2D Rectangular Lattice Research Articles

Uniform flows in rectangular lattice networks

The general problem of describing the flow distribution in manifolds feeding parallel channels has been addressed in a number of earlier publications. Such systems are used for example as heat exchanger plates, as distributors for fuel cells, or as solar collectors. We address here the "inverse problem", i.e. of finding a design, which provides uniform flow distribution among the channels, for which much less literature is available. We approach that problem by considering the steady-state flow-rates in the branches of 2D rectangular lattice networks with a single inlet and a single outlet, of which the classical parallel channels (ladder) network can be viewed as a special case. The purpose of this paper is to show how to organize the flow resistances on the periphery of the lattice so as to achieve flow uniformity in all inside channels. This is done under the assumption of a resistive network, with linear flow/pressure behavior (pure viscous Stokes flow, neglecting the inertial terms, neglecting or linearizing the effect of branching singularities) allowing completely analytical solutions and comparison of different configurations. The compatibility of uniform flow distribution and uniform Residence-Time-Distribution is discussed. Examples of new conceptual designs of heat exchanger plates, of fluid distributors and of mixers are proposed, based on this possibility of uniform distribution.

Novel non-discoid chiral copper(II)-salen type [N2O2] donor Schiff base complexes with a cyclohexane diamine spacer: synthesis, electrochemistry, columnar mesomorphism and DFT study

A series of new non-discoid chiral copper(ii)-salen type complexes, [Cu((4-C n H2n+1O)2salen)], n = 14, 16, 18 and salen = N,N′-cyclohexane bis (salicylideneiminato), containing 4-substituted alkoxy tails in the side aromatic rings and a cyclohexane spacer, have been prepared and their mesogenic properties investigated. The compounds were characterised by Fourier transform infra-red spectroscopy, 1H and 13C nuclear magnetic resonance spectroscopy, UV-vis spectroscopy and fast atom bombardment mass spectrometry. Mesomorphic properties of these compounds were studied by polarising optical microscopy, differential scanning calorimetry and X-ray diffraction (XRD). The ligands are non-mesogenic but the complexes exhibited enantiotropic rectangular columnar mesophases (Colr) with extended temperature range (111–220°C). Molecular models for the organisation in the mesophases are proposed on the basis of the small-angle XRD diffraction patterns. Molecules with their planes tilted are stacked in columns which self-organise in a 2D-rectangular lattice with C2mm symmetry. Cyclic voltammetry carried out in CH2Cl2 solution revealed a single electron quasireversible response for the Cu(i)/Cu(ii) redox couple (E 1/2 = –0.226V, ΔE p = 453 mV). The more negative redox potentials indicated a planar coordination environment. Density functional theory calculations performed using the DMol3 program at the BLYP/DNP level revealed a near-square-planar arrangement around the metal centre with the cyclohexane ring tilted relative to the plane of the [N2O2] core of the Schiff base ligand.

Liquid-crystalline functional carbazole and naphthalene platforms

Two novel families of highly functionalized molecules consisting of a central carbazole or naphthalene core carrying actionable acid functions and linked to gallate platforms equipped with three long alkoxy chains have been rationally designed. The presence of amide tethers and chelating rigid fragments generate a new class of luminescent mesomorphic materials. Platforms bearing long paraffin chains (n = 12 and 16) self-organize into columnar liquid crystalline phases in which the columns are arranged according to a 2D oblique, rectangular or hexagonal lattice symmetry as deduced from powder XRD experiments. FT-IR studies confirm that the common driving force for aggregation and micro-segregation in the mesophase is due to the occurrence of a tight intermolecular H-bonded network. Spectroscopic measurements show that these platforms are luminescent in solution and in the solid state. Due to the synthetic availability of carbazole and naphthalene rigid cores and the simplicity by which these platforms can be graphed, this methodology represents a practical alternative to the production of fluorescent and mesomorphic materials.

