Thermal Expansion Coefficients of Thin Crystal Films

The formulas for atomic displacements and Hamiltonian of a thin crystal film in phonon occupation number representation are obtained with the aid of Green's function theory. On the basis of these results, the formulas for thermal expansion coefficients of the thin crystal film are derived with the perturbation theory, and the numerical calculations are carried out. The results show that the thinner films have larger thermal expansion coefficients.

The Approach of Compensation of Air Refractive Index in Thermal Expansion Coefficients Measurement Based on Laser Feedback Interferometry

We present the thermal expansion coefficient （TEC） measurement technology of compensating for the effect of variations in the refractive index based on a Nd： YA G laser feedback system, the beam frequency is shifted by a pair of aeousto-optic modulators and then the heterodyne phase measurement technique is used. The sample measured is placed in a muffle furnace with two coaxial holes opened on the opposite furnace walls. The measurement beams hit perpendicularly and coaxially on each surface of the sample. The reference beams hit on the reference mirror and the high-refiectivity mirror, respectively. By the heterodyne configuration and computing, the influences of the vibration, distortion of the sample supporter and the effect of variations in the refractive index are measured and largely minimized. For validation, the TECs of aluminum samples are determined in the temperature range of 29-748K, confirming not only the precision within 5 × 10-7 K-1 and the accuracy within 0.4% from 298K to 448K but also the high sensitivity non-contact measurement of the lower reflectivity surface induced by the sample oxidization from 448 K to 748 K.

Effect of colouring green stage zirconia on the adhesion of veneering ceramics with different thermal expansion coefficients

This study evaluated the adhesion of zirconia core ceramics with their corresponding veneering ceramics, having different thermal expansion coefficients （TECs）, when zirconia ceramics were coloured at green stage. Zirconia blocks （N=240; 6 mm x 7 mm x 7 mm） were manufactured from two materials namely, ICE Zirconia （Group 1） and Prettau Zirconia （Group 2）. In their green stage, they were randomly divided into two groups. Half of the specimens were coloured with colouring liquid （shade A2）, Three different veneering ceramics with different TEC （ICE Ceramic, GC Initial Zr and IPS e.max Ceram） were fired on both coloured and non-coloured zirconia cores. Specimens of high noble alloys （Esteticor Plus） veneered with ceramic （VM 13） （n= 16） acted as the control group. Core-veneer interface of the specimens were subjected to shear force in the Universal Testing Machine （0.5 mm-min-1）. Neither the zirconia core material （P=0.318） nor colouring （P=0.188） significantly affected the results （three-way analysis of variance, Tukey＇s test）. But the results were significantly affected by the veneering ceramic （P=0.000）. Control group exhibited significantly higher mean bond strength values （45.7__.8） MPa than all other tested groups （（27.1__.4.1）-（39.7__.4.7） and （27.4__.5.6）-（35.9___4.7） MPa with and without colouring, respectively） （P^0.001）. While in zirconia-veneer test groups, predominantly mixed type of failures were observed with the veneering ceramic covering ~ 1/3 of the substrate surface, in the metal-ceramic group, veneering ceramic was left adhered 1/3 of the metal surface. Colouring zirconia did not impair adhesion of veneering ceramic, but veneering ceramic had a significant influence on the core-veneer adhesion. Metal-ceramic adhesion was more reliable than all zirconia-veneer ceramics tested.

Measurement Method of Compressibility and Thermal Expansion Coefficients for Density Standard Liquid at 2329 kg/m^3 based on Hydrostatic Suspension Principle

The accurate measurement on the compressibility and thermal expansion coefficients of density standard liquid at 2329kg/m3（DSL-2329） plays an important role in the quality control for silicon single crystal manufacturing. A new method is developed based on hydrostatic suspension principle in order to determine the two coefficients with high measurement accuracy. Two silicon single crystal samples with known density are immersed into a sealed vessel full of DSL-2329. The density of liquid is adjusted with varying liquid temperature and static pressure, so that the hydrostatic suspension of two silicon single crystal samples is achieved. The compression and thermal expansion coefficients are then calculated by using the data of temperature and static pressure at the suspension state. One silicon single crystal sample can be suspended at different state, as long as the liquid temperature and static pressure function linearly according to a certain mathematical relationship. A hydrostatic suspension experimental system is devised with the maximal temperature control error ±50 μK; Silicon single crystal samples can be suspended by adapting the pressure following the PID method. By using the method based on hydrostatic suspension principle, the two key coefficients can be measured at the same time, and measurement precision can be improved due to avoiding the influence of liquid surface tension. This method was further validated experimentally, where the mixture of 1, 2, 3-tribromopropane and 1,2-dibromoethane is used as DSL-2329. The compressibility and thermal expansion coefficients were measured, as 8.5′10–4 K–1 and 5.4′10–10 Pa–1, respectively.

