Centered Rectangular Lattice Research Articles

Low-temperature ordering of the dimer phase of a two-dimensional model of core-softened particles.

Purely pairwise interactions of the core-softened type, i.e., featuring a soft repulsion followed by a hard-core interaction at shorter distance, give rise to nontrivial equilibrium structures entirely different from the standard close packing of spheres. In particular, in a suitable low-temperature region of their phase diagram, such interactions are well known to favor a transition from a fluid to a cluster crystal. The residual mutual interaction between individual clusters can lead to the formation of patterns of their reciprocal orientations. In this work, we investigate two examples of such models in two dimensions, at the density most appropriate to the dimer phase, whereby clusters consist of just two particles, studying them with optimization techniques and Monte Carlo simulations. We focus on the dimer crystal, and unveil a second phase transition at extremely low temperature. This transition leads from a triangular dimer lattice with randomly disordered dimer orientations at high temperature to a reduced-symmetry ground state with nematic orientational order and a slightly distorted structure characterized by a centered-rectangular lattice at low temperature.

Self-Assembly of 2D Gold Nanoparticle Superlattice in a Polymer Vesicle Layer Driven by Hydrophobic Interaction.

Self-assembly of gold nanoparticles (AuNPs) into highly ordered superstructures provides a promising route toward fabricating materials with new functionalities or enhanced physical properties. Although self-assembly of AuNPs has garnered significant research attention recently, a highly ordered superlattice of AuNPs under a low concentration in a confined geometry formed by nonfunctionalized materials has not been reported. Herein, we investigate the self-assembly of a 2D AuNPs superlattice in a polymer vesicle layer using hydrophobic interactions, which exhibits centered rectangular lattice symmetry. To create the highly ordered AuNPs superlattice, the P(EGx-b-iPGEy) block copolymers that form the thickness of the hydrophobic vesicle layer comparable to the size of the AuNP are used as a template to control the AuNP degree of freedom. To the best of our knowledge, this study provides the first demonstration of a centered rectangular structure formation of AuNPs at the vesicle layer in 2D confined geometry.

X-ray Fiber Diffraction and Computational Analyses of Stacked Hexads in Supramolecular Polymers: Insight into Self-Assembly in Water by Prospective Prebiotic Nucleobases

Aqueous solutions of equimolar mixtures of 2,4,6-triaminopyrimidine (TAP) and carboxylic acid substituted cyanuric acid (CyCo6 or R-4MeCyCo6) monomers self-assemble into gel-forming supramolecular polymers. Macroscopic fibers drawn from these mixtures were analyzed by X-ray diffraction to determine their molecular structures. Computational methods were used to explore the intrinsic intermolecular interactions that contribute to the structure and stability of these assemblies. Both polymers are formed by the stacking of hexameric rosettes, (TAP/CyCo6)3 or (TAP/R-4MeCyCo6)3, respectively, into long, stiff, twisted stacks of essentially planar rosettes. Chiral, left-handed supramolecular polymers with a helical twist angle of -26.7° per hexad are formed when the pure enantiomer R-4MeCyCo6 is used. These hexad stacks pack into bundles with a hexagonal crystalline lattice organization perpendicular to the axis of the macroscopic fiber. Polymers formed from TAP and CyCo6, both of which are achiral, assemble into macroscopic domains that are packed as a centered rectangular lattice. Within these domains, the individual polymers exist as either right-handed or left-handed helical stacks, with twist angles of +15° or -15° per hexad, respectively. The remarkable ability of TAP and cyanuric acid derivatives to self-assemble in water, and the structural features of their supramolecular polymers reported here, provide additional support for the proposal that these heterocycles could have served as recognition units for an early form of nucleic acids, before the emergence of RNA.

Morphology of poly(lactide)‐block‐poly(dimethylsiloxane)‐block‐polylactide high‐χ triblock copolymer film studied by grazing incidence small‐angle X‐ray scattering

AbstractThe morphologies of poly(lactide)‐block‐poly(dimethylsiloxane)‐block‐polylactide (PLA‐b‐PDMS‐b‐PLA) triblock copolymer films were studied using a combination of grazing‐incidence small‐angle X‐ray scattering, X‐ray reflectivity and scanning electron microscopy. This block copolymer is characterized by a high Flory–Huggins interaction parameter which leads to the self‐assembly of periodic high‐resolution nanodomains. In this article, we performed a detailed analysis of GISAXS patterns, in the frame of the Distorted Wave Born Approximation, in order to determine the morphology of blocks and their spatial arrangement. For a low volume fraction of PLA (17%), a three‐dimensional hexagonal lattice of PLA spherical blocks is revealed, while, for a PLA fraction twice larger, in‐plane (parallel) PLA lying cylinders adopt a two‐dimensional centered rectangular lattice. Moreover, the in‐depth electron density profile of the polymer film for the cylindrical morphology was extracted from the XRR data, revealing the presence of interfacial layers at the top surface and at the interface with the Si substrate.

