Centered Rectangular Lattice Research Articles

Polarization sensitive near-complete reflection from photonic crystal slab in centered rectangular lattice

In this study, reflective properties have been systematically calculated for two-dimensional photonic crystal slabs in centered rectangular lattices with elliptical patterns for transverse electric polarization along perpendicular x- and y-directions. The slab structures can be geometrically optimized for a near-complete reflection over a broad wavelength range. The reflection from the photonic crystal slabs is polarization selective, allowing a complete reflection for one polarization and complete transmission for the other polarization, which could be useful for a laser cavity mirror without using Brewster windows. The dependence of the reflection on the slab thickness and the structural parameters demonstrates the effect of guided-mode resonances on the reflection.

Transfer, Amplification, and Inversion of Helical Chirality Mediated by Concerted Interactions of C3-Supramolecular Dendrimers

The synthesis, structural, and retrostructural analysis of two libraries containing 16 first and second generation C(3)-symmetric self-assembling dendrimers based on dendrons connected at their apex via trisesters and trisamides of 1,3,5-benzenetricarboxylic acid is reported. A combination of X-ray diffraction and CD/UV analysis methods demonstrated that their C(3)-symmetry modulates different degrees of packing on the periphery of supramolecular structures that are responsible for the formation of chiral helical supramolecular columns and spheres self-organizable in a diversity of three-dimensional (3D) columnar, tetragonal, and cubic lattices. Two of these periodic arrays, a 3D columnar hexagonal superlattice and a 3D columnar simple orthorhombic chiral lattice with P222(1) symmetry, are unprecedented for supramolecular dendrimers. A thermal-reversible inversion of chirality was discovered in helical supramolecular columns. This inversion is induced, on heating, by the change in symmetry from a 3D columnar simple orthorhombic chiral lattice to a 3D columnar hexagonal array and, on cooling, by the change in symmetry from a 2D hexagonal to a 2D centered rectangular lattice, both exhibiting intracolumnar order. A first-order transition from coupled columns with long helical pitch, to weakly or uncorrelated columns with short helical pitch that generates a molecular rotator, was also discovered. The torsion angles of the molecular rotator are proportional to the change in temperature, and this effect is amplified in the case of the C(3)-symmetric trisamide supramolecular dendrimers forming H-bonds along their column. The structural changes reported here can be used to design complex functions based on helical supramolecular dendrimers with different degree of packing on their periphery.

Creation of Nanoscale Two-Dimensional Patterns of ZnO Nanorods using Laser Interference Lithography Followed by Hydrothermal Synthesis at 90 °C

Arrayed ZnO nanorods were fabricated on photoresist coated ZnO/Si(100) substrates. Five different two-dimensional (2D) periodic nanopatterns, including a square lattice, rectangular lattice, centered rectangular lattice, oblique lattice, and hexagonal lattice, of opening windows in photoresist were fabricated by laser interference lithography using a two-step exposure process. The ZnO nanorods were grown through the patterned area by a hydrothermal method. The morphology of 2D nanopatterns and ZnO nanorods was investigated by atomic force microscopy, scanning electron microscopy and transmission electron microscopy. The optical property of ZnO nanorods was explored by room temperature photoluminescence. The height and diameter of ZnO nanorods were approximately 500 nm and 150 nm, respectively. The crystallinity, microstructure, and optical property of ZnO nanorods array could be controlled by the proper use of a surfactant chemical, that is, Na-citrate in this report.

