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Calculations of Internal Oxidation Rate Equations and Boundary Conditions

between Internal and External Oxidation in Silicon Containing Steels

Takashi Onishi, Shouhei Nakakubo and Mikako Takeda

Materials Research Laboratory, Kobe Steel, Ltd., Kobe 651-2271, Japan

The rate constants for internal oxidation of Si containing steels (Fe-Si alloys) at 850�C were calculated in order to clarify the formationmechanism of fayalite scale, which can form as a ‘‘sub-scale’’ in Si containing steels. The diffusion coefficient of oxygen in the oxide layer, DO,and the oxygen concentration at specimen surface, NO

(s), which are constituents of the internal oxidation rate constant, (2DONO(s)=NB

(O)n), werecalculated under various oxidation conditions, and the rate equation for the internal oxide layer was derived. Comparing the calculated and themeasured values of (2DONO

(s)=NB(O)n), we confirmed that the rate equation determined for the internal oxide layer was reasonable. The

conditions at the boundary between internal to external oxidation of Si containing steels (Fe-Si alloys) at 850�C were also calculated bysubstituting the calculated values of DO and NO

(s) at the boundary into the rate equation. [doi:10.2320/matertrans.M2009256]

(Received July 24, 2009; Accepted December 4, 2009; Published January 27, 2010)

Keywords: internal-oxidation, fayalite, scale, silicon containing steel, high-temperature oxidation, transition from internal to external

oxidation, rate constant, oxygen permeability

1. Introduction

The iron oxide scale that forms on billets and slabs of hot-rolled steel is usually removed using a hydraulic descalingprocess. However, any residual primary or secondary scalethat forms after the descaling process can remain on thesurface of the final product through subsequent hot- and cold-working steps. This residual scale greatly influences thesurface quality of the steel by modifying its mechanicalproperties. Deformation, fracture and spalling of the steel canoccur. Hence, it is of great importance to understand thephysical and mechanical properties of iron oxide scale inorder to control its formation and properties, and ultimatelyto improve the surface quality of the steel.

The composition of the oxide scale that forms on steel isprincipally Fe2O3, Fe3O4, and FeO, which forms in lamellarstrata from the substrate towards the outer layer. In thecase of Si-containing steels, which are widely used in themanufacture of automobile bodies and frames in the formof high-tensile steel sheets, fayalite scale (Fe2SiO4) can formas a ‘‘sub-scale’’ beneath the external wustite (FeO) layer.Since the bottommost layer of scale is fayalite scale, theproperties and formation mechanism of this need to beclarified in order to optimize the properties of iron oxide scaleon Si containing steels.1–8)

This paper focuses on internal oxidation, which is the firststage of sub-scale formation, and on calculating the internaloxidation rate constants of Si containing steels in orderto clarify the formation mechanism of fayalite scale. Thevalidity of their calculations was confirmed by comparing theresults with experimental values. Furthermore, the conditionsat the boundary between internal to external oxidation werealso calculated on the basis of the kinetics theory of internaloxidation.

2. Internal Oxidation Kinetics

2.1 Internal oxidation rateInternal oxidation can be defined as the formation of a

dispersed metal oxide layer near to the surface of an alloy.This layer consists of very fine metal oxide particlescomposed of one or more alloy elements and oxygen. Inthe case of Si containing steels (Fe-Si alloys), because theaffinity for oxygen of Si is higher than that of Fe, the Sioxidizes preferentially and very fine SiO2 particles becomedispersed near the surface forming an internal oxide layer.

Rate equations for the internal oxidation of flat specimenshave been derived using the quasi-steady state approxima-tion. Figure 1 shows approximate concentration profiles foroxygen and Si in the internal oxide layer of Fe-Si alloys.

In flat specimens composed of an A-B binary alloy (Fe-Sibinary alloy), it is assumed that B (Si) is a dilute solute andthe affinity of B for oxygen is higher than that of A (Fe).It is also assumed that the oxygen partial pressure in thisatmosphere is higher than the equilibrium oxygen pressureof SiO2 and is lower than that of FeO. Assuming the oxygenconcentration at the specimen surface varies linearly, with

Fig. 1 Approximate concentration profiles for oxygen and silicon in the

internal-oxide layer of Fe-Si alloys.

