Statistical Computation of Radii of Gyration and Mean Internal Dimensions of Polymer Molecules

A Monte Carlo numerical method has been employed for the computation of the mean-square separation between pairs of elements i and j, 〈Rij2〉, and the mean-square radius of gyration, 〈Sn2〉, for an n-segment coiling type polymer molecule. A restricted random walk was used as a model for the polymer, and the average dimensions in question were calculated for random samples of nonintersecting walks generated by means of a high-speed digital computer. 〈Rij2〉 has been found to be a rather complicated function of i/n and j/n, depending upon the position in the chain as well as upon the contour length of the segment between i and j. The mean-square radius of gyration was found to be proportional to the average square end-to-end distance, as for simple unrestricted random walks, but the constant of proportionality differs from the value of ⅙ which is applicable in the absence of the excluded volume effect. The exact value of the ratio of the two quantities depends upon whether the molecule is constrained to 2 or 3 dimensions.