Elastic Constants C11 Research Articles

Melting temperature and elastic constants of disordered nonstoichiometric cubic TiCy, ZrCy and HfCy carbides

Based on the analysis of phase diagrams of carbide-forming systems M–C (M = Ti, Zr, Hf), an empirical relationship is proposed between the elastic stiffness constants cij of nonstoichiometric cubic carbides of titanium, zirconium and hafnium and their melting temperature. The dependences of the melting temperatures of nonstoichiometric cubic carbides TiCy, ZrCy and HfCy on their composition in homogeneity regions are calculated using the elastic stiffness constants c11(y) and c44(y) of these carbides. The calculated maximum melting temperatures are observed for carbides ~TiC0.80, ~ZrC0.82 and ~ HfC0.94–0.95 and are equal to 3345, 3708 and 4192 K, respectively. There is a qualitative correlation between the concentration dependences of the melting temperatures Tm(y) of TiCy, ZrCy and HfCy carbides and the anisotropy of the elastic properties of these carbides.

Anisotropy of elastic properties of disordered cubic titanium monoxide TiOy

For the first time, the elastic constants c11, c12, and c44 of disordered cubic titanium monoxide TiOy were determined depending on the oxygen content y in the homogeneity region from TiO0.80 to TiO1.25. It has been established that the values of the elastic moduli depend on the oxygen content y and the crystallographic direction [hkl ]. Large changes in the elastic characteristics of TiOy depending on the [hkl ] direction indicate a strong anisotropy of the elasticity of disordered cubic titanium monoxide.

The physical properties of crystalline iron-rich silicides

The formation enthalpy, electronic structure, lattice dynamics and elasticity of several complex ordered iron-rich silicides (Fe11Si5, Fe13Si3, Fe17Si12 and Fe24Si5) are calculated from the first-principles calculations. It is found that the Debye temperature ΘD of the system decreases with the increase of the ratio of the number of iron atoms to the number of silicon atoms in the same structure. The softening of low-energy acoustic phonons is more obvious with the increase of iron atoms. The elastic constants C11 and C44, Young's modulus E and shear modulus G of the system decrease with the increase of iron atoms. An anomalous phenomenon occurs in the high frequency region of Fe24Si5. The contribution of Fe1 atoms with smaller magnetic moment to the phonon density of states is much larger than that of the Si and other Fe atoms. This indicates that the local metal bond strength between Fe1-Fe1 atoms is stronger.

A DFT study to explore structural, elastic, mechanical, electronic and optical properties of inverse perovskite SbPMg3 for solar cell applications

This study presents the structural, mechanical and optoelectronic properties of inverse perovskite SbPMg3 for optoelectronic applications. The structural optimization is carried out by employing Generalized Gradient Approximation of Wu–Cohen, Perdew–Burke–Ernzerhof and PBE revised for solids. The optimized lattice constant and bulk modulus of SbPMg3 is 4.85 Å and 54.20 GPa with WC-GGA scheme. The positive phonon frequencies are indicating that SbPMg3 is dynamically stable. The mechanical properties were calculated by obtaining elastic constants C11, C12 and C44. The structural stability is evaluated according to Born–Huang stability criteria. The band gap, static dielectric constant and static refractive index is 1.70 eV, 8.23 and 2.87 calculated with WC-mBJ exchange and correlation functional. The band structures show that inverse perovskite SbPMg3 possesses direct band gap nature. The absorption peaks in visible and ultraviolet energy region anticipates that SbPMg3 is promising novel material for optoelectronic applications.

