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Atomic-size effect and solid solubility of multicomponent alloys
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Atomic-size effect and solid solubility of multicomponent alloys ⇑
Zhijun Wang,a,b Yunhao Huang,b Yong Yang,a Jincheng Wangb and C.T. Liua, a
Center of Advanced Structural Materials, Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong, China b State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China Received 27 August 2014; accepted 14 September 2014
In physical metallurgy, solid solubility of alloys is known to play a vital role in determining their physical/mechanical properties. Hume–Rothery rules show the great effect of size difference between solvent and solute atoms on the solid solubility of binary alloys. However, modern multicomponent systems, such as high-entropy alloys, defy the classic atomic size effect due to the absence of solvent and/or solute atoms. Here, we propose an effective atomic size parameter by considering atomic packing misfitting in multicomponent systems. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Alloy solubility; High entropy alloys; Atomic-size effect; Atomic packing
The solubility of a solute element in an alloy is an important aspect of alloy design. Through the addition of soluble elements, one may obtain a solid solution effect that can significantly improve the alloy’s performance. For a substitutional binary alloy, the Hume–Rothery rules provide the key guidance to evaluating whether a solid solution can be formed [1]. These include: (i) the relative atomic size difference between the solute and solvent elements should be less than 15%; (ii) the formation of stable intermediate compounds should be restricted by carefully choosing the combination of metallic elements; and (iii) the electron concentration of the constituent elements should be tuned in favor of the formation of solid solutions, not otherwise. The Hume–Rothery rules have been used extensively for more than half a century [2–6]. However, the rules are poses a challenge with the recent design of multicomponent alloys [7–10], in some of which the molar fractions of the constituent elements are equal or nearly equal, such as high-entropy alloys (HEAs) [9–13]. As a result, there is no distinct identity of “solvent” or “solute” atoms in such alloys. Therefore, the atomic size factor as defined by the Hume–Rothery rules cannot be applied directly to HEAs. Despite this, some empirical parameters have been suggested as extensions of the Hume–Rothery rules to explain the solid solubility of HEAs [14]; however, they do not do so satisfactorily when compared with experimental data [15]. The atomic size difference is of primary importance in determining the solubility of alloys. The critical value of 15% atomic size difference in binary alloys was confirmed
⇑ Corresponding cityu.edu.hk
author. Tel.: +852 3442 7213; e-mail: chainliu@
by the continuum elastic theory [16]. However, it is difficult to define an atomic size parameter in relation to solubility of HEAs. For simplicity, the polydispersity resulting from the atomic size difference has been commonly used for predicting the solid solubility of a given multicomponent alloy [14], which is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi n X Xn 2 ; r ¼ d¼ c ð1 r = ci r i ð1Þ r Þ i i i i
where ci is the atomic concentration of the ith atom and ri is the atomic radius of the ith atom. It has been widely applied in the systems of hard-sphere fluids [17], metallic glasses [18] and HEAs [14,15]. In the hard-sphere fluid, d ¼ 0:06 is the criterion used to predict crystalline structure formation, and this criterion was also claimed in the HEA investigation. However, this parameter does not describe the solubility of HEAs very well; as a result, many intermetallics have been detected around d ¼ 0:06 [15]. Basically, the d parameter takes the average effect of the atomic size difference of all elements in the alloy. However, the solid solution instability may essentially be determined by the largest and smallest atoms in multicomponent alloy systems. Moreover, the physical meaning of d in determining the solubility is also not well understood, and it cannot return to the Hume–Rothery limit. Therefore it is necessary to explore the new and physically acceptable parameter of the atomic size effect on the solubility of multicomponent alloys, especially HEAs, which have received increasing attention recently from the materials community [9,11]. In this letter, we address this issue by considering the atomic size effect based on atomic packing behavior. The proposed parameter not only has a clear physical meaning,
