Analytical model of selective external oxidation of Fe-Mn binary alloys during isothermal annealing treatment

Journal Pre-proof Analytical Model of Selective External Oxidation of Fe-Mn Binary Alloys during Isothermal Annealing Treatment L. Gong (Investigation) (Methodology) (Validation) (Visualization) (Writing - original draft), N. Ruscassier (Investigation), M. Ayouz (Investigation), P. Haghi-Ashtiani (Investigation), M.-L. Giorgi (Conceptualization) (Methodology) (Validation) (Writing - review and editing) (Supervision)

PII:

S0010-938X(19)32164-X

DOI:

https://doi.org/10.1016/j.corsci.2020.108454

Reference:

CS 108454

To appear in:

Corrosion Science

Received Date:

14 October 2019

Revised Date:

8 January 2020

Accepted Date:

9 January 2020

Please cite this article as: Gong L, Ruscassier N, Ayouz M, Haghi-Ashtiani P, Giorgi M-L, Analytical Model of Selective External Oxidation of Fe-Mn Binary Alloys during Isothermal Annealing Treatment, Corrosion Science (2020), doi: https://doi.org/10.1016/j.corsci.2020.108454

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Analytical Model of Selective External Oxidation of Fe-Mn Binary Alloys during Isothermal Annealing Treatment

L. Gong 1, N. Ruscassier 1, M. Ayouz 1, P. Haghi-Ashtiani 2, M.-L. Giorgi 1, * 1

LGPM, CentraleSupélec, Université Paris Saclay, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette cedex,

France. 2

MSSMat, CNRS, CentraleSupélec, Université Paris Saclay, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette

*

Corresponding

author.

Tel.:

+33

1

75

31

61

72;

E-mail

address:

marie-

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[email protected]

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cedex, France.

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Graphical abstract

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Highlights 

An analytical model was developed to describe the kinetics of external selective oxidation of binary Fe-Mn alloys during isothermal annealing.



The model predicts the heterogeneous nucleation and growth of MnO oxides on the Fe-Mn surface.



The model calculates the size of the oxides and the surface area fraction covered by them.

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The model results were validated by measurements conducted on annealed Fe-Mn alloys;



The model can be applied to continuous industrial annealing of steels.

Abstract An analytical model was developed to describe the kinetics of external selective oxidation of Fe-Mn alloys during isothermal industrial annealing. This model describes the surface reactions and

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thermodynamics involving water vapor, the pre-oxidation and diffusion of oxygen and manganese and the heterogeneous nucleation and growth of MnO oxides on the surface. Theoretical equations were established and solved to calculate the critical radius and nucleation rate of MnO embryos, the size and surface coverage fraction of MnO oxides and the manganese concentration profiles as a function of annealing time. The model results were validated by annealing experiments at 800 °C.

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Keywords: A. alloy, B. modelling studies, B. SEM, C. kinetic parameters, C. selective oxidation.

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Nomenclature

Notes: all data are given for Fe-Mn (1 wt.%) alloy annealed at 800 °C in an atmosphere of N2 and 5

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vol.% H2 with a dew point of -40 °C.

′ 𝑎𝑀𝑛 , 𝑎𝑂′

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𝑎𝑀𝑛 , 𝑎𝑂 , 𝑎𝑀𝑛𝑂 Activities of Mn and O in iron (reference state: pure Mn and O2) and MnO Activities of Mn and O in iron (reference state of Henry’s iron solution containing 1

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wt.% Mn or 1 wt.% O)

𝑏𝑢𝑙𝑘 𝑏𝑢𝑙𝑘 𝑐𝑀𝑛 , 𝑐𝑂 𝑒𝑞𝑢𝑖

𝑒𝑞𝑢𝑖

Concentrations of Mn and O in the alloy bulk (mol‧m-3)

𝑐𝑀𝑛 , 𝑐𝑂

Equilibrium concentrations of Mn and O at the oxide / alloy interface (mol‧m-3)

𝐷𝑀𝑛 , 𝐷𝑂

Diffusion coefficients for Mn (9.09 × 10-16 m2‧s-1) [20] and O (8.02 × 10-12 m2‧s-1) [20] in α-Fe at 800 °C

𝑔∗

Critical oxide volume fraction (0.3 [24,26])

𝐼ℎ𝑒𝑡

Heterogeneous nucleation rate of MnO oxides on the alloy surface (m-2‧s-1)

𝐽𝑀𝑛 , 𝐽𝑂

Diffusion fluxes of Mn and O (mol‧m-2‧s-1)

𝑘𝐵

Boltzmann’s constant (1.38 × 10-23 J‧K-1)

𝐾𝑀𝑛𝑂

Solubility product of MnO in α-Fe at 800 °C (7.21 ppm2)

𝐿𝑐,𝑀𝑛 , 𝐿𝑐,𝑂

Characteristic diffusion lengths of Mn and O (m) Quantity of Mn (mol‧m-2) in all MnO nuclei

𝑀𝑀𝑛 , 𝑀𝑂

Molar mass of Mn (54.94 g‧mol-1) and O (16.0 g‧mol-1)

𝑀𝑀𝑛𝑂

Molar mass of MnO oxide (70.94 g‧mol-1)

𝑁𝑛𝑢𝑐𝑙𝑒𝑖

Total number of MnO nuclei per unit area (3.5×1012 m-2)

𝑛𝑀𝑛𝑂

Total quantity of MnO oxides (mol‧m-2) present on unit area of the alloy surface

𝑁𝐴

Avogadro number (6.022×1023 mol-1)

𝑁𝑠

Number of sites where a nucleus can form per unit area (m-2)

𝑅

Ideal gas constant (8.314 J‧mol-1·K-1)

𝑟

Radius of curvature of a MnO embryo (m)

𝑟∗

Critical radius of a MnO embryo (m)

𝑡𝑐−𝑛𝑢𝑐𝑙

Time to complete the MnO nucleation (s)

𝑡𝑠−𝑛𝑢𝑐𝑙

Time when MnO nucleation begins (s) 𝑒𝑞𝑢𝑖

Weight fractions of Mn and O in the Fe-Mn alloy in equilibrium with gas atmosphere

𝑡

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𝑤𝑀𝑛 , 𝑤𝑂

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𝑒𝑞𝑢𝑖

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𝑚𝑀𝑛

𝑇

Temperature (°C)

𝑥

Distance perpendicular to the steel surface (where x = 0)

𝑋𝑐𝑜𝑣

Surface area fraction covered by MnO particles on the alloy surface

𝑥𝑐𝑟𝑖𝑡 𝑀𝑛

Critical mole fraction of Mn for the transition from internal to external oxidation

𝑉𝑎𝑙𝑙𝑜𝑦

Molar volume of the binary alloy (7.22×10-6 m3 mol-1 [60])

𝑏𝑢𝑙𝑘 𝑤𝑀𝑛 , 𝑤𝑂𝑏𝑢𝑙𝑘 Weight fractions of Mn (0.01) and O (0) in the alloy

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Annealing time (s)

𝑉𝑀𝑛𝑂

Molar volume of MnO (1.33×10-5 m3 mol-1 [60])

𝑍

Zeldovich factor

𝛽∗

Growth rate of a cluster with critical radius 𝑟 ∗ (s-1)

𝛾𝐹𝑒 , 𝛾𝑀𝑛𝑂

Surface energies of Fe and MnO (J‧m-2)

𝛾𝐹𝑒/𝑀𝑛𝑂

Interface energy between MnO and ferrite (J‧m-2)

∆𝐺 ∗

Gibbs free energy of formation of a spherical cap-shaped MnO embryo with critical radius 𝑟 ∗ Gibbs free energies of Mn and O dissolution in solid iron

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∆𝑟 𝐺𝑑𝑖𝑠𝑠,𝑀𝑛 , ∆𝑟 𝐺𝑑𝑖𝑠𝑠,𝑂

Gibbs free energy of reaction (R-4)

° (𝑇) ∆𝑟 𝐺𝑀𝑛𝑂

Standard Gibbs free energy of reaction (R-4) at a given temperature T

𝜃

Contact angle between the MnO particle and alloy

𝜌𝑎𝑙𝑙𝑜𝑦

Fe-Mn (1 wt.%) alloy density (assumed to be equal to that of iron because the alloy is

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mainly composed of Fe) (kg‧m-3)

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∆𝑟 𝐺𝑀𝑛𝑂

Density of solid iron at 800 °C (7736 kg‧m-3 [60])

𝜌𝑀𝑛𝑂

Density of MnO oxide at 800 °C (5350 kg‧m-3 [60])

Ω1

Volume of a MnO molecule (m3)

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𝜌𝐹𝑒

1. Introduction

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The field of study presented here is the use of advanced high strength steels in car bodies to reduce their weight and, consequently, fuel consumption and CO2 emissions [1]. These steels must be coated with a zinc layer to protect them from corrosion, which is generally done by hot-dip galvanizing. Before immersion in the molten zinc bath, the steel strip is annealed at high temperature (about 800 °C). The main objectives of such a treatment are to recrystallize the microstructure after coldrolling and reduce native oxides to improve wettability with liquid zinc [1-4], which is achieved using a reducing atmosphere composed mainly of nitrogen and hydrogen (about 5 vol.%). However, the selective oxidation of minor alloying elements, such as manganese, silicon, aluminum and chromium, also occurs due to traces of water present in the gas atmosphere (dew point of about -40 °C

corresponding to a partial water pressure of 19 Pa). During this process, the oxygen atoms saturate the steel surface, and the oxidizable alloying elements diffuse outwards and react with solute oxygen atoms [2,3,5,6]. Oxides precipitate at the surface or sub-surface alloy region. Selective oxidation during continuous annealing of steels corresponds to the cases of internal or external oxidation with discrete oxide particles or films, both in the absence of external iron oxide scale formation. Several expressions have been derived for internal oxidation in the case of the absence of external scale formation. Wagner [7] proposed the first model in 1959, and a good summary of this theory was given by Rapp [8]. In this model, the kinetics of internal oxide formation is assumed to be controlled by diffusion of oxygen and/or solute metal that oxidizes in the base metal. The depth of the oxidation front is a parabolic function of time and the velocity of the reaction front motion depends on the

