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A predictive model to reflect the final stage of spark plasma sintering of submicronic α-alumina

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A predictive model to reflect the final stage of spark plasma sintering of submicronic α-alumina
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  Short communication A predictive model to re fl ect the fi nal stage of spark plasma sintering of submicronic  α -alumina Charles Manière a,b,c , Lise Durand b , Alicia Weibel a , Claude Estournès a,c, n a Université de Toulouse, Institut Carnot Cirimat, UMR 5085 CNRS, Université Toulouse III Paul-Sabatier-INPT, 118 Route de Narbonne, 31062 ToulouseCedex 9, France b CNRS, CEMES, UPR 8011 and Université de Toulouse, 29 Rue Jeanne Marvig, 31055 Toulouse, France c CNRS, Institut Carnot CIRIMAT, 118 Route de Narbonne, 31602 Toulouse Cedex 9, France a r t i c l e i n f o  Article history: Received 13 January 2016Received in revised form5 February 2016Accepted 7 February 2016 Keywords: Spark plasma sinteringModelingGrain growthDensi fi cation a b s t r a c t The grain growth/densi fi cation interaction is known to strongly decrease the shrinkage rate during the fi nal stages of sintering. This phenomenon was studied for α -alumina to provide more accurate sinteringmodels for ceramics. Isothermal interrupted experiments were conducted to identify the parameters of the grain growth law for the model. &  2016 Elsevier Ltd and Techna Group S.r.l. All rights reserved. In both pressureless and pressurized sintering processes, likeHot Pressing (HP) and Spark Plasma Sintering (SPS), the grain sizeis known to have a strong in fl uence on the  fi nal stage of sintering.According to Rahaman [1], the attainment of high relative densityrequires minimization of the grain growth phenomenon. For ex-ample, transparent alumina ceramics (porosity lower than 0.1%)were obtained at low temperature or low heating rate [2,3], which allows densi fi cation without grain growth. These results can beexplained considering the grain size dependence of the diffusioncreep mechanisms, Nabarro-Herring (lattice diffusion) [4,5] and Coble (grain boundary diffusion) [6]. In both of them, the dis-placement is due to atomic motion over varying distances de-pending on the grain size. The larger the grain, the lower the creeprate, because the atomic diffusion distances increase. Thus, in bothlattice and grain boundary diffusion creep, the grain size termappears in the denominator of the creep rate equation: εσ    ̇ =( ) CDG kT   1 c nm where  ̇ c   is the creep rate,  C   a constant, the stress,  D  the diffusioncoef  fi cient,  k  the Boltzmann constant,  G  the grain size,  T   the absolutetemperature,  m  and n the grain size and stress exponent, respectively.In this expression, the creep law appears to be strongly in fl u-enced by the grain size exponent. Nabarro and Herring [4,5] determined, for lattice diffusion creep, a value for the grain sizeexponent  m ¼ 2. On the other hand, Coble [6] established a grainsize exponent  m ¼ 3 for grain boundary diffusion creep. Conse-quently, the determination of the  m  exponent allows the identi- fi cation of the diffusional creep mechanism. Most widespreadsintering models are built on porous creep behavior law, such asOlevsky's sintering model [7,8] for which the creep law is coupled to a grain growth law.In a previous publication [9] on SPS densi fi cation modeling of apure submicronic  α -alumina powder, we showed that taking intoaccount the grain growth phenomenon allows correction of therelative density values at the  fi nal stage of sintering. We showedthat, by inverse analysis, it is possible to determine an approx-imate parameter for the grain growth law using the experimentaldata obtained at the  fi nal stage of sintering. Thus, using this value,the model obtained appears to be reliable since the  fi nal averagegrain diameter calculated is in good agreement with the experi-mentally observed one. Moreover, the sintering laws identi fi ed invarious pressure and heating rate conditions converged to a linear( n ¼ 1) creep behavior. A result similar to that reported by Langeret al. [10] on the same alumina powder.The aim of the present work is to determine the parameters of the alumina grain growth law by isothermal interrupted tests andto compare the values obtained to those estimated by reverseanalysis of our previous study. The other objective is to discuss thegrain growth mechanisms and to study the densi fi cation/graingrowth interaction during the  fi nal stage of sintering in the SPSprocess.Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ceramint Ceramics International http://dx.doi.org/10.1016/j.ceramint.2016.02.0480272-8842/ &  2016 Elsevier Ltd and Techna Group S.r.l. All rights reserved. n Corresponding author at: CNRS, Institut Carnot CIRIMAT, 118 Route de Nar-bonne, 31602 Toulouse Cedex 9, France. Please cite this article as: C. Manière, et al., A predictive model to re fl ect the  fi nal stage of spark plasma sintering of submicronic  α -alumina, Ceramics International (2016), http://dx.doi.org/10.1016/j.ceramint.2016.02.048i Ceramics International  ∎  ( ∎∎∎∎ )  ∎∎∎ – ∎∎∎  The sintering experiments were conducted on the Dr. Sinter2080 SPS machine (SPS Syntex INC JapanCo. Ltd., Japan) of thePlateforme Nationale CNRS de Frittage Flash located at the Uni-versité Toulouse III-Paul Sabatier inToulouse. For each experiment,1 g of powder ( α -alumina 99.99%, reference TM-DAR, TaimeiChemicals Co. Ltd, median particle size 0.14  m m) was introducedand consolidated within an 8 mm inner diameter graphite die.Each experiment was performed under vacuum ( o 10 Pa) with apulse sequence of 12  2 (12 pulses and 2 dead times, each pulseand dead time having a duration of 3.3 ms). For easy removal of the sample graphite foil (e.g. Papyex from Mersen) was introducedat the sample/die and sample/punch interfaces. The heating ratewas 100 K/min. The pressure was kept constant at 100 MPa duringthe whole cycle. In order to identify all the terms of the graingrowth law, isothermal tests were performed at three pro-grammed temperatures (1100  ° C, 1200  ° C and 1300  ° C) and withdifferent dwell times (0, 1, 5 and 15 min). The real temperature of the sample was determined in other experiments, performed insimilar conditions, with a sacri fi cial thermocouple located in thepowder bed. An empiric function obtained by calibrating the twotemperatures at the beginning of the sintering cycle was used toextrapolate the temperature of the dwell to respectively 1305,1439 and 1575  ° C. The huge differences between setpoint andextrapolated sample temperatures observed are explained by ahigh thermal contact resistance present at the punch/die andsample/die interfaces. In previous studies (C. Arnaud et al. [11], C.Manière et al. [12]) these temperature differences where con- fi rmed by thermal images taken during experiments performedwith an open die. The fracture surfaces of the pellets were ob-served by  fi eld emission-gun scanning electron microscopy (FES-EM, JEOL JSM 6700 F). FESEM images of the fracture surfaces of thesintered materials are presented in Fig. 1. Based on the work of Horovistiz et al. [13] the average grain size was determined fromsuch fracture images considering about a hundred grains. TheMendelson [14] stereological factor of 1.56 was used to transformthe 2D average intercept length into 3D average grain size. Asso-ciated standard deviation (Table 1) represents the grain size dis-tribution, the error on the determination of the grain size is 7 0.05  m m. As expected, grain size depends much more on thetemperature than on the dwell time (Table 1). It is to be noted thatthere is a very high standard deviation at high temperatures(1439  ° C and 1575  ° C) which is due to the presence of the smallgrains.The analytic Olevsky's sintering model [7,8] for uni-axial die compaction can be summarized by Eq. (2): ⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟ ( )  ρ ρσ ψ φ ρ   ̇=( ) + ( )( ) + − − K T G ,2  z nnn 231 12 nn 12 where  ρ     ̇  is the densi fi cation rate,  ρ  is the relative density,  σ   z   theapplied stress,  φ  and  ψ   the shear and bulk moduli,  K  ( T,G ) the creepconsistency factor depending on the grain size  G  and the tem-perature  T  , and the stress exponent  n .Based on our previous work [9] the expression of the con-sistency factor is: ⎛⎝⎜⎞⎠⎟⎛⎝⎜⎞⎠⎟ ( ) =( ) K T G GG AT  Q nRT  , 1exp3 onn 0211 Where  G 0  is the initial grain diameter,  A 0  a constant of 0.873 K.Pa  n s  1 ,  Q   is the activation energy of 179 kJ/mol and  n the stress exponent equal to 1 [9].The grain growth law mainly depends on the temperature [1]but mayalso depends on both the applied pressure ( P  ) [15 – 17] andthe relative density (  ρ ) [18]. As we will see later, the grain growthmodel considered, which does not take into account any pressureor porosity dependence, gives good agreement with experiment,therefore in our experimental domain these dependences ( P   and  ρ )can be ignored. In the present study, expression (4) [19,20] T sample= 1305 °C 10 µm10 µm T sample= 1439 °CT sample= 1575 °C0 s60 s300 s900 s Dwell:1 µm1 µm10 µm10 µm10 µm10 µm10 µm10 µm1 µm2 µm Fig. 1.  