The First-Principles Design of Ductile Refractory Alloys

Refractory Metals Research Summary The First-Principles Design of Ductile Refractory Alloys

Michael C. Gao, Ömer N. Doğan, Paul King, Anthony D. Rollett, and Michael Widom

The purpose of this work is to pre- chromium, niobium, and molybdenum at elevated temperatures (≥1,000ºC).3 dict elastic and thermodynamic prop- are 1,863ºC, 2,469ºC, and 2,623ºC, re- More importantly, chromium is inex- erties of chromium-based alloys based spectively. In particular, chromium al- pensive compared to the other refrac- on first-principles calculations and to loys are attractive because they have tory metals because it is more abun- demonstrate an appropriate computa- low density, high thermal conductiv- dant. However, its low-temperature tional approach to develop new mate- ity, and high strength at elevated tem- (e.g., at room temperature) brittleness rials for high-temperature applications peratures. Chromium generally forms a and embrittlement from nitrogen con- in energy systems. In this study, Poisson dense surface scale of Cr2O3 that pos- tamination at elevated temperatures ratio is used as a screening parameter sesses excellent corrosion resistance at have prevented it from major engineer- to identify ductilizing additives to the high temperatures (≤900–1,100ºC de- ing applications.4 (“Low temperature” refractory alloys. The results predict pending on oxygen partial pressure). in this report refers to low homologous that elements such as Ti, V, Zr, Nb, Hf, In addition, strategies have been devel- temperature [e.g., <0.3Tm]). In fact, the and Ta show potential as ductilizers in oped for the chromium alloys to main- lack of low-temperature ductility (e.g., Cr while Al, Ge, and Ga are predicted tain acceptable oxidation resistance high ductile-to-brittle transition tem- to decrease the ductility of Cr. Experi- perature [DBTT]) is a common weak- mental evidence, where available, vali- How would you . . . ness of some refractory metals, such as dates these predictions. chromium, molybdenum and tungsten, describe the overall significance of and their alloys.5 Therefore, studying INTRODUCTION this paper? how to improve the ductility of refrac- This paper describes an alloying In order to reduce environmental strategy to improve intrinsic tory metal alloys is important and yet emissions in fossil power generation, ductility of chromium-based challenging. more efficient energy-generating tech- alloys at low temperatures using There are two main difficulties in first-principles density functional developing refractory alloys: first, a nologies such as oxy-fuel gas turbines, theory calculations. Experimental hydrogen turbines, and syngas turbines evidence, where available, validates lack of basic experimental data on the are being developed. One common bar- the predictions made in this work. thermodynamics and mechanical and rier in the development of these differ- describe this work to a materials physical properties of most of these ent technologies for future energy gen- science and engineering alloy systems, and second, difficulties erating systems is an insufficiency of professional with no experience in associated with processing of these al- your technical specialty? existing materials at high temperatures loys. In order to avoid traditional tri- This paper uses Poisson’s ratio as (>1,150ºC) and aggressive atmospheres the screening parameter to identify al-and-error experiments that are also (e.g., steam, oxygen, CO2). Even the potential ductilizing additives to time consuming and expensive, it has highly alloyed and costly nickel-based the refractory elements such as become essential to develop theoretical superalloys do not have the desired chromium. First-principles density modeling to guide experimental alloy functional theory calculations are properties for these applications since used to predict Poisson’s ratio of development. Such theoretical model- they soften at ~1,100ºC. To enable the various chromium binary alloys. ing can be multiscale in nature, which development of these new technolo- The results indicate that Poisson’s includes first-principles density func- gies, new materials with high strength, ratio can be a good indicator of tional theory (DFT) calculations, and intrinsic ductility of metals and good ductility and fracture toughness, alloys. atomistic, mesoscale, and continuum and resistance against creep, high-tem- simulations. Due to their interpretative describe this work to a layperson? and predictive capacities, first-princi- perature corrosion, wear, and thermal The goal of this work is to fatigue have been sought. accelerate the design of new ples calculations are widely employed Alloys of body-centered cubic (bcc) materials using quantum to study alloy lattice stability, interfa- refractory metals with high melting mechanical calculations. cial energies, defect structures, etc.6–16 1,2 This work has identified several points are promising candidate ma- potentially powerful ductilizing This report presents first-principles terials for these structural applications. elements to chromium. calculations on a series of chromium-

For example, the melting points (Tm) of based binary alloys for initial screening

