Journal of Materials Science

, Volume 53, Issue 8, pp 5799–5825 | Cite as

Implementation of a phase field model for simulating evolution of two powder particles representing microstructural changes during sintering

Interface Behavior


In this work, we present a multiphysics phase field model for capturing microstructural evolution during solid-state sintering processes. The model incorporates modifications of phase field equations to include rigid-body motion, elastic deformation, and heat conduction. The model correctly predicts consolidation of powder particles during sintering because of two competing mechanisms—neck formation and grain growth. The simulations show that the material undergoes three distinctive stages during the sintering process—stage I where neck or grain boundary between two particles is formed, stage II in which neck length stabilizes and growth or shrinkage of individual particles initiates, and finally stage III with rapid grain growth leading to disappearance of one of the grains. The driving forces corresponding to different mechanisms are found to be dependent on the radius of the particles, curvature at the neck location, surface energy, and grain boundary energy. In addition, variation in temperature is found to significantly influence the microstructure evolution by affecting the diffusivity and grain boundary mobility of the sintered material. The model is also used to compare sintering simulation results in 2D and 3D. It is observed that due to higher curvature in 3D, model predicts faster microstructural evolution in 3D when compared to 2D simulations under identical boundary conditions.



We acknowledge the funding received from DoE-NETL supporting this research work. We are grateful to the MOOSE developers, especially Cody Permann and Michael Tonks, and the user community for helping with technical discussions to overcome modeling hurdles.


