Isothermal Elastic Constants Research Articles

Elastic and transport coefficients of the perfect hard-sphere crystal from the poles of the hydrodynamic spectral functions

The elastic and transport coefficients of a perfect face-centered cubic crystal of hard spheres are computed from the poles of the dynamic structure factor and of the spectral functions of transverse momentum density fluctuations. For such crystals, the relevant coefficients are the three isothermal elastic constants (C11T,C12T,C44T) , the heat conductivity κ, and the three viscosities (η11,η12,η44) (in Voigt’s notations), which are directly computed using molecular dynamics simulations. The elastic and transport coefficients are then compared to the values of the same coefficients obtained with the method of Helfand moments, showing good agreement and providing strong support for the microscopic hydrodynamic theory of perfect crystals based on the local-equilibrium approach.

Hydrodynamic properties of the perfect hard-sphere crystal: microscopic computations with Helfand moments

Within the framework of the local equilibrium approach, the equilibrium and nonequilibrium properties relevant to the hydrodynamics of the perfect hard-sphere crystal were obtained through molecular dynamics simulations using the Helfand moments associated with momentum and energy transport. Because this crystal is face-centered cubic, the hydrodynamic properties we considered were hydrostatic pressure, isothermal bulk modulus, specific heat capacities and their ratio, three isothermal elastic constants (C11T,C12T,C44T) , heat conductivity, and three viscosities (η11,η12,η44) (in Voigt’s notation). These properties were computed as a function of the particle density. The pressure and transport coefficients diverge near the close-packing density, as the collision frequency per particle also diverges.

Thermodynamic and elastic properties of tungsten and tungsten silicide

We present thermodynamic and elastic theory for BCC metal and binary interstitial alloy established by the statistical moment method (SMM) and perform numerical calculations for the silicon concentration, pressure and temperature dependences of nearest neighbor distance, volume, thermal expansion coefficient, heat capacities at constant volume and at constant pressure, Debye temperature, Gruneisen parameter, isothermal elastic modulus, bulk modulus, shearing modulus, elastic constants and longitudinal wave velocity for W and WSi under temperature up to 3600 K, pressure up to 100 GPa and silicon atom concentration up to 5%. The SMM numerical calculations for W are compared with experiments and other calculations. The SMM calculated results for WSi are predictive, orienting experimental results in the future.

Continuum mechanics of nonideal crystals: Microscopic approach based on projection-operator formalism.

We present a microscopic derivation of the laws of continuum mechanics of nonideal ordered solids including dissipation, defect diffusion, and heat transport. The starting point is the classical many-body Hamiltonian. The approach relies on the Zwanzig-Mori projection operator formalism to connect microscopic fluctuations to thermodynamic derivatives and transport coefficients. Conservation laws and spontaneous symmetry breaking, implemented via Bogoliubov's inequality, determine the selection of the slow variables. Density fluctuations in reciprocal space encode the displacement field and the defect concentration. Isothermal and adiabatic elastic constants are obtained from equilibrium correlations, while transport coefficients are given as Green-Kubo formulas, providing the basis for their measurement in atomistic simulations or colloidal experiments. The approach to the linearized continuum mechanics and results are compared to others from the literature.

Isothermal and adiabatic elastic constants from virial fluctuations.

We derive expressions for classical isothermal and adiabatic elastic constants for periodic systems with the boundary contributions included explicitly. The potential-dependent part of these expressions is written in terms of potential energies of atomic groups that make up the total potential energy. It is shown that in the thermodynamic limit, the Born term, which depends on the second derivatives of potential energy, can be expressed exactly in terms of equilibrium averages that involve two types of atomic-group virials. As a result, the new form of the Born term involves only first derivatives of either atomic-group or total potential energies. The derived elastic constant expressions involving the two forms of the Born terms are tested and compared using molecular-dynamics simulations of crystalline argon and silicon. For both materials, the elastic constants obtained using the two forms of the Born term are in good agreement. In particular, the new form of the Born term converges to the same value as the original Born term but at a slower rate. The results for silicon also agree well with the results from the previous molecular-dynamics studies.