FESEM images of the fracture surfaces of the materials sintered at different temperatures and dwell times. C. Manière et al. / Ceramics International  ∎  ( ∎∎∎∎ )  ∎∎∎ – ∎∎∎ 2 Please cite this article as: C. Manière, et al., A predictive model to re fl ect the  fi nal stage of spark plasma sintering of submicronic  α -alumina, Ceramics International (2016), http://dx.doi.org/10.1016/j.ceramint.2016.02.048i  considered for the grain growth rate (  ̇ G ) is only temperaturedependent: ( )   ̇=( ) − Gk expG  4 Q RT m 0  g  where  k 0  is a constant,  Q   g   the grain growth activation energy and  R the gas constant.Eq. (4) can be transformed into its logarithmic form (Eq. (5)): ( )  ( )   ̇= − * − * ( )( ) ln G ln k Q R T  m ln G 15  g  0 at constant temperature the following term of Eq. (5) is a constant: ( ) = − *( ) ln k Q R T  Cst 16  g  0 As there is not enough available data in the grain growth curveto determine  ̇ G  we have considered a  fi t (i.e. the interpolationcurves reported in Fig. 2).All of the linear regressions (Fig. 3) give straight lines which,over the experimental domain explored, validate the temperaturedependent grain growth model (4).The slope of the plot of   ( ̇)= ( ) ln G f ln G  at a given temperature(Fig. 3a) gives an  m  exponent ranging from 1.89 to 2.12. Con-sidering Eq. (6) and a mvalue equal to 2,  k 0  constant and activationenergy  Q   g   were determined (Fig. 3b) and are equal to5.53  10  4 m 3 /s and 530 kJ/mol, respectively.For pure materials undergoing normal grain growth, an mvalueequal to 2 corresponds to a grain boundary control mechanism[21]. The grain growth activation energy of 530 kJ/mol is in goodagreement with that (520 kJ/mol) determined in previous worksby calibration on the  fi nal stage of sintering [9]. Olevsky et al. [7] identi fi ed a higher grain growth activation energy (570 kJ/mol) forsubmicronic  α -alumina probably because of the larger initial grainsize of the powder used (0.38  m m  vs  0.14  m m).To highlight the effect of grain growth on sintering, the graingrowth law parameters were used to model previously publishedsintering data at 1400  ° C [9]. The result is reported in Fig. 4. The green curve converging at the end of the sintering to a full  Table1 Average grain size, standard deviation and relative density for the differenttemperatures and dwell times.Dwell tempera-ture ( ° C)Dwelltime (s)Average grainsize ( m m)Standard de-viation ( m m)Relative den-sity ( 7 0.5%)1305 0 0.31 0.08 94.860 0.46 0.11 99.3300 1.08 0.29 98.6900 1.94 0.55 98.31439 0 4.47 1.50 97.360 4.95 2.18 98.2300 5.38 2.22 98.1900 5.89 2.44 98.01575 0 6.06 1.49 97.960 7.23 2.25 96.8300 8.81 2.90 96.9900 12.07 3.77 98.6 020040060080010000246810121416 Exp. T = 1305°CExp. T = 1439°CExp. T = 1575°Cmodel at 1305°Cmodel at 1439°Cmodel at 1575°C    G  r  a   i  n   S   i  z  e   (  m   ) Time (s)      µ   Fig. 2.  Model/experimental isothermal plot of grain size versus time. Fig. 3.  Grain growth model identi fi cation a) determination of the m exponent (b)  k 0  constant and  Q   determination. 0 200 400 600 800 10000,550,600,650,700,750,800,850,900,951,00 ExperimentalWith grain growthWithout grain growth    R  e   l  a   t   i  v  e   D  e  n  s   i   t  y Time (s) 020040060080010001200140016001800Sample temperature    T  e  m  p  e  r  a   t  u  r  e   (   °   C   ) Fig. 4.  