Defects in lamellar diblock copolymers: Chevron- and Ω-shaped tilt boundaries

The lamellar phase in diblock copolymer systems appears as a result of a competition between molecular and entropic forces, which selects a preferred periodicity of the lamellae. Grain boundaries are formed when two grains of different orientations meet. We investigate the case where the lamellae meet symmetrically with respect to the interface. The form of the interface strongly depends on the angle, θ, between the normals of the grains. When this angle is small, the lamellae transform smoothly from one orientation to the other, creating the chevron morphology. As θ increases, a gradual transition is observed to an omega morphology characterized by a protrusion of the lamellae along the interface between the two phases. We present a theoretical approach to find these tilt boundaries in two-dimensional systems, based on a Ginzburg-Landau expansion of the free energy, which describes the appearance of lamellae. Close to the tips at which lamellae from different grains meet, these lamellae are distorted. To find this distortion for small angles, we use a phase variation ansatz in which one assumes that the wave vector of the bulk lamellar phase depends on the distance from the interface. Minimization of the free energy gives an expression for the order parameter [Formula Presented] The results describe the chevron morphology very well. For larger angles, a different approach is used. We linearize φ around its bulk value [Formula Presented] and expand the free energy to second order in their difference. Minimization of the free energy results in a linear fourth-order differential equation for the distortion field, with proper constraints, similar to the Mathieu equation. The calculated monomer profile and line tension agree qualitatively with transmission electron microscope experiments, and with full numerical solution of the same problem.