investigation of dynamic behaviour in the α/γ/α phase transformations in steels

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Materials Science Group

Master Thesis

High-temperature in-situ EBSD

investigation of dynamic behaviour in the α/γ/α phase transformations in steels

M.S.B. (Maxens) van Daalen S2050668

supervisors:

prof.dr. J.Th.M. (Jeff) de Hosson dr.ir. V. (Václav) Ocelik

ir. G. (Gerrit) Zijlstra

September 18, 2016

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ples was investigated using high-temperature in-situ electron backscatter diffraction (HT-EBSD).

Nucleation was observed to be heterogeneous and occurred at either grain boundaries or triple points. The majority of new grains showed the Kurdjumov-Sachs orientation relationship to one parent phase grain at the least. Measurement of phase boundaries speeds was possible for specific stages of the phase transformations using a novel measurement technique. In the case of a constant heating rate (0.5 °C/min), speeds between 1.4-4.0 nm/s were found. Constant cooling (1.0 °C/min) lead to speeds between 5.0-6.4 nm/s. An isothermal transformation at 840 °C gave speeds of 0.4- 1.7 nm/s. Stages of accelerated boundary speed, described as jerky motion, were observed in the constant cooling/heating experiments. These moments of accelerated motion were to ambiguous to quantify precisely, but lower speed limits were found. In the isothermal transformation this jerky motion was not found.

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Acknowledgement

This thesis would not have been possible without the help and guidance of several people.

Foremost, I would like to thank Prof. Jeff de Hosson for giving me the opportunity to conduct my research at the Materials Science Group. I would also like to thank him for his help and advice during the making of this thesis.

I also want to express my gratitude to dr. Václav Ocelik for all his help and cooperation during the conducting and discussion of experiments.

Finally, I would like to thank ir. Gerrit Zijlstra for his continuous guidance during the thesis.

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List of Figures 2

List of Figures 3

1 Introduction 6

2 Theory 8

2.1 Electron backscatter diffraction . . . 8

2.2 Phase transformations . . . 10

2.3 Grain boundaries . . . 10

2.3.1 Low-angle grain boundaries . . . 11

2.3.2 High-angle grain boundaries . . . 12

2.4 Phase boundaries . . . 13

2.5 Diffusion . . . 14

2.6 Nucleation . . . 16

2.7 Growth . . . 21

2.8 JMAK equation . . . 23

3 Experimental 25 4 Results & discussion 28 4.1 Test experiment . . . 28

4.2 Isochronal 1.5 °C/min heating . . . 34

4.3 Isochronal 0.5 °C/min heating . . . 39

4.4 Isothermal . . . 51

5 Summary & conclusions 62 6 Recommendations 64 6.1 Drift correction . . . 64

6.2 PRIAS . . . 65

7 Bibliography 66

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List of Figures

2.1 Schematic representation of diffraction of electron beam on tilted sample. Sample unit cell diffracting cones indicated. (Adapted from Schwartz et al. [22].) . . . 9 2.2 Hough transform procedure. (a) Kikuchi band and points in (x,y) space. (b) Locus

of points from (a) in (ρ,ϕ) space. (c) Final Hough transform. (Adapted from Randle [25].) . . . 9 2.3 Iron carbon phase diagram. (Adapted from Massalski et al. [28].) . . . 11 2.4 Low-angle grain boundary (left) and high-angle grain boundary (right) in simple

cubic structure, according to a dislocation model. (Adapted from Gleiter [31].) . . 12 2.5 Direct exchange between atoms. (Adapted from Mittemeijer [38].) . . . 15 2.6 Substitutional diffusion in lattice and corresponding energy barrier. (Adapted from

Mittemeijer [38].) . . . 15 2.7 Interstitial diffusion. (Adapted from Mittemeijer [38].) . . . 15 2.8 Free energy change and energy contributions versus radius. The activation energy

barrier is given as ∆G^∗. (Adapted from Porter, Easterling and Sherif [40].) . . . . 17 2.9 Nucleation of new phase β at grain boundary between two α phase grains. The

energies of the interfaces γαα and γαβ are shown together with the dihedral angle θ. (Adapted from Clemm and Fisher [41].) . . . 18 2.10 Figure to determine the coeffiecients a, b and c for nucleation on a grain boundary.

(Adapted from Clemm and Fisher [41].) . . . 19 2.11 Nucleation of new phase at triple junction (left) and grain corner. (Adapted from

Clemm and Fisher [41].) . . . 19 2.12 Energy barrier ratio as a function of cosine of angle for heterogeneous nucleation.