Orientational vibrations of 2D molecular layers with quadrupolar interaction

AbstractMolecular layers with rotational degrees of freedom and quadrupolar interaction between linear molecules are investigated theoretically. Rotation of the molecules in the plane of an adsorbing surface is considered; centers of gravity of the molecules are pinned to sites of a rectangular lattice. The energy of the system contains an additional parameter ρ (ratio of the lattice constants of a 2D system), in comparison with the 1D case. It is found that alternating orientation of the adsorbed molecules along both sides of the rectangular unit cell corresponds to the minimum of the energy. Equations of rotational movement are derived for linear oscillations of molecules near the equilibrium positions in a 2D lattice. Dispersion relations are obtained and analyzed. The Brillouin zone cannot be divided as in the 1D case after unit‐cell doubling. The parameter B (k), which describes the topology of the dispersion surfaces, is introduced. It is shown that both branches of the spectrum have optical behavior. The gaps and bandwidths are found as functions of ρ. The method of calculation of the density of states (DOS) for translational oscillations of a 2D rectangular lattice with one atom per unit cell is generalized for rotational oscillations of a 2D two‐sublattice molecular lattice. The diagram describing three regions with different analytical properties of the DOS which correspond to different topologies of the isofrequency lines is introduced. This diagram is a portrait in the plane of the parameters B and ρ. The DOS is found in terms of complete elliptic integrals of the first kind. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Treating the Independent Set Problem by 2D Ising Interactions with Adiabatic Quantum Computing

We construct a nearest-neighbor interaction whose ground states encode the solutions to the NP-complete problem “independent set” for cubic planar graphs. The important difference to previously used Hamiltonians in adiabatic quantum computing is that our Hamiltonian is spatially local. Due to its special structure our Hamiltonian can be easily simulated by Ising interactions between adjacent particles on a 2D rectangular lattice. We describe the required pulse sequences. Our methods could help to implement adiabatic quantum computing by “physically reasonable” Hamiltonians like short-range interactions. Therefore, this “universal resource Hamiltonian” can be used for different graphs by applying suitable control operations. This is in contrast to a previous proposal where the Hamiltonians have to be wired in hardware for each graph. PACS: 03.67.Lx

Anderson localization in two-dimensional disordered systems

Using the transfer matrix method we have calculated the localization length in a 2D (two dimension) rectangular lattice with both on-site and off-diagonal disorder. Using finite size scaling we show that systems with off-diagonal disorder are much more sensitive to disorder than the system with on-site disorder, e.g. at the same amount of disorder ( W= w=2) the localization length for on-site disorder is 10 4 times longer than for off-diagonal disorder. We consider both isotropic and anisotropic systems, where the latter can be considered as a model for an organic crystal. In the anisotropic case the maximum localization length is at the band center, the isotropic system has a maximum away from the band center.

High-brightness mid-infrared photonic-crystal distributed-feedback lasers

The far-field emission and spectral characteristics of a photonic-crystal distributed-feedback (PCDFB) laser with an antimonide type-II ‘W’ active region were studied theoretically and experimentally. A 2nd-order grating was defined on a 2D rectangular lattice tilted by 20° relative to the facet normal, using optical lithography and dry etching. For pulsed optical pumping, the emission line centred on λ = 4.6–4.7 μm was up to an order of magnitude narrower (7–10 nm) than those of Fabry-Pérot and angled-grating DFB (α-DFB) lasers fabricated from the same wafer. The PCDFB beam quality was also substantially enhanced, for example, by a factor of 5 compared to the α-DFB at a pump-stripe width of 200 μm. Theoretical simulations based on a self-consistent time-domain Fourier-transform simulation confirm significant advantages of the PCDFB laser over an α-DFB geometry.

Rectangular lattices of permalloy nanoparticles: Interplay of single-particle magnetization distribution and interparticle interaction

Regular 2D rectangular lattices of permalloy nanoparticles (40 nm in diameter) were prepared by the method of the electron lithography. The magnetization curves were studied by Hall magnetometry with the compensation technique for different external field orientations at 4.2K and 77K. The shape of hysteresis curves indicates that there is magnetostatic interaction between the particles. The main peculiarity is the existence of remanent magnetization perpendicular to easy plain. By numerical simulation it is shown, that the character of the magnetization reversal is a result of the interplay of the interparticle interaction and the magnetization distribution within the particles (vortex or uniform).

Magnetization curves for two-dimensional rectangularlattices of permalloy nanoparticles: experimental investigation andnumerical simulation

This work is concerned with the behaviour of a regular 2D rectangularlattice of Ni3Fe nanoparticles, determined by the dipoleinteraction between them. The samples under study, prepared byelectron-beam lithography, consisted of about 105 particlesapproximately 50 nm in size. The magnetization curves were studiedby Hall magnetometry for different external magnetic fieldorientations at 4.2 K and 77 K. The results indicate a collectivebehaviour of the system. The magnetization curves depend on theexternal magnetic field direction and temperature; the systemexhibits multistability. A model system of interacting 3D magneticdipoles forming a rectangular lattice was numerically simulated bysolving a system of stochastic Landau-Lifshitz equations. Themultistability of the system and steps in the magnetizationcurves were obtained. It is shown analytically that the shape ofthe magnetization curves depends on the character of the interactionin the system.

