Chain in an array of fixed obstacles

Chains in polymer melts and entangled polymer solutions form an effective 

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 

In the second line of Eq. (9.49), the term kTT Nb l(2 ya ) was added and-subtracted so as to complete the square inside the square brackets, in 

The constant term kTT Nj 2 yNe) in Eq. (9.49) does not affect the dependence of the free energy F L) on the contour length L of the primitive 

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Other computer simulations, such as the Evans Edwards model of a chain in an array of fixed obstacles (described in detail in Section 9.6.2) exhibit fluctuations of the tube length and also find stronger molar mass dependences of relaxation time t and diffusion coefficient... 

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. 

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... 

If a step strain is imposed on a system of an unattached chain in an array of fixed obstacles, such as an ideal network, the resulting stress decays as the chain reptates into an undeformed configuration. The cells of the network that sustain stress are those from the original tube that has never been vacated by the chain since the time of the step strain. 

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

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