Estimation of linear thermal expansion coefficient from cohesive energy obtained by ab-initio calculation of metals and ceramics

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1 aaaaa Journal of the Ceramic Society of Japan 118 [3] Paper Estimation of linear thermal expansion coefficient from cohesive energy obtained by ab-initio calculation of metals and ceramics Yusuke TSURU, *,**, Yoshifumi SHINZATO, ** Yuki SAITO, ** Megumi SHIMAZU, * Mitsunobu SHIONO * and Masahiko MORINAGA ** * TOTO LTD., 2-1-1, Kitakyushu-shi ** Graduate School of Engineering, Department of Materials Science and Engineering, Nagoya University, Chikusa-ku, Nagoya, The cohesive energy was found to be nearly proportional to the melting point that has correlated empirically with the linear thermal expansion coefficient of solids. So, the relationship between the linear thermal expansion coefficient and the cohesive energy was examined at 295, 500, 1000 and 1200 K by using the experimental data of various metals, oxides, borides, carbides and nitrides. As a result, it was revealed that the linear thermal coefficient is inversely proportional to the cohesive energy. Therefore, there is a great possibility of evaluating the linear thermal expansion coefficient of these materials from the abinitio calculation of the cohesive energy. This proposed method will be very practical and effective in designing new materials The Ceramic Society of Japan. All rights reserved. Key-words : Thermal expansion coefficient, Cohesive energy, Melting point, Density functional theory, Ab-initio calculation [Received November 11, 2009; Accepted January 15, 2010] 2010 The Ceramic Society of Japan 1. Introduction The thermal expansion coefficient is an important factor to control the hetero-epitaxial growth, since a large difference of the thermal expansion coefficients between heterogeneous materials leads to the formation of defects such as cracks and misfit dislocations at the boundary. Therefore, accurate and rapid prediction of the thermal expansion coefficients is strongly required in developing new materials, especially in case when the difference in the thermal expansion coefficients is large. Recently, the simulation has been widely used in every field of materials science. In general, classical molecular dynamics calculation has been applied to the prediction of the thermal expansion coefficient. 1) The linear thermal expansion coefficient is expressed as Eq. (1): Corresponding author: Y. Tsuru; yusuke.tsuru@toto.co.jp α L = 1 Δl (1) l0 Δt where l, t and l 0 are the lattice constant, temperature and the lattice constant at standard temperature to be used as a reference, respectively. Since a very large unit cell composed of hundreds of atoms is usually employed in the calculation, long computing time would be inevitably required even in use of the advanced computer. In the classical molecular dynamics calculation, the empirical interatomic potentials where electronic effects are incorporated implicitly are used to shorten the calculation time. However, it would be problematic to develop the potential parameters that describe accurately the physical properties of materials. Ab-initio molecular dynamics calculation is able to calculate accurately all the elements listed in the periodic table without using the empirical interatomic potentials. 2) But the calculation time is too long when the calculation model including the trace elements is large. It is known that the average linear thermal expansion coefficient (α L) of any material correlates well with the melting point (M p). 3) Its correlation could be separated into two material groups. One is the inversely proportional relation expressed as α LM p = that is valid for the materials with the CsCl structure, NaCl structure, fluorite (CaF 2) structure and perovskite structure. The other is expressed as α LM p = that is valid for the materials with the close-packed anion structures such as WC structure, wurtzite structure, corundum structure and ilmenite structure as well. In this study, instead of the melting point, the cohesive energy was used for the estimation of the linear thermal expansion coefficients of materials. The relation between the melting point and the cohesive energy was first