High temperature oxidation of HSLA steel under vapor conditions

High temperature oxidation of HSLA steel under vapor conditions

Pushkar Dashrath Devkate

Materials Engineering, master's level (120 credits) 2020

Luleå University of Technology

Department of Engineering Sciences and Mathematics

1 Acknowledgements

First and foremost, I would like to express my gratitude towards SSAB for the opportunity given to carry out my master thesis in their research institute and for all the support afforded during the project.

I would especially like to thank my supervisors Pr. Esa Vuorinen and Dr. Rosa M. Pineda for their guidance and support during this semester.

This project would not have been possible without the technical help of the technical staff at Luleå University of Technology (LTU), therefore, I also want to thank Mr. Lars Frisk and Mr. Johnny Grahn for taking their time, either to operate the machines or to seek other technical information.

Finally, I would like to thank Swerea MEFOS and everyone that has contributed

to the development of this project, whereas technicians at Swerea MEFOS,

friends (special thanks to my friend MOUMNI Fahd), and colleagues at both

MEFOS and LTU, who not only contributed with productive discussions but also

have been there to support me along the way.

2 Abstract

Manufactured continuous casted slabs are covered by an oxide layer. The oxide layer growth depends on the temperature, on the time subjected to air, and on the water vapor content of the atmosphere. The oxide layer consists of different oxides; hematite, magnetite and Wüstite. The iron-oxygen phase stability diagram and its intermediate compounds with other metal oxides are established from 1000°C to1200°C, and the oxide phase composition within different scale layers and content of alloying elements have been investigated. At higher water partial pressures, the stable oxide phases are Wüstite and hematite. The magnetite phase amount increases with an increase of alloying elements content in the steel, whereas only the Spinel phase is stable for different alloying elements content at lower water partial pressures. For alloyed steel, hematite and magnetite can be formed inside the scale layer in addition to Wüstite during continuous casting.

For that, three different cases are considered, i.e. sample with as cast, clean, and

covered with casting powder. The thickness after the oxide layer formation at

different temperatures and at different conditions has also been measured.

3 List of figures

Figure 1. Schematic illustration of a casting machine showing different main sections [1]. .... 6

Figure 2. Heat transfer in the mould, secondary cooling zone and formation of solid shell. Mushy zone and liquid core can also be seen [6]. ... 8

Figure 3. Oxidation reaction to oxide layer with air and water droplet [19]. ... 12

Figure 4. The oxidation growth can ideally be divided into the following classes: Logarithmic (x log (t)), parabolic (x2t) and linear (x t), where x represents weight gain and t time [19]. ... 14

Figure 5. Three common metal oxide crystal structures: (a) rock-salt structure,(b) ... 14

Figure 6. Oxidation and reduction exchanging atoms [20]... 15

Figure 7. Interfacial reactions and transport processes for high temperature oxidation ... 16

Figure 8. The iron-oxygen equilibrium phase diagram. ... Error! Bookmark not defined. Figure 9. The different diffusion steps and interfacial reactions of the oxidation of iron above 570 °C [23]. ... 19

Figure 10. The reaction of carbon with iron at high temperature [29]. ... 21

Figure 11. TGA experimental weight loss profiles obtained from the thermal reduction of hematite at different temperatures using 20% H2[37]. ... 24

Figure 12. Sample lay-out from steel slab used for oxidation under water vapor. ... 25

Figure 13. Experimental setup for oxidation under water vapor atmosphere. ... 26

Figure 14 .Cutting machine Struers model Discotom 100. ... 27

Figure 15. JEOL JSM-IT300 used for microstructural analysis. ... 29

Figure 16. Malvern Analytical Empyrean diffractometer. ... 30

Figure 17. Etched with Citric acid and NaSCn and Chloral, beige color shade for magnetite and blue reveal hematite. ... 33

Figure 18. Oxide layer after etching Wüstite magnetite and hematite. ... 33

Figure 19. LOM image of etched with chloral and citric acid and sodium thiocyanate three layers of Wüstite, magnetite and hematite. Error! Bookmark not defined. Figure 20. LOM at higher magnification structure a) Mixture of magnetite and hematite b) hematite layer. ... 34

Figure 21. SEM image of cast sample heated at 1000°c ... 35

Figure 22.Experimental oxide scale thickness as a function of temperature obtained for specimens oxidized at 1000 ℃, 1100 ℃ and 1200 ℃ after 45 minutes holding time treated with various surface conditions (As cast, clean and covered with casting powder). ... 36

Figure 23. SEM at high magnification with corundum structure hematite at 1200°C cleaned sample. ... 37

4

Figure 24. SEM micrographs showing the oxide scale thickness formed on specimens oxidized at 1000℃, 1100 ℃ and 1200 ℃ after 45 minutes holding time and various surface

conditions: (a) as cast, (b) clean and (c) covered with casting powder. ... 38

Figure 25. EDS electron image of oxide layer to spectrum point from top to near to substrate. ... 39

Figure 26. EDS spectrum result of amount of element in spectrum 1. ... 39

Figure 27.Graph of carbon content in the oxide layer with distance near substrate to outer layer for temperature a) 1000 °C b) 1100 °C c) 1200 °C. ... 41

Figure 28. XRD diffraction pattern comparison of as cast samples treated at 1000 °C 1100 °C 1200 °C. ... 42

Figure 29. XRD diffraction pattern comparison of samples with clean metallic surfaces treated at 1000 °C, 1100 °C, and 1200°C. ... 43

Figure 30. XRD diffraction pattern comparison of samples metallic surfaces coated with powder and treated at 1000 °C, 1100 °C, 1200 °C. ... 44

Figure 31. Graph of concentration vs distance with respect to time. ... 47

Figure 32. Conideration of Fick’s second law model to derive equation ... 48

Figure 33. Concentration profile for non-steady diffusion as per eqn (16). ... 52

Figure 34. Oxide model for each phase for diffusion of oxygen from surrounding through inside the oxide. ... 54

Figure 35 Diffusion coefficient in function of time at a) 1000 °C b) 1100 °C c) 1200 °C... 56

Figure 36. Oxide scale model for a fixed concentration of oxygen in chamber ... 57

Figure 37. Graph of sample thickness vs. time with the equations of the linear functions for each temperature. ... 57

5 List of Tables

Table 1. Defects present in continuous casting [8]. ... 10

Table 2. Parameters for hot mounting... 27

Table 3. Polishing and grinding steps. ... 28

Table 4. Thickness measurement of oxide for each condition... 31

Table 5. Etchants for the different oxide layers. ... 32

Table 6. Percentage of phases formed in the oxide scale obtained by XRD analysis. ... 45