Ribbon Phase of Dendrimer–Surfactant Complexes

The cylindrical micelles formed by amphiphilic surfactants in aqueous media generally organize in two-dimensional (2D) hexagonal lattice because of their largely circular cross section. Here we demonstrate that the packing habit of the cylindrical micelles of anionic sodium dodecyl sulfate (SDS) was perturbed by the electrostatic binding with a positively charged poly(amidoamine) (PAMAM) G4 dendrimer via cooperative deformation for charge matching, leading to the ribbon phase with nonhexagonal 2D lattices. The cylindrical micelles with elliptic cross section packed in the centered rectangular lattice with orthorhombic cmm symmetry over a broad range of surfactant-to-dendrimer amine group molar ratio. The packing structure transformed to the oblique lattice with monoclinic p2 symmetry at the stoichiometric composition. Detailed analysis of the geometry of the dendrimers accommodated in the interstitial tunnels surrounded by the SDS cylinders revealed that the dendrimer molecules were elongated into prolate...

Magnetic Properties and Critical and Compensation Temperatures in Mixed Spin-1/2–Spin-1 Ferrimagnetic-Centered Rectangular Structure Using Monte Carlo Simulation

The Monte Carlo simulation has been used to investigate the magnetic properties of a mixed spin-1 (sublattice A) and spin-1/2 (sublattice B), where each sublattice has a rectangular structure and the whole lattice system forms a 2-D centered rectangular lattice. Five possible interactions are considered. Namely, two S–S interactions and two $\sigma $ – $\sigma $ interactions depend on crystal direction, and an antiferromagnetic interaction between the two sublattices is taken. Our results show that the critical temperature increases with increasing the variation of any exchange interactions. The existence of the compensation temperature depends on the balance between the S–S interactions along [10] and $\sigma $ – $\sigma $ interactions along [01] directions.

The Magnetic Properties of a Centered Rectangular Lattice with Mixed Spins Using XY Model

The XY model has been used in a centered rectangular lattice structure consists of two sub-lattices of spin 1–1/2 with different ferromagnetic couplings on each lattice and anti-ferromagnetic interaction between the two sub-lattices. Utilizing the Monte Carlo simulation, we have studied the energy per site, the specific heat, the total magnetization, the initial susceptibility, and hysteresis effects. Our results revealed that existences of two peaks on the specific heat and a single loop historic behavior. The Binder cumulant and initial susceptibility both confirmed the existence of critical temperature where the system possesses a phase transition to paramagnetism. Such behaviors are monitoring by the different coupling constants adopted in this study.

Nonequilibrium solid-solid phase transition in a lattice of liquid jets

Solid-solid phase transitions are commonly encountered at the atomic scale in alloys and in superatomic mesoscopic systems of colloidal particles. Here we investigate a solid-solid phase transition occurring at the macroscopic scale between lattices of liquid jets with different symmetries generated by convection in a horizontal layer of a binary liquid mixture. In the absence of a shear stress, upwelling and downwelling jets arrange into two staggered square lattices with a spacing of approximately 3 mm. Applying a shear stress triggers a phase separation of the square patterns into two centered-rectangular lattices drifting into opposite directions, each lattice being made either by upwelling or downwelling jets. This structural phase transition is reversible. The macroscopic nature of the system allows us to investigate the kinetics of the transition by direct visualization with shadowgraphy. The mechanism of the transition depends on the path followed. It occurs through a nucleation and growth mechanism when the shear stress is imposed, and through a martensitic transformation of the lattice when the stress is removed.

