End-to-end distance vector distribution with fixed end orientations for the wormlike chain model

Abstract

We find exact expressions for the end-to-end distance vector distribution function with fixed end orientations for the wormlike chain model. This function in Fourier-Laplace space adopts the form of infinite continued fractions, which emerges upon exploiting the hierarchical structure of the moment-based expansion. Our results are used to calculate the root-mean-square end displacement in a given direction for a chain with both end orientations fixed. We find that the crossover from rigid to flexible chains is marked by the root-mean-square end displacement slowly losing its angular dependence as the coupling between chain conformation and end orientation wanes. However, the coupling remains strong even for relatively flexible chains, suggesting that the end orientation strongly influences chain conformation for chains that are several persistence lengths long. We then show the behavior of the distribution function by a density plot of the probability as a function of the end-to-end distance vector for a wormlike chain in two dimensions with one end pointed in a fixed direction and the other end free (in its orientation). As we progress from high to low rigidity, the distribution function shifts from being peaked at a location near the full contour length of the chain in the forward direction, corresponding to a straight configuration, to being peaked near zero end separation, as in the Gaussian limit. The function exhibits double peaks in the crossover between these limiting behaviors.