, either neighbor must go down one, making the average leveling of these instantaneous fluctuations a diffusion problem along the tube. If the time for transfer of a monomer is r, then the terminal relaxation time is and there is a Rouse-like spectrum of shorter relaxation times down to t. ... 

Exercise Prove this result using the fact that the probability of a long unentangled loop (zero tube length) goes like [with a prefactor which we... 

Elastomer is a general term used for cross-linked polymeric materials such as rubber with elastic properties described above. However, elastomers are not purely elastic materials and they exhibit damping, that is, fractional energy loss per cycle in, say, an oscillatory experiment. The energy loss is due primarily to stress relaxation (viscous dissipation) of inelastic pendant chains and unentangled loops in the network. In the case of a prescribed stress sinusoidal deformation, the applied stress is expressed as... 

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