First-Principles Study On Nonlinear Elasticity And Thermoelasticity Of Graphene And Rare-Earth Intermetallics

Elastic properties are the fundamental solid state properties. The elastic behaviorsof solid are determined by the elastic constants, which are defined as the second-, third-,or high-order derivatives of the Helmholtz free energy or internal energy, named assecond-, third-, or high-order elastic constants. In the linear theory of elasticity, theelastic deformation is very small and the second-order elastic constants are sufficient todescribe the elastic stress-strain response in solids. Currently, modernelectronic-structure methods based on density functional theory can treat thezero-temperature （i.e., T=0K） elastic constants exactly, but treating the correspondingtemperature dependence of elasticity is formidable challenge. The traditional theory ofelasticity is established on linear elasticity and zero-temperature. For the past few years,because of the development of engineering application and the emergence of novelmaterials continually, the linear theory of elasticity have exhibited more and moreboundedness. For example, graphene is a novel two-dimensional （2D） sheet material ofcovalently bonded carben atoms and has found in2004, and it has been confirmed as thestrongest material ever to measured being able to reversible elastic deformations inexcess of20%; the novel intermetallics YAg and YCu exhibit high ductility besides theexcellent properties of the traditional intermetallics; rare-earth magnesium MgRE andrare-earth aluminum AlRE intermetallics have good high-temperature strength andstability in comparison with the traditional magnesium and aluminum alloys. Therefore,the high-temperature and high-pressure applications and design of the novel materialsrequire expanding the linear and zero-temperature elasticity to the case of finite strainand finite temperature, i.e., the nonlinear theory of elasticity and thermoelasticity.The key points of the nonlinear elasticity are obtaining the complete set of third-and higher-order elastic constants, which can be determined from the lattice symmetry.The introducing of temperature needs to calculate the phonon spectra based on thelattice dynamics, and the accurate calculations of the temperature-dependent elasticconstants must consider the relations between phonon specta and strains. Hence, asimple and efficient approach of calculating the thermoelasticity is very important intheory. In this paper, we have furtherly developed the frist-principles approach tocalculate the third-order elastic constants in the nonlinear theory of elasticity and thetemperature-dependent elastic constants based on the density functional theory. Then, the method has been performed to investigate the nonlinear elastic properties andthermoelasticity of the novel materials, such as graphene, YAg and YCu,rare-earth-magnesium MgRE, and rare-earth-aluminum AlRE intermetallics. The Lotsof the calculating results are important for the applications and design of the novalmaterials. The main work and results as in the following:（1） The nonlinear elasticity of graphene. Experimentally, graphene has the largeintrinsic breaking strength and exhibites the nonlinear elasticity strongly. In this paper,the independent second-and third-order elastic constants of graphene have been derivedon the basis of its lattice symmetry, and the relations between the density strain energyand strain have also been obtained. The calculated results agree well with the dataobtained from the previous tight-binding atomistic simulations and experiment, andshow that linear approximation is not sufficient for strain larger than approximately3.0%and must consider nonlinear elasticity. In the framework of the relation ofnonlinear stress-strain, we have obtained the value of the effective third-order elasticmodulus D is-610Nm^-1that improves the coincidence rate of the value-583Nm^-1obtained from tight-binding atomistic simulations by Cadelano et.al. and theexperimental value-690±120Nm^-1. The internal relaxation is the very importantphenomenon when the compound lattice undergoes a macro-strain. In this paper, weprovide a novel and efficient approach to determined the internal relaxation, and thismethod has been applied to calculate the internal relaxation displacements as thefunction of strain. In the region of small strain, the relations between the internalrelaxation and strain are linear. In order to confirme the correcting of our results, wehave also demonstrated that the symmetry of the relationship between the internaldisplacement and the infinitesimal strains can be satisfied.