One-dimensional Rods Research Articles

Asymptotic Analysis and Modeling of the Jointing of a Massive Body with Thin Rods

Asymptotic formulas are obtained for solutions of the anisotropic elasticity problem for a body with cavities into which thin rods are inserted, the outer ends of the rods being rigidly fixed. The surface of the body and the lateral surface of the rods are assumed load-free, but the entire elastic junction is subject to mass forces. The elastic materials are inhomogeneous and the stiffness of the rods may differ greatly from that of the body, their ratio being of the order hγ with an arbitrary exponent γ ∈ ℝ; for γ = 0, the junction is homogeneous. Together with the asymptotic formulas, we construct and justify an asymptotic model of the junction. This model is applicable for a wide range of the exponent γ and preserves the parameter h in the conjugation conditions but is represented by a regularly perturbed problem. Since the leading asymptotic term involves fields with strong singularities, we have to give correct statements of the limit problem for a body with one-dimensional rods. For this purpose, we use the theory of self-adjoint extensions of operators or the technique of weighted spaces with separated asymptotics. The justification of our asymptotic expansions utilizes weighted anisotropic Korn inequalities, which take into account the mutual position of the rods and provide the best possible a priori estimates of the solutions. In contrast to other investigations, we describe, in explicit terms, the dependence of the bounds in the error estimates on the right-hand sides of the original problem. We also discuss the relationship between the asymptotic ansatz formulas and the weighted norms in the asymptotically precise Korn inequality.

Dynamics in a Rod Model of Solid Flame Waves

We consider gasless solid fuel combustion in a cylinder of radius $\tilde{R}$, associated with the SHS (self-propagating high-temperature synthesis) process, which employs combustion waves to synthesize materials. A powder mixture of reactants is cold pressed into a sample, typically a cylinder, and ignited at one end. A high-temperature combustion wave then propagates along the sample, converting the unreacted powder into the desired product. It has recently become popular to model systems governed by partial differential equations as an array of interacting oscillators. Here, we extend this approach by considering an array of interacting rods, each of which supports propagating waves. Thus, we employ an array of interacting one-dimensional (1D) rods connected via heat transfer. The heat transfer terms correspond to a discretization of the transverse Laplacian. Both the full 3D model and the rod model allow for a uniformly propagating planar combustion wave. The dispersion relation for this solution is determined for both models and shown to be equivalent when certain parameters in the two models are identified. The rod model is able to describe a number of the features of the 3D model, thus allowing numerical simulations with significantly reduced computational resources. In this paper we consider a rod model consisting of an outer ring of three rods equally spaced along the ring, together with an axial rod. Clearly, this limits the 3D modes of wave propagation which can be described, and the results below have to be considered within this basic limitation. The 3/1 model admits analogues of spin and radial modes which are known to exist experimentally and as solutions of the 3D model. We propose that the new modes of solution behavior that we find are also related to modes of the 3D model. We determine solution behavior as a function of R, the nondimensionalized cylindrical radius. We consider three cases characterized by the Zeldovich number $Z=N(1-\sigma)/2$, where N is a nondimensionalized activation energy and $\sigma$ is the ratio of the unburned to the burned temperature. For Z sufficiently small, the uniformly propagating planar solution is stable to planar, i.e., rod independent, perturbations. In this case, analysis of the dispersion relation predicts that for sufficiently small and sufficiently large R, only the uniformly propagatingsolution is stable. Spin modes, seen in experiments as hot spots spinning periodically around the cylinder as the wave propagates, occur for small R, and radial modes, periodically oscillating solutions independent of the cylindrical angle, occur for larger R. We find analogues of these modes for the rod model and describe the transition from spin to radial modes via a family of quasiperiodic (QP) modes, manifested by periodic variations in the temperature and rotation speed of the spot. For a larger value of Z, the uniformly propagating solution is unstable to planar perturbations.In this case, we find that for sufficiently small values of R, the singly periodic pulsating planar (PP) solution is the only stable solution. For larger values of R, there is bistability between PP solutions and spin solutions. As R increases further, both solutions lose stability, and the only stable solution is a QP mode. This mode is a combination of spin and radial behavior. Unlike the smaller Z case, stable radial solutions are not found. As R increases, the two generator frequencies of the QP solutions converge to the dominant frequency of the PP solution. In the time domain the solution in each rod is approximately the PP solution on intermediate time scales, with a long time envelope corresponding to the small difference between the two generator frequencies. There is a phase difference between adjacent outer rods which is essentially constant over intermediate time scales, but which varies over longer time scales. The intermediate time scale increases as R increases. Thus, we find QP spin-like behavior which on intermediate time scales appears as a spinning manifestation of the PP solution. We refer to this behavior as spinning PP (SPP) behavior. We also consider a yet larger value of Z. We find a PP solution which is now period doubled. In addition, we find spin and QP spin solutions as before. We also find an interval in R where spinning-type solutions reverse direction in a periodic or QP fashion.