shown by using experimental data. The cohesive energy is the most fundamental physical quantity to show the strength of the chemical bond between atoms in solids. It is the energy gained by arranging the atoms in a crystalline solid, as compared with the gas state. The anharmonic atomic vibrations are involved in the thermal expansion, but it is still expected that the material does not expand so easily when the cohesive energy is large. To show this, the relation between the linear thermal expansion coefficient and the cohesive energy was investigated using experimental data of various metals, oxides, borides, carbides and nitrides. The cohesive energies are able to be obtained by the ab-initio calculation based on the density functional theory without using a very large unit cell composed of hundreds of atoms. 4) In fact, following the ab-initio calculation, all kinds of elements listed in the periodic table can be treated without assuming interatomic potential, because the energy and atomic forces are derived by solving the density functional theory equations. This is in contrast to the classical molec- 241

2 JCS-Japan Tsuru et al.: Estimation of linear thermal expansion coefficient from cohesive energy obtained by ab-initio calculation of metals and ceramics ular dynamics calculation where empirical potential is adopted. In this study, we first revealed a possibility of estimating the linear thermal expansion coefficient from the ab-initio calculation of the cohesive energy, and then proposed a calculation method of the linear thermal expansion coefficient that is complementary to experiments. 2. Calculation procedure In the calculation of the cohesive energies, CASTEP code based on the density functional theory was used and all the calculations were performed at absolute zero temperature. 5) The generalized gradient approximation (GGA) of Perdew-Wang (PW91) functional was chosen for the exchange-correlation functional approximation. 6) Ultrasoft pseudopotentials were used with the maximum cutoff energy of the plane wave basis set of 380 ev. 7) The Brillouin zone sampling was carried out using the Monkhorst-Pack scheme. 8) Electronic energy minimization was performed with a self-consistent field tolerance of ev/ atom. The BFGS geometry optimization was used for the refinement of lattice constants and atom positions in the material. The converge tolerance of energy of ev/atom, the maximum force of 0.05 ev/å, the maximum stress of 0.1 GPa, the maximum displacement of Å were chosen in the present calculation. 9) 3. Proportional relation between melting point and cohesive energy The experimental values of the cohesive energy for (a) metals and (b) oxides, borides, nitrides and carbides are plotted against the melting point in Fig. 1. The cohesive energy increases linearly with the melting point. Since the linear thermal expansion coefficient, α L, decreases with increasing melting point of materials, 3) the α L value will decrease with increasing cohesive energy. The experimental values of the cohesive energies were calculated from the heat of formation and the cohesive energies of pure metals or other elements. 10),11) For example, the cohesive energy of A xb y compound is calculated by Eq. (2): EA x+ EB y ΔH Ecoh = (2) N Here, E coh is the cohesive energy of A xb y, E A and E B are the cohesive energies of pure metals or other elements, A and B, respectively. ΔH is the heat of formation of A xb y. x and y are compositions of A and B atoms. N is the total number of atoms in A xb y. In the next section by referring to the literature values of the cohesive energies and the linear thermal expansion coefficients, their correlation will be further investigated. Fig. 1. Correlation of the cohesive energy with the melting point of (a) Fig. 2. Correlation of the cohesive energy with the linear thermal metals and (b) oxides, brides, nitrides and carbides. 10),11) expansion coefficient measured at 295 K. 10) 13) 242

