Nanovoid dynamics based on temperature dependent Young modulus and void formation energy in Nickel: a phase field study

Document Type : Research Article

Authors

1 Mechanical Engineering Group, Pardis College, Isfahan University of Technology,

2 Isfahan University of Technology, Department of Mechanical Engineering

Abstract

In the present work, a phase field method is used to study the growth/annihilation of nanovoids under thermal and mechanical loadings. To do so, the coupled system of the Cahn-Hilliard and elasticity equations is solved using the nonlinear finite element method in 2 dimensional. This coupling is due to the presence of elastic energy in the Cahn-Hilliard free energy and the dependence of total strain on the void misfit strain. The novel point in the present physical model is including the temperature dependence of elastic properties and void formation energy. Then, examples of nanovoid structure evolution are presented consisting of planar gas-solid interface formation and evolution, growth/annihilation of circular nanovoids at different temperatures, growth/annihilation of nanovoids under biaxial compression and at different temperatures and nanovoid structure evolution with initially, randomly distributed void pattern. The obtained results show a faster growth with larger amounts of void concentration at lower temperatures. Also, the stress field significantly varies during nanovoids growth/ annihilation especially inside the solid-gas interface and its value depends on the nanovoid size and the concentration.

Keywords

Main Subjects


[1] T. Davis, D. Healy, A. Bubeck, R. Walker, Stress concentrations around voids in three dimensions: The roots of failure, Journal of Structural Geology, 102 (2017) 193-207.
[2] D. Norris, Voids in nickel irradiated with electrons after previous argon ion bombardment, Nature, 227(5260) (1970) 830.
 [3] D. Norris, Voids in irradiated metals (Part I), Radiation Effects, 14(1-2) (1972) 1-37.
[4] D. Norris, Voids in irradiated metals (Part II), Radiation Effects, 15(1-2) (1972) 1-22.
[5] J.L. Katz, H. Wiedersich, Nucleation of voids in materials supersaturated with vacancies and interstitials, The Journal of Chemical Physics, 55(3) (1971) 1414-1425.
 [6] K. Russell, Thermodynamics of gas-containing voids in metals, Acta Metallurgica, 20(7) (1972) 899-907.
 [7] R. Mayer, L. Brown, Nucleation and growth of voids by radiation: II. Differential equations, Journal of Nuclear Materials, 95(1-2) (1980) 58-63.
[8] M. Imada, Void Lattice formation-spinodal decomposition of vacancies, Journal of the Physical Society of Japan, 45(5) (1978) 1443- 1448.
[9] K. Krishan, Void ordering in metals during irradiation, Philosophical Magazine A, 45(3) (1982) 401-417.
[10]  A. Semenov, C. Woo, Void lattice formation as a nonequilibrium phase transition, Physical Review B, , 74(2) (2006) 024108.
[11] A. Brailsford, L. Mansur, Time dependent rate theory for diffusional defect processes, Acta Metallurgica, 33(8) (1985) 1425-1437.
[12] N. Doan, G. Martin, Elimination of irradiation point defects in crystalline solids: sink strengths, Physical Review B, 67(13) (2003) 134107.
[13] W.J. Boettinger, J.A. Warren, C. Beckermann,A. Karma, Phase-field simulation of solidification, Annual review of materials research, 32(1) (2002) 163-194.
[14] H. Henry, H. Levine, Dynamic instabilities of fracture under biaxial strain using a phase field model, Physical review letters, 93(10) (2004) 105504.
[15] Y.U. Wang, Computer modeling and simulation of solid-state sintering: A phase field approach, Acta materialia, 54(4) (2006) 953-961.
[16] N. Moelans, B. Blanpain, P. Wollants, Quantitative analysis of grain boundary properties in a generalized phase field model for grain growth in anisotropic systems, Physical Review B, 78(2) (2008) 024113.
[17] V.I. Levitas, M. Javanbakht, Advanced phase-field approach to dislocation evolution, Physical Review B, 86(14) (2012) 140101.