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� — Antiferromagnetic 300 — Cal. � — Nonmagnetic — Exp. 250 � –9.40 Bulk Modulus of Cr 200 –9.45 � � � –9.50 Figure 1. (a) Total energy vs. � � 150 –9.55 � � volume for bcc chromium in � � � antiferromagnetic and non- –9.60 � � � Bulk Modulus (GPa) 100 �� magnetic states; (b) a compar- �� otal Energy (eV/atom) –9.65 ison between calculated and T 50 –9.70 experimental bulk modulus for 10 11 12 13 14 0 selected pure elements. Volume (Å3/atom) Al Zr Hf Nb Cr Ta Mo W a b

of alloying elements to improve the in- um with 6.25 at.% vacancies was also dopotentials are used as supplied with trinsic ductility of chromium. calculated. VASP. This study uses the Perdew- It is well known that the Poisson ra- Burke–Ernzerhof25 gradient approxi- COMPUTATIONAL DETAILS tio is well correlated with ductility of mation to the exchange-correlation AND METHODOLOGY crystalline alloys17,18 and amorphous functional. metals.19,20 The higher the Poisson ratio The first-principles calculations use Reciprocal space (k-point) meshes is, the better ductility the crystalline or the plane-wave code VASP22,23 which are increased to achieve convergence to amorphous metal has at low tempera- solves for the electronic band structure a precision of better than 1 meV/at. All tures. For example, gold has a Poisson using electronic density functional the- structures are fully relaxed (both lattice ratio of 0.42 and it has an elongation ory. Projector augmented-wave24 pseu- parameters and atomic coordinates) of 50%; niobium has a Poisson ratio of until energies converge to a precision 0.40 and it has an elongation of 44% at of 0.25 meV/at. A “high precision” set- room temperature. Other ductile met- Equations ting is used since the derivative of total als (e.g., silver, palladium, and copper) energy is required for calculation of  ∂P ∂2E also have high values of Poisson ratio. BV= − = V (1) elastic properties. The plane-wave en-   2  ∂V  E ∂V E In contrast, commonly known brittle min mn ergy cutoff is held constant across each metals have low values of Poisson ratio. binary system at 500 eV. For example, beryllium has a Poisson   The semi-core 3p, 4p, and 5p elec- ratio of 0.08 and its tensile elongation δ 0 0  trons of selected transition metals are  ε = 0 −δ 0  (2) is only 1%; chromium has a Poisson ra- o   explicitly treated as valence electrons. tio of 0.21 and it is very brittle below its  0 0 δ2  Spin polarization with collinear magne-  1 2  DBTT, which is about 150ºC. Similar  ()− δ  tization or anti-ferromagnetism is con- trends are also observed in wholly or sidered in all calculations since chromi- 19,20 partially amorphous metallic alloys. ∆EE()δ= ∆ () − δ =EE()0 − ()δ um is known to be anti-ferromagnetic Therefore, Poisson ratio is chosen as at its ground state. To examine the sub- =()CC − VOδ2+ [] δ 4 (3) the first screening tool to gauge ductil- 11 12 0 stitutional effect, a 2×2×2 bcc supercell ity in this project. Moreover, it can be is built and then one chromium atom evaluated completely from first-prin-   is replaced with one alloying element.  0δ / 2 0  ciples calculations with virtually no  Thus the alloy composition is fixed at ε = δ / 2 0 0  (4) empirical information. m   Cr X (X = 6.25 at.%) in the present 2 15 1 A survey of established chromium-  0 0 δ  study.  ()4 − δ2  based binary phase diagrams21 indi- For a material with cubic symmetry, cates that feasible alloying elements there are three independent single-crys- are Ti, V, Fe, Co, Ni, Zr, Nb, Mo, Ru, ∆EE()δ= ∆ () − δ tal elastic constants: the bulk modulus EE()0 () 1 CV 2O[] 4 (5) 1 Rh, Pd, Hf, Ta, W, Re, Os, Ir, Pt, Al, Si, = − δ=2 44 0 δ + δ BC=3 ()11 + 2C12 , the tetragonal shear 1 Ga, and Ge. These elements are soluble modulus CC′ =2 ()11 − C12 , and the to varying extents in bcc chromium up trigonal shear modulus C . All three 9BC+ 4 ′ 3CB()+ 4C′ 44 to very high temperatures, whereas all G3 + G2 − 44 G elastic constants must be positive in 8 8 other elements in the periodic table ex- order for the structure to be mechani- 3CC′B hibit essentially negligible solubility. − 44 = 0 (6) cally stable. In the present study, we 4 Therefore, all 22 elements were evalu- obtained these elastic constants using ated as potential substitutional alloying 3BG− 2 the approach proposed by M.J. Mehl et ν = (7) 6 elements in this study. For comparison 6BG+ 2 al. purposes, the elasticity of pure chromi- To obtain the equilibrium unit cell

