A Computer Program for the Distribution of End-to-End Distances in Polymer Molecules
free and;andom'rotationis allowed ahout the bonds.2 The program enables students to compare how the distribution of distances is affected by varying the number of chains included, the length of the chains, and the angle hetween segments. In addition, an optional provision has been made for an excluded volume approximation. The program generates the randomly coiled chains hy means of a series of operations involving a 3 X 3 matrix, B, which is a function of the constant bond angle 8, and a 3 X 3 matrix, A, as a function of the random rotation angle, Q. The form of the equation is
X, = ( I + BA, + B A I B A ~+ . . . + B A I B A ~... BA,-I)R where X, is a 3 X 1matrix giving the coordinates of the end of the nth segment, I is the identity matrix, and R represents a unit vector. Any number of chains of any length may be generated. We have found that the expected Gaussian distribution of end-to-end distances is approximated quite closely for all values of 0 , if a sufficiently large number of chains is included (several thousand). The excluded volume approximation results from an optional subroutine which disallows any value of 4 that would bring a newly generated chain segment within a specified distance of any other part of the chain. A resulting increase in the root-mean-square end-to-end distance is observed. The output of the program includes statistical dataon the distrihution of distances, an optiunallist of distance values, and a distribution plot of variable size, giving of frequency of occurrence versus the end-to-end distance. A listing of the program, sample output, and comments on its use are available and may he obtained by writing the authors.
' To whom correspondence should be addressed. 5 Flory, P. J., "Principles of Polymer Chemistry", Cornell University Press, Ithsca, NY, 1953, pp 399-431.
Calvin College Grand Rapids, Michigan 49506
William Van Doorne' Jack Kuipers William Hoekstra
Volume 53. Number 6,June 1976 / 353
