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Diffusion in Ni-Based Single Crystal Superalloys with Density Functional Theory and Kinetic Monte Carlo Method

Published online by Cambridge University Press:  31 August 2016

Min Sun ,
Zi Li ,
Guo-Zhen Zhu ,
Wen-Qing Liu ,
Shao-Hua Liu  and
Chong-Yu Wang
Min Sun*
Affiliation:
Central Iron & Steel Research Institute, Beijing 100081, China
Zi Li
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
Guo-Zhen Zhu
Affiliation:
State Key Laboratory of Metal Matrix Composites, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
Wen-Qing Liu
Affiliation:
Key Laboratory for Microstructures, Shanghai University, Shanghai 200444, China
Shao-Hua Liu
Affiliation:
School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China
Chong-Yu Wang*
Affiliation:
Central Iron & Steel Research Institute, Beijing 100081, China Department of Physics, Tsinghua University, Beijing 100084, China
*
*Corresponding author. Email addresses:cywang@mail.tsinghua.edu.cn (C.-Y.Wang), ilphysics@163.com (M. Sun)
*Corresponding author. Email addresses:cywang@mail.tsinghua.edu.cn (C.-Y.Wang), ilphysics@163.com (M. Sun)
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Abstract

In the paper, we focus on atom diffusion behavior in Ni-based superalloys, which have important applications in the aero-industry. Specifically, the expressions of the key physical parameter – transition rate (jump rate) in the diffusion can be given from the diffusion theory in solids and the kinetic Monte Carlo (KMC) method, respectively. The transition rate controls the diffusion process and is directly related to the energy of vacancy formation and the energy of migration of atom from density functional theory (DFT). Moreover, from the KMC calculations, the diffusion coefficients for Ni and Al atoms in the γ phase (Ni matrix) and the γʹ phase (intermetallic compound Ni3Al) of the superalloy have been obtained. We propose a strategy of time stepping to deal with the multi-time scale issues. In addition, the influence of temperature and Al concentration on diffusion in dilute alloys is also reported.

Keywords

Diffusion behaviortransition ratedensity functional theorykinetic Monte Carlo methodsuperalloy
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 3 , September 2016 , pp. 603 - 618
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Reed, R. C.. The Superalloys: Fundamentals and Applications. New York: Cambridge University Press; 2006.CrossRefGoogle Scholar
[2] Shewmon, P.. Diffusion in Solids. 2nd TMS, Warrendale, PA; 1989, p.223.Google Scholar
[3] Wang, Y., Liu, Z. K. and Chen, L. Q.. Thermodynamic properties of Al, Ni, NiAl, and Ni3Al from first-principles calculations. Acta Mater. 2004; 52:2665–2671.CrossRefGoogle Scholar
[4] Zhu, T., Wang, C. Y., Gan, Y., Effect of Re in γ phase, γʹ phase and γ/γʹ interface of Ni-based single-crystal superalloys, Acta Mater. 2010; 58:2045–2055.CrossRefGoogle Scholar
[5] Ziegler, A., Idrobo, J. C., Cinibulk, M. K., Kisielowski, C., N. D.|Browning, Ritchie, R. O.. Interface structure and atomic bonding characteristics in silicon nitride ceramics. Science 2004; 306:1768–1770.CrossRefGoogle ScholarPubMed
[6] Srinivasan, R., Banerjee, R., Hwang, J. Y., Viswanathan, G. B., Tiley, J., Dimiduk, D. M., Fraser, H. L.. Atomic scale structure and chemical composition across order-disorder interfaces. Phys. Rev. Lett. 2009; 102:086101.CrossRefGoogle ScholarPubMed
[7] Yasuda, H. Y., Nakajima, H., Koiwa, M.. Diffusion in L12-type intermetallic compounds. Defect Diffus. Forum. 1993; 95:823–830.CrossRefGoogle Scholar
[8] Krčmar, M., Fu, C. L., Janotti, A., Reed, R. C.. Diffusion rates of 3D transition metal solutes in nickel by first-principles calculations. Acta Mater. 2005; 53:2369–2376.CrossRefGoogle Scholar
[9] Mantina, M., Wang, Y., Chen, L. Q., Liu, Z. K., Wolverton, C.. First principles impurity diffusion coefficients, Acta Mater. 2009; 57:4102–4108.CrossRefGoogle Scholar
[10] Zhang, X., Wang, C. Y.. First-principles study of vacancy formation and migration in clean and Re-doped γʹ-Ni3Al. Acta Mater. 2009; 57:224–231.CrossRefGoogle Scholar
[11] Yu, X. X., Wang, C. Y.. The effect of alloying elements on the dislocation climbing velocity in Ni: A first-principles study. Acta Mater. 2009; 57:5914–5920.CrossRefGoogle Scholar
[12] Zhu, T., Wang, C.-Y.. Misfit dislocation networks in the γ/γʹ phase interface of a Ni-Based single-crystal superalloys: Molecular dynamics simulations. Phys. Rev. B 2005; 72:014111.CrossRefGoogle Scholar
[13] Du, J. P., Wang, C. Y., Yu, T.. Construction and application of multi-elements EAM potential (Ni-Al-Re) in γ/γʹ Ni-based single crystal superalloys. Modelling Simul. Mater. Sci. Eng. 2013; 21:015007.CrossRefGoogle Scholar
[14] Liu, Z.-G., Wang, C.-Y., Yu, T.. Molecular dynamics simulations of influence of Re on lattice trapping and fracture stress of cracks in Ni. Comput. Mater. Sci. 2014; 83:196–205.CrossRefGoogle Scholar
[15] Liu, S. L., Wang, C. Y., Yu, T.. Influence of the alloying elements Re, Co and W on the propagation of the Ni/Ni3Al interface crack. RSC Advances 2015; 5:52473–52480.CrossRefGoogle Scholar
[16] Wang, C., Wang, C.-Y.. Ni/Ni3Al interface: A density functional theory study. Applied Surface Science 2009; 255:3669–3675.CrossRefGoogle Scholar
[17] Wang, Y.-J., Wang, C.-Y.. A comparison of the ideal strength between L12Co3(Al,W) and Ni3Al under tension and shear from first-principles calculations. Applied Physics Letters 2009; 94:261090.CrossRefGoogle Scholar
[18] Wu, X., Wang, C. Y.. Density functional theory study of the thermodynamic and elastic properties of Ni-based superalloys. J. Phys.: Condensed Matter 2015; 27:295401.Google ScholarPubMed
[19] Mehrer, H.. Diffusion in solids: Fundamentals, methods, materials, diffusion-controlled processes. Springer Science & Business Media, 2007, p.56–65, 70-72, 80-82, 98-99, 327-331, 355-357.CrossRefGoogle Scholar
[20] Koiwa, M., Numakura, H., Ishioka, S.. Diffusion in L12 type intermetallic compounds. Defect and Diffusion Forum. 1997; 143:209–222.CrossRefGoogle Scholar
[21] Koiwa, M., Ishioka, S.. Random walks and correlation factor in diffusion in a three-dimensional lattice with coordination number 8. Philos. Mag. A 1983; 48:1–9.CrossRefGoogle Scholar
[22] Voter, A. F.. Radiation Effects in Solids. Edited by Sickafus, K. E., Kotomin, E. A., Uberuaga, B. P.. Dordrecht: Springer, NATO Publishing Unit, 2006; p. 1–24.Google Scholar
[23] Vineyard, G. H.. Frequency factors and isotope effects in solid state rate processes. J. Phys. Chem. Solids, 1957; 3:121–127.CrossRefGoogle Scholar
[24] Fichthorn, K. A., Weinberg, W. H.. Theoretical foundations of dynamical Monte Carlo simulations. J. Chem. Phys., 1991; 95:1090.CrossRefGoogle Scholar
[25] Young, W. M., Elcock, E. W.. Monte Carlo studies of vacancy migration in binary ordered alloys: I. Proc. Phys. Soc. 1966; 89:735.CrossRefGoogle Scholar
[26] Kinzel, W., Selke, W., Binder, K.. Phase transitions on centered rectangular lattice gases: A model for the adsorption of H on Fe(110). Surf. Sci. 1982; 121:13.CrossRefGoogle Scholar
[27] Sahni, P. S., Grest, G. S., Safran, S. A.. Temperature dependence of domain kinetics in two dimensions. Phys. Rev. Lett., 1983; 50:60.CrossRefGoogle Scholar
[28] Stepanyuk, V. S., Negulyaev, N. N., Niebergall, L., Longo, R. C., Bruno, P.. Adatom self-organization induced by quantum confinement of surface electrons. Phys. Rev. Lett., 2006; 97:186403.CrossRefGoogle ScholarPubMed
