High-accurate and efficient numerical algorithms for the self-consistent field theory of liquid-crystalline polymers

TL;DR

The paper tackles the challenge of accurately solving high-dimensional ($6$-D) SCFT equations for liquid-crystalline polymers modeled with wormlike chains and Maier–Saupe interactions. It introduces a comprehensive numerical toolkit, including high-order time integrators (RK, BDF, OS, Upwind, and SDC), Fourier pseudospectral spatial discretization, and spherical-harmonic orientation treatments, coupled with adaptive nonlinear solvers (adaptive Anderson mixing) and a cascadic multilevel strategy. An automatic computational-domain optimization is developed to identify optimal periodicities and minimize the saddle-point free energy. Extensive 4D–6D simulations demonstrate rich self-assembly behavior (nematic, smectic-C, and 3D lattice phases) with significant efficiency gains, enabling detailed exploration of high-dimensional LC polymer morphologies and more precise free-energy calculations.

Abstract

Self-consistent field theory (SCFT) is one of the most widely-used framework in studying the equilibrium phase behaviors of inhomogenous polymers. For liquid crystalline polymeric systems, the main numerical challenges of solving SCFT encompass efficiently solving plenty of six dimensional partial differential equations (PDEs), precisely determining the subtle energy difference among self-assembled structures, and developing effective iterative methods for nonlinear SCF iteration. To address these challenges, this work introduces a suite of high-order and efficient numerical methods tailored for SCFT of liquid-crystalline polymers. These methods include various advaced PDE solvers, an improved Anderson iteration algorithm to accelerate SCFT calculations, and an optimization technique of adjusting the computational domain during the SCF iterations. Extensive numerical tests demonstrate the efficiency of the proposed methods. Based on these algorithms, we further explore the self-assembly behavior of liquid crystalline polymers through simulations in four, five, and six dimensions, uncovering intricate three-dimensional spatial structures.

Figures (13)

• Figure 1: Schematic of the flexible-semiflexible chain containing a flexible block $A$ (red) and a semiflexible block $B$ (blue).
• Figure 2: The flowchart of SCFT iteration process.
• Figure 3: Comparison of ten time discretization schemes in Sec. \ref{['sec:numerical']}, with model parameter $\lambda=100$.
• Figure 4: Numerical behavior of free energy $H$ in the RK4 method, the IMEX3 method and the EXUP method, with the parameter $\lambda=100$.
• Figure 5: Comparison of the ADI method, AM method and AAM method, with parameters $\chi N=16$, $\eta N=8$, $f=0.6$, $\lambda=100$, $\beta=2$, $N_{s}=200$, $N_{\bm{r}}=32$, $N_{\theta}=8$ and $N_{\varphi}=17$.