Elastic Anisotropy of 1,3,5‐Triamino‐2,4,6‐Trinitrobenzene as a Function of Temperature and Pressure: A Molecular Dynamics Study

AbstractThe equation of state of the triclinic compound 1,3,5‐triamino‐2,4,6‐trinitrobenzene (TATB) as well as its second‐order isothermal elastic tensor were computed through classical molecular dynamics simulations under various temperature and pressure conditions. Hydrostatic pressures similar to previous diamond anvil cell experiments were imposed up to 60 GPa and temperatures chosen between 100 and 900 K in conjunction with the most recent version of an all‐atom fully flexible molecule force field. The isothermal elastic constants were computed using the generalized Hooke's law by fitting Cauchy stress vs. linear strain curves for pressures below 50 GPa. Along isobaric loadings, TATB single crystal stiffnesses are found to undergo softening, less pronounced at high pressure, while maintaining its elastic anisotropy. On the other hand, along an isothermal loading, a non‐linear increase is observed in the elastic constants with respect to pressure with a significant decrease in anisotropy. Towards a precise mesoscopic modelling of TATB single crystal mechanical behavior, we provide “ready to plug‐in” analytical formulations of the P, V, T equation of state and pressure‐temperature dependent non‐linear elasticity.

Quasi-harmonic temperature dependent elastic constants: applications to silicon, aluminum, and silver

We present ab-initio calculations of the quasi-harmonic temperature dependent elastic constants. The isothermal elastic constants are calculated at each temperature as second derivatives of the Helmholtz free energy with respect to strain and corrected for finite pressure effects. This calculation is repeated for a grid of geometries and the results interpolated at the minimum of the Helmholtz free energy. The results are compared with the quasi-static elastic constants. Thermodynamic relationships are used to derive the adiabatic elastic constants that are compared with the experimental measurements. These approaches are implemented for cubic solids in the thermo_pw code and are validated by applications to silicon, aluminum, and silver.

Elastic properties of single crystal hydrogen sulfide: A Brillouin scattering study under high pressure-temperature

We have performed high pressure-temperature Brillouin scattering measurements on single crystal hydrogen sulfide using externally heated diamond anvil cell techniques. The pressure dependences of the acoustic velocities, isothermal elastic constants, and moduli of single crystal hydrogen sulfide have been determined along four isotherms. Both elastic constants and moduli increase monotonously with pressure along each isotherm, while they show a decreased tendency with temperature elevated under the same pressure points. The experimental equation of state of single crystal hydrogen sulfide is obtained by fitting with a third-order Birch-Murnaghan and Tait equation. It is proposed that the effect of hydrogen bonds contributes to the unique tendency of elastic anisotropy in single crystal hydrogen sulfide. Through our work, we have extended the melting curve and phase diagram of hydrogen sulfide up to 12 GPa and 580 K.

Molecular Dynamics Simulations of the Elastic Anisotropy of Pd at Extreme Conditions**Supported by the National Natural Science Foundation of China (41574076), the Basic Research of Technology Program of China under Grant No. JSHS2014404B002, the Young Core Teacher Scheme of Henan Province under Grant No. 2014GGJS-108, and key project of science and technology research of Henan Provincial Education Department under Grant

It is very interesting to discover the elastic properties of engineering material palladium, especially its elastic anisotropy along Hugoniot states. We here investigate the evolution of its high pressure and temperature (PT) elastic ansotropy along Hugoniot using molecular dynamics simulations based on accurate classical interatomic potential. In order to testify the validity of the interatomic potential of Pd in describing the high PT elastic properties, we calculate its isothermal and adiabatic elastic moduli using molecular dynamics method. The obtained data are in good agreement with experimental data. From the isothermal elastic constants, we deduce the Hugoniot acoustic velocities and find that the resulting data are in good agreement with experimental acoustic velocity data. Based on the reliable elastic constants, we further investigate the spacial elastic ansotropy along Hugoniot PT states. It is found that the spacial elastic anisotropy of Pd increases along Hugoniot states.

Elastic properties of solids at high pressure

This review examines the elastic response of solids under load. The definitions of isothermal and adiabatic elastic constants of () for a loaded crystal are given. For the case of hydrostatic pressure, two techniques are proposed for calculating the second-, third-, and fourth-order elastic constants from the energy-strain and stress-strain relations. As an example, using the proposed approach within the framework of the density functional theory, the second- to fourth-order elastic constants of bcc tungsten are calculated for the pressure range of 0–600 GPa.

