The order-disorder transition and the disordered micelle regime in sphere-forming block copolymer melts

The order-disorder transition in a melt of asymmetric micelle-forming diblock copolymers is studied analytically in the strong segregation limit. In contrast to previous calculations by Semenov and by self-consistent mean-field theory, both the translational entropy of the micelles in a disordered micelle regime and the intermicelle free energy are taken into account. This enables us to locate the order-disorder transition, ODT, where the long-range order (or lattice) of micelles disappears. In agreement with some experimental observations, the ODT occurs between body-centered-cubic spheres and disordered micelles. A face-centered-cubic sphere phase remains thermodynamically unstable, as it is superseded by the disordered micelle regime due to the gain in the translational entropy of micelles. The ODT is accompanied by a decrease in the micelle volume fraction and, more importantly, in the number density of micelles. The aggregation number and the average distance between micelles also change at the ODT, but only slightly. At higher temperatures the number of micelles decreases, but some small fraction of micelles may persist to very high temperatures. A "critical micelle temperature" (cmt) may be estimated, which separates the disordered micelle regime (with a finite micelle concentration) from a disordered melt (with exponentially small micelle fraction), but it is not a true phase transition. Thus, the disordered micelle regime is part of the disordered phase. The composition dependence of the ODT is also analyzed. We compare the predictions of this model with experimental data for poly(styrene-b-isoprene) diblock copolymers, with encouraging agreement.