Entangled random walks

The connection between problem of determination of the knot entropy and statistics of entangled random walks is schematically shown in the Table 1. We argue that both these topological questions could be considered from one general point of view of random walk on non commutative groups. (Some preliminary remarks concerning this connection one can find in [2].)... 

Vincent analyzed the tensile fracture stress o, of a broad range of polymers as a function of the number of backbone bonds per cross sectional area ( 2) and found a nearly linear relation, o 2, as shown in Fig. 12. 2 is related to via the theory of entanglements for random walk chains as [74]... 

The reason why simple scaling considerations do not lead to a unique result for the exponent a is due to the fact that the entanglement problem as a geometrical phenomenon contains two independent lengths the lateral distance between the chains s = (L/V) 1/2 and the step width of the random walk... 

For separations in CGE the pore size of such a polymer solution is interesting. The chain segment between two points of entanglement can be regarded as an independent subunit which can undergo random walk per se. The volume en-... 

The diameter, a, of the tube corresponds to the entanglement spacing, Mg. That is, a strand of polymer having molecular weight Mg spans a random walk end-to-end distance a (Fig. 3-24). Thus, = a M/Mg, and... 

The tube can be thought of as being composed of NjN sections of size a, with each section containing monomers. The chain can be considered as either a random walk of entanglement strands NjNc strands of size a) or a random walk of monomers (A monomers of size b). 

The tube diameter in the melt, a(l) b /Ne(l), is given by Eq. (9.1) in terms of the number of Kuhn monomers in an entanglement strand in the melt Ne (1). Notice that a (l)>b, which makes a > at all concentrations. Since the chain is a random walk of correlation blobs on scales larger than the entanglement strand is a random walk of correlation blobs, as depicted in Fig. 9.6. 

Owing to the fact that the tube diameter is always larger than the correlation length (fl > 0, the entanglement strand is a random walk of correlation volumes in any solvent ... 

The dynamics of an entangled chain in an array of fixed obstacles can also be studied by Monte Carlo simulations. An initial unrestricted random walk conformation of a chain on a lattice (representing a chain in a melt) could be obtained using the method of section 9.6.2.2. The topological entanglement net of surrounding chains is represented by obstacles, sketched as solid circles in the middle of each elementary cell in Fig. 9.32. 

The ellipsoidal, random-walk molecules of a polymeric amorphous zone caught together by their entanglements. 

Entanglements are characterized by the tube diameter a (the mesh size of the temporary entanglement network see Figure 40). The polymer strand between entanglements is a random walk of Ne/g( (a/f correlation blobs, where iVe is... 

Rather remarkably for a network of a given chain density one can actually calculate Vj since when a chain moves in a lattice of other chains the paths available to it are equivalent to a Cayley tree i.e. to a random walk which always retraces its steps. Details are given in refs 4 and 5 (see also ref 6). Since it appears the only tractable representation available to us, we adopt (3.7) and take entanglements to be equivalent to slip links i.e. are characterized by an Y and are quadrifunctional. 

Assume that we are dealing with polymers in good solvents and in the semidilute solution. If r is a scale to measure, then the chain entanglement shows the following properties. At r >, that is, outside the blob, the repulsive interactions between monomers are screened out by other chains in the solution so that the whole chain is composed of blobs connected in an ordinary random walk without excluded volume effect. Overall, the chain follows Gaussian statistics. At r <, that is, within the blob, the chain does not interact with other chains, but there is a strong excluded volume effect. 

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