Anisotropic Thermal Expansion Coefficients Research Articles

Investigation of the anisotropic thermal expansion of PbIn6Te10 crystal by high temperature x-ray diffraction measurements

The PbIn6Te10 crystal, an IR laser frequency conversion material, grown via the Bridgman method with dimensions ϕ35 mm × 90 mm in a four-zone furnace, was subject to the investigation of its thermal expansion behavior using high-temperature x-ray diffraction in the range of 25–450 °C. Based on the obtained data, the average thermal expansion coefficients of 15.21 × 10−6 K−1 for α a and 6.44 × 10−6 K−1 for α c were determined utilizing the least square method. The study revealed that the linear and volume thermal expansion coefficients of PbIn6Te10 crystals satisfy the relationships α a > α c > 0, α V = 2α a + α c , and α a increased while α c decreased with increasing temperature, accentuating a substantial anisotropy in thermal expansion between the crystal’s principal axes. A detailed exploration pinpointed that the variations in the PbTe6 octahedron primarily governed the changes in the PbIn6Te10 unit cells, with further investigation uncovering its association with variations in the nearest neighboring bonds, which is mainly related to Pb-Te4 and Pb-Te2 bonds. Additionally, the determination of temperature-dependent anisotropic thermal expansion coefficient α was complemented by calculating Grüneisen parameter γ using Quasi-harmonic Debye model. Remarkably, these parameters also exhibited anisotropic behaviors ( γ⊥ increases with temperature whereas γ// decreases, α ⊥ > α ∥), contributing additional insights into the crystal thermal characteristics.

Anisotropy of microstructure, mechanical properties and thermal expansion in Invar 36 alloy fabricated via laser powder bed fusion

Invar 36 is a renowned iron-nickel alloy with an ultra-low coefficient of thermal expansion (CTE), which is widely used in aerospace and precision instruments. Laser powder bed fusion (LPBF) as a prevalent metal additive manufacturing technique could break through the limitations of traditional processes, such as low material utilization and difficulties in forming complex structures. In this work, Invar 36 alloy specimens were fabricated via LPBF process in 0° (H0 specimen), 45° (D45 specimen), and 90° (V90 specimen) build orientations. The microstructure, tensile properties, thermal expansion, and magnetic properties of the three orientations specimens were comprehensively investigated, and the anisotropy of microstructure, yield strength and CTE were analyzed. The V90 specimen exhibited the highest elongation (62.70%), while the D45 specimen owned the highest yield strength (496 MPa). The anisotropic mechanical properties can be attributed to variations in dislocation density, grain size, and intrinsic yield strength. The CTE of LPBF-fabricated Invar 36 alloy specimens were all lower than that of traditional manufacturing process at all temperature ranges. Since CTE is more related to magnetic properties, not sensitive to microstructure, the anisotropy of CTE is not significant in three build orientations. This study has significant implications to produce Invar 36 alloy devices with exceptional properties and 3D dimensional stability via the LPBF process.

A series of auxetic metamaterials with negative thermal expansion based on L-shaped microstructures

Mechanical metamaterials are facing various multi-functional requirements. In this study, a series of auxetic metamaterials with enhanced negative thermal expansion and improved stiffness are proposed. These metamaterials consist of four novel L-shaped microstructures connected in various clockwise or counterclockwise directions. Two length ratios and four angle parameters are utilized to define the geometry of these metamaterials. Both analytical analysis and validated numerical homogenization methods are employed to determine their effective thermoelastic properties, including effective Young's modulus, Poisson's ratio, and coefficient of thermal expansion. It is demonstrated that tailoring these microstructural geometries can modulate the effective thermoelastic properties of these metamaterials accordingly. By appropriately tailoring the geometric parameters, these metamaterials exhibit increased effective stiffness as well as more pronounced auxeticity and negative thermal expansion compared to traditional triangle and trapezoid microstructures. Notably, the in-plane effective anisotropic or isotropic Poisson's ratio and coefficient of thermal expansion can be adjustable over a broader range from negative to positive values. High stiffness, High strength, and near-zero thermal expansion can be achieved concurrently in these metamaterials, which is very conducive for structural designing in thermal environments. These metamaterials with a coupled design of stiffness, thermal expansion, and Poisson's ratio hold significant potential for advancing applications in satellite platforms, space structures, composite sandwiches, and precision equipment.

