Three-dimensional Counterparts Research Articles

Transformation plasmonics

Abstract Surface plasmons polaritons (SPPs) at metal/dielectric interfaces have raised lots of expectations in the on-going quest towards scaling down optical devices. SPP optics offers a powerful and flexible platform for real two-dimensional integrated optics, capable of supporting both light and electrons. Yet, a full exploitation of the features of SPPs is conditioned by an accurate control of their flow. Most efforts have so far focused on the extrapolation of concepts borrowed from guided optics. This strategy has already led to many important breakthroughs but a fully deterministic control of SPP modes remains a challenge. Recently, the field of optics was stimulated by a novel paradigm, transformation optics, which offers the capability to control light flow in any desired fashion. While it has already significantly contributed to the design of metamaterials with unprecedented optical properties, its versatility offers new opportunities towards a fully deterministic control of SPPs and the design of a new class of plasmonic functionalities. Here, we review recent progress in the application of transformation optics to SPPs. We first briefly describe the theoretical formalism of transformation plasmonics, focusing on its specificities over its three-dimensional optical counterpart. Numerical simulations are then used to illustrate its capability to tame SPP flows at a metal interface patterned with a dielectric load. Finally, we review recent experimental implementations leading to unique SPP functionalities at optical frequencies.

Comment on: Size effects on yield strength and strain hardening for ultra-thin Cu films with and without passivation: A study by synchrotron and bulge test techniques

Copper sub-micron films are structured as columnar arrays of grains and are virtual two-dimensional polycrystalline solids. By modelling their plastic deformation by a two-dimensional flow of sliding deformable grains, a new perspective of the mechanical tests of Gruber et al. [Acta Mater. 56 (2008) 2318] is attained that reveals remarkable regularities which are missed by the more conventional analysis of the authors. The model shows that two-dimensional plasticity has significant qualitative differences from the three-dimensional counterpart.

B12H n and B12F n : planar vs icosahedral structures

Using density functional theory and quantum Monte Carlo calculations, we show that B12Hn and B12Fn (n = 0 to 4) quasi-planar structures are energetically more favorable than the corresponding icosahedral clusters. Moreover, we show that the fully planar B12F6 cluster is more stable than the three-dimensional counterpart. These results open up the possibility of designing larger boron-based nanostructures starting from quasi-planar or fully planar building blocks.

Shell models of two-dimensional Coulomb crystals: Assessment and comparison with the three-dimensional case

Three shell models, differing in accuracy and computational cost, are formulated for two-dimensional Coulomb crystals. Offering a new means of predicting and analyzing properties of these species, the new models also provide new insights into their previously derived three-dimensional counterparts. In particular, analysis of the individual components of the energy error points out to the neglect of the positional relaxation as the main source of the differences between the approximate and exact energies. Within the realm of shell models, the two-dimensional case turns out to be somewhat more challenging than the three-dimensional one. Due to the lack of exact closed-form expressions for the optimal shell radii, it is computationally more expensive and the energy predictions at the same level of approximation are less accurate (as indicated by the maximum relative energy error of 0.15% vs. that of 0.03% found for spherical Coulomb crystals).

Birman—Wenzl—Murakami Algebra and Topological Basis

In this paper, we use entangled states to construct 9 × 9-matrix representations of Temperley—Lieb algebra (TLA), then a family of 9 × 9-matrix representations of Birman—Wenzl—Murakami algebra (BWMA) have been presented. Based on which, three topological basis states have been found. And we apply topological basis states to recast nine-dimensional BWMA into its three-dimensional counterpart. Finally, we find the topological basis states are spin singlet states in special case.

Effective wave numbers for thermo-viscoelastic media containing random configurations of spherical scatterers

The dispersion relation is derived for the coherent waves in fluid or elastic media supporting viscous and thermal effects and containing randomly distributed spherical scatterers. The formula obtained is the generalization of Lloyd and Berry’s [Proc. Phys. Soc. London 91, 678–688 (1967)], the latter being limited to fluid host media, and it is the three-dimensional counterpart of that derived by Conoir and Norris [Wave Motion 47, 183–197 (2010)] for cylindrical scatterers in an elastic host medium.

