On the shape and configuration of polymer molecules

An attempt is made to estimate the distribution of the length of a polymer molecule when the excluded volume effect is taken into account. The model of a self-avoiding walk on a lattice is used, and exact enumerations are undertaken on a computer of walks of up to 18 steps on the two-dimensional simple quadratic lattice (nearly 125 million walks) and up to 13 steps on the three-dimensional simple cubic lattice (nearly 950 million walks). It is conjectured that the circular and spherical symmetry properties of the limiting space distributions for simple random walks are not changed by the self-avoiding condition. The limiting distribution of a rectangular coordinate is estimated, and is found to differ appreciably from the Gaussian distribution of a simple random walk. Instead the distribution can be well fitted by a function of the form exp(-|x|ν) dx where ν is equal to 4 in the two-dimensional case and 2.5 in the three-dimensional case. The distributions of length corresponding to these functions are then calculated, and are significantly sharper than those corresponding to randomly linked units.