ATERIALS GENGE S ENGIEERING ELSEVIER Materials Science and Engineering A262(1999)16-24 Modeling of oxidation behavior of sic-reinforced ceramic matrix composites P. Mogilevsky * A. Zangvil Unicersity of Illinois at Urbana-Champaign, Frederick Seitz Materials Research Laboratory, 104 South Goodwin Avenue, Urbana IL 61801 USA Received 15 June 1998: received in revised form 6 October 1998 Abstract Internal oxidation of Sic reinforcement is a major factor affecting the environmental stability of sic reinforced ceramic matrix omposites(CMCs) for high temperature applications. a simple phenomenological model describing the unidirectional oxidation of Sic reinforced oxide CMCs is presented. The model allows to calculate the thickness of the silica layer formed on a Sic reinforcement as a function of its location(depth beneath the surface)and time, if the oxygen permeabilities of silica and the matrix are known. The oxidation mode can thereby be predicted. Alternatively, the model allows to evaluate the oxygen permeabilities of silica and the matrix from the experimental oxidation data. Moreover, the expected mode of oxidation, I or II can be predicted depending on oxygen permeabilities and volume fraction of the reinforcement phase. Application of the model to the results of the microscopic study of the oxidation of Sic reinforced mullite-zirconia matrix composites allowed to oxygen permeabilities of the matrix and of the growing silica layer on the Sic particles. It was found that while ermeability of the silica layer on the Sic particles may depend significantly on the type of Sic reinforcement, it is rea close to the values obtained from the experiments on direct oxidation of Sic and permeation through vitreous silica permeability of the mullite -ZrO, matrix showed a dependence on the microstructure and composition of the matrix. o 1999 Elsevier Science S.A. All rights reserved. Keywords: Environmental stability: Oxygen permeabilities: Oxidation; Silicon carbide: Ceramic matrix composites 1. Introduction [7-9]. Two oxidation modes of Sic reinforced oxide matrix composite materials have been described, Fig. I Ceramic matrix composite(CMC) materials rein- [8, 9]. Mode I is defined as the case where oxygen reacts forced with SiC particles, whiskers, or platelets have with the whole Sic particle before it diffuses farther received increasing attention due to their potentially into the matrix, resulting in a clear boundary separating high fracture toughness and strength [1-5]. Internal a layer with completely oxidized SiC from the underly oxidation of SiC reinforcement is a major factor affect- ing composite with unoxidized SiC Mode II is the case ng the environmental stability of Sic reinforced CMCs where oxygen can deeply penetrate into the matrix for high temperature applications. Hence, alumina and before Sic particles in the outer region are completely mullite have been selected as matrix materials due to oxidized, leaving a long range of partially oxidized Sic heir excellent high-temperature stability and slow oxy particles behind. Mode I was found in mullite-SiC gen permeation Oxidation of Sic has been studied in detail [6]. The composites, while mode II was observed in mullite- study of oxidation of Sic in alumina, mullite, and ZrO2-SiC composites [10. However, no model conve- mullite-zirconia matrices was performed both micro- niently linking the observed oxidation behavior of Sic topically and via conventional weight gain method reinforced CMCs with the diffusion characteristics of the composite components has been applied to analyze the results. In the present study we propose a simple author.Tel:+1-217-3332367;fax:l-217 phenomenological model for mode II of the oxidation 278: e-mail: mogilevsky@mrlxp2 mrl uiuc. edu of sic reinforced CMcs 0921-5093/99/S- see front matter c 1999 Elsevier Science S.A. All rights reserved. PI:s0921-5093(98)01029-6
Materials Science and Engineering A262 (1999) 16–24 Modeling of oxidation behavior of SiC-reinforced ceramic matrix composites P. Mogilevsky *, A. Zangvil Uni6ersity of Illinois at Urbana-Champaign, Frederick Seitz Materials Research Laboratory, 104 South Goodwin A6enue, Urbana, IL 61801, USA Received 15 June 1998; received in revised form 6 October 1998 Abstract Internal oxidation of SiC reinforcement is a major factor affecting the environmental stability of SiC reinforced ceramic matrix composites (CMCs) for high temperature applications. A simple phenomenological model describing the unidirectional oxidation of SiC reinforced oxide CMCs is presented. The model allows to calculate the thickness of the silica layer formed on a SiC reinforcement as a function of its location (depth beneath the surface) and time, if the oxygen permeabilities of silica and the matrix are known. The oxidation mode can thereby be predicted. Alternatively, the model allows to evaluate the oxygen permeabilities of silica and the matrix from the experimental oxidation data. Moreover, the expected mode of oxidation, I or II, can be predicted depending on oxygen permeabilities and volume fraction of the reinforcement phase. Application of the model to the results of the microscopic study of the oxidation of SiC reinforced mullite–zirconia matrix composites allowed to evaluate oxygen permeabilities of the matrix and of the growing silica layer on the SiC particles. It was found that while oxygen permeability of the silica layer on the SiC particles may depend significantly on the type of SiC reinforcement, it is reasonably close to the values obtained from the experiments on direct oxidation of SiC and permeation through vitreous silica. Oxygen permeability of the mullite–ZrO2 matrix showed a dependence on the microstructure and composition of the matrix. © 1999 Elsevier Science S.A. All rights reserved. Keywords: Environmental stability; Oxygen permeabilities; Oxidation; Silicon carbide; Ceramic matrix composites 1. Introduction Ceramic matrix composite (CMC) materials reinforced with SiC particles, whiskers, or platelets have received increasing attention due to their potentially high fracture toughness and strength [1–5]. Internal oxidation of SiC reinforcement is a major factor affecting the environmental stability of SiC reinforced CMCs for high temperature applications. Hence, alumina and mullite have been selected as matrix materials due to their excellent high-temperature stability and slow oxygen permeation. Oxidation of SiC has been studied in detail [6]. The study of oxidation of SiC in alumina, mullite, and mullite–zirconia matrices was performed both microscopically and via conventional weight gain method [7–9]. Two oxidation modes of SiC reinforced oxide matrix composite materials have been described, Fig. 1 [8,9]. Mode I is defined as the case where oxygen reacts with the whole SiC particle before it diffuses farther into the matrix, resulting in a clear boundary separating a layer with completely oxidized SiC from the underlying composite with unoxidized SiC. Mode II is the case where oxygen can deeply penetrate into the matrix before SiC particles in the outer region are completely oxidized, leaving a long range of partially oxidized SiC particles behind. Mode I was found in mullite–SiC composites, while mode II was observed in mullite– ZrO2 –SiC composites [10]. However, no model conveniently linking the observed oxidation behavior of SiC reinforced CMCs with the diffusion characteristics of the composite components has been applied to analyze the results. In the present study we propose a simple phenomenological model for mode II of the oxidation of SiC reinforced CMCs. * Corresponding author. Tel.: +1-217-3332367; fax: 1-217- 2442278; e-mail: mogilevsky@mrlxp2.mrl.uiuc.edu. 0921-5093/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S0921-5093(98)01029-6
P. Mogilecsky, 4. Zanguil Materials Science and Engineering 4262(1999)16-24 Oxygen Partial Pressure Let us consider a single Sic platelet parallel to the sample surface situated within the matrix at a depth z below the sample surface(Fig. 2). Assume that the platelets are loosely dispersed within the matrix, such that other platelets do not ' the platelet under onsideration. Assume also that the platelets are large enough such that unidirectional diffusion can be as sumed. Then the oxygen flux through the growing silica layer can be written out as: J PI)-( where Ps is the oxygen permeability of silica, P, is the oxygen partial pressure at the matrix-silica interface, p Fig. 2. Growth of the oxide layer on a Sic platelet in the oxide is the oxygen partial pressure at the silica-Sic inter- ax face, and n is the parameter which depends on the mechanism of oxygen dissolution in silica. It is believed that oxygen dissolves in, and diffuses through, fused This equation can be solved analytically only for n=l ilica as diatomic molecule [11, 12]. For this case n and n=2. For n=l(molecular permeation of O2) the and, assuming P >>p, we can write solution is. P P The oxygen flux through the matrix towards the platelet can be similarly written out The growth rate of the silica layer is connected with the A=22- oxygen flux through the mass balance equation where Pm is the oxygen permeability of the matrix and Po, is the oxygen partial pressure at the sample surface where C a3.6x10-3 mol- cm-3 is the concentration w equalizing Jm and Js, we can obtain an equation of oxygen in silica, and a is the number of moles of for the unknown oxygen partial pressure at the matrix oxygen required to produce I mol of silica. This num- silica interface, p ber depends on the particular type of the oxidation reaction. For example, if the oxidation of Sic proceeds P by the reaction 2SiC+30,→2SiO,+2CO where the value of a is 1.5 P h Combining Eqs.(2),(6)and(7), and taking the initial (5) condition h(, 0)=0 we obtain after integration 2P or, changing the variables to h'=h√ t and 2'=z/√ (h )+2P2_, 1-=0 0o。。。 which yields: 2P sPOt 會會:會 Eq(11)allows to calculate the thickness of silica as a function of the platelet location (depth beneath the surface)and time, if the oxygen permeabilities of Fig. 1. Oxidation modes of SiC reinforced oxide matrix composites: and the matrix, Ps and Pm, are known. Alternat (a) mode I;(b)modeⅡ fitting the experimental values of h'(z) for given
P. Mogile6sky, A. Zang6il / Materials Science and Engineering A262 (1999) 16–24 17 2. Model Let us consider a single SiC platelet parallel to the sample surface situated within the matrix at a depth z below the sample surface (Fig. 2). Assume that the platelets are loosely dispersed within the matrix, such that other platelets do not ‘screen’ the platelet under consideration. Assume also that the platelets are large enough such that unidirectional diffusion can be assumed. Then the oxygen flux through the growing silica layer can be written out as: Js=Ps[(pi) 1/n−(p%) 1/n ] h (1) where Ps is the oxygen permeability of silica, pi is the oxygen partial pressure at the matrix–silica interface, p% is the oxygen partial pressure at the silica–SiC interface, and n is the parameter which depends on the mechanism of oxygen dissolution in silica. It is believed that oxygen dissolves in, and diffuses through, fused silica as diatomic molecule [11,12]. For this case n=1 and, assuming pip%, we can write Js=Ps pi h (2) The oxygen flux through the matrix towards the platelet can be similarly written out: Jm=Pm[(pO2 ) 1/n−(pi) 1/n ] z (3) where Pm is the oxygen permeability of the matrix and pO2 is the oxygen partial pressure at the sample surface. Now equalizing Jm and Js, we can obtain an equation for the unknown oxygen partial pressure at the matrix– silica interface, pi: pi pO2 1/n + 1 x pi pO2 =1 (4) where x=Pmh Psz (pO2 ) (1−n)/n (5) Fig. 2. Growth of the oxide layer on a SiC platelet in the oxide matrix. This equation can be solved analytically only for n=1 and n=2. For n=1 (molecular permeation of O2) the solution is: pi= pO2 1+ Psz Pmh (6) The growth rate of the silica layer is connected with the oxygen flux through the mass balance equation: aCs dh dt =Js (7) where Cs:3.6×10−3 mol · cm−3 is the concentration of oxygen in silica, and a is the number of moles of oxygen required to produce 1 mol of silica. This number depends on the particular type of the oxidation reaction. For example, if the oxidation of SiC proceeds by the reaction 2SiC+3O22SiO2+2CO (8) the value of a is 1.5. Combining Eqs. (2), (6) and (7), and taking the initial condition h(z,0)=0 we obtain after integration: h2+ 2Ps Pm zh−2Ps pO2 aCs t=0 (9) or, changing the variables to h%=h/ t and z%=z/ t: (h%) 2+ 2Ps Pm z%h%−2Ps pO2 aCs t=0 (10) which yields: h%=z%Ps Pm +' z%Ps Pm 2 + 2Ps pO2 aCs (11) Eq. (11) allows to calculate the thickness of silica as a function of the platelet location (depth beneath the surface) and time, if the oxygen permeabilities of silica and the matrix, Ps and Pm, are known. Alternatively, fitting the experimental values of h%(z%) for given oxidaFig. 1. Oxidation modes of SiC reinforced oxide matrix composites: (a) mode I; (b) mode II.
P. Mogilensky, 4. Zanguil Materials Science and Engineering 4262(1999)16-24 n=2 10 n=1 Fig 3. Approximation of Eq (4): (a)the dependence Of p Po, on x for n=l(),n=2(0), n=4() and n=6(A) and its best approximation /(1+arr m)(solid curves);(b)deviation of the approximate values of p,(x) from the exact ones. tion times with Eq(11), the oxygen permeabilities of silica and the matrix can be obtained where However, n= l is not typical for oxide ceramics. In (2一b) most cases oxygen diffuses through its sublattice in the (14) oxide, and the concentration of diffusing species is not Note that for n=l an,=bn=l, and Eq. (13)reduces to proportional to the external oxygen pressure. For ex- Eq.(10) ample, the values of n=6 and n= 4 have been reported o find the parameters Ps and Pm from the ex for AlOs and zrOz, respectively[3-16). In such cases, mental values of h'=h(z), Eq. (13)should be q(4)cannot be solved analytically. An approximate nized as follows analytical solution of Eq.(4)is, therefore, needed in order to find an expression of h as a function of z and (2y2(h9-)=clpo2) 1=P"(2PPo2 2C-(h)2 To find such an approximate solution, one must first know the possible range of values that the parameterx Accordingly, the parameters Ps and Pm can be found n Eq(4)can have. As shown in Appendix A, we found from the linear fit of (2 m(,)2-u) plotted against that for the considered systems and specific measure- (h). Since, in general, n is not known, the values of ments,the possible range of the values of x can be (=0(1)2-b) should be plotted against (h)?for differ safely assumed to be 10-3 to 10 ent values of n. Linear dependence will indicate the It can be shown that in this interval, Eq.(4)may be correct choice of the parameter n. The values of P, and approximated with reasonable precision by p can then be determined as PO2 1+ax-b (12) P= (16) K Pm=P(Po,m( (17) n n. Fig. 3a shows this approximation for different values of n. The deviation of the approximated values where K and A, respectively, are the slope of, and the pilpo. from the exact ones is shown in Fig. 3b. It can be segment cut on the ordinate axis by, the fitting line seen that in the aforementioned interval this deviation does not exceed 15%. The values of the parameters a Table I and bn obtained from the best fit of Eq.(4)with E Parameters a, and b for n=1, 2, 4, and 6 (12)are given in Table 1 Now combining Eqs. (2),(7)and(12)we can obtain a solution for h(=) (h)2+ 「oa o)“()(2 Ps Po 3.5636 (13)
18 P. Mogile6sky, A. Zang6il / Materials Science and Engineering A262 (1999) 16–24 Fig. 3. Approximation of Eq. (4): (a) the dependence Of pi/pO2 on x for n=1 (), n=2 ( ), n=4 ("), and n=6 (), and its best approximation with the functions pi/pO2 =1/(1+anx−bn ) (solid curves); (b) deviation of the approximate values of pi(x) from the exact ones. tion times with Eq. (11), the oxygen permeabilities of silica and the matrix can be obtained. However, n=1 is not typical for oxide ceramics. In most cases oxygen diffuses through its sublattice in the oxide, and the concentration of diffusing species is not proportional to the external oxygen pressure. For example, the values of n=6 and n=4 have been reported for Al2O3 and ZrO2, respectively [13–16]. In such cases, Eq. (4) cannot be solved analytically. An approximate analytical solution of Eq. (4) is, therefore, needed in order to find an expression of h as a function of z and t. To find such an approximate solution, one must first know the possible range of values that the parameter x in Eq. (4) can have. As shown in Appendix A, we found that for the considered systems and specific measurements, the possible range of the values of x can be safely assumed to be 10−3 to 103 . It can be shown that in this interval, Eq. (4) may be approximated with reasonable precision by: pi pO2 = 1 1+anx−bn (12) where an and bn are the adjustment parameters depending on n. Fig. 3a shows this approximation for different values of n. The deviation of the approximated values pi/pO2 from the exact ones is shown in Fig. 3b. It can be seen that in the aforementioned interval this deviation does not exceed 15%. The values of the parameters an and bn obtained from the best fit of Eq. (4) with Eq. (12) are given in Table 1. Now combining Eqs. (2), (7) and (12) we can obtain a solution for h%(z%): (h%) 2+ 1 cn Ps Pm (pO2 ) n−1 n nbn (z%) bn (h%) (2−bn ) −2Ps pO2 aCs =0 (13) where cn=(2−bn) 2an (14) Note that for n=1 an=bn=1, and Eq. (13) reduces to Eq. (10). To find the parameters Ps and Pm from the experimental values of h%=h%(z%), Eq. (13) should be reorganized as follows: (z%) bn ·(h%) (2−bn ) =cn Pm Ps (pO2 ) 1−n n nbn 2Ps pO2 aCs −(h%) 2 (15) Accordingly, the parameters Ps and Pm can be found from the linear fit of (z%) bn ·(h%) (2−bn ) plotted against (h%) 2 . Since, in general, n is not known, the values of (z%) bn ·(h%) (2−bn ) should be plotted against (h%) 2 for different values of n. Linear dependence will indicate the correct choice of the parameter n. The values of Ps and Pm can then be determined as: Ps= − aCs 2pO2 · A K (16) Pm=Ps(pO2 ) n−1 n −K cn 1/bn (17) where K and A, respectively, are the slope of, and the segment cut on the ordinate axis by, the fitting line. Table 1 Parameters an and bn for n=1, 2, 4, and 6 n an bn 1 1 1 2 1.5965 0.91894 4 0.87734 2.6278 6 3.5636 0.86348
P. Mogilensky, 4. Zanguil Materials Science and Engineering 4262(1999)16-24 the silica layer on individual Sic particles by means of scanning and transmission electron microscopy [8, 10 Brief descriptions of the materials and oxidation condi tions are given in Table 2 a星 6 show the results of the analysis performed for the data on oxidation of two mullite ZrO, -SiC reinforced CMC materials. It is believed that in these materials Pm is dominated by Zro2, which possesses one of the highest values of oxygen perme- ability among ceramic compounds [13], and is also known to have the value of n=4 [13-16. However, fitting of the experimental data with n=4(Fig 5a and Fig. 6a)results in non linear dependence for small values of h. A much better linearization is achieved if the same data are plotted with n=l. However, n=I Fig 4. The effect of the parameter n on the linearization of Eq (13). would mean a diatomic molecular diffusion of oxygen The original dependence h'(=) fits Eq(13) with n= l in the matrix. This is unlikely as long as diffusion through pores and/or massive formation of a glassy phase in the matrix are excluded. Fig 5b and Fig 6b o Fig 4 exemplifies the effect of the correct choice of n show the original experimental data and the oxidation linearization of the experimental data. This figure profiles calculated for n= l and n=4. It can be seen shows that for h'> 0.7hmax (where hmax is the value that in both cases, for n= 4 the calculated curv of hat z=0) the dependence is nearly linear with the ates systematically upwards in the tail of the same slope regardless of the choice of n. Therefore, this a possible reason for this phenomenon may tion of the experimental data can be used to deter- the model considers each single particle independently mine the permeability values, even if linearization over of all others. In reality, however, the oxygen flux reach- the whole range of the experimental data is not ing a particle would be reduced by oxidation of other It should be noted that. as follo cles located closer to Irface. This (16)and(17), such fitting does not affect the value of effect intensifies for large :(and for higher volume Ps, but will produce an incorrect result for Pm. The fraction of the reinforcement)and must lead to smaller difference between the values of p obtained for differ values of h as compared to the calculated ones in the ent values of n depends on the external oxygen partial tail of the profile. This may plausibly explain the ob- pressure. According to Eq(17), for Po,=0. 