in Advanced High Strength Steels
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op woensdag 10 september 2014 om 12:30 uur
door
Vahid AGHAEI LASHGARI
Master of Science in Materials Science and Engineering Delft University of Technology, The Netherlands
Copromotor: Dr.ir. W.G. Sloof
Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof.dr. B.J. Thijsse, Technische Universiteit Delft, promotor Dr.ir. W.G. Sloof, Technische Universiteit Delft, copromotor Prof. Dr.-Ing. H.U. Krupp, Hochschule Osnabrück
Prof.dr. Y.T. Pei, University of Groningen Prof. Dr.-Ing. W.J. Quadakkers, Forschungszentrum Jülich Prof.dr.ir. J. Sietsma, Technische Universiteit Delft
Dr. W. Melfo, TATA Steel IJmuiden
Prof.dr. I.M. Richardson, Technische Universiteit Delft, reservelid
Dr.ir. W.G. Sloof heeft als begeleider in belangrijke mate aan de totstandkoming van het proefschrift bijgedragen.
ISBN 978-94-91909-13-9
Copyright © 2014 by Vahid Aghaei Lashgari
All rights reserved. No part of the material protected by this copy right notice may be reproduced or utilized in any form or by any means, electronically or mechanically, including photocopying, recording or by any information storage and retrieval system, without written permission from the author.
in Advanced High Strength Steels
This research was carried out under project number MC7.07295a in the framework of the Research Program of the Materials innovation institute (M2i) (www.m2i.nl).
Front cover image after:
Mataigne, J.M., M. Lamberigts, and V. Leroy. Selective oxidation of cold-rolled steel during recrystallization annealing. in Developments in the annealing of sheet steels. Cincinnati, Ohio, 1991, p.511.
I
Chapter 1: Introduction 1
1.1. Background ……….. 1
1.2. Outline of the thesis ……….. 7
References ………... 9
Chapter 2: Internal vs. External oxidation revisited 11 2.1. Oxidation modes ……….. 12
2.2. Internal oxidation in the absence of an external oxide scale ……… 15
2.3. Internal oxidation in the presence of an external scale ……… 22
2.4. Oxygen diffusion ……….. 26
2.5. Solute enrichment ………. 27
2.6. Transition from internal to external oxidation ………. 28
2.7. Direct formation of an external scale ………... 30
2.8. Nucleation of internal oxide precipitates ………. 31
2.9. Internal precipitates of an oxide with high solubility product …………. 38
References ………... 43
Chapter 3: Transition from internal to external oxidation of Mn steel alloys 47 3.1. Introduction ……….. 48
3.2. Experimental procedures ……….. 49
3.2.1. Materials and sample preparation ………. 49
3.2.2. Oxidation experiment ……… 50
3.3. Microstructure and composition analysis ………. 51
3.4. Background ……….. 53
3.5. Results and discussion ……….. 55
3.5.1. Oxide precipitation behaviour ………... 55
3.5.2. Critical volume fraction of internal oxide precipitates ……….. 57
3.6. Conclusions ……….. 64
II
Chapter 4: Kinetics of internal to external oxidation of Mn-steel alloys 69
4.1. Introduction ……….. 70
4.2. Experimental procedures ……….. 71
4.2.1. Materials and sample preparation ………. 71
4.2.2. Oxidation experiment ……… 71
4.2.3. Microstructure and composition analysis ……….. 73
4.3. Theoretical background of internal oxidation ……….. 74
4.4. Modelling of internal oxidation ……… 76
4.5. results and discussions ………. 79
4.5.1. Parabolic rate constant ……..……… 79
4.5.2. Diffusion coefficient of oxygen in the austenitic iron ………... 81
4.5.3. Diffusion coefficient of manganese in the austenitic iron ……. 82
4.5.4. Solubility product of MnO in the austenitic iron ……….. 85
4.5.5. Depth profiles, modelling vs. experimental data ………... 87
4.6. Simulated internal oxidation composition depth profiles ……… 89
4.6.1. Effect of Mn concentration and dew point …………... 90
4.6.2. Effect of solubility product …..………. 92
4.6.3. Effect off phase transformation ………. 92
4.7. Conclusions ……….. 94
Acknowledgments ………... 95
References ….……….. 96
Chapter 5: Composition of oxides formed during annealing of Mn steels in N2- H2-H2O atmosphere 99 5.1. Introduction ………..……… 100
5.2. Experimental procedures …..……… 101
5.2.1. Materials and sample preparation ….……… 101
5.2.2. Oxidation experiments ……….. 102
III
5.3.3. Electron microscopy and X-ray microanalysis (morphology) .. 105
5.3.4. X-ray photoelectron spectroscopy ….……… 106
5.4. Results and discussion ….………. 106
5.4.1. Oxide composition and defect structure ……… 107
5.4.2. Effects of oxidation conditions on the oxide composition …… 115
5.5. Conclusions ….………. 118
Acknowledgments …..………. 119
References ……….……….. 120
Appendices 125 Appendix A. Dew point ……….. 125
Appendix B. Auxiliary functions ….………... 126
Appendix C. Gibbs free energy ………... 127
Appendix D. Oxygen solubility in Fe ….……… 128
Appendix E. Spatial size distribution of internal oxide precipitates ………... 129
Appendix F. Auger parameter ….……… 130
References ………... 131
Summary 133
Samenvatting 137
Publications 141
1
Chapter 1
Introduction
1.1. Background
There are different ways to classify steels, one important classification is based on the strength of steels including low strength steels, conventional high strength steels (HSS) and advanced high strength steels (AHSS). HSS have yield strength from 210 to 550 MPa and tensile strengths from 270 to 700 MPa, whereas AHSS have yield and tensile strengths greater than 550 and 700 MPa, respectively. In contrast to these arbitrary defined ranges that suggest discontinuous change of strength from one category to another, strength of steels show a continuum across the entire steel strengths. Hence, many steel grades have wide ranges of strength that covers more than one category [1].
AHSS have been used extensively in the automotive industries. The main characteristic of these steels is combination of high strength and enhanced formability that makes them very attractive for automotive applications. These applications result in a production of safer and lighter automobiles. Hence, less fuel is consumed and besides the economical importance of that, it causes a reduction in the emission of greenhouse gases (GHG) such as carbon dioxide. According to the life cycle assessment (LCA) of the World Steel Association [2], there would be a 156 million tonnes reduction in the CO2 amount in 2008,
if the body structures of all the cars produced worldwide were made of AHSS. Replacing conventional steels with AHSS in the car body, results in 17 to 25 % mass saving for body in white (BIW) which equals to 117 kg or 9 % overall weight reduction for a typical five-passenger family car. This corresponds to a lifetime saving of 2.2 tonnes of CO2 per
vehicle. This saving in the emission of CO2 gas is more than the total amount of the CO2
2
a 1.9 to 8.2 % improvement in the fuel economy (litres of fuel per 100 km driving distance) for every 10 % reduction in the vehicle weight. For a typical five-passenger family car the resulting fuel economy is about 5.1 %. Furthermore, the use of the vehicles is responsible for about 10 % of the total man-made GHG emissions and for every 1 kg of AHSS used in the vehicle, total life cycle saving of 8 kg GHG are achieved. This is a 5.7 % reduction in GHG emissions over the full life cycle of the vehicle; cf. Figure 1.1.
