Crossover scaling in biased self-avoiding walks

We discuss biased self-avoiding walks defined as self-avoiding random walks with anisotropic or different weights for steps in the same direction as the previous step and those in a different direction. This problem is a discrete analog of the stiff ‘‘wormlike’’ chain in polymer physics but with the excluded-volume effect included. Our approach extends that of Halley, Nakanishi, and Sundararajan to include three dimensions. The scaling analyses of our Monte Carlo simulations for the simple cubic and square lattices suggest qualitative differences in the effect of excluded volume on the chain conformation in the stiff limit betwen two and three dimensions in a manner similar to a suggestion made by Petschek. In the limit of gauche weight p→0 and contour length N→∞, we find scaling for the mean-square end-to-end distance 〈$R^2$〉 with the crossover exponent one as before; however, the scaling function in three dimensions closely matches the random stiff-chain result with no excluded volume while that in two dimensions exhibits marked deviation from the random limit.