Influence of average radii of RE3+ ions on phase structures and thermal expansion coefficients of high-entropy pyrosilicates

High-entropy pyrosilicate element selection is relatively blind, and the thermal expansion coefficient (CTE) of traditional β-type pyrosilicate is not adjustable, making it difficult to meet the requirements of various types of ceramic matrix composites (CMCs). The following study aimed to develop a universal rule for high-entropy pyrosilicate element selection and to achieve directional control of the thermal expansion coefficient of high-entropy pyrosilicate. The current study investigates a high-entropy design method for obtaining pyrosilicates with stable β-phase and γ-phase by introducing various rare-earth (RE) cations. The solid-phase method was used to create 12 different types of high-entropy pyrosilicates with 4–6 components. The high-entropy pyrosilicates gradually transformed from β-phase to γ-phase with an increase in the average radius of RE^(3+) ions ( r¯(RE^(3+))). The nine pyrosilicates with a small r¯(RE^(3+)) preserve β-phase or γ-phase stability at room temperature to the maximum of 1400 ℃. The intrinsic relationship between the thermal expansion coefficient, phase structure, and RE–O bond length has also been found. This study provides the theoretical background for designing high-entropy pyrosilicates from the perspective of r¯(RE^(3+)). The theoretical guidance makes it easier to synthesize high-entropy pyrosilicates with stable β-phase or γ-phase for the use in environmental barrier coatings (EBCs). The thermal expansion coefficient of γ-type high-entropy pyrosilicate can be altered through component design to match various types of CMCs.

ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS

Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.

Fourier-Mellin expansion coefficients of scaled pupils

Orthogonal polynomials over the interior of a unit circle are widely used in aberration theory and in describing ocular wavefront in ophthalmic applications. In optics, Zernike polynomials （ZPs） are commonly applied for the same purpose, and scaling their expansion coefficients to arbitrary aperture sizes is a useful technique to analyze systems with different pupil sizes. By employing the orthogonal Fourier-Mellin polynomials and their properties, a new formula is established based on the same techniques used to develop the scaled pupil sizes. The description by the orthogonal Fourier-Mellin polynomials for the aberration functions is better than that by the ZPs in terms of the wavefront reconstruction errors.

Determination of thermal expansion coefficients for unidirectional fiber-reinforced composites

In the present work, the coefficients of thermal expansion（CTEs） of unidirectional（UD）fiber-reinforced composites are studied. First, an attempt is made to propose a model to predict both longitudinal and transverse CTEs of UD composites by means of thermo-elastic mechanics analysis. The proposed model is supposed to be a concentric cylinder with a transversely isotropic fiber embedded in an isotropic matrix, and it is subjected to a uniform temperature change. Then a concise and explicit formula is offered for each CTE. Finally, some finite element（FE） models are created by a finite element program MSC. Patran according to different material systems and fiber volume fractions. In addition, the available experimental data and results of other analytical solutions of CTEs are presented. Comparisons are made among the results of the cylinder model,the finite element method（FEM）, experiments, and other solutions, which show that the predicted CTEs by the new model are in good agreement with the experimental data. In particular, transverse CTEs generally offer better agreements than those predicted by most of other solutions.

Microstructure evolution and thermal expansion of Cu-Zn alloy after high pressure heat treatment

The thermal expansion coefficients of Cu-Zn alloy before and after high pressure treatment were measured by thermal expansion instrument in the temperature range of 25？700 ℃,and the microstructure and phase transformation of the alloy were examined by optical microscope,X-ray diffractometer（XRD） and differential scanning calorimeter（DSC）.Based on the experimental results,the effects of high pressure treatment on the microstructure and thermal expansion of Cu-Zn alloy were investigated.The results show that the high pressure treatment can refine the grain and increase the thermal expansion coefficient of the Cu-Zn alloy,resulting in that the thermal expansion coefficient exhibits a high peak value on the α-T curve,and the peak value decreases with increasing the pressure.

Mixed Alkali-zinc Effects on Thermo-mechanical Properties in Borosilicate Glasses

A series of mixed alkali-zinc borosilicate glasses with various r values(r=molar ratio of[ZnO]/([R^(2)O]+[ZnO]))from 0.00 to 1.00 were fabricated to probe the mixed alkali-zinc effects on thermo-mechanical properties.The nonlinear evolution of glass transition temperature(T_(g))with the addition of ZnO is ascribed to the competition of two converse factors,i e,the T_(g)depression as one of the colligative properties for a solution,on the one hand,and the enhancement of T_(g)due to the higher field strength of zinc cations compared to that of alkali ions.However,the nonlinear evolution of elastic moduli and coefficients of thermal expansion with r is attributed to the variance of intermediate-range clusters,which is confirmed by infrared and Raman scattering spectra.These findings are very helpful in tailoring the performance of borosilicate glasses.

