Two-dimensional Unit Cell Research Articles

Dirichlet-to-Neumann map method for analyzing crossed arrays of circular cylinders

An efficient and accurate computational method is developed for analyzing finite layers of crossed arrays of circular cylinders, including woodpile structures as special cases. The method relies on marching a few operators (approximated by matrices) from one side of the structure to another. The marching step makes use of the Dirichlet-to-Neumann (DtN) maps for two-dimensional unit cells in each layer where the structure is invariant in the direction of the cylinder axes. The DtN map is an operator that maps two wave field components to their normal derivatives on the boundary of the unit cell, and they can be easily constructed by vector cylindrical waves. Unlike existing numerical methods for crossed gratings, our method does not require a discretization of the structure. Compared with the multipole method that uses vector cylindrical wave expansions and scattering matrices, our method is relatively simple since it does not need sophisticated lattice sums techniques.

Density and spatial distribution of charge carriers in the intrinsicn-typeLaAlO3-SrTiO3interface

In order to establish the density and spatial distribution of charge carriers intrinsic to the $n$-type ${\text{LaAlO}}_{3}{\text{-SrTiO}}_{3}$ heterointerface, we carry out first-principles calculations on the ${({\text{LaAlO}}_{3})}_{n}{({\text{SrTiO}}_{3})}_{15}$ slab model with $n=2--10$. As the thickness of the ${\text{LaAlO}}_{3}$ layer increases, the charge transfer from ${\text{LaAlO}}_{3}$ to ${\text{SrTiO}}_{3}$ converges to half an electron per two-dimensional unit cell. It is found that the electrons in the conduction band of ${\text{SrTiO}}_{3}$ consist of various types of interface-bound states. The mobile electrons evaluated by excluding those states tightly bound to the interface within 2 nm or having large effective masses are in good agreement with the experimental carrier densities for all ${\text{LaAlO}}_{3}$ thicknesses, suggesting that the loosely bound states play a major role in the transport property. A large calculation including up to ${({\text{LaAlO}}_{3})}_{5}{({\text{SrTiO}}_{3})}_{30}$ shows that about 70% of electrons are confined within 3 nm from the interface, which is in good comparison with the experiments. It is found that the transferred electrons decay exponentially at short distances from the interface, but there is a crossover to an algebraically decaying region at $\ensuremath{\sim}4\text{ }\text{nm}$.

Numerical modeling of geosynthetic-encased stone column-reinforced ground

This paper presents the results of a comparative study on different finite element modeling approaches for modeling geosynthetic-encased stone column-reinforced ground for use in rapid embankment construction. The specific models considered include: (1) an axisymmetric unit cell; (2) a three-dimensional (3D) column; and (3) a full 3D model. The validity of the unit cell model was tested by comparison with the results from the 3D models. The applicability of continuum elements for modeling the geosynthetic encasement was also investigated. The results show that the 3D column model yielded practically identical results when compared with those of the full 3D model whereas the two-dimensional axisymmetric unit cell model tended to yield results that were 10 to 20% larger in terms of the vertical effective stress and lateral deformation of the stone column. It is also shown that a layer(s) of continuum elements can be used to model the geosynthetic encasement instead of membrane elements which are not readily available in commercial software for geotechnical analysis, provided that the axial stiffness of the geosynthetic encasement is taken into consideration. Based on the results of analysis, the effect of geosynthetic encasement on the performance of stone columns installed in soft ground under embankment loading is also discussed.

Thermal expansion behaviour of aluminum matrix composites with densely packed SiC particles

Abstract The coefficient of thermal expansion (CTE) of Al-based metal matrix composites containing 70 vol.% SiC particles (AlSiC) has been measured based on the length change from room temperature (RT) to 500 °C. In the present work, the instantaneous CTE(T) of AlSiC is studied by thermo-elastic models and micromechanical simulation using finite element analysis in order to explain abnormalities observed experimentally. The CTE(T) is predicted according to analytical thermo-elastic models of Kerner, Schapery and Turner. The CTE(T) is modelled for heating and cooling cycles from 20 °C to 500 °C considering the effects of microscopic voids and phase connectivity. The finite element analysis is based on a two-dimensional unit cell model comparing between generalized plane strain and plane stress formulations. The thermal expansion behaviour is strongly influenced by the presence of voids and confirms qualitatively that they cause the experimentally observed decrease of the CTE(T) above 250 °C.

