Order-disorder transitions and free energies in asymmetric diblock copolymers

Simulations of simple bead-spring models of asymmetric diblock copolymers are used to study the dependence of order-disorder transitions and free energies upon the invariant degree of polymerization N¯ and the fraction f of beads in the minority block. Well-tempered metadynamics is used to determine values of (χN)ODT along the lamellar-disorder and hexagonal-disorder transitions over the range 0.1875 ≤ f ≤ 0.5 for two models with different values of N¯ = 480 and 1920, where χ is an effective Flory-Huggins interaction parameter, N is the degree of polymerization, and (χN)ODT is a value of χN at the order-disorder transition (ODT). More extensive studies are performed for systems with f = 1/4, which undergo a hexagonal-disorder transition. Equivalent results for both phase boundaries and free energies are obtained for one pair of systems with different numbers of beads per chain but matched values of f = 1/4 and N¯, in agreement with the corresponding state hypothesis. Comparison of results for (χN)ODT for systems with f = 1/4 and several values for N¯ show a systematic decrease in (χN)ODT with an increase N¯, consistent with the expected approach to the self-consistent field (SCFT) prediction as N¯ → ∞. Results for the free energy per chain in the disordered and hexagonal phases of systems with f = 1/4 show that SCFT gives rather accurate predictions for the free energy in the ordered hexagonal phase but that the random-mixing approximation underlying SCFT significantly overestimates the free energy of the disordered phase.