Anisotropic thermal expansion based on a novel metamaterial

Metamaterials exhibiting exceptional thermoelastic properties are emerging as promising candidates for scientific and engineering applications. Previous studies on mechanical metamaterials with negative thermal expansion (NTE) have primarily focused on enhancing isotropic NTE, often resulting in a high coefficient of thermal expansion (CTE) but low stiffness that characterizes bending-dominated NTE metamaterials. In this study, we proposed a novel mechanical metamaterial with significant anisotropic thermal expansion behavior and substantial stiffness based on the cross structure, re-entrant structure, and triangular structure with NTE. The programmable CTE and effective elastic modulus were analytically derived using the unit load method and stiffness matrix method, respectively, which were validated by numerical results obtained from finite element analysis (FEA). Drawing inspiration from the body-centered cubic and face-centered cubic strategies, two types of extended three-dimensional metamaterials can be obtained. The results demonstrate that a larger length ratio λ and angle θ3 can enhance the NTE effect, while reducing the Young's modulus. The maximum magnitudes of NTE and positive thermal expansion (PTE) are achieved at approximately θ2 = 90°, resulting in a decreased stiffness and slight negative Poisson's ratio (NPR). By adjusting the angles θ2 of re-entrant structures, six distinct anisotropic thermal expansion behaviors can be obtained in this planar metamaterial. Furthermore, a wider range of isotropic CTE as well as an extended range of anisotropic CTE can be achieved in the present metamaterial compared to its original metamaterials. The proposed metamaterial can be combined with traditional triangle metamaterials to enhance thermoelastic properties. The analysis also provides valuable insights into designing metamaterials with large CTE, optimal stiffness, and lightweight properties simultaneously for engineering applications.

K2 PbX][Ga7 S12 ] (X=Cl, Br, I): The First Lead-Containing Cationic Moieties with Ultrahigh Second-Harmonic Generation and Band Gaps Exceeding the Criterion of 2.33eV.

In contrast to anionic group theoryofnonlinearoptical(NLO)materials that second-harmonic generation (SHG) responses mainly originate from anionic groups, structural regulation on the cationic groups of salt-inclusionchalcogenides(SICs)is performed to make them also contribute to the NLOeffects. Herein, the stereochemically active lone-electron-pair Pb2+ cation is first introduced to the cationic groups of NLO SICs, and the resultant [K2 PbX][Ga7 S12 ] (X=Cl, Br, I) are isolated via solid-state method. The features of their three-dimensional structures comprise highly oriented [Ga7 S12 ]3- and [K2 PbX]3+ frameworks derived from AgGaS2 , which display the largest phase-matching SHG intensities (2.5-2.7×AgGaS2 @1800nm) among all SICs. Concurrently, three compounds manifest band gap values of 2.54, 2.49, and 2.41eV(exceedingthe criterion of 2.33eV),whichcan avoid two-photon absorption under the fundamental laser of 1064nm, along with the relatively low anisotropy of thermal expansion coefficients, leading to improved laser-induceddamagethresholds(LIDTs) values of 2.3, 3.8, and 4.0 times that of AgGaS2 . In addition, the density of states and SHG coefficient calculations demonstrate that the Pb2+ cations narrow the band gaps and benefit SHG responses.

Ferroelastic tetragonal-monoclinic phase transition and anisotropic thermal expansion of LuNbO4 ceramics

Rare-earth tantalates (RETaO4) and niobates (RENbO4) are widely studied as promising thermal barrier coatings based on the ferroelastic tetragonal-monoclinic (t-m) phase transition. We studied the t-m transition process, anisotropic thermal expansion coefficients (TECs), and the corresponding spontaneous strain of LuNbO4 by powder high-temperature X-ray diffraction. The results proved that the unit cell volume and TECs continuously increased with increasing temperature, and the dominant mechanisms of thermal expansion were proposed. Two different orientation states (S1 and S2) caused by the t-m transition led to formation of the ferroelastic domains, and the corresponding spontaneous strain was determined. The relationship between the t-m transition and spontaneous strain was proposed for further property tailoring. This work provided an in-depth analysis of the ferroelastic t-m phase transition, and it would advance the applications of ferroelastic ceramics.

