Statistical Mechanics of Wormlike Chains. II. Excluded Volume Effects

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Abstract
The mean‐square end‐to‐end distance and the second virial coefficient are evaluated for stiff chains with excluded volume. The calculation is based on a wire‐bead model whose backbone in the unperturbed state obeys wormlike‐chain statistics. In order to establish the necessary distribution functions, results derived in the second Daniels approximation in the previous paper are used, and suitable extrapolations from the coil region to the high‐stiffness (low temperature) limit are made. The ring closure probability in this limit is also derived by applying a variation principle to a functional‐integral representation of the partition function in terms of the bending elastic energy of the chain. The result is used for the first‐order perturbation calculation of the mean‐square end‐to‐end distance. Evaluation of the second virial coefficient is carried out in the double‐contact approximation. The interpenetration function ψ is also calculated in an approximate fashion, and is shown to be appreciably smaller than for the coil limit over the ordinary range of interest for typical stiff chains. This result is consistent with experimental data for cellulose nitrates.