Temperature dependence of the isothermal bulk modulus of TiB 2 H. Özkan Department of Physics, Middle East Technical University, Inönü Bulvarı, 06531 Ankara, Turkey article info Article history: Received 21 January 2010 Received in revised form 13 October 2010 Accepted 20 December 2010 Available online 11 January 2011 Keywords: A. Intermetallics, miscellaneous B. Mechanical properties at high temperatures B. Thermal properties abstract The temperature dependencies of the isothermal bulk modulus of TiB 2 above room temperature were calculated by using the equation for the Anderson- Grüneisen parameter (d T ) and its solution by Garai and Laugier (J. Appl. Phys.101, 023514 (2007)). The present calculations utilize the relevant experimental data for the pressure derivative of the isothermal bulk modulus, K 0 as d T in addition to the volume thermal expansion coefficients (a V ). The temperature derivatives of the isothermal bulk modulus of TiB 2 obtained in 300e900 K, using the latest published thermal expansion coefficients are, 0.012e0.013 GPa/ K, in good agreement with the corresponding experimental values. The results verify the present approach as a practical way to predict the bulk moduli of materials at high temperatures. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Titanium diboride (TiB 2 ) is a well-known intermetallic compound for its high strength, high wear resistance and high melting point. It has relatively low density, low electrical resistivity and good thermal conductivity. Combinations of these properties make TiB 2 an inter- esting high temperature material. The temperature dependencies of the mechanical and thermal properties of TiB 2 are important for the high temperature applications. In a recent first principle study, the density functional theory (DFT) was applied to calculate the elastic and thermodynamic properties of TiB 2 at high temperatures (up to 2000 K) and high pressures (up to 170 GPa) [1]. The isothermal bulk modulus (K T ) at room temperature and its pressure derivative reported in Ref. [1] (K T : 245 GPa, and (dK T /dP) T : about 4) are close to the experimental values, (K T : 233e276 GPa and (dK T /dP) T : 3.9) [2,3]. But, the temperature derivatives of the isothermal bulk modulus ((dK T /dT) P ; À0.032/À0.038 GPa/K in 293e1773 K) are 3e4 times larger in magnitude than the experimental values (À0.010/ À0.012 GPa/K) at and above room temperature [2,4]. The volume thermal expansion coefficients (a V ) of TiB 2 obtained by DFT [1] (21e35 Â 10 À6 /K in 293 Ke1273 K) are quite larger than the earlier [4,5], and the latest published experimental a V data [6] (22.2e25.8 Â 10 À6 /K in 293e1273 K and 13.3e13.5 Â 10 À6 /K in 300e900 K, respectively). Note that the earlier a V data [4,5] are quite larger than the more recently reported ones [6,7]. The large theo- retical values for (dK T /dT) P and a V at high temperatures reported in Ref. [1] for TiB 2 and their statements; “it is the rapid variation of volume which makes the rapid decrease of bulk modulus” are not justified by the experimental observations. In fact, such rapid decreases of the bulk modulus and increases of the unit cell volume with temperature were not expected for a very hard, stiff material as TiB 2 . For example, zirconium diboride (ZrB 2 ), a structurally similar but slightly less stiff compound have smaller values for (dK T /dT) P (À0.020) and a V (18e22 Â 10 À6 /K in 300e1073 K), respectively [8]. These data and arguments point out the need for more relevant values for the temperature dependencies of the isothermal bulk modulus of TiB 2 at high temperatures. For many solids, assuming the quasi-harmonic approximation the isothermal Anderson-Grüneisen parameter (d T ) is equivalent to the pressure derivative of the bulk modulus (K 0 ) at temperatures near the Debye temperature (q D ) [9,10]. In 2007, Garai and Laugier reported a theoretical expression for the temperature dependen- cies of the isothermal bulk modulus [11]. Recently, the present author evaluated the temperature dependencies of the isothermal bulk modulus of zircon (ZrSiO 4 ) [12] by utilizing the K 0 data as d T in the theoretical expression. The results were in good agreement with the experimentally obtained temperature derivatives of the isothermal bulk modulus. In the present study, the temperature dependencies of the isothermal bulk modulus of TiB 2 above room temperature were computed by utilizing the relevant K 0 data as d T in the expressions for K T [10,11]. The values obtained for the temperature dependen- cies of the isothermal bulk modulus for TiB 2 agree well with the corresponding experimental values [2,4]. The present results for TiB 2 and the previous ones for ZrSiO 4 [12] reveal good correlations E-mail address: hozkan@metu.edu.tr. Contents lists available at ScienceDirect Intermetallics journal homepage: www.elsevier.com/locate/intermet 0966-9795/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2010.12.013 Intermetallics 19 (2011) 596e598