On the Volume-Diffusion Governed Termination-Migration Assisted Globularization in Two-Phase Solid-State Systems: Insights from Phase-Field Simulations

  • P. G. Kubendran AmosEmail author
  • Ephraim Schoof
  • Daniel Schneider
  • Britta Nestler
Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)


Morphology of the constituent phase in a microstructure influences the mechanical properties of the materials similar to the chemical composition, crystal structure and volume fraction of the phases. Often to enhance the mechanical properties, heat treatment techniques that exclusively facilitate the morphological evolution without any phase transformation are adopted. In the present work, the mechanism and the kinetics of this morphological transformation governed by the inherent curvature-difference introduced by the shape, referred to shape instabilities, are analysed through phase-field simulations. By monitoring the temporal evolution of the elliptical plate, a finite three-dimensional structure associated with two-phase titanium alloys, it is observed that the transformation is dictated by the recession of the edges, or termination migration. It is identified that the elliptical structure transforms to a rod before assuming an ellipsoidal shape at the midpoint and ultimately, evolves into a spheroid. The results of this work are compared with the existing analytical approach and the deviations introduced by the geometrical approximations of the theoretical study are discussed. Furthermore, for the first time, it is shown that the hitherto assumed monotonic decrease in the driving force is interrupted by a series of ‘aberrations’ predominantly in the first half of the transformation, which intensifies with increase in aspect ratio. An analytical prediction which includes this non-monotonic evolution of the curvature is presented from the outcomes of the simulation.



PGKA thanks the financial support of the German Research Foundation (DFG) under the project AN 1245/1. This work was performed on the computational resource ForHLR II, funded by the Ministry of Science, Research and Arts of Baden-Wuerttemberg and the DFG. Authors acknowledge the primary guidance of Prof. Kumar Ankit and Dr. Avisor Bhattacharya.


