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

Area of Science:

• Polymer physics
• Statistical mechanics
• Computational chemistry

Background:

• The wormlike chain model is crucial for understanding polymer behavior.
• Characterizing polymer chain conformations and end-to-end distributions is fundamental.

Purpose of the Study:

• To derive exact expressions for the end-to-end distance vector distribution function for the wormlike chain model with fixed end orientations.
• To investigate the influence of fixed end orientations on chain conformation and the crossover between rigid and flexible chain behaviors.

Main Methods:

• Utilizing Fourier-Laplace space to represent the distribution function.
• Exploiting the hierarchical structure of moment-based expansions to derive infinite continued fractions.
• Calculating root-mean-square end displacement for chains with fixed end orientations.

Main Results:

• Exact expressions for the end-to-end distance vector distribution function were obtained.
• The root-mean-square end displacement shows a crossover from angular dependence (rigid chains) to independence (flexible chains) as chain rigidity decreases.
• Strong coupling between chain conformation and end orientation persists even for relatively flexible chains.

Conclusions:

• End orientation significantly impacts polymer chain conformation, particularly for chains several persistence lengths long.
• The distribution function transitions from a peak near contour length (straight chain) to near zero separation (Gaussian limit) with decreasing rigidity.
• Crossover behavior in chain flexibility is characterized by the emergence of double peaks in the distribution function.