P. Mogilersky, 4. Zanguil Materials Science and Engineering 4354(2003)58-66 are known or can be assumed with sufficient accuracy, second column of Table Al. Furthermore, the time the rate of the oxidation can be calculated. The model needed for the complete oxidation of a particle is predicts essentially parabolic kinetics of the oxidation. determined by its smallest dimension (Table Al, third Reinforcement particle size may have an effect on the column). As has been mentioned above, Eq.(12)was oxidation kinetics only if the composite matrix and the derived for unidirectional oxidation of coated sub- product of the reinforcement oxidation have signifi strates. Accordingly, it does not adequately describe cantly different dependence of oxygen permeation on the oxidation kinetics of reinforcement particles, for oxygen partial pressure. The effect of the volume which a corresponding correction for the particle fraction of the reinforcement on the oxidation kinetics symmetry should be introduced. Such a correction is depends on the actual mechanism of oxygen transport made easier by the fact that for the purpose of this during oxidation. If oxygen transport proceeds mainly study, we do not need to know the entire kinetics of through the phase formed due to the oxidation of the oxidation of individual particles, but only the total time reinforcement, i.e. the product of the reinforcement needed for their complete oxidation oxidation or a new phase resulting from the reaction of Consider a unidirectional oxidation of a flat sample this product with the matrix material, the kinetics of with constant oxygen partial pressure at the surface. The oxidation is essentially independent of the volume growth of the oxide layer then is described by the fraction. If, however, the transport of oxygen mainly parabolic law: occurs hrough the original matrix, or equ ally idation h=2Kp' all the phases present in the oxidized layer, the oxidation rate becomes a function of the volume fraction and where Kp is the parabolic growth constant, and h is the greatly accelerates at low volume fraction of the thickness of the oxide. The oxidation under identical reinforcement. The model has been applied to the conditions of a spherical particle of radius Ro made of experimental results on oxidation mode I of a number the same material is then described by the equation first of alumina matrix composites reinforced with SiC, derived by Pirani and Sandor for carburization of described in the literature. Good correlation with the tungsten spheres [27] perimental results from the literature was found In accordance with the predictions of the model, even in R similar materials the kinetics of the oxidation can show 6 completely different dependence on such system para meters as volume fraction of the reinforcement, depend where r is the radius of the unreacted core of the ing on the actual path and mechanism of the oxidation particle. As follows from Eq(A2), the time needed for complete oxidation (r=0)is (A3) Acknowledgements Substituting this into Eq(Al), we obtain that during This work was supported in part by the Air Force the same period of time an oxide layer of thickness h Research Laboratory, Materials and Manufacturing R/ 3 would grow on a flat substrate. Taking B into Directorate, under Air Force Contract No. F33615-96- account,R=kR/(3v3)should then allow Eq. (9)and C-5258 the following equations to work for spherical reinforce- ment. A similar ysis for cylindrical shapes can be done based on the equation developed by Andrews and Appendix a Dushman [28] for diffusion of carbon into tungsten The form factor B can be readily evaluated for basic reinforcement shapes such as platelets, spherical parti R cles, or cylinders(whiskers or fibers)and is shown in the Table Al Effective size of oxidation R for various types of reinforcement Type of reinforcement Critical dimension Platelet pherical particles Radius, R o Fibers or whiskers k122 kR/(2v2are known or can be assumed with sufficient accuracy, the rate of the oxidation can be calculated. The model predicts essentially parabolic kinetics of the oxidation. Reinforcement particle size may have an effect on the oxidation kinetics only if the composite matrix and the product of the reinforcement oxidation have signifi￾cantly different dependence of oxygen permeation on oxygen partial pressure. The effect of the volume fraction of the reinforcement on the oxidation kinetics depends on the actual mechanism of oxygen transport during oxidation. If oxygen transport proceeds mainly through the phase formed due to the oxidation of the reinforcement, i.e. the product of the reinforcement oxidation or a new phase resulting from the reaction of this product with the matrix material, the kinetics of oxidation is essentially independent of the volume fraction. If, however, the transport of oxygen mainly occurs through the original matrix, or equally through all the phases present in the oxidized layer, the oxidation rate becomes a function of the volume fraction and greatly accelerates at low volume fraction of the reinforcement. The model has been applied to the experimental results on oxidation mode I of a number of alumina matrix composites reinforced with SiC, described in the literature. Good correlation with the experimental results from the literature was found. In accordance with the predictions of the model, even in similar materials the kinetics of the oxidation can show completely different dependence on such system para￾meters as volume fraction of the reinforcement, depend￾ing on the actual path and mechanism of the oxidation reaction. Acknowledgements This work was supported in part by the Air Force Research Laboratory, Materials and Manufacturing Directorate, under Air Force Contract No. F33615-96- C-5258. Appendix A The form factor b can be readily evaluated for basic reinforcement shapes such as platelets, spherical parti￾cles, or cylinders (whiskers or fibers) and is shown in the second column of Table A1. Furthermore, the time needed for the complete oxidation of a particle is determined by its smallest dimension (Table A1, third column). As has been mentioned above, Eq. (12) was derived for unidirectional oxidation of coated sub￾strates. Accordingly, it does not adequately describe the oxidation kinetics of reinforcement particles, for which a corresponding correction for the particle symmetry should be introduced. Such a correction is made easier by the fact that for the purpose of this study, we do not need to know the entire kinetics of oxidation of individual particles, but only the total time needed for their complete oxidation. Consider a unidirectional oxidation of a flat sample with constant oxygen partial pressure at the surface. The growth of the oxide layer then is described by the parabolic law: h22Kpt (A1) where Kp is the parabolic growth constant, and h is the thickness of the oxide. The oxidation under identical conditions of a spherical particle of radius R0 made of the same material is then described by the equation first derived by Pirani and Sandor for carburization of tungsten spheres [27]: R2 0 6
13  r R0 2 2  r R0 3 Kpt (A2) where r is the radius of the unreacted core of the particle. As follows from Eq. (A2), the time needed for complete oxidation (r/0) is: t R2 0 6Kp (A3) Substituting this into Eq. (A1), we obtain that during the same period of time an oxide layer of thickness h/ R0// ffiffiffi 3 p would grow on a flat substrate. Taking b into account, R/k2/3R0/(3/ ffiffiffi 3 p ) should then allow Eq. (9) and the following equations to work for spherical reinforce￾ment. A similar analysis for cylindrical shapes can be done based on the equation developed by Andrews and Dushman [28] for diffusion of carbon into tungsten filaments: R2 0 4
1  r R0 2 2  r R0 2 ln r R0 Kpt (A4) Table A1 Effective size of oxidation R for various types of reinforcement Type of reinforcement b Critical dimension R Platelets 1 Thickness, H H/2 Spherical particles k2/3/3 Radius, R0 k2/3R0/(3/ ffiffiffi 3 p ) Fibers or whiskers k1/2/2 Radius R0 k1/2R0/(2/ ffiffiffi 2 p ) P. Mogilevsky, A. Zangvil / Materials Science and Engineering A354 (2003) 58
/66 65