Elastic properties of solids at high pressure

This review examines the elastic response of solids under load. The definitions of isothermal and adiabatic elastic constants of () for a loaded crystal are given. For the case of hydrostatic pressure, two techniques are proposed for calculating the second-, third-, and fourth-order elastic constants from the energy-strain and stress-strain relations. As an example, using the proposed approach within the framework of the density functional theory, the second- to fourth-order elastic constants of bcc tungsten are calculated for the pressure range of 0–600 GPa.

A fluctuation method to calculate the third order elastic constants in crystalline solids

This paper derives exact expressions of the isothermal third order elastic constants (TOE) in crystalline solids in terms of the kinetic and potential energies of the system. These expressions reveal that the TOE constants consist of a Born component and a relaxation component. The Born component is simply the third derivative of the system's potential energy with respect to the deformation, while the relaxation component is related to the non-uniform rearrangements of the atoms when the system is subject to a macroscopic deformation. Further, based on the general expressions derived here, a direct (fluctuation) method of computing the isothermal TOE constants is developed. Numerical examples of using this fluctuation method are given to compute the TOE constants of single crystal iron.

Thermoelastic properties ofα-iron from first-principles

We calculate the thermomechanical properties of $\alpha$-iron, and in particular its isothermal and adiabatic elastic constants, using first-principles total-energy and lattice-dynamics calculations, minimizing the quasi-harmonic vibrational free energy under finite strain deformations. Particular care is made in the fitting procedure for the static and temperature-dependent contributions to the free energy, in discussing error propagation for the two contributions separately, and in the verification and validation of pseudopotential and all-electron calculations. We find that the zero-temperature mechanical properties are sensitive to the details of the calculation strategy employed, and common semi-local exchange-correlation functionals provide only fair to good agreement with experimental elastic constants, while their temperature dependence is in excellent agreement with experiments in a wide range of temperature almost up to the Curie transition.

First principles investigation of the finite temperature dependence of the elastic constants of zirconium, magnesium and gold

The purpose of this work is to investigate how well the temperature dependence of the elastic constants of single crystal zirconium, magnesium and gold are reproduced by ab initio density functional theory (DFT). The modelling was conducted via the quasi-harmonic approximation with the exchange–correlation functional based on the local density approximation. For gold and magnesium, the low and high temperature dependency agrees well with measurements, whereas the transition between low and high temperature ranges occurs over a wider range of temperature than observed experimentally. For zirconium, the simulations qualitatively predict the temperature dependence of the isentropic elastic constants, where C12 and C13 increase with increasing temperature. Because this behaviour is absent for the isothermal elastic constants, the increase can be attributed to the transformation from isothermal to isentropic elastic constants.

External temperature and pressure effects on thermodynamic properties and mechanical stability of yttrium chalcogenides YX (X=S, Se and Te)

The full potential linearized augmented plane wave method within the framework of density functional theory is employed to investigate the structural, thermodynamic and elastic properties of the yttrium chalcogenides (YX: X=S, Se, and Te) in their low-pressure phase (Fm3¯m) and high-pressure phase (Pm3¯m). The exchange-correlation potential is treated with the generalized gradient approximation of Perdew–Burke–Ernzerhof (GGA-PBE). Temperature dependence of the volume and both adiabatic and isothermal bulk moduli is predicted for a temperature range from 0to1200K for the both phases of the herein considered materials. Furthermore, we have analyzed the thermodynamic properties such as the heat capacities, CV and CP, thermal expansion, α, and Debye temperature, ΘD, under variable pressure and temperature. We have calculated the isothermal elastic constants CijT of the YX monochalcogenides in both NaCl-B1 and CsCl-B2 phases at zero pressure and a temperature range 0−1200K. The results show that rare earth yttrium monochalcogenides are mechanically stable at high temperature. The elastic anisotropy of all studied materials in the two phases has been studied using three different methods.

