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

Abstract

The lamellar phase in diblock copolymer systems appears as a result of a competition between molecular and entropic forces which select 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 φ(x, y). The results describe the chevron morphology very well. For larger angles, a different approach is used. We linearize φ around its bulk value φL 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.

Measuring Relative Grain-Boundary Energies in Block-Copolymer Microstructures

“…Since x increases with decreasing , we hypothesize that the predicted scaling EðÞ $ 3 might indeed exist in the vicinity of ¼ 0; however, it cannot be resolved by our method due to its limitations in the analysis of small tilt-angle boundaries. Note that our results indicate that the 3 regime is narrower than predicted by prior simulations-an effect that we interpret as a consequence of the increased degree of segregation [19,20]. A plateau regime is observed for angles exceeding ¼ crit ffi 85 , in which the boundary energy is found to be independent (within the experimental error) of the misorientation.…”

“…One parameter that is of particular interest is the energy penalty associated with GB formation as it provides insight into both the mechanism and driving force of grain coarsening during annealing. The current understanding of GB energies in BCPs is limited to simulation studies, for example, by Schick and co-workers as well as Matsen who-using a GinzburgLandau free-energy and self-consistent field approach, respectively-evaluated the surface energy of various types of tilt GB structures in weakly segregated lamellar BCP [18][19][20]. While these studies have established the principal trends of the GB energy for various GB types and also provided a rationale for experimental observations such as the transition from chevron-to -type morphologies with increasing misorientation of symmetric tilt GBs, a more quantitative understanding of the structure evolution in BCP systems will require the experimental determination of GB energies.…”

Interfacial Phenomena of Solvent-Diluted Block Copolymers

“…Omega-shaped tilt grain boundaries (figure 5) and Tjunction grain boundaries (figure 6) are shown as examples of possible grain boundaries between lamellar phases and agree well with previous works on pure BCP systems [13,15]. Due to the wavenumber doubling effect one can see in both figures "bulbs" of higher density at the grain boundaries, which is more pronounced for the ρ patterns than for the φ ones.…”

Existence and Stability of Spherically Layered Solutions of the Diblock Copolymer Equation

“…The perfect lamellar pattern consists micro-domains separated by parallel flat planes, Figure 3. However one often observes the lamellar pattern with topological defects such as dislocations, disclinations, grain boundaries, and tilt boundaries [44]. In this section we consider the spherical lamellar pattern, Figure 4, which we view as a defective lamellar pattern.…”

“…This pattern may be viewed as a defective lamellar pattern. Other defective patterns are considered in [44] where a model of a forth order differential equation is used. Given the number of interfaces we look for a solution that consists of spherical layers of micro-domains separated by narrow interfaces.…”