In-situ synchrotron x-ray diffraction and thermal expansion of TiB2 up to ∼3050 °C

There is an increasing interest in understanding the performance and properties of ultra-high temperature ceramics due to their high melting points (<3000 °C) that make them promising for extreme environment applications. In-situ high temperature X-ray diffraction experiments were performed on TiB2 beads up to ∼3050 °C. For these experiments, TiB2 powders were fabricated into spherical beads via gel casting methods and densified in a high temperature graphite furnace. These sample beads were then levitated in a conical nozzle levitator with reducing atmosphere (3% H2 -Ar) while being heated using a 400 W CO2 laser. During levitation a collimated synchrotron X-ray source was used to perform in-situ, temperature-dependent structural characterizations. The anisotropic coefficients of thermal expansion of TiB2 were characterized as a function of temperature up to ∼3050 °C. Elucidation of these properties are critical for the advancement of TiB2 ceramics and other transition metal di-borides for use in high temperature applications such as hypersonic platforms, nuclear reactors, and atmospheric re-entry.

Frequency-domain probe beam deflection method for measurement of thermal conductivity of materials on micron length scale.

Time-domain thermoreflectance and frequency-domain thermoreflectance (FDTR) have been widely used for non-contact measurement of anisotropic thermal conductivity of materials with high spatial resolution. However, the requirement of a high thermoreflectance coefficient restricts the choice of metal coating and laser wavelength. The accuracy of the measurement is often limited by the high sensitivity to the radii of the laser beams. We describe an alternative frequency-domain pump-probe technique based on probe beam deflection. The beam deflection is primarily caused by thermoelastic deformation of the sample surface, with a magnitude determined by the thermal expansion coefficient of the bulk material to measure. We derive an analytical solution to the coupled elasticity and heat diffusion equations for periodic heating of a multilayer sample with anisotropic elastic constants, thermal conductivity, and thermal expansion coefficients. In most cases, a simplified model can reliably describe the frequency dependence of the beam deflection signal without knowledge of the elastic constants and thermal expansion coefficients of the material. The magnitude of the probe beam deflection signal is larger than the maximum magnitude achievable by thermoreflectance detection of surface temperatures if the thermal expansion coefficient is greater than 5 × 10-6K-1. The uncertainty propagated from laser beam radii is smaller than that in FDTR when using a large beam offset. We find a nearly perfect matching of the measured signal and model prediction, and measure thermal conductivities within 6% of accepted values for materials spanning the range of polymers to gold, 0.1-300 W/(m K).

Effect of Sn Grain Orientation on Reliability Issues of Sn-Rich Solder Joints

Sn-rich solder joints in three-dimensional integrated circuits and their reliability issues, such as the electromigration (EM), thermomigration (TM), and thermomechanical fatigue (TMF), have drawn attention related to their use in electronic packaging. The Sn grain orientation is recognized as playing an important role in reliability issues due to its anisotropic diffusivity, mechanical properties, and coefficient of thermal expansion. This study reviews the effects of the Sn grain orientation on the EM, TM, and TMF in Sn-rich solder joints. The findings indicate that in spite of the failure modes dominated by the Sn grain orientation, the size and shape of the solder joint, as well as the Sn microstructures, such as the cycling twining boundary (CTB), single crystals, and misorientations of the Sn grain boundary, should be considered in more detail. In addition, we show that two methods, involving a strong magnetic field and seed crystal layers, can control the Sn grain orientations during the solidification of Sn-rich solder joints.

