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Constitutive modeling of alumina sintering: grain-size effect on dominant densification mechanism
Computational Materials Science 32 (2005) 196–202 www.elsevier.com/locate/commatsci
Constitutive modeling of alumina sintering: grain-size effect on dominant densification mechanism Zeming He *, J. Ma School of Materials Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Received 25 March 2004; received in revised form 3 August 2004; accepted 25 August 2004
Abstract In the present work, alumina powders with the initial grain sizes of 0.9 and 7.0 lm, respectively, were sintered at different temperatures. Constitutive laws for densification were employed to model the sintering process of alumina ceramics. Based on the constitutive laws employed and the experimental results obtained, the dominant densification mechanism was identified and the effect of grain size on dominant densification mechanism was discussed. The activation energy for densification was also evaluated. In the investigated sintering temperature range, interface reaction was identified as the controlling process in sintering of alumina powders with the initial grain size of 0.9 lm, while grainboundary diffusion was identified as the dominant process in sintering of alumina powders with the initial grain size of 7.0 lm. The activation energies for densification of the finer and coarser grain size alumina ceramics were determined as 342 and 384 kJ mol1, respectively, which provided a strong support on the densification mechanism investigation. Ó 2004 Elsevier B.V. All rights reserved. PACS: 65.70.+y; 66.10.Cb Keywords: Constitutive modeling; Densification; Grain size; Grain-boundary diffusion; Interface reaction; Activation energy
1. Introduction Ceramic products are generally fabricated from the powder metallurgy route. Sintering is one of
*
Corresponding author. Tel.: +65 6790 4590; fax: +65 6790 9081. E-mail address: [email protected] (Z. He).
the key procedures in the process for achieving materials with good performance. If residual stress or density inhomogeneity is present during densification, the formed defects could result in the property deterioration of the prepared components. Therefore, good sintering process control is necessary. It requires a full understanding of the constitutive laws which govern the densification behavior.
0927-0256/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.08.006
Z. He, J. Ma / Computational Materials Science 32 (2005) 196–202
A sintering constitutive law is a mathematic model used to describe materials densification under a specific mechanism. Several mechanisms are involved in the densification process, and the exact mechanism in operation, however, depends on the sintering conditions such as temperature, pressure, and material parameters such as grain size and porosity. It is well accepted that diffusional mechanisms dominate the sintering process of ceramic component at high temperature and low pressure. It is also noted that the dominant path for diffusion is usually along the grain boundary [1]. The classical grain-boundary diffusion mechanism assumes that grain boundaries act as perfect sources and sinks for the diffusing atoms during the diffusion process and the energy provided is all available to drive the diffusional flux along the grain boundaries. In practice, however, some energy is expended for materials to be added to, or removed from, a grain boundary, that is, an interface reaction occurs [2,3]. If the process of interface reaction is taken into account, the rate of densification can then be controlled either by the rate of diffusional transport between the sources and sinks, or the rate at which the sources and sinks can provide, or accept, material for the diffusional process. Furthermore, it is noted that the grain size has the important correlation with the dominance of the two processes [4]. To investigate the effect of grain size on the dominant densification mechanism, alumina powders with the initial grain size of 0.9 and 7.0 lm were pressurelessly sintered at different temperatures in the present work. Constitutive laws for densification [5–7] were employed to model the sintering process of alumina ceramics. Based on the constitutive laws employed and the experimental results obtained, materials dominant densification mechanism, grain-boundary diffusion or interface reaction, is identified and the effect of grain size on dominant densification mechanism is discussed. The activation energies for densification of the prepared alumina ceramics were also evaluated. The sintering stage investigated in the present work is limited to Stage 1, where the pores form interconnected channels when the relative density of the component is less than 0.9 [8].
