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. I1700 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