Table 7.The Error Function... 52

Table 8.Concentration of oxygen in per phase. ... 53

Table 9. diffusion coefficient for each phase by using error function. ... 55

Table 10. Diffusion coefficient by using fixed concentration of oxygen in chamber ... 58

6 1. Introduction

Steel has become the backbone material of the modern World because of its many unique and irreplaceable characteristics and properties and will remain the foundation for future industrial development and progress [1, 2]. According to Steel Statistical Yearbook 2016, the percentage of continuously cast steel has reached 95.9% in comparison with the total amount of steel produced in the World [3].In continuous casting, molten steel is poured from the tundish in the water cooled mould and partially solidified bloom/billet or slab (hereafter called strand) is withdrawn from the bottom of the mould so that solidified bloom/billet or slab is produced constantly and continuously. Continuous casting is widely adopted by steelmakers.

The essential components of a continuous casting machine are tundish, water- cooled mould, water spray and torch cutters. Torch cutter cuts the required length of the strand. In Fig. 1, the arrangement of tundish, mould and water spray is shown.

Figure 1. Schematic illustration of a casting machine showing different main sections [1].

7 The tundish acts as a reservoir for the molten steel and it supplies the liquid steel continuously through the caster machine at a controlled rate. A layer of slag is usually formed on the top of the molten steel. The flow rate is maintained constant by keeping the same steel bath height in the tundish through teeming of molten steel from it. The number of moulds is either one or more than one. Normally, bloom and billet casting machines are multi‐strand meaning that the number of moulds is either 4 or 6 or 8. Slab casters usually have either one or two of them. During sequence casting and ladle change-over periods, tundish supplies molten steel to the mould. Solidification of steel begins in the mould. In the water cooled mould, molten stream enters from the tundish into the mould in presence of flux through the submerged nozzle immersed in the liquid steel. The casting powder is added onto the top of the molten steel where it melts and penetrates between the surface of the mould and the solidifying shell to minimize the friction. Controlling the height of the molten steel in the mould is crucial for the success of the continuous casting.

The solidification begins from the meniscus of steel level in the mould. Mould level sensors are used to control the level in the mould [4].

Secondary Cooling

Below the partial mould solidification strand is water sprayed to complete the solidification. There is a number of primary parameters influencing the rate of heat extraction such as [4]:

• Water drop flux.

• Mean drop size.

• Droplet velocity hitting the strand surface.

• Wetting effects.

A water vapor blanket forms on the strand surface which prevents the direct

contact of water droplets with the strand surface. Velocity of droplets should be

such that it allows the penetration of the vapor layer so that they can wet the

8 surface in order to cool it down. In secondary cooling, the number of nozzles is distributed over the surface of the moving strand. Overlapping of spray may occur [4].

Heat transfer in continuous casting

Heat transfer in continuous casting takes place in the mould and in the secondary cooling by a combination of conduction, convection and radiation [5]. Fig. 2 shows the heat transfer occurring in the mould as the secondary cooling.

Figure 2. Heat transfer in the mould, secondary cooling zone and formation of solid shell.

Mushy zone and liquid core can also be seen [6].

The formation of a mould air gap influences the heat transfer. The higher the heat flux in the mould the higher the casting speeds. Heat flux depends on the composition of steel and the following variables:

• The mould taper.

• The type of lubricant.

• The type of mould (straight or curve).

• The casting speed.

9 The major requirements for the secondary cooling are:

• The partially solidified strand must have a sufficiently strong shell at the exit of the mould to avoid a breakout due to the liquid pressure. The liquid core should be bowl shaped.

• The solidification must be completed before the withdrawal roll. Water spray must be distributed uniformly on the moving strand so that the reheating of the strand does not occur. A non‐uniform cooling leads to the generation of thermal stresses on the surface that may induce the appearance of cracks.

• The outer surface temperature should be higher than 850 °C to avoid a volumetric expansion accompanying the transformation of austenite structure to ferrite [7].

Mist spray cooling, i.e. mixture of air + water provides a uniform cooling. A high pressure of the mixture is sprayed on the metal surfaces. Some advantages are:

a)

Uniform cooling.

b)

Lower water requirement.

c)

Reduced surface cracking.

Casting defects

Currently, steels are casted continuously into slabs to be used as flat products (plates, sheets…) and as blooms and billets for structural products (rails, bars…).

Defects in continuous casting, as shown in Table.1, originate from several

factors such as mould oscillation, mould flux, segregation coefficient of solute

elements; phase transformation etc. [8].

10

Table 1. Defects present in continuous casting [8].

2. Objectives

Oxidation is relevant for high-temperature processes since it plays a key role on the formation of defects in micro-alloyed steels. The effects include changes on heat transfer during solidification and subsequent cooling for several metallurgical processes. Oxide formation itself is a hindrance for processing because the oxide has to be removed from as-cast surfaces before subsequent processing. Therefore, a thorough understanding of its formation mechanisms is needed for an adequate control of the oxide formation during a hot processing.

The main objectives of this master thesis are:

1. Characterization of oxide scale formed at different temperatures and surface treatments by using different characterization techniques (i.e.

SEM, OLM, EDS).

2. Analysis of the influence of casting powder on oxidized specimens.

3. Identification of phases by using XRD technique.

Defects

Internal Surface Shape

• Midway cracks. • Longitudinal midface • Rhomboidity.

• Longitudinal depression orality.

• Triple point cracks.

• Center line cracks.

and corner cracks.

• Transverse midface.

• Diagonal cracks.

• Center segregation and porosity.

• Casting flux inclusion.

• Blowholes

and corner cracks.

• Deep oscillation masks.

11 2.1 Background

The present work is carried out in collaboration with a steel-making company in Finland and it aims to study the oxidation mechanisms for micro-alloyed steels concerning different stages of the casting process (e.g. secondary cooling and final as-cast product). Thus, the control of oxide formation by changing casting parameters such as cooling rate, water exposure, etc., would have an impact that ultimately improves the yield strength, the surface quality and the value of the cast products.

3. Oxide formation

Oxide scale generated on the steel surface during and after the rolling is classified as tertiary scale, [9,10] which is usually coiled with the strip and retained at room temperature. During the cooling step after coiling, the scale may grow further if oxygen is available and will also undergo structural changes. The final scale structures developed on the strip are greatly affected by the coiling temperature and the coil-cooling conditions [11,12].

The oxidation behavior of steel in ambient air differs significantly from

that of an atmosphere in a reheat furnace [11]. Each of the elements present

in the steel may behave differently from iron. Generally, the existence of

chromium, aluminum, and silicon, which are less noble than iron, provide

a certain level of oxidation resistance for steel, but the protective effect

becomes insignificant if their levels are very low [9]. Residual elements,

such as copper, nickel and tin, which are nobler than iron, are usually

accumulated at the scale–substrate interface and have little effect on the

steel oxidation behavior. High-carbon steel may suffer from

decarburization during steel oxidation [13][14]. The reactions between iron

and oxygen are exothermic in nature and, as a result, an over- temperature

phenomenon [15] is present during the initial oxidation stage, as the heat

generated by the rapid initial reactions cannot be quickly conducted away

12 when small samples are used. The temperature is, therefore, higher for smaller samples [15]. Over temperature also increases with higher oxidation temperatures. Despite the rapid initial reactions, the longer-term oxidation rate of iron under isothermal-oxidation conditions is quite steady and usually follows the parabolic rate law. Within the parabolic-oxidation regime, the scale developed comprises an extremely thin outermost hematite layer, a thin intermediate magnetite layer, and a thick inner Wüstite layer [15].

This proportion of oxide phases reflects the fact that the diffusion coefficient of iron in Wüstite is higher than in magnetite and that the diffusion of oxygen and iron through the hematite layer are extremely slow [16][17][18].

3.1 Oxidation in air and water at high temperature

Figure 3. Oxidation reaction to oxide layer with air and water droplet [19].

In Fig. 3 sketch of a water droplet (after Ebbing), the oxidizing iron supplies

electrons at the edge of the droplet to reduce oxygen from the air. The iron surface

inside the droplet acts as the anode for the process [19]:

13 Fe(s) → + Fe

(aq) + 2e

………...……….(1)

The electrons can move through the metallic iron to the outside of the droplet where:

O

(g) + 2H

O(l) + 4e

→ 4OH

(aq)……….………. (2)

Within the droplet, the hydroxide ions can move inward to react with the iron (II) ions moving from the oxidation region. Iron (II) hydroxide is precipitated.

Fe

(aq) + 2OH

(aq) →Fe (OH)

(s)……….. (3) Oxide is then quickly produced by the oxidation of the precipitate [19].

4Fe (OH)

(s) + O

(g) → 2Fe

O

•H

O(s) + 2H

O(l)… (4) Then formation of hematite takes place with a release of water vapor.

A study of the oxide growth rate gives important information about the oxidation mechanisms. Two different methods to measure the oxide growth were used. The oxide growth can ideally mainly be divided into three categories: Logarithmic (x log (t)), parabolic (x2t) and linear (x t), where “x” represents the weight gain and

“t” the time, see Fig. 4 [19]. The logarithmic growth is initially very rapid and

decreases then to a very low value. It is common for metals at lower temperatures,

typically below 300ºC. The rate-limiting step is known to be the

tunneling/diffusion of electrons or ions through the oxide scale (the classical

Wagner mechanism), resulting in a parabolic growth rate. A linear reaction rate

is constant in time independent of the scale thickness. One example of linear

growth rate is when the surface reaction is the rate-limiting step. However, for

most alloys, the growth rate is a combination of the ideal cases.

14

Figure 4. The oxidation growth can ideally be divided into the following classes: Logarithmic (x log (t)), parabolic (x2t) and linear (x t), where x represents weight gain and t time [19].

3.2 Phases in oxide scales

Figure 5. Three common metal oxide crystal structures: (a) rock-salt structure, (b) Spinel structure, (c) corundum structure [18].

15 In order to understand the oxidation behavior of an alloy it is important to analyze the properties of the elements with highest affinity for oxygen in the alloy, i.e the elements that will form the oxide scale either as a solid solution or as different layers [18].The alloys in this project can mainly form three types of oxide structures. They are rock-salt, Spinel and corundum structures, see Fig.7. In the following sections, the properties of the most important oxides for the materials investigated in this thesis will be described.

3.3 Oxidation in Iron

Oxidation can be a spontaneous process or may start artificially in the presence of certain oxidizing agents. Chemically it can be defined as a gain of oxygen or loss of hydrogen or even loss of electrons.

Figure 6. Oxidation and reduction exchanging atoms [20]

Elements which can easily lose electrons are said to “oxidize” easily, such as iron,

potassium, sodium: few reactive metals which can oxidize easily, meaning they

are good reducing agents. On the other hand, non-metals oxidize hardly as they

are more reluctant to lose electrons and hold them tightly. They cannot oxidize

16 easily like nitrogen, oxygen, and chlorine. Oxidation reaction, with the help of reactions, occurs during the formation of rust on iron metal surface. The metal surface reacts with atmospheric moisture and oxygen that oxidize the metal atoms to Fe

ions [20].The first reaction begins at the metal/gas interface and this reaction forms an intermediate layer between the alloy and the gas. The possible forms this layer can create are numerous; and that may alter the reaction process.

In order for the reaction to proceed further, one or both reactants must penetrate the scale, i.e. either metal must be transported through the oxide to the oxide/gas interface and react there,

Figure 7. Interfacial reactions and transport processes for high temperature oxidation mechanisms[21].

or oxygen must be transported to the oxide/metal interface and react there,

resulting later in a high temperature oxidation. Since all metal oxides are ionic in

nature, it is clear that ions and electrons must migrate in order for the reaction to

17 proceed. The transporting step of the reaction mechanism links the two-phase boundary reactions, as it can be seen in Fig. 7 [21].

The oxidation of alloys used in high temperature applications are generally complex. It is therefore an advantage to work with simplified systems in order to build knowledge about the oxidation process in steps. This can be achieved by studying the impact of a simple environment, e.g. oxygen with water vapor, or by using pure metals or model alloys.

In Fig. 4 M= Fe (iron) and the reaction with oxygen in air forms an iron oxide FeO.

Fe= Fe

+2e

………. (5) 1/2O

+2e

=O

……….... (6) 2Fe

+ O

→ 2FeO……….…….. (7)

Iron oxide (FeO) is thermodynamically unstable below 575 °C, tending to disproportionate to metal and Fe

O

[23].

