Stability of Two-Dimensional Dodecagonal Quasicrystalline Phase of Block Copolymers

Design of Linear Block Copolymers and ABC Star Terpolymers That Produce Two Length Scales at Phase Separation

Quasicrystals (materials with long-range order but without the usual spatial periodicity of crystals) were discovered in several soft matter systems in the last 20 years. The stability of quasicrystals has been attributed to the presence of two prominent length scales in a specific ratio, which is 1.93 for the 12-fold quasicrystals most commonly found in soft matter. We propose design criteria for block copolymers such that quasicrystal-friendly length scales emerge at the point of phase separation from a melt, basing our calculations on the Random Phase Approximation. We consider two block copolymer families: linear chains containing two different monomer types in blocks of different lengths, and ABC star terpolymers. In all examples, we are able to identify parameter windows with the two length scales having a ratio of 1.93. The models that we consider that are simplest for polymer synthesis are, first, a monodisperse ALBASB melt and, second, a model based on random reactions from a mixture of AL, AS, and B chains: both feature the length scale ratio of 1.93 and should be relatively easy to synthesize.

Self-assembly of dodecagonal and octagonal quasicrystals in hard spheres on a plane

Hard spheres are one of the most fundamental model systems in soft matter physics, and have been instrumental in shedding light on nearly every aspect of classical condensed matter. Here, we add one more important phase to the list that hard spheres form: quasicrystals. Specifically, we use simulations to show that an extremely simple, purely entropic model system, consisting of two sizes of hard spheres resting on a flat plane, can spontaneously self-assemble into two distinct random-tiling quasicrystal phases. The first quasicrystal is a dodecagonal square-triangle tiling, commonly observed in a large variety of colloidal systems. The second quasicrystal has, to our knowledge, never been observed in either experiments or simulations. It exhibits octagonal symmetry, and consists of three types of tiles: triangles, small squares, and large squares, whose relative concentration can be continuously varied by tuning the number of smaller spheres present in the system. The observed tile composition of the self-assembled quasicrystals agrees very well with the theoretical prediction we obtain by considering the four-dimensional (lifted) representation of the quasicrystal. Both quasicrystal phases form reliably and rapidly over a significant part of parameter space. Our results demonstrate that entropy combined with a set of geometrically compatible, densely packed tiles can be sufficient ingredients for the self-assembly of colloidal quasicrystals.

Inverse Design of Complex Block Copolymers for Exotic Self-Assembled Structures Based on Bayesian Optimization

Variable chain topologies of multiblock copolymers provide great opportunities for the formation of numerous self-assembled nanostructures with promising potential applications. However, the consequent large parameter space poses new challenges for searching the stable parameter region of desired novel structures. In this Letter, by combining Bayesian optimization (BO), fast Fourier transform-assisted 3D convolutional neural network (FFT-3DCNN), and self-consistent field theory (SCFT), we develop a data-driven and fully automated inverse design framework to search for the desired novel structures self-assembled by ABC-type multiblock copolymers. Stable phase regions of three exotic target structures are efficiently identified in high-dimensional parameter space. Our work advances the new research paradigm of inverse design in the field of block copolymers.

The Largest Quasicrystalline Tiling with Dodecagonal Symmetry from a Single Pentablock Quarterpolymer of the AB1CB2D Type

A quasicrystalline tiling pattern with tile size of ca. 60 nm has been discovered in the bulk state of a four-component pentablock polymer molecule of the AS₁IS₂P type, where A, S, I, and P denote poly(4-vinylbenzyldimethylamine), polystyrene, polyisoprene, and poly(2-vinylpyridine), respectively. The polymer samples used were prepared by anionic polymerizations and have narrow molecular weight distribution. The sample films were obtained by an extremely slow solvent-cast process from dilute solutions of tetrahydrofuran for 14 days. It has been found by TEM observation that the quarterpolymer, AS₁IS₂P-4 (M_n = 149 kg/mol, ϕ_A/ϕ_S1/ϕ_I/ϕ_S2/ϕ_P = 0.12/0.27/0.20/0.29/0.13), reveals two final stable structures, i.e., a 3.3.4.3.4 periodic tiling pattern as a minor component and a quasicrystalline (QC) tiling with dodecagonal symmetry as a major component, where the former includes a triangle/square number ratio of 2 and the latter has one of approximately 2.28, which is close enough to the ideal ratio, 4/ 3 ≑ 2.31, for the triangle/square random tiling of the dodecagonal QC tiling (DDQC). Two structures were also clearly proved by SAXS diffraction patterns. Here, it should be noted this QC structure having a tile side length of ca. 60 nm was created with a single block polymer molecule.

