function of heat treatment

2009:129 CIV

M A S T E R ' S T H E S I S

Quantitative phase analysis of duplex stainless steels as a

function of heat treatment

Anders Säfsten

Luleå University of Technology MSc Programmes in Engineering

Engineering Physics

Department of Applied Physics and Mechanical Engineering Division of Physics

2009:129 CIV - ISSN: 1402-1617 - ISRN: LTU-EX--09/129--SE

Quantitative phase analysis of duplex stainless steels as a function of heat treatment

Anders Säfsten

M.Sc. thesis in Engineering physics

Luleå University of Technology, dept of Applied physics and engineering mechanics.

Supervisor Niklas Lehto.

Accomplished 2009 at Sandvik Materials Technology, Sandviken.

Supervisor Ping Liu.

Abstract

During heat treatments the microstructure of metals changes and heat treatment times, heating rates and cooling rates all affect the final microstructure of metals. In the duplex stainless steels chromium nitrides have been seen after quenching from higher temperatures. The precipitates have negative effects on the material properties and decrease the fracture toughness at lower temperatures and reduce pitting corrosion resistance. To study the precipitates a method for quantification is needed and a method for this will be developed.

The methods used are x-ray diffraction, magnetic balance, scanning differential calorimetry and electron microscopy with image analysis. Due to too small volume fraction the β-Cr ₂ N phase could not be detected with x-rays or in the differential scanning calorimetry. However, with electron microscopes they could be quantified and this method together with image analysis gave adequate results.

Samples of SAF3207HD were heat treated with different heat treatment solution temperatures and cooling rates. Higher solution heat treatment temperatures resulted in higher fractions of chromium nitrides, but an increased cooling rate has an even larger effect on the volume fraction. Phase fractions up to 2.33 vol% were registered.

Depending on the solution heat treatment temperature a minimum cooling rate exists which produces β-Cr ₂ N precipitates and these rates are strongly dependent in the solution heat treatment temperature. The nitrides form in three shapes, as needles, clouds or along grain boundaries. Faster cooling cause clouds to form while at slower cooling the chromium nitrides are preferably formed in grain boundaries.

Finally the results are plotted in two diagrams, where the heat treatment temperature,

the cooling rate and the fraction of precipitates are the three parameters. If two of these

three parameters are fixed, the third can be predicted from these diagrams. These

diagrams can be a very useful tool when working with chromium nitrides.

iii

Contents

1 Introduction ... 1

2 Background ... 3

2.1 Duplex stainless steels ... 3

2.1.1 Common alloying elements ... 4

2.1.2 σ-phase ... 5

2.1.3 Chromium nitride (β-Cr

N) ... 6

2.2 Diffusion of N and Cr ... 7

2.3 Previous work ... 8

3 Methods ... 11

3.1 Electron microscopy ... 11

3.1.1 Scanning electron microscopy (SEM) ... 12

3.1.2 Transmission electron microscopy (TEM) ... 12

3.1.3 Image analysis ... 13

3.2 X-ray diffraction (XRD) ... 14

3.3 Magnetic balance... 15

3.4 Differential scanning calorimetry (DSC) ... 16

4 Experiments ... 17

4.1 Samples ... 17

4.2 Magnetic balance... 18

4.3 X-ray diffraction (XRD) ... 18

4.4 Differential scanning caliometry ... 18

4.5 Scanning transmission electron microscopy (STEM) ... 19

4.6 Scanning electron microscope (SEM) ... 19

4.7 Image analysis ... 21

5 Results ... 25

5.1 Magnetic balance and x-ray diffraction (XRD) ... 25

5.1.1 Magnetic balance ... 25

5.1.2 X-ray diffraction (XRD)... 26

5.2 Differential scanning calorimetry (DSC) ... 27

5.3 Scanning transmission electron microscopy (STEM) ... 28

5.4 Scanning electron microscopy (SEM) ... 28

6 Discussion ... 31

6.1 Formation of β-Cr

N... 31

6.1.1 Construction of a continuous cooling transformation diagram ... 35

6.1.2 Types of precipitates... 36

6.2 XRD and DSC ... 38

6.2.1 X-ray and magnetic balance ... 38

6.2.2 Differential scanning calorimetry ... 39

6.3 Image analysis ... 40

6.4 Statistics ... 41

6.5 Flow chart for quantification method... 44

6.6 Suggestion for further work ... 45

7 Conclusions ... 47

8 Acknowledgement ... 49

9 References ... 51

Appendix ... 53

A.1 Solution heat treatment example curve ... 53

A.2 Results from different magnifications in SEM ... 54

A.3 Sampling results ... 55

A.4 Image analysis ... 57

A.5 Statistical investigations ... 60

A.5.1 References ... 70

1

1 Introduction

Sandvik Materials Technology produces stainless steels of many kinds with different compositions for different purposes. The steels are used in wires, pipes, knives and razor blades for example. The products have to maintain high quality and keep excellent levels of chemical and mechanical properties such as corrosion resistance, hardness and fracture toughness.

The microstructure of the steels is important to study and understand since it determines the material properties. Different microstructures are present after different thermal and mechanical treatments, and each structure have its special properties. The duplex stainless steels consist of a mixture of two microstructures, austenite and ferrite. Within these microstructures, or phases, small amounts of other phases can form after certain heat treatments. The small phases formed (precipitates) can have major effects on the materials mechanical and chemical properties. One of these phases formed are the chromium nitrides that forms in high chromium duplex steels such has SAF3207HD. The precipitates cause a decrease in the fracture toughness at low temperatures and hence they are undesirable.

In order to develop the high chromium steels it is necessary to be able to study these heat treatment effects on the microstructure. The chromium nitrides have not been deeply studied and there is no satisfying method for quantification of the precipitates.

As a main goal in this paper a method that quantifies the precipitates should be developed, and further to use the method to study the effect of heat treatment on the fraction of precipitates formed.

Three different methods are suggested as possible for quantification and then investigated.

The first method is x-ray diffraction, in which the diffraction of x-rays is used to

identify and quantify different microstructures. As a second method the previous

mentioned x-ray results are used in combination with magnetic balance, which

measures the fraction of magnetic phase. Differential scanning calorimetry, the third

method, identifies phase transitions by heating and continuously studying the heat

flow. As a last attempt images analysis is used to quantify the precipitates with images

obtained by electron microscopes.