[29] Ziegler, M., Kröger, J., Berndt, R., Filinov, A., Bonitz, M.. Scanning tunneling microscopy and kinetic Monte Carlo investigation of cesium superlattices on Ag(111). Phys. Rev. B 2008; 78:245427.CrossRefGoogle Scholar
[30] Chen, Z., Kioussis, N., Tu, K.-N., Ghoniem, N., Yang, J.-M.. Inhibiting adatom diffusion through surface alloying. Phys. Rev. Lett. 2010; 105:015703.Google ScholarPubMed
[31] Watkins, P. K., Walker, A. B., and Verschoor, G. L. B.. Dynamical Monte Carlo modelling of organic solar cells: The dependence of internal quantum efficiency on morphology. Nano Lett. 2005; 5:1814.CrossRefGoogle ScholarPubMed
[32] Zhang, X., Li, Z., Lu, G.. First-principles simulations of exciton diffusion in organic semiconductors. Phys. Rev. B 2011; 84:235208.CrossRefGoogle Scholar
[33] Voter, A. F.. Classically exact overlayer dynamics: Diffusion of rhodium clusters on Rh(100). Phys. Rev. B 1986; 34:6819.CrossRefGoogle ScholarPubMed
[34] DeLorenzi, G., Flynn, C. P., Jacucci, G.. Effect of anharmonicity on diffusion jump rates. Phys. Rev. B 1984; 30:5430.CrossRefGoogle Scholar
[35] Sørensen, M. R., Voter, A. F.. Temperature-accelerated dynamics for simulation of infrequent events. J. Chem. Phys. 2000; 112:9599.CrossRefGoogle Scholar
[36] Srinivasan, R., Banerjee, R., Hwang, J. Y., Viswanathan, G. B., Tiley, J., Dimiduk, D. M., Fraser, H. L.. Atomic scale structure and chemical composition across order-disorder interfaces. Phys. Rev. Lett. 2009; 102:086101.CrossRefGoogle ScholarPubMed
[37] Reed, R. C., Yeh, A. C., Tin, S., Babu, S. S., Miller, M. K.. Identification of the partitioning characteristics of ruthenium in single crystal superalloys using atom probe tomography. Scripta Materialia 2004; 51:327–331.CrossRefGoogle Scholar
[38] Yu, X. X., Wang, C. Y., Zhang, X. N., Yan, P., Zhang, Z.. Synergistic effect of rhenium and ruthenium in nickel-based single-crystal superalloys, J. Alloys Compd. 2014; 582:299–304.CrossRefGoogle Scholar
[39] Yeh, A. C., Tin, S.. Effects of Ru and Re additions on the high temperature flow stresses of Ni-base single crystal superalloys, Scripta Materialia 2005; 52:519–524.CrossRefGoogle Scholar
[40] Henkelman, G., Uberuaga, B. P., Jónsson, H.. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys. 2000; 113:9901–9904.CrossRefGoogle Scholar
[41] Wert, C. A.. Diffusion Coefficient of C in α-Iron. Phys. Rev. 1950; 79:601.CrossRefGoogle Scholar
[42] Eyring, H.. The activated complex in chemical reactions. J. Chem. Phys. 1935; 3:107–115.CrossRefGoogle Scholar
[43] Li, Z., Zhang, X., Cristiano, F., Lu, W. G.. Understanding molecular structure dependence of exciton diffusion in conjugated small molecules. Appl. Phys. Lett. 2014; 104:143303.Google Scholar
[44] Kresse, G., Hafner, J.. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 1993; 47:558.CrossRefGoogle ScholarPubMed
[45] Kresse, G., Furthmüller, J.. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 1996; 54:11169.CrossRefGoogle ScholarPubMed
[46] Blöchl, P. E.. Projector augmented-wave method. Phys. Rev. B 1994; 50:17953.CrossRefGoogle ScholarPubMed
[47] Kresse, G., Joubert, D.. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 1999; 59:1758.CrossRefGoogle Scholar
[48] Kohn, W., Sham, L. J.. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965; 140:A1133-A1138.CrossRefGoogle Scholar
[49] Lomer, W. M.. Vacancies and other point defects in metals and alloys. Institute of Metals, London, 1958; 79.Google Scholar
[50] Herzig, C., Divinski, S.. Diffusion in Intermetallic Compounds in: Diffusion Processes in Advanced Technological Materials. William Andrew, Inc., 2005.Google Scholar
[51] Zhang, P., Xu, Y., Zheng, F., Wu, S. Q., Yang, Y., Zhu, Z. Z.. Ion diffusion mechanism in PnNaxLi2-xMnSiO4. Cryst. Eng. Comm. 2015; 17:2123–2128.CrossRefGoogle Scholar
