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Ab initio molecular dynamics study on thermal expansion of solid-solution compounds in MAX phase
Computational Materials Science 103 (2015) 200–203
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Ab initio molecular dynamics study on thermal expansion of solid-solution compounds in MAX phase Hai Hu a, Xinzhu Chen a, Xiujian Zhao a, Neng Li a,b,c,⇑ a
State Key Laboratory of Silicate Materials for Architectures, Wuhan University of Technology, 122 Luoshi Road, Wuhan 430070, PR China Center for Photovoltaics and Solar Energy, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, PR China c Department of Physics and Astronomy, University of Missouri-Kansas City, Kansas City, MO 64110, USA b
a r t i c l e
i n f o
Article history: Received 24 November 2014 Received in revised form 18 March 2015 Accepted 24 March 2015 Available online 15 April 2015 Keywords: Ab initio molecular dynamics Coefficient of thermal expansion High temperature MAX phases
a b s t r a c t The thermal expansion (CTE) of the Cr-based solid-solution compounds in Mn+1AXn phases is explored by ab initio molecular dynamics (AIMD) method. The calculated CTE and thermal expansion anisotropy (TEA) agree well with experimental measurement. The anisotropy is reduced considerably as exemplified in the case of Cr2(Al0.667Ge0.333)C where the CTE of c axis is reduced from 11.2 10 6/K to 10.3 10 6/K. For (Cr1.833Ti0.167)AlC and (Cr1.333Ti0.667)AlC, the calculated results show that the Ti doping does reduce the average CTE of Cr2AlC from 12 to 11. This study demonstrates that a solid-solution approach is a route for tuning a physical property like the thermal expansion. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction The Mn+1AXn phases (or ‘‘MAX phases’’, n = 1–3), as one group of inherently nanolaminated compounds, where M is an early transition metal, A is an A-group element, and X is either C or N [1–4]. The MAX phases have attracted immense attentions in the recent years not only in order to shed light on their machineable and suitable for bulk materials application but also owing to their good coefficient of thermal expansion (CTE) under high temperature. In another side, MAX phases are also attracted much more and more attentions on elastic and thermodynamic properties and their behaviors under high pressure [5–13]. Moreover, MAX phases is also well-known as a promising strategy to tailor properties over tunneling element on the M, A and/or X sites forming solid solutions, which is good for many applications requiring further alloying designs with optimizing properties such as CTE and thermal expansion anisotropy (TEA). For example, the CTEs of Ti2AlC and Nb2AlC are the same (8.7 10 6 K 1), but marginally lower than that of their 50–50% solid solution at 8.9 10 6 K 1 [14]. Similarly, it has been reported that substituting Ge for Si in Ti3SiC2 did not substantially affect the CTEs of a 50–50% Si and Ge solid solution relative to the end members [15]. Although, there are already some experimental works to ⇑ Corresponding author at: State Key Laboratory of Silicate Materials for Architectures, Wuhan University of Technology, 122 Luoshi Road, Wuhan 430070, PR China. E-mail addresses: [email protected], [email protected] (N. Li). http://dx.doi.org/10.1016/j.commatsci.2015.03.034 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
elucidate the alloying effect, but it is critical and yet it is also difficult, time-consuming and expensive. Thus, there is need a theoretical method to not only maintain an isotropic but also reduce average CTE of solid solutions in MAX phases with high temperature. In this regard, ab initio molecular dynamics (AIMD) offer an effective method to obtain the critical properties liking CTE and TEA under high temperature. The main purpose of this work is to use AIMD approach to tailor the thermal expansions of the selected Cr-based MAX phases. The rest of this article is organized as follows: The computational details are given in Section 2. In Section 3, we present the results and discussions. Finally, a brief summary is presented in Section 4.
