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Theoretical analysis of experimental densification kinetics in final sintering stage of nano-sized zirconia
Journal of the European Ceramic Society xxx (xxxx) xxx–xxx
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Original Article
Theoretical analysis of experimental densification kinetics in final sintering stage of nano-sized zirconia ⁎
Byung-Nam Kim , Tohru S. Suzuki, Koji Morita, Hidehiro Yoshida, Ji-Guang Li, Hideaki Matsubara1 National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki, 305-0047, Japan
A R T I C LE I N FO
A B S T R A C T
Keywords: Densification Kinetics Grain-growth Sintering Pore structure
The experimental densification kinetics of 7.8 mol% Y2O3-stabilized zirconia was analyzed theoretically during isothermal sintering in the final stage. By taking concurrent grain growth into account, a possible value of the grain-size exponent n was examined. The Coble’s corner-pore model recognized widely was found not to be applicable for explaining the densification kinetics. The corner-pore model of n = 4 shows a significant divergence in the kinetics at different temperatures. Microstructural observation shows that most pores are not located at grain corners and have a size comparable to the surrounding grains. The observed pore structure is similar to the diffusive model where single pore is surrounded by dense body. The diffusive model combined with theoretical sintering stress predicts n = 1 or n = 2, which shows a good consistence to the measured densification kinetics. During sintering of nano-sized powder, it is found that the densification kinetics can be explained distinctively by the diffusive single-pore model.
1. Introduction Sintering is a process of densification or pore shrinkage in powder compacts. The process has been divided into three stages: initial, intermediate and final stages. Most densification occurs in the intermediate stage where open pores are formed and shrink mainly through particle re-arrangement. In the final stage, pores are closed and shrink mainly through diffusion. The sintering stage may be distinguished by pore structure [1,2] or by densification mechanism [3,4]. The rate equation of densification in the intermediate and final stages can be represented as [5,6]
exp (−Q/RT ) f (D) D˙ = A (1) T Gn where D˙ is the densification rate, A is a constant, R is the gas constant, T is the absolute temperature, Q is the activation energy of densification, n is an exponent of the grain size G, and f(D) is an unspecified function of the relative density D. In order to understand the densification behavior, all the sintering parameters in Eq. (1), that is, Q, n and f(D), should be evaluated. A representative is the method of Wang-Raj [5,6] and of master sintering curve [7]. In the Wang-Raj method, Q and n are evaluated at a fixed density, and in the method of master sintering
curve, an average value of Q is evaluated in a given density range. In both methods, however, f(D) remains unknown, and the experimental densification kinetics has not been discussed in terms of f(D). f(D) is a mixed function of the bulk viscosity and the sintering stress, and is related to both pore structure and densification mechanism [8]. Physically, f(D) is linearly proportional to the densification rate, when no grain growth occurs, as in Eq. (1). In order to understand and predict the densification behavior of powder compacts, the evaluation of f(D) is indispensable, whereas most studies on sintering have been focused on the densification mechanism through evaluating n and Q. In the previous study [9], f(D) was evaluated experimentally in the intermediate stage during isothermal sintering of zirconia, and compared with theoretical predictions. Although some theoretical models showed partial consistence in a limited density range, no models were consistent with the experimentally determined f(D) in the entire density range of the intermediate stage. This is mainly due to a difference in the pore structure between models and experiments. Whereas the pores in theoretical models have a simple structure, the actual pores during sintering are very complicated particularly in the intermediate stage. Unlikely in the intermediate stage, the pore structure in the final stage is relatively simple. Typical is the corner-pore model where a spherical pore is located at grain corners and the pore shrinkage occurs
⁎
Corresponding author. E-mail address: [email protected] (B.-N. Kim). 1 Tohoku University, 6-6 Aoba, Aramaki, Aoba-ku, Sendai 980-8579, Japan. https://doi.org/10.1016/j.jeurceramsoc.2018.12.007 Received 19 September 2018; Received in revised form 26 November 2018; Accepted 1 December 2018 0955-2219/ © 2018 Elsevier Ltd. All rights reserved.
Please cite this article as: Kim, B.-N., Journal of the European Ceramic Society, https://doi.org/10.1016/j.jeurceramsoc.2018.12.007
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was observed using a scanning electron microscope (SEM, SU-8000, Hitachi). Before SEM observation, the polished surface was etched in air for 1 h at the temperature lower than the sintering temperature by 150 °C, and was coated with Pt. For one sample, 5–7 photographs were taken, each of which includes more than 400 grains. On each photograph, the number and the area of largest grains covering 20∼21% of the total area were measured using a software of Photoshop. The grain size in this study was defined as an average size of the largest grains. The number of the largest grains for each photograph is 40∼65, and the present grain size is about 2.1∼2.4 times the size measured by the conventional mean-intercept method for selected samples. Though the present grain size is different from the conventionally defined one, the two sizes would be linearly proportional, if normal grain growth occurs. The grain size was examined at D≥0.85 with an interval of ΔD = 0.02 or 0.03.
by a mechanism of grain-boundary diffusion [1,10–15]. The cornerpore model was originally proposed by Coble [1]. Though the densification rate by grain-boundary diffusion was not derived in his original study, Kang and Jung [13] showed later that D˙ is independent of the density, indicating a constant value of f(D). Zhao and Harmer [18] modified the Coble model, to show the densification rate proportional to the number of pores per grain (N), that is, f(D)∝N. Since N is dependent on the density, f(D) may be represented as a function of the density in their modified model. The corner-pore model was also analyzed by using a constitutive equation. Wilkinson [10], Svoboda et al. [11], Pan and Cocks [12], Kim et al. [14] and Delannay [15] analyzed the viscous compressive deformation during densification by introducing a mechanical concept of the bulk viscosity and the sintering stress. Riedel and co-workers [11,16,17] also analyzed the shrinkage of pores locating at two-grain junction. In these constitutive models, f(D) is represented as a function of the density. In addition to the corner-pore model, stochastic and diffusive models have been proposed for the final sintering stage. The stochastic model [8] is based on a continuum theory, and the diffusive model [3] treats the pores surrounded by dense polycrystals, which structure is different from the corner pores. For the two models, f(D) is also represented as a function of the density. In this study, the experimental densification kinetics with concurrent grain growth is analyzed by evaluating the density-dependent f (D) and the grain-size exponent n, so as to predict the kinetics theoretically. Firstly, the typical corner-pore model of n = 4 is applied to explain the experimental kinetics. During sintering of nano-sized ZrO2 powder, however, it is found that the pore structure is different from corner pores, and that the corner-pore model is not applicable for explaining the present kinetics. Based on the densification kinetics, a theoretical model is explored applicable for sintering of nano-sized powder. Finally, it is shown that the diffusive model which pore structure is similar to the observed one can effectively explain the experimental kinetics of n = 1 or n = 2.
3. Results and discussion 3.1. Densification Usually, the density range in the final stage has widely been recognized to be D > 0.9. If the sintering stage is distinguished with a criterion of pore shape, the density range in the final stage would be D > 0.92∼0.95, where all open pores are closed [2,22]. In the previous study [9], however, the sintering stage was experimentally divided at D = 0.85: the intermediate stage at D < 0.85 and the final stage at D > 0.85. For convenience, the sintering stage was distinguished from a remarkable change in the kinetics of both densification and grain growth during isothermal sintering of zirconia. Although the phenomenological criterion has a weak physical basis, the density range of D > 0.85 is regarded as the final stage in this study. The densification behavior of 8YSZ in the final stage is shown in Fig. 1. In the density range of 0.85 < D < 0.97, the densification rate D˙ can empirically be represented as ∼ρa, where ρ ( = 1-D) is the porosity and a is a constant. The value of a increases with increasing temperature from 1.27 at 1200 °C to 1.32 at 1250 °C and to 1.64 at 1300 °C. Though the change in the value of a is not remarkable, the increasing tendency with temperature is apparent within our repeated measurements. Such temperature-dependent D˙ –D relationship was also observed in the intermediate stage [9]. Providing that the relationship between grain size and density is independent of the temperature during densification, as assumed in the existing methods of Wang-Raj [5,6] and master sintering curve [7], the grain size G in Eq. (1) can be replaced with the density term, and then the densification rate can be represented as a function of the density only during isothermal sintering. The density function should be independent of the temperature, which indicates that the value of a
2. Experimental Commercial ZrO2 powder containing 7.8 mol% Y2O3 (TZ-8Y, Tosoh; 8YSZ) with a specific surface area of 13 m2/g and a crystallite size of 25 nm was used as a raw material. The as-received powder was pressed uniaxially at < 1 MPa into a compact of 30 mm diameter, and then pressed isostatically at 392 MPa in water. The powder compact has a green density of 3.13 g/cm3, corresponding to a relative density of 0.53 with an absolute density of 5.90 g/cm3 for 8YSZ. From the powder compact, rectangular specimens of 4 mm × 4 mm × 10 mm were cut out, and the two surfaces of 4 mm × 4 mm were carefully machined to be parallel. The isothermal shrinkage of the specimen was measured in air using a dilatometer (DIL 402C, Netzsch) equipped with an alumina rod and holder. The alumina rod contacted to the specimen at 0.25 N, corresponding to a stress of ∼10 kPa that is sufficiently lower than the intrinsic sintering stress of conventional ceramics [19–21]. The shrinkage measurement was conducted at 1200, 1250 and 1300 °C. The specimen was heated to the sintering temperature at a rate of 10 °C/min. The measured shrinkage was corrected with the thermal expansion of the fully densified 8YSZ, which had been measured separately, and then converted to the relative density. The shrinkage measurement was continued until the densification reached a rate of < 1 × 10−6 /min. At above 1250 °C, when the densification rate reached 4 × 10−6 /min, the relative density was about 0.99. At each sintering temperature, 2–3 specimens were tested. After the densification, the density of the specimen was measured by three different methods: the Archimedes method, a change in the specimen length and the dilatometer. For all samples, the three densities are within an error of 1%, which confirms the validity of the shrinkage data. The polished surface of the sample after prescribed densification
Fig. 1. Densification rate decreasing with densification. 2
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Fig. 2. Microstructures during isothermal sintering at 1250 °C; a) D = 0.85, b) D = 0.90 and c) D = 0.95.
D = 0.90 and to 17% at D = 0.95, due to pore shrinkage and grain growth. Though the pores with a coordination of 3 increases relatively, not all the pores surrounded by 3 grains on cross-section are corner pores. Tunnel-like pores located along triple line can appear with 3 surrounding grains on cross-section. Thus, the observation shows that most pores in the final stage are not corner pores but edge pores, face pores and/or grain pores. A discussion of the pore characteristics is continued in the next section of densification kinetics. The grain-growth kinetics is shown in Fig. 4(a). The grain-growth exponent m in the growth kinetics of Gm∼t, decreases from 3.61 at 1200 °C to 2.22 at 1300 °C with increasing temperature. At 1250 °C, the growth exponent of 2.44 in the final stage is smaller than the value (∼4.2) in the intermediate stage [9], which indicates a change in the growth mechanism with densification. In the intermediate stage of low densities (0.65 < D < 0.85), the growth mechanism for m∼4 was attributed to grain rotation. At high densities (D > 0.85), the increased number of contacting grains may restrict the grain rotation. Then, the grain growth would be controlled mainly by the diffusional process of both boundary migration and pore drag. In the final stage of sintering, the measured grain-growth exponent of 2.22∼3.61 would be a mixture of the boundary-control (m = 2) and the pore-control (m = 2–4) [24]. With increasing temperature, the boundary-control mechanism may become significant to yield the decreasing growth exponent. For sintering of 3 mol% Y2O3-stabilized tetragonal ZrO2, Matsui et al. [25] reported a similar value of m ( = 3) and attributed the mechanism to a solute drag. During densification in the final stage, it was found that the relationship between the grain size G and the relative density D is dependent on the temperature, as shown in Fig. 4(b). As a function of the porosity ρ, the grain size G can be represented as G∼ρ−0.22, ∼ρ-0.38 and ∼ρ-0.46 at 1200 °C, 1250 °C and 1300 °C, respectively. With increasing temperature, the density- or porosity-dependence of the grain size increases, and the temperature-dependent G-D relationship is responsible for the variant value of a in Fig. 1. At D = 0.98, the grain size (1.04 μm) for 1300 °C is about 1.6 times that (0.65 μm) for 1200 °C. This tendency of larger grain sizes for higher temperatures is opposite to the case in the intermediate stage [9], that is, larger grain sizes for lower temperatures at a given density. Whereas the density dependence of the grain size is weaker at higher temperatures in the intermediate stage, it is reversed in the final stage. The temperature-dependent G-D relationship may be understood by considering the contribution of various mechanisms. At high temperatures, several mechanisms contribute simultaneously to the grain growth, such as the boundary control (boundary energy, solute drag) and the pore control (surface diffusion, lattice diffusion, vapor transport). Since the activation energy for each mechanism is not identical, the different contribution of the mechanisms at different temperatures would yield a change in the grain-growth kinetics [9], as shown in Fig. 4. The temperature-dependent growth kinetics and G-D relationship in Fig. 4 may be caused by the segregation of Y3+ ions on grain boundaries. For grain growth in Y2O3-stabilized zirconia, the distribution of Y3+ ions plays an important role. Whereas the segregation for 2∼3 mol % Y2O3-stabilized zirconia suppresses the grain growth by a mechanism
should also be constant independently of the temperature. Thus, under an assumption of the temperature-independent G-D relationship, the variant value of a at different temperatures in Fig. 1 cannot be explained. It is expected, therefore, that the temperature dependence of the densification behavior in the final stage be caused by the temperature-dependent G-D relationship, as in the intermediate stage [9]. At D > 0.97, D˙ deviates from the empirical relationship of ρa and the decreasing rate increases with densification, as shown in Fig. 1. At high densities, all the pores are closed, and the pressure of the entrapped gas increases with densification, which lowers the densification rate [23]. According to Spusta et al. [22], all open pores are closed at around ρ = 0.06 for 8YSZ. Hence, the effect of the gas pressure on D˙ would appear at porosities lower than 0.06. The gas pressure is also relaxed by diffusion as time proceeds. In Fig. 1, the decreasing rate of D˙ at ρ < 0.03 is apparently lower for 1200 °C compared for 1300 °C. At low temperatures, densification occurs slowly, so that the diffusioninduced relaxation may weaken the gas-pressure effect on D˙ . 3.2. Microstructure The microstructural evolution during isothermal sintering at 1250 °C is shown in Fig. 2, where densification and grain growth proceed concurrently. It is noteworthy that most pores observed in the final stage are not corner pores. It has widely been recognized that a typical pore in the final stage is located at grain corners and surrounded by 4 grains [1]. On two-dimensional cross-section, the corner pores should appear with 3 surrounding grains. In Fig. 2, however, such pores with a coordination of 3 are scarcely observed, and some pores have a size comparable to the surrounding grains. The number density of pores for each pore coordination number is shown in Fig. 3. In the final stage, most pores are surrounded by more than 4 grains. With densification, the average pore coordination decreases gradually from 5.2 at D = 0.85 to 4.7 at D = 0.90 and to 4.5 at D = 0.95, and the number fraction of pores with a coordination of 3 increases from 8% at D = 0.85 to 11% at
Fig. 3. Number density of pores for each pore coordination during isothermal sintering at 1250 °C. 3
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Fig. 4. Grain growth behavior with respect to a) time and b) porosity.
of the relative density D. In their study, the number density N decreased with densification. Thus, f(D) must not be a constant but be a function of D, which decreases with densification. Based on the theoretical analysis of the bulk viscosity and the sintering stress, several densification rates in the corner-pore model were calculated in numerical and analytical ways [10–12,14,15]. Among them, the reproducible rate equations were proposed by Wilkinson [10], Pan and Cocks [12] and Kim et al. [14] as
of solute drag, the relatively uniform distribution on grain boundaries for 8YSZ has a weak effect of solute drag resulting in higher growth rate [26,27]. With grain growth, the grain-boundary area decreases, and then a degree of the segregation would be increased. The degree of the segregation would also be dependent on the temperature. Such change in the segregation of Y3+ ions would affect the diffusion coefficient, and resultantly may cause the temperature-dependent growth kinetics and G-D relationship in Fig. 4. In this study, the definition of grain size is used in order for conformity with our previous study in the intermediate sintering stage [9,28]. For nano-sized grains at low densities in the intermediate stage, the boundary between pores and grains on polished surfaces is unclear, so that it was difficult to measure the grain size by the conventional intercept method. The agglomeration of nano-sized grains also makes the intercept size difficult to be measured. Furthermore, even for the microstructure with a bimodal size distribution of grains at reducing atmosphere, the densification kinetics could well be explained by the present definition of grain size [9,28]. Since a ratio of the present grain size to the mean-intercept size for selected samples was almost constant, the adoption of the conventional mean-intercept size would yield grain-growth kinetics and G-D relationship similar to the above. On the other hand, the present experiments were conducted in a limited temperature range of 1200 °C and 1300 °C. At the higher temperature of 1350 °C, the density is about 0.80 when the isothermal sintering begins, and increases rapidly to 0.90 in 7 min and to 0.95 in 17 min. The short time interval cannot guarantee the validity of the grain size measured with respect to the sintering time and the density. Moreover, at the lower temperature of 1150 °C, the densification rate is lower than 5 × 10−6/min when the density reaches 0.86 after a sintering time of 13 days. To obtain such G-D relationship in the final stage as in Fig. 4(b), it takes too long time, probably more than half year. Even at 1200 °C, the density reached 0.988 after a sintering time of 190 h. This is the reason why the present temperature is limited to a range of 1200 °C and 1300 °C.
˙ 4 = cf (D) = c (1 − ρ2/3)[3ρ2/3 − (1 + ρ2/3) lnρ − 3]−1Pd (D) DG
(3)
˙ 4 = cf (D) = cD 2 [4.98ρ2/3 − 1.92ρ4/3 − 1.28lnρ − 3.20]−1Pd (D) DG
(4)
and
˙ 4 = cf (D) = cD 2 [4.96ρ2/3 − 1.87ρ4/3 − 1.10lnρ − 3.14]−1Pd (D) DG
(5)
respectively. Delannay [15] also analyzed the influence of dihedral angle and grain coordination on the densification in the corner-pore model (the rate equation is not shown because of long expressions). Here, Pd(D) is the density term of the sintering stress. Pd(D) can be calculated according to the equation (Eq. (10) in Ref. 15]) proposed by Delannay [15] for spherical pores and tetrakaidecahedral grains. The calculated Pd(D) is well fitted to ∼ρ−0.352. In Fig. 5, the experimental D˙ G4 is represented and compared with the theoretical f(D) of Eqs. (2)–(5) and of Delannay [15]. Here, the experimental G-D relationships shown in Fig. 4(b) were used for continuous D˙ G4 curves. All the D˙ G4- and f(D)-values were normalized with the values at ρ = 0.15. The experimental values of D˙ G4 at 1250 °C and 1300 °C are apparently different from those at 1200 °C. Providing that the pore structure and the densification mechanism are invariant in this temperature range, all the D˙ Gn or f(D) curves at different temperatures should be convergent for true n-value. The divergent D˙ G4
3.3. Densification kinetics 3.3.1. Sintering model of n = 4 The densification behavior in the final stage has typically been described by the shrinkage of corner pores under a mechanism of grainboundary diffusion. By analyzing the diffusion in the corner-pore model of Coble [1], Kang and Jung [13] proposed an equation of the densification kinetics as
˙ 4 = cf (D) = const. DG
(2)
where c is a constant, the grain-size exponent n is 4 and f(D) is constant. Harmer and co-workers [18,29] modified Eq. (2) by introducing the number of pores per grain N for Al2O3. Though their measured data showed n = 3.2 in Eq. (1) with constant f(D), the data modified by the number density (f(D)∼N) showed n = 4 and D˙ G4/N was independent
Fig. 5. Comparison of experimental D˙ G4 (n = 4) with theoretical f(D) of cornerpore model. f(D) of Delannay [15] was calculated for a dihedral angle of 180˚ (spherical pore) and a grain coordination of 14. 4
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curves at 1200 °C and 1300 °C indicate n ≠ 4. Moreover, the sintering models of Eqs. (2)–(5) with n = 4 cannot explain the experimental densification kinetics at different temperatures. All the existing cornerpore models predict the n-value of 4 under a mechanism of grainboundary diffusion, so that other sintering models of n ≠ 4 are required for theoretical understanding of the densification kinetics in the final stage. The other notable point in Fig. 5 is that the experimental D˙ G4 or f (D) for both 1250 °C and 1300 °C increases with densification at D < 0.97. Delannay [30] showed that the increasing grain coordination with densification may result in an increase in f(D). This increasing tendency, however, seems unnatural. As mentioned above, f(D) is linearly proportional to the densification rate, in the absence of grain growth. At D = 1, the bulk viscosity increases infinitely and densification eventually stops. Then, f(D) is considered to approach zero gradually, whereas in the Delannay’s model [30], densification is accelerated at around D = 1. If the pore structure and the densification mechanism are invariant in the sintering stage, it is expected that f(D) would show a monotonous behavior which probably decreases toward zero with densification. If f(D) shows an inflection in the behavior, additional sintering stage would be required for characterizing densification after the inflection, indicating a change in pore structure and/ or densification mechanism. It is considered, therefore, that the increase in D˙ G4 was caused by the incorrect n-value ( = 4).
Fig. 6. Comparison of experimental D˙ G2 (n = 2) with theoretical f(D) of Eq. (7).
coincident with the Olevsky-Molinari model [32]. Since the diffusioncontrolled deformation of fully dense materials shows a linear viscous behavior, the sintering stress of Olevsky and Molinari [32] seems suitable for analyzing theoretically the present densification kinetics with the diffusive model. f(D) of Eq. (7) is compared with the experimental D˙ G2 in Fig. 6. The entire tendency of the decreasing D˙ G2 with densification is well consistent with the model prediction. Although the deviation of the experimental D˙ G2 from the theoretical f(D) increases with densification at 1250 °C and 1300 °C, the deviation is much smaller than the case for n = 4 in Fig. 5. A notable point is that the difference between the experimental D˙ G2 at different temperatures is remarkably reduced compared to the case of n = 4. Considering an experimental error, the difference between the D˙ G2 curves in Fig. 6 seems allowable. It is concluded, therefore, that the grain-size exponent (n) in the final stage for the present 8YSZ may be determined to be 2 and then the densification kinetics of n = 2 can be explained theoretically by using the diffusive model. In the above theoretical f(D) of Eq. (7), the sintering stress of ∼D/G proposed by Olevsky and Molinari [32] was employed. For the sintering stress, various theoretical equations have been proposed as a function of both density and grain size [8]. For the pore structure of the diffusive model, the theoretical sintering stress increases to a finite value with densification and is inversely proportional to the grain size [8,32,33]. Since f(D) is proportional to the sintering stress, the selection of proper sintering stress is important for evaluating the theoretical densification rate. For example, with the sintering stress of ∼D2/G proposed by Skorohod [8], the value of f(D) at D = 0.92 in Eq. (6) is larger by 8% than that for ∼D/G.
3.3.2. Sintering model of n = 2 The present 8YSZ in the final stage exhibits the pore structure different from the corner-pore model, as shown in Figs. 2 and 3. Whereas relatively small corner pores were observed during sintering of large grains (> 10 μm) [1,18,29], for small grains (< 1 μm), the pore size in the final stage is comparable to the size of surrounding grains. The corner-pore model was proposed in 1960s for explaining the densification kinetics of coarse powder [1]. At present, however, most functional and engineering ceramics are sintered using fine or nano-sized powder. During sintering of nano-sized powder, as in the present study, most pores are not located at grain corners, and the sizes are comparable to the surrounding grains. Thus, a different pore structure is required for explaining the final-stage sintering behavior of nano-sized powder. Kim et al. [3] proposed the diffusive model where single spherical pore is surrounded by dense polycrystals and the pore shrinkage occurs by grain-boundary diffusion from the surrounding matrix. The pore structure is identical to that of the Mackenzie-Shuttleworth model where single spherical pore is surrounded by homogeneous incompressible material [31]. In the diffusive model, the densification rate is represented as
˙ = cρ−2/3D5/3g (ρ)−1P DG
(6) −1
-2/3
-1/3
1/3
where g(ρ) is a function of the porosity, (3ρ -8ρ +6ρ -ρ ). In Eq. (6), the pore size rp in the original equation (Eq. (10) in Ref. [3]) was replaced with (ρ/D)1/3, by using a relationship of constant rp3D/ρ for volume conservation of matter during pore shrinkage. Here, the sintering stress P of ∼D/G proposed by Olevsky and Molinari [32] can be employed, because their model has the same pore structure with the diffusive model. Then, the theoretical f(D) is represented as
˙ 2 = cf (D) = cρ−2/3D8/3g (ρ)−1 DG
3.3.3. Sintering model of n = 1 The sintering stress measured experimentally was reported to increase with densification and decrease after a peak [19–21]. The decrease in the sintering stress may be caused by the increasing gas pressure in closed pores and also by the accelerated grain growth at high densities. The net effect of the grain size, however, on the sintering stress has not been examined experimentally. Moreover, all theoretical sintering stresses were derived in the absence of grain growth. During sintering, densification and grain growth occur simultaneously, and their driving force is the reduction of pore-surface energy and of grainboundary energy. The two energy sources contribute to densification and grain growth simultaneously. For simplicity, however, the contribution can be separated. It may be considered that the driving force due to the reduction of grain-boundary energy acts mainly to grain growth rather than to densification, and that the grain-size effect on the sintering stress may be less significant than the theoretical predictions. In the following, the theoretical densification rate is examined under an
(7)
with n = 2. Unlikely the corner-pore model of n = 4, the diffusive model with P∼D/G gives n = 2, despite the identical mechanism of grain-boundary diffusion. In the previous study [9], the sintering stress proposed by Skorohod [8] was employed for analyzing the densification kinetics in the final stage. Recently, however, Giuntini and Olevsky [33] derived the sintering stress for porous body where the fully dense material surrounding spherical pores deforms in a nonlinear viscous manner, and showed that the sintering stress for a linear viscous material is 5
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Fig. 7. (a) Variation of specific surface area measured using small-angle neutron scattering and (b) comparison of experimental D˙ G (n = 1) with theoretical f(D) of Eq. (6).
account, the experimental kinetics of densification was analyzed with the existing theoretical models. Since the grain-size exponent n was unknown for the present sintering, a possible n-value was explored to explain the densification kinetics on theoretical basis. From the analysis, it is concluded that the n-value in the final stage must be around 1∼2 for the present nano-sized ZrO2 powder. This conclusion contradicts the conventional prediction of n = 4 for the Coble’s corner-pore model under a mechanism of grain-boundary diffusion. The experimental kinetics of densification could be explained by using the diffusive model, and the microstructural observation also supported the pore structure in the diffusive model.
assumption that the sintering stress is solely dependent on the surface energy of pores. Then, the theoretical densification kinetics of n = 1 can be derived from Eq. (6) and be compared to the possible kinetics of n = 1 for the present sintering. The sintering stress depending on the surface energy only can be estimated from the variation of the specific surface area Sv [8]. Fig. 7(a) shows the specific surface area measured using small-angle neutron scattering during sintering of Al2O3 and Al2O3/ZrO2, which were reproduced from the reported data [34,35]. For the two cases, Sv can be fitted to ∼ρ1.75 and ∼ρ1.3, respectively, and the sintering stress P proportional to (dSv/dρ)D2 [8] can be represented as ∼ρ0.75D2 and ∼ρ0.3D2. Then, the densification kinetics of Eq. (6) in the diffusive model yields D˙ G =cf(D) with n = 1. Eq. (6) with P∼ρ0.75D2 or P∼ρ0.3D2 is compared with the experimental D˙ G in Fig. 7(b). The experimental curves of D˙ G at three different temperatures are convergent, and well consistent with the theoretical kinetics. Thus, the grain-size exponent (n) in the final stage may be determined to be 1, and if that’s the case, the densification kinetics of n = 1 can be explained theoretically by using the diffusive model of Eq. (6). Though the sintering stress for the present ZrO2 was estimated from the data for Al2O3 [34] and Al2O3/ZrO2 [35], it was assumed that the pore structure or the variation of Sv is similar during sintering of such powder. In the above, a possible value of n has been explored to explain the observed densification kinetics on theoretical basis. It was found that the grain-size exponent of 4, which has widely been accepted to characterize the final sintering stage, cannot explain the present densification kinetics of nano-sized ZrO2 powder. D˙ Gn or f(D) for n = 4 shows an increasing tendency with densification and a divergence at different temperatures. f(D) for n = 3 also shows a significant divergence at different temperatures, though it was not shown. For n = 0, f(D) shows a good convergence, though it was not shown also. n = 0, however, indicates that D˙ is independent of the grain size. This seems not true, because D˙ usually decreases with increasing grain size [6]. From the convergence of the experimental f(D) curves and their consistence with the diffusive model, a plausible value of n for the present ZrO2 must be around 1∼2. This conclusion contradicts the existing corner-pore model under a mechanism of grain-boundary diffusion. Experiments for other system of nano-sized powder are required.
Acknowledgments This work was partly supported by the Structural Materials for Innovation of the Cross-ministerial Strategic Innovation Promotion Program (SIP) of Japan Science and Technology (JST). References [1] R.L. Coble, Sintering crystalline solids. I. Intermediate and final state diffusion models, J. Appl. Phys. 32 (1961) 787–792. [2] G. Okuma, D. Kadowaki, T. Hondo, S. Tanaka, F. Wakai, Interface topology for distinguishing stages of sintering, Sci. Rep. 7 (2017) 11106. [3] B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, H. Zhang, Diffusive model of pore shrinkage in final-stage sintering under hydrostatic pressure, Acta Mater. 59 (2011) 4079–4087. [4] B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, Y. Sakka, Y.-J. Park, Grain-boundary sliding model of pore shrinkage in late intermediate sintering stage under hydrostatic pressure, Acta Mater. 61 (2013) 6661–6669. [5] J. Wang, R. Raj, Estimate of the activation energies for boundary diffusion from rate-controlled sintering of pure alumina, and alumina doped with zirconia or titania, J. Am. Ceram. Soc. 73 (1990) 1172–1175. [6] J. Wang, R. Raj, Activation energy for the sintering of two-phase alumina/zirconia ceramics, J. Am. Ceram. Soc. 74 (1991) 1959–1963. [7] H. Su, D.L. Johnson, Master sintering curve: a practical approach to sintering, J. Am. Ceram. Soc. 79 (1996) 3211–3217. [8] E.A. Olevsky, Theory of sintering: from discrete to continuum, Mater. Sci. Eng. R23 (1998) 41–100. [9] B.-N. Kim, T.S. Suzuki, K. Morita, H. Yoshida, J.-G. Li, H. Matsubara, Evaluation of densification and grain-growth behavior during isothermal sintering of zirconia, J. Ceram. Soc. Jpn. 125 (2017) 357–363. [10] D.S. Wilkinson, The Mechanisms of Pressure Sintering, Ph.D. Thesis, Univ. Cambridge, 1978. [11] J. Svoboda, H. Riedel, H. Zipse, Equilibrium pore surfaces, sintering stresses and constitutive equations for the intermediate and late stages of sintering –I. Computation of equilibrium surfaces, Acta Metall. Mater. 42 (1994) 435–443. [12] J. Pan, A.C.F. Cocks, A constitutive model for stage 2 sintering of fine grained materials – I. Grain-boundaries act as perfect sources and sinks for vacancies, Acta Metall. Mater. 42 (1994) 1215–1222. [13] S.-J.L. Kang, Y.-I. Jung, Sintering kinetics at final stage sintering: model calculation and map construction, Acta Mater. 52 (2004) 4573–4578. [14] B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, H. Zhang, Shrinkage of pores located at grain corners by grain-boundary diffusion, J. Am. Ceram. Soc. 94 (2011) 982–984. [15] F. Delannay, Influence of dihedral angle and grain coordination on densification rate during intermediate and final sintering stages, J. Am. Ceram. Soc. 98 (2015) 3469–3475. [16] H. Riedel, H. Zipse, J. Svoboda, Equilibrium pore surfaces, sintering stresses and
4. Summary and conclusions In the final sintering stage (D > 0.85) of nano-sized 8YSZ powder, the experimental relationship was obtained between the densification rate D˙ and the porosity ρ as D˙ ∼ρa. The value of a is temperaturedependent and increases gradually with increasing temperature. The non-constant a-value must be related with the temperature-dependent kinetics of grain growth, because the G-D relationship was also found to be temperature-dependent. By taking the concurrent grain growth into 6
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[17] [18] [19] [20] [21] [22] [23] [24] [25]
[26]
constitutive equations for the intermediate and late stages of sintering –II. Diffusional densification and creep, Acta Metall. Mater. 42 (1994) 445–452. H. Riedel, V. Kozak, J. Svoboda, Densification and creep in the final stage of sintering, Acta Metall. Mater. 42 (1994) 3093–3103. J. Zhao, M.P. Harmer, Sintering kinetics for a model final-stage microstructure: a study of Al2O3, Philos. Mag. Lett. 63 (1991) 7–14. M.N. Rahaman, L.C. De Jonghe, R.J. Brook, Effect of shear stress on sintering, J. Am. Ceram. Soc. 69 (1986) 53–58. O. Guillon, J. Rödel, R.K. Bordia, Effect of green-state processing on the sintering stress and viscosity of alumina compacts, J. Am. Ceram. Soc. 90 (2007) 1637–1640. T. Ikegami, N. Iyi, I. Sakaguchi, Influence of magnesia on sintering stress of alumina, Ceram. Int. 36 (2010) 1143–1146. T. Spusta, J. Svoboda, K. Maca, Study of pore closure during pressure-less sintering of advanced oxide ceramics, Acta Mater. 115 (2016) 347–353. D.S. Wilkinson, A pressure-sintering model for the densification of polar firn and glacier ice, J. Glaciol. Geocryol. 34 (1988) 40–45. M.N. Rahaman, Ceramic Processing and Sintering, 2nd ed., Taylor & Francis, New York, 2003. K. Matsui, H. Yoshida, Y. Ikuhara, Isothermal sintering effects on phase separation and grain growth in yttria-stabilized tetragonal zirconia polycrystal, J. Am. Ceram. Soc. 92 (2009) 467–475. K. Matsui, H. Yoshida, Y. Ikuhara, Grain-boundary structure and microstructure development mechanism in 2-8 mol% yttria-stabilized zirconia polycrystals, Acta
Mater. 56 (2008) 1315–1325. [27] J. Chevalier, L. Gremillard, A.V. Virkar, D.R. Clarke, The tetragonal-monoclinic transformation in zirconia: lessons learned and future trends, J. Am. Ceram. Soc. 92 (2009) 1901–1920. [28] B.-N. Kim, T.S. Suzuki, K. Morita, H. Yoshida, Y. Sakka, H. Matsubara, Densification kinetics during isothermal sintering of 8YSZ, J. Eur. Ceram. Soc. 36 (2016) 1269–1275. [29] J. Zhao, M.P. Harmer, Effect of pore distribution on microstructure development: III, model experiments, J. Am. Ceram. Soc. 75 (1992) 830–843. [30] F. Delannay, Sintering kinetics across pore closure transition accounting for continuous increase in grain coordination with density, J. Am. Ceram. Soc. 98 (2015) 3476–3482. [31] J.K. Mackenzie, R. Shuttleworth, A phenomenological theory of sintering, Proc. Phys. Soc. (Lond.) B 62 (1949) 833–852. [32] E.A. Olevsky, A. Molinari, Instability of sintering of porous bodies, Int. J. Plast. 16 (2000) 1–37. [33] D. Giuntini, E.A. Olevsky, Sintering stress of nonlinear viscous materials, J. Am. Ceram. Soc. 99 (2016) 3520–3524. [34] S. Krueger, G.G. Long, Characterization of the densification of alumina by multiple small-angle neutron scattering, Acta Cryst. A47 (1991) 282–290. [35] Y.M. Pan, R.A. Page, G.G. Long, S. Krueger, Role of zirconia addition in pore development and grain growth in alumina compacts, J. Mater. Res. 14 (1999) 4602–4614.