197
2. Constitutive laws When grain-boundary diffusion is the controlling process, the volumetric strain rate, e_ v , is expressed by the model [5] as 3 e_ 0b L0 e_ v ¼ 9 fb ðqÞðrm rs Þ ð1Þ r0 L for Stage 1 sintering, where the subscript ÔbÕ indicates grain-boundary diffusion controlled sintering. e_ 0b is the uniaxial strain rate under an applied stress r0 for a material of the initial grain size L0. L is grain size of the material, fb(q) is the dimensionless function of the relative density q, rm is the mean stress and rs is the sintering stress. Following the model [5], the volumetric strain rate in Stage 1 sintering for interface reaction controlled sintering is expressed as " 32 3 # L0 re j3ðrm rs Þj 2 þ fr ðqÞ fr ðqÞ cr ðqÞ e_ v ¼ 3_e0r r0 L r0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0 3ðrm rs Þ ð2Þ j3ðrm rs Þj r0
where the subscript ÔrÕ indicates interface reaction controlled sintering and re is effective stress. fr(q) and cr(q) are dimensionless functions of the relative density. Eqs. (1) and (2) are general expressions to describe material densification behavior when grainboundary diffusion and interface reaction are the dominant mechanisms respectively. However, specific forms of the variables shown in these equations are required for modeling the sintering process. The dimensionless density functions are mechanism dependent. For the grain-boundary diffusion controlled sintering, Du and Cocks [6] proposed the expression for fb(q) during Stage 1 sintering fb ðqÞ ¼
0:54ð1 q0 Þ qðq q0 Þ
2
2
ð3Þ
where q0 is the green density of the compact. For the interface reaction controlled sintering, the dimensionless density functions fr(q) and cr(q) for Stage 1 sintering proposed by Du and Cocks [7] are fr ðqÞ ¼
1 ð1 q0 Þ 3 q32 q12 ðq q Þ 0 0
ð4Þ
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Z. He, J. Ma / Computational Materials Science 32 (2005) 196–202
and cr ðqÞ ¼
2 ð1 q0 Þ 3 q32 q12 ðq q Þ 0 0
ð5Þ
It is considered that the sintering stress is the function of relative density and grain size, but independent of densification mechanism. In Stage 1, the expression of the sintering stress is to be [6] 3ð2cs cb Þ 2 2q q0 q rs ¼ ð6Þ L 1 q0 where cs and cb are surface energy per unit area and grain-boundary energy per unit area respectively. The relationship between the rate constant for densification, e_ 0 (_e0b , or e_ 0r ), and the absolute sintering temperature can be expressed as Qd e_ 00 _e0 ¼ exp ð7Þ RT RT where e_ 00 is a material constant independent of temperature, and Qd is the activation energy for densification. T and R are absolute sintering temperature and the gas constant, respectively.
3. Experimental procedure The raw materials used in the experiment were 99.9% a-alumina powders (Baikowski, USA) with initial grain sizes of 0.9 and 7.0 lm, respectively. An Enerpac hydraulic press was used to form the powders into pellets under a uniaxial pressure of 200 MPa. After dry pressing, the green compact was subjected to sintering in a high-temperature furnace (Carbolite, UK). The sintering profile started with a heating rate of 2 K min1 up to 600 °C, and then progressed at 15 K min1 until the holding sintering temperature. The sintering temperatures chosen were 1400, 1450, and 1500 °C for 0.9 lm powders, and 1600, 1650, and 1700 °C for 7.0 lm powders. At a given sintering temperature, different holding time was set for achieving the samples with a certain density range. Finally, the compacts were cooled at a rate of 20 K min1 to room temperature.
The as-sintered sample was weighed using a Scientech SA 510 analytical balance and its diameter and the thickness were measured by a vernier calliper and a digital micrometer. The relative density was calculated using the measured density divided by the theoretical density of alumina (3.98 g cm3). The thermally-etched sample was gold coated and imaged using a JSM-5310 scanning electron microscope (SEM) (JEOL, Japan). The grain size was determined by counting the number of grains traversed by different straight lines of known length. The final mean grain size was taken as 1.56 times the average intercept length [9]. The densification rate of the component at a given sintering temperature was obtained from the gradient of the curve relative density against holding time. Software Mathcad (Mathsoft, USA) was used to do the modeling.
4. Results and discussion As an illustration, the SEM image of the sample sintered at 1450 °C for 300 min is shown in Fig. 1. As stated, the grain sizes of the samples were measured from the SEM images. Fig. 2 shows the grain size-relative density trajectories of the samples sintered at different temperatures. For both the finer and the coarser ceramics, it is noted that the grain sizes obtained at different sintering temperatures but at a fixed relative density are approximately the same. This phenomenon has been previously
Fig. 1. SEM image of sample with the initial grain size of 0.9 lm sintered at 1450 °C for 300 min.