2 bar the served deviation of the calculated oxidation profiles ratio of the Pm values calculated for n=4 and n=l is from the experimental ones 0.85 bar/, while for Po.=I bar this ratio is 2.87 bar/4. The values of the permeabilities of silica, Ps, and the matrix, Pm, calculated for both materials assuming n= l and n=4 are given in Table 3. It should be noted 3. Results and discussion first that for all calculations a factor P/P(Po)-nyn in the range 5 x 10-3 x 102 was obtained. It justifies the Though the model was originally developed for large application of Eq.(12)as an approximate solution for Sic platelets loosely dispersed in the matrix, an attempt was made to apply it to previously obtained experimen It is interesting to compare the results of the present tal results on the oxidation of SiC particle and whisker analysis with the literature data on oxidation of Sic reinforced mullite-zirconia CMCs. The process of oxi- materials and oxygen permeation through silica. It dation was studied through measuring the thickness of should be noted first, however, that the values of Ps Table 2 Description of CMC materials Material Reinforcement Oxidation Ref Volume fraction, f, T.°C Time. h Atm MZY30P Mullite-30%ZrO 3%Y,O, SiC particles 30 25-500 MZY35Wb Mullite-35%ZrO -1.5%Y, O, SiC whiskers 25-500 [8 Hot pressing of mullite, zirconia, and yttria powders mixed with SiC particles b Hot pressing of alkoxide derived mullite-zirconia-yttria composition mixed with SiC whiskers
P. Mogile6sky, A. Zang6il / Materials Science and Engineering A262 (1999) 16–24 19 Fig. 4. The effect of the parameter n on the linearization of Eq. (13). The original dependence h%(z%) fits Eq. (13) with n=1. the silica layer on individual SiC particles by means of scanning and transmission electron microscopy [8,10]. Brief descriptions of the materials and oxidation conditions are given in Table 2. Fig. 5 and Fig. 6 show the results of the analysis performed for the data on oxidation of two mullite– ZrO2-SiC reinforced CMC materials. It is believed that in these materials Pm is dominated by ZrO2, which possesses one of the highest values of oxygen permeability among ceramic compounds [13], and is also known to have the value of n=4 [13–16]. However, fitting of the experimental data with n=4 (Fig. 5a and Fig. 6a) results in non linear dependence for small values of h%. A much better linearization is achieved if the same data are plotted with n=1. However, n=1 would mean a diatomic molecular diffusion of oxygen in the matrix. This is unlikely as long as diffusion through pores and/or massive formation of a glassy phase in the matrix are excluded. Fig. 5b and Fig. 6b show the original experimental data and the oxidation profiles calculated for n=1 and n=4. It can be seen that in both cases, for n=4 the calculated curve deviates systematically upwards in the tail of the profile. A possible reason for this phenomenon may be that the model considers each single particle independently of all others. In reality, however, the oxygen flux reaching a particle would be reduced by oxidation of other particles located closer to the surface. This ‘screening’ effect intensifies for large z (and for higher volume fraction of the reinforcement) and must lead to smaller values of h% as compared to the calculated ones in the tail of the profile. This may plausibly explain the observed deviation of the calculated oxidation profiles from the experimental ones. The values of the permeabilities of silica, Ps, and the matrix, Pm, calculated for both materials assuming n=1 and n=4 are given in Table 3. It should be noted first that for all calculations a factor Pm/Ps(pO2 ) (1−n)/n in the range 5×101 –3×102 was obtained. It justifies the application of Eq. (12) as an approximate solution for Eq. (4). It is interesting to compare the results of the present analysis with the literature data on oxidation of SiC materials and oxygen permeation through silica. It should be noted first, however, that the values of Ps Fig. 4 exemplifies the effect of the correct choice of n on linearization of the experimental data. This figure shows that for h%\ :0.7h% max (where h% max is the value of h% at z%=0) the dependence is nearly linear with the same slope regardless of the choice of n. Therefore, this portion of the experimental data can be used to determine the permeability values, even if linearization over the whole range of the experimental data is not achieved. It should be noted that, as follows from Eqs. (16) and (17), such fitting does not affect the value of Ps, but will produce an incorrect result for Pm. The difference between the values of Pm obtained for different values of n depends on the external oxygen partial pressure. According to Eq. (17), for pO2 =0.2 bar the ratio of the Pm values calculated for n=4 and n=1 is 0.85 bar3/4 , while for pO2 =1 bar this ratio is 2.87 bar3/4 . 3. Results and discussion Though the model was originally developed for large SiC platelets loosely dispersed in the matrix, an attempt was made to apply it to previously obtained experimental results on the oxidation of SiC particle and whisker reinforced mullite–zirconia CMCs. The process of oxidation was studied through measuring the thickness of Table 2 Description of CMC materials Material Matrix Reinforcement Oxidation Ref. Type Atm. Volume fraction, fv T, °C Time, h Mullite–30%ZrO SiC particles [10] MZY30P 2–3%Y2O3 a 30 1000 25–500 Air MZY35W Mullite–35%ZrO2–1.5%Y2O3 SiC whiskers 30 1200 25–500 Air [8] b a Hot pressing of mullite, zirconia, and yttria powders mixed with SiC particles. b Hot pressing of alkoxide derived mullite–zirconia–yttria composition mixed with SiC whiskers