Modification of the trapped field in bulk high-temperature superconductors as a result of the drilling of a pattern of artificial columnar holes

The trapped magnetic field is examined in bulk high-temperature superconductors that are artificially drilled along their c-axis. The influence of the hole pattern on the magnetization is studied and compared by means of numerical models and Hall probe mapping techniques. To this aim, we consider two bulk YBCO samples with a rectangular cross-section that are drilled each by six holes arranged either on a rectangular lattice (sample I) or on a centered rectangular lattice (sample II). For the numerical analysis, three different models are considered for calculating the trapped flux: (i), a two-dimensional (2D) Bean model neglecting demagnetizing effects and flux creep, (ii), a 2D finite-element model neglecting demagnetizing effects but incorporating magnetic relaxation in the form of an E – J power law, and, (iii), a 3D finite element analysis that takes into account both the finite height of the sample and flux creep effects. For the experimental analysis, the trapped magnetic flux density is measured above the sample surface by Hall probe mapping performed before and after the drilling process. The maximum trapped flux density in the drilled samples is found to be smaller than that in the plain samples. The smallest magnetization drop is found for sample II, with the centered rectangular lattice. This result is confirmed by the numerical models. In each sample, the relative drops that are calculated independently with the three different models are in good agreement. As observed experimentally, the magnetization drop calculated in the sample II is the smallest one and its relative value is comparable to the measured one. By contrast, the measured magnetization drop in sample (1) is much larger than that predicted by the simulations, most likely because of a change of the microstructure during the drilling process.

Maximum and overlapped photonic band gaps in both transverse electric and transverse magnetic polarizations in two-dimensional photonic crystals with low symmetry

In this paper, photonic band gaps have been systematically calculated for two-dimensional photonic crystals in centered rectangular lattices with elliptical patterns for both transverse electric and transverse magnetic polarizations. Two-dimensional centered rectangular lattices can be considered to be a stretched or compressed form of the hexagonal lattice with lower symmetry. For infinitely thick two-dimensional photonic crystals, the maximum overlapped photonic band gaps for both transverse electric and transverse magnetic polarizations occur in the well-studied hexagonal lattice. However, the maximum overlapped band gap happens in centered rectangular lattices with elliptical patterns for two-dimensional photonic crystal slabs, supporting other studies that reducing symmetry can open overlapped photonic band gaps.

From anatase (1 0 1) surface to TiO 2 nanotubes: Rolling procedure and first principles LCAO calculations

A simple procedure of 1D nanotubes (NT) construction based on the supercell of 2D (1 0 1) slab rolling and subsequent cylindrical coordinate system introduction is suggested. This procedure is applicable for any of five 2D lattices as well as both single- and multi-wall nanotubes provided that the chirality and translation vectors orthogonality condition is satisfied. The procedure suggested is applied to the centered rectangular 2D lattice, formed by (1 0 1) sheet of the bulk anatase. It is shown that ( n, 0) and (0, m) nanotubes can be constructed by rolling an anatase (1 0 1) sheet along the [ 1 ¯ 0 1 ] and [0 1 0] directions respectively. The orthogonal to chirality vector translation vector does not exist for n ≠ m ≠ 0 general case. The first principles LCAO calculations of ( n, 0) and (0, m) NT’s are made with hybrid HF/KS (PBE0) exchange-correlation functional for systems containing up to 180 atoms ( n = 6, 9, 12, 15; m = 3−6). It is demonstrated that the energy band gap increases (from 4.7 to 5.4 eV) when the NT radius changes from 3.61 to 9.92 Å. The strain energy is larger for ( n, 0) than for (0 , m) nanotubes of a similar radius. The changes of the unit cell periodicity and radius of titania nanotubes after the structure optimization are negligible, however the atomic relaxations are noticeable.

Negative refraction behaviors in two-dimensional elliptical-rod photonic crystals in a centered rectangular lattice

It is shown that negative refractions exist in two-dimensional photonic crystals consisting of centered rectangular lattices of elliptical dielectric rods in air. These lattices can be considered as deviated structures from well-studied, strict hexagonal lattices. There is a tolerance of negative refractions to deviations of structural parameters from the hexagonal lattice, indicating certain flexibilities for fabrication process for these photonic crystals including the hexagonal one.

