Two-dimensional Unit Cell Research Articles

Micromechanics simulations of glass–estane mock polymer bonded explosives

Polymer bonded explosives (PBXs) are particulate composites containing explosive particles and a continuous binder. The elastic modulus of the particles, at room temperature and higher, is often three to four orders of magnitude higher than that of the binder. Additionally, the explosive particles occupy high volume fractions, often greater than 90%. Both experimental and numerical determination of macroscopic properties of these composites is difficult. High modulus contrast mock PBXs provide a means of relatively inexpensive experimentation and validation of numerical approaches to determine properties of these materials. The goal of this investigation is to determine whether the effective elastic properties of monodisperse glass–estane mock PBXs can be predicted from two-dimensional micromechanics simulations using the finite element (FEM) method. In this study, the effect of representative volume element (RVE) size on the prediction of two-dimensional properties is explored. Two-dimensional estimates of elastic properties are compared with predictions from three-dimensional computations and with experimental data on glass–estane composites containing three different volume fractions of spherical glass beads. The effect of particle debonding on the effective elastic properties is also investigated using contact analyses. Results show that two-dimensional unit cells containing 10–20 circular particles are adequate for modelling glass–estane composites containing less than 60% glass particles by volume. No significant difference is observed between properties predicted by the two- and three-dimensional models. FEM simulations of RVEs, containing particles that are perfectly bonded to the binder, produce estimates of Young's modulus that are higher than the experimental data. Incorporation of debonding between particles and the binder causes the effective Young's modulus to decrease. However, the results suggest that cracks in the composite may play a significant role in determining the effective properties of mock polymer bonder explosives composed of glass and estane. The FEM simulations indicate that two-dimensional models that incorporate debonds and cracks can be used to obtain accurate estimates of the effective properties of glass–estane composites and possibly of PBXs.

Second-harmonic generation in reflection and diffraction by a GaAs photonic-crystal waveguide

Nonlinear reflection and diffraction measurements have been performed on a GaAs/AlGaAs photonic-crystal waveguide patterned with a square lattice: The basis in the two-dimensional unit cell consists of rings of air in the dielectric matrix. The measured angles of diffracted second-harmonic beams agree with those predicted for nonlinear diffraction conditions. Results for second-harmonic intensities as a function of incidence angle, polarization, and pump wavelength show that the reflected second-harmonic signal is dominated by the crystalline symmetry of GaAs, whereas nonlinear diffraction is determined by the photonic-crystal structure.

Two-dimensionally-periodic diffractive optical elements: limitations of scalar analysis.

The range of validity of the scalar diffraction analysis is quantified for the case of two-dimensionally-periodic diffractive optical elements (crossed gratings). Three canonical classes of two-dimensionally-periodic grating structures are analyzed by using the rigorous coupled-wave analysis as well as the scalar diffraction analysis. In all cases the scalar-analysis diffraction efficiencies are compared with the exact diffraction efficiencies. The error in using the scalar analysis is then determined as a function of the grating-period(s)-to-wavelength ratio(s), the minimum feature size, the grating depth, the refractive index of the grating, the incident polarization, and the number of phase levels. The three classes of two-dimensional (2-D) unit cells are as follows: (1) a rectangular pillar, (2) an elliptical pillar, and (3) an arbitrarily pixellated multilevel 2-D unit cell that is representative of more complicated diffractive optical elements such as computer-generated holograms. In all cases a normally incident electromagnetic plane wave is considered. It is shown that the error of the scalar diffraction analysis in the case of two-dimensionally-periodic diffractive optical elements is greater than that for the corresponding one-dimensionally-periodic counterparts. In addition, the accuracy of the scalar diffraction analysis degrades with increasing refractive index, grating thickness, and asymmetry of the 2-D unit cell and with decreasing grating-period-to-wavelength ratio and feature size.