20 P. Mogilensky, 4. Zanguil Materials Science and Engineering 4262(1999)16-24 MZY30P n=1 E (h)2,μm2/sec MZY30P calculated. n-g 110 0.050.10.15 Fig. 5. Experimental data [10 for the MZY30P material: (a)linear fitting for n= l and n=4; (b) the same data plotted as h=h(=) with oxidation curves calculated for n=l and n=4 obtained in this study are not necessarily the true related to the parabolic growth constant k, by the of oxygen permeability in silica. The process of ox equation: permeation may be controlled by oxygen mobility, as is the case in the experiments on oxygen permeation Ps CS (18) hrough a silica membrane, and, apparently, oxidation of silicon, but may also be(partially) controlled by Such a comparison with the data from [11, 17-2 other factors. such as the diffusion of co. or an shown in Fig. 7. It can be seen that the value of p for interface reaction during the oxidation of SiC [6]. In MZY35W material reinforced with single crystal SiC such a case, the values of Ps obtained from the oxida- whiskers is in a good agreement with the data of tion experiments reflect the actual permeation of oxy- Costello[17] and Zheng[23] for oxidation of Sic single gen under given conditions, as opposed to the crystals. The value of Ps for the MZY30P material permeability as a measure of oxygen mobility, and are reinforced with polycrystalline Sic particles is reason
20 P. Mogile6sky, A. Zang6il / Materials Science and Engineering A262 (1999) 16–24 Fig. 5. Experimental data [10] for the MZY30P material: (a) linear fitting for n=1 and n=4; (b) the same data plotted as h%=h%(z%) with oxidation curves calculated for n=1 and n=4. obtained in this study are not necessarily the true values of oxygen permeability in silica. The process of oxygen permeation may be controlled by oxygen mobility, as is the case in the experiments on oxygen permeation through a silica membrane, and, apparently, oxidation of silicon, but may also be (partially) controlled by other factors, such as the diffusion of CO, or an interface reaction during the oxidation of SiC [6]. In such a case, the values of Ps obtained from the oxidation experiments reflect the actual permeation of oxygen under given conditions, as opposed to the permeability as a measure of oxygen mobility, and are related to the parabolic growth constant Kp by the equation: Ps=aKpCs 2pO2 (18) Such a comparison with the data from [11,17–25] is shown in Fig. 7. It can be seen that the value of Ps for MZY35W material reinforced with single crystal SiC whiskers is in a good agreement with the data of Costello [17] and Zheng [23] for oxidation of SiC single crystals. The value of Ps for the MZY30P material reinforced with polycrystalline SiC particles is reason-
P. Mogilensky, 4. Zanguil Materials Science and Engineering 4262(1999)16-24 ably close to the reported value of oxygen permeability the data obtained by Reetz et al. [26] for alumina of vitreous silica [ll] and the values which can be (2.4 x 10-15 mol- cm-s- at 1000C, when recalcu evaluated from the data on oxidation of Si single lated for a pressure difference of l bar assuming n=6 crystals [17-20. It should be noted that large variations for pure alumina). Assuming that the oxygen permeabil in the reported oxidation kinetics of Si containing ity of mullite is comparable to or slightly greater than materials have been attributed to the differences in the that of alumina [7, 27], that would mean that the oxygen materials used in each investigation depending on the permeability of the mullite-30% ZrO2 matrix is the amount and type of impurities [17]. same as, or close to, that of pure mullite. This, however The rate of oxygen permeation through the matrix contradicts the results [28] obtained by weight gain btained for the MZY30P material (Pm=[1. 2- method. We have to assume, therefore, that the oxygen 1.8]10-Imol-cm-Is-I for the pressure difference 1 permeability of mullite in this system must be signif bar) is close to the value which can be evaluated from cantly lower than that reported for alumina in [26 3106g(a) MZY35W 210 L⊥1 0 5108 25104r(b) MZY35W 104 0 0.15 0.25 0.3 Fig. 6. Experimental data [8] MZY35W material: (a) linear fitting for n= I and n=4; (b) the same data plotted as h=h(=) with oxidation curves alculated fe I and n=4
P. Mogile6sky, A. Zang6il / Materials Science and Engineering A262 (1999) 16–24 21 ably close to the reported value of oxygen permeability of vitreous silica [11] and the values which can be evaluated from the data on oxidation of Si single crystals [17–20]. It should be noted that large variations in the reported oxidation kinetics of Si containing materials have been attributed to the differences in the materials used in each investigation depending on the amount and type of impurities [17]. The rate of oxygen permeation through the matrix obtained for the MZY30P material (Pm=[1.2– 1.8]·10−15 mol · cm−1 · s−1 for the pressure difference 1 bar) is close to the value which can be evaluated from the data obtained by Reetz et al. [26] for alumina (2.4×10−15 mol · cm−1 · s−1 at 1000°C, when recalculated for a pressure difference of 1 bar assuming n=6 for pure alumina). Assuming that the oxygen permeability of mullite is comparable to or slightly greater than that of alumina [7,27], that would mean that the oxygen permeability of the mullite–30% ZrO2 matrix is the same as, or close to, that of pure mullite. This, however, contradicts the results [28] obtained by weight gain method. We have to assume, therefore, that the oxygen permeability of mullite in this system must be signifi- cantly lower than that reported for alumina in [26]. Fig. 6. Experimental data [8] MZY35W material: (a) linear fitting for n=1 and n=4; (b) the same data plotted as h%=h%(z%) with oxidation curves calculated for n=1 and n=4
P. Mogileusky, 4. Zanguil/ Materials Science and Engineering 4262(1999)16-24 8104 Calculated values of the permeabilities of silica(Ps)and matrix(Pm) MZY30P. n=I MZY30P MZY35W 6104 P,Mol·s-1.8.4×10-178.1×10-1797×10-181.1×10-17 18×10-151.2×10-157.5×10-165.2×10-16 m/P, P/P=21. 4(best fit A rather surprising finding is the lower value of matrix permeability obtained for the MzY35w mate- rial, which has higher ZrO, content and was oxidized at P /P=5 higher temperature, than for the MZY30P material 210 This may be partially due to the difference in the mullite stoichiometry and microstructure resulted from differ- ent preparation and processing procedures employed in the studies [8, 10]. However, the most plausible reason for the decrease of the apparent matrix permeability is 0+LLLiLiLrcL-er 0050.10.150.20.250.3 -TC 150014001300120011001000 Fig. 8. The effect of matrix oxygen permeability on the oxidation profile according to the present model. 