Magnetized stratified rotating shear waves

We present a spectral linear analysis in terms of advected Fourier modes to describe the behavior of a fluid submitted to four constraints: shear (with rate S), rotation (with angular velocity Ω), stratification, and magnetic field within the linear spectral theory or the shearing box model in astrophysics. As a consequence of the fact that the base flow must be a solution of the Euler-Boussinesq equations, only radial and/or vertical density gradients can be taken into account. Ertel's theorem no longer is valid to show the conservation of potential vorticity, in the presence of the Lorentz force, but a similar theorem can be applied to a potential magnetic induction: The scalar product of the density gradient by the magnetic field is a Lagrangian invariant for an inviscid and nondiffusive fluid. The linear system with a minimal number of solenoidal components, two for both velocity and magnetic disturbance fields, is eventually expressed as a four-component inhomogeneous linear differential system in which the buoyancy scalar is a combination of solenoidal components (variables) and the (constant) potential magnetic induction. We study the stability of such a system for both an infinite streamwise wavelength (k(1) = 0, axisymmetric disturbances) and a finite one (k(1) ≠ 0, nonaxisymmetric disturbances). In the former case (k(1) = 0), we recover and extend previous results characterizing the magnetorotational instability (MRI) for combined effects of radial and vertical magnetic fields and combined effects of radial and vertical density gradients. We derive an expression for the MRI growth rate in terms of the stratification strength, which indicates that purely radial stratification can inhibit the MRI instability, while purely vertical stratification cannot completely suppress the MRI instability. In the case of nonaxisymmetric disturbances (k(1) ≠ 0), we only consider the effect of vertical stratification, and we use Levinson's theorem to demonstrate the stability of the solution at infinite vertical wavelength (k(3) = 0): There is an oscillatory behavior for τ > 1+|K(2)/k(1)|, where τ = St is a dimensionless time and K(2) is the radial component of the wave vector at τ = 0. The model is suitable to describe instabilities leading to turbulence by the bypass mechanism that can be relevant for the analysis of magnetized stratified Keplerian disks with a purely azimuthal field. For initial isotropic conditions, the time evolution of the spectral density of total energy (kinetic + magnetic + potential) is considered. At k(3) = 0, the vertical motion is purely oscillatory, and the sum of the vertical (kinetic + magnetic) energy plus the potential energy does not evolve with time and remains equal to its initial value. The horizontal motion can induce a rapid transient growth provided K(2)/k(1)>>1. This rapid growth is due to the aperiodic velocity vortex mode that behaves like K(h)/k(h) where k(h)(τ)=[k(1)(2) + (K(2) - k(1)τ)(2)](1/2) and K(h) =k(h)(0). After the leading phase (τ > K(2)/k(1)>>1), the horizontal magnetic energy and the horizontal kinetic energy exhibit a similar (oscillatory) behavior yielding a high level of total energy. The contribution to energies coming from the modes k(1) = 0 and k(3) = 0 is addressed by investigating the one-dimensional spectra for an initial Gaussian dense spectrum. For a magnetized Keplerian disk with a purely vertical field, it is found that an important contribution to magnetic and kinetic energies comes from the region near k(1) = 0. The limit at k(1) = 0 of the streamwise one-dimensional spectra of energies, or equivalently, the streamwise two-dimensional (2D) energy, is then computed. The comparison of the ratios of these 2D quantities with their three-dimensional counterparts provided by previous direct numerical simulations shows a quantitative agreement.