Sintering model with and without grain growth (heating rate 100 K/min,8 mm diameter sample [9]). (For interpretation of the references to color in this fi gure, the reader is referred to the web version of this article.). C. Manière et al. / Ceramics International  ∎  ( ∎∎∎∎ )  ∎∎∎ – ∎∎∎  3 Please cite this article as: C. Manière, et al., A predictive model to re fl ect the  fi nal stage of spark plasma sintering of submicronic  α -alumina, Ceramics International (2016), http://dx.doi.org/10.1016/j.ceramint.2016.02.048i  densi fi cation is the one given by the sintering model (Eq. (2))without taking into account the grain growth law. The blue curvegiving a full description of the evolution of the relative densityduring the  fi nal stages of sintering was obtained using the com-plete sintering model ((Eqs. (2) and (4)) with the grain growth effect.In conclusion, the grain growth law parameters of a pure sub-micronic α -alumina were determined using isothermal treatmentsat three temperatures. In the experimental domain considered, thegrain growth mechanism suggested by the m exponent is a grainboundary control mechanism. Introducing the parameters ob-tained in a mechanical model of the sintering process enabled agood description of the relative density curve and in particular of the non-attainment of full densi fi cation at the end of the cycle.  Acknowledgments Electron microscopy was performed at the  “ Centre de micro-caractérisation Raimond Castaing ” –  UMS 3623, Toulouse. References [1] M.N. Rahaman, Sintering of Ceramics, CRC Press, Boca Raton, FL, 2008.[2] B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, Spark plasma sintering of trans-parent alumina, Scr. Mater. 57 (2007) 607.[3] B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, Effects of heating rate on micro-structure and transparency of spark-plasma-sintered alumina, J. Eur. Ceram.Soc. 29 (2009) 323.[4] F.R.N. Nabarro, Deformation of crystals by the motion of single ions, in: Pro-ceedings of the Int Conf on the Strength of solids, Physical Society London,1948, pp. 75.[5] C. Herring, Diffusional viscosity of a polycristalline solid, J. Appl. Phys. 21(1950) 437.[6] R.L. Coble, A model for boundary diffusion controlled creep in polycristallinematerials, J. Appl. Phys. 34 (1963) 1679.[7] E.A. Olevsky, C. Garcia-Cardona, W.L. Bradbury, C.D. Haines, D.G. Martin,D. Kapoor, Fundamental aspects of spark plasma sintering: II. Finite elementanalysis of scalability, J. Am. Ceram. Soc. 95 (2012) 2414 – 2422.[8] E.A. Olevsky, Theory of sintering: from discrete to continuum, Mater. Sci. Eng.R23 (1998) 41 – 100.[9] C. Maniere, L. Durand, A. Weibel, C. Estournès, Spark-plasma-sintering and fi nite element method: from the identi fi cation of the sintering parameters of asubmicronic  α -alumina powder to the development of complex shapes, ActaMater. (2016) 169 – 175.[10] J. Langer, M.J. Hoffmann, O. Guillon, Direct comparison between hot pressingand electric  fi eld assisted sintering of submicron alumina, Acta Mater. 57(2009) 5454 – 5465.[11] C. Arnaud, C. Manière, G. Chevallier, C. Estournès, R. Mainguy, F. Lecouturier,et al., Dog-bone copper specimens prepared by one-step spark plasma sin-tering, J. Mater. Sci. 50 (2015) 7364 – 7373.[12] C. Manière, A. Pavia, L. Durand, G. Chevallier, K. Afanga, C. Estournès, Finite-element modeling of the electro-thermal contacts in the spark plasma sin-tering process, J. Eur. Ceram. Soc. 36 (2016) 741 – 748.[13] A.L. Horovistiz, J.R. Frade, L.R.O. Hein, Comparison of fracture surface andplane section analysis for ceramic grain size characterisation, J. Eur. Ceram.Soc. 24 (2004) 619 – 626.[14] M.I. Mendelson, Average grain size in polycrystalline ceramics, J. Am. Ceram.Soc. 52 (1969) 443 – 446.[15] M. Clark, T. Alden, Deformation enhanced grain growth in a superplastic Sn-1%Bi alloy, Acta Metal. 21 (1973) 1195 – 1206.[16] H. Tagai, T. Zisner, High-temperature creep of polycrystalline magnesia: I, ef-fect of simultaneous grain growth, J. Am. Ceram. Soc. 51 (1968) 303 – 310.[17] K.R. Venkatachari, R. Raj, Superplastic  fl ow in  fi ne-grained alumina, J. Am.Ceram. Soc. 69 (1986) 135 – 138.[18] M.A. Spears, A.G. Evans, Microstructure development during  fi nal/ inter-mediate stage sintering  –  II. Grain and pore coarsening., Acta Metal. 30 (1982)1281 – 1289.[19] J. Besson, M. Abouaf, Grain growth enhancement in Alumina during hot iso-static pressing, Acta Metall. Mater. 39 (1991) 2225 – 2234.[20] H.V. Atkinson, Overview no. 65: theories of normal grain growth inpure singlephase systems, Acta Metall. 36 (1988) 469.[21] D. Bernache-Assollant, M. Soustelle, C. Monty, J.M. Chaix, Chimie-physique dufrittage, Hermes (1993). C. Manière et al. / Ceramics International  ∎  ( ∎∎∎∎ )  ∎∎∎ – ∎∎∎ 4 Please cite this article as: C. Manière, et al., A predictive model to re fl ect the  fi nal stage of spark plasma sintering of submicronic  α -alumina, Ceramics International (2016), http://dx.doi.org/10.1016/j.ceramint.2016.02.048i
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