Frank-Kasper Phases Research Articles

Two-Dimensional Frank−Kasper Z Phase with One Unit-Cell Thickness

Z phase is one of the three basic units by which the Frank-Kasper (F-K) phases are generally assembled. Compared to the other two basic units, that is, A15 and C15 structures, the Z structure is rarely experimentally observed because of a relatively large volume ratio among the constituents to inhibit its formation. Moreover, the discovered Z structures are generally the three-dimensional ordered Gibbs bulk phases to conform to their thermodynamic stability. Here, we confirmed the existence of a metastable two-dimensional F-K Z phase that has only one unit-cell height in the crystallography in a model Mg-Sm-Zn system, using atomic-scale scanning transmission electron microscopy combined with the first-principles calculations. Self-adapted atomic shuffling can convert the simple hexagonal close-packed structure to the topologically close-packed F-K Z phase. This finding provides new insight into understanding the formation mechanism and clustering behavior of the F-K phases and even quasicrystals in general condensed matters.

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.

Self-organisation of rhombitruncated cuboctahedral hexagonal columns from an amphiphilic Janus dendrimer

Frank-Kasper phases are available in both hard and soft complex matter. They have been discovered in metal alloys and subsequently in self-organisations of supramolecular spheres from self-assembling dendrons, dendrimers and dendronized polymers. Recently, they were found in block copolymers, lipids, surfactants, giant surfactants, nanoparticles, DNA particles and even in condensed small molecules such as N2, O2, and CO. Here we report the discovery of an amphiphilic Janus dendrimer, that self-assembles in water into vesicles known as dendrimersomes, self-organised also in bulk state into helical columns and spheres forming columnar hexagonal and a Frank-Kasper A15 phase known as Pm n. These self-organisations display a supramolecular orientational memory (SOM) effect that is induced by an epitaxial nucleation at the transition from columns to spheres and during the reverse process. This SOM effect is mediated by the closed contact supramolecular spheres from the A15 phase. In this case a rhombitruncated cuboctahedral hexagonal columnar arrangement, that to the best of our knowledge is not known in biology or synthetic chemistry, was self-organised. We believe that the addition of amphiphilic Janus dendrimers to the classes of molecules displaying Frank-Kasper phases and the SOM effect will broaden our ability to discover additional complex soft condensed matter morphologies.

Quasicrystals and Their Approximants in a Crystalline–Amorphous Diblock Copolymer

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 Mₙ = 8300 g/mol, Đ = Mw/Mₙ = 1.10, and volume composition fO = 0.21. On heating at ambient temperature, this polymer assembles sequentially into five distinct morphologies prior to melt disordering at TODT = 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.

Assembly of Structural Types of Frank–Kasper Phases Cr3Si, MgCu2 and MgZn2 from a Universal Building Unit

Assembly of Structural Types of Frank–Kasper Phases Cr3Si, MgCu2 and MgZn2 from a Universal Building Unit

Complex structures arising from the self-assembly of a simple organic salt.

Molecular self-assembly is the spontaneous association of simple molecules into larger and ordered structures1. It is the basis of several natural processes, such as the formation of colloids, crystals, proteins, viruses and double-helical DNA2. Molecular self-assembly has inspired strategies for the rational design of materials with specific chemical and physical properties3, and is one of the most important concepts in supramolecular chemistry. Although molecular self-assembly has been extensively investigated, understanding the rules governing this phenomenon remains challenging. Here we report on a simple hydrochloride salt of fampridine that crystallizes as four different structures, two of which adopt unusual self-assemblies consisting of polyhedral clusters of chloride and pyridinium ions. These two structures represent Frank-Kasper (FK) phases of a small and rigid organic molecule. Although discovered in metal alloys4,5 more than 60 years ago, FK phases have recently been observed in several classes of supramolecular soft matter6-11 and in gold nanocrystal superlattices12 and remain the object of recent discoveries13. In these systems, atoms or spherical assemblies of molecules are packed to form polyhedra with coordination numbers 12, 14, 15 or 16. The two FK structures reported here crystallize from a dense liquid phase and show a complexity that is generally not observed in small rigid organic molecules. Investigation of the precursor dense liquid phase by cryogenic electron microscopy reveals the presence of spherical aggregates with sizes ranging between 1.5 and 4.6 nanometres. These structures, together with the experimental procedure used for their preparation, invite interesting speculation about their formation and open different perspectives for the design of organic crystalline materials.

