Thermal Equation of State of Rhenium Diboride by High Pressure-Temperature Synchrotron X-Ray Studies
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PHYSICAL REVIEW B 78, 224106 ͑2008͒ Thermal equation of state of rhenium diboride by high pressure-temperature synchrotron x-ray studies Yuejian Wang,* Jianzhong Zhang, Luke L. Daemen, Zhijun Lin, and Yusheng Zhao† LANSCE Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Liping Wang Mineral Physics Institute, State University of New York, Stony Brook, New York 11794, USA ͑Received 10 September 2008; published 16 December 2008͒ ͑ ͒ The unit-cell volume of rhenium diboride ReB2 has been measured by synchrotron x-ray diffraction at pressures and temperatures of up to 7.5 GPa and 1100 K with a cubic anvil apparatus. From the pressure ͑ ͒ ͑ ͒ ͑ ͒ P -volume V -temperature T measurements, thermoelastic parameters were derived for ReB2 based on a modified high-T Birch-Murnaghan equation of state and a thermal-pressure approach. With the pressure de- Ј ͑ ͒ rivative of the bulk modulus, K0, fixed at 4.0, we obtain: the ambient bulk modulus K0 =334 23 GPa, −1 -T͒P =−0.064͑6͒ GPa K , volumetric therץ/Kץ͑ temperature derivative of bulk modulus at constant pressure −1 −5 −1 −8 −2 mal expansivity ␣T͑K ͒=a+bT with a=1.33͑25͒ϫ10 K and b=1.48͑64͒ϫ10 K , pressure derivative −7 −1 −1 P͒T =−5.76͑95͒ϫ10 GPa K , and temperature derivative of bulk modulus atץ/␣ץ͑ of thermal expansion −1 -T͒V =−0.049͑11͒ GPa K . The ambient bulk modulus derived from this work is comץ/Kץ͑ constant volume parable to previous experimental and theoretical results. These results, including the ambient bulk modulus and other thermoelastic parameters determined in present study, extend our knowledge of the fundamental thermo- physical properties on ReB2 and are important to the development of theoretical and computational modelings of hard materials. DOI: 10.1103/PhysRevB.78.224106 PACS number͑s͒: 62.50.Ϫp, 64.30.Ϫt, 61.05.cp I. INTRODUCTION Materials with three-dimensional covalent-bonded net- works of light elements, including boron, carbon, nitrogen, and oxygen, are generally expected to exhibit simultaneously low compressibility, good thermodynamic and chemical sta- bilities, large hardness, and high wear resistance.1–4 Typi- cally, these materials ͑e.g., diamond, cubic boron nitride, and ͒ 2–4 BC2N have been synthesized at high pressure. This places severe limitations on the production volume and the cycle rate of the apparatus and hence presents disadvantages in large scale industrial manufacturing. An alternative path to the development of ultrahard mate- rials is to make use of high electron densities in noble metals in combination with the short and strong covalent bonds nor- mally formed by B, N, and O. This approach restricts the creation and propagation of defects, which would in turn lead to higher resistance to plastic deformation.5 As a result a number of noble-metal borides, nitrides, and oxides have been investigated by optimizing the valence-electron density in metals and bonding properties of light elements.6–9 Re- cently much theoretical and experimental attention has been focused on the synthesis and characterization of rhenium di- 10–14 boride, ReB2, at ambient pressure. Thermoelastic properties are of fundamental importance to understand the mechanical properties of materials. Though there is no direct correlation between the elastic bulk modu- lus and a material’s hardness, superhard materials usually have a high bulk modulus, e.g., diamond with K0 FIG. 1. Representative x-ray diffraction patterns used for the =442 GPa and c-BN with K0 =400 GPa. In the case of refinement of unit-cell parameters of ReB2 under high P-T condi- ReB2, previous studies reported a value of 360 GPa for the tions. The peaks marked with stars are fluorescence lines of Re. The bulk modulus, suggesting that ReB2 is potentially a super- BN peaks were observed because of the BN cylinder surrounding hard material.10 However all previous theoretical and experi- the sample. 1098-0121/2008/78͑22͒/224106͑5͒ 224106-1 ©2008 The American Physical Society WANG et al. PHYSICAL REVIEW B 78, 224106 ͑2008͒ ͑ ͒ FIG. 2. P-V-T data measured for ReB2. The curves represent FIG. 3. Thermal pressure Pth of ReB2 as a function of tem- results of the least-squares fitting using Eq. ͑1͒. The ambient unit- perature. The spread of the data points at any given temperature ͑ ͒ cell volume V0 determined from the P-V-T fit is in good agree- corresponding to thermal pressures at different volumes, which is ment with the reported value in the JCPDS file 00-011-0581 ͑Ref. plotted in detail in the inset. The dashed lines in the inset plot show 15͒. approximate constant values of thermal pressure for a given tem- perature, indicating that thermal pressure is independent of volume mental