Theoretical Study of the Competition Mechanism of Alloying Elements in L12-(Nix1Crx2Cox3)3Al Precipitates

The impact of variations in the content of single alloying element on the properties of alloy materials has been extensively discussed, but the influence of this change on the content of multiple alloying elements in the alloy materials has been disregarded, as the performances of alloy materials should be determined by the collective influence of multiple alloying elements. To address the aforementioned issue, the present study conducted a comprehensive investigation into the impact of variations in the content of alloying elements, namely Ni, Cr, and Co, on the structural and mechanical properties of L12-(Nix1Crx2Cox3)3Al precipitates using the high-throughput first-principles calculations and the partial least squares (PLS) regression, and the competitive mechanism among these three elements was elucidated. The findings demonstrate that the same alloying element may exhibit opposite effects in both single element analysis and comprehensive multi-element analysis, for example, the effect of Ni element on elastic constant C11, and the influence of Cr element on Vickers hardness and yield strength. The reason for this is that the impact of the content of other two alloying elements is ignored in the single element analysis. Meanwhile, the Co element demonstrates a significant competitive advantage in the comparative analysis of three alloying elements for different physical properties. Therefore, the methodology proposed in this study will facilitate the elucidation of competition mechanisms among different alloy elements and offer a more robust guidance for experimental preparation.

The mechanical properties of Al3Sc single crystal investigated by nanoindentation using Hertzian elasticity theory method and Oliver-Pharr method

The crystal structure, orientation, reduced modulus, and hardness of Al3Sc single crystal were studied using X-ray diffraction (XRD), electron backscatter diffraction (EBSD) and nanoindentation techniques, combined with Hertzian elastic theory method (Hertz) and Oliver-Pharr method (O&P). Millimeter-sized Al3Sc single crystals were prepared using quasi-equilibrium solidification of hypereutectic Al-15(wt %) Sc alloy. The XRD results showed that the samples were composed of L12-Al3Sc and Al phases. The EBSD and nanoindentation results showed that for five Al3Sc single crystals with different orientations, the reduced moduli measured by Hertz method were 146.6 ± 0.4–148.3 ± 0.2 GPa; while measured by O&P method they were 147 ± 2–150 ± 3 GPa. The corresponding hardnesses were varied between 3.8 ± 0.1 and 3.9 ± 0.1 GPa. The Poisson's ratio and elastic constants were determined through iterative calculation based on the relationship between the reduced modulus and Young’s modulus at various orientations. The polycrystal elastic properties, bulk modulus, shear modulus and Young's modulus, were determined. The elastic constants c11, c12, and c44 were 179.2 ± 0.4, 44.1 ± 0.1, and 68.3 ± 0.3 GPa, respectively; the Poisson's ratio was 0.196 ± 0.001. The Young's modulus of five Al3Sc single crystals oriented in different directions was in the narrow range 161–164 GPa. The anisotropy factor of Al3Sc crystal was 1.011 ± 0.002, confirming that it is elastically isotropic. These results provide fundamental information for the research and development of the Al3Sc intermetallic compound and related alloys in future.

Graphenyldiene: A new sp2-graphene-like nanosheet

The race and the discovery of novel two-dimensional (2D) carbon-based materials have been intensified because many are suitable for energy storage systems, thermoelectric devices, and catalysis applications. Therefore, this study introduces to the scientific community a novel 2D nanosheet named graphenyldiene (GPD), which is formed by arranging cyclobutadiene and bi-phenyl groups to create a monolayer with octadecagonal, hexagonal and tetragonal rings. The cohesive energy of GPD is only 1.37 and 0.65 eV/atom higher than graphene and biphenylene, respectively. Molecular dynamics simulations confirmed its structural and thermal stability. The GPD monolayer remains stable, with no significant deformations at around 1000 K, and the disintegration of the geometry occurs only at a temperature of 1500 K, which is characterized by the formation of an amorphous graphdiyne. The GPD electronic structure shows a direct band gap transition, 1.26 eV, at the Γ point. GPD is a promising alternative to electronic devices due to its carrier mobility of around 103.cm2/V.s. Also, the GPD satisfies the Born-Huang criterion for mechanical stability with elastic constants C11 = 157.62 N/m, C12 = 53.66 N/m and C66 = 51.98 N/m. The Bader's topological analysis indicated that all bonds have strong shared shell characteristics. Finally, the vibrational analysis identified 54 modes, where 21 are Raman active, with A1g and E2g modes dominating the spectrum at 1347, 1685 and 1697 cm−1.

