Abstract
Applying the thermodynamic extremal principle, a model for grain growth and densification in the final stage of sintering of doped ceramics was derived, with segregation-dependent interfacial energies and mobilities (or diffusivities). The model demonstrated an interdependence between the driving forces of grain growth and densification during sintering evolution, observed because the surface energy contributes positively to the driving force of grain growth while the GB energy negatively to the driving force of densification. The model was tested in alumina as a host system, and calculations demonstrate that dopants with more negative GB (or surface) segregation enthalpy or which cause lower GB diffusion coefficient can induce higher relative densities at a given grain size. Comparatively studying yttria- and lanthana-doped alumina, the lanthana doping showed significantly enhanced sintering attributed to the larger La3+ radius causing a more negative GB segregation energy. This present model is expected to help dopant designing to improve control over sintering.
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Notes
Here, ignoring the difference in atomic volume between the three different regions, then the molar fraction will be equivalent to the volume fraction.
For ceramics (ionic compounds), the diffusion path of the dopant cation should be consistent with the host cation, since the cations generally occupy the same sub-lattice. For the case of isovalent doping, where the same electrostatic interaction acts on these two types of cations, the variation of the surrounding environment due to segregation would produce a similar effect on the diffusion of the host cation and that of the dopant cation. For aliovalent doping, the dopant cation may exhibit different dependence of diffusion on segregation compared to the host cation since, although the dopant cation passes through the same diffusion path as the host cation, the difference in electrostatic interactions due to their own charge differences may have a more pronounced effect on diffusion.
The relative density during the final stage of sintering is usually above 90%.
Model calculations show that, at moderate levels of GB segregation and surface segregation (ΔH GBseg = ΔH sseg = −40 kJ mol−1) with T = 1300 °C and x B = 1×10−3, when D GBB0 > 0.02D GB0, solute drag will not exert any observable effect on grain growth and densification.
References
Jorgensen PJ (1965) Modification of sintering kinetics by solute segregation in Al2O3. J Am Ceram Soc 48:207–210
Harmer MP, Brook RJ (1980) The effect of MgO additions on the kinetics of hot pressing in Al2O3. J Mater Sci 15:3017–3024. doi:10.1007/BF00550370
Soni KK, Thompson AM, Harmer MP, Williams DB, Chabala JM, Setti RL (1995) Solute segregation to grain boundaries in MgO-doped alumina. Appl Phys Lett 66:2795–2797
Fang JX, Thompson AM, Harmer MP, Chan HM (1997) Effect of yttrium and lanthanum on the final-stage sintering behavior of ultrahigh-purity alumina. J Am Ceram Soc 80:2005–2012
Tekeli S, Erdogan M, Aktas B (2004) Influence of α-Al2O3 addition on sintering and grain growth behaviour of 8 mol% Y2O3-stabilised cubic zirconia (c-ZrO2). Ceram Int 30:2203–2209
Tekeli S, Erdogan M, Aktas B (2004) Microstructural evolution in 8 mol% Y2O3-stabilized cubic zirconia (8YSCZ) with SiO2 addition. Mater Sci Eng A 386:1–9
Averback RS, Höfler HJ, Hahn H, Logas JC (1992) Sintering and grain growth in nanocrystalline ceramics. Nanostruct Mater 1:173–178
Li JG, Ikegami T, Mori T (2004) Low temperature processing of dense samarium-doped CeO2 ceramics: sintering and grain growth behaviors. Acta Mater 52:2221–2228
Bowen P, Carry C (2002) From powders to sintered pieces: forming, transformations and sintering of nanostructured ceramic oxides. Powder Technol 128:248–255
Theunissen GSAM, Winnubst AJA, Burggraaf AJ (1993) Sintering kinetics and microstructure development of nanoscale Y-TZP ceramics. J Eur Ceram Soc 11:315–324
Chang CH, Gong MM, Dey S, Liu F, Castro RHR (2015) Thermodynamic stability of SnO2 nanoparticles: the role of interface energies and dopants. J Phys Chem C 119:6389–6397
Wu LJ, Aguiar JA, Dholabhai PP, Holesinger T, Aoki T, Uberuaga BP, Castro RHR (2015) Interface energies of nanocrystalline doped ceria: effects of manganese segregation. J Phys Chem C 119:27855–27864
Dey S, Chang CH, Gong MM, Liu F, Castro RHR (2015) Grain growth resistant nanocrystalline zirconia by targeting zero grain boundary energies. J Mater Res 30:2991–3002