http://dx.doi.org/10.1016/j.scriptamat.2014.09.010 1359-6462/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Please cite this article in press as: Z. Wang et al., Scripta Mater. (2014), http://dx.doi.org/10.1016/j.scriptamat.2014.09.010
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but is also consistent with the 15% atomic size difference in the Hume–Rothery rules. In particular, this parameter is more effective in describing the solubility of HEAs. In solid solutions, the atomic size difference influences the topological instability of atomic packing. Egami [19] presented the topological instability of atomic packing in dilute binary alloys to discuss the structural instability involving glasses, and Miracle et al. [20] discussed the atomic packing efficiency in metallic glasses. Random packing in colloidal suspensions has also been studied extensively [21]. To date, however, there has been no report on the random atomic packing in HEAs. The instability of atomic packing is more complicated in multicomponent solid solutions with difference atomic sizes. However, the atoms with the largest and smallest sizes certainly play a dominant role in determining the stability of a lattice. Therefore, the packing state around the atoms with the largest and smallest sizes in HEAs should be the most important factor used to reveal how far away from the ideal case the atomic packings are. Hence, the packing states around the largest and smallest atoms affect the stability of the solid solutions. The solid angles of atomic packing for the elements with the largest and smallest atomic sizes are chosen to quantitatively describe the atomic packing effect in multicomponent alloys. The solid angles around the largest and smallest atoms in respect to the surrounding atoms are described geometrically by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrS þ rÞ2 r2 ðrL þ rÞ2 r2 xL ¼ 1 ¼ 1 ; x ð2Þ S ðrS þ rÞ2 ðrL þ rÞ2 where rL and rs are the radii of the largest and smallest atoms (see Fig. 1). A normalized parameter of the geometric packing state should be a good candidate to reveal the atomic packing instability. Here, we chose the ratio between the solid angles of the smallest and largest atoms c ¼ xS =xL sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ðrS þ rÞ2 r2 ðrL þ rÞ2 r2 1 ¼ 1 2 ðrS þ rÞ ðrL þ rÞ2
ð3Þ
as an indictor to reveal the atomic packing misfitting and topological instability. It is also important to point out that the Hume–Rothery rule of 15% of the atomic size difference in binary alloys corresponds to a critical value of packing misfitting of c ¼ 1:167. (a)
1.5
Solid solution Intermetallics Metallic glass
1.4 1.3
γ
2
γ=1.175
1.2
δ=0.06 1.1 1.0 0.00
0.04
0.08
0.12
δ
Figure 2. Statistics of the atomic packing parameter c and the polydispersity parameter d of atomic size difference from representative experimental results on the phase selection in HEAs. c ¼ 1:17 clearly distinguishes the solid solutions from the intermetallics. All the alloys are from Table 1 in Ref. [9].
Figure 2 shows the c d plot from a statistical analysis of representative experimental results regarding the phase selection in HEAs reported recently by Guo et al. [15]. All of the alloys analyzed were prepared by suction and injection casting in metal molds by arc or induction melting. The figure shows that there are many solid solutions and intermetallics coexisting in the region of 0:04 < d < 0:08. Even taking into consideration the mixing enthalpy, it is still impossible to distinguish the two different kinds of phases in the coexistence region [15]. Therefore, d is clearly not a good parameter for separating the solid solutions and intermetallics. It appears that d is not competent enough to describe the solubility in HEAs. On the other hand, Figure 2 shows that the atomic packing parameter c can clearly distinguish solid solutions from multiphase regions with intermetallics. All of the solid solutions are in the region of c < 1:175. Most of the multiphase regions with intermetallics together with metallic glasses are all distributed in the region of c > 1:175. Note that metallic glasses are in a metastable state, prepared by relatively fast cooling, the glass phases of which mainly change to intermetallic phases during equilibrium treatments. A total of 95 kinds of HEAs are included in Figure 2. The critical value of c ¼ 1:175 can distinguish nearly all the 59 solid solution alloys. The previous parameter, d ¼ 0:06, misses six solid solutions, with a 10% error for
(b)
r
r
ωL rL
ωS rS
Figure 1. Sketch of the atomic packing around an atom via a solid angle: (a) around a largest atom; (b) around a smallest atom. r is the average atomic radius.
