A Polymer Chain in the Lattice of Obstacles

It is noteworthy that for investigation of properties of real polymer systems with topological constraints it is not enough to be able to calculate the statistical characteristics of chains in the lattice of obstacles. It is also necessary to be able to compare any concrete physical system with the unique lattice of obstacles, which is a much more complicated problem than the first task. In this way, the model polymer chain in an array of obstacles is an intermediate between the microscopical and phenomenological approaches. The direct investigation of the PCAO-model was fulfilled in Refs. [16-25]. 

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). 

In a number of papers [16-23,25], the discrete variant of the PCAO-model is considered the chain is modeled by a random walk on the lattice with spacing a and the topological constraints are placed on the dual lattice with period c. 

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

We have presented only a few basic physical results obtained within the framework of the PCAO-model the mathematical methods for investigating equilibrium and dynamic properties of a test polymer chain of different topology placed in the lattice of obstacles will be reviewed below. 

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