10157a ormation of zircon(ZrSio4) in the reaction be- ZrO, and SiO,, which becomes important at [28]. In this reaction, high permeability Zro phase in the matrix is replaced with zircon which has very low oxygen permeability [29-31]. This may result in a drastic decrease of the matrix oxygen permeability Fig. 8 shows the effect of matrix oxygen permeability 91- OI 9 MZY30P on the oxidation profile according to the present model using the example of the previously discussed experi mental data for the mzy30P material. The value of parameter n= l and the corresponding value of silica oxygen permeability from Table 3, P,=8.4 x 10-17 molcm-Is-l bar-I were used for the calculations of the oxidation curves for different values of matrix oxygen permeability, Pm. This figure shows that the oxidation profiles are sensitive to the value of matrix oxygen permeability and can be used for experimenta determination of P. On the other hand, this means 5.56.06.5707,58.08.59.0 that calculated permeability values, or even experimen- tal data measured for similar materials, should be used Fig. 7. Comparison of the calculated values of Ps from the present with caution for predictions of oxidation behavior, analysis with the literature data on oxidation of Si and Sic materia since relatively small variation of the matrix permeabil- in dry oxygen and oxygen permeation through silica: . MZY35W; ity can result in significant change of the oxidation MZY30P: 1, oxygen permeation through vitreous silica [11]: 2, si profile. single crystal [17]: 3, Si single crystal [18]: 4, Si single crystal (19:5, Si single crystal [20]: 6, SiC single crystal [21, 22]: 7a, SiC single crystal Based on the above model, a simple criterion which (0001)[17] 7b, SiC single crystal (0001)[17]: 8a, SiC single crystal determines whether the oxidation follows mode i or (0001)[23: 8b, SiC single crystal (0001)(23]: 9, hot pressed Sic [17 mode ii can be evaluated. It should be noted first 10, CVD SIC [24]: Il, CVD SiC [25]. however, that in the framework of this simple model
22 P. Mogile6sky, A. Zang6il / Materials Science and Engineering A262 (1999) 16–24 Table 3 Calculated values of the permeabilities of silica (Ps) and matrix (Pm). MZY30P MZY35W n=1 n=4 n=1 n=4 8.4×10−17 1.1×10−17 8.1×10−17 Ps, Mol · s−1 · 9.7×10−18 cm−1 · bar−1 1.8×10−15 1.2×10−15 7.5×10−16 Pm, Mol · 5.2×10−16 s−1 · cm−1 · bar1/n ) Fig. 8. The effect of matrix oxygen permeability on the oxidation profile according to the present model. A rather surprising finding is the lower value of matrix permeability obtained for the MZY35W material, which has higher ZrO2 content and was oxidized at higher temperature, than for the MZY30P material. This may be partially due to the difference in the mullite stoichiometry and microstructure resulted from different preparation and processing procedures employed in the studies [8,10]. However, the most plausible reason for the decrease of the apparent matrix permeability is Fig. 7. Comparison of the calculated values of Ps from the present analysis with the literature data on oxidation of Si and SiC materials in dry oxygen and oxygen permeation through silica: , MZY35W; ", MZY30P; 1, oxygen permeation through vitreous silica [11]; 2, Si single crystal [17]; 3, Si single crystal [18]; 4, Si single crystal [19]; 5, Si single crystal [20]; 6, SiC single crystal [21,22]; 7a, SiC single crystal (0001() [17]; 7b, SiC single crystal (0001) [17]; 8a, SiC single crystal (0001() [23]; 8b, SiC single crystal (0001) [23]; 9, hot pressed SiC [17]; 10, CVD SiC [24]; 11, CVD SiC [25]. the formation of zircon (ZrSiO4) in the reaction between ZrO2 and SiO2, which becomes important at 1200°C [28]. In this reaction, high permeability ZrO2 phase in the matrix is replaced with zircon which has very low oxygen permeability [29–31]. This may result in a drastic decrease of the matrix oxygen permeability. Fig. 8 shows the effect of matrix oxygen permeability on the oxidation profile according to the present model, using the example of the previously discussed experimental data for the MZY30P material. The value of parameter n=1 and the corresponding value of silica oxygen permeability from Table 3, Ps=8.4×10−17 mol · cm−1 · s−1 · bar−1 , were used for the calculations of the oxidation curves for different values of matrix oxygen permeability, Pm. This figure shows that the oxidation profiles are sensitive to the value of matrix oxygen permeability and can be used for experimental determination of Pm. On the other hand, this means that calculated permeability values, or even experimental data measured for similar materials, should be used with caution for predictions of oxidation behavior, since relatively small variation of the matrix permeability can result in significant change of the oxidation profile. Based on the above model, a simple criterion which determines whether the oxidation follows mode I or mode II can be evaluated. It should be noted first, however, that in the framework of this simple model
P. Mogilensky, 4. Zanguil Materials Science and Engineering 4262(1999)16-24 observed for both considered materials. Taking into Is also clear why mode I was experimentally observed in mullite-SiC composites [101 It should be noted that activation energy for oxygen permeation in such oxides as alumina, and mullite is typically much higher than that for molecular perme- ation of oxygen through silica. Therefore, according to Fig 9. Evaluation of the criterion for oxidation mode I according to the present mode Eqs. (22) and(23), the oxidation of the same material can proceed as mode I at low temperatures, but can change to mode II if the temperature is raised. Also, he oxidation mode I in its pure form cannot take for n+ l the oxidation mode can depend on the exter place: according to Eq (13), some degree of oxidation nal oxygen pressure. Thus, oxidation of a material always takes place even for a particle located at what- proceeding by mode II rding to eq ever large distance below the surface. Therefore, to (22), change to mode I if performed, for example, in introduce the notion of mode i into this model. a practical limit when a particle can be considered not oxidized should be selected, for example when the thickness of the oxide film does not exceed 10% of the 4. Conclusions 9. For this case, the following criterion of mode i can A simple model describing the oxidation mode Il of be evaluated from the present model Sic reinforced oxide CMCs was presented. The model allows a semi-quantitative analysis of the oxidation process and evaluation of the composite parameters <0.1bn (19)(such as matrix oxygen permeability and the rate of the reinforcement oxidation) using experimental mea- where D is the mean particle diameter, and L is the surements of the thickness of the oxide layer on rein- average distance between nearest-neighbor pairs of forcement particles as a function of the depth beneath particles in a volume. The distance L is related to the the composite surface. For systems where permeability mean number of particles per unit volume, Pv, by the values are known or can be assumed with sufficient following expression [32] accuracy, the extent of reinforcement oxidation versus L=0.554P=13 depth can be predicted in a semiquantitative manner. (20) Moreover, the expected mode of oxidation, I or II Defining now the mean particle diameter D as an can be predicted depending on oxygen permeabilities equivalent sphere diameter, we can obtain and volume fraction of the reinforcement phase The model has been applied to the results of the 