Sterically Induced Shape and Crystalline Phase Control of GaP Nanocrystals

We demonstrate a novel synthetic scheme that can be used to control the crystalline phase and shape of GaP semiconductor nanocrystals. Our study shows that steric effects of surfactant ligands can modulate the crystalline phases and control the shapes of nanocrystals. The shape of the nanocrystals obtained varies from zero-dimensional spheres to one-dimensional rods via controlling the ratio between primary and tertiary alkylamines. III-V semiconductors (in our case: GaP) under 10 nm in width are first reported, and unique optical properties due to shape anisotropy are also observed.

Existence and nature of a freezing transition inside three-dimensional arrays of narrow channels.

The density functional for matter confined inside a three-dimensional array of narrow channels is constructed. It is shown that the effect of long range interactions amounts to a three-dimensional quadratic perturbation on an array of one-dimensional hard rods. Hence, it follows that the recently proposed freezing transition of atoms confined in nanotube arrays is continuous. This conclusion is validated beyond mean field theory by Monte Carlo simulations in the NPT ensemble.

The Structure Directing Effect of Hydrogen Bonding in the Novel Polymeric Thioantimonate Mn2(H2N(CH2)2NH2)2Sb2S5

A new transition metal thioantimonate(III) with composition Mn2(H2N(CH2)2NH2)Sb2S5 has been synthesised under solvothermal conditions. Two trigonal SbSb3 pyramids and two octahedrally coordinated Mn atoms are interconnected to form Mn2SbS4 heterocubanes as secondary building units (SBU’s). The SBU’s are covalently linked into linear infinite onedimensional rods. Long Sb-S bonds connect the rods to form layers. The two N atoms of the ethylenediamine molecule (en) are chelating one Mn(II) ion. The amino hydrogen atoms of the en ligand are engaged in hydrogen bonding which is responsible for the different structure pattern of the title compound compared to a previously reported series of Mn-amino-thioantimonates(III) with analogous stoichiometry built up from different amino ligands.

Dynamics of a one-dimensional granular gas with a stochastic coefficient of restitution

Recently, we have modelled inelastic collisions of one-dimensional rods [1,2] by the absorption of translational energy E tr through internal degrees of freedom, in particular elastic vibrations. We arrived at a stochastic description of collision processes, characterised by a stochastic coefficient of restitution ε. In this paper, we construct an analytic approximation for the transition probability E tr→ E tr′=(1− ε 2) E tr. This allows us to perform much longer simulations of large, strongly inelastic granular systems and study relaxation to the true equilibrium state. If the internal vibrations are undamped, equilibrium is characterised by propagating sound waves. In the case of damping, the system develops towards a final state which consists of one big cluster, containing all particles at rest.