2

3

2 Background

2.1 Duplex stainless steels

Usually when people think of stainless steels it is referred to iron with a concentration of carbon in the order of 1%. This is not the case these days, since the carbon content is only about 0.01-0.03%. Instead several elements are added to improve the physical, mechanical and chemical properties. The term duplex relates to the fact that the steel is based on two phases, usually α-ferrite (bcc structure, Fig. 2.1a) and γ-austenite (fcc structure, Fig. 2.1b). The ferrite has higher concentration of chromium, molybdenum and silicon while austenite has a higher concentration of nickel and manganese. At higher temperatures, above 1300°C, the concentration of these elements is more uniform through the steel [1].

(a) (b)

Figure 2.1. Body-centered cubic structure (a) and face-centered cubic structure (b).

The presence of different phases in steels depends on factors such as temperature and chemical composition. In Tab. 2.1 the chemical composition of SAF3207HD is listed and in Fig. 2.2 a phase diagram calculated with Thermo-Calc for SAF3207HD is shown.

Table 2.1. Chemical composition (%) of SAF3207HD, Ch 508017 [7].

Cr Ni Mo C Mn Si N

31.38 7.07 3.50 0.014 0.82 0.25 0.48

4

Figure 2.2. Phase diagram of SAF3207HD calculated with Thermo-Calc.

The equilibrium phase composition at a temperature can be read from the vertical axis.

The phase diagram in Fig. 2.2 is an equilibrium diagram obtained by computer calculations. It is however noteworthy that the Cr ₂ N maximum equilibrium temperature lies around 1050°C-1100°C, and also that the equilibrium relation between ferrite and austenite changes around 1000°C-1100°C. After these temperatures the volume fraction of the ferrite increases and the austenite decreases, and below these temperatures it is opposed. The ferrite eventually disappears below 900°C. However, these temperatures are calculated from theory and should only be taken as approximate.

2.1.1 Common alloying elements Chromium (Cr)

The addition of chromium increases the localized corrosion resistance. The chromium content though reaches an upper limit, since too high chromium content enhance the precipitation of intermetallic phases. These precipitates can deteriorate mechanical and corrosion properties. Also, chromium stabilizes the ferrite phase and increases nitrogen solubility [2].

Molybdenum (Mo)

Molybdenum increases the resistance to pitting and crevice corrosion in chloride solutions. The content of molybdenum is up to 4%, since above this level the formation of the sigma phase is enhanced [2].

γ-austenite α-ferrite

β-Cr

N

Liquid

σ-phase

5 Nickel (Ni)

Like chromium stabilizes ferrite, nickel stabilizes austenite. The level of content for nickel is dependent on the chromium content. Too much nickel will lead to increased content of chromium and molybdenum in ferrite, which could result in the formation of intermetallic phases at temperatures of 650-950°C. Nickel also has some positive corrosion effects [2].

Nitrogen (N)

Crevice corrosion resistance, pitting resistance and strength are among the enhanced properties by adding of nitrogen as well as an increased content of austenite. The increase in austenite content arises from its solubility of nitrogen, which is higher than for ferrite. In duplex alloys the formation of intermetallic phases, such as the sigma phase, is inhibited by reduction of Cr-partitioning. Even the formation of nitride is inhibited since these forms in the ferrite, and the ferrite content is reduced as a consequence of the fact the austenite content increase [2].

Manganese (Mn)

Increases in wear and abrasion resistance as well as increase in strength are effects from addition of manganese. It also increases the solid solubility of nitrogen.

However, high manganese concentration reduces the pitting resistance when combined with nitrogen [2].

Silicon (Si)

One important effect of silicon is the increased high temperature oxidation resistance.

The silicon level is limited to about 1% due to the ability to enhance the formation of sigma phase [2].

2.1.2 σ-phase

The atomic structure of sigma(σ)-phase, shown in Fig. 2.3, is rich of chromium and molybdenum and causes embrittlement and reduction in impact properties and corrosion resistance in the material. It forms between 650-1000°C, with its peak around 900°C. The additives Cr, Mo, Mn and Si all facilitate the formation of σ-phase.

Nickel also enhances the formation but at the same time it reduces the volume fraction

of ferrite. This is because it increases the production of austenite results in more Cr,

Mo and Si in the ferrite, which enhances the formation of σ–phase. At temperatures

close to the decomposition temperature the nucleation rate is low, while the diffusion

rate is high, resulting in fast growth. When temperature is lower, the nucleation rate

increases and diffusion rate decreases [1]. Cold working increases the rate of

production of σ-phase and the phase is usually formed in stainless steels with more

than 16% chromium. In SAF 2906 the σ-phase composition is 30% Cr, 57% Fe and

13% Mo [8].

6

Figure 2.3. The crystal structure of sigma phase. Image from the Naval Research Laboratory [3]

2.1.3 Chromium nitride (β-Cr ₂ N)

At high temperatures, above 1040°C, the fraction of ferrite increases and if the temperature is high enough only the ferrite phase exists. The solubility of nitrogen is much higher in austenite than in ferrite at lower temperatures, so as the solid is cooled the ferrite gets supersaturated with nitrogen. Chromium nitrides are then formed if cooling rate is high enough to prevent diffusion of nitrogen into austenite. The atomic structure is shown in Fig. 2.5. When cooling is slow the nitrogen diffuses to the austenite instead where it is solvable. The nitrogen left in the ferrite grains that were unable to reach the austenite is prompt to precipitate. The presence of nitrogen in duplex stainless steels is important since it increase the pitting resistance equivalent, PRE = %Cr+3,3·{%Mo}+16·{%N}. Typically this value is greater than 40. [1]

Table 2.2 Structure information and lattice parameters of the phases. [5 ]

Phase Crystal system Space group Lattice parameters [Å]

α-ferrite Body-centered cubic Im 3 m ^a=2.876

γ-austenite Face-centered cubic Fm 3 m ^a=3.5911

σ-phase Tetragonal P4

/mnm a=8.7995, c=4.5442

β-Cr

N Hexagonal P 3 1m a=4.8113, c=4.4841

7

Figure 2.4. Hexagonal structure. Figure 2.5. The β-Cr

N structure. Image from the Naval Research Laboratory [4].

2.2 Diffusion of N and Cr

Diffusion is usually related to difference in concentration between areas, volumes, phases etc. If a concentration of a substance or element is lower in some region diffusion usually occurs to this region from a region of higher concentration. This proceeds until the concentrations are leveled out and equilibrium is reached. However, there are other factors that also controls diffusion and may even cause reverse directions of diffusion, i.e. from lower concentration to higher concentration. The chemical potential is such a property, and is defined by the Gibbs free energy (G) [6].