Correlation between d-orbital bandwidth and local coordination environment in RE2SiO5 compounds with implications in minimizing the coefficient of thermal expansion anisotropy (RE = Sc, Y, La)

Y2SiO5 is one of the promising environmental barrier coating (EBC) materials that protect the gas turbine engine components from unfavorable reactions at higher temperatures. The Y2SiO5 compound forms in the monoclinic crystal structure (C2/c space group), and one of the drawbacks is its appreciable coefficient of thermal expansion (CTE) anisotropy, which adversely affects its lifetime as the EBC material. The objective of this work is to uncover previously unknown correlation between the electronic structure and crystal structure of RE2SiO5 compounds in the equilibrium and hypothetical C2/c structures (where RE = Sc, Y, or La). Our density functional theory calculations reveal a trend in the RE-cation d-orbital bandwidth as a function of the RE electronic configuration, local RE–O coordination environment, and unit cell volume. We predict that the Y-4d orbital bandwidth can become narrower when Y2SiO5 forms in an open structure with a reduced Y–O effective coordination number. We conjecture that a narrow Y-4d orbital bandwidth may give rise to smaller CTE anisotropy compared to Y2SiO5 in its equilibrium structure. The outcome of this work has potential implications in the rational design of Y2SiO5-based EBCs for use under extreme temperature environments.

Asymmetrical Cross-Sectional Buckling in Arc-Prepared Multiwall Carbon Nanotubes Revealed by Iodine Filling

We report the intercalation of iodine chains in highly crystalline arc-discharge multiwalled carbon nanotubes (MWCNTs), not in the central cavity but instead between the concentric graphene shells. High-resolution transmission electron microscopy demonstrated that the intercalation was asymmetric with respect to the longitudinal axis of the nanotubes. This filling is explained through the existence of asymmetric intershell channels which formed as the tubes shrank upon cooling after growth. Shrinkage occurred because the geometrically constrained equilibrium intershell spacing was higher at growth than room temperature, due to the highly anisotropic coefficient of thermal expansion of graphite (or graphene stacks). Computational modelling supported the formation of such cavities and explained why they all formed on the same side of the tubes. The graphene shells were forced to bend outward, thereby opening aligned intergraphene nanocavities, and subsequently allowing the intercalation with iodine once the tube ends were opened by oxidative treatment. These observations are specific to catalyst-free processes because catalytic processes use too low temperatures, but they are generally applicable in geometrically closed carbon structures grown at high temperatures and so should be present in all arc-grown MWCNTs. They are likely to explain multiple observations in the literature of asymmetric interlayer spacings in multiple-shell graphenic carbon structures.

Prediction of the spring-in of cylindrically orthotropic media and cross-ply laminates

The equation for prediction of the spring-in angle of a cylindrically orthotropic segment is shown to be independent of all material properties except for the anisotropic coefficients of thermal expansion and a stress-free state is insured for the corresponding unconstrained deformation. In contrast, the complete cylindrical geometry is shown to provide constraint to thermal deformation and thereby induce thermal residual stresses in the form of a moment. The method of superposition is demonstrated whereby traction-free conditions yield stress-free cylindrical elements with corresponding angular displacements at the element free boundaries. The first derivation of the spring-in equation is attributed to Radford, in contrast to the widely accepted view that the equation was first developed by Spencer et al. Finite-element methods, combined with the superposition approach, further validate the accuracy of the Radford equation for cylindrically orthotropic segments and explore its limitations for multiaxial composite laminates.

Measurement of thermal expansion coefficient of rock minerals using XRD and its implications to thermal damage mechanism

The measurement of thermal expansion coefficient (TEC) of rocks is of vital importance to various geotechnical engineering applications, such as high-level radioactive waste disposal, enhanced geothermal energy exploitation, ultra-deep borehole stability, liquid nitrogen storage, etc. However, previous researchers mainly focused on the thermal expansion coefficient of various rocks, instead of the constituent minerals. In this manuscript, we measured the both temperature-dependent volume and linear thermal expansion coefficients of two rock-forming minerals (quartz and albite) using both high-temperature X-ray diffraction (XRD) as well as low-temperature XRD. It is found that there exist strong anisotropic linear thermal expansion coefficients of the two minerals, which can provide strong reference for the relevant engineering applications and numerical simulations.

New method to calculate the anisotropies of polarizability and thermal expansion of uniaxial liquid crystals.