Bond-order and charge-density waves in the isotropic interacting two-dimensional quarter-filled band and the insulating state proximate to organic superconductivity

We report two surprising results regarding the nature of the spatial broken symmetries in the two-dimensional (2D), quarter-filled band with strong electron-electron interactions. First, in direct contradiction to the predictions of one-electron theory, we find a coexisting ``bond-order and charge density wave'' (BCDW) insulating ground state in the 2D rectangular lattice for all anisotropies, including the isotropic limit. Second, we find that the BCDW further coexists with a spin-density wave (SDW) in the range of large anisotropy. Further, in contrast to the interacting half-filled band, in the interacting quarter-filled band there are two transitions: first, a similar singlet-to-AFM/SDW transition for large anisotropy and second, an AFM/SDW-to-singlet transition at smaller anisotropy. We discuss how these theoretical results apply to the insulating states that are proximate to the superconducting states of 2:1 cationic charge-transfer solids (CTS). An important consequence of this work is the suggestion that organic superconductivity is related to the proximate Coulomb-induced BCDW, with the SDW that coexists for large anisotropies being also a consequence of the BCDW, rather than the driver of superconductivity.

Magnetic-Field-Induced Directional Localization in a 2D Rectangular Lattice

We study the effect of a perpendicular uniform magnetic field on the dissipative conductivity of a rectangular lattice with anisotropic hopping, $t_x\neq t_y $. We show that the magnetic field may enhance dramatically the directional anisotropy in the conductivity. The effect is a measurable physical realization of Aubry's duality in Harper systems.

A Monte Carlo simulation of submonolayer homoepitaxial growth on Ag(110) and Cu(110)

The low-coverage ( θ≤0.05) growth of Ag/Ag(110) and Cu/Cu(110) is studied by kinetic Monte Carlo simulations. Our model includes deposition, diffusion and fully reversible aggregation on a 2D rectangular lattice with both anisotropic diffusion barriers and anisotropic attachment. The barriers for the diffusion processes are calculated by many-body RGL potentials, and compared to the available data with good agreement. Depending on T, in both metals the model shows morphology transitions to 1D in-channel strips and then to 2D compact islands. The latter transition is due to the activation of in-channel detachment followed by terrace-mediated reshaping and not to direct corner rounding. Even if the transition takes place in the same temperature range for both metals, the 1D strips obtained in Cu are much longer. At the transition, a break in the slope of the island density versus 1/ T is obtained, due to the onset of ripening during growth. There, the island density presents a maximum at very low coverages, followed by a decrease due to island dissolution and not to coalescence. The possibility of extracting the diffusion barriers by measuring the island density is discussed, finding that it is possible to extract a reliable value for the in-channel diffusion barrier in Ag, where the diffusion anisotropy is stronger. In Cu, an average between the in-channel and the cross-channel barriers is extracted.

Quantum spin liquid on 2D rectangular frustrated AF lattice

The state of a 2D rectangular spin lattice with first, second and third nearest-neighbour anisotropic antiferromagnetic interactions is investigated. In a special line of parameter space, it is proved that the quantum spin liquid is the exact eigenstate of the system. For general parameter values, Green's function method with cut-off approximation is used to determine the phase diagram at zero temperature, and it is found that the quantum spin liquid appears on a rectangular lattice when the third nearest-neighbour interaction in the direction with shorter lattice spacing is introduced. The relations to the possibility of mixture of the spin liquid and the Fermi liquid in high-Tc superconductors are discussed.

Variant of Green's function nodal method for neutron diffusion.

A variant of the Green's function nodal method derived from the boundary integral form of the multigroup neutron diffusion equation in rectangular geometry is presented. As usual in the nodal methods, the multi-dimensional diffusion equation is integrated in the transverse direction. The resulting 1D diffusion equation is solved following the Boundary Element technique in one dimension. In this way a weighted residual method is obtained, with a Green's function for weighting, but with different boundary conditions than normally applied in the Green's function nodal methods. Mathematical formulation of the method is given and the iteration procedure is described. A computer program BINDIF has been designed, based on the new method. Its capabilities include the solution of the multigroup neutron diffusion equation of 1D, 2D and 3D rectangular lattices. The BINDIF program has been checked against other methods used for global reactor calculations on benchmark problems, representative of realistic power reactor cores. The results indicate that the method is attractive to design highly efficient algorithms for a large mainframe, a personal computer or a parallel processor.