Figure 1.1 Advantages of the advanced high strength steels (AHSS) compared to the conventional steels. Here the body in white (BIW), mass, fuel consumption and life cycle assessment (LCA) greenhouse gases (GHG) emissions are considered [2].
Other lightweight materials like aluminium, magnesium, polymers and carbon fibers have been considered as replacement for steels in many applications including those in the automotive industry. However, extraordinary fatigue strength of steels with respect to other lightweight materials and their recyclability, makes steel an excellent material for automotive applications [3]. In addition, many of the most potent green house gases such as perfluorocarbons and sulphurhexafluoride, have a much stronger global warming effect than CO2 and they are emitted during the production of aluminium and magnesium [2].
Moreover, GHG emission during the first and secondary material production is not the same for different materials. In fact, in contrast to other alternative materials, steel’s primary and secondary production emit 5 and 2 times, respectively, less GHG emissions than aluminium [4].
3 Figure 1.2 Equivalent emissions from the production of materials used in the vehicle manufacture, including carbon fiber reinforced plastics (FRP) [4].
The use of AHSS offers savings in both GHG emissions and cost with respect to the aluminium. However, use of aluminium has advantages in terms of the BIW mass, vehicle mass and fuel consumption, cf. Figure 1.3 [4].
Figure 1.3 GHG emissions and costs increase if aluminium is used instead of advanced high strength steels. The baseline for the comparison is the emissions and costs of AHSS [4].
The benefits of using steels instead of other materials are not limited to the body parts of an automobile. In life-cycle assessment reported by the TATA Steel [5], the front-end
4
module1 from a C-class car was chosen. This module can be designed in a variety of materials such as galvanized steel, aluminium and two steel/glass fiber reinforced plastics (referred to as composite-1 and composite-2). Evaluation of mass, cost and life-cycle carbon footprint for these different material solutions is summarized in Figure 1.4 [5]. While the steel design is not the lightest, it has significant benefits in terms of manufacturing costs and life-cycle carbon footprint.
Figure 1.4 Assessment of mass, cost and life-cycle carbon footprint of steel, two steel/glass fiber reinforced plastics (referred to as composite-1 and composite-2) and aluminium for front-end module of a C-class car [5].
Besides the benefits of using AHSS in automotive applications, the major drawback is their poor corrosion resistance. To guarantee the use of AHSS sheets as the car body parts, the corrosion is of high importance and is mainly assured by hot-dip galvanizing [6, 7]. The continuous hot-dip galvanizing process applies a zinc coating to the surface of a continuous ribbon of steel sheet as it passes through a zinc bath [8]. During the hot-dip galvanizing process (cf. Figure 1.5), many different heat treatments and processes are used in which the steels are exposed to high temperature oxidizing environments and either inert atmosphere. In these processes, the alloying elements and impurities are
1 This module is positioned at the front of the vehicle to support the radiator pack, fan, ducting, bumper beam, bumper moulding and the headlamps.
5 enriched at the steel surface which strongly affect the properties and the reactivity of steel surface [9].
Figure 1.5 Schematic view of the annealing parts of the continuous hot-dip galvanizing line [10]. The gas atmosphere in DFF and RTF is oxidizing for Fe and reducing for Fe-oxide (wüstite), respectively. Abbreviations: PHF (Pre Heating Furnace); TC (Top Chamber); DFF (Direct Fire Furnace); RTF (Radiant Tube Furnace); and JCF (Jet Cooled Furnace).
Figure 1.6 displays the stability diagram for iron and some of the alloying elements in terms of temperature versus the oxidizing power of the environment expressed as dew point (see Appendix A). Each line in the figure separates the stability region of the alloying element and its corresponding oxide. It is clear that for the industrial working conditions, most of the alloying elements used in AHSS form very stable oxides. Although the annealing atmosphere of the continuous galvanizing line (CGL) is oxidizing for the alloying elements such as Al, Si, Mn and Cr, it is reducing for iron oxide (cf. RTF section in Figure 1.5).
6
Figure 1.6 Stability diagram for iron and alloying elements, in this diagram mole fraction was considered as an activity (ideal behaviour) and dew point was given on the basis of N2 with 5 vol.% H2 gas mixture.
Thus, the alloying elements in AHSS tend to segregate to the surface upon annealing and form stable oxides [11]. Presence of the oxides of the alloying elements at the surface reduces the wettability between the molten zinc and the steel [12, 13]. Poor wetting between the liquid zinc and the steel surface causes delamination of the zinc coating (cf. Figure 1.7) and occurrence of the bare spots (cf. Figure 1.8).
Figure 1.7 Delamination of the zinc coating (courtesy of TATA steel).
Figure 1.8 Visual appearance of galvanized CMnSi transformation induced plasticity (TRIP) steel [14].
7 As clearly shown in Figure 1.6, formation of the stable oxides on the surface is inevitable and reduction of them is not feasible from industrial and economical point of view. Hence, an alternative solution is to screen the formed external oxides with a barrier such as iron. By increasing the oxidizing power of the annealing line beyond the equilibrium oxygen partial pressure for wüstite (FeO) formation (DFF section in Figure 1.5), wüstite can be formed at the steel surface. The wüstite layer can be reduced completely during the subsequent annealing line (i.e. RTF section in Figure 1.5). Consequently, the steel surface is covered with a layer of iron at the end of the process and can be galvanized satisfactorily, as depicted in Figure 1.9.
(a) (b)
Figure 1.9 Schematic representations of changes of the surface region in a successful industrial annealing process, (a) initial oxidation of the steel surface to form the wüstite layer, (b) selective reduction of the wüstite layer.
Thus, understanding the general aspects of the oxidation phenomena in terms of the thermodynamics and kinetics, evaluating the oxidation modes of the alloying elements in AHSS and finding the appropriate conditions are critical in order to provide an oxide-free surface prior to galvanizing.
1.2. Outline of the thesis
In this thesis, the basic thermodynamic and kinetic aspects of internal oxidation are reviewed in Chapter 2. The transition between internal and external oxidation, specifically the critical volume fraction of the internal oxide precipitates, will be
8
discussed in more detail in Chapter 3. The kinetics of internal oxidation of Mn-steel alloys and the corresponding depth profiles are dealt with in Chapter 4. The chemical composition of the formed oxides in Mn-steel alloys will be addressed in Chapter 5. Useful thermodynamic data related to the oxidation are given in the Appendices.
