Vegard's Rule Research Articles

Simulation of ionic transport in concentrated solutions using bi-velocity method

The work is devoted to the theoretical study of transport processes in the concentrated electrolyte solutions using bi-velocity method. The Nernst-Planck equations in the approximation of solution electroneutrality and the Vegard rule are used as the mathematical model. For one-dimensional case (a plane electrode in the stagnant or intensively stirred electrolyte solutions), a scheme of numerical solution based on the conservative finite volume method is developed. Within the framework of bi-velocity method, the equations for the diffusion-migration fluxes of the solution components and for the drift velocity are obtained. The results of simulation are presented and compared with the results obtained using the known method for dilute solutions. It is shown that ignoring the velocity induced by the ionic transport leads to a considerable error in the limiting current density up to tens of percent.

Existence, uniqueness and properties of global weak solutions to interdiffusion with Vegard rule

We consider the diffusional transport in an $r$-component solid solution. The model is expressed by the nonlinear system of strongly coupled parabolic differential equations with initial and nonlinear boundary conditions. The techniques involved are the local mass conservation law for fluxes, which are a sum of the diffusional and Darken drift terms, and the Vegard rule. The considered boundary conditions allow the physical system to be not only closed but also open. The theorems on existence, uniqueness and properties of global weak solutions are proved. The main tool used in the proof of the existence result is the Galerkin approximation method. The agreement between the theoretical results, numerical simulations and experimental data is shown.

SYNTHESIS AND X-RAY INVESTIGATION OF Cu2CdSn(SxSe1–x)4 SOLID SOLUTIONS

The quaternary semiconductors Cu2CdSnS4, Cu2CdSnSe4 and Cu2CdSn(SxSe1–x)4 solid solutions were synthesized by the one-temperature method from the elementary components. The X-ray diffraction method showed that the obtained polycrystalline samples are single-phased. The unit cell parameters of the synthesized compounds and Cu2CdSn(SxSe1–x)4 solid solutions were determined from diffraction spectra by the full-profile analysis using the Rietveld method with the Fullprof software package. It has been established that with an increase in sulfur concentration, the unit cell parameters decrease smoothly linearly in accordance with the Vegard rule, which indicates the formation of a continuous series of solid solutions in the Cu2CdSn(SxSe1–x)4 system within the range 0 ≤ x ≤ 1. The parameter of crystal lattice tetragonal distortions h of the investigated compounds is calculated. The h values are close to 1 for all the compositions studied, which indicates a small crystal lattice distortion of the obtained samples.

Investigation of material properties and defect behavior in In-doped CdO films

In-doped CdO (InxCd1-xO) thin films were prepared by magnetron sputtering and characterized in detail. Results indicated that the electron mobility improves with small amount of In doping (x = 0.012) to CdO, but then decreases with further In addition. It was proposed that the initial increase in mobility is because the oxygen vacancy concentration decreases and the grain size increases, and that the later degradation is due to the increased scattering by the In ions and the decreased grain size. It was also observed that the carrier concentration, while generally increased with the addition of the In donors, decreases with high amount of In doping (x = 0.072). It was proposed that this is because the solubility limit of In in CdO film is reached and excess In ions form non-conducting In-oxide clusters. It was observed that the In doping widens the energy band gap of CdO film via the Burnstein-Moss effect and the Vegard's rule.

Atomic Charges and Chemical Bonding in Y-Ga Compounds

A negative deviation from Vegard rule for the average atomic volume versus yttrium content was found from experimental crystallographic information about the binary compounds of yttrium with gallium. Analysis of the electron density (DFT calculations) employing the quantum theory of atoms in molecules revealed an increase in the atomic volumes of both Y and Ga with the increase in yttrium content. The non-linear increase is caused by the strengthening of covalent Y-Ga interactions with stronger participation of genuine penultimate shell electrons (4d electrons of yttrium) in the valence region. Summing the calculated individual atomic volumes for a unit cell allows understanding of the experimental trend. With increasing yttrium content, the polarity of the Y-Ga bonding and, thus its ionicity, rises. The covalency of the atomic interactions in Y-Ga compounds is consistent with their delocalization from two-center to multi-center ones.