2. Computational details In the present work, for all calculations, we used the AIMD approach as implemented in Vienna ab initio simulation package (VASP) [16,17]. For pure Cr2AlC and Cr2GeC, 3 3 1 supercells with 72 atoms are adopted for modeling the dopant effect. Such a size is adequate owing to the relatively small number of atoms in the unit cell (8 atoms). For the other three solid solution systems Cr2(Al0.667Ge0.333)C, (Cr1.333Ti0.667)AlC, and (Cr1.833Ti0.167)AlC are built as fellow, respectively: six Al atoms are replaced by the Ge atoms [Cr36(Al12Ge6)C18], twelve Cr atoms are replaced by the Ti atoms [(Cr24Ti12)Al18C18], and three Cr atoms are replaced by the Ti atoms [(Cr33Ti03)Al18C18], respectively (see Fig. 1). The crystal structures are optimized with AIMD method for each temperature,
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Fig. 1. Side view of (a) the 3 3 1 supercell of Cr2AlC (Cr36Al18C18), (b) Cr2(Al0.667Ge0.333)C (Cr36Al12Ge06C18), (c) (Cr1.833Ti0.167)AlC (Cr33Ti03Al18C18).
and used to extract the lattice constant under high-temperature. AIMD calculations in this work within isothermal–isobaric ensemble (NPT ensemble) with each ionic MD time step of 2 femtosecond (fs) for duration up to 5 picoseconds (ps). The more details of the calculated here as following parameters: (1) The projector augmented wave with Perdew–Burke–Ernzerhof (PAW-PBE) potentials [18] with the generalized gradient approximation (GGA); (2) electronic convergence criterion is set at 10 4 eV; (3) a high energy cutoff of 500 eV; (4) 2 2 1 k-point sampling is used.
3. Results and discussion The CTE of Cr2GeC is the highest among all known MAX phases with an expansion along the c axis higher than in the basal plane. In contrast, in Cr2AlC, expansion along a axis is higher than along c axis [19]. It is therefore reasonable to assume that somewhere along the composition domain; the two thermal expansions should cross. One could thus obtain a solid solution with isotropic thermal
expansion and ideally eliminate the issues with residual strain upon cooling. To this end, we have design a few solid solution systems Cr2(Al0.667Ge0.333)C, and (Cr1.833Ti0.167)AlC (see Fig. 1), and determined doping effect on their structural parameters and thermal expansion coefficients in the wide temperature range from 298 K to 1473 K. Thermal expansion is a good indicator to validate our method. Many previous works [20–22] had assumed linear thermal expansion within the temperature range they have considered. We also consider a linear fit for the thermal expansion of the a-axis and c-axis when we calculate the CTE. The calculated CTE and TEA is tabulated in Table 1 along with the literatures’ results for the MAX phases [21,22]. As Table 1 shows that our results are good agreement with the experimental measurement and other theoretical results. The anisotropy ratio ac/aa, listed in column 6 of Table 1, for the most part, is 1, as might be anticipated since the c direction involves the relatively weaker M–A bonds. This value may have been underestimated the experimental results, but the linearity of the change in lattice parameters and the trend in anisotropy are in agreement with the experimental results. Thus, AIMD can be used to tailor the alloying strategy for the MAX phase so as to optimize its thermal properties. While there appears to be a good correlation between the ratio of axial contribution to the Gruneisan parameter (ac/aa), one cannot apply this approach rigorously especially for alloying design where the change in TEA, while it has a very important consequence, is nevertheless quite small in magnitude. For example, there has been a recent work showing evidence that the thermal expansion anisotropy of Cr2AlC (<1) can be made almost isotropic by partly replacing the Al with Ge [24]. This approach was successful due to the fact that the fact that Cr2GeC has a TEA of >1 and thus by substituting part of Al with Ge in solid solution along the Cr2AlC–Cr2GeC compositions [Cr2(Al0.667Ge0.333)C], the overall TEA will approach 1. This strategy will be in contradiction to a strategy to be based on the calculated c/a ratio as the ac/aa for Cr2GeC was actually lower than that of Cr2AlC, quite the opposite of the trend in ac/aa. Fig. 2a and b shows the compositional dependence of the a and c lattice parameters with room temperature (T = 298 K), respectively. These results show that the c parameter increases with increasing Al content (Fig. 2a) while the a parameter decreases (Fig. 2b). Fig. 2c shows the relative changes in a, c, c/a and unit cell volumes, Vuc, as a function of increasing Al content x. From these results it is clear that while c and c/a increase with x, Vuc remains essentially constant and the a lattice parameter decreases. Furthermore, the linear evolution of a, c and c/a, with increasing Al content (Fig. 2c) is not a general trend in MAX phases even if it is also the case for (Ti1 x,Vx)2AlC and (V1 x,Crx)2AlC solid solutions [25]. For instance, the a lattice parameter varies linearly as a function of x in Ti2Al(CxN1 x) [26] but not the c lattice parameter.
