Primitive path models

The free energy and the elastic force for simple elongation or compression in the primitive path model is62)... 

The harmonic pipe model has been discussed exhaustively by Heinrich et Let us now discuss more basic features of the entanglement models. There are various versions of the mathematical formulation but in physical content they are all the same. One of the most visible is the primitive path model, now widely studied. ... 

The word W consisting of a sequence of letters corresponding to different entanglements (introduced in Sect. 2.21) plays a role of full topological invariant for the PCAO-model. It is closely connected with the concept of the primitive path obtained by means of roughing of the microscopic chain trajectory up to the scale of the lattice cell and by exclusion of all loop fragments not entangled with the obstacles (Fig. 5). 

Despite these complications, there are now numerous evidences that the tube model is basically con-ect. The signatory mark that the chain is trapped in a tube is that the chain ends relax first, and the center of the chain remains unrelaxed until relaxation is almost over. Evidence that this occurs has been obtained in experiments with chains whose ends are labeled, either chemically or isotopically (Ylitalo et al. 1990 Russell et al. 1993). These studies show that the rate of relaxation of the chain ends is distinctively faster than the middle of the chain, in quantitative agreement with reptation theory. The special role of chain ends is also shown indirectly in studies of the relaxation of star polymers. Stars are polymers in which several branches radiate from a single branch point. The arms of the star cannot reptate because they are anchored at the branch point (de Gennes 1975). Relaxation must thus occur by the slower process of primitive-path fluctuations, which is found to slow down exponentially with increasing arm molecular weight, in agreement with predictions (Pearson and Helfand 1984). 

The Doi-Edwards model has been extended to allow processes of primitive-path fluctuations, constraint release, and tube stretching. These extensions of the theory allow accurate prediction of many steady-state and time-dependent phenomena, including shear thinning, stress overshoots, and so on. Predictions of strain localization and slip at walls... 

Fig. 8. A network strand blowing path-occupying and surplus porticms as rqvesented by the primitive segment model. Tte length of the arrows indicated the mesh size in the system...

That assumption simplifies the analysis of primitive path rearrangement. Local path jumps now correspond to random flips in the Orwoll-Stockmayer chain model, and we can apply these results directly. For our case the local jump distance is the path step length a, and the average time between jumps is 2r , where r is the mean waiting time for release of a constraint which allows a length preserving jump. The average number of such suitably situated constraints per cell is z(z < zo), and we assume for simplicity that all cells have z such constraints. 

The Doi-Edwards theory treats monodisperse linear chain liquids by a model which suppresses fluctuations and assumes a topologically invariant medium. Two parameters are required, the monomeric friction coefficient which characterizes the local dynamics and the primitive path step length a which characterizes the topology of the medium. The step length is related to the entanglement molecular weight of earlier theories, = cRqT/Gn, by Eqs. 1 and 37 ... 

Each monomer is constrained to stay fairly close to the primitive path, but fluctuations driven by the thermal energy kT are allowed. Strand excursions in the quadratic potential are not likely to have free energies much more than kT above the minimum. Strand excursions that have free energy kT above the minimum at the primitive path define the width of the confining tube, called the tube diameter a (Fig. 7.10). In the classical affine -and phantom network models, the amplitude of the fluctuations of a... 

The reptation model assumes the contour length of the primitive path is fixed at its average value (L). In reality, the primitive path length... 

L fluctuates in time as the chain (or snake) moves. A full description of chain dynamics requires knowledge of the probability distribution of the primitive path lengths. This problem has been solved exactly by Helfand and Pearson in 1983 for a lattice model of a chain in a regular array of... 

Lattice model of a chain in an array of fixed topological obstacles. Thick line—primitive path. 

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