Pressure dependence of c-axis elastic parameters of oriented graphite

The elastic c-axis moduli C₃₃ and C₄₄ of compression-annealed pyrolytic graphite and their derivatives with hydrostatic pressure have been measured at room temperature over a pressure range from 1 to 7000 bars (700 MNm^?2) using an ultrasonic pulse-echo technique. The experiments were also made with material irradiated with fast neutrons. The pressure dependence of the c-axis compressibility of these materials was calculated from the experimental data, enabling pressure derivatives of the c-axis moduli to be also expressed as c-axis strain derivatives.     

The pressure dependence of elastic moduli of NiF2 to 10 kbar

At room temperature, the pressure dependence of the elastic stiffness moduli of NiF₂ have been measured by pulse superposition method to 10 kbar. The observed strong decreasing behavior in the modulus $\text{C}{}_{\mathrm{<mtext>s</mtext>}}\text{=}\text{1}\text{2}\text{(C}{}_{11}\text{? C}{}_{12}\text{)}$ is discussed.     

MgO和Cu的弹性常数随温度及压强变化关系的研究

从Anderson-Grüneisen参数定义式和热膨胀系数定义式出发,根据Tallon的普遍化理论,推导出了一个用于计算弹性常数的理论模型,用所得到的理论模型分别计算了MgO和Cu两种固体材料的弹性常数C_(11)、C_(12)和C_(44)随压强以及温度变化的理论数值,并将理论计算结果与Kumar模型计算结果以及相关实验数据进行了比较,对所得到的理论模型计算结果与Kumar模型计算结果之间的差异性进行了分析和探讨.研究结果表明:所推导的理论模型计算结果与实验数据更加符合,并且由于所用的推导方法不依赖于晶体的结构,因此该模型具有更好的合理性和普适性.     

MgO和Cu的弹性常数随温度及压强变化关系的研究

从Anderson-Grüneisen参数定义式和热膨胀系数定义式出发,根据Tallon的普遍化理论,推导出了一个用于计算弹性常数的理论模型,用所得到的理论模型分别计算了MgO和Cu两种固体材料的弹性常数 、 和 随压强以及温度变化的理论数值,并将理论计算结果与Kumar模型计算结果以及相关实验数据进行了比较,对所得到的理论模型计算结果与Kumar模型计算结果之间的差异性进行了分析和探讨。研究结果表明：所推导的理论模型计算结果与实验数据更加符合,并且由于所用的推导方法不依赖于晶体的结构,因此该模型具有更好的合理性和普适性。     

MgO和Cu的弹性常数随温度及压强变化关系的研究

从Anderson-Grüneisen参数定义式和热膨胀系数定义式出发,根据Tallon的普遍化理论,推导出了一个用于计算弹性常数的理论模型,用所得到的理论模型分别计算了MgO和Cu两种固体材料的弹性常数 、 和 随压强以及温度变化的理论数值,并将理论计算结果与Kumar模型计算结果以及相关实验数据进行了比较,对所得到的理论模型计算结果与Kumar模型计算结果之间的差异性进行了分析和探讨。研究结果表明：所推导的理论模型计算结果与实验数据更加符合,并且由于所用的推导方法不依赖于晶体的结构,因此该模型具有更好的合理性和普适性。     

The elastic moduli and their pressure and temperature derivatives for calcium oxide

The room temperature adiabatic elastic moduli of CaO were obtained using a pulse superposition technique to measure the ultrasonic velocities. The elastic moduli were found to be (in GPa) at 298 K.

     

9.

