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 devel- oped 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 = MA/(MA + MB) = 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 {\rho}/{\rho}\ast = 4,5 and for several cubic structures as FCC,BCC,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 sys- tem prefers to crystallize in less packed structures. At {\rho}/{\rho}\ast = 4 close to melting, the Lindemann ratio is fairly high (~ 0.29) and the concentration of vacancies is roughly 6%. At {\rho}/{\rho}\ast = 5 the mechanical stability of the stable BCC structure increases and the concentration of vacancies ac- cordingly 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.

💡 Research Summary

This paper presents a comprehensive thermodynamic and structural investigation of micellar crystals formed by an athermal model of AB‑diblock copolymers in solution. By employing a previously developed coarse‑graining scheme, each copolymer chain is represented as a soft dumbbell consisting of two interacting beads that mimic the hydrophobic A‑block (core) and the solvophilic B‑block (corona). This drastic reduction of degrees of freedom enables simulations of very large aggregated systems, allowing the authors to explore the phase behavior at the mesoscopic scale.

The study focuses on a specific block‑length ratio f = MA/(MA + MB) = 0.6 and examines two reduced densities, ρ/ρ* = 4 and 5, which lie above the order‑disorder transition of the micellar fluid. Three candidate cubic lattices—face‑centered cubic (FCC), body‑centered cubic (BCC), and the complex A15 structure—are considered as possible equilibrium arrangements of the micelles. Because a micelle can occupy a lattice site with multiple polymer chains (multiple‑occupancy crystals), the conventional single‑occupancy free‑energy formalism is insufficient. The authors therefore adopt the multiple‑occupancy crystal methodology introduced by Mladek et al. (Phys. Rev. Lett. 99, 235702, 2007). In this framework, both the lattice constant a and the average number of micelles per lattice site N_s are treated as variational parameters; the equilibrium state is identified by minimizing the unconstrained free energy F(N_s, a) with respect to both variables.

Free‑energy calculations reveal that, at both densities, the BCC lattice possesses the lowest free energy, making it the thermodynamically stable structure among those examined. The A15 lattice is nearly degenerate with BCC, indicating a slight preference for less densely packed arrangements, while FCC consistently exhibits higher free energy and is therefore unstable. The authors also quantify mechanical stability through the Lindemann ratio (L) and the concentration of vacancies (ν). At ρ/ρ* = 4, close to the melting point, L≈0.29—a relatively large value compared with atomic crystals—signaling substantial thermal motion, and ν≈6 % indicating a noticeable population of vacant lattice sites. At the higher density ρ/ρ* = 5, both L and ν decrease, reflecting enhanced mechanical rigidity and a reduced defect concentration as the system becomes more compressed.

A key validation step compares the simulation outcomes with experimental data for poly(styrene‑b‑isoprene) (22‑12) in an isoprene‑selective solvent. Experiments report that this system crystallizes in a BCC lattice and that the ratio of corona thickness to core radius matches the values obtained from the coarse‑grained model. This agreement confirms that the athermal dumbbell representation captures essential physical features of real diblock copolymer micelles.

In summary, the paper delivers four major contributions: (1) a highly efficient coarse‑graining strategy that enables large‑scale micellar crystal simulations; (2) the application of a multiple‑occupancy crystal formalism to correctly determine equilibrium lattice parameters and free energies; (3) a detailed analysis of defect concentrations and vibrational amplitudes as functions of density, providing insight into the mechanical stability of soft micellar crystals; and (4) quantitative validation against experimental measurements, establishing the model’s relevance to real polymer systems. The methodology and findings are broadly applicable to the design and prediction of ordered phases in complex block‑copolymer and soft‑matter systems, especially where multiple occupancy and soft interactions play a decisive role.