Ultrasoft primitive model of polyionic solutions: Structure, aggregation, and dynamics

We introduce an ultrasoft core model of interpenetrating polycations and polyanions, with continuous Gaussian charge distributions, to investigate polyelectrolyte aggregation in dilute and semi-dilute salt-free solutions. The model is studied by a combination of approximate theories (random phase approximation and hypernetted chain theory) and numerical simulations. The calculated pair structure, thermodynamics, phase diagram, and polyion dynamics of the symmetric version of the model (the "ultrasoft restricted primitive model" or UPRM) differ from the corresponding properties of the widely studied "restricted primitive model" (RPM) where ions have hard cores. At sufficiently low temperatures and densities, oppositely charged polyions form weakly interacting, polarizable neutral pairs. The clustering probabilities, dielectric behavior, and electrical conductivity point to a line of sharp conductor-insulator transitions in the density-temperature plane. At very low temperatures, the conductor-insulator transition line terminates near the top of a first order coexistence curve separating a high-density liquid phase from a low-density vapor phase. The simulation data hint at a tricritical behavior, reminiscent of that observed for the two-dimensional Coulomb gas, which contrasts with the Ising criticality of its three-dimensional counterpart, the RPM.

Determination of 3D flame surface density variables from 2D measurements: Validation using direct numerical simulation

Three-dimensional compressible direct numerical simulation (DNS) data of freely propagating statistically planar turbulent premixed flames have been used to assess the accuracy of the isotropy- derived correction factors, which relate the two-dimensional projections of the different terms of the Reynolds averaged flame-surface density (FSD) transport equation with their corresponding actual three-dimensional counterparts for different values of Karlovitz number Ka, Lewis number Le, heat release parameter τ, and turbulent Reynolds number Ret. It is shown that the isotropic distribution of the surface area weighted probability density function (pdf) of the angle φ between the normal vectors on the measurement plane and on the flame surface provides a simple algebraic relation between the generalised FSDs evaluated in two and three dimensions (i.e., Σ2D and Σ3D), irrespective of the values of Ka,τ,Le, and Ret. Isotropic relations between two-dimensional and three-dimensional counterparts of the surface-averaged curvature and the FSD propagation term are also found to work well for all the values of Ka,τ,Le, and Ret considered in this study. However, the relations between the value obtained from two-dimensional projection and the actual three-dimensional value for the tangential strain rate and curvature terms in the FSD transport equation work well only for the high values of turbulent Reynolds number. The reasons behind the disagreement between the predictions of the relations derived based on isotropy arguments for both the tangential strain rate and curvature terms of the FSD transport equation are explained in detail. It is found that the threshold value of Ret above which the assumption of isotropy yields an accurate relation between two-dimensional projection and three-dimensional values for the tangential strain rate and curvature terms of the FSD transport equation depends on the values of heat release parameter, Lewis number, and the regime of the prevailing combustion process.

Density functional theory beyond the linear regime: Validating an adiabatic local density approximation

We present a local density approximation (LDA) for one-dimensional (1D) systems interacting via the soft-Coulomb interaction based on quantum Monte Carlo calculations. Results for the ground-state energies and ionization potentials of finite 1D systems show excellent agreement with exact calculations obtained by exploiting the mapping of an $N$-electron system in $d$ dimensions onto a single electron in $N\ifmmode\times\else\texttimes\fi{}d$ dimensions, properly symmetrized by the Young diagrams. We conclude that 1D LDA is of the same quality as its three-dimensional (3D) counterpart, and we infer conclusions about 3D LDA. The linear and nonlinear time-dependent responses of 1D model systems using LDA, exact exchange, and the exact solution are investigated and show very good agreement in both cases, except for the well-known problem of missing double excitations. Consequently, the 3D LDA is expected to be of good quality beyond the linear response. In addition, the 1D LDA should prove useful in modeling the interaction of atoms with strong laser fields, where this specific 1D model is often used.