Many-atom interactions in the theory of higher order elastic moduli: A general theory     

I. A. Osipenko  O. V. Kukin  A. Yu. Gufan  Yu. M. Gufan 《Physics of the Solid State》2013,55(12):2405-2412

The total potential energy of a crystal U({r _ik }) as a function of the vectors r _ik connecting centers of equilibrium positions of the ith and kth atoms is assumed to be represented as a sum of irreducible interaction energies in clusters containing pairs, triples, and quadruples of atoms located in sites of the crystal lattice A2: U({r _ik }) ≡ Σ _N=1 ⁴ E _N ({r _ik }). The curly brackets denote the “entire set.” A complete set of invariants {I _j ({r _ik })} _N , which determine the energy of each individual cluster as a function of the vectors connecting centers of equilibrium positions of atoms in the cluster E _N ({r _ik }) ≡ E _N ({I _j ({r _ik })} _N ), is obtained from symmetry considerations. The vectors r _ik are represented in the form of an expansion in the basis of the Bravais lattice. This makes it possible to represent the invariants {I _j ({r _ik })} _N in the form of polynomials of integers multiplied by τ ₂ ^m . Here, τ₂ is one-half of the edge of the unit cell in the A2 structure and m is a constant determined by the model of interaction energy in pairs, triples, and quadruples of atoms. The model interaction potential between atoms in the form of a sum of the Lennard-Jones interaction potential and similarly constructed interaction potentials of triples and quadruples of atoms is considered as an example. Within this model, analytical expressions for second-order and third-order elastic moduli of crystals with the A2 structure are obtained.     

10.

The elastic interaction between the intercalants in graphite — A two-dimensional model     

S. Ohnishi  S. Sugano 《Solid State Communications》1980,36(9):823-826

The anisotropic nature of the elastic interaction between the intercalants in graphite is discussed theoretically based on a two-dimensional model It is shown that the intercalant—intercalant interaction in the c-axis direction is also attractive when the x-z component in the force dipole tensor is taken into account. The elastic interaction between the sheet-like domains of the intercalants is also discussed.     

11.

Interface stresses and their effects on the elastic moduli of metallic multilayers     

Gumbsch P  Daw MS 《Physical review. B, Condensed matter》1991,44(8):3934-3938

     

12.

Elastic moduli of ordered solid solutions and elastic interaction of superdislocations with the walls     

A. E. Ginzburg  É. V. Kozlov 《Russian Physics Journal》1972,15(4):479-484

     

13.

The size dependence of equilibrium elastic strain in finite epitaxial islands     

W.A. Jesser  J.H. van Der Merwe 《Surface science》1972

The equilibrium elastic strain in epitaxial islands of small size is shown to have a sawtooth dependence on island width by minimizing the systems energy as calculated from a periodic interaction potential between substrate and overgrowth. Such a sawtooth variation is consistent with Vincent's observations and idea that a given misfit between island and substrate may not be entirely accommodated by an integral number of identical misfit dislocations, and hence, the remaining misfit, which depends in part on island width, will be accommodated by residual elastic strain. The present calculations of mechanical equilibrium support these ideas and further reveal that in some cases misfit dislocations may over compensate for the misfit, thereby introducing an elastic strain of opposite sign to that normally expected. Other results of the calculations are in agreement with earlier theoretical and experimental observations.     

14.

Strongly anisotropic elastic moduli of nematic elastomers: Analytical expressions and nonlinear temperature dependence     

Z. Zeng  L. Jin  Y. Huo 《The European physical journal. E, Soft matter》2010,32(1):71-79

Exact formulae for the elastic moduli of the nematic elastomers are obtained by the implicit function method based on somewhat general energy functions. The formulae indicate that both the moduli parallel and perpendicular to the director of the nematic elastomers are smaller than the modulus of the classical elastomers because of the mechanical-nematic coupling. Moreover, the moduli are generally anisotropic due to the biaxiality induced by stretching the nematic elastomers perpendicular to the director. Then we get the explicit analytical expressions of the parallel and perpendicular moduli by making use of the Landau-de Gennes free energy and the neo-classical elastic energy. Very different from the classical elastomers, they are both strongly nonlinear functions of the temperature in the nematic phase. Furthermore, their ratio, the degree of anisotropy, changes with the temperature as well. The results agree qualitatively with some experiments. Better quantitative agreement is obtained by some modifications of the constitutive relation of the elastic energy.     

15.