Asymmetrical LaO9 polyhedron inducing anisotropic positive thermal expansion and moderate lattice thermal conductivity in LaBO3 with isolated planar [BO3] group

Thermal physical properties of inorganic borates are important in application of optical device working under high temperature. Herein, we investigate the temperature-dependent thermoelasticity, thermal expansion(α), and lattice thermal conductivity(κ) of simple orthohomibic LaBO3 with a two dimension like crystal structure though first principles for the first time. The calculated rate of temperature-dependent second elastic constant C11 is significantly larger than that of C22 and C33, and the theoretical slopes of corresponding bulk, shear, and Young's moduli in the range of 300–900 K are −0.0289 GPa•K−1, −0.0156 GPa•K−1, and −0.0398 GPa•K−1, respectively. The predicted α is positive and increases with the rise of temperature. The anisotropic ratio αc/αa or αc/αb is approximately 2 at 300 K, which might be attributed to the difference of elastic compliance S12 and S13. Besides, the anisotropic lattice thermal conductivity κa, κb and κc of LaBO3 is predicted to be 1.43, 1.40 and 0.80 W/m-K at 300 K via Debye-Callaway model. Most importantly, the atomic projected Grüneisen parameters γ reveal that asymmetrical LaO9 polyhedra in the crystal lattice dominate the thermal expansion and lattice thermal conductivity. Our results indicate that LaBO3 has good thermal physical properties working under high temperature.

Structural, elastic, mechanical, electronic, and magnetic properties of In2NbX6 (X = Cl, Br) variant perovskites

AbstractThis article is about the study of structural, elastic, mechanical, electronic, and magnetic properties of two variant perovskites In2NbX6 (X = Cl, Br) using density functional theory. To the best of our information all calculations were carried out for the first time. The results established the cubic state stability of the present compounds in the FM phase. The thermodynamic and perovskites phase stabilities are respectively confirmed from negative values of formation energies and tolerance factor. The calculated elastic constants C11, C12, and C44 confirmed the mechanical stability of In2NbX6 compounds. The mechanical properties confirmed ductile nature of In2NbCl6 and brittle nature of In2NbBr6. The electronic properties showed that these compounds were metallic in spin up state and semiconductor in spin down state. The magnetic moments were calculated as 1 μB for both the compounds majorly associated with the Nb atom. The spin dependent electrical conductivity was calculated via BoltzTrap code. Besides half metallic nature, high spin dependent electrical conductivities reveals that In2NbX6 variant perovskites compounds are suitable for spintronic applications.

Elastic constants of equiatomic Fe–Ni Invar alloy single crystal

In a disordered fcc structure, the equiatomic Fe–Ni alloy is of great interest as a soft magnetic Invar alloy with a near zero or low thermal expansion coefficient. In the L10 ordered phase, it is of interest as a potential high-performance permanent magnet alloy. In this work, the first detailed measurements of the elastic constants in a [001]-oriented Fe-50 at. % Ni alloy single crystal under a no magnetic field condition were made using the resonant ultrasound spectroscopy technique. The elastic constants measured allow one to understand the nature of atomic bonding in a material and the mechanical response to applied stress. The single crystal sample having a rectangular parallelepiped shape was prepared from a single crystal grown using the vertical Bridgman technique. The elastic constants C11, C12, and C44 were obtained from the resonant frequencies observed over a frequency range of 100–900 kHz. Using these C11, C12, and C44 values, Young’s modulus, shear modulus, bulk modulus, and Poisson’s ratio were calculated to be 99.1 GPa, 73.4 GPa, 176.7 GPa, and 0.20, respectively. The large anisotropy factor of 3.29 highlights the strong directional dependence of the material’s mechanical behavior.

First-principles investigation of the structural stability and mechanical properties of TM3Al2C(TM=Mo, Cr and W) carbides