4FeO → Fe + Fe

O

……...……….. (8)

3.4 Iron-Oxygen Phase diagram

There are three different types of iron oxides, Wüstite (FeO), magnetite (Fe3O4)

and hematite (Fe2O3) [22]. The oxides form a layered scale at high temperatures,

i.e Wüstite as closest to the metal (above 570 ºC), magnetite in the middle, and

hematite as the outer layer. Therefore, Wüstite forms at the lowest partial pressure

of oxygen and hematite at its highest. Wüstite is known to be the fastest growing

iron oxide [22]. It has a rock-salt structure, see Fig. 7a, and is classified as a p-

type oxide. The phase diagram for Fe-O shows that Wüstite is not stable below

570 ºC, see Fig. 8. Magnetite has an inverse Spinel structure and contains both

Fe3+ and Fe2+ ions in the structure. It is classified as an n-type oxide [22].

18 Hematite has a corundum structure, see Fig. 7c, and it is classified as an n-type oxide [22].

Magnetite typically forms a thicker layer than hematite [23]. The different diffusion steps and interfacial reactions during the growth of a layered iron oxide are shown in Fig. 9.

Figure 8. The iron-oxygen equilibrium phase diagram.

19

Figure 9. The different diffusion steps and interfacial reactions of the oxidation of iron above 570 °C [23].

4 Effect of Carbon during Steel Oxidation

During steel oxidation, carbon from the steel substrate can be oxidized, causing decarburization of the steel when the rate of carbon oxidation exceeds that of the iron. Decarburization is normally observed above 700°C, particularly for steels containing relatively higher levels of carbon [24]. Simultaneous scaling and decarburization has been studied [25], and the general consensus is that during steel oxidation, carbon reacts with the scale via the following reaction [25]:

FeO+ C →Fe + CO……… (9)

Where “C” denotes carbon in solution in steel. Further reaction between CO and the scale may produce CO

via the following reaction:

CO+FeO→ Fe+CO

……….. (10)

In equations (9) and (10) reactions can proceed only when CO and CO

can escape

through the scale. These gases could escape through micro-channels, such as

micro cracks or pores [26]. If the transport through micro-channels is too slow or

prohibited, then the pressure of CO gas would build up. Once it exceeds a certain

level, it can cause blistering or rupture of the scale [27].

20 As showed earlier in Fig.2 , a large number of vacancies are left at regions at, or near, the scale–steel interface with the continuous outward diffusion of iron through the scale. These vacancies may condense to form voids at the interface, providing a space to store the CO and CO

gases produced by reactions (9) and (10). The CO

, CO mixture would be in local equilibrium with the scale. The existence of an oxygen-activity gradient in the scale would induce similar gradient in the gas phase within the void, which results in the scale formation at the bottom surface of the void but also in the scale decomposition at the outer surface [27].

The carbon-bearing gases generated during steel oxidation can also be transported via the outward movement of voids originated at the scale–steel interface [28] . The overall effect of this is the simultaneous migration of the voids through the scale and loss of carbon from the steel substrate [28].

It is unlikely that the oxidizing atmosphere enters into the blister because the steel surface is not oxidized at the nucleation stage. The oxygen partial pressure inside the blister is the value that equilibrates between iron (Fe) and Wüstite (FeO). The partial pressure of oxygen (O) is estimated to be 2×10

atm. at 950 °C [28]. If the internal oxidizing gas is O

, the oxygen partial pressure is too low to oxidize for such a short time.

At this stage, the gases inside are CO, CO

and N (Fig. 10). It is reported that oxidation is caused by the stress generated during scale formation and gas release at the scale/steel interface [29].

During the oxidation, magnified in-situ surface observation technique helps to

see the oxide nucleation process when the surface phase is changing. Such

magnified observation may be useful for obtaining any information concerning

the blister nucleation site. Steel surfaces inside the initiated blisters are oxidized.

21

Figure 10. The reaction of carbon with iron at high temperature [29].

This process explains that detached oxide scale oxidizes the separated metal surface (Fig. 10). When H–OH are contained in a space such as a void located inside a scale, hydrogen (H) is oxidized into water (H

O) at the bottom surface of the detached scale and the water reaches the metal surface to oxidize the metal surface. The water is then reduced to hydrogen. This process also works in the CO–CO

system although gas measurement inside the blisters detects both CO and CO

. It is estimated that a similar process occurs in the oxidation inside the blisters. Here, the oxidation inside the blister occurs for several seconds because the steel surface is not oxidized at the next growth stage. That means the oxidizing gases of CO and CO

disappear or are gradually consumed. One possibility to explain it is that gases inside the blisters permeate through the scale.

The scale has gas permeability and the oxidizing atmosphere can enter the

blisters. This is inconsistent with the result that the steel surface is not oxidized

at the stage of nucleation and at the growth stage. Carbon diffusion mechanism is

proposed here and schematically shown in Fig. 10.The steel surface under the

oxide scale is decarburized if the scale has CO gas permeability. However, it is

supposed that gas permeability of the scale is low as described above. In such

22 case, carbon is enriched at the steel surface. As scale grows, carbon activity at the surface of the steel increases. As shown in Eq. (9) and (10) as the partial pressure of CO increase, CO

’s pressure also does. Therefore the ratio of the CO partial pressure keeps increasing CO

until the state of equilibrium.

4.1 Oxidation under water vapor

The presence of water vapor is known to change the oxidation behavior of metals and alloys compared to oxidation in dry oxygen. It is generally known to increase the corrosion of the environment. Researchers have presented several studies treating the water vapor effect on iron and on Fe-Cr (-Ni) steels over the years [30].

Exposure to O

/H

O mixtures in the temperature range of 1000 to 1400°C, have been reported to result in chromium evaporation. According to Ebbinghaus [25], the most probable vaporizing specie is CrO

(OH)

in this temperature range [21].

High strength steel alloys have been investigated in the presence of water vapor

in this temperature range [23]. Asteman et al investigated the water vapor effect

[3].This shows that the oxidation kinetics change in the presence of water vapor,

causing breakaway oxidation. Tang et al, on the other hand, performed a

microstructural investigation of steel exposed to dry and wet oxygen [4]. A

breakdown of the protective oxide scale was shown to cause the formation of an

outward growing and inward growing of oxide, one important conclusion from

these investigations was that evaporation could lead to depletion of the protective

oxide resulting in the formation of a poorly protective Fe-rich oxide, causing an

increased oxidation rate [31]. The ability of an alloy to endure vaporization

without losing the protective properties of the oxide is expected to depend on the

rate of supply from the alloy to the oxide scale. The material transport

mechanisms have been reported to increase, in the presence of water vapor

[32,33]. Hydrogen has been reported to become incorporated into the oxide scale,

changing the defect-dependent properties [34].

23 4.2 Reversible reaction

When the reactants form products that, in turn, react together to give the reactants back, reversible reactions reach an equilibrium point where the concentrations of the reactants and products no longer change.

The same kind of reaction could be reached with a temperature increase: it would reduce the weight of hematite and form Wüstite again [35]. In case of iron oxide hematite reduction is usually described as a three-step mechanism, i.e., Fe

O

→ Fe

O

→ FeO → Fe, [36] rather than a two-step mechanism, Fe

O

→ Fe

O

→ Fe, when reacting with gases, such as CO and H

. However, Slagtern et al [36]

suggested a two-step reduction mechanism: Fe

O

→ FeO → Fe, when reacting with H

. A three step reaction would form Wüstite, meaning that it would become a reversible reaction. In graph Fig.10. it is shown that the hematite phase forms Wüstite as the temperature increase, even if the Wüstite phase is triggered at 550°C: Due to the reversible reaction, the weight is reduce as hematite induces Wüstite forming [37].

TGA experimental weight loss profiles obtained from the thermal reduction of hematite at different temperatures using 20% H2.

24

Figure 11. TGA experimental weight loss profiles obtained from the thermal reduction of hematite at different temperatures using 20% H2[37].

In graph, at temperatures between 700 and 950 °C, the weight reduction may proceed according to the following reactions:

3𝐹𝑒₂𝑂₃+ 𝐻₂→ 2𝐹𝑒¹ ₃𝑂₄+ 𝐻₂𝑂………. (11)

2𝐹𝑒₃𝑂₄+ 2𝐻₂→ 6𝐹𝑒𝑂 + 2𝐻¹ ₂𝑂………(12)

According to equations (11) and (12) the hematite formed magnetite and then

Wüstite. Furthermore as the temperature increases, the reaction continues and

reverses, then as long as the hematite phase is formed, it becomes a reversible

reaction and forms Wüstite [37].

25 5 Experimental Procedure

Microstructural investigations, have been performed with the use of a scanning electron microscope (SEM) equipped with energy dispersive X-ray spectrometer (EDS) and an electron energy loss spectrometer (EELS), enabling qualitative and quantitative measurements, elemental distribution mapping, and chemical state analyses in the micro-size observation areas. The microstructure has then been linked to the exposure conditions in order to further understand the oxidation mechanisms. The SEM was also used to investigate the surface morphology and to determine the qualitative analysis of elements by EDX of the oxides through EDX mapping.

The experimental procedure includes a full characterization of oxide scale formed at different temperatures and times. The characterization of the oxide scale, required a microstructural analysis performed by using different techniques such as a scanning electron microscope (SEM), but also an optical microscope, an EDX and an XRD that were applied for chemical and phase analysis.

Oxidation process

Oxidation under water vapor was performed in a pilot plant at Swerim, research center. Specimens from a micro-alloyed steel slab were sectioned with dimensions of 5 cm x 7.5 cm, as it can be seen in Fig.12.

Figure 12. Sample lay-out from steel slab used for oxidation under water vapor.

26 Oxidation process was performed in an industrial furnace, N 87/H with dimensions of 340 x 1040 x400 mm. Specimens were introduced in the furnace in a separate box, where a thermocouple was installed in one of the specimens for monitoring temperature during oxidation. The furnace was connected to a steam generator where the water vapor was carried by argon gas into the isolated box providing a saturated vapor atmosphere. Besides, Nitrogen was injected into the furnace for corrosion protection. The experimental set-up can be seen in Fig.13.

Figure 13. Experimental setup for oxidation under water vapor atmosphere.

Specimens were oxidized at temperatures of 1000 °C, 1100 °C and 1200 °C, during the constant time of 45 minutes for studying the effect of surface conditions. Furthermore, for the study of oxidation kinetics, specimens were heated up to temperatures of 1000 °C, 1100 °C and 1200 °C respectively during the holding times of 20, 30 and 60 min respectively . Thermal cycles of these experiments can be observed in Fig.13.

5.2 Sample preparation

In order to analyze the casting pieces, a proper sample preparation is necessary.

Samples were prepared by cutting, grinding, and polishing. Cutting was

27 performed by using Discotom 100. Due to the high hardness of the formed material (steel), a speed of 2750 rpm was applied.

Figure 14 .Cutting machine Struers model Discotom 100.

The hot mounting step was done with a Buehler equipment model Simplest 1000 by using a phenolic thermoset resin called Phenocure. The parameters used in each hot mounting operation are included in the Table 2.

Table 2. Parameters for hot mounting.

Heat time (min)

Cool time (min) Pressure (bar) Mould size (mm) Temperature (°C)

1 3 290 40 150

After several trials of grinding and polishing, the most suitable procedure was

found. In the Table 3, all the details of each grinding and polishing step are

included:

28

Table 3. Polishing and grinding steps.

Step Surface Suspension Lubricant Rotation speed (rpm)

Time (min)

1 SiC 600 - Water 300 0.5

2 SiC 1200 - Water 300 0.5

3 Microcloth Diamond 9 µm Kemet W2 150 2

4 Microcloth Diamond 6 µm Kemet W2 150 2

5 Microcloth Diamond 3 µm Kemet W2 150 2

6 Microcloth Diamond 1 µm Kemet W2 150 2

7 Microcloth MasterMet Water 150 4

The first two steps were carried out to grind the surface by using silicon carbide papers, while all the following steps correspond to the polishing. The polishing was achieved by using a series of micro cloths and suspensions with a finer particle size each time.

Regarding the rotation sense of the samples, in case of the polishing steps, the samples were rotating counter-clockwise, while in the grinding steps the samples were static.

The lubricant which commercial name is Kemet W2 was used during most of the polishing steps as it can be seen in the table. This lubricant consists in a low viscosity water based fluid especially developed for use with Diamond suspensions.

5.2.1 Microstructural analysis

The microstructural analysis was performed by using a Scanning Electron

Microscopy (SEM). For this purpose, a JEOL model JSM-IT300 shown in

Fig.15 equipment was used. High vacuum was reached during these analyses, and

the images were taken by using the Secondary Electrons Detector (SED). This

detector was enough to obtain good images of the microstructures. In regard of

the accelerating voltage and the probe current, in most of the cases a voltage of

20 kV and a high current of 1 µA were used. The observation was carried out

with a working distance in between 11 and 12 mm.

29

Figure 15. JEOL JSM-IT300 used for microstructural analysis.

5.2.2 Chemical and phase analysis

In order to identify the elemental composition of the resulting material of the oxide scale. Energy Dispersive X-ray Spectroscopy (EDS) and X-ray Diffraction (XRD) were used.

EDS was performed by the use of the previously mentioned SEM equipment (JEOL JSM-IT300) shown in Fig.16 which also had this tool integrated. This technique provides a qualitative analysis of different compositional maps where the presence of different elements could be localized.

On the other hand, XRD was used to identify phases present in the oxide. For this

purpose, a Malvern Analytical model Empyrean diffractometer, shown in

Fig.16, was used. This equipment is provided with a PIXcel3D detector, which

allows high-resolution results.

30

Figure 16. Malvern Analytical Empyrean diffractometer.

5.3 Thickness measurement

To obtain the thickness of the oxide layers, various parameters are modified in

the sample preparation. For three different temperatures and three different

conditions, the oxide layer thickness is measured. The average values of these

measurements are listed in Table 4. From the results, a growth of thickness is

observable from clean to powder condition and from lowest to highest

temperature. Thicknesses were measured on the overall length of each sample

with a step of 1 mm of distance, and then the average values were calculated.

31

Table 4. Thickness measurement of oxide layers for clean, cast and powder conditions, for the temperatures 1000 ℃, 1100 ℃ and 1200 ℃.

5.4

Metallographic Techniques

Different metallographical techniques are used for study of microstructures in metals and alloys.

For the analysis it is necessary to etch the sample. For that, some chemical combination is used for detecting the phases that are supposed to be present in the oxide layer. Table 5 shows the etchants used for the detection of Wüstite, hematite, and magnetite

The main purpose to realize etching of the oxide layers, is to distinguish the different iron oxides from each other, how they appear and how they are attached to the metal surface.

Temperature Condition Average Thickness

1000 °C

Clean 441 µm

Cast 462 µm

Powder 558 µm

1100 °C

Clean 829 µm

Cast 923 µm

Powder 1064 µm

1200 °C

Clean 1373 µm

Cast 1724µm

Powder 1809µm

32

Table 5. Etchants for the different oxide layers.

Etchant Chemicals Time Phases

Citric acid and Sodium thiocyanate[30]

Citric acid (10%) Sodium thiocyanate

(10%)

30 sec Wüstite(FeO) Magnetite(Fe

O

)

Chloral[29] HCl (1%) with ethanol

30 sec Hematite(Fe

O

)

The effect of different etchants for the revealing of different phases can be seen in Fig.17. The beige color is obtained after etching with citric acid and sodium thiocyanate for the magnetite phase. In Fig.17 an orange arrow shows the middle layer of oxide with magnetite.

After etching for the magnetite the sample is cleaned with ethanol for the next

etching step in order to reveal the hematite phase by using chloral. Samples were

submerged in solution for 30 sec and then cleaned with ethanol and dried. After

this etching, the hematite could be identified by its white color.

33

Figure 17. Etched with Citric acid and NaSCn and Chloral, beige color shade for magnetite and blue reveal hematite.

6 Results and discussions

Light optical microscopy (LOM) can be used for the study of the different oxides phases formed. In Fig. 18 the oxide layers formed on the substrate, FeO (Wüstite), Fe

O

(magnetite), Fe

O

(hematite) are shown .

Figure 18. Oxide layers after etching Wüstite, magnetite and hematite.

Basically iron oxide has Wüstite (FeO), magnetite (Fe

O

) and hematite (Fe

O

) on that basis different reagents are used to reveal the phases. In LOM, three different temperatures are applied: 1000°C, 1100 °C, and 1200 °C for cast, clean

Fe2O3 (Oxygen deficient)Hematite

FeO Wüstite Metal Deficient Fe3O4

Magnetite

34 and powder samples. As we see the thickness proportion of each cases’

attachment of oxide layer from the surface in Fig. 19, the oxide which is near to be substracted is Wüstite which is in low amount. Then in the outer most part the white in color is hematite. After that it seems that the white patches in the upper part are a mixture of hematite and magnetite. As seen in the iron oxygen diagram in the range of oxygen percentage there is a mixture of hematite and magnetite, so it is clearly visible that the oxygen is between 27-28%. In Fig. 19, at higher magnification we clearly see the hematite layer on the outer surface and the mixture of magnetite and hematite. It is used to investigate for other condition phases and their amount.

Figure 19. LOM image of etched with chloral and citric acid and sodium thiocyanate three layers of Wüstite, magnetite and hematite.

35

Figure 20. LOM at higher magnification structure a) Mixture of magnetite and hematite b) hematite layer.

In order to see how the oxide layer forms, it is needed to have an oxide layer investigation at a higher magnification. As temperature increases, the thickness increases and from SEM images it looks like :

Figure 21. SEM image of cast sample heated at 1000

°C

The thickness of the oxide layer increases but inside the oxide layer gaps are

formed. As temperature increases, cracks are formed in the oxide layer close to

the substrate, in Fig. 21, the SEM-image of a cast sample heated at 1000 °C shows

interface cracks located parallel with the scale-metal interface, caused by internal

stresses. Small cracks in this orientation can be seen in the magnetite region in

Fig. 21.

36

Figure 22.Experimental oxide scale thickness as a function of temperature obtained for specimens oxidized at 1000 ℃, 1100 ℃ and 1200 ℃ after 45 minutes holding time treated

with various surface conditions (As cast, clean and covered with casting powder).

In Fig. 22, it is understandable that if the temperature increases the oxide scale thickness also increases for constant holding times; at 1000 °C, 1100 °C and 1200

°C. In case of clean surface condition, the thickness is smaller than for other conditions. In the condition of cast samples, the thickness is slightly larger than for clean surface samples, as this thickness is measured all along the length of the sample.

0 200 400 600 800 1000 1200 1400 1600 1800 2000

950 1000 1050 1100 1150 1200 1250

Thicknessµm

Temperature °C

Thickness vs temperature

clean cast powder

37 In addition, after etching, SEM study shows the hematite and magnetite microconstituents, see Fig. 23, with the corundum constituent (large blocks) as the hematite.

Figure 23. SEM at high magnification with corundum structure hematite at 1200°C cleaned sample.

SEM images reveal that the propagation of cracks increases with the temperature.

In Fig. 24 defects as cracks and void gaps are formed for all cases with increasing

temperature. At 1000 °C it is clearly visible that for the clean surface the oxide is

attached to the surface. In case of a cast sample cracks occur and void gaps are

visible. While in the casting powder sample the oxide layer is well attached to the

surface but it has more void gaps.

38

Figure 24. SEM micrographs showing the oxide scales formed on specimens oxidized at 1000

℃, 1100 ℃ and 1200 ℃ after 45 minutes holding time and various surface conditions: (a) as cast, (b) clean and (c) covered with casting powder.

Fig. 24, shows that when the temperature increases, more cracks and also void gaps are created at 1100 °C in comparison with 1000 ℃. The gaps generate more pores and the oxide layer is well attached in powder condition case. At 1200 °C the thickness of the oxide layers gets larger in comparison to lower temperatures and it is clearly visible that the void gaps and cracks multiply.

6.3 Chemical and Phase analysis

SEM showed a different behavior for each condition, with the casting powder

case resulting in the largest thicknesses. In order to distinguish the presence of

elements in the oxide layer, EDX analysis was conducted. An EDS electron

image of the oxide layer from the outer layer to close to the substrate is shown in

Fig. 25.

39

Figure 25. EDS electron image of oxide layer to spectrum point from top to near to substrate.

As previously seen, a carbon reaction with the oxide tends to form another oxide, consequently, the most abundant elements would be Iron, Oxygen, an Carbon. In Fig. 26, the “Spectrum 1” of the outer layer of the oxide scale shows the presence of carbon, thus, it affects the thickness increase as carbon reacts with the oxide layer.

Figure 26. EDS spectrum result of the amount of elements in spectrum 1.

From the EDS analysis we calculate the carbon in each condition. As from eq. (9)

and (10) it is seen that carbon is reacting with iron oxide and forming CO

as a

40 gas. It seems that the void gaps are the origins of the CO

release. But, to investigate the amount of carbon EDS is performed for all samples from near substrate to outer layer oxide.

In Fig. 27, which have the graphs of carbon content in each condition: at 1000 °C 1100 °C, and 1200 °C, it is seen that at 1000 °C, the carbon content reaches a maximum near the substrate and it decreases with increasing distance.

Oppositely, in the case of the powder condition, the carbon percentage is very

low while it is higher in the cleaned sample case.

41

Figure 27. Graph of carbon content in the oxide layer with distance near substrate to outer layer for temperature a) 1000 °C b) 1100 °C c) 1200 °C.

6.4 XRD Analysis

After the microstructure analysis of the oxide, the next step is to calculate the amount of phases in the oxide layer. Phase analysis was performed on the top of the oxide layer of each condition 1000 °C, 1100 °C, 1200°C. XRD analysis shows

11; 14,8

11; 11,4

6 8 10 12 14 16 18

0 5 10 15

Carbon %

Top to near substrate

1000 °C

Cast clean powder

6 8 10 12 14 16 18 20

0 5 10 15

Carbon %

Top to near substrate

1100 °C

Cast Clean Powder

6 16 26 36

0 5 10 15

Carbon %

Top to near substrate

1200 °C

Cast Clean Powder

42 that Wüstite (FeO), Magnetite (Fe

O

) and Hematite (Fe

O

) phases are present in the diffraction patterns. The Fig. 28 compares the XRD diffraction patterns of as cast samples for different temperatures. At 1000 °C, Wüstite peaks appear with the smallest percentage as the amount of magnetite is 36.5 % and the hematite is 51.3 %. An interpretation of this can be that Wüstite, which has the highest Fe content of the oxides, and which is formed on the steel surface, can still at 1000℃

be detected at the oxide surface but now together with Magnetite, which has grown on top of the Wüstite. With increased time and temperature will the oxygen amount increase at the oxide surface and this will increase the magnetite amount but also the hematite content because it has an even higher oxygen content. When the temperature increases the Wüstite disappears at 1100 °C and the amount of magnetite is reduced to 14.1 %. At 1200 °C the Wüstite phase reappears and it can be explained by an hematite oxygen deficiency which would allow the formation of water molecules and the transformation of some hematite phase into Wüstite phase as described with eqs. (11) and (12), moreover, the Wüstite is unstable below 570°C.

Figure 28. XRD diffraction pattern comparison of as cast samples treated at 1000 °C 1100 °C 1200 °C.

43 In Fig. 29, a comparison is established between clean-surface samples at 1000

°C, 1100 °C, 1200 °C. At 1000°C they contain Wüstite but again with a very low percentage of 12.5 %. Magnetite and hematite are also present with percentages of 34.4 % and 53.2 % respectively. When temperature increases to 1100 °C, the Wüstite forms again magnetite mainly, and the oxygen deficient hematite also reacts inside the oxide to form magnetite. When it reaches 1200°C it only contains magnetite phase.

Figure 29. XRD diffraction pattern comparison of samples with clean metallic surfaces treated at 1000 °C, 1100 °C, and 1200°C.

44 In Fig. 30, the powder condition shows that at 1000 °C only magnetite and hematite are present with amounts of 39.6 % and 60.4 % respectively. At 1100

°C, the Wüstite phase suddenly appears with a percentage of 32.2% while the magnetite totally vanishes, surely as it transforms mainly into Wüstite. Then at 1200°C, a reversible reaction takes place as the Wüstite transforms mainly into magnetite to reach a percentage of 40.1 % and a percentage of 59.9% for the hematite.

Figure 30. XRD diffraction pattern comparison of samples metallic surfaces coated with powder and treated at 1000 °C, 1100 °C, 1200 °C.

45 All the results obtained from the different XRD diffraction pattern comparisons of samples in function of the various parameters and conditions are listed in Table.6.

Table 6. Percentage of phases formed in the oxide scale obtained by XRD analysis.

XRD results clearly show that the Wüstite phase is present at high temperatures even though it is unstable below 570 °C. Due to the reversible reaction the Wüstite phase may come from magnetite, hematite or even the Fe ions from substrate.

Temperature Condition

Wüstite (FeO)

Magnetite (Fe3O4)

Hematite (Fe2O3)

1000°C

Clean 12.5% 34.4% 53.2%

Cast 12.1% 36.5% 51.3%

Powder - 39.6% 60.4%

1100°C

Clean - 42.9% 57.1%

Cast - 14.1% 85.9%

Powder 32.2% - 67.8%

1200°C

Clean - - 98%

Cast 17.1% - 82.1%

Powder - 40.1% 59.9%

46 7 Modeling

Experimentally the oxide layer formation has been characterized by different experiments and analyze methods. To understand the formation of oxide layer and the diffusion of elements in oxide layer, it is necessary to evaluate and calculate the diffusion coefficient using Fick’s second law and consider the water concentration by making a model of the different oxide phases. The diffusion coefficient has to be calculated for each phase.

7.1 Diffusion Coefficient

Ficks established laws governing the diffusion of atoms and molecules, which can be applied to the diffusion processes in metals and alloys. He suggested two laws, the first for steady-state condition and unidirectional flow of atoms, and the second law which deals with time dependence of concentration gradient; and the flow of atoms in all directions.

For this case of oxidation Fick’s second law can be used in order to find the diffusion coefficient by using the concentration of oxygen in the oxide.

Fick's second law predicts how diffusion causes the concentration to change with respect to time. It is a partial differential equation which in one dimension reads [38]:

𝜕𝐶

𝜕𝑡

= −𝐷

𝜕𝑥²

………..(13)

Where,

𝜕𝑡

= 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ 𝑡𝑖𝑚𝑒 D=Diffusion coefficient assumed to be constant.

For non-steady state processes, at the same cross- section, the flux is not the same

at different times. Hence the concentration-distance profile (Fig. 31) changes with

time (t).

47

Figure 31. Graph of concentration vs distance with respect to time.

The differential form of Fick’s second law is the basic equation for the study of isothermal diffusion.

7.1.1Derivation of Fick’s Second Law

Consider an elemental slab of thickness Δx along the diffusion distance x (see Fig.32). Let the slab X-section be perpendicular to x and its area be unity. The volume of the slab is then ∆x*x*1 = ∆x.

Let, J / x = Rate of change of flux with distance, and

C / t = Rate of change of concentration per unit volume in the elementary slab.

Under non-steady state condition, the flux entering into the slab J

is not equal to the flux coming out of the slab, J

.

Hence there will be a net accumulation (or depletion) of flux within the volume

of the elementary slab which is given by (C / t)* volume of elementary slab =

(C / t) * ∆x .

48

Figure 32. Consideration of Fick’s second law model to derive equation

If matter is consumed this must be equal to the difference of fluxes in and out of the slab, and it can be expressed as

( 𝜕𝐶

𝜕𝑡 ) ∆𝑥 = 𝐽𝑥 − 𝐽(𝑥 + ∆𝑥) = 𝐽𝑥 − {𝐽𝑥 + ( 𝜕𝐽

𝜕𝑥 )∆𝑥}

Or(

𝜕𝑡

) =-

𝜕𝑥

……….(14)

From the first law

J=-D

……… (15)

From equation (15) is Fick’s second law of unidirectional flow under non-steady state condition. If D is independent of concentration equation(14) reduces to

𝜕𝐶

𝜕𝑡

= −𝐷

𝜕𝑥²

………..………(16)

Solution to this expression (concentration is terms of both position and time) is

possible when physically meaningful boundary conditions are specified. Most

practical diffusion situations are non-steady ones. That is, the diffusion flux and

the concentration gradient at some particular point in a solid vary with time, with

a net accumulation or depletion of the diffusing species resulting.

49 7.1.2 Solution to Fick’s Second Law of Diffusion

One practically important solution is for semi-infinite solid (A bar of solid is considered to be semi-infinite if none of the diffusing atoms reaches the bar end during the time over which the diffusion takes place. A bar of length I is considered to be semi-infinite when I > 10 √D*t), in which the surface concentration is held constant. Frequently the source of the diffusing species is a gas phase, the partial pressure of which is maintained at constant value.

Furthermore, the following assumptions are made:

(i) Prior to diffusion, any of the diffusing atoms in the solid are uniformly distributed with a concentration of C

.

(ii) At the surface, the value of x is zero and it increases with distance into the solid.

(iii) At the instant before the diffusion process begins, the time is taken to be zero.

The general diffusion equation for one-dimensional analysis under non- steady state condition is defined by Fick’s second law, eq. (16).

Let D be a constant and use the function y=f(x,t) be defined by

𝑦 =

2√𝐷𝑡

(17) Thus, the partial derivatives of eq.17 are

𝜕𝑦

𝜕𝑥

=

2√𝐷𝑡

And

=

4√𝐷𝑡³

(18) By definition

𝜕𝐶

𝜕𝑡

=

𝑑𝑦

𝜕𝑦

𝜕𝑡

=

4√𝐷𝑡³ 𝑑𝐶

𝑑𝑦

(19)

𝜕²𝐶

𝜕𝑥²

=

𝜕𝑥

[

𝑑𝑦

(

𝜕𝑥

)] =

4𝐷𝑡 𝑑²𝐶

𝑑𝑦²

(20) Substituting eqn. (19) and (20) into (16) yields

𝑑𝐶

𝑑𝑦

= −

𝑥 𝑑²𝐶

𝑑𝑦²

(21) Combining eq. (17) and (21) gives

𝑑𝐶

𝑑𝑦

= −

2𝑦 𝑑²𝐶

𝑑𝑦²

(22)

50 Now, let so that eq. (22) becomes

𝑧 = −

2𝑦 𝑑𝑧

𝑑𝑦

(23) -2 ∫ 𝑦𝑑𝑦 = ∫

(24) Then,

−𝑦

= ln 𝑧 − ln 𝐵 (25)

Where B is an integration constant. Rearranging eq. (25) yields 𝑧 = 𝐵𝑒𝑥𝑝(−𝑦

) (26)

And

∫ 𝑑𝐶 = 𝐵 ∫ exp (−𝑦

)𝑑𝑦 (27)

The function represents the so-called “Bell-Shaped Curves”.

The solution of the integrals are based on a set of boundary conditions.

A.1 First Boundary condition

In order to solve integrals given by eq. (A6) a set of boundary conditions the concentration and the parameter are necessary. This boundary condition are just the integral limits. Thus,

𝐶 = {𝐶

= 𝐶

𝑓𝑜𝑟 𝑦 = 0 𝑎𝑡 𝑡 > 0 𝑎𝑛𝑑 𝑥 = 0

{𝐶

= 𝐶

𝑓𝑜𝑟 𝑦 = 𝑎𝑡 𝑡 = 0 𝑎𝑛𝑑 𝑥 > 0

∫

𝑑𝐶 = 𝐵 ∫ exp (−𝑦

)𝑑𝑦 (28) 𝐶

− 𝐶

= 𝐵 ∫ exp (−𝑦

)𝑑𝑦 (29)

Use the following integral definition and properties of the error function erf(y)

∫ exp(−𝑦

) 𝑑𝑦 =

2

∞

0

(a)