**4. Critical concentration of solute required for the transformation of internal to external oxidation as a function of grain size**

Oxidation of engineering alloys is very complex as the components of the alloys has different affinities for the oxygen, and reacting atoms do not diffuse at the same rates in the oxides or alloy substrates. Various types of oxides can be formed on and in the alloy. Atomic ratios of the elements in the oxide scale may differ significantly from those in the alloy. When oxygen and metal atoms diffuse and react at the surface of an alloy, an external oxide layer is formed on the surface and this is termed as the "external oxidation". For external oxidation, outward flow of metal atoms must exceed the inward flow of oxygen, whereas, when inward flow of oxygen exceeds the outward flow of the metal atoms, oxygen diffuses inside the metal and oxidation takes places within the alloy. This process is termed as "internal oxidation" which leads to catastrophic loss in the material property [44,45,109]. Figures 2 and 3 schematically show internal and external oxidation for alloy A-B under the conditions where only B oxidises and both A and B oxidise, respectively.

$$N\_B\{\mathcal{C}rit1\}\; = \left(\frac{g.\pi}{3}, \frac{\nu\_{\text{AB}}}{\nu\_{\text{BO}\_\forall}}, \frac{N\_O^{\langle s\rangle}D\_O}{D\_{\text{B}}}\right)^{1/2} \tag{8}$$

$$N\_B(Crit2) \ = \frac{V\_{AB}}{V\_{BO\_V}} \left(\frac{\pi k\_p}{2D\_B}\right)^{1/2} \tag{9}$$

$$
\alpha^2 = 2k\_p t \tag{10}
$$

$$N\_B\{Crit1\} = A\{D\_b + \frac{2\mathcal{S}}{d}D\_{gb}\}^{-1/2} \tag{11}$$

$$A = (\frac{g \pi N\_O^{(s)} D\_O V\_{AB}}{2V\_{BO\_V}})^{1/2} \text{ }.$$

$$N\_B\{crit2\} = B\left(D\_b + \frac{2\delta}{d}D\_{gb} - \frac{2\delta}{d}D\_b\right)^{-\frac{1}{2}} = B\left(\frac{\sqrt{d}}{\sqrt{2\delta\left(D\_{gb} - D\_b\right) + d.D\_b}}\right) \tag{12}$$

$$\mathbf{B} = \frac{\mathbf{V}\_{\text{AB}}}{\mathbf{V}\_{\text{BO}\_V}} \left(\frac{\pi \mathbf{k}\_{\text{p}}}{2}\right)^{1/2} \text{ .}$$

$$N\_{oB} \text{(crit2)} = B. \text{(}D\_{\text{b}}\text{)}^{-\frac{1}{2}} \text{ and } N\_{oB} \text{(}crit1\text{)} = A. \text{(}D\_{\text{b}}\text{)}^{-\frac{1}{2}} \text{} \tag{13}$$

$$X = \frac{N\_{\rm B}(crlt1)}{N\_{\rm oB}(crlt1)} = \frac{N\_{\rm B}(crlt2)}{N\_{\rm oB}(crlt2)} = \sqrt{\frac{d}{2\delta \binom{D\_{\rm gb}}{D\_{\rm b}} - 1}} = \sqrt{\frac{1}{f\binom{D\_{\rm gb}}{D\_{\rm b}} - 1}}\tag{14}$$

$$X' = \frac{N\_B}{\nu} = \sqrt{\frac{d(2\delta \overline{\{D\_{gb} - D\_b\} + d\_0 D\_b})}{d\_0(2\delta \overline{\{D\_{gb} - D\_b\} + d.D\_b})}}\tag{15}$$


Oxidation Resistance of Nanocrystalline Alloys 227

 nc Fe10Cr mc Fe10Cr

Fig. 5a. Oxidation kinetics of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys at 300°C as represented by weight-gain vs time plots for 3120 minutes [12,37,39]. Inset shows the zoom of the region representing initial periods of oxidation (up to 240 minutes of oxidation).

> R2 =0.98

 nc Fe10Cr mc Fe10Cr

0 500 1000 1500 2000 2500 3000 3500

Time (min)

0.0

0.1

0.2

0.3

0 50 100 150 200 250

Fig. 5b. Oxidation kinetics of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys oxidised at 300°C: weight-gain2 with time (up to 240 minutes) suggesting parabolic kinetics

0 50 100 150 200 250

Time (min)

R2 =0.93

for both mc and nc alloys [12,37,39]

0.00

0.02

0.04

(Weight gain per unit area)2

(mg/cm2

)

2

0.06

0.08

0.10

0.0

0.2

0.4

0.6

Weight gain per unit area

(mg/cm2

)

0.8

1.0

1.2

Figures 5-7 [12,38,39]. Oxidation kinetics at 300°C shows the microcrystalline alloy to be oxidizing at a considerably greater rate than the nanocrystalline alloy. After 3120 minutes of oxidation, weight gain of microcrystalline Fe-10Cr alloy was found to be nearly seven times greater than that of nanocrystalline alloy of same chemical composition.

Besides the considerably higher weight gain of the microcrystalline alloy, the evolution of oxidation kinetics was also different. Both nanocrystalline and microcrystalline Fe10Cr alloys follow parabolic kinetics for the first 240 minutes of oxidation (as evident in the weight-gain2 versus time plot in Figures 5b and 6b). However during subsequent oxidation, nanocrystalline Fe10Cr alloy show considerable deviation from the parabolic behaviour whereas, microcrystalline alloy of same chemical composition continued to follow the parabolic kinetics (Figures 5c and 6c). The marked deviation of the nanocrystalline Fe-10Cr alloy from the parabolic behaviour is accounted for the insignificant increase in the weightgain of this material after the first 240 minutes of oxidation (Figures 5a, 6a, 5b and 6c). This behaviour could be attributed to some critical change in the chemical characteristic of the oxide scale formed on both nanocrystalline and microcrystalline alloys before and after 240 minutes of oxidation which was described by Gupta et al using SIMS [12,38,39] analysis of oxide formed during various period of oxidation.

Oxidation kinetics of nanocrystalline and microcrystalline Fe-10Cr alloys at 350 and 400°C are presented in Figures 6 and 7. The trend of greater oxidation rate of the microcrystalline alloy, as seen at 300oC is also followed at the two higher temperatures. However, the influence of nanocrystalline structure in improving the oxidation resistance was extraordinarily enhanced at these higher temperatures as indicated by the comparative weight gains after 3120 minutes of oxidation: weight gain of microcrystalline Fe-10Cr alloy was found to be 18 times greater than that of the nanocrystalline Fe-10Cr alloy at 350°C, and nearly 17 times greater at 400°C.

A close observation of the data as presented in Figures 5-7 show that both nanocrystalline and microcrystalline Fe-10Cr alloys follow parabolic kinetics ,i.e., (weight grain per unit area)2 = kt. The rate constants (k) in nanocrystalline alloy changes with time (Table 2). Oxidation kinetics of nanocrystalline Fe-10Cr can be divided in the two stages, each stage characterized by a unique k value (Table 2). Microcrystalline alloy, on the other hand follow a single parabolic rate constant. As presented in the Table 2, k value for microcrystalline (mc) Fe10Cr alloys are more than an order of magnitude greater than either of the k values for nanocrystalline (nc) Fe10Cr alloy at the three temperatures.


Table 2. Parabolic oxidation rate constants (k) values in g2cm-4s-1[12]

Figures 5-7 [12,38,39]. Oxidation kinetics at 300°C shows the microcrystalline alloy to be oxidizing at a considerably greater rate than the nanocrystalline alloy. After 3120 minutes of oxidation, weight gain of microcrystalline Fe-10Cr alloy was found to be nearly seven times

Besides the considerably higher weight gain of the microcrystalline alloy, the evolution of oxidation kinetics was also different. Both nanocrystalline and microcrystalline Fe10Cr alloys follow parabolic kinetics for the first 240 minutes of oxidation (as evident in the weight-gain2 versus time plot in Figures 5b and 6b). However during subsequent oxidation, nanocrystalline Fe10Cr alloy show considerable deviation from the parabolic behaviour whereas, microcrystalline alloy of same chemical composition continued to follow the parabolic kinetics (Figures 5c and 6c). The marked deviation of the nanocrystalline Fe-10Cr alloy from the parabolic behaviour is accounted for the insignificant increase in the weightgain of this material after the first 240 minutes of oxidation (Figures 5a, 6a, 5b and 6c). This behaviour could be attributed to some critical change in the chemical characteristic of the oxide scale formed on both nanocrystalline and microcrystalline alloys before and after 240 minutes of oxidation which was described by Gupta et al using SIMS [12,38,39] analysis of

Oxidation kinetics of nanocrystalline and microcrystalline Fe-10Cr alloys at 350 and 400°C are presented in Figures 6 and 7. The trend of greater oxidation rate of the microcrystalline alloy, as seen at 300oC is also followed at the two higher temperatures. However, the influence of nanocrystalline structure in improving the oxidation resistance was extraordinarily enhanced at these higher temperatures as indicated by the comparative weight gains after 3120 minutes of oxidation: weight gain of microcrystalline Fe-10Cr alloy was found to be 18 times greater than that of the nanocrystalline Fe-10Cr alloy at 350°C, and

A close observation of the data as presented in Figures 5-7 show that both nanocrystalline and microcrystalline Fe-10Cr alloys follow parabolic kinetics ,i.e., (weight grain per unit area)2 = kt. The rate constants (k) in nanocrystalline alloy changes with time (Table 2). Oxidation kinetics of nanocrystalline Fe-10Cr can be divided in the two stages, each stage characterized by a unique k value (Table 2). Microcrystalline alloy, on the other hand follow a single parabolic rate constant. As presented in the Table 2, k value for microcrystalline (mc) Fe10Cr alloys are more than an order of magnitude greater than either of the k values

300°C 5.65×10-13 (1st stage) and 7.42×10-14 (2nd stage) 7.74×10-12

350°C 1.04×10-12 (1st stage) and 1.7×10-13 (2nd stage) 1.46 ×10-10

400°C 1.34×10-12 (1st stage) and 5.69×10-13(2nd stage) 2.53×10-10

nanocrystalline Fe10Cr microcrystalline

Fe10Cr

greater than that of nanocrystalline alloy of same chemical composition.

oxide formed during various period of oxidation.

for nanocrystalline (nc) Fe10Cr alloy at the three temperatures.

Table 2. Parabolic oxidation rate constants (k) values in g2cm-4s-1[12]

nearly 17 times greater at 400°C.

Fig. 5a. Oxidation kinetics of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys at 300°C as represented by weight-gain vs time plots for 3120 minutes [12,37,39]. Inset shows the zoom of the region representing initial periods of oxidation (up to 240 minutes of oxidation).

Fig. 5b. Oxidation kinetics of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys oxidised at 300°C: weight-gain2 with time (up to 240 minutes) suggesting parabolic kinetics for both mc and nc alloys [12,37,39]

Oxidation Resistance of Nanocrystalline Alloys 229

 nc Fe10Cr mc Fe10Cr

Fig. 6b. Oxidation kinetics of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys oxidised at 350°C: weight-gain2 with time (up to 240 minutes) suggesting parabolic kinetics

> R2 =0.989

0 50 100 150 200 250

Time (min)

R2 =0.95

> nc Fe10Cr mc Fe10Cr

R2 =0.98

Fig. 6c. Oxidation kinetics of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys, oxidised at 350°C: weight-gain2 with time, suggesting parabolic kinetics for mc alloy but

0 500 1000 1500 2000 2500 3000 3500

Time (min)

R2 =0.71

departure from parabolic kinetics for nc alloy [12,39].

for both mc and nc alloys [12,39]

(Weight gain per unit area)2

(mg/cm2

)

2

0.0

0.4

0.8

(Weight gain per unit area)2

(mg/cm2

)

2

1.2

1.6

2.0

Fig. 5c. Oxidation kinetics of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys, oxidised at 300°C: weight-gain2 with time, suggesting parabolic kinetics for mc alloy but departure from parabolic kinetics for nc alloy [12,37,39].

Fig. 6a. Oxidation kinetics (weight-gain vs time plot) of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys, during oxidation at 350°C for 3120 min in air [12,39]. Inset shows the zoom of the region showing initial periods of oxidation (up to 240 minutes of oxidation).

Fig. 5c. Oxidation kinetics of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys, oxidised at 300°C: weight-gain2 with time, suggesting parabolic kinetics for mc alloy but

R2 =0.993

0 500 1000 1500 2000 2500 3000 3500

Time (min)

R2 =0.65  nc Fe10Cr mc Fe10Cr

Fig. 6a. Oxidation kinetics (weight-gain vs time plot) of nanocrystalline (nc) and

of oxidation).

microcrystalline (mc) Fe-10Cr alloys, during oxidation at 350°C for 3120 min in air [12,39]. Inset shows the zoom of the region showing initial periods of oxidation (up to 240 minutes

0 500 1000 1500 2000 2500 3000 3500

Time (min)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0 50 100 150 200 250

departure from parabolic kinetics for nc alloy [12,37,39].

 nc Fe10Cr mc Fe10Cr

0.0

0

1

2

3

Weight gain per unit area

(mg/cm2

)

4

5

6

0.2

0.4

0.6

0.8

(Weight gain per unit area)2

(mg/cm2

)

2

1.0

1.2

1.4

1.6

1.8

Fig. 6b. Oxidation kinetics of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys oxidised at 350°C: weight-gain2 with time (up to 240 minutes) suggesting parabolic kinetics for both mc and nc alloys [12,39]

Fig. 6c. Oxidation kinetics of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys, oxidised at 350°C: weight-gain2 with time, suggesting parabolic kinetics for mc alloy but departure from parabolic kinetics for nc alloy [12,39].

Oxidation Resistance of Nanocrystalline Alloys 231

Fig. 8. SIMS depth profiles for Cr the in the oxide scale developed during oxidation of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys at 300°C for in air for 3120 minutes, using a Cameca ims (5f) dynamic SIMS instrument. SIMS parameters were: Cs+ ion

0 500 1000 1500 2000 2500 3000

Sputtering time (s)

 nc Fe10Cr mc Fe20Cr

 nc Fe10Cr mc Fe20Cr

Fig. 9. SIMS depth profiles for Cr the in the oxide scale developed during oxidation of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys at 350°C for in air for 3120 min, using a Cameca ims (5f) dynamic SIMS instrument. SIMS parameters were: Cs+ ion primary

0 500 1000 1500 2000 2500 3000

Sputtering time (s)

beam (10 nA), depth profiling of craters of 250 µm × 250 µm area [12,39].

primary beam (10 nA), depth profiling of craters of 250 µm × 250 µm area [12,39].

0

0

1x106

2x106

Counts per s

3x106

4x106

1x10<sup>6</sup>

2x10<sup>6</sup>

Counts per s

3x10<sup>6</sup>

4x10<sup>6</sup>

Fig. 7. Oxidation kinetics (weight-gain vs time plot) of nanocrystalline (nc) and microcrystalline (mc) Fe-10Cr alloys, during oxidation at 400°C for 3120 min in air [12,39]. Inset shows a zoom of the region of initial periods of oxidation (up to 240 minutes of oxidation).