9
References
1. World Steel Association. AHSS application guidelines. 2009; 4.1: Available from:
http://www.worldautosteel.org/projects/AHSSGuidelines/AHSS-application-guidelines-version-4.aspx.
2. World Steel Association. Environmental case study Automotive. 2008; Available from:
http://www.worldsteel.org/pictures/programfiles/Automotive%20case%20study.p df.
3. Silva, F., N.I.A. Lopes, and D.B. Santos, Microstructural characterization of the C–Mn multiphase high strength cold rolled steel. Materials Characterization, 2006.
56(1): p. 3-9.
4. World Auto Steel. 2009; Available from: http://www.worldautosteel.org/.
5. TATA-Steel. Sustainable steel for cars. 2013; Available from:
http://www.tatasteelautomotive.com/file_source/StaticFiles/Automotive/new-2013/sustainable-steel-for-cars-case-study.pdf.
6. Mahieu, J., J. Maki, B.C. De Cooman and S. Claessens, Phase transformation and mechanical properties of Si-free CMnAl transformation-induced plasticity-aided steel. Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science, 2002. 33(8): p. 2573-2580.
7. Maki, J., J. Mahieu, B.C. De Cooman and S. Claessens, Galvanisability of silicon free CMnAl TRIP steels. Materials Science and Technology, 2003. 19(1): p. 125-131.
8. GalvInfo Center. Available from: http://www.galvinfo.com/galv_info_notes.htm. 9. Grabke, H.J., V. Leroy, and H. Viefhaus, Segregation on the surface of steels in
heat treatment and oxidation. ISIJ International, 1995. 35(2): p. 95-113. 10. TATA Steel. Available from: http://www.tatasteel.nl/.
11. Song, G.M., W.G. Sloof, T. Vystavl and J.T.M. De Hosson, Interface microstructure and adhesion of zinc coatings on TRIP steels. Materials Science Forum, 2007. 539-543: p. 1104-1109.
12. Van De Putte, T., D. Loison, J. Penning and S. Claessens, Selective oxidation of a CMnSi steel during Heating to 1000 °C: reversible SiO2 oxidation. Metallurgical
10
and Materials Transactions A: Physical Metallurgy and Materials Science, 2008.
39(12): p. 2875-2884.
13. Suzuki, Y., T. Yamashita, Y. Sugimoto, S. Fujita and S. Yamaguchi, Thermodynamic analysis of selective oxidation behavior of Si and Mn-added steel during recrystallization annealing. ISIJ International, 2009. 49(4): p. 564-573. 14. Mahieu, J., S. Claessens, and B.C. De Cooman, Galvanizability of high-strength
steels for automotive applications. Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science, 2001. 32(11): p. 2905-2908.
11
Chapter 2
Internal vs. External oxidation revisited
Abstract
The analytical description of internal oxidation of binary A-B alloys and the transition to external oxidation are based on the classical theory that was first developed by C. Wagner [1]. However, this theory is limited to oxide precipitation with a low solubility of the oxide in the parent matrix and binary alloys that are single phase and single crystalline. Also these alloys are considered to behave as ideal solid solution. In this chapter, aspects and consequences will be addressed of the presence of an external oxide scale, nucleation of new oxide precipitates and supersaturation at the oxidation front on the analytical description of the alloy oxidation. These aspects and consequences will be illustrated with examples pertaining to the oxidation of Fe-Mn alloys.
Keywords
12
2.1. Oxidation modes
When a homogenous binary alloy A-B is exposed to an oxidizing atmosphere, preferential oxidation may occur if the oxygen partial pressure is sufficient to oxidize B, but not A.
Consider a single-phase binary alloy A-B reacting with oxygen at temperature T and pressure
2
O
p to form pure oxide phase(s) A AO and/or B BO according to [2]: ( ) 2 ( ) ( ) 2 2 M s g s M M M O MO
M A B,
(2.1) with 0 0 B A BO AO G G ,where 0 M MO G denotes the standard Gibbs free energy of the formation of an oxide
M
MO per mole of O2 and M is stoichiometry of the concerning oxide. The equilibrium partial pressure of oxygen for which the oxide is in local equilibrium with the alloy at the oxide/alloy interface is given by:
2 2 0 exp M M M M MO MO MO O M a G p a RT (2.2) where M MO
a and aM are activities of metal oxide and metal, respectively,R is the gas
constant and T is the absolute temperature. In general, the activity of component i is related to its mole fraction according to:
i i i
a N (2.3)
where i and Ni are the activity coefficient and mole fraction of the component i in the
system, respectively.
Furthermore, high oxygen partial pressures are defined as the oxygen partial pressures above the equilibrium oxygen partial pressure of the less stable oxide, i.e. the oxide of the most noble element (matrix A). On the contrary low oxygen partial pressures are denoted
13 as the oxygen partial pressures above the equilibrium oxygen partial pressure of the most stable oxide, i.e. the oxide of the most reactive element (alloying element B) and below that of matrix.
Since 0 0
B A
BO AO
G G
, according to Eq. (2.2) it holds that:
2 2
B A
BO AO
O O
p p (2.4)
If the solubility of oxygen in the matrix (A) is appreciable, then the formation of B-oxide may take place inside the metal. This phenomenon is called internal oxidation, cf. Figure 2.1a. However, if B atoms diffuse through the solid solution to the surface and form B-oxide at the surface then the oxidation is called external oxidation, cf. Figure 2.1b [3].
Figure 2.1 (a) Schematic representation of internal oxidation of the alloying element B in A-B alloy.
Figure 2.1 (b) Schematic representation of external oxidation of the alloying element B in A-B alloy.