Rectangular Cylinders Formed by Compositionally Bidisperse ABC Triblock Terpolymer Blends: A Self-Consistent Field Theory Study

Compared with traditional cylinders that have circular cross-sections, cylinders with rectangular cross-sections can endow nanomaterials with various novel optical properties and functions. In this work, the formation of the rectangular cylinders self-assembled by compositionally bidisperse ABC triblock terpolymer blends has been investigated via numerical simulations based on self-consistent field theory. The specially designed blending systems are composed of two types of linear ABC triblock terpolymers that have the same total chain lengths and the middle B block chain lengths, but different chain lengths of the side A/C blocks. By tuning the chain length fractions and the interactions between different blocks, rectangular cylinders with a fourfold symmetry pattern were successfully obtained in our simulations. Each rectangular phase domain is self-assembled together by the short and long side blocks of the same species. The simulation results indicate that the selective aggregation of the short side blocks determines the formation of the rectangular cylindrical phase, i.e., the short side blocks prefer to aggregate at the four corners within a rectangular cylindrical phase domain. This simulation result reveals a formation mechanism that is different from the mechanism proposed in previous experiments [Asai ACS Macro Lett., 2014, 3, 166-169]. Moreover, under different middle B block chain length fractions, phase diagrams as a function of the interaction parameter between different blocks and the short side block chain length fraction have been constructed. The phase diagrams show that the parameter window of the rectangular cylinders is considerably expanded by increasing the chain length fraction of the middle B blocks. Our simulation works can provide a theoretical basis for molecular design to regulate and fabricate nanomaterials with nontraditional phase domains in future experiments.

Reevaluation of Poly(ethylene-alt-propylene)-block-Polydimethylsiloxane Phase Behavior Uncovers Topological Close-Packing and Epitaxial Quasicrystal Growth

Reanalysis of an asymmetric poly(ethylene-alt-propylene)-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.

Frank-Kasper Phases Self-Assembled from a Linear A1B1A2B2 Tetrablock Copolymer

The Frank-Kasper (FK) phases self-assembled from block copolymer systems have attracted abiding interest. In this work, the formation mechanism of the complex FK phases from the self-assembly of simple A₁B₁A₂B₂ tetrablock copolymers is investigated using self-consistent field theory (SCFT). For a typical set of parameter spaces, we utilize SCFT to construct a number of phase diagrams. In these phase diagrams, the FK phases exhibit a notable stability region. The stable region of the FK phases reveals that the distribution of A₁ and A₂ blocks can be precisely regulated by tuning the ratio of the A₁/A₂ block, wherein the long A₁ blocks can aggregate within the "core" while the short A₂ blocks can form the "shell" of a spherical domain in the FK phases, respectively, to accommodate the sizes and shapes of the spherical domains in the complex spherical packing phases. Besides, we also demonstrate that the existence of the B₂ block plays a crucial factor to stabilize the FK phases.