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Z. Wang et al. / Scripta Materialia xxx (2014) xxx–xxx
all 59 solid solution alloys. There are alloys with the same d but different c around d ¼ 0:06, indicating that the alloying elements may have the same distribution of atomic size differences but with different atomic packing statuses. Furthermore, it is important to note that the criterion of 1.175 in HEAs agrees very well with the 15% limit of the Hume–Rothery rules for binary alloys. Comparison of the parameters of d and c reveals clearly that c is more effective in distinguishing solid solutions from multiphases with intermetallics. From the viewpoint of physical foundation, the atomic packing parameter c has a more distinct meaning, referring to the topological instability, than the polydispersity parameter d. Firstly, c presents the normalized packing density discrepancy between the smallest and largest atoms which predominantly determine the atomic packing instability, whereas d reveals only the average effect of atomic size differences on the phase selection. Secondly, c also reasonably considers the atomic size effect of atoms
3
with medium sizes through atomic packing with the average atomic radius r, as shown in Eq. (3). Finally, its criterion of c < 1:175 can return to the 15% limit of the Hume–Rothery rules in binary alloys. Note that we carefully analyzed other atomic size effects in this study; however, no other size parameters have the merit of the c parameter. We explored several dozen superalloys in order to further confirm the application of the effect of this new atomic packing parameter on the solid solubility of multicomponent alloys. We considered 83 kinds of commercial superalloys, as shown in Table 1. As is known, superalloys generally contain disordered face-centered cubic (fcc) (solid solution) and ordered fcc (L12) structures. The results are shown in Figure 3, where alloying elements with an atomic fraction larger than 1% are considered. The figure shows that the criterion of c ¼ 1:175 still works very well in distinguishing the solid solution phases in the superalloys, much better than d parameter. There are two exceptions in the
Table 1. The superalloys presented in Figure 3. Superalloys (solid solution)
d
c
Superalloys (solid solution + precipitation)
d
c
Haynes 188 Haynes25 MAR_M918 MP159 MP35N s-816 Stellite B UMCo-50 Incoloy 801 Illim G Inconel 690 Hastelloy B Hastelloy B-2 Hastelloy C-276 Hastelloy C-4 Hastelloy G Hastelloy G-3 Hastelloy N Hastelloy S Hastelloy W Hastelloy X Haynes 214 Haynes 230 Nimonic 86 Haynes HR-120 Haynes HR-160 Inconel 600 Inconel 601 Inconel 617 Inconel 625 Inconel 690 Nimonic 75 RA333 19–9D Haynes 556 Incoloy 800 Incoloy 800H Incoloy 800HT Hastelloy X Incoloy 802 N-155(Multimet) REX-78
0.04727 0.03617 0.03046 0.03806 0.02204 0.06003 0.08298 0.02862 0.02709 0.04175 0.01367 0.0417 0.03999 0.03243 0.03787 0.03134 0.02788 0.03775 0.03018 0.04663 0.04847 0.048 0.03946 0.0299 0.02884 0.0259 0.02351 0.03072 0.03698 0.0329 0.01554 0.03216 0.02446 0.04898 0.02485 0.03391 0.03391 0.0349 0.03472 0.05785 0.04034 0.02477
1.1115 1.11153 1.1644 1.19866 1.10313 1.16859 1.11228 1.00868 1.19982 1.10336 1.03229 1.10745 1.10746 1.11136 1.16791 1.10808 1.10798 1.10779 1.10776 1.10758 1.17099 1.17069 1.1116 1.10321 1.10819 1.00869 1.00695 1.17146 1.16605 1.11154 1.06592 1.00694 1.10817 1.09664 1.10809 1.17169 1.1717 1.17169 1.10807 1.1719 1.10817 1.10817
Jetalloy1570 A-286 Discaloy Incoloy 903 Incoloy 907 Incoloy 909 Pyromet CTX-1 Thermo-Span V-57 W-545 AM-1 AM-3 B-1900 C-1023 Ford 406 GMR-235 Inconel 100 Inconel 713LC Inconel 713C Inconel 718 Inconel 738 Inconel 792 M22 M-252 MAR-M 200 MAR-M 246 MAR-M 421 MC2 MS2 N-4 PWA 1480 Nimonic 105 Jetalloy1650 Incoloy 901 Incoloy 925 Inconel 783 Inconel 939 Inconel 706 Inconel 718 Udimet 630 C-263
0.05189 0.05958 0.03562 0.03844 0.03538 0.03575 0.04032 0.03424 0.04344 0.0446 0.05446 0.05621 0.07968 0.06605 0.0528 0.05969 0.0681 0.05642 0.06097 0.0397 0.0638 0.06749 0.06448 0.0532 0.06456 0.06447 0.06058 0.05202 0.05437 0.05575 0.05598 0.05391 0.05856 0.04388 0.03195 0.04898 0.05719 0.0341 0.04309 0.04472 0.03991
1.19912 1.19993 1.19954 1.19926 1.19929 1.19932 1.19908 1.19922 1.19936 1.19949 1.19229 1.1922 1.60503 1.19256 1.19766 1.19864 1.19227 1.16492 1.16497 1.19902 1.19291 1.19274 1.16468 1.19348 1.19764 1.19246 1.19288 1.19263 1.19229 1.1923 1.19222 1.19293 1.19354 1.19924 1.19929 1.17022 1.19857 1.19942 1.19906 1.19871 1.19381
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4
Z. Wang et al. / Scripta Materialia xxx (2014) xxx–xxx 1.2
γ
1.15 1.1 1.05
1
solid solution+precipitation solid solution 0.02
0.04
0.06
0.08
0.10
δ
Figure 3. Application of the atomic packing parameter on the solid solubility of superalloys.
