Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry

Fischer, Steffen; Exner, Alexander; Zielske, Kathrin; Perlich, Jan; Deloudi, Sofia; Steurer, Walter; Lindner, Peter; Förster, Stephan

doi:10.1073/pnas.1008695108

Cited by 236 publications

(208 citation statements)

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“…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.…”

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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.

234

362

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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...

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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.

234

362

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“…More recently, quasicrystalline order has been found in an expanding variety of soft materials including self-assembling wedge-shaped dendrimers, mixtures of ligand coated nanoparticles, concentrated solutions of surfactant and block polymer micelles, multicomponent polymer blends containing ABC-type miktoarm star polymers, and an ABA′C-type linear multiblock polymer (3)(4)(5)(6)(7)(8). The underlying principles that govern the formation of quasicrystals in soft materials remain largely unresolved.…”

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confidence: 99%

Dodecagonal quasicrystalline order in a diblock copolymer melt

Gillard

Lee

Bates

2016

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

188

264

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We report the discovery of a dodecagonal quasicrystalline state (DDQC) in a sphere (micelle) forming poly(isoprene-b-lactide) (IL) diblock copolymer melt, investigated as a function of time following rapid cooling from above the order-disorder transition temperature (T ODT = 66°C) using small-angle X-ray scattering (SAXS) measurements. Between T ODT and the order-order transition temperature T OOT = 42°C, an equilibrium body-centered cubic (BCC) structure forms, whereas below T OOT the Frank-Kasper σ phase is the stable morphology. At T < 40°C the supercooled disordered state evolves into a metastable DDQC that transforms with time to the σ phase. The times required to form the DDQC and σ phases are strongly temperature dependent, requiring several hours and about 2 d at 35°C and more than 10 and 200 d at 25°C, respectively. Remarkably, the DDQC forms only from the supercooled disordered state, whereas the σ phase grows directly when the BCC phase is cooled below T OOT and vice versa upon heating. A transition in the rapidly supercooled disordered material, from an ergodic liquid-like arrangement of particles to a nonergodic soft glassy-like solid, occurs below ∼40°C, coincident with the temperature associated with the formation of the DDQC. We speculate that this stiffening reflects the development of particle clusters with local tetrahedral or icosahedral symmetry that seed growth of the temporally transient DDQC state. This work highlights extraordinary opportunities to uncover the origins and stability of aperiodic order in condensed matter using model block polymers.quasicrystals | Frank-Kasper phases | block polymers T he discovery of aperiodic order, today referred to as quasicrystalline order, in rapidly cooled Al-Mn alloys by Shechtman and coworkers in 1984 heralded a paradigm shift in the understanding of what constitutes long-range order in matter (1, 2). More recently, quasicrystalline order has been found in an expanding variety of soft materials including self-assembling wedge-shaped dendrimers, mixtures of ligand coated nanoparticles, concentrated solutions of surfactant and block polymer micelles, multicomponent polymer blends containing ABC-type miktoarm star polymers, and an ABA′C-type linear multiblock polymer (3-8). The underlying principles that govern the formation of quasicrystals in soft materials remain largely unresolved.Block polymers are uniquely tunable self-assembling soft materials in which the nanoscale morphology, characteristic length scale, chemical and physical properties, and dynamics can be precisely controlled through the judicious choice of block repeat unit chemistries, block molecular weights, and molecular architecture (9, 10). Such exquisite control over structure, and the associated dynamics, makes block polymers ideal model materials for fundamental inquiries into the formation and stability of soft quasicrystals. In this article, we report the discovery of a long-lived metastable dodecagonal quasicrystalline phase (DDQC) in a (nominally) single-component poly(1,4-is...

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“…However, QCs have recently been found in nanoparticles [2], mesoporous silica [3], and soft matter [4] systems. The latter include micellar melts [5,6] formed, e.g., from linear, dendrimer or star block copolymers. Recently, three-dimensional (3D) icosahedral QCs have been found in molecular dynamics simulations of particles interacting via a three-well pair potential [7].…”

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confidence: 99%

“…The local (linear) stability of the equilibria is obtained by linearizing the amplitude equations (6). The regions of local stability extend beyond the lines demarcating the boundaries of the regions of global stability, and many locally stable structures can coexist at given parameter combinations.…”

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Three-Dimensional Icosahedral Phase Field Quasicrystal

et al. 2016

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We investigate the formation and stability of icosahedral quasicrystalline structures using a dynamic phase field crystal model. Nonlinear interactions between density waves at two length scales stabilize threedimensional quasicrystals. We determine the phase diagram and parameter values required for the quasicrystal to be the global minimum free energy state. We demonstrate that traits that promote the formation of two-dimensional quasicrystals are extant in three dimensions, and highlight the characteristics required for three-dimensional soft matter quasicrystal formation. DOI: 10.1103/PhysRevLett.117.075501 Periodic crystals form ordered arrangements of atoms or molecules with rotation and translation symmetries, and possess discrete x-ray diffraction patterns, or equivalently, discrete spatial Fourier spectra. In contrast, quasicrystals (QCs) lack the translational symmetries of periodic crystals, yet also display discrete spatial Fourier spectra. QCs made from metal alloys were discovered in 1982 [1] and attracted the Nobel prize for chemistry in 2011. QCs can be quasiperiodic in all three dimensions (e.g., with icosahedral symmetry), or can be quasiperiodic in two (or one) directions while being periodic in one (or two). The vast majority of the QCs discovered so far are metallic alloys (e.g., Al/Mn or Cd/Ca). However, QCs have recently been found in nanoparticles [2], mesoporous silica [3], and soft matter [4] systems. The latter include micellar melts [5,6] formed, e.g., from linear, dendrimer or star block copolymers. Recently, three-dimensional (3D) icosahedral QCs have been found in molecular dynamics simulations of particles interacting via a three-well pair potential [7].In recent years, model systems in two dimensions (2D) have been studied in order to understand soft matter QC formation and stability [8][9][10][11][12]. Phase field crystal models have been employed to simulate the growth of 2D QCs [13] and the adsorption properties on a quasicrystalline substrate [14]. The ingredients for 2D quasipattern formation are, first, a propensity towards periodic density modulations with two characteristic wave numbers k 1 and k 2 [15-18]. The ratio k 2 =k 1 must be close to certain special values; e.g., for dodecagonal QCs the value is 2 cosðπ=12Þ. Second, strong reinforcing (i.e., resonant) nonlinear interactions between these two characteristic density waves are required [17,19,20]. Earlier work on quasipatterns observed in Faraday wave experiments reveals similar requirements [19,[21][22][23]. We demonstrate here, following Mermin and Troian [24], that these same requirements suffice to stabilize icosahedral QCs in 3D. In contrast, nonlinear resonant interactions between density waves at a single wavelength are important in stabilizing simple crystal structures, such as body-centered cubic (bcc) crystals [25] although, with the right coupling, QCs can also be stabilized [26].We consider a 3D phase field crystal (PFC) model, appropriate for soft matter systems, that generates modulations with two l...

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Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry

Cited by 236 publications

References 22 publications

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

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

Dodecagonal quasicrystalline order in a diblock copolymer melt

Three-Dimensional Icosahedral Phase Field Quasicrystal

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