Table 1 Lattice parameters and thermal expansions for the four the selected MAX-phases. Compound Cr2AlC Cr2(Al0.667Ge0.333)C Cr2GeC (Cr1.833Ti0.167)AlC (Cr1.333Ti0.667)AlC Ti2AlC a b c
This work. Ref. [23]. Ref. [19].
a (Å)
aa (10
c (Å) b
b
2.8571(2) 2.9212a 2.9616a 2.9500(2)b 2.9991a 2.9290a 2.9408a
12.8208(16) 12.4201a 12.323a 12.1010(10)b 12.0190a 12.5280a 12.8390a
3.0730a
13.6440a
6 b
12.8(3) 12.7(6)a 11.2(0)a 12.9(1)b 10.6(0)a 12.6(0)a 14.7(0)a 7.1(3)c 8.9(2)a
K
1
)
ac (10
6 b
12.1(1) 12.0(1)a 14.5(1)a 17.6(2)b 16.2(3)a 9.61(1)a 8.72(0)a 10.0(5)c 9.5(1)a
K
1
)
ac/aa 0.94(5)b 0.94(0)a 1.29(6)a 1.37(1)b 1.53(1)a 0.76(3)a 0.59(3)a 1.41(0)c 1.06(6)a
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Fig. 2. For Cr2AlxGe1 xC compounds with room temperature T = 298 K, evolution as a function of the Al content (x) of the c and a lattice parameters ((a) and (b) respectively), the relative change of a, c, c/a and the unit cell volume (c), deduced linear thermal expansion coefficients along the a and c axis (aa and ac) (d).
Fig. 3. Evolution of the a and c lattice parameters, (a) and (b), as a function of the temperature.
Neither a nor c vary linearly with x for (Ti1 x,Crx)2AlC compounds [25]. Therefore, there are no general simple rules or empiric laws that predict the evolution of the lattice parameters. Fig. 2d as a function of composition. From these results it is clear that the thermal expansion along [0 0 1] in Cr2AlC is lower than in Cr2GeC. In Fig. 3a and b, the temperature dependencies of the a and c lattice parameters, respectively, are plotted as a function of Cr2AlC, Cr2GeC and composition Cr2(Al0.667Ge0.333)C. These results were least-squares-fitted and converted to CTEs along the a and c
directions aa and ac respectively. From these results it is clear that the effect of increasing the Al-content on ac is higher than on aa, the latter remaining almost constant. Significantly, the two values are equal around the Cr2(Al0.667Ge0.333)C composition. Fig. 4a and b shows the temperature dependencies of the other two solid solutions in MAX phases: (Cr1.833Ti0.167)AlC and (Cr1.333Ti0.667)AlC. As Fig. 4a and b shows that Ge substitutes for Al reduces the TEA, but the CTE values increase compared to the pure Cr2AlC phase. It is therefore reasonable to assume that a small
H. Hu et al. / Computational Materials Science 103 (2015) 200–203
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Fig. 4. Evolution of the a and c lattice parameters of (Cr1.833Ti0.167)AlC and (Cr1.333Ti0.667)AlC, (a) and (b), as a function of the temperature.