Interface and thermal expansion of carbon fiber reinforced aluminum matrix composites

Two kinds of unidirectional PAN M40 carbon fiber （55%, volume fraction） reinforced 6061Al and 5A06Al composites were fabricated by the squeeze-casting technology and their interface structure and thermal expansion properties were investigated. Results showed that the combination between aluminum alloy and fibers was well in two composites and interface reaction in M40/5A06Al composite was weaker than that in M40/6061Al composite. Coefficients of thermal expansion （CTE） of M40/Al composites varied approximately from （1.45-2.68）×10^-6 K^-1 to （0.35-1.44）×10^-6 K^-1 between 20℃ and 450℃, and decreased slowly with the increase of temperature. In addition, the CTE of M40/6061Al composite was lower than that of M40/SA06Al composite. It was observed that fibers were protruded significantly from the matrix after thermal expansion, which demonstrated the existence of interface sliding between fiber and matrix during the thermal expansion. It was believed that weak interracial reaction resulted in a higher CTE. It was found that the experimental CTEs were closer to the predicted values by Schapery model.

PREPARATION OF HIGH THERMAL EXPANSION COEFFICIENT PORCELAINS FUSED TO METALS

Usually the thermal expansion coefficients (TEC) of metals are higher than that of porcelains. In order to match the TECs in the case of coating porcelains on metals, high TEC porcelains are needed. In this research, the high TEC phase leucite (KAlSi2 O6) in the high TEC porcelain was prepared by sol-gel method. The crystal size of leucite made by sol-gel is about 77nm through controlling the process parameters. The process from xerogel to leucite was investigated by means of DSC (differential scanning calorimetry), TG (thermogravimetry), XRD ( X-ray diffraction) and IR (infrared absorption spectrum). Leucite had been detected after the gel was treated at 900℃, this formation temperature is about 250℃ lower than that of melting method. The porcelain made from 50% of the leucite powder and 50% of low fused temperature frit has an average TEC of 19.2×10-6/℃ C from room temperature to 450℃, which is much higher than the common porcelains.

DEVELOPMENT OF INCONEL~ ALLOY 783，A LOW THERMAL EXPANSION，CRACK GROWTH RESISTANT SUPERALLOY

Low thermal expansion superalloys have been used for a number of years in a variety of applications, including gas turbine engines. The low thermal expansion characteristics of the most widely used class of materials are derived from the ferromagnetic characteristics of Ni, Fe, and Co-based austenitic matrices containing little or no Cr.Alloy developments have been aimed at improving the oxidation resistance and stress accelerated grain boundary oxygen (SAGBO) attack.INCONEL alloy 783 is an oxidation resistant, low coefficient of thermal expansion superalloy developed for gas turbine applications. Alloy 783 represents a culmination in the development, of an alloy system with very high alumtnum content that, in addition to forming γ′,causes βaluminide phase precipitation in the austenitic matrix.This type of structure can be processed to resist both SAGBO and general oxidation,while providing low thermal expansion and useful mechanical properties up to 700℃.Key aspects of the alloy's development are presented.

A novel implementation algorithm of asymptotic homogenization for predicting the effective coefficient of thermal expansion of periodic composite materials

Asymptotic homogenization (AH) is a general method for predicting the effective coefficient of thermal expansion (CTE) of periodic composites. It has a rigorous mathematical foundation and can give an accurate solution if the macrostructure is large enough to comprise an infinite number of unit cells. In this paper, a novel implementation algorithm of asymptotic homogenization (NIAH) is developed to calculate the effective CTE of periodic composite materials. Compared with the previous implementation of AH, there are two obvious advantages. One is its implementation as simple as representative volume element (RVE). The new algorithm can be executed easily using commercial finite element analysis (FEA) software as a black box. The detailed process of the new implementation of AH has been provided. The other is that NIAH can simultaneously use more than one element type to discretize a unit cell, which can save much computational cost in predicting the CTE of a complex structure. Several examples are carried out to demonstrate the effectiveness of the new implementation. This work is expected to greatly promote the widespread use of AH in predicting the CTE of periodic composite materials.