Comparison of 2D and 3D density-weighted displacement speed statistics and implications for laser based measurements of flame displacement speed using direct numerical simulation data

In a recent study, a light sheet imaging approach has been proposed (Hartung et al., J. Appl. Phys. B 96 (2009) 843–862) which permits measurement of a quantity S d ∗ 2 D , which is the two-dimensional projection of the actual density-weighted displacement speed S d ∗ for turbulent premixed flames. Here the statistics of S d ∗ and S d ∗ 2 D are compared using a direct numerical simulation database of statistically planar turbulent premixed flames. It is found that the probability density functions (pdfs) of S d ∗ 2 D approximate the pdfs of S d ∗ satisfactorily for small values of root-mean-square turbulent velocity fluctuation u′, though the S d ∗ 2 D pdfs are wider than the S d ∗ pdfs. Although the agreement between the pdfs and the standard-deviations of S d ∗ 2 D and S d ∗ deteriorate with increasing u′, the mean values of S d ∗ 2 D correspond closely with the mean values of S d ∗ for all cases considered here. The pdfs of two-dimensional curvature κ m 2 D and the two-dimensional tangential-diffusion component of density-weighted displacement speed S t ∗ 2 D are found to be narrower than their three-dimensional counterparts (i.e. κ m and S t ∗ respectively). It has been found that the pdfs, mean and standard-deviation of π / 2 × κ m 2 D and π / 2 × S t ∗ 2 D faithfully capture the pdfs, mean and standard-deviation of the corresponding three-dimensional counterparts, κ m and S t ∗ respectively. The combination of wider S d ∗ 2 D pdfs in comparison to S d ∗ pdfs, and narrower S t ∗ 2 D pdfs in comparison to S t ∗ pdfs, leads to wider S r ∗ + S n ∗ 2 D = S d ∗ 2 D - S t ∗ 2 D pdfs than the pdfs of combined reaction and normal-diffusion components of density-weighted displacement speed S r ∗ + S n ∗ . This is reflected in the higher value of standard-deviation of S r ∗ + S n ∗ 2 D , than that of its three-dimensional counterpart S r ∗ + S n ∗ . However, the mean values of S r ∗ + S n ∗ 2 D remain close to the mean values of S r ∗ + S n ∗ . The loss of perfect correlation between two and three-dimensional quantities leads to important qualitative differences between the S r ∗ + S n ∗ 2 D - κ m 2 D and S r ∗ + S n ∗ - κ m , and between the S d ∗ 2 D - κ m 2 D and S d ∗ - κ m correlations. For unity Lewis number flames, the S d ∗ - κ m correlation remains strongly negative, whereas a weak correlation is observed between S d ∗ 2 D and κ m 2 D . The study demonstrates the strengths and limitations of the predictive capabilities of the planar imaging techniques in the context of the measurement of density-weighted displacement speed, which are important for detailed model development or validation based on experimental data.

Magnetic quantum critical point and dimensionality trend in cerium-based heavy-fermion compounds

We present realistic Kondo-lattice simulation results for the recently-discovered heavy-fermion antiferromagnet CePt2In7 comparing with its three-dimensional counterpart CeIn3 and the less two-dimensional ones, Ce-115's. We find that the distance to the magnetic quantum critical point is the largest for CeIn3 and the smallest for Ce-115's, and CePt2In7 falls in between. We argue that the trend in quasi-two-dimensional materials stems from the frequency dependence of the hybridization between Cerium 4f-electrons and the conduction bands.

Numerical modelling of the pull-out of hooked steel fibres from high-strength cementitious matrix, supplemented by experimental results

The paper deals with the numerical modelling of hooked steel fibres embedded in high-strength cementitious matrix. The pull-out of the fibres is first studied by means of accurate three-dimensional models which take into account the non linearities that are present in the physical model. The bonding properties of the fibre–matrix interface considered in the numerical model are based on experimental results of pull-out tests on straight fibres. The results of the numerical model are compared with respect to the results of experimental tests on hooked fibres. Then, the paper studies the effect of different fibre geometries to the load–displacement pull-out curves. It is concluded that fibres having a doubly bended shape have a superior behaviour compared to fibres having a single bended part. Finally, a simplified two-dimensional model is proposed that yield fast results which are similar to the ones obtained by its three-dimensional counterpart.