Dynamic viscous permeability of an open-cell aluminum foam: Computations versus experiments

Is it possible to find a two-dimensional (2D) periodic unit cell representative of the dynamic viscous dissipation properties of a real porous media? This is a challenging question addressed in this paper through a review of tools and methods of experimental and computational micro(poro)mechanics. The combination of advanced experimental imaging and numerical homogenization techniques provides a unique opportunity to understand and assess the limits of two-dimensional models of microstructures, as a potential basis for the engineering prediction of macroscopic properties of acoustical materials. This is illustrated for a real sample of open-cell aluminum foam. The conclusion, based on this analysis, is that the 2D periodic foam model geometry provides a reliable estimate of the dynamic permeability, except in the low frequency range. This is not surprising because in the 2D periodic foam model geometry, ligaments are always perpendicular to the flow direction, thus decreasing artificially the static permeability of the viscous flow.

Superconducting graphite intercalation compounds with calcium

In the graphite–lithium–calcium system, four well-defined intercalation compounds were synthesised. Two of them, CaC 6 and Li 3Ca 2C 6, exhibit superconducting properties at 11.5 K and 11.15 K, respectively, the highest critical temperatures among those of graphite intercalation compounds. The samples are synthesised using a liquid–solid method allowing the preparation of pure bulk samples, auspicious for crystallographic and magnetic measurements. The crystal structure of CaC 6 was entirely specified; this compound crystallises in the R-3 m space group. The two-dimensional unit cell of Li 3Ca 2C 6 is hexagonal and commensurate with that of graphite and the intercalated sheets, very rich in metal, are seven-layered. The magnetic properties of these phases were studied with an applied field parallel and perpendicular to the graphene sheets. In both cases the magnetic phase diagram indicates that these compounds are type II superconducting materials slightly anisotropic in spite of their lamellar structure. In the case of CaC 6, in-plane magnetic penetration depth measurements show a clear exponential behaviour at low temperatures, consistent with an s-wave symmetry of the gap function, well fitted by the standard BCS theory in the dirty limit.

Polycrystalline domain structure of pentacene thin films epitaxially grown on a hydrogen-terminated Si(111) surface

Single-monolayer high pentacene (Pn) dendrites grown on a hydrogen-terminated Si(111) surface [H-Si(111)] under ultrahigh vacuum were observed by low-energy electron microscopy and microbeam low-energy electron diffraction analyses. We determined the epitaxial structure (type I) inside a unique polycrystalline domain structure of such dendrites, each of which has six equivalent epitaxial orientations of Pn two-dimensional (2D) unit cells. There are three sets of these cells, which are rotated $\ifmmode\pm\else\textpm\fi{}120\ifmmode^\circ\else\textdegree\fi{}$ relative to each other. Domain boundaries inside each dendrite were successfully observed by scanning tunneling microscopy. In addition, we found another epitaxial relation (type II): the polycrystalline domain structure and lattice parameters are similar to those of the type-I dendrite; however, the 2D unit cells of the type-II dendrite are rotated approximately 90\ifmmode^\circ\else\textdegree\fi{} relative to those of the type-I dendrite. These results suggest that the crystal structure of the dendrites on H-Si(111) is determined mainly by the interaction between Pn molecules. Each dendrite is composed of domains that are exclusively of type I or II. The so-called point-on-line coincidences are found between the Pn 2D lattices of types I and II, and H-Si(111). The higher commensurability of the type-I dendrites than the type-II dendrites results in a higher probability of type-I dendrite formation. Moreover, for both the type-I and type-II dendrites, we found supercell structures. We estimated the minimum interface energy between the dendrite and H-Si(111) from an island's free energy, which is necessary to reproduce the growth of a single-monolayer high dendrite.

Microstructure and Acoustical Macro-Behavior: Approach by Reconstruction of a Representative Elementary Cell

The fundamental issue of determining acoustic properties of porous media from their local geometry is examined in this Ph.D. dissertation thesis, thanks to a sample of open-cell aluminium foam analyzed by axial computed microtomography. Various geometric properties are measured to characterize the experimental sample at the cell size level. This is done in order to reconstruct a porous medium by means of idealized three- and two-dimensional unit cells. The frequency dependent thermal and velocity fields governing the propagation and dissipation of acoustic waves through rigid porous media are computed by Brownian motion simulation and the finite element method, respectively. Macroscopic behavior is derived by spatial averaging of the local fields. Our results are compared to experimental data obtained from impedance tube measurements. First, this approach leads to the identification of the macroscopic parameters involved in Pride and Lafarge semiphenomenological models. Second, it yields a direct access to thermal and viscous dynamic permeabilities. However, the bidimensional model underestimates the static viscous permeability as well as the viscous characteristic length; what thus require a three-dimensional implementation.