The relation between grain-boundary structure and sliding resistance

Abstract During sliding, the grain-boundary (GB) energy depends on the atomic structures produced during relative translation of the two grains. The variation in the GB energy within the two-dimensional boundary unit cell (BUC) constitutes the GB-γ surface. Maxima in the slope of the γ surface determines the sliding resistance, that is the stress required to move the system over the lowest saddle points along a particular path within the BUC. In this paper we present the results of an atomistic study of the γ surfaces for two types of boundaries in a fcc metal. One of the boundaries is a Σg = 11, <110 > {131} which is a low-energy boundary and has a simple γ surface with a single stable configuration located at the corners and centre of the BUC. The resistance to sliding was determined by chain-of-states methods along four shear vectors connecting equivalent states within the BUC and is found to be very high in all cases. The asymmetric, Σ = 11, <110 > {252}-{414} GB, has a higher GB energy and its γ surface is much more complex, with distinctly different structures appearing at various locations in the BUC. At certain locations, more than one structure is found for the asymmetric GB. Although complex, a chain-of-states calculation along one path across the BUC suggests that the shear strength of this GB is also quite high. Extrinsic GB dislocations are found to lower the resistance to shear considerably and, therefore, to perform the same role in shear of GBs as do glide dislocations in slip of the lattice. The existence of multiple configurations has significant implications for the interaction of lattice dislocations with GBs, the core structure of GB dislocations, the temperature dependence of GB properties, and the GB sliding resistance, which we discuss.

The relation between grain-boundary structure and sliding resistance

Abstract During sliding, the grain-boundary (GB) energy depends on the atomic structures produced during relative translation of the two grains. The variation in the GB energy within the two-dimensional boundary unit cell (BUC) constitutes the GB-γ surface. Maxima in the slope of the γ surface determines the sliding resistance, that is the stress required to move the system over the lowest saddle points along a particular path within the BUC. In this paper we present the results of an atomistic study of the γ surfaces for two types of boundaries in a fcc metal. One of the boundaries is a Σg = 11, <110 > {131} which is a low-energy boundary and has a simple γ surface with a single stable configuration located at the corners and centre of the BUC. The resistance to sliding was determined by chain-of-states methods along four shear vectors connecting equivalent states within the BUC and is found to be very high in all cases. The asymmetric, Σ = 11, <110 > {252}-{414} GB, has a higher GB energy and its γ surface is much more complex, with distinctly different structures appearing at various locations in the BUC. At certain locations, more than one structure is found for the asymmetric GB. Although complex, a chain-of-states calculation along one path across the BUC suggests that the shear strength of this GB is also quite high. Extrinsic GB dislocations are found to lower the resistance to shear considerably and, therefore, to perform the same role in shear of GBs as do glide dislocations in slip of the lattice. The existence of multiple configurations has significant implications for the interaction of lattice dislocations with GBs, the core structure of GB dislocations, the temperature dependence of GB properties, and the GB sliding resistance, which we discuss.

Langmuir and Langmuir-Blodgett films of amphiphilic hexa-peri-hexabenzocoronene: new phase transitions and electronic properties controlled by pressure.

We present the synthesis as well as the structural and electronic properties of an amphiphilic derivative of hexaalkylhexa-peri-hexabenzocoronene (HBC), which contains one alkyl substituent that is terminated with a carboxylic acid group. The molecules form well-defined Langmuir films when spread from a solution at the air-water interface. Grazing-incidence X-ray diffraction (GIXD) and X-ray reflectivity studies of the Langmuir monolayer reveal two crystallographic phases at room temperature which depend on the surface pressure applied to the film. Scattering from very well-ordered (zeta = 200-400 A) pi-stacked lamellae of HBC molecules tilted approximately 45 degrees relative to the surface normal is observed in the low-pressure phase. In this phase, the HBC molecules pack in a rectangular two-dimensional unit cell with a = 22.95 A and b = 4.94 A. In the high-pressure phase, coherence from the pi stack is lost. This is a consequence of stress induced by the crystallization of the substituent alkyl chains into a hexagonal lattice, which has a trimerized superstructure in one direction: a = 3 x b = 15.78 A, b = 5.26 A, gamma = 120 degrees, A = 71.9 A2 = 3 x 23.9 A2. Thin monolayer films can be transferred to solid supports by the Langmuir-Blodgett (LB) technique. Atomic force microscopy (AFM) with atomic resolution reveals the crystalline packing of alkyl chains in the high-pressure phase. Kelvin force microscopy (KFM) shows a clear potential difference between the high- and low-pressure phases. This is discussed in terms of orbital delocalization (band formation) in the highly coherent low-pressure phase, which is in contrast to the localized molecular orbitals present in the high-pressure phase. The highly coherent pi stack is expected to sustain a very high charge-carrier mobility.

