【作者】 王锐；

【导师】 王少峰；

【作者基本信息】 重庆大学 ， 凝聚态物理学， 2012， 博士

【摘要】 固体的弹性性质是固体的基本物理属性，固体的弹性行为一般由固体的弹性常数来描述。弹性常数为系统的Hemholtz自由能或内能对Lagrange应变的二阶、三阶或者更高阶导数，即二阶、三阶或者更高阶弹性常数。在线性弹性理论中，弹性形变很小，这时二阶弹性常数可以精确描述固体中的应力—应变关系。目前，基于密度泛函理论的电子结构方法能够能精确计算零温（即T=0K）时的弹性常数，但要获得弹性常数随温度的变化关系，仍面临着巨大的挑战。传统上，对弹性理论的第一性原理研究主要限于线性弹性和零温弹性。近年来，随着工程应用的拓展和新型材料的不断涌现，线性和零温弹性理论越来越显示出它们的局限性。例如，2004年发现的新型二维材料石墨烯（graphene），在形变小于20%的范围内都可以为弹性形变；新型金属间化合物YAg和YCu除了具有传统的金属间化合物的优良属性外还具有高延展性；稀土镁MgRE和稀土铝AlRE金属间化合物则具有了传统镁铝合金所不曾有的高温稳定性。因此，对新型材料高压高温的应用与设计，则必须把线性弹性和零温弹性拓展到有限应变和有限温度的情况，而这就需要考虑三阶或者更高阶的弹性常数以及弹性常数随温度的变化规律，即非线性弹性和热弹性。进行非线性弹性研究的关键在于获得完备的三阶弹性常数以及更高阶的弹性常数，这可以根据晶格的对称性来确定。温度的引入则需要在晶格动力学的基础上计算声子谱；另外，弹性常数与温度的关系的精确计算还需要考虑声子谱与应变的关系，这使得热弹性性质的计算变得很复杂。因此，在理论上找到一种计算热弹性性质的简单且有效的方法具有重要意义。本文在密度泛函理论的基础上进一步发展了计算非线性三阶弹性常数和弹性常数随温度变化规律的第一性原理方法，并研究了graphene、延展性稀土金属间化合物YAg和YCu以及稀土镁MgRE和稀土铝AlRE金属间化合物的非线性弹性和热弹性性质，并且计算结果对新型材料的应用和设计具有重要意义。主要内容如下：（1）Graphene的非线性弹性研究。实验表明，graphene具有很高的本征断裂强度，展现出了显著的非线性弹性性质。本文首先通过晶格对称性导出了graphene的独立二阶和三阶弹性常数，并在此基础上确定了应变能密度与应变的解析关系。计算结果显示，我们计算的二阶弹性常数与先前的理论值和实验值相符合。本文获得的有效非线性（三阶）弹性模量D的值为-610Nm^-1，与Cadelano等人用紧束缚原子模拟的值-583Nm^-1相比，我们的计算值与实验值-690±120Nm^-1更符合。通过对数值结果的非线性弹性拟合和线性弹性拟合的比较发现，当对graphene施加的应变大于3.0%时必须考虑非线性弹性效应。内弛豫是复式晶格发生宏观应变时伴随着的重要现象，本文结合密度泛函理论找到了一种新的有效方法来确定晶格的内弛豫位移，并用该方法详细计算了graphene的内弛豫位移与应变的关系。在小应变区域，内弛豫位移与应变呈线性关系。此外，我们还根据对称性原理论证了计算结果的正确性。（2）稀土金属间化合物的非线性弹性计算。金属间化合物具有有序的超点阵结构，保持着很强的金属键结合。它们的物理性质尤其是力学性质优于普通金属，具有高强度、高熔点、低密度以及良好的抗腐蚀能力等，因此，金属间化合物被广泛应用于各工程领域。本文首先结合B2型金属间化合物的对称性获得了完备的二阶和三阶弹性常数，然后根据第一性原理方法和有限应变连续弹性理论计算了12种B2型金属间化合物YAg和YCu、MgRE（RE=Y，Tb，Dy，Nd）以及AlRE（RE=Y,Pr, Nd, Tb, Dy, Ce）的三阶弹性常数。