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diffusion coefficients of oxygen and solute metal, the concentration of the solute metal and the oxygen partial pressure. Although Wagner’s model made several restricting hypotheses, such as consideration of binary alloys only, formation of a single oxide with a very low solubility and small fractions of oxides formed, his pioneering study facilitated subsequent works (see for example [9-18]). Major improvements include, for example, the consideration of non-zero solubility products [9,11], degree of

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supersaturation necessary for the nucleation of new oxide particles [13], oxidation of both elements in a binary alloy [12] or one [14] or two elements [10, 15-17] in a ternary alloy, and the higher diffusion

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coefficient in the grain boundary than in the grain [18]. Douglass [19] made a comprehensive discussion of Wagner’s model and improved some variants. The most general models to date [20,21]

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took into account an arbitrary number of diffusing chemical elements and precipitate phases with arbitrary compositions and the dependence of diffusion coefficients and solubility products on temperature. In addition, the diffusion coefficients of the metallic matrix elements can be modified in

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the case of precipitates present in large fractions [21]. The diffusion coefficients were chosen to decrease by a factor equal to the volume fraction of the metallic matrix [21,22]. These works generally focused on selective internal oxidation but can be used to study the transition

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from internal to external oxidation. Indeed, Wagner [7] proposed that this transition should take place when the volume fraction of oxides reaches a critical value near the surface. The oxides formed then

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prevent the diffusion of oxygen into the material. The Wagner model makes it possible to calculate the volume fraction of internal oxides in a single crystal of a binary alloy from the flux of the element that oxidizes at the precipitation front. The Wagner criterion was also modified to take into account the grain boundary diffusion or the interaction of several oxidizable elements [23]. However, these models do not predict the value of the critical volume fraction of oxides that results in the transition from internal to external oxidation. There is only one experimental value of this critical oxide volume fraction: Rapp [24] measured that it is equal to 0.3 for Ag-In alloys at 550°C. Leblond et al. [25,26] estimated it analytically from Wagner’s model by writing that the oxygen diffusion coefficient is a decreasing function of the volume fraction [25,26] and the shape of oxides [26]. The oxide shape,

chosen to be an oblate spheroid, is modelled by a shape factor W which compares the lengths of the major and minor axes of the spheroid. This model makes it possible to obtain the critical volume fraction of the oxides resulting in the transition. It is a function of the specific volumes of the matrix and oxide and the shape factor W [26]. The order of magnitude obtained is in very good agreement with the experimental Rapp’s value. To our knowledge, there is not yet a model to quantitatively interpret the external oxidation behavior and predict external oxides’ coverage in the field of continuous annealing. However, the formation of external oxides can result in problems of wettability and zinc coating defects during galvanizing. In order to evaluate the galvanizability of advanced high strength steels and to try to minimize the formation of oxide particles on the steel surface, it is necessary to be able to evaluate the

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surface area fraction covered by the oxides according to the annealing parameters. As it is difficult to carry out measurements in industrial plants, it is necessary to develop a kinetic model of selective external oxidation in order to predict the nucleation and growth of oxides on the steel surface.

Numerous annealing studies have been made on the advanced high strength steels to explore the

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effect of steel composition, annealing temperature and dew point of atmosphere on the selective oxidation behavior, including both laboratory materials and industrial products (see for example the literature review in [27] and [28]). The particles formed were then composed of several oxides. Owing

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to the diversity and complexity of oxidation behavior, we first focus on the selective oxidation of model systems composed of iron and manganese only, Mn being one of the main alloying elements

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(typically 1.0 to 2.5 wt.% Mn) in advanced high strength steels used in automotive bodies. Studies conducted on continuous annealing of binary Fe-Mn alloys (0.3 - 5.0 wt.% Mn) show that the iron oxides spontaneously formed in air are completely reduced in an atmosphere of N2 and 5 vol.% H2

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with a dew point of -80 to 0 °C and at a temperature from 700 to 820 °C [29-37]. The metallic iron surface obtained is faceted. Mn is selectively oxidized and precipitates as small oxide particles observed on ferrite grains (preferentially aligned on facets) and at grain boundaries (diffusion short-

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circuits) [32,33,35,37]. The oxide most frequently detected is MnO [29,30,32-37], and Mn3O4 was also occasionally analyzed [29].

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The main objective of this work is to establish an analytical model to describe the nucleation mechanism and growth kinetics of MnO oxides and to calculate the precipitate amount as a function of annealing time. We have chosen to develop an analytical model to easily calculate orders of magnitude characteristic of external selective oxides in order to (i) fully understand and evaluate the limiting mechanisms of the oxidation reaction and (ii) quickly evaluate the process parameters for modifying these selective oxides. The annealing atmosphere considered here contains high purity N2 and 5 vol.% H2 with a dew point of -40 °C conducted at 800 °C, corresponding to an oxygen partial pressure of 5.77×10-19 Pa [20]. In this condition, MnO particles appear on the Fe-Mn (1 wt.%) alloy surface after

annealing at 800 °C for 60 s [30,32]. The simplifying assumptions of the modelling will be based on these experimental results. The approach used consists of four steps: (1) experiments carried out to measure the mean size of the oxide particles formed; (2) overview of reactions occurring on the alloy surface; (3) construction of a kinetic model for external oxidation reactions; (4) solution and validation of the model based on experimental data.

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2. General description of the oxidation reactions occurring at the steel surface 2.1 Experiments

The Fe-Mn binary alloy (delivered by Goodfellow) used in this work was completely recrystallized and its composition was 0.0013 wt.% C, 1.043 wt.% Mn, 0.002 wt.% Si, 0.002 wt.% P and balance Fe.

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The crystal structure is body-centered cubic ferrite and Mn forms a substitutional solid solution with Fe for temperature less than 840 °C [38]. Prior to the annealing experiments, the 20 mm × 20 mm × 1 mm samples were first polished with SiC papers to remove the iron oxides layer and then mirror-

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polished up to 1 µm diamond suspension, resulting in an average roughness of a few nanometers. The obtained samples were cleaned in an ultrasonic bath with ethanol and completely dried. The annealing

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experiments were conducted in an infrared radiation furnace (Ulvac Sinku-Riko) under conditions relevant to galvanizing line practice. The gas atmosphere used in the furnace chamber was N2 + 5 vol.% H2 gas mixture (Air Liquide with less than 3 ppm of H2O and 2 ppm of O2) with a dew point of (-40  (1)

. Each sample was first heated to 800 °C with about

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2) °C (corresponding frost point of -36.5 °C)

6 °C s-1 and then kept at this temperature for 30, 60, 120 and 180 s before cooling down to room temperature. The annealing of some samples was also interrupted during the temperature rise (650,

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700, 800 °C) to detect the beginning of nucleation of oxide particles. The rapid cooling process (about 4.5 °C s-1 down to 650 °C) acts as a quench for selective oxidation reactions. Further details of the

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experimental setup are described elsewhere [27]. After the annealing treatment, the sample surfaces were observed by a field emission gun scanning

electron microscope (FEG-SEM Leo 1530). As shown in Fig. 1, the substrate surfaces were partially covered with small particles of a few hundred of nanometers. The oxides had different shapes and sizes depending on the ferrite grain where they are formed. For the validation of the analytical oxidation model, we need an average of the surface density and surface fraction covered by the (1)

The saturation vapor pressure is a function of temperature only. When the temperature is below 0 °C as here, this temperature is called the dew point for the saturation vapor pressure of water with respect to supercooled water and the frost point for the saturation vapor pressure of water with respect to ice.

particles per annealing conditions. To obtain reliable data, at least 10 random images were taken on each sample to quantitatively measure these geometric parameters. Nucleation begins at the ferrite grain boundaries at 650 °C and occurred on all grains at 700 °C. The surface density of MnO particles is estimated by counting the number of MnO crystals obtained in the experiments. It is almost constant over time for 120 s and equal to 3.5×1012 m-2 at 800 °C. Secondary nucleation occurs for annealing time longer than 180 s and very small particles are observed between the large particles that have previously formed. We have also included the experimental results at 180 s for validation purposes (Section 4.3) because the contribution of small particles is negligible at 180 s (growth of large particles is the main phenomena during 180 s). It should be noted that the surface density of the oxide 12 -2 particles depends strongly on the ferrite grain orientation (3.5+5 −3 × 10 m ). However, since we want

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to create a tool to predict the average behavior of the material in relation to external oxidation, we have chosen the mean value for the calculations.

The chemical composition of these particles was determined by quantitative EDX analysis in a scanning / transmission electron microscope (TEM/STEM-FEI Titan3 G2 60-300) performed on FIB thin foils, which were extracted using a focused ion beam microscope (FIB-FEI Helios NanoLab 660).

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The particles appear yellow in the analysis, a mixture of O in red and Mn in green (Fig. 2). The quantitative analysis showed that the particles mainly contain Mn and O in the same atomic

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proportions, i.e. MnO. These cross-section observations also show that oxidation is essentially external. The red line at the top of the sample is rich in O and correspond to iron oxide formed in air after the

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annealing.

The kinetics and mechanisms of oxide’s formation on the surface of Fe-Mn alloy in the presence of water vapor should take into account two main reactions occurring at the interface between the steel

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surface and the atmosphere: (i) surface reactions between water vapor or oxygen and ferrite and (ii)

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nucleation and growth of MnO particles.

