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 paramK.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-