Impact of Architectural Asymmetry on Frank-Kasper Phase Formation in Block Polymer Melts.

In recent decades, the discoveries of complex low-symmetry phases in soft matter have inspired advances in molecular and materials design. However, understanding the mechanisms underlying symmetry selection across soft matter remains an important challenge in materials science. Block polymers represent attractive model materials because they permit wide synthetic tunability and provide access to multiple length scales (1-100 nm). However, to date the block polymer design space has been largely limited to variations in molecular weight, block volume fraction, and conformational asymmetry. The molecular architecture-the way in which chains are connected-offers rich potential but remains relatively unexplored in experimental block polymers. Our work bridges this gap, connecting molecular architecture, space-filling demands, and symmetry selection in block polymer self-assembly. Three series of block polymers were synthesized by living polymerization, tuning the architectural asymmetry across the linear-b-linear and linear-b-bottlebrush limits. The bottlebrush architecture amplifies two key ingredients for the formation of Frank-Kasper phases: high conformational asymmetry and high self-concentration. Analysis by small-angle X-ray scattering provides insight into the impact of architectural asymmetry on block polymer self-assembly. Increasing the asymmetry between blocks opens the complex phase window, expanding opportunities to tune symmetry selection in block polymer melts.

Discrete Block Copolymers with Diverse Architectures: Resolving Complex Spherical Phases with One Monomer Resolution.

This work describes the first rigorous example of a single-component block copolymer system forming unconventional spherical phases. A library of discrete block polymers with uniform chain length and diverse architectures were modularly prepared through a combination of a step-growth approach and highly efficient coupling reactions. The precise chemical structure eliminates all the molecular defects associated with molar weight, dispersity, and compositional ratio. Complex spherical phases, including the Frank–Kasper phase (A15 and σ) and quasicrystalline phase, were experimentally captured by meticulously tuning the composition and architectures. A phase portrait with unprecedented accuracy was mapped out (up to one monomer resolution), unraveling intriguing details of phase behaviors that have long been compromised by inherent molecular weight distribution. This study serves as a delicate model system to bridge the existing gaps between experimental observations and theoretical assessments and to provide insights into the formation and evolution of the unconventional spherical phases in soft matter systems.

Symmetry breaking in particle-forming diblock polymer/homopolymer blends

Compositionally asymmetric diblock copolymers provide an attractive platform for understanding the emergence of tetragonally close-packed, Frank-Kasper phases in soft matter. Block-polymer phase behavior is governed by a straightforward competition between chain stretching and interfacial tension under the constraint of filling space at uniform density. Experiments have revealed that diblock copolymers with insufficient conformational asymmetry to form Frank-Kasper phases in the neat-melt state undergo an interconversion from body-centered cubic (bcc) close-packed micelles to a succession of Frank-Kasper phases (σ to C14 to C15) upon the addition of minority-block homopolymer in the dry-brush regime, accompanied by the expected transition from bcc to hexagonally packed cylinders in the wet-brush regime. Self-consistent field theory data presented here qualitatively reproduce the salient features of the experimental phase behavior. A particle-by-particle analysis of homopolymer partitioning furnishes a basis for understanding the symmetry breaking from the high-symmetry bcc phase to the lower-symmetry Frank-Kasper phases, wherein the reconfiguration of the system into polyhedra of increasing volume asymmetry delays the onset of macroscopic phase separation.

Emergence of a C15 Laves Phase in Diblock Polymer/Homopolymer Blends.

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 (Mn = 33.3 kg/mol, Đ = Mw/Mn = 1.08, fB = 0.18) with two different poly(1,4-butadiene) (B) homopolymer additives. When the B additive Mn 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 Mn 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 Mn 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.

Supramolecular spheres assembled from covalent and supramolecular dendritic crowns dictate the supramolecular orientational memory effect mediated by Frank–Kasper phases

A supramolecular orientational memory (SOM) effect that provides access to complex columnar hexagonal architectures that cannot be designed by any other methodology was recently reported by our laboratory. This SOM concept is encountered upon heating and cooling a hexagonal columnar periodic array via a Frank–Kasper phase that is generated from supramolecular spheres. Here, a library of twenty-four supramolecular spheres self-organizing into Frank–Kasper phases and columnar hexagonal arrays was investigated for SOM. Twenty of these supramolecular spheres, that were self-assembled from conical shaped dendrons, did not exhibit SOM. The four supramolecular spheres assembled from covalent and supramolecular dynamic crowns displayed SOM. The supramolecular dynamic crowns forming spheres by a strong and directional H-bonding process, generated via SOM, a previously unencountered distorted dodecahedral architecture constructed from hexagonally ordered supramolecular columns. Therefore, SOM can discriminate between mechanisms of self-assembly of supramolecular spheres, a process that cannot be accomplished by analysis with X-ray diffraction methods alone. These results will facilitate access to the design of new complex columnar hexagonal and other periodic columnar array architectures via SOM.

Small-Molecule Modulation of Soft-Matter Frank-Kasper Phases: A Method for Adding Function to Form.

Incorporation of small amounts of α-tocopherol (vitamin E) in blends with the cellobiose-triazole-linked atactic poly(4-methyl-1-pentene) (CB-aPMP) sugar-polyolefin conjugate can be used to exert external control over thermotropic phase behavior and provide access to non-canonical soft matter Frank-Kasper A15 and σ phases. These results establish a paradigm that can be used for the further design and development of scalable quantities of soft matter FK phases of increased structural complexity and functional capability.

Small‐Molecule Modulation of Soft‐Matter Frank–Kasper Phases: A Method for Adding Function to Form

AbstractIncorporation of small amounts of α‐tocopherol (vitamin E) in blends with the cellobiose–triazole‐linked atactic poly(4‐methyl‐1‐pentene) (CB‐aPMP) sugar–polyolefin conjugate can be used to exert external control over thermotropic phase behavior and provide access to non‐canonical soft matter Frank–Kasper A15 and σ phases. These results establish a paradigm that can be used for the further design and development of scalable quantities of soft matter FK phases of increased structural complexity and functional capability.

A15, σ, and a Quasicrystal: Access to Complex Particle Packings via Bidisperse Diblock Copolymer Blends.

A renewed focus on the phase behavior of nominally single-component, compositionally asymmetric diblock copolymers has revealed a host of previously unanticipated Frank-Kasper (FK) and quasicrystalline phases. However, these periodic and aperiodic particle packings have thus far only been reported in low molecular weight, highly conformationally asymmetric diblock copolymers, leaving researchers with a relatively small library of polymers in which these phases can be studied. In this work, we report on a simple approach to access these morphologies: blending two diblock copolymers with the same corona block length and varied core block lengths. Compositionally symmetric and asymmetric polystyrene-b-1,4-polybutadiene (SB) diblock copolymers with constant corona block lengths were blended together and shown via small-angle X-ray scattering and transmission electron microscopy to order into the FK A15 and σ phases, as well as a dodecagonal quasicrystal, providing a route to various particle packings in high molecular weight diblock copolymer melts.

An Exceptionally Stable and Scalable Sugar-Polyolefin Frank-Kasper A15 Phase.

"One-component" soft material Frank-Kasper (FK) phases are an intriguing structural form of matter that possess periodically ordered structures arising from the self-reconfiguration and close packing of an initial assembly of identical "deformable" spheres into two or more size- or shape-distinct sets of particles. Significant challenges that must still be addressed to advance the field of soft matter FK phases further, however, include their rare and unpredictable occurrence, uncertain mechanisms of solid-state assembly, and low thermodynamic stability. Here we show that a readily-accessible sugar-polyolefin conjugate quantitatively produces an exceptionally stable solid-state FK A15 phase through a rapid and irreversible thermotropic order-order transition, which contrary to other prevailing proposed mechanisms, does not require mass transfer between particles or large structural reorganization in the bulk to establish unit cell non-equivalency. Our results provide the basis for a realistic strategy for obtaining practical and scalable quantities of a diverse range of sugar-polyolefin FK A15 phases with unique intrinsic physical properties and chemical reactivities not previously seen in such systems.

