Statistical Computation of Mean Dimensions of Macromolecules. III

Abstract

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, 〈r_n²〉_Av/n, where 〈r_n²〉_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.