  1. 1.
    Langer J, Hoffmann MJ, Guillon O (2009) Direct comparison between hot pressing and electric field-assisted sintering of submicron alumina. Acta Mater 57(18):5454–5465CrossRefGoogle Scholar
  2. 2.
    Stanciu LA, Kodash VY, Groza JR (2001) Effects of heating rate on densification and grain growth during field-assisted sintering of α-Al2O3 and MoSi2 powders. Metall Mater Trans A 32(10):2633–2638CrossRefGoogle Scholar
  3. 3.
    Chaim R et al (2008) Sintering and densification of nanocrystalline ceramic oxide powders: a review. Adv Appl Ceram 107(3):159–169CrossRefGoogle Scholar
  4. 4.
    Dahl P et al (2007) Densification and properties of zirconia prepared by three different sintering techniques. Ceram Int 33(8):1603–1610CrossRefGoogle Scholar
  5. 5.
    Guyot P et al (2014) Hot pressing and spark plasma sintering of alumina: discussion about an analytical modelling used for sintering mechanism determination. Scripta Mater 84–85:35–38CrossRefGoogle Scholar
  6. 6.
    Groza JR, Cirtis JD, Krämer M (2000) Field assisted sintering of nanocrystalline titanium nitride. J Am Ceram Soc 83(5):1281–1283CrossRefGoogle Scholar
  7. 7.
    Groza JR (2000) Sintering activation by electrical field. Mater Sci Eng 287:8CrossRefGoogle Scholar
  8. 8.
    Muccillo R, Muccillo ENS (2013) An experimental setup for shrinkage evaluation during electric field-assisted flash sintering: application to yttria-stabilized zirconia. J Eur Ceram Soc 33(3):515–520CrossRefGoogle Scholar
  9. 9.
    Grigoryev E (2011) High voltage electric discharge consolidation of tungsten carbide - cobalt powder. In: Cuppoletti J (ed) Nanocomposites with unique properties and applications in medicine and industry. InTechGoogle Scholar
  10. 10.
    Groza JR, Garcia M, Schneider JA (2001) Surface effect in field assisted sintering. J Mater Res 16(01):286–292CrossRefGoogle Scholar
  11. 11.
    Vanmeensel K et al (2005) Modelling of the temperature distribution during field assisted sintering. Acta Mater 53(16):4379–4388CrossRefGoogle Scholar
  12. 12.
    Tiwari D, Basu B, Biswas K (2009) Simulation of thermal and electric field evolution during spark plasma sintering. Ceram Int 35(2):699–708CrossRefGoogle Scholar
  13. 13.
    Maizza G et al (2007) Relation between microstructure, properties and spark plasma sintering (SPS) parameters of pure ultrafine WC powder. Sci Technol Adv Mater 8(7–8):644–654CrossRefGoogle Scholar
  14. 14.
    Kraft T, Riedel H (2004) Numerical simulation of solid state sintering; model and application. J Eur Ceram Soc 24(2):345–361CrossRefGoogle Scholar
  15. 15.
    Olevsky E, Froyen L (2006) Constitutive modeling of spark-plasma sintering of conductive materials. Scripta Mater 55(12):1175–1178CrossRefGoogle Scholar
  16. 16.
    Braginsky M, Tikare V, Olevsky E (2005) Numerical simulation of solid state sintering. Int J Solids Struct 42(2):621–636CrossRefGoogle Scholar
  17. 17.
    Tikare V, Braginsky M, Olevsky EA (2003) Numerical simulation of solid-state sintering: I, sintering of three particles. J Am Ceram Soc 86(1):49–53CrossRefGoogle Scholar
  18. 18.
    Tikare V et al (2010) Numerical simulation of microstructural evolution during sintering at the mesoscale in a 3D powder compact. Comput Mater Sci 48(2):317–325CrossRefGoogle Scholar
  19. 19.
    Bjørk R et al (2015) Modeling the microstructural evolution during constrained sintering. J Am Ceram Soc 98(11):3490–3495CrossRefGoogle Scholar
  20. 20.
    Boettinger WJ et al (2002) Phase-field simulation of solidification. Annu Rev Mater Res 32(1):163–194CrossRefGoogle Scholar
  21. 21.
    Loginova I, Amberg G, Agren J (2001) Phase-field simulations of non-isothermal binary alloy solidification. Acta Mater 49(4):573–581CrossRefGoogle Scholar
  22. 22.
    Osório WR, Freire CMA, Garcia A (2005) Dendritic solidification microstructure affecting mechanical and corrosion properties of a Zn4Al alloy. J Mater Sci 40(17):4493–4499. CrossRefGoogle Scholar
  23. 23.
    Uehara T, Tsujino T (2005) Phase field simulation of stress evolution during solidification. J Cryst Growth 275(1–2):e219–e224CrossRefGoogle Scholar
  24. 24.
    Grafe U et al (2000) Simulations of the initial transient during directional solidification of multicomponent alloys using the phase field method. Modell Simul Mater Sci Eng 8(6):871–879CrossRefGoogle Scholar
  25. 25.
    Hu S, Henager CH Jr (2009) Phase-field modeling of void lattice formation under irradiation. J Nucl Mater 394(2–3):155–159CrossRefGoogle Scholar
  26. 26.
    Hu SY, Henager CH Jr (2010) Phase-field simulation of void migration in a temperature gradient. Acta Mater 58(9):3230–3237CrossRefGoogle Scholar
  27. 27.
    Li Y et al (2011) Phase-field modeling of void evolution and swelling in materials under irradiation. Sci China Phys Mech Astron 54(5):856–865CrossRefGoogle Scholar
  28. 28.
    Millett PC et al (2011) Phase-field simulation of irradiated metals: part I: void kinetics. Comput Mater Sci 50(3):949–959CrossRefGoogle Scholar
  29. 29.
    Millett PC et al (2009) Void nucleation and growth in irradiated polycrystalline metals: a phase-field model. Modell Simul Mater Sci Eng 17(6):064003CrossRefGoogle Scholar
  30. 30.
    Millett PC, Tonks M (2011) Application of phase-field modeling to irradiation effects in materials. Curr Opin Solid State Mater Sci 15(3):125–133CrossRefGoogle Scholar
  31. 31.
    Ahmed K et al (2014) Phase field simulation of grain growth in porous uranium dioxide. J Nucl Mater 446(1–3):90–99CrossRefGoogle Scholar
  32. 32.
    Anderson MP et al (1984) Computer simulation of grain growth—I. Kinetics. Acta Metall 32(5):783–791CrossRefGoogle Scholar
  33. 33.
    Kazaryan A et al (2000) Generalized phase-field model for computer simulation of grain growth in anisotropic systems. Phys Rev B 61(21):14275–14278CrossRefGoogle Scholar