TS PV E

G = + −

E is the energy, P the pressure, V the volume, T the temperature and S the entropy. If a system is in equilibrium, any change of the Gibbs free energy is positive (∆G≥0), i.e.

the Gibbs free energy at equilibrium corresponds to the minimum value [6]. This is principally the same as the potential energy in mechanics: if an object is lifted its potential energy is increased. Spontaneously, the object will fall down again and reduce its energy, while it would not spontaneously lift itself since that causes an increase of potential energy.

The chemical potential µ _i for a substance i in a solid solution of n _j moles is described by [6]

i n V i S n

T i P i

j j

n E n

G G  = µ





 



∂

= ∂

 



 



∂

= ∂

, , ,

,

.

The chemical potential describes what happens when a small amount of a component is added to a system, without changing the amount of the already present components.

If the energy change is positive the added component struggles to leave the system.

The chemical potentials of nitrogen and chromium differ in austenite and ferrite.

Nitrogen has a lower potential in austenite and the potential of chromium is lower in (1)

(2)

8

ferrite. This causes diffusion of nitrogen to austenite and diffusion of chromium to ferrite. However, nitrogen prefers chromium over iron as its surroundings, so an increased content of chromium in ferrite causes the chemical potential of nitrogen in ferrite to be decreased. This results in diffusion of nitrogen into the ferrite.

Corresponding situation applies for chromium, i.e. its potential is decreased in austenite due to the presence of nitrogen. This means that for nitrogen steels, such as SAF3207HD, the amounts of chromium and nitrogen in ferrite and austenite is changed until equilibrium is reached. Nitrogen can according to this theory diffuse from ferrite to austenite, even though its concentration in austenite is higher than in ferrite, thanks to the lower chemical potential [6].

As will be described in the next section, the diffusion rate of nitrogen is significantly higher than of chromium in ferrite. This will imply that the formation of Cr ₂ N is controlled by the nitrogen diffusion, due to the combination of its diffusion rate and the fact that its chemical potential in ferrite is increased when the temperature is decreased.

The sigma phase is controlled by the chromium diffusion in ferrite, which is significantly slower than nitrogen diffusion. This requires longer annealing times temperatures around 900°C for precipitation of sigma phase to occur [7].

2.3 Previous work

The duplex stainless steels consist of approximately 50% ferrite (α) and 50% austenite (γ). The impact strength of austenite decreases at lower temperatures, and the ferrite experiences a ductile-to-brittle transition. This transition temperature is transferred to lower temperature when mixed with austenite. A large reduction in toughness for SAF2205 (22% Cr, 5% Ni) was measured and many possible explanations for this were suggested, such as composition, Cr concentration and homogeneity of microstructure, though none of these were satisfying. Therefore the microstructure was studied to find a relation to the fracture toughness for SAF2205. It was discovered that different annealing temperatures resulted in different fracture toughness, especially at low temperatures around the transition temperature -50°C. Formations of chromium nitrides (β-Cr2N) were found along the {110} _α planes. These precipitates reduce the mobility of dislocations in the ferrite, and the ductile-to-brittle transition is dependent on this mobility [9].

Later a new investigation showed that at high annealing temperature the fracture toughness increased, but a cooling rate rapid enough to avoid precipitates was not possible [10].

In the most recent experiments three different heat treatments were carried out on

SAF3207HD. The first sample was annealed at 1200°C for 5 min and cooled at

339°C/s to room temperature. Investigations showed β-Cr ₂ N precipitates at grain

boundaries and in ferrite grains, negligible were found in austenite grains. Moreover, it

was frequently observed that there was a precipitation–free zone with a width of 2 µ m

9

along the ferritic boundaries. The second sample was annealed at 1120°C for 15 min and at 1000°C for 16 min, no precipitates were found. Third sample was annealed at 1120°C for 15 min and at 700°C for 32 min, and this showed precipitates of σ phase but no β-Cr ₂ N. However, a distinction between the two phases was not confirmed [7].

(a) (b)

Figure 2.6. (a) Solid solubility of nitrogen in ferrite [7]. At 700°C the solubility is almost zero.(b) Diffusion coefficient of nitrogen in ferrite [7].The coefficient, that affects the diffusion rate, reduces to less than a tenth when temperature lowers from 1200°C to 700°C

The solubility for chromium in ferrite reduces approximately from 0.32 wt% to 0.23 wt% between 1000°C and 700°C, while the solubility for nitrogen in ferrite reduces from 12·10 ^-4 wt% to zero when temperature decreases from 1200°C to 700°C (Fig. 2.6a). The solubility of nitrogen in austenite is higher and that cause the nitrogen to diffuse to austenite grains. This is an explanation why the β-Cr ₂ N precipitates at temperature above 700°C. Also, the diffusivity for nitrogen (Fig. 2.6b) is about a thousand times higher than for chromium. Hence, the precipitation of β-Cr ₂ N is controlled by the nitrogen solubility and diffusivity in ferrite [7].

Higher solution temperature increased the amount of precipitates and was attributed to the higher solubility of nitrogen, i.e. more nitrogen is present in ferrite at higher temperature. At higher cooling rates the supersaturation of nitrogen in ferrite is increased which results in an increased driving force for β-Cr ₂ N formation [7].

Several attempts have been made to quantify β-Cr ₂ N. One was to visualize the amount

of precipitates formed by five different levels that were specified by reference images,

and grading of a sample should be done by comparing its images with the reference

images. Though this methods works, it is strongly dependent on the observer even

when the number of levels is as low as five, and it is desirable to have a more exact

and independent method to quantify the precipitates [11].

10

A method for quantification is not obvious since the precipitates form in different shapes, needle-like and in clusters of particles of varied size. Quantification by image analysis is complex due to the very small precipitates, and phase quantification by x- ray diffraction is most likely not possible since the volume fraction of β-Cr ₂ N is too low, probably less than 3 vol%, and XRD is useful to quantify phases present in a volume fraction of at least 5 vol%.

Figure 2.7. Chromium nitrides observed in an optical microscope

with, magnification 1000X. All particles are not precipitates but only

scrap or etching effects.

11

3 Methods

3.1 Electron microscopy

The physics behind the electron microscopy method is complicated and theoretical, and this is only a brief description of the method. The sample is bombarded with an electron beam, and when the beam hit the sample the electrons interact with the crystal lattice and the atomic potentials. At certain incidence angles, the electrons are elastically diffracted in a constructive way. This happens when the Bragg condition is fulfilled,

λ θ ⁿ d

sin =

2 .