solid solution superalloys – the MP159 and Incoloy 801 superalloys – which contain 3.7 and 1.3 at.% Ti, respectively. These two superalloys have single solid solution phases at high temperatures, which are retained after cooling to room temperature, possibly due to the sluggish phase transformation. In conclusion, we have revealed a new parameter of the atomic size difference, c, based on the random atomic packing in multicomponent alloy systems of HEAs and superalloys. The proposed parameter has a clear physical meaning in determining the solubility of these multicomponent alloys, and is consistent to the 15% limit of Hume–Rothery rules commonly used in binary alloys. Also, the application of c to phase selection in HEAs and superalloys indicates that it is more efficient at distinguishing solid solutions from intermetallics than the previous polydispersity parameter of atomic size difference d. Through a comprehensive analysis of the existing data, it is shown that the newly proposed parameter is able to capture the general trend of phase selection in multicomponent alloys much better than the classic one. It should be noted that c < 1:175 is one of the necessary conditions for determining the solubility of multicomponent alloys but not a sufficient one, and that the mixing enthalpy, electronegativity and electron concentration should also be considered in order to confirm the solubility of multicomponent alloys. This research is supported by the Research Grant Council (RGC), the Hong Kong Government, through the General Research Fund (GRF) under project number CityU/521411. Z.W. is also supported by the Hong Kong Scholar Program and the National Science Foundation for Post-doctoral Scientists of China under project number 2013M542385.
[1] R.W. Cahn, P. HaasenPhysical Metallurgy, vol. 1, NorthHolland, Amsterdam., 1996. [2] W. Hume-Rothery, B.R. Coles, The transition metals and their alloys, Adv. Phys. 3 (1954) 149. [3] J. Chelikowsky, Solid solubilities in divalent alloys, Phys. Rev. B 19 (1979) 686. [4] J. Alonso, S. Simozar, Prediction of solid solubility in alloys, Phys. Rev. B 22 (1980) 5583. [5] V.A. Singh, A. Zunger, Phenomenology of solid solubilities and ion-implantation sites: an orbital-radii approach, Phys. Rev. B 25 (1982) 907. [6] B.-W. Zhang, Semi-empirical theory of solid solubility in binary alloys, Z. Metallkd. 76 (1985) 264–270. [7] B. Smola, I. Stulı´kova´, F. von Buch, B.L. Mordike, Structural aspects of high performance Mg alloys design, Mater. Sci. Eng. A 324 (2002) 113–117. [8] B. Geddes, Superalloys: Alloying and Performance, ASM International, Materials Park, OH, 2010. [9] Y. Zhang, T.T. Zuo, Z. Tang, M.C. Gao, K.A. Dahmen, P.K. Liaw, Z.P. Lu, Microstructures and properties of highentropy alloys, Prog. Mater. Sci. 61 (2014) 1–93. [10] B. Cantor, Multicomponent and high entropy alloys, Entropy 16 (2014) 4749–4768. [11] M.-H. Tsai, J.-W. Yeh, High-entropy alloys: a critical review, Mater. Res. Lett. (2014) 1–17. [12] J.W. Yeh, S.K. Chen, S.J. Lin, J.Y. Gan, T.S. Chin, T.T. Shun, C.H. Tsau, S.Y. Chang, Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes, Adv. Eng. Mater. 6 (2004) 299–303. [13] B. Cantor, I.T.H. Chang, P. Knight, A.J.B. Vincent, Microstructural development in equiatomic multicomponent alloys, Mater. Sci. Eng. A 375–377 (2004) 213–218. [14] Y. Zhang, Y.J. Zhou, J.P. Lin, G.L. Chen, P.K. Liaw, Solidsolution phase formation rules for multi-component alloys, Adv. Eng. Mater. 10 (2008) 534–538. [15] S. Guo, Q. Hu, C. Ng, C.T. Liu, More than entropy in highentropy alloys: forming solid solutions or amorphous phase, Intermetallics 41 (2013) 96–103. [16] J.D. Eshelby, The continuum theory of lattice defects, in: S. Frederick, T. David (Eds.), Solid State Physics, Academic Press, New York, 1956, pp. 79–144. [17] P. Pusey, The effect of polydispersity on the crystallization of hard spherical colloids, J. Phys. 48 (1987) 709–712. [18] S. Fang, X. Xiao, L. Xia, W. Li, Y. Dong, Relationship between the widths of supercooled liquid regions and bond parameters of Mg-based bulk metallic glasses, J. Non-Cryst. Solids 321 (2003) 120–125. [19] T. Egami, Universal criterion for metallic glass formation, Mater. Sci. Eng. A 226–228 (1997) 261–267. [20] D.B. Miracle, W.S. Sanders, O.N. Senkov, The influence of efficient atomic packing on the constitution of metallic glasses, Philos. Mag. 83 (2003) 2409–2428. [21] M. Clusel, E.I. Corwin, A.O.N. Siemens, J. Brujic, A “granocentric” model for random packing of jammed emulsions, Nature 460 (2009) 611–615.
