Numerical simulations of the phase behaviors of rod-coil diblock copolymers confined on a spherical surface

The self-assembly of rod-coil diblock copolymers has garnered significant attention owing to their intricate phase behavior. One of the most successful theories to investigate these phase behaviors is the self-consistent field theory (SCFT). In this work, we employ SCFT to study the phase behavior of rod-coils when these polymers are confined on a spherical surface. While such confinement enriches the diversity of possible phases, it also increases the complexity of the numerical simulations with SCFT. To address this challenge, we develop a novel pseudo-spectral method that employs spherical harmonics, the Wigner D-matrix, and a time-splitting approach to efficiently solve the Fokker-Planck equations in SCFT. The interplay between the liquid crystalline nature of the rod-like units, microphase separation, and spherical confinement results in a remarkable diversity of morphologies. The choice of initial fields plays a crucial role in determining the final converged states. To facilitate the generation of initial states for various striped structures, we propose a cut-and-rotate method. Using this numerical framework, we systematically investigate the phase behavior of rod-coil diblock copolymers over a range of sphere radii. For rod-coils with symmetric composition, we observe a sequence of stable striped patterns, including lamellar, single-helical, and double-helical structures, which alternate as the sphere radius increases. For asymmetric compositions, spotted patterns emerge as stable configurations. These findings provide new insights into the self-assembly of rod-coil copolymers under spherical confinement and underscore the utility of advanced numerical techniques for exploring complex phase behaviors.