Phase retrieval methods for surface x-ray diffraction

We develop an iterative input–output feedback method for the phasing of surface x-ray diffraction (SXRD) amplitudes that relies on successive operations in real and reciprocal space. We demonstrate its use for the recovery of the real and positive electron density of a surface unit cell from simulated SXRD intensities. We have successfully recovered the entire surface electron density in a case where the two-dimensional surface unit cell is the same as that of the bulk and also in one where the surface unit cell is four times larger than that of the bulk. We show that the exponential modelling algorithm for structure completion derived earlier from maximum entropy theory may be regarded as a special case of an input–output phasing algorithm with a particular form of object-domain operations.

Comparisons between three-dimensional and two-dimensionalmulti-particle unit cell models for particle reinforced metal matrixcomposites

Three-dimensional and two-dimensional unit cell models for describing themechanical behaviour of particle reinforced metal matrix composites (MMCs) arecompared by assessing predictions obtained from microgeometries consisting of20 randomly positioned elastic particles embedded in an elastoplastic matrix.The elastic response to uniaxial loading predicted by the three-dimensionalunit cells is found to comply with the appropriate three-point bounds.Predictions for the elastoplastic regime are somewhat less satisfactory,indicating that configurations containing a higher number of particles will berequired to resolve the regions of concentrated plastic strains that developin inhomogeneous materials. This implies that in the nonlinear range the sizeof reference volume elements depends on material behaviour. Comparisons ofresults obtained from planar and three-dimensional multi-particle unit cellsshow clear differences in terms of both the overall stiffnesses and phaseaverages as well as the standard deviations of the microscale stress andstrain fields. These differences are much more pronounced in the elastoplasticrange, where planar analyses do not adequately describe the overall strainhardening behaviour of particle reinforced MMCs and tend to markedlyunderpredict the equivalent stresses and maximum principal stresses in theparticles.

A scanning tunneling microscopy study on the monolayer epitaxy of [cyano(ethoxycarbonyl) methylene]-2-ylidene-4,5-dimethyl-1,3-dithiole on graphite and its dynamic feature

The monolayer film of [cyano(ethoxycarbonyl)methylene]-2-ylidene-4,5-dimethyl-1,3-dithiole (CEMYDD) vacuum-deposited on a (0001) graphite surface was investigated by scanning tunneling microscopy (STM). High-resolution STM images showed that all of the molecules formed the same two-dimensional unit cell in each domain and that the unit cell was almost equivalent to that in the (011) plane of the second-harmonic-generation active β-form crystal. Therefore the CEMYDD molecules were arranged as polar domains with a size of several hundred nm. The crystallographic orientation of the unit cell in the film with respect to the graphite surface was found to have commensurate epitaxial coincidence. This commensurate structure was stable under STM measurements, except near the domain boundaries where the moiré-like, long-range, periodic contrast modulation due to the commensurate coincidence often varied while scanning with the STM tip. Although the arrangement of molecular images themselves moved only slightly, such a collective dynamic feature at boundaries could be detected from changes in the moiré-like contrast modulation.

Direct observation of organic–organic heteroepitaxy: perylene-tetracarboxylic-dianhydride on hexa-peri-benzocoronene on highly ordered pyrolytic graphite

We investigate the heteroepitaxial growth of two films of large planar molecules on an inorganic substrate: for the particular system [perylene-tetracarboxylic-dianhydride (PTCDA) on hexa-peri-benzocoronene (HBC) on highly ordered pyrolytic graphite (HOPG)], large domains of several hundred of nanaometers in size with interfaces that are flat and almost defect-free on the molecular scale are observed using scanning tunneling microscopy (STM). The images obtained represent, to the best of our knowledge, the first STM observation of the ordered growth of large planar molecules in a heteroepitaxially stacked film. The mutual lattice alignment and epitaxial relations between the two organic layers are determined from the pronounced moiré pattern in the STM images. Beside these findings, the intramolecular contrast of HBC on HOPG is used to determine the molecular orientation of the molecules within the two-dimensional unit cell by STM.