level of substitution may be OK, but lost of [Al] is not good for oxidation resistance. Thus, we construct the solid solution using Ti small substitutes for Cr, which may also reduce the CTE values, minimizing thermal residual stresses. This approach also benefit of no reduction in [Al] content in MAX phase. As Table 1 shows that, for (Cr1.833Ti0.167)AlC and (Cr1.333Ti0.667)AlC, there are need improve for both minimizing thermal residual stresses and keeping oxidation resistance. 4. Conclusion In conclusion, the most important result of the work is the demonstration of compositional tuning of the thermal expansion coefficients by AIMD’s NPT simulations, so as to render them virtually isotropic for the Cr2(Al0.667Ge0.333)C, (Cr1.833Ti0.167)AlC and (Cr1.333Ti0.667)AlC compositions. These compounds should thus respond to temperature variations like a cubic solid, which has the key benefit of absence of residual stresses at room temperature when cooling from higher temperatures. Moreover, we now can tailor the CTE of higher order and/or alloyed MAX phases through theoretical means, which is be able to accelerate the adaptation of MAX phase as protective coatings for high-temperature applications by lowering the average and at the same maintaining a reasonably isotropic CTEs. Acknowledgments This work is supported by the Natural Science Foundation of Shenzhen under Grant No. JCYJ20130402113127530. This work is also supported National Natural Science Foundation (NSFC) with No. 51032005, Natural Science Foundation (NSF) of Hubei Province with No. 2013CFA008, Doctoral Fund of Ministry of education priority development projects with No. 20130143130002, and the key technology innovation project of Hubei Province with No. 2013AAA005. The author acknowledges the help and support of Prof. Ridwan Sakidja, Prof. Wai-Yim Ching and ESG group in the Department of Physics and Astronomy, University of Missouri-Kansas City.
References [1] M.W. Barsoum, Prog. Solid State Chem. 28 (2000) 201–280. [2] P. Eklund, M. Beckers, U. Jansson, H. Högberg, L. Hultman, Thin Solid Films 518 (2010) 1851–1878. [3] M.W. Barsoum, M.M. Radovic, Annu. Rev. Mater. Res. 41 (2011) 195–227. [4] Z.M. Sun, Int. Mater. Rev. 56 (2011) 143–166. [5] Z.J. Yang, L. Tang, A.M. Guo, X.L. Cheng, Z.H. Zhu, X.D. Yang, J. Appl. Phys. 114 (2013) 083506. [6] Z.J. Yang, J. Li, R.F. Linghu, X.S. Song, X.L. Cheng, Z.H. Zhu, X.D. Yang, Eur. Phys. J. B 86 (2013) 208. [7] Z.J. Yang, J. Li, R.F. Linghu, X.S. Song, X.L. Cheng, X.D. Yang, Can. J. Phys. 91 (2013) 1–4. [8] Z.J. Yang, J. Li, R.F. Linghu, X.L. Cheng, X.D. Yang, J. Alloys Compd. 574 (2013) 573–579. [9] Z.J. Yang, J. Li, R.F. Linghu, X.L. Cheng, X.D. Yang, J. Alloys Compd. 551 (2013) 435–439. [10] Z.J. Yang, Q. Liu, J. Li, Z. Wang, A.M. Guo, R.F. Ling, X.L. Cheng, X.D. Yang, China Phys. Mech. 56 (5) (2013) 916–924. [11] Z.J. Yang, Y.D. Guo, R.F. Linghu, X.D. Yang, Chin. Phys. B 21 (3) (2012) 036301. [12] Z.J. Yang, Y.D. Guo, R.F. Linghu, X.L. Cheng, X.D. Yang, Chin. Phys. B 21 (5) (2012) 056301. [13] Z.J. Yang, R.F. Linghu, X.L. Cheng, X.D. Yang, Acta Phys. Sin. 61 (4) (2012) 046301 (in Chinese). [14] M.W. Barsoum, I. Salama, T. El-Raghy, J. Golczewski, W.D. Porter, H. Wang, Metall. Mater. Trans. 