（2） Calculation of the nonlinear elasticity of rare-earth intermetallics. Theintermetallics have the ordered superlattice, and maintain the strong metallic bond. Theyintensely attracted considerable attention, due to their chemical, physical, electronic,and especially mechanical properties that are often superior to ordinary metals, such ashigh strength, melting at high temperature, low specific weight, and good corrosionresistance. In this paper, the second-and third-order elastic constants of the sixteennovel intemetallics YAg and YCu, MgRE （RE=Y, Tb, Dy, Nd） and AlRE （RE=Y, Pr, Nd,Tb, Dy, Ce） with B2-type structure have been calculated from the first-principlesmothed based on the density functional theory （DFT） and on the nonlinear theory ofelasticity. In order to confirme the correcting of the calculated results, the tests of convergence have been carefully checked. When the applied strain is arriving at8.0%the nonlinear relationships between strain energy and strain are still consistent with thecalculated data, but the critical value of the approach of the linear elasticity is3.0%. Asthe application of the third-order elastic constants, the pressure dependence of thesecond-order elastic constants has been obtained in the framework of the nonlineartheory of elasticity. The derivatives of the bulk modulus have good agreement with thevalues obtained from the fitting of the Vinet and Rose equation of state, and the resultsdemonstrate the correction and application of the calculated third-order elastic constantsfurther.（3） The lattice dynamics and the thermodynamics of YAg and YCu, and MgREintermetallics. YAg and YCu are a new class of ductile intermetallic compounds withB2-type structure. However, no theoretical study of thermodynamic properties has beenreported. In this paper, the supercell approach has been employed to calculate dynamicsof lattice, and the real-space force constants of supercell are calculated in thedensity-functional perturbation theory （DPFT）. The thermodynamic properties, such asthermal expansion, heat capacity at constant volume and constant pressure, and theisothermal bulk modulus are obtained from the quasiharmonic approximation （QHA）.To judge that our computational method is reasonable, NiAl has also been investigatedin comparisons with experimental data and previous theoretical results. The latticedynamics of YAg and YCu shows that the density of states is mostly composed of Ystates at high frequencies, and that is mostly composed of Ag or Cu at low frequencies.The thermal expansion coefficient of YAg is larger than that of YCu, and the resultsdemonstrate that the intermetallics with better ductility have larger thermal expansions.In this paper, we have also investigated the lattice dynamics and thermodynamics ofrare-earth-magnesium MgRE （RE=Y, Dy, Pr, Tb） since they have the goodhigh-temperature strength and stabiltity. Our results show that the contribution of REatoms is dominant in phonon frequency. In comparison with YAg and YCu, we haveconsidered the contributions of thermal electrons on the free energy in the MgRE. Theelectronic heat capacity is discussed, and is found to be important for the calculatedMgRE intermetallics.（4）The thermoelasticity of the YAg and YCu, and MgRE intermetallics. The elasticconstants dependent on temperature are important to predict and understand the strengthof mechanics, stability, transition, etc. According to the experimental data of thetemperature-dependent elastic constants, we present an approach to calculate the temperature dependence of the elastic constants. In this method, we assume that theelastic constants C_IJ（T）can be approximated as the zero-temperature elastic constantsfrom the corresponding equilibrium volume V（T）. The comparisons between predictedresults and the available experimental data for some benchmark materials Ag, Cu andNiAl provide good agreements. In further study, the thermoelasticity of theintermetallics YAg and YCu, MgRE （RE=Y, Dy, Pr, Sc, Tb） have been investigated. Ourresults show that the elastic constants of all calculated intermetallcis follow a normalbehavior with temperature, i.e., decrease with increasing temperature and approachlinearity at higher temperature and zero slope around zero temperature （T=0K）. In thecalculated ranges of temperature, the elastic constants statisfy the stability conditionsfor B2-type structures, i.e., C₁₁-C₁₂>0, C₁₁>0, C₄₄>0. For the ductile intermetallics YAgand YCu, the ductility as a function of temperature is discussed by the concept ofCauchy pressure. In addition, the sound velocities as a function of temperature forMgRE have also been investigated.