Homogenization procedure for a metamaterial and local violation of the second principle of thermodynamics

Classical theory of crystals states that a medium to be considered homogeneous must satisfy the following requirements (a) the dimension of the elementary cell must be much smaller than the incident wavelength; (b) the sample must contain a large number of elementary cells, i.e. it must be macroscopic with respect to wavelength. Under these conditions, macroscopic quantities can be introduced in order to describe the optical response of the medium. We analytically demonstrate that for a symmetric elementary cell those requirements can be relaxed, and it is possible to assign a permittivity and a permeability to a composite structure, even if the metamaterial cannot be considered homogeneous under the requirements stated above. However, the effective permittivity and permeability in some cases may give rise to unphysical, effective behaviors inside the medium, notwithstanding the fact that they satisfy requirements like being Kramers-Kronig pairs, for example, and are consistent with all the linear properties outside the structure (i.e. reflection, transmission, and absorption at all frequencies). In some situations the medium is assigned a magnetic response even though the medium is not magnetically active. In particular, we demonstrate that the homogenization procedure can lead to a medium that locally violates the second principle of thermodynamics. We also show that, in the non-homogeneous regime, it is not possible to describe the nonlinear behavior of the structure using an effective parameters approach, despite the possibility to assign an effective linear refractive index.     

The thermodynamics of Pd-Au-H ternary solid solutions

The solubility of hydrogen in PdAuH ternary solutions in equilibrium with H gas at atmospheric pressure has been measured in the temperature range 625–1250 K and in Pd-Au “binary solvents” containing up to 80 At % of Au Concomitant elastic measurements have provided data which enable the partial thermodynamic functions of the H atoms deduced from the solubility measurements to be converted so as to refer to a hypothetical Pd-Au lattice of constant specific volume The resulting “volume corrected” functions have been discussed in terms of the cell model for ternary solutions and have been shown to vary with temperature and Au concentration in a manner in accord with this model.     

The thermodynamics of solid solutions of argon and methane

The primary concern of this paper is with the estimation of the excess Gibbs energy G^E,S for solid solutions of two molecularly simple components which are completely miscible in the solid state. The method depends on combining information on the excess thermodynamic functions of liquid mixtures of the two components with a knowledge of the liquidus and solidus lines on the temperature-composition phase diagram. It is applied to the particular case of argon-methane. For this system, unit cell sizes and some heat of fusion measurements are also available, from which V^E,S and H^E,S have been calculated.A solid solution of argon and methane departs much more from ideality than does a liquid mixture of the same composition at the same temperature, the ratio r, = G^E,S/G^E,L, being about 3. Moreover, the concentration dependence of G^E,S is less symmetrical than that of G^E,L, and the ratio r increases markedly with increasing argon mole fraction. A dilute solution of methane (which has the larger molecules) in argon has a larger G^E,S than the corresponding dilute solution of argon in methane.For a solid solution at 71 K with an argon mole fraction of 0.60, H^E,S is ≈4801 Jmol^?1. This gives TS^E,S ≈ 220 J mol^?1, which is about the same as G^E,S. The solid solutions cannot therefore be regarded as even approximating to regular solutions.From the calculated G^E,S results, it is predicted that the face-centred cubic solid solutions of argon and methane should separate into two phases on cooling. The calculated coordinates of the upper critical solution point are T = 67 K and an argon mole fraction of 0.63, in reasonable agreement with the experimental values of 63 K and 0.65 respectively.     

On the density of surface acoustic waves and surface thermodynamics in a piezoelectric

Piezoelectricity introduces new modes and modifies part of the elastic vibrational spectrum for both bulk and surface waves. This has a direct bearance on surface thermodynamics. A simple phenomenological model is studied analytically and exactly. This bears out the basic physics involved and allows for a quick and easy estimate of the changes to be expected, which turn out to be substantial.     

Modelling of solid oxide proton conducting reactor-cells: thermodynamics and kinetics

A model is prepared for hydro- and dehydrogenation reactions that are carried out in solid electrolyte cell-reactors. Hydrogen is supplied in the form of H⁺ through the proton conducting wall and the thermodynamic activity of hydrogen on the catalyst surface is controlled electrochemically. Three reactor types are considered, a plug flow reactor, a well-mixed reactor and a “single-chamber” cell in which both electrodes are exposed to the same reacting mixture. Under certain conditions, the yield of the cell-reactor is orders of magnitude higher than that of the corresponding conventional catalytic reactor. The key parameters that affect the product yield as well as the conditions that must be met, are discussed. Experimental results for the reaction of methane dimerization studied with a strontia-ceria-ytterbia electrolyte on palladium electrodes, are presented and compared to model predictions. Paper presented at the 3rd Euroconference on Solid State Ionics, Teulada, Sardinia, Italy, Sept. 15–22, 1996     