(Adapted from Cahn [42].) . . . 20 2.13 Formation of low energy interface during nucleation on grain boundary. (Adapted

from Porter, Easterling and Sherif [40].) . . . 20 2.14 Energy change of system upon jump of atom from parent phase to product phase.

(Adapted from Mittemeijer [38].) . . . 22 3.1 Setup for high temperature in-situ EBSD in Tescan Lyra chamber. . . 25 3.2 Heating profile in Kammrath & Weiss DDS software. . . 26 3.3 SEM image of low carbon steel sample secured by a tungsten spring and shielded

with tantalum plates. . . 27 3.4 SEM image of sample showing feature, creep drift direction and other drift causes. 27 4.1 Temperature profile of the first experiment. . . 29 4.2 Phase map of low carbon steel sample at 840 °C showing ferrite (red) and austenite

(green). Highlighting shows nucleation at triple points (solid blue circle) and grain boundaries (dotted blue circle). The bold yellow line indicates phase boundaries that fulfill the Kurdjumov-Sachs orientation relationship. . . 30 4.3 Fraction of austenite as a function of time. . . 31 4.4 . . . 32 4.5 [001] inverse pole figure maps (top) and phase maps (bottom) showing key moments

of transformation from ferrite (red) to austenite (green). . . 32

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4.6 . . . 33 4.7 [001] inverse pole figure maps (top) and phase maps (bottom) showing key moments

of transformation from ferrite (red) to austenite (green). . . 33 4.8 Temperature profile of isochronal experiment with 1.5 °C/min heating. . . 34 4.9 Phase map of low carbon steel sample at 895.5 °C showing nucleation at triple point

(solid blue circle) and grain boundaries (dotted blue circle). The bold yellow line indicates the special orientation relationship. . . 35 4.10 Overview phase map at room temperature after cooling. . . 35 4.11 . . . 37 4.12 [001] inverse pole figure maps (top) and phase maps (bottom) showing key moments

of transformation from ferrite (red) to austenite (green) with 1.5 °C/min heating. . 37 4.13 . . . 38 4.14 [001] inverse pole figure maps (top) and phase maps (bottom) showing key moments

of transformation from ferrite (red) to austenite (green) with 1.5 °C/min heating. . 38 4.15 Fraction of austenite as a function of temperature and time. . . 39 4.16 Temperature profile of isochronal experiment with 0.5 °C/min heating and 1 °C/min

cooling. . . 40 4.17 Fraction of austenite as a function of temperature and time. . . 40 4.18 Effective activation energy versus austenite fraction. . . 41 4.19 Effective activation energy versus temperature for 0.5 °C/min and 1.5 °C/min heating. 41 4.20 Growth exponent versus temperature. . . 42 4.21 . . . 43 4.22 [001] inverse pole figure maps (top) and phase maps (bottom) showing key moments

of transformation from ferrite (red) to austenite (green) with 0.5 °C/min heating. . 43 4.23 . . . 44 4.24 [001] inverse pole figure maps (top) and phase maps (bottom) showing key moments

of transformation from ferrite (red) to austenite (green) with 0.5 °C/min heating. . 44 4.25 Schematic showing novel measurement technique. The red arrows give the growing

direction, the red dotted lines show the average positions of the boundaries at t0, t1

and t2 and the solid red lines the distance interval. . . 45 4.26 Boundary speed measurement technique example showing average position of the

boundaries (dotted lines), distance interval (solid lines) and growing direction (arrow). 45 4.27 Overlaid phase maps indicating phase boundary growing direction and misorientation. 46 4.28 Position of boundaries plotted as a function of temperature and normalised time.

Misorienation and labeling correspond to figure 4.27. . . 47 4.29 Fraction of austenite as a function of temperature and time. . . 48 4.30 . . . 49 4.31 [001] inverse pole figure maps (top) and phase maps (bottom) showing key moments

of transformation from austenite (green) to ferrite (red) with 1 °C/min cooling. . . 49 4.32 . . . 50 4.33 [001] inverse pole figure maps (top) and phase maps (bottom) showing key moments

of transformation from austenite (green) to ferrite (red) with 1 °C/min cooling. . . 50 4.34 Position of boundaries plotted as function of normalised time for the austenite to

ferrite transformation. . . 51 4.35 Temperature profile of the fourth experiment. . . 52 4.36 . . . 54 4.37 [001] inverse pole figure maps (top) and phase maps (bottom) showing key moments

of transformation from austenite (green) to ferrite (red). . . 54 4.38 . . . 55 4.39 [001] inverse pole figure maps (top) and phase maps (bottom) showing key moments

of transformation from austenite (green) to ferrite (red). . . 57 4.44 . . . 58

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LIST OF FIGURES LIST OF FIGURES

