Bulk modulus of osmium, 4–300 K

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Acta Materialia 57 (2009) 544–548 www.elsevier.com/locate/actamat

Bulk modulus of osmium, 4–300 K C. Pantea a,*, I. Mihut a, H. Ledbetter b, J.B. Betts a, Y. Zhao a, L.L. Daemen a, H. Cynn c, A. Migliori a b

a Los Alamos National Laboratory, Los Alamos, NM 87545, USA Mechanical Engineering, University of Colorado, Boulder, CO 80309, USA c Lawrence Livermore National Laboratory, Livermore, CA 94550, USA

Received 25 July 2008; received in revised form 24 September 2008; accepted 24 September 2008 Available online 24 October 2008

Abstract We determined osmium’s 0–300 K monocrystal and polycrystal bulk modulus using two ultrasonic methods: resonant ultrasound spectroscopy and a novel pulse-echo method. All other recent bulk modulus measurements on osmium have used high-pressure X-ray diffraction, a method with known lower accuracy than ultrasonic methods. Our study was motivated by current theoretical and measurement disputes over osmium’s bulk modulus and whether it exceeds that of diamond. At all temperatures, we found osmium’s bulk modulus to be much lower (by 7–8%) than diamond’s. The room-temperature bulk modulus for osmium is 405 GPa; extrapolation to 0 K gives 410 GPa. Osmium’s 0–300 K bulk modulus change is unusually small: about 1%. Temperature behavior is smooth, fitting closely to an Einstein-oscillator model. Published by Elsevier Ltd on behalf of Acta Materialia Inc. Keywords: Bulk modulus; Acoustic methods; Transition metals; Acoustic properties

1. Introduction The bulk modulus, the only elastic modulus possessed by all states of matter, reveals much about interatomic bonding strength. Consider the room-temperature bulk moduli for a range of elements with different bonding types and bonding strengths: diamond 442 GPa, iron 167, copper 138, aluminum 76, lead 45, sodium 6 [1], solid helium 0.02 (value at 0 K) [2], covering four orders of magnitude. The bulk modulus is the easiest elastic constant to calculate from the interatomic potential. It is also the most often cited elastic constant to compare interatomic bonding strength among various materials. It occurs in many formulas describing disparate mechanical–physical properties, varying from point defect properties to mechanical deformation. Along with atomic volume and binding energy, it represents one of a solid’s three basic cohesion properties.

*

Corresponding author. Tel.: +1 505 665 7598; fax: +1 505 665 4292. E-mail address: [email protected] (C. Pantea).

1359-6454/$34.00 Published by Elsevier Ltd on behalf of Acta Materialia Inc. doi:10.1016/j.actamat.2008.09.034

Based on both measurement and theory, Cynn and colleagues reported that osmium possesses a higher bulk modulus (lower compressibility) than diamond [3]. The enormous implications of this for understanding interatomic bonding strength and related physical properties in d-electron metals prompted new osmium studies, including the present one. Table 1 summarizes most previous results chronologically. The Cynn et al. results were not unreasonable because, among metals, osmium shows the highest mass density, the highest hardness, the highest bulk modulus, probably the highest shear modulus and a high melting temperature. Before the Cynn et al. study, several Russian sources suggested that osmium’s bulk modulus approximates diamond’s. Numerous modern quantum mechanical calculations predict a higher than diamond stiffness for osmium (see Table 1). And at least one study (see Table 1) reported a higher bulk modulus based on a measured polycrystal Young’s modulus and an estimated polycrystal shear modulus [13]. Friedel’s tight-bindingapproximation d-electron model predicts a high bulk modulus for osmium [18]. Finally, Grossman and colleagues

C. Pantea et al. / Acta Materialia 57 (2009) 544–548 Table 1 Osmium’s bulk modulus from various theory and measurement sources. B (GPa)

Source

Method

Theory 476 419 445 436, 457, 461 409–510 437, 382 454, 401 403

Fast et al. [4] Grossman et al. [5] Cynn et al. [3] Joshi et al. [6] Hebbache and Zemzemi [7] Occelli et al. [8] Sahu and Kleinman [9] Zheng [10]

