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 volume1702 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)