Resonance frequencies of piezoelectric plates surrounded by solid and fluid half-spaces

The design of ultrasound transducers, resonators and other piezoelectric devices usually requires the calculation of the resonance frequencies of piezoelectric plates. Recent studies have shown that the resonance frequencies for plates in vacuum correspond to frequencies where the waveguide group velocity vanishes (zero-group-velocity points). However, those studies are limited to vacuum boundary conditions. The objective of the present study is to analyze the resonance frequencies of layered piezoelectric plates in contact with solid and fluid half-spaces and their relation to the dispersion behavior of the elastic guided wave propagation. Theoretical analysis using partial-wave approach of leaky Lamb waves is performed to study wave propagation in, and resonance behavior of, multilayered plates in contact with solid and fluid half-spaces. A novel observation resulted from this analysis is that, for plates in contact with solid and fluid half-spaces, the resonance frequencies occur at points where the magnitude of the wavenumber reaches a minimum. This frequency is named as a ‘transition frequency’. Such observations are important because they allow an easy identification of resonance frequencies with high amplitude response directly from the dispersion curves. This study will be helpful for the design of piezoelectric components used for resonators and sensors.     

Parametric excitation of acoustic waves in piezoelectric solid state magnetoplasma

Absolute instability of the acoustic wave (excited parametrically) has been explored on the basis of the Bers and Briggs criterion in the presence of a large transverse static magnetic field. In the magnetoplasma threshold the electric field decreases to about 0.4 times the value at zero magnetic field.     

The thermodynamics of palladium-based solid solutions containing a noble-metal substitutional solute and hydrogen

A cell model for tenary solid solutions containing a substitutional and an interstitial solute species is considered. The interstitial solute is restricted to cells (interstitial sites) in the f.c.c. solvent lattice in which it has either zero or one nearest neighbor substitutional solute atom. The effects of the dilation of the solvent lattice by the substitutional solute are taken into account in an appropriate manner using the currently limited elastic data available.It is shown that the model is in good accord with the measured partial enthalpies and excess entropies in the Pd-Ag-H system, in which the Ag atoms cause a positive lattice dilation, but not for the case of Pd-Cu-H, where there is a contraction.     

Reflection and transmission of plane waves from a fluid-porous piezoelectric solid interface

The reflection and transmission of plane waves from a fluid-porous piezoelectric solid interface is studied. The porous piezoelectric solid, having 6 mm symmetry, is supposed to be filled with viscous fluid. The expressions for amplitude ratios and energy ratios corresponding to reflected wave and transmitted waves are derived analytically. The Christoffel equation of a leaky wave propagating along the surface of a porous piezoelectric solid is derived. The effects of the angle of incidence, frequency, porosity, piezoelectric interaction, and anisotropy on the reflected and transmitted energy ratios are studied numerically for a particular model BaTiO(3). The porous piezoelectric solid half space is assumed to be loaded with water. The effects of porosity and frequency on the leaky wave velocity are also studied.     

A spread in electrophysical parameters of ferroelectric piezoelectric solid solutions and its minimization

A spread in electrophysical parameters of solid solutions based on PZT and niobium oxides is considered for a wide range of component concentration. It is shown that composition fluctuations and their associated solution imperfection cause a significant deviation of the parameters from their mean values. The fluctuations are most pronounced when the crystal structure is unstable (e.g., at morphotropic transitions and near the solubility limits of the components). The optimization of methods for solid solution preparation greatly reduces the effect of process variables on parameter reproducibility.     

Volume effects in the thermodynamics of binary substitutional solid solutions at high temperatures

Recent measurements of the variation with temperature and composition of the elastic properties of Fe-based solid solution containing the substitutional solutes Mn, Ni and Cr have enabled calculations to be made of the effect of the composition-dependent variation of the specific volume of the lattice upon the partial thermodynamic functions of the solute species. The calculations show that, in contrast to the case of binary interstitial solid solutions, the effects are of the same order, or less, as the normal uncertainty in the experimental determination of the partial thermodynamic functions of solid solutions at high temperatures (1273 K).     