2.2 Surface reactions between water vapor or oxygen and ferrite

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As indicated above, the annealing conditions considered here are a gas composed of high purity N2 and 5 vol.% H2 with a dew point of -40 °C and a temperature of 800 °C. The partial pressure is then 19 Pa [39] and the partial oxygen pressure is equal to 5.77×10-19 Pa in equilibrium [20]. This corresponds to about 1.3×1021 molecules of H2O and 39 molecules of O2 per m3 of gas (calculated with the assumption of the ideal gas). This means that the surface reactions on ferrite mainly involve the dissociation of water. Surface reactions of water vapor on ferrite were studied on single crystals of Fe(100), Fe(110) and Fe(111) under ultra-high vacuum with low energy electron diffraction (EELS), Auger electron

spectroscopy (AES) and temperature-programmed desorption. It has been shown that the water dissociation on clean iron is irreversible. All the results are summarized in two general reviews on the interaction of water with solid surfaces [40,41]. The thermal reaction pathway by which water can dissociate on solid iron at temperatures higher than 310 K is as follows [42-46]: H2O(g) → H2O(ad) → OH(ad) + H(ad) → O(ad) + H2(g)

(R-1)

First, water molecules are physically adsorbed on the steel surface. Their activation energy is sufficient to dissociate into an adsorbed hydroxide group and hydrogen atom. The former can be further transformed by giving an adsorbed oxygen atom and releasing a dihydrogen molecule. The case we are studying is different since the annealing of the iron-rich alloy does not occur under

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high vacuum but in a gas atmosphere of N2-H2 with a partial pressure of water vapor of 19 Pa. Usually, it is assumed that the adsorption reactions are the same as those studied under vacuum [47]. During annealing, the Fe / Mn surface becomes covered with MnO particles. Water dissociation on MnO occurs only at defect sites, unlike ferrite (where it occurs over the entire surface) [40]. Moreover, the surface area fraction covered by MnO particles is less than 30% for an annealing of 180 s under our

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experimental conditions (section 2.1, Fig. 7). Considering these facts, it is assumed here that water dissociation occurs mainly on ferrite.

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Finally, the oxygen atoms adsorbed on ferrite can dissolve into the superficial ferrite lattice. The surface reaction is as follows:

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O(ad) → O(dis)

(R-2)

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2.3 Nucleation and growth of MnO particles on ferrite

The schematic representation for oxidation reactions occurring on the alloy surface is shown in Fig. 3. The main steps are the transport of H2O to the alloy surface, its subsequent adsorption and

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dissociation (section 2.2) and the nucleation and growth of MnO.

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Let us first consider the nucleation of surface oxides. The elementary steps of the initial oxidation of metallic alloys have generally been studied with an ultra-high vacuum surface science approach [40]. The pure metals and metallic alloys studied in the literature and representative of the Fe-Mn alloys studied here are Fe and FeCr-based alloys [48-50] (as well as references in [48]), FeAl [51], and intermetallic compounds containing Ti and Al [52]. The irreversible decomposition of water (section 2.2) with reactive metal alloys generally results in the oxidation of the most reactive metal, accompanied by the nucleation of its oxides at the solid-vacuum interface [40]. This is the case of FeAl(100) where alumina was detected by X-ray photoelectron spectroscopy (XPS) at the temperature above 250 K. The formation mechanism can be described by the R-1 and R-3 reactions [51].

y O(ad) + x Al  AlxOy

(R-3)

The first steps of oxidation have been particularly well described in the case of α2-Ti3Al and -TiAl in contact with O2 partial pressure of 10-7 mbar at 650 °C [52]. A pre-oxidation step occurs characterized by the adsorption / dissociation of O2 and the adsorption of O atoms in the subsurface lattice of the alloy. The concentration of dissolved O below the alloy surface increases to saturation (equal to O solubility limit in the alloy). When the saturation is reached, the further oxygen uptake leads to the selective oxidation of Al, i.e. the critical O concentration for the nucleation of aluminum oxide is reached on the alloy surface. The oxide islands that partially cover the substrate grow laterally to form a continuous layer. The oxide growth is limited by the diffusion of Al in the alloy to the oxide

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/ metal interface and is accompanied by the formation of an Al-depleted zone under the oxide layer. To our knowledge, no studies on the dissolution of water or oxygen have been carried out with FeMn alloys. However, in our opinion, the cases of Fe-Al alloy [51] and intermetallic compounds containing Ti and Al [52] are representative of the Fe-Mn alloy studied here, since Mn is more oxidizable than Fe. Therefore, from the results obtained in Refs. [51,52], we can deduce the different

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steps of selective oxidation of the Fe-Mn alloy (Fig. 3): (i) a pre-oxidation step occurs characterized by the adsorption / dissociation of H2O and the adsorption of O atoms in the subsurface lattice of the

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alloy (section 2.2); the concentration of dissolved O below the alloy surface increases to saturation; (ii) when saturation is reached, the additional oxygen uptake leads to nucleation of MnO embryos on the alloy surface; (iii) the oxide islands that partially cover the substrate grow laterally with the diffusion

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of Mn from the alloy bulk to the surface.

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3 Construction of the kinetics model of external selective oxidation A theoretical model for the selective oxidation of the Fe-Mn alloys, as shown in Fig. 3, will be constructed here. In order to compare the results with existing studies, the different physical

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parameters will be taken from Ref. [20] and the references cited in Ref. [20].

3.1 Model assumptions The first approach to simplify the kinetic model is to find the limiting steps of the oxidation process.

3.1.1 External transport of H2O The external transport of H2O from the gas atmosphere to the alloy surface is assumed to be fast compared to the other elementary steps of the oxidation process. In industrial annealing process, the

rate of the external transport of H2O must be evaluated taking into account the gas flow conditions in the vicinity of the moving steel strip. 3.1.2 Pre-oxidation step After the external transport of H2O, its adsorption / dissociation and the adsorption of O atoms in the subsurface lattice of the alloy occur. This is the so-called pre-oxidation stage. To our knowledge, the overall rate of surface water reactions has not been accurately determined. However, the kinetics of the pre-oxidation step has been precisely studied in the case of α2-Ti3Al and -TiAl in contact with 10-7 mbar of O2 at 650 °C [52]. It has been shown that the duration of the pre-oxidation step depends on the alloy: it is shorter for -TiAl (20 s) than for α2-Ti3Al (6 min), probably due to the lower O saturation

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value in the first alloy (2 at.%) than in the second (16 at.%). Of course, the oxidation conditions are very different in our case, but the O solubility in ferrite at equilibrium with the partial pressure of oxygen in the gas atmosphere used is even lower (1.410-6 wt.% or 4.810-6 at.% in pure Fe, calculated for a saturation vapor pressure of water corresponding to -

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36.5 °C in Ref. [20]), which leads us to conclude that the pre-oxidation step could be also very fast compared to the nucleation and growth of MnO oxide. This has also been demonstrated in the case of oxidation of Fe- and FeCr-based alloys in O2 [48,49]. However, it seems that the oxidation of iron in

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water vapor progresses more slowly than in oxygen [50]. This leads us to further analyze the role of water. The first O-H bond of the free water molecule has a dissociation energy of 498 kJ‧ mol-1 [41].

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The dissociation barrier is lowered to 40 kJ‧ mol-1 due to the catalytic effect of the iron surface [53]. All these results support us in assuming that the pre-oxidation step is really fast compared to the oxidation step. Therefore, it is assumed that the instantaneous local thermodynamic equilibrium

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between water vapor and dissolved oxygen is reached when the alloy sheet is brought into contact with the annealing atmosphere.

This leads us to estimate the O weight fraction dissolved in the Fe-Mn alloy in equilibrium with the

𝑒𝑞𝑢𝑖

= 1.4 × 10−6 𝑤𝑡. %

Eq. (1)

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𝑤𝑂

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gas atmosphere considered at 800 °C [20]:

3.1.3 Diffusion of O and Mn in the Fe-Mn alloy For the diffusion in a semi-infinite medium, the characteristic diffusion length of element i (i = Mn

or O) is given by: 𝐿𝑐,𝑖 = √𝐷𝑖 . 𝑡

Eq. (2)

where Di is the diffusion coefficient of element i and t is the annealing time. The characteristic lengths are equal to 510-2 mm for O and 510-4 mm for Mn at 800 °C after 300 s annealing treatment. The thickness 𝑒 of steel strips is generally in the order of 1 mm. Therefore, 𝐿𝑐,𝑂 and 𝐿𝑐,𝑀𝑛 ≪

𝑒

Eq. (3)

2

The inequality remains valid as long as the annealing time is less than 300 s. Since the continuous industrial annealing treatment always takes about 60 s [3], it can be assumed that the diffusion of O and Mn in the alloy studied here can be modelled by diffusion in a semi-infinite medium. Generally speaking, O diffuses from the surface to the bulk and Mn diffuses from the bulk to the surface in the alloy. In order to evaluate the respective weight of O and Mn diffusion in the selective

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oxidation process studied, it is necessary to compare the diffusion flux of O and Mn.

The diffusion flux of each element 𝑖 will be estimated on the alloy surface, in the case when the limiting step is solid state diffusion, i.e. when the thermodynamic equilibrium is reached at the gas / alloy interface for O and at the oxide / alloy interface for Mn. The diffusion flux of the element i on

𝑒𝑞𝑢𝑖

− 𝑐𝑖𝑏𝑢𝑙𝑘 ). √

𝐷𝑖

Eq. (4)

𝜋.𝑡

re

𝐽𝑖 (𝑥 = 0, 𝑡) = (𝑐𝑖

-p

the alloy surface in the case of diffusion in a semi-infinite medium is given by:

𝑒𝑞𝑢𝑖

and 𝑐𝑖𝑏𝑢𝑙𝑘 are the 𝑖 equilibrium concentration at the oxide / alloy interface and the 𝑖

where 𝑐𝑖

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concentration in the alloy bulk. The gas / alloy interface is located at x = 0. As the alloy is diluted with Mn and O, the 𝑖 concentration is approximated by 𝑐𝑖 = 𝑤𝑖 .