Robust edge states of planar phononic crystals beyond high-symmetry points of Brillouin zones

Robust edge states of the phononic crystals (PnCs) with the Dirac points fixed at corners or centers of the Brillouin zones (BZs) have received wide attentions. Here, we show that the Dirac points can degenerate at the high symmetrical boundaries of the first irreducible BZs. The mechanism of such Dirac dispersion is systematically discussed in the square lattice, the rectangular lattice, the centered rectangular lattice and the triangular lattice. These degenerated points, characterized by the vortex structure in a momentum space, are attributed to the band crossing in the presence of the crystalline mirror reflection symmetries. Variations of the geometric parameters can shift the positions of these Dirac points, but not lift them. By breaking the mirror reflection symmetries, the corresponding Dirac points will be lifted, yielding the directional or complete bandgaps. The phononic bandgaps efficiently block the propagation of the acoustic waves inside the bulks of the phononic systems, while their edge states localized at the domain walls between the PnCs with the distinct topological phases ensure the robust propagation of the sound energy flows. These results are unambiguously verified and determined by the experimental tests on the PnCs with the crystalline planar group P3m1.

Lamellar Liquid Crystals of In-Plane Lying Rod-Like Mesogens with Designer Side-Chains: The Case of Sliding versus Locked Layers.

The dimensionality of self-assembled nanostructures plays an essential role for their properties and applications. Herein, an understanding of the transition from weakly to strongly coupled layers in soft matter systems is provided involving in-plane organized π-conjugated rods. For this purpose, bolaamphiphilic triblock molecules consisting of a rigid biphenyl core, polar glycerol groups at the ends, and a branched (swallow-tail) or linear alkyl or semiperfluoroalkyl chain in lateral position have been synthesized and investigated. Besides weakly coupled lamellar isotropic (LamIso ), lamellar nematic (LamN ) and sliding lamellar smectic phases (LamSm ), a sequence of three distinct types of strongly coupled (correlated) lamellar smectic phases with either centered (c2mm) or non-centered rectangular (p2mm) lattice and an intermediate oblique lattices (p2) were observed depending on chain length, chain branching and degree of chain fluorination. This new sequence is explained by the strengthening of the layer coupling and the competition between energetic packing constraints and the entropic contribution of either longitudinal or tangential fluctuations. This example of directed side chain engineering of small generic model compounds provides general clues for morphological design of two-dimensional and three-dimensionally coupled lamellar systems involving larger π-conjugated molecular rods and molecular or supramolecular polymers, being of actual interest in organic electronics and nanotechnology.

First-principles study on Jahn-Teller effect in Cr monolayer film

Computational physics has been used in many scientific research fields, in which first-principles calculation based on density functional theory has made brilliant achievements. Unlike three-dimensional materials, low-dimensional materials present fantastic physical effect, due to the reduction of material dimensions. With the rapid development of two-dimensional materials, people have a more in-depth understanding of them. Requirements for high performance of two-dimensional materials are raised for potential applications, so the exploration of some effects affecting the stability of two-dimensional materials becomes more and more important. Based on the pioneers' work, Jahn-Teller effect is found to have a certain influence on the stabilities of two-dimensional structure of some elements. In the present paper, we explain the stable structure of Cr monolayer film through theoretical calculation, providing a guidance for experimental synthesis. Using first-principles calculation, we study a series of two-dimensional structures (rectangular, square, hexagonal, oblique and centered rectangular) of Cr monolayer film, focusing on the structural stability and electronic properties. Firstly, the equilibrium lattice constant and cohesive energy of each structure are calculated. Then, the bond angle and lattice constant dependence of the total energy are analyzed in detail. Finally, we investigate the energy band structures, total electronic densities of states, charge densities and electron occupation numbers of orbitals. The results show that low-symmetry oblique and centered rectangular lattice are stable in the two-dimensional system of Cr, while high-symmetry square and hexagonal lattices are not stable and the adhesive energy of the rectangular lattices is very small. Two stable structures of Cr monolayer sheet are formed due to hexagonal structure distortion. The hexagonal structure can shape into a centered rectangular structure with the increase of bond angle, while it changes into an oblique structure with the decrease of bond angle. Because of Jahn-Teller effect, the degenerate energy level spontaneously splits. Then the structure deforms into two reduced-symmetry structures, resulting in a stable system. Therefore, we can infer that the Jahn-Teller effect plays a crucial role in the structural stability of monolayer sheet.

Nanopatterned Adhesive, Stretchable Hydrogel to Control Ligand Spacing and Regulate Cell Spreading and Migration.