Structural studies on polymer whiskers by transmission electron microscopy. II. Unit cell of poly(p-hydroxybenzoic acid) whisker crystal at room temperature

A set of conspicuous reflections, which are not indexed using the cell constants reported so far for poly(p-hydroxybenzoic acid), were found in the electron diffraction (ED) patterns of its whisker crystal. A new unit cell [a = 1.487 nm, b = 0.572 nm, and c (chain axis) = 1.258 nm, α = β = γ = 90°] was proposed to explain the reflections in question, based on a set of the ED patterns viewed along the different zone axes, which were obtained by tilting the whisker around its long axis. The density of the whisker was experimentally measured to be 1.500 g/cm3, showing that four molecular stems are running through the new unit cell. From the extinction rule, the highest space group was assumed to be Pbca. In this case, the projection of the new unit cell onto the ab-plane gives a two-dimensional unit cell with pgm and is basically in agreement with those proposed so far. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 2456–2462, 1999

Multiphase, Multicomponent Fluid Flow in Homogeneous and Heterogeneous Porous Media

The flow of several components and several phases through a porous medium is generally described by introducing macroscopic mass-balance equations under the form of generalized dispersion equations. This model raises several questions that are discussed in this paper on the basis of results obtained from the volume averaging method, coupled with pore-scale simulations of the multiphase flow. The study is limited to a binary, two-phase system, and we assume that the momentum equations can be solved independently from the diffusion/advection equations. The assumption of local-equilibrium is discussed and several length-scale and time-scale constraints are provided. A key issue concerns the impact on the dispersion tensors of the pore-scale equilibrium condition at the interface between the different phases. Our results show that this phenomenon may lead to significant variations of the dispersion coefficients with respect to passive dispersion, i. e. , dispersion without interfacial mass fluxes. Macroscopic equations are then obtained in the general case, and several local closure problems are provided that allow one to calculate the dispersion tensors and others properties, from the pore-scale geometry, velocities, and fluid characteristics. Examples of solutions of these closure problems are given in the case of two-dimensional representative unit cells. The two-phase flow equations are solved in two different ways : a boundary element technique, or a modified lattice Boltzmann approach. Solutions of the closure problems associated with the dispersion equations are then given using a finite volume element formulation of the partial differential equations. The results show the influence of velocity and saturation on the effective parameters. They emphasize the importance of geometry on the behavior of the dispersion tensor. Extension of these results to a larger-scale including the effect of heterogeneities is proposed in a limited case corresponding to the flow of one phase, the other phase being at residual saturation. A new large-scale dispersion equation is provided, which features a large-scale dispersion tensor that can be determined from the heterogeneity characteristics through a set of closure problems. Results are extended to a more general two-phase flow problem, when the large-scale two-phase flow can be assumed to be quasi-static. Indications are given on the difficulties associated with flow under large-scale dynamic conditions, with abnormal dispersion.

Structural Characterization of Valinomycin and Nonactin at the Air−Solution Interface by Grazing Incidence X-ray Diffraction

Grazing incidence X-ray diffraction (GID) has been applied to the study of structural characteristics of two-dimensional crystallites of the ionophores valinomycin (VM) and nonactin (NA) when complexed with various cations at the air-solution interface. The VM complexes assume a bracelet shape that packs in a two-dimensional hexagonal unit cell. Other crystalline phases were formed on potassium iodide and barium perchlorate solutions. The presence of particular lipids induced ordered stacking of VM−potassium chloride complexes into three to four layers. NA packs in a pseudotetragonal unit cell on solutions of NH4SCN and KSCN. Upon compression of the NA−NH4SCN film, crystallites seven to eight layers thick were detected and the GID data enabled the determination of the structure. The tendency of these complexed ionophores to form multilayer crystallites at interfaces may have bearing on ion transport through membranes, via stacking.

