A cyclic process for the nitridation of Si powder

Materials Science and Engineering A 408 (2005) 85–91

A cyclic process for the nitridation of Si powder MinKee Kim a,∗ , JongKu Park b , Hae-Weon Lee b , Shinhoo Kang a a

School of Materials Science and Engineering, Seoul National University, Kwanak-ku, Seoul 151-742, South Korea b Nano-Materials Research Center, Korea Institute of Science and Technology, Seoul 136-791, South Korea Received in revised form 15 July 2005; accepted 10 August 2005

Abstract The presence of coarse Si particles contributes a hindrance to achieving fully reacted Si3 N4 by a reaction-bonded silicon nitride (RBSN) technique for use in real net-shape parts, such as turbo-charger rotors. A new process is described, in which the nitridation rate is enhanced by cyclic heating and the cyclic and isothermal heating processes were compared. When the nitridation was conducted at various temperatures and times (1370–1420 ◦ C for times up to 60 h), the reaction rate of the cyclic process was three times faster than that of the isothermal reaction. This can be attributed to the cracking of the reacted Si3 N4 shell during nitridation, which exposes unreacted liquid Si in the core. Cracking was found to occur by the internal stress resulting from the volume mismatch between Si3 N4 and liquid Si. © 2005 Elsevier B.V. All rights reserved. Keywords: Si3 N4 ; Cyclic reaction; Internal stress; Stress intensity factor

1. Introduction It is well known that silicon nitride has a high hardness, good wear resistance and can be used in high-temperature engineering applications. In order to extend the applications of silicon nitride, some difficulties must be resolved relative to some problems that would occur during shaping and sintering. For this purpose, reaction-bonded silicon nitride (RBSN) and gas pressure sintered-reaction-bonded silicon nitride (GPSed-RBSN) have been designed as alternatives. These approaches permit the easy fabrication of complex shapes and provide smaller shrinkage than hot isostatic pressed silicon nitride (HIPSN), hot pressed silicon nitride (HPSN) and gas pressure sintered silicon nitride (GPSSN) [1–7]. To date, most studies on Si3 N4 have focused on the effects of temperature, gases, particle size, specimen size and impurities on the nitridation of Si. According to these studies, the appropriate temperature is in the range from 1300 to 1400 ◦ C. In the case of the direct nitridation of Si, the reaction gases are composed of nitrogen and hydrogen. Nitridation can be improved by decreasing the particle size, which leads to an increased the surface area. On the other side, a higher green density can be achieved by using fine particle and results in protecting the continuous ∗

Corresponding author. Tel.: +82 2880 8333; fax: +82 2884 1578. E-mail address: [email protected] (M. Kim).

0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.08.061

gas flow through the interstitial pores between the particles as the process proceeds. To sustain these pores, the use of coarse particle could be considered. However, in this case, the reaction is controlled by diffusion. During the nitridation of silicon powder, the growth of ␣and ␤-Si3 N4 occurs [2,8]. The ␣-Si3 N4 phase is a polymorph that grows from the vapor phase reaction. It has been generally accepted that alpha needles of Si3 N4 are formed by a vapor–liquid–solid (VLS) mechanism. ␤-Si3 N4 can form as the result of direct nitridation between solid Si and N2 or through the agency of a liquid phase. In such studies, a long time is required to achieve fully nitrided bulk and the focus of those studies has been on the reaction conditions. However, little effort has been made to shorten the nitridation time and to identify the relevant mechanisms. In this study, assuming that the presence of coarse particle is the main cause of the delayed reaction, we focused on the effect of particle size. We report herein on a methodology for enhancing the nitridation process for turbo-charger-rotors and attempt to explain the mechanism, by comparing the phases that are formed during each reaction process. 2. Experimental procedures In order to measure the time for complete nitridation at a given temperature and atmosphere, specimens were prepared by

M. Kim et al. / Materials Science and Engineering A 408 (2005) 85–91

86

Fig. 1. Heating schedules for: (a) isothermal and (b) cyclic process for Si nitridation. The isothermal reactions were performed at 1370 ◦ C and the maximum and minimum temperatures for the cyclic process were 1400 and 1420 ◦ C.

an injection molding process. Si powders (Herman C. Starck), 8.3 ␮m in average diameter were used and the solid loading was in the range of 55–60 vol%. The reaction temperature was fixed at 1370 ◦ C with a flowing gas mixture of 5% H2 and 95% N2 (purity > 99.999%). A mixture of large-size Si (∼25 ␮m) and Si3 N4 of ∼0.2 ␮m (SN-E10, Ube Industries Inc., Japan, ␣-rich (>95%) phase) powders was also investigated under the same conditions, to understand the effect of particle size on the process. The temperatures were maintained between 1370 and 1420 ◦ C for 0–60 h (Fig. 1). The powder was mixed by dry milling in a polyethylene container for 6 h. The composition was 70 wt.% Si3 N4 –30 wt.% Si. The powder mixture was uni-axially pressed at 40 MPa. A cyclic reaction process was employed as shown in Fig. 1 to reduce the overall reaction time. This cyclic process consists of heating to a temperature above the melting point of silicon (>1413 ◦ C) and then cooling down to a designated isothermal (1370 ◦ C). The degree of nitridation (D.N.) was calculated as follows: D.N. (%) =

wf − ws × 100 0.665ws

(SEM, Philips) and transmission electron microscopy (CM20, Jeol, USA). 3. Results and discussion 3.1. Isothermal reaction process Table 1 and Fig. 2(a) show the degree of nitridation and XRD results, respectively, for the isothermal reaction of fine Si particles. In the table, ␣-Si3 N4 was the major phase produced in the initial nitriding stage of fine particles and ␤-Si3 N4 levels were increased as the reaction proceeded. Table 2 and Fig. 2(b) show D.N. and XRD results for the mixture of coarse Si and Si3 N4 , respectively. Comparing FI2 and CI1 or CI2 in Tables 1 and 2, the degree of nitridation for FI2 as well as α/β ratio are higher than those for CI1 and CI2. Thus, it is clear that the production of ␣-Si3 N4 from coarse Si particles is significantly lower than that from fine Si, even after a long nitridation time. In general, however, the α/β ratio decreases with an increase in D.N., as shown in Tables 1 and 2, i.e. the growth of ␤-Si3 N4 is promoted in both systems with nitridation time. The appearance of ␣-Si3 N4 in the initial stage can be attributed to the somewhat different formation process of this phase. It is known that ␣-Si3 N4 can be formed by solid–gas or liquid–gas reactions. More specifically, it occurs by a reaction between volatile SiO and N2 [2,10,11]. Since the oxygen content of metallic silicon is in the range of 0.3–3 wt.%, the content of SiO2 would be 1–12 mol, assuming that all the oxygen is consumed [12]. Volatile SiO is formed through the reduc-

(1)

where ws and wf are the weights of starting and final product, respectively. The crystalline phases were determined by X-ray diffractometer (XRD, 30 kV, 30 mA, Rigaku Co., Japan). The weight fraction of ␤-Si3 N4 in the reacted body was calculated on the basis of the two highest XRD peaks, corresponding to ␣- and ␤Si3 N4 as proposed by previous work [9]. Microstructures were analyzed by optical microscopy, scanning electron microscopy Table 1 Processing conditions and results for the reaction of Si fine particle Label

Composition

Avgerage Si particle size (␮m)

Fa I b 1 FI2 FI3 FI4 FI5 Fc1

Si

8.3

Reaction time (h)

D.N. (%)

0 2 30 60 90 31

6.6 54.9 72.1 92.6 97.8 98.6

Specimens were prepared by injection molding. Nitridation was performed after a debinding process to remove wax. a F: fine. b I: isothermal, c: cyclic.

α (%) – 84.7 65 62.5 59 77.9

β (%)

α/β

– 15.3 35 37.5 41 22.1

– 5.5 1.9 1.7 1.4 3.5

M. Kim et al. / Materials Science and Engineering A 408 (2005) 85–91

87

Fig. 2. XRD results of: (a) injection-molded Si fine powder and (b) mixture of coarse Si powder with Si3 N4 .

