Skip to main content
SpringerLink
Log in

Particle curvature effects on microstructural evolution during solid-state sintering: phenomenological insights from phase-field simulations

  • Computation & theory
  • Published:
Journal of Materials Science Aims and scope Submit manuscript

Abstract

Local curvatures have a profound influence on sintered microstructure. Here, using phase-field simulations, particle curvature effects were phenomenologically investigated by using geometrical configurations of two, three, and four particles, and by systematically varying particle curvatures. Some geometries, involving two, three and four particles, exhibited the expected smooth neck-length evolution, where the maximum neck length was determined by grain boundary (GB) energy (\(\gamma _{GB}\)) rather than surface energy (\(\gamma _{S}\)). In contrast, triangular arrangement of particles with unequal radii manifested a secondary necking event in form of a step during neck evolution. The secondary necking event coincided with internal pore collapse, and only specific range of particle radius ratios manifested such a mechanism. \(\gamma _{S}\) played a dominant role in triggering the secondary necking event, while \(\gamma _{GB}\) determined the remnant microstructure. Broadly, the geometries employed here allow us to computationally examine the sintering of particles that display wide variation in shapes and size distributions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11

Similar content being viewed by others

Data availability

Data can be made available upon reasonable request. C++ codes developed for numerically solving equations (1)–(5) are available at https://github.com/DeepChoudhuri/Phase-field-modeling-of-Sintering.

References

  1. Kang S-JL (2004) Sintering: densification, grain growth and microstructure. Elsevier, London

    Google Scholar 

  2. Cavaliere P, Sadeghi B, Shabani A (2019) Spark plasma sintering: process fundamentals. Spark plasma sintering of materials. Springer, New York, pp 3–20

    Chapter  Google Scholar 

  3. Lóh N, Simão L, Faller C, De Noni Jr A, Montedo O (2016) A review of two-step sintering for ceramics. Ceram Int 42(11):12556–12572

    Article  Google Scholar 

  4. Blackford JR (2007) Sintering and microstructure of ice: a review. J Phys D: Appl Phy 40(21):R355

    Article  CAS  Google Scholar 

  5. Cordier A, Kleitz M, Steil MC (2012) Welding of Yttrium-doped zirconia granules by electric current activated sintering (ecas): protrusion formation as a possible intermediate step in the consolidation mechanism. J European Ceram Soc 32(8):1473–1479

    Article  CAS  Google Scholar 

  6. Diao K, Xiao Z, Zhao Y (2015) Specific surface areas of porous cu manufactured by lost carbonate sintering: measurements by quantitative stereology and cyclic voltammetry. Mat Chem Phys 162:571–579

    Article  CAS  Google Scholar 

  7. Olakanmi E, Cochrane R, Dalgarno K (2011) Densification mechanism and microstructural evolution in selective laser sintering of Al-12Si powders. J Mat Process Technol 211(1):113–121

    Article  CAS  Google Scholar 

  8. Albiter A, Leon C, Drew R, Bedolla E (2000) Microstructure and heat-treatment response of Al-2024/TiC composites. Mat Sci Eng: A 289(1–2):109–115

    Article  Google Scholar 

  9. Reimanis I, Kleebe H-J (2009) A review on the sintering and microstructure development of transparent spinel (mgal2o4). J American Ceram Soc 92(7):1472–1480

    Article  CAS  Google Scholar 

  10. Bajpai I, Han Y-H, Yun J, Francis J, Kim S, Raj R (2016) Preliminary investigation of hydroxyapatite microstructures prepared by flash sintering. Adv Appl Ceram 115(5):276–281

    Article  CAS  Google Scholar 

  11. Gregorová E, Pabst W, Uhlířová T, Nečina V, Veselỳ M, Sedlářová I (2016) Processing, microstructure and elastic properties of mullite-based ceramic foams prepared by direct foaming with wheat flour. J European Ceram Soc 36(1):109–120

    Article  Google Scholar 

  12. Sun Y, Luo G, Zhang J, Wu C, Li J, Shen Q, Zhang L (2018) Phase transition, microstructure and mechanical properties of TC4 titanium alloy prepared by plasma activated sintering. J Alloys Compd 741:918–926

