P. Mogilersky, 4. Zanguil Materials Science and Engineering 4354(2003)58-66 Combining Eqs. (4)and( 8)then yields: its value when the oxidation of a particle is complete, the R lete oxidation of the f where R is the effective particle size R(see Appendix A). 4r R2+A(1+b)= R(2-b) B. confined to the layer Az, the balance between the influx ygen through the surface of the oxidized lay the rate of its incorporation into the newly formed oxide A=o-b(1+w)/P (Po. )6-1//476. on the reinforcement particles can be written out as (14) follows Jo=Js=fs Combining Eqs.(9)and (13), the rate of the propaga or, taking into account Eq (7) tion of the oxidation front can be then calculated as. J。=J (11) dz△z B That means that with the choice of the layer thickness R+A(1+b,)=b,R(l-b, (16) z according to Eq.(), the average oxygen partial pressure at the particle/matrix interface within this layer With the initial condition z(0)=0, this gives after becomes equal to that at the interface between the oxide integration film on a flat substrate under an oxide coating of thickness :(Fig. Ib). Having addressed the issue of ER+AR(-b_(+,=Bt (17) particle geometry by defining the effective particle size or, in reduced coordinates R, the time needed for complete oxidation of a particle equation derived for unidirectional oxidation of a flat x+Ax(+b, )Bt within the layer Az can be, therefore, obtained from the non-oxide substrate under an oxide coating. The latter case for a Sic substrate has been analyzed in detail where x=Z/R elsewhere [17]. The equation developed there can be We can conclude, therefore, that initially(small z)the further extended to any non-oxide material growth of the oxidized layer on the composite surface follows the linear kinetics and is controlled by the rate of 2 P b(2-b)=2po2) he reinforcement oxidation 2-bP 2P(o) (12) (19) CssR where h=h/vr, i=/vt, Po, is the external oxygen The growth rate of the oxidized layer during the linear partial pressure, Po and Ps are oxygen permeabilities of tage is predictably higher for smaller particle size and the coating and the oxidation product, respectively, o volume fraction of the reinforcement and ns are the pressure dependence exponents for the For larger thickness of the oxidized layer, the process oxygen permeation in the coating and the oxidation turns to the approximately parabolic kinetics product, respectively, which depend on the mechanism of oxygen diffusion in each substance, n=nd/ns, and an,/=\(1-b, Bt tabulated for different values of n[17]. Cs is the volume concentration of oxygen in the product of reinforcement and is essentially controlled by the oxygen diffusion oxidation, and a is the number of moles of oxygen through the oxidized layer. Since the values of b,are required to produce I mol of this product. Thus, for generally close to 1 [17], the deviation from th oxidation of Sic to amorphous silica Cs 3.6 x 10-3 parabolic kinetics may not be readily apparent when mol cm-3, and a can be 1, 1.5, or 2 depending on the is plotted against t even for relatively large values of z degree of oxidation of carbon (i.e. >500 um for particle size of 5 um), Fig. 2a. Eq(12)describes an oxidation process that is jointly However, the apparent parabolic constants may differ controlled by oxygen diffusion through the coating and significantly. The value zc at which the linear kinetics the growing oxide film at the interface. USing Eq(12) expires can be found approximately by equating the two for the oxide thickness on individual particles and R as terms in the left side of Eq.(18). This yieldCombining Eqs. (4) and (8) then yields: Dzobh fs R fs (9) where R is the effective particle size R (see Appendix A). Note that since the oxidation is assumed to be confined to the layer Dz, the balance between the influx of oxygen through the surface of the oxidized layer and the rate of its incorporation into the newly formed oxide on the reinforcement particles can be written out as follows: JoJs Dz V¯ r f S¯ r (10) or, taking into account Eq. (7), JoJs (11) That means that with the choice of the layer thickness Dz according to Eq. (7), the average oxygen partial pressure at the particle/matrix interface within this layer becomes equal to that at the interface between the oxide film on a flat substrate under an oxide coating of thickness z (Fig. 1b). Having addressed the issue of particle geometry by defining the effective particle size R, the time needed for complete oxidation of a particle within the layer Dz can be, therefore, obtained from the equation derived for unidirectional oxidation of a flat non-oxide substrate under an oxide coating. The latter case for a SiC substrate has been analyzed in detail elsewhere [17]. The equation developed there can be further extended to any non-oxide material: h¯ 2 2an 2  bn
Ps Po (pO2 ) (no ns)=nons bn z¯ bn h¯ (2bn) 2Ps(pO2 ) 1=ns aCs (12) where h¯h= ffiffi t p ; z¯z= ffiffi t p ; pO2 is the external oxygen partial pressure, Po and Ps are oxygen permeabilities of the coating and the oxidation product, respectively, no and ns are the pressure dependence exponents for the oxygen permeation in the coating and the oxidation product, respectively, which depend on the mechanism of oxygen diffusion in each substance, n/no/ns, and an and bn are numerical parameters that have been tabulated for different values of n [17]. Cs is the volume concentration of oxygen in the product of reinforcement oxidation, and a is the number of moles of oxygen required to produce 1 mol of this product. Thus, for oxidation of SiC to amorphous silica Cs:/3.6/103 mol cm3 , and a can be 1, 1.5, or 2 depending on the degree of oxidation of carbon. Eq. (12) describes an oxidation process that is jointly controlled by oxygen diffusion through the coating and the growing oxide film at the interface. Using Eq. (12) for the oxide thickness on individual particles and R as its value when the oxidation of a particle is complete, the time necessary for the complete oxidation of the reinforcement particles in the layer Dz is: DtR2 A(1 bn)zbnR(2bn) Bfs (13) where A 2an (2  bn)(1 bn)
Ps Po (pO2 ) (nons)=nons bn (14) B2Ps(pO2 ) 1=ns aCsfs (15) Combining Eqs. (9) and (13), the rate of the propaga￾tion of the oxidation front can be then calculated as: dz dt : Dzo Dt B R A(1 bn)zbnR(1bn) (16) With the initial condition z(0)/0, this gives after integration: zRAR(1bn) z(1bn) Bt (17) or, in reduced coordinates, xAx(1bn) Bt R2 (18) where x/z/R. We can conclude, therefore, that initially (small z) the growth of the oxidized layer on the composite surface follows the linear kinetics and is controlled by the rate of the reinforcement oxidation: z2Ps(pO2 ) 1=ns aCsfsR t (19) The growth rate of the oxidized layer during the linear stage is predictably higher for smaller particle size and volume fraction of the reinforcement. For larger thickness of the oxidized layer, the process turns to the approximately parabolic kinetics: z2 z R (1bn) Bt A (20) and is essentially controlled by the oxygen diffusion through the oxidized layer. Since the values of bn are generally close to 1 [17], the deviation from the parabolic kinetics may not be readily apparent when z is plotted against t 1/2 even for relatively large values of z (i.e. /500 mm for particle size of 5 mm), Fig. 2a. However, the apparent parabolic constants may differ significantly. The value zc at which the linear kinetics expires can be found approximately by equating the two terms in the left side of Eq. (18). This yields: 60 P. Mogilevsky, A. Zangvil / Materials Science and Engineering A354 (2003) 58
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