Crystallographic Thermal Expansion Research Articles

77 new thermodynamic identities among crystalline elastic material properties leading to a shear modulus constitutive law in isotropic solids

Thermodynamics of crystalline materials is analyzed using strain volumes, an incremental tensorial state variable which is the volume per unit mass multiplied by the incremental strain. It is shown that the derivatives of the strain volume variables with respect to the stresses are reduced to conventional well known isothermal, elastic, and crystallographic compliances and crystallographic thermal expansion coefficients. The formulism is extended to all third order derivatives which establish 77 new thermodynamic identities: 27 are thermal and 50 are among selected stress components of elastic compliances. The stress dependence of heat capacity is typically coupled into the crystallographic thermal expansion coefficient's temperature dependence; the temperature dependence of the elastic compliances is related to the stress dependence of the thermal expansion coefficients; stress dependent relationships among elastic compliances are also found. This paper emphasizes internal self-consistency. General triclinic identity relations which are of academic interest are applied to specific higher symmetries including orthorhombic, tetragonal, and cubic; identities in higher symmetry crystals are very useful. A new generalized, constitutive law is found and applied to isotropic polycrystalline materials; the concept is based on shear thermal expansion coefficients. The constitutive modulus relation describes thermal and pressure properties only through an isochoric modulus.

Inverse analysis of giant macroscopic negative thermal expansion of Ca2RuO4−y ceramics based on elasticity and structural topology optimization

Ca2RuO4−y ceramics exhibit a large volumetric negative thermal expansions (NTE), although the crystallographic volume contraction on heating is much smaller than the NTE. Therefore, we examine the differences in the mechanisms underlying the volumetric thermal expansion for ruthenate ceramics and crystals in the context of the elasticity. We identify the possible microstructure of ruthenate ceramics composed of crystal grains and cavities using structural topology optimization. We conclude that the measured large volumetric NTE of ruthenate ceramics is certainly possible via anisotropic crystallographic thermal expansion through an elastic mechanism.

High temperature in-situ synchrotron-based XRD study on the crystal structure evolution of C/C composite impregnated by FLiNaK molten salt

An in-situ real-time synchrotron-based grazing incidence X-ray diffraction was systematically used to investigate the crystal structural evolution of carbon fiber reinforced carbon matrix (C/C) composite impregnated with FLiNaK molten salt during the heat-treatment process. It was found that the crystallographic thermal expansion and contraction rate of interlayer spacing d002 in C/C composite with FLiNaK salt impregnation is smaller than that in the virgin sample, indicating the suppression on interlayer spacing from FLiNaK salt impregnated. Meanwhile the crystallite size LC002 of C/C composite with FLiNaK salt impregnation is larger than the virgin one after whole heat treatment process, indicating that FLiNaK salt impregnation could facilitate the crystallization of C/C composite after heat treatment process. This improved crystallization in C/C composite with FLiNaK salt impregnation suggests the synthetic action of the salt squeeze effect on crooked carbon layer and the release of internal residual stress after heating-cooling process. Thus, the present study not only contribute to reveal the interaction mechanism between C/C composite and FLiNaK salt in high temperature environment, but also promote the design of safer and more reliable C/C composite materials for the next generation molten salt reactor.

On EXAFS Debye-Waller factor and recent advances

The effects of structural and vibrational disorder on the EXAFS signals are parameterized in terms of the Debye Waller (DW) factor. Here the vibrational contribution is addressed, which for most systems can be singled out by studying the temperature dependence of the EXAFS DW factor, which corresponds to a good accuracy to the parallel mean square relative displacement (MSRD) around the inter-atomic equilibrium distance. By comparing the first-shell EXAFS thermal expansion with the crystallographic thermal expansion one can evaluate the perpendicular MSRD. The results of recent measurements on copper and on several tetrahedral semiconductors are here critically compared and some properties of the MSRDs are discussed, such as the dependence of correlation, force constants and vibrational anisotropy on crystal structure and bond ionicity as well as the relative merits of the correlated Debye and Einstein models. The anharmonic contribution to the parallel MSRD of CdTe has been evaluated and a quasi-harmonic analysis has been attempted, leading to an estimation of the bond Grüneisen parameter.

Local thermal expansion in copper: Extended x-ray-absorption fine-structure measurements and path-integral Monte Carlo calculations

A combined approach has been used to study thermal effects on the extended x-ray absorption fine-structure (EXAFS) of copper between 4 and 500 K. A phenomenological data analysis shows that the thermal expansions measured from the first and third cumulants significantly differ between each other and from the crystallographic thermal expansion. Path-integral Monte Carlo calculations of EXAFS cumulants have been performed, using a many-body potential. The good reproduction of experimental values validates the phenomenological analysis and opens more perspectives for applications to more complex systems. It is shown that the reproduction of EXAFS parameters allows for a test of the interaction potentials with regard to anharmonicity.

