Akash Arora

Thermal processing of diblock copolymer melts mimics metallurgy

Ordered phases commonly found in metal alloys are formed in diblock copolymer melts through innovative thermal treatments. When polymers behave like metals Diblock copolymers, in which two dissimilar chains are chemically linked, can show a rich array of morphologies. These are usually attained by slow cooling to give the chains time to find their thermodynamically preferred arrangements. Rather than using slow cooling, Kim et al. rapidly quenched their materials from the disordered state and then annealed at low to moderate temperatures (see the Perspective by Stein). Different processing routes drove assembly into a variety of low-dimensional phases more typical of metal alloys. Science, this issue p. 520; see also p. 487 Small-angle x-ray scattering experiments conducted with compositionally asymmetric low molar mass poly(isoprene)-b-poly(lactide) diblock copolymers reveal an extraordinary thermal history dependence. The development of distinct periodic crystalline or aperiodic quasicrystalline states depends on how specimens are cooled from the disordered state to temperatures below the order-disorder transition temperature. Whereas direct cooling leads to the formation of documented morphologies, rapidly quenched samples that are then heated from low temperature form the hexagonal C14 and cubic C15 Laves phases commonly found in metal alloys. Self-consistent mean-field theory calculations show that these, and other associated Frank-Kasper phases, have nearly degenerate free energies, suggesting that processing history drives the material into long-lived metastable states defined by self-assembled particles with discrete populations of volumes and polyhedral shapes.

Origins of low-symmetry phases in asymmetric diblock copolymer melts

Significance We demonstrate that low-molecular weight asymmetric diblock copolymer melts can form multiple metastable liquid states at a common temperature, dependent on the processing history. Formation of ordered self-assembled micelles at low temperatures shapes the number density of the mesoscopic particles, which is preserved upon heating above the order–disorder transition temperature. Cooling returns the liquid to the same crystalline state reflecting a memory—a type of hidden symmetry—imprinted in the fluid. These surprising results are explained based on the large energetic penalty associated with fusing or fragmenting micelles in the highly structured liquid state. This work reveals concepts related to spontaneous symmetry breaking in self-assembled soft materials including surfactant-based systems. Cooling disordered compositionally asymmetric diblock copolymers leads to the formation of nearly spherical particles, each containing hundreds of molecules, which crystallize upon cooling below the order–disorder transition temperature (TODT). Self-consistent field theory (SCFT) reveals that dispersity in the block degrees of polymerization stabilizes various Frank–Kasper phases, including the C14 and C15 Laves phases, which have been accessed experimentally in low-molar-mass poly(isoprene)-b-poly(lactide) (PI-PLA) diblock copolymers using thermal processing strategies. Heating and cooling a specimen containing 15% PLA above and below the TODT from the body-centered cubic (BCC) or C14 states regenerates the same crystalline order established at lower temperatures. This memory effect is also demonstrated with a specimen containing 20% PLA, which recrystallizes to either C15 or hexagonally ordered cylinders (HEXC) upon heating and cooling. The process-path–dependent formation of crystalline order shapes the number of particles per unit volume, n/V, which is retained in the highly structured disordered liquid as revealed by small-angle X-ray scattering (SAXS) experiments. We hypothesize that symmetry breaking during crystallization is governed by the particle number density imprinted in the liquid during ordering at lower temperature, and this metastable liquid is kinetically constrained from equilibrating due to prohibitively large free energy barriers for micelle fusion and fission. Ordering at fixed n/V is enabled by facile chain exchange, which redistributes mass as required to meet the multiple particle sizes and packing associated with specific low-symmetry Frank–Kasper phases. This discovery exposes universal concepts related to order and disorder in self-assembled soft materials.

Accelerating self-consistent field theory of block polymers in a variable unit cell

Self-consistent field theory (SCFT) is one of the most widely used tools to study the equilibrium phase behavior of block polymers. We have extended an existing version of the Anderson-mixing iteration scheme to solve the highly nonlinear SCFT equations while simultaneously optimizing the unit-cell dimensions. This improved scheme substantially increases the computational efficiency compared to existing schemes.

Stable Frank–Kasper phases of self-assembled, soft matter spheres

Significance Formation of complex Frank–Kasper phases in soft matter systems confounds intuitive notions that equilibrium states achieve maximal symmetry, owing to an unavoidable conflict between shape and volume asymmetry in space-filling packings of spherical domains. Here we show the structure and thermodynamics of these complex phases can be understood from the generalization of two classic problems in discrete geometry: the Kelvin and Quantizer problems. We find that self-organized asymmetry of Frank–Kasper phases in diblock copolymers emerges from the optimal relaxation of cellular domains to unequal volumes to simultaneously minimize area and maximize compactness of cells, highlighting an important connection between crystal structures in condensed matter and optimal lattices in discrete geometry. Single molecular species can self-assemble into Frank–Kasper (FK) phases, finite approximants of dodecagonal quasicrystals, defying intuitive notions that thermodynamic ground states are maximally symmetric. FK phases are speculated to emerge as the minimal-distortional packings of space-filling spherical domains, but a precise measure of this distortion and how it affects assembly thermodynamics remains ambiguous. We use two complementary approaches to demonstrate that the principles driving FK lattice formation in diblock copolymers emerge directly from the strong-stretching theory of spherical domains, in which a minimal interblock area competes with a minimal stretching of space-filling chains. The relative stability of FK lattices is studied first using a diblock foam model with unconstrained particle volumes and shapes, which correctly predicts not only the equilibrium σ lattice but also the unequal volumes of the equilibrium domains. We then provide a molecular interpretation for these results via self-consistent field theory, illuminating how molecular stiffness increases the sensitivity of the intradomain chain configurations and the asymmetry of local domain packing. These findings shed light on the role of volume exchange on the formation of distinct FK phases in copolymers and suggest a paradigm for formation of FK phases in soft matter systems in which unequal domain volumes are selected by the thermodynamic competition between distinct measures of shape asymmetry.