Predicting disorder-order phase transitions in polymeric micelles

The structure observed in concentrated polymeric micelles results from interactions between coronal chains that develop as micelles are brought to approach distances where the chains either compress or interdigitate. One powerful model for polymeric micelles comprises spherical particles with chains tethered to their core at a specified surface density. This treatment combined with self-consistent field theory provides an estimate of the pair interaction potential between micelles. These pair interaction potentials allow modeling of the structure and thermodynamic properties that depend on the overall micelle concentration. We perform neutron scattering experiments to measure the short-range correlations in the liquid, through the static structure factor S(q), and compare these results with models that rely on a solution of the Ornstein-Zernike equation subject to a Rogers-Young closure. A description of the homogeneous liquid serves as the basis for employing density functional theory (DFT) to estimate the free energy of the solid. In this investigation, we use the modified weighted density approximation of Denton and Ashcroft [Phys. Rev. A 39, 4701 (1989)] to estimate the free energy of the solid for each of our micellar systems to predict the liquid-solid phase transition. Although we experimentally observe transitions to face-centered-cubic (fcc) and body-centered-cubic (bcc) crystals depending on the length of the corona relative to the core, we only predict a simple liquid-fcc transition with the DFT method. The nature of the transition suggests a simple perturbation result using the hard sphere as the reference system. Despite the inability to predict the bcc lattice type, both DFT and hard-sphere models accurately predict coexistence over the entire range of our experiments.