Statistical Computation of Mean Dimensions of Macromolecules. II

Abstract

Random walks subject to the excluded volume effect have been generated by means of a high‐speed electronic digital computer for four different two‐dimensional lattices. A semiempirical theory has also been developed for interpreting the statistical data. From the results obtained it is possible to show for several two‐dimensional lattices that the quotient, 〈r_n²〉_Av/n, diverges as n→ ∞, where 〈r_n²〉_Av is the mean square length of permissible walks of n steps. This conclusion was reached by using an appropriate difference equation in conjunction with data on the mean square lengths of successful walks and the mean square lengths of failures resulting from ring closures. Integration of that difference equation indicates that if n is sufficiently large, 〈r_n²〉_Av is proportional to n^A, where A depends upon the lattice system.