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.
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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
- 2
Cavaliere P, Sadeghi B, Shabani A (2019) Spark plasma sintering: process fundamentals. Spark plasma sintering of materials. Springer, New York, pp 3–20
- 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
- 4
Blackford JR (2007) Sintering and microstructure of ice: a review. J Phys D: Appl Phy 40(21):R355
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 16
Zhang Y, Nie J, Chan JM, Luo J (2017) Probing the densification mechanisms during flash sintering of ZnO. Acta Mater 125:465–475
- 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
- 18
Banerjee A, Bandyopadhyay A, Bose S (2007) Hydroxyapatite nanopowders: synthesis, densification and cell-materials interaction. Mat Sci Eng C 27(4):729–735
- 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
- 20
Szabo D, Schneebeli M (2007) Subsecond sintering of ice. Appl Phys Lett 90(15):151916
- 21
Kingery W (1960) Regelation, surface diffusion, and ice sintering. J Appl Phys 31(5):833–838
- 22
Coble RL (1961) Sintering crystalline solids. i. intermediate and final state diffusion models. J Appl Phys 32(5):787–792
- 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
- 24
Coble R, Kingery W (1956) Effect of porosity on physical properties of sintered alumina. J American Ceram Soc 39(11):377–385
- 25
Colbeck S (1998) Sintering in a dry snow cover. J Appl Phys 84(8):4585–4589
- 26
Colbeck S (2001) Sintering of unequal grains. Journal of Applied Physics 89(8):4612–4618
- 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
- 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
- 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
- 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
- 32
Wang YU (2006) Computer modeling and simulation of solid-state sintering: A phase field approach. Acta Mat 54(4):953–961
- 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
- 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
- 35
Herring C (1950) Diffusional viscosity of a polycrystalline solid. J appl phys 21(5):437–445
- 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
- 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
- 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
- 39
Groza JR (1999) Nanosintering. Nanostruct mat 12(5–8):987–992
- 40
Chen L-Q (2002) Phase-field models for microstructure evolution. Annu rev mat res 32(1):113–140
- 41
Provatas N, Elder K (2011) Phase-field methods in materials science and engineering. John Wiley & Sons
- 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
- 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
- 44
Hötzer J, Seiz M, Kellner M, Rheinheimer W, Nestler B (2019) Phase-field simulation of solid state sintering. Acta Mat 164:184–195
- 45
Shinagawa K (2014) Simulation of grain growth and sintering process by combined phase-field/discrete-element method. Acta Mat 66:360–369
- 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
- 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
- 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
- 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
- 50
Chakrabarti T, Mukherjee R (2019) Effect of heterogeneous particle size on nanostructure evolution: A phase-field study. Comput Mat Sci 169:109115
- 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
- 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
- 55
Kathuria Y (1999) Microstructuring by selective laser sintering of metallic powder. Surf Coatings Technol 116:643–647
- 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
- 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
- 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
- 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
- 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
- 61
Zhang X, Liao Y (2018) A phase-field model for solid-state selective laser sintering of metallic materials. Powder Technol 339:677–685
- 62
Grossmann C, Roos H-G, Stynes M (2007) Numerical treatment of partial differential equations, vol 154. Springer
- 63
LeVeque RJ, Leveque RJ (1992) Numerical methods for conservation laws, vol 3. Springer
- 64
Biner SB (2017) Programming phase-field modeling. Springer
- 65
Mullins W, Shewmon P (1959) The kinetics of grain boundary grooving in copper. Acta Metall 7(3):163–170
- 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
- 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
- 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
- 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
- 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
- 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
- 72
Chen P, Tsai YL, Lan C (2008) Phase field modeling of growth competition of silicon grains. Acta Mat 56(15):4114–4122
- 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
- 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
- 76
Spears M, Evans A (1982) Microstructure development during final/intermediate stage sintering-ii. grain and pore coarsening. Acta Metall 30(7):1281–1289
- 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
- 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
- 79
Wakai F (2006) Modeling and simulation of elementary processes in ideal sintering. J American Ceram Soc 89(5):1471–1484
- 80
Brakke KA (1992) The surface evolver. Exp math 1(2):141–165
- 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
- 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
- 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
- 84
Murr LE (1975) Interfacial phenomena in metals and alloys. Addison Wesley Publishing Company
- 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
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.
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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
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