Synthesis and thermal expansion hysteresis of Ca1-xSrxZr4P6O24

The low thermal expansion ceramic system, Ca1-xSr{x}Zr4P6O24, for the compositions with x = 0, 0.25, 0.50, 0.75 and 1 was synthesized by solid-state reaction. The sintering characteristics were ascertained by bulk density measurements. The fracture surface microstructure examined by scanning electron microscopy showed the average grain size of 2.47 μm for all the compositions. The thermal expansion data for these ceramic systems over the temperature range 25–800°C is reported. The sinterability of various solid solutions and the hysteresis in dilatometric behaviour are shown to be related to the crystallographic thermal expansion anisotropy. A steady increase in the amount of porosity and critical grain size with increase in x is suggested to explain the observed decrease in the hysteresis.

On the cumulant analysis of EXAFS in crystalline solids.

The analysis of temperature-dependent EXAFS spectra based on the cumulant expansion is critically reviewed, seeking for accurate relations between EXAFS parameters and physical properties of crystals. The treatment is based on the distinction between the real and effective distribution of distances, and is divided into three logical steps. (a) The connection between lattice dynamics and cumulants C(n)* of the real distribution is studied and the extent of the usual approximations are numerically evaluated. Atomic vibrations perpendicular to the bond direction make the EXAFS thermal expansion larger than the crystallographic one; the difference is connected to a shift of the effective pair potential rather than to its asymmetry. Peculiar information on lattice dynamics of crystals can be obtained from accurate EXAFS measurements and their cumulant analysis. (b) The differences between cumulants of the real and effective distribution (C(n)* and C(n), respectively) are calculated for various physically realistic distributions. The largest discrepancy concerns the first cumulant: C(1)* measures the thermal expansion of the interatomic bond, while C(1) is a better estimate of the crystallographic thermal expansion. (c) The convergence properties of the cumulant series are discussed and some phenomenological procedures are suggested to monitor and possibly work out the connected failures of the cumulant method. Benefits and risks of the use of an effective pair potential are at last debated.

Thermal Expansion of Ti5Si3 with Ge, B, C, N, or O Additions

The crystallographic thermal expansion coefficients of Ti5Si3 from 20 to 1000 °C as a function of B, C, N, O, or Ge content were measured by high-temperature x-ray diffraction using synchrotron sources at Cornell University (Cornell High Energy Synchrotron Source; CHESS) and Argonne National Laboratory (Advanced Photon Source; APS). Whereas the ratio of the thermal expansion coefficients along the c and a axes was approximately 3 for pure Ti5Si3, this ratio decreased to about 2 when B, C, or N atoms were added. Additions of O and Ge were less efficient at reducing this thermal expansion anisotropy. The extent by which the thermal expansion was changed when B, C, N, or O atoms were added to Ti5Si3 correlated with their expected effect on bonding in Ti5Si3.

Effect of Heat Treatment on Grain Size, Phase Assemblage, and Mechanical Properties of 3 mol% Y‐TZP

The effect of heat treatment on the grain size, phase assemblage, and mechanical properties of a 3 mol% Y‐TZP ceramic was investigated. Specimens were initially sintered for 2 h at 1450°C to near theoretical density; some specimens were then heat‐treated at 1550°, 1650°, 1750°, or 1850°C to coarsen the microstructure. The average grain size increased with heat treatment from <0.5 to ∼10 μ‐m. Phase analyses revealed predominantly tetragonal and cubic phases below 1750°C, with a significant decrease in tetragonal content and increase in monoclinic content for temperatures >1750°C. The maximum fraction of tetragonal phase that transformed during fracture corresponded with the largest tetragonal grain size of ∼5–6 μm. Strength was on the order of 1 GPa, and was surprisingly insensitive to heat‐treatment temperature and grain size, contrary to previous studies. The fracture toughness increased from 4 to 10 MPa.m1/2 with increasing grain size, owing to an increasing transformation zone size. Grain sizes larger than 5–6 μm spontaneously transformed to monoclinic phase during cooling. Such critical grain sizes are much larger than those found in past investigations, and may be due to the greater fraction of cubic phase present which decreases the strain energy arising from crystallographic thermal expansion anisotropy of the tetragonal phase.

