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P. Mogilersky, 4. Zanguil Materials Science and Engineering 4354(2003)58-66 2P(po /m 3. Application to experimental results: Al2O3/SiC (35) C, Let us now consider how the above analysis can be Finally, if both phases react to form a single new applied to the experimental results on the oxidation phase with permeability Po, Eq.(25)will retain its Sic reinforced oxide matrix composites obtained by Lin original form, with the dependence on the volume et al. [6]and Luthra and Park [8]. In these studies, fraction the same as in Eq (35 ). In the last three cases, AlO / SiC composites containing 1-50 vol. of Sic the oxidation rate becomes a function of the volume whiskers [6] and 8-50 vol. of Sic particles [8] were fraction of the reinforcement, and a rapid acceleration studied. The mechanism of oxidation, however,wa of the oxidation kinetics is to be expected for composites significantly different in the two cases. In the study of with low volume fraction of the reinforcement Luthra and Park [8], the reaction between silica and the To conclude the discussion on the proposed model, surrounding matrix(alumina)was apparently very fast, one should note that the present model predicts resulting in the formation, in addition to the products infinitely high oxidation rate when the volume fraction expected from the phase diagram, of non-equilibrium fs(or becomes infinitely small. This apparent absurd- aluminosilicate glass. The microscopic investigation y does not point to a possible fault in the model. On revealed that the aluminosilicate glass was present one hand, the continuum diffusion equations which throughout the cross-section of the oxidation product were put into the foundation of the present model [8]. The amount of the aluminosilicate liquid formed assume that concentration(oxygen partial pressure) is a during the reaction must have been proportional to the continuous function of time and coordinates, as well as volume fraction of silica in the oxidation product its first time derivative and first and second coordinate Assuming that the aluminosilicate glassy phase has the derivatives. As a result, they a non-zero solution highest value of oxygen permeability among the phases for the concentration of the diffusing species(oxygen) present [9, 18, 19), oxidation must have proceeded ac- Aor any time t>0, even at the infinitely large distance cording to Eq(33), i.e. with the parabolic constant from the surface. On the other hand, infinitely small independent of the volume fraction of Sic in the volume fraction may be realized as an assembly of either composite. Indeed, at each temperature nearly the infinitely small particles with finite point density in the same values of the parabolic constant Kp were reported matrix, or particles of a finite size with an infinitely low for all the materials in this study, Fig. 3 point density. Keeping in mind the aforementioned In the study of Lin et al., the reaction between silica nature of the diffusion equations, the first case(infi- and the matrix was apparently delayed, such that the nitely small particles), in fact, means that all particles oxidized layer consisted of two sublayers. In the outer will be oxidized immediately, regardless of their distance sublayer where the reaction was complete, the equili from the surface, which is equivalent to infinite oxida- brium phases(alumina and mullite, or mullite and silica, tion rate. The second case(finite particles with infinitely depending on the initial composition) were observed small point density)can not be accommodated within The inner sublayer, in which the reaction between silica the framework of this model. Let us recall that the value and the matrix had not yet occurred, contained the Az(Fig. la)in Eq. (1)was assumed to be small enough original matrix and silica formed during the oxidation of to ignore the variation of oxygen partial pressure within SiC whiskers. In this case, the process of oxidation was this layer, but at the same time large enough to contain a apparently controlled by the oxygen diffusion through statistically representative number of reinforcement the inner sublayer. Since silica has very low value of particles. When the point density of the particles oxygen permeability, the oxidation must have proceeded becomes too low, both conditions can not be met at according to Eq(34), with the parabolic constant K the same time and all subsequent development becomes proportional to(1-fs)/s. In Fig. 3b the values of Kp invalid. In addition, the whole notion of mode I as the calculated from the data presented in [6] are plotted as a case when a sharp boundary exists between completely function of(1-fs)/fs. The graph shows nearly perfect oxidized and unoxidized material becomes poorly de- linear fit in accordance with the prediction of the present fined when the average inter particle distance become many times larger than the particles size. It should be Unfortunately, in the study [6] the dependence of the noted, however, that very high oxidation rates have been oxidation kinetics on the oxygen partial pressure was eported for composites with low volume fraction of the not studied. If the oxidation proceeds according to the reinforcement. Thus Lin et al. who studied the oxidation described mechanism, with oxygen diffusing mainly kinetics of Al2O /SiC composites containing 1-50 vol. through the alumina matrix(no=lm=6[24]) and the of Sic whiskers reported that the oxidation rate of the reaction between silica and the matrix delayed(ns composites with I and 4%of the reinforcement was too [25, 26], Eq(23)predicts that the parabolic constant Kp high to be measured [6] should be proportional toKp2Ps(pO2 ) 1=m