2_1.1089≈/B (21) microscopic study of the oxidation of Sic reinforced mullite-zirconia CMCs. It was found that the oxygen where fy is the volume fraction of the reinforcement. permeability of the silica layer which grows on SiC Then, Eq(19)becomes particles is significantly higher than that for whiskers As could be expected, it was also found that oxygen permeation through the mullite <0.1 pending on the preparation and processing procedures. (cn) It should be noted that an experimental assembly or, for n= l, simply hich fits best the proposed model is a planar sample such Sic substrate covered with a layer of the 0.2/13 (23) matrix material (a ceramic oxide). It is supposed that application of the proposed model to such planar sam- For typical values fv=l and n=l. the condi ples would allow better evaluation and understanding ion for oxidation mode becomes Pm/P, <0.3- of the mechanisms and rate controlling factors(oxy- corresponds to the gen diffusion through the matrix composites with higher volume fraction of the rein- reaction, evacuation of CO) of the process. This work, forcement Comparing this with the data from as well as further development of the model, are now Table 3, it is clear why mode II was experimentally in progress
P. Mogile6sky, A. Zang6il / Materials Science and Engineering A262 (1999) 16–24 23 . Fig. 9. Evaluation of the criterion for oxidation mode I according to the present model. observed for both considered materials. Taking into account very low permeability of pure mullite, it is also clear why mode I was experimentally observed in mullite–SiC composites [10]. It should be noted that activation energy for oxygen permeation in such oxides as alumina, and mullite is typically much higher than that for molecular permeation of oxygen through silica. Therefore, according to Eqs. (22) and (23), the oxidation of the same material can proceed as mode I at low temperatures, but can change to mode II if the temperature is raised. Also, for n"1 the oxidation mode can depend on the external oxygen pressure. Thus, oxidation of a material proceeding by mode II in air may, according to Eq. (22), change to mode I if performed, for example, in pure oxygen. 4. Conclusions A simple model describing the oxidation mode II of SiC reinforced oxide CMCs was presented. The model allows a semi-quantitative analysis of the oxidation process and evaluation of the composite parameters (such as matrix oxygen permeability and the rate of the reinforcement oxidation) using experimental measurements of the thickness of the oxide layer on reinforcement particles as a function of the depth beneath the composite surface. For systems where permeability values are known or can be assumed with sufficient accuracy, the extent of reinforcement oxidation versus depth can be predicted in a semiquantitative manner. Moreover, the expected mode of oxidation, I or II, can be predicted depending on oxygen permeabilities and volume fraction of the reinforcement phase. The model has been applied to the results of the microscopic study of the oxidation of SiC reinforced mullite–zirconia CMCs. It was found that the oxygen permeability of the silica layer which grows on SiC particles is significantly higher than that for whiskers. As could be expected, it was also found that oxygen permeation through the mullite matrix can vary, depending on the preparation and processing procedures. It should be noted that an experimental assembly which fits best the proposed model is a planar sample such as a SiC substrate covered with a layer of the matrix material (a ceramic oxide). It is supposed that application of the proposed model to such planar samples would allow better evaluation and understanding of the mechanisms and rate controlling factors (oxygen diffusion through the matrix or silica, chemical reaction, evacuation of CO) of the process. This work, as well as further development of the model, are now in progress. the oxidation mode I in its pure form cannot take place: according to Eq. (13), some degree of oxidation always takes place even for a particle located at whatever large distance below the surface. Therefore, to introduce the notion of mode I into this model, a practical limit when a particle can be considered not oxidized should be selected, for example when the thickness of the oxide film does not exceed 10% of the particle size (radius or another typical dimension), Fig. 9. For this case, the following criterion of mode I can be evaluated from the present model: Pm Ps B0.1 2−bn bn < (pO2 ) n−1 n (cn) 1/bn = 2L D (19) where D is the mean particle diameter, and L is the average distance between nearest-neighbor pairs of particles in a volume. The distance L is related to the mean number of particles per unit volume, Pv, by the following expression [32]: L=0.554Pv −1/3 (20) Defining now the mean particle diameter D as an equi6alent sphere diameter, we can obtain 2L D =1.108 3 fv '3 p 6 :f v −1/3 (21) where fv is the volume fraction of the reinforcement. Then, Eq. (19) becomes Pm Ps B0.1 2−bn bn < (pO2 ) n−1 n (cn) 1/bn = f v −1/3 (22) or, for n=1, simply Pm Ps B0.2f v −1/3 (23) For typical values fv=10–40% and n=1, the condition for oxidation mode I then becomes Pm/PsB0.3– 0.4, where the lower limit corresponds to the composites with higher volume fraction of the reinforcement phase. Comparing this with the data from Table 3, it is clear why mode II was experimentally
P. Mogilensky, 4. Zanguil Materials Science and Engineering 4262(1999)16-24 MZY30P from about unity for such matrices as alumina and mullite (very low values of oxygen permeability),to 0-3 x 10 for zirconia, which has one of the high values of oxygen permeability among oxide ceramics [13] Addition of more than 25% of Zro, to mullite matrix is known to strongly increase the oxygen permeability of he matrix [8-10]. For a mullite matrix containing about 30%ZrO2 a factor P/P(Po of about 10-10 can be assumed(Po. x l bar). As for the ratio h/z, it depend on the permeabilities of the matrix and silica, as well as 103 on the specific interval where the measurements were performed. Fig. 10 shows the ratio h/= for the MZY30P material as a function of the variable z=z/vt. It show that the values of h/= for the collected experimental data fall within the range of 10-4-10-I. Based on the above discussion, we found that for the considered systems and specific measurements, the possible range of the values of x can be safely assumed to be 103-10 ⊥⊥⊥⊥-」 References 00050.10.150.20.250.3 [1 P F. Becher, G.C. Wei, J. Am. Ceram Soc. 67(1984) 2C. Nischik, M. M. Seibold, N.A. Travitzky, N. Claussen, J. Am Fig. 10. Range of h/z ratios for the MZY30P material. Ceran.Soc.74(1991)2464. 3Y.-s. Chou, D.J. Green, J. Am. Ceram Soc. 75(1992)3346 Y.S.Chou, D.J. Green, J. Am. Ceram Soc. 76(1993)1452. YS Chou, D.J. Green, J Am Ceram Soc. 76(1993)1985 [6K.L Luthra, J. Am. Ceram Soc. 74(1991)1095. Acknowledgements [7 K.L. Luthra. H D. Park, J. Am. Ceram. Soc. 73(1990)1014 8 CC. Lin, Ph D. Thesis, University of Illinois at Urbana-Cham- This work is supported by the U.s. Department of paign, 1991 Energy, Division of Materials Sciences, under Grant No 9 A. Zangvil, Y. Xu, G. Fu, Ceram. Trans. 