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Original Article
Theoretical analysis of experimental densification kinetics in final sintering stage of nano-sized zirconia ⁎
Byung-Nam Kim , Tohru S. Suzuki, Koji Morita, Hidehiro Yoshida, Ji-Guang Li, Hideaki Matsubara1 National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki, 305-0047, Japan
A R T I C LE I N FO
A B S T R A C T
Keywords: Densification Kinetics Grain-growth Sintering Pore structure
The experimental densification kinetics of 7.8 mol% Y2O3-stabilized zirconia was analyzed theoretically during isothermal sintering in the final stage. By taking concurrent grain growth into account, a possible value of the grain-size exponent n was examined. The Coble’s corner-pore model recognized widely was found not to be applicable for explaining the densification kinetics. The corner-pore model of n = 4 shows a significant divergence in the kinetics at different temperatures. Microstructural observation shows that most pores are not located at grain corners and have a size comparable to the surrounding grains. The observed pore structure is similar to the diffusive model where single pore is surrounded by dense body. The diffusive model combined with theoretical sintering stress predicts n = 1 or n = 2, which shows a good consistence to the measured densification kinetics. During sintering of nano-sized powder, it is found that the densification kinetics can be explained distinctively by the diffusive single-pore model.
1. Introduction Sintering is a process of densification or pore shrinkage in powder compacts. The process has been divided into three stages: initial, intermediate and final stages. Most densification occurs in the intermediate stage where open pores are formed and shrink mainly through particle re-arrangement. In the final stage, pores are closed and shrink mainly through diffusion. The sintering stage may be distinguished by pore structure [1,2] or by densification mechanism [3,4]. The rate equation of densification in the intermediate and final stages can be represented as [5,6]
exp (−Q/RT ) f (D) D˙ = A (1) T Gn where D˙ is the densification rate, A is a constant, R is the gas constant, T is the absolute temperature, Q is the activation energy of densification, n is an exponent of the grain size G, and f(D) is an unspecified function of the relative density D. In order to understand the densification behavior, all the sintering parameters in Eq. (1), that is, Q, n and f(D), should be evaluated. A representative is the method of Wang-Raj [5,6] and of master sintering curve [7]. In the Wang-Raj method, Q and n are evaluated at a fixed density, and in the method of master sintering
curve, an average value of Q is evaluated in a given density range. In both methods, however, f(D) remains unknown, and the experimental densification kinetics has not been discussed in terms of f(D). f(D) is a mixed function of the bulk viscosity and the sintering stress, and is related to both pore structure and densification mechanism [8]. Physically, f(D) is linearly proportional to the densification rate, when no grain growth occurs, as in Eq. (1). In order to understand and predict the densification behavior of powder compacts, the evaluation of f(D) is indispensable, whereas most studies on sintering have been focused on the densification mechanism through evaluating n and Q. In the previous study [9], f(D) was evaluated experimentally in the intermediate stage during isothermal sintering of zirconia, and compared with theoretical predictions. Although some theoretical models showed partial consistence in a limited density range, no models were consistent with the experimentally determined f(D) in the entire density range of the intermediate stage. This is mainly due to a difference in the pore structure between models and experiments. Whereas the pores in theoretical models have a simple structure, the actual pores during sintering are very complicated particularly in the intermediate stage. Unlikely in the intermediate stage, the pore structure in the final stage is relatively simple. Typical is the corner-pore model where a spherical pore is located at grain corners and the pore shrinkage occurs
⁎
Corresponding author. E-mail address: [email protected] (B.-N. Kim). 1 Tohoku University, 6-6 Aoba, Aramaki, Aoba-ku, Sendai 980-8579, Japan. https://doi.org/10.1016/j.jeurceramsoc.2018.12.007 Received 19 September 2018; Received in revised form 26 November 2018; Accepted 1 December 2018 0955-2219/ © 2018 Elsevier Ltd. All rights reserved.
Please cite this article as: Kim, B.-N., Journal of the European Ceramic Society, https://doi.org/10.1016/j.jeurceramsoc.2018.12.007
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was observed using a scanning electron microscope (SEM, SU-8000, Hitachi). Before SEM observation, the polished surface was etched in air for 1 h at the temperature lower than the sintering temperature by 150 °C, and was coated with Pt. For one sample, 5–7 photographs were taken, each of which includes more than 400 grains. On each photograph, the number and the area of largest grains covering 20∼21% of the total area were measured using a software of Photoshop. The grain size in this study was defined as an average size of the largest grains. The number of the largest grains for each photograph is 40∼65, and the present grain size is about 2.1∼2.4 times the size measured by the conventional mean-intercept method for selected samples. Though the present grain size is different from the conventionally defined one, the two sizes would be linearly proportional, if normal grain growth occurs. The grain size was examined at D≥0.85 with an interval of ΔD = 0.02 or 0.03.
by a mechanism of grain-boundary diffusion [1,10–15]. The cornerpore model was originally proposed by Coble [1]. Though the densification rate by grain-boundary diffusion was not derived in his original study, Kang and Jung [13] showed later that D˙ is independent of the density, indicating a constant value of f(D). Zhao and Harmer [18] modified the Coble model, to show the densification rate proportional to the number of pores per grain (N), that is, f(D)∝N. Since N is dependent on the density, f(D) may be represented as a function of the density in their modified model. The corner-pore model was also analyzed by using a constitutive equation. Wilkinson [10], Svoboda et al. [11], Pan and Cocks [12], Kim et al. [14] and Delannay [15] analyzed the viscous compressive deformation during densification by introducing a mechanical concept of the bulk viscosity and the sintering stress. Riedel and co-workers [11,16,17] also analyzed the shrinkage of pores locating at two-grain junction. In these constitutive models, f(D) is represented as a function of the density. In addition to the corner-pore model, stochastic and diffusive models have been proposed for the final sintering stage. The stochastic model [8] is based on a continuum theory, and the diffusive model [3] treats the pores surrounded by dense polycrystals, which structure is different from the corner pores. For the two models, f(D) is also represented as a function of the density. In this study, the experimental densification kinetics with concurrent grain growth is analyzed by evaluating the density-dependent f (D) and the grain-size exponent n, so as to predict the kinetics theoretically. Firstly, the typical corner-pore model of n = 4 is applied to explain the experimental kinetics. During sintering of nano-sized ZrO2 powder, however, it is found that the pore structure is different from corner pores, and that the corner-pore model is not applicable for explaining the present kinetics. Based on the densification kinetics, a theoretical model is explored applicable for sintering of nano-sized powder. Finally, it is shown that the diffusive model which pore structure is similar to the observed one can effectively explain the experimental kinetics of n = 1 or n = 2.