Effect of Unsaturation in Fatty Acids on the Main Characteristics of Langmuir Monolayers

The main characteristics of unsaturated acid monolayers deviate considerably from those of the corresponding saturated fatty acids. The monolayers of an unsaturated cis and trans fatty acid with the same hydrocarbon chain length having the double bond at the same position (ω9, Δ13) are compared to obtain detailed information on the effect of the cis and trans double bonds. The studies were performed with erucic acid chosen for the cis form and brassidic acid chosen for the trans form. Some additional studies performed with the homologous cis-nervonic acid (ω9, Δ15) and trans-elaidic acid (ω9, Δ9) allow information about the effect of the hydrocarbon chain length. Phase behavior, domain morphology, and two-dimensional lattice structure of the cis and trans unsaturated fatty acid monolayers are compared, using surface pressure−molecular area (π-A) isotherms, Brewster angle microscopy (BAM), and grazing incidence X-ray diffraction (GIXD) techniques. The π-A isotherms of the unsaturated fatty acids studied have a fluid/condensed phase coexistence region (plateau) over the measured temperature range. The thermodynamic analysis, performed on the basis of the π-A isotherms, shows that the introduction of unsaturation into the hydrocarbon chain affects the thermodynamic characteristics of the monolayer in a similar way as shortening the alkyl chain length, and it is much larger for the cis isomers rather than for the trans isomers. The morphology of the condense phase domains formed in the fluid/condensed coexistence region reveals considerable differences between the cis and the trans conformations at the same molecular structure of the unsaturated fatty acid. Completely different to the behavior of saturated fatty acid monolayers, for all cis and trans unsaturated fatty acids studied, only a centered rectangular lattice with alkyl chains tilted toward the nearest neighbor (NN) direction is found, and no indication for a phase transition between different condensed phases could be observed independent of different surface pressures and temperatures. The conformity of the cis and trans unsaturated fatty acid monolayers to only one lattice type having a rectangular centered unit cell with alkyl chains tilted in the NN direction correlates obviously with the obstructive effect of the double bond in the alkyl chain on the molecular ordering.

Steric Communication of Chiral Information Observed in Dendronized Polyacetylenes

Structural and retrostructural analysis of helical dendronized polyacetylenes (i.e., self-organizable polyacetylenes containing first generation dendrons or minidendrons as side groups) synthesized by the polymerization of minidendritic acetylenes with [Rh(nbd)Cl]2 (nbd = 2,5-norbornadiene) reveals an approximately 10% change in the average column stratum thickness (l) of the cylindrical macromolecules with a chiral periphery, through which a strong preference for a single-handed screw-sense is communicated. The cylindrical macromolecules reversibly interconvert between a three-dimensional (3D) centered rectangular lattice (Phi r-c,k) exhibiting long-range intracolumnar helical order at lower temperatures and a two-dimensional (2D) hexagonal columnar lattice (Phi h) with short-range helical order at higher temperatures. A polymer containing chiral, nonracemic peripheral alkyl tails is found to have a larger l as compared to the achiral polymers. In methyl cyclohexane solution, the same polymer exhibits an intense signal in circular dichroism (CD) spectra, whose intensity decreases upon heating. The observed change in l indicates that the chiral tails alter the polymer conformation from that of the corresponding polymer with achiral side chains. This change in conformation results in a relatively large free energy difference (DeltaGh) favoring one helix-sense over the other (per monomer residue). The capacity to distort the polymer conformation and corresponding free energy is related to the population of branches in the chiral tails and their distance from the polymer backbone by comparison to recently reported first and second generation dendronized polyphenylacetylenes.

Self-Assembly, Structural, and Retrostructural Analysis of Dendritic Dipeptide Pores Undergoing Reversible Circular to Elliptical Shape Change

The synthesis of dendritic dipeptides (4-3,4-3,5-4)12G2-CH(2)-Boc-L-Tyr-L-Ala-OMe and (4-3, 4-3,5-4)12G2-CH(2)-Boc-D-Tyr-D-Ala-OMe is described. These dendritic dipeptides self-assemble into porous elliptical and circular columns that in turn self-organize into centered rectangular columnar and hexagonal columnar periodic arrays. The transition from porous elliptical to porous circular columns is mediated in a reversible or irreversible way by the thermal history of the sample. A method to determine the dimensions of hollow elliptical and circular columns by the reconstruction of the small-angle powder X-ray diffractograms of the centered rectangular or hexagonal columnar lattices was elaborated. This technique together with wide-angle X-ray experiments performed on aligned fibers provided access to the structural and retrostructural analysis of elliptical supramolecular pores. This procedure is general and can be adapted for the determination of the dimensions of pores of any columnar shape.