    Article  CAS  Google Scholar 

  13. Reis RM, Barbosa AJ, Ghussn L, Ferreira EB, Prado MO, Zanotto ED (2019) Sintering and rounding kinetics of irregular glass particles. J American Ceram Soc 102(2):845–854

    CAS  Google Scholar 

  14. Razavi-Tousi S, Yazdani-Rad R, Manafi S (2011) Effect of volume fraction and particle size of alumina reinforcement on compaction and densification behavior of al-al2o3 nanocomposites. Mat Sci Eng A 528(3):1105–1110

    Article  Google Scholar 

  15. Razavi-Tousi S, Yazdani-Rad R, Manafi S (2011) Production of Al nanocomposite reinforced by Fe-Al intermetallic, Al\(_4\)C\(_3\) and nano-Al\(_2\)O\(_3\) particles using wet milling in toluene. J alloys compd 509(22):6489–6496

    Article  CAS  Google Scholar 

  16. Zhang Y, Nie J, Chan JM, Luo J (2017) Probing the densification mechanisms during flash sintering of ZnO. Acta Mater 125:465–475

    Article  CAS  Google Scholar 

  17. Zhang X, Zhang Z, Wang W, Che H, Zhang X, Bai Y, Zhang L, Fu Z (2017) Densification behaviour and mechanical properties of B\(_4\)C-SiC intergranular/intragranular nanocomposites fabricated through spark plasma sintering assisted by mechanochemistry. Ceram Int 43(2):1904–1910

    Article  CAS  Google Scholar 

  18. Banerjee A, Bandyopadhyay A, Bose S (2007) Hydroxyapatite nanopowders: synthesis, densification and cell-materials interaction. Mat Sci Eng C 27(4):729–735

    Article  CAS  Google Scholar 

  19. Chaim R, Levin M, Shlayer A, Estournès C (2008) Sintering and densification of nanocrystalline ceramic oxide powders: a review. Adv Appl Ceram 107(3):159–169

    Article  CAS  Google Scholar 

  20. Szabo D, Schneebeli M (2007) Subsecond sintering of ice. Appl Phys Lett 90(15):151916

    Article  Google Scholar 

  21. Kingery W (1960) Regelation, surface diffusion, and ice sintering. J Appl Phys 31(5):833–838

    Article  CAS  Google Scholar 

  22. Coble RL (1961) Sintering crystalline solids. i. intermediate and final state diffusion models. J Appl Phys 32(5):787–792

    Article  CAS  Google Scholar 

  23. Kingery WD, Berg M (1955) Study of the initial stages of sintering solids by viscous flow, evaporation-condensation, and self-diffusion. J Appl Phys 26(10):1205–1212

    Article  CAS  Google Scholar 

  24. Coble R, Kingery W (1956) Effect of porosity on physical properties of sintered alumina. J American Ceram Soc 39(11):377–385

    Article  Google Scholar 

  25. Colbeck S (1998) Sintering in a dry snow cover. J Appl Phys 84(8):4585–4589

    Article  CAS  Google Scholar 

  26. Colbeck S (2001) Sintering of unequal grains. Journal of Applied Physics 89(8):4612–4618

    Article  CAS  Google Scholar 

  27. Kuczynski GC (1990) Self-diffusion in sintering of metallic particles, in: Sintering Key Papers, Springer, New York pp. 509–527

  28. Lange FF (1989) Powder processing science and technology for increased reliability. J American Ceram Soc 72(1):3–15

    Article  CAS  Google Scholar 

  29. Lange FF, Kellett BJ (1989) Thermodynamics of densification: II, grain growth in porous compacts and relation to densification. J American Ceram Soc 72(5):735–741

    Article  CAS  Google Scholar 

  30. Tikare V, Braginsky M, Olevsky EA (2003) Numerical simulation of solid-state sintering: I, sintering of three particles. J American Ceram Soc 86(1):49–53

    Article  CAS  Google Scholar 

  31. Kumar V, Fang Z, Fife P (2010) Phase field simulations of grain growth during sintering of two unequal-sized particles. Mat Sci Eng: A 528(1):254–259