62 www.tms.org/jom.html JOM • July 2008 GPa, respectively. 0.8 The effect of alloying elements on bcc Cr15X1 the Poisson ratio of bcc Cr15X1 is shown 0.6 in Figure 6. All elements increase the /atom) 3 Poisson ratio of Cr except Al, Ge, and 0.4 Ga. Titanium increases it by 21%, fol-

(Å Volume lowed by V, Ta, Zr, Hf, and Nb. It is worth noting that vacancy substitution 0.2 increases the Poisson ratio by 25% because vacancy substitution lowers the 00 shear modulus by 36 GPa (Figure 5). Change in Atomic Present theoretical calculations pre- –0.2 dict that all the transition metals select- Al Si Ga Ge Vac Ti V Fe Co Ni Zr Nb Mo Ru Rh Pd Hf Ta W Re Os Ir Pt ed tend to increase the Poisson ratio

Figure 2. The effect of alloying elements on the atomic volume of bcc Cr15X1. The label moderately for the 6.25 at.% composi- “Vac” signifies monovacancy. tions, thus improving the ductility of chromium. The results are supported by several experimental findings.3,27–31 volume and bulk modulus, the total Cr15X1 substitution solid solution while For example, our calculations predict energies were calculated at 15 differ- all other elements increase it. Calcu- that the Cr-6.25 at.%V alloy enhance ent volumes and then fitted to the basic lations on the enthalpy of mixing in the Poisson ratio of chromium by 15%, 27 equation of state while preserving the forming bcc Cr15X1 solid solution show and H. Kurishita et al. found that a shape of bcc lattice (see Equation 1; all that all the elements have a repulsive properly processed V-52Cr-1.8Y equations are shown in the table). reaction with chromium in bcc lattice (wt.%) alloy achieved a yield strength An example is shown in Figure 1a for except Al, Si, and V. No correlation is of 610–740 MPa and a total elongation chromium. It is found that the antifer- found between atomic volume and en- of 10–19% at room temperature. This romagnetism of chromium is critically thalpy of mixing. study predicts that a Cr-6.25 at.%Re al- important to predict correct physical The effect of alloying elements on loy can increase the Poisson ratio of properties of chromium. Thus, all calcu- the bulk modulus and shear modulus pure chromium by 10%, and it was re- lations on Cr15X1 structures are initiated in the bcc Cr15X1 is shown in Figures ported that alloying of Re (≥20 at.%) to with an antiferromagnetic structure. To 4 and 5, respectively. Calculations pre- chromium can significantly improve obtain the shear moduli C´ and C44, the dict that 11 elements plus the 6.25 at.% the low-temperature ductility and fabri- authors applied volume-conserving or- monovacancy lower the bulk modulus cability of chromium.28 Recently, M.P. thorhombic and monoclinic strains to of Cr, including Al, Si, Ga, Ge, Ni, Zr, Brady et al.29 found that a eutectic mi- the bcc lattice,6 respectively, as shown Nb, Rh, Pd, Hf, Pt, and Zr. In contrast, crostructure that consists of an iron- in Equations 2–5. alloying with all 22 elements and va- rich bcc matrix and brittle strengthen-