Chen PL, Chen IW (1994) Role of defect interaction in boundary mobility and cation diffusivity of CeO2. J Am Ceram Soc 77:2289–2297
Rahaman MN, Manalert R (1998) Grain boundary mobility of BaTiO3 doped with aliovalent cations. J Eur Ceram Soc 18:1063–1071
Gong MM, Dey S, Wu LJ, Chang CH, Li H, Castro RHR, Liu F (2017) Effects of concurrent grain boundary and surface segregation on the final stage of sintering: the case of Lanthanum doped yttriastabilized zirconia. J Mater Sci Technol 33:251–260
Brook RJ (1982) Fabrication principles for the production of ceramics with superior mechanical properties. Proc Br Ceram Soc 32:7–24
Kingery WD (1984) Segregation phenomena at surfaces and at grain boundaries in oxides and carbides. Solid State Ion 12:299–307
Nowotny J (1989) Surface and grain boundary segregation in metal oxides. In: Dufour L-C, Monty C, Petot-Ervas G (eds) Surfaces and interfaces of ceramic materials. Springer, Netherlands, pp 205–239
Powers JD, Glaeser AM (1998) Grain boundary migration in ceramics. Interface Sci 6:23–39
Glaeser AM (1984) Microstructure development in ceramics: the role of grain growth. J Ceram Assoc Jpn 92:537–546
Cahn JW (1962) The impurity-drag effect in grain boundary motion. Acta Metall 10:789–798
Kingery WD, Francois B (1965) Grain growth in porous compacts. J Am Ceram Soc 48:546–547
Nichols FA (1966) Theory of grain growth in porous compacts. J Appl Phys 37:4599–4602
Nichols FA (1968) Further comments on the theory of grain growth in porous compacts. J Am Ceram Soc 51:468–469
Brook RJ (1969) Pore-grain boundary interactions and grain growth. J Am Ceram Soc 52:56–57
Brook RJ (1969) Pores and grain growth kinetics. J Am Ceram Soc 52:339–340
Riedel H, Svoboda J (1993) A theoretical study of grain growth in porous solids during sintering. Acta Metall Mater 41:1929–1936
Readey DW (1966) Mass transport and sintering in impure ionic solids. J Am Ceram Soc 49:366–369
Readey DW (1966) Chemical potentials and initial sintering in pure metals and ionic compounds. J Appl Phys 37:2309–2312
Gong MM, Chang CH, Wu LJ, Dey S, Castro RHR, Liu F (2017) Modeling the grain growth kinetics of doped nearly fully dense nanocrystalline ceramics. Ceram Int 43:6677–6683
Svoboda J, Riedel H (1992) Pore-boundary interactions and evolution equations for the porosity and the grain size during sintering. Acta Metall Mater 40:2829–2840
Svoboda J, Turek I, Fischer FD (2005) Application of the thermodynamic extremal principle to modeling of thermodynamic processes in material sciences. Philos Mag 85:3699–3707
Svoboda J, Fischer FD, Gamsjäger E (2002) Influence of solute segregation and drag on properties of migrating interfaces. Acta Mater 50:967–977
Svoboda J, Fischer FD, Leindl M (2011) Transient solute drag in migrating grain boundaries. Acta Mater 59:6556–6562
Gong MM, Castro RHR, Liu F (2015) Modeling grain growth kinetics of binary substitutional alloys by the thermodynamic extremal principle. J Mater Sci 50:4610–4621. doi:10.1007/s10853-015-9010-4
Svoboda J, Fischer FD, Fratzl P (2006) Diffusion and creep in multi-component alloys with non-ideal sources and sinks for vacancies. Acta Mater 54:3043–3053
Svoboda J, Fischer FD, Fratzl P, Kozeschnik E (2004) Modelling of kinetics in multi-component multi-phase systems with spherical precipitates I: theory. Mater Sci Eng A 385:166–174
Kozeschnik E, Svoboda J, Fratzl P, Fischer FD (2004) Modelling of kinetics in multi-component multi-phase systems with spherical precipitates II: numerical solution and application. Mater Sci Eng A 385:157–165
Coble RL (1961) Sintering crystalline solids. I. Intermediate and final state diffusion models. J Appl Phys 32:787–792
Hillert M (2007) Phase equilibria, phase diagrams and phase transformations: their thermodynamic basis, 2nd edn. Cambridge University Press, New York, pp 361–363
Hillert M, Sundman B (1976) A treatment of the solute drag on moving grain boundaries and phase interfaces in binary alloys. Acta Metall 24:731–743
Blendell JE, Handwerker CA (1986) Effect of chemical composition on sintering of ceramics. J Cryst Growth 75:138–160
Hwang SL, Chen IW (1990) Grain size control of tetragonal zirconia polycrystals using the space charge concept. J Am Ceram Soc 73:3269–3277
Johnson WC (1977) Grain boundary segregation in ceramics. Metall Trans A 8:1413–1422