For a binary alloy A-B (A: more noble metal than B, i.e. B as reactive metal), if the solubility of oxygen in the matrix is appreciable then B-oxide may be formed inside the alloy (denoted as [BO], by reaction of dissolved B atoms with O atoms, according to:
[ ]B [ ] [O BO] (2.5)
where the standard Gibbs free energy of this reaction is written as:
[ ] 0 [ ] [ ] [ ] ln BO BO B O a G RT N N (2.6)14
and where N[ ]B and N[ ]O are the mole fraction of the dissolved alloying element and the dissolved oxygen in the matrix, respectively. is the number of oxygen atoms per one atom of the alloying element in the formed oxide compound (for simplicity the stoichiometry for the oxide of the alloying element is denoted as instead of B throughout this thesis). The product of the concentration of the dissolved alloying element and oxygen is also known as solubility product Ksp, hence:
[ ] [ ]
sp B O
K N N (2.7)
and thus (cf. Eq. (2.6)):
[ ] 0 [ ] ln BO BO sp a G RT K (2.8)
The standard Gibbs free energy of the reaction in Eq. (2.5) can be obtained by considering the following reactions:
( )s [ ] ( in solid A) B B [ ] [ ] [ ] ln [ ] xs B B B B G H T S RT N (2.9) 2 ( ) ( ) 1 [ ] 2O g O in solid A [ ] [ ] [ ] ln [ ] xs O O O O G H T S RT N (2.10) ( ) 2 ( ) ( ) 2 s g s B O BO G0BO (2.11) ( )s [ ] BO BO 0 p G (2.12) Then: ( ) 0 0 0 [ ] [ ] [BO] B O BO s p G G G G G (2.13) where 0 p G
is the free energy change related to the precipitation and has contributions from volume change, interface energy, misfit strain energy and defect site annihilation [4]. The mole fraction is considered as the standard state for the dissolved alloying element and dissolved oxygen. The partial molar enthalpy and the partial molar excess entropy of mixing of component i in the solid solution are expressed as Hi and
xs i S
15 The mole fraction of dissolved oxygen in the alloy and at its surface ( )s
O
N , see Eq. (2.10), is related to the partial pressure of oxygen at the surface according to the Sievert’s law [5]:
21/ 2 ( )s
O p O
N K p (2.14)
where the equilibrium reaction constant Kp is related to the standard Gibbs free energy
change as:
0 ln
p
G RT K
(2.15)
In the absence of any external oxide layer, the partial pressure of oxygen at the surface equals to the partial pressure of oxygen in the ambient. However, in the presence of an external oxide layer, the partial pressure of oxygen at the metal/oxide interface is controlled by the local equilibrium between the oxide and the metal (alloy). Hence, the maximum solubility of oxygen in a binary A-B alloy is then determined by the equilibrium between alloy and the less stable oxide at the metal/oxide interface.
2.2. Internal oxidation in the absence of an external oxide scale
For a binary alloy A-B, if the solubility of oxygen in the matrix is appreciable then reaction of dissolved B atoms with O atoms results in the formation of B-oxide inside the alloy. This phenomenon is known as internal oxidation.
For internal oxidation, two requirements must be met [3]: (i) sufficiently higher affinity of oxygen to B than to A
0 0
B A
BO AO
G G
, and (ii) higher diffusivity of O in A than B in A
DO DB
.The fluxes of solute element and oxygen in the A-B alloy associated with internal oxidation are described with Fick’s first law, which in one dimension reads:
i i i C J D x ; i[ ]O or i[ ]B (2.16)
16
, where J is the flux (net rate) per unit area of a plane oriented perpendicular to the diffusion direction x, C the concentration and D the diffusion coefficient of diffusing species i. Consequently, the flux depends besides the diffusion coefficient on the concentration gradient.
The distribution of oxygen and the selectively oxidized element (B) in the binary alloy can be resolved by adopting Fick’s second law, which in one dimension reads:
2 2 i i i C C D t x ; i[ ]O or i[ ]B (2.17)
Since the concentration of an element Ci , is defined as:
i i A B m N C V (2.18) , where A B m
V is the molar volume of the alloy, Fick’s second law may also be expressed in terms of mole fraction Ni, hence:
2 2 i i i N N D t x ; i[ ]O or i[ ]B (2.19)
Then, the molar volume of the alloy is considered constant, which is not always the case. Assuming that the previously precipitated oxide particles, BO , do not interfere with the oxygen diffusion in the matrix, Eq. (2.19) can be solved for the following boundary conditions, cf. Figures 2.2 and 2.3:
( )s O O N N for x0, t0 (2.20) 0 O N for x, t0 (2.21) (0) B B N N for x0, t0 (2.22) 0 B N for x, t0 (2.23)
, wherex is the distance from the outer surface, is the depth of internal oxidation zone (IOZ), NO is the mole fraction of oxygen in the base metal, NO( )s is the mole fraction of
17 oxygen at the external surface, NB is the mole fraction of the alloying element, and (0)
B N
is the mole fraction of the alloying element in the bulk alloy [6]. Then the solutions of Eq. (2.19) are:
( ) 1 / 2 O s O O erf x D t N N erf for 0 x (2.24)
(0) 1/ 2 / 2 1 B B B erfc x D t N N erfc for x (2.25) , with: O B D D (2.26), where is a dimensionless time-dependent kinetic parameter, erf is the error function and erfc is the complementary error function [5, 6].
If the precipitation of BO takes place solely at the reaction front, i.e. at x, then the flux of the O atoms arriving from the external surface must equal to the flux of B atoms arriving from the bulk of the alloy. In other words, the mass balance at the reaction front indicates that the fluxes of O and B towards the interface are equal [5], thus:
0
lim JO x JB x
(2.27)
Using Eqs. (2.16) and (2.18) reads:
0 lim O B O B x x N N D D x x (2.28)
Here, is a very small increment in x, used to indicate that the fluxes are evaluated very close to, but on the opposite sides of the reaction front [5].
18
Substitution of Eqs. (2.24) and (2.25) into Eq. (2.28) and differentiation leads to:
2 ( ) (0) 1/ 2 2 1/ 2 exp exp s O B erf N N erfc (2.29)The parameter can be obtained numerically from this equation, since the mole fraction of oxygen at surface ( )s
O
N (see Eq. (2.14)) and the mole fraction of the alloying element in the bulk (0)
B
N , as well as the ratio of their respective diffusion coefficients (see Eq. (2.26)) are known.
For convenience two auxiliary functions F and G are used to express Eq. (2.29):
( ) (0) 1/ 2 s O B G N N F (2.30)The auxiliary functions are defined as, respectively:
2
exp F u u u erfc u (2.31)
exp
2
G u u u erf u (2.32)The key factor that determines the oxidation mode is the permeability of oxygen as well as of the permeability of the alloying element. Here, the permeability is defined as the product of the diffusion coefficient and the concentration (in mole fraction) of the element concerned. When considering the permeability of oxygen and of the alloying element, two limiting conditions can be identified:
i. a much higher permeability of oxygen, ( )s (0)
O O B B
N D N D
ii. a much higher permeability of the alloying element, (0) ( )s
B B O O
N D N D
These two cases lead to the simplified solutions for the concentration depth profiles of the oxygen and the alloying element. Note that in these limiting cases, both permeabilities are very small, i.e. ( )s 1
O O
N D and (0) 1
B B
N D , respectively.
19 In the limiting conditions where:
( )s (0)
O O B B
N D N D (2.33)
then, it can be shown that (cf. Appendix B): 1 and 1/2 1, hence Eq. (2.29) can
be approximated as: ( ) 2 (0) 2 s O B N N (2.34)
In the other limiting conditions where diffusion of the alloying element is significant,
(0) ( )s
B B O O
N D N D (2.35)
then, it can be shown that (cf. Appendix B): 1 and 1/21, hence Eq. (2.29) can
be approximated by: ( ) (0) 1/ 2 2 s O B N N (2.36)
The concentration profiles of oxygen and the alloying element for these two limiting conditions are plotted in Figure 2.2 and Figure 2.3, respectively.
Figure 2.2 Concentration profiles of oxygen and alloying element after internal oxidation for the case that the oxygen permeability is much higher than of the alloying element. Nrepresents the mole fraction and xis the distance measured with respect to the original surface. The depth of internal oxidation zone ( ) is indicated.
20
Figure 2.3 Concentration profiles of oxygen and alloying element after internal oxidation for the case that the permeability of the alloying element is much higher than that of the oxygen. Nrepresents the mole fraction and xis the distance measured with respect to the original surface. The depth of internal oxidation zone ( ) is indicated.
As indicated in Figs. 2.2 and 2.3, the concentration of oxygen at the front of the IOZ is assumed to be zero for a very low solubility product oxide, hence:
0
O
N at x (2.37)
Considering that the concentration of oxygen at the surface is not zero (i.e. ( )s 0 O
N ) and combining Eqs. (2.24) and (2.37), it follows that:
/ 2 1 erf D tO 0 erf (2.38)After rearranging, it holds that:
2 D tO
(2.39)
The diffusion coefficient of oxygen and therefore kinetics of internal oxidation are affected by the presence of internal oxide precipitates (see Section 2.4).
21 As generally holds for diffusion-controlled phenomena, also the kinetics of internal oxidation is a parabolic function of time, see Eq. (2.39). By defining ( )i
p
k as the parabolic growth rate constant for internal oxidation, Eq. (2.39) becomes:
2 2 ( )i p k t
(2.40)
The kinetics of internal oxidation is obtained by calculating the value of from the general solution, i.e. Eq. (2.29). However, for the two limiting conditions the values of are obtained from Eqs. (2.34) and (2.36), respectively. Then, the following parabolic rate constants are obtained:
( ) ( ) (0) s i O O p B N D k N
(i) much higher permeability of the oxygen (2.41)
2 ( ) ( ) (0) 2 s i O O p B B N D k D N (ii)
much higher permeability of the alloying
element (2.42)
For Fe-Mn binary alloys containing 1.7, 3.5 and 7.0 wt% Mn at 950 °C, Figure 2.4 shows the ratio of the gamma’s general value (obtained from Eq. (2.29)) to the simplified value (obtained from Eqs. (2.34) and (2.36)) as a function of the oxygen partial pressure expressed as dew point (see Appendix A).
At very high or low dew points the deviation between the general and simplified solution is negligible. However, a minimum exists for the ratio of the gamma’s general value to the simplified value, which corresponds to the largest error when using the simplified relations. This minimum is located within the internal oxidation domain of these alloys (cf. Section 2.6).
22
Figure 2.4 Deviation of simplified solutions of gamma value from general solution of that as a function of dew point for Fe-Mn binary alloys at 950 °C.
2.3. Internal oxidation in the presence of an external scale
When the oxygen partial pressure in the ambient is higher than the equilibrium oxygen partial pressure of the matrix element of the A-B alloy as specified in Section 2.1, i.e.
2 2
A
AO ambient
O O
p p , the surface of the alloy may be covered with a scale of A
AO .
The partial pressure of oxygen required for formation of an
A
AO scale will be sufficient to oxidize B as well, since:
2 2 2
B A
BO AO ambient
O O O
p p p (2.43)
Considering Gibbs free energy of formation for MnO and FeO [7], equilibrium partial pressures of oxygen at 950 °C for binary Fe-Mn alloys studied in this thesis are presented in Table 2.1. The activities of Fe and Mn in the alloys are calculated using the Thermo-Calc PKP v1 database [8].
23 Table 2.1 Equilibrium partial pressure of oxygen for formation of MnO and FeO at 950 °C for the Fe-Mn binary alloys. Mn (wt%) 2 MnO O p (atm.) 2 FeO O p (atm.) 1.7 1.44×10-22 1.67×10-16 3.5 3.44×10-23 1.73×10-16 7.0 8.83×10-24 1.86×10-16
In some other practical cases, internal oxidation occurs in the presence of an external scale. For example, dilute Ni-Cr, Ni-Al, Fe-Cr and Fe-Al alloys form external scales together with internal oxides of Cr or Al [5]. A schematic view of this type of oxidation and the related concentration profiles are shown in Figure 2.6.
Figure 2.5 shows an electron image from the cross-section of Fe - 1.7 wt% Mn alloy oxidized at 950 °C for 10 minutes. The oxidation was carried out at a dew point of 50 °C (equivalent to oxygen partial pressure of 2.38×10-15 atm.), which favours formation of wüstite, cf. Table 2.1. The wüstite layer and internal oxide precipitates are seen in Figure 2.5.
Figure 2.5 Formation of internal oxide precipitates underneath the wüstite layer. This steel alloy contains 1.7 wt% Mn and oxidized at 950 °C at a dew point of 50 °C for 10 minutes.
24
Figure 2.6 Schematic representation an A-B alloy with internal oxidation BOin the presence of an external
A
AO scale. The concentration profiles of oxygen and the alloying element B are indicated as NO
and NB respectively. The original alloy surface is considered as the origin of the xaxis. Xc and are depth of external scaling and internal oxidation zone, respectively. Both depths are measured with respect to the original gas/alloy interface. Here for a general case the concentration profiles for oxygen and the alloying element are shown [5]. However, when the solubility product (cf. Eq. (2.7)) is very small (i.e. stable oxide precipitates), then complete precipitation occurs in the IOZ; hence, NO at the internal oxidation front and NB in the IOZ are virtually zero.
The distribution of oxygen and the selectively oxidized element (B) in the binary alloy can be resolved also in this case using Fick’s second law, see Eq. (2.19). However, the boundary conditions must be adopted to the case of internal oxidation in the presence of an external scale (see Figure 2.6), thus:
2 2 O O O N N D t x For Xc x (2.44) 2 2 B B B N N D t x For x (2.45)
The boundary conditions are:
( )s O O N N For x X c, t0 (2.46) 0 O N For x, t0 (2.47) (0) B B N N For x0, t0 (2.48)
25 0
B
N For x, t0 (2.49)
All the distances in the x direction are measured with respect to the original alloy surface, i.e. at x0.
Usually, the recession of the external oxide layer follows a parabolic rate law [9], hence:
2 2
c c
X k t (2.50)
In which the parabolic rate constant can be written as:
2 A A B m c p AO m V k k V (2.51)
, where kpis the parabolic growth rate constant for external oxidation of A, A B m
V molar volume of the alloy and AOA
m
V molar volume of the oxide.
