Triangular Phases Research Articles

Directing the formation of tunable superlattice crystalline phases from anisotropic nanoparticles

The ability to precisely control the self-assembly of colloidal nanoparticles into desired and tunable superstructures holds immense potential, but achieving this remains a challenge due to intricate interplay of various controlling factors that govern this process at the nanoscale. Here, we demonstrate the controlled fabrication of co-existing smectic and triangular crystalline phases of gold nanorods, separated by domain boundary, by fine tuning the interparticle and particle–substrate interaction. The interaction forces between nanorods is tailored via the addition of surfactant, cetyltrimethylammonium bromide (CTAB), which in turn affects the particle–substrate attraction through screened electrostatic potential due to surfactant adsorption on the substrate. The emergence of the resulting phases is discussed via simulation of a coarse-grained model which considers the nanorod superlattices as nematic liquid crystals. It accounts for the interplay of repulsive, attractive forces, and particle–substrate interactions. The co-existing superlattice structures are obtained at low particle concentration over a large length scale, paving the way for various potential applications that specifically require both smectic and vertical rods on the same substrate.

Highly branched bolapolyphilic liquid crystals with a cubic A15 network at the triangle-square transition.

Rod-like bolapolyphiles with highly branched carbosilane-based side-chains self-assemble into several honeycomb structures if the oligo(p-phenylene ethynylene) core is polyfluorinated, whereas for the non-fluorinated series an A15 type cubic network of rod-bundles was observed instead, suggesting a brand new pathway for the transition between triangular and square honeycomb phases.

Transformations of spike and burst oscillations in the stochastic Rulkov model

We study the problem of modeling and analysis of complex oscillatory regimes in neuron dynamics on the basis of the two-dimensional Rulkov model. We focus on the parameter zone in which sharp changes in the type of oscillatory activity occur: burst oscillations with an active phase of a triangular shape are transformed into bimodal bursts combining triangular and trapezoidal phases. Mechanisms of such transformations are clarified geometrically by the technique of critical curves. For the stochastically forced model, we study two phenomena: (i) noise-induced splitting of tonic spiking; (ii) noise-induced transitions from the triangular mono-bursting to the more complicated bi-bursting. The phenomena of coherence and anti-coherence resonance are revealed and discussed.

Phases of surface-confined trivalent colloidal particles.

Patchy colloids promise the design and modelling of complex materials, but the realization of equilibrium patchy particle structures remains challenging. Here, we assemble pseudo-trivalent particles and elucidate their phase behaviour when confined to a plane. We observe the honeycomb phase, as well as more complex amorphous network and triangular phases. Structural analysis performed on the three condensed phases reveals their shared structural motifs. Using a combined experimental and simulation approach, we elucidate the energetics of these phases and construct the phase diagram of this system, using order parameters to determine the phase coexistence lines. Our results reveal the rich phase behaviour that a relatively simple patchy particle system can display, and open the door to a larger joined simulation and experimental exploration of the full patchy-particle phase space.

Phase-field crystal method for multiscale microstructures with cubic term

One mode phase-field crystal model is introduced to describe morphological transition in the presence of a cubic term. The application of the field leads to a comprehensive pattern formation with stripe, triangular and liquid phases. Combining with amplitude phase-field crystal model, some steady solution of amplitudes and wave number are easily obtained. The convergence between growth morphology and size domain is investigated. The results indicate that the parameter of cubic term is important to the crystalline phase, especially the triangular phase, in the 2D phase diagrams and it can obtain different crystal shapes by the point from 2D phase diagram. The competition growth of multi grains on larger square computational domain was also investigated. In addition, the method can be extended to other solidification microstructures.

Strain-induced stripe phase in charge-ordered single layer NbSe2

Charge density waves are ubiquitous phenomena in metallic transition metal dichalcogenides. In NbSe2, a triangular 3 × 3 structural modulation is coupled to a charge modulation. Recent experiments reported evidence for a triangular-stripe transition at the surface, due to strain or accidental doping and associated to a 4 × 4 modulation. We employ ab initio calculations to investigate the strain-induced structural instabilities in a pristine single layer and analyse the energy hierarchy of the structural and charge modulations. Our results support the observation of phase separation between triangular and stripe phases in 1H-NbSe2, relating the stripe phase to compressive isotropic strain, favouring the 4 × 4 modulation. The observed wavelength of the charge modulation is also reproduced with a good accuracy.