Crystallographic characterization of laser-generated, polymer-stabilized 4 nm silver-gold alloyed nanoparticles

Monometallic silver and gold nanoparticles and bimetallic silver-gold (AgAu) nanoparticles were prepared by laser ablation in liquids in the atomic composition range of Ag:Au from 0:100 to 100:0 with steps of 10 at% and colloidally stabilized with poly(N-vinylpyrrolidone) (PVP). As metallic bulk targets for laser ablation, pure silver, pure gold, and alloyed AgAu foils with the desired composition were used. Size separation by centrifugation and freeze-drying gave monodisperse spherical nanoparticles with a diameter of 4 nm as determined by differential centrifugal sedimentation (DCS) and transmission electron microscopy (TEM). A crystallographic characterization of the nanoparticles was carried out by X-ray powder diffraction (XRD) and Rietveld refinement, leading to highly precise cubic lattice parameters (fcc crystal system) and crystallite sizes. For comparison, the same analysis including the determination of the microstrain was carried out for the bulk target materials (AgAu alloys in the full concentration range). Both nanoparticles and bulk target materials obeyed Vegard's rule, with only slight deviations. The fact that the crystallite size as determined by XRD was identical to the hydrodynamic diameter by DCS and the Feret diameter by TEM indicates that the particles consist of only one domain, i.e. they are single crystals. The combination of UV-vis spectroscopy with energy-dispersive X-ray spectroscopy (EDX) as line scan along the nanoparticle showed a homogenous distribution of the gold and silver inside the nanoparticles, indicating solid solution alloys, in contrast to what was observed earlier for chemically prepared AgAu nanoparticles by reduction of metal ions in water.

Non-equilibrium solid solution of molybdenum and sodium: Atomic scale experimental and first principles studies

We report a combined experimental and first principles study of an extremely immiscible alloy of Mo with 1 and 2 at.% Na, which was produced by high-energy ball milling. The microstructure of the as-milled and annealed state were examined by various methods, including atom-probe tomography (APT), transmission electron microscopy, and energy-dispersive (EDX) analysis. Despite the complete immiscibility of the Mo-Na system in the solid and even in the liquid state, APT measurements clearly evidence the formation of a true nanocrystalline solid-solution microstructure with insignificant Na clustering for samples with 1 at.% Na. In agreement with our x-ray diffraction experiments, first principles calculations expose that the Na atoms do not expand the Mo lattice, which is in contrast to predictions using Vegard's rule. Heating at 700 °C induces only slight grain growth while the solid solution remains remarkably stable without any decomposition. On the contrary, after annealing at 900 °C first Na segregations at triple junctions and significant grain growth are observable, although the solid solution still retains most of the dissolved Na.

Synthesis and anode properties of corundum-type structured (Fe2O3)1-x(Al2O3)x solid solutions in the whole compositional range

We have successfully synthesized corundum-type structured solid solutions of (Fe2O3)1-x(Al2O3)x in the whole compositional range (0≤x≤1) by means of mechanochemical synthesis route starting from γ-Fe2O3 and γ-Al2O3 mixture. The lattice parameters of the obtained solid solution coincide with the calculated values assuming Vegard's rule based on the JCPDS card data of α-Fe2O3 (33-664) and α-Al2O3 (46-1212), indicating that the composition of the obtained solid solution is essentially equivalent to the initial Fe2O3/Al2O3 ratio in the starting materials. The test cells using the obtained solid solution as the active material exhibited good cycle performances. Especially in the compositional range in 0.33≤x≤0.45, after a gradual decrease in charge-discharge capacity down to ~300mAhg−1 for around 100cycles, the capacity recovers again. Even for x=0.95, the capacity retains at least 100mAhg−1 after 260th cycles, which is larger than the estimated from the Fe2O3 amount.

Influence of Eu substitution for Gd on the structure and photoluminescent properties of (Gd1−xEux)2Ti2O7 pyrochlore solid solutions

This paper presents a study of the synthesis and structural properties of the pyrochlore-type titanates (Gd1−xEux)2Ti2O7. Six compositions with 0≤x≤1 were prepared by solid-state reaction with thermal treatments at 1000 and 1200°C under atmospheric pressure conditions. All the products were systematically characterized by X-ray powder diffraction (XRD), Raman spectroscopy, photoluminescence (PL) and photoluminescent excitation spectra (PLE). Structural refinements of X-ray powder diffraction data using Rietveld method show that all compounds of the (Gd1−xEux)2Ti2O7 solid solution crystallize in a pyrochlore structure. The lattice parameters increase linearly with increasing Eu content in agreement with Vegard's rule. PL spectra show that the characteristic peaks correspond to the f–f transition 5D0–7FJ (J=1, 2, 3 and 4) of Eu3+. The effect of Eu3+ ions concentration on the optical properties, namely, photoluminescence emission, is measured and discussed. The oxides exhibit higher PL intensity with the Eu3+ ion concentration and strong orange emission at 589nm (5D0–7F1) corresponding to the magnetic dipole transition.