When both internal oxidation and external scaling follow a parabolic rate law, Eq. (2.40) and Eq. (2.50), respectively, then solving the Fick’s second law (cf. Eqs. (2.44)-(2.45)) for the relevant boundary conditions (cf. Eqs. (2.46)-(2.49)) results in the evolution of the concentration profiles of the oxygen and the solute element [10].
Hence:
( ) / 2 / 2 O s O O c O erf erf x D t N Nerf erf k D for Xc x (2.52)
(0) 1/ 2 / 2 1 B B B erfc x D t N N erfc for Xcx (2.53)The mass balance at the reaction front, see Eq. (2.28), is also obeyed for the case considered here. Now, inserting Eqs. (2.52) and (2.53) into Eq. (2.28) and considering Eq. (2.39), it is obtained that [10]:
26
2 ( ) (0) 1/ 2 2 1/ 2 exp / 2 exp s c O O B erf erf k D N N erfc (2.54)Note that in the absence of an external scale, i.e., kc0, the derived equations will
reduce to the equations for the case in which only the internal oxide is present, compare Eqs. (2.52), (2.53) and (2.54) with Eqs. (2.24), (2.25) and (2.29), respectively.
2.4. Oxygen diffusion
Internal oxidation studies have been proven to be a useful method for determining the diffusivity of oxygen in metals and alloys [11-17]. In general, the diffusion of oxygen in the internal oxidation zone DOis influenced by the presence of internal oxide precipitates and consequently the oxide/metal phase boundaries. By defining fBO as the volume fraction of internal oxide particles, diffusion of oxygen increases with fBOas:
matrix
O O BO
D D bf
(2.55)
where b is a constant and matrix O
D is the diffusion coefficient of oxygen in the precipitate free alloy.
The increase of the diffusion coefficient of oxygen with the volume fraction of formed internal oxides, can be explained by the enhanced diffusion of oxygen along the internal oxide-matrix interface. Generally, diffusion along an interface is more rapid than that through a lattice. Hence, calculated diffusion of oxygen from Eq. (2.39) is affected by the presence of internal oxide precipitates. Consequently, by extrapolating the diffusion data to zero volume fraction of internal oxide precipitates, the diffusion coefficient of oxygen in the matrix can be obtained. Enhanced diffusion of oxygen along the internal oxide-matrix interface is significant in the case of elongated rod shape internal oxide precipitates, such as those in Ni-Al alloys [18]. For Fe-Mn alloys with MnO precipitates, the diffusion coefficient of oxygen was determined at 950 °C and equals to 3.58×10-7 cm2sec-1 (see Chapter 5).
27
2.5. Solute enrichment
When virtually no diffusion of the alloying element happens, cf. Figure 2.2, a negligible enrichment of the internal oxidation zone (IOZ) by the alloying element as oxide takes place. However, if the diffusion of B is appreciable, cf. Figure 2.3, then a significant enrichment of the IOZ by B as BOν will occur and consequently depletion of solute
element from the unoxidized alloy will result. The degree of solute enrichment (B) in
the IOZ is defined as [1]:
(0) BO B B N N (2.56)
, where NBO is the mole fraction of the BO precipitates in the IOZ.
If NBO
represents the mole fraction of BO in the IOZ and
A B m
V is the molar volume of
the alloy, then / A B
BO m
N V
will be the molar concentration per unit volume. When the
internal oxidation front with an area A advances by a distance d with time dt then the number of moles in a volume element Ad will be
/ A B
BO m
N V Ad
. This quantity
must be equal to the number of moles of B arriving at the internal oxidation front by diffusion from within the alloy, i.e. from x. Hence [1, 6, 19]:
0 lim BO B A B m N Ad AJ dt V (2.57), which with Eqs. (2.16) and (2.18) reads:
0 lim BO B B A B A B x m m N D N Ad A dt V V x (2.58)
Now, by considering the derivatives of Eqs. (2.25) and (2.39), it holds that:
(0) 1/ 2 2 1/ 2 1 exp BO B N N erfc (2.59)28
Then, using the definitions of an enrichment factor (see Eq. (2.56)) and the auxiliary function, cf. Eq. (2.31) it follows that:
1/ 2
1 B F (2.60)The mole fraction of internal oxide precipitates in the IOZ can be determined from electron probe X-ray microanalysis (EPMA). Then, from these measurement data of internally oxidized Fe-Mn alloys, the degree of Mn enrichment in the IOZ (cf. Eq. (2.56)) and the diffusion coefficient of Mn (cf. Eq. (2.60)) are obtained; see Table 2.2.
Table 2.2 The degree of solute enrichment and obtained diffusion coefficient of Mn for Fe-Mn alloys upon oxidation at 950 °C, at a dew point of 10 °C for 20 hours.
Mn (wt%) Mn DMn (cm2sec-1) 1.7 1.06 4.44×10-12 7.0 1.07 1.24×10-12
2.6. Transition from internal to external oxidation
Upon formation of internal oxide particles, diffusion of oxygen within these oxide precipitates is negligible. Hence, diffusion of oxygen takes place through the bulk and along the interface between the oxide particles and the matrix. When the volume fraction of internal oxide precipitates reaches a critical value, i.e. g, these internal oxide particles act as the diffusion barriers for oxygen and therefore internal oxidation ceases. Consequently, oxidation proceeds at the alloy surface, which is then called external oxidation. This phenomenon is commonly known as the transition from internal to external oxidation [1, 20].
The volume fraction of internal oxide precipitates can be defined as:
BO BO m A B m N V g V (2.61)
29 Next, by combining Eqs. (2.56), (2.60) and (2.61), it is obtained that:
(0) 1/ 2 A B m B BO m V N g F V (2.62)When the volume fraction of internal oxide precipitates reaches a critical value, g, then
the transition from internal to external oxidation takes place [1]. Consequently, from Eq. (2.62) the critical mole fraction of the alloying element for this transition equals:
(0) 1/ 2 A B m B critical BO m V N g F V (2.63)This equation is known as the general criterion for the transition from internal to external oxidation [21]. However, for the limiting condition that favours the transition, i.e. higher permeability of the alloying element (cf. Eq. (2.35)), the auxiliary function can be simplified as, (cf. Appendix B):
1/ 2
1/ 2F (2.64)
Now, substitution of Eqs. (2.36) and (2.64) into Eq. (2.63) results in:
1/ 2 ( ) (0) 2 A B s m O O B critical BO m B g V N D N V D (2.65)
This equation is known as the simplified criterion for the transition from internal to external oxidation [1, 20, 22].