tion of SiO2 on the surfaces of silicon particles in the presence of H2 [4,10,13]. Thus, the formation of ␣-Si3 N4 is controlled by the interfacial reaction between volatile SiO and N2 as long as SiO2 remains. After the consumption of surface oxide, ␤Si3 N4 forms by the diffusion of nitrogen into the Si particles. These results are shown in Tables 1 and 2 with FI3–5 and CI2. On the other hand, Atkinson et al. [5,6] proposed that gas–gas reactions between Si vapor and N2 occur through the nitride shell. It is evident that the reaction occurs between volatile SiO (or Si) and N2 through the presence of pores that are present around the interfaces between Si and the reaction product (Si3 N4 ). This is known to be the formation process for the alpha matte phase at the outer side of Si particles. The TEM images in Fig. 3 show the phases in this RBSN. The nitrided shell is composed of both ␣- and ␤-Si3 N4 , but the electron diffraction study shows that the inner part is only composed of ␤-Si3 N4 (Fig. 3). Fig. 4 shows the weight gain in fine Si powders as a function of nitridation time. These data show that more than 100 h are required to achieve full nitridation by the isothermal process. Fine particles provide a larger surface area than the coarse particles. Thus, fine particles are initially nitrided faster than coarse particles. In the case of FI2, 55% nitridation was achieved within 2 h. However, the pores in this fine system, which act as channels for the reaction gas, become closed with an increase in reaction time, thus slowing the reaction. For example, 30 h are required to achieve an additional 5% nitridation for FI4 beyond 92.6%. In the case of the coarse particle system, large-size pores are created. However, the nitriding reaction depends largely on the diffusion of nitrogen through the large Si particles, making the reaction slow.

Fig. 3. TEM images of the Si3 N4 shell structure which is composed of ␣- and ␤-Si3 N4 at the surface of a particle and ␤-Si3 N4 formed in the silicon inner region. The white area represents unreacted Si in which gray and dark areas show reacted Si3 N4 reacted.

3.2. Cyclic reaction process When the cyclic process was used, the nitridation was promoted, reaching over 70% in 31 h (Cc2). Moreover, the injectionmolded specimen (Fc1 in Table 1) could be fully nitrided in a short time compared to the isothermally processed ones (FI3). This is in contrast to the isothermal process with coarse particles (CI1 and CI2) that resulted in ∼40% nitridation. Thus, the

Table 2 Processing conditions and results for the reaction of Si coarse particle Label

Composition (wt.%)

Ca Ib 1 CI2 Cc1 Cc2

Si +70Si3 N4 d

Average Si particle size (␮m)

Reaction time (h)

D.N. (%)

α (%)

β (%)

α/β

25

10 30 31 31

38.1 42.2 48.1 73.8

90.6 (20.4)c 75.4 (8.3) 67.2 (0.7) 65.4 (−1.8)

9.4 (5.0) 24.6 (19.7) 32.8 (31.2) 34.6 (50.9)

9.6 (4.09) 3.1 (0.42) 2.0 (0.02) 1.9

70 wt.% Si3 N4 was used for matrix to observe behaviors of large particles. a C: coarse. b I: isothermal, c: cyclic. c With respect to Si. d Si N is composed of 95% ␣-phase and <5% ␤-phase. 3 4

88

M. Kim et al. / Materials Science and Engineering A 408 (2005) 85–91

Fig. 4. Weight gain through nitridation with respect to time (h) for the isothermal and cyclic processes.

rate of nitridation was remarkably improved, when the cyclic nitridation method was used. Further, the major phases formed from Si were different, that is, ␣- and ␤-Si3 N4 formed in the fine and coarse particle systems, respectively (Fc1 and Cc2). According to the results in Table 2, the formation of ␤-Si3 N4 is clearly preferred in the coarse particles even though ␣-Si3 N4 is formed in the initial stage. This indicates that ␤-Si3 N4 is formed, when the nitridation takes place through a diffusion process. This is also supported by the fine particle system, which showed an increase in the amount of ␤-Si3 N4 at the late stage of nitridation, as shown in Table 1. The formation of ␤-Si3 N4 was enhanced further when the cyclic process was used. A comparison between CI2 and Cc2 proves this. Once Li et al. reported that the interface stress

developed between Si and Si3 N4 in RBSN [14]. If the cyclic nitridation is performed below the melting point of Si < 1400 ◦ C, the stress at the reaction interface might not be high enough to cause fracturing during the formation of ␤-Si3 N4 . However, if performed across the melting point, the repeated transformation from dense liquid to solid would increase the stress intensity, facilitating the formation of interface fractures. Since the density of liquid Si (2.533 g/cm3 at mp) is higher than that of solid Si (2.33 g/cm3 at 300 K), the volume contraction and expansion can effectively lower the fracture resistance of the ␤-Si3 N4 phase. This can be seen from the result for the Cc1 samples, which were processed between 1370 and 1400 ◦ C (∼50% D.N.). For the coarse particle system, the high-temperature phase, ␤Si3 N4 , grew inwardly toward the unreacted silicon cores from the Si–Si3 N4 interface without fracture. This is illustrated in Fig. 5(a). When the nitridation was performed between 1370 and 1420 ◦ C, fracturing occurred, resulting in high nitridation, as shown for Cc2 (∼74% D.N.). The microstructure obtained from this process was unique, showing hollow holes in the nitrided shell while the isothermally reacted microstructure had no holes in the shell (Fig. 5(b and c)). A fracture allows unreacted silicon to be exposed to the nitriding atmosphere through the crevices (Fig. 5(d)). This process enabled the reaction to be completed in a short time. By repeating of the cycles, the shell became fractured and nitridation was enhanced along with Si melt rearrangement. In this process, if the fracture occurs in the early stage of reaction, the Si melt would fill up the pores among the particles and interrupt the flow of reaction gases. Therefore, further fracturing of the shells will be hampered after a certain extent of nitridation. From this study, it was found that a delayed initiation of fracture (cyclic process) led to full nitridation of the bulk. The critical level of nitridation for initiating the cyclic process

Fig. 5. Microstructures of cyclically processed specimens from coarse particles: (a) processed at 1400 ◦ C after isothermal reaction at 1370 ◦ C for 10 h; (b and c) processed at 1420 ◦ C after isothermal reaction at 1370 ◦ C for 10 h; (d) isothermally processed specimens.

M. Kim et al. / Materials Science and Engineering A 408 (2005) 85–91

equation can be obtained:
 1 dNA 1 − = 4πDCAg − dt rc R

89

(4)

And then, the size of unreacted core can be considered. As the core shrinks, the product layer becomes thicker, which causes a decrease in the rate of diffusion of A. Consequently, for integrating of Eq. (4) with respect to time and other variables, one of three variables, t, NA and rc must be eliminated or written in terms of the other variables through the relationship as followed [15]: −dNB = −bdNA = −ρB dV = −4πρB rc2 drc

Fig. 6. Representation of a reacting particle when product diffusion is the controlling resistance, the reaction being A (g) + bB (s) → products.

was ∼40% based on calculations and experimental results using coarse particles. 3.3. Fracture of shell structure In this experiment, diffusion through product would be considered as the rate-controlling step. For simplification, the relationship between particle size and reaction time, the particle was thought to be spherical as shown in Fig. 6 [15]. Considering a partially reacted particle as shown in Fig. 6, both reactant A and boundary of unreacted core move toward the center of the particle. It is reasonable to assume that the unreacted core is stationary, which means quasi steady-state condition for diffusing A at any time for any radius of unreacted core. The rate of reaction of reactant A at any instant is given by its rate of diffusion into the particle through a product shell of any radius r in the product [15], −

dNA = 4πr 2 QA = 4πR2 QAs = 4πrc2 QAc = constant dt

(2)

where NA is moles of fluid reactant A, t the reaction time (s) and other parameters are defined in Fig. 6. For convenience, the flux of A within the product layer is expressed by Fick’s law as following: QA = D

dCA dr

(3)

where D is the diffusion coefficient and CA is concentration of A within the product layer. By integrating across the product layer from R to rc after combining Eqs. (2) and (3) [15], below

(5)

where ρB is the density of the solid reactant. Eq. (5) shows that the decrease in volume or radius of unreacted core results in the disappearance of dNB moles of solid reactant or dNA moles of fluid reactant. By replacing Eq. (5) in Eq. (4), separating variables and integrating [15], we obtain:   r 2  r 3  ρB R2 c c t= 1−3 (6) +2 6bDCAg R R The parameter b is the stoichiometric constant ratio equal to 1.5 [16] and R is the final radius of particle, which completes a reaction at the degree of x as expressed by Eq. (7):
1/3 0.7x R= 1+ , R0 = A1/3 R0 (7) 2.34 + 0.86x Eq. (7) means that the radius of particle in reaction is changed because the conversion of Si to Si3 N4 accompanies the volume expansion about 22% [5]. The experimental results of this study show that it takes about 100 h (360,000 s) for the reaction to reach x > 0.95 when the initial radius of particle, R0 is 4 ␮m. Using this result for spherical particles, the empirical relationship among the degree of nitridation, the initial radius of particle and time for reaction completion can be found from Eq. (8): 
2
  ρSi A2/3 R20 3 rc 2 rc 3 t= 1 − 2/3 (8) + A R0 A R0 1.542 × 10−12 where ρSi = 2.34 g/cm3 and rc /R0 = (1 − x)1/3 . Fig. 7(a) summarizes the estimated values, D.N. from Eq. (8) along with some experimental results. The difference in D.N. between the predicted and experimental values for R0 = 4 ␮m must be due to the effect of particle shape and size distribution. In case of cyclic nitridation, the reaction time is about onethird of that for the isothermal reaction. The short reaction time can be attributed to the continuous exposure of unreacted Si by fracture of the product shell. The fracture behavior can be explained in terms of the resistance of the nitrided shell to fracture. The internal pressure, P, developed at the reaction interface can be expressed as in Eq. (9) with physical constants of Si and Si3 N4 [17]: p=

αT (1 + νSN )/2ESN + (1 − 2νSi )/ESi

(9)

90

M. Kim et al. / Materials Science and Engineering A 408 (2005) 85–91

Further, the relationship between stress intensity factor, KI , and internal pressure, P, is shown along with particle and flaw sizes as in Eq. (10) [18]:
 2pR2 √ a RSi KI = 2 πaF , (10) l l R − R2Si where Ri , R and RSi (same as rc ) are the radii of initial, final and unreacted silicon particle, respectively. F (a/l, RSi /l) can be expressed as below [18]:   a 2  a 4  F = 1.1 + A 4.951 (11) + 1.092 l l where a and l are the length of flaw and thickness of reacted products, respectively. It was assumed for the analysis that the hoop stress for the cross section of the particle is approximately equal to that of a pressurized cylinder. The constant, A, is expressed differently from Eqs. (12) and (13) within given boundary conditions [18]:
0.25 RSi RSi A = 0.2 , for − 1.0 < 5 or > 10 (12) l l
0.25 RSi RSi A = 0.125 − 0.25 , for 5 ≤ ≤ 10 (13) l l As shown in Eqs. (12) and (13), the boundary condition can be defined by the ratio of the radius of unreacted silicon (RSi ) to the thickness, l, of the reacted product, which is the difference between R and RSi . The thickness of the reacted product can be expressed by l = R(1 − (1 − x)1/3 ) [1]. Through this relationship, RSi /l can be expressed as below: (1 − x)1/3 RSi = l 1 − (1 − x)1/3

Fig. 7. (a) Plot of reaction time vs. degree of nitridation. It shows experimental and calculated results for different initial particle sizes is shown. (b) Plots of the estimated Kl for coarse particle and various defect size in the initial stage of nitridation. (c) In the intermediate stage, gray and small black areas in (b and c) indicate that the D.N. region, where a fracture of a shell could occur.

where α and T are the difference in linear thermal expansion coefficients of Si and Si3 N4 and the magnitude of the temperature fluctuation, respectively, and νSi , νSi3 N4 , Poisson’s ratio and ESi , ESi3 N4 Young’s Moduli for Si and Si3 N4 , respectively. The values for αSi and αSi3 N4 are 3.92 × 10−6 , 3.0 × 10−6 and Poisson’s ratio, νSi , νSi3 N4 are 0.3 (for isotropy), 0.18 (for ␣Si3 N4 ), respectively. Young’s Modulus, ESi and ESi3 N4 are 170 and 350 GPa. If T is 50 ◦ C as in this study, the internal pressure, P, developed at the interface amounts to ∼11.4 MPa.

(14)

In the initial stage, the ratio, RSi /l, could be larger than 10. Also, it will become smaller than 5 in the final stage. KIC of Si3 N4 is known to be in the range 2–14 MPa m1/2 . For this boundary condition, the maximum of degree of nitridation (x) can be calculated as ∼0.25 using Eq. (8) and min. KIC (2 MPa m1/2 ) in the initial stage. In addition, a fracture is known to develop along pre-existing cracks. Therefore, the minimum for the degree of nitridation (x) can be calculated using max. KIC (14 MPa m1/2 ) and maximum flaw size since the thickness of the product must be larger than the flaw size. For the intermediate stage, RSi /l is in the range of 5–10. Similarly, the maximum and minimum for the degree of nitridation were found to be 0.42 and 0.25, respectively, under the limitation of the KIC values and crack size. Fig. 7(b) shows the relationship between the degree of nitridation and KI in the initial stage. This illustrates the change in KI with respect to the degree of nitridation for a given flaw size. The initial particle size was assumed to be 25 ␮m along with various sizes of pre-existing flaws (0.1, 0.3 and 1.0 ␮m). Each arrow indicates the range of degree of nitridation that could cause a fracture in the shell. Fig. 7(c) illustrates the same estimation for the intermediate stage. In this stage, the nitridation range, x,