Photoresponsive and Redox-Active Supramolecular Systems with Rigid Sp Carbon Chain Spacers

The construction and investigation of photoresponsive and redox-active supramolecular systems with acetylenyl (polyyne) and cumulenic sp hybridized carbon chains (Cn) are reported here. Novel types of one-dimensional rigid molecular rods and macrocyclic molecular squares with controllable and yet fixed chain lengths or cavity sizes are explored. Such systems will have the desired features such as synthetic accessibility, coordination versatility, rigidity, directionality, photostability, electronic conductivity, controllable distances between donors and acceptors, and controllable cavity sizes in macrocyclic molecular squares. These features are extremely important to the future applications of these new rigid supramolecular systems in molecular photonic and electronic devices, light-energy conversion, molecular recognition, and host-guest chemistry. In addition to the exploration of synthetic features, the solution state redox characteristics and molecular photophysics of the metal-containing supramolecular systems with sp carbon chains are also under investigation. This includes the modulation of electronic communication between multiple redox centers spanned by rigid spacers of various chain lengths and the study of long-range, directional photoinduced energy/electron transfer across such spacers. In addition, covalent surface confinement of systems containing sp carbon chains is also investigated here. Specifically, UV/vis and X-ray photoelectron spectroscopy have been used as spectroscopic tools to monitor the growth of mono-and multilayer thin films.

Grain Structure Evolution in 1-d Rods and 2-d Strips of Polycrystalline Aluminium

The paper presents data on the motion of grain boundary system with individual triple junction in aluminium and data on the grain structure evolution in one-dimensional (1-D) rods (bamboo structure without triple junctions) and two-dimensional (2-D) strips (columnar structure) of polycrystalline aluminium. Influence of triple junction motion on the grain growth in polycrystalline aluminium is discussed.

Predicting adsorption in one-dimensional zeolite pores with the exact theory of one-dimensional hard rods

The exact statistical mechanical theory of a one-dimensional hard-rod fluid in an arbitrary external field is applied to the adsorption of simple molecules into nearly one-dimensional zeolitic pores, using a zeolite-sorbent interaction potential based on the system of xenon in mordenite. The effect is explored of changing the zeolite-sorbent interaction on adsorption. By optimizing one parameter in the model, the results of this theory compare well with experimental isotherms of xenon in mordenite.

Structure of CdCl2.C3H7NO2

catena-Poly[cadmium-di-μ-chloro-μ-L-alanine-O:O'], CdCl 2 .C 3 H 7 NO 2 , has been synthesized and structurally characterized. The Cd atom occupies a distorted octahedral environment of four Cl atoms and two O atoms. These octahedra are linked through Cl...Cl edges and bridged by carboxyl groups of the alanine molecules to form one-dimensional rods that extend along [010]. Crystallization in C2 with a=16.240, b=7.272, c=7.987A, β=116.44°, Z=4, R=0.063

Molecular dynamics and Monte Carlo simulations in the grand canonical ensemble: Local versus global control

In inhomogeneous systems the diffusion paths available to particles are often restricted by the topology of the system itself. In search of a realistic scheme to simulate such inhomogeneous systems, we developed a new method, of local control of the chemical potential. The local control method, which limits creation and destruction of particles to a control volume, can be successfully implemented in both Monte Carlo (MC) and molecular dynamics (MD) simulations of the grand canonical ensemble (GC). We present applications of local GCMC calculations of the one-dimensional lattice gas and hard rods confined by hard walls, as well as local GCMC and GCMD of a three-dimensional bulk Lennard-Jones fluid system. The simulation results compare successfully with analytical solutions and equation of state approximations.

Crystallization of one-dimensional hard rods with a periodic interaction

A one-dimensional model of hard rods with a long-range cosine interaction is solved exactly in some canonical ensemble cases. It exhibits a crystalline phase at low temperatures as a result of a spontaneous symmetry breaking of the order parameter which represents an effective collective mode. When the wavelength of the attractive potential is shorter than and commensurate with the rod length, the rods behave like point particles in a reduced volume; when the wavelength is comparable to the system size, the effective potential is equivalent to a quenched random field.

Applications of modified weighted density functional theory: Freezing of simple liquids

A new formulation of the weighted density approximation, the so-called modified weighted density approximation (MWDA), has been developed recently by Denton and Ashcroft [Phys. Rev. A39, 4701 (1989)]. In the present article, the MWDA theory is examined by applying it to the freezing of simple liquids in low dimensions such as one-dimensional hard rods and two-dimensional hard disks, and also applying it to three-dimensional adhesive hard spheres and classical one-component plasma systems. Comparisons with previous density functional investigations and computer simulations are also presented.