Z. He, J. Ma / Computational Materials Science 32 (2005) 196–202 16 F1400 F1450 F1500 C1600 C1650 C1700
14 12 10
L (µm) 8 6 4 2 0 0.5
0.6
0.7
ρ
0.8
0.9
1
Fig. 2. Grain size-relative density trajectories of samples sintering at different temperatures (ÔFÕ refers to the sample with the initial grain size of 0.9 lm and ÔCÕ refer to the sample with the initial grain size of 7.0 lm).
described by Gupta [10], and it also occurred in our earlier work of sintering of PZT ceramics [11]. In Eqs. (1) and (2), re and rm are zero for pres_ when sureless sintering. The densification rate, q, grain-boundary diffusion and interface reaction dominant are derived as 3 L0 q_ ¼ Ab q fb ðqÞrs ð8Þ L and
L0 2 q_ ¼ Ar q f ðqÞr2s L r
ð9Þ
respectively, where Ab and Ar are constants. Software Mathcad was used to perform the modeling. A simple Euler scheme was adopted, where the change of the relative density after a time step, Dt, can be expressed as qtþDt ¼ qt þ q_ t Dt
ð10Þ
In Eq. (10), the specific forms of densification rate are expressed by Eqs. (8) and (9), because both the grain-boundary diffusion and the interface reaction mechanisms are assumed and later one of them will be identified. During modeling, the green densities were taken as 0.5083 for finerand 0.5500 for coarser-grained ceramics. The values of cs and cb were taken as 1 and 0.5 J/m2, respectively [12]. To avoid the added complication of the selection of the grain-growth law, the grain
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size was converted to the function of the relative density, with respect to Fig. 2. The constant Ab and Ar, however, cannot be chosen directly. They can be obtained from the modeling process, i.e. the appropriate rate constant can be determined by least square fitting of the experimental data to the constitutive model [11]. Figs. 3 and 4, respectively, show the modeling curves for the densification behavior (relative density versus holding time) of the finer and the coarser powders sintered at different temperatures. For comparison, the experimental points are also shown in the corresponding figures. For the densification behavior of 0.9 lm alumina powder in the investigated sintering temperature range, it is found that the grain-boundary diffusion mechanism always overestimates the densification rate as holding time increases, while the interface reaction mechanism provides a good agreement between modeling and experiment (Fig. 3). On the contrary, for the alumina samples with the initial grain size of 7.0 lm, a reverse trend is shown. That is, as holding time increases, interface reaction mechanism always overestimates the densification rate, while grain-boundary diffusion gives good prediction to the experimental results (Fig. 4). From the modeling results, it is considered that interface reaction is a more appropriately dominant densification mechanism for the sintering of fine-grained (with the initial grain size of 0.9 lm) alumina in the sintering temperature range of 1400–1500 °C. On the other hand, grain-boundary diffusion is a more appropriately dominant densification mechanism for the sintering of coarsegrained (with the initial grain size of 7.0 lm) alumina in the sintering temperature range of 1600–1700 °C. Grain-boundary diffusion and interface reaction are two sequential processes in the densification of materials. Therefore, the slower mechanism would dominate the sintering process. The grain-size effect on the dominant densification mechanism could be explained by considering the distance of the diffusion path from matter source to sink. The smaller the grain size is, the shorter the transport path and hence the faster the grainboundary diffusion. Therefore, the interface reaction becomes the controlling process for sintering
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1
1
0.9
0.9 0.8
0.8
ρ
ρ
0.7
0.7 Experiment
0.6
Experiment
0.6
Model GB
Model GB
Model IR
Model IR
0.5 0
500
1000
1500
t (min)
(a)
0.5
0
500
(b)
1000
1500
t (min)
1 0.9 0.8
ρ 0.7 Experiment
0.6
Model GB Model IR
0.5
0
200
400
600
t (min)
(c)
Fig. 3. Modeling results of densification process of 0.9 lm initial grain size samples sintered at different temperatures: (a) 1400 °C (b) 1450 °C and (c) 1500 °C (ÔGDÕ and ÔIRÕ refer to grain-boundary diffusion and interface reaction respectively).
of finer grain size. As grain size increases, that is as the distance that the materials have to travel along grain boundary increases, the grain-boundary diffusion becomes more important. With respect to Fig. 2, it is noted that the difference in grain sizes at different sintering temperatures but at a given relative density is minimal. If the grain-size difference is ignored, the relationship between the densification rate at a fixed density and the activation energy for densification is expressed from Eq. (7) as _ Þ¼ lnðqT
Qd þC RT
where C is a constant.
ð11Þ
Fig. 5 shows the variations of the logarithm of _ with the reciprocal of the absolute sintering qT temperature of the two investigated alumina ceramic series. The activation energies for densification were evaluated from the slopes in the figure. The values determined are 342 kJ mol1 for alumina with 0.9 lm initial grain size and 384 kJ mol1 for alumina with 7.0 lm initial grain size. The activation energy for densification of the fine-grained ceramics is close to that measured by Ma [13] with 325 kJ mol1, when interface reaction is the dominant mechanism. The energy value of the coarse-grained ceramics is close to that reported by Frost and Ashby [14] with 419 kJ mol1, when grain-boundary diffusion is the dominant mechanism.