[52] Hanggi, P., Talkner, P., Borkovec, M.. Reaction-rate theory: Fifty years after Kramers. Rev. Mod. Phys. 1990; 62:251.CrossRefGoogle Scholar
[53] Gillespie, D. T.. Multiscale stochastic simulation algorithm with stochastic partial equilibrium assumption for chemically reacting systems. J. Comp. Phys. 1976; 22: 403.CrossRefGoogle Scholar
[54] Fichthorn, K. A., Weinberg, W. H.. Theoretical foundations of dynamical Monte Carlo simulations. J. Chem. Phys. 1991; 95: 1090.CrossRefGoogle Scholar
[55] Perdew, J. P., Burke, K., Ernzerhof, M.. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996; 77:3865.CrossRefGoogle ScholarPubMed
[56] Harada, H., Ishida, A., Murakami, Y., Bhadeshia, H. K. D. H., Yamazaki, M.. Atom-probe microanalysis of a nickel-base single crystal superalloy. Appl. Surf. Sci. 1993; 67:299–304.CrossRefGoogle Scholar
[57] Monkhorst, H. J., Pack, J. D.. Special points for Brillouin-zone integrations. Phys. Rev. B 1976; 13:5188.CrossRefGoogle Scholar
[58] Lee, B. J., Shim, J. H., Baskes, M. I.. Semiempirical atomic potentials for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, Al, and Pb based on first and second nearest-neighbor modified embedded atom method. Phys. Rev. B 2003; 68:144112.CrossRefGoogle Scholar
[59] Smedskjaer, L. C., Fluss, M. J., Legnini, D. G., Chason, M. K., Siegel, R. W.. Positron Annihilation. Amsterdam: North-Holland; 1982.Google Scholar
[60] Maier, K., Peo, M., Saile, B., Schaefer, H. E., Seeger, A.. High-temperature positron annihilation and vacancy formation in refractory metals. Philos. Mag. A 1979; 40:701–728.CrossRefGoogle Scholar
[61] Wycisk, W., Kniepmeier, M. F.. Quenching experiments in high purity Ni. J. Nucl. Mater. 1978; 69:616–619.CrossRefGoogle Scholar
[62] Ortega, M. G., Ramos de Debiaggi, S. B., Monti, A. M.. Self-Diffusion in FCC Metals: Static and dynamic simulations in aluminium and nickel. Phys. Status Solidi (b) 2002; 234:506.3.0.CO;2-Q>CrossRefGoogle Scholar
[63] Smithells, C. J.. Metals Reference Book, seventh ed. Oxford: Butterworth-Heinemann; 1992.Google Scholar
[64] Wu, Q., Li, S. S., Ma, Y., Gong, S. K.. First principles calculations of alloying element diffusion coefficients in Ni using the five-frequency model. Chin. Phys. B 2012; 21:109102.CrossRefGoogle Scholar
[65] Gust, W., Hintz, M. B., Loddwg, A., Odelius, H., Predel, B.. Impurity diffusion of Al in Ni single crystals studied by secondary ion mass spectrometry (SIMS). Phys. Status Solidi (a) 1981; 64:187–194.CrossRefGoogle Scholar
[66] Gong, X. F., Yang, G. X., Fu, Y. H., Xie, Y. Q., Zhuang, J., Ning, X. J.. First-principles study of Ni/Ni3Al interface strengthening by alloying elements. Comput. Mater. Sci. 2009; 47:320–325.CrossRefGoogle Scholar
[67] Cardellini, F., Cleri, F., Mazzone, G., Montone, A., Rosato, V.. Experimental and theoretical investigation of the order-disorder transformation in Ni3Al. J. Mater. Res. 1993; 8:2504–2509.CrossRefGoogle Scholar
[68] Badura-Gergen, K., Schaefer, H. E.. Thermal formation of atomic vacancies in Ni3Al. Phys. Rev. B 1997; 56:3032.CrossRefGoogle Scholar
[69] Wang, T.M., Shimotomai, M., Doyama, M.. Study of vacancies in the intermetallic compound Ni3Al by positron annihilation. J. Phys. F: Metal Phys. 1984; 14:37.CrossRefGoogle Scholar
[70] Chen, G. X., Wang, D. D., Zhang, J. M., Huo, H. P., Xu, K. W.. Self-diffusion of Ni in the intermetallic compound Ni3Al. Physica B 2008; 403:3538–3542.CrossRefGoogle Scholar
[71] Delley, B. An all-electron numerical method for solving the local density functional for polyatomic molecules. J. Chem. Phys. 1990; 92:508–517.CrossRefGoogle Scholar
[72] Delley, B. Analytic energy derivative in the numerical local-density-functional approach. J. Chem. Phys. 1991; 94:7245–7250.CrossRefGoogle Scholar