The anisotropy of molecular polarizability in liquid crystals is linked to the birefringence in these substances. The classic methods to compute the polarizabilities of liquid crystals assume an average number density of molecules that is equal in all directions. In the present work, a new model is proposed for the anisotropic molar polarization based on a virtual anisotropy of the number density of molecules in the liquid-crystalline material. This new strategy hence allows for the computation of both the anisotropic polarizabilities and the anisotropic thermal-expansion coefficients of liquid crystals. The model is applied to the liquid crystals 4-n-pentyl-4'-cyanobiphenyl and N-(4-methoxybenzylidene)-4-butylaniline, yielding polarizabilities similar to those reported for these materials. For these nematic liquid crystals, the results imply the existence of a positive thermal-expansion coefficient in the direction perpendicular to the director vector throughout the entire nematic temperature range and a negative thermal-expansion coefficient parallel to the director vector near the temperature of the nematic-isotropic transition. At the isotropization temperature, there exists divergent and critical behavior of the anisotropic thermal-expansion coefficients, consistent with the typical discontinuity of volume in first-order transitions.

Thermomechanical and thermochemical stability of HfSiO4 for environmental barrier coating applications

AbstractThermomechanical and thermochemical stability of hafnon (HfSiO4) was evaluated for application as an environmental barrier coating (EBC) candidate for SiC/SiC CMCs. High‐temperature X‐ray diffraction (XRD) analysis showed that hafnon has an excellent coefficient of thermal expansion (CTE) match with SiC as well as minimal CTE anisotropy, which supports hafnon as an EBC candidate. Alternatively, high‐velocity water vapor testing at 1,200‐1,400°C showed large amounts of silica depletion from chemical reaction at all velocities and excessive material erosion at steam velocities greater than 190 m/s. HfSiO4 is therefore not suitable as an EBC for turbine applications due to insufficient thermochemical stability in water vapor‐containing combustion environments.

Grain-resolved temperature-dependent anisotropy in hexagonal Ti-7Al revealed by synchrotron X-ray diffraction

Hexagonal metals have anisotropic coefficients of thermal expansion causing grain-level internal stresses during heating. High energy x-ray diffraction microscopy, a non-destructive, in situ, micromechanical and microstructural characterization technique, has been used to determine the anisotropic coefficients of thermal expansion (CTEs) for Ti-7Al. Two samples of polycrystalline α-phase Ti-7Al were continuously heated from room temperature to 850 °C while far-field HEDM scans were collected. The results showed a change in the ratio of the CTEs in the ‘a’ and ‘c’ directions which explains discrepancies found in the literature. The CTE additionally appears to be affected by the dissolution of α2 precipitates. Analysis of the grain-resolved micromechanical data also shows reconfiguration of the grain scale stresses likely due to anisotropic expansion driving crystallographic slip.

Topology and Shape Optimization of 2-D and 3-D Micro-Architectured Thermoelastic Metamaterials Using a Parametric Level Set Method

2-D and 3-D micro-architectured multiphase thermoelastic metamaterials are designed and analyzed using a parametric level set method for topology optimization and the finite element method. An asymptotic homogenization approach is employed to obtain the effective thermoelastic properties of the multiphase metamaterials. The -constraint multi-objective optimization method is adopted in the formulation. The coefficient of thermal expansion (CTE) and Poisson’s ratio (PR) are chosen as two objective functions, with the CTE optimized and the PR treated as a constraint. The optimization problems are solved by using the method of moving asymptotes. Effective isotropic and anisotropic CTEs and stiffness constants are obtained for the topologically optimized metamaterials with prescribed values of PR under the constraints of specified effective bulk modulus, volume fractions and material symmetry. Two solid materials along with one additional void phase are involved in each of the 2-D and 3-D optimal design examples. The numerical results reveal that the newly proposed approach can integrate shape and topology optimizations and lead to optimal microstructures with distinct topological boundaries. The current method can topologically optimize metamaterials with a positive, negative or zero CTE and a positive, negative or zero Poisson’s ratio.

