The observation of complex, Frank-Kasper (FK) particle packings in diblock polymer melts has until recently been limited to low molecular weight, conformationally asymmetric polymers. We report temperature-dependent small-angle X-ray scattering (SAXS) studies of blends of a sphere-forming poly-(styrene-block-1,4-butadiene) (SB) diblock polymer (M n = 33.3 kg/mol, Đ = M w /M n = 1.08, f B = 0.18) with two different poly(1,4butadiene) (B) homopolymer additives. When the B additive M n is the same as that of the diblock core-forming B segment, these blends remarkably form tetrahedrally close-packed FK σ and Laves C14 and C15 phases with increasing B content. However, binary blends in which the B additive M n is 60% of that of the diblock B segment form only the canonical body-centered cubic (BCC) particle packing and hexagonally-packed cylinders (HEXc). The observed phase behavior is rationalized in terms of "wet" and "dry" brush blending, whereby higher B M n drives stronger localization of the homopolymer in the particle cores while preserving the interfacial area per SB diblock chain. The consequent packing constraints in these blends destabilize the BCC packing, and FK phases emerge as optimal minimal surface solutions to filling space at constant density while maximizing local particle sphericity.

The water-driven self-assembly of homologous dianionic surfactants into lyotropic liquid crystals (LLCs) is investigated, with a focus on understanding how surfactant headgroup and counterion identities guide supramolecular spherical mesophase selection. Using temperature-dependent small-angle X-ray scattering (SAXS), we demonstrate that 2-alkylmalonate surfactants (CMal-M) with n = 8 (octyl) or 10 (decyl) and M = K, Cs, or (CH)N form both simple and complex micelle packings. Observed spherical morphologies include body-centered cubic (BCC), hexagonally closest-packed (HCP), and tetrahedrally closest-packed Frank-Kasper (FK) A15 and σ phases (Pm3(-)n and P4/mnm symmetries, respectively). Previously observed in only one other minimally hydrated surfactant, the σ phase is a rare LLC morphology comprising a low-symmetry unit cell containing 30 sub-2-nm quasispherical micelles, each of which belongs to one of five symmetry-equivalent classes with discrete aggregation numbers. Temperature versus water concentration phase maps for CMal-M LLCs reveal that σ-phase formation depends sensitively on the size and polarizability of the surfactant counterion and the length of the surfactant alkyl tail. These observations are rationalized in terms of a delicate interplay between global packing symmetry and local particle symmetry, and the extent to which counterion-headgroup correlations enforce the latter structures in these LLC phases.

Discoveries of Frank–Kasper phases and closely related dodecagonal quasicrystals (DDQCs) in soft, mesoscopic systems have galvanized efforts to unveil the fundamental mechanisms that drive the formation of these remarkably complex micellar packings. Toward this end, we report temperature-dependent small-angle X-ray scattering analyses of a crystalline–amorphous poly­(ethylene oxide)-block-poly­(2-ethyl hexylacrylate) (OA) diblock copolymer with M n = 8300 g/mol, Đ = M w/M n = 1.10, and volume composition f O = 0.21. On heating at ambient temperature, this polymer assembles sequentially into five distinct morphologies prior to melt disordering at T ODT = 69 °C: semicrystalline lamellae (Lc), a liquid-like packing (LLP) of particles lacking translational order, an aperiodically ordered DDQC, a periodic FK σ phase, and a body-centered cubic (BCC) packing of particles. Detailed investigations of thermal processing conditions that foster DDQC formation reveal that this metastable morphology only forms in melts exhibiting LLP characteristics arising from either melting the Lc phase at low temperature or quenching a high-temperature disordered state, and that this DDQC eventually evolves into a σ approximant phase. Cooling a well-ordered BCC phase induces direct formation of a σ phase with no evidence of DDQC formation, suggesting the critical importance of particle size distribution of the disorganized yet segregated LLP state in triggering the emergence of a DDQC.

Small-angle X-ray scattering analyses reveal that the hydrated diblock oligomer n -C 16 H 23 (OCH 2 CH 2 ) 20 -OH (C 16 E 20 or Brij 58) forms lyotropic liquid crystals (LLCs) exhibiting face-centered cubic (FCC), body-centered cubic (BCC), Frank–Kasper (FK) A15, and cylindrical (H I ) morphologies over the concentration range 30–65 wt % amphiphile. Heating LLCs comprising 54–59 wt % C 16 E 20 drives the temperature-dependent phase transition sequence: A15 → BCC → H I . However, rapidly quenching the resulting H I phase from 70 to 25 °C initially forms a BCC phase that isothermally transforms into a complex, tetragonal FK σ phase comprising 30 quasispherical micelles. The metastability of this micellar σ phase is shown to depend on the sample cooling rate, thermal quench depth, and isothermal annealing temperature. We rationalize the preference for the A15 structure at 25 °C in terms of minimizing unfavorable water/hydrophobic contacts, while maximizing local particle sphericity. The symmetry breaking transition kinetics in these micellar LLCs apparently stem from the temperature-dependent activation barriers for phase nucleation and growth, which are intimately coupled to the time scales for micelle reconfiguration by amphiphile chain exchange and their spatial rearrangement. These findings highlight how thermal processing influences nucleation and growth of the self-assembled morphologies of intrinsically reconfigurable, soft spherical particles.

Reanalysis of an asymmetric poly(ethylene-altpropylene)-block-polydimethylsiloxane (PEP-PDMS) diblock copolymer first investigated in 1999 has revealed a rich phase behavior including a dodecagonal quasicrystal (DDQC), a Frank−Kasper σ phase, and a body-centered cubic (BCC) packing at high temperature adjacent to the disordered state. On subjecting the sample to large amplitude oscillatory shear well below the σ-BCC order−order transition temperature (T OOT ), small-angle X-ray scattering evidenced the emergence of a twinned BCC phase that, on heating, underwent a phase transition to an unusually anisotropic DDQC state. Surprisingly, we observe no evidence of this apparent epitaxy on heating or cooling through the equilibrium σ-BCC transition. We rationalize these results in terms of a shear-induced order−order transition and an apparent BCC-DDQC epitaxy favored by micelle translation-mediated ordering dynamics far below T OOT .