结果显示，我们计算的二阶弹性常数与先前的计算值以及实验数据符合得很好。对于非线性弹性区域，当施加的应变的最大值达到8.0%时应变与能量的关系仍和数值计算结果符合得很好，这远大于线性弹性的临界应变值2.5%。此外，计算的三阶弹性常数可以方便地用来讨论压强下的弹性行为。我们计算了二阶弹性常数与压强的关系，在非线性弹性理论框架下获得的体模量对压强的一阶导数值与用Vinet态方程和Rose态方程拟合的值一致。（3）YAg和YCu以及MgRE金属间化合物的晶格动力学和热力学性质研究。YAg和YCu是一类新型的具有高延展性的稀土金属间化合物。本文采用第一性原理声子计算讨论了YAg和YCu的晶格动力学和热力学性质。在具体计算中，对晶格动力学计算利用了超晶胞近似，并结合密度泛函微扰理论计算了实空间的力常数。热力学性质，如热膨胀、等容和等压比热、等温体模量等都通过准谐近似求出。计算的基准材料NiAl的声子谱和热膨胀与先前的理论值和实验值相符合。研究显示，YAg和YCu的高频声子主要来自于Y原子振动，而低频声子则来自于Ag或Cu原子。通过对比具有延展性的金属间化合物和脆性金属间化合物的声子谱发现，脆金属间化合物NiAl的光学支和声学支间具有带隙，而延展性金属间化合物YAg和YCu没有带隙。YAg的热膨胀系数大于YCu，这说明高延展性的YAg具有更好的热膨胀。新型稀土镁金属间化合物MgRE具有良好的高温强度和高温稳定性。与YAg和YCu不同的是，MgRE的Fermi能级处于电子态密度的峰值位置且具有大的态密度。因此，对于MgRE必须考虑热电子激发对自由能的贡献。在计算中，我们首先采用超晶胞方法计算MgRE的晶格动力学性质，结果表明，稀土元素RE支配着MgRE的低频声子（声学声子）振动。通过定性分析MgRE的热力学性质与电子结构的关系发现，MgRE的热膨胀随温度的变化关系与RE元素的外层电子排布相关。相对于传统金属，本文的计算结果表明MgRE的电子比热在高温时也扮演着重要角色。（4）YAg和YCu以及MgRE金属间化合物的热弹性性质计算。材料的弹性常数与温度的关系对预测和理解晶格的力学强度、稳定性和相变具有重要意义。在总结弹性常数随温度变化的实验数据的基础上，本文提出了一种新的方法来计算二阶弹性常数与温度的关系。我们假定在温度T时的弹性常数C_IJ（T）与该温度下的平衡体积V （T）对应的零温弹性常数相等。计算中，我们首先选取了Ag、Cu和NiAl作为基准材料来验证方法的正确性，得到的结果与实验值符合。在进一步讨论中，我们计算和分析了金属间化合物YAg和YCu、MgRE（RE=Y，Dy，Pr，Sc，Tb）的二阶弹性常数随温度的变化行为，结果显示，所有计算的金属间化合物的弹性常数随温度的变化规律都符合正常行为，即弹性常数随温度的升高而降低。高温时，弹性常数基本与温度成线性关系，而在T=0K附近，弹性常数则基本保持为常数。研究显示，弹性常数随温度的变化满足C₁₁-C₁₂>0、C₁₁>0和C₄₄>0，这说明此类材料在计算温度范围内都具有稳定相。我们用Cauchy压强随温度的变化讨论了YAg和YCu的延展性与温度的关系。结果表明，YAg和YCu的延展性随温度的升高而升高，并且YAg的延展性始终高于YCu。此外，我们还讨论了稀土镁金属间化合物MgRE的声速与温度的变化关系。更多还原

【Abstract】 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.更多还原