Growth modes of quasicrystals involving intermediate phases and a multistep behavior studied by phase field crystal model

Understanding the microscopic kinetics of quasicrystal formation via nucleation and growth is crucial. Here, we report unusual pathways to nucleation and growth of dodecagonal quasicrystals via a phase field crystal model incorporating a two-length-scale potential. Under certain thermodynamic parameters, both quasicrystal growths via heterogeneous and homogeneous nucleation may be associated with a multistep behavior and the transient appearance of triangular and intermediate phases, different from classical nucleation pathways. The metastable intermediate phase spontaneously occurs to bridge the triangular phase and quasicrystal nuclei of different orientations to reduce the total free energy of the system. Decomposition of an undercooled fluid phase into quasicrystal phase shows a multistep pathway wherein the triangular phase and the intermediate phase may occur faster than the quasicrystal phase, when the growth rate of one length-scale ordering is significantly different from the other and the subsequent competing and coupling of both length scales are involved. The calculated structure factor, radial distribution function, and the aperiodic tiling structure of the intermediate phase explain why it appears during the quasicrystal formation.

α-Cyanostilbene and fluorene based bolaamphiphiles: synthesis, self-assembly, and AIEE properties with potential as white-light emissive materials and light-emitting liquid crystal displays

Reported bolaamphiphiles self-assembled into triangular honeycomb phases with unique molecular arrangements, and showed the AIEE effect. Their potentials as both WLED and LE-LCD devices have been initially demonstrated.

Bolapolyphiles via Click Reaction: Effect of Flexible Polyaromatic Backbone Involving Polar Hydrogen Bonding Units on Their Self‐assembly

AbstractThree series of bolapolyphiles consisting of a flexible diphenylthiophene polyaromatic backbone involving hydrogen bonding triazole units and two terminal glycerol groups as well as lateral alkyl chains (being either linear or branched) have been synthesized by CuAAC click reaction. The effect of the flexible polyaromatic backbone, as well as the length, number and branching or linearity of the lateral chain on their self‐assembly was investigated. The backbone flexibility led to the formation of structures intermediate between the well defined honeycombs of the rigid bolapolyphiles and the conventional flexible amphiphiles. T‐shaped bolapolyphiles with one lateral chain occupied small volume (one branched chain or one short linear chain) display triangular honeycomb phases. Whereas T‐shaped or π‐shaped bolapolyphiles with one or two lateral chain occupied larger volume (one longer linear or two linear chain) display square honeycomb phases with diffused or less defined cylinder wall structures. Further these compounds can self assemble into organogels in organic solvents. Their photophysical properties are affected greatly by the type of lateral chains, and they can act as chemosensors towards Fe3+.

Influence of Ni and ZrO2 contents on sintering and mechanical properties of WC-2wt.%ZrO2-1wt.%Ni composites

WC-2wt.%ZrO2-1wt.%Ni composites were prepared by vacuum pressureless sintering (VPS) and post-hot isostatic pressing (post-HIP). The microstructure, phase composition, densification, hardness, fracture toughness, and flexural strength of composites with different Ni and ZrO2 contents were systematically investigated. The results show that WC-Ni and WC-ZrO2-Ni composites prepared by VPS can be densified by the addition of a small amount of Ni as the binder phase. Moreover, the densities of the composites can be further enhanced i.e. the composites are rendered nearly fully dense following HIP while the grains remain fine without obvious growth. The binder Ni and ZrO2 phases are uniformly distributed in the WC matrix and exhibit high bonding strength with it. The hardness, fracture toughness, and flexural strength of the WC-ZrO2-Ni composites following HIP could reach 22.4 GPa, 12.0 MPa m1/2, and 1101.2 MPa, respectively. Based on the influence of the Ni and ZrO2 contents on the microstructures and mechanical properties of the WC-2wt.%ZrO2-1wt.%Ni composites, the fracture mechanism was determined to be governed by the phase transformation of ZrO2 that led to the development of some micro-cracks followed by deflection, bridging, and branching of the cracks to improve fracture toughness. The composites are mainly composed of elongated triangular prismatic WC grains and ZrO2 phases, and hence the fracture mode can change such that transgranular fracture becomes the main fracture mode accompanied by a small amount of intergranular fracture. Thus, the flexural strength of the composites can be improved.