Negative refraction in the visible spectrum in photonic crystals: search for focusing by fluorescent quantum dots inside synthetic opals

The negative refraction inside two dimensional (2D) photonic crystals (PC) and homogeneous structures was studied numerically by inserting a point source inside the negative index part of these materials. The three-dimensional (3D) counterparts of these structures, the synthetic opal, were examined experimentally by infiltrating them with CdS quantum dots (QDs). Confocal microscopy measurements in which we tracked the infiltration of the QDs inside the opal indicate the focusing of light emitted by QDs, which can be due to negative refraction occurring at the opal/glass interface. The formation of a focus can be an indication of the negative refraction happening in these synthetic opals in the [111] direction in its higher photonic band, above the photonic band gap (PBG). This result is very promising because, until now, negative refraction has not been seen in 3D photonic crystals in the visible region of light. This result was made possible due to the use of infiltrated QDs as internal light sources inside the porous photonic crystal, which appears to be a very useful technique for the study of other negative-index materials (NIM) effects. Finally, the possibility of self-focusing of second harmonic by QD in nonlinear opal was developed.

Three-dimensional discrete-time Lotka–Volterra models with an application to industrial clusters

We consider a three-dimensional discrete dynamical system that describes an application to economics of a generalization of the Lotka–Volterra prey–predator model. The dynamic model proposed is used to describe the interactions among industrial clusters (or districts), following a suggestion given by [23]. After studying some local and global properties and bifurcations in bidimensional Lotka–Volterra maps, by numerical explorations we show how some of them can be extended to their three-dimensional counterparts, even if their analytic and geometric characterization becomes much more difficult and challenging. We also show a global bifurcation of the three-dimensional system that has no two-dimensional analogue. Besides the particular economic application considered, the study of the discrete version of Lotka–Volterra dynamical systems turns out to be a quite rich and interesting topic by itself, i.e. from a purely mathematical point of view.

Velocity and energy profiles in two- versus three-dimensional channels: Effects of an inverse- versus a direct-energy cascade

In light of some recent experiments on quasi two-dimensional (2D) turbulent channel flow we provide here a model of the ideal case, for the sake of comparison. The ideal 2D channel flow differs from its three-dimensional (3D) counterpart by having a second quadratic conserved variable in addition to the energy and the latter has an inverse rather than a direct cascade. The resulting qualitative differences in profiles of velocity V and energy K as a function of the distance from the wall are highlighted and explained. The most glaring difference is that the 2D channel is much more energetic, with K in wall units increasing logarithmically with the Reynolds number Re_{tau} instead of being Re_{tau} independent in 3D channels.

Defect-induced effects on carrier migration through one-dimensional poly(para-phenylenevinylene) chains

Defects in one-dimensional (1D) systems can be intrinsically distinct from its three-dimensional counterparts, and polymer films are good candidates for showing both extremes that are difficult to individuate in the experimental data. We study theoretically the impact of simple hydrogen and oxygen defects on the electron transport properties of one-dimensional poly(para-phenylenevinylene) chains through a multiscale technique, starting from classical structural simulations for crystalline films to extensive ab initio calculations within density functional theory for the defects in single crystalline-constrained chains. The most disruptive effect on carrier transport comes from conjugation breaking imposed by the overcoordination of a carbon atom in the vinyl group independently from the chemical nature of the defect. The particular case of the $[\text{C}=\text{O}]$ (keto-defect) shows in addition unexpected electron-hole separation, suggesting that the experimentally detected photoluminescence bleaching and photoconductivity enhancement could be due to exciton dissociation caused by the 1D characteristics of the defect.