    Article  Google Scholar 

  32. Wang YU (2006) Computer modeling and simulation of solid-state sintering: A phase field approach. Acta Mat 54(4):953–961

    Article  CAS  Google Scholar 

  33. Biswas S, Schwen D, Tomar V (2018) Implementation of a phase field model for simulating evolution of two powder particles representing microstructural changes during sintering. J Mat Sci 53(8):5799–5825

    Article  CAS  Google Scholar 

  34. Ahmed K, Yablinsky CA, Schulte A, Allen T, El-Azab A (2013) Phase field modeling of the effect of porosity on grain growth kinetics in polycrystalline ceramics. Model Simulation Mat Sci Eng 21(6):065005

    Article  Google Scholar 

  35. Herring C (1950) Diffusional viscosity of a polycrystalline solid. J appl phys 21(5):437–445

    Article  Google Scholar 

  36. Shilan ST, Mazlan SA, Ido Y, Hajalilou A, Jeyadevan B, Choi S-B, Yunus NA (2016) A comparison of field-dependent rheological properties between spherical and plate-like carbonyl iron particles-based magneto-rheological fluids. Smart Mat Struct 25(9):095025

    Article  Google Scholar 

  37. Mostafaei A, Kimes KA, Stevens EL, Toman J, Krimer YL, Ullakko K, Chmielus M (2017) Microstructural evolution and magnetic properties of binder jet additive manufactured Ni-Mn-Ga magnetic shape memory alloy foam. Acta Mat 131:482–490

    Article  CAS  Google Scholar 

  38. Mapley M, Pauls JP, Tansley G, Busch A, Gregory SD (2019) Selective laser sintering of bonded magnets from flake and spherical powders. Scripta Mat 172:154–158

    Article  CAS  Google Scholar 

  39. Groza JR (1999) Nanosintering. Nanostruct mat 12(5–8):987–992

    Article  Google Scholar 

  40. Chen L-Q (2002) Phase-field models for microstructure evolution. Annu rev mat res 32(1):113–140

    Article  CAS  Google Scholar 

  41. Provatas N, Elder K (2011) Phase-field methods in materials science and engineering. John Wiley & Sons

    Google Scholar 

  42. Biswas S, Schwen D, Wang H, Okuniewski M, Tomar V (2018) Phase field modeling of sintering: Role of grain orientation and anisotropic properties. Comput Mat Sci 148:307–319

    Article  Google Scholar 

  43. Dzepina B, Balint D, Dini D (2019) A phase field model of pressure-assisted sintering. J European Ceram Soc 39(2–3):173–182

    Article  CAS  Google Scholar 

  44. Hötzer J, Seiz M, Kellner M, Rheinheimer W, Nestler B (2019) Phase-field simulation of solid state sintering. Acta Mat 164:184–195

    Article  Google Scholar 

  45. Shinagawa K (2014) Simulation of grain growth and sintering process by combined phase-field/discrete-element method. Acta Mat 66:360–369

    Article  CAS  Google Scholar 

  46. Yang Y, Ragnvaldsen O, Bai Y, Yi M, Xu B-X (2019) 3D non-isothermal phase-field simulation of microstructure evolution during selective laser sintering. NPJ Comput Mat 5(1):1–12

    Google Scholar 

  47. Shinagawa K, Maki S, Yokota K (2014) Phase-field simulation of platelike grain growth during sintering of alumina. J European Ceram Soc 34(12):3027–3036

    Article  CAS  Google Scholar 

  48. Biswas S, Schwen D, Singh J, Tomar V (2016) A study of the evolution of microstructure and consolidation kinetics during sintering using a phase field modeling based approach. Extreme Mech Lett 7:78–89

    Article  Google Scholar 

  49. Zhang Z, Yao X, Ge P (2020) Phase-field-model-based analysis of the effects of powder particle on porosities and densities in selective laser sintering additive manufacturing. Int J Mech Sci 166:105230

    Article  Google Scholar 

  50. Chakrabarti T, Mukherjee R (2019) Effect of heterogeneous particle size on nanostructure evolution: A phase-field study. Comput Mat Sci 169:109115