It is noted that these general or simplified criteria hold when the solubility product of the oxide precipitates (BO), Eq. (2.7), is very small, since then both NO and NB at the front of IOZ are virtually zero (see Eqs. (2.21) and (2.23)).
As an example, the oxidation domains for binary Fe-Mn alloys are given in Figure 2.7, which was depicted with the help of Eqs. (2.63) and (2.65) [23]. Increasing the solute concentration (vertical axis) changes an oxidation mode from internal to external, at constant temperature and oxygen partial pressure. Increasing oxygen potential in the
30
oxidant gas, corresponds to an increase in dew point (horizontal axis), shifts oxidation from the external towards the internal region, at constant temperature and solute concentration. The overall effect of increasing the temperature is that the external oxidation region expands.
Figure 2.7 Oxidation domains for binary Fe-Mn alloys oxidizing at 950 °C in a H2 + N2 gas mixture
containing 5 vol.% hydrogen. The regions above and below the transition line corresponds to external and internal oxidation modes, respectively. The activity of iron is taken unity and the activity of Mn is calculated with Thermo-Calc using PKP v1 database [8]. The molar volume of MnO is taken 13.21 cm3mol -1 [24]. The molar volumes of the Fe-Mn alloys were calculated with Thermo-Calc using TCFE 5 database,
but neglecting the small effect of temperature. The diffusion coefficient of oxygen and manganese in iron were taken from [25] and [26], respectively. The partial pressure of water vapour is expressed in terms of a dew point (cf. Appendix A). The Gibbs free energies for the formation of MnO, FeO and H2O are given in
the Appendix C. The solubility of oxygen in iron is given in Appendix D. The critical volume fraction of internal oxide precipitates for transition is assumed to be 0.3 [20].
2.7. Direct formation of an external scale
The growth of an external (BO) oxide scale on a A-B alloy without preceding formation of internal oxide precipitates, requires a higher concentration of the alloying element than
31 the critical concentration for internal-external oxidation, Eq. (2.65). The concentration of the alloying element should be high enough to sustain the scale growth [19, 27]:
1/2 (0) min. 2 A B p m B BO m AB k V N V D (2.66)
, where kp is the parabolic growth rate constant for external oxidation of B (cf. Eq. (2.67)), and DABis the interdiffusion coefficient in the A-B alloy and assumed to be independent of the concentration.
2 2
p
X k t (2.67)
, where X is the scale thickness.
Oxidation at 950 °C of 7.0 wt% Mn alloy with an extremely low dew point (-45 °C) leads to the MnO scale formation. From the oxide scale thickness as observed in cross-section images, the parabolic growth rate constant is estimated to be 1.14×10-13 cm2sec-1 (cf. Chapter 4). When for the interdiffusion coefficient (cf. Eq. (2.66), DAB) the diffusion coefficient of Mn in fcc iron is taken, the minimum concentration of Mn to sustain the scale growth equals to about 23 wt% using Eq. (2.66).
It is noted that for such a high Mn concentration the theory on the transition of internal to external oxidation breaks down, because there is no solution for the auxiliary function F(u) in the transition criterion, Eq. (2.63), when the Mn concentration is about 23 wt%. However, for the Fe-Mn alloys studied here, the Mn concentration is so low that the direct scaling will not occur.
2.8. Nucleation of internal oxide precipitates
In order to nucleate a new internal precipitate, an excess amount of oxidant is required. The need for this supersaturation has been recognized already [28], but is not
32
incorporated into the description of internal oxidation. This supersaturation can be expressed as [5], cf. Eq. (2.7): 1/ [ ] [ ] sp O B K N N (2.68)
In the previous description of internal oxidation, the solubility product of the internal oxide is assumed to be very small, cf. Eqs. (2.21) and (2.23). Furthermore, the nucleation-precipitation of the internal oxide is assumed to be extremely rapid. Consequently, the internal oxidation front is considered as a plane parallel to the original alloy surface and there the concentration of oxygen and alloying element can be neglected, cf. Eqs. (2.21) and (2.23). However, the precipitation of internal oxides with a high solubility product requires a finite degree of supersaturation for the nucleation of new oxide particles behind the internal oxidation front [29, 30]. Consequently, the precipitation does not occur along a sharp front inside the alloy, but rather within a region of small, albeit of finite width, wherein the concentration profiles of O and B change with time as new oxide particles nucleate and grow, see Figure 2.8.
Figure 2.8 Schematic concentration profiles for internal oxide precipitation showing a supersaturated region ahead of the internal oxidation zone.
33 Then, at the position xX/ (where x0 denotes the original alloy surface) the
concentrations of the two reactants are /
B
N and /
O
N , cf. Figure 2.8. At the position x X
where supersaturation is achieved, the concentrations of the two reactants are NB and
O N.
Hence, the following relations hold [29, 30]:
/ / B O sp N N K at xX/ (2.69)
B O N N S at x X (2.70) / B B N N (2.71) / O O N N (2.72) sp S K (2.73)The degree of supersaturation is defined as:
sp S K
(2.74)
Now, Fick’s second law of diffusion for oxygen and solute element, Eq. (2.19), is solved for the relevant boundary conditions, Eqs. (2.75)-(2.79):
( )s O O N N for x0, t0 (2.75) 0 O N for x0, t0 (2.76) 0 B N for x0, t0 (2.77) (0) B B N N for x0, t0 (2.78) (0) B B N N for x , t0 (2.79)
For parabolic internal oxidation kinetics, i.e. Eq. (2.40), the diffusion equation solutions are:
( ) ( ) / 2 s s O O O O O N N N N erf x D t erf for x X t ( ) (2.80)34
(0) (0) 1/ 2 / 2 B B B B B N N N N erfc x D t erfc for x X t ( ) (2.81), with and as defined in Eqs. (2.26) and (2.39), respectively.
Considering the mass balance at the IOZ front, cf. Eq. (2.28), a general equation is obtained:
( ) / (0) 1/2 (0) / s O O B B B B B B G N N N N G N N F N N (2.82)where F u
and G u
were defined in Eqs. (2.31) and (2.32), respectively.The parameter may be calculated from Eq. (2.82) when the values of the other relevant parameters are known. In the particular case of (0) ( )s 1
B B O O N D N D , wherein 1/2 1 , 1, and considering * ( )s O O
N N , then kinetic parameter from Eq. (2.82) is approximated by [29, 30]:
( ) 2 (0) / 2 s O B B N N N (2.83)For extremely low solubility product oxides, i.e. neglecting /
B
N with respect to (0)
B
N , Eq. (2.83) equals to Eq. (2.34) that was derived under the same simplifying assumptions, i.e. Eq. (2.33).