3. Results and discussion 3.1. Densification Usually, the density range in the final stage has widely been recognized to be D > 0.9. If the sintering stage is distinguished with a criterion of pore shape, the density range in the final stage would be D > 0.92∼0.95, where all open pores are closed [2,22]. In the previous study [9], however, the sintering stage was experimentally divided at D = 0.85: the intermediate stage at D < 0.85 and the final stage at D > 0.85. For convenience, the sintering stage was distinguished from a remarkable change in the kinetics of both densification and grain growth during isothermal sintering of zirconia. Although the phenomenological criterion has a weak physical basis, the density range of D > 0.85 is regarded as the final stage in this study. The densification behavior of 8YSZ in the final stage is shown in Fig. 1. In the density range of 0.85 < D < 0.97, the densification rate D˙ can empirically be represented as ∼ρa, where ρ ( = 1-D) is the porosity and a is a constant. The value of a increases with increasing temperature from 1.27 at 1200 °C to 1.32 at 1250 °C and to 1.64 at 1300 °C. Though the change in the value of a is not remarkable, the increasing tendency with temperature is apparent within our repeated measurements. Such temperature-dependent D˙ –D relationship was also observed in the intermediate stage [9]. Providing that the relationship between grain size and density is independent of the temperature during densification, as assumed in the existing methods of Wang-Raj [5,6] and master sintering curve [7], the grain size G in Eq. (1) can be replaced with the density term, and then the densification rate can be represented as a function of the density only during isothermal sintering. The density function should be independent of the temperature, which indicates that the value of a
2. Experimental Commercial ZrO2 powder containing 7.8 mol% Y2O3 (TZ-8Y, Tosoh; 8YSZ) with a specific surface area of 13 m2/g and a crystallite size of 25 nm was used as a raw material. The as-received powder was pressed uniaxially at < 1 MPa into a compact of 30 mm diameter, and then pressed isostatically at 392 MPa in water. The powder compact has a green density of 3.13 g/cm3, corresponding to a relative density of 0.53 with an absolute density of 5.90 g/cm3 for 8YSZ. From the powder compact, rectangular specimens of 4 mm × 4 mm × 10 mm were cut out, and the two surfaces of 4 mm × 4 mm were carefully machined to be parallel. The isothermal shrinkage of the specimen was measured in air using a dilatometer (DIL 402C, Netzsch) equipped with an alumina rod and holder. The alumina rod contacted to the specimen at 0.25 N, corresponding to a stress of ∼10 kPa that is sufficiently lower than the intrinsic sintering stress of conventional ceramics [19–21]. The shrinkage measurement was conducted at 1200, 1250 and 1300 °C. The specimen was heated to the sintering temperature at a rate of 10 °C/min. The measured shrinkage was corrected with the thermal expansion of the fully densified 8YSZ, which had been measured separately, and then converted to the relative density. The shrinkage measurement was continued until the densification reached a rate of < 1 × 10−6 /min. At above 1250 °C, when the densification rate reached 4 × 10−6 /min, the relative density was about 0.99. At each sintering temperature, 2–3 specimens were tested. After the densification, the density of the specimen was measured by three different methods: the Archimedes method, a change in the specimen length and the dilatometer. For all samples, the three densities are within an error of 1%, which confirms the validity of the shrinkage data. The polished surface of the sample after prescribed densification
Fig. 1. Densification rate decreasing with densification. 2
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Fig. 2. Microstructures during isothermal sintering at 1250 °C; a) D = 0.85, b) D = 0.90 and c) D = 0.95.
D = 0.90 and to 17% at D = 0.95, due to pore shrinkage and grain growth. Though the pores with a coordination of 3 increases relatively, not all the pores surrounded by 3 grains on cross-section are corner pores. Tunnel-like pores located along triple line can appear with 3 surrounding grains on cross-section. Thus, the observation shows that most pores in the final stage are not corner pores but edge pores, face pores and/or grain pores. A discussion of the pore characteristics is continued in the next section of densification kinetics. The grain-growth kinetics is shown in Fig. 4(a). The grain-growth exponent m in the growth kinetics of Gm∼t, decreases from 3.61 at 1200 °C to 2.22 at 1300 °C with increasing temperature. At 1250 °C, the growth exponent of 2.44 in the final stage is smaller than the value (∼4.2) in the intermediate stage [9], which indicates a change in the growth mechanism with densification. In the intermediate stage of low densities (0.65 < D < 0.85), the growth mechanism for m∼4 was attributed to grain rotation. At high densities (D > 0.85), the increased number of contacting grains may restrict the grain rotation. Then, the grain growth would be controlled mainly by the diffusional process of both boundary migration and pore drag. In the final stage of sintering, the measured grain-growth exponent of 2.22∼3.61 would be a mixture of the boundary-control (m = 2) and the pore-control (m = 2–4) [24]. With increasing temperature, the boundary-control mechanism may become significant to yield the decreasing growth exponent. For sintering of 3 mol% Y2O3-stabilized tetragonal ZrO2, Matsui et al. [25] reported a similar value of m ( = 3) and attributed the mechanism to a solute drag. During densification in the final stage, it was found that the relationship between the grain size G and the relative density D is dependent on the temperature, as shown in Fig. 4(b). As a function of the porosity ρ, the grain size G can be represented as G∼ρ−0.22, ∼ρ-0.38 and ∼ρ-0.46 at 1200 °C, 1250 °C and 1300 °C, respectively. With increasing temperature, the density- or porosity-dependence of the grain size increases, and the temperature-dependent G-D relationship is responsible for the variant value of a in Fig. 1. At D = 0.98, the grain size (1.04 μm) for 1300 °C is about 1.6 times that (0.65 μm) for 1200 °C. This tendency of larger grain sizes for higher temperatures is opposite to the case in the intermediate stage [9], that is, larger grain sizes for lower temperatures at a given density. Whereas the density dependence of the grain size is weaker at higher temperatures in the intermediate stage, it is reversed in the final stage. The temperature-dependent G-D relationship may be understood by considering the contribution of various mechanisms. At high temperatures, several mechanisms contribute simultaneously to the grain growth, such as the boundary control (boundary energy, solute drag) and the pore control (surface diffusion, lattice diffusion, vapor transport). Since the activation energy for each mechanism is not identical, the different contribution of the mechanisms at different temperatures would yield a change in the grain-growth kinetics [9], as shown in Fig. 4. The temperature-dependent growth kinetics and G-D relationship in Fig. 4 may be caused by the segregation of Y3+ ions on grain boundaries. For grain growth in Y2O3-stabilized zirconia, the distribution of Y3+ ions plays an important role. Whereas the segregation for 2∼3 mol % Y2O3-stabilized zirconia suppresses the grain growth by a mechanism
should also be constant independently of the temperature. Thus, under an assumption of the temperature-independent G-D relationship, the variant value of a at different temperatures in Fig. 1 cannot be explained. It is expected, therefore, that the temperature dependence of the densification behavior in the final stage be caused by the temperature-dependent G-D relationship, as in the intermediate stage [9]. At D > 0.97, D˙ deviates from the empirical relationship of ρa and the decreasing rate increases with densification, as shown in Fig. 1. At high densities, all the pores are closed, and the pressure of the entrapped gas increases with densification, which lowers the densification rate [23]. According to Spusta et al. [22], all open pores are closed at around ρ = 0.06 for 8YSZ. Hence, the effect of the gas pressure on D˙ would appear at porosities lower than 0.06. The gas pressure is also relaxed by diffusion as time proceeds. In Fig. 1, the decreasing rate of D˙ at ρ < 0.03 is apparently lower for 1200 °C compared for 1300 °C. At low temperatures, densification occurs slowly, so that the diffusioninduced relaxation may weaken the gas-pressure effect on D˙ . 3.2. Microstructure The microstructural evolution during isothermal sintering at 1250 °C is shown in Fig. 2, where densification and grain growth proceed concurrently. It is noteworthy that most pores observed in the final stage are not corner pores. It has widely been recognized that a typical pore in the final stage is located at grain corners and surrounded by 4 grains [1]. On two-dimensional cross-section, the corner pores should appear with 3 surrounding grains. In Fig. 2, however, such pores with a coordination of 3 are scarcely observed, and some pores have a size comparable to the surrounding grains. The number density of pores for each pore coordination number is shown in Fig. 3. In the final stage, most pores are surrounded by more than 4 grains. With densification, the average pore coordination decreases gradually from 5.2 at D = 0.85 to 4.7 at D = 0.90 and to 4.5 at D = 0.95, and the number fraction of pores with a coordination of 3 increases from 8% at D = 0.85 to 11% at
Fig. 3. Number density of pores for each pore coordination during isothermal sintering at 1250 °C. 3
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Fig. 4. Grain growth behavior with respect to a) time and b) porosity.
of the relative density D. In their study, the number density N decreased with densification. Thus, f(D) must not be a constant but be a function of D, which decreases with densification. Based on the theoretical analysis of the bulk viscosity and the sintering stress, several densification rates in the corner-pore model were calculated in numerical and analytical ways [10–12,14,15]. Among them, the reproducible rate equations were proposed by Wilkinson [10], Pan and Cocks [12] and Kim et al. [14] as
of solute drag, the relatively uniform distribution on grain boundaries for 8YSZ has a weak effect of solute drag resulting in higher growth rate [26,27]. With grain growth, the grain-boundary area decreases, and then a degree of the segregation would be increased. The degree of the segregation would also be dependent on the temperature. Such change in the segregation of Y3+ ions would affect the diffusion coefficient, and resultantly may cause the temperature-dependent growth kinetics and G-D relationship in Fig. 4. In this study, the definition of grain size is used in order for conformity with our previous study in the intermediate sintering stage [9,28]. For nano-sized grains at low densities in the intermediate stage, the boundary between pores and grains on polished surfaces is unclear, so that it was difficult to measure the grain size by the conventional intercept method. The agglomeration of nano-sized grains also makes the intercept size difficult to be measured. Furthermore, even for the microstructure with a bimodal size distribution of grains at reducing atmosphere, the densification kinetics could well be explained by the present definition of grain size [9,28]. Since a ratio of the present grain size to the mean-intercept size for selected samples was almost constant, the adoption of the conventional mean-intercept size would yield grain-growth kinetics and G-D relationship similar to the above. On the other hand, the present experiments were conducted in a limited temperature range of 1200 °C and 1300 °C. At the higher temperature of 1350 °C, the density is about 0.80 when the isothermal sintering begins, and increases rapidly to 0.90 in 7 min and to 0.95 in 17 min. The short time interval cannot guarantee the validity of the grain size measured with respect to the sintering time and the density. Moreover, at the lower temperature of 1150 °C, the densification rate is lower than 5 × 10−6/min when the density reaches 0.86 after a sintering time of 13 days. To obtain such G-D relationship in the final stage as in Fig. 4(b), it takes too long time, probably more than half year. Even at 1200 °C, the density reached 0.988 after a sintering time of 190 h. This is the reason why the present temperature is limited to a range of 1200 °C and 1300 °C.
˙ 4 = cf (D) = c (1 − ρ2/3)[3ρ2/3 − (1 + ρ2/3) lnρ − 3]−1Pd (D) DG
(3)
˙ 4 = cf (D) = cD 2 [4.98ρ2/3 − 1.92ρ4/3 − 1.28lnρ − 3.20]−1Pd (D) DG
(4)
and
˙ 4 = cf (D) = cD 2 [4.96ρ2/3 − 1.87ρ4/3 − 1.10lnρ − 3.14]−1Pd (D) DG
(5)
respectively. Delannay [15] also analyzed the influence of dihedral angle and grain coordination on the densification in the corner-pore model (the rate equation is not shown because of long expressions). Here, Pd(D) is the density term of the sintering stress. Pd(D) can be calculated according to the equation (Eq. (10) in Ref. 15]) proposed by Delannay [15] for spherical pores and tetrakaidecahedral grains. The calculated Pd(D) is well fitted to ∼ρ−0.352. In Fig. 5, the experimental D˙ G4 is represented and compared with the theoretical f(D) of Eqs. (2)–(5) and of Delannay [15]. Here, the experimental G-D relationships shown in Fig. 4(b) were used for continuous D˙ G4 curves. All the D˙ G4- and f(D)-values were normalized with the values at ρ = 0.15. The experimental values of D˙ G4 at 1250 °C and 1300 °C are apparently different from those at 1200 °C. Providing that the pore structure and the densification mechanism are invariant in this temperature range, all the D˙ Gn or f(D) curves at different temperatures should be convergent for true n-value. The divergent D˙ G4
3.3. Densification kinetics 3.3.1. Sintering model of n = 4 The densification behavior in the final stage has typically been described by the shrinkage of corner pores under a mechanism of grainboundary diffusion. By analyzing the diffusion in the corner-pore model of Coble [1], Kang and Jung [13] proposed an equation of the densification kinetics as
˙ 4 = cf (D) = const. DG
(2)
where c is a constant, the grain-size exponent n is 4 and f(D) is constant. Harmer and co-workers [18,29] modified Eq. (2) by introducing the number of pores per grain N for Al2O3. Though their measured data showed n = 3.2 in Eq. (1) with constant f(D), the data modified by the number density (f(D)∼N) showed n = 4 and D˙ G4/N was independent
Fig. 5. Comparison of experimental D˙ G4 (n = 4) with theoretical f(D) of cornerpore model. f(D) of Delannay [15] was calculated for a dihedral angle of 180˚ (spherical pore) and a grain coordination of 14. 4
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curves at 1200 °C and 1300 °C indicate n ≠ 4. Moreover, the sintering models of Eqs. (2)–(5) with n = 4 cannot explain the experimental densification kinetics at different temperatures. All the existing cornerpore models predict the n-value of 4 under a mechanism of grainboundary diffusion, so that other sintering models of n ≠ 4 are required for theoretical understanding of the densification kinetics in the final stage. The other notable point in Fig. 5 is that the experimental D˙ G4 or f (D) for both 1250 °C and 1300 °C increases with densification at D < 0.97. Delannay [30] showed that the increasing grain coordination with densification may result in an increase in f(D). This increasing tendency, however, seems unnatural. As mentioned above, f(D) is linearly proportional to the densification rate, in the absence of grain growth. At D = 1, the bulk viscosity increases infinitely and densification eventually stops. Then, f(D) is considered to approach zero gradually, whereas in the Delannay’s model [30], densification is accelerated at around D = 1. If the pore structure and the densification mechanism are invariant in the sintering stage, it is expected that f(D) would show a monotonous behavior which probably decreases toward zero with densification. If f(D) shows an inflection in the behavior, additional sintering stage would be required for characterizing densification after the inflection, indicating a change in pore structure and/ or densification mechanism. It is considered, therefore, that the increase in D˙ G4 was caused by the incorrect n-value ( = 4).
Fig. 6. Comparison of experimental D˙ G2 (n = 2) with theoretical f(D) of Eq. (7).
coincident with the Olevsky-Molinari model [32]. Since the diffusioncontrolled deformation of fully dense materials shows a linear viscous behavior, the sintering stress of Olevsky and Molinari [32] seems suitable for analyzing theoretically the present densification kinetics with the diffusive model. f(D) of Eq. (7) is compared with the experimental D˙ G2 in Fig. 6. The entire tendency of the decreasing D˙ G2 with densification is well consistent with the model prediction. Although the deviation of the experimental D˙ G2 from the theoretical f(D) increases with densification at 1250 °C and 1300 °C, the deviation is much smaller than the case for n = 4 in Fig. 5. A notable point is that the difference between the experimental D˙ G2 at different temperatures is remarkably reduced compared to the case of n = 4. Considering an experimental error, the difference between the D˙ G2 curves in Fig. 6 seems allowable. It is concluded, therefore, that the grain-size exponent (n) in the final stage for the present 8YSZ may be determined to be 2 and then the densification kinetics of n = 2 can be explained theoretically by using the diffusive model. In the above theoretical f(D) of Eq. (7), the sintering stress of ∼D/G proposed by Olevsky and Molinari [32] was employed. For the sintering stress, various theoretical equations have been proposed as a function of both density and grain size [8]. For the pore structure of the diffusive model, the theoretical sintering stress increases to a finite value with densification and is inversely proportional to the grain size [8,32,33]. Since f(D) is proportional to the sintering stress, the selection of proper sintering stress is important for evaluating the theoretical densification rate. For example, with the sintering stress of ∼D2/G proposed by Skorohod [8], the value of f(D) at D = 0.92 in Eq. (6) is larger by 8% than that for ∼D/G.
3.3.2. Sintering model of n = 2 The present 8YSZ in the final stage exhibits the pore structure different from the corner-pore model, as shown in Figs. 2 and 3. Whereas relatively small corner pores were observed during sintering of large grains (> 10 μm) [1,18,29], for small grains (< 1 μm), the pore size in the final stage is comparable to the size of surrounding grains. The corner-pore model was proposed in 1960s for explaining the densification kinetics of coarse powder [1]. At present, however, most functional and engineering ceramics are sintered using fine or nano-sized powder. During sintering of nano-sized powder, as in the present study, most pores are not located at grain corners, and the sizes are comparable to the surrounding grains. Thus, a different pore structure is required for explaining the final-stage sintering behavior of nano-sized powder. Kim et al. [3] proposed the diffusive model where single spherical pore is surrounded by dense polycrystals and the pore shrinkage occurs by grain-boundary diffusion from the surrounding matrix. The pore structure is identical to that of the Mackenzie-Shuttleworth model where single spherical pore is surrounded by homogeneous incompressible material [31]. In the diffusive model, the densification rate is represented as
˙ = cρ−2/3D5/3g (ρ)−1P DG
(6) −1
-2/3
-1/3
1/3
where g(ρ) is a function of the porosity, (3ρ -8ρ +6ρ -ρ ). In Eq. (6), the pore size rp in the original equation (Eq. (10) in Ref. [3]) was replaced with (ρ/D)1/3, by using a relationship of constant rp3D/ρ for volume conservation of matter during pore shrinkage. Here, the sintering stress P of ∼D/G proposed by Olevsky and Molinari [32] can be employed, because their model has the same pore structure with the diffusive model. Then, the theoretical f(D) is represented as
˙ 2 = cf (D) = cρ−2/3D8/3g (ρ)−1 DG
3.3.3. Sintering model of n = 1 The sintering stress measured experimentally was reported to increase with densification and decrease after a peak [19–21]. The decrease in the sintering stress may be caused by the increasing gas pressure in closed pores and also by the accelerated grain growth at high densities. The net effect of the grain size, however, on the sintering stress has not been examined experimentally. Moreover, all theoretical sintering stresses were derived in the absence of grain growth. During sintering, densification and grain growth occur simultaneously, and their driving force is the reduction of pore-surface energy and of grainboundary energy. The two energy sources contribute to densification and grain growth simultaneously. For simplicity, however, the contribution can be separated. It may be considered that the driving force due to the reduction of grain-boundary energy acts mainly to grain growth rather than to densification, and that the grain-size effect on the sintering stress may be less significant than the theoretical predictions. In the following, the theoretical densification rate is examined under an
(7)
with n = 2. Unlikely the corner-pore model of n = 4, the diffusive model with P∼D/G gives n = 2, despite the identical mechanism of grain-boundary diffusion. In the previous study [9], the sintering stress proposed by Skorohod [8] was employed for analyzing the densification kinetics in the final stage. Recently, however, Giuntini and Olevsky [33] derived the sintering stress for porous body where the fully dense material surrounding spherical pores deforms in a nonlinear viscous manner, and showed that the sintering stress for a linear viscous material is 5
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Fig. 7. (a) Variation of specific surface area measured using small-angle neutron scattering and (b) comparison of experimental D˙ G (n = 1) with theoretical f(D) of Eq. (6).