Materials Transactions, Vol. 51, No. 3 (2010) pp. 482 to 487#2010 The Japan Institute of Metals

http://dx.doi.org/10.2320/matertrans.M2009256

depth, the flux of oxygen permeating into the internaloxidation layer, JO, can be expressed by

JO ¼ DONO(s)=Vmx ð1Þ

where DO is the diffusion coefficient of oxygen in the oxidelayer, NO

(s) is the oxygen concentration at the specimensurface, Vm is the molar volume of the A-B binary alloy(Fe-Si binary alloy), and x is the thickness of the internaloxide layer. JO can be also expressed as the followingequation because the amount of oxygen accumulated at theinterface of the internal oxide layer and the A-B alloy (Fe-Sialloy) is equivalent to JO.

JO ¼ DONO(s)=Vmx ¼ ðNB

(O)=VmÞðdx=dtÞ ð2Þ

By transposing and integrating eq. (2), we obtain a rateequation for internal oxidation in the absence of an externaloxide layer as follows

x ¼ ð2DONO(s)=NB

(O)nÞ1=2t1=2 ð3Þ

where NB(O) is the initial concentration of the alloy element,

B (the initial Si concentration of the Fe-Si alloy) in the oxidelayer, and n is the atomic ratio of oxygen to B (Si) in theoxide. From eq. (3), the thickness of the internal oxidationzone, x, is estimated to be a parabolic function of time, t.Substituting values for the constants DO, NO

(s), NB(O) and n in

the internal oxidation rate constant, (2DONO(s)=NB

(O)n), weobtain the rate equation for internal oxidation of an A-Bbinary alloy (Fe-Si binary alloy).

2.2 Transition from internal to external oxidationAs shown in eq. (3), the internal oxidation rate decreases

with decreasing NO(s) or increasing NB

(O). Above a criticalconcentration of NO

(s), continuous scale layers composed ofBOn form on A-B binary alloys, and internal oxidation issuppressed. The transition from internal to external oxidationbegins to occur in this stage.

Wagner’s theory for the transition from internal to externaloxidation is developed as follows.9) The ratio of NO

(s) toNB

(O) is expressed by the following equation

NO(s)=NB

(O)n ¼ expð�2Þ erfð�Þ=�1=2 expð�2�Þ erfcð��1=2Þ ð4Þ

where n is the atomic ratio of oxygen to B (Si) in the oxide, �is the ratio of the diffusion coefficient of oxygen, DO, to thediffusion coefficient of B (Si), DB, and � is a dimensionlessparameter defined by x ¼ 2�ðDOtÞ1=2.

In the case of low Si alloys (i.e. NB(O) is small),

� � 1 and ��1=2 � 1; DB=DO < NO(s)=NB

(O) < 1;

the internal oxidation rate is expressed by eq. (3).On the other hand, in case of high Si alloys (i.e. NB

(O) islarge),

� � 1 and ��1=2 � 1; NO(s)=NB

(O) < DB=DO < 1

whereupon eq. (3) becomes

x ¼ ð�1=2DONO(s)=NB

(O)DB1=2nÞt1=2: ð5Þ

In this case, B concentrates as BOn in the oxide layer, and themolar concentration, f , of BOn is given by

f =NB(O) ¼ 2nNB

(O)DB=�NO(s)DO: ð6Þ

Furthermore, the volume fraction of BOn, g�, can beexpressed as

g� ¼ f ðVOX=VÞ ð7Þ

where V is the molar volume of the A-B alloy and VOX is themolar volume of BOn.

If the value of g� exceeds the critical value of 0.3, theinternal oxidation begins to change to external oxidation.10)

Substituting eq. (7) into eq. (6), the concentration of the alloyelement, B (Si), at which external oxidation begins to occurcan be found and is given by

NB(O)

= ð�g�NO(s)DOV=2nDBVOXÞ1=2: ð8Þ

2.3 Boundary conditions between internal and externaloxidation

If the internal diffusion rate of oxygen is higher than theexternal diffusion rate of the alloy element, B (Si), internaloxidation occurs, and if internal diffusion rate of oxygen islower or NB

(O) is higher, the external oxidation is disruptedby internal oxidation. Equation (9) is obtained by rearrangingeq. (8).

NO(s) ¼ 2nðNB

(O)Þ2VOXDX=�g�VDO ð9Þ

NO(s) seems to follow Sievert’s low depending on temper-

ature and is expressed as eq. (10).

NO(s) ¼ AðTÞPO2

1=2 ð10Þ

where A(T) is the function fixed by specifying temperature.The boundary conditions between internal and externaloxidation can be obtained by combining eqs. (9) and (10).The most appropriate g� value is reported to be 0.3.10)

3. Experimental Procedure

Ingots of Fe-Si alloy were prepared using vacuuminduction furnace. The chemical compositions of these alloysare shown in Table 1. The ingots were soaked at 1100�C for10 h, hot-forged at 1100�C for 2 h, hot-rolled at 1100�C for1.5 h, and descaled by sand blasting. Cylindrical sampleswith a diameter of 6mm and height of 10mm were preparedfor use in the oxidation experiments. The sides of thespecimens were polished with emery paper and finished bybuffing. Each specimen was cleaned ultrasonically in acetoneand ethanol before being oxidized.

Internal oxidation of Fe-Si alloys occurs below theequilibrium oxygen pressure of Fe/FeO (2:76� 10�13 Paat 850�C). The tests were conducted under a fixedpartial oxygen pressure at 850�C, and the partial oxygenpressure was controlled by using N2-3%H2 gas with thedew point.

Table 1 Chemical compositions of Si containing steels. (mass%)

Steel C Si Cr Mn P S

Fe-0.2mass%Si 0.09 0.20 <0:01 <0:03 <0:005 0.0023

Fe-0.5mass%Si 0.10 0.50 <0:01 <0:03 <0:005 0.0006

Fe-1.0mass%Si 0.11 1.00 <0:01 <0:03 <0:005 0.0031

Fe-2.0mass%Si 0.10 2.01 <0:01 <0:03 <0:005 0.0018

Calculations of Internal Oxidation Rate Equations and Boundary Conditions between Internal and External Oxidation 483

The oxidised cylindrical specimens were mounted andsectioned perpendicular to the cylinder axis, and the exposedsections were polished to a mirror finish with abrasive paper(#220–#1500) and alumina paste. The depth of internaloxidation was measured using a SEM.

4. Results and Discussion

4.1 Calculation of internal oxidation rate constantsWhereas the values of NB

(O) and n are known, the values ofDO and NO

(s) are unknown, so the most appropriate valueswere substituted in the internal oxidation rate constant,(2DONO

(s)=NB(O)n), to obtain the most accurate internal

oxidation rates. Takeda et al. reported the following equationfor DO by measuring the diffusion coefficients of oxygen inFe-0.06-90.274mass%Si alloys.11) In this study, the value ofDO was obtained by using eq. (11).

DO ¼ 2:91� 10�7 expf�89:5 ðkJ/molÞ=RTg ðm2/sÞ ð11Þ

Figure 2 shows SEM images of internally oxidized Fe-0.2mass%Si alloy specimens oxidized for 10min at 850�C infixed oxygen partial pressures. The internal oxide layers, inwhich very fine SiO2 particles are dispersed, can be observednear in the surfaces of specimens. The thickness of theseinternal oxide layers increases with increasing oxygen partialpressure. From the results, it is estimated that NO

(s), increaseswith increasing oxygen partial pressure.11–14)

The equation of NO(s) is set up as follows. The solubility

of oxygen in Fe seems to follow Sievert’s low depending ontemperature and is expressed as eq. (10). Takeda et al.evaluated NO

(s) on the basis of thermodynamic data andreported that the temperature dependence of NO

(s) underthe equilibrium oxygen pressure with Fe and FeO isshown to be given by eq. (12).14) Substituting the equi-librium oxygen pressure with Fe and FeO given in Table 2for PO2

in eq. (10), and combining eqs. (10) and (12), anexpression for A(T) can be obtained. This is given ineq. (13).12,13)

NO(s) ¼ 0:381 expf�104 ðkJ/molÞ=RTg ð12Þ

AðTÞ ¼ 9:67� 10�5 expf161:95 ðkJ/molÞ=RTg ð13Þ

Therefore, the oxygen concentration near to the surface ofSi containing steels (Fe-Si alloys), NO

(s), can be calculatedusing eqs. (10) and (13).

