Stoichiometry Effects on Paralinear Oxidation

Paralinear oxidation models provide a description of parabolic scale growth combined with linear loss , as might occur for scales forming volatile oxide, hydroxide, chloride, or fluoride scales. Classic weight change exhibits an initial parabolic oxygen gain, a maximum ( Δ W max at t max ), then a linear loss. The magnitude of these features is determined by the parabolic growth rate, k p , the linear volatility rate, k v , and the stoichiometric constant of the reaction, S (fixed by the atomic weights and stoichiometry of the reaction). Model curves were generated (at constant k p and k v ) to show that, for typical oxides, increases in S only moderately decrease Δ W max and t max , but directly increase the rate of mass loss. Furthermore, it is shown that, on average, k p  4.06 ( Δ W max ) 2 / t max and k v  1.15 ( Δ W max )/ t max . These relations apply for a broad spectrum of scale molecular weights, ranging from low mass SiO 2 to high mass Ta 2 O 5 oxides. Oxidation of carbides and nitrides may release C and N elements and thus increase the effective S eff , with concomitant effects on the paralinear curves. The above analysis was facilitated by COSP for Windows software.


Introduction
The high temperature oxidative behavior of metal elements generally entails the development of a continuous, healing oxide scale that controls further oxidation via transport through the thickening scale.The subsequent growth rate is inversely proportional to the thickness of the scale, leading to parabolic rate laws.In some environments, the scale may also be subject to evaporative losses by forming volatile species.This may occur directly by 'evaporation' of the scale or upon further reaction with the gaseous environment.The latter phenomena do not depend on the thickness of the scale and present themselves as a linear loss rate factor.The combined growth/volatility phenomena leads to paralinear kinetic behavior, as proposed for Pb-PbCl2 (Wajszel) and Cr-CrO3 (Tedmon) substrates-volatile species.[1][2] The initial kinetic laws were developed for oxide thickness data (x): Paralinear curves based on microstructural scale thickness are not subject to stoichiometry effects.However, such studies are rarely complete enough to exhibit a precise maximum characteristic of paralinear behavior and entail numerous, laborintensive measurements.
Subsequent efforts addressed oxidative weight changes, which are more conveniently measured, but require stoichiometric corrections to produce oxygen gain -oxide lost.[2][3][4][5] A basic form of the paralinear weight change response was demonstrated to be highly dependent on the parabolic and linear oxidation rate parameters, kp and kv.That is, the weight gain proceeded initially 'parabolically' to a maximum, then lost weight, approaching a 'linear' loss rate.The time to reach maximum is also seen to be ¼ the time to reach 'zero' weight change.The response curve is also scaled by another factor: stoichiometry (mass of oxide formed/mass of oxygen reacted), usually a constant.
In order to broadly demonstrate general paralinear kinetic responses via graphical analyses of weight change data, a cyclic oxidation computer program, COSP for Windows, [6] was recently employed.[7] Though originally developed for cyclic oxidation, where spallation was believed to increase with scale thickness, the basic algorithm was adjusted here to fix a constant rate of loss.This is accomplished by setting the spall exponent, , to -1: wspall = Q0(wretained) (α+1) ; and thus (2) wspall = Q0, per cycle time, Δt, (often set to 1 h for rates in convenient mg/cm 2 /h) where wretained is the amount of oxide present prior to a 'spalling' cycle; wspall is the weight of the scale removed per cycle, Δt; and Q0 is the spall constant.The 'spalling' geometry is defined here as the removal of a uniform outer layer of scale.Paralinear growth and volatility constants are provided by kp and kv, where kv (or kl ) is given as Q0/Δt.Generally, 1 h was used as an incremental time unit and was small enough to approximate a continuous (isothermal) process and produce smooth, high resolution model curves.Other parameters from the program have the usual meaning (see Glossary).However, it is noted that the stoichiometric constant (S), to obtain the amount of new oxide growth per cycle from the amount of oxygen gained, must sometimes be adjusted (Seff) to account for release of any elements not contained in the oxide.This occurs for C and N losses in the oxidation of SiC, Si3N4, and BN.
In-depth analysis of published experimental paralinear curves for Pb, Cr, NiCr, SiC, Si3N4, and BN was accomplished by using COSP iterations to fit experimental curves.[7] Good agreement (shown in Table 1 below) was obtained for kp and kv with those determined by the published mathematical techniques.Here the differential between reported and COSP fits of the four parameters were below 5%.approximations above simplify estimates of kp and kv from simple weight change curves.
In a related matter, it can be seen that the overall paralinear response is sensitive to the atomic weight of the components.Simply put, more severe weight losses are registered for metal atoms whose atomic weights (AW) are much greater that the weight of the oxygen component.Also, the paralinear behavior of light atom oxides can be more significantly affected by C or N losses (from B4C, SiC, Si3N4) B than heavy atom oxides (from TaC, NbC).An example of the reaction stoichiometry (S) is given below for SiC oxidation, where one C atom is released for every unit of SiO2 formed: To extend this perspective, multiple curves were generated by COSP for Windows, utilizing oxides comprised of metals ranging from low atomic weight Be (at.wt.9.0 gmatom) to Ta (180.9 gm-atom).Common carbides and nitride compounds were also addressed.While most studies have focused on single systems, the response to stoichiometric changes is shown broadly here for Be, B, Si, Ti, Cr, Mn, Fe, Co, Ni, Nb, and Ta alloys or compounds.Most of these have been associated volatile oxides or hydroxides, such as BeO, Be(OH)2, HBO2, SiO, Si(OH)4, CrO3, CrO2(OH)2, TiO(OH)2, etc.In order to emphasize just S or Seff, the kp and kv were fixed at 0.01 mg 2 /cm 4 /h and 0.01 mg/cm 2 /h for all systems.Thus, the analysis is strictly conceptual, with no direct relation to actual kp and kv experimental data, as had been addressed previously.The magnitudes of kp and kv are otherwise known to vary widely with temperature, total gas pressure, p(O2), p(H2O), and gas velocity, depending on the system and environment.Generally, those parametric variations obscure or overshadow the relative role of S in the paralinear response.Thus, the purpose of this exercise is to isolate the effects of oxide stoichiometry on paralinear behavior.

Results
A list of the stoichiometric factors for the systems evaluated is presented in Table 2.
These are needed to generate model paralinear curves.Here, S is defined as the gmatom ratio of the scale formed/reactant (oxide formed/oxygen reacted) in the oxidation of the metal atom.Additionally, the ratio Seff, for carbide and nitride systems, is that of oxide formed/(oxygen reacted -carbon (or nitrogen) released.These are the factors needed in the COSP algorithm to calculate the retained oxide weight (in oxygen gained) after a unit time interval: wretained (as oxide) = S (kpt) 1/2 , or, (7a) The results for a full range of S are shown in Figure 1.It can be seen that for S increasing to 2.0, (Fig. 1a) the weight change more clearly exhibits a maximum, and eventually crosses zero in the time frame for 200 1-h 'cycles'.Further increases in S to 20 (Fig. 1b) produce ever increasing severity of the weight loss slope and deepening total losses.The final weight loss changes from about -0.15 to -0.55 mg/cm 2 as S increases from 2.0 to 20.Convergence of behaviors is seen for higher S curves.successfully.[7] However, kp and kv varied widely between material, temperature, and test condition so as to mask any sensitivity to stoichiometry.
The formal relationships between the weight and time at maximum were recalled as functions of kp, kv, and S (or Seff) [3][7]:   Alternatively, kp and kv can be solved from eqn. 5 and 6 above to yield: The complex conversion factors, Fw and Ft, and terms containing S are summarized in Table 2.For illustration purposes, kp and kv were again fixed at 0.01 mg 2 /cm 4 /h and 0.01 mg/cm 2 /h, respectively, and ΔWmax and tmax, were calculated as functions of S, shown in Figures 6a, 6b.(Note, there is no physical meaning for S  1.13, (e.g., for at.no. 1, H2O), and little potential for S  15.Accordingly, S or Seff for the systems studied in this work ranged only from ~1.5 -12).The limit of Fw as S approaches 1 is 1.From eqn. 14, ΔWmax decreases from an endpoint value of 0.5 mg/cm 2 at S =1.For S>10, it is seen empirically that Fw approaches ½ and ΔWmax approaches { Thus Figure 6 provides perspective on the moderate sensitivity of paralinear behavior to stoichiometry: over the range of S = 2 to 5, for the most common oxide scales, it is seen that ΔWmax decreases only by ~ 15 % and tmax decreases by about 30%.This is consistent with the approximations found previously, [7] where: These numerical coefficients in eqns.18, 19 are now confirmed more precisely as 3.61 and 1.62, when averaged over S = 1.5 to 12 in Table 2 .
By contrast, the (negative) terminal slope increases by ~ 70% over the same range of S = 2 to 5 and is obtained from kv: kv =  (T.S.) 20a kv = (S/(S-1)) (T.S.) 20b where  is the MW scale/MW compound stoichiometric ratio, given by S/(S-1).The corresponding steady-state negative slopes of the weight change curve predicted from kv (fixed at 0.01 mg/cm 2 /h) are shown in Figure 6c.They are seen to increase from 0 to 0.01 mg/cm 2 /h, asymptotically.For complex oxides, S can be calculated or adjusted to represent any combination of oxides, such as 2.760 for NiAl2O4 spinel or 4.249 for AlTaO4 tapiolite.Also, though fixed throughout the examples presented here, it is recalled that kp, kv variations will, of course, provide direct, sizeable effects.Finally, kp can be obtained simply and directly from (ΔWmax ) 2 /tmax , and kv can be obtained from (ΔWmax )/tmax using the coefficients indicated in eqn's.16 and 17.These coefficients can be approximated as 4.06  0.07 and 1.15  0.11 (horizontal dashed lines, Figure 7), averaged, for oxides of common elements or compounds having S (or Seff ) between 1.5 and 12.Only a few lower atomic weight elements (e.g., 9 B) exist which may decrease S to 1.45 and so increase these kp and kv estimates (6% and 27 %, respectively).On the other hand, higher extremes of atomic weight (e.g., 262 Lr) would only increase S to 11.9.The estimates for kp and kv would then only decrease by less than 1.4% and 13%, respectively.(It is seen graphically and mathematically (solid lines) that the coefficients increase to an undefined level at S = 1.0 and decrease to asymptotes of 4.0 and 1.0 for increasing S).More precise calculations of kp and kv can be obtained from the specific value of S (solid lines) for that reaction system, as shown in Table 2. Also, if  the terminal slope is considered to be linear and has reached steady-state, kv can also be determined from S, T.S. and eqn.20 directly.
Paralinear curves are of course influenced by the kinetic parameters kp and kv.The latter are related experimentally to scale diffusivity, temperature, volatile species, thermodynamics, p(O2), p(H2O), p(Cl2), v, etc.Generally, these parameters have been studied for single systems where stoichiometric variations need not be addressed.Paralinear curves based on microstructural scale thickness would not be subject to S or Seff.However, such studies are rarely complete enough to exhibit a precise maximum and entail numerous, labor-intensive measurements.

Summary and Conclusions
Model paralinear oxidation responses were investigated for a wide spectrum of oxide scales and corresponding stoichiometry.Stoichiometry is shown to slightly affect the characteristic paralinear oxidation feature (Wmax, tmax), but significantly increase the final negative slope (TS).To isolate the effect of oxide stoichiometry, model curves were generated by COSP for Windows.Weight gain and time to reach maximum correlated with kp and kv.The reverse was also true, allowing rate constants to be easily extracted from the paralinear curves: These simplifications were provided for oxides ranging from the low value of S = 1.9 for low mass 28 SiO2, to a high value of 5.5 for high mass 180 Ta2O5.(Corrections to may be required to accommodate gaseous losses of C or N for starting compounds of carbide or nitride).Some positive deviation from the average is seen for the few oxides with S below 2. Only slight negative deviation occurs for S > 5. Normalizing coefficients above approach asymptotes of 4.0 and 1.0 for kp and kv; the terminal (steady-state) loss rate (TS) approaches an asymptote equal to kv.

Fw
And where, from [Barrett/Pressler][3]: A = (a 2 /2) kp (10) B = (a/b) kv (11) with stoichiometric ratios for gas and metal reactants in the scale: a = MW Metal/gas (= b -1) (12) b = MW scale product/gas (i.e., S or Seff in COSP)(13)Combining relations (8-13) above, complex functions of S (Fw and Ft) yield predictions for ΔWmax and tmax: and Ft can be considered dimensionless values of ∆  and tmax, respectively, using weight and time constants, respectively.[8]Similarly, the terminal slope can be normalized by kv, yielding (S-1)/S.These generalized quantities (W*max, t*max , and T.S.* can be visualized as dimensionless functions of S in Figure 5. Asymptotes are indicated at 0.5 for W*max, 0.5 for t*max , and 1.0 for T.S.*, the latter only for very high S  20.Intermediate values of W*max, t*max , and T.S.* are apparent for the most important values of S, between about 2 to 5.
0.25 mg/cm 2 , asymptoticly.Similarly, Ft in eqn.15 is undefined at S = 1, and tmax decreases rapidly from an undefined high value.That condition for S= 1 defines behavior with no maximum, Figure1a.For S>10, it is seen empirically that Ft approaches ½ and tmax decreases to the asymptote at { or 25 h as indicated in Figure1bfor S = 20.

Table 1 .
Comparison of COSP fit paralinear parameters with published values for five materials.(W/Amax, tmax, kp and kv).
k v  1.15 W max /t max k p  4.06 (W max ) 2 /t max