    Article  CAS  Google Scholar 

  51. Wood M, Gao X, Shi R, Heo TW, Espitia JA, Duoss EB, Wood BC, Ye J, Exploring the relationship between solvent-assisted ball milling, particle size, and sintering temperature in garnet-type solid electrolytes, Journal of Power Sources 484 229252

  52. Asp K, Ågren J (2006) Phase-field simulation of sintering and related phenomena-a vacancy diffusion approach. Acta Mat 54(5):1241–1248

    Article  CAS  Google Scholar 

  53. Shemon P (1964) Diffusion in solids, McGraw- 33. De Schepper L., Knuyt G. and Stals L.. Phys. Stat Solidi 64

  54. Porter DA, Easterling KE (2009) Phase transformations in metals and alloys (revised reprint). CRC Press

    Book  Google Scholar 

  55. Kathuria Y (1999) Microstructuring by selective laser sintering of metallic powder. Surf Coatings Technol 116:643–647

    Article  Google Scholar 

  56. Williams JM, Adewunmi A, Schek RM, Flanagan CL, Krebsbach PH, Feinberg SE, Hollister SJ, Das S (2005) Bone tissue engineering using polycaprolactone scaffolds fabricated via selective laser sintering. Biomaterials 26(23):4817–4827

    Article  CAS  Google Scholar 

  57. Duan B, Wang M, Zhou WY, Cheung WL, Li ZY, Lu WW (2010) Three-dimensional nanocomposite scaffolds fabricated via selective laser sintering for bone tissue engineering. Acta biomat 6(12):4495–4505

    Article  CAS  Google Scholar 

  58. Xie F, He X, Cao S, Qu X (2013) Structural and mechanical characteristics of porous 316l stainless steel fabricated by indirect selective laser sintering. J Mat Process Technol 213(6):838–843

    Article  CAS  Google Scholar 

  59. Chen A-N, Li M, Wu J-M, Cheng L-J, Liu R-Z, Shi Y-S, Li C-H (2019) Enhancement mechanism of mechanical performance of highly porous mullite ceramics with bimodal pore structures prepared by selective laser sintering. J Alloys Compd 776:486–494

    Article  CAS  Google Scholar 

  60. Du L, Yang S, Zhu X, Jiang J, Hui Q, Du H (2018) Pore deformation and grain boundary migration during sintering in porous materials: a phase-field approach. J Mat Sci 53(13):9567–9577

    Article  CAS  Google Scholar 

  61. Zhang X, Liao Y (2018) A phase-field model for solid-state selective laser sintering of metallic materials. Powder Technol 339:677–685

    Article  CAS  Google Scholar 

  62. Grossmann C, Roos H-G, Stynes M (2007) Numerical treatment of partial differential equations, vol 154. Springer

    Book  Google Scholar 

  63. LeVeque RJ, Leveque RJ (1992) Numerical methods for conservation laws, vol 3. Springer

    Book  Google Scholar 

  64. Biner SB (2017) Programming phase-field modeling. Springer

    Book  Google Scholar 

  65. Mullins W, Shewmon P (1959) The kinetics of grain boundary grooving in copper. Acta Metall 7(3):163–170

    Article  CAS  Google Scholar 

  66. Saylor DM, Rohrer GS (1999) Measuring the influence of grain-boundary misorientation on thermal groove geometry in ceramic polycrystals. J American Ceram Soc 82(6):1529–1536

    Article  CAS  Google Scholar 

  67. Amram D, Klinger L, Gazit N, Gluska H, Rabkin E (2014) Grain boundary grooving in thin films revisited: the role of interface diffusion. Acta Mat 69:386–396

    Article  CAS  Google Scholar 

  68. Schölhammer J, Baretzky B, Gust W, Mittemeijer E, Straumal B (2001) Grain boundary grooving as an indicator of grain boundary phase transformations. Interf Sci 9(1–2):43–53

    Article  Google Scholar 

  69. Erk KA, Deschaseaux C, Trice RW (2006) Grain-boundary grooving of plasma-sprayed yttria-stabilized zirconia thermal barrier coatings. J American Ceram Soc 89(5):1673–1678

    Article  CAS  Google Scholar 

  70. Kelly MN, Rheinheimer W, Hoffmann MJ, Rohrer GS (2018) Anti-thermal grain growth in srtio3: Coupled reduction of the grain boundary energy and grain growth rate constant. Acta Mat 149:11–18