33a (2002) 2775–2779. [15] P. Finkel, B. Seaman, K. Harrell, J. Palma, J.D. Hettinger, S.E. Lofland, Phys. Rev. B 70 (2004) 085104. [16] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. [17] G. Kresse, J. Furthmüller, Comput. Mater. Sci. 6 (1996) 15. [18] P.E. Blochl, Phys. Rev. B 50 (24) (1994) 17953; J.P. Perdew, Phys. Rev. Lett. 55 (1985) 1665. [19] T.H. Scabarozi, S. Amini, O. Leaffer, A. Ganguly, S. Gupta, W. Tambussi, J. Appl. Phys. 105 (2009) 013543. [20] F. Chu, D.J. Thoma, K. McClellan, P. Peralta, Y. He, Intermetallics 7 (1999) 611. [21] H.L. Zhao, M.J. Kramer, M. Akinc, Intermetallics 12 (2004) 493. [22] C.J. Rawn, J.H. Schneibel, C.M. Hoffmann, C.R. Hubbard, Intermetallics 9 (2001) 209. [23] T. El-Raghy, M. Ali, M.W. Barsoum, Metall. Mater. Trans. A 31A (2000) 1857. [24] T. Cabioch, P. Eklund, V. Mauchamp, M. Jaouen, M.W. Barsouma, J. Eur. Ceram. Soc. 33 (2013) 897–904. [25] J.C. Schuster, H. Nowotny, C. Vaccaro, J. Solid State Chem. 32 (1980) 213–219. [26] T. Cabioch, P. Eklund, V. Mauchamp, M. Jaouen, J. Eur. Ceram. Soc. 32 (2012) 1803–1811.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Ab initio molecular dynamics study on thermal expansion of solid-solution compounds in MAX phase Hai Hu a, Xinzhu Chen a, Xiujian Zhao a, Neng Li a,b,c,⇑ a
State Key Laboratory of Silicate Materials for Architectures, Wuhan University of Technology, 122 Luoshi Road, Wuhan 430070, PR China Center for Photovoltaics and Solar Energy, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, PR China c Department of Physics and Astronomy, University of Missouri-Kansas City, Kansas City, MO 64110, USA b
a r t i c l e
i n f o
Article history: Received 24 November 2014 Received in revised form 18 March 2015 Accepted 24 March 2015 Available online 15 April 2015 Keywords: Ab initio molecular dynamics Coefficient of thermal expansion High temperature MAX phases
a b s t r a c t The thermal expansion (CTE) of the Cr-based solid-solution compounds in Mn+1AXn phases is explored by ab initio molecular dynamics (AIMD) method. The calculated CTE and thermal expansion anisotropy (TEA) agree well with experimental measurement. The anisotropy is reduced considerably as exemplified in the case of Cr2(Al0.667Ge0.333)C where the CTE of c axis is reduced from 11.2 10 6/K to 10.3 10 6/K. For (Cr1.833Ti0.167)AlC and (Cr1.333Ti0.667)AlC, the calculated results show that the Ti doping does reduce the average CTE of Cr2AlC from 12 to 11. This study demonstrates that a solid-solution approach is a route for tuning a physical property like the thermal expansion. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction The Mn+1AXn phases (or ‘‘MAX phases’’, n = 1–3), as one group of inherently nanolaminated compounds, where M is an early transition metal, A is an A-group element, and X is either C or N [1–4]. The MAX phases have attracted immense attentions in the recent years not only in order to shed light on their machineable and suitable for bulk materials application but also owing to their good coefficient of thermal expansion (CTE) under high temperature. In another side, MAX phases are also attracted much more and more attentions on elastic and thermodynamic properties and their behaviors under high pressure [5–13]. Moreover, MAX phases is also well-known as a promising strategy to tailor properties