X-Ray Study of the Mesophase-Mesophase Transition of Octaalkyloxyorthocyclophanes

Previous studies of the mesomorphic octaether derivatives of cyclotetraveratrylene, CTTV-I- n (n is the number of carbon atoms per ether chain), have shown two columnar mesophases for 9<n<16. In the higher temperature mesophase the columns are packed in a two-dimensional hexagonal unit cell, while in the lower phase the packing is two-dimensional centered rectangular. DSC showed no heat of transition between the two mesophases. For n=13, the transition occurred over a wide temperature range. We find this wide transition region is a third mesophase, with an oblique net intermediate between the other two. The upper boundary of this phase marks breaking of hexagonal symmetry by the appearance of a unit cell having γ=120°, but with a≠b. With decreasing temperature, b is essentially constant while a and γ increase until -2b cosγ=a, marking the transition to the primitive cell for the two dimensional centered rectangular net. Similar behavior is observed for n=14.

Intralayer and interlayer spin-singlet pairing and energy gap functions with different possible symmetries in high-Tclayered superconductors

Anisotropy and the wave-vector dependence of the energy gap function determine many important properties of a superconductor. Starting from first principles, we present here a complete analysis of possible symmetries of the superconducting gap function E{sub g}({bold k}) at the Fermi surface in high-T{sub c} layered superconductors with either a simple orthorhombic or a tetragonal unit cell. This is done within the framework of Gorkov{close_quote}s mean-field theory of superconductivity in the so-called {open_quotes}layer representation{close_quotes} introduced by us earlier. For N conducting cuprate layers, J=1,2,{hor_ellipsis},N, in each unit cell, the spin-singlet order parameters {Delta}{sub JJ}({bold k}) can be expanded in terms of possible basis functions of all the irreducible representations relevant to layered crystals, which are obtained here. In layered materials, the symmetry is restricted to the translational lattice periodicity in the direction perpendicular to the layers and the residual point group and translational symmetries for the two-dimensional unit cell in each layer of the three-dimensional unit cell. We derive an exact general relation to determine different branches of the energy gap function E{sub g}({bold k}) at the Fermi surface in terms of {Delta}{sub JJ}({bold k}), which include both intralayer and interlayer order parameters. For N=2, we also obtain an exactmore » expression for quasiparticle energies E{sub p}({bold k}), p=1,2, in the superconducting state in the presence of intralayer and complex interlayer order parameters as well as complex tunneling matrix elements between the two layers in the unit cell, which need not be equivalent. The form of the possible basis functions are also listed in terms of cylindrical coordinates k{sub t},{phi},k{sub z} to take advantage of the orthogonality of functions with respect to {phi} integrations. (Abstract Truncated)« less

Columnar mesophases of octa-alkyloxydibenzopyrenes and their charge transfer complexes: Synthesis, X-ray and NMR

Abstract The mesomorphic properties of five homologues of the 1,2,5,6,8,9,12,13-octa-alkyloxydibenzopyrene series (DBPn, where n is the number of carbons in each of the side chains) and their charge transfer complexes with 2,4,7-trinitro-9-fluorenone (TNF) are described. The neat compounds exhibit a single mesophase which has been identified by X-ray diffractometry and optical microscopy as hexagonal columnar (Dhd). Two-dimensional unit cell parameters are reported for the various homologues as functions of temperature within the mesophase region. Deuterium NMR of labelled DBPn was used to determine the nature and rate of the molecular reorientation in the mesophase. Addition of the electron acceptor TNF to the donor DBPn mesogens increases the stability range of the mesophases, with a maximum stability for the equimolar donor/acceptor mixtures. X-ray and deuterium NMR measurements on these charge transfer stabilized mesophases are also reported. Besides being thermally more stable, they are more highly ordered and even exhibit well defined intramolecular stacking order, classifying them as Dho. Under certain conditions, a diffraction peak at a position corresponding to twice the normal intracolumnar distance is also observed, indicating that the donor and acceptor molecules are stacked alternately in the columns. When alkanes are added to the 1:1 charge transfer complexes, a lyotropic nematic phase is induced. It is shown that these nematics are of the type Nc, with columnar structures as the mesogenic units.