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ScienceDirect Scripta Materialia xxx (2014) xxx–xxx www.elsevier.com/locate/scriptamat
Atomic-size effect and solid solubility of multicomponent alloys ⇑
Zhijun Wang,a,b Yunhao Huang,b Yong Yang,a Jincheng Wangb and C.T. Liua, a
Center of Advanced Structural Materials, Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Kowloon, Hong Kong, China b State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China Received 27 August 2014; accepted 14 September 2014
In physical metallurgy, solid solubility of alloys is known to play a vital role in determining their physical/mechanical properties. Hume–Rothery rules show the great effect of size difference between solvent and solute atoms on the solid solubility of binary alloys. However, modern multicomponent systems, such as high-entropy alloys, defy the classic atomic size effect due to the absence of solvent and/or solute atoms. Here, we propose an effective atomic size parameter by considering atomic packing misfitting in multicomponent systems. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Alloy solubility; High entropy alloys; Atomic-size effect; Atomic packing
The solubility of a solute element in an alloy is an important aspect of alloy design. Through the addition of soluble elements, one may obtain a solid solution effect that can significantly improve the alloy’s performance. For a substitutional binary alloy, the Hume–Rothery rules provide the key guidance to evaluating whether a solid solution can be formed [1]. These include: (i) the relative atomic size difference between the solute and solvent elements should be less than 15%; (ii) the formation of stable intermediate compounds should be restricted by carefully choosing the combination of metallic elements; and (iii) the electron concentration of the constituent elements should be tuned in favor of the formation of solid solutions, not otherwise. The Hume–Rothery rules have been used extensively for more than half a century [2–6]. However, the rules are poses a challenge with the recent design of multicomponent alloys [7–10], in some of which the molar fractions of the constituent elements are equal or nearly equal, such as high-entropy alloys (HEAs) [9–13]. As a result, there is no distinct identity of “solvent” or “solute” atoms in such alloys. Therefore, the atomic size factor as defined by the Hume–Rothery rules cannot be applied directly to HEAs. Despite this, some empirical parameters have been suggested as extensions of the Hume–Rothery rules to explain the solid solubility of HEAs [14]; however, they do not do so satisfactorily when compared with experimental data [15]. The atomic size difference is of primary importance in determining the solubility of alloys. The critical value of 15% atomic size difference in binary alloys was confirmed
⇑ Corresponding cityu.edu.hk
author. Tel.: +852 3442 7213; e-mail: chainliu@
by the continuum elastic theory [16]. However, it is difficult to define an atomic size parameter in relation to solubility of HEAs. For simplicity, the polydispersity resulting from the atomic size difference has been commonly used for predicting the solid solubility of a given multicomponent alloy [14], which is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi n X Xn 2 ; r ¼ d¼ c ð1 r = ci r i ð1Þ r Þ i i i i