3 Journal of the Ceramic Society of Japan 118 [3] JCS-Japan 4. Correlation between linear thermal expansion coefficient and cohesive energy The correlation between the cohesive energy (E coh) and the linear thermal expansion coefficient (α L) is shown in Figs. 2, 3, 4 and 5 using the measured α L values at 295, 500, 1000 and 1200 K, respectively. 12),13) The cohesive energies (E coh) were obtained using Eq. (2). In each figure, all the data of metals, oxides, borides, carbides and nitrides are summarized in (a), and the data of these materials but excluding metals are shown in an enlarged scale in (b). The total number of data shown in Figs. 2, 3, 4 and 5 decreases as the temperature rises, because the materials reach the melting point. The E coh values of metals are distributed over a wide range of 1 9 ev/atom in these figures. But the E coh values are distributed in the energy range of 3 8 ev/atom for the oxides, 5 9 ev/atom for the carbides, 6 8 ev/atom for the nitrides and around 7 ev/atom for the borides. At any temperatures, the linear thermal expansion coefficients decrease with increasing cohesive energies. There is a clear tendency that the linear thermal expansion coefficient (α L) is inversely proportional to the cohesive energy (E coh). Following the relation of inverse proportion, α LE coh = C, the linear thermal expansion coefficient can be estimated directly from the cohesive energy if the constant C is known. The C values were determined from the experimental values of the cohesive energies and the linear thermal expansion coefficients to be (0.72), (0.68), (0.62) and (0.54) [10 6 ev/k] at 295, 500, 1000 and 1200 K respectively, where the value in the parenthesis is a standard deviation of the least squares analysis. The constant value of C is expected to increase with temperature, because the linear thermal expansion coefficient generally increases with temperature. Using the relation of α LE coh = C, the linear thermal coefficient, α L, can work out from the cohesive energy, E coh. It is well known that the cohesive energy is calculated by the ab-initio method. 14) The calculated cohesive energy per atom is defined as Eq.(3): ( ) E = E E N coh atom total where, E coh is the cohesive energy, E total is the total energy of the material calculated by the BFGS geometry optimization method, E atom is the total energy of a neutral atom, and N is the number of atoms in the material. Figure 6 shows the comparison in the cohesive energy between the ab-initio calculation and the experiment for metals, oxides, borides, carbides and nitrides. The calculation models are shown in Fig. 7(a) for Ca and Al with cubic Fm-3m structure, (b) for Na and Nb with cubic Im-3m structure, and (c) for Ti and Zr with hexagonal P63/mmc structure. The calculation models are shown in Fig. 8(a) for CdO, MgO, TiC, HfC, TiN and ZrN with Fm-3m structure, (b) for BeO with hexagonal P63mc structure, (c) for Al 2O 3 with rhombohedral R-3c, and (d) for TiB 2 and ZrB 2 with cubic F-43m structure. (3) Fig. 3. Correlation of the cohesive energy with the linear thermal Fig. 4. Correlation of the cohesive energy with the linear thermal expansion coefficient measured at 500 K. 10) 13) expansion coefficient measured at 1000 K. 10) 13) 243

4 JCS-Japan Tsuru et al.: Estimation of linear thermal expansion coefficient from cohesive energy obtained by ab-initio calculation of metals and ceramics (a) (b) (c) Fig. 7. Calculation models of (a) Na, Ca and Al with cubic Fm-3m structure, (b) Na and Nb with cubic Im-3m structure and (c) Ti, Zr with hexagonal P63/mmc structure. (a) (b) (c) (d) Fig. 5. Correlation of the cohesive energy with the linear thermal expansion coefficient measured at 1200 K. 10) 13) Fig. 8. Calculation models of (a) CdO, MgO, TiC, HfC, TiN and ZrN with Fm-3m structure where the white atoms are Cd, Mg, Ti and Hf, and black atoms are O, C and N, (b) BeO with hexagonal P63/mmc structure where the white atoms are Be and the black atoms are O, (c) Al 2O 3 with rhombohedral R-3c structure where the white atoms are Al and the black atoms are O and (d) TiB 2 and ZrB 2 with cubic F-43m structure where the white atoms are Ti and Zr, and the black atoms are B. Fig. 6. Comparison in the cohesive energy between calculation and experiment. 10),11) The average discrepancy between the experimental and calculated values was in the range of ± 14%. In this way, the cohesive energy (E coh) of any material will be obtained by the ab-initio calculation. The geometry optimization of these materials spent the calculation time in the range of 12.8 s for HfC to 32.0 min for Al 2O 3 by using the Intel Xeon 3.0 GHz cluster system of eight processers. In the previous work, we revealed the calculation method of the linear thermal expansion coefficient using abinitio molecular dynamics. 