KRIGING FRACTURE INDICATORS TO ESTIMATE RECOVERY IN DIMENSIONAL STONE EXPLORATION

A study was conducted for a granite quarry producing dimensional stone in the Laurentides mountains (Québec, Canada) to forecast the recovery and waste amounts of blocks due to the presence of fractures. In this paper, the term “fracture” refers to any kind of natural structural discontinuities, i.e. fractures, joints, faults, veins, dykes. Equal area stereographic projection of fracture pole density was used to identify four sets of structural discontinuities in the quarry. For each of these sets, four binary indicator functions were defined as different levels in the quality of the rock mass. The value of these binary functions is deduced from two fracture density parameters calculated in two-dimensional unit cell: the number of fracture intersections and the cumulative fractured length. The database underlying the estimation of these parameters was obtained from a systematic field survey of fracture traces observed on the accessible working faces and floors.A quality index was estimated on a regular grid of elementary blocks according to a systematic production plan. The predicted values of the quality index for each set were combined to estimate a recovery ratio in three target areas. It was found that the geostatistical procedure presented in this paper can be used to optimize both the extent and the orientation of the working faces. The recovery index was estimated at 15% for the first quality blocks and 15% for the second quality blocks in a 40m x 30m x 6m volume of rock. This result indicates very high ratio for the building stone industry, but the selected volume is not sufficient to ensure quarry operation for several years. Other target areas were rejected and an exploration program is currently yield on the same property, but outside the actual limits of the quarry to find additional granite reserves.

Molecular Packings−Photopolymerization Behavior Relationship of Diacetylene Langmuir−Blodgett Films

The formation processes of polydiacetylene (PDA) Langmuir−Blodgett (LB) films in the blue and the bluish-green forms were investigated from the viewpoint of the molecular lateral packings evaluated by electron diffraction (ED) analysis. The PDA blue and bluish-green forms exhibit absorption peaks at 640 and 704 nm in the visible absorption spectrum, respectively. During the formation process of the PDA blue form, the two-dimensional unit cell of the monomeric LB film shrunk due to the covalent bond formation among the molecules. On the other hand, during the formation process of the PDA bluish-green form, there was no change in the molecular packings. Also, the ED analysis revealed the difference in the repeating unit distances of main chains between both forms, implying that both forms had distinct resonance backbone structures. It was concluded from comparison of the ED analysis and the molecular mechanics calculation that the bluish-green form was composed of the butatrienic type main chains rather tha...

Basic structural types of binary niobates and tantalates with 1∶3 stoichiometry of cations. Variants of cation arrangements in planar nets

For planar (two-dimensional) unit cells containing 4 and 8 nodes, we have found 3 and 19 variants of arrangements of two cation types with 1∶3 stoichiometry in the 36 net, respectively, and suggested notation for these variants. Virtually all structures analyzed have either an F- or I-complex of close-packed cation planes and the corresponding cation sublattices. In La2ThTaO6Cl3, the Cl− anion occupies some sites of the cation sublattice. The cation sublattices La3NbO7 (weberite structural type) and Sm2ScTaO7=Sm2(Se, Ta)2O7 (pyrochlore) have 44 nets with identical or different cation packings, which alternate in the structures. In some structures with 1∶1 general stoichiometry of cations (e.g., Na2Ta2O5F2), we observe alternation of the 66 dense nets having 1∶3 and 3∶1 stoichiometries

Micromechanical modeling of reinforcement fracture in particle-reinforced metal-matrix composites

Finite element analyses of the effect of particle fracture on the tensile response of particle-reinforced metal-matrix composites are carried out. The analyses are based on two-dimensional plane strain and axisymmetric unit cell models. The reinforcement is characterized as an isotropic elastic solid and the ductile matrix as an isotropically hardening viscoplastic solid. The reinforcement and matrix properties are taken to be those of an Al-3.5 wt pct Cu alloy reinforced with SiC particles. An initial crack, perpendicular to the tensile axis, is assumed to be present in the particles. Both stationary and quasi-statically growing cracks are analyzed. Resistance to crack growth in its initial plane and along the particle-matrix interface is modeled using a cohesive surface constitutive relation that allows for decohesion. Variations of crack size, shape, spatial distribution, and volume fraction of the particles and of the material and cohesive properties are explored. Conditions governing the onset of cracking within the particle, the evolution of field quantities as the crack advances within the particle to the particle-matrix interface, and the dependence of overall tensile stress-strain response during continued crack advance are analyzed.