FPLMTO-LDA Pseudopotential-LDA FPLMTO-LDA LDA, GGA DFT, pseudopotential LDA, GGA LDA, GGA Pseudopotential

Measurements 388 419 459 431 441 462 ± 12 411 ± 6 395 ± 15 435 ± 19 397 ± 20 393 ± 20 405 ± 5 406 ± 5 410 ± 5 414 ± 5

Ko¨ster [11] Gschneidner [12] Narayana and Swamy [13] Savitsky et al. [14] Savitsky [15] Cynn et al. [3] Occelli et al. [8] Takemura [16] Voronin et al. [17] Present, 300 K Present, 300 K Present, 300 K Present, 300 K Present, 0 K Present, 0 K

Polycrystal resonance Review Debye temp, estimate Handbook Handbook P, X-ray diffraction P, X-ray diffraction P, X-ray diffraction P, X-ray diffraction Pulse-echo, monocrystal Pulse-echo, polycrystal RUS, monocrystal RUS, polycrystal RUS, monocrystal RUS, polycrystal

Bold entries represent values that are the same as or higher than diamond’s bulk modulus (442 GPa). P denotes pressure, RUS denotes resonance ultrasonic spectroscopy.

pointed out that among the d-electron elements they considered theoretically, osmium showed the largest charge density in the interstitial regions, indicating a tendency toward strong covalent bonding and high bulk modulus [5]. Diamond, the paradigm covalent material, has the highest measured bulk modulus [19], as we confirm here. Throughout most of the current osmium bulk modulus research, there persists a wrong idea: the bulk modulus B provides the best elastic modulus indicator of hardness. Actually, the shear modulus provides a better hardness indicator [20,21]. This occurs because hardness depends on dislocation mobility, and dislocation mobility depends on the shear modulus G [21]. The bulk modulus offers a viable surrogate only when comparing materials with the same Poisson ratio m, their well-known interconnectivity being the following: m¼

1 3B  2G : 2 3B þ G

ð1Þ

The osmium and diamond Poisson ratios differ dramatically, at 0.23 and 0.07, reflecting their respective strong metallicity and strong covalency, very different interatomic bonding types. 2. Bulk modulus lore Current texts tend to omit important historical dates. However, the bulk modulus was measured at least as early as 1877 by Bauschinger, who studied practical materials

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such as Bessemer steel, cast iron and wrought iron [22]. Earlier, Stokes identified the bulk modulus as one of the two fundamental solid-state elastic constants (the second being the shear modulus) [23]. In the early twentieth century, physicists began to focus on the bulk modulus. Einstein connected the bulk modulus with what we now call the Einstein frequency and the Einstein temperature [24,25]. Since then, the bulk modulus has been connected with the core of quantum theory: the Heisenberg uncertainty principle [26]. The importance of the bulk modulus for progress in measurement, theory and equations of state received a clear review in Gru¨neisen’s Handbuch der Physik review in 1926 [27]. In more recent times, Gilman, in considering solid-state interatomic bonding, devoted an entire chapter to the bulk modulus [28]. Also, Zener (for cubic symmetry crystals) dispensed with Voigt’s late-1800s choices for the Cij (C11, C12, C44) and used instead three elastic constants that included the more physical bulk modulus B [29]. Similar reasoning could be extended to lower symmetry crystals such as osmium. Probably the most research on the bulk modulus came from Bridgman at Harvard, which resulted in the 1946 Nobel prize for physics. As summarized in his book Physics of High Pressure, he measured many elements and compounds, often with surprising results. His student John Slater (later at MIT) also focused on the bulk modulus, especially for alkali halides. 3. Measurements Because all recent measurements but ours were direct measurements that used high-pressure X-ray diffraction, known to occasionally yield spurious results, and with consistently lower resolution than acoustic-based techniques, we chose two alternative high-precision indirect measurement approaches: one, resonant ultrasound spectroscopy (RUS), the many advantages of which include a single temperature run on a single specimen to get the complete elastic stiffness tensor Cij, high accuracy (typically a small fraction of a percent) and very high precision [30,31]. The other was a novel digital pulse-echo overlap measurement of the time of flight of an acoustic pulse [32]. Indeed, Cynn and colleagues emphasized the need for measuring osmium’s elastic constants using acoustic methods. We acquired monocrystals (oriented along the c-axis) from Accumet Materials Co. (Briarcliff Manor, New York). Polycrystalline chilled shot were obtained from Alfa Products, Thiokol/Ventron Division (Danvers, MA). We polished all specimens to a 1–2 mm parallelepiped with flat and parallel faces, and sharp corners and edges. Laue X-ray diffraction confirmed the orientation and structure of the monocrystals (along the c-axis and hexagonal, respectively). Pulse-echo experiments provide the sound speed in a studied material by determining the two-way travel time of a train of pulses. Details of the method occur elsewhere [32]. LiNbO3 40 MHz transducers, acquired from Boston PiezoOptics (Bellingham, MA), were used for both compressional