Z. He, J. Ma / Computational Materials Science 32 (2005) 196–202
0.8
0.8
0.7
0.7
ρ
201
ρ 0.6
0.6 Experiment
Experiment
Model GB
Model GB
Model IR
0.5
0
150
Model IR
300
450
t (min)
(a)
0.5
0
150
300
450
t (min)
(b)
0.8
0.7
ρ 0.6 Experiment Model GB Model IR
0.5
0
100
200
300
t (min)
(c)
Fig. 4. Modeling results of densification process of 7.0 lm initial grain size samples sintered at different temperatures: (a) 1600 °C (b) 1650 °C and (c) 1700 °C (ÔGDÕ and ÔIRÕ refer to grain-boundary diffusion and interface reaction respectively).
2
1
In (ρT) 0 -1 -1 (min K ) •
Finer grain size Coarser grain size Linear fitting (Coarser grain size) Linear fitting (Finer grain size)
-1
-2 4.8
5.2
5.6
6 -1
From the values of the activation energies determined in the present work, it is noted that the results provide a strong support on the dominant densification mechanism investigation. Interface reaction is the appropriate mechanism for describing the densification process of fine-grained alumina, and grain-boundary diffusion is the appropriate mechanism for describing the densification process of coarse-grained alumina.
6.4
-1
10000×T (K )
_ with reciprocal of Fig. 5. Variations of logarithm of qT absolute sintering temperature at relative density of 0.8 in finer-grained samples and of 0.7 in coarser-grain samples.
5. Conclusions Interface reaction was identified as the controlling process in sintering of alumina powders with
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Z. He, J. Ma / Computational Materials Science 32 (2005) 196–202
the initial grain size of 0.9 lm in the temperature range of 1400–1500 °C. Grain-boundary diffusion was identified as the dominant mechanism in sintering of alumina powders with the initial grain size of 7.0 lm in the temperature range of 1600– 1700 °C. It was noted that grain size was an indication of the distance of the diffusion path and it had important effect on dominant densification mechanism. The mechanism could transit from interface reaction to grain-boundary diffusion as grain size increased. The activation energies for densification of the investigated fine- and coarse-grained alumina ceramics were determined as 342 and 384 kJ mol1, respectively, which provided the strong supporting data for dominant densification mechanism investigation.
References [1] W.D. Kingery, H.K. Bowen, D.R. Uhlmann, Introduction to Ceramics, John Wiley and Sons, New York, 1975. [2] M.F. Ashby, On interface-reaction control of Narbarro– Herring creep and sintering, Scripta Metall. 3 (1969) 837– 842. [3] B. Burton, Interface reaction controlled diffusional creep: a consideration of grain boundary dislocation climb sources, Mater. Sci. Eng. 10 (1972) 9–13.
[4] J. Ma, A.C.F. Cocks, A constitutive model for the sintering of fine grain alumina, in: N.A. Fleck, A.C.F. Cocks (Eds.), IUTAM Symposium on Mechanics of Granular and Porous Materials, Cambridge, UK, 1996, pp. 129–138. [5] A.C.F. Cocks, The structure of constitutive laws for the sintering of fine grained materials, Acta Metall. Mater. 42 (1994) 2191–2210. [6] Z.Z. Du, A.C.F. Cocks, Constitutive models for the sintering of ceramic components: I. Material models, Acta Metall. Mater. 40 (1992) 1969–1979. [7] Z.Z. Du, A.C.F. Cocks, Sintering of fine-grained materials by interface reaction controlled grain boundary diffusion, Int. J. Solids Struct. 31 (1994) 1429–1445. [8] M.F. Ashby, Sintering and isostatic pressing diagrams, Technical report, University of Cambridge, Cambridge, 1990. [9] M.N. Rahaman, L.C. De Jonghe, R.J. Brook, Effect of shear stress on sintering, J. Am. Ceram. Soc. 69 (1986) 53– 58. [10] T.K. Gupta, Possible correlation between density and grain size during sintering, J. Am. Ceram. Soc. 55 (1972) 276–277. [11] Z. He, J. Ma, Constitutive modeling of the densification of PZT ceramics, J. Phys. Chem. Solids 64 (2003) 177–183. [12] J. Zhao, M.P. Harmer, Effect of pore distribution on microstructure development: I. Matrix pores, J. Am. Ceram. Soc. 71 (1988) 113–120. [13] J. Ma, Constitutive modeling of the densification of porous ceramic components, Ph.D. thesis, University of Cambridge, Cambridge, 1997. [14] H.J. Forst, M.F. Ashby, Deformation Mechanism Maps, Pergamon, Oxford, 1982.
Constitutive modeling of alumina sintering: grain-size effect on dominant densification mechanism Zeming He *, J. Ma School of Materials Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore Received 25 March 2004; received in revised form 3 August 2004; accepted 25 August 2004
Abstract In the present work, alumina powders with the initial grain sizes of 0.9 and 7.0 lm, respectively, were sintered at different temperatures. Constitutive laws for densification were employed to model the sintering process of alumina ceramics. Based on the constitutive laws employed and the experimental results obtained, the dominant densification mechanism was identified and the effect of grain size on dominant densification mechanism was discussed. The activation energy for densification was also evaluated. In the investigated sintering temperature range, interface reaction was identified as the controlling process in sintering of alumina powders with the initial grain size of 0.9 lm, while grainboundary diffusion was identified as the dominant process in sintering of alumina powders with the initial grain size of 7.0 lm. The activation energies for densification of the finer and coarser grain size alumina ceramics were determined as 342 and 384 kJ mol1, respectively, which provided a strong support on the densification mechanism investigation. Ó 2004 Elsevier B.V. All rights reserved. PACS: 65.70.+y; 66.10.Cb Keywords: Constitutive modeling; Densification; Grain size; Grain-boundary diffusion; Interface reaction; Activation energy
1. Introduction Ceramic products are generally fabricated from the powder metallurgy route. Sintering is one of
*
Corresponding author. Tel.: +65 6790 4590; fax: +65 6790 9081. E-mail address: [email protected] (Z. He).
the key procedures in the process for achieving materials with good performance. If residual stress or density inhomogeneity is present during densification, the formed defects could result in the property deterioration of the prepared components. Therefore, good sintering process control is necessary. It requires a full understanding of the constitutive laws which govern the densification behavior.
0927-0256/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.08.006
Z. He, J. Ma / Computational Materials Science 32 (2005) 196–202
A sintering constitutive law is a mathematic model used to describe materials densification under a specific mechanism. Several mechanisms are involved in the densification process, and the exact mechanism in operation, however, depends on the sintering conditions such as temperature, pressure, and material parameters such as grain size and porosity. It is well accepted that diffusional mechanisms dominate the sintering process of ceramic component at high temperature and low pressure. It is also noted that the dominant path for diffusion is usually along the grain boundary [1]. The classical grain-boundary diffusion mechanism assumes that grain boundaries act as perfect sources and sinks for the diffusing atoms during the diffusion process and the energy provided is all available to drive the diffusional flux along the grain boundaries. In practice, however, some energy is expended for materials to be added to, or removed from, a grain boundary, that is, an interface reaction occurs [2,3]. If the process of interface reaction is taken into account, the rate of densification can then be controlled either by the rate of diffusional transport between the sources and sinks, or the rate at which the sources and sinks can provide, or accept, material for the diffusional process. Furthermore, it is noted that the grain size has the important correlation with the dominance of the two processes [4]. To investigate the effect of grain size on the dominant densification mechanism, alumina powders with the initial grain size of 0.9 and 7.0 lm were pressurelessly sintered at different temperatures in the present work. Constitutive laws for densification [5–7] were employed to model the sintering process of alumina ceramics. Based on the constitutive laws employed and the experimental results obtained, materials dominant densification mechanism, grain-boundary diffusion or interface reaction, is identified and the effect of grain size on dominant densification mechanism is discussed. The activation energies for densification of the prepared alumina ceramics were also evaluated. The sintering stage investigated in the present work is limited to Stage 1, where the pores form interconnected channels when the relative density of the component is less than 0.9 [8].
197
2. Constitutive laws When grain-boundary diffusion is the controlling process, the volumetric strain rate, e_ v , is expressed by the model [5] as 3 e_ 0b L0 e_ v ¼ 9 fb ðqÞðrm rs Þ ð1Þ r0 L for Stage 1 sintering, where the subscript ÔbÕ indicates grain-boundary diffusion controlled sintering. e_ 0b is the uniaxial strain rate under an applied stress r0 for a material of the initial grain size L0. L is grain size of the material, fb(q) is the dimensionless function of the relative density q, rm is the mean stress and rs is the sintering stress. Following the model [5], the volumetric strain rate in Stage 1 sintering for interface reaction controlled sintering is expressed as " 32 3 # L0 re j3ðrm rs Þj 2 þ fr ðqÞ fr ðqÞ cr ðqÞ e_ v ¼ 3_e0r r0 L r0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0 3ðrm rs Þ ð2Þ j3ðrm rs Þj r0
where the subscript ÔrÕ indicates interface reaction controlled sintering and re is effective stress. fr(q) and cr(q) are dimensionless functions of the relative density. Eqs. (1) and (2) are general expressions to describe material densification behavior when grainboundary diffusion and interface reaction are the dominant mechanisms respectively. However, specific forms of the variables shown in these equations are required for modeling the sintering process. The dimensionless density functions are mechanism dependent. For the grain-boundary diffusion controlled sintering, Du and Cocks [6] proposed the expression for fb(q) during Stage 1 sintering fb ðqÞ ¼
0:54ð1 q0 Þ qðq q0 Þ
2
2
ð3Þ
where q0 is the green density of the compact. For the interface reaction controlled sintering, the dimensionless density functions fr(q) and cr(q) for Stage 1 sintering proposed by Du and Cocks [7] are fr ðqÞ ¼
1 ð1 q0 Þ 3 q32 q12 ðq q Þ 0 0
ð4Þ
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Z. He, J. Ma / Computational Materials Science 32 (2005) 196–202
and cr ðqÞ ¼
2 ð1 q0 Þ 3 q32 q12 ðq q Þ 0 0
ð5Þ
It is considered that the sintering stress is the function of relative density and grain size, but independent of densification mechanism. In Stage 1, the expression of the sintering stress is to be [6] 3ð2cs cb Þ 2 2q q0 q rs ¼ ð6Þ L 1 q0 where cs and cb are surface energy per unit area and grain-boundary energy per unit area respectively. The relationship between the rate constant for densification, e_ 0 (_e0b , or e_ 0r ), and the absolute sintering temperature can be expressed as Qd e_ 00 _e0 ¼ exp ð7Þ RT RT where e_ 00 is a material constant independent of temperature, and Qd is the activation energy for densification. T and R are absolute sintering temperature and the gas constant, respectively.
3. Experimental procedure The raw materials used in the experiment were 99.9% a-alumina powders (Baikowski, USA) with initial grain sizes of 0.9 and 7.0 lm, respectively. An Enerpac hydraulic press was used to form the powders into pellets under a uniaxial pressure of 200 MPa. After dry pressing, the green compact was subjected to sintering in a high-temperature furnace (Carbolite, UK). The sintering profile started with a heating rate of 2 K min1 up to 600 °C, and then progressed at 15 K min1 until the holding sintering temperature. The sintering temperatures chosen were 1400, 1450, and 1500 °C for 0.9 lm powders, and 1600, 1650, and 1700 °C for 7.0 lm powders. At a given sintering temperature, different holding time was set for achieving the samples with a certain density range. Finally, the compacts were cooled at a rate of 20 K min1 to room temperature.
The as-sintered sample was weighed using a Scientech SA 510 analytical balance and its diameter and the thickness were measured by a vernier calliper and a digital micrometer. The relative density was calculated using the measured density divided by the theoretical density of alumina (3.98 g cm3). The thermally-etched sample was gold coated and imaged using a JSM-5310 scanning electron microscope (SEM) (JEOL, Japan). The grain size was determined by counting the number of grains traversed by different straight lines of known length. The final mean grain size was taken as 1.56 times the average intercept length [9]. The densification rate of the component at a given sintering temperature was obtained from the gradient of the curve relative density against holding time. Software Mathcad (Mathsoft, USA) was used to do the modeling.
4. Results and discussion As an illustration, the SEM image of the sample sintered at 1450 °C for 300 min is shown in Fig. 1. As stated, the grain sizes of the samples were measured from the SEM images. Fig. 2 shows the grain size-relative density trajectories of the samples sintered at different temperatures. For both the finer and the coarser ceramics, it is noted that the grain sizes obtained at different sintering temperatures but at a fixed relative density are approximately the same. This phenomenon has been previously
Fig. 1. SEM image of sample with the initial grain size of 0.9 lm sintered at 1450 °C for 300 min.
Z. He, J. Ma / Computational Materials Science 32 (2005) 196–202 16 F1400 F1450 F1500 C1600 C1650 C1700
14 12 10
L (µm) 8 6 4 2 0 0.5
0.6
0.7
ρ
0.8
0.9
1
Fig. 2. Grain size-relative density trajectories of samples sintering at different temperatures (ÔFÕ refers to the sample with the initial grain size of 0.9 lm and ÔCÕ refer to the sample with the initial grain size of 7.0 lm).
described by Gupta [10], and it also occurred in our earlier work of sintering of PZT ceramics [11]. In Eqs. (1) and (2), re and rm are zero for pres_ when sureless sintering. The densification rate, q, grain-boundary diffusion and interface reaction dominant are derived as 3 L0 q_ ¼ Ab q fb ðqÞrs ð8Þ L and
L0 2 q_ ¼ Ar q f ðqÞr2s L r
ð9Þ
respectively, where Ab and Ar are constants. Software Mathcad was used to perform the modeling. A simple Euler scheme was adopted, where the change of the relative density after a time step, Dt, can be expressed as qtþDt ¼ qt þ q_ t Dt
ð10Þ
In Eq. (10), the specific forms of densification rate are expressed by Eqs. (8) and (9), because both the grain-boundary diffusion and the interface reaction mechanisms are assumed and later one of them will be identified. During modeling, the green densities were taken as 0.5083 for finerand 0.5500 for coarser-grained ceramics. The values of cs and cb were taken as 1 and 0.5 J/m2, respectively [12]. To avoid the added complication of the selection of the grain-growth law, the grain
199
size was converted to the function of the relative density, with respect to Fig. 2. The constant Ab and Ar, however, cannot be chosen directly. They can be obtained from the modeling process, i.e. the appropriate rate constant can be determined by least square fitting of the experimental data to the constitutive model [11]. Figs. 3 and 4, respectively, show the modeling curves for the densification behavior (relative density versus holding time) of the finer and the coarser powders sintered at different temperatures. For comparison, the experimental points are also shown in the corresponding figures. For the densification behavior of 0.9 lm alumina powder in the investigated sintering temperature range, it is found that the grain-boundary diffusion mechanism always overestimates the densification rate as holding time increases, while the interface reaction mechanism provides a good agreement between modeling and experiment (Fig. 3). On the contrary, for the alumina samples with the initial grain size of 7.0 lm, a reverse trend is shown. That is, as holding time increases, interface reaction mechanism always overestimates the densification rate, while grain-boundary diffusion gives good prediction to the experimental results (Fig. 4). From the modeling results, it is considered that interface reaction is a more appropriately dominant densification mechanism for the sintering of fine-grained (with the initial grain size of 0.9 lm) alumina in the sintering temperature range of 1400–1500 °C. On the other hand, grain-boundary diffusion is a more appropriately dominant densification mechanism for the sintering of coarsegrained (with the initial grain size of 7.0 lm) alumina in the sintering temperature range of 1600–1700 °C. Grain-boundary diffusion and interface reaction are two sequential processes in the densification of materials. Therefore, the slower mechanism would dominate the sintering process. The grain-size effect on the dominant densification mechanism could be explained by considering the distance of the diffusion path from matter source to sink. The smaller the grain size is, the shorter the transport path and hence the faster the grainboundary diffusion. Therefore, the interface reaction becomes the controlling process for sintering
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1
1
0.9
0.9 0.8
0.8
ρ
ρ
0.7
0.7 Experiment
0.6
Experiment
0.6
Model GB
Model GB
Model IR
Model IR
0.5 0
500
1000
1500
t (min)
(a)
0.5
0
500
(b)
1000
1500
t (min)
1 0.9 0.8
ρ 0.7 Experiment
0.6
Model GB Model IR
0.5
0
200
400
600
t (min)
(c)
Fig. 3. Modeling results of densification process of 0.9 lm initial grain size samples sintered at different temperatures: (a) 1400 °C (b) 1450 °C and (c) 1500 °C (ÔGDÕ and ÔIRÕ refer to grain-boundary diffusion and interface reaction respectively).
of finer grain size. As grain size increases, that is as the distance that the materials have to travel along grain boundary increases, the grain-boundary diffusion becomes more important. With respect to Fig. 2, it is noted that the difference in grain sizes at different sintering temperatures but at a given relative density is minimal. If the grain-size difference is ignored, the relationship between the densification rate at a fixed density and the activation energy for densification is expressed from Eq. (7) as _ Þ¼ lnðqT
Qd þC RT
where C is a constant.
ð11Þ
Fig. 5 shows the variations of the logarithm of _ with the reciprocal of the absolute sintering qT temperature of the two investigated alumina ceramic series. The activation energies for densification were evaluated from the slopes in the figure. The values determined are 342 kJ mol1 for alumina with 0.9 lm initial grain size and 384 kJ mol1 for alumina with 7.0 lm initial grain size. The activation energy for densification of the fine-grained ceramics is close to that measured by Ma [13] with 325 kJ mol1, when interface reaction is the dominant mechanism. The energy value of the coarse-grained ceramics is close to that reported by Frost and Ashby [14] with 419 kJ mol1, when grain-boundary diffusion is the dominant mechanism.
Z. He, J. Ma / Computational Materials Science 32 (2005) 196–202
0.8
0.8
0.7
0.7
ρ
201
ρ 0.6
0.6 Experiment
Experiment
Model GB
Model GB
Model IR
0.5
0
150
Model IR
300
450
t (min)
(a)
0.5
0
150
300
450
t (min)
(b)
0.8
0.7
ρ 0.6 Experiment Model GB Model IR
0.5
0
100
200
300
t (min)
(c)
Fig. 4. Modeling results of densification process of 7.0 lm initial grain size samples sintered at different temperatures: (a) 1600 °C (b) 1650 °C and (c) 1700 °C (ÔGDÕ and ÔIRÕ refer to grain-boundary diffusion and interface reaction respectively).
2
1
In (ρT) 0 -1 -1 (min K ) •
Finer grain size Coarser grain size Linear fitting (Coarser grain size) Linear fitting (Finer grain size)
-1
-2 4.8
5.2
5.6
6 -1
From the values of the activation energies determined in the present work, it is noted that the results provide a strong support on the dominant densification mechanism investigation. Interface reaction is the appropriate mechanism for describing the densification process of fine-grained alumina, and grain-boundary diffusion is the appropriate mechanism for describing the densification process of coarse-grained alumina.
6.4
-1
10000×T (K )
_ with reciprocal of Fig. 5. Variations of logarithm of qT absolute sintering temperature at relative density of 0.8 in finer-grained samples and of 0.7 in coarser-grain samples.
5. Conclusions Interface reaction was identified as the controlling process in sintering of alumina powders with
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the initial grain size of 0.9 lm in the temperature range of 1400–1500 °C. Grain-boundary diffusion was identified as the dominant mechanism in sintering of alumina powders with the initial grain size of 7.0 lm in the temperature range of 1600– 1700 °C. It was noted that grain size was an indication of the distance of the diffusion path and it had important effect on dominant densification mechanism. The mechanism could transit from interface reaction to grain-boundary diffusion as grain size increased. The activation energies for densification of the investigated fine- and coarse-grained alumina ceramics were determined as 342 and 384 kJ mol1, respectively, which provided the strong supporting data for dominant densification mechanism investigation.
References [1] W.D. Kingery, H.K. Bowen, D.R. Uhlmann, Introduction to Ceramics, John Wiley and Sons, New York, 1975. [2] M.F. Ashby, On interface-reaction control of Narbarro– Herring creep and sintering, Scripta Metall. 3 (1969) 837– 842. [3] B. Burton, Interface reaction controlled diffusional creep: a consideration of grain boundary dislocation climb sources, Mater. Sci. Eng. 10 (1972) 9–13.
[4] J. Ma, A.C.F. Cocks, A constitutive model for the sintering of fine grain alumina, in: N.A. Fleck, A.C.F. Cocks (Eds.), IUTAM Symposium on Mechanics of Granular and Porous Materials, Cambridge, UK, 1996, pp. 129–138. [5] A.C.F. Cocks, The structure of constitutive laws for the sintering of fine grained materials, Acta Metall. Mater. 42 (1994) 2191–2210. [6] Z.Z. Du, A.C.F. Cocks, Constitutive models for the sintering of ceramic components: I. Material models, Acta Metall. Mater. 40 (1992) 1969–1979. [7] Z.Z. Du, A.C.F. Cocks, Sintering of fine-grained materials by interface reaction controlled grain boundary diffusion, Int. J. Solids Struct. 31 (1994) 1429–1445. [8] M.F. Ashby, Sintering and isostatic pressing diagrams, Technical report, University of Cambridge, Cambridge, 1990. [9] M.N. Rahaman, L.C. De Jonghe, R.J. Brook, Effect of shear stress on sintering, J. Am. Ceram. Soc. 69 (1986) 53– 58. [10] T.K. Gupta, Possible correlation between density and grain size during sintering, J. Am. Ceram. Soc. 55 (1972) 276–277. [11] Z. He, J. Ma, Constitutive modeling of the densification of PZT ceramics, J. Phys. Chem. Solids 64 (2003) 177–183. [12] J. Zhao, M.P. Harmer, Effect of pore distribution on microstructure development: I. Matrix pores, J. Am. Ceram. Soc. 71 (1988) 113–120. [13] J. Ma, Constitutive modeling of the densification of porous ceramic components, Ph.D. thesis, University of Cambridge, Cambridge, 1997. [14] H.J. Forst, M.F. Ashby, Deformation Mechanism Maps, Pergamon, Oxford, 1982.