investigations were conducted without concerning the for ReB2. The scattering of the data points at room temperature is temperature dependence of compressibility. This dependence representative of the uncertainties in thermal-pressure calculations is embedded in the thermoelastic equation of state ͑EOS͒, from the present P-V-T measurements. which describes the relationship between pressure, volume, and temperature ͑P-V-T͒. The EOS gives access to II. SAMPLE PREPARATION AND HIGH-PRESSURE temperature-dependent properties such as the temperature EXPERIMENTAL METHOD derivative of the bulk modulus and the pressure derivative of At ambient conditions, ReB2 has a primitive hexagonal ͑ / ͒ thermal expansion, which are very important to understand structure space group P63 mmc, No. 194 with two ReB2 better correlations between crystal structure and mechanical units per unit cell. This structure can be depicted in terms of behavior, as well as to benchmark theoretical and computa- alternative layers of rhenium ͑Re͒ and boron ͑B͒ atoms.15 tional modelings. The starting ReB2 powders were synthesized by using the In the present study, we performed an in situ high P-T methodologies published in Ref. 10. The mixture of Re and x-ray diffraction experiment to investigate the thermoelastic amorphous B powders with molar ratio of 1:5 was first properties of ReB2. Thermal EOS parameters, such as bulk pressed into tablets and sealed into a silica tube under modulus, temperature derivative of the bulk modulus, volu- vacuum. The tube was then heated at 1200 °C for 5 days, metric thermal expansion, and pressure derivative of thermal followed by cooling to room temperature. X-ray diffraction expansion, were derived by fitting the P-V-T data sets to a data of the reacted powders reveal a single phase material modified high-temperature Birch-Murnaghan EOS. A with a diffraction pattern that is consistent with JCPDS file thermal-pressure approach was also used to produce the tem- 00-011-0581.15 At room temperature and ambient pressure, perature derivative of the bulk modulus at constant the lattice parameters were determined to be a=2.900 Å and volume—a thermoelastic parameter that is experimentally c=7.478 Å after a Rietveld refinement of the data. difficult to measure. From these analyses an internally con- The high P-T experiment was conducted using a cubic sistent thermal equation of state is obtained for ReB2. anvil apparatus at beamline X17B2 of the National Synchro- ͒ ץ/␣ץ͑ TABLE I. Summary of thermoelastic parameters for ReB2. Except for P T, the numbers in paren- theses are standard deviations from the least-squares fits and refer to the last digit͑s͒ of the parameter values. ͒ ץ/ ץ͑ ͒ ץ/␣ץ͑ For P T, the uncertainties are estimated from the error propagation of K0 and K T P. ͒ ץ/ ץ͑ ͒ ץ/␣ץ͑ −1͒ ͑ ␣ ͒ ץ/ ץ͑ K0 K T P T K =a+bT P T K T V ͑ ͒ Ј ͑ −1͒ −5 −8 ͑ −7 −1 −1͒͑ −1͒ Ref. GPa K0 GPa K a,10 b,10 10 GPa K GPa K This worka 334͑23͒ 4.0 −0.064͑6͒ 1.33͑25͒ 1.48͑64͒ −5.76Ϯ0.95 This workb 334͑23͒ 4.0 −0.071͑11͒ −6.36Ϯ1.32 −0.049͑11͒ 10 360 4.0 14 359 4.09 aBased on the measured P-V-T data and Eqs. ͑1͒ and ͑2͒. bThermal-pressure approach based on the measured data and Eqs. ͑3͒ and ͑4͒. 224106-2 THERMAL EQUATION OF STATE OF RHENIUM DIBORIDE… PHYSICAL REVIEW B 78, 224106 ͑2008͒ tron Light Source, Brookhaven National Laboratory. The general form of this modified EOS is formulated by white radiation from the superconducting wiggler magnet P =3K f͑1+2f͒5/2͓1−3/2͑4−KЈ͒f + ...͔, ͑1͒ was used for energy-dispersive measurements. The diffracted T x-rays were collected with a 13-element detector at a fixed where Bragg angle of 2=6.4757°. The cell assembly used in the ͒ ͑͒ ץ/ ץ͑ present experiment has been described elsewhere.16 Briefly, a KT = KTo + K T T − 300 , cube made up of a mixture of amorphous B and epoxy resin ,Pץ/Kץ = was employed as pressure-transmitting medium, and amor- KЈ phous carbon was used as furnace material to achieve high and temperatures. The ReB2 sample, sandwiched by NaCl pow- der, was packed into a cylindrical container of boron nitride 1 / f = ͓͑V /V ͒2 3 −1͔, ͑BN͒, 1.0 mm inner diameter and 2.0 mm length. 2 T PT In the present study NaCl was used as an internal pressure marker and the sample pressure was calculated from Deck- 17 ͫ͵ ␣͑ ͒ ͬ er’s EOS for NaCl. At each experimental condition, four VT = V0 exp 0,T dT . NaCl diffraction peaks, 111, 200, 220, and 420, were used ͑ ͒ for the determination of pressure. The uncertainty in pressure In Eq. 1 , KTo and KT represent the isothermal bulk modulus T͒ andץ/Kץ͑ measurements is mainly due to the statistical variation in the at 300 K and a higher temperature T, and P͒ stand for the temperature and pressure derivativesץ/Kץ͑ positions of different diffraction peaks and is less than 0.2 GPa in the P-T range investigated here.