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and shear mode pulse-echo measurements. The following elastic moduli were determined: (i) for the polycrystalline specimens, we determined C11 (compressional mode, or Pwave) and C44 (shear mode, or S-wave); (ii) for the monocrystalline specimens, we determined only four independent elastic moduli from a total of five (C11, C12, C13, C33 and C44), using the C13 value obtained from RUS for further calculations. The four Cijs were determined as follows (i) C11: P-wave propagation perpendicular to the c-axis; (ii) C12: obtained from determining C66 = ½(C11  C12) with S-wave propagation perpendicular to the c-axis, polarization perpendicular to the c-axis; (iii) C33: P-wave propagation parallel to the c-axis; (iv) C44: S-wave propagation parallel to the c-axis, arbitrary orientation. In contrast, RUS needs only one set of measurements to determine the full Cij tensor. The transducers used were LiNbO3, 30 MHz, compressional mode, acquired from the same source as above. A frequency sweep reveals the mechanical resonances of the specimen, which relate directly to the Cij values (also, we must know the mass density and specimen dimensions). The fitting errors were less than 0.2% for up to 50 macroscopic resonances.

Fig. 1. Osmium’s bulk modulus from 0 to 300 K. We view slight irregularities as measurement artifacts, not as departures from smooth temperature changes. The line represents an Einstein-oscillator fit to the measurements. Note the exceptionally small change, about only 1%, compared with about 3.5% for the average metal.

C ijkl ¼ ðo2 U =ogij ogkl Þ:

4. Results Table 2 shows the mass density, atomic volume and bulk modulus at both 300 and 0 K for our polycrystal and monocrystal specimens. Thermal expansion corrections were applied according to Finkel et al. [33]. Room-temperature density was calculated from measured mass and dimensions of the parallelepiped specimens, and later corrected to a value of 22.61 g cm3 [34]. Several mass–density values for Os appear in the literature, ranging from 22.57–22.89 g cm3 [33–40]. We used the value of the bulk modulus obtained using these different densities to estimate our error as ±5 GPa. Fig. 1 shows the change of B from about ambient temperature to zero temperature. The change in temperature was achieved in a flow cryostat. Experimental details of the RUS technique can be found in Ref. [41]. The curve in Fig. 1 represents an Einstein-oscillator fit to the measurements [42,43]. 5. Discussion The connection between elastic stiffness components and B is indirect through the internal energy U [44]: Table 2 Physical properties of the four studied osmium specimens as determined by RUS. Bulk modulus Temperature Mass density Atomic ˚ 3) (GPa) volume (A (K) (g cm3) Monocrystal 300 Monocrystal 0 Polycrystal 300 Polycrystal 0

22.61 22.70 22.61 22.71

13.993 13.930 13.993 13.930

405 ± 5 410 ± 5 406 ± 5 414 ± 5

ð2Þ

Here gij denotes a Lagrangean strain. For the bulk modulus, Eq. (2) becomes B ¼ j1 ¼ V ðoP =oV Þ ¼ V ðo2 U =o2 V Þ:

ð3Þ

Here, j denotes compressibility; V, volume; P, pressure. For hexagonal symmetry [45], B1 ¼ 2S 11 þ S 33 þ 2ðS 12 þ 2S 13 Þ:

ð4Þ

Here, Sij denotes the elastic compliance tensor, the tensor inverse of Cijkl. Osmium’s zero-temperature bulk modulus assumes importance for three principal reasons. First, most modern quantum mechanical theories predict only the zero-temperature elastic constants (they fail to consider entropy effects). Secondly, full comparison with diamond requires the zero-temperature value, which could exceed considerably the 300 K value. Because of diamond’s high Debye temperature (2244 K) [19] and low Gru¨neisen parameter, cooling diamond to 0 K increases the bulk modulus only 1 part in 1000 [41]. This is not so, however, for osmium. Thirdly, many d-electron metals show odd temperature behavior, thus precluding the confident estimation of the zero-temperature value from the ambient value [46]. The room-temperature bulk modulus of monocrystalline osmium determined by RUS from two independent monocrystalline samples was 403 and 407 GPa, respectively, with an average of 405 ± 5 GPa. This value is about 2–4% larger than the pulse-echo result. Two main factors may explain the difference: (i) scatter, a consequence of required five separate pulse-echo measurement sequences is needed to determine all moduli (unlike RUS, which requires only one measurement); (ii) Laue X-ray diffraction on polycrystals revealed that the polycrystalline specimens

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contained slightly misoriented regions, due to the large grain size and a slightly anisotropic specimen (any such anisotropy must be small; otherwise the RUS inversion analysis would fail). Fig. 1, which shows the monocrystal bulk modulus temperature dependence, shows immediately that at all temperatures osmium’s bulk modulus falls well below that of diamond (about 7–8%). Thus, we need not modify all the textbooks that report diamond as nature’s least-compressible material. That the measured bulk modulus [3] was too high should have been suspected from the accompanying too-low bulk modulus pressure derivative dB/dP = 2.4 [3], the value for diamond being about 4. Whether osmium permits modification to surpass diamond’s stiffness remains an open question. Recently, scientists at UCLA succeeded in synthesizing OsB2 (orthorhombic), which has a hardness that exceeds that of sapphire [47]. (On the Mohs hardness scale, sapphire stands second to diamond.) One might conjecture that high pressure boosts osmium’s stiffness above diamond’s. Against this thinking, Grossman and colleagues showed that, unlike many d-electron metals, adding carbon or nitrogen to osmium lowers its bulk modulus, probably because these elements disrupt osmium’s strong interstitial covalent-like interatomic bonding [5]. Elsewhere [48], we discussed osmium’s Debye temperature, 477 K (obtained from elastic moduli extrapolated to zero temperature), which is much lower than that of diamond because the Debye temperature depends much more on the shear modulus than on the bulk modulus. Later, we shall report the full Cij elastic tensor between 300 and 0 K, including an estimate of the thermodynamic Gru¨neisen parameter. Perhaps the most remarkable feature shown in Fig. 1 is the small bulk modulus change between 300 and 0 K, which, expressed as a ratio, is 1.012. From handbooks, for six common metals, we find the following ratios: Al 1.043, Cu 1.036, Mg 1.045, Ti 1.025, Ni 1.022 and Fe 1.117. Thus, for these six metals, there is an average increase of 3.33%, essentially three times that shown by Os. Osmium’s small change must arise from either d-electron effects or electronic changes within Os that compensate partially for the usual temperature-induced lattice softening. This point requires further focus. Based on a relationship given by Ledbetter [49], the small bulk modulus change predicts a small Gru¨neisen parameter c, this parameter being the quintessential single parameter for characterizing anharmonic physical properties. Thus, Os shows low anharmonicity. 6. Conclusions In conclusion, ultrasonic studies on both polycrystal and monocrystal osmium in the 0–300 K temperature range show that osmium’s bulk modulus fails to exceed diamond’s, or even closely approximate it, being always 7–8% lower. This finding disputes several recent measurements and theoretical calculations, and upholds the long-

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