《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_fiber110

Availableonlineatwww.sciencedirect.com SCIENCE DIRECT E噩≈S ELSEVIER Joumal of the European Ceramic Society 25(2005)1699-1704 www.elsevier.comlocate/jeurcerams Ceramic matrix composite corrosion models Klaus. nickel University Tubingen, Institute for Geosciences, Applied Mineralogy. WilheImstr: 56. D-72074 Tuebingen, Germany ailable online 8 January 2005 Abstract This paper discusses physical and empirical models for the description of corrosion processes. Physical strict models are advantageous for simple ceramics and simple composites. In additive-containing ceramics and composites with such matrices, the transport properties will vary with time, in these cases simple physical models alone are not adequate. The use of an empirical equation of the form x kit+Kvi+ klog log(r), fitted by simple multiple linear regression, is capable to describe many versions of corrosion processes, if k allowed to become negative or positive(: scale thickness or specific mass change, t: time). The equation is recommended for complex cases, but the variability of the corrosion function makes it often necessary to have more than one parameter of evaluation of the material to deduce the most important engineering parameter, penetration depths O 2004 Elsevier Ltd. All rights reserved Keyords: Corrosion; Kinetics; Composite 1. Introduction partial pressures of oxygen calculated for the coexistence of substrate and scale material at high temperatures. An exam- Ceramic matrix composites are by definition multiphase ple is the Po, of approximately 10-28 bar at 1000.C for an materials. Every material has a specific reaction to the envi- equilibrium between Si and Sioz, calculated by a thermo- ronment it is exposed to and therefore the behavior is bound chemical program or straight from the tabulated values of to be more complex, unless a component may be treated as In this paper, we review a number of models for the de- scription of corrosion processes in view of their usefulness This situation is a classic diffusion problem for oxygen through a growing layer of silica. Relating the flux of the oxi- dant through the scale to reflect the growth in scale thickness x with time t, we derive the basic parabolic relation already 2. Models for simple mass gain or loss from Ficks first law(: effective diffusion coefficient, We begin the discussion with models for simple one-phase oxygen concentration at the scale top= solubility) materials. For oxidation as a special case of the corrosion of dr=D o=0 →|xdx=/(Dco)dr non-oxides, the modeling is often done by assuming a simple parabolic law. A simple parabolic law is the result of assum ing steady-state conditions with a constant oxygen partial (2) pressure in the atmosphere and at the gas-scale interface ap- proaching zero at the scale-substrate interface. The last as- Incorporating the factor 2 into kp we have x'=kpt,the sumption seems justified in view of the very low equilibrium simplest form of the parabolic law, which is often used in the analysis of oxidation and corrosion data Tel:+4970712976802. Deal and Grove showed that this analysis does not take E-mail address: klaus. nickelauni-tueb into account the effects of gas phase transport and chemical 0955-2219/S- see front matter 2004 Elsevier Ltd. All rights reserved doi: 10.1016/j- jeurceramsoc. 2004. 12.010

Journal of the European Ceramic Society 25 (2005) 1699–1704 Ceramic matrix composite corrosion models Klaus G. Nickel∗ University T ¨ubingen, Institute for Geosciences, Applied Mineralogy, Wilhelmstr. 56, D-72074 Tuebingen, Germany Available online 8 January 2005 Abstract This paper discusses physical and empirical models for the description of corrosion processes. Physical strict models are advantageous for simple ceramics and simple composites. In additive-containing ceramics and composites with such matrices, the transport properties will vary with time; in these cases simple physical models alone are not adequate. The use of an empirical equation of the form x = klt + k p √t + klog log(t), fitted by simple multiple linear regression, is capable to describe many versions of corrosion processes, if kl is allowed to become negative or positive (x: scale thickness or specific mass change, t: time). The equation is recommended for complex cases, but the variability of the corrosion function makes it often necessary to have more than one parameter of evaluation of the material to deduce the most important engineering parameter, penetration depths. © 2004 Elsevier Ltd. All rights reserved. Keywords: Corrosion; Kinetics; Composite 1. Introduction Ceramic matrix composites are by definition multiphase materials. Every material has a specific reaction to the envi￾ronment it is exposed to and therefore the behavior is bound to be more complex, unless a component may be treated as inert. In this paper, we review a number of models for the de￾scription of corrosion processes in view of their usefulness for composite evaluation. 2. Models for simple mass gain or loss We begin the discussion with models for simple one-phase materials. For oxidation as a special case of the corrosion of non-oxides, the modeling is often done by assuming a simple parabolic law. A simple parabolic law is the result of assum￾ing steady-state conditions with a constant oxygen partial pressure in the atmosphere and at the gas–scale interface ap￾proaching zero at the scale–substrate interface. The last as￾sumption seems justified in view of the very low equilibrium ∗ Tel.: +49 7071 2976802. E-mail address: klaus.nickel@uni-tuebingen.de. partial pressures of oxygen calculated for the coexistence of substrate and scale material at high temperatures. An exam￾ple is the PO2 of approximately 10−28 bar at 1000 ◦C for an equilibrium between Si and SiO2, calculated by a thermo￾chemical program3 or straight from the tabulated values4 of the reaction: Si + O2 ⇔ SiO2 (1) This situation is a classic diffusion problem for oxygen through a growing layer of silica. Relating the flux of the oxi￾dant through the scale J to reflect the growth in scale thickness x with time t, we derive the basic parabolic relation already from Fick’s first law (D: effective diffusion coefficient, c0: oxygen concentration at the scale top = solubility): J = dx dt = Dc0 − 0 x ⇒
x dx =
(Dc0) dt ⇒ x2 = 2kpt (2) Incorporating the factor 2 into kp we have x2 = kpt, the simplest form of the parabolic law, which is often used in the analysis of oxidation and corrosion data. Deal and Grove5 showed that this analysis does not take into account the effects of gas phase transport and chemical 0955-2219/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jeurceramsoc.2004.12.010

K.G. Nickel/Journal of the European Ceramic Sociery 25(2005)1699-1704 reaction rate, which must be acting and which have to be con- Particulate composites, in which the reinforcing parti- trolling parameters at least in the beginning of an oxidation cles are oxidized within an oxide matrix, were addressed by process. Their detailed analysis yielded the equation ogilevsky and Zan anvil. ,7 Their analysis introduced an ef- fective grain size of the particles R and the value x is now x'+Ax= b(t +r) ( the propagation of the oxidation front divided by R.The in which the factors A and B incorporate the physical transport oxygen partial pressures and particle fraction and are thus not parameter and t represents the shift in the time coordinate identical to A and B of Deal and grove,but the form is very which allows for the presence of a pre-existing oxide scale similar to Eq(3) of thickness x. Numerically it was found that at long times the behavior approached simple parabolic relations, while at short times we have almost linear kinetics x+Ar(l+bn) Bt For practical use we can transform Eq (3)by the definition Eq. (5) contains the parameter bn, which adjusts for non- of parabolic and linear rate constant kp=B and k=B/A to molecular diffusion through the oxide matrix For the purpose of this paper, it is only necessary to ob- serve the change of the function of corrosion front with time kp ky (4) in differing situations. This is done in Fig. 2, where the likely values for bn act on a set of fixed arbitrarily chosen A=0.1 This is a useful analytical form of the Deal and Grove re- and B=0.5 parameters with R=l. A value of bn=l yields a lationship, because now t, 1/kp and I/k may be obtained quadratic relation similar to Deal and Grove's Eq(3);chang from a simple multiple linear regression. The action of the ing bn to lower plausible values steepens the function parameters is illustrated in Fig. 1, where an arbitrarily chosen It should be noted that the deviation from parabolic value of 0.5 um2/cm is linked to widely varying values of k. kinetics with decreasing bn is towards linear behavior and It is obvious from Fig. 1 that k values smaller than kp do increasing absolute x and values, i.e. acceleration of the cause growth retardation and significant linearization of the corrosion process. This is very different from the processes curve, while high k values induce an almost perfect parabolic described in Fig. I, where a linearization meant process retar- behavior. The higher h is the more it is insignificant. Physi dation cally this makes sense because the chemical reaction rate has The corrosion of ceramics and glasses by liquids is often to be high to allow the scale-substrate interface to get towards modeled empirically. The problem encountered is that two equilibrium with very little oxygen processes, the leaching of components of a glass phase and The negative counterpart of this relation, the loss of mate the complete dissolution of a glass phase, occur simular al to the atmosphere in a composite, has been investigated ously at a given time. Dissolution in a steady-state situation with the aid of Eq. (4)by Eckel et al. b The modeled com- Is usually limited by the dissolution(reaction)rate, which posite was a reactive carbon fiber within an inert matrix of should induce a linear process with time. Leaching produces alumina. In this case t becomes 0 and the parabolic rate con- a residue and diffusion through this growing residue can be- stant is predictable from known gas kinetics. The diameter of come rate controlling. When both rates operate at comparable the oxidized carbon fiber is decisive for this situation: small velocities, the dissolution is reducing the residue thickness fibers at relatively low temperatures had a k small enough to Hence, a deviation from parabolic kinetics occurs. The equa- have a significant effect with process retardation and kinetics becoming linear bo klmg/(cm:h)l *1000 Fig. 2. Influence of varying bn in Eq. (5)for fixed values of A and B from Fig. 1. Influence of varying h for a fixed value of kp=0.5 um-/cr the model of Ref. I

1700 K.G. Nickel / Journal of the European Ceramic Society 25 (2005) 1699–1704 reaction rate, which must be acting and which have to be con￾trolling parameters at least in the beginning of an oxidation process. Their detailed analysis yielded the equation x2 + Ax = B(t + τ) (3) in which the factorsAandBincorporate the physical transport parameter and τ represents the shift in the time coordinate, which allows for the presence of a pre-existing oxide scale of thickness xi. Numerically it was found that at long times the behavior approached simple parabolic relations, while at short times we have almost linear kinetics. For practical use we can transform Eq.(3) by the definition of parabolic and linear rate constant kp = B and kl = B/A to derive x2 kp + x kl − τ = t (4) This is a useful analytical form of the Deal and Grove re￾lationship, because now τ, 1/kp and 1/kl may be obtained from a simple multiple linear regression. The action of the parameters is illustrated in Fig. 1, where an arbitrarily chosen value of 0.5
m2/cm is linked to widely varying values of kl. It is obvious from Fig. 1 that kl values smaller than kp do cause growth retardation and significant linearization of the curve, while high kl values induce an almost perfect parabolic behavior. The higher kl is the more it is insignificant. Physi￾cally this makes sense because the chemical reaction rate has to be high to allow the scale-substrate interface to get towards equilibrium with very little oxygen. The negative counterpart of this relation, the loss of mate￾rial to the atmosphere in a composite, has been investigated with the aid of Eq. (4) by Eckel et al.6 The modeled com￾posite was a reactive carbon fiber within an inert matrix of alumina. In this case τ becomes 0 and the parabolic rate con￾stant is predictable from known gas kinetics. The diameter of the oxidized carbon fiber is decisive for this situation: small fibers at relatively low temperatures had a kl small enough to have a significant effect with process retardation and kinetics becoming linear. Fig. 1. Influence of varying kl for a fixed value of kp = 0.5
m2/cm. Particulate composites, in which the reinforcing parti￾cles are oxidized within an oxide matrix, were addressed by Mogilevsky and Zangvil.1,7 Their analysis introduced an ef￾fective grain size of the particles R and the value x is now the propagation of the oxidation front z divided by R. The parameters A and B in their equation contain permeabilities, oxygen partial pressures and particle fraction and are thus not identical to A and B of Deal and Grove,5 but the form is very similar to Eq. (3): x + Ax(1+bn) = Bt R2 (5) Eq. (5) contains the parameter bn, which adjusts for non￾molecular diffusion through the oxide matrix. For the purpose of this paper, it is only necessary to ob￾serve the change of the function of corrosion front with time in differing situations. This is done in Fig. 2, where the likely values for bn act on a set of fixed arbitrarily chosen A = 0.1 and B = 0.5 parameters with R = 1. A value of bn = 1 yields a quadratic relation similar to Deal and Grove’s Eq. (3); chang￾ing bn to lower plausible values steepens the function. It should be noted that the deviation from parabolic kinetics with decreasing bn is towards linear behavior and increasing absolute x and z values, i.e. acceleration of the corrosion process. This is very different from the processes described in Fig. 1, where a linearization meant process retar￾dation. The corrosion of ceramics and glasses by liquids is often modeled empirically.8 The problem encountered is that two processes, the leaching of components of a glass phase and the complete dissolution of a glass phase, occur simultane￾ously at a given time. Dissolution in a steady-state situation is usually limited by the dissolution (=reaction) rate, which should induce a linear process with time. Leaching produces a residue and diffusion through this growing residue can be￾come rate controlling. When both rates operate at comparable velocities, the dissolution is reducing the residue thickness. Hence, a deviation from parabolic kinetics occurs. The equa￾Fig. 2. Influence of varying bn in Eq. (5) for fixed values of A and B from the model of Ref. 1

KG. Nickel/ournal of the European Ceramic Sociery 25 (2005)1699-1704 0,001 0,01 400600 020406080100 3000 Time(.u) 2500 time [h] Fig. 4. Effect of increasing k values on an arbitrary chosen value of kp in Eqs. (7)(left)and (8)(right) Fig 3. Effect of increasing ki values on an arbitrary chosen value of kp in Eq(6) retardation, but at long times or higher k it turns into near linear mass loss. This behavior was termed tion, which is often successful in describing this behavior, and is physically well constrained As examples for fiber-reinforced ceramics we have in x=k*t+k√n SiC/C and Sic/BN systems the recession of fibers in com bination with SiC oxidation. The formation of annular holes Eq(6)looks very similar to Eq (4), but this is deceptive. In around fibers from the active oxidation of their interface ma Eq (6)the dependent variable of fitting is x, while in Eq (4) terials is retarded and eventually stopped by silica growth, it is t. In Eq- (6)any increase in k will automatically increase which seals the pathway for oxygen. Models for these cases the total ofx, i.e. it accelerates the process(Fig 3), while the have been presented by. 2, 3 The latter calculates the sealing opposite is observed in Eq (4)(Fig. 1). The deviation from time by an application of Eq (3), the former via numerical ( parabolic kinetics is here more akin to the case of Ea integration of differential equations We have presented a modeling scheme, which is an exten- sion of the empirical model of Eq. (6): 14 3. Models for combined mass gain and loss x=Kt+kpVt+ klog log(r) In this equation, there is a third term(hog). The physical basis In composites we often have the problem of simultaneous behind this term is the behavior of materials with asymptotic rocesses acting in differing directions. An example is the passivation. If a corrosion product is completely blocking its evaporation of a scale material, which is formed on oxidation substrate from further attack, we have a simple reduction in of a substrate. Thus, growth and recession are the opposing effective exposed area approaching 100%. Mathematically factors in terms of a scale thickness and mass gain and loss this is a retardation function with a constant scaled by the in terms of the total mass function of time It has been shown that the so called Tedmon equation is One or two constants in Eq (9)can become zero for a given capable of modeling this behavior. 9- The two forms of the data set. In these cases Eq. (9)reduces to simpler forms of Tedmon equation for scale thickness and mass change are corrosion equations. In view of the reality of corrosion processes with combined mass gain and loss, Eq( 9)may be used including negative 2( p alues for k. With this change to the original proposal the function is very variable in fitting corrosion processes. Typi- cal variations are shown in Fig. 5 △U1 △ By varying the parameters it is possible to describe ac celeration or retardation of the process by Eq (9), including paralinear characteristics In those equations a and B are stoichiometric factors, which account for the mass balance of the appropriate reactions. The form of the functions with varying k is illustrated in Fig. 4. 4. Discussion In Fig. 4 it can be seen that now an increase in k implies a different behavior: for scale thickness it means a retardation The physical models reviewed above work well for the of process velocity, but this time approaching an asymptotic simple systems for which they have been developed and are behavior. Mass change with small k is also seen as a process certainly of great value to find the physical border param

K.G. Nickel / Journal of the European Ceramic Society 25 (2005) 1699–1704 1701 Fig. 3. Effect of increasing k∗ l values on an arbitrary chosen value of k∗ p in Eq. (6). tion, which is often successful in describing this behavior, is x = k∗ l t + k∗ p √t (6) Eq. (6) looks very similar to Eq. (4), but this is deceptive. In Eq. (6) the dependent variable of fitting is x, while in Eq. (4) it ist. In Eq. (6) any increase in k∗ l will automatically increase the total of x, i.e. it accelerates the process (Fig. 3), while the opposite is observed in Eq. (4) (Fig. 1). The deviation from simple parabolic kinetics is here more akin to the case of Eq. (5). 3. Models for combined mass gain and loss In composites we often have the problem of simultaneous processes acting in differing directions. An example is the evaporation of a scale material, which is formed on oxidation of a substrate. Thus, growth and recession are the opposing factors in terms of a scale thickness and mass gain and loss in terms of the total mass function. It has been shown that the so called Tedmon equation is capable of modeling this behavior.9–11 The two forms of the Tedmon equation for scale thickness and mass change are t = k p 2(k l ) 2 −2k l x k p − ln 1 − 2k l x k p  (7) and t= α2kp 2k2 l  −2klw1 αkp − ln  1−2klw1 αkp   −∆w2 βkl (8) In those equations α and β are stoichiometric factors, which account for the mass balance of the appropriate reactions. The form of the functions with varying kl is illustrated in Fig. 4. In Fig. 4 it can be seen that now an increase in kl implies a different behavior: for scale thickness it means a retardation of process velocity, but this time approaching an asymptotic behavior. Mass change with small kl is also seen as a process Fig. 4. Effect of increasing kl values on an arbitrary chosen value of k∗ p in Eqs. (7) (left) and (8) (right). retardation, but at long times or higher kl it turns into near linear mass loss. This behavior was termed “para-linear”10 and is physically well constrained. As examples for fiber-reinforced ceramics we have in SiC/C and SiC/BN systems the recession of fibers in com￾bination with SiC oxidation. The formation of annular holes around fibers from the active oxidation of their interface ma￾terials is retarded and eventually stopped by silica growth, which seals the pathway for oxygen. Models for these cases have been presented by.12,13 The latter calculates the sealing time by an application of Eq. (3), the former via numerical integration of differential equations. We have presented a modeling scheme, which is an exten￾sion of the empirical model of Eq. (6): 14 x = k l t + k p √t + klog log(t) (9) In this equation, there is a third term (klog). The physical basis behind this term is the behavior of materials with asymptotic passivation. If a corrosion product is completely blocking its substrate from further attack, we have a simple reduction in effective exposed area approaching 100%. Mathematically this is a retardation function with a constant scaled by the logarithm of time. One or two constants in Eq.(9) can become zero for a given data set. In these cases Eq. (9) reduces to simpler forms of corrosion equations. In view of the reality of corrosion processes with combined mass gain and loss, Eq. (9) may be used including negative values for kl. With this change to the original proposal14 the function is very variable in fitting corrosion processes. Typi￾cal variations are shown in Fig. 5. By varying the parameters it is possible to describe ac￾celeration or retardation of the process by Eq. (9), including paralinear characteristics. 4. Discussion The physical models reviewed above work well for the simple systems for which they have been developed and are certainly of great value to find the physical border param-

K.G. Nickel/Journal of the European Ceramic Sociery 25(2005)1699-1704 000000000 0 EE品55E 10 0, 1F.Maeda data q得4}eq{9} 1500 2500 020040060080010001200 me [h] Time [hI kp and klog y.7g sing k' values on an arbitrary chosen set of values of Fig. 5. Effect of AIN-SiC-ZrB2 composites at high temperature(this volume) eter for the best use of these materials under those condi- and shows the strong deviation from parabolic behavior tions. However, if we deal with ceramic matrix composites There have been other attempts to model non-parabolic of higher complexity they are not likely to give reliable an- behavior. In particular, Nygren and coworkers 21, 2 have de- swers veloped models which include an arctan-function of time Most ceramics based on Si3 N4 and an increasing num- This was successful for a number of siaion ceramics and ber of liquid phase sintered SiC ceramics contain additive allowed them to model retardation problems. However, with which change the oxidation behavior drastically. There is a more data available, more parameters were added to their great wealth of literature data, which is collected in text equation, because in those ceramics linearization can also books on the simple base ceramics and illustrates the decisive 15-18 occur. A later version was then23 fluence However, from those textbook data it is also clear that x=aarctan b(t+)+cr+g+kt (10) for many cases the behavior is distinctly non-parabolic. The physical reason behind this is not just found in the deduction Eq. (10) has similarities to Eq (9)by consisting of terms of Eq (3)by Ref. 5. The main reason is the change of prop- for linear and parabolic terms plus a retardation term. Eq erties of scales with time. In additive-containing systems we (10) has more constants(a, b, c, q and k) to be fitted,which have a continuous change of chemistry and crystallization makes the fitting procedure more cumbersome Ogbuji- in- state with time and we may additionally find self-destruction of the protective character of a scale with time. None of the found that an increasing number of fit parameters did not help physical models described above is capable of handling these to get a better agreement with the data and that Eq (9)was the most consistently useful approach. In earlier papers, 20 we have attempted to model some In particulate composites with a matrix containing sinter features on a physical basis. These studies showed how the ng additives, such as Si3N4-TiN composites, the situation changes introduced by crystallization and scale chemistry does not become simpler. On top of all the complications change influence the function of mass change or scale growth with time. The complex mathematical solution gave curves which showed both the possibility of deviation from parabol icity towards a linear acceleration as well as towards asymp- totic retardation, because the effective diffusion coefficient is Experimental data Eq.(9)is capable of reproducing these features. Fig. 6 shows that long-time experiments on silicon nitride ceramics do deviate from a simple parabolic behavior and that it much better described by Eq(9)than by Eq(4). This not Multiple regression surprising, because the model according to Eq(4)allows only for a time shift to reflect an initial first scale. Eq.(9) correctly describes the early relative fast period and the de- 400 1600 celeration process A recent example for the application of Eq (9)to compos Fig. 7. Application of Eq (9)to AlN-SiC-ZrB2 composites(Brach et al ites is shown in Fig. 7. The example comes from a paper on this volume

1702 K.G. Nickel / Journal of the European Ceramic Society 25 (2005) 1699–1704 Fig. 5. Effect of increasing k l values on an arbitrary chosen set of values of k p and klog in Eq. (9). eter for the best use of these materials under those condi￾tions. However, if we deal with ceramic matrix composites of higher complexity they are not likely to give reliable an￾swers. Most ceramics based on Si3N4 and an increasing num￾ber of liquid phase sintered SiC ceramics contain additives, which change the oxidation behavior drastically. There is a great wealth of literature data, which is collected in text￾books on the simple base ceramics and illustrates the decisive influence.15–18 However, from those textbook data it is also clear that for many cases the behavior is distinctly non-parabolic. The physical reason behind this is not just found in the deduction of Eq. (3) by Ref. 5. The main reason is the change of prop￾erties of scales with time. In additive-containing systems we have a continuous change of chemistry and crystallization state with time and we may additionally find self-destruction of the protective character of a scale with time. None of the physical models described above is capable of handling these processes. In earlier papers19,20 we have attempted to model some features on a physical basis. These studies showed how the changes introduced by crystallization and scale chemistry change influence the function of mass change or scale growth with time. The complex mathematical solution gave curves, which showed both the possibility of deviation from parabol￾icity towards a linear acceleration as well as towards asymp￾totic retardation, because the effective diffusion coefficient is changing with time. Eq. (9) is capable of reproducing these features. Fig. 6 shows that long-time experiments on silicon nitride ceramics2 do deviate from a simple parabolic behavior and that it is much better described by Eq. (9) than by Eq. (4). This not surprising, because the model according to Eq. (4) allows only for a time shift to reflect an initial first scale. Eq. (9) correctly describes the early relative fast period and the de￾celeration process. A recent example for the application of Eq. (9) to compos￾ites is shown in Fig. 7. The example comes from a paper on Fig. 6. Long-time experimental data of a Si3N4 ceramic (M of Ref. 2 and fits by Eqs. (4) and (9)). AlN–SiC–ZrB2 composites at high temperature (this volume) and shows the strong deviation from parabolic behavior. There have been other attempts to model non-parabolic behavior. In particular, Nygren and coworkers21,22 have de￾veloped models which include an arctan-function of time. This was successful for a number of SiAlON ceramics and allowed them to model retardation problems. However, with more data available, more parameters were added to their equation, because in those ceramics linearization can also occur. A later version was then23 x = a arctan b(t + q) + c t + q + klt (10) Eq. (10) has similarities to Eq. (9) by consisting of terms for linear and parabolic terms plus a retardation term. Eq. (10) has more constants (a, b, c, q and kl) to be fitted, which makes the fitting procedure more cumbersome. Ogbuji24 in￾vestigated the problem of non-parabolic oxidation of SiC and found that an increasing number of fit parameters did not help to get a better agreement with the data and that Eq. (9) was the most consistently useful approach. In particulate composites with a matrix containing sinter￾ing additives, such as Si3N4–TiN composites, the situation does not become simpler. On top of all the complications Fig. 7. Application of Eq. (9) to AlN–SiC–ZrB2 composites (Brach et al., this volume)

K.G. Nickel/Journal of the European Ceramic Sociery 25 (2005)1699-1704 1703 discussed above for the matrix ceramic there is an additional Allowing the linear term in Eq. (9)to become negative it oxidation of the reinforcing phase, in this case to yield TiO handles a great variety of differing mechanisms and processes and an initial amount of this oxide present in the starting ma- and is the preferred tool also for composite corrosion terial. It is highly unlikely that TiO2 is not interacting with the oxides of the matrix system and the newly formed Ox- ides from the matrix oxidation, because there is a solubility for it. TiO2 is also a known opacifier, i. e. an agent to induce Acknowledgements crystallization of silicate glasses. Therefore, it cannot be ex- ed that We gratefully acknowledge funding by the Deutsche which is the base for all the physical models, holds for such Forschungsgemeinschaft (DFG) under contract number Even stronger effects are expected for composites with Corrosion"HPRN-Ct-2000-00044) econd phases, which become volatile upon oxidation. Ex- amples are the boride reinforcement systems. Certainly the paralinear behavior of oxide formation and evaporation can References be handled by physically strict models such as Eqs. (7)and ( 8). But in a reactive matrix with silicate glass and/or borate I. Mogilevsky, P. and Zangvil, A, Kinetics of oxidation in oxide ceramic formation the diffusion coefficients must change with time matrix composites. Mater: Sci. Eng, 2003, A354, 58-66. and temperature. Examples for strongly non-parabolic cor- 2. Maeda, M., Nakamura, K. and Yamada, M., Oxidation resistance of rosion-both in oxidation as well as in liquid corrosion-can silicon nitride ceramics with various additives. J. Mater: Sci., 1990, 5,3790-3794 be found in this volume 3. Hack, K, ChemSage-a computer program for the calculation of As long as such complex changes cannot be treated ad complex chemical equilibria. Met. Trans., 1990, 21B, 1013-1023 equately in a strict physical model, their use is no real ad 4. Chase, M. W, Davies, C. A, Downey, J.R, Frurip, D J, McDonald, vantage over empirical models in those systems. It is thus R. A. and Syverud, A. N, JANAF Thermochemical tables-third dition. J. Phys. Chem. Ref. Data, 1985, 14(Suppl. 1) concluded that Eq. (9)is a simple and robust empirical ap- 5. Deal, B. E. and Grove, A.S., General relationship for the proach to model the corrosion behavior and should be useful mal oxidation of silicon. J. Appl. Phys., 1965, 36(12) for many ceramics and composites thereof The freedom of shapes of corrosion curves implied by 6. Eckel, A J, Cawley, J D and Parthsarathy, T. A, Oxidation kinetics the empirical model necessitates in practice that we need more independent information about the process to be able 7. mogilevsky, P and Zangvil, A, modeling to evaluate it correctly. The translation of mass data into SiC-reinforced ceramic matrix composite er: Sci. Eng, 1999 scale thickness, penetration depths, and component size A262,16-24 ice versa is not straightforward for the 8. Clark, D. E. and Zoitos, B. K, Corrosion of Glass, Ceramics an Each system has to be calibrated by detailed investiga- Ceramic Superconductors. Noyes Publications, ParkRidge, NJ, USA tions using a multiplicity of investigation techniques, aiming 9. Opila, E I and Jacobson, N.S., Sio (g) formation from Sic in mixed for the most important engineering parameter, penetration xidising-reducing gases. Oxid. Metals, 1995, 44(3/4), 1-17 10. Pila, E.J. and Hann Jr, R. E, Paralinear oxidation of CVD SiC in water vapor. J. Am. Ceram. Soc., 1997, 80(1), 197-205 lI J, Smialek, J. L, Robinson, R. C. Fox, D. S and Jacobson N.s., SiC recession caused by Sio scale volatility under combustion 5. Conclusions conditions: Il, thermodynamics and gaseous-diffusion model. J. Am. Ceram soc.,1999,827),1826-1834. Physically strict models for oxidation and corrosion can 12. Filipuzzl, L. and Naslain, R, Oxidation mechanism and kinetics of be successful for simple ceramics and composites. In the ap- D-SiC/C/SiC composite materials: Il, modeling. J Am Ceram Soc. propriate systems they should be used and will allow the best 1994,77(2),467-480 13. Jacobson, N. S, Morscher, G. N, Bryant, D. R. and Tressler, extrapolation outside of the experimental range R. E, High-temperature oxidation of boron nitride: Il, boron ni Complex systems, in particular those involving additive- tride layers in composites. J. Am. Ceram. Soc., 1999, 82(6), 1473 containing matrices, will have transport properties, which 482. change with time. It is a challenge to handle this in phy 14 K. G, Multiple law modelling for the ical models. First results indicate that this is possible with of Adanced Ceramics-Measurement and d. K complex models and a good deal of detailed information on Nickel. Kluwer Academic Pub., Dordrecht, NI For the purpose of describing, comparing, ranking and 15. Nickel,K. G. and Quirmbach, P, Gask developing materials it is often sufficient- and far less ex- keramische Werkstoffe. In Technische K Werkstoffe, ed pensive in terms of time, effort and money-to use empirical J. Kriegesmann. Koln, Deutscher Wirtscha [chapter 5.4.1.1- models. A relatively simple and robust form of such an em- 16. Gogotsi, Y G and Lavrenko, V. A, Corrosion of High-Performance al model is given in Eq (9) Ceramics. Springer Verlag, Berlin, 1992, p. 181

K.G. Nickel / Journal of the European Ceramic Society 25 (2005) 1699–1704 1703 discussed above for the matrix ceramic there is an additional oxidation of the reinforcing phase, in this case to yield TiO2, and an initial amount of this oxide present in the starting ma￾terial. It is highly unlikely that TiO2 is not interacting with the oxides of the matrix system and the newly formed ox￾ides from the matrix oxidation, because there is a solubility for it. TiO2 is also a known opacifier, i.e. an agent to induce crystallization of silicate glasses. Therefore, it cannot be ex￾pected that the assumption of constant diffusion coefficients, which is the base for all the physical models, holds for such composites. Even stronger effects are expected for composites with second phases, which become volatile upon oxidation. Ex￾amples are the boride reinforcement systems. Certainly the paralinear behavior of oxide formation and evaporation can be handled by physically strict models such as Eqs. (7) and (8). But in a reactive matrix with silicate glass and/or borate formation the diffusion coefficients must change with time and temperature. Examples for strongly non-parabolic cor￾rosion – both in oxidation as well as in liquid corrosion – can be found in this volume. As long as such complex changes cannot be treated ad￾equately in a strict physical model, their use is no real ad￾vantage over empirical models in those systems. It is thus concluded that Eq. (9) is a simple and robust empirical ap￾proach to model the corrosion behavior and should be useful for many ceramics and composites thereof. The freedom of shapes of corrosion curves implied by the empirical model necessitates in practice that we need more independent information about the process to be able to evaluate it correctly. The translation of mass data into scale thickness, penetration depths, and component size or vice versa is not straightforward for the complex cases. Each system has to be calibrated by detailed investiga￾tions using a multiplicity of investigation techniques, aiming for the most important engineering parameter, penetration depth.25 5. Conclusions Physically strict models for oxidation and corrosion can be successful for simple ceramics and composites. In the ap￾propriate systems they should be used and will allow the best extrapolation outside of the experimental range. Complex systems, in particular those involving additive￾containing matrices, will have transport properties, which change with time. It is a challenge to handle this in phys￾ical models. First results indicate that this is possible with complex models and a good deal of detailed information on the systems. For the purpose of describing, comparing, ranking and developing materials it is often sufficient – and far less ex￾pensive in terms of time, effort and money – to use empirical models. A relatively simple and robust form of such an em￾pirical model is given in Eq. (9). Allowing the linear term in Eq. (9) to become negative it handles a great variety of differing mechanisms and processes and is the preferred tool also for composite corrosion. Acknowledgements We gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG) under contract number Ni299/7 and the EU (Research Training Network “Composite Corrosion” HPRN-Ct-2000-00044). References 1. Mogilevsky, P. and Zangvil, A., Kinetics of oxidation in oxide ceramic matrix composites. Mater. Sci. Eng., 2003, A354, 58–66. 2. Maeda, M., Nakamura, K. and Yamada, M., Oxidation resistance of silicon nitride ceramics with various additives. J. Mater. Sci., 1990, 25, 3790–3794. 3. Hack, K., ChemSage—a computer program for the calculation of complex chemical equilibria. Met. Trans., 1990, 21B, 1013–1023. 4. Chase, M. W., Davies, C. A., Downey, J. R., Frurip, D. J., McDonald, R. A. and Syverud, A. N., JANAF Thermochemical tables—third edition. J. Phys. Chem. Ref. Data, 1985, 14(Suppl. 1). 5. Deal, B. E. and Grove, A. S., General relationship for the ther￾mal oxidation of silicon. J. Appl. Phys., 1965, 36(12), 3770– 3778. 6. Eckel, A. J., Cawley, J. D. and Parthsarathy, T. A., Oxidation kinetics of a continuous carbon phase in a nonreactive matrix. J. Am. Ceram. Soc., 1995, 78(4), 972–980. 7. Mogilevsky, P. and Zangvil, A., Modeling of oxidation behavior of SiC-reinforced ceramic matrix composites. Mater. Sci. Eng., 1999, A262, 16–24. 8. Clark, D. E. and Zoitos, B. K., Corrosion of Glass, Ceramics and Ceramic Superconductors. Noyes Publications, ParkRidge, NJ, USA, 1992, 672. 9. Opila, E. J. and Jacobson, N. S., SiO (g) formation from SiC in mixed oxidising-reducing gases. Oxid. Metals, 1995, 44(3/4), 1–17. 10. Opila, E. J. and Hann Jr., R. E., Paralinear oxidation of CVD SiC in water vapor. J. Am. Ceram. Soc., 1997, 80(1), 197–205. 11. Opila, E. J., Smialek, J. L., Robinson, R. C., Fox, D. S. and Jacobson, N. S., SiC recession caused by SiO2 scale volatility under combustion conditions: II, thermodynamics and gaseous-diffusion model. J. Am. Ceram. Soc., 1999, 82(7), 1826–1834. 12. Filipuzzi, L. and Naslain, R., Oxidation mechanism and kinetics of 1D-SiC/C/SiC composite materials: II, modeling. J. Am. Ceram. Soc., 1994, 77(2), 467–480. 13. Jacobson, N. S., Morscher, G. N., Bryant, D. R. and Tressler, R. E., High-temperature oxidation of boron nitride: II, boron ni￾tride layers in composites. J. Am. Ceram. Soc., 1999, 82(6), 1473– 1482. 14. Nickel, K. G., Multiple law modelling for the oxidation of ad￾vanced ceramics and a model-independent figure-of-merit. In Cor￾rosion of Advanced Ceramics—Measurement and Modelling, ed. K. G. Nickel. Kluwer Academic Pub., Dordrecht, NL, 1994, pp. 59– 72. 15. Nickel, K. G. and Quirmbach, P., Gaskorrosion nichtoxidischer keramischer Werkstoffe. In Technische Keramische Werkstoffe, ed. J. Kriegesmann. Koln, Deutscher Wirtschaftsdienst, 1991, pp. 1–76 ¨ [chapter 5.4.1.1]. 16. Gogotsi, Y. G. and Lavrenko, V. A., Corrosion of High-Performance Ceramics. Springer Verlag, Berlin, 1992, p. 181

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