This depends on the incident angle θ, the interplanar spacing d _hkl of the atomic planes and the electron energy (which is expressed through its wavelength λ by the de Broglie relation). In principle this happens when the path difference between two electrons diffracted from different atomic planes of distance d equals in integer number n of its wavelength. When this happen the Bragg condition is fulfilled and θ is called the Bragg angle [12].

The high energy electrons of the beam scatter against the atoms in the specimen.

Incident electrons can scatter elastically with the atoms (or mostly the nucleus) and back-scatter from the sample with still high energy; these electrons are called the backscattered electrons (BSE). The incident electrons can also scatter inelastically with the atomic electrons and transfer a small fraction of its momentum. The atomic electron can receive enough energy to scatter out of the sample and is then called a secondary electron (SE). There is also a probability that the atomic electron does not leave the atom but gets excitated, and thus a photon is emitted when deexcitated. This photon is characteristic of the element and can be used for qualitative studies in electron microscopes equipped with an x-ray detector [12].

The probability of elastic scattering is proportional to the atomic number and inversely proportional to the energy. At the specimen surface the incident electrons have high energy and low probability for elastic scattering, and hence the largest fraction proceeds into the specimen. However, they do scatter inelastically and looses some of their energy. The scattering angles for inelastic scattering are smaller than for elastic scattering, and if they scatter inelastic their change of direction is small. As a result the interaction volume is shaped like a pear, since when they have lost some energy they start to scatter more elastic and hence the spread increases [12].

For topographic imaging it is the secondary electrons that are of interest and these only comes from the top few nanometer of the specimen. This can be understood by the fact that the low energy secondary electrons emitted at a larger distance from the surface scatter mainly elastic and hence have a hard time to reach the surface. Near the specimen surface the incident electron have high energy and scatter more inelastic, as mentioned above [12].

(3)

12

The angle at which the electrons hit the surface strongly affects the signal of SE. The pear shape then is angled from a vertical pear to a smaller angle with the surface. As more of the interaction volume lies closer to the surface more secondary electrons are able to be emitted from the specimen [12].

3.1.1 Scanning electron microscopy (SEM)

An important application is the scanning electron microscopy, SEM. The specimen is bombarded with electrons and placed above the specimen are detectors for SE, BSE and x-rays. It gives high resolution images, which can not be given by an optical microscope. The reason for this high resolution is the short electron wavelength, compared to the wavelength of visible light, which makes it possible to resolve small objects. SEM normally uses an acceleration voltage of 1-20 kV. The contrast of an SE image in SEM can be increased if the SE detector is placed behind the objective lens, a so called InLens detector. This eliminates the noise coming from BSE, and also the undesirable SE electrons emitted when BSE strikes the pole piece.

3.1.2 Transmission electron microscopy (TEM)

Unlike in SEM, in TEM the transmitted electrons are registered by detectors placed under the specimen. The specimen is hit by the electrons which have higher energies than in SEM, usually 100-300 keV. When TEM is used, the specimen must be very thin, which demands careful sample preparation to make the object to be measured thin enough, only a thickness of a few 100 nm. The thickness is limited by the mean free path of the electrons. Common preparations methods are for example electro chemical polishing, chemical and mechanical polishing and ion milling.

The TEM detectors are placed below the specimen. Detectors placed directly under the specimen, and thus they detect electrons with small scattering angles, are used for bright field (BF) imaging. The detectors that register electrons with a larger scattering angle are used for dark field (DF) imaging. There can be several factors that cause differences in contrast. The specimen thickness is one; as the specimen thickness increases the electrons transmitted have larger scattering angles. This cause dark areas in the BF image and bright areas in a DF image. Higher atomic numbers have the same effect since the probability for elastic scattering increases with an increased atomic number. These contrasts are most useful for amorphous specimens and chemical analysis [13].

The Bragg law applies for TEM too, of course. The Bragg reflection is as well a source

of contrast. When the Bragg condition is fulfilled the number of electrons at that angle

will be increased, and as a consequence the number of electrons will be decreased in

another angle. How to handle this contrast is complicated and will not be described

here since it is not of importance for this paper [13]. If equipped with x-ray detectors

the TEM can also perform qualitative chemical analysis [14].

13 Scanning transmission electron

microscopy (STEM)

When using STEM, the beam is focused on a spot as small as possible and then scanned over an area of the specimen, measuring for some time interval at each spot to achieve a desired intensity. The best result is obtained when the TEM is equipped with a field emission gun (FEG), a FEG- STEM. An ultra high vacuum is needed and because of this the specimen is not contemned and the counting time can be increased for better resolution [15].

Using secondary electron imaging (SEI) in TEM results in lower

background noise than in SEM. The reason for this is the placement of the SE detector.

In the STEM it is placed inside the pole piece, which eliminates unwanted SE signals from backscattered electrons which when they hit the pole piece causes emission of SE electrons (SE3) (i.e. they do not carry any information from the sample). In SEM SE is also emitted from the final probe-forming aperture (SE4) which is placed by the final lens. In STEM this is not the case since this function is accomplished elsewhere and not yielding any SE signal. The SE from the specimen (SE1) are confined by the magnetic field at the upper pole piece, while the BSE are not due to their higher energy. The fourth kind of SE comes from backscattered electrons in the specimen which causes emission from regions surrounding the beam (SE2). See Fig. 3.1.

Because of the higher acceleration voltage the brightness of the source is higher in STEM and this increases the SE signal [13]. For comparison, in this paper the acceleration voltage in STEM is 200 kV and in SEM it is 7 kV.

3.1.3 Image analysis

The images received by STEM are used for quantification by image analysis. The principle of image analysis is simple. The images are in greyscale, where 0 is black and 255 is white, i.e. there are 256 levels of grey. The grey levels of the areas of interest for quantification are determined. The number of pixels at this level, or more likely at this level interval, is then counted and compared to the total number of pixels in the image. The result is the area fraction of the phase of interest, which in general can be assumed to also represent the volume fraction. Unfortunately, this method is not so straight forward as it seems. In an image several areas may have the same grey level but they represent a different phase, and it is not obvious where to set the limit for the grey level of interest.

Figure 3.1. Different types of secondary electrons (SE1- 4). In STEM the SE detector is placed inside the pole piece and hence avoids signals from SE3 and SE4.

SE3

SE4

SE1

Pole piece Incident beam

SE detector BSE

SE2

Specimen

14

The contrast in an image is very important for image analysis. For our eye to see contrast the level of contrast mush exceed 5%, but it is difficult to see contrast levels below 10% [15]. Hence it is desirable to work with high contrast images with a high S/N ratio (signal to noise). See Fig. 3.2.

Figure 3.2. Comparison between high and low contrast. Large differences between intensity levels create high contrast.