Spatial molecular patterning enables the regulation of adhesion receptor clustering and can thus play a pivotal role in multiple biological activities such as cell adhesion, viability, proliferation, and differentiation. A wide range of nanopatterned, adhesive interfaces have been designed to decipher the essence of molecular-scale interactions between cells and the adhesive interface. Although an interligand spacing of less than 70 nm is a proven prerequisite for the formation of stable focal adhesions, there is a paucity of data concerning how cells behave on substrates featuring heterogeneous adhesiveness. In this study, a stretchable hydrogel functionalized with a quasi-hexagonally arranged nanoarray was stretched along one direction, resulting in ligands periodically arranged in a pattern resembling a centered rectangular lattice with an interligand spacing smaller than 70 nm in one direction and greater than 70 nm in the orthogonal direction. This substrate was utilized to modulate interligand spacing and investigate cell adhesion and migration. An interligand spacing larger than 70 nm-even in just one direction-prevented the establishment of stable focal adhesions. The stretched interface promoted dynamic remodeling at cell contacts, resulting in higher cellular mobility. Our nanopatterned stretchable hydrogel permits reversible control over cell adhesion and migration on nanopatterned ligand interfaces.

Hofstadter butterfly evolution in the space of two-dimensional Bravais lattices

The self-similar energy spectrum of a particle in a periodic potential under a magnetic field, known as the Hofstadter butterfly, is determined by the lattice geometry as well as the external field. Recent realizations of artificial gauge fields and adjustable optical lattices in cold atom experiments necessitate the consideration of these self-similar spectra for the most general two-dimensional lattice. In a previous work, we investigated the evolution of the spectrum for an experimentally realized lattice which was tuned by changing the unit cell structure but keeping the square Bravais lattice fixed. We now consider all possible Bravais lattices in two dimensions and investigate the structure of the Hofstadter butterfly as the lattice is deformed between lattices with different point symmetry groups. We model the optical lattice by a sinusoidal real space potential and obtain the tight binding model for any lattice geometry by calculating the Wannier functions. We introduce the magnetic field via Peierls substitution and numerically calculate the energy spectrum. The transition between the two most symmetric lattices, i.e. the triangular and the square lattice displays the importance of bipartite symmetry featuring deformation as well as closing of some of the major energy gaps. The transition from the square to rectangular and from the triangular to centered rectangular lattices are analyzed in terms of coupling of one-dimensional chains. We calculate the Chern numbers of the major gaps and Chern number transfer between bands during the transitions. We use gap Chern numbers to identify distinct topological regions in the space of Bravais lattices.

Epitaxial Electrodeposition of Fe on Au(111): Structure, Nucleation, and Growth Mechanisms

Iron epitaxial electrodeposition on Au(111) substrates is investigated using in situ scanning tunneling microscopy (STM) and ex situ X-ray diffraction (XRD). STM observations show that Fe grows quasi layer-by-layer at sufficiently negative potentials. XRD results indicate that Fe layers thicker than 3 ML are bcc and present the epitaxial relationship Fe(110) [1–10]||Au(111) [11–2]. They also show that the Fe lattice is uniaxially in-plane strained along Fe[1–10] and that the strain is progressively relieved with increasing layer thickness. The growth of Fe on a Ni layer deposited on Au(111) leads to strain free Fe layers with the same epitaxial relationship. The specific shape of Fe of monatomic islands suggests that the first Fe monolayer deposited on Au(111) presents a centered rectangular lattice similar to that of bcc Fe(110) but is stretched along Fe[1–10] by more than 8%.

Noncircular skyrmion and its anisotropic response in thin films of chiral magnets under a tilted magnetic field

We study the equilibrium and dynamical properties of skyrmions in thin films of chiral magnets with oblique magnetic field. The shape of an individual skyrmion is non-circular and the skyrmion density decreases with the tilt angle from the normal of films. As a result, the interaction between two skyrmions depends on the relative angle between them in addition to their separation. The triangular lattice of skyrmions under a perpendicular magnetic field is distorted into a centered rectangular lattice for a tilted magnetic field. For a low skyrmion density, skyrmions form a chain like structure. The dynamical response of the non-circular skyrmions depends on the direction of external currents.

Effects of surface pressure on the properties of Langmuir monolayers and interfacial water at the air-water interface.

The effects of surface pressure on the physical properties of Langmuir monolayers of palmitic acid (PA) and dipalmitoylphosphatidic acid (DPPA) at the air/water interface are investigated through molecular dynamics simulations with atomistic force fields. The structure and dynamics of both monolayers and interfacial water are compared across the range of surface pressures at which stable monolayers can form. For PA monolayers at T = 300 K, the untilted condensed phase with a hexagonal lattice structure is found at high surface pressure, while the uniformly tilted condensed phase with a centered rectangular lattice structure is observed at low surface pressure, in agreement with the available experimental data. A state with uniform chain tilt but no periodic spatial ordering is observed for DPPA monolayers on a Na(+)/water subphase at both high and low surface pressures. The hydrophobic acyl chains of both monolayers pack efficiently at all surface pressures, resulting in a very small number of gauche defects. The analysis of the hydrogen-bonding structure/dynamics at the monolayer/water interface indicates that water molecules hydrogen-bonded to the DPPA head groups reorient more slowly than those hydrogen-bonded to the PA head groups, with the orientational dynamics becoming significantly slower at high surface pressure. Possible implications for physicochemical processes taking place on marine aerosols in the atmosphere are discussed.