Although Mo3Al2C MAX phase is a promising high temperature ceramics because of the high temperature strength and excellent oxidation resistance, the structural and mechanical properties of Mo3Al2C-type carbide is unknown. Here, we apply the first-principles method to study the structural stability, mechanical and thermodynamic properties of TM3Al2C MAX phase. Three refractory metals (TM=Mo, Cr and W) are considered. The calculated results show that the formation enthalpy follows the sequence of Mo3Al2C<Cr3Al2C<W3Al2C. Naturally, the structural stability of TM3Al2C carbides is determined by the formation of TM-Al bond, TM-C bond and TM-TM bond. Importantly, it is found that three TM3Al2C carbides are mechanical stability because the calculated elastic constants of these carbides meet the Born stability criteria. The calculated elastic constant C11 of W3Al2C is 321.0 GPa, which is larger than the elastic constant C11 for Mo3Al2C and Cr3Al2C. Importantly, the bulk modulus of W3Al2C is 249.9 GPa, which is larger than the elastic modulus of Mo3Al2C and Cr3Al2C. In addition, three TM3Al2C carbides show better ductility. Naturally, the bulk modulus of TM3Al2C is related to the TM-C bond and TM-TM bond in a a-b plane. However, the strength of shear modulus of TM3Al2C depends on the TM-C bond and TM-Al bond along the b- axis.

Ab initio study of structural, mechanical and electronic properties of 3d transitional metal carbide in cubic rocksalt (rs), zincblende (zb), and cesium chloride (cc) structures by using LDA and GGA Approximation.

This study rigorously investigates three 3d transition metal carbide (TMC) structures via LDA and GGA approximations. It examines cohesive energy (Ecoh), Vickers hardness (Hv), mechanical stability, and electronic properties. Notably, most 3d TMCs exhibit higher cohesive energy than nitrides, and rs-TiC demonstrates a Vickers hardness of 25.66 GPa, outperforming its nitride counterpart. The study employs theoretical calculations to expedite research, revealing mechanical stability in CrC and MnC (GGA) and CrC (LDA in cc structure), while all 3d TMCs in rs and seven in zb structures show stability. Charge transfer and bonding analysis reveal enhanced covalency along the series, influenced by the interplay between p orbitals of carbon and d orbitals of the metal. Most 3d TMCs exhibit metallic properties, excluding zb-TiC and zb-FeC in all phases. An inverse correlation between elastic constant C44 and electronic states near the Fermi level (EF) emerges, guiding applications and design. This study efficiently uncovers 3d TMC properties, offering insights for applications and design. We employed the Vienna ab initio Simulation software (VASP) to perform computations based on density functional theory (DFT). Our approach incorporated both the projector augmented wave (PAW) and PW91 general gradient approximation (GGA) methods within the local density approximation (LDA).

Engineering Ferroelectricity and Large Piezoelectricity in h-BN.

Hexagonal boron nitride (h-BN) is a well-known layered van der Waals (vdW) material that exhibits no spontaneous electric polarization due to its centrosymmetric structure. Extensive density functional theory (DFT) calculations are used to demonstrate that doping through the substitution of B by isovalent Al and Ga breaks the inversion symmetry and induces local dipole moments along the c-axis, which promotes a ferroelectric (FE) alignment over antiferroelectric. For doping concentrations below 25%, a "protruded layered" structure in which the dopant atoms protrude out of the planar h-BN layers is energetically more stable than the flat layered structure of pristine h-BN or a wurtzite structure similar to w-AlN. The computed polarization, between 7.227 and 21.117 μC/cm2, depending on dopant concentration and the switching barrier (16.684 and 45.838 meV/atom) for the FE polarization reversal are comparable to that of other well-known FEs. Interestingly, doping of h-BN also induces a large negative piezoelectric response in otherwise nonpiezoelectric h-BN. For example, we compute d33 of -24.214 pC/N for Ga0.125B0.875N, which is about 5 times larger than that of pure w-AlN (5 pC/N), although the computed e33 (-1.164 C/m2) is about 1.6 times lower than that of pure w-AlN (1.462 C/m2). Because of the layered structure, the rather small elastic constant C33 provides the origin of the large d33. Moreover, doping makes h-BN an electric auxetic piezoelectric. We also show that ferroelectricity in doped h-BN may persist down to its trilayer, which indicates high potential for applications in FE nonvolatile memories.

