P. Mogilersky, 4. Zanguil Materials Science and Engineering 4354(2003)58-66 decrease in the overall oxygen permeability of the matrix will be observed practically from the beginning of the [13-15], since zircon has a very low oxygen permeability process. For this stage, Eq.(27)produces similar results value [20-22 as Eq.(17), and therefore, all the analysis made there- If the product of the reinforcement oxidation may after, with the exception of the dependence of the uickly dissolve in the surrounding matrix or the parabolic constant on the particle size, fully applies to reinforcement does not form a solid oxide at all(e. g. this case carbon reinforcement), an oxide 'envelope'around Before a discussion of the application of the described individual reinforcement particles will not form. In this model to experimental results on oxidation of real case, the oxidation of the composite will be controlled composites, we have first to address the issue of the by the diffusion of oxygen through the oxidized layer oxygen permeability Po of the oxidized layer that grows and by the rate of the reaction of reinforcement on the surface of a composite. If there is no interaction oxidation. In such situation, Eq.(12)is no longer valid. between the matrix and the product of the reinforcement The corresponding set of equations, however, can be oxidation, this layer will consist of two component solved using the same technique that was used in the original matrix and the oxidized reinforcement. Even for development of Eq.(12)[17]. This results in the this case, evaluation of permeability of such composit following equation media is not straightforward and can be additionally K(po)'/mt complicated by percolation. It has been observed, for C+a1(k/P(pa2)-1 (26) example, that in Al2O3/ZrOz and mullite/ZrOz compo- sites, percolation occurs at about 25 vol. of ZrO2, where Ah is the recession of the reinforcement due to causing a sharp increase in the oxygen permeability of oxidation, Ks and ns are the rate constant and the order e composite [10, 12]. For the present analysis, we will of the reaction of oxidation, respectively, other para assume that the permeability of a composite media meters having the same meaning as in Eq.(12).A follows the rule of mixtures, an assumption which is corresponding solution for the thickness of the oxidized often justified when diffusion in two-phase media is layer z on the composite surface can then be obtained considered [23]. In this case, we can rewrite Eq.(25 =+A=(+,=Br 2[P2(1-f)+Pfo,)m with the parameters A and B redefined now as: x Cf 1+ Oxidation mode I is expected first of all when oxygen diffusion through the product of the reinforcement K oxidation is much faster than through the matrix. In his case, the contribution of this phase to the overall permeability of the oxidized layer must be significant, or This is similar to Eq(17). Note, however, that in this even dominant, and Eq. (32)simply reduces to the ase the particle size does not affect the kinetics of the equation for the oxidation of pure reinforcement process at any stage. Again, the kinetics of the linear stage is determined by the rate of the reinforcement 2Ps(Po) oxidation(this time reaction controlled ): and the oxidation kinetics becomes independent of the volume fraction of the reinforcement The linear stage is followed by the approximately If, however, the diffusion of oxygen through the parabolic kinetics controlled by the oxygen transport matrix is faster than through the product of th through the oxidized layer, with the transition occurring enforcement oxidation (Pm >> Ps)the permeation of oxygen through the oxidized layer will be dominated by diffusion through the original matrix, and Eq. (32) 3/n 1+bn1 (31) omes R A/b, K Again, for the oxidation mode I to sustain, the K,=m Pn(Po )/m 1-fs (34) f for the particles of given size to be completely oxidized before the oxidation front moves farther into the In the intermediate case, when the permeability values material, which results in the values =c of the same of all the phases present in the oxidized layer are close to order as the particle size. The parabolic kinetics, thus, each other (Pm A Ps), Eq (32) becomesdecrease in the overall oxygen permeability of the matrix [13
/15], since zircon has a very low oxygen permeability value [20