where ci is the atomic concentration of the ith atom and ri is the atomic radius of the ith atom. It has been widely applied in the systems of hard-sphere fluids [17], metallic glasses [18] and HEAs [14,15]. In the hard-sphere fluid, d ¼ 0:06 is the criterion used to predict crystalline structure formation, and this criterion was also claimed in the HEA investigation. However, this parameter does not describe the solubility of HEAs very well; as a result, many intermetallics have been detected around d ¼ 0:06 [15]. Basically, the d parameter takes the average effect of the atomic size difference of all elements in the alloy. However, the solid solution instability may essentially be determined by the largest and smallest atoms in multicomponent alloy systems. Moreover, the physical meaning of d in determining the solubility is also not well understood, and it cannot return to the Hume–Rothery limit. Therefore it is necessary to explore the new and physically acceptable parameter of the atomic size effect on the solubility of multicomponent alloys, especially HEAs, which have received increasing attention recently from the materials community [9,11]. In this letter, we address this issue by considering the atomic size effect based on atomic packing behavior. The proposed parameter not only has a clear physical meaning,
http://dx.doi.org/10.1016/j.scriptamat.2014.09.010 1359-6462/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Please cite this article in press as: Z. Wang et al., Scripta Mater. (2014), http://dx.doi.org/10.1016/j.scriptamat.2014.09.010
Z. Wang et al. / Scripta Materialia xxx (2014) xxx–xxx
but is also consistent with the 15% atomic size difference in the Hume–Rothery rules. In particular, this parameter is more effective in describing the solubility of HEAs. In solid solutions, the atomic size difference influences the topological instability of atomic packing. Egami [19] presented the topological instability of atomic packing in dilute binary alloys to discuss the structural instability involving glasses, and Miracle et al. [20] discussed the atomic packing efficiency in metallic glasses. Random packing in colloidal suspensions has also been studied extensively [21]. To date, however, there has been no report on the random atomic packing in HEAs. The instability of atomic packing is more complicated in multicomponent solid solutions with difference atomic sizes. However, the atoms with the largest and smallest sizes certainly play a dominant role in determining the stability of a lattice. Therefore, the packing state around the atoms with the largest and smallest sizes in HEAs should be the most important factor used to reveal how far away from the ideal case the atomic packings are. Hence, the packing states around the largest and smallest atoms affect the stability of the solid solutions. The solid angles of atomic packing for the elements with the largest and smallest atomic sizes are chosen to quantitatively describe the atomic packing effect in multicomponent alloys. The solid angles around the largest and smallest atoms in respect to the surrounding atoms are described geometrically by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrS þ rÞ2 r2 ðrL þ rÞ2 r2 xL ¼ 1 ¼ 1 ; x ð2Þ S ðrS þ rÞ2 ðrL þ rÞ2 where rL and rs are the radii of the largest and smallest atoms (see Fig. 1). A normalized parameter of the geometric packing state should be a good candidate to reveal the atomic packing instability. Here, we chose the ratio between the solid angles of the smallest and largest atoms c ¼ xS =xL sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ðrS þ rÞ2 r2 ðrL þ rÞ2 r2 1 ¼ 1 2 ðrS þ rÞ ðrL þ rÞ2
ð3Þ
as an indictor to reveal the atomic packing misfitting and topological instability. It is also important to point out that the Hume–Rothery rule of 15% of the atomic size difference in binary alloys corresponds to a critical value of packing misfitting of c ¼ 1:167. (a)
1.5
Solid solution Intermetallics Metallic glass
1.4 1.3
γ
2
γ=1.175
1.2
δ=0.06 1.1 1.0 0.00
0.04
0.08
0.12
δ
Figure 2. Statistics of the atomic packing parameter c and the polydispersity parameter d of atomic size difference from representative experimental results on the phase selection in HEAs. c ¼ 1:17 clearly distinguishes the solid solutions from the intermetallics. All the alloys are from Table 1 in Ref. [9].
