Microdomain morphology of star copolymers in the strong-segregation limit

Abstract

The strong-segregation limit of microphase separation in star diblock copolymers is treated with a mean-field theory originally introduced for linear diblocks by Leibler and extended for the strong segregation limit by Ohta and Kawasaki. In addition to the usual morphologies of spheres, cylinders, and lamellae, the ordered bicontinuous double-diamond (OBDD) structure discovered in experiments in our group is included in the free energy competition. The structure factor for a star copolymer is that derived by de la Cruz and Sanchez, and the long-range contribution to the free energy is computed by a numerical summation over reciprocal lattice vectors. Constant-mean-curvature surfaces with double-diamond symmetry, calculated by a finite element method, are used to define model OBDD structures, and the form factors for the model structures so defined are calculated analytically. These cpnstant-mean-curvature surfaces yield lower free energies than either surfaces determined by assemblies of cylindrical struts or surfaces displaced a constant distance from the Schwarz diamond minimal surface. The conclusion that the constant-mean-curvature surface is the best description of the microscopic interface is also supported by comparisons of calculated projections with actual TEM data. Use of these surfaces of constant mean curvature is thus crucial in proper modeling of the OBDD morphology. The predicted lattice parameters for the OBDD structures observed in the star diblocks for arm numbers above five by Thomas and co-workers and in the linear diblocks by Hashimoto and co-workers are in very good agreement with experiment, but the calculated free energy of the OBDD morphology does not become less than that of the cylindrical morphology in either the star or linear diblock cases as is indicated by the experiments. This is most likely due to the inadequacy of Gaussian chain statistics in the microphase-separated state-particularly with respect to the higher arm stars.