Lamellar diblock copolymer grain boundary morphology .3. Helicoid section twist boundary energy

The helicoid section morphology allows a diblock copolymer lamellar phase to maintain microphase separation across a twist grain boundary. The interface between the two microphases in the grain boundary region approximates a stack of sections of the helicoid minimal surface. Grain boundary energies were calculated for the helicoid section morphology both as a function of diblock chain characteristics and as a function of grain boundary twist angle. The basic approach to grain boundary energy calculation is to formulate a general expression for local free energy density as a function both of chain characteristics and of the local curvature of the interface. The local energy density is then integrated over the mathematical model for the Scherk grain boundary. Two general methods of calculation were used, and the results where then compared. First, a self-consistent field model was formulated in which average energies per chain were calculated for all the possible interfacial curvature environments encountered by diblocks in the helicoid section morphology. A second general approach utilized a continuum (Helfrich) model for interfacial deformation in which moduli are used to impose energetic penalties for curvature of the interface in the grain boundary region. The helicoid section grain boundary energies were compared to energies of a competing twist boundary morphology, the Scherk surface, which was analyzed in the preceding paper of this series. It was found that the energies of both the Scherk morphology and the helicoid section increase with increasing twist. The Scherk and helicoid section energies are comparable at low twist angles, less than about 15 degrees. Both morphologies are observed in this twist range. For higher twist angles, where only the Scherk morphology is observed, the helicoid section boundary energy becomes prohibitively high due to a compression of the lamellar layers.