The initial growth of PTCDA on Cu(111) studied by STM

The initial growth of 3,4,9,10-perylene-tetracarboxylic-dianhydride (PTCDA) wasanalysed. Ultrathin films with coverages of up to two layers were prepared on a (111)orientated copper single crystal by means of vapour deposition in an ultrahigh-vacuumchamber. The films were characterized in situ by scanning tunnelling microscopy(STM). Within the first layer, two different structures were found. Both exhibit aherringbone-like arrangement of the molecules, which is also found in the (102) plane of theα and β bulk phases. The two-dimensional unit cell is given by two molecules which are rotated by about90°. As an effect of the interaction with the substrate, a voltage-dependent moiré pattern wasobserved for one of these phases. For the second layer, a herringbone phase was found thatis denser than the phases of the first layer but less dense than the bulk phases.

Observation of a Complex Nanoscale Magnetic Structure in a Hexagonal Fe Monolayer

We have observed a novel magnetic structure in the pseudomorphic Fe monolayer on Ir(111). Using spin-polarized scanning tunneling microscopy we find a nanometer-sized two-dimensional magnetic unit cell. A collinear magnetic structure is proposed consisting of 15 Fe atoms per unit cell with 7 magnetic moments pointing in one and 8 moments in the opposite direction. First-principles calculations verify that such an unusual magnetic state is indeed lower in energy than all solutions of the classical Heisenberg model. We demonstrate that the complex magnetic structure is induced by the strong Fe-Ir hybridization.

Computational characterization of micro- to mesoscopic deformation behavior of semicrystalline polymers

In the present study, we clarify the micro- to mesoscopic deformation behavior of semicrystalline polymers by the finite element homogenization method. The crystalline plasticity theory using a penalty method for the inextensibility constraint in the chain direction and the nonaffine molecular chain network theory were used to the representation of the deformation behavior of crystalline and amorphous phases, respectively, in the composite microstructure of a semicrystalline polymer. Various directional tensions are applied to the two-dimensional plane-strain unit cell model of a composite microstructure. The results reveal a highly anisotropic deformation behavior caused by the rotation of the chain direction and lamella interface, which depends on tensile direction and manifests as substantial hardening/softening in the early stage of deformation. The mesoscopic structure of a semicrystalline polymer was modeled using a voronoi polygon comprised of composite microstructures with different lamella interface directions. The initial isotropy of the response of the mesoscopic scale was verified. Due to their interaction with the surrounding grains, the individual grains of the mesoscopic scale show a conservative response as compared with that of the unit cell, and a very nonuniform response depending on the location of the respective grain is observed; these are typical of the mesoscopic response of semicrystalline polymers.

Numerical Simulation of Viscous Sintering by a Periodic Lattice of a Representative Unit Cell

In this paper numerical simulations of the viscous sintering phenomenon are presented, i.e., of the process that occurs (for example) during the densification of a porous glass heated to such a high temperature that it becomes a viscous fluid. The numerical approach consists of simulating the shrinkage of a two-dimensional unit cell which is in some sense representative for the porous glass. Hence it is assumed that the microstructure of the glass can be described by a periodic continuation in two directions of this unit cell. In this way it is possible to obtain insights into the viscous sintering process with respect to both pore size and pore distribution of the material. In particular this model is able to examine the consequences of microstructures on the evolution of the pore size distribution. The major finding is that the pores vanish in order of size one after another-the smallest pores first, followed by the larger ones. Moreover, it is shown that pores with concave boundary parts may initially grow before they start shrinking at a later stage.

