Markoff Chains and Excluded Volume Effect in Polymer Chains

In most of the theoretical work on statistical properties of polymer chains one includes (for mathematical simplicity) configurations which correspond to several monomers located at the same point. Here a mathematical method based on the theory of Markoff chains is used to eliminate short-range excluded volume effects in polymer chains formed on space lattices. Those aspects of the theory of Markoff chains which are important in this problem are briefly reviewed. If as(s=j—k) is the correlation coefficient between the x component of the position of the jth monomer and that of the kth, then the root-mean-square distance between ends of a chain of degree of polymerization N is proportional to N½ (as N→∞) provided that as→0 more rapidly than A/s1+ε (as s→∞). Here A is a constant and ε is an arbitrarily small positive constant. This condition is satisfied in a chain in which any group of n0 successive monomers do not overlap each other or any monomers in neighboring groups of n0 provided n0≪N. A detailed analysis of a polymer chain on a square lattice is given.