3.2 X-ray diffraction (XRD)

The diffraction of x-rays in a crystal is extremely useful for determination of the crystal structure and lattice parameters. X-rays are electromagnetic waves, and these waves scatter against the atoms, called scattering points. Though, the x-rays will cancel out each other unless the path difference is an integer number (n) of the wavelength (λ). Only under this condition there will be a diffracted beam, where the x- rays are in phase and there is constructive interference. The condition for diffraction is called the Bragg law and is defined in equation (3) in section 3.1. If the angle 2θ is measured the distance between the reflecting planes can be calculated, if the wavelength is known. From this result the Miller indices (hkl) of the reflecting planes can be derived and lattice parameters calculated, and hence the crystal structure determined [16].

The software used to calculate the fraction of phases in the present study is called Topas (by Bruker AXS). Many different parameters can be adjusted in Topas. For example preferred directions, background subtraction, grain size and so on. This makes it possible to fit the results in many ways and hence many different results are given dependent on the parameters chosen. Great care must be taken when Topas is used and the parameters are chosen. The accuracy of the results obtained by XRD and Topas is seen to be better for samples with a ferrite fraction of about 50% [17].

Intensity

Distance

S/N ratio High contrast

Low contrast

15

3.3 Magnetic balance

Ferrite is a magnetic phase; while neither austenite nor chromium nitrides are magnetic. This makes it possible to detect the ferrite phase separately with magnetism.

The magnetic balance method measures the force that arises from an applied magnetic field through the sample, and from this the magnetic saturation is calculated. A theoretical value of the magnetic saturation (σ _m ) can be calculated or taken from a table. Two different formulas can be used, Salwen or Hoselitz, which use the amount of some elements in the whole sample and in the magnetic phase, respectively.

According to Khokhar [17] the Hoselitz formula shows better results for steels with roughly 50% ferrite than the Salwen formula. Hence, in this paper the Hoselitz formula is used. The respective formulas are as following:

Salwen σ

= 213.3 − 2.76 ⋅ %Cr − 2.8 ⋅ %Ni − 2.58 ⋅ %Mo ,

Hoselitz

%Ni 0.75

%N 6

%Cu 2.3

%S 7

%P 3

%Al 2.6

%Mo 1.2

%Cr 3

%Mn 1.9

%Si 2.4

%C 12 217.75 σ

⋅

−

⋅

−

⋅

−

⋅

−

⋅

−

⋅

−

⋅

−

−

⋅

−

⋅

−

⋅

−

⋅

−

= .

The theoretical value gives the magnetic saturation that the sample would have if only the magnetic phase existed. The experimental value (σ _s ) is the magnetic moment measured and is defined as

z H m µ σ F

x z

s

= ⋅ ⋅ ∂ ∂ .

F _z is the force caused by the applied magnetic field in the z-direction, H _x is the magnetic field in the x-direction, m is the sample mass and µ is the magnetic permeability. The only magnetic phase present in the samples in this paper is ferrite, and so the weight percent of ferrite is easily determined,

m

s

σ

= σ phase magnetic of

fraction volume

This method is easy and fast and delivers stable results, and the main limitation relevant in this case is the specimen weight which should be 0.3-1 g. The use of a theoretical value could be a source of error since it is based on previous measurements.

For specimens with amounts of ferrite that exceeds about 20% the relative error is increased and must be taken into consideration [17].

(4a)

(4b)

(5)

(6)

16

3.4 Differential scanning calorimetry (DSC)

Differential scanning calorimetry, DSC, is a widely used method to study variations in materials that are caused by change of temperature. For metals, these variations have their origin in phase transitions, for example when precipitates are formed or dissolved.

DSC is a thermal method, and the principle is to heat up the sample at a constant rate,

i.e. constant increase of temperature. Along with the sample a known reference sample

is also heated at exactly the same rate. The analysis is then performed by examination

of the amount of heat flow into the specimen. This amount is constant as long as the

heat capacity of the sample remains the same. When, for example, a phase

transformation occurs this reaction is either endothermic or exothermic, which results

in either an increase or decrease in the heat flow. An endothermic reaction requires

more energy and hence an increased heat flow is needed to keep the constant

temperature increase, and the opposite applies to the exothermic reaction.

17

4 Experiments

4.1 Samples

The sample of SAF3207HD is received as a piece of tube with outer diameter 114.3 mm and thickness 6.35 mm (chart No 508017). Samples are then machined to be turned into small sticks of length 10 mm and a diameter of 3 mm in order to fit the dilatometer.

To determine the initial phase composition the samples are run in the magnetic balance, and after this the samples are heat treated in the dilatometer. The samples are then once again run in the magnetic balance to determine the new phase fraction of ferrite.

Next step is to prepare the samples for measurement. The sample preparations are described below for each method.

In order to decide a suitable temperature and time for heat treatment in dilatometer, some experiments are carried out. A suitable heat treatment is one that doesn’t change the amounts of ferrite and austenite significantly, since this could affect the amounts of precipitates formed. Six test samples are heat treated at 1200°C, 1150°C and 1100°C for 5 min or 2 min. The results are tabled in Tab. 4.1.

Table 4.1. The ferrite fraction before and after. The difference between the fractions after heat treatment also included. Higher temperatures and longer time increases the change in weight%

of ferrite.

Temperature Time Weight % before Weight % after Change weight %

1200°C 5 min 50.0 61.8 +11.8

1200°C 2 min 49.9 60.3 +10.4

1150°C 5 min 50.3 54.7 +4.4

1150°C 2 min 50.9 55.0 +4.1

1100°C 5 min 51.6 53.0 +1.4

1100°C 2 min 50.8 51.4 +0.6

The ferrite fraction is measured by magnetic balance before and after heat treatment in order to determine the change in ferrite fraction. These indicated that the heat treatment temperature should not exceed 1150°C in order to avoid an increase of ferrite fraction, since a significant increased ferrite fraction is observed for the samples treated at 1200°C. The highest heat treatment temperature is set to 1120°C for 5 min since no significant effect of heat treatment time is revealed from these experiments.

5 min is chosen over 2 min since the whole sample should be heated uniformly; the temperature in the dilatometer is measured at the end of the samples and not in the middle inside. For this temperature several cooling rates will be tested. One sample is however treated at 1200°C for solution heat treatment temperature comparisons.

In earlier reports chromium nitrides were observed in SAF3207HD heat treated at

1120°C for 5 min and the cooled for 40°C/s. However, no precipitates were observed

18

in the sample cooled for 5°C/s and this indicates that there exists a minimum cooling rate at which precipitation occurs [11].