An anisotropic Ehrenfest equation for orthorhombic shear strains in YBa2Cu3O7

A (110) shear transformation has previously been shown to account for the crystallographic thermal expansion coefficient's anomalies or discontinuities on both the a and b crystal axes. The mean-field jump in the thermal expansion coefficient of the normalized orthorhombic strain is shown here to be this same shear strain transformation. The anomaly at Tc, assuming piece-wise continuous orthorhombic strains as determined from x-ray and neutron diffraction data, agrees with single-crystal thermal expansion jumps measured exactly at Tc. The shear thermal expansion coefficient measured from the shear strains of (b-a)/(b+a), at temperatures well above Tc, is constant. Finally, Ehrenfest's equations have been integrated and an exponential form relating Tc to the applied shear stress and anomalies in the shear thermal expansion coefficient and heat capacity is presented. The Ehrenfest anomaly is shown here to include the inverse thermal expansion coefficient discontinuity with a characteristic temperature of approximately 9*10-5 K. The shear stress is compared to a structural contribution to the superconducting onset across the superconducting phase boundary of 23 GPa in YBa2Cu3O7; this value is compared to the estimated in-plane shear modulus of 58-98 GPa.

Tetragonal symmetry from superconductivity in orthorhombic YBa2Cu3O7

A shear transformation has previously been shown to account for the discontinuities or jumps in the crystallographic thermal expansion coefficients across the normal to superconducting transition temperature. The crystallographic stress dependence of the superconducting transition temperature is shown here to also support a shear transformation; d T c/d τ=4K/GPa. The shear motion of the lattice restricts the elastic compliances involved in superconductivity to tetragonal symmetry while considering all nine elastic interactions with the orthorhombic superconducting crystal. The shear transformation has isothermal elastic interactions with the lattice through the orthorhombic shear compliance changes of ( Δs 11 + Δs 22 − 2 Δs 12). The onset of superconductivity restricts to a higher symmetry the shear motion of the orthorhombic lattice with Δs 11 = Δs 22 and Δs 23 = Δs 13 across th e superconducting phase boundary. Crystallographic second order thermodynamics predicts that isothermally Δs 11 = - Δs 12 = 1.7 × 10 -15/ Pa. It is shown that the jumps in the elastic compliance matrix are unique functions of the thermal expansion matrix jumps and the heat capacity jumps for any second order phase transition.

A shear transformation in the a-b plane of YBa 2Cu 3O 7−δ from the superconducting onset

The discontinuities or jumps in the crystallographic thermal expansion coefficients across the superconducting to normal phase boundary have been reported to contain opposite signs and nearly equal magnitudes between the a and b crystal axes. A shear transformation is shown to account for almost all of the thermal expansion jumps in a coordinate system that is rotated by π 4 about the c axis. The heat capacity, the shear thermal expansion coefficient and the elastic compliance matrix jumps in this rotated coordinate are related using second order phase transformation thermodynamics for a solid. The lattice motion for the shear transformation involves a single coupled displacement. A shear stress that gives a pure shear strain displacement will change T c in this crystal. The thermal expansion jumps when restricted to contain no volume terms have been emphasized in this paper. This shear thermal expansion is found to interact with the lattice through the orthorhombic shear compliance of ( s 11+ s 22−2 s 12).

Crystallographic thermal expansion and elasticity across the superconducting transition in YBa2Cu3O7- delta.

A thermodynamic treatment for a second-order superconducting phase transformation has been developed which incorporates crystallographic effects. The temperature dependence of x-ray-measured lattice constants in polycrystalline $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ has been used to determine the orthorhombic, crystallographic thermal-expansion coefficient matrix. The principal thermal strains were curvefit above and below the superconducting transition temperature to deduce the thermal-expansion jumps at the phase boundary. The thermodynamic arguments show that the three elastic shear stiffness elements do not contribute to the superconducting specific-heat jump. Also, the three independent shear stresses at the superconducting transition point are zero. The specific-heat jump is, however, related to the three normal stresses at the phase boundary. The experimental thermal-expansion data suggest that a compressive stress aligned with the oxygen-deficient crystallographic \^a axis changes the transition temperature by 0.3 K/kbar; the $\stackrel{\mathrm{^}}{\mathrm{b}}$ axis stress is 0.07 K/kbar, while the \ifmmode \hat{c}\else \^{c}\fi{} is 0.1 K/kbar, and all stresses are in compression. The isothermal elastic compliance jumps can be predicted by assuming zero jumps in the adiabatic elastic compliances. The specific-heat jump is calculated from this assumption and crystallographic thermodynamics.