Figure 2 shows the c d plot from a statistical analysis of representative experimental results regarding the phase selection in HEAs reported recently by Guo et al. [15]. All of the alloys analyzed were prepared by suction and injection casting in metal molds by arc or induction melting. The figure shows that there are many solid solutions and intermetallics coexisting in the region of 0:04 < d < 0:08. Even taking into consideration the mixing enthalpy, it is still impossible to distinguish the two different kinds of phases in the coexistence region [15]. Therefore, d is clearly not a good parameter for separating the solid solutions and intermetallics. It appears that d is not competent enough to describe the solubility in HEAs. On the other hand, Figure 2 shows that the atomic packing parameter c can clearly distinguish solid solutions from multiphase regions with intermetallics. All of the solid solutions are in the region of c < 1:175. Most of the multiphase regions with intermetallics together with metallic glasses are all distributed in the region of c > 1:175. Note that metallic glasses are in a metastable state, prepared by relatively fast cooling, the glass phases of which mainly change to intermetallic phases during equilibrium treatments. A total of 95 kinds of HEAs are included in Figure 2. The critical value of c ¼ 1:175 can distinguish nearly all the 59 solid solution alloys. The previous parameter, d ¼ 0:06, misses six solid solutions, with a 10% error for
(b)
r
r
ωL rL
ωS rS
Figure 1. Sketch of the atomic packing around an atom via a solid angle: (a) around a largest atom; (b) around a smallest atom. r is the average atomic radius.
Please cite this article in press as: Z. Wang et al., Scripta Mater. (2014), http://dx.doi.org/10.1016/j.scriptamat.2014.09.010
Z. Wang et al. / Scripta Materialia xxx (2014) xxx–xxx
all 59 solid solution alloys. There are alloys with the same d but different c around d ¼ 0:06, indicating that the alloying elements may have the same distribution of atomic size differences but with different atomic packing statuses. Furthermore, it is important to note that the criterion of 1.175 in HEAs agrees very well with the 15% limit of the Hume–Rothery rules for binary alloys. Comparison of the parameters of d and c reveals clearly that c is more effective in distinguishing solid solutions from multiphases with intermetallics. From the viewpoint of physical foundation, the atomic packing parameter c has a more distinct meaning, referring to the topological instability, than the polydispersity parameter d. Firstly, c presents the normalized packing density discrepancy between the smallest and largest atoms which predominantly determine the atomic packing instability, whereas d reveals only the average effect of atomic size differences on the phase selection. Secondly, c also reasonably considers the atomic size effect of atoms
3
with medium sizes through atomic packing with the average atomic radius r, as shown in Eq. (3). Finally, its criterion of c < 1:175 can return to the 15% limit of the Hume–Rothery rules in binary alloys. Note that we carefully analyzed other atomic size effects in this study; however, no other size parameters have the merit of the c parameter. We explored several dozen superalloys in order to further confirm the application of the effect of this new atomic packing parameter on the solid solubility of multicomponent alloys. We considered 83 kinds of commercial superalloys, as shown in Table 1. As is known, superalloys generally contain disordered face-centered cubic (fcc) (solid solution) and ordered fcc (L12) structures. The results are shown in Figure 3, where alloying elements with an atomic fraction larger than 1% are considered. The figure shows that the criterion of c ¼ 1:175 still works very well in distinguishing the solid solution phases in the superalloys, much better than d parameter. There are two exceptions in the
Table 1. The superalloys presented in Figure 3. Superalloys (solid solution)
d
c
Superalloys (solid solution + precipitation)
d
c