Neutron diffraction of Cu–Zn–Sn ternary alloys: non-invasive assessment of the compositions of historical bronze/brass copper ternary alloys

Neutron diffraction can be used as a tool for the characterization of metal materials in a totally non-invasive mode. In binary alloys with two elements in solid solution, crystallographic structure analysis provides information on the overall element compositions of the metal, based on the linear relationship between elemental fractions and lattice parameters known as Vegard's rule. However, for ternary solid-solution alloys the derivation of the overall metal composition is not straightforward because the problem is mathematically underdetermined. A number of artificially produced samples in the ternary system Cu–Zn–Sn, widely used in antiquity for gunmetal, were investigated by time-of-flight neutron diffraction, inductively coupled plasma mass spectroscopy, scanning electron microscopy with energy-dispersive X-ray spectroscopy and electron microprobe analysis. The multi-analysis approach allows definition of the limits and capabilities of neutron diffraction for obtaining the overall composition of a small sample set of ternary alloys, and thus moves the methodical approach a step forward even though it is applicable to the present sample set only. A relation showing an increasing Cu and Sn fraction counterbalanced by decreasing Zn content is presented, which allows the determination of the δ-phase composition from a lattice parameter measurement. Furthermore, the observed Zn loss up to 1.8 wt% for each melting step is of significance for the reconstruction of ancient technologies.

The effect of Ce dilution on the ferromagnetic ordering and Kondo behavior of CeRuPO

The structural, electronic and magnetic properties and Kondo behavior of Ce1−xLaxRuPO(x=0, 0.25, 0.5, 0.75 and 1) alloys are investigated using density functional theory by utilizing Wien2k package. The exchange-correlation potential is treated with the generalized gradient approximation (GGA). Moreover, the GGA+U approach (where U is the Hubbard correlation term) is employed to treat the f-electrons properly. We also present a comparative study between the electronic structure and magnetic properties of these alloys within GGA and GGA+U approaches. The calculated lattice parameters and bulk moduli of these alloys as a function of x are in the best agreement with Vegard's linear rule. The total and partial electron density of states and linear coefficient of electronic specific heat of these alloy within GGA and GGA+U are investigated and compared. The effect of La substitution on the Kondo behavior of CeRuPO compound is investigated.

Zintl Phases K4–xNaxSi4 (1 ≤ x ≤ 2.2) and K7NaSi8: Synthesis, Crystal Structures, and Solid‐State NMR Spectroscopic Investigations

The Zintl phases K4–xNaxSi4 (1 ≤ x ≤ 2.2) and K7NaSi8 are the first representatives of the K–Na–Si system, and both contain tetrahedral [Si4]4– clusters and a charge‐balancing number of K+ and Na+ cations. All phases of K4–xNaxSi4 (1 ≤ x ≤ 2.2) crystallize in a new structure type with space group P21/n, as determined by single‐crystal X‐ray diffraction analysis of the parent phase K3NaSi4. Rietveld refinement of the X‐ray diffraction data showed that the solid solutions follow Vegard's rule. K7NaSi8 can only be synthesized by diffusion‐controlled reaction of binary precursors and is isostructural with known A7A′T8 (A = Na–Cs; A′ = Li, Na; T = Si, Ge) phases. A combination of solid‐state NMR investigations and quantum mechanical calculations served to show the anisotropic chemical bonding behavior of all the atoms in K7NaSi8, which is additionally compared with the related phases A7NaSi8 (A = Rb, Cs).

Work function bowing in Si1−xGex heterostructures: Ab initio results

A systematic theoretical study of the work function behavior for Si1−xGex heterostructures over the whole composition range, from Si (x = 0) to Ge (x = 1), is presented. Our results, obtained through Density Functional Theory calculations and in good agreement with experimental evidences, show that increasing the Ge content lowers the work function value. We find that in order to exactly reproduce this behaviour in relation to the work function of pure Ge and Si systems and their concentrations, a deviation from the linear Vegard's rule is necessary. However, the calculated bowing parameter is very small, thus making the simple linear interpolation a valid approximation to obtain the work function of complex SiGe alloys.