48(1994)1003 C.Y. Tsai, CC. Lin, J. Am. Ceram Soc. (In press). DEFG0296ER45439 [l F.J. Norton, Nature 191(1961)701. [2R. H Doremus, J Phys. Chem. 80(1976)1773. [13] W.D. Kingery, Introduction to Ceramics, Wiley, New York, 1960,p.236. X A [14 S.F. Palguev, V K. Gilderman, A D. Neujmin, J. Electrochem. Soc.122(1975)745 The value of x is determined by the oxygen permeabil- [5 A.w. Smith, F.W.Meszaros, C D. Amata, J. Am. Ceram. Soc ities of the matrix, Pm and silica, Ps and by the ratio h/z The value of Pm can vary strongly depending on the R Hartung. H.H. Mobius,Z.Phys.Chem 243(1970)133 7 J.A. Costello, R.E. Tressler, J. Am. Ceram Soc. 69(1986)674 composition of the matrix. Thus, oxygen permeability of [18] J. Jorgensen, J. Chem. Phys. 37(1962)874 ZrO,. about 6x 10-11 mol-cm bar-1/4 at [19] B.H. Claussen, M. Flower, J Electrochem. Soc. 110(1963)983 1200C, can be evaluated from the data in [14 .Oxygen (21R.CA. Harris, R.L. Call, in: R.C. Marshall, I W.Faust, Jr through grain boundaries [7, 26] and was found to follow C E. Ryan(Eds ) Silicon Carbide, University of South Carolina the formula P=3.2 exp(-40 300/T)(g cm s )for the R.C.A. Harris, J. Am. Ceram. Soc. 58(1975) oxygen partial pressure difference of x 0.2 X 10 Pa[26] 223Z. Zheng, R.E. Tressler, K E Spear, J. Electrochem. Soc. 137 hich yields P=1.3x10mol t120 (1990)854 24 E.J. Opila, J. Am. Ceram Soc. 77(1994)730 and 1.8 x 10-15 mol cm-Is-I at 1000 C. Not much (25]L. U.J.T. Ogbuji,EJ.Opila,J Soc.142(1995)730 is known about oxygen permeability of mullite, but it has (26]T. Reetz, U. Lohse, K Teske, mInt.71(1994)631 been assumed to be of the same order of magnitude as rch. eisenhuettenwes of alumina [7, 27]. The value of the oxygen permeability 42(1971)873. 28 C.Y. Tsai, CC Lin, A. Zangvil, A K. Li, J. Am. Ceram. Soc. 8 of silica at 1200oC, extrapolated from the data of Norton (1998)2413. 11], is Ps=2 x10-mol-cm-.s. bar. Accord- (291 ingly, for the experiments conducted in air Carbon 33(1995) at atmospheric pressure (Po, I bar), the factor Pm/ [32j E.E. Underwood, Quantitative Stereology, Addison Wesley P n)/n in our systems should be within the range Reading, MA, 1970, P. 8
24 P. Mogile6sky, A. Zang6il / Materials Science and Engineering A262 (1999) 16–24 Fig. 10. Range of h/z ratios for the MZY30P material. from about unity for such matrices as alumina and mullite (very low values of oxygen permeability), to 103 –3×104 for zirconia, which has one of the highest values of oxygen permeability among oxide ceramics [13]. Addition of more than 25% of ZrO2 to mullite matrix is known to strongly increase the oxygen permeability of the matrix [8–10]. For a mullite matrix containing about 30% ZrO2 a factor Pm/Ps(pO2 ) of about 101 –104 can be assumed (pO2 :1 bar). As for the ratio h/z, it depends on the permeabilities of the matrix and silica, as well as on the specific interval where the measurements were performed. Fig. 10 shows the ratio h/z for the MZY30P material as a function of the variable z%=z/ t. It shows that the values of h/z for the collected experimental data fall within the range of 10−4 –10−1 . Based on the above discussion, we found that for the considered systems and specific measurements, the possible range of the values of x can be safely assumed to be 10−3 –103 . References [1] P.F. Becher, G.C. Wei, J. Am. Ceram. Soc. 67 (1984) 267. [2] C. Nischik, M.M. Seibold, N.A. Travitzky, N. Claussen, J. Am. Ceram. Soc. 74 (1991) 2464. [3] Y.-S. Chou, D.J. Green, J. Am. Ceram. Soc. 75 (1992) 3346. [4] Y.-S. Chou, D.J. Green, J. Am. Ceram. Soc. 76 (1993) 1452. [5] Y.-S. Chou, D.J. Green, J. Am. Ceram. Soc. 76 (1993) 1985. [6] K.L. Luthra, J. Am. Ceram. Soc. 74 (1991) 1095. [7] K.L. Luthra, H.D. Park, J. Am. Ceram. Soc. 73 (1990) 1014. [8] C.C. Lin, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 1991. [9] A. Zangvil, Y. Xu, G. Fu, Ceram. Trans. 48 (1994) 1003. [10] C.Y. Tsai, C.C. Lin, J. Am. Ceram. Soc.(In press). [11] F.J. Norton, Nature 191 (1961) 701. [12] R.H. Doremus, J. Phys. Chem. 80 (1976) 1773. [13] W.D. Kingery, Introduction to Ceramics, Wiley, New York, 1960, p. 236. [14] S.F. Palguev, V.K. Gilderman, A.D. Neujmin, J. Electrochem. Soc. 122 (1975) 745. [15] A.W. Smith, F.W. Meszaros, C.D. Amata, J. Am. Ceram. Soc. 49 (1966) 240. [16] R. Hartung, H.-H. Mobius, Z. Phys. Chem 243 (1970) 133. [17] J.A. Costello, R.E. Tressler, J. Am. Ceram. Soc. 69 (1986) 674. [18] P.J. Jorgensen, J. Chem. Phys. 37 (1962) 874. [19] B.H. Claussen, M. Flower, J. Electrochem. Soc. 110 (1963) 983. [20] B.E. Deal, A.S. Grove, J. Appl. Phys. 36 (1965) 3770. [21] R.C.A. Harris, R.L. Call, in: R.C. Marshall, J.W. Faust, Jr., C.E. Ryan (Eds.), Silicon Carbide, University of South Carolina Press, Columbia, SC, 1974, p. 329. [22] R.C.A. Harris, J. Am. Ceram. Soc. 58 (1975) 7. [23] Z. Zheng, R.E. Tressler, K.E. Spear, J. Electrochem. Soc. 137 (1990) 854. [24] E.J. Opila, J. Am. Ceram. Soc. 77 (1994) 730. [25] L.U.J.T. Ogbuji, E.J. Opila, J. Electrochem. Soc. 142 (1995) 730. [26] T. Reetz, U. Lohse, K. Teske, Ceram. Forum Int. 71 (1994) 631. [27] P.A. Cherkasoy, W.A. Ficher, C. Pieper, Arch. Eisenhuettenwes. 42 (1971) 873. [28] C.Y. Tsai, C.C. Lin, A. Zangvil, A.K. Li, J. Am. Ceram. Soc. 81 (1998) 2413. [29] K.M. Trappen, R.A. Eppler, J. Am. Ceram. Soc. 72 (1989) 882. [30] L.M. Manocha, S.M. Manocha, Carbon 33 (1995) 435. [31] O. Yamamoto, T. Sasamoto, M. Ingaki, Carbon 33 (1995) 359. [32] E.E. Underwood, Quantitative Stereology, Addison Wesley, Reading, MA, 1970, p. 85. Acknowledgements This work is supported by the U.S. Department of Energy, Division of Materials Sciences, under Grant No. DE-FG02-96ER45439. Appendix A The value of x is determined by the oxygen permeabilities of the matrix, Pm and silica, Ps and by the ratio h/z. The value of Pm can vary strongly depending on the composition of the matrix. Thus, oxygen permeability of ZrO2, about 6×10−11 mol · cm−1 · s−1 · bar−1/4 at 1200°C, can be evaluated from the data in [14]. Oxygen permeation through polycrystalline fused alumina occurs through grain boundaries [7,26] and was found to follow the formula P=3.2 exp(−40 300/T) (g cm s−1 ) for the oxygen partial pressure difference of :0.2×105 Pa [26], which yields P=1.3×10−13 mol · cm−1 · s−1 at 1200°C and 1.8×10−15 mol · cm−1 · s−1 at 1000°C. Not much is known about oxygen permeability of mullite, but it has been assumed to be of the same order of magnitude as of alumina [7,27]. The value of the oxygen permeability of silica at 1200°C, extrapolated from the data of Norton [11], is Ps=2×10−15 mol · cm−1 · s−1 · bar−1 . Accordingly, for the experiments conducted in air or in oxygen at atmospheric pressure (pO2 :1 bar), the factor Pm/ Ps(pO2 ) (1−n)/n in our systems should be within the range