  34. 34.
    Tikare V, Holm EA (1998) Simulation of grain growth and pore migration in a thermal gradient. J Am Ceram Soc 81(3):480–484CrossRefGoogle Scholar
  35. 35.
    Wang YU (2006) Computer modeling and simulation of solid-state sintering: a phase field approach. Acta Mater 54(4):953–961CrossRefGoogle Scholar
  36. 36.
    Deng J (2012) A phase field model of sintering with direction-dependent diffusion. Mater Trans 53(2):385–389CrossRefGoogle Scholar
  37. 37.
    Ahmed K et al (2013) Phase field modeling of the effect of porosity on grain growth kinetics in polycrystalline ceramics. Modell Simul Mater Sci Eng 21(6):065005CrossRefGoogle Scholar
  38. 38.
    Chanthapan S et al (2012) Sintering of tungsten powder with and without tungsten carbide additive by field assisted sintering technology. Int J Refract Metal Hard Mater 31:114–120CrossRefGoogle Scholar
  39. 39.
    Biswas S et al (2016) A study of the evolution of microstructure and consolidation kinetics during sintering using a phase field modeling based approach. Extreme Mech Lett 7:78–89CrossRefGoogle Scholar
  40. 40.
    Gaston DR, Peterson JW, Permann CJ, Andrš D, Slaughter AE, Miller JM (2014) Continuous integration for concurrent computational framework and application development. J Open Res Softw 2(1):e10. CrossRefGoogle Scholar
  41. 41.
    Gaston D et al (2009) MOOSE: A parallel computational framework for coupled systems of nonlinear equations. Nucl Eng Des 239(10):1768–1778CrossRefGoogle Scholar
  42. 42.
    Tonks M, Gaston D, Millett P, Andrš D, Talbot P (2012) An object-oriented finite element framework for multiphysics phase field simulations. Comput Mater Sci 51(1):20–29CrossRefGoogle Scholar
  43. 43.
    Novascone SR et al (2015) Evaluation of coupling approaches for thermomechanical simulations. Nucl Eng Des 295:910–921CrossRefGoogle Scholar
  44. 44.
    Tonks MR et al (2016) Development of a multiscale thermal conductivity model for fission gas in UO2. J Nucl Mater 469:89–98CrossRefGoogle Scholar
  45. 45.
    Moelans N, Blanpain B, Wollants P (2008) An introduction to phase-field modeling of microstructure evolution. Calphad 32(2):268–294CrossRefGoogle Scholar
  46. 46.
    Verma D et al (2016) Relating interface evolution to interface mechanics based on interface properties. JOM 69(1):30–38CrossRefGoogle Scholar
  47. 47.
    Kumar V, Fang ZZ, Fife PC (2010) Phase field simulations of grain growth during sintering of two unequal-sized particles. Mater Sci Eng, A 528(1):254–259CrossRefGoogle Scholar
  48. 48.
    Zhang R-J et al (2014) Thermodynamic consistent phase field model for sintering process with multiphase powders. Trans Nonferrous Met Soc China 24(3):783–789CrossRefGoogle Scholar
  49. 49.
    Permann CJ et al (2016) Order parameter re-mapping algorithm for 3D phase field model of grain growth using FEM. Comput Mater Sci 115:18–25CrossRefGoogle Scholar
  50. 50.
    Khachaturyan A-G (1983) Theory of structural transformations in solids. Wiley, HobokenGoogle Scholar
  51. 51.
    Hu SY, Chen LQ (2001) A phase-field model for evolving microstructures with strong elastic inhomogeneity. Acta Mater 49(11):1879–1890CrossRefGoogle Scholar
  52. 52.
    Yongsheng L et al (2014) Effects of temperature gradient and elastic strain on spinodal decomposition and microstructure evolution of binary alloys. Modell Simul Mater Sci Eng 22(3):035009CrossRefGoogle Scholar
  53. 53.
    Tonks M et al (2010) Analysis of the elastic strain energy driving force for grain boundary migration using phase field simulation. Scripta Mater 63(11):1049–1052CrossRefGoogle Scholar
  54. 54.
    Zhang L et al (2013) A quantitative comparison between and elements for solving the Cahn–Hilliard equation. J Comput Phys 236:74–80CrossRefGoogle Scholar
  55. 55.
    Grujicic M, Zhao H, Krasko GL (1997) Atomistic simulation of Sigma 3 (111) grain boundary fracture in tungsten containing various impurities. Int J Refract Metal Hard Mater 15(5–6):341–355CrossRefGoogle Scholar
  56. 56.
    Lassner E, Schubert W-D (1999) Tungsten: properties, chemistry, technology of the element, alloys, and chemical compounds. Springer, New YorkCrossRefGoogle Scholar
  57. 57.
    Johnson RA (1982) Point-defect calculations for tungsten. Phys. Rev. B 27(4):2014–2018CrossRefGoogle Scholar
  58. 58.
    Lee JS, Minkwitz C, Herzig C (1997) Grain boundary self-diffusion in polycrystalline tungsten at low temperatures. Phys Stat Sol 202:931–940CrossRefGoogle Scholar
  59. 59.
    Kumar V (2011) Simulations and modeling of unequal sized particles sintering. In: Department of metallurgical engineering. The University of UtahGoogle Scholar
  60. 60.
    Schwen D et al (2017) Rapid multiphase-field model development using a modular free energy based approach with automatic differentiation in MOOSE/MARMOT. Comput Mater Sci 132:36–45CrossRefGoogle Scholar
  61. 61.
    Millett PC et al (2012) Phase-field simulation of intergranular bubble growth and percolation in bicrystals. J Nucl Mater 425(1–3):130–135CrossRefGoogle Scholar
  62. 62.
    Chen L-Q (2002) Phase-field models for microstructure evolution. Annu Rev Mater Res 32(1):113–140CrossRefGoogle Scholar
  63. 63.
    Cahn JW, Hilliard JE (1958) Free energy of a nonuniform system. I. Interfacial free energy. J Chem Phys 28(2):258–267CrossRefGoogle Scholar
  64. 64.
    Gao Z et al (2012) Kinetics of densification and grain growth of pure tungsten during spark plasma sintering. Metall Mater Trans B 43(6):1608–1614CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Aeronautics and AstronauticsPurdue UniversityWest LafayetteUSA
  2. 2.Fuels Modeling and SimulationIdaho National LaboratoryIdaho FallsUSA

Personalised recommendations