Hexagonally Packed Cylindrical Structures with Multiple Satellites from Pentablock Quarterpolymers of the AB1CB2D Type and Their Blends with Homopolymers

Three kinds of hexagonally packed cylindrical (HPC) structures with 10/8 satellites have been found from a neat pentablock quarterpolymer of the AS₁IS₂P type, composed of poly(4-vinylbenzyldimethylamine) (A), polystyrene (S), polyisoprene (I), and poly(2-vinylpyridine) (P), and block polymer/homopolymer blends. AS₁IS₂P-2 (M = 276k, ϕ_A/ϕ_S1/ϕ_I/ϕ_S2/ϕ_P = 0.08/0.17/0.12/0.33/0.30) shows HPC of P with 10 single-arrayed I satellites and 6 A subsatellites in the S matrix, where the I/P ratio number is 5, while AS₁IS₂P-2 (M = 325k, ϕ_A/ϕ_S1/ϕ_I/ϕ_S2/ϕ_P = 0.07/0.31/0.14/0.18/0.30)/S_h(M;11k) = 90/10 blend reveals another HPC of P with 10 I satellites, giving an I/P ratio of 6, whose array of I is a single/double mixture. Moreover, AS₁IS₂P-1/S_h/A_h(M;9k) = 90/5/5 blend represents two enantiomeric HPC packings with 8 double-arrayed I satellites having an I/P ratio of 7, which can be considered to be approximants of two enantiomeric 3.3.3.3.6 Archimedean tilings. These structures were conceived to be formed by the enhanced miscibility of tail A-chains with added homopolymers, resulting in an increase of the (A + S)/I interaction leading to the split of I domains.

Order and Disorder in ABCA' Tetrablock Terpolymers

Self-assembly of poly(styrene)-block-poly(isoprene)-block-poly(lactide)-block-poly(styrene) (PS-PI-PLA-PS' or SILS') tetrablock terpolymers, where the volume fractions of the first three blocks are nearly equivalent, was studied both experimentally and using the self-consistent field theory (SCFT). SCFT indicates that addition of the terminal PS' chain to a low-molecular-mass, hexagonally packed cylinders forming, SIL precursor can produce a disordered state due to preferential mixing of the polystyrene end-blocks with the PI and PLA midblocks in the SILS' tetrablock, alleviating the unfavorable contact between the highly incompatible PI and PLA segments. In contrast, SCFT predicts that higher-molar-mass triblock precursors will maintain an ordered morphology upon addition of the terminal PS' block due to stronger overall segregation strengths. These predictions were tested using three sets of SILS' polymers that were synthesized based on three precursor SIL triblock polymers differing in total molar mass (14, 30, and 47 kg mol^-1) and varying the length of the terminal PS' chain. In the lowest-molar-mass set of tetrablock polymers, the shift from order to disorder was observed in the materials at ambient temperature as the molar mass of the terminal PS' block was increased, consistent with SCFT calculations. Disorder with longer S' chain lengths was not found in the two higher-molar-mass polymer sets; the medium-molar-mass set showed both microphase separation and long-range order based on transmission electron microscopy (TEM) and small-angle X-ray scattering (SAXS), while the largest of these block polymers microphase separated but showed limited long-range order. The combination of the experimental and theoretical results presented in this work provides insights into the self-assembly of ABCA'-type polymers and highlights potential complications that arise from frustration in accessing well-ordered materials.

Periodic and Aperiodic Tiling Patterns from a Tetrablock Terpolymer System of the A1BA2C Type

Two tetrablock terpolymers of the S₁IS₂P type, where S, I, and P denote polystyrene, polyisoprene, and poly(2-vinylpyridine), respectively, were prepared anionically. S₁IS₂P-1 (S₁/I/S₂/P = 0.35/0.16/0.34/0.15, four numbers being volume fractions of S₁, I, S₂, and P block chains) has a structure with double hexagonal cylinders, while S₁IS₂P-2 (S₁/I/S₂/P = 0.47/0.15/0.22/0.16) has an unusual double tetragonal structure. Moreover, 13 binary blends were prepared from these two parent polymers. Among them, five blends with α(= φ_S1/φ_S2) covering the range 1.50 ≤ α ≤ 1.86 were confirmed to have a 3.3.4.3.4 Archimedean tiling structure, in which their P domains adopt five satellite I domains, while four blends with α covering the range 1.37 ≤ α ≤ 1.48 were revealed to be a quasicrystalline tiling structure with dodecagonal symmetry.