𝜌𝐹𝑒 𝑀𝑖

where 𝜌𝐹𝑒 is the density of solid iron at

na

800 °C, 𝑀𝑖 is the molar mass of element 𝑖 and 𝑤𝑖 is the weight fraction of element 𝑖 in the alloy. The ratio between the O and Mn diffusion fluxes is then equal to: 𝐽𝑂 (𝑥=0,𝑡) 𝐽𝑀𝑛 (𝑥=0,𝑡)

|=

𝑒𝑞𝑢𝑖

𝑏𝑢𝑙𝑘 −𝑤𝑂 𝑒𝑞𝑢𝑖 𝑏𝑢𝑙𝑘 −𝑤𝑀𝑛 +𝑤𝑀𝑛

𝑤𝑂

.√

𝐷𝑂

𝐷𝑀𝑛

∙

𝑀𝑀𝑛 𝑀𝑂

= 5 × 10−4 ≪ 1

Eq. (5)

ur

|

In Eq. (5), the weight fraction of O and Mn in the alloy bulk are 0 and 0.01 respectively and the value 𝑒𝑞𝑢𝑖

𝑒𝑞𝑢𝑖

𝑒𝑞𝑢𝑖

Jo

of 𝑤𝑀𝑛 can be estimated with the solubility product of MnO [20] such as 𝐾𝑀𝑛𝑂 = 𝑤𝑀𝑛 ∙ 𝑤𝑂

at

800 °C (Eq. (14)). This calculation shows that the O flux is very small in front of the Mn flux. Furthermore, using Wagner’s model [7,8], the critical mole fraction of Mn 𝑥𝑐𝑟𝑖𝑡 𝑀𝑛 necessary for the transition from internal to external oxidation is given by 𝑥𝑐𝑟𝑖𝑡 𝑀𝑛 = (

𝑠𝑢𝑟𝑓

𝜋∙𝑔∗ ∙𝑥𝑂

∙𝐷𝑂 ∙𝑉𝑎𝑙𝑙𝑜𝑦

2∙𝐷𝑀𝑛 ∙𝑉𝑀𝑛𝑂

0.5

)

Eq.(6)

𝑠𝑢𝑟𝑓

where 𝑔∗ is the critical oxide volume fraction, 𝑥𝑂

is the oxygen mole fraction at the alloy surface

𝑒𝑞𝑢𝑖

(related to 𝑤𝑂 ), 𝑉𝑎𝑙𝑙𝑜𝑦 and 𝑉𝑀𝑛𝑂 are the molar volumes of the binary alloy and MnO respectively. 𝑥𝑐𝑟𝑖𝑡 𝑀𝑛 is evaluated as 0.01, which is equal to the composition of the binary alloy studied here. In conclusion, the ratio between the O and Mn diffusion fluxes is very low and the composition of the binary alloy is of the same order of magnitude as the critical mole fraction of Mn required for external oxidation. Therefore, the selective oxidation of Mn can be considered as mainly external [7]. This has been observed in our experiments (section 2.1). However, some internal oxidation has also been observed (Fig. 2f), since the oxygen diffusion into the alloy cannot be completely stopped. From our TEM observations, the amount of Mn consumed by internal particles (with a mean size of 40 nm and a

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density of 3.10-9 nm-3) can be estimated to be less than 8 mol/m3 which is much lower than its bulk concentration (1408 mol/m3). Therefore, the formation of internal MnO oxides is not sufficient to affect the Mn concentration in the bulk, and thus the Mn diffusion from the alloy bulk to the surface. Finally, only the diffusion of Mn in the alloy is modelled.

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3.1.4 Nucleation and growth of MnO

In the following sections, a theoretical model for the nucleation and growth of MnO particles will

validated as the model is presented.

re

be established on the alloy surface. The necessary additional assumptions will be described and

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The model finally aims to describe the following reaction mechanisms: (1) the adsorbed oxygen reacts preferentially with the surface Mn atoms to form the MnO nuclei on the steel surface; (2) Mn diffuses from the steel bulk to the surface; (3) the manganese that reaches the alloy surface reacts with the adsorbed oxygen and participates in the growth of MnO oxides; (4) the growth of the MnO oxides

na

is accompanied by the formation of an area depleted with Mn under the oxides.

ur

3.2 Model description

The model for the kinetics of the oxidation reactions is based on the solution of generalized

Jo

equations for the diffusion of manganese to the atmosphere / alloy interface (section 3.2.3). To model the formation of the external oxides, it is assumed that the nucleation and growth of MnO oxides represent two distinct stages completely separated in time, as the nucleation process is generally very fast and the surface density of oxide particles remains constant for annealing time less than or equal to 120 s. This means that the growth of the first nuclei formed during nucleation process is neglected. With this assumption, the nucleation (section 3.2.2) and growth (section 3.2.4) of the oxide particles can be described consecutively. Of course, as long as oxidation occurs, we will need to evaluate the thermodynamics of the system under study. We therefore begin this section with a description of the

thermodynamics of the MnO / Mn (diluted in ferrite) equilibrium in the selected N2-H2 atmosphere (section 3.2.1). 3.2.1 Thermodynamic aspects The chemical reaction between dissolved oxygen and manganese in the alloy is as follows: Mn (dis) + O (dis) ↔ MnO (s)

(R-4)

The stability of MnO depends on the value of the Gibbs free energy, ∆𝑟 𝐺𝑀𝑛𝑂 , of reaction (R-4). If the reference state chosen is pure solid Mn and pure O2 at standard pressure of 1.013 bar, the law of mass action is written as [54]: 𝑎𝑀𝑛𝑂

Eq. (7)

𝑎𝑀𝑛 .𝑎𝑂

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° (𝑇) + 𝑅𝑇ln ∆𝑟 𝐺𝑀𝑛𝑂 = ∆𝑟 𝐺𝑀𝑛𝑂

° where ∆𝑟 𝐺𝑀𝑛𝑂 is the standard Gibbs free energy of reaction (R-4) at a given temperature T (in K), and

𝑎𝑖 is the activity of substance 𝑖. As a first approximation, the oxide formed MnO can be considered pure, i.e., 𝑎𝑀𝑛𝑂 = 1. Finally, MnO precipitates when: ° ∆𝑟 𝐺𝑀𝑛𝑂 − 𝑅𝑇𝑙𝑛(𝑎𝑀𝑛 . 𝑎𝑂 ) ≤ 0

-p

Eq. (8)

re

As Mn and O are present in small quantities in the alloy, i.e. in the form of a dilute solution, the interaction between them can be neglected and Henry’s law can be used to express their activities in the alloy. In steel industry, the reference state of dilute elements in steel is generally chosen to be

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Henry’s solution containing 1 wt.% element, instead of pure solid element or pure O 2 [55]. In this case, with the reference state of Henry’s solution containing 1 wt.% element, a new scale of activities can be defined:

na

𝑎𝑖′ = 𝑤𝑖

Eq. (9)

where 𝑤𝑖 is the mass fraction of i (i being Mn or O) in the alloy (in wt.%).

ur

To be able to use the reference state of Henry’s law containing 1 wt.% element in the law of mass action (Eq. (7)), it is necessary to complete the thermochemical data with the Gibbs free energy of Mn

Jo

and O dissolution in solid iron for the following transformations: pure Mn(s) ⇆ Mn (Henry ′ s solution 1 wt. % in solid iron)

∆𝑟 𝐺𝑑𝑖𝑠𝑠,𝑀𝑛

(R-5)

1

∆𝑟 𝐺𝑑𝑖𝑠𝑠,𝑂

(R-6)

2

O2 (pure gas) ⇆ O (Henry ′ s solution 1 wt. % in solid iron)

The link between both activities, 𝑎𝑖 and 𝑎′𝑖 , is then given by: 𝑅𝑇𝑙𝑛 𝑎𝑖 = 𝑅𝑇𝑙𝑛 𝑎𝑖′ + ∆𝑟 𝐺𝑑𝑖𝑠𝑠,𝑖 Therefore, Eq. (8) can be written as:

Eq. (10)

∆𝑟 𝐺 0 − 𝑅𝑇𝑙𝑛(𝑤𝑀𝑛 . 𝑤𝑂 ) ≤ 0

Eq. (11)

° with ∆𝑟 𝐺 0 = ∆𝑟 𝐺𝑀𝑛𝑂 − ∆𝑟 𝐺𝑑𝑖𝑠𝑠,𝑀𝑛 − ∆𝑟 𝐺𝑑𝑖𝑠𝑠,𝑂 .

At equilibrium, the solubility product of MnO oxide, 𝐾𝑀𝑛𝑂 , is expressed as follows: 𝐾𝑀𝑛𝑂 = 𝑒𝑥𝑝 ( 𝑒𝑞𝑢𝑖

where 𝑤𝑖

∆𝑟 𝐺 0 𝑅𝑇

𝑒𝑞𝑢𝑖

𝑒𝑞𝑢𝑖

) = 𝑤𝑀𝑛 ∙ 𝑤𝑂

Eq. (12)

is the concentration of element i in solid iron (in wt.%) at equilibrium with MnO oxide

(reaction R-4). Combining Eq. (11) and (12), MnO oxide precipitates if: 𝑤𝑀𝑛 ∙ 𝑤𝑂 ≥ 𝐾𝑀𝑛𝑂

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Eq. (13)

with the solubility product of MnO expressed in ppm2 summarized by Huin et al. [20]: 𝑙𝑜𝑔 𝐾𝑀𝑛𝑂 = 10.95 −

10830

Eq. (14)

𝑇

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As discussed in Section 3.1.2, the local thermodynamic equilibrium between water vapor and dissolved oxygen is established instantly. The corresponding weight fraction of O dissolved in the

re

alloy at 800 °C is given by Eq. (1). Under these conditions, the minimum weight fraction of Mn 𝑒𝑞𝑢𝑖

necessary for the formation of MnO can be calculated with Eq. (13): 𝑤𝑀𝑛 = 0.053 wt.%, which is much smaller than the Mn content of the alloy studied. This means that the surface alloy is

3.2.2 Nucleation of MnO oxides

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oversaturated with dissolved oxygen and manganese compared to MnO formation.

na

As the adsorption and dissociation of water vapor on the alloy surface is very rapid, the equilibrium between dissolved oxygen and water vapor in the annealing atmosphere is expected to be achieved instantaneously. Thus, it can be assumed that nucleation starts at the instant 𝑡𝑠−𝑛𝑢𝑐𝑙 with 𝑡𝑠−𝑛𝑢𝑐𝑙 ≈ 0

ur

(section 3.1.2). In addition, this equilibrium state is maintained throughout the nucleation process. In the case of the external oxidation studied in this work, a simplified configuration is considered