Effect of Fluorination of the Hydrophilic Heads on Morphology and Molecular Structure of Langmuir Monolayers of Long-Chain Ethers

Brewster angle microscopy (BAM) and grazing incidence X-ray diffraction (GIXD) were used to compare morphology and molecular structure of Langmuir monolayers of docosyl trifluoroethyl ether (DFEE) and docosyl ethyl ether (DEE) on water. When spread at a molecular area of 70 Å2/molecule both substances form islands of condensed phase and suspension of small 2D microcrystals attaching at the borderlines under compression. The DEE islands and the DEE monolayer at low surface pressure exhibit strong in-plane optical anisotropy, while the DFEE islands and the compact DFEE film are isotropic. The anisotropy of the DEE monolayer disappears at 10 mN/m, and above this surface pressure the monolayers with fluorinated and nonfluorinated heads are visually the same. The GIXD analysis identifies a L2‘ phase with NNN molecular tilt and NN lattice distortion in the DEE islands and DEE monolayer at low surface pressure. The maximum tilt angle at zero compression is 22.6°. The islands and the compact DFEE film consist of closely packed upright molecules forming an S phase. At high surface pressure both monolayers have the same upright molecular arrangement with centered rectangular lattice and practically the same unit cell parameters and lattice distortion. Under such conditions the DFEE and DEE molecules occupy the same area of the water surface Axy = A0 = 19.1 Å2. Since the van der Waals radius of the CF3 group is by 16% larger as compared to that of the CH3 group, the same A0 value of the two films means a significantly thinner hydration shell of the fluorinated head. Substitution of the terminal CH3 group in the DEE head by the CF3 group in the DFEE head removes the low-pressure tilted phase and causes formation of the condensed phase of upright fluorinated molecules even at no compression. The decreased headgroup hydration of DFEE could be responsible for the upright versus tilted orientation of the DFEE and DEE molecules at low surface pressure. We suppose that the mechanism of the continuous tilting transition causing erection of the DEE chains is related to an increasing dehydration of the headgroups under compression.

Effect of hydroxyl group position and system parameters on the features of hydroxystearic Acid monolayers.

The characteristic features of hydroxystearic acid monolayers OH-substituted in the mid position of the alkyl chain deviate considerably from those of the usual nonsubstituted stearic acid. The phase behavior, domain morphology, and two-dimensional lattice structure of 9-, 11-, and 12-hydroxystearic acids are studied, using pi-A isotherms, Brewster angle microscopy (BAM), and grazing incidence X-ray diffraction (GIXD), to obtain detailed information on the effect of the exact position of the OH-substitution. The pi-A isotherms of all three hydroxyoctadecanoic acids have an extended flat plateau region, the extension of which only slightly decreases with the increase of temperature. At the same temperature, the extension of the plateau region increases and the plateau pressure decreases from 9-hydroxyoctadecanoic acid to 12-hydroxyoctadecanoic acid. The absolute -DeltaH and -DeltaS values for the phase transition increase slightly from 9-hydroxyoctadecanoic acid to 12- hydroxyoctadecanoic acid and indicate differences in the ordering of the condensed phase under consideration of the special reorientation mechanism of these bipolar amphiphiles at the fluid/condensed phase transition. The morphology of the condensed phase domains formed in the fluid/condensed coexistence region is specific for the position of the OH-substitution of the alkyl chain, just as the lattice structures of the condensed monolayer phase. 11-hydroxyoctadecanoic acid monolayers form centered rectangular lattices with the chain tilt toward the NNN (next nearest neighbor) direction, and 12-hydroxyoctadecanoic acid monolayers have an oblique lattice over the entire pressure range. A special feature of 9-hydroxystearic acid monolayers is the phase transition between two condensed phases observed in the pi-A isotherm of 5 degrees C at approximately 18 mN/m, where the centered rectangular lattice shows a NNN/NN transition. The morphology of the condensed phase domains formed in the fluid/condensed coexistence region, just as the lattice structures of the condensed monolayer phase, reveal the high specifity of the monolayer feature of the bipolar hydroxystearic acids OH-substituted in the mid position.