Ultrasonic investigation of the elastic properties of the approximant GdCd6

We have investigated complementarily the magneto-elastic coupling in the approximant GdCd6 by measuring the temperature and magnetic field dependence of the longitudinal elastic constant C11. A pronounced elastic softening toward the phase transition temperature Ts= 156 K was observed in the temperature dependence of the C11, suggesting the presence of a structural phase transition. Furthermore, anomalous elastic behavior was observed in the temperature dependence of the C11 below 20 K, closely related to the already known consecutive magnetic phase transitions. The H–T magnetic phase diagram deduced from elastic anomalies appearing at the different phase transitions are consistent with our previous data of the other independent elastic constants CE= (C11−C12)/2 and C44 for a cubic crystal. We argue the origin of the structural phase transition and low temperature elastic property including a possible origin of such a rich variety of magnetic phases.

Effects of lattice distortion and amount of d electrons on the pressure-driven structural phase transition in ZnO: A comparative study of V-, Mn-, and Co-doped ZnO

The pressure-driven phase transition from fourfold würtzite (B4) to sixfold coordinated rock salt (B1) structure in ZnO, in particular in transition metal (TM) doped ones, exhibited very scattered results and remained as an outstanding issue. The in situ angle-dispersive x-ray diffraction revealed that TM doping can result in significant reduction on the onset pressure at which the pressure-driven B4-to-B1 phase transition starts to emerge, albeit the transition path appeared to be very much dependent on the doping transition metal element. Systematic comparisons on a series of V-, Mn-, and Co-doped ZnO materials indicated that, in addition to ionic sized difference-induced lattice distortion, the amount of the 3d electrons carried by the doped TM atom may also play a prominent role in the underlying mechanism of the pressure-driven phase transition. Specifically, the existence of 3d electrons may have resulted in increased screening of the Coulomb forces between the cations at increasing pressure, leading to significant softening of the c44 and c66 elastic constants, which, in turn, lowers the onset pressure in triggering the B4-to-B1 phase transition in ZnO. Our results also indicate that the initial lattice distortion induced by the ionic size difference may change the landscape of energy barrier and alter the phase transition path.

The influence of site preference on the elastic properties of FCC_CoCrFeNi multi-principal element alloy

The influence of the existing site preference of the constituent elements on the sublattice on elastic properties is rarely explored in multi-principal element alloys (MPEAs). In this work, in order to explore the influence of the site preference of constituent atoms on the thermodynamic properties and elastic properties, the ordered configurations based on the previously predicted site occupying fractions (SOFs) and the disordered configuration based on a special quasi-random structure (SQS) were established, and the thermodynamic properties and elastic properties were predicted using the quasi-harmonic approximation (QHA) method first. It was found that the site preferring behaviors of atoms on the sublattices will not only improve the stability of the structure but will also lead to bigger elastic properties than its ideal disordered structure. Although the prediction of the temperature-dependent elastic properties using the QHA method shortens the time costs by only considering the effects of thermal expansion, it ignores the significant effect of lattice vibration on elastic properties at high temperatures. In order to explore the influence of lattice vibration on the temperature-dependent elastic properties further, the elastic properties of the ordered FCC_CoCrFeNi MPEA were predicted using the ab initio molecular dynamics (AIMD) method at several selective temperatures. The results show that except for the elastic constant C12 and bulk modulus, the lattice vibration reduces other elastic properties on the basis of thermal expansion. The predicted temperature-dependent shear and Young's moduli agree well with the limited experimental literature data. Thus, the temperature-dependent elastic properties can be reasonably predicted using the AIMD method based on site preference. The predicted polycrystalline elastic moduli B, G, and E of FCC_CoCrFeNi MPEA at 973 K are 146.68, 53.79, and 143.78 GPa, respectively.

Experimental-computational approach to investigate elastic properties of struvite.

The research presented in the paper concerns the elastic properties of struvite. The article combines theoretical and experimental research. Experimental studies were carried out on struvite single crystals grown in sodium metasilicate gel by single diffusion. This unique method leads to obtaining crystals of sufficiently large size to conduct, for the first time, experimental measurement of elastic properties of monocrystalline struvite. Using the nanoindentation method, the Ez = 29.1 ± 0.7GPa value of the component of Young's modulus was determined for a struvite single crystal. In addition, the elastic constants C11, C22, and C33 were determined using micro-Brillouin spectroscopy. Theoretical calculations of the abovementioned properties have been carried out by employing density functional theory methods. Scaling of the theoretical elastic constants leads to obtaining good agreement with the experimental values. Values of the Ex and Ey components of the Young's modulus, not available from the experimental nanoindentation technique, have been determined theoretically as 23GPa and 27GPa, respectively. Differences in the values of elastic components and Young's modulus components are related to the layered crystal structure of struvite and directional character of the hydrogen-bonding pattern.