The present study adopted A.V. Her- cancy lowers the shear modulus. The ing Cr2Ta intermetallics at a relatively shey’s averaging method.26 Accord- hexagonal close-packed (hcp) elements large volume fraction (Cr-30Fe-6.3Ta- ing to this method, the polycrystalline (Zr, Hf, and Ti) have a pronounced ef- 4Mo-0.5Ti-0.3Si-0.1La) exhibited a shear modulus is obtained by solving fect on the shear modulus of Cr; they toughness of 20 MPa√m at room tem- Equation 6. lower it by 30 GPa, 25 GPa, and 19 perature and a yield strength of 350 Finally, for cubic materials, the Pois- son ratio is calculated as Equation 7. 120 RESULTS AND bcc Cr15X DISCUSSIONS 100 1 80 Figure 1b shows the comparison between calculated bulk modulus and 60 experimental values for selected pure 40 elements. Clearly the agreement is ex- 20 cellent and the difference in all falls within 2.5%. This validates the calcu- 0 lation method. –20 of Mixing (meV/atom) Enthalpy The effect of alloying elements on –40 the atomic volume and heat of mixing in the bcc structure of Cr X is shown –60 15 1 Al Si Ga Ge Vac Ti V Fe Co Ni Zr Nb Mo Ru Rh Pd Hf Ta W Re Os Ir Pt in Figures 2 and 3, respectively. Ele- Figure 3. The effect of alloying elements on the heat of mixing of bcc Cr X . ments Si, V, Fe, and Co lower the aver- 15 1 age atomic volume when forming bcc

2008 July • JOM www.tms.org/jom.html 63 MPa at 1,000ºC. The present study pre- and composition. In order to enhance lent agreement with the present predic- dicts that iron is a moderate ductilizer the oxidation resistance by forming tion that shows the bcc Cr15Al1 alloy because the Cr Fe alloy increases the 15 1 Al2O3 surface scale, alloying chromium has a 7% lower Poisson ratio than pure Poisson ratio by 7%. The other advan- with aluminum was practiced by sev- chromium. tages of choosing iron are that iron is eral research groups,3,30,31 but it was The present study also predicts that very inexpensive, and that iron and found that the addition of hcp metals (Ti, Zr, Hf) have a potent chromium forms an extensive solid so- 5 at.% Al30 or 10 at.%Al3,31 deteriorates ductilizing effect. This agrees with an- lution over a wide range of temperature ductility significantly. This is in excel- other theoretical study12 that used the Rice–Thompson parameter32 as the ductility prediction parameter for mo- 30 lybdenum alloys. However, these hcp bcc Cr15X1 metals have low solubility in chromi- 20 um, so alloying strategies to enhance 10 their solubility in ternary and higher- order systems will be needed. Again, 0 there is little information on phase diagrams for these systems, which moti- –10 vates further theoretical calculations. –20 CONCLUSIONS Change in Bulk Modulus (GPa) –40 Based on the first-principles DFT calculations on the Poisson ratio of 22 –60 Al Si Ga Ge Vac Ti V Fe Co Ni Zr Nb Mo Ru Rh Pd Hf Ta W Re Os Ir Pt bcc Cr15X1 alloys, one can conclude that on an atom-for-atom basis, Ti, V, Figure 4. The effect of alloying elements on the bulk modulus of bcc Cr X . 15 1 Zr, Nb, Hf, and Ta are predicted to be potent ductilizers in chromium. Recent experiments by H. Kurishita et al.27 Al Si Ga Ge Vac Ti V Fe Co Ni Zr Nb Mo Ru Rh Pd Hf Ta W Re Os Ir Pt demonstrated that alloying vanadium 0 to chromium with proper processing improved the room-temperature ductil- –10 ity of chromium significantly. Rhenium and iron are predicted to be moderate ductilizers in chromium. There is ex- –20 perimental evidence for ductilization of chromium by rhenium and iron.28,29 In addition, Al, Ge, and Ga are predicted –30 to embrittle Cr. Again, there is experi-

Change in Shear Modulus (GPa) mental evidence that aluminum de- bcc Cr15X1 grades the ductility of chromium.3,30,31 –40 Finally, the calculated elastic properties Figure 5. The effect of alloying elements on the shear modulus of bcc Cr X . are found to be in good agreement with 15 1 reported experiments, indicating that Poisson ratio can be used as a screening 30 parameter for alloy development. bcc Cr15X1 25 ACKNOWLEDGEMENT

20 This technical effort was performed in support of the National Energy 15 Technology Laboratory’s (NETL) on- 10 going program in materials research under the RDS contract DE-AC26- 5 04NT41817, and the Pittsburgh Super- 0 computing Center (PSC) with Grant Ratio (%) Change in Poisson No. DMR070016P and Teragrid Grant –5 No. DMR070065N. M.C. Gao is grate- –10 ful for kind help from Dr. Yang Wang Al Si Ga Ge Vac Ti V Fe Co Ni Zr Nb Mo Ru Rh Pd Hf Ta W Re Os Ir Pt at PSC. The authors also acknowledge Figure 6. The effect of alloying elements on the Poisson ratio of bcc Cr X . 15 1 useful discussions with Dr. Chris Cow- en and Dr. Paul Jablonski at NETL.