Molecular dynamics study of the polymer clay nanocomposites (PCNs): Elastic constants and basal spacing predictions

Micromechanical approaches seem to be limited in the prediction of polymer nanocomposite properties, since the active molecular interactions between nanofillers and polymer matrix as well as those between nanofillers themselves, are far from being explicitly taken into consideration. The molecular interactions between nanofillers and polymer matrix lead to the so-called interphase region, where polymer chains mobility is reduced in the vicinity of nanofillers interface. Many assumptions are made on this interphase region to incorporate implicitly nanofillers/polymer interactions in micromechanical models. Experimental characterizations of this interphase and the confined interlayer polymer in the intercalated structure are not available to date. In this work, we have used isothermal–isostress (NσT) molecular dynamics simulation to predict the isothermal elastic constants of nylon-6 clay nanocomposites in the case of intercalated structure. The effect of the number of monomers in the nylon-6 chains on the basal spacing is evaluated. Isothermal bulk modulus is determined via isothermal–isobaric (NPT) molecular dynamics simulation. We also provide through this work an insight into molecular interactions in intercalated nylon-6 clay nanocomposites.

Calculations of isothermal elastic constants in the phase-field crystal model

The phase-field crystal (PFC) method is an emerging coarse-grained atomistic model that can be used to predict material properties. In this work, we describe procedures for calculating isothermal elastic constants using the PFC method. We find that the conventional procedures used in the PFC method for calculating the elastic constants are inconsistent with those defined from a theory of thermoelasticity of stressed materials. Therefore we present an alternative procedure for calculating the elastic constants that are consistent with the definitions from the thermoelasticity theory, and show that the two procedures result in different predictions. Furthermore, we employ a thermodynamic formulation of stressed solids to quantify the differences between the elastic constants obtained from the two procedures in terms of thermodynamic quantities such as the pressure evaluated at the undeformed state.

Elastic constants of solids at high pressures

The isothermal and adiabatic nth-order (n ≥ 2) elastic constants of a loaded crystal are defined. These constants fully determine the behavior of solids at an arbitrary load and are controlled by both an interatomic interaction and an applied load. Expressions that relate these constants (of the second, third, and fourth order) to Brugger elastic constants of the corresponding order, which are only determined by an inter-atomic interaction, are found for cubic symmetry crystals under hydrostatic pressure. These expressions are used to calculate the equation of state and the second- and third-order elastic constants of bcc tantalum at T = 0 K over a wide pressure range (0–600 GPa) using an electron density functional method. The results of calculating the equation of state and the second-order elastic constants agree with available experimental data and the calculation results obtained in other works.

First-principles calculations for elastic properties of the rocksalt structure MgO

The thermoelasticity of rocksalt (RS) structure MgO is investigated by ab initio plane-wave pseudopotential density functional theory method and the quasi-harmonic Debye model. The obtained lattice constant, aggregate elastic moduli, pressure derivative of isothermal bulk modulus and elastic constants are well consistent with the experimental data and other theoretical results. Moreover, the pressure dependences of individual elastic modulus at 300 and 800 K are also discussed. It is shown that, at 300 K, the Debye temperatures from our obtained elastic constants are well consistent with other theoretical results. The Debye temperature increases nonlinearly with increasing pressure.

Dynamic Acoustic Anomalies in KSCN Near the Improper Ferroelastic Phase Transition

The tensor of the elastic stiffness constants Cij of potassium thiocyanate (KSCN) has been measured in the temperature range 290 K to 420 K, including the order-disorder phase transition at 415 K, by the ultrasonic pulse echo overlap method at frequencies f = 20 MHz. Using the measured values for Cij the isothermal elastic constants Cij T were calculated from the data on specific heat Cp and the thermal expansion coefficients αi. It turns out that there is a huge difference between the ultrasonic and the isothermal elastic constants in the low symmetry phase. The ultrasonic elastic constants exhibit only small positive anomalies which are proportional to the square of the order parameter, while the isothermal ones tend to diminish on approaching Tc . Measurements of the elastic constants of KSCN at ultralow frequencies (f = 0.1 Hz–50 Hz) show two dispersion regimes. One is due to the isothermal-adiabatic crossover (between 0.1 Hz and 50 Hz) and the second (between 50 Hz and 20 MHz) results from the order parameter dynamics.