Synthesis and thermal expansion of 4J36/ZrW_2O_8 composites

Gas atomized 4J36 alloy powder was milled for 72 h then mixed with ZrW2O8 powder and sintered at 600℃ for 4 h under argon atmosphere. 4J36/ZrW2O8 composites containing 10 vol.%, 20 vol.%, 30 vol.%, and 40 vol.% ZrW2Os were fabricated, the relative density of which ranged from 70% to 80%. Thermal expansion coefficients of the composites decreased as the amount of ZrW2O8 increased, in agreement with the rule of the mixture. The coefficient of thermal expansion of the 4J36/40 vol.%ZrW2O8 composite in 25-100℃ is 0.55 × 10^＋6/℃.

Effects of Chemical Treatments on Thermal Expansion Properties of Cordierite Ceramics

Cordierite honeycomb ceramics was treated with 1.5 M HNO3 , followed with 1.5 M NaOH at 93 ℃. The combination of acid treatment with alkali treatment significantly diminished the rebounding of coefficient of thermal expansion （CTE） caused by heat treatment, a phenomenon observed in samples treated solely with acid. Inductively coupled plasma （ICP） analysis results reveal that the alkali treatment preferentially dissolved ＂free＂ SiO2 left in the acid-treated samples, which is considered to be a key factor responsible for the CTE rebounding.

Numerical investigation of multi-beam laser heterodyne measurement with ultra-precision for linear expansion coefficient of metal based on oscillating mirror modulation

This paper proposes a novel method of multi-beam laser heterodyne measurement for metal linear expansion coefficient. Based on the Doppler effect and heterodyne technology, the information is loaded of length variation to the frequency difference of the multi-beam laser heterodyne signal by the frequency modulation of the oscillating mirror, this method can obtain many values of length variation caused by temperature variation after the multi-beam laser heterodyne signal demodulation simultaneously. Processing these values by weighted-average, it can obtain length variation accurately, and eventually obtain the value of linear expansion coefficient of metal by the calculation. This novel method is used to simulate measurement for linear expansion coefficient of metal rod under different temperatures by MATLAB, the obtained result shows that the relative measurement error of this method is just 0.4%.

Anisotropy of Expansion Coefficient and Slag Resistance of Spinel Carbon Bricks

Effects of the pressure direction on the thermal expansion and slag corrosion resistance were investigated and anisotropic microstructures of flaky graphite in spinel carbon bricks were examined. The experimental results show that slag corrosion velocities in the direction parallel to the pressure direction display a decrease of 34% compared to those in the vertical direction. Meantime, the linear expansion coefficient in the direction parallel to the pressure direction is 2.45 times as large as that in the vertical pressure direction. Slag corrosion velocities of spinel carbon bricks soaked in the AOD melting slag display a 46%-47% decrease compared to those of magnesia carbon bricks. The microstructure observation shows that spinel carbon bricks have a high degree of preferred orientation.

Effects of heat treatment temperature on crystallization and thermal expansion coefficient of Li_2O-Al_2O_3-SiO_2

The basic glass of Li2O-Al2O3-SiO2 system using P2O5 as nucleator was prepared by means of conventional melt quenching technology, and the heat-treatment process was determined by using differential thermal analysis. The crystalline phases and the microstructure of the glass-ceramics were investigated by using X-ray diffraction and scanning electron microscopy. The results show that the glass based on Li2O-Al2O3-SiO2 oxides using P2O5 as nucleator can be prepared at lower melt temperature of 1 450 ℃ and the glass-ceramics with lower thermal expansion coefficient of 21.6×10-7 ℃-1 can also be obtained at 750 ℃. The glass-ceramics contain a few crystal phases in which the main crystal phase is β-quartz solid solution and the second crystal phase is β-spodumene solid solution. When the heat treatment temperature is not higher than 650 ℃, the transparent glass-ceramics containing β-quartz solid solution can be prepared. β-quartz solid solution changes into β-spodumene solid solution at about 750 ℃. And the appearance of the glass-ceramics changes from translucent, part opaque to complete opaque with increasing （temperature.）

High Performance ZrNbAl Alloy with Low Thermal Expansion Coefficient

Thermal expansion is a common phenomenon in both metals and alloys, which is important for metallic material applications in modern industry, especially in nuclear and aerospace industries. A lower thermal expansion coefficient may cause lower thermal stress and higher accuracy. A new Zr-based alloy is developed and presented.The XRD diffraction results demonstrate that only a close-packed hexagonal phase（α or α＇ phase） exists in the microstructure. The thermal expansion and mechanical properties are studied. According to the experimental results, the new Zr-based alloy presents a low thermal expansion coefficient and good mechanical properties.Also,its thermal expansion coefficient is stable through solution treatment.