Simplified Paralinear Oxidation Analyses

Abstract

Paralinear oxidative behavior, i.e., concurrent parabolic scale growth (k_p) and linear scale volatility (k_l or k_v), was analyzed by an alternative to the Tedmon equation. A convenient COSP for Windows cyclic oxidation program analyzed published data for Cr, NiCr, Pb, SiC, Si₃N₄, and BN. All of these exhibit scale volatility due to CrO₃, CrO₂(OH)₂, PbCl₂, Si(OH)₄, or HBO₂. The ‘cyclic’ model used an iterative constant outer layer loss formalism, whereby a normalized spall constant, Q₀ /Δt, defined the scale volatility rate, k_l (or k_v). Optimized trial inputs (fitting maximum mass gain (ΔW_max at t_max) and time to reach ΔW = 0) generated accurate k_p and k_l values. COSP models replicated ideal paralinear form. Simple approximations for k_p and k_l are also found as ~ 4.1 (ΔW_max²/ t_max) and ~ 1.2 (ΔW_max/t_max), respectively, for most scales. Alternatively, ΔW_max and t_max can be predicted by k_p and k_l. High or low volatility extremes (e.g., burner tests, short times, or Al₂O₃ scales) may mask classic paralinear weight change features.

Appendices

Appendix 1

Analytical Predictions for Paralinear Features

$\overline{W}$= ΔW_max at maximum weight gain and $\overline{t}$= t_max (at maximum weight gain) in COSP) can be obtained from the original paralinear solution espoused by Wajszel[15], as utilized by Barrett/Presler [5]

$$\overline{W} = \Delta W_{max} = \frac{A}{B} \cdot \alpha ;\quad {\text{where }}\alpha = \frac{1}{a} - ln\frac{b}{a}$$

(11)

$$\overline{t}={t}_{max}=\frac{A}{{B}^{2}}\cdot \beta ;\quad \mathrm{where }\;\beta =-\frac{1}{b}+ln\frac{b}{a}$$

(12)

And where, from Barrett/Pressler,[5] p.3:

$$A = \, \left( {a^{2} /2} \right)k_{p}$$

(13)

$$B = \, (a/b) \, k_{l}\;({\text{or}}\;k_{v} )$$

(14)

with stoichiometric ratios for gas and metal reactants in the scale:

$$a \, = \, M_{{\text{e}}} /{\text{gas}}\left( { = b - { 1}} \right)$$

(15)

$$b = {\text{scale product}}/{\text{gas}}\left({{\text{i.e.,}}\,S\, {\text{or}}\, S_{{{\text{eff}}}} {\text{ in COSP}}} \right)$$

(16)

Using these conversions and k_p, k_{l(or v)} from the published paralinear and fitted COSP parameters, ΔW_max and t_max are calculated and compared in Table 3.

Lastly, the terminal or final slope can be obtained from k_l(v), as stated in Eq. 4:

$$k_{{\text{l}}} = \gamma\left( {T.S.} \right)$$

(17)

where γ is the MW scale/MW compound stoichiometric ratio. Thus, the steady-state slopes of the weight change curve predicted from scale volatility rates are shown for the published paralinear and fitted COSP k_l(v) in Table 3.

Also, the A, B, a, b Wajszel parameters above are used to obtain conventional paralinear k_p, k_l(v) for Pb/PbCl₂ in Table 2 according to:

A = 4.2E-11 g2/cm4/s	= 0.1512	mg2/cm4h
B = 2.8E-9 g/cm2/s	= 0.01008	mg/cm2h
kp = 2A/a2	= 0.0356	mg2/cm4h
kl,v) = bB/a	= 0.0135	mg/cm2h
a = metal/gas (MW)	= 2.922
b = scale/gas (MW)	= 3.922

In general, the paralinear rate constants can also be determined from knowing the stoichiometric factor b of the oxide, (S or S_eff), and the data point coordinates at maximum (ΔW_max, t_max). By combining Eqs. 11–16, one obtains the paralinear rate constants as simple functions of the coordinates at maximum:

$$k_{{\text{p}}} = \frac{2}{{a^{2} }}\left( {\frac{\beta }{{\alpha^{2} }}} \right)\frac{{\Delta W_{{{\text{max}}}}^{2} }}{{t_{{{\text{max}}}} }} = F_{p} \frac{{\Delta W_{{{\text{max}}}}^{2} }}{{t_{{{\text{max}}}} }} \cong 4.06\frac{{\Delta W_{{{\text{max}}}}^{2} }}{{t_{{{\text{max}}}} }}$$

(18)

$$k_{{\text{v}}} = \left( \frac{b}{a} \right)\left( {\frac{\beta }{\alpha }} \right)\frac{{\Delta W_{max}^{{}} }}{{t_{max} }} = F_{v} \frac{{\Delta W_{max}^{{}} }}{{t_{max} }} \cong 1.16\frac{{\Delta W_{max}^{{}} }}{{t_{max} }}$$

(19)

The stoichiometric factor coefficients, F_p and F_v, preceding the max point quotients are tabulated in Table 4. A regular progression of model oxide stoichiometries and those for a selection of actual oxides are listed (for S only, not S_eff). It is found that the F_p and F_v factors remain relatively tight over the range considered in the present analyses. An average factor of F_p ~ 4.1 ± 1.1% is obtained for the parabolic rate constant, k_p in Eq. 18, and F_v ~ 1.2 ± 7.5% for the linear volatility constant, k_v in Eq. 19. More precisely, F_p and F_v are equal to 4.13 and 1.28, respectively, for low MW oxides with S ≤ 3 and equal to 4.01, 1.07 for high MW oxides with S ≥ 3. The overall consistency of both Eqns. 18 and 19 relations is displayed in Figs. 11 and 12 for all the cases examined. A log–log presentation was used to illustrate details across the full range of paralinear parameters examined. It is seen that the dependencies do not differ appreciably between relations using exact and average F coefficients. The average and precise F_p data points are seen to essentially coincide in Fig. 11, whereas the average and precise Fv data points sometimes show minor vertical offsets in Fig. 12. Furthermore, these F factors do not vary at all with k_p or k_l. Regression analyses of these model cases (log–log plots) yielded:

$${F}_{\mathrm{p}}\frac{{\Delta W}_{\mathrm{max}}^{2}}{{t}_{\mathrm{max}}} = 1.00\,k_{p}\,1.0046 ; \quad r2 = 1.000$$

(20)

$${F}_{\mathrm{p},\mathrm{ avg}}\frac{{\Delta W}_{\mathrm{max}}^{2}}{{t}_{\mathrm{max}}}= 1.01\,k_{p}\,1.0049;\quad r2 = 1.000$$

(21)

$${F}_{\mathrm{v}}\frac{{\Delta W}_{\mathrm{max}}}{{t}_{\mathrm{max}}} =0.96\,k_{V}\,0.9958 ;\quad r2 = 0.999$$

(22)

$${F}_{\mathrm{v},\mathrm{avg}}\frac{{\Delta W}_{\mathrm{max}}}{{t}_{\mathrm{max}}}= 1.11\,k_{V}\,1.0196;\quad r2 = 0.997$$

(23)

thus, verifying that k_p and k_v are linearly equivalent to and can be reasonably estimated from ${F}_{\mathrm{p}}\frac{{\Delta W}_{\mathrm{max}}^{2}}{{t}_{\mathrm{max}}}$ and ${F}_{\mathrm{v}}\frac{{\Delta W}_{\mathrm{max}}}{{t}_{\mathrm{max}}}$, respectively. While these were generated from model COSP curves, they also apply to experimental data, insofar as the data reflect ideal behavior, Figs. 7 and 8.

More specifically, the paralinear analysis of Cr oxidized in O₂-10%H₂O (e.g., CrO₂(OH)₂) was shown to produce rate constants as a function of ΔW_max and t_max (Pujilaksono et al.) [8]. The derivations were constructed using thickness parameters, so direct comparisons with weight change curves were indirect and problematic. Thus, the data summarized in Table 5, indicating max data point coordinates from plots and rate parameters, are shown both as-reported and converted to various units. Conversion of scale volatility from thickness to weight could be made easily using the density of the Cr₂O₃ scale, ~ 5.22 g/cm³. Linear volatility, labeled k_s, was converted and equivalent to k_l (k_v). But conversion of scale growth is less transparent in that the accrued weight is only from oxygen gain, whereas the scale thickness is composed of Cr₂O₃. Thus, the conversion of parabolic growth, labeled k_d, then k´´, to traditional k_p for weight change was not readily apparent. The same conversion factor, implied for k_d conversion to k´´ from the data obtained in dry air, was therefore used. The final results for k_l (k_v) and k_p in mg/cm²h and mg²/cm⁴/h are seen to agree within 10–15% of those obtained by typical COSP analyses. The relative discrepancies are likely due in part to the extremely low values of overall weight change and low values of t_max.

Appendix 2

COSP and the Tedmon Study

The original Tedmon paralinear study [1] derived the analytical, transcendental solution and compared it to experimental data for Fe–Cr and Cr alloys. The data were presented as log (weight gain) vs log (time), or weight change vs log (time). These are less-familiar presentations than typical weight change vs time linear plots. Such data are shown for Cr-0.2%Y, Fig. 

Fig. 13

Comparisons for paralinear and COSP fits to ΔW/A vs log t results for Cr-Cr₂O₃ oxidation at 1200 °C (Tedmon [1]). COSP Model #3 is a better fit for most long-time data (> 30–280 h). Model #4 (~ Tedmon curve) fits better for short times (1–10 h): (1) solid, black: parabolic: k_p = 0.54 mg²/cm⁴/h, also used in #4, #5. (2) Large circles, blue: experimental data points. (3) Small circles, teal: data matched at Δw_(max), Δw_(final) using k_p = 0.34 mg²/cm⁴/h, k_v = Q₀ /Δt = 0.0273 mg/cm²/h. (4) Long dash, blue: Tedmon analytical curve matched, but using k_v = Q₀ /Δt = 0.048 mg/cm²/h. (5) Short dash, red: using k_v = Q_o /Δt = 0.058 mg/cm²/h from [Hagel, 1966][25], but curve underestimates data

13. The weight gain data exhibit a maximum, then decrease in units of mg/cm² vs log time (s). Parabolic kinetics were extracted from the initial portion of the curve, then combined with k_v from the Hagel 1963 volatility study for bulk Cr₂O₃: [25]

$$\begin{gathered} k_{{\text{p}}} = { 1}.{\text{5 x 1}}0^{{ - {1}0}} {\text{g}}^{{2}} /{\text{cm}}^{{4}} {\text{s }}\left( {0.{\text{54 mg}}^{{2}} /{\text{cm}}^{{4}} {\text{h}}} \right),{\text{ and}} \hfill \\ k_{{\text{v}}} = \, 0.{\text{214 exp }}\left( { - {\text{ 48 kcal}}/{\text{RT}}} \right){\text{ g}}/{\text{cm}}^{{2}} {\text{s }}({\text{yielding }}0.0{\text{58 mg}}/{\text{cm}}^{{2}} {\text{h at 12}}00^\circ {\text{C}}), \hfill \\ \end{gathered}$$

The calculated paralinear curve shown was similar in form but did not fit the data closely. Unlike the data, the curve produced a lower maximum in weight gain, at shorter times, and approached 0 mg/cm² at the end of test.

Appropriate COSP model curves are provided in a similar mg/cm² vs log time (s) format, Fig. 13. They were based on Tedmon/Hagel parameters or specific characteristics of the plots in Tedmon [1], with stoichiometric factors given in Table 1. The shape of the COSP model curves is consistent with the Tedmon plot, but the characteristic parameters show some discrepancies. The results are summarized in Table

Table 6 Summary of COSP parameters fitting original data for Cr-0.2 wt.%Y at 1200 °C, from Fig. 5 in Tedmon [1]

6. Curve 1 just represents parabolic growth with no volatility based on the initial portion of the curve. Curve 2, open circles, is the original data. Curve 3 (small, filled circles, teal) is the best COSP fit to the experimental data, but with lower k_p = 0.34 mg²/cm⁴h and k_v = 0.0273 mg/cm²h than those stated in Tedmon. COSP curve 4, large dash, blue) matched the paralinear curve in Tedmon, but needed k_v = 0.048 mg/cm²h; and the lowest curve 5, small dash, red) used the k_p and k_v prescribed by Tedmon (0.540 mg²/cm⁴h, 0.058 mg/cm²h), which underpredicts the data.

This suggests inconsistencies in the original analysis. It appears that the experimental data correspond to slightly lower k_p and k_v parameters:

• (1) An effective reduction in k_p can be generated by cubic-linear behavior, where grain growth over time simultaneously decreases scale growth controlled by short circuit paths. This produces subparabolic behavior. Attempts at a COSP cubic-linear fit produced an improved fit but did not match the entire curve, nor did the ‘perturbation’ solution offered by Chen et al., in their Fig. 5[24]. Barrett and Presler [5] found best fits for the Tedmon k_p to vary from 0.29 to 0.44 mg²/cm⁴h, depending on the portion of the curve being fit, in general agreement with present findings (0.34 mg²/cm⁴h).

• (2) The k_v for the scale may be different than that measured by Hagel for bulk Cr₂O₃ [25]. This can occur for test conditions containing different gas flow rates or moisture content, both of which affect k_v. With little experimental detail available, no further speculation is offered. The COSP model curves parallels the Tedmon plot but implies k_p and k_v parameters ~ 50% less than those quoted. Barrett and Presler [5] found best fits for the Tedmon k_v to vary from 0.023 to 0.034 mg/cm²h, depending on the portion of the curve being fit, in general agreement with present findings (k_v = 0.027 mg/cm²h).