Sample designation and heat treatment are given in Tab. 4.2. A cooling rate set to

“max” means that the cooling time at the dilatometer is set to zero.

Table 4.2 Sample names and their heat treatment.

Sample designation Heat treatment

Temperature [°C] Heating rate [°C/s] Time [min] Cooling rates [°C/s]

1200/1/max 1200 50 1 max

1120/1/130, 40, 20 1120 50 1 130, 40 ,20

1120/5/max, 90 1120 50 5 max, 90

1050/1/max 1050 50 1 max

The samples in Tab. 4.2 should give quantitative information of precipitation dependent on heat treatment temperature and cooling rate. Heat treatments are performed in the dilatometer to achieve a controlled temperature and cooling rate. The sample heat treated at 1120°C for 5 min (and with maximum cooling rate and 90°C/s) was the first to be heat treated and it is seen in the magnetic scale that the ferrite fraction was increased (see section 5.1.1). Therefore, the time is reduced to one minute for the forthcoming samples.

4.2 Magnetic balance

The Hoselitz formula has previously been shown to give more exact values of the magnetic saturation constant than the Salwen formula. The software in the equipment for magnetic balance the Salwen value is the constant in use for SAF3207HD. The chemical composition in ferrite is therefore experimentally analyzed with an electron probe microanalyser (EPMA), since its composition differs from sample as a whole.

The Hoselitz value is then derived from this composition (See section 3.3).

Each specimen is weightened with an accuracy of a hundredth microgram. The magnetic balance then measures the fraction of magnetic phase and the result is then compared with the theoretical value to obtain the ferrite weight fraction in the sample.

4.3 X-ray diffraction (XRD)

After the disks for STEM are cut out, the samples are mounted in Bakelite, grinded and then scanned with the XRD. Chromium radiation of wavelength 2.28970 Å is applied. For the purpose of determination of volume fractions of ferrite and austenite x-ray powder diffraction is carried out between 64° to 167° (2θ angle), the step size is 0.1° and the scanning time 10 seconds. Evaluation was then performed with Topas (see section 3.4).

4.4 Differential scanning caliometry

Disks of thickness 1.0-1.5 mm are cut from a dilatometer sample heat treated at

1200°C for 2 min and then quenched. The disks are then heated in the DSC and the

19

heat flow is registered continuously. A detailed description of the procedure is described in section 5.2 together with the results.

4.5 Scanning transmission electron microscopy (STEM)

The samples for STEM are small disks that are cut out from the dilatometer samples;

these are about a millimeter thick. The disks are grinded down to a thickness of around 50 µm and one side are polished 3-1 µ and then etched in 1:5 saltpeter acid (HNO ₃ ) with 2.0 V for about half a second. The TEM voltage is set to 200 kV.

The magnification and the number of reads are set after images taken with different levels of these settings are evaluated. A proper resolution is found to be 1024 x 1024 and a suitable number of reads is set to 1000. The magnification is set to 10 000 X, since this is large enough to discern the smallest particles and small enough to show a sample area large enough to avoid a requirement of too many image samples.

4.6 Scanning electron microscope (SEM)

The samples are mounted in conductive polyfast and then grinded. Polishing is done with 9, 6, 3 and 1 µm diamond suspension and followed by Silica polishing. The samples are then electrochemically etched in saltpeter acid (1:5 HNO ₃ -H ₂ O) with 2.0 V for ~1 second.

The settings in SEM when the images are taken are of significant importance, since different settings result in a different image. The first decision to be made is which type of detector gives the best image, i.e. highest contrast. For image analysis it is desirable to achieve a good contrast to enhance the quantitative results. The detectors available are:

• SE1, Secondary electron detector

• BSE, Backscattered electron detector

• QBSE, Backscattered electron detector (solid state detector)

• InLens (secondary electrons, detector placed in the pole piece)

All of these are tested with different settings of the following SEM parameters:

• Acceleration voltage (EHT)

• Working distance (WD)

• Aperture size

• Brightness

• Contrast

• Collectors bias (for BSE)

The outstanding detector to achieve a good contrast is the InLens detector, which is

chosen for the present investigation. For comparison, images from the BSE, QBSD

and InLens detector are shown in Fig. 4.1a-c. In Fig. 4.1d the image is a combination

of the InLens detector and QBSD detector.

20

(a) (b)

(c) (d)

Figure 4.1. (a) ESB detector, (b) QBSD detector, (c) InLens detector and (d) InLens and QBSD detector combined.

The best contrast is obtained with the InLens detector.

After the InLens detector is chosen more detailed tests are carried out with varied settings of the instrument parameters. For the best contrast the following parameters is decided:

• Acceleration voltage: 7.0 kV

• Working distance: 6.5 mm

• Aperture size: 20 µm

• Brightness: 0.0%

• Contrast: 45.5-46.0%

The level of contrast sometimes needs to be changed but fortunately within a very

small interval. Next step is to find a suitable magnification. This is an extremely

important factor, since the statistical result will be strongly dependent on this. The

smallest chromium nitride particles are only around 10 nm, and for these to be

identified a large magnification is needed. But, as a consequence of large

magnification the sample area is small which increases the number of images required

21

for adequate statistics. Each image takes time, both at SEM and at image analysis, so the time factor must also be taken into consideration.

The precipitates appear in many different shapes, the disks, at grain boundaries and as small dust particles in a “cloud”. For a particle to be registered in the image the pixel size has to be maximum half the particle size, otherwise it is filtered as noise. The pixel size depends on both the magnification and the resolution. In order to use a lower magnification a high resolution is chosen, 2048 x 1536. Before the decision of which magnification to choose, it should be realized that a particle of say 10 nm can result in four pixels in an image of more than three million pixels. The precipitates in disks and at the grain boundaries are of hundreds or thousands of pixels and it is realized that these small dust particles does not have much influence on the final result.

Table 4.3. Results from SEM from the same sample but with different magnifications. The complete set is shown in Appendix A.2 Tab. A.1.

1120/5/max 6000 X 8000 X 10000 X 13000 X

Average 2.53 3.45 2.63 3.56

Standard dev. 3.86 4.01 3.19 4.53

As can been seen from Tab. 4.3 above, the difference in average value does not differ significantly between the magnifications. The mean value of all four magnifications is 3.04. Since no correct answer is known this mean value is the best available indication of the true value. The magnifications closest to this value are 8000 X and 10000 X, so one of these two should be the magnification chosen for this method. The latter of these has the lowest standard deviation, although this result is not to be taken as definite but only for comparison between these four series. From this data the magnification is set to 10000 X.