An Exceptionally Stable and Scalable Sugar–Polyolefin Frank–Kasper A15 Phase

Abstract“One‐component” soft material Frank–Kasper (FK) phases are an intriguing structural form of matter that possess periodically ordered structures arising from the self‐reconfiguration and close packing of an initial assembly of identical “deformable” spheres into two or more size‐ or shape‐distinct sets of particles. Significant challenges that must still be addressed to advance the field of soft matter FK phases further, however, include their rare and unpredictable occurrence, uncertain mechanisms of solid‐state assembly, and low thermodynamic stability. Here we show that a readily‐accessible sugar–polyolefin conjugate quantitatively produces an exceptionally stable solid‐state FK A15 phase through a rapid and irreversible thermotropic order–order transition, which contrary to other prevailing proposed mechanisms, does not require mass transfer between particles or large structural reorganization in the bulk to establish unit cell non‐equivalency. Our results provide the basis for a realistic strategy for obtaining practical and scalable quantities of a diverse range of sugar–polyolefin FK A15 phases with unique intrinsic physical properties and chemical reactivities not previously seen in such systems.

Dissipative particle dynamics for directed self-assembly of block copolymers.

The dissipative particle dynamics (DPD) simulation method has been shown to be a promising tool to study self-assembly of soft matter systems. In particular, it has been used to study block copolymer (BCP) self-assembly. However, previous parameterizations of this model are not able to capture most of the rich phase behaviors of BCPs in thin films nor in directed self-assembly (chemoepitaxy or graphoepitaxy). Here, we extend the applicability of the DPD method for BCPs to make it applicable to thin films and directed self-assembly. Our new reparameterization not only is able to reproduce the bulk phase behavior but also manages to predict thin film structures obtained experimentally from chemoepitaxy or graphoepitaxy. A number of different complex structures, such as bilayer nanomeshes, 90° bend structures, circular cylinders/lamellae and Frank-Kasper phases directed by trenches, and post arrays or chemically patterned substrates, have all been reproduced in this work. This reparameterized DPD model should serves as a powerful tool to predict BCP self-assembly, especially in some complex systems where it is difficult to implement self-consistent field theory.

A cornerstone of complex crystals.

Frank–Kasper phases are topologically close-packed structures typically found in ordered mixtures of ‘hard particles’ such as metallic alloys. Now, a shape amphiphile has been shown to self-assemble into a Frank–Kasper Z phase, which had so far remained elusive in soft materials.

Identification of a Frank-Kasper Z phase from shape amphiphile self-assembly.

Frank-Kasper phases, a family of ordered structures formed from particles with spherical motifs, are found in a host of materials, such as metal alloys, inorganic colloids and various types of soft matter. All the experimentally observed Frank-Kasper phases can be constructed from the basic units of three fundamental structures called the A15, C15 and Z phases. The Z phase, typically observed in metal alloys, is associated with a relatively large volume ratio between its constituents, and this constraint inhibits its formation in most self-assembled single-component soft-matter systems. We have assembled a series of nanosized shape amphiphiles that comprise a triphenylene core and six polyhedral oligomeric silsesquioxane cages grafted onto it through linkers to give a variety of unconventional structures, which include the Z phase. This structure was obtained through fine tuning of the linker lengths between the core and the peripheral polyhedral oligomeric silsesquioxane cages, and exhibits a relatively large volume asymmetry between its constituent polyhedral particle motifs.

Frank–Kasper Phases Identified in PDMS‐b‐PTFEA Copolymers with High Conformational Asymmetry

In the search for the formation of Frank-Kasper phases from diblock copolymer self-assembly, a series of compositionally asymmetric poly(dimethylsiloxane)-b-poly(2,2,2-triflouroethyl acrylate)s (PDMS-b-PTFEAs) are synthesized to produce PDMS-rich phases with PDMS volume fractions (fPDMS ) ranging from 0.746 to 0.869. As determined by small-angle X-ray scattering analysis, the Frank-Kasper σ and C14 phases are identified at fPDMS = 0.796 and 0.851, respectively, plausibly due to high conformational asymmetry (ε ≈ 2.20) between the two blocks. Intriguingly, the σ phase develops during heating from a short-range liquid-like packing (LLP) state, whereas the C14 phase is achieved at room temperature, which are both followed by a disordering at higher temperatures. Based on thermal experiments from a super cooled disordered state, the findings further provide compelling evidence of an LLP-hexagonally packed cylinder-σ transition and a direct pathway to the C14 phase during heating from an LLP state.