Elastic moduli of superhard rhenium diboride

The elastic moduli of polycrystalline rhenium diboride are measured as a function of temperature between 5 and 325 K. The room temperature results show that ReB2 has very high values for both the bulk and shear modulus, confirming the incompressible and superhard nature of this material. With decreasing temperature, the moduli increase, with a hint of softening below 50 K. The search for superhard materials, indispensable for industrial applications such as scratch-resistant coatings and cutting tools, has recently led to the identification of rhenium diboride (ReB2) as a potential superhard and incompressible solid. Although the synthesis at ambient pressure was established in 1962 [1] its scientifically interesting mechanical properties were revealed only recently, through hardness and incompressibility measurements [2], reporting a maximum hardness of 55.5 GPa under a load of 0.49 N. Since the hardness is deduced from the size of the indentation after deformation, a hard material typically requires a high bulk modulus (in order for the material to support the volume decrease created by the applied pressure), and a low Poisson ratio or high shear modulus (such that the material will not deform in a direction different from the applied load), and the materials must have minimal plastic deformation [3]. Diamond, the archetypal hard material, has a bulk modulus of 443 GPa (the record incompressibility until the bulk modulus of osmium was measured to be 462 GPa) [4], and a Poisson ratio of 0.08. Because of the strong correlation between the hardness of a material and its elastic moduli, multiple theoretical studies of ReB2 have focused on the calculation of its elastic moduli, predicting values of the shear modulus in the range 290–310 GPa, and values of the bulk modulus between 340 and 370 GPa [5–12]. However, experimental studies of the elastic moduli have thus far not been reported. In this paper, we present the elastic moduli for polycrystalline ReB2, measured using resonant ultrasound spectroscopy (RUS) [13–16] as a function of temperature between 5 and 325 K. A polycrystalline ingot of ReB2 was synthesized by arc melting the appropriate amounts of Re (99.9%) and B11 (99.9%) in an Ar atmosphere. Coarse powders of the elements were first mixed together and then pressed into a pellet with an initial composition of ReB2.5. The pellet was arc-melted on a water-cooled copper hearth and flipped 6–7 times to ensure chemical homogeneity. Assuming only B is lost during the melting process, the final weight of the ingot corresponded to a composition of ReB2. To confirm the phase purity of the obtained sample, powder x-ray spectroscopy was performed. As is shown in figure 1, the diffraction pattern for our sample matches that of ReB2, confirming successful synthesis under atmospheric pressure and the absence of significant impurities. All lines in the pattern can be indexed using a hexagonal unit cell with a = 2.8998(2)Å and c = 7.4763(8)Å. Three specimens of approximately 2 × 2.5 × 3 mm3 were cut out of the sample and prepared for RUS measurements. RUS is based on the measurement of the resonances of a freely vibrating body [13–16]. Whereas the mechanical resonances can be calculated for a sample with known dimensions, density and elastic tensor, RUS uses an inverse procedure, in which the mechanical resonances of a freely vibrating solid of known shape are measured, and an iteration procedure is used to ‘match’ the measured lines with the calculated spectrum. RUS allows the determination of all elastic constants of the solid from a single frequency scan, which gives the technique a clear advantage over more conventional methods: there is no need for separate measurements to probe different moduli. RUS also eliminates the need for bonding agents, as the sample is lightly held between two transducers. Another advantage lies in the ability of RUS to work with small samples: whereas conventional techniques can demand a sample size up to a centimetre, RUS measurements can be made on millimetressized samples. The RUS data reported here were carried out 0022-3727/09/095414+04$30.00 1 © 2009 IOP Publishing Ltd Printed in the UK J. Phys. D: Appl. Phys. 42 (2009) 095414 M R Koehler et al Figure 1. Observed x-ray pattern for polycrystalline ReB2. All lines in the pattern are indexed using a hexagonal unit cell with a = 2.8998(2)Å and c = 7.4763(8)Å. Table 1. Measured bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), Poisson’s ratio σ and Debye temperature θD of ReB2 compared with theoretical data reported by others. L (GPa) G (GPa) B (GPa) E (GPa) σ θD (K) Experiment 300 K 685 276 317 642 0.163 738 5 K 693 280 320 650 0.161 738 Hao [5, 7] LDA 762.4 294.9 369.2 698.7 0.1846 GGA 750.4 289.4 364.5 682.5 0.1791 695 264 343 630 0.1937 716 Wang [6] LDA 776 313 359 696 0.22 GGA 749 304 344 642 0.21 774 Liang [9] LDA 772 310 359 725 0.171 782 763a 305a 356a 714a 0.172a Zhou [10] GGA 727 283 350 669 0.182 749 a These calculations include spin–orbit coupling in Re. as a function of temperature using a custom designed probe that was inserted in a commercial quantum design physical properties measurement system (PPMS). Polycrystalline materials are elastically isotropic, i.e. they have only two independent elastic moduli. Whereas the bulk modulus B (with the strains perpendicular to the stress directions all equal) and the shear modulus G (with the strains perpendicular to the stress directions all zero) are usually considered the fundamental moduli for isotropic solids [17], acoustic measurements will yield the longitudinal modulus L and the shear modulus G, which are directly linked to the longitudinal and transverse sound velocities, vL and vT, in the material: vL = (L/ρ)1/2 and vT = (G/ρ)1/2. The bulk modulus (B), Young’s modulus (E) and Poisson’s ratio (σ) are connected to L and G through the following relations [17]: