MIATERIALS SENE S ENGEERING ELSEVIER Materials Science and Engineering A354(2003)58-66 Kinetics of oxidation in oxide ceramic matrix composites P. Mogilevsky a, b, * A. Zangvil Air Force Research Laboratory/MLLN, Wright-Patterson Air Base. OH 45433. USA UES Inc.. 4401 Dayton-Xenia Rd, Dayton, OH 45432. US.A Ceramic Shield Ltd. Misgar Carmiel Technology Incubator, M. P. Misgav 20179, Israel Received 25 June 2002. received in revised form 4 Novem ber 2002 Oxidation of sic reinforcement is or factor affecting the environmental stability of Sic reinforced ceramic matrix omposites(CMCs) for high temperature applications. a new quantitative model for the oxidation of oxide CMCs with non-oxide reinforcements is described. The proposed model is applied to the experimental results from the literature on oxidation of Al2O3/Sic composites. C 2003 Elsevier Science B V. All rights reserved. Keywords: Kinetics: Oxidation; Oxide ceramic matrix 1. Introduction situated deeper into composite would occur. This mode results in a sharp interface separating the surface layer Ceramic matrix composite (CMC) materials rein- of completely oxidized material from the underlying forced with Sic particles, whiskers, fibers, or platelets virgin composite Mode II is the case where oxygen can have high fracture toughness and strength [1-5]. In deeply penetrate into the matrix before reinforcement oxide matrix CMCs, high temperature oxidation of Sic particles in the outer region are completely oxidized, reinforcement affects the environmental stability of the leaving behind a region of partially oxidized reinforce- CMC. Alumina and mullite have been considered as ment Mode I was experimentally found in alumina/Sic matrix materials for oxide CMCs due to their excellent and mullite/SiC composites [6, 8, 14], while mode II was high-temperature stability and, in particular, slow oxy- observed in mullite/Zro2/SiC composites [7, 10, 12-14 gen permeation, minimizing the oxidation of the re- A semi-quantitative model was proposed by the present inforcement authors for the oxidation mode ii of sic reinforced Oxidation of sic reinforced alumina. mullite, and oxide CMCs[17]. Based on this model, a qualitative link mullite/zirconia matrix composites has been studied between the diffusion characteristics of the components both microscopically and via conventional weight gain ind structural characteristics of the composite(such as method [6-16]. Two oxidation modes of Sic reinforced the volume fraction of the non-oxide reinforcement) nd the oxidation mode of the composite, was estab- [7. 8, 10, 11, 14. Mode I was defined as the case when Sic lished. a quantitative model describing the oxidation icles at a particular depth become completely mode I was first suggested by Luthra and Park [8]. In he present study the approach of [17] is extended and ized before any observable oxidation of particles used as a basis to develop a new model for the oxidation mode i of oxide Cmcs. that allows to examine the s Corresponding author. Tel: +1-937-255-9855: fax: +1-937-656- oxidation kinetics in detail. The proposed model is plied to the expe E-mail address: pavel mogilevsky @wpafb af mil(P Mogilevsky) AlO3/SiC composites available in the literature
Kinetics of oxidation in oxide ceramic matrix composites P. Mogilevsky a,b,*, A. Zangvil c a Air Force Research Laboratory/MLLN, Wright-Patterson Air Force Base, OH 45433, USA b UES Inc., 4401 Dayton-Xenia Rd., Dayton, OH 45432, USA c Ceramic Shield Ltd. Misgav Carmiel Technology Incubator, M.P. Misgav 20179, Israel Received 25 June 2002; received in revised form 4 November 2002 Abstract Oxidation of SiC reinforcement is a major factor affecting the environmental stability of SiC reinforced ceramic matrix composites (CMCs) for high temperature applications. A new quantitative model for the oxidation of oxide CMCs with non-oxide reinforcements is described. The proposed model is applied to the experimental results from the literature on oxidation of Al2O3/SiC composites. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Kinetics; Oxidation; Oxide ceramic matrix 1. Introduction Ceramic matrix composite (CMC) materials reinforced with SiC particles, whiskers, fibers, or platelets have high fracture toughness and strength [1/5]. In oxide matrix CMCs, high temperature oxidation of SiC reinforcement affects the environmental stability of the CMC. Alumina and mullite have been considered as matrix materials for oxide CMCs due to their excellent high-temperature stability and, in particular, slow oxygen permeation, minimizing the oxidation of the reinforcement. Oxidation of SiC reinforced alumina, mullite, and mullite/zirconia matrix composites has been studied both microscopically and via conventional weight gain method [6/16]. Two oxidation modes of SiC reinforced oxide matrix composite materials have been reported [7,8,10,11,14]. Mode I was defined as the case when SiC particles at a particular depth become completely oxidized before any observable oxidation of particles situated deeper into composite would occur. This mode results in a sharp interface separating the surface layer of completely oxidized material from the underlying virgin composite. Mode II is the case where oxygen can deeply penetrate into the matrix before reinforcement particles in the outer region are completely oxidized, leaving behind a region of partially oxidized reinforcement. Mode I was experimentally found in alumina/SiC and mullite/SiC composites [6,8,14], while mode II was observed in mullite/ZrO2/SiC composites [7,10,12/14]. A semi-quantitative model was proposed by the present authors for the oxidation mode II of SiC reinforced oxide CMCs [17]. Based on this model, a qualitative link between the diffusion characteristics of the components and structural characteristics of the composite (such as the volume fraction of the non-oxide reinforcement), and the oxidation mode of the composite, was established. A quantitative model describing the oxidation mode I was first suggested by Luthra and Park [8]. In the present study the approach of [17] is extended and used as a basis to develop a new model for the oxidation mode I of oxide CMCs, that allows to examine the oxidation kinetics in detail. The proposed model is applied to the experimental results on oxidation of Al2O3/SiC composites available in the literature. * Corresponding author. Tel.: /1-937-255-9855; fax: /1-937-656- 4296. E-mail address: pavel.mogilevsky@wpafb.af.mil (P. Mogilevsky). Materials Science and Engineering A354 (2003) 58/66 www.elsevier.com/locate/msea 0921-5093/02/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-5093(02)00872-9
P. Mogilersky, 4. Zanguil Materials Science and Engineering 4354(2003)58-66 2. Kineties of oxidation mode i during the oxidation of the reinforcement. For example, for the oxidation of Sic to amorphous silica, k a 2. 19 Consider a composite containing non-oxide reinforce- Disregarding the volume change associated with the ment particles in an oxide matrix, oxidizing in mode I, possible reaction between the matrix and the product of Fig. la. During oxidation, an oxide film with the the reinforcement oxidation, complete oxidation of the permeability Ps grows on each reinforcement particle, reinforcement within the layer Az will increment the resulting in the formation on the composite surface of thickness of the oxidized layer by an oxidized layer(consisting of the matrix, the product of the reinforcement oxidation, and, possibly, the =△[l+f(k-1) product of a reaction between them) with effective which along with Eq(2)gives thickness 4△ of unoxidized material adiacent to the△,=△M(=△ oxidation front(currently located at depth z beneath the 1+f(k-1) rface), Fig. la. At this stage, no assumptions regard where ing the thickness of this layer is made other than that the this layer and that it is thick enough to contain a 51+f-0 statistically representative number of reinforcement particles. Since the oxidation proceeds in mode I,we is the volume fraction of the product of reinforcement can assume at the same time that this layer is thin oxidation in the oxidized layer enough such that the variation of oxygen partial On the other hand, AVs can be expressed through the pressure within it is negligible hickness, h, of the oxide on completely oxidized Such a layer contains a total volume of the reinforce- particles: ment(per unit area of the composite surface): △=B△Sh △V=△zf where f is the volume fraction of the reinforcement where S. and v. are the average surface area and phase. After complete oxidation it will transform into a volume of the reinforcement particles, and B is a form volume AVs of the oxidation product, given by factor that depends on the particle shape(see Appendi △V=kAzf Let us now choose the value az such that the total surface area of the reinforcement particles contained in here k is the coefficient of the volumetric change this layer is equal to the area of the sample surface(or in other words, such that the layer will have a unit surface I For clarity, in the following treatment the term'oxide film'is used area of the reinforcement particles per unit area of the mposite surface during its oxidation. While the former is generally P JS, =/urface) to describe the oxide growing on individual particles of reinforcement during their oxidation inside the matrix. The term oxidized layer is A used to describe the completely oxidized material growing on the single phase(e.g. silica in case of Sic reinforcement), the latter may ontain a number of phases including the original matrix, the product This will reduce Eq(6)to of reinforcement oxidation, and/or possible products of the reaction △V=Bh Oxide. h Coating 8°8 0:自:0::合 合:Q Fig. I. Growth of the oxidized layer during oxidation mode I of CMCs(a)and oxidation of a fat substrate under an oxide coating(b)
2. Kinetics of oxidation mode I Consider a composite containing non-oxide reinforcement particles in an oxide matrix, oxidizing in mode I, Fig. 1a. During oxidation, an oxide film with the permeability Ps grows on each reinforcement particle, resulting in the formation on the composite surface of an oxidized layer (consisting of the matrix, the product of the reinforcement oxidation, and, possibly, the product of a reaction between them) with effective oxygen permeability Po. 1 Consider now a layer of the thickness Dz of unoxidized material adjacent to the oxidation front (currently located at depth z beneath the surface), Fig. 1a. At this stage, no assumptions regarding the thickness of this layer is made other than that the process of actual oxidation takes place entirely within this layer and that it is thick enough to contain a statistically representative number of reinforcement particles. Since the oxidation proceeds in mode I, we can assume at the same time that this layer is thin enough such that the variation of oxygen partial pressure within it is negligible. Such a layer contains a total volume of the reinforcement (per unit area of the composite surface): DV Dzf (1) where f is the volume fraction of the reinforcement phase. After complete oxidation it will transform into a volume DVs of the oxidation product, given by: DVskDzf (2) where k is the coefficient of the volumetric change during the oxidation of the reinforcement. For example, for the oxidation of SiC to amorphous silica, k :/2.19. Disregarding the volume change associated with the possible reaction between the matrix and the product of the reinforcement oxidation, complete oxidation of the reinforcement within the layer Dz will increment the thickness of the oxidized layer by DzoDz[1f (k1)] (3) which along with Eq. (2) gives: DVs Dzokf 1 f (k 1) Dzofs (4) where fs kf 1 f (k 1) (5) is the volume fraction of the product of reinforcement oxidation in the oxidized layer. On the other hand, DVs can be expressed through the thickness, h, of the oxide on completely oxidized particles: DVsbDzf S¯ rh V¯ r (6) where S¯ r and V¯ r are the average surface area and volume of the reinforcement particles, and b is a form factor that depends on the particle shape (see Appendix A). Let us now choose the value Dz such that the total surface area of the reinforcement particles contained in this layer is equal to the area of the sample surface (or in other words, such that the layer will have a unit surface area of the reinforcement particles per unit area of the composite’s surface): Dz V¯ r f S¯ r1 (7) This will reduce Eq. (6) to DVsbh (8) Fig. 1. Growth of the oxidized layer during oxidation mode I of CMCs (a) and oxidation of a flat substrate under an oxide coating (b). 1 For clarity, in the following treatment the term ‘oxide film’ is used to describe the oxide growing on individual particles of reinforcement during their oxidation inside the matrix. The term ‘oxidized layer’ is used to describe the completely oxidized material growing on the composite surface during its oxidation. While the former is generally single phase (e.g. silica in case of SiC reinforcement), the latter may contain a number of phases including the original matrix, the product of reinforcement oxidation, and/or possible products of the reaction between the two. P. Mogilevsky, A. Zangvil / Materials Science and Engineering A354 (2003) 58/66 59
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 yield
Combining 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 propagation 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) 58/66
P. Mogilersky, 4. Zanguil Materials Science and Engineering 4354(2003)58-66 100 the ' parabolic stage. Eq(22) indicates a slight depen dence of the apparent parabolic constant on the size of he reinforcement particles. The dependence in Eq (22) is shown graphically in Fig. 2b for different values of n It Is seen tha deviation from the parabolic kinetics increases with time(as also seen from Fig 2a), and this increase is more profound for smaller particle size and n deviating significantly from 1. This deviation should be -n=1 taken into account if the permeability values are calculated from the apparent parabolic constants Note that the initial value of the apparent parabolic constant does not depend on the reinforcement particle size or rate of the reinforcement oxidation, as is expected ⊥⊥⊥ if the process is entirely controlled by the diffusion through the oxidized layer. If n,=ns=m, then n and bn=an=l [17]. in which case the preceding R equations simplify significantly 2 Time- (24) (o,) For large z the kinetics in this case is true parabolic and the process is entirely controlled by the diffusion through the oxidized layer It is intuitively clear that if there is no reaction 二-:D= between the matrix and the product of the reinforcement oxidation, the oxidation mode I can only occur if Ps is considerably higher than oxygen permeability of the matrix, Pm. In this case, Po can not be greater than Ps and therefore, in normal conditions the value of zc will be of the same order or less as the size of Fig-2. Deviation from the parabolic law(a)and the dependence of the practice it willn es, R(Eq reinforcement particles, R(Eq(21). This means that apparent parabolic constant Kp on the particle size(b)for n+ I stage experimentally. However, a reaction between the matrix and the product of the reinforcement oxidation hay he kinetics of the oxidation process, as is the case for the oxidation of Sic RA1/=p(o,(-b1+b in Al,O3 and mullite/ZrO, matrices. For instance, the new phase can have the permeability value significantly higher than Ps, particularly if it is a glassy phase Substituting Eqs. (14)and(15) into Eq(20)and (aluminosilicate). In such a case, both constantA(21), we obtain for the apparent SiC oxidation and oxygen diffusion through the oxi- dized layer can be considerably accelerated K2=[/R] [8,9, 15, 18, 19](note that in this case Ps in the preceding equations must refer to the permeability not of silica but that of the aluminosilicate glass). In this case, the arly Kp B 2P(Po. )'no[(2-b)(1+bm)7. occur at larger =, and the linear stage can possibly be (23) observed experimentally. On the other hand, in matrices ontaining ZrO2, formation ZrSiO4,can is the apparent parabolic constant in the beginning of occur. This reaction is known to cause a very sharp
zc R 1 A1=bn Po Ps (pO2 ) (ns no)=nons (2 bn)(1 bn) 2an 1=bn (21) Substituting Eqs. (14) and (15) into Eq. (20) and taking note of Eq. (21), we obtain for the apparent parabolic constant Kp: Kp K0 p z=R A1=bn (1bn) (22) where K0 p B A1=bn 2Po(pO2 ) 1=no aCsfs (2 bn)(1 bn) 2an 1=bn (23) is the apparent parabolic constant in the beginning of the ‘parabolic’ stage. Eq. (22) indicates a slight dependence of the apparent parabolic constant on the size of the reinforcement particles. The dependence in Eq. (22) is shown graphically in Fig. 2b for different values of n. It is seen that the deviation from the parabolic kinetics increases with time (as also seen from Fig. 2a), and this increase is more profound for smaller particle size and n deviating significantly from 1. This deviation should be taken into account if the permeability values are calculated from the apparent parabolic constants. Note that the initial value of the apparent parabolic constant does not depend on the reinforcement particle size or rate of the reinforcement oxidation, as is expected if the process is entirely controlled by the diffusion through the oxidized layer. If no/ns/m, then n/1 and bn /an /1 [17], in which case the preceding equations simplify significantly: zc RPo Ps (24) and Kp2Po(pO2 ) 1=m aCsfs (25) For large z the kinetics in this case is true parabolic and the process is entirely controlled by the diffusion through the oxidized layer. It is intuitively clear that if there is no reaction between the matrix and the product of the reinforcement oxidation, the oxidation mode I can only occur if Ps is considerably higher than oxygen permeability of the matrix, Pm. In this case, Po can not be greater than Ps, and therefore, in normal conditions the value of zc will be of the same order or less as the size of the reinforcement particles, R (Eq. (21)). This means that in practice it will not be possible to observe the linear stage experimentally. However, a reaction between the matrix and the product of the reinforcement oxidation may have a profound effect on the kinetics of the oxidation process, as is the case for the oxidation of SiC in A12O3 and mullite/ZrO2 matrices. For instance, the new phase can have the permeability value significantly higher than Ps, particularly if it is a glassy phase (aluminosilicate). In such a case, both the process of SiC oxidation and oxygen diffusion through the oxidized layer can be considerably accelerated [8,9,15,18,19] (note that in this case Ps in the preceding equations must refer to the permeability not of silica, but that of the aluminosilicate glass). In this case, the transition from linear to nearly parabolic kinetics will occur at larger z, and the linear stage can possibly be observed experimentally. On the other hand, in matrices containing ZrO2, formation of zircon, ZrSiO4, can occur. This reaction is known to cause a very sharp Fig. 2. Deviation from the parabolic law (a) and the dependence of the apparent parabolic constant Kp on the particle size (b) for n "/1. P. Mogilevsky, A. Zangvil / Materials Science and Engineering A354 (2003) 58/66 61
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) becomes
decrease 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 parameters 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 thereafter, 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 composites, 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) 58/66
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 to
Kp2Ps(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
P. Mogilersky, 4. Zanguil Materials Science and Engineering 4354(2003)58-66 Volume fraction, f 0.5 0 0.08 -------- →==-- ●-1375°C -■--1575°C (1f Volume fraction, f 0.5 0.1 1500°C 所 Fig 3. Dependence of the parabolic growth constant Kp on the volume fraction of the reinforcement in the studies [8](a)and [6](b). The parameter f s was calculated assuming k=2.2 for the transformation of Sic into amorphous silica. p~(po0y-6)≈(p2 0.146 (36) indicate for the aluminosilicate glass the value of n in the range 4.2-3.3, respectively Luthra and Park [8] studied the oxidation kinetics in flowing oxygen and Ar-1%O2 mixture. The exact effect of the partial oxygen pressure in this study is not 4. Conclusions possible to predict, since the parameter n for the aluminosilicate liquid, which apparently was the main a quantitative model for the oxidation mode I of path for oxygen diffusion in that study, is unknown oxide CMCs was developed, which allows an analysis of Without the reaction between alumina and silica, Eq. the oxidation kinetics, depending on the oxygen perme- 6) would predict the ratio of the parabolic constants ability values, external oxygen pressure, and structural for these two atmospheres close to 2. Experimentally, characteristics such as phase composition of the oxida- however, the parabolic constant ratio about 3-4 ticle eported, which, according to the present model, would size and volume fraction. For systems where such data
Kp(pO2 ) (1(5=6) b6 ) :(pO2 ) 0:146 (36) Luthra and Park [8] studied the oxidation kinetics in flowing oxygen and Ar/1% O2 mixture. The exact effect of the partial oxygen pressure in this study is not possible to predict, since the parameter n for the aluminosilicate liquid, which apparently was the main path for oxygen diffusion in that study, is unknown. Without the reaction between alumina and silica, Eq. (36) would predict the ratio of the parabolic constants for these two atmospheres close to 2. Experimentally, however, the parabolic constant ratio about 3/4 was reported, which, according to the present model, would indicate for the aluminosilicate glass the value of n in the range 4.2/3.3, respectively. 4. Conclusions A quantitative model for the oxidation mode I of oxide CMCs was developed, which allows an analysis of the oxidation kinetics, depending on the oxygen permeability values, external oxygen pressure, and structural characteristics such as phase composition of the oxidation product, phase distribution, reinforcement particle size and volume fraction. For systems where such data Fig. 3. Dependence of the parabolic growth constant Kp on the volume fraction of the reinforcement in the studies [8] (a) and [6] (b). The parameter fs was calculated assuming k/2.2 for the transformation of SiC into amorphous silica. 64 P. Mogilevsky, A. Zangvil / Materials Science and Engineering A354 (2003) 58/66
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/(2v2
are 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 significantly 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 parameters as volume fraction of the reinforcement, depending 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 particles, 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 substrates. 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 reinforcement. 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
P. Mogilersky, 4. Zanguil Materials Science and Engineering 4354(2003)58-66 which,in combination with B yields R=k Ro/(22) 9 M. Backhaus-Ricoult, J. Am. Ceram Soc. 74(1991)1793. l0 CC. Lin, Ph D. thesis, University of Illinois at Urbana-Cham The results of the above discussion are summarized in paign, 1991 Table al [ A. Zangvil, Y Xu, G. Fu, Ceram. Trans. 48(1994)1003. [12 C.Y. Tsai, CC. Lin, J. Am. Ceram Soc. 81(1998)3150. [13] C.Y. Tsai, CC. Lin, A. Zangvil, A.K. Li, J. Am. Ceram Soc. 81 (1998)2413 References [14 CC. Lin, A. Zangvil, R Ruh, Acta Mater. 47(1977)1999 15]CC Lin, A Zangvil, R Ruh, J Am Ceram Soc. 82(1999)283 F Becher, G.C. Wei, J. Am. Ceram Soc. 67(1984)C-267. 116 C.C. Lin, A. Zangvil, R Ruh, J Am Ceram Soc. 83 (2000)1797 22] C. Nischik, M. M. Seibold, N.A. Travitzky, N. Claussen, J. Am. [17 P. Mogilevsk Ceram.Soc.74(1991)2464. [18R F. Davis, J.A. Pask, J Am Ceram Soc. 55(1972)525 3Y..s. Chou, D.J. Green, J. Am. Ceram Soc. 75(1992)3346 [19 E.J. Opila, J. Am. Ceram Soc. 77(1994)730 [4Y.S Chou, D.J. Green, J. Am. Ceram Soc. 76(1993)14 o K.M. Trappen, R.A. Eppler, J. Am. Ceram Soc. 72(1989)882. 5Y..S. Chou, D.J. Green, J Am Ceram Soc. 76(1985)1993. 1] L.M. Manocha, S M. Manocha, Carbon 33(1995)435 [6 F. Lin, T. Marieb, A. Morrone, S Nutt, MRS Symp. Proc. 120 [22 O. Yamamoto, T Sasamoto, M. Ingaki, Carbon 33(1995)359 23J. Philibert, Atom movements. Diffusion and mass transport in 门CC.Lin, Stability and Oxidation ds, Les Editions de Physique, 1991 ehaviour of d at the 24 W.D. Kingery, Introduction to Ceramics, Wiley, 1960, p. 236 91st Annual Me c Ceramic 5 F.J. Norton, Nature 191(1961)701. lis, IN, 23-27 April 1989, Engineering Ceramic Division, Paper [26]R H. Doremus, J Phys. Chem. 80(1976)1773 27 M. Pirani, I. Sandor, J Inst Metals 73(1947)384. [8K.L Luthra, H D. Park, J Am Ceram Soc. 73(1990)1014 228M. Andrews, S. Dushman, J. Phys. Chem. 29(1925)462
which, in combination with b yields R/k1/2R0/(2/ ffiffiffi 2 p ): The results of the above discussion are summarized in Table A1. References [1] F. Becher, G.C. Wei, J. Am. Ceram. Soc. 67 (1984) C-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 (1985) 1993. [6] F. Lin, T. Marieb, A. Morrone, S. Nutt, MRS Symp. Proc. 120 (1988) 323. [7] C.C. Lin, A. Zangvil, R. Ruh, Phase Stability and Oxidation Behaviour of Mullite/SiC Whisker Composites, Presented at the 91st Annual Meeting of the American Ceramic Society, Indianapolis, IN, 23/27 April 1989, Engineering Ceramic Division, Paper 7-JII-89. [8] K.L. Luthra, H.D. Park, J. Am. Ceram. Soc. 73 (1990) 1014. [9] M. Backhaus-Ricoult, J. Am. Ceram. Soc. 74 (1991) 1793. [10] C.C. Lin, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1991. [11] A. Zangvil, Y. Xu, G. Fu, Ceram. Trans. 48 (1994) 1003. [12] C.Y. Tsai, C.C. Lin, J. Am. Ceram. Soc. 81 (1998) 3150. [13] C.Y. Tsai, C.C. Lin, A. Zangvil, A.K. Li, J. Am. Ceram. Soc. 81 (1998) 2413. [14] C.C. Lin, A. Zangvil, R. Ruh, Acta Mater. 47 (1977) 1999. [15] C.C. Lin, A. Zangvil, R. Ruh, J. Am. Ceram. Soc. 82 (1999) 2833. [16] C.C. Lin, A. Zangvil, R. Ruh, J. Am. Ceram. Soc. 83 (2000) 1797. [17] P. Mogilevsky, A. Zangvil, Mater. Sci. Eng. A262 (1999) 16. [18] R.F. Davis, J.A. Pask, J. Am. Ceram. Soc. 55 (1972) 525. [19] E.J. Opila, J. Am. Ceram. Soc. 77 (1994) 730. [20] K.M. Trappen, R.A. Eppler, J. Am. Ceram. Soc. 72 (1989) 882. [21] L.M. Manocha, S.M. Manocha, Carbon 33 (1995) 435. [22] O. Yamamoto, T. Sasamoto, M. Ingaki, Carbon 33 (1995) 359. [23] J. Philibert, Atom movements. Diffusion and mass transport in solids, Les E` ditions de Physique, 1991. [24] W.D. Kingery, Introduction to Ceramics, Wiley, 1960, p. 236. [25] F.J. Norton, Nature 191 (1961) 701. [26] R.H. Doremus, J. Phys. Chem. 80 (1976) 1773. [27] M. Pirani, I. Sandor, J. Inst. Metals 73 (1947) 384. [28] M. Andrews, S. Dushman, J. Phys. Chem. 29 (1925) 462. 66 P. Mogilevsky, A. Zangvil / Materials Science and Engineering A354 (2003) 58/66