M. Kim et al. / Materials Science and Engineering A 408 (2005) 85–91

for fracture becomes very narrow. Based on these results, cyclic nitridation was initiated after ∼40% of D.N. was achieved. According to previous results [19,20], the critical thickness for fracture is in the range of 1/50–1/20 of the initial particle size. This is consistent with our experimental results. It was also claimed that a fracture could occur during the cooling stage and that the nitridation occurs primarily by the spallation of the nitride layers [20]. If, however, the fracture occurs by spalling, the cracks of which are parallel to the interface between the reacted product and Si, this would not be effective for opening the surface of the unreacted Si cores. Even though spallation could continue, a much longer time would be required for the nitridation of large particles. Thus, the effective nitridation by cyclic reaction can be attributed to the fracturing of the nitride layer, which generates cracks perpendicular to the Si/Si3 N4 interface. 4. Summary and conclusions The presence of coarse particles makes the nitridation of silicon difficult by an isothermal process. This is because the reaction occurs by diffusion through the nitride layer, which is formed on the surface of Si particles. In this study, a new process was developed to enhance the nitridation rate by cyclic heating and the cyclic and isothermal heating processes were compared. When the nitridation was conducted at various temperatures and time, a significant reduction in time for full nitridation time was achieved. The reaction rate of the cyclic process was three times faster than that of the isothermal reaction. This can be attributed to the cracking of the reacted Si3 N4 shell during nitridation, exposing unreacted liquid Si in the cores. The shell fracture originated from the mismatch in thermal expansion coefficients between Si and the newly formed Si3 N4 and to the volumetric change in the silicon core when Si melts from a solid. Thus, we conclude that the cyclic reaction is more effective for the nitridation of large Si particles and large dimensional specimens, such as turbo-charger rotors than the isothermal reaction.

91

Acknowledgements This research was supported by the HAN Project (1998) through Ministry of Science and Technology, Republic of Korea and partially supported by the Brain Korea 21 of Ministry of Education, Republic of Korea. References [1] R.E. Carter, J. Chem. Phys. 34 (6) (1961) 2010–2015. [2] H.M. Jennings, M.H. Richman, J. Mater. Sci. 10 (1976) 2087–2098. [3] G. Ziegler, J. Heinrich, G. Wotting, J. Mater. Sci. 22 (1987) 3041– 3086. [4] A.J. Moulson, J. Mater. Sci. 14 (1979) 1017–1051. [5] A. Atkinson, P.J. Leatt, A.J. Moulson, E.W. Roberts, J. Mater. Sci. 9 (1974) 981–984. [6] A. Atkinson, A.J. Moulson, E.W. Roberts, J. Am. Ceram. Soc. 59 (1976) 285. [7] M.W. Lindley, D.P. Elias, B.F. Jones, K.C. Pitman, J. Mater. Sci. 14 (1979) 70–85. [8] P. Longland, A.J. Moulson, J. Mater. Sci. 13 (1978) 2279–2280. [9] C.P. Gazzara, D.R. Messier, Am. Ceram. Soc. Bull. (1977) 381–392. [10] M.W. Lindley, D.P. Elias, B.F. Jones, K.C. Pitman, J. Mater. Sci. 14 (1979) 70–85. [11] D. Campos-Loriz, F.L. Riley, J. Mater. Sci. 14 (1979) 1007. [12] F.F. Lange, Int. Met. Rev. 1 (1980) 1–20. [13] F.W. Chang, T.H. Liou, F.M. Tsai, Thermochim. Acta 354 (2000) 71–80. [14] W.B. Li, B.Q. Lei, T. Lindback, R. Warren, J. Eur. Ceram. Soc. 19 (1995) 277–283. [15] Levenspiel, Octave, Chemical Reaction Engineering, John Wiley and Sons Inc., pp. 344–348. [16] W. Ku, O.J. Gregory, H.M. Jennings, J. Am. Ceram. Soc. 73 (2) (1990) 286–296. [17] R.W. Davidge, Mechanical Behaviour of Ceramics, Cambridge University Press, pp. 81–90. [18] T.L. Anderson, Fracture Mechanics: Fundamentals and Applications, second ed., CRC Press, p. 636. [19] M. Maalmi, A. Varma, W.C. Strieder, Chem. Eng. Sci. 53 (4) (1998) 679–689. [20] J. Koike, S. Kimura, J. Am. Ceram. Soc. 79 (2) (1996) 365–370.