A shallow-water theory for annular sections of Keplerian disks

A scaling argument is presented that leads to a shallow water theory of non-axisymmetric disturbances in annular sections of thin Keplerian disks. To develop a theoretical construction that will aid in physically understanding the relationship of known two-dimensional vortex dynamics to their three-dimensional counterparts in Keplerian disks. Using asymptotic scaling arguments varicose disturbances of a Keplerian disk are considered on radial and vertical scales consistent with the height of the disk while the azimuthal scales are the full $2\pi$ angular extent of the disk. The scalings lead to dynamics which are radially geostrophic and vertically hydrostatic. It follows that a potential vorticity quantity emerges and is shown to be conserved in a Lagrangian sense. Uniform potential vorticity linear solutions are explored and the theory is shown to contain an incarnation of the strato-rotational instability under channel flow conditions. Linearized solutions of a single defect on an infinite domain is developed and is shown to support a propagating Rossby edgewave. Linear non-uniform potential vorticity solutions are also developed and are shown to be similar in some respects to the dynamics of strictly two-dimensional inviscid flows. Based on the framework of this theory, arguments based on geophysical notions are presented to support the assertion that the strato-rotational instability is in a generic class of barotropic/baroclinic potential vorticity instabilities. Extensions of this formalism are also proposed. The shallow water formulation achieved by the asymptotic theory developed here opens a new approach to studying disk dynamics.

Attitude Dynamics of Rigid Bodies in the Vicinity of the Lagrangian Points

I N THE last 30 years, several sun-monitoring spacecraft have orbited around the sun–Earth L1 point, and several space observatories will be placed near the sun–EarthL2 in the next decade. Although a large body of literature exists on the orbital dynamics of spacecraft in the restricted three-body problem, very few have explored the attitude dynamics of a rigid Lagrangian-point spacecraft. Kane and Marsh [1] considered the attitude dynamics of an axial symmetric satellite that is rotating about its axis of symmetry, which is normal to the primary bodies’ orbital plane. Robinson [2] first studied the attitude dynamics of a dumbbell satellite located at a triangular point. The same author [3] later determined the equilibrium attitudes of an arbitrary shaped satellite located at a collinear point or at a triangular point and constructed a linear stability diagram about one of the equilibrium configurations. Misra andBellerose [4] studied the librational dynamics of a tethered satellite located at the Earth–moon Lagrangian points and obtained the libration frequencies. The rigid spacecraft is assumed to be held at the Lagrangian points in all of these studies. In practice, however, collinear point satellites are not located directly at the Lagrangian point but in periodic orbits around the point. This note extends the work done by Robinson [3] and Misra and Bellerose [4] and addresses the question of how the translational motions of the spacecraft affect its attitude dynamics. The attitude dynamics of the satellite are studied while it is in a planar Lyapunov orbit and its three-dimensional counterpart. A triangular point spacecraft was previously demonstrated to have two equilibrium configurations, and linear stability diagrams about both equilibrium configurations for a L4 spacecraft are also presented. II. Problem Formulation

Interfacial rheology of stable and weakly aggregated two-dimensional suspensions

The interfacial rheological properties of stable and weakly aggregated two-dimensional suspensions are studied experimentally using a magnetic rod interfacial rheometer. Particle monolayers with well controlled structures were prepared. Charged polystyrene particles create two-dimensional colloidal crystals at the water-decane interface over a wide range of concentrations. Under similar conditions a predominantly liquid structure is obtained at the water-air interface for the same particles. The addition of appropriate combinations of the anionic surfactant sodiumdodecylsulfate (SDS) and sodium chloride (NaCl) to the aqueous subphase leads to a destabilization of these monolayers with the formation of fractal aggregates at low concentrations and a heterogeneous gel forming as the surface coverage is increased. After the structures have been built up a reproducible structure can be obtained, of which the interfacial rheological properties can be investigated using a magnetic rod stress rheometer. In all cases, numerical calculations were used to assess the importance of instrumental artifacts and the effect of the coupling between surface and subphase flows. The rheology of aggregated suspensions was compared to the reference case of a colloidal crystal. The two-dimensional aggregated suspensions display rheological features which are similar to their three-dimensional counterparts. These include an elastic response with small linearity limits, a power law dependence on surface coverage and a dependence on the strength of attraction. The results shed some light on the possible role of interfacial rheology on the stability of particle laden high interface systems. Additionally, the 2D suspensions could present fundamental insights in the rheological properties of dense colloidal suspensions.