Two-and three-dimensional ordered structures formed by electro-magnetorheological colloids

We have observed two- and three-dimensional ordered structures in electro-magnetorheological fluids by using uniform-sized microspheres coated with an inner layer of nickel and an outer layer of insulating material. The coated microspheres, when suspended in oil, exhibit electro-magnetorheological responses. The 2D structures were formed by floating the microspheres on a liquid meniscus. A planar hexagonal structure was observed, with variable lattice constant as a function of applied normal magnetic field. Other planar symmetries, such as oblique, centered rectangular, and square lattices, were also obtained by tilting the external magnetic field away from the normal. The balance between the magnetic interaction and the “attractive” interaction, due to the weight of the particles projected along the surface tangent, is the basic underlying physical mechanism responsible for the planar structures. Three-dimensional ordered structures, with body-centered tetragonal or face-centered cubic symmetries, were obtained by dispersing the coated microspheres in liquid and applying crossed electric and magnetic fields. A martensitic transition between these two crystal symmetries was observed as a function of the ratio between the magnetic field and the electric field.

Highly Aligned Lamellar Lipid Domains Induced by Macroscopic Confinement

We have carried out neutron diffraction experiments on a lipid mixture in a macroscopically confined sample geometry (1 mm gap) and have observed the transformation of bilayered micelles into two distinct orientationally aligned domains of perforated lamellae seemingly littered with defects. Depending on the separation of the confining surfaces, the defects can lie on a two-dimensional centered rectangular lattice where the angle, φ, between lamellar domains can vary. The particular lamellar phase is shown to be metastable and eventually relaxes, either in time or by perturbation of the sample by centrifugation, into aligned multibilayer stacks of single orientation.

Grazing-incidence x-ray diffraction study of Langmuir films of amphiphilic monodendrons.

We have used pressure-area isotherms and grazing-incidence x-ray diffraction to study structures of Langmuir films of first-generation monodendrons with two or three peripheral alkyl chains. Unlike the structures observed in their bulk liquid crystalline mesophases, these multichain monodendrons form either a centered rectangular lattice with molecular axes tilted toward nearest neighbors or an oblique lattice with molecular axes tilted in low-symmetry directions.

Quenched spin tunneling and diabolical points in magnetic molecules. II. Asymmetric configurations

The perfect quenching of spin tunneling first predicted for a model with biaxial symmetry, and recently observed in the magnetic molecule Fe_8, is further studied using the discrete phase integral (or Wentzel-Kramers-Brillouin) method. The analysis of the previous paper is extended to the case where the magnetic field has both hard and easy components, so that the Hamiltonian has no obvious symmetry. Herring's formula is now inapplicable, so the problem is solved by finding the wavefunction and using connection formulas at every turning point. A general formula for the energy surface in the vicinity of the diabolo is obtained in this way. This formula gives the tunneling apmplitude between two wells unrelated by symmetry in terms of a small number of action integrals, and appears to be generally valid, even for problems where the recursion contains more than five terms. Explicit results are obtained for the diabolical points in the model for Fe_8. These results exactly parallel the experimental observations. It is found that the leading semiclassical results for the diabolical points appear to be exact, and the points themselves lie on a perfect centered rectangular lattice in the magnetic field space. A variety of evidence in favor of this perfect lattice hypothesis is presented.