2) In the case of Ti and Al, the calculation times of the cohesive energies were only 60.6 and s where that of the linear thermal expansion coefficients by ab-initio molecular dynamics were 7.6 and 10.2 h with same calculation models, respectively. 2) Thus, the calculation times were greatly shorten. Then, by utilizing the experientially determined inverse proportional relation, α LE coh = C, the linear thermal expansion coefficient (α L) will be predicted from the calculated cohesive energy alone. 244

5 Journal of the Ceramic Society of Japan 118 [3] JCS-Japan 5. Conclusion By using experimental values, the cohesive energy was shown to be nearly proportional to the melting point. The relation between the linear thermal expansion coefficient (α L) and the cohesive energy (E coh) was examined with metals, oxides, borides, carbides and nitrides. It was found that the linear thermal expansion coefficient is inversely proportional to the cohesive energy. In the relation of α LE cho = C, the constants C were determined to be 48.14, 46.73, and [10 6 ev/k] at 295, 500, 1000 and 1200 K, respectively. These results suggest a great possibility of evaluating the linear thermal expansion coefficient simply from the calculation of the cohesive energy. The classical molecular dynamics calculation necessitates the use of the potential parameters and hence it is applicable to very restricted materials depending on the constituent elements. On the contrary, the ab-initio molecular dynamics calculation is applicable to every material containing any kinds of elements in the periodic table. However, its calculation is time consuming in use of the large calculation model containing trace elements. On the other hand, our proposed method for obtaining the linear thermal expansion coefficient from the cohesive energy can be worked out in metals, oxides, borides, carbides and nitrides by doing the ab-initio calculation once for the geometry optimization. The simulation time is greatly shortened in this calculation as compared with the ab-initio molecular dynamics calculation where many repetitions of the geometry optimized calculations are required inevitably. For example, in the case of Ti and Al, the respective simulation times were only 60.6 and s, which were much shorter than 7.6 and 10.2 h to be needed for the abinitio molecular dynamics. 2) Thus, our prediction method for the linear thermal expansion coefficient using the ab-initio calculation is very useful for new materials design. References 1) J. Adachi, K. Kurosaki, M. Uno and S. Yamanaka, J. Alloys Comp., 396, 260 (2005). 2) Y. Tsuru, M. Shimazu, M. Shiono and M. Morinaga, J. Appl. Phys., 48, (2009). 3) L. G. Van Uitert, H. M. O Brayan, M. E. Lines, H. J. Guggenheim and G. Zydzik, Mat. Res. Bull., 12, (1977). 4) W. Kohn and L. J. Sham, Phys Rev. A, 140, (1965). 5) S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. J. Probert, K. Refson and M. C. Payne, Zeitschrift fuer Kristallographie, 220[5,6], (2005). 6) J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, Phys. Rev. B, 46, (1992). 7) D. Vanderbilt, Phys. Rev. B, 41, (1990). 8) H. J. Monkhorst and J. D. Pack, Phys. Rev. B, 13, (1976). 9) B. G. Pfrommer, M. Cote, S. G. Louie and M. L. Cohen, J. Comput. Phys., 131, (1997). 10) R. C. Weast, M. J. Astle and W. H. Beyer, CRC handbook of Chemistry and Physics, A Ready-reference Book of Chemical and Physical Data (84th ed.), CRC press (2003). 11) L. Brewer, The Cohesive Energies of the Elements, Lawrence Berkeley Laboratory Report LBL-3720 (1977). 12) Y. S. Touloukian, R. K. Kirby, R. E. Taylor and P. D. Desai, Thermophysical Properties of Matter vol.12 - Metallic Elements and Alloys (Ann Arbor, Mich. : University Microfilms International, 1975). 13) Y. S. Touloukian, R. K. Kirby, R. E. Taylor and T. Y. R. Lee, Thermophysical Properties of Matter vol.13 Nonmetallic Solids (Ann Arbor, Mich. : University Microfilms International, 1977). 14) P. Goudochnikov and A. J. Bell, J. Phys.: Condens. Matter, 19, (2007). 245