Statistical Computation of Mean Dimensions of Macromolecules. III

Random walks subject to the excluded volume effect have been generated by means of a high-speed electronic digital computer for a number of different three-dimensional lattices and for one four-dimensional lattice. In contrast to results obtained for two-dimensional lattices discussed in an earlier article, the present results are compatible with the view that the ratio, 〈rn2〉Av/n, where 〈rn2〉Av is the mean square length of permissible walks of n steps, converges as n→ ∞ for three-dimensional and four-dimensional lattices. This conclusion is obtained through an analysis of an appropriate difference equation employing the methods developed earlier. Convergence to a limiting value of the critical ratio mentioned above appears relatively rapid for the four-dimensional lattice investigated. Possible convergence for the three-dimensional lattices that have been investigated to date appears to be very slow. Thus, it appears that the limiting behavior will not be approached experimentally for values of n within the usual range of the degrees of polymerization of real polymer molecules.