  1. 1.
    Martin, J.W., Martin, J.W., Doherty, R.D., Cantor, B.: Stability of Microstructure in Metallic Systems. Cambridge University Press, Cambridge (1997)Google Scholar
  2. 2.
    Wang, Y.-T., Adachi, Y., Nakajima, K., Sugimoto, Y.: Quantitative three-dimensional characterization of pearlite spheroidization. Acta Mater. 58(14), 4849–4858 (2010)CrossRefGoogle Scholar
  3. 3.
    Malzahn Kampe, J.C., Courtney, T.H., Leng, Y.: Shape instabilities of plate-like structures-I. Experimental observations in heavily cold worked in situ composites. Acta Metall. 37(7), 1735–1745 (1989)CrossRefGoogle Scholar
  4. 4.
    Xu, J., Zeng, W., Jia, Z., Sun, X., Zhou, J.: Static globularization kinetics for Ti-17 alloy with initial lamellar microstructure. J. Alloy. Compd. 603, 239–247 (2014)CrossRefGoogle Scholar
  5. 5.
    Cline, H.E.: Shape instabilities of eutectic composites at elevated temperatures. Acta Metall. 19(6), 481–490 (1971)CrossRefGoogle Scholar
  6. 6.
    Ho, E.: Coarsening of lamellar structures at elevated temperatures. J. Jpn. Inst. Met. 15(2), 114–120 (1974)Google Scholar
  7. 7.
    Nichols, F.A.: On the spheroidization of rod-shaped particles of finite length. J. Mater. Sci. 11(6), 1077–1082 (1976)CrossRefGoogle Scholar
  8. 8.
    Ramanujan, R.V., Maziasz, P.J., Liu, C.T.: The thermal stability of the microstructure of \(\gamma \)-based titanium aluminides. Acta Mater. 44(7), 2611–2642 (1996)CrossRefGoogle Scholar
  9. 9.
    Nichols, F.A., Mullins, W.W.: Surface-(interface-) and volume-diffusion contributions to morphological changes driven by capillarity. Trans. Metall. Soc. 233, 1840–1848 (1965)Google Scholar
  10. 10.
    Tian, Y.L., Kraft, R.W.: Kinetics of pearlite spheroidizations. Metall. Trans. A 18(8), 1359–1369 (1987)CrossRefGoogle Scholar
  11. 11.
    Courtney, T.H., Malzahn Kampe, J.C.: Shape instabilities of plate-like structures-II. Anal. Acta Metall. 37(7), 1747–1758 (1989)CrossRefGoogle Scholar
  12. 12.
    Stefansson, N., Semiatin, S.L.: Mechanisms of globularization of Ti-6Al-4V during static heat treatment. Metall. Mater. Trans. A 34(4), 691–698 (2003)CrossRefGoogle Scholar
  13. 13.
    Qian, M., Baicheng, L., Runqi, L.: Thermodynamic considerations of the instability of rod morphologies. Acta Metall. Mater. 42(12), 4087–4089 (1994)CrossRefGoogle Scholar
  14. 14.
    Qian, M.A.: Non-linear capillary shape evolution of rod morphologies via interfacial diffusion. Acta Mater. 46(5), 1669–1681 (1998)CrossRefGoogle Scholar
  15. 15.
    Tian, L., Russell, A.: Phase field study of interfacial diffusion-driven spheroidization in a composite comprised of two mutually insoluble phases. J. Chem. Phys. 140(12), 124706 (2014)CrossRefGoogle Scholar
  16. 16.
    Kubendran Amos, P.G., Mushongera, L.T., Nestler, B.: Phase-field analysis of volume-diffusion controlled shape-instabilities in metallic systems-I: 2-Dimensional plate-like structures. Comput. Mater. Sci. 144, 363–373 (2018)CrossRefGoogle Scholar
  17. 17.
    Kubendran Amos, P.G., Mushongera, L.T., Mittnacht, T., Nestler, B.: Phase-field analysis of volume-diffusion controlled shape-instabilities in metallic systems-II: Finite 3-dimensional rods. Comput. Mater. Sci. 144, 374–385 (2018)CrossRefGoogle Scholar
  18. 18.
    Joshi, C., Abinandanan, T.A., Choudhury, A.: Phase field modelling of Rayleigh instabilities in the solid-state. Acta Mater. 109, 286–291 (2016)CrossRefGoogle Scholar
  19. 19.
    Steinbach, I.: Phase-field models in materials science. Modell. Simul. Mater. Sci. Eng. 17(7), 073001 (2009)CrossRefGoogle Scholar
  20. 20.
    Caginalp, G.: Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations. Phys. Rev. A 39(11), 5887 (1989)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Chen, L.-Q.: Phase-field models for microstructure evolution. Ann. Rev. Mater. Res. 32(1), 113–140 (2002)CrossRefGoogle Scholar
  22. 22.
    Nichols, F.A., Mullins, W.W.: Morphological changes of a surface of revolution due to capillarity-induced surface diffusion. J. Appl. Phys. 36(6), 1826–1835 (1965)CrossRefGoogle Scholar
  23. 23.
    Plapp, M.: Unified derivation of phase-field models for alloy solidification from a grand-potential functional. Phys. Rev. E 84(3), 031601 (2011)CrossRefGoogle Scholar
  24. 24.
    Garcke, H., Nestler, B., Stoth, B.: A multiphase field concept: numerical simulations of moving phase boundaries and multiple junctions. SIAM J. Appl. Math. 60(1), 295–315 (1999)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Oono, Y., Puri, S.: Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling. Phys. Rev. A 38(1), 434 (1988)CrossRefGoogle Scholar
  26. 26.
    Hötzer, J., Tschukin, O., Said, M.B., Berghoff, M., Jainta, M., Barthelemy, G., Smorchkov, N., Schneider, D., Selzer, M., Nestler, B.: Calibration of a multi-phase field model with quantitative angle measurement. J. Mater. Sci. 51(4), 1788–1797 (2016)CrossRefGoogle Scholar
  27. 27.
    Park, C.H., Won, J.W., Park, J.-W., Semiatin, S.L., Lee, C.S.: Mechanisms and kinetics of static spheroidization of hot-worked Ti-6Al-2Sn-4Zr-2Mo-0.1 Si with a lamellar microstructure. Metall. Mater. Trans. A 43(3), 977–985 (2012)CrossRefGoogle Scholar
  28. 28.
    Semiatin, S.L., Stefansson, N., Doherty, R.D.: Prediction of the kinetics of static globularization of Ti-6Al-4V. Metall. Mater. Trans. A 36(5), 1372–1376 (2005)CrossRefGoogle Scholar
  29. 29.
    Semiatin, S.L., Kirby, B.C., Salishchev, G.A.: Coarsening behavior of an alpha-beta titanium alloy. Metall. Mater. Trans. A 35(9), 2809–2819 (2004)CrossRefGoogle Scholar
  30. 30.
    Choudhury, A., Nestler, B.: Grand-potential formulation for multicomponent phase transformations combined with thin-interface asymptotics of the double-obstacle potential. Phys. Rev. E 85(2), 021602 (2012)CrossRefGoogle Scholar
  31. 31.
    Sharma, G., Ramanujan, R.V., Tiwari, G.P.: Instability mechanisms in lamellar microstructures. Acta Mater. 48(4), 875–889 (2000)CrossRefGoogle Scholar
  32. 32.
    Cline, H.E.: Shape instabilities of eutectic composites at elevated temperatures. Acta Metall. 19(6), 481–490 (1971)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • P. G. Kubendran Amos
    • 1
    Email author
  • Ephraim Schoof
    • 1
    • 2
  • Daniel Schneider
    • 1
    • 2
  • Britta Nestler
    • 1
    • 2
  1. 1.Institute of Applied Materials - Computational Materials Science (IAM-CMS)Karlsruhe Institute of Technology (KIT)KarlsruheGermany
  2. 2.Institute of Digital Materials Science (IDM)Karlsruhe University of Applied SciencesKarlsruheGermany

Personalised recommendations