Jo

for heterogeneous nucleation: spherical cap-shaped MnO clusters with radius 𝑟 are formed on the supposed flat surface. The classical nucleation theory [56-59] assumes that three key parameters are then involved in heterogeneous nucleation, namely the surface energies of MnO and Fe respectively (𝛾𝑀𝑛𝑂 and 𝛾𝐹𝑒 ) and the interface energy between MnO and ferrite (𝛾𝐹𝑒/𝑀𝑛𝑂 ). If 𝛾𝑀𝑛𝑂 is isotropic, the equilibrium shape of MnO clusters is indeed spherical cap-shaped. However, the MnO nuclei can be faceted crystals (Fig. 1 and 2). In this case, 𝛾𝑀𝑛𝑂 is anisotropic and depends on the orientation of MnO crystal faces. It is then advised to use an average surface energy corresponding to an equivalent

spherical cap-shaped cluster, having the same volume and surface energy as the actual one [58]. The link between the three surface energies and the contact angle θ is given by Young’s equation: 𝛾𝐹𝑒 = 𝛾𝑀𝑛𝑂 ∙ cos 𝜃 + 𝛾𝐹𝑒/𝑀𝑛𝑂

Eq. (15)

Using Becker-Döring nucleation theory, the heterogeneous nucleation rate of MnO oxides on the alloy surface is given by [57-58]: 𝐼ℎ𝑒𝑡 = 𝑁𝑆 𝛽 ∗ 𝑍𝑒𝑥𝑝 (

−∆𝐺 ∗ 𝑘𝐵 𝑇

)

Eq. (16)

where 𝑁𝑆 (m-2) is the number of sites where a nucleus can form per unit area, 𝛽 ∗ (s-1) is the growth rate of a cluster with critical radius 𝑟 ∗ , Z is the Zeldovich factor, ∆𝐺 ∗ is the Gibbs free energy of formation

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of a spherical cap-shaped MnO embryo with critical radius 𝑟 ∗ , and 𝑘𝐵 is the Boltzmann’s constant.

The critical radius of an embryo is defined as the minimum radius above which the embryo’s growth is energetically favorable [57-58]. 2∙𝛾𝑀𝑛𝑂 ∙𝑀𝑀𝑛𝑂 ∆𝑟 𝐺𝑀𝑛𝑂 ∙𝜌𝑀𝑛𝑂

and ∆𝐺 ∗ =

16𝜋𝛾𝑀𝑛𝑂 3

2 ∆ 𝐺 𝜌 3( 𝑟 𝑀𝑛𝑂 𝑀𝑛𝑂 ) 𝑀𝑀𝑛𝑂

∙

(2+cos 𝜃)∙(1−cos 𝜃)2 4

Eq. (17)

-p

𝑟∗ = −

𝑁𝑆 is approximated by the average number of manganese atoms present on the alloy surface per unit

𝑏𝑢𝑙𝑘 𝑤𝑀𝑛 ∙𝜌𝑎𝑙𝑙𝑜𝑦 ∙𝑁𝐴

𝑁𝑆 = (

)

2⁄ 3

Eq. (18)

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𝑀𝑀𝑛

re

area.

where 𝜌𝑎𝑙𝑙𝑜𝑦 is the alloy density (assumed to be equal to that of iron because the alloy is mainly composed of Fe, i.e. 7.736×106 g‧ m-3 at 800 °C [60]), 𝑁𝐴 is the Avogadro number (6.022×1023 mol-1),

na

and 𝑀𝑀𝑛 is the Mn molar mass (54.938 g‧ mol-1).

The Zeldovich factor is used to describe cluster fluctuations around the critical size, and in

3

4√𝜋𝑘𝐵 𝑇

∙

(∆𝑟 𝐺𝑀𝑛𝑂 ⁄𝑁𝐴 )2 3 (𝐴∙𝛾𝑀𝑛𝑂 ) ⁄2

Eq. (19)

Jo

𝑍=

ur

particular the probability for a stable nucleus to be redissolved [57-58].

where 𝐴 is described as 1⁄3

A = (9𝜋Ω1 2 )

2−cos 𝜃(1+cos 𝜃) (2+cos 𝜃)2⁄3 (1−cos 𝜃)1/3

Eq. (20)

where Ω1 is the volume of a MnO molecule. Ω1 =

𝑀𝑀𝑛𝑂 𝜌𝑀𝑛𝑂 ∙𝑁𝐴

Eq. (21)

where 𝜌𝑀𝑛𝑂 is the MnO density (5.352×106 g‧ m-3 at 800 °C [60]), and 𝑀𝑀𝑛𝑂 is the MnO molar mass. The limiting step of MnO growth must be determined in order to estimate the growing rate 𝛽 ∗ of the critical cluster. O adatoms are present on the alloy surface in equilibrium with the gas atmosphere (section 3.1.2). The growth can be controlled by the surface diffusion of O adatoms (0.73 eV) or Mn adatoms (0.7 eV), the volume diffusion of O atoms (2.48 eV) or Mn atoms (2.82 eV), the extraction of Mn atoms from the alloy lattice and its adsorption on the surface (1.62 eV) and the formation rate of MnO. The energy barriers of these elementary steps were estimated by DFT (Density Functional Theory) calculations (see the values in brackets in eV / atom [32,61]). The reaction rate at the MnO / ferrite interface has not been evaluated but can be considered fast compared to diffusion phenomena as is usually done for solid / solid reactions [58,59]. Based on the energy barrier values, the rate-limiting

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step is therefore the long-range diffusion of Mn atoms in ferrite (strictly speaking, we must also consider the pre-exponential factors to calculate the different constants at the temperature of interest but these pre-exponential factors are not known to our knowledge except for the diffusion coefficients of Mn and O in ferrite). In this case, 𝛽 ∗ is obtained by solving the classical diffusion problem

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associated with a growing particle [58,59]. This has already been presented in the literature for a spherical particle isolated in an infinite medium (Appendix B in Ref. [59]). Applying this approach to

𝛽 ∗ = 4𝐷𝑀𝑛

𝜌𝑎𝑙𝑙𝑜𝑦 ∙𝑁𝐴 𝑀𝑀𝑛

𝑒𝑞𝑢𝑖

re

a spherical cap-shaped MnO clusters with radius r (Fig. 3) gives (Appendix A): 𝑏𝑢𝑙𝑘 (𝑤𝑀𝑛 − 𝑤𝑀𝑛 )𝑟 ∗ sin 𝜃

Eq. (22)

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Nucleation ceases at the instant 𝑡𝑐−𝑛𝑢𝑐𝑙 when the total number of MnO nuclei per unit area is equal to 𝑁𝑛𝑢𝑐𝑙𝑒𝑖 (≈ 3.5×1012 m-2), which is estimated by counting the number of MnO crystals obtained in

na

the experiments (section 2.1). 𝑡

𝑁𝑛𝑢𝑐𝑙𝑒𝑖 = ∫0 𝑐−𝑛𝑢𝑐𝑙 𝐼ℎ𝑒𝑡 (𝑡)𝑑𝑡 = 𝐼ℎ𝑒𝑡 ∙ 𝑡𝑐−𝑛𝑢𝑐𝑙

Eq. (23)

𝑁𝑛𝑢𝑐𝑙𝑒𝑖 Ω1 ∙𝑁𝐴

∙

𝜋∙𝑟 ∗ 3

3

∙ (2 − 3 ∙ cos 𝜃 + 𝑐𝑜𝑠 3 𝜃)

Eq. (24)

Jo

𝑚𝑀𝑛 =

ur

At the end of the nucleation step, the quantity of Mn (mol‧ m-2) in the nuclei can be estimated with:

𝑚𝑀𝑛 is negligible compared to the amount of Mn in the alloy (Section 4.1). We can therefore assume that the Mn concentration at the steel surface remains constant until 𝑡𝑐−𝑛𝑢𝑐𝑙 when the nucleation is completed. 𝑏𝑢𝑙𝑘 𝑐𝑀𝑛 (𝑥 = 0, 𝑡 ≤ 𝑡𝑐−𝑛𝑢𝑐𝑙 ) = 𝑐𝑀𝑛

Eq. (25)

Once the nucleation step is complete, the growth of oxide particles can begin. As explained above, the rate-limiting step of Mn selective oxidation is the long-range diffusion of Mn atoms in ferrite

(since MnO is formed as small discrete particles and a Mn depletion in steel has been measured at the MnO / steel interface under the same operating conditions as ours [32], it has been assumed that the Mn diffusion into the MnO particles is not the rate-limiting step of oxidation). From now on, the model is divided into two parts: (i) studying the diffusion of Mn atoms from the bulk to the surface of ferrite (section 3.2.3) and (ii) calculating the evolution of the size of external oxides as a function of annealing time (section 3.2.4).

3.2.3 Diffusion of Mn in the alloy The model for the kinetics of the oxidation reactions is based on the solution of generalized

𝐷𝑀𝑛

𝜕2 𝑐𝑀𝑛 𝜕𝑥 2

=

𝜕𝑐𝑀𝑛

Eq. (26)

𝜕𝑡

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equations for the diffusion of Mn to the atmosphere / alloy interface:

where 𝐷𝑀𝑛 is the diffusion coefficient of Mn in the Fe-Mn alloy assumed to be constant. As it is

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difficult to define it precisely from current published researches, it is assumed that this diffusion coefficient does not vary with the concentration of elements in iron and is taken equal to the one found

re

for α-Fe. We also assume that the diffusion occurs in the direction x perpendicular to the alloy surface, i.e., the model does not take into account different diffusion rates in the grains and in the grain

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boundaries of the alloy.

With the initial and boundary conditions described below, this partial differential equation can be solved. The solution allows to calculate the Mn concentration profile 𝑐𝑀𝑛 as a function of annealing

Initial condition

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time 𝑡 and distance 𝑥 perpendicular to the steel surface.

When the Fe-Mn alloy is brought into contact with the annealing atmosphere, the Mn concentration

ur

in the alloy is assumed to be constant and corresponds to the composition of alloy bulk. In addition, the Mn concentration profile remains constant until the end of oxide nucleation.

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𝑏𝑢𝑙𝑘 𝑐𝑀𝑛 (𝑥 ≥ 0, 𝑡 ≤ 𝑡𝑐−𝑛𝑢𝑐𝑙 ) = 𝑐𝑀𝑛

Eq. (27)

Boundary condition at depth in the alloy In Section 3.1.3, it was demonstrated that the characteristic diffusion length of Mn is negligible

compared to half the thickness of the alloy sheet. Consequently, the diffusion of Mn in the alloy can be modelled by diffusion in a semi-infinite medium, which means that the boundary condition can be written as: 𝑏𝑢𝑙𝑘 𝑐𝑀𝑛 (𝑥 = +∞, 𝑡 ≥ 𝑡𝑐−𝑛𝑢𝑐𝑙 ) = 𝑐𝑀𝑛

Eq. (28)

Boundary condition at the atmosphere / alloy interface As said before (section 3.2.2), the crystal growth is controlled by the manganese diffusion from the alloy bulk to the atmosphere / alloy interface. This implies that the interface between MnO and ferrite is at thermodynamic equilibrium. The concentration of dissolved oxygen in the ferrite surface is also at equilibrium with the water vapor of annealing atmosphere before a continuous oxide film is formed. The equilibrium concentration of O and Mn at the atmosphere / alloy interface can therefore be determined using the thermodynamic equations presented in section 3.2.1. Finally, the boundary condition at the atmosphere / alloy interface during nucleation and growth stages can be written as: 𝑒𝑞𝑢𝑖

𝑐𝑀𝑛 (𝑥 = 0, 𝑡 > 𝑡𝑐−𝑛𝑢𝑐𝑙 ) = 𝑐𝑀𝑛 𝑒𝑞𝑢𝑖

𝜌𝑎𝑙𝑙𝑜𝑦 𝑀𝑀𝑛

, in mol‧ m-3.

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𝑒𝑞𝑢𝑖

where 𝑐𝑀𝑛 = 𝑤𝑀𝑛 ∙

Eq. (29)

Solution of the system of diffusion equations

The system of equations to be solved is composed with Eqs. (26), (27), (28) and (29). The

𝑒𝑞𝑢𝑖

𝑒𝑞𝑢𝑖

𝑏𝑢𝑙𝑘 𝑐𝑀𝑛 (𝑥, 𝑡 > 𝑡𝑐−𝑛𝑢𝑐𝑙 ) = 𝑐𝑀𝑛 + (𝑐𝑀𝑛 − 𝑐𝑀𝑛 ) ∙ 𝑒𝑟𝑓 (

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concentration profile as a function of x and t is given by [62]: 𝑥

)

Eq. (30)

re

2∙√𝐷𝑀𝑛 ∙(𝑡−𝑡𝑐−𝑛𝑢𝑐𝑙 )

Since the Mn concentration is very low in the Fe-Mn alloy, the same equation applies for the weight

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fraction of Mn.

The Mn flux at the atmosphere / alloy interface is then: 𝐷𝑀𝑛

𝜋∙(𝑡−𝑡𝑐−𝑛𝑢𝑐𝑙 )

𝑒𝑞𝑢𝑖

𝑏𝑢𝑙𝑘 ∙ (𝑐𝑀𝑛 − 𝑐𝑀𝑛 )

Eq. (31)

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𝐽𝑀𝑛 (𝑥 = 0, 𝑡 > 𝑡𝑐−𝑛𝑢𝑐𝑙 ) = −√ 3.2.4 Growth of MnO particles

To facilitate calculations during the growth phase, it is essential to introduce some simplifying

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assumptions. First of all, all spherical cap-shaped nuclei have the same critical radius 𝑟 ∗ , and are uniformly distributed over the alloy surface. Secondly, the nuclei grow in the same spherical cap-

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shape with the same contact angle and extend laterally until they completely cover the surface (during industrial annealing treatment, the holding at high temperature is not long enough to obtain a continuous oxide film). During the elementary time 𝑑𝑡, the volume of the spherical cap-shaped crystals increases by 𝑑𝑉 given by: 𝑑𝑉 = 𝜋 ∙ 𝑟 2 ∙ (2 − 3 𝑐𝑜𝑠 𝜃 + 𝑐𝑜𝑠 3 𝜃) ∙ 𝑑𝑟

Eq. (32)

If the growth of MnO particles is controlled by Mn diffusion in the alloy, the previously estimated Mn diffusion flux (Eq. (31)) corresponds to the number of moles of Mn and then MnO included in the MnO crystals per unit surface area and time. This Mn diffusion flux is supposed to be uniformly distributed to the crystals leading to the equation. |𝐽𝑀𝑛 (𝑥=0,𝑡>𝑡𝑐−𝑛𝑢𝑐𝑙 )| 𝑁𝑛𝑢𝑐𝑙𝑒𝑖

𝜌𝑀𝑛𝑂

∙ 𝑑𝑡 =

𝑀𝑀𝑛𝑂

∙ 𝑑𝑉

Eq. (33)

This equation can be integrated to calculate the radius of the spherical cap-shaped crystal as a function of time: 𝐷𝑀𝑛

√𝜋∙(𝑡−𝑡 𝑐−𝑛𝑢𝑐𝑙

𝑐−𝑛𝑢𝑐𝑙

∙ )

𝑒𝑞𝑢𝑖

𝑏𝑢𝑙𝑘 𝑐𝑀𝑛 −𝑐𝑀𝑛

𝑁𝑛𝑢𝑐𝑙𝑒𝑖

𝑟 𝜌𝑀𝑛𝑂 𝑀𝑀𝑛𝑂

∙ 𝑑𝑡 = ∫𝑟 ∗

∙ 𝜋 ∙ 𝑟 2 ∙ (2 − 3 cos 𝜃 + 𝑐𝑜𝑠 3 𝜃) ∙ 𝑑𝑟 Eq. (34)

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𝑡

∫𝑡

And finally after integration: 𝑟3 − 𝑟∗3 = 6 ∙

𝑒𝑞𝑢𝑖

𝑏𝑢𝑙𝑘 𝑐𝑀𝑛 −𝑐𝑀𝑛

𝑁𝑛𝑢𝑐𝑙𝑒𝑖

𝐷𝑀𝑛 𝑀𝑀𝑛𝑂

∙√

𝜋3

∙

𝜌𝑀𝑛𝑂

∙

1 2−3 cos 𝜃+𝑐𝑜𝑠 3 𝜃

∙ √𝑡 − 𝑡𝑐−𝑛𝑢𝑐𝑙

Eq. (35)

In the hot-dip galvanizing process, since oxides are not wetted by liquid zinc, an important

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parameter for wettability is 𝑋𝑐𝑜𝑣 , the surface area fraction covered by oxides on the alloy surface (section 1). 𝑋𝑐𝑜𝑣 can be deduced from 𝑟:

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𝑋𝑐𝑜𝑣 = 𝜋 ∙ 𝑟 2 ∙ 𝑠𝑖𝑛2 𝜃 ∙ 𝑁𝑛𝑢𝑐𝑙𝑒𝑖

Eq. (36)

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The initial covered surface fraction is 𝜋 ∙ 𝑟 ∗ 2 ∙ 𝑠𝑖𝑛2 𝜃 ∙ 𝑁𝑛𝑢𝑐𝑙𝑒𝑖 , and the continuous oxide film is formed when 𝑋𝑐𝑜𝑣 = 1.

calculated: 𝜌𝑀𝑛𝑂 𝑀𝑀𝑛𝑂

. 𝑁𝑛𝑢𝑐𝑙𝑒𝑖 .

𝜋𝑟 3 3

∙ (2 − 3 cos 𝜃 + 𝑐𝑜𝑠 3 𝜃)

Eq. (37)

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𝑛𝑀𝑛𝑂 =

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The total quantity of MnO oxides (mol‧ m-2) present on unit area of the alloy surface can also be

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4 Prediction of the kinetics model of selective external oxidation This model was applied to isothermal annealing at 800 °C (T = 1073.15 K) in the atmosphere of N2

and 5 vol.% H2 with a dew point of -40 °C. The physico-chemical parameters involved in the selected operating conditions have been given in the Nomenclature section.

4.1 Nucleation of MnO particles

Since it is assumed that the thermodynamic equilibrium between dissolved oxygen in ferrite and water vapor in the annealing atmosphere is reached very quickly, the first step in the selective oxidation of Mn is the nucleation of MnO. The annealing treatment considered here is isothermal at 800 °C. The relevant nucleation parameters are then estimated at 800 °C. These nucleation parameters involve 𝛾𝑀𝑛𝑂 , ∆𝑟 𝐺𝑀𝑛𝑂 and 𝜃 which are determined as follows. As we do not know which crystallographic planes compose the faces of the MnO particles and even if these oxide particles are faceted at the nucleus scale, we use the smallest value of the surface energy of the MnO oxide that can be found in the literature. This is because the crystallographic planes most likely to be formed are those with the lowest surface energy to minimize the total energy of the particle. The surface energies for MnO(100) and MnO(110) planes are calculated by DFT [63], and the MnO(100) plane provided by literature [63,64] is 0.89 J‧ m-2.

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value of 𝛾𝑀𝑛𝑂 obtained for MnO(100) is smaller than MnO(110) plane. The average value for

The Gibbs free energy, ∆𝑟 𝐺𝑀𝑛𝑂 of reaction (R-4) can be calculated by combining Eqs. (7) to (12): ∆𝑟 𝐺𝑀𝑛𝑂 = 𝑅𝑇𝑙𝑛𝐾𝑀𝑛𝑂 − 𝑅𝑇𝑙𝑛(𝑤𝑀𝑛 ∙ 𝑤𝑂 )

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Eq. (38)

𝑒𝑞𝑢𝑖

with the solubility product 𝐾𝑀𝑛𝑂 given by Eq. (14), 𝑤𝑀𝑛 = 104 ppm and 𝑤𝑂 = 𝑤𝑂

= 1.4×10-2 ppm.

Thus, the value of ∆𝑟 𝐺𝑀𝑛𝑂 at the annealing temperature of 800 °C is -26258 J‧ mol-1. Due to the

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shape of MnO particles observed, the contact angle 𝜃 is less than 𝜋⁄2 (Fig. 2). However, its exact value is not known. In the model presented here, the contact angle is the only parameter evaluated

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from our experimental results. All other physical parameters are based on independent data in the literature. The approach consists in calculating the time 𝑡𝑐−𝑛𝑢𝑐𝑙 at 800 °C when the total number of MnO nuclei per unit area is reached as a function of 𝜃 (Eq. (23), Fig. 4(a)). 𝑡𝑐−𝑛𝑢𝑐𝑙 varies greatly

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when the contact angle is increased: 2.92×10-5 s for 𝜃 = 40 deg to 1.23×106 s for 𝜃 = 60 deg. In the experiments (section 2.1 and [27,65]), it was observed that all the nuclei were formed at about 700 °C, i.e., 𝑡700°C = 113 s after the start of annealing (heating rate of 6 °C s-1). In the isothermal annealing

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treatment at higher temperature (800 °C), it is therefore expected that the nuclei will form in less time. To obtain an order of magnitude of the time required for nucleation at 800 °C, it is assumed that the

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characteristic diffusion length of Mn at 800 °C after 𝑡𝑐−𝑛𝑢𝑐𝑙 is the same as the characteristic diffusion length of Mn from 22 to 700 °C during 𝑡700°C , calculated using the mean diffusion coefficient of Mn over this temperature range and then: 𝑡

𝑡𝑐−𝑛𝑢𝑐𝑙 (800 °C)~

∫0 700°C 𝐷𝑀𝑛 𝑑𝑡 𝐷𝑀𝑛 (800 °C)

= 0.41 s

Eq. (39)

This corresponds to a contact angle of 49.9 deg, as shown in Fig. 4(b). With the physical parameters discussed above and those given in the Nomenclature section, the critical radius of MnO nuclei is equal to 0.90 nm. The volume Ω1 of a MnO molecule (Eq. (21)) being

equal to 2.2×10-29 m3, the critical clusters contain about 12 MnO molecules. The Gibbs free energy ∆𝐺 ∗ of formation of a spherical cap-shaped MnO embryo with critical radius 𝑟 ∗ is 2.5×10-19 J. The Zeldovich factor 𝑍 is 0.12, the number of nucleation sites per unit area 𝑁𝑆 is 9.0×1017 m-2, and the growth rate 𝛽 ∗ is 2008 s-1, leading to a pre-factor 𝑁𝑆 𝛽 ∗ 𝑍 of the heterogeneous nucleation rate of 2.1×1020 m-2‧ s-1 (same order of magnitude as the pre-factor estimated in Ref. [66] for heterogeneous nucleation). Thus the nucleation rate 𝐼ℎ𝑒𝑡 of MnO particles at 800 °C annealing treatment is 8.5×1012 m-2‧ s-1. At the end of nucleation step, the amount of Mn contained in the MnO nuclei is 6.72×10-11 mol‧ m-2. With respect to the Mn content of the alloy (1 wt.% or 1408 mol‧ m-3), this means that the Mn atoms present in the nuclei just come from the extreme surface of the alloy (thickness of 5×10-14 m). This

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justifies the hypothesis that the Mn concentration at the alloy surface remains constant until the end of the nucleation step (section 3.2.3).

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4.2 Mn diffusion in the alloy

The Mn concentration profiles during the isothermal annealing treatment are given as a function of depth and annealing time by Eq. (30). Fig. 5(a) shows the evolution of Mn concentration as a function

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of x from 0 corresponding to the alloy surface to 3000 nm, for samples annealed at 800 °C for 10, 30, 60, 120 and 180 s. As expected, the diffusion distance of Mn increases with longer annealing time.

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The effective diffusion distance 𝑥𝐷 from the alloy surface required to recover 99.9% of the bulk concentration can be evaluated with the Mn concentration profiles. After annealing at 800 °C for 60 s (blue line in Fig. 5(a)), the calculated effective diffusion distance of Mn is about 1150 nm, which

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corresponds quite well to the experimental results of Chen [32] (1800 nm) and Ollivier-Leduc et al. [67] (1600 nm). As an example, Fig. 5(b) compares the calculated Mn profile with previous experimental results (annealing at 800 °C for 60 s). The experimental Mn profile was measured for an

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Fe-Mn (1 wt.%) alloy annealed under exactly the same operating conditions as here [32]. The two Mn concentration profiles are in good agreement. The differences are due to: (i) the Mn diffusion coefficient chosen for the calculation is not perfectly suited to the real alloy; (ii) it takes about 129 s

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before heating to 800 °C in the experiment [32], which means longer annealing time and thus larger effective diffusion distance. Despite this, we have decided to keep the physical parameters found in the independent references in the literature because we know that some parameters, such as diffusion coefficients, may depend on the microstructure of the materials used (role of the grain boundaries for example).

4.3 Growth of MnO particles

The radius of the MnO particles during isothermal treatment is given as a function of annealing time by Eq. (35). As shown in Fig. 6, for samples annealed at 800 °C for 30, 60, 120 and 180 s, the corresponding radius of MnO oxide is 139, 156, 175 and 187 nm, respectively. It can be seen that oxides grow more and more slowly with a longer annealing time, especially when the annealing time is more than 180 s (the radius increment from 180 to 300 s is almost equal to that from 120 to 180 s). This corresponds to the decrease in the Mn flux at the alloy surface as the annealing time increases (Eq. (31), Fig. 5(a)). To compare the calculated and experimental results, the surface area fraction 𝑋𝑐𝑜𝑣 covered by the MnO particles (Fig. 7(a)) and the mean diameter of the MnO / ferrite interface for a crystal (Fig. 7(b)) are given as a function of annealing time. The experimental data for 𝑋𝑐𝑜𝑣 are obtained by image

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analysis on FEG-SEM images (section 2.1). Uncertainty on 𝑋𝑐𝑜𝑣 is relatively high due to its dependence on the ferrite grain orientation (e.g., 10.4 ± 1.0 % for Fe(111), 25.7 ± 1.3 % for Fe(110) and 18.2 ± 0.8 % for Fe(100) at 800 °C). But we propose to compare the result of the calculation with an average value of 𝑋𝑐𝑜𝑣 . The average diameter of the MnO / ferrite interface for a crystal is calculated

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using the experimental values of 𝑋𝑐𝑜𝑣 with Eq. (36). The uncertainty on this average diameter is also deduced from Eq. (36) and is given by: 2

𝑋𝑐𝑜𝑣

∙ (2𝑟𝑠𝑖𝑛𝜃)

Eq. (40)

re

1 ∆𝑋𝑐𝑜𝑣

∆(2𝑟𝑠𝑖𝑛𝜃) = ∙

As said in Section 2.1, secondary nucleation occurs for annealing times longer than 180 s but the

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contribution of the very small particles remains negligible at 180 s. That is why we have also included the experimental point at 180 s for validation purposes.

As shown in Fig. 7, the calculated and experimental results show the same trend for MnO particle

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growth, namely the particle size increases and the growth rate decreases as the annealing time increases. The mean experimental surface area fraction covered and the mean diameter of the MnO / ferrite interface are slightly higher than those given by the calculation. It can be said that both agree

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relatively well with the model, given the measurement uncertainty. The differences between the experimental data and the calculation results can be the results of three

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main causes: (1) The surface area fraction covered by oxides strongly depends on the ferrite grain orientation and this is shown by the error bars in Fig. 7; (2) Annealing is not isothermal but is carried out with rapid heating to 800 °C before holding at 800 °C and then cooling to room temperature. Theses heating and cooling steps extend the annealing time. The corresponding characteristic duration for these two steps can be estimated by the method presented in section 4.1 and Eq. (39) with the mean diffusion coefficient of Mn over the temperature range [20-800 °C]. The characteristic time at 800 °C corresponding to the heating and cooling steps is about 13 s. This means that the experimental points must be shifted by 13 s (see blue points in Fig. 7); (3) The Mn diffusion coefficient (𝐷𝑀𝑛 ) may vary

with the change of manganese concentration. The two main parameters of the model, namely 𝛾𝑀𝑛𝑂 and 𝐷𝑀𝑛 , are not precisely known. In order to quantify their influence, a variation of +/-20% of 𝛾𝑀𝑛𝑂 or 𝐷𝑀𝑛 was studied using the same procedure as explained above (Sections 3.2.2, 3.2.3 and 3.2.4). The results obtained are presented in Fig. 8, which shows the surface area fraction covered by MnO particles (Fig. 8a) and the diameter of the MnO / ferrite interface (Fig. 8b) as a function of annealing time. The influence of the variation of 𝛾𝑀𝑛𝑂 is greater than that of 𝐷𝑀𝑛 on 𝑋𝑐𝑜𝑣 and 2𝑟𝑠𝑖𝑛𝜃. The estimated contact angle with the procedure explained in Section 3.2.2 also varies more for the variation of 𝛾𝑀𝑛𝑂 (from 42.8 to 60.8°) than for the variation of 𝐷𝑀𝑛 (from 49.7 to 50.1°). The main conclusion of this uncertainty analysis is that the model works well, compared to the uncertainty of the experimental points.

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This analytical model can also predict the Mn concentration profile as a function of annealing time at the given depth, the total quantity of MnO oxides present on the unit area of the alloy surface as a function of annealing time, and the Mn and O content consumed by the growth of MnO oxides.