    Article  CAS  Google Scholar 

  71. Bouville M, Hu S, Chen L-Q, Chi D, Srolovitz DJ (2006) Phase-field model for grain boundary grooving in multi-component thin films. Model Simulation Mat Sci Eng 14(3):433

    Article  CAS  Google Scholar 

  72. Chen P, Tsai YL, Lan C (2008) Phase field modeling of growth competition of silicon grains. Acta Mat 56(15):4114–4122

    Article  CAS  Google Scholar 

  73. Kim H-K, Kim SG, Dong W, Steinbach I, Lee B-J (2014) Phase-field modeling for 3d grain growth based on a grain boundary energy database. Model Simulation Mat Sci Eng 22(3):034004

    Article  Google Scholar 

  74. Laxmipathy VP, Wang F, Selzer M, Nestler B (2020) Phase-field simulations of grain boundary grooving under diffusive-convective conditions. Acta Mat 116497

  75. Hsueh C, Evans A, Coble R (1982) Microstructure development during final/intermediate stage sintering-i. pore/grain boundary separation. Acta Met 30(7):1269–1279

    Article  Google Scholar 

  76. Spears M, Evans A (1982) Microstructure development during final/intermediate stage sintering-ii. grain and pore coarsening. Acta Metall 30(7):1281–1289

    Article  Google Scholar 

  77. Riedel H, Svoboda J (1993) A theoretical study of grain growth in porous solids during sintering. Acta Metall et Materialia 41(6):1929–1936

    Article  CAS  Google Scholar 

  78. Wakai F, Akatsu T, Shinoda Y (2006) Shrinkage and disappearance of a closed pore in the sintering of particle cluster. Acta Mat 54(3):793–805

    Article  CAS  Google Scholar 

  79. Wakai F (2006) Modeling and simulation of elementary processes in ideal sintering. J American Ceram Soc 89(5):1471–1484

    Article  CAS  Google Scholar 

  80. Brakke KA (1992) The surface evolver. Exp math 1(2):141–165

    Article  Google Scholar 

  81. Velmurugan C, Senthilkumar V, Biswas K, Yadav S (2018) Densification and microstructural evolution of spark plasma sintered NiTi shape memory alloy. Adv Powder Technol 29(10):2456–2462

    Article  CAS  Google Scholar 

  82. Nersisyan HH, Yoo BU, Kim YM, Son HT, Lee KY, Lee JH (2016) Gas-phase supported rapid manufacturing of Ti-6Al-4V alloy spherical particles for 3D printing. Chem Eng J 304:232–240

    Article  CAS  Google Scholar 

  83. Devaraj S, Sankaran S, Kumar R (2013) Influence of spark plasma sintering temperature on the densification, microstructure and mechanical properties of Al-4.5 wt.% Cu alloy. Acta Metall Sinica (English Letters) 26(6):761–771

    Article  CAS  Google Scholar 

  84. Murr LE (1975) Interfacial phenomena in metals and alloys. Addison Wesley Publishing Company

    Google Scholar 

  85. Choudhuri D, Banerjee R, Srinivasan S (2017) Interfacial structures and energetics of the strengthening precipitate phase in creep-resistant Mg-Nd-based alloys. Scientific reports 7:40540

    Article  CAS  Google Scholar 

Download references

Acknowledgements

DC acknowledges support from New Mexico Tech’s faculty startup and computer time on the Bridges2 ocean cluster through the XSEDE allocation TG-MAT200006.

Author information

Authors and Affiliations

  1. Department of Materials and Metallurgical Engineering, New Mexico Institute of Mining and Technology, Socorro, New Mexico, 87801, USA

    Deep Choudhuri & Logan Blake

Authors
  1. Deep Choudhuri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Logan Blake
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Deep Choudhuri.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Handling Editor: M. Grant Norton.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Choudhuri, D., Blake, L. Particle curvature effects on microstructural evolution during solid-state sintering: phenomenological insights from phase-field simulations. J Mater Sci 56, 7474–7493 (2021). https://doi.org/10.1007/s10853-021-05802-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10853-021-05802-8

Search

Navigation