Frank–Kasper, quasicrystalline and related phases in liquid crystals

Ungar, Goran; Zeng, Xiangbing

doi:10.1039/b502443a

Cited by 198 publications

(172 citation statements)

References 79 publications

Supporting

Mentioning

162

Contrasting

Unclassified

Order By: Relevance

“…The inclusion of a second length scale provides greater control over the self-assembly of the clusters than what can be achieved with a single minimum only. We demonstrate how this enables one to generate a wide range of novel periodic and aperiodic cluster crystal structures.Our findings come at a time when an ever-growing number of soft-matter systems are found to exhibit phases with quasiperiodic long-range order-mostly showing dodecagonal symmetry [12]. First discovered in liquid crystals made of amphiphilic dendritic macromolecules [13], self-assembled soft quasicrystals have since appeared in ABC-star polymers [14], in sphere-forming block PRL 113, 098304 (2014) P H Y S I C A L…”

mentioning

confidence: 79%

Controlled Self-Assembly of Periodic and Aperiodic Cluster Crystals

2014

View full text Add to dashboard Cite

Soft particles are known to overlap and form stable clusters that self-assemble into periodic crystalline phases with density-independent lattice constants. We use molecular dynamics simulations in two dimensions to demonstrate that, through a judicious design of an isotropic pair potential, one can control the ordering of the clusters and generate a variety of phases, including decagonal and dodecagonal quasicrystals. Our results confirm analytical predictions based on a mean-field approximation, providing insight into the stabilization of quasicrystals in soft macromolecular systems, and suggesting a practical approach for their controlled self-assembly in laboratory realizations using synthesized soft-matter particles. DOI: 10.1103/PhysRevLett.113.098304 PACS numbers: 82.70.Dd, 61.44.Br, 64.70.D-, 64.75.Yz Particles interacting via pair potentials with repulsive cores, which are either bounded or only slowly diverginglike those found naturally in soft matter systems [1]-can be made to overlap under pressure to form clusters [2], which then self-assemble to form crystalline phases [3]. The existence of such cluster crystals was recently confirmed in amphiphilic dendritic macromolecules using monomer-resolved simulations [4], and in certain bosonic systems [5]. They occur even when the particles are purely repulsive, and typically exhibit periodic fcc or bcc structures. Here we employ molecular dynamics (MD) simulations in two dimensions, guided by analytical insight, to show how isotropic pair potentials can be designed to control the self-assembly of the clusters, suggesting a practical approach that could be applied in the laboratory. We obtain novel phases, including a striped (lamellar) phase and a hexagonal superstructure, as well as decagonal (tenfold) and dodecagonal (twelvefold) quasicrystals.Given a system of N particles in a box of volume V, interacting via an isotropic pair potential UðrÞ with a repulsive core, a sufficient condition for the formation of a cluster crystal is a negative global minimumŨ min ¼Ũðk min Þ < 0 in the Fourier transform of the potential [6]. This condition implicitly requires the potential not to diverge too strongly, so that the Fourier transform exists. The wave number k min determines the length scale for the order in the system by setting the typical distance between neighboring clusters. Above a sufficiently high mean particle densityc ¼ N=V, a further increase ofc increases the number of overlapping particles within each cluster, but does not change the distance between their centers. It also determines the spinodal temperature k B T sp ¼ −Ũ minc [3,6], below which the liquid becomes unstable against crystallization, where k B is the Boltzmann constant.As the particles form increasingly larger clusters, the system becomes well characterized by a continuous coarse-grained density function cðrÞ. The thermodynamic behavior can then be described in a mean-field approximation, which becomes exact in the high-density hightemperature limit [7]. Using MD simulations, we ex...

show abstract

mentioning

confidence: 79%

Controlled Self-Assembly of Periodic and Aperiodic Cluster Crystals

2014

View full text Add to dashboard Cite

show abstract

“…The scientific literature is replete with examples of Frank-Kasper phases in hard materials, particularly in the area of intermetallics (7)(8)(9), but also in a few complex elemental crystals, including manganese (10,11) and uranium (12). Recently, this class of crystalline order has cropped up in a host of soft materials, including dendrimers (13), surfactant solutions (14), and block polymers (15,16), often in close proximity to QC phases (17)(18)(19). To the best of our knowledge the principles underlying the formation of Frank-Kasper phases across both categories of materials have not been established, presenting enticing challenges to scientist and engineers bent on controllably arranging atoms and molecules for specific materials applications.…”

mentioning

confidence: 99%

Sphericity and symmetry breaking in the formation of Frank–Kasper phases from one component materials

Lee

Leighton

Bates

2014

Proc. Natl. Acad. Sci. U.S.A.