[18] V.I. Levitas, M. Javanbakht, Interaction between phase transformations and dislocations at the nanoscale. Part ١. General phase field approach, Journal of the Mechanics and Physics of Solids, 82(2015)287-319.
[19] M. Javanbakht, V.I. Levitas, Interaction between phase transformations and dislocations at the nanoscale. Part ٢: Phase field simulation examples, Journal of the Mechanics and Physics of Solids, 82 (2015) 164-185.
[20] M. Javanbakht, V. Levitas, Phase Field Method to the Interaction of Phase Transformations and Dislocations at Nanoscale, AUT Journal of Mechanical Engineering, 1(2) (2017) 243-246.
[21] H.-C. Yu, W. Lu, Dynamics of the self-assembly of nanovoids and nanobubbles in solids, Acta Materialia, 53(6) (2005) 1799-1807.
[22] S. Hu, C.H. Henager Jr, Phase-field modeling of void lattice formation under irradiation, Journal of Nuclear Materials, 394(2-3) (2009) 155-159..
[23] S.Y. Hu, C. Henager Jr, Phase-field simulation of void migration in a temperature gradient, Acta materialia, 58(9) (2010) 3230-3237.
[24] S. Rokkam, A. El-Azab, P. Millett, D. Wolf, Phase field modeling of void nucleation and growth in irradiated metals, Modelling and simulation in materials science and engineering, 17(6) (2009) 064002.
[25] P.C. Millett, A. El-Azab, S. Rokkam, M. Tonks, D. Wolf, Phase-field simulation of irradiated metals: Part I: Void kinetics, Computational materials science, 50(3) (2011) 949-959.
[26] P.C. Millett, A. El-Azab, D. Wolf, Phase- field simulation of irradiated metals: Part II:Gas bubble kinetics, Computational Materials Science, 50(3) (2011) 960-970.
[27] P.C. Millett, M. Tonks, Application of phase-field modeling to irradiation effects in materials, Current Opinion in Solid State and Materials Science, 15(3) (2011) 125-133.
[28] Y. Li, S. Hu, X. Sun, F. Gao, C.H. Henager Jr, M. Khaleel, Phase-field modeling of void migration and growth kinetics in materials under irradiation and temperature field, Journal of Nuclear Materials, 407(2) (2010) 119-125..
[29] Z. Xiao, A. Semenov, C. Woo, S. Shi, Single void dynamics in phase field modeling, Journal of nuclear materials, 439(1-3) (2013) 25-32.
[30] A. Semenov, C. Woo, Interfacial energy in phase-field emulation of void nucleation and growth, Journal of nuclear materials, 411(1-3) (2011) 144-149.
[31] A. Semenov, C. Woo, Phase-field modeling of void formation and growth under irradiation, Acta Materialia, 60(17) (2012) 6112-6119.
[32] Y. Li, S. Hu, R. Montgomery, F. Gao, X. Sun, Phase-field simulations of intragranular fission gas bubble evolution in UO٢ under post-irradiation thermal annealing, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 303 (2013) 62-67.
[33] I.W. Vance, P.C. Millett, Phase-field simulations of pore migration and morphology change in thermal gradients, Journal of Nuclear Materials, 490 (2017) 299-304.
[34] Y. Gao, Y. Zhang, D. Schwen, C. Jiang, C. Sun, J. Gan, X.-M. Bai, Theoretical prediction and atomic kinetic Monte Carlo simulations of void superlattice self-organization under irradiation, Scientific reports, 8(1).(2018) 6629.
[35]  W. Wang, C.-l. Yi, K.-q. Fan, Molecular dynamics study on temperature and strain rate dependences of mechanical tensile properties of ultrathin nickel nanowires, Trans. Nonferrous Met. Soc. China, 23(3353) (2013) 3361.
[36] ] Y. Gong, B. Grabowski, A. Glensk, F. Körmann, J. Neugebauer, R.C. Reed, Temperature dependence of the Gibbs energy of vacancy formation of fcc Ni, Physical Review B, 97(21) (2018) 214106.
[37]  D. Schwen, L.K. Aagesen, J.W. Peterson, M.R. Tonks, Rapid multiphase-field model development using a modular free energy based approach with automatic differentiation in MOOSE/MARMOT, Computational Materials Science, 132 (2017) 36-45.
[38] M. Shaikh, K. Ehrlich, Swelling in nickel-carbon and nickel-silicon alloys, Pakistan Inst. of Nuclear Science and Technology, 1990.