Theoretical Analysis of Anion Exclusion and Diffusive Transport Through Platy-Clay Soils

Diffusive transport through geosynthetic clay liners and engineered compacted clay landfill liners is the primary mechanism for mass transport from well-engineered modern landfills. For this reason, accurate estimates of diffusion coefficients for clay soils are essential for the design of engineered liner systems. A long-standing theoretical problem is the role of anion exclusion on the estimation of diffusion coefficients for ionic solutes migrating through charged porous media. This paper describes the steady-state solution of a fully coupled set of transport equations modeling ion movement through a permanently charged platy-clay soil. The microscale analysis takes into account the actual diffusion coefficient for each ion species, ion-pairing (as required by electroneutrality of the solution), as well as anion exclusion and cation inclusion ,arising from the permanent charge on clay particles. To render the problem tractable, the theoretical analysis focuses on an extremely small two-dimensional unit cell in an ideal, saturated, two-phase porous medium. The analysis presented here is limited to a particular geometrical example, but this example is sufficiently general for characteristic behaviours of systems of this kind to be identified. Most importantly, new insight is gained into the mechanism of ion migration through a charged platy-clay soil. The numerical results obtained from this study show that the identification of macroscopic transport quantities such as effective diffusion coefficients and membrane potentials from diffusion cell tests using standard diffusion theory only hold for a specific system. While ion exclusion behaviours are often referred to in the literature, as far as the authors are aware there has been no previous detailed microscale analysis of their role in steady-state diffusion through a charged platy-clay soil.

Two-dimensional and axisymmetric unit cell models in the analysis of composite materials

Unit cell models have been widely used for investigating fracture mechanisms and mechanical properties of composite materials assuming periodically arrangement of inclusions in matrix. It is desirable to clarify the geometrical parameters controlling the mechanical properties of composites because they usually contain randomly distributed particulate. To begin with a tractable problem this paper focuses on the effective Young’s modulus E of heterogeneous materials. Then, the effect of shape and arrangement of inclusions on E is considered by the application of FEM through examining three types of unit cell models assuming 2D and 3D arrays of inclusions. It is found that the projected area fraction and volume fraction of inclusions are two major parameters controlling effective elastic modulus of inclusions.

A crystal structure of ultra-dispersed form of polytetrafluoroethylene based on X-ray powder diffraction data

An X-Ray powder diffraction study of ultra-dispersed polytetrafluoroethylene was carried out. As well as a regular polytetrafluoroethylene the ultra-dispersed form contents a high proportion of the crystalline phase. The X-ray diffraction pattern could be described with two-dimensional hexagonal unit cell [a=5.685(1) Å, symmetry group p6mm]. Structural modeling with a continuous electron density approach as well as with a discrete disordered atoms distribution was accomplished. The model was refined using the Rietveld method. The structure is characterized by a spiral arrangement of polymers (CF2-)n along the z-axis with complete mutual disordering by rotational displacement around z, as well as a partial molecular translation along the z-axis. Molecular disordering results in a systematic absence of reflections with 1≠0 and as a sequence in two-dimensional unit cell effect. The presence of complete rotational disordering distinguishes the ultra-dispersed form of polytetrafluoroethylene from the standard one (fluoroplast-4), where only partial disordering is observed.

Probing crystallization of calcium oxalate monohydrate and the role of macromolecule additives with in situ atomic force microscopy.

Kidney stones are crystal aggregates, most commonly containing calcium oxalate monohydrate (COM) microcrystals as the primary constituent. Macromolecules, specifically proteins rich with anionic side chains, are thought to play an important role in the regulation of COM growth, aggregation, and attachment to cells, all key processes in kidney stone formation. The microscopic events associated with crystal growth on the [010], [121], and [100] faces have been examined with in situ atomic force microscopy (AFM). Lattice images of each face reveal two-dimensional unit cells consistent with the COM crystal structure. Each face exhibits hillocks with step sites that can be assigned to specific crystal planes, enabling direct determination of growth rates along specific crystallographic directions. The rates of growth are found to depend on the degree of supersaturation of calcium oxalate in the growth medium, and the growth rates are very sensitive to the manner in which the growth solutions are prepared and introduced to the AFM cell. The addition of macromolecules with anionic side chains, specifically poly(acrylic acid), poly(aspartic acid), and poly(glutamic acid), results in inhibition of growth on the hillock step planes. The magnitude of this effect depends on the macromolecule structure, macromolecule concentration, and the identity of the step site. Poly(acrylic acid) was the most effective inhibitor of growth. Whereas poly(aspartic acid) inhibited growth on the (021) step planes of the (100) hillocks more than poly(glutamic acid), the opposite was found for the same step planes on the (010) hillocks. This suggests that growth inhibition is due to macromolecule binding to both planes of the step site or pinning of the steps due to binding to the (100) and (010) faces alone. The different profiles observed for these three macromolecules argue that local binding of anionic side chains to crystal surface sites governs growth inhibition rather than any secondary polymer structure. Growth inhibition by cationic macromolecules is negligible, further supporting an important role for proteins rich in anionic side chains in the regulation of kidney stone formation.