aCsfs (35) Finally, if both phases react to form a single new phase with permeability Po, Eq. (25) will retain its original form, with the dependence on the volume fraction the same as in Eq. (35). In the last three cases, the oxidation rate becomes a function of the volume fraction of the reinforcement, and a rapid acceleration of the oxidation kinetics is to be expected for composites with low volume fraction of the reinforcement. To conclude the discussion on the proposed model, one should note that the present model predicts infinitely high oxidation rate when the volume fraction fs (or f) becomes infinitely small. This apparent absurdity does not point to a possible fault in the model. On one hand, the continuum diffusion equations which were put into the foundation of the present model assume that concentration (oxygen partial pressure) is a continuous function of time and coordinates, as well as its first time derivative and first and second coordinate derivatives. As a result, they yield a non-zero solution for the concentration of the diffusing species (oxygen) for any time t/0, even at the infinitely large distance from the surface. On the other hand, infinitely small volume fraction may be realized as an assembly of either infinitely small particles with finite point density in the matrix, or particles of a finite size with an infinitely low point density. Keeping in mind the aforementioned nature of the diffusion equations, the first case (infinitely small particles), in fact, means that all particles will be oxidized immediately, regardless of their distance from the surface, which is equivalent to infinite oxidation rate. The second case (finite particles with infinitely small point density) can not be accommodated within the framework of this model. Let us recall that the value Dz (Fig. 1a) in Eq. (1) was assumed to be small enough to ignore the variation of oxygen partial pressure within this layer, but at the same time large enough to contain a statistically representative number of reinforcement particles. When the point density of the particles becomes too low, both conditions can not be met at the same time and all subsequent development becomes invalid. In addition, the whole notion of mode I as the case when a sharp boundary exists between completely oxidized and unoxidized material becomes poorly defined when the average inter particle distance becomes many times larger than the particles size. It should be noted, however, that very high oxidation rates have been reported for composites with low volume fraction of the reinforcement. Thus Lin et al. who studied the oxidation kinetics of Al2O3/SiC composites containing 1/50 vol.% of SiC whiskers reported that the oxidation rate of the composites with 1 and 4% of the reinforcement was too high to be measured [6]. 3. Application to experimental results: Al2O3/SiC composites Let us now consider how the above analysis can be applied to the experimental results on the oxidation of SiC reinforced oxide matrix composites obtained by Lin et al. [6] and Luthra and Park [8]. In these studies, Al2O3/SiC composites containing 1/50 vol.% of SiC whiskers [6] and 8/50 vol.% of SiC particles [8] were studied. The mechanism of oxidation, however, was significantly different in the two cases. In the study of Luthra and Park [8], the reaction between silica and the surrounding matrix (alumina) was apparently very fast, resulting in the formation, in addition to the products expected from the phase diagram, of non-equilibrium aluminosilicate glass. The microscopic investigation revealed that the aluminosilicate glass was present throughout the cross-section of the oxidation product [8]. The amount of the aluminosilicate liquid formed during the reaction must have been proportional to the volume fraction of silica in the oxidation product. Assuming that the aluminosilicate glassy phase has the highest value of oxygen permeability among the phases present [9,18,19], oxidation must have proceeded according to Eq. (33), i.e. with the parabolic constant independent of the volume fraction of SiC in the composite. Indeed, at each temperature nearly the same values of the parabolic constant Kp were reported for all the materials in this study, Fig. 3a. In the study of Lin et al., the reaction between silica and the matrix was apparently delayed, such that the oxidized layer consisted of two sublayers. In the outer sublayer where the reaction was complete, the equilibrium phases (alumina and mullite, or mullite and silica, depending on the initial composition) were observed. The inner sublayer, in which the reaction between silica and the matrix had not yet occurred, contained the original matrix and silica formed during the oxidation of SiC whiskers. In this case, the process of oxidation was apparently controlled by the oxygen diffusion through the inner sublayer. Since silica has very low value of oxygen permeability, the oxidation must have proceeded according to Eq. (34), with the parabolic constant Kp proportional to (1/fs)/fs. In Fig. 3b the values of Kp calculated from the data presented in [6] are plotted as a function of (1/fs)/fs. The graph shows nearly perfect linear fit in accordance with the prediction of the present model. Unfortunately, in the study [6] the dependence of the oxidation kinetics on the oxygen partial pressure was not studied. If the oxidation proceeds according to the described mechanism, with oxygen diffusing mainly through the alumina matrix (no/nm/6 [24]) and the reaction between silica and the matrix delayed (ns/1 [25,26]), Eq. (23) predicts that the parabolic constant Kp should be proportional to: P. Mogilevsky, A. Zangvil / Materials Science and Engineering A354 (2003) 58/66 63