On the kinetic equations of a system of one-dimensional hard rods

The hydrodynamical approximation to an infinite system of one-dimensional identical hard rods interacting through elastic collisions, is shown to be an integrable system possessing a one-parameter family of nonlinear Hamiltonian structures.

Pressure and stress tensor in fluids in a periodic potential.

For fluids in a periodic potential the grand-canonical potential per unit volume (negative ``thermodynamic pressure'') differs from the diagonal components of the average stress tensor. This is in contrast to homogeneous fluids. The thermodynamic meaning and the microscopic origin of this difference are discussed within the framework of classical statistical mechanics. For one-dimensional hard rods and for an ideal gas of diatomic molecules the two quantities are calculated exactly. General expressions are derived in a cluster expansion which can be applied to low-density classical fluids with more realistic interactions.

Transmission and dissipation of stress wave energy at a percussive drill rod joint

Transmission and dissipation of stress wave energy at a percussive drill rod joint is determined for three joint types, three joint preload levels, three hammer lengths (∼ incident wave lengths) and a range of impact velocities (∼ incident wave amplitudes). The energies of the incident, reflected and transmitted waves are evaluated from measured strains, and then the dissipated energy is determined as the deficit of energy in the two latter waves compared with the first. The accuracy needed is obtained by introducing a compensation factor and determining its value from the requirement of momentum conservation. The experimental results for threaded standard joints are compared with experimental results for threadless dummy joints, made in one piece, and also with theoretical results. The latter are based on one model which represents the joint as a well on a one- dimensional elastic rod (characteristics impedance or CI model) and on another model which represents the joint as a rigid mass between one-dimensional elastic rods (rigid mass or RM model). For the joints 70–100% of the incident wave energy is transmitted and 0–20% is dissipated. The corresponding figures for the dummies are 90–100% and (±)1%, repectively. The latter figure indicates the inaccuracy in the measurement and evaluation procedures. The two models agree fairly well with the dummies. Under certain conditions they also give reasonably accurate predictions for the energy transmitted through a joint. As they sometimes give quite inaccurate results for energy transmission and furthermore fail to predict energy dissipation they need be refined.

Monte Carlo simulation of adhesive disks

A new method has been developed that permits the Monte Carlo simulation of systems in which the intermolecular potential contains a well which is both infinitely deep and infinitesimally wide. Adhesive potentials of this type are analytically tractable and have been used in a number of applications. The simulation algorithm combines the generation and acceptance steps of the conventional Metropolis method to overcome the effect of the singularity in the potential. The method is applied to a two-dimensional system of adhesive disks. Results are reported for the equation of state and the radial distribution function, which has delta function peaks not present in the three-dimensional Percus–Yevick solution. We also present an exact solution for one-dimensional adhesive rods.

Rigorous statistical thermodynamics of a model binary solution: One-dimensional soft rods with hard rods

From canonical ensemble theory, the thermodynamic functions A, F (the one-dimensional pressure), μ, S, G, and H are obtained in the thermodynamic limit for a system of soft rods (two equal- mass points between which there is a potential proportional to separation) and for a system of hard rods of the same total mass and arbitrary length σ, on a circle. Introduction of a tangent “interchange loop” allows the evaluation of the partition function of an arbitrary mixture of physically indistinguishable soft rods with physically indistinguishable hard rods. All circular sequences of the two types of particles give the same previously evaluated integral; the number of sequences is related to the number of different linear arrangements in a way exact in the thermodynamic limit. The same thermodynamic functions are evaluated for the mixture. The fugacities reveal that the two non-ideal components surprisingly form a perfect solution with each other, regardless of the volume fraction, often invoked in three-dimensional solution theory.

Hydrodynamics for one-dimensional identical elastic hard rods

Some recent results are reported, by which the general idea of the hydrodynamic limit is made into a rigorous proof for the case of an infinite system of one-dimensional elastic hard rods. A law of large numbers for particle trajectories on a macroscopic scale is proved, and, as a consequence, the “Euler equation” for the system is deduced. In addition an existence and uniqueness theorem for the equation is given.

Virial coefficients for one-dimensional hard rods

Virial coefficients for one-dimensional hard rods