Crystalline free energies of micelles of diblock copolymer solutions

We report a characterization of the relative stability and structural behavior of various micellar crystals of an athermal model of AB-diblock copolymers in solution. We adopt a previously developed coarse-graining representation of the chains which maps each copolymer on a soft dumbbell. Thanks to this strong reduction of degrees of freedom, we are able to investigate large aggregated systems and for a specific length ratio of the blocks f = M(A)∕(M(A) + M(B)) = 0.6, to locate the order-disorder transition of the system of micelles. Above the transition, mechanical and thermal properties are found to depend on the number of particles per lattice site in the simulation box, and the application of a recent methodology for multiple occupancy crystals [B. M. Mladek et al., Phys. Rev. Lett. 99, 235702 (2007)] is necessary to correctly define the equilibrium state. Within this scheme we have performed free energy calculations at two reduced density ρ∕ρ∗ = 4, 5 and for several cubic structures such as fcc, bcc, and A15. At both densities, the bcc symmetry is found to correspond to the minimum of the unconstrained free energy, that is to the stable symmetry among the few considered, while the A15 structure is almost degenerate, indicating that the present system prefers to crystallize in less packed structures. At ρ∕ρ∗ = 4 close to melting, the Lindemann ratio is fairly high (∼0.29) and the concentration of vacancies is roughly 6%. At ρ∕ρ∗ = 5 the mechanical stability of the stable bcc structure increases and the concentration of vacancies accordingly decreases. The ratio of the corona layer thickness to the core radius is found to be in good agreement with experimental data for poly(styrene-b-isoprene)(22-12) in isoprene selective solvent which is also reported to crystallize in the bcc structure.