Spherical-harmonic Expansion of the Modified Diffusion Equation for Wormlike Chain in Curvilinear Coordinates

TL;DR

This paper develops a spherical-harmonic expansion method to simplify the modified diffusion equation for wormlike chains, enabling more efficient analysis of polymer conformations in curvilinear coordinates.

Contribution

It introduces a spherical-harmonic series expansion of the MDE for wormlike chains, simplifying the equation to depend only on spatial variables for easier computation.

Findings

Derived simplified coupled equations depending solely on spatial variables.

Compared three orientation-setting methods and selected the simplest for calculations.

Validated the expansion approach in cylindrical and spherical coordinates.

Abstract

We investigate the wormlike polymer chains using self-consistent field theory and take into account the Onsager excluded-volume interaction between polymer segments. The propagator of polymer chain is one of the essential physical quantities used to study the conformation of polymers, which satisfies the modified diffusion equation (MDE) for wormlike chain. The propagator of wormlike chain is not only dependent on the spatial variables, but also on the orientation. We separate the variables of propagator by using spherical-harmonic series and then simplify the MDE to a coupled set of equations only depends on spatial variables in this paper. We expand the MDE by spherical-harmonic functions in cylindrical coordinates and spherical coordinates, respectively. We find that there are three ways to set the orientation, no matter in cylindrical coordinates or spherical coordinates. But for the convenience of calculation, we compare these three forms and choose the simplest one to simplify the MDE. And we get a coupled set of equations only depends on spatial variables.