Two-dimensional TiOx nanostructures on Au(111): a scanning tunneling microscopy and spectroscopy investigation

We investigated the growth of titanium oxide two-dimensional nanostructures on Au(111), produced by Ti evaporation and post-deposition oxidation. Scanning tunneling microscopy and spectroscopy (STM and STS) and low-energy electron diffraction measurements characterized the morphological, structural and electronic properties of the observed structures. Five distinct TiOx phases were identified: the honeycomb and pinwheel phases appear as monolayer films wetting the gold surface, while nanocrystallites of the triangular, row and needle phases grow mainly over the honeycomb or pinwheel layers. Density Functional Theory investigation of the honeycomb structure supports a structural model based on a Ti-O bilayer having Ti2O3 stoichiometry. The pinwheel phase was observed to evolve, for increasing coverage, from single triangular crystallites to a well-ordered film forming a superstructure, which can be interpreted within a moiré-like model. Structural characteristics of the other three phases were disclosed from the analysis of high-resolution STM measurements. STS measurements revealed a partial metallization of honeycomb and pinwheel and a semiconducting character of row and triangular phases.

Light-induced phase transitions of colloidal monolayers with crystalline order

We experimentally study the phase behavior of a charge-stabilized two-dimensional colloidal crystal which is subjected to a one-dimensional periodic light field. Such light fields are created by a scanned optical line tweezer which allows the variation of the periodicity without optical realignments. In order to realize a wide range of line spacings relative to the lattice constant, we use a suspension of silica particles in bromobenzene. This colloidal system has a Debye screening length of about 4.6 μm which results in the formation of crystals with lattice constants up to 20 μm. Because the refractive index of bromobenzene is larger than that of the colloids, optical gradient forces lead to the attraction of particles at regions where the intensity is smallest. Depending on the depth and periodicity of the optical potential, we observe the light-induced assembly of colloids into triangular, rhombic and square phases.

Surface ordering of molecular structures by dispersion forces

The ubiquitous dispersion forces still escape both an accurate experimental determination and a complete theoretical formulation. As a case in point, here it is presented a numerical study about the formation of ordered structures of ${\text{C}}_{60}$ molecules on the Si:B(111) surface by means of atomic-scale computer simulations. Order-order transitions between one linear and two triangular surface phases are found, which fit well the results of various experimental observations. It is shown that the transitions are driven by subtle differences between intermolecular- and molecule-surface forces. Eventually, combined molecular dynamics and lattice Monte Carlo models can be compared to experimental data as a unique method to deduce the relative and absolute intensities of dispersion interactions.

Freezing and Melting of a Colloidal Adsorbate on a 1D Quasicrystalline Substrate

Using Monte Carlo simulations and an extended Landau-Alexander-McTague theory, we demonstrate that colloids in a one-dimensional quasicrystalline potential order in triangular and rhombic-alpha crystalline phases. Increasing the strength of the potential further, a new type of light-induced melting is discovered that has its origin in the nonperiodicity of the potential. In contrast to reentrant melting in periodic potentials, the quasicrystalline potential melts the crystalline phases even when they already exist at zero potential.

Global phase diagram for the honeycomb potential

We calculate the global phase diagram using classical statistical mechanics for an isotropic pair potential that has been previously [Rechtsman et al., Phys. Rev. Lett. 95, 228301 (2005)] shown to produce the low-coordinated two-dimensional honeycomb crystal as the ground-state structure. Low-coordinated crystals are of practical interest because they have desirable photonic band-gap properties. The phase diagram is obtained from Helmholtz free energies calculated using thermodynamic integration and Monte Carlo simulations. Our results show that the honeycomb crystal remains stable in the global phase diagram even after temperature effects are taken fully into account. Other stable phases in the phase diagram are high and low density triangular phases and a fluid phase. We find no evidence of gas-liquid or liquid-liquid phase coexistence.

QUANTUM PHASES IN PARTIALLY FILLED LANDAU LEVELS

We compare the energies of different electron solids, such as bubble crystals with triangular and square symmetry and stripe phases, to those of correlated quantum liquids in partially filled intermediate Landau levels. Multiple transitions between these phases when varying the filling of the top-most partially filled Landau level explain the observed reentrance of the integer quantum Hall effect. The phase transitions are identified as first-order. This leads to a variety of measurable phenomena such as the phase co-existence between a Wigner crystal and a two-electron bubble phase in a Landau-level filling-factor range 4.15≲ν≲4.26, which has recently been observed in transport measurements under micro-wave irradiation.

Structure and phase behavior of a two-dimensional system with core-softened and long-range repulsive interactions.

The structure and phase behavior of a two-dimensional system with purely repulsive core-softened and long-range interactions are studied using Monte Carlo computer simulations. The pair interactions are of the form, u(r)=4epsilon[(sigma/r)(12)-(sigma/r)(6)]+epsilon(')(sigma/r)(3), with the energy parameter, epsilon(')=(8sqrt[6]/9)epsilon chosen to give a stationary point of inflection in the pair potential at r=6(1/6)sigma. This potential approximates the effective interparticle interactions for a two-dimensional dipolar system in a strong field aligned perpendicular to the plane. The low-temperature portion of the phase diagram is sketched out, and the static properties of the various phases are analyzed in some detail. At low temperatures a variety of interesting states are in evidence, including: fluids with chainlike, striped, and 6-10 sided polygon structural motifs; low-density and high-density triangular crystalline phases; and defective Kagomé lattices. It is shown that clustering is driven by the presence of the repulsive shoulder in the pair potential. Other features, such as the presence of a disordered phase with a network structure, are due to the long-range (1/r(3)) repulsive tail in the potential. The relevance of the simulation results to experimental work, including materials synthesis, is briefly discussed.

Trigonometry of the quantum state space, geometric phases and relative phases

A complete set of invariants for three states in the quantum space of states is obtained together with a complete set of relationships linking them. This is done in a way that preserves the self-duality of and leads to a clear geometric description of invariants (distances, lateral phases; Hermitian angles, angular phases; and two purely triangular phases). Some of these invariants appear here for the first time. Symplectic area (and hence the triangle geometric phase) is proportional to a 'mixed phase excess', thus extending to the relation 'area-angular excess' in the real sphere. The new triangle lateral phases provide a description, intrinsic to , of relative phases in a superposition. This approach also provides closed expressions for the triangle holonomy associated with the usual Fubini–Study metric in , as well as many other expressions for similar 'loop' operators along the triangle, including closed and exact expressions for the triangle Aharonov–Anandan geometric phase.

Domain textures of multi-q modulated phases

Modulated phases in solids may exhibit different textural organizations and phase transitions between them. The incommensurate phase of quartz is taken as an example to illustrate the approach to multi-q phases in terms of domain textures and to explain the phase transition between the equilateral- and elongated- triangular phases, recently observed by transmission electron microscopy. The macroscopic properties of these phases are discussed. Other systems are also briefly mentioned.

Exact results for interacting electrons in high Landau levels.

We study a two-dimensional electron system in a magnetic field with a fermion hardcore interaction and without disorder. Projecting the Hamiltonian onto the n-th Landau level, we show that the Hartree-Fock theory is exact in the limit n \rightarrow \infty, for the high temperature, uniform density phase of an infinite system; for a finite-size system, it is exact at all temperatures. In addition, we show that a charge-density wave arises below a transition temperature T_t. Using Landau theory, we construct a phase diagram which contains both unidirectional and triangular charge-density wave phases. We discuss the unidirectional charge-density wave at zero temperature and argue that quantum fluctuations are unimportant in the large-n limit. Finally, we discuss the accuracy of the Hartree-Fock approximation for potentials with a nonzero range such as the Coulomb interaction.