An efficient hybrid spectral-compact difference scheme for rod–coil diblock copolymers in slit confinement

Abstract

Self-consistent field theory simulations of rod–coil diblock copolymers in slit confinement present significant numerical challenges due to sharp density gradients near hard walls. To rigorously resolve these systems utilizing the Gaussian and wormlike chain models, a hybrid spectral-compact finite difference scheme is developed on a non-uniform Chebyshev–Gauss–Lobatto grid. Shen’s Chebyshev spectral method is employed for the flexible blocks. For the semiflexible blocks, a second-order upwind compact scheme together with an L-stable TR-BDF2 contour-stepping algorithm is adopted. This hybrid framework effectively suppresses spurious numerical oscillations. This unconditionally stable formulation strictly preserves propagator non-negativity and achieves up to a two-orders-of-magnitude speedup over uniform-grid implementations while maintaining linear spatial scaling. Simulations utilizing this advanced framework under neutral wall conditions reveal that the confining walls naturally induce preferential wetting of the semiflexible blocks at the impenetrable boundaries. As the incompressibility penalty increases, the compressible system progressively approaches the incompressible limit. For the selected physical parameters, decreasing the slit width induces a sequence of structural transitions from a smectic-C morphology with three internal periods (SC3) to morphologies with two and one internal periods (SC2 and SC1), and ultimately to a highly compressed smectic-P morphology (SP1). The equilibrium thickness of these confined structures deviates from exact integer multiples of the bulk spatial period. This deviation arises from the volume compensation associated with boundary depletion layers, together with adjustments in the molecular tilt angle and the degree of molecular interdigitation.