A Finite Volume Method for Self-Consistent Field Theory of Brush in Cylindrical and Spherical Coordinate Systems

Abstract

A method to obtain self-consistent mean field solution for polymeric systems is to solve partial differential equations for partition functions of the polymer chain. One widely used numerical scheme to solve these equations is the finite difference method (FDM), but it is well known that FDM has a problem in keeping the amount of material in the system, especially when curvilinear coordinate systems are adopted. Here, I provide a numerical framework applying two dimensional finite volume method (FVM) to treat this conservation problem.

For the purpose of checking material conservation of various numerical algorithms used in the self-consistent field theory (SCFT), I develop an algebraic method using matrix and bra-ket notation, which traces the symmetry of the product of the volume and evolution matrices. Algebraic tests reveal that when Crank-Nicolson method is adopted, FVM is the only way to conserve material perfectly for an arbitrary shape of potential field, initial conditions of partition functions and the segment number. I also find that Alternating direction implicit (ADI) methods combined with FVM cannot conserve material in the cylindrical and spherical coordinate systems, though it is still a very good candidate after considering speed and accuracy simultaneously.

For the numerical test, I consider molten polymer brushes with one end of each polymer chain is grafted to the spherical or cylindrical surface while the other chain end is free, and the volume of the polymeric system is filled with polymers. Because polymer chains are tethered, the brushes are strongly inhomogeneous, and thus they are good model systems to test the accuracy, speed and mass conservation of SCFT algorithms.

The FVM I develop is primarily for the SCFT calculation, but this method is versatile in that it is applicable to other parabolic problems in the cylindrical and spherical coordinate systems, and the algebraic method developed to test material conservation can be very helpful for polymer scientists implementing various types of SCFT.