The temperature dependence of the elastic constants for highly oriented pyrolytic graphite determined by ultrasonic techniques     

D.M. Hwang 《Solid State Communications》1983,46(2):177-181

Sound velocities along the c-axis of highly oriented pyrolytic graphite between 4 and 325 K were measured with an ultrasonic pulse-echo phase-comparison method. The echo delay times were determined with an accuracy of 0.05 and 0.15%, respectively, for the longitudinal and transverse modes. The temperature dependence of the c-axis elastic constants C₃₃ and C₄₄ was deduced with the correction of thermal expansion. C₃₃ was found to decrease by 5.8% from 4 to 300 K and can be interpreted by a model based on the simple Lennard-Jones interlayer interaction. C₄₄ and its temperature variation are found to be strongly sample dependent.     

16.

Determination of the dynamic elastic moduli and internal friction using thin resonant bars     

Guo Q  Brown DA 《The Journal of the Acoustical Society of America》2000,108(1):167-174

Analysis of the data from the resonant-bar technique for determining wave speed and internal friction is presented. Internal friction has been included in the longitudinal and torsional wave equations and the analytical solution has been obtained. In determining the acoustical constants of lossy materials, a broad frequency spectrum is used that includes many resonances and not just data at or near the individual resonances. Corrections due to the mass, length, stiffness, and damping of the transducers are also presented. The theoretical solutions are compared with the measured magnitude and phase response data for torsional, longitudinal, and flexural measurements and are shown to be in good agreement.     

17.

The pressure and temperature derivatives of the elastic moduli of lithium hydride     

D. Gerlich  C.S. Smith 《Journal of Physics and Chemistry of Solids》1974,35(12):1587-1592

The pressure and temperature derivatives of the elastic moduli of single crystal LiH have been determined at room temperature, by the ultrasonic pulse echo method. From the pressure data, the mode Grüneisen parameters as well as the low and high temperature limiting values of the Grüneisen constant were computed; the latter were only slightly different. An equation of state has been constructed for LiH, and found to be in agreement with the experimental data. Based on a non-central force model, the values of the elastic moduli and their pressure derivatives have been calculated theoretically, and the latter values compared with the experimental data.     

18.

The polarizing tensor of the elastic coefficients and moduli for α-quartz     

K. Hruška  V. Kazda 《Czechoslovak Journal of Physics》1968,18(4):500-503

     

19.

The temperature dependences of the elastic moduli of solid C60     

N. P. Kobelev  R. K. Nikolaev  N. S. Sidorov  Ya. M. Soifer 《Physics of the Solid State》2002,44(3):429-431

This paper reports on the determination of the temperature dependences of the complete set of the elastic moduli of solid C₆₀ from sound-velocity measurements made along different crystallographic directions in single-crystal samples within the 100-to 300-K range. Substantial differences in their behavior were revealed, which are accounted for by different relative contributions from relaxation processes to various elastic moduli.     

20.

Electronic structure and elastic moduli of the simple cubic fullerite C24—A new allotropic carbon modification     

V. L. Bekenev  V. V. Pokropivny 《Physics of the Solid State》2006,48(7):1405-1410

The energy band structure, equation of state, density of states, and elastic moduli of a new allotropic carbon modification, namely, fullerite C₂₄ with a simple cubic lattice (known previously as cubic graphite), are calculated by the full-potential linearized augmented-plane-wave (FLAPW) method with geometry optimization for the first time. The dependence of the total energy on the lattice constant exhibits a minimum for a ₀ = 0.60546 nm. In this case, the lengths of the C-C bonds between fullerene molecules, the lengths of the 6,6-bonds shared by hexagons, and the lengths of the 4,6-bonds shared by a square and a hexagon are equal to 0.1614, 0.1503, and 0.1637 nm, respectively. An analysis of the energy band structure and the density of states demonstrates that the simple cubic fullerite C₂₄ is a direct-band-gap insulator or a semiconductor with a band gap of 1.6 eV. The calculated bulk modulus B ₀ = 196 GPa and the elastic moduli C ₁₁ = 338 GPa, C ₁₂ = 139 GPa, and C ₄₄ = 30 GPa indicate that the fullerite under investigation is a mechanically stable material. The inference is made that the simple cubic fullerite C₂₄ is a new diamond-like molecular zeolite with a unique combination of properties, such as the porosity and nonpolarizability, on the one hand, and the mechanical strength, chemical inertness, and high thermal conductivity, on the other hand. The simple cubic fullerite C₂₄ can be considered a promising low-dielectric-constant (low-k) material (?₀ < 5.7) for use in fabricating interconnections and substrates intended for integrated circuits and nanoelectronics.     

c11	c12	c44
226.2 ±0.9	62.4 ±0.9	80.6 ±0.3 The pressure derivatives of the elastic moduli, which were measured at and below room temperature, decrease with decreasing temperature. The temperature derivatives of the elastic moduli were also obtained for several temperatures below room temperature.

Many-atom interactions in the theory of higher order elastic moduli: A general theory

The total potential energy of a crystal U({r _ik }) as a function of the vectors r _ik connecting centers of equilibrium positions of the ith and kth atoms is assumed to be represented as a sum of irreducible interaction energies in clusters containing pairs, triples, and quadruples of atoms located in sites of the crystal lattice A2: U({r _ik }) ≡ Σ _N=1 ⁴ E _N ({r _ik }). The curly brackets denote the “entire set.” A complete set of invariants {I _j ({r _ik })} _N , which determine the energy of each individual cluster as a function of the vectors connecting centers of equilibrium positions of atoms in the cluster E _N ({r _ik }) ≡ E _N ({I _j ({r _ik })} _N ), is obtained from symmetry considerations. The vectors r _ik are represented in the form of an expansion in the basis of the Bravais lattice. This makes it possible to represent the invariants {I _j ({r _ik })} _N in the form of polynomials of integers multiplied by τ ₂ ^m . Here, τ₂ is one-half of the edge of the unit cell in the A2 structure and m is a constant determined by the model of interaction energy in pairs, triples, and quadruples of atoms. The model interaction potential between atoms in the form of a sum of the Lennard-Jones interaction potential and similarly constructed interaction potentials of triples and quadruples of atoms is considered as an example. Within this model, analytical expressions for second-order and third-order elastic moduli of crystals with the A2 structure are obtained.     

The elastic interaction between the intercalants in graphite — A two-dimensional model

The anisotropic nature of the elastic interaction between the intercalants in graphite is discussed theoretically based on a two-dimensional model It is shown that the intercalant—intercalant interaction in the c-axis direction is also attractive when the x-z component in the force dipole tensor is taken into account. The elastic interaction between the sheet-like domains of the intercalants is also discussed.     

The size dependence of equilibrium elastic strain in finite epitaxial islands

The equilibrium elastic strain in epitaxial islands of small size is shown to have a sawtooth dependence on island width by minimizing the systems energy as calculated from a periodic interaction potential between substrate and overgrowth. Such a sawtooth variation is consistent with Vincent's observations and idea that a given misfit between island and substrate may not be entirely accommodated by an integral number of identical misfit dislocations, and hence, the remaining misfit, which depends in part on island width, will be accommodated by residual elastic strain. The present calculations of mechanical equilibrium support these ideas and further reveal that in some cases misfit dislocations may over compensate for the misfit, thereby introducing an elastic strain of opposite sign to that normally expected. Other results of the calculations are in agreement with earlier theoretical and experimental observations.     

Strongly anisotropic elastic moduli of nematic elastomers: Analytical expressions and nonlinear temperature dependence

Exact formulae for the elastic moduli of the nematic elastomers are obtained by the implicit function method based on somewhat general energy functions. The formulae indicate that both the moduli parallel and perpendicular to the director of the nematic elastomers are smaller than the modulus of the classical elastomers because of the mechanical-nematic coupling. Moreover, the moduli are generally anisotropic due to the biaxiality induced by stretching the nematic elastomers perpendicular to the director. Then we get the explicit analytical expressions of the parallel and perpendicular moduli by making use of the Landau-de Gennes free energy and the neo-classical elastic energy. Very different from the classical elastomers, they are both strongly nonlinear functions of the temperature in the nematic phase. Furthermore, their ratio, the degree of anisotropy, changes with the temperature as well. The results agree qualitatively with some experiments. Better quantitative agreement is obtained by some modifications of the constitutive relation of the elastic energy.     