Ab initiothermodynamics of oxide surfaces:O2onFe2O3(0001)

The oxygen coverage, structure, and thermodynamic stability of (0001) surfaces of ${\mathrm{Fe}}_{2}{\mathrm{O}}_{3}$ (hematite) as a function of temperature and oxygen pressure are investigated by ab initio density functional theory with the generalized gradient approximation. Spin-polarized total energy and force calculations are performed using the projector augmented wave method as implemented in the Vienna ab initio simulation package. At high chemical potentials of oxygen (i.e., high pressure or low temperature), the most stable (0001) surface of hematite is completely covered with oxygen atoms. At low chemical potentials, a structure with one surface iron atom per two-dimensional unit cell is found to be the most stable surface termination. Around 800 K at an oxygen partial pressure of 0.2 bar, this reactive surface iron atom can bind and release an oxygen atom, thus switching between formal oxidation states (III) and (V), i.e., between stoichiometric and ferrate-like states. The fully reduced (iron terminated) surface is found to be thermodynamically unstable and dissociates adsorbed oxygen molecules spontaneously.

A high-mobility electron gas at the LaAlO3/SrTiO3 heterointerface.

Polarity discontinuities at the interfaces between different crystalline materials (heterointerfaces) can lead to nontrivial local atomic and electronic structure, owing to the presence of dangling bonds and incomplete atomic coordinations. These discontinuities often arise in naturally layered oxide structures, such as the superconducting copper oxides and ferroelectric titanates, as well as in artificial thin film oxide heterostructures such as manganite tunnel junctions. If polarity discontinuities can be atomically controlled, unusual charge states that are inaccessible in bulk materials could be realized. Here we have examined a model interface between two insulating perovskite oxides--LaAlO3 and SrTiO3--in which we control the termination layer at the interface on an atomic scale. In the simple ionic limit, this interface presents an extra half electron or hole per two-dimensional unit cell, depending on the structure of the interface. The hole-doped interface is found to be insulating, whereas the electron-doped interface is conducting, with extremely high carrier mobility exceeding 10,000 cm2 V(-1) s(-1). At low temperature, dramatic magnetoresistance oscillations periodic with the inverse magnetic field are observed, indicating quantum transport. These results present a broad opportunity to tailor low-dimensional charge states by atomically engineered oxide heteroepitaxy.

Epitaxial relation and island growth of perylene-3.4.9.10-tetracarboxylic dianhydride (PTCDA) thin film crystals on a hydrogen-terminated Si(1 1 1) substrate

Surface of an off-cut Si(1 1 1) substrate with an average step distance of 100 Å was terminated with monohydride. On this substrate, thin film crystals of organic semiconductor PTCDA were grown by molecular beam epitaxy (MBE). Surface of the thin film crystals on the hydrogen-terminated Si(1 1 1) surface (H-Si(1 1 1)) were observed using ultra-high vacuum scanning tunneling microscopy (UHV-STM) and atomic force microscopy (AFM). The PTCDA thin film crystals had an island shape of Volmer–Weber type. From the direction of the vicinal steps of the substrate [1 1 ̄ 0] , we determined the size and orientation of the PTCDA thin film crystals on the H-Si(1 1 1). Long-axis of the two-dimensional (2D) unit cell of the thin film crystals matches the vector (6, 2) of the H-Si(1 1 1) surface. We proposed two conceivable epitaxial relations: one is 6 2 2/3 11/3 and the other is point-on-line coincidence. 2D unit cells of the thin film crystals have widely stretched structure (2–10%), and the island growth of Volmer–Weber type is probably due to this large lattice misfit.

Naphtho[2,3-a]pyrene forms chiral domains on Au111.

Chiral domains have been prepared by evaporation of a two-dimensionally chiral molecule, naphtho[2,3-a]pyrene (NP), onto the hexagonal Au(111) surface in an ultrahigh vacuum environment. High-resolution UHV scanning tunneling microscopy (STM) showed that NP formed chiral domains consisting of only one 2D enantiomer rather than racemic two-dimensional unit cells. A structural model is proposed that agrees with the STM observations. Chiral pockets between adsorbed molecules may be useful for binding a specific enantiomer of a 3D chiral molecule.