account, the experimental kinetics of densification was analyzed with the existing theoretical models. Since the grain-size exponent n was unknown for the present sintering, a possible n-value was explored to explain the densification kinetics on theoretical basis. From the analysis, it is concluded that the n-value in the final stage must be around 1∼2 for the present nano-sized ZrO2 powder. This conclusion contradicts the conventional prediction of n = 4 for the Coble’s corner-pore model under a mechanism of grain-boundary diffusion. The experimental kinetics of densification could be explained by using the diffusive model, and the microstructural observation also supported the pore structure in the diffusive model.
assumption that the sintering stress is solely dependent on the surface energy of pores. Then, the theoretical densification kinetics of n = 1 can be derived from Eq. (6) and be compared to the possible kinetics of n = 1 for the present sintering. The sintering stress depending on the surface energy only can be estimated from the variation of the specific surface area Sv [8]. Fig. 7(a) shows the specific surface area measured using small-angle neutron scattering during sintering of Al2O3 and Al2O3/ZrO2, which were reproduced from the reported data [34,35]. For the two cases, Sv can be fitted to ∼ρ1.75 and ∼ρ1.3, respectively, and the sintering stress P proportional to (dSv/dρ)D2 [8] can be represented as ∼ρ0.75D2 and ∼ρ0.3D2. Then, the densification kinetics of Eq. (6) in the diffusive model yields D˙ G =cf(D) with n = 1. Eq. (6) with P∼ρ0.75D2 or P∼ρ0.3D2 is compared with the experimental D˙ G in Fig. 7(b). The experimental curves of D˙ G at three different temperatures are convergent, and well consistent with the theoretical kinetics. Thus, the grain-size exponent (n) in the final stage may be determined to be 1, and if that’s the case, the densification kinetics of n = 1 can be explained theoretically by using the diffusive model of Eq. (6). Though the sintering stress for the present ZrO2 was estimated from the data for Al2O3 [34] and Al2O3/ZrO2 [35], it was assumed that the pore structure or the variation of Sv is similar during sintering of such powder. In the above, a possible value of n has been explored to explain the observed densification kinetics on theoretical basis. It was found that the grain-size exponent of 4, which has widely been accepted to characterize the final sintering stage, cannot explain the present densification kinetics of nano-sized ZrO2 powder. D˙ Gn or f(D) for n = 4 shows an increasing tendency with densification and a divergence at different temperatures. f(D) for n = 3 also shows a significant divergence at different temperatures, though it was not shown. For n = 0, f(D) shows a good convergence, though it was not shown also. n = 0, however, indicates that D˙ is independent of the grain size. This seems not true, because D˙ usually decreases with increasing grain size [6]. From the convergence of the experimental f(D) curves and their consistence with the diffusive model, a plausible value of n for the present ZrO2 must be around 1∼2. This conclusion contradicts the existing corner-pore model under a mechanism of grain-boundary diffusion. Experiments for other system of nano-sized powder are required.
Acknowledgments This work was partly supported by the Structural Materials for Innovation of the Cross-ministerial Strategic Innovation Promotion Program (SIP) of Japan Science and Technology (JST). References [1] R.L. Coble, Sintering crystalline solids. I. Intermediate and final state diffusion models, J. Appl. Phys. 32 (1961) 787–792. [2] G. Okuma, D. Kadowaki, T. Hondo, S. Tanaka, F. Wakai, Interface topology for distinguishing stages of sintering, Sci. Rep. 7 (2017) 11106. [3] B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, H. Zhang, Diffusive model of pore shrinkage in final-stage sintering under hydrostatic pressure, Acta Mater. 59 (2011) 4079–4087. [4] B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, Y. Sakka, Y.-J. Park, Grain-boundary sliding model of pore shrinkage in late intermediate sintering stage under hydrostatic pressure, Acta Mater. 61 (2013) 6661–6669. [5] J. Wang, R. Raj, Estimate of the activation energies for boundary diffusion from rate-controlled sintering of pure alumina, and alumina doped with zirconia or titania, J. Am. Ceram. Soc. 73 (1990) 1172–1175. [6] J. Wang, R. Raj, Activation energy for the sintering of two-phase alumina/zirconia ceramics, J. Am. Ceram. Soc. 74 (1991) 1959–1963. [7] H. Su, D.L. Johnson, Master sintering curve: a practical approach to sintering, J. Am. Ceram. Soc. 79 (1996) 3211–3217. [8] E.A. Olevsky, Theory of sintering: from discrete to continuum, Mater. Sci. Eng. R23 (1998) 41–100. [9] B.-N. Kim, T.S. Suzuki, K. Morita, H. Yoshida, J.-G. Li, H. Matsubara, Evaluation of densification and grain-growth behavior during isothermal sintering of zirconia, J. Ceram. Soc. Jpn. 125 (2017) 357–363. [10] D.S. Wilkinson, The Mechanisms of Pressure Sintering, Ph.D. Thesis, Univ. Cambridge, 1978. [11] J. Svoboda, H. Riedel, H. Zipse, Equilibrium pore surfaces, sintering stresses and constitutive equations for the intermediate and late stages of sintering –I. Computation of equilibrium surfaces, Acta Metall. Mater. 42 (1994) 435–443. [12] J. Pan, A.C.F. Cocks, A constitutive model for stage 2 sintering of fine grained materials – I. Grain-boundaries act as perfect sources and sinks for vacancies, Acta Metall. Mater. 42 (1994) 1215–1222. [13] S.-J.L. Kang, Y.-I. Jung, Sintering kinetics at final stage sintering: model calculation and map construction, Acta Mater. 52 (2004) 4573–4578. [14] B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, H. Zhang, Shrinkage of pores located at grain corners by grain-boundary diffusion, J. Am. Ceram. Soc. 94 (2011) 982–984. [15] F. Delannay, Influence of dihedral angle and grain coordination on densification rate during intermediate and final sintering stages, J. Am. Ceram. Soc. 98 (2015) 3469–3475. [16] H. Riedel, H. Zipse, J. Svoboda, Equilibrium pore surfaces, sintering stresses and
4. Summary and conclusions In the final sintering stage (D > 0.85) of nano-sized 8YSZ powder, the experimental relationship was obtained between the densification rate D˙ and the porosity ρ as D˙ ∼ρa. The value of a is temperaturedependent and increases gradually with increasing temperature. The non-constant a-value must be related with the temperature-dependent kinetics of grain growth, because the G-D relationship was also found to be temperature-dependent. By taking the concurrent grain growth into 6
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Mater. 56 (2008) 1315–1325. [27] J. Chevalier, L. Gremillard, A.V. Virkar, D.R. Clarke, The tetragonal-monoclinic transformation in zirconia: lessons learned and future trends, J. Am. Ceram. Soc. 92 (2009) 1901–1920. [28] B.-N. Kim, T.S. Suzuki, K. Morita, H. Yoshida, Y. Sakka, H. Matsubara, Densification kinetics during isothermal sintering of 8YSZ, J. Eur. Ceram. Soc. 36 (2016) 1269–1275. [29] J. Zhao, M.P. Harmer, Effect of pore distribution on microstructure development: III, model experiments, J. Am. Ceram. Soc. 75 (1992) 830–843. [30] F. Delannay, Sintering kinetics across pore closure transition accounting for continuous increase in grain coordination with density, J. Am. Ceram. Soc. 98 (2015) 3476–3482. [31] J.K. Mackenzie, R. Shuttleworth, A phenomenological theory of sintering, Proc. Phys. Soc. (Lond.) B 62 (1949) 833–852. [32] E.A. Olevsky, A. Molinari, Instability of sintering of porous bodies, Int. J. Plast. 16 (2000) 1–37. [33] D. Giuntini, E.A. Olevsky, Sintering stress of nonlinear viscous materials, J. Am. Ceram. Soc. 99 (2016) 3520–3524. [34] S. Krueger, G.G. Long, Characterization of the densification of alumina by multiple small-angle neutron scattering, Acta Cryst. A47 (1991) 282–290. [35] Y.M. Pan, R.A. Page, G.G. Long, S. Krueger, Role of zirconia addition in pore development and grain growth in alumina compacts, J. Mater. Res. 14 (1999) 4602–4614.
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