After internal oxidation of Fe-Mn alloys, the concentration of remained Mn in the IOZ is measured with EPMA; see Table 2.3. Since equilibrium exists between the internal oxide precipitates and the matrix, these values are considered as /
B
N . Then the value of can be obtained from Eq. (2.83); see Table 2.3. However, when it is assumed that all Mn in the IOZ is precipitated as oxide (i.e. low Ksp), also a value for can be obtained from Eq.
(2.34); see Table 2.3. The difference between these two values for is about 6%, which leads to a corresponding error in the determination of e.g. the diffusion coefficient of oxygen (cf. Eq. (2.39)).
35 Table 2.3 Effect of remained Mn concentration on the gamma value (cf. Eq. (2.83)) for Fe-Mn binary alloys upon oxidation for 20 hours at 950 °C with a dew point of 10 °C.
(0) Mn N / Mn N / 0 B N / 0 B N 1.73×10-2 0.22×10-2 1.09×10-2 1.02×10-2 7.11×10-2 0.72×10-2 5.29×10-3 5.01×10-3
The calculation of a value for using Eq. (2.82) requires not only a knowledge of (0)
B
N ,
( )s O
N and (which may be available from independent measurements), but also of /
B N , B N , / O N and O
N . These parameters can be obtained from measurement of the density of internal oxide particles at the internal oxidation front as a function of its distance from the alloy-gas interface [29, 30], which will be demonstrated in the sequel.
The relative width of internal precipitation region, i.e. X X/ (with X X*X/), is
related to various atom fractions in the internally oxidized alloy (cf. Figure 2.8) according to the following three independent equations as [30]:
( ) (0) / s O B O B B N N X X N N N (2.84) ( ) / / (0) / s O B B O O B B N N N X X N N N N (2.85)
( ) (0) (0) / / s O B B B B B B N N N X X N N N N (2.86)As a measure for the width of internal precipitation region, the mean distance between the oxide precipitates is taken. Alternatively, a value of relative width of internal precipitation region, X X/ , can be evaluated from the range of the log Z vs. log X curve where the slope is approximately -3 (cf. Eqs. (2.87)-(2.88)) and Z is the number of oxide particles per unit volume [29].
36
NO( )s 3 Z X X (2.87) with 3 ( ) 1 s O X N X (2.88)If X X/ is known from experimental data, Eqs. (2.84)-(2.86) allow to calculate the associated concentrations. However, for the system for which (0) /
B B B
N N N , by considering Eq. (2.86) and simplifying, it follows that [29]:
(0) (0) ( ) 1 B B B B s O O N N D N X X D N (2.89)
and from Eq. (2.84), it holds that [29]:
( ) (0) s B O O B N N X N N X (2.90)
Application of this analysis to the observed internal oxide precipitation in an Fe – 1.7 wt% Mn alloy after internal oxidation (see Figure 2.9), leads to the desired atom fractions at the internal oxidation front. Subsequently, the results can be used to determine the solubility product and supersaturation, respectively. To determine the relative width of the internal precipitation region (X X/ ), the number of internal oxide precipitates as a function of depth is resolved see Figure 2.10. Next, the value of X X/ was determined using Eqs. (2.87) and (2.88).
37 Figure 2.9 Backscattered electron image from the cross-section of an internally oxidized Fe – 1.7 wt% Mn alloy. The oxidation experiment was performed for 20 hours at 950 °C and a dew point of 10 °C. The internal oxide precipitates are the dark dots in the image.
Figure 2.10 Logarithmic representation of the number of internal oxide precipitates, Z, versus depth, X,
with respect to the gas/alloy interface of an internally oxidized Fe – 1.7 wt% Mn alloy, cf. Figure 2.9.
Finally, the concentration of dissolved Mn and O where the supersaturation occurs, are determined using the general relations (i.e. Eqs. (2.84)-(2.86)) as well as the simplified relations (i.e. Eqs. (2.89)-(2.90)). The results pertaining to the internally oxidized Fe – 1.7 wt% Mn alloy are given in Table 2.4.
38
Table 2.4 Concentration of dissolved Mn and O at the supersaturation Ni*, the equilibrium concentration of dissolved O at the precipitation front NO/ and the degree of supersaturation of the internally oxidized Fe – 1.7 wt% Mn alloy for 20 hours at 950 °C and at a dew point of 10 °C. The mean distance between the internal oxide precipitates was derived from Figure 2.10. The equilibrium concentration of dissolved Mn at the precipitation front NMn/ was measured with EPMA and equals to 2.2×10-3.
Equation (2.89) (2.90) (2.86) (2.84) (2.85) (2.74) Simplified Eqs. General Eqs.
(0) Mn N X X * Mn N * O N * Mn N * O N / O N 1.73×10-2 37.29 1.17×10-2 6.47×10-8 1.28×10-2 8.12×10-8 1.48×10-7 3.18
According to Eq. (2.69), the solubility product of MnO in the fcc-Fe at 950 °C becomes 3.26×10-10, which is in good agreement with that derived from Thermo-Calc [8], i.e.
7.75×10-10. Also the value of the solubility product of MnO obtained from experimental data is in agreement with the value calculated with Eqs. (2.7)-(2.12) using thermo-chemical data [7, 25, 31] which equals 1.46×10-10. The difference between the two values for (cf. Table 2.3) and the supersaturation needed for the nucleation of Mn-oxides in fcc-Fe (cf. Table 2.4) indicate that the solubility product of MnO in fcc-Fe can be considered as low.
2.9. Internal precipitates of an oxide with high solubility product
When the solubility product of the internal oxide precipitates is very small, then the concentration of internal oxides across the IOZ is constant, cf. Figs. 2.2 and 2.3 [6, 19]. However, for the high solubility product oxides, precipitation occurs continuously through the IOZ as the IOZ forms [32]. For this type of precipitation it holds that: (i) The concentration of the solute element does not become zero in the IOZ, but decreases from the precipitation front to the surface. (ii) The concentration of the oxygen does not become zero either at the reaction front, but decreases monotonically into the alloy beyond the reaction front. (iii) The mole fraction of the precipitates varies across the IOZ
39 [12, 33]. For the high solubility product oxide, the concentration profiles of the oxygen, alloying element and precipitates are schematically shown in Figure 2.11 [12, 34].
Figure 2.11 Concentration profiles for high solubility product oxide formed during internal oxidation [12, 34].
The aspects of high solubility oxides were examined by Laflamme and Morral [32] and Ohriner and Morral [33] and will be reviewed briefly here.
The equilibrium fraction of precipitate can be related to the composition via the lever rule as [33]:
, (0)
, , B B p B B N N x t F x t N N x t (2.91) , where p BN and NB are local values of the solute in the precipitate and matrix, respectively. Eq. (2.91) is valid when there is no long-range diffusion of solute ( (0)
B
N is
the local average concentration) and the solution is dilute ( p
B B N N ). Then, Eq. (2.91) becomes:
