Interfacial free energy anisotropy driven faceting of precipitates

References

• M. Fährmann, P. Fratzl, O. Paris, E. Fährmann, and W.C. Johnson, Influence of coherency stress on microstructural evolution in model Ni--Al--Mo alloys, Acta Metall. Mater. 43 (1995), pp. 1007–1022.
   Google Scholar
• T.B. Massalski, W.A. Soffa, and D.E. Laughlin, The nature and role of incoherent interphase interfaces in diffusional solid--solid phase transformations, Metall. Mater. Trans. A 37 (2006), pp. 825–831.
   Google Scholar
• W.Z. Zhang, X.F. Gu, and F.Z. Dai, Faceted interfaces: A key feature to quantitative understanding of transformation morphology, npj Comput. Mater. 2 (2016), p. 16021.
   Google Scholar
• M.K. Santala, V. Radmilovic, R. Giulian, M.C. Ridgway, R. Gronsky, and A.M. Glaeser, The orientation and morphology of platinum precipitates in sapphire, Acta Mater. 59 (2011), pp. 4761–4774.
   Web of Science ®Google Scholar
• M. Backhaus-Ricoult, Growth and equilibrium morphology of metal precipitates formed inside an oxide matrix, Inter. Sci. 4 (1997), pp. 285–302.
   Google Scholar
• A. van de Walle, Q. Hong, L. Milijacic, C. Balaji Gopal, S. Demers, G. Pomrehn, A. Kowalski, and P. Tiwary, Ab initio calculation of anisotropic interfacial excess free energies, Phys. Rev. B 89 (2014), p. 184101.
   Web of Science ®Google Scholar
• G.H. Deaf, The coarsening process of Ge precipitates in an Al-4 wt.% Ge alloy, Phys. B: Cond. Matter. 348 (2004), pp. 115–120.
   Web of Science ®Google Scholar
• J. He, I.D. Blum, H.Q. Wang, S.N. Giard, J. Doak, L.D. Zhao, J.C. Zheng, G. Casillas, C. Wolverton, M. Jose-Yacaman, D.N. Seidman, M.G. Kanatzidis, and V.P. Dravid, Morphology control of nanostructures; Na-doped PbTe--PbS system, Nanoletters 12 (2012), pp. 5979–5984.
   Web of Science ®Google Scholar
• E.A. Marquis and D.N. Seidman, Nanoscale structural evolution of Al3Sc precipitates in Al(Sc) alloys, Acta Mater. 49 (2001), pp. 1909–1919.
   Web of Science ®Google Scholar
• W. Rheinheimer, M. Bäurer, H. Chien, G.S. Rohrer, C.A. Handwerker, J.E. Blendell, and M.J. Hoffmann, The equilibrium crystal shape of strontium titanate and its relationship to the grain boundary plane distribution, Acta Mater. 82 (2015), pp. 32–40.
   Web of Science ®Google Scholar
• G.H. Kim and C.V. Thompson, Effect of surface energy anisotropy on Rayleigh-like solid-state dewetting and nanowire stability, Acta Mater. 84 (2015), pp. 190–201.
   Web of Science ®Google Scholar
• W. Jiang, Y. Wang, Q. Zhao, D.J. Srolovitz, and W. Bao, Solid-state dewetting and island morphologies in strongly anisotropic materials, Scr. Mater. 115 (2016), pp. 123–127.
   Web of Science ®Google Scholar
• L.Q. Chen, Phase-field models for microstructure evolution, Ann. Rev. Mat. Res. 32 (2002), pp. 113–140.
   Web of Science ®Google Scholar
• W.J. Boettinger, J.A. Warren, C. Beckermann, and A. Karma, Phase-field simulation of solidification, Ann. Rev. Mat. Res. 32 (2002), pp. 163–194.
   Web of Science ®Google Scholar
• I. Steinbach, Phase-field models in materials science, Mod. Sim. Mat. Sci. Engg. 17 (2009), pp. 073001–31pp.
   Web of Science ®Google Scholar
• K. Thornton, J. Ågren, and P.W. Voorhees, Modelling the evolution of phase boundaries in solids at the meso- and nano-scales, Acta Mater. 51 (2003), pp. 5675–5710.
   Web of Science ®Google Scholar
• N. Moelans, B. Blanpain, and P. Wollants, An introduction to phase-field modeling of microstructure evolution, Comp. Coup. Phase Dia. Thermochem. 32 (2008), pp. 268–294.
   Web of Science ®Google Scholar
• J.S. Langer, Models of pattern formation in first-order phase transitions, in Directions in Condensed Matter Physics: Memorial Volume in Honor of Shang-Keng Ma, G. Grinstein and G. Mazenko, eds., World Scientific, Singapore, 1986.
   Google Scholar
• A.A. Wheeler, B.T. Murray, and R.J. Schaefer, Computation of dendrites using a phase field model, Physica D: Nonlin. Phenom. 66 (1993), pp. 243–262.
   Web of Science ®Google Scholar
• J.B. McFadden, A.A. Wheeler, R.J. Braun, S.R. Coriell, and R.F. Sekerka, Phase-field models for anisotropic interfaces, Phys. Rev. E 48 (1993), pp. 2016–2024.
   PubMed Web of Science ®Google Scholar
• A.A. Wheeler and G.B. McFadden, A x-vector formulation of anisotropic phase-field models: 3D asymptiotics, Eur. J. App. Math. 7 (1996), pp. 367–381.
   Google Scholar
• R.J. Braun, J.W. Cahn, G.B. McFadden, and A.A. Wheeler Anisotropy of interfaces in an ordered alloy: A multiple-order-parameter model, Phil. Trans.: Math., Phys. Engg. Sci. 355 (1997), pp. 1787–1833.
   Google Scholar
• R.J. Braun, J.W. Cahn, G.B. McFadden, H.E. Rushmeier, and A.A. Wheeler, Theory of anisotropic growth rates in the ordering of an f.c.c. alloy, Acta Mater. 46 (1998), pp. 1–12.
   Web of Science ®Google Scholar
• A. Karma and W.J. Rappel, Quantitative phase-field modeling of dendritic growth in two and three dimensions, Phys. Rev. E 57 (1998), pp. 4323–4349.
   Web of Science ®Google Scholar
• B. Nestler and A.A. Wheeler, Anisotropic multi-phase-field model: Interfaces and junctions, Phys. Rev. E 57 (1998), pp. 2602–2609.
   Web of Science ®Google Scholar
• A.A. Wheeler, Cahn--Hoffman ξ-vector and its relation to diffuse interface models of solidification, J. Stat. Phys. 95 (1999), pp. 1245–1280.
   Web of Science ®Google Scholar
• J.W. Cahn, S.C. Han, and G.B. McFadden, Anisotropy of interfaces in an ordered hcp binary alloy, J. Stat. Phys. 95 (1999), pp. 1337–1360.
   Web of Science ®Google Scholar
• A. Kazaryan, Y. Wang, S.A. Dregia, and B.R. Patton, Generalized phase-field model for computer simulation of grain growth in anisotropic systems, Phys. Rev. B 61 (2000), pp. 14275–14278.
   Web of Science ®Google Scholar
• T.A. Abinandanan and F. Haider, An extended Cahn--Hilliard model for interfaces with cubic anisotropy, Philos. Mag. A 81 (2001), pp. 2457–2479.
   Web of Science ®Google Scholar
• J.J. Eggleston, G.B. McFadden, and P.W. Voorhees, A phase-field model for highly anisotropic interfacial energy, Physica D: Nonlin. Phenom. 150 (2001), pp. 91–103.
   Web of Science ®Google Scholar
• J.M. Debierre, A. Karma, F. Celestini, and R. Guérin, Phase-field approach for faceted solidification, Phys. Rev. E 68 (2003), pp. 416041–4160413.
   Web of Science ®Google Scholar
• I. Loginova, J. Ågren, and G. Amberg, On the formation of Widmanstätten ferrite in binary Fe--C -- Phase-field approach, Acta Mater. 52 (2004), pp. 4055–4063.
   Web of Science ®Google Scholar
• S.G. Kim and W.T. Kim, Phase field modeling of dendritic growth with high anisotropy, J. Cryst. Growth 275 (2005), pp. e355–e360.
   Web of Science ®Google Scholar
• A.A. Wheeler, Phase-field theory of edges in an anisotropic crystal, Proc. Roy. Soc. A: Math. Phys. Engg. Sci. 462 (2006), pp. 3363–3384.
   Web of Science ®Google Scholar
• T. Haxhimali, A. Karma, G. Gonzales, and M. Rappaz, Orientation selection in denditic evolution, Nat. Mater. 5 (2006), pp. 660–664.
   PubMed Web of Science ®Google Scholar
• N. Moelans, B. Blanpain, and P. Wollants, Quantitative phase-field approach for simulating grain growth in anisotropic systems with arbitrary inclination and misorientation dependence, Phys. Rev. Lett. 101 (2008), pp. 025502–1–025502–4.
   Web of Science ®Google Scholar
• R.S. Qin and H.K.D.H. Bhadeshia, Phase-field model study of the effect of interface anisotropy on the crystal morphological evolution of cubic metals, Acta Mater. 57 (2009), pp. 2210–2216.
   Web of Science ®Google Scholar
• R.S. Qin and H.K.D.H. Bhadeshia, Phase-field model study of the crystal morphological evolution of hcp metals, Acta Mater. 57 (2009), pp. 3382–3390.
   Web of Science ®Google Scholar
• I.M. McKenna, M.P. Gururajan, and P.W. Voorhees, Phase field modeling of grain growth: Effect of boundary thickness, triple junctions, misorientation, and anisotropy, J. Mat. Sci. 44 (2009), pp. 2206–2217.
   Google Scholar
• S. Torabi, J. Lowengrub, A. Voigt, and S. Wise A new phase-field model for strongly anisotropic systems, Proc. Roy. Soc. A: Math. Phys. Engg. Sci. 465 (2009), pp. 1337–1359.
   Web of Science ®Google Scholar
• S. Torabi and J. Lowengrub, Simulating interfacial anisotropy in thin-film growth using an extended Cahn--Hilliard model, Phys. Rev. E 85 (2012), p. 041603.
   Web of Science ®Google Scholar
• E.S. Nani and M.P. Gururajan, On the incorporation of cubic and hexagonal interfacial energy anisotropy in phase field models using higher order tensor terms, Philos. Mag. 94 (2014), pp. 3331–3352.
   Web of Science ®Google Scholar
• J. Choi, S.K. Park, H.Y. Hwang, and J.Y. Huh, A comparative study of dendritic growth by using the extended Cahn--Hilliard model and the conventional phase-field model, Acta Mater. 84 (2015), pp. 55–64.
   Web of Science ®Google Scholar
• W. Liu, N. Wang, Y. Ji, P. Song, C. Zhang, Z. Yang, and L. Chen, Effects of surface energy anisotropy on void evolution during irradiation: A phase-field model, J. Nuclear Mater. 479 (2016), pp. 316–322.
   Web of Science ®Google Scholar
• I.H. Shames and C.L. Dym, Energy and Finite Element Methods in Structural Mechanics, New Age International, New Delhi, 1995.
   Google Scholar
• L.Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations, Comput. Phys. Commun. 108 (1998), pp. 147–158.
   Web of Science ®Google Scholar
• J.W. Cahn and J.E. Hilliard, Free energy of a non-uniform system I. Interfacial free energy, J. Chem. Phys. 28 (1958), pp. 258–267.
   Web of Science ®Google Scholar
• D.A. Porter and K.E. Easterling, Phase Transformations in Metals and Alloys, Chapman & Hall, London, 1996.
   Google Scholar
• A.R. Roosen, R.P. McCormack, and W.C. Carter, Wulffman: A tool for the calculation and display of crystal shapes, Comput. Mat. Sci. 11 (1998), pp. 16–26.
   Web of Science ®Google Scholar
• A. Roy and M.P. Gururajan, 3D growth Kinetics of precipitates with anisotropic interfacial free energy: A phase-field study, Trans. Ind. Inst. Met. 68 (2015), pp. 177–183.
   Web of Science ®Google Scholar
• D.W. Hoffman and J.W. Cahn, A vector thermodynamics for anisotropic surfaces. I. Fundamentals and application to plane surface junctions, Surf. Sci. 31 (1972), pp. 368–388.
   Web of Science ®Google Scholar
• A. Karma and W.J. Rappel, Numerical simulation of three-dimensional dendritic growth, Phys. Rev. Lett. 77 (1996), pp. 4050–4053.
   PubMed Web of Science ®Google Scholar
• A. Braun, Quantitative model for anisotropy and reorientation thickness of the magnetic moment in thin epitaxially strained metal films, Physica B: Cond. Mat. 373 (2006), pp. 346–354.
   Web of Science ®Google Scholar
• M.D.Graef and M.E.McHenry, Structure of Materials: An Introduction to Crystallography, Diffraction, and Symmetry, Cambridge University Press, Cambridge, 2007.
   Google Scholar