Kinetic Monte Carlo study of fractal growth with edge diffusion and reversible aggregation on a bcc(1 1 0) surface

The low-coverage growth of Cu islands deposited on W(110) is studied by kinetic Monte Carlo simulation. The model includes deposition, diffusion of monomers, diffusion along island edges and reversible aggregation on centered rectangular lattice with anisotropy of diffusion barriers. The simulations are performed at temperatures from 150 to 280K with fluxes changing from 10−7 to 10−3 Ml/s. Island morphology ranges from ramified structures (fractals) at low temperatures to compact shapes at higher temperatures. The low temperature growth regime consists of standard diffusion limited aggregation regime and extended fractal regime. The existence of EF regime at bcc(110) surface results from the inhibition of island equilibration by two neighboring bonds of edge adatom to an island in a centered rectangular lattice. The fractals obtained in EF regime have arms that widen with increasing temperature as it is observed in experiments for other systems. The submonolayer morphology diagram in the (flux, temperature) plane is presented and the region of possible fractal observation is indicated.

Diabolical points in magnetic molecules: An exactly solvable model

The magnetic molecule Fe_8 has been observed to have a rich pattern of degeneracies in its magnetic spectrum as the static magnetic field applied to the molecule is varied. The points of degeneracy, or diabolical points in the magnetic field space, are found exactly in the simplest model Hamiltonian for this molecule. The points are shown to form a perfect centered rectangular lattice, and are shown to be multiply diabolical in general. The multiplicity is found. An earlier semiclassical solution to this problem is thereby shown to be exact in leading order in 1/J where J is the spin.

Herringbone ordering and lattice distortions in a planar-molecule model for langmuir monolayers

A model of planar molecules, made up of "atoms" interacting by Lennard-Jones potentials and arranged to mimic the cross section of alkyl chains, is used to study the problem of backbone plane ordering in Langmuir monolayers. It is shown that two minima of the interaction energy are reached if molecules lie on the sites of a centered rectangular lattice in a herringbone configuration with two different dihedral angles. These orientationally ordered phases can be related to the so-called herringbone and pseudoherringbone structures, whose lattice distortions qualitatively agree with those determined by means of grazing incidence x-ray diffraction experiments on Langmuir monolayers. A third energy minimum is obtained for a configuration of parallel molecules on an oblique lattice, which has also been observed in some experiments. The competition between the three phases is investigated, upon varying geometric parameters of the model molecules and surface pressure. The effect of temperature is analyzed in a mean field approximation, by taking into account the orientational entropy contribution on a lattice system with variable unit cell parameters. In this framework the transition to an orientationally disordered phase is also pointed out.

Structural assembly of Cd-arachidate molecules in multilayers

The three-dimensional (vertical and lateral) structure of Cd-arachidate multilayers prepared on two different substrates, glass and silicon, is studied simultaneously by grazing incidence x-ray reflection/diffraction. The grazing incidence reflectivity studies indicate the formation of a well-ordered layered structure on both the substrates. The reflectivity simulations show that the bilayer spacing in the case of a glass substrate is 5.54 nm while that on a silicon substrate is only 5.44 nm, indicating a vertical tilt of the Cd-arachidate molecules on a glass substrate by about 9.5°. The interface roughness is found to be 0.3 nm for multilayers on a glass substrate and 0.4 nm on the silicon substrate. The interface topography, determined by studying the nonspecular scattering behavior, is found to be replicated between the various interfaces in the multilayers, independent of the type of substrate. The lateral topography of the interfaces in multilayers on a glass substrate shows a self-affine nature with no lateral length limit for the roughness fluctuations. The interfaces in multilayers on a silicon substrate, however, exhibit a saturation behavior for the roughness fluctuations. In the plane of the multilayers the Cd-arachidate molecules on a glass substrate have a centered rectangular lattice arrangement while they have a distorted hexagonal lattice arrangement on a silicon substrate.