(585h) Geometry of Stable Spherical Phases in Diblock Copolymer Melts

Frank-Kasper (FK) phases are complex spherical particle packings known to form in single-component AB diblock copolymer melts. They exhibit low lattice symmetry compared to classical close-packed spherical phases, contradicting the maxim that nature favors high symmetry configurations. Understanding the driving forces for a polymer melt to select an FK phase over a close-packed phase remains an area of active investigation.

While the free energy of an ordered packing of polymer micelles arises from the sum of chain-stretching and interfacial tension, it is desirable to connect these molecular contributions to a macromolecular geometric argument about the shape and organization of the micelles. In this context, sphericity of the lattice Wigner-Seitz (WS) cells, which tends to increase with the number of particles in the unit cell, has been hypothesized to be closely related to the stability of FK phases. However, WS cell geometry is a lattice property that does not change with polymer system parameters, and thus cannot fully explain changes in stability across parameter space.

Informed by previous studies of a “diblock foam model”, we hypothesize that a more nuanced approach, considering the geometries of both the micelle core (i.e. the A/B interface) and the WS cell, will reveal an intuitive and more complete physical picture of FK phase stability. To investigate this hypothesis, we use self-consistent field theory and in-house numerical analysis techniques to study a large dataset of equilibrium polymer melts in the sphere-forming region of the diblock copolymer phase diagram. We report findings about the relationship between micelle packing and geometry in stable phases that illuminate a clear and non-trivial distinction between complex and classical spherical phases.