The temperature dependence of the elastic constants for highly oriented pyrolytic graphite determined by ultrasonic techniques

Sound velocities along the c-axis of highly oriented pyrolytic graphite between 4 and 325 K were measured with an ultrasonic pulse-echo phase-comparison method. The echo delay times were determined with an accuracy of 0.05 and 0.15%, respectively, for the longitudinal and transverse modes. The temperature dependence of the c-axis elastic constants C₃₃ and C₄₄ was deduced with the correction of thermal expansion. C₃₃ was found to decrease by 5.8% from 4 to 300 K and can be interpreted by a model based on the simple Lennard-Jones interlayer interaction. C₄₄ and its temperature variation are found to be strongly sample dependent.     

Determination of the dynamic elastic moduli and internal friction using thin resonant bars

Analysis of the data from the resonant-bar technique for determining wave speed and internal friction is presented. Internal friction has been included in the longitudinal and torsional wave equations and the analytical solution has been obtained. In determining the acoustical constants of lossy materials, a broad frequency spectrum is used that includes many resonances and not just data at or near the individual resonances. Corrections due to the mass, length, stiffness, and damping of the transducers are also presented. The theoretical solutions are compared with the measured magnitude and phase response data for torsional, longitudinal, and flexural measurements and are shown to be in good agreement.     

The pressure and temperature derivatives of the elastic moduli of lithium hydride

The pressure and temperature derivatives of the elastic moduli of single crystal LiH have been determined at room temperature, by the ultrasonic pulse echo method. From the pressure data, the mode Grüneisen parameters as well as the low and high temperature limiting values of the Grüneisen constant were computed; the latter were only slightly different. An equation of state has been constructed for LiH, and found to be in agreement with the experimental data. Based on a non-central force model, the values of the elastic moduli and their pressure derivatives have been calculated theoretically, and the latter values compared with the experimental data.     

The temperature dependences of the elastic moduli of solid C60

This paper reports on the determination of the temperature dependences of the complete set of the elastic moduli of solid C₆₀ from sound-velocity measurements made along different crystallographic directions in single-crystal samples within the 100-to 300-K range. Substantial differences in their behavior were revealed, which are accounted for by different relative contributions from relaxation processes to various elastic moduli.     

Electronic structure and elastic moduli of the simple cubic fullerite C24—A new allotropic carbon modification

The energy band structure, equation of state, density of states, and elastic moduli of a new allotropic carbon modification, namely, fullerite C₂₄ with a simple cubic lattice (known previously as cubic graphite), are calculated by the full-potential linearized augmented-plane-wave (FLAPW) method with geometry optimization for the first time. The dependence of the total energy on the lattice constant exhibits a minimum for a ₀ = 0.60546 nm. In this case, the lengths of the C-C bonds between fullerene molecules, the lengths of the 6,6-bonds shared by hexagons, and the lengths of the 4,6-bonds shared by a square and a hexagon are equal to 0.1614, 0.1503, and 0.1637 nm, respectively. An analysis of the energy band structure and the density of states demonstrates that the simple cubic fullerite C₂₄ is a direct-band-gap insulator or a semiconductor with a band gap of 1.6 eV. The calculated bulk modulus B ₀ = 196 GPa and the elastic moduli C ₁₁ = 338 GPa, C ₁₂ = 139 GPa, and C ₄₄ = 30 GPa indicate that the fullerite under investigation is a mechanically stable material. The inference is made that the simple cubic fullerite C₂₄ is a new diamond-like molecular zeolite with a unique combination of properties, such as the porosity and nonpolarizability, on the one hand, and the mechanical strength, chemical inertness, and high thermal conductivity, on the other hand. The simple cubic fullerite C₂₄ can be considered a promising low-dielectric-constant (low-k) material (?₀ < 5.7) for use in fabricating interconnections and substrates intended for integrated circuits and nanoelectronics.