Algebraic Test of Material Conservation in Self-Consistent Mean Field Theory of Polymers

Abstract

In the standard mean field theory of Gaussian polymer chains, the partition function of a polymer chain is obtained by solving partial differential equations which are essentially modified diffusion equations. Even though this statistical mechanical method is a perfectly accurate theory, the process of numerical solution often has problems in keeping the amount of polymer materials in the system. Recently, for the purpose of tracking material conservation in the selfconsistent field theory (SCFT), our research group has developed an algebraic test which uses matrix and bra-ket notation. The test reveals that when Crank-Nicolson method is adopted, finite volume method (FVM) is the only way to conserve material perfectly in the cylindrical and spherical coordinate systems. Alternating direction implicit method combined with FVM cannot conserve material, though it is still a good candidate after considering speed and accuracy simultaneously. We also confirm that the widely used pseudospectral method in the Cartesian coordinate system has the ability to conserve material.