/22]. If the product of the reinforcement oxidation may quickly dissolve in the surrounding matrix or the reinforcement does not form a solid oxide at all (e.g. carbon reinforcement), an oxide ‘envelope’ around individual reinforcement particles will not form. In this case, the oxidation of the composite will be controlled by the diffusion of oxygen through the oxidized layer and by the rate of the reaction of reinforcement oxidation. In such situation, Eq. (12) is no longer valid. The corresponding set of equations, however, can be solved using the same technique that was used in the development of Eq. (12) [17]. This results in the following equation: Dh Ks(pO2 ) 1=ns t aCs[1 an(zKs=Po(pO2 ) (nons)=nons ) bn ] (26) where Dh is the recession of the reinforcement due to oxidation, Ks and ns are the rate constant and the order of the reaction of oxidation, respectively, other para￾meters having the same meaning as in Eq. (12). A corresponding solution for the thickness of the oxidized layer z on the composite surface can then be obtained: zAz(1bn) Bt (27) with the parameters A and B redefined now as: A an 1 bn
Ks Po (pO2 ) (no ns)=nons bn (28) BKs(pO2 ) 1=ns aCsfs (29) This is similar to Eq. (17). Note, however, that in this case the particle size does not affect the kinetics of the process at any stage. Again, the kinetics of the linear stage is determined by the rate of the reinforcement oxidation (this time reaction controlled): zKs(pO2 ) 1=ns aCsfs t (30) The linear stage is followed by the approximately parabolic kinetics controlled by the oxygen transport through the oxidized layer, with the transition occurring roughly at zc R 1 A1=bn Po Ks (pO2 ) (no ns)=nons
1 bn an 1=bn (31) Again, for the oxidation mode I to sustain, the reaction of reinforcement oxidation must be fast enough for the particles of given size to be completely oxidized before the oxidation front moves farther into the material, which results in the values zc of the same order as the particle size. The parabolic kinetics, thus, will be observed practically from the beginning of the process. For this stage, Eq. (27) produces similar results as Eq. (17), and therefore, all the analysis made there￾after, with the exception of the dependence of the parabolic constant on the particle size, fully applies to this case. Before a discussion of the application of the described model to experimental results on oxidation of real composites, we have first to address the issue of the oxygen permeability Po of the oxidized layer that grows on the surface of a composite. If there is no interaction between the matrix and the product of the reinforcement oxidation, this layer will consist of two components, the original matrix and the oxidized reinforcement. Even for this case, evaluation of permeability of such composite media is not straightforward and can be additionally complicated by percolation. It has been observed, for example, that in Al2O3/ZrO2 and mullite/ZrO2 compo￾sites, percolation occurs at about 25 vol.% of ZrO2, causing a sharp increase in the oxygen permeability of the composite [10,12]. For the present analysis, we will assume that the permeability of a composite media follows the rule of mixtures, an assumption which is often justified when diffusion in two-phase media is considered [23]. In this case, we can rewrite Eq. (25): Kp2[Pm(1  fs) Psfs](pO2 ) 1=m aCsfs (32) Oxidation mode I is expected first of all when oxygen diffusion through the product of the reinforcement oxidation is much faster than through the matrix. In this case, the contribution of this phase to the overall permeability of the oxidized layer must be significant, or even dominant, and Eq. (32) simply reduces to the equation for the oxidation of pure reinforcement: Kp2Ps(pO2 ) 1=m aCs (33) and the oxidation kinetics becomes independent of the volume fraction of the reinforcement. If, however, the diffusion of oxygen through the matrix is faster than through the product of the reinforcement oxidation (Pm/Ps) the permeation of oxygen through the oxidized layer will be dominated by diffusion through the original matrix, and Eq. (32) becomes: Kp2Pm(pO2 ) 1=m aCs 1  fs fs (34) In the intermediate case, when the permeability values of all the phases present in the oxidized layer are close to each other (Pm:/Ps), Eq. (32) becomes: 62 P. Mogilevsky, A. Zangvil / Materials Science and Engineering A354 (2003) 58
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