Sharp organic interface of molecular C60 chains and a pentacene derivative SAM on Au(788): A combined STM & DFT study

Controlling the molecular structure of the donor–acceptor interface is essential to overcoming the efficiency bottleneck in organic photovoltaics. We present a study of self-assembled fullerene (C60) molecular chains on perfectly ordered 6,13-dichloropentacene (DCP) monolayers forming on a vicinal Au(788) surface using scanning tunneling microscopy in conjunction with density functional theory calculations. DCP is a novel pentacene derivative optimized for photovoltaic applications. The molecules form a brick-wall patterned centered rectangular lattice with the long axis parallel to the monatomic steps that separate the 3.9nm wide Au(111) terraces. The strong interaction between the C60 molecules and the gold substrate is well screened by the DCP monolayer. At submonolayer C60 coverage, the fullerene molecules form long parallel chains, 1.1nm apart, with a rectangular arrangement instead of the expected close-packed configuration along the upper step edges. The perfectly ordered DCP structure is unaffected by the C60 chain formation. The controlled sharp highly-ordered organic interface has the potential to improve the conversion efficiency in organic photovoltaics.

Salt Effects in the Formation of Self-Assembled Lithocholate Helical Ribbons and Tubes

The formation of self-assembled nanotubes is usually accounted for by anisotropic elastic properties of membranelike precursors. We present experimental data as evidence of the role played by electrostatics in the formation of self-assembled tubes in alkaline aqueous suspensions of lithocholic acid (LCA). Striking salt effects are characterized by comparing the rheological, dynamical, and scattering properties of systems prepared either in stoichiometric neutralization conditions (SC) of LCA or in a large excess of sodium hydroxide (EOC, experimentally optimized conditions) and finally, in two steps: stoichiometric neutralization followed by an appropriate addition of NaCl (AISC). The SC liquid system is originally made up of loose helical ribbons (previous transmission electron microscopy data), and upon aging they exhibit both intra- and interordering processes. Initially, the helical ribbons are loose and progressively wind around a cylinder (R = 330 Å) with their edges exposed to the solvent. They can be temporarily organized in a centered rectangular two-dimensional lattice (pgg, a = 224 Å, b = 687 Å). Upon further aging, the ribbons wind into more compact helical ribbons (or tubes with helical grooves): their edges are less-exposed and their ordering vanishes. Upon addition of NaCl salt (as in the AISC systems), the specific screening of the intra-aggregate electrostatic repulsions induces the closure of the ribbons into tubes (R(ext) = 260 Å, R(int) = 245 Å as in the EOC systems). Simultaneously with the closure of the ribbons into plain tubes, a drastic enhancement of their interconnectivity through van der Waals attractions develops. Eventually, gels are obtained with networks having hexagonal bundles of tubes.

Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode

A general coupled-wave theory analysis is presented for centered-rectangular lattice photonic crystal (PC) lasers with transverse-electric-like polarization. This analysis affords a realistic approach to the treatment of the full three-dimensional structure of the laser devices by incorporating surface emission and high-order coupling effects. Band structures and radiation constants of band-edge modes are calculated and discussed for different lattice angles, and their accuracies are verified via experiments and numerical simulations. Thus, the proposed analysis facilitates comprehensive understanding of the physics of PC surface-emitting lasers (PC-SELs), and it reveals extra degrees of structural flexibility for developing novel PC-SEL devices.

Centered-rectangular lattice photonic-crystal surface-emitting lasers

We investigate the effects of lattice structure on the properties of photonic-crystal surface-emitting lasers. We analyze a general type of crystalline geometry known as the centered-rectangular lattice, which includes both square and triangular lattices. We theoretically and experimentally investigate the laser cavity characteristics of devices based on such photonic crystals, including the band structure and beam patterns, in order to explore the feasibility of two-dimensional (2D) lasing oscillation. The more general type of crystalline geometry represented by the centered-rectangular lattice should enable a comprehensive understanding of the 2D cavity characteristics to be achieved and also lead to further possibilities in the field of 2D photonic-crystal lasers.