Molecular dynamics simulations of thermodynamics,elastic constants and solid solution strengths for Mg-Gd alloys

The thermal and mechanical properties of Mg-Gd intermetallics MgGd, MgGd₂ and MgGd₃ are studied using a modified analytical embedded atom method. Calculated results agree well with the available experimental data and other theoretical results. The results on the elastic constants suggest that thermal softening behavior is observed as the temperature increases and the bulk moduli of ordered phases are larger than that of elemental Mg above 300 K. The heat capacities of MgGd, Mg₂Gd and Mg₃Gd are 22.91, 23.04 and 23.09 J mol K^-1, respectively, at 300 K. Furthermore, the addition of Gd gives rise to an increase of c/a. With the same content of Gd, the ratio c/a remains unchanged with increasing temperature, whereas this phenomenon does not occur in pure Mg, which indicates that the temperature-independent c/a restrains the occurrence of non-basal slip and twinning. Hence the addition of Gd can enhance the strength of Mg, in good agreement with experimental observation.     

Predominant factors affecting the dielectric and piezoelectric properties of bismuth-containing complex perovskite solid solution

To study the factors affecting the dielectric and piezoelectric properties of bismuth-containing complex perovskites, the solid solution (1−x)Pb(Mg_1/3Nb_2/3)O₃-xBi(Mg_2/3Nb_1/3)O₃ was prepared by the solid state reaction method and its dielectric and piezoelectric properties were investigated. It is found that (1) at room temperature, the nonlinearity of the DE-loop for Pb(Mg_1/3Nb_2/3)O₃ is completely suppressed at a rather low x (<5%); (2) dielectric constant versus temperature curves deviate from the Curie-Weiss law at a temperature T_d much higher than the dielectric constant peak temperature T_m and T_m−T_d decreases considerably with increasing x; and (3) frequency dispersion ΔT_m=T_m (1 MHz)−T_m (10 kHz) increases with increasing x. Possible factors responsible for the variation of the dielectric and piezoelectric properties with x are discussed.     

Immiscibility regions in iron based ferritic solid solutions and their relevance to thermodynamics and kinetics of nitriding

In the process of equilibration of a ferritic iron-based alloy with gaseous nitriding atmosphere, the inwardly diffusing N into the ferritic solid-solution from the gas atmosphere may bring the evolving solid-solution chemistry into immiscible region of the corresponding phase diagram. Distinct kinetic mechanisms of nitriding are operative in different alloy systems, depending on whether the alloy system has a region of immiscibility or not and whether the applied chemical potential of N in the nitriding atmosphere allows the evolving chemistry of the alloy to sample the immiscibility region or not. With this new kind of thermodynamic interpretation, it is now possible to precisely understand the experimental results reported in the literature pertaining to nitrided iron-based binary alloys. Slow kinetics of nitride precipitation in nitrided ferritic Fe-Si and Fe-Al alloys has been attributed to the absence of immiscibility region in ferritic Fe-Si-N and Fe-Al-N systems whereas the slower precipitation of nitrides in ferritic Fe-Mo alloys has been attributed to the implausibility of sampling the immiscibility region of Fe-Mo-N system for typically applied nitriding conditions and Mo contents. Faster kinetics of nitride precipitation in Fe-Ti, Fe-V, Fe-Cr alloys has been attributed to the presence of immiscibility in Fe-Ti-N, Fe-V-N and Fe-Cr-N systems and the applied nitriding conditions allowing sampling of this immiscibility region. Recognising the role of ‘miscibility gap’ in alloy systems to realise rapid kinetics during nitridation treatments opens up a new, unexplored alloy design strategy for the development of steels with favourable nitriding response.     

Forced responses of solid axially polarized piezoelectric ceramic finite cylinders with internal losses

A method is presented to determine the forced responses of piezoelectric cylinders using weighted sums of only certain exact solutions to the equations of motion and the Gauss electrostatic conditions. One infinite set of solutions is chosen such that each field variable is expressed in terms of Bessel functions that form a complete set in the radial direction. Another infinite set of solutions is chosen such that each field variable is expressed in terms of trigonometric functions that form a complete set in the axial direction. Another solution is used to account for the electric field that can exist even when there is no vibration. The weights are determined by using the orthogonal properties of the functions and are used to satisfy specified, arbitrary, axisymmetric boundary conditions on all the surfaces. Special cases including simultaneous mechanical and electrical excitation of cylinders are presented. All numerical results are in excellent agreement with those obtained using the finite element software ATILA. For example, the five lowest frequencies at which the conductance and susceptance of a stress-free cylinder, of length 10 mm and radius 5 mm, reach a local maximum or minimum differ by less than 0.01% from those computed using ATILA.     

Interaction of a piezoelectric ceramic based on lead-zirconium-titanium solid solution with different solders

Thermodynamic estimation of the interaction of a piezoelectric ceramic based on the lead-zirconium-titanium (PZT) solid solution with lead, tin, copper, silver, palladium, and their alloys has been performed. The results of thermodynamic calculations have been experimentally verified by estimating the adhesion and capillary properties of solders on a PZT piezoelectric ceramic.