By using eqs. (10), (11) and (13), the calculated valuesof DO, NO

(s) and (2DONO(s)=NB

(O)n) of Fe-0.1mass%C-0.2mass%Si alloy in oxygen partial pressures of 3:42�10�15 Pa and 1:88� 10�14 Pa at 850�C are summarized inTable 3. DO is 1:99� 10�11 m2/s at 850�C. The calculatedvalues of NO


(O)n) are 6:096� 10�7

mol/mol and 3:033� 10�15 m2/s respectively, in an oxygenpartial pressure of 3:42� 10�15 Pa. In the same way, thecalculated values of NO


(O)n) are1:432� 10�6 mol/mol and 7:124� 10�15 m2/s respectively,in an oxygen partial pressure of 1:88� 10�14 Pa.

4.2 Comparison of calculated values with experimentalvalues for internal oxidation rate

Figures 3 to 6 show SEM images of internally oxidizedFe-0.2mass%Si alloy and Fe-0.5mass%Si alloy specimensoxidized at 850�C in oxygen partial pressures of 7:61�10�14 Pa and 2:76� 10�13 Pa. Very fine SiO2 particlesdispersed in the internal oxide layers are observed near thesurfaces in all cases. The thickness of the internal oxide layerincreases parabolically with increasing oxidation time.

Figures 7 to 10 show the internal oxidation rates in Fe-Sialloys. The calculated rates are also shown in Figs. 7 to 10.

Fig. 2 SEM images of internally oxidized Fe-0.2mass%Si alloy specimens oxidized for 10min at 850�C in fixed oxygen partial pressures

of (a) PO2¼ 2:03� 10�16 Pa, (b) PO2

¼ 3:42� 10�15 Pa and (c) PO2¼ 1:88� 10�14 Pa.

Table 2 Equilibrium oxygen pressures of Fe/FeO.

Temperature, T/�C 800 825 850 875 900

Equilibrium oxygen pressure1:96� 10�14 7:65� 10�14 2:80� 10�13 9:68� 10�13 3:17� 10�12

of Fe/FeO, PO2/Pa

Table 3 Calculated values of DO, NO(s) and (2DONO

(s)=NBð0Þn) of

Fe-0.1mass%C-0.2mass%Si alloy in oxygen partial pressures of

3:42� 10�15 Pa and 1:88� 10�14 Pa at 850�C.

PO2(Pa) DO (m2/s) N(s)

O (mol/mol) 2DON(s)O =N(O)

B n (m2/s)

3:42� 10�15 1:99� 10�11 6:096� 10�7 3:033� 10�15

1:88� 10�14 1:99� 10�11 1:432� 10�6 7:124� 10�15

484 T. Onishi, S. Nakakubo and M. Takeda

Fig. 3 SEM images of internally oxidized Fe-0.2mass%Si alloy specimens oxidized at 850�C for (a) 10min, (b) 30min and (c) 60min in

oxygen partial pressures of PO2¼ 7:61� 10�14 Pa.








Close agreement between the experimental and calculatedvalues is obtained for both Fe-0.2mass%Si alloy and Fe-0.5mass%Si alloy.

From these results, it is concluded that the calculationof the internal oxidation rate constant, (2DONO

(s)=NB(O)n),

using eqs. (10), (11) and (13) is valid, and the followingvalues of DO and N(s)

O are appropriate for internal oxidation at850�C.

DO ¼ 1:99� 10�11 ðm2/sÞNO

(s) ¼ 3:32� 103 ðPO2=P0Þ1=2

4.3 Calculation of boundary conditions between inter-nal and external oxidation

As described in the section 2. 3, the boundary conditionsbetween internal and external oxidation can be calculated bysolving eqs. (9) and (10) simultaneously. The boundaryconditions at 850�C are calculated using the two parametersof NB

(O) and PO2.11–14) The following values of the atomic

ratio of oxygen to Si, n, the molar volume of SiO2, VOX, thediffusion coefficient of Si in Fe, DX , the value of g� and themolar volume of Fe, V , were used in this calculation: n ¼ 2,VOX ¼ 2:311� 10�5 m3/mol, DX ¼ 4:28� 10�15 m2/s,g� ¼ 0:3, V ¼ 7:096� 10�6 m3/mol.

The calculated boundary conditions between internaland external oxidation in Fe-Si alloys at 850�C are shownin Fig. 11. The oxygen partial pressure is plotted on theordinate and the composition of the Fe-Si alloy is plotted

Fig. 7 Time dependence of thickness of internal-oxide layer in Fe-

0.2mass%Si alloy specimens in oxygen partial pressures of PO2¼

7:61� 10�14 Pa at 850�C.



2:76� 10�13 Pa at 850�C.



7:61� 10�14 Pa at 850�C.



2:76� 10�13 Pa at 850�C.

Fig. 11 Oxygen partial pressure vs Si concentration to illustrate transition

from internal to external oxidation in Fe-Si alloys at 850�C; exhibiting

internal oxidation; exhibiting internal and external oxidation;

exhibiting external oxidation; exhibiting outer scale formation.

486 T. Onishi, S. Nakakubo and M. Takeda

on the abscissa. The range of oxygen partial pressure inwhich internal oxidation occurs decreases with increasingSi concentration, and vanishes for Si concentrations of1.0mass% or more.

Figure 12 shows schematic representations of the scalestructure of specimens for the regions labeled in Fig. 11.Internal oxidation is restricted to region (c); externaloxidation occurs in region (d) in which the oxygen partialpressure is lower and the diffusion rate of oxygen slower thanin region (c). A very thin SiO2 layer forms on the alloysurface in region (d), because, in this region, the diffusionrate of Si is relatively high compared with that of oxygen. Nooxidation occurs in region (e), in which the oxygen partialpressure is lower than in region (d). On the other hand,fayalite, Fe2SiO4, or wustite, FeO, form in regions (b) and(a), respectively. No internal oxidation occurs in regions (b)and (a), in which the oxygen partial pressure is lower than inregion (c).

Thus, oxidation behavior of Fe-Si alloy strongly dependson the Si concentration and the oxygen partial pressure. Theoxidation situation determined by metallographic observa-tion is also shown in Fig. 11. Internal oxidation is limited toregion (c), and external oxidation or surface oxidation islimited to region (d) and (b). From these results, the boundaryconditions between internal and external oxidation as shownas Fig. 11 are confirmed to be valid.

5. Conclusion

In the present study, the rate constants of internal oxidationof Si containing steels (Fe-Si alloys) at 850�C werecalculated in order to clarify the formation mechanismof fayalite scales, which can form as a ‘‘sub-scale’’ in Sicontaining steels. The internal oxidation rate equation,x ¼ ð2DONO

(s)=NB(O)nÞ1=2t1=2, was used. The diffusion co-

efficient of oxygen in the oxide layer, DO, and the oxygenconcentration at the specimen surface, NO

(s), which areconstituents of internal oxidation rate constant, (2DONO

(s)=NB

(O)n), were selected, and the internal oxidation rates werecalculated under various oxidation conditions. The validity ofthe values of DO and NO

(s) was confirmed by comparing thecalculated results with experimental data.

The boundary conditions between internal and externaloxidation were calculated using the values of two parameters,DO and NO

(s), determined in this study. From the results, itwas confirmed that internal oxidation occurs in a limitedoxygen partial pressure range and is limited to Si concen-trations below 1.0mass%. The oxygen partial pressure range,in which internal oxidation can occur, decreases withincreasing Si concentration, and vanishes if the Si concen-tration is 1.0mass% or more.

Acknowledgements

The authors wish to express their thanks to ProfessorT. Maruyama and Assistant Professor M. Ueda at TokyoInstitute of Technology for their useful advice on this work.

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Fig. 12 Schematic representation of several scale structures of Fe-Si

alloys. (a) formation of FeO scale, (b) formation of internal-oxide layer

of SiO2 and Fe2SiO4 scale, (c) formation of internal-oxide layer of SiO2,

(d) formation of external scale of SiO2, (e) no oxidation.


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