Terwilliger CD, Chiang YM (1995) Size-dependent solute segregation and total solubility in ultrafine polycrystals: Ca in TiO2. Acta Metall Mater 43:319–328
Tschöpe A (2005) Interface defect chemistry and effective conductivity in polycrystalline cerium oxide. J Electroceram 14:5–23
Colbourn EA, MacKrodt WC, Tasker PW (1983) The segregation of calcium ions at the surface of magnesium oxide: theory and calculation. J Mater Sci 18:1917–1924. doi:10.1007/BF00554983
Mclean D (1957) Grain boundaries in metals. Oxford University Press, Oxford, pp 15–43
Lejček P, Hofmann S, Janovec J (2007) Prediction of enthalpy and entropy of solute segregation at individual grain boundaries of α-iron and ferrite steels. Mater Sci Eng A 462:76–85
Hillert M (1965) On the theory of normal and abnormal grain growth. Acta Metall 13:227–238
Kang SJL, Jung Y (2004) Sintering kinetics at final stage sintering: model calculation and map construction. Acta Mater 52:4573–4578
Johnson DL (1970) a general model for the intermediate stage of sintering. J Am Ceram Soc 53:574–577
Hansen JD, Rusin RP, Teng MH, Johnson DL (1992) Combined-stage sintering model. J Am Ceram Soc 75:1129–1135
KaKar AK (1968) Sintering kinetics based on geometric models. J Am Ceram Soc 51:236
Gibbs JW (1928) The collected work of J. W. Gibbs. Longmans, Green & Co, New York, pp 55–56
Weissmüller J (1993) Alloy effects in nanostructures. Nanostruct Mater 3:261–272
Kirchheim R (2002) Grain coarsening inhibited by solute segregation. Acta Mater 50:413–419
Liu F, Kirchheim R (2004) Nano-scale grain growth inhibited by reducing grain boundary energy through solute segregation. J Cryst Growth 264:385–391
Krill CE, Ehrhardt H, Birringer R (2005) Thermodynamic stabilization of nanocrystallinity. Z Metallkunde 96:1134–1141
Trelewicz JR, Schuh CA (2009) Grain boundary segregation and thermodynamically stable binary nanocrystalline alloys. Phys Rev B 79:094112-1–094112-13
Darling KA, VanLeeuwen BK, Semones JE, Koch CC, Scattergood RO, Kecskes LJ, Mathaudhu SN (2011) Stabilized nanocrystalline iron-based alloys: guiding efforts in alloy selection. Mater Sci Eng A 528:4365–4371
Saber M, Kotan H, Koch CC, Scattergood RO (2013) Thermodynamic stabilization of nanocrystalline binary alloys. J Appl Phys 113:063515-1–063515-10
Wu LJ, Dey S, Gong MM, Liu F, Castro RHR (2014) Surface segregation on manganese doped ceria nanoparticles and relationship with nanostability. J Phys Chem C 118:30187–30196
Wynblatt P, Ku RC (1977) Surface energy and solute strain energy effects in surface segregation. Surf Sci 65:511–531
Wynblatt P, Chatain D (2006) Anisotropy of segregation at grain boundaries and surfaces. Metall Mater Trans A 37:2595–2620
Saber M, Kotan H, Koch CC, Scattergood RO (2013) A predictive model for thermodynamic stability of grain size in nanocrystalline ternary alloys. J Appl Phys 114:103510
Chookajorn T, Schuh CA (2014) Thermodynamics of stable nanocrystalline alloys: a Monte Carlo analysis. Phys Rev B 89:064102-1–064102-10
Borisov VT, Golikov VM, Scherbedinskiy GV (1964) Relation between diffusion coefficients and grain boundary energy. Fiz Met Metalloved 17:881–885
Burke JE, Turnbull D (1952) Recrystallization and grain growth. Prog Met Phys 3:220–292
Bernardini J, Gas P, Hondros ED, Seah MP (1982) The role of solute segregation in grain boundary diffusion. Proc R Soc Lond A 379:159–178
Chiang YM, Birnie DP, Kingery WD (1997) Physical ceramics: principles for ceramic science and engineering. Wiley, New York, pp 413–421
Acknowledgements
The authors are grateful to the financial support of National Basic Research Program of China (No. 2011CB610403), the Natural Science Foundation of China (Nos. 51134011 and 51431008), the Fundamental Research Fund of Northwestern Polytechnical University (No. JC20120223), and the China National Funds for Distinguished Young Scientists (No. 51125002). M. M. Gong is thanked for the financial supports of the Doctorate Foundation of Northwestern Polytechnical University (CX201204) and of China Scholarship Council. R. H. R. Castro is thanked for the financial support of the National Science Foundation (DMR 1609781).
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Gong, M.M., Castro, R.H.R. & Liu, F. Modeling the final sintering stage of doped ceramics: mutual interaction between grain growth and densification. J Mater Sci 53, 1680–1698 (2018). https://doi.org/10.1007/s10853-017-1617-1
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DOI: https://doi.org/10.1007/s10853-017-1617-1