Accelerated Pseudospectral Method of Wormlike-Chain Self-Consistent Field Theory

Abstract Self-consistent field theory (SCFT) has become one of the most successful methods for the study of block copolymers. Although SCFT based on a universal wormlike chain (WLC) model has been established for many years, efficiently solving its equations has long been a major challenge. The lack of efficient numerical algorithms hinders the understanding of the self-assembly mechanisms of many important semirigid or rigid macromolecules. One of the most efficient existing methods, the pseudospectral method based on the third-order backward differentiation formula, remains computationally expensive and memory-intensive, primarily due to the frequent solution of large linear systems. Here, we propose reordering the coefficient matrices and avoiding explicit matrix inversion to preserve matrix sparsity, and further introduce Boolean matrices and symbolic LU decomposition to fully exploit the sparsity. By computing and storing only the nonzero elements of the sparse matrices, we dramatically reduce the memory cost and substantially accelerate the solution of the WLC-SCFT equations, enabling high-precision calculation of the free energies of three-dimensional structures and thereby the accurate construction of the phase diagrams of semirigid AB diblock copolymers. These accurate phase diagrams not only correct some results in previous phase diagrams but also lead to new findings. Specifically, we find that the tetragonal cylinder phase, which was previously reported to be stable, is only metastable. Moreover, the stability window of O70 with respect to the volume fraction first widens, then narrows and widens again as χN increases. O70 remains stable over a wide range of χN mainly because semirigid chains favor its lower mean curvature than the competing double-gyroid phase. Applying our accurate and highly efficient method to other macromolecular systems containing semirigid chains may provide further physical insights into their self-assembly behavior.