Of each sample 25 images are taken. To get indications of how stable the method is 25 + 25 further images is obtained from the 1200/1/max sample.

4.7 Image analysis

For the determination of the area fraction of β-Cr ₂ N in the images from the electron microscope, image analysis is performed with the software AxioVision 4.4. The nitrides appear bright and the principle of the method is to calculate the fraction of bright pixels in the image. What is defined as bright, i.e. what are nitrides, has to be defined. The images consist of pixels on a grayscale from 0 (black) to 255 (white). A lower level is set when the lower threshold is decreased and frozen when the precipitates are covered so that only their edges are visible. One has to be aware of the fact that not all white areas are precipitates. The different morphologies of precipitates were studied in previously [7]. Fig. 4.2 shows precipitate forms; needles, cloud of needles, clouds of plates and grain boundary precipitates.

It is important to note the directions of the needle precipitates, since this has been used

as characteristic chromium nitrides. The needles are concentrated to two directions

which are preferably oriented to the ferrite matrix [7].

22

The images are then transformed from a grayscale to a binary image of only black and white pixels, where the nitrides are identified as white. The fraction of white particles is then calculated. The procedure of image analysis can be found in more detail in Appendix A.4 and section 6.5.

(a)

(b)

23 (c)

Figure 4.2. Examples of β-Cr

N structure. (a) Grain boundary, needles and

clouds. (b) Grain boundary, needles and clouds. (c) Needles and clouds,

note both the plate and disk shape of the cloud precipitates.

24

25

5 Results

For the samples cooled at maximum rate their respective cooling rate is shown in Tab. 5.1. The cooling time in the dilatometer is set to zero, but the cooling rates achieved are calculated from the measured time-temperature curve (see Appendix A.1). The rates are round off since they are measured as a slope of a curve and can not be taken as precise. The remaining cooling rates closely match the planned cooling rate. The samples in Tab. 5.1 are hereafter called 1200/1/400, 1120/5/360 and 1050/1/360, respectively.

Table 5.1. The achieved cooling rates for the samples cooled with maximum rate.

1200/1/max 1120/5/max 1050/1/max Cooling rate 400°C/s 360°C/s 360°C/s

5.1 Magnetic balance and x-ray diffraction (XRD)

5.1.1 Magnetic balance

The chemical composition analyzed by EPMA is given in Tab. 5.2 below and the results from the magnetic balance are shown below in Tab. 5.3. The sample heat treated at 1200°C shows larger increase in ferrite fraction than those treated at 1050- 1120°C. Ferrite fractions are plotted in Fig. 5.1 as function of cooling rate in order compare the two solution treatment times at the same temperature. As can be seen from Fig. 5.1 a larger increase of ferrite fraction occurred for the samples with 5 min solution time.

Table 5.2. Chemical composition in ferrite.

Si Mn Cr Mo Cu N Ni σ

(Hoselitz)

0.13% 0.88% 32.96% 3.98% 0.03% 0.05% 5.97% 107.56 [J·kg/T·m

]

Table 5.3 Ferrite fractions before and after heat treated. The samples heat treated at high temperature and/or long time have a larger increase in ferrite fraction.

Sample Ferrite before Ferrite after Change after HT

1200/1/400 49,28 64,74 15,46

1120/5/360 49,36 62,10 12,74

1120/1/130 50,45 56,77 6,32

1120/5/90 49,50 61,63 12,13

1120/1/40 50,18 55,75 5,57

1120/1/20 49,36 55,61 6,25

1050/1/360 49,57 52,06 2,49

26

0 10 20 30 40 50 60 70

0 50 100 150 200 250 300 350

Cooling rate [°C/s]

Ferrite content [%] 1120/1 before

1120/5 before 1120/1 after 1120/5 after 1120/1 diff 1120/5 diff

Figure 5.1. Ferrite content before and after heat treatment. Blue dots heat treated for one minute and pink dots for five minutes. The difference in fraction is smaller for the samples heat treated for one minute.

5.1.2 X-ray diffraction (XRD)

The results obtained by Topas are shown below in Tab. 5.4. It can be seen that much higher fractions of ferrite are obtained than in the magnetic balance.

Table 5.4. Ferrite and austenite fraction obtained with x-ray diffraction.

Sample Ferrite Austenite Austenite/Ferrite

1120/5/360 71,6 28,4 0,397

1120/1/130 77,4 22,6 0,292

1120/5/90 69,9 30,1 0,430

Figure 5.2. Scan profile from XRD. The blue curve represents the measured intensity and the red curve represents the by Topas calculated intensity.

211α polyfast

110α

111γ

200α 220γ

polyfast 200γ

27

5.2 Differential scanning calorimetry (DSC) The result of the DSC is shown in Fig. 5.3.

Figure 5.3. Results from DSC. The sample was first heat treated in dilatometer at 1200°C for 2 min and then quenched. The DSC procedure is as follows. The sample is heated at 100°C/min to 1220°C and then cooled to room temperature in 60 min. The sample is then heated to 850°C and held for 30 min and again cooled to room temperature in 60 min. Finally, the first step is repeated, the samples is heated to 1220°C and then cooled to room temperature.

The first peak in the curve is caused by a non-equilibrium state between the austenite and ferrite which was created in the prior heat treatment. The phase fractions at equilibrium vary with temperature and so the equilibrium is different at the heat treatment temperature and at room temperature. When the material is reheated above 400°C a temperature correct equilibrium could be established.

Initially the sample is heated to 1220°C and two peaks are observed, at 774.7°C and 1046.0°C. Both these are suspected sigma phase peaks, namely the formation and dissolute temperature, respectively. To verify these suspicions, the sample is cooled and then heated to 850°C for 30 min to create sigma phase. The first procedure is then repeated and peaks are detected similar to the peaks observed in the first round. During the first cooling a peak is marked at 781.9°C and is assumed to belong to sigma phase formation.

The second and third peak is likely sigma phase which is formed and dissolved,

respectively, as a consequence of the slow heating rate. The position of this peak is

known from previous experiments. No other peak can be observed.

28

5.3 Scanning transmission electron microscopy (STEM)

Due to technical problems this quantification could not be performed with STEM images, since the intensity obtained is too low. Although some images are taken, the image quality is unfortunately not good enough for image analysis. For comparison an image from both STEM and FEG-SEM is shown below in Fig. 5.4. It is however observed that the contrast obtained by STEM is very good.

(a) (b)

Figure 5.4. (a) Image with InLens in FEG-SEM and (b) image with STEM.

5.4 Scanning electron microscopy (SEM)

The results from the sampling of electron microscope images and image analysis are shown in Tab. 5.5 and the complete data set can be seen in Tab. A.2 in Appendix A.3.

The mean area fraction of chromium nitrides as a function of cooling rate is plotted in Fig. 5.6 and the spread of measured values in each sample is illustrated in Fig. 5.5. The 1120/5/90 sample was unfortunately destroyed during SEM sample preparation and hence no further data on that sample is accessible.

Table 5.5. Mean and standard deviation values of β-Cr

N from the samples.

1200/1/400 1120/5/360 1120/1/130 1120/1/40 1120/1/20 1050/1/360

Mean 2.33 1.78 1.08 0.65 0.14 0.53

Stand dev 2.14 2.12 1.62 0.93 0.33 0.90

29

0 1 2 3 4 5 6 7 8

A re a f ra c ti o n

1200/1/400 1120/5/360 1120/1/130 1120/1/40 1050/1/360 Mean values

Figure 5.5. The spread of measured values. The density of measure points is higher at lower area fractions.

In Fig. 5.5 the spread of the measured values are illustrated. Samples with higher mean values also have larger standard deviation (see Tab. 5.5). In all samples but the 1200°C/1min the standard deviation is larger than the mean value, and this is not a desirable result since under no circumstances can there be a measured value below zero. This can also be an indication that the standard deviation is not a proper measurement of uncertainty.

0 0,2 0,4 0,6 0,8 1 1,2

0 1 2 3 4 5 6 7 8

Area fraction

C u m u la ti v e f ra c ti o n

1200/1/400 1120/5/360 1120/1/130 1120/1/40 1050/1/360 W eibull γ=1 W eibull γ=0,5 W eibull γ=2 W eibull α=2

Figure 5.6. The area fraction vs the cumulative fraction. The pattern indicates that the data

might arise from an exponential distribution.

30

In Fig. 5.6 the mean area fraction is plotted against the cumulative fraction of samples.

A study of Fig. 5.5 and Fig. 5.6 indicates that an exponential distribution is likely. The density of measurements is larger for lower values and the mean value is not in the middle of the spread but shifted to lower regions. A proper distribution could be the Weibull distribution, which includes the exponential distribution. The Weibull cumulative distribution is expressed as [18]

( )

γ

α µ



 



 −

−

=

x

e x

F 1 .

If γ is changed different shapes of the curve will appear and the techniques for estimation of γ and the scale parameter α are shown in Appendix A.5. The location parameter µ is set to zero in this case.

The Weibull distribution hypothesis is tested for valibility with the Chi-square test and the Anderson-Darling test, see Appendix A.5. The tests are done without the 0- samples; instead it is done with these values excluded and only the non-zero values are considered. The test results reject the Weibull hypothesis except for the 1200/1/400 and 1120/5/360 samples.

The results of the twice repeated sampling of 1200/1/max sample are given in Tab. 5.6. Same statistical tests as above are done on these total 75 sample images and also for the second and third sequence separately (Appendix A.5). The spread is illustrated in Fig. 5.7.

Table 5.6. The mean value from three sets from the same sample. The complete set is shown in Appendix A.3 Tab A.3.

1200/1/max First set Second set Third set Total set

Mean value 2.33 2.23 2.30 2.30

0 1 2 3 4 5 6 7 8 9

A re a f ra c ti o n

First set Second set Thrid set Mean values

Figure 5.7. The spread of measured values of the three sets from 1200/1/400 sample.

(7)

31

6 Discussion

6.1 Formation of β-Cr ₂ N

From Fig. 6.1a it is easy to see that the increase of cooling rates increase the volume fraction of precipitates. So is the solid solution temperature: the increase of the temperature increases the volume fraction. These results are not surprising if the previous experiments and the theory are considered. There are four samples heat treated at 1120°C but cooled at different rates. An attempt is made to connect these four points with a logarithmic curve since the shape of the points indicates this type of function. The calculated mean area fraction of precipitates as function of cooling rate using Excel is shown in Fig. 6.1a. A plot with a logarithmic scale is also shown in Fig 6.1b.

If the straight line is extrapolated to higher rates, one can see that to produce a 2.5%

area fraction the cooling rate should be 1000°C/s, which is a practically impossible rate to perform. The slowest cooling which should produce chromium nitrides is 14.4°C/s if the trend line is accepted. Unfortunately only one point for the other two heat treatment temperatures is determined. However, the corresponding lines for these temperatures should be approximately parallel to the 1120°C-line. Then from a temperature of 1050°C a cooling rate of just over 133°C/s is the minimum rate for precipitation of β-Cr ₂ N. If the solution temperature is increased to 1200°C this minimum cooling rate is 5.2°C/s.

The assumption that the lines for other temperatures are parallel can be motivated by the conclusion mentioned in section 2.3 that the formation of β-Cr ₂ N is controlled by nitrogen diffusion. Diffusion is described by Fick’s law,



 



−

=

∂

− ∂

=

RT E_A

e D D

D x J

0

φ

J is the diffusion flux, Φ the concentration and x the position. D is the diffusion coefficient described by (8b) where D ₀ is a constant, E _A the activation energy, R the universal gas constant and T the temperature. If E _A , D ₀ and R are all constants the diffusion coefficient only depends on the temperature T. It does not depend on at which temperature the cooling begins, and hence it is always the same at a certain temperature. Since the formation of β-Cr ₂ N is controlled by the nitrogen diffusion it is reasonable to assume that the slope of the 1200°C and 1050°C lines in Fig. 6.1b are parallel to the 1120°C line.

(8a)

(8b)

32

y = 0,5372Ln(x) - 1,4324

0 0,5 1 1,5 2 2,5

0 50 100 150 200 250 300 350 400 450

Cooling rate

M e a n a re a f ra c ti o n

1120°C 1200°C 1050°C Log. (1120°C)

(a)

y = 0,5372Ln(x) - 1,4324

0 0,5 1 1,5 2 2,5

1 10 100 1000

Cooling rate

M e a n a re a f ra c ti o n 1120°C

1200°C 1050°C Log. (1200°C) Log. (1050°C) Log. (1120°C)

(b)

Figure 6.1. (a) The mean area fraction of β-Cr

N as a function of cooling rate. (b) The mean area fraction of β-Cr

N as a function of cooling rate with a logarithmic scale.

If the lines from all temperatures are assumed to be parallel, the heat treatment

temperature can be plotted against the minimum cooling rate determined from the

cross of horizontal axis and the line in a diagram such as in Fig 6.1b. With a