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5 Conclusions

An analytical model has been developed to describe the selective external oxidation of binary Fe-

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Mn alloys during isothermal annealing. The elementary reaction mechanisms evaluated in the discussion are the adsorption and dissociation of water vapor on the alloy surface, the thermodynamics

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of oxidation reactions, the heterogeneous nucleation mechanism of the MnO embryos, the growth kinetics of the MnO oxides, and the diffusion of manganese in the alloy. The model was solved with analytical expressions under the general conditions of high temperature annealing in N2 and H2 with

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traces of water. These expressions are valid as long as the Mn oxidation remains external. In practice, this concerns Fe-Mn alloys with Mn content less than 2 wt.% annealed at 780 to 860 °C, in N2 and 5 to 15 vol.% H2 with dew point below -30 °C [29,30,32-37]. The model is able to calculate the critical

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radius of the MnO embryos, the nucleation rate and time required, the Mn concentration profiles as a function of annealing time and alloy depth, the growth rate, the size and surface coverage of MnO oxides as a function of annealing time. The model was validated with experimental results conducted

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at the laboratory scale with Fe-Mn (1 wt.%) alloys annealed at 800 °C in N2-5 vol.% H2 with -40 °C dew point for annealing time less than 180 s. The experimental size of the MnO particles formed is very well predicted by the model. The analytical model is therefore a versatile and practical tool for studying the influence of annealing temperature, annealing atmosphere composition, dew point and manganese content of steels. However, in the practical annealing line, there must be heating before isothermal annealing and cooling afterwards. Another numerical model, taking into account heating and cooling, to describe external oxidation during annealing should be developed in the future using the same theoretical ideas.

Data availability The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.

Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Sample CRediT author statement

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Li Gong: Investigation, Methodology, Validation, Visualization, Writing - Original Draft. Nathalie Ruscassier : Investigation. Mehdi Ayouz : Investigation. Paul Haghi-Ashtiani : Investigation. Marie-Laurence Giorgi : Conceptualization, Methodology, Validation, Writing - Review & Editing, Supervision.

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Acknowledgements

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This work was supported by the French “Agence Nationale de la Recherche” through the “Investissements d’avenir” program (ANR-10-EQPX-37 MATMECA) and the China Scholarship Council (201604490035). The authors are extremely grateful to Mikhael Balabane for his valuable

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assistance and Jean-Michel Mataigne for fruitful discussions.

Appendix A: Growth rate of a cluster of critical radius 𝒓∗

The growth rate can be calculated for the simplified configuration given in Fig. A1. The growth of the MnO cluster is controlled by the long-range diffusion of Mn atoms in ferrite. In case of a cluster isolated on the surface of an infinite medium, the system of equations to be solved in cylindrical coordinates is given by:

𝜕2 𝑐𝑀𝑛 𝜕𝑟 2

+

1 𝜕𝑐𝑀𝑛 𝑟 𝜕𝑟

+

𝜕2 𝑐𝑀𝑛 𝜕𝑥 2

=0

𝑏𝑢𝑙𝑘 𝑐𝑀𝑛 (𝑟 ≥ 0, 𝑥 → ∞) = 𝑐𝑀𝑛 𝑏𝑢𝑙𝑘 𝑐𝑀𝑛 (𝑟 → ∞, 𝑥 ≥ 0) = 𝑐𝑀𝑛

𝑐𝑀𝑛 (𝑟 ≤ 𝑅, 𝑥 = 0) = {

𝜕𝑐𝑀𝑛 𝜕𝑥

Eq. (A1)

𝑒𝑞𝑢𝑖 𝑐𝑀𝑛

(𝑟 > 𝑅, 𝑥 = 0) = 0 𝑒𝑞𝑢𝑖

𝑏𝑢𝑙𝑘 where 𝑐𝑀𝑛 , 𝑐𝑀𝑛 and 𝑐𝑀𝑛 are the Mn concentrations (mol‧ m-3) in the alloy, in the alloy far from the

oxide particles and in the alloy in equilibrium with MnO. The solution of this system can be found in Ref. [62].

𝜕𝑥

(𝑟, 𝑥 = 0) =

2 𝜋

∙

𝑒𝑞𝑢𝑖

𝑏𝑢𝑙𝑘 𝑐𝑀𝑛 −𝑐𝑀𝑛

Eq. (A2)

√𝑅2 −𝑟 2

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𝜕𝑐𝑀𝑛

The diffusion flux of Mn atoms at the interface (mol‧ m-2‧ s-1) is then: 𝑁𝑀𝑛 = −𝐷𝑀𝑛 ∙

𝜕𝑐𝑀𝑛 𝜕𝑥

(𝑟, 𝑥 = 0)

Eq. (A3)

integral: 𝑒𝑞𝑢𝑖

𝑟=𝑅 𝑟

𝑏𝑢𝑙𝑘 𝐽𝑀𝑛 = −4 ∙ 𝐷𝑀𝑛 ∙ (𝑐𝑀𝑛 − 𝑐𝑀𝑛 ) ∫𝑟=0

𝑅

∙

1 2

√1−( 𝑟 )

𝑑𝑟

𝑒𝑞𝑢𝑖

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And finally

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𝑏𝑢𝑙𝑘 𝐽𝑀𝑛 = −4 ∙ (𝑐𝑀𝑛 − 𝑐𝑀𝑛 ) ∙ 𝑟 ∗ ∙ 𝑠𝑖𝑛 𝜃

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Eq. (A4)

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𝑅

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The total diffusion flux of Mn atoms under the oxide particles (mol‧ s-1) can be calculated with the

Eq. (A4)

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[13] F. Gesmundo, P. Castello, F. Viani, C. Roos, The effect of supersaturation on the internal oxidation of binary alloys, Oxid Met. 49 (1998) 237-260. https://doi.org/10.1023/A:1018834525388. [14] F. Gesmundo, Y. Niu, The internal oxidation of ternary alloys I: the single oxidation of the mostreactive

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Figure Captions:

Fig. 1 SEM images of samples annealed at 800 °C for (a) 30 s, (b) 60 s and (s) 180 s. The oxides had different shapes and sizes depending on the ferrite grain where they are formed. Fig. 2 HAADF-STM images ((a), (c) and (e)) and corresponding EDX mappings with O, Mn and Fe ((b), (d) and (f)) of samples annealed at 800 °C for 60 s. The small yellow particles analyzed mainly contain Mn and O in the same atomic proportions. Fig. 3 Schematic representation for the oxidation reactions: (i) pre-oxidation characterized by the adsorption / dissociation of H2O and the absorption of O atoms in the subsurface lattice of the alloy

growth of MnO islands.

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(section 2.2); (ii) nucleation of MnO embryos when saturation in dissolved O is reached; (iii) lateral

Fig. 4 Evolution of the time 𝑡𝑐−𝑛𝑢𝑐𝑙 when the total number of nuclei is reached as a function of the contact angle: (a) 40-60 deg and (b) 48-51 deg. An order of magnitude of the nucleation time at

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800 °C is 0.41 s, corresponding to a contact angle of 49.9 deg.

Fig. 5 Calculated Mn concentration profiles as a function of depth, obtained for the case of Fe-Mn (1

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wt.%) alloys annealed at 800 °C for 10, 30, 60, 120 and 180 s (Fig. 5(a)), comparison with an experimental profile measured by GDOES for an Fe-Mn (1 wt.%) alloy annealed at 800 °C for 60 s

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under exactly the same operating conditions [32] (Fig. 5(b)).

Fig. 6 Calculated radius of MnO particles as a function of annealing time obtained for the case of Fe-

s and radius in nm.

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Mn (1 wt.%) alloys annealed at 800 °C. The values in parenthesis correspond to the annealing time in

Fig. 7 Calculated surface area fraction covered by MnO particles (a) and diameter of the MnO / ferrite interface for a crystal (b) as a function of annealing time compared with the experimental points

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measured by image analysis on FEG-SEM images.

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Fig. 8 Calculated surface area fraction covered by MnO particles (a) and diameter of the MnO / ferrite interface for a crystal (b) as a function of annealing time compared with the experimental points when 𝛾𝑀𝑛𝑂 and 𝐷𝑀𝑛 vary by +/-20%. The conclusion of this uncertainty analysis is that the model works well compared to the uncertainty of the experimental points. Fig. A1 Simplified configuration considered to determine the growth rate of a cluster of critical radius 𝑟 ∗ in cylindrical coordinates.

Fig. 1 SEM images of samples annealed at 800 °C for (a) 30 s, (b) 60 s and (s) 180 s. The oxides had different shapes and sizes depending on the ferrite grain where they are formed.

Fig. 2 HAADF-STM images ((a), (c) and (e)) and corresponding EDX mappings with O, Mn and Fe

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((b), (d) and (f)) of samples annealed at 800 °C for 60 s. The small yellow particles analyzed mainly

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contain Mn and O in the same atomic proportions.

Fig. 3 Schematic representation for the oxidation reactions: (i) pre-oxidation characterized by the adsorption / dissociation of H2O and the absorption of O atoms in the subsurface lattice of the alloy (section 2.2); (ii) nucleation of MnO embryos when saturation in dissolved O is reached; (iii) lateral

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growth of MnO islands.

Fig. 4 Evolution of the time 𝑡𝑐−𝑛𝑢𝑐𝑙 when the total number of nuclei is reached as a function of the

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contact angle: (a) 40-60 deg and (b) 48-51 deg. An order of magnitude of the nucleation time at

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800 °C is 0.41 s, corresponding to a contact angle of 49.9 deg.

Fig. 5 Calculated Mn concentration profiles as a function of depth, obtained for the case of Fe-Mn (1 wt.%) alloys annealed at 800 °C for 10, 30, 60, 120 and 180 s (Fig. 5(a)), comparison with an experimental profile measured by GDOES for an Fe-Mn (1 wt.%) alloy annealed at 800 °C for 60 s

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under exactly the same operating conditions [32] (Fig. 5(b)).

Fig. 6 Calculated radius of MnO particles as a function of annealing time obtained for the case of Fe-

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Mn (1 wt.%) alloys annealed at 800 °C. The values in parenthesis correspond to the annealing time in

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s and radius in nm.

Fig. 7 Calculated surface area fraction covered by MnO particles (a) and diameter of the MnO / ferrite interface for a crystal (b) as a function of annealing time compared with the experimental points

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measured by image analysis on FEG-SEM images.

Fig. 8 Calculated surface area fraction covered by MnO particles (a) and diameter of the MnO / ferrite interface for a crystal (b) as a function of annealing time compared with the experimental points when

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𝛾𝑀𝑛𝑂 and 𝐷𝑀𝑛 vary by +/-20%. The conclusion of this uncertainty analysis is that the model works

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well compared to the uncertainty of the experimental points.

Fig. A1 Simplified configuration considered to determine the growth rate of a cluster of critical radius

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𝑟 ∗ in cylindrical coordinates.