over tunneling element on the M, A and/or X sites forming solid solutions, which is good for many applications requiring further alloying designs with optimizing properties such as CTE and thermal expansion anisotropy (TEA). For example, the CTEs of Ti2AlC and Nb2AlC are the same (8.7 10 6 K 1), but marginally lower than that of their 50–50% solid solution at 8.9 10 6 K 1 [14]. Similarly, it has been reported that substituting Ge for Si in Ti3SiC2 did not substantially affect the CTEs of a 50–50% Si and Ge solid solution relative to the end members [15]. Although, there are already some experimental works to ⇑ Corresponding author at: State Key Laboratory of Silicate Materials for Architectures, Wuhan University of Technology, 122 Luoshi Road, Wuhan 430070, PR China. E-mail addresses: [email protected], [email protected] (N. Li). http://dx.doi.org/10.1016/j.commatsci.2015.03.034 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
elucidate the alloying effect, but it is critical and yet it is also difficult, time-consuming and expensive. Thus, there is need a theoretical method to not only maintain an isotropic but also reduce average CTE of solid solutions in MAX phases with high temperature. In this regard, ab initio molecular dynamics (AIMD) offer an effective method to obtain the critical properties liking CTE and TEA under high temperature. The main purpose of this work is to use AIMD approach to tailor the thermal expansions of the selected Cr-based MAX phases. The rest of this article is organized as follows: The computational details are given in Section 2. In Section 3, we present the results and discussions. Finally, a brief summary is presented in Section 4.
2. Computational details In the present work, for all calculations, we used the AIMD approach as implemented in Vienna ab initio simulation package (VASP) [16,17]. For pure Cr2AlC and Cr2GeC, 3 3 1 supercells with 72 atoms are adopted for modeling the dopant effect. Such a size is adequate owing to the relatively small number of atoms in the unit cell (8 atoms). For the other three solid solution systems Cr2(Al0.667Ge0.333)C, (Cr1.333Ti0.667)AlC, and (Cr1.833Ti0.167)AlC are built as fellow, respectively: six Al atoms are replaced by the Ge atoms [Cr36(Al12Ge6)C18], twelve Cr atoms are replaced by the Ti atoms [(Cr24Ti12)Al18C18], and three Cr atoms are replaced by the Ti atoms [(Cr33Ti03)Al18C18], respectively (see Fig. 1). The crystal structures are optimized with AIMD method for each temperature,
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Fig. 1. Side view of (a) the 3 3 1 supercell of Cr2AlC (Cr36Al18C18), (b) Cr2(Al0.667Ge0.333)C (Cr36Al12Ge06C18), (c) (Cr1.833Ti0.167)AlC (Cr33Ti03Al18C18).
and used to extract the lattice constant under high-temperature. AIMD calculations in this work within isothermal–isobaric ensemble (NPT ensemble) with each ionic MD time step of 2 femtosecond (fs) for duration up to 5 picoseconds (ps). The more details of the calculated here as following parameters: (1) The projector augmented wave with Perdew–Burke–Ernzerhof (PAW-PBE) potentials [18] with the generalized gradient approximation (GGA); (2) electronic convergence criterion is set at 10 4 eV; (3) a high energy cutoff of 500 eV; (4) 2 2 1 k-point sampling is used.
3. Results and discussion The CTE of Cr2GeC is the highest among all known MAX phases with an expansion along the c axis higher than in the basal plane. In contrast, in Cr2AlC, expansion along a axis is higher than along c axis [19]. It is therefore reasonable to assume that somewhere along the composition domain; the two thermal expansions should cross. One could thus obtain a solid solution with isotropic thermal
expansion and ideally eliminate the issues with residual strain upon cooling. To this end, we have design a few solid solution systems Cr2(Al0.667Ge0.333)C, and (Cr1.833Ti0.167)AlC (see Fig. 1), and determined doping effect on their structural parameters and thermal expansion coefficients in the wide temperature range from 298 K to 1473 K. Thermal expansion is a good indicator to validate our method. Many previous works [20–22] had assumed linear thermal expansion within the temperature range they have considered. We also consider a linear fit for the thermal expansion of the a-axis and c-axis when we calculate the CTE. The calculated CTE and TEA is tabulated in Table 1 along with the literatures’ results for the MAX phases [21,22]. As Table 1 shows that our results are good agreement with the experimental measurement and other theoretical results. The anisotropy ratio ac/aa, listed in column 6 of Table 1, for the most part, is 1, as might be anticipated since the c direction involves the relatively weaker M–A bonds. This value may have been underestimated the experimental results, but the linearity of the change in lattice parameters and the trend in anisotropy are in agreement with the experimental results. Thus, AIMD can be used to tailor the alloying strategy for the MAX phase so as to optimize its thermal properties. While there appears to be a good correlation between the ratio of axial contribution to the Gruneisan parameter (ac/aa), one cannot apply this approach rigorously especially for alloying design where the change in TEA, while it has a very important consequence, is nevertheless quite small in magnitude. For example, there has been a recent work showing evidence that the thermal expansion anisotropy of Cr2AlC (<1) can be made almost isotropic by partly replacing the Al with Ge [24]. This approach was successful due to the fact that the fact that Cr2GeC has a TEA of >1 and thus by substituting part of Al with Ge in solid solution along the Cr2AlC–Cr2GeC compositions [Cr2(Al0.667Ge0.333)C], the overall TEA will approach 1. This strategy will be in contradiction to a strategy to be based on the calculated c/a ratio as the ac/aa for Cr2GeC was actually lower than that of Cr2AlC, quite the opposite of the trend in ac/aa. Fig. 2a and b shows the compositional dependence of the a and c lattice parameters with room temperature (T = 298 K), respectively. These results show that the c parameter increases with increasing Al content (Fig. 2a) while the a parameter decreases (Fig. 2b). Fig. 2c shows the relative changes in a, c, c/a and unit cell volumes, Vuc, as a function of increasing Al content x. From these results it is clear that while c and c/a increase with x, Vuc remains essentially constant and the a lattice parameter decreases. Furthermore, the linear evolution of a, c and c/a, with increasing Al content (Fig. 2c) is not a general trend in MAX phases even if it is also the case for (Ti1 x,Vx)2AlC and (V1 x,Crx)2AlC solid solutions [25]. For instance, the a lattice parameter varies linearly as a function of x in Ti2Al(CxN1 x) [26] but not the c lattice parameter.
Table 1 Lattice parameters and thermal expansions for the four the selected MAX-phases. Compound Cr2AlC Cr2(Al0.667Ge0.333)C Cr2GeC (Cr1.833Ti0.167)AlC (Cr1.333Ti0.667)AlC Ti2AlC a b c
This work. Ref. [23]. Ref. [19].
a (Å)
aa (10
c (Å) b
b
2.8571(2) 2.9212a 2.9616a 2.9500(2)b 2.9991a 2.9290a 2.9408a
12.8208(16) 12.4201a 12.323a 12.1010(10)b 12.0190a 12.5280a 12.8390a
3.0730a
13.6440a
6 b
12.8(3) 12.7(6)a 11.2(0)a 12.9(1)b 10.6(0)a 12.6(0)a 14.7(0)a 7.1(3)c 8.9(2)a
K
1
)
ac (10
6 b
12.1(1) 12.0(1)a 14.5(1)a 17.6(2)b 16.2(3)a 9.61(1)a 8.72(0)a 10.0(5)c 9.5(1)a
K
1
)
ac/aa 0.94(5)b 0.94(0)a 1.29(6)a 1.37(1)b 1.53(1)a 0.76(3)a 0.59(3)a 1.41(0)c 1.06(6)a
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Fig. 2. For Cr2AlxGe1 xC compounds with room temperature T = 298 K, evolution as a function of the Al content (x) of the c and a lattice parameters ((a) and (b) respectively), the relative change of a, c, c/a and the unit cell volume (c), deduced linear thermal expansion coefficients along the a and c axis (aa and ac) (d).
Fig. 3. Evolution of the a and c lattice parameters, (a) and (b), as a function of the temperature.
Neither a nor c vary linearly with x for (Ti1 x,Crx)2AlC compounds [25]. Therefore, there are no general simple rules or empiric laws that predict the evolution of the lattice parameters. Fig. 2d as a function of composition. From these results it is clear that the thermal expansion along [0 0 1] in Cr2AlC is lower than in Cr2GeC. In Fig. 3a and b, the temperature dependencies of the a and c lattice parameters, respectively, are plotted as a function of Cr2AlC, Cr2GeC and composition Cr2(Al0.667Ge0.333)C. These results were least-squares-fitted and converted to CTEs along the a and c
directions aa and ac respectively. From these results it is clear that the effect of increasing the Al-content on ac is higher than on aa, the latter remaining almost constant. Significantly, the two values are equal around the Cr2(Al0.667Ge0.333)C composition. Fig. 4a and b shows the temperature dependencies of the other two solid solutions in MAX phases: (Cr1.833Ti0.167)AlC and (Cr1.333Ti0.667)AlC. As Fig. 4a and b shows that Ge substitutes for Al reduces the TEA, but the CTE values increase compared to the pure Cr2AlC phase. It is therefore reasonable to assume that a small
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203
Fig. 4. Evolution of the a and c lattice parameters of (Cr1.833Ti0.167)AlC and (Cr1.333Ti0.667)AlC, (a) and (b), as a function of the temperature.
level of substitution may be OK, but lost of [Al] is not good for oxidation resistance. Thus, we construct the solid solution using Ti small substitutes for Cr, which may also reduce the CTE values, minimizing thermal residual stresses. This approach also benefit of no reduction in [Al] content in MAX phase. As Table 1 shows that, for (Cr1.833Ti0.167)AlC and (Cr1.333Ti0.667)AlC, there are need improve for both minimizing thermal residual stresses and keeping oxidation resistance. 4. Conclusion In conclusion, the most important result of the work is the demonstration of compositional tuning of the thermal expansion coefficients by AIMD’s NPT simulations, so as to render them virtually isotropic for the Cr2(Al0.667Ge0.333)C, (Cr1.833Ti0.167)AlC and (Cr1.333Ti0.667)AlC compositions. These compounds should thus respond to temperature variations like a cubic solid, which has the key benefit of absence of residual stresses at room temperature when cooling from higher temperatures. Moreover, we now can tailor the CTE of higher order and/or alloyed MAX phases through theoretical means, which is be able to accelerate the adaptation of MAX phase as protective coatings for high-temperature applications by lowering the average and at the same maintaining a reasonably isotropic CTEs. Acknowledgments This work is supported by the Natural Science Foundation of Shenzhen under Grant No. JCYJ20130402113127530. This work is also supported National Natural Science Foundation (NSFC) with No. 51032005, Natural Science Foundation (NSF) of Hubei Province with No. 2013CFA008, Doctoral Fund of Ministry of education priority development projects with No. 20130143130002, and the key technology innovation project of Hubei Province with No. 2013AAA005. The author acknowledges the help and support of Prof. Ridwan Sakidja, Prof. Wai-Yim Ching and ESG group in the Department of Physics and Astronomy, University of Missouri-Kansas City.
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