Realization of diversity in physical properties of Zr2Se(B1‐xSex) MAX phases through DFT approach

AbstractThe discovery of a series of MAX phases, Zr2Se(B1‐xSex), with Se at both A‐ and X‐sites, drives a new chemical diversity to the MAX family. Here, we employed the density functional theory (DFT) approach to realize the diversity in physical properties of Zr2Se(B1‐xSex). All compositions of Zr2Se(B1‐xSex) are mechanically stable and the dynamical stability of the end member Zr2SeSe is confirmed. The elastic constant C33 and bulk moduli B show a decrease almost monotonically with Se‐content x while other constants and moduli change irregularly. All elastic constants and moduli except C12 and C13 are highest for the end member Zr2SeB. Additionally, the Vickers hardness, Debye temperature, minimum thermal conductivity, and lattice thermal conductivity are highest for Zr2SeB. The increase of Se‐content x at X‐site reduces most of the properties of Zr2Se(B1‐xSex). The electronic band structures change drastically with increasing Se‐content x. This diversity in electronic band structures is mainly the reason for the diversity in physical properties of Zr2Se(B1‐xSex). All compositions of Zr2Se(B1‐xSex) have the potential to be thermal barrier coating materials, and Zr2SeB has the potential to be etched into 2D MXene, Zr2B.

A systematic description of the thermodynamic, elastic and mechanical properties of binary Zr-based bcc alloys from first principles

The effects of the dissolution of 3d, 4d, and 5d metals, as well as Al, In, and Sn in the bcc lattice of Zr, were studied within the framework of the electron density functional theory. Using the EMTO-CPA method, we calculated the lattice parameters, enthalpies of mixing, single-crystal elastic constants C11, C12, C44, and C', polycrystalline elastic moduli E, G, and plasticity characteristics of disordered Zr-based bcc alloys over a wide concentration range. The PAW-SQS method was used to study the effects of alloying on the specified properties of bcc Zr-X alloys, where X is a series of 4d elements Nb, Mo, Tc, Ru, Rh and 5d elements Ta, W, Re, Os, Ir for concentration cuts 6.25, 25 and 50 at.%. The analysis of concentration and periodic dependences of properties of alloys, their stability is carried out.

Influence of Halides on Elastic and Vibrational Properties of Mixed-Halide Perovskite Systems Studied by Brillouin and Raman Scattering.

The relationship between halogen content and the elastic/vibrational properties of MAPbBr3-xClx mixed crystals (x = 1.5, 2, 2.5, and 3) with MA = CH3NH3+ has been studied using Brillouin and Raman spectroscopy at room temperature. The longitudinal and transverse sound velocities, the absorption coefficients and the two elastic constants C11 and C44 could be obtained and compared for the four mixed-halide perovskites. In particular, the elastic constants of the mixed crystals have been determined for the first time. A quasi-linear increase in the sound velocity and the elastic constant C11 with increasing chlorine content was observed for the longitudinal acoustic waves. C44 was insensitive to the Cl content and very low, indicating a low elasticity to shear stress in mixed perovskites regardless of the Cl content. The acoustic absorption of the LA mode increased with increasing heterogeneity in the mixed system, especially for the intermediate composition where the Br and Cl ratio was 1:1. In addition, a significant decrease in the Raman-mode frequency of the low-frequency lattice modes and the rotational and torsional modes of the MA cations was observed with decreasing Cl content. It clearly showed that the changes in the elastic properties as the halide composition changes were correlated with the lattice vibrations. The present findings may facilitate a deeper understanding of the complex interplay between halogen substitution, vibrational spectra and elastic properties, and may also pave the way for optimizing the operation of perovskite-based photovoltaic and optoelectronic devices by tailoring their chemical composition.