The structural evolution and spectral blue shift of solid solution phosphors Sr3−mCamB2O6:Eu2+

An unexpected spectral blue shift of the emission from yellow to green is observed for the solid solution phosphors Sr3−mCamB2O6:Eu2+ (m = 0–3), when larger Sr2+ ions are substituted by smaller Ca2+ ions in Sr3B2O6 lattices. The powder X-ray patterns of all the phases indicate a structural similarity to the end compounds and show a smooth variation of the structural parameters with composition in the full range. The linear structural evolution of the iso-structural Sr3−mCamB2O6 solid solutions obeying Vegard's rule has also been examined and verified by Rietveld refinement, high resolution transmission electron microscopy and Fourier transform infrared spectroscopy. The average bond lengths of the central cations to the coordinating anions and the distortion indexes of the coordination polyhedra are investigated. As a result, the symmetry of the coordination crystal field is found to be higher with increasing Ca content in the solid solutions phosphors, which should be primarily responsible for the continuous blue shift of the emission spectra of Sr2.96−mCamB2O6:0.04Eu2+ with the growth of the m value.

Optical study of narrow band gapInAsxSb1−x(x=0, 0.25, 0.5, 0.75, 1) alloys

The structural, electronic, and optical properties of InAs, InSb, and their ternary alloys ${\mathrm{InAs}}_{x}{\mathrm{Sb}}_{1\ensuremath{-}x}$ ($x=0.25$, 0.5, 0.75) are investigated within density functional theory utilizing the wien2k package. We find that the lattice constants and bulk moduli as a function of $x$ are in best agreement with Vegard's linear rule. When computing the electronic band structures with the modified Becke-Johnson exchange-correlation functional (mBJLDA), our results for the band gaps of InAs, InSb, and their ternary alloys are in good agreement with the available experimental results while the conventional Wu-Cohen generalized gradient approximation (GGA) functional leads to zero or close to zero band gaps. In particular, our mBJLDA results confirm experimental evidence that the minimum band gap occurs for As concentrations around $x\ensuremath{\approx}0.3$. Furthermore, we investigate the dielectric function of these compounds within the random phase approximation using both the Wu-Cohen GGA and the mBJLDA functionals. While the mBJLDA results of our fully first-principles calculations show good agreement of the peak positions in ${\ensuremath{\epsilon}}_{2}(\ensuremath{\omega})$ with experiments, the peaks in the optical spectra based on the Wu-Cohen GGA band structure appear redshifted compared to experiment. We further identify the interband transitions responsible for the structures in the spectra. Looking at the optical matrix element, we note that the major peaks are dominated by transition from the Sb $5p$ (As $4p$) states to In $s$ states for InSb and ${\mathrm{InAs}}_{0.25}{\mathrm{Sb}}_{0.75}$ (InAs, ${\mathrm{InAs}}_{0.75}{\mathrm{Sb}}_{0.25}$, and ${\mathrm{InAs}}_{0.5}{\mathrm{Sb}}_{0.5}$).

Lattice parameters and Raman-active phonon modes of β-(AlxGa1−x)2O3

We present X-ray diffraction and Raman spectroscopy investigations of a (100)-oriented (AlxGa1–x)2O3 thin film on MgO (100) and bulk-like ceramics in dependence on their composition. The thin film grown by pulsed laser deposition has a continuous lateral composition spread allowing to determine precisely the dependence of the phonon mode properties and lattice parameters on the chemical composition. For x < 0.4, we observe the single-phase β-modification. Its lattice parameters and phonon energies depend linearly on the composition. We determined the slopes of these dependencies for the individual lattice parameters and for nine Raman lines, respectively. While the lattice parameters of the ceramics follow Vegard's rule, deviations are observed for the thin film. This deviation has only a small effect on the phonon energies, which show a reasonably good agreement between thin film and ceramics.

A first-principles study on three-dimensional covalently-bonded hexagonal boron nitride nanoribbons

We studied three-dimensional honeycomb-structure boron nitride (BN) allotrope using first-principles calculations and the tight-binding method. Interconnected by sp3-bonding at the vertices, hexagonal BN nanoribbons construct highly-porous, covalently-bonded hexagonal BN nanoribbons (CBBNs). We investigated the structural and mechanical properties of CBBNs with various sizes, compared with those of carbon and other BN allotropes. The mechanical and thermal stabilities are also checked. Our calculations show that, despite the high porosity and low mass density, CBBNs are stable and mechanically hard materials as cubic BN. Moreover, our calculated results suggest that CBBNs can be regarded as a binary alloy of sp2- and sp3-bonded BNs following the Vegard's rule in average bond lengths and bulk moduli. Calculated band structures show that the band gap of CBBNs has similar variation upon increasing size as BN nanoribbons and is also limited by the second-neighbor interaction between the pz states of sp2-bonded atoms in adjacent nanoribbons.

First-principles Study of Structural Properties of MgxZn1-xO ternary alloys

First-principles calculations have been carried out to investigate the structural and electronic properties of MgxZn1-xO ternary alloys. The calculations are performed using the full potential linearized augmented plane wave (FP-LAPW) method within the density functional theory(DFT). We conclude that the structural properties of these materials, in particular the composition dependence on the lattice constant and the band gap is found to be linear. The a-axis length in the lattice gradually increases, while the c-axis length decreases with the increase in Mg doping concentration, and be corresponded with the Vegard's law linear rule. The lattice parameters of the MgxZn1-xO ternary alloys are consistent with experimental data and other theoretical results. We found in the conduction band portion, the Mg 2p 2s states are moved to high energy region as the Mg content increases, so the band gap increases.

Antiferromagnetic Kondo lattice to intermediate valence transition in Ce(Au1−xNix)2Si2 ([formula omitted

Transition effects from magnetically ordered Kondo lattice (KL) behaviour in CeAu2Si2 to intermediate valence (IV) behaviour in CeNi2Si2 have been investigated by substitution of Au with Ni. These investigations were done through measurements of the lattice parameters, electrical resistivity (ρ(T)), magnetisation (M(μ0H)) and magnetic susceptibility (χ(T)) on the Ce(Au1−xNix)2Si2 (0≤x≤1) series. Lattice parameters as derived from XRD measurements deviate from Vegard's rule around x=0.6–0.8. ρ(T) data indicate KL behaviour in the presence of a crystal field for x≤0.6, the occurrence of magnetic ordering for x≤0.2 and valence fluctuation for x≥0.8. χ(T) data at high temperatures, follow the Curie–Weiss relation for alloys in the concentration 0≤x≤0.6 (KL region) and give effective magnetic moment values μeff close to that expected for free Ce3+ ion. The low temperature χ(T) data indicate the onset of antiferromagnetic ordering for x≤0.2. For alloys in the concentration range 0.8≤x≤1, the χ(T) behaviour is reminiscent of intermediate valence (IV) systems. M(μ0H) measurements indicate metamagnetic behaviour for alloys in the concentration range 0≤x≤0.1.

Electronic structure and mechanical properties of ternary ZrTaN alloys studied byab initiocalculations and thin-film growth experiments

The structure, phase stability, and mechanical properties of ternary alloys of the Zr-Ta-N system are investigated by combining thin-film growth and ab initio calculations. ${\mathrm{Zr}}_{1\ensuremath{-}x}{\mathrm{Ta}}_{x}\mathrm{N}$ films with $0\ensuremath{\le}x\ensuremath{\le}1$ were deposited by reactive magnetron cosputtering in $\mathrm{Ar}+{\mathrm{N}}_{2}$ plasma discharge and their structural properties characterized by x-ray diffraction. We considered both ordered and disordered alloys, using supercells and special quasirandom structure approaches, to account for different possible metal atom distributions on the cation sublattice. Density functional theory within the generalized gradient approximation was employed to calculate the electronic structure as well as predict the evolution of the lattice parameter and key mechanical properties, including single-crystal elastic constants and polycrystalline elastic moduli, of ternary ${\mathrm{Zr}}_{1\ensuremath{-}x}{\mathrm{Ta}}_{x}\mathrm{N}$ compounds with cubic rocksalt structure. These calculated values are compared with experimental data from thin-film measurements using Brillouin light scattering and nanoindentation tests. We also study the validity of Vegard's empirical rule and the effect of growth-dependent stresses on the lattice parameter. The thermal stability of these ${\mathrm{Zr}}_{1\ensuremath{-}x}{\mathrm{Ta}}_{x}\mathrm{N}$ films is also studied, based on their structural and mechanical response upon vacuum annealing at 850 \ifmmode^\circ\else\textdegree\fi{}C for 3 h. Our findings demonstrate that ${\mathrm{Zr}}_{1\ensuremath{-}x}{\mathrm{Ta}}_{x}\mathrm{N}$ alloys with Ta fraction $0.51\ensuremath{\leqslant}x\ensuremath{\leqslant}0.78$ exhibit enhanced toughness, while retaining high hardness \ensuremath{\sim}30 GPa, as a result of increased valence electron concentration and phase stability tuning. Calculations performed for disordered or ordered structures both lead to the same conclusion regarding the mechanical behavior of these nitride alloys, in agreement with recent literature findings [H. Kindlund, D. G. Sangiovanni, L. Martinez-de-Olcoz, J. Lu, J. Jensen, J. Birch, I. Petrov, J. E. Greene, V. Chirita, and L. Hultman, APL Materials 1, 042104 (2013)].