Haynes 188 Haynes25 MAR_M918 MP159 MP35N s-816 Stellite B UMCo-50 Incoloy 801 Illim G Inconel 690 Hastelloy B Hastelloy B-2 Hastelloy C-276 Hastelloy C-4 Hastelloy G Hastelloy G-3 Hastelloy N Hastelloy S Hastelloy W Hastelloy X Haynes 214 Haynes 230 Nimonic 86 Haynes HR-120 Haynes HR-160 Inconel 600 Inconel 601 Inconel 617 Inconel 625 Inconel 690 Nimonic 75 RA333 19–9D Haynes 556 Incoloy 800 Incoloy 800H Incoloy 800HT Hastelloy X Incoloy 802 N-155(Multimet) REX-78
0.04727 0.03617 0.03046 0.03806 0.02204 0.06003 0.08298 0.02862 0.02709 0.04175 0.01367 0.0417 0.03999 0.03243 0.03787 0.03134 0.02788 0.03775 0.03018 0.04663 0.04847 0.048 0.03946 0.0299 0.02884 0.0259 0.02351 0.03072 0.03698 0.0329 0.01554 0.03216 0.02446 0.04898 0.02485 0.03391 0.03391 0.0349 0.03472 0.05785 0.04034 0.02477
1.1115 1.11153 1.1644 1.19866 1.10313 1.16859 1.11228 1.00868 1.19982 1.10336 1.03229 1.10745 1.10746 1.11136 1.16791 1.10808 1.10798 1.10779 1.10776 1.10758 1.17099 1.17069 1.1116 1.10321 1.10819 1.00869 1.00695 1.17146 1.16605 1.11154 1.06592 1.00694 1.10817 1.09664 1.10809 1.17169 1.1717 1.17169 1.10807 1.1719 1.10817 1.10817
Jetalloy1570 A-286 Discaloy Incoloy 903 Incoloy 907 Incoloy 909 Pyromet CTX-1 Thermo-Span V-57 W-545 AM-1 AM-3 B-1900 C-1023 Ford 406 GMR-235 Inconel 100 Inconel 713LC Inconel 713C Inconel 718 Inconel 738 Inconel 792 M22 M-252 MAR-M 200 MAR-M 246 MAR-M 421 MC2 MS2 N-4 PWA 1480 Nimonic 105 Jetalloy1650 Incoloy 901 Incoloy 925 Inconel 783 Inconel 939 Inconel 706 Inconel 718 Udimet 630 C-263
0.05189 0.05958 0.03562 0.03844 0.03538 0.03575 0.04032 0.03424 0.04344 0.0446 0.05446 0.05621 0.07968 0.06605 0.0528 0.05969 0.0681 0.05642 0.06097 0.0397 0.0638 0.06749 0.06448 0.0532 0.06456 0.06447 0.06058 0.05202 0.05437 0.05575 0.05598 0.05391 0.05856 0.04388 0.03195 0.04898 0.05719 0.0341 0.04309 0.04472 0.03991
1.19912 1.19993 1.19954 1.19926 1.19929 1.19932 1.19908 1.19922 1.19936 1.19949 1.19229 1.1922 1.60503 1.19256 1.19766 1.19864 1.19227 1.16492 1.16497 1.19902 1.19291 1.19274 1.16468 1.19348 1.19764 1.19246 1.19288 1.19263 1.19229 1.1923 1.19222 1.19293 1.19354 1.19924 1.19929 1.17022 1.19857 1.19942 1.19906 1.19871 1.19381
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Z. Wang et al. / Scripta Materialia xxx (2014) xxx–xxx 1.2
γ
1.15 1.1 1.05
1
solid solution+precipitation solid solution 0.02
0.04
0.06
0.08
0.10
δ
Figure 3. Application of the atomic packing parameter on the solid solubility of superalloys.
solid solution superalloys – the MP159 and Incoloy 801 superalloys – which contain 3.7 and 1.3 at.% Ti, respectively. These two superalloys have single solid solution phases at high temperatures, which are retained after cooling to room temperature, possibly due to the sluggish phase transformation. In conclusion, we have revealed a new parameter of the atomic size difference, c, based on the random atomic packing in multicomponent alloy systems of HEAs and superalloys. The proposed parameter has a clear physical meaning in determining the solubility of these multicomponent alloys, and is consistent to the 15% limit of Hume–Rothery rules commonly used in binary alloys. Also, the application of c to phase selection in HEAs and superalloys indicates that it is more efficient at distinguishing solid solutions from intermetallics than the previous polydispersity parameter of atomic size difference d. Through a comprehensive analysis of the existing data, it is shown that the newly proposed parameter is able to capture the general trend of phase selection in multicomponent alloys much better than the classic one. It should be noted that c < 1:175 is one of the necessary conditions for determining the solubility of multicomponent alloys but not a sufficient one, and that the mixing enthalpy, electronegativity and electron concentration should also be considered in order to confirm the solubility of multicomponent alloys. This research is supported by the Research Grant Council (RGC), the Hong Kong Government, through the General Research Fund (GRF) under project number CityU/521411. Z.W. is also supported by the Hong Kong Scholar Program and the National Science Foundation for Post-doctoral Scientists of China under project number 2013M542385.
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Please cite this article in press as: Z. Wang et al., Scripta Mater. (2014), http://dx.doi.org/10.1016/j.scriptamat.2014.09.010