64 www.tms.org/jom.html JOM • July 2008 References 29 (2005), pp. 163–211. 26. A.V. Hershey, J. Appl. Mech., 21 (1954), pp. 236– 14. M.C. Gao, A.D. Rollett, and M. Widom, CALPHAD, 240. 1. B.P. Bewlay et al., Metall. Mater. Trans. A, 34A (2003), 30 (2006), pp. 341–348. 27. H. Kurishita, T. Kuwabara, and M. Hasegawa, Mat. pp. 2043–2052. 15. M.C. Gao, A.D. Rollett, and M. Widom, Phys. Rev. B, Sci. Eng. A, 433 (2006), pp.32–38. 2. P. Jehanno et al., Metall. Mater. Trans. A, 36A (2005), 75 (2007), art. No. 174120 28. C.S. Wukusick, Refractory Metals and Alloys IV— pp. 515–523. 16. M.C. Gao et al., Metall. Mater. Trans. A, 38A (2007), Research and Development (New York: Gordon and 3. Ö.N. Dog˘an, Oxidation of Metals, 69 (2008), pp. pp. 2540–2551. Breach Science Publishers, 1967), pp. 231–245. 233–247. 17. S.F. Pugh, Phil. Mag., 45 (1954), pp. 823–843. 29. M.P. Brady et al., Scripta Mater., 52 (2005), pp. 4. W.D. Klopp, J. Metals, 21 (1969), pp. 23–32. 18. A.H. Cottrell, Advances in Physical Metallurgy, ed. 815–819. 5. Y.F. Gu, H. Harada, and Y. Ro, JOM, 56 (9) (2004), J.A. Charles and G.C. Smith (London: Institute of Met- 30. U. Holzwarth and H. Stamm, J. Nucl. Mater., 300 pp. 28–33. als, 1990), pp. 181–187. (2002), pp. 161–177. 6. M.J. Mehl et al., Phys. Rev. B, 41 (1990), pp. 10311– 19. J. Schroers and W.L. Johnson, Phys. Rev. Lett., 93 31. M.P. Brady, private communications (2008). 10323. (2004), art. No. 255506. 32. J.R. Rice and R. Thomson, Phil. Mag., 19 (1974), 7. U.V. Waghmare et al., Model. Simul. Mater. Sci. Eng., 20. J.J. Lewandowski, W.H. Wang, and A.L. Greer, Phil. pp. 73–97. 6 (1998), pp. 493–506. Mag. Lett., 85 (2005), pp. 77–87. 8. W.T. Geng, Phys. Rev. B, 68 (2003), art. No. 233402. 21. T.B. Massalski et al., Binary Alloy Phase Diagrams Michael C. Gao, Ömer N. Dog˘an, and Paul King 9. N.I. Medvedeva, Y.N. Gornostyrev, and A.J. Free- (Materials Park, OH: ASM International, 1995). are with National Energy Technology Laboratory, man, Phys. Rev. B, 67 (2003), art. No. 134204. 22. G. Kresse and J. Hafner, Phys. Rev. B, 47 (1993), 1450 Queen Ave. SW, Albany, OR 97321; Anthony 10. L. Vitos, P.A. Korzhavyi, and B. Johansson, Nature pp. 558–561. D. Rollett is with the Department of Materials Sci- Mater., 2 (2003), pp. 25–28. 23. G. Kresse and J. Furthmueller, Phys. Rev. B, 54 ence and Engineering, and Michael Widom is with 11. M.C. Gao et al., Metall. Mater. Trans. A, 36A (2005), (1996), pp. 11169–11186. the Department of Physics, Carnegie Mellon Uni- pp. 3269–3279. 24. P.E. Blöchl, Phys. Rev. B, 50 (1994), pp. 17953– versity, Pittsburgh, PA 15213. Dr. Gao is also with 12. C.B. Geller et al., Scripta Mater., 52 (2005), pp. 17979. Parsons, P.O. Box 618, South Park, PA 15129. Dr. 205–210. 25. J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Gao can be reached at (541) 967-5869; fax: (541) 13. S. Curtarolo, D. Morgan, and G. Ceder, CALPHAD, Lett., 77 (1996), pp. 3865–3868. 967-5958; e-mail [email protected].

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