First-principles study of the anisotropic thermal expansion and thermal transport properties in h-BN

The thermal expansion coefficient (TEC) and thermal conductivity (k) of thermal fillers are key factors for designing thermal management and thermal protection composite materials. Due to its unique advantages, hexagonal boron nitride (h-BN) is one of the most commonly used thermal fillers. However, its TEC and k values are still unclear due to the inconsistency of characterization techniques and sample preparations. In this work, these disputes were addressed using the quasi-harmonic approximation (QHA) method and phonon Boltzmann transport equation (BTE) theory based on the density functional theory (DFT), respectively. The accuracy of our calculated TEC and k values was confirmed by previously reported experimental results, and the underlying physical principles were analyzed from the phonon behaviors. Our TEC results show that the h-BN has small in-plane negative value and large cross-plane positive value, which are −2.4A—10−6 and 36.4A—10−6 K−1 at 300 K, respectively. And the anisotropic TEC is mainly determined by the anisotropic isothermal bulk modulus and the low-frequency out-of-plane longitudinal phonon modes. We found that the convergence of cutoff radius and q-grid size have significant effect on the accuracy of k of h-BN. Our results show that the in-plane k is much higher than the cross-plane k, and the values at 300 K are 286.6 and 2.7 W m−1 K−1, respectively. The anisotropic phonon group velocity arising from the vibration behaviors of acoustic phonon modes should be primarily responsible for the anisotropic k. Our calculated TEC and k values will provide important references for the design of h-BN composite materials.

Evolution of microstructure and thermal conductivity of multifunctional environmental barrier coating systems

Environmental barrier coating (EBC) systems are applied to the surface of silicon-based composites exposed to high temperature combustion gas flow paths in gas turbine engines. They reduce the rate of composite oxidation, its volatilization by reactions with water vapor, and the temperature of the composite as their thermal conductivity decreases. Current EBC systems consist of a silicon bond coat covered by an ytterbium disilicate (Yb2O3⋅2SiO2: YbDS) permeation resistant and low silica activity barrier. When applied by atmospheric-plasma spray deposition, this outer layer contains 10–15% of a secondary ytterbium monosilicate (Yb2O3⋅3SiO2: YbMS) phase. YbDS has a coefficient of thermal expansion (CTE) of 4–5×10−6 °C−1, similar to those of the silicon and the silicon-based composite, but a relatively high thermal conductivity of 5–7 W m−1 K−1. YbMS has a higher steam volatility resistance than YbDS, and it has a much lower thermal conductivity (∼2–2.5 W m−1 K−1) at ambient temperature compared to YbDS, but its higher, highly anisotropic CTE (3–11×10−6 °C−1) results in channel cracking which reduces environmental protection. Here, we use a combination of scanning electron beam and laser-based thermoreflectance methods to spatially map the distribution of the silicate phases and thermal conductivity at ambient temperature in a Si-ytterbium disilicate EBC system exposed to thermal cycling in water vapor. We show that during thermal cycling, diffusion of silica from the thermally grown oxide on the Si bond coat surface to nearby YbMS regions transforms this phase to YbDS, thereby reducing the risk of thermomechanical coating failure but decreasing its effective thermal resistance. We also show that as silica is volatilized at the water vapor–ytterbium silicate interface, YbDS is transformed to YbMS, restoring some of the thermal protection of the coating system lost by its reduction in thickness and the YbMS to YbDS transformation near the bond coat.

The crystal structure of TlMgCl3 from 290 K to 725 K.

The title compound, thallium magnesium trichloride, has been identified as a scintillator with both moderate gamma-stopping power and moderate light yield. Knowledge of its crystal structure is needed for further development. This work determines the crystal structure of TlMgCl3 to be hexa-gonal P63/mmc (No. 194) and isostructural with RbMgCl3, contrary to previously reported data. This structure was obtained by single-crystal X-ray diffraction and was further confirmed by neutron diffraction measurements. Extending neutron diffraction measurements to high temperature, the data show that TlMgCl3 maintains this crystal structure from 290 K up through 725 K, approaching the melting point of 770 K. Anisotropic thermal expansion coefficients increase over this temperature range, from 31 to 38 × 10-6 K-1 along the a axis and from 19 to 34 × 10-6 K-1 along the c axis.