230

362

View full text Add to dashboard Cite

Frank-Kasper phases are tetrahedrally packed structures occurring in numerous materials, from elements to intermetallics to selfassembled soft materials. They exhibit complex manifolds of Wigner-Seitz cells with many-faceted polyhedra, forming an important bridge between the simple close-packed periodic and quasiperiodic crystals. The recent discovery of the Frank-Kasper σ-phase in diblock and tetrablock polymers stimulated the experiments reported here on a poly(isoprene-b-lactide) diblock copolymer melt. Analysis of small-angle X-ray scattering and mechanical spectroscopy exposes an undiscovered competition between the tendency to form self-assembled particles with spherical symmetry, and the necessity to fill space at uniform density within the framework imposed by the lattice. We thus deduce surprising analogies between the symmetry breaking at the body-centered cubic phase to σ-phase transition in diblock copolymers, mediated by exchange of mass, and the symmetry breaking in certain metals and alloys (such as the elements Mn and U), mediated by exchange of charge. Similar connections are made between the role of sphericity in real space for polymer systems, and the role of sphericity in reciprocal space for metallic systems such as intermetallic compounds and alloys. These findings establish new links between disparate materials classes, provide opportunities to improve the understanding of complex crystallization by building on synergies between hard and soft matter, and, perhaps most significantly, challenge the view that the symmetry breaking required to form reduced symmetry structures (possibly even quasiperiodic crystals) requires particles with multiple predetermined shapes and/or sizes.symmetry breaking | sphericity | Frank-Kasper phases | block polymers T he discovery of materials with aperiodic order, often referred to as "quasicrystals," 30 years ago (1, 2) heralded new and promising vistas for designing materials endowed with unique properties. In the 1950s Frank and Kasper (3, 4) recognized complex tetrahedral atomic-and molecular-packing geometries that bridge the familiar close-packed crystals [e.g., face-centered cubic (FCC), hexagonally close-packed (HCP), and body-centered cubic (BCC) structures] characterized by periodic order, and quasiperiodic crystals (QCs) that extend crystallography beyond the 230 space groups relevant to periodic crystals (5, 6). The scientific literature is replete with examples of Frank-Kasper phases in hard materials, particularly in the area of intermetallics (7-9), but also in a few complex elemental crystals, including manganese (10, 11) and uranium (12). Recently, this class of crystalline order has cropped up in a host of soft materials, including dendrimers (13), surfactant solutions (14), and block polymers (15, 16), often in close proximity to QC phases (17)(18)(19). To the best of our knowledge the principles underlying the formation of Frank-Kasper phases across both categories of materials have not been established, presenting enticing challenges to sc...

show abstract

“…Although squishy spheres do form BCC and FCC crystals, they also form tetrahedrally closest-packed Frank-Kasper (FK) phases that contain combinations of 12-, 14-, 15-, and 16-coordinate lattice sites (11)(12)(13)(14). The first FK A15 (15) and C15 (16) phases in ionic surfactant and lipidic LLCs were identified over 30 y ago, yet the principles governing their formation remain poorly understood (17).…”

mentioning

confidence: 99%

Low-symmetry sphere packings of simple surfactant micelles induced by ionic sphericity

Kim

Jeong

Yethiraj

et al. 2017

Proc. Natl. Acad. Sci. U.S.A.

118

137

View full text Add to dashboard Cite

Supramolecular self-assembly enables access to designer soft materials that typically exhibit high-symmetry packing arrangements, which optimize the interactions between their mesoscopic constituents over multiple length scales. We report the discovery of an ionic small molecule surfactant that undergoes water-induced selfassembly into spherical micelles, which pack into a previously unknown, low-symmetry lyotropic liquid crystalline Frank-Kasper σ phase. Small-angle X-ray scattering studies reveal that this complex phase is characterized by a gigantic tetragonal unit cell, in which 30 sub-2-nm quasispherical micelles of five discrete sizes are arranged into a tetrahedral close packing, with exceptional translational order over length scales exceeding 100 nm. Varying the relative concentrations of water and surfactant in these lyotropic phases also triggers formation of the related Frank-Kasper A15 sphere packing as well as a common body-centered cubic structure. Molecular dynamics simulations reveal that the symmetry breaking that drives the formation of the σ and A15 phases arises from minimization of local deviations in surfactant headgroup and counterion solvation to maintain a nearly spherical counterion atmosphere around each micelle, while maximizing counterion-mediated electrostatic cohesion among the ensemble of charged particles.self-assembly | liquid crystals | surfactants | Frank-Kasper phases | lyotropic phase M olecular self-assembly provides a facile means of constructing a plethora of multifunctional soft materials, with mesoscopic structures that dictate their tailored properties and performance applications. Driven by noncovalent interactions between constituents, block polymers (1), giant shape amphiphiles (2), thermotropic liquid crystals (LCs) (3), lyotropic liquid crystals (LLCs) (4), and colloids (5) exemplify soft matter systems that spontaneously form periodic 1D lamellar phases, 2D columnar structures, and 3D packings of spherical particles. Columnar and spherical phases are useful as templates for mesoporous heterogeneous catalysts (6) and as microscale photonic bandgap materials (7). Manipulating supramolecular self-assembly to achieve specific materials morphologies and functions requires a fundamental understanding of the interplay between the structure and symmetry of the constituents and their multibody interactions.Although the packing of spherical objects (e.g., oranges and billiard balls) seems intuitively simple, point particles form a dizzying array of periodic crystals, quasicrystals (QCs), and structurally disordered glasses. Metallic elements typically form high-symmetry body-centered cubic (BCC), hexagonally closest-packed, and facecentered cubic (FCC) structures, due to the isotropy of metallic cohesion mediated by itinerant electrons (8). A few pure elements (e.g., Mn and U) form low-symmetry crystals with large and complex unit cells that maximize metallic cohesion against local constraints, such as maximization of Fermi surface sphericity (9).Sphere-forming soft mater...

show abstract

Frank–Kasper, quasicrystalline and related phases in liquid crystals

Cited by 198 publications

References 79 publications

Controlled Self-Assembly of Periodic and Aperiodic Cluster Crystals

Controlled Self-Assembly of Periodic and Aperiodic Cluster Crystals

Sphericity and symmetry breaking in the formation of Frank–Kasper phases from one component materials

Low-symmetry sphere packings of simple surfactant micelles induced by ionic sphericity

Contact Info

Product

Resources

About