4.45 [001] inverse pole figure maps (top) and phase maps (bottom) showing key moments of transformation from austenite (green) to ferrite (red). . . 58 4.46 750 °C and 820 °C [001] inverse pole figure maps showing points of recystallization

(dotted circle) and decreasing γ grain size (solid circle). . . 59 4.47 Overlaid [001] inverse pole figure maps showing growth at constant temperatures. . 59 4.48 Overlaid [001] inverse pole figure maps indicating phase boundary growing direction

and misorientation at 840 °C. . . 60 4.49 Position of boundaries plotted as a function of normalised time. Misorienation and

labeling correspond to 4.48. . . 61 6.1 Drift velocity (nm/s) as a function of temperature. . . 64 6.2 [001] inverse pole figure map and PRIAS differentiation maps at 845 °C. . . 65

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Introduction

In materials science and engineering phase transformations are employed as a practical method for manipulating the microstructure to obtain desired material properties. For iron alloys and steels, the austenite (FCC) to ferrite (BCC) (γ/α) phase transformation is the key transformation for microstructure manipulation. Understanding of the transformation is therefore of great importance and it has received a lot of attention over the years. Various experimental techniques have been used to study the transformation. Especially in-situ observations provide great insight into the nature of the transformation. The phase transformation has been studied in-situ using techniques such as optical microscopy [1,2], transmission electron microscopy (TEM) [3], X-ray and neutron diffraction [4] and electron backscatter diffraction (EBSD) [5,6].

It is this electron backscatter diffraction that has received increasing attention in the in-situ studies of phase transformations. Such in-situ EBSD studies have for example been performed on the α/β phase transformation in Ti [7], the β/α transformation in Co [8], the α/γ phase transformation in Fe-Ni alloys [9,10] and the α/γ/α phase transformations in low carbon steel [11–13]. Although several of these studies have been done, little is still known about the dynamics of phase transformations.

In dynamics the time derivative of acceleration is known as jerk, i.e. the third derivative of distance with respect to time. In physics literature, there is little attention for the concept, as it does not appear in expressions for force, momentum or energy [14]. Although inherently present in nature, jerkiness is not always apparent due to its length and time-scale. Phenomena which appear to go smoothly on larger scales, are truly jerky on smaller scales.

In recrystallization in metals, jerkiness has been observed in the grain boundary motion. For example in in-situ TEM experiments on thin Au film, where grain boundary migration of sub- sections of 30-500 nm length was observed to occur by brief spurts of tens of nm/s, spaced by intervals of a few minutes with no observed motion [15]. It could, however, not be proved this was an intrinsic effect. Similar experiments on thin Cu-Au film were done and discrete steps in diffusion-induced grain boundary migration were observed with a velocity around 1-10 µm/min, separated by longer periods of inactivity [16].

Both these experiments suggest grain boundary motion is jerky on the microscopic scale. Evi- dence suggesting jerky motion occurs on macroscopic scale has also been found [17,18].

Phase transformations are thought to belong to the world of jerky phenomena. Periods of accelerated and decelerated behaviour, called jerky motion or simply jerkiness, are expected to be observed on specific length and time scales. Such behaviour has been confirmed experimentally for the γ/α phase transformation. In isothermal hot stage TEM experiments on steel (0.4 wt% C and 0.71 wt% C) the γ/α interface velocity was seen to be discontinuous and showed acceleration and deceleration that can be distinguished from jerky motion by pinning effects [3]. As a possible explanation the interaction between driving force (difference in Gibbs free energy) and opposing force (strain energy) at the interface was given. Where intervals of deceleration were accompanied by stress build up and intervals of acceleration by stress relieve, due to vacancy diffusion [19].

These fluctuations in interface velocity correspond with observations in isochronal high-resolution dilatometry on pure iron [20] and ultralow-carbon steel (0.01 at.% C) [21]. In the case of cooling, the transformation proceeded until the deformation-induced strain energy was the same as the chemical driving force and there was no longer a net driving force. A net driving force occurred again when the material was further cooled, which was due to the increasing chemical driving force

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CHAPTER 1. INTRODUCTION

with decreasing temperature. The transformation then continued till the driving force was again equal to the strain energy, which lead to jerky behavior of the interface velocity.

The main motivation for this thesis is to expose combinations of length and time scales leading to jerkiness in the α/γ and γ/α transformations in a low carbon steel. These observations will lead to a better understanding of the phase transformation and reveal possible valuable insights into how this can be used in the production of steels with improved performance. High-temperature in- situEBSD will be used, which will mainly involve observation of the dynamic behaviour to look for correlation length and time scales and possibly quantify these into phase boundary speeds. EBSD is chosen as an experimental method as it provides information on phase identity and distribution, as well as grain orientation and local texture, which are believed to correlate to jerky motion.

The method has not been used before to monitor dynamic behaviour of phase transformations on the minute time scale. The optimization of experimental parameters will therefore be a point of interest. These will be presented as recommendations for future use of the technique.

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Theory

This chapter will provide the theoretical background to understand the nature of dynamic behaviour in phase transformations and the experimental technique used to observe it. Several concepts and definitions will be introduced. First, the experimental technique electron backscatter diffraction (EBSD) will be discussed in greater detail. This will give insight into the possibilities and strengths of the technique and why it was chosen for the experiments. A thermodynamical discussion follows that explains what drives phase transformations. Then the specifics of the α and γ phases are given, together with the steel phase diagram. In steels, atoms are arranged in an ordered periodic structure, called crystal structure. A real steel is formed by regions of different crystal direction or phase, called grains. The complete structure formed by these grains is called the microstructure. It is this microstructure that has a strong influence on behaviour during a phase transformation. The concepts of grain boundaries and phase boundaries will be introduced.

These are closely related to the microstructure and are essential for the processes happening during phase transformations. Then, the time related concepts of diffusion, nucleation and growth are discussed. This is where the atomic structure and thermodynamics come together and where dynamic behaviour emerges. Finally, the JMAK equation is discussed, which is a representation of the kinetics of phase transformations. With a JMAK representation of the experimental data it is possible to derive some parameters of the phase transformations, as explained in this section.

2.1 Electron backscatter diffraction

Electron Backscatter Diffraction (EBSD) is a technique used to analyze crystalline materials. The surface of a bulk sample can be analyzed to provide information on the orientation of individual grains, local texture, phase identity and phase distribution. An electron beam is focused on the surface of a sample and the surface normal is tilted at approximately 70° to the direction of the electron beam. Backscattered electrons, from depths around 20 nm, give a pattern that becomes visible by fluorescence of a phosphor screen and which is captured by a CCD camera. These patterns are called Kikuchi patterns and their geometry is a gnomonic projection of the crystal structure on the flat phosphor screen, see figure 2.1.

The pattern consists of a number of parallel lines, with different thicknesses, that cross with other pairs of lines. These lines follow from the Bragg’s condition of reflection. Parallel lines represent lattice planes and the thickness is inversely proportional to the interplanar spacing, which follows from the condition that the angular width is twice the Bragg angle. The electron wavelength can be altered by changing the accelerating voltage, a decreasing voltage causes an increasing wavelength and hence increasing line spacing, which follows approximately θhkl ∼ 1/V^1/2. To determine the microstructure of the analyzed volume, the Kikuchi pattern needs to be indexed.

This indexing seems straightforward, but is not trivial to automate. Therefore, the lines are transformed with a method called the Hough transform [23]. It was not until indexing became fully automated using this Hough transform, that EBSD became a truly powerful technique [24].

The Hough transform is generated by taking an equation for a straight line in (x,y) space

ρ = x cos ϕ + y sin ϕ (2.1)

with ρ the distance from the origin to the line and ϕ the angle between the line and the x-axis.

In practice, a number of coordinates (x,y) is taken on the centre line of the Kikuchi band, see

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2.1. ELECTRON BACKSCATTER DIFFRACTION CHAPTER 2. THEORY

Figure 2.1: Schematic representation of diffraction of electron beam on tilted sample. Sample unit cell diffracting cones indicated. (Adapted from Schwartz et al. [22].)

figure 2.2a. These points are then transformed to (ρ,ϕ) space, as shown in figure 2.2b. In this space the points on the line are sinusoidal curves that intersect in a single point and Kikuchi bands are now represented as peaks of high intensity. The detection of such a single spot in Hough space is much easier for a computer than the detection of a line in the Kikuchi pattern. Then a masking technique is applied to the butterfly shaped spots to make them sharper and represent the centre lines of the bands more accurately, see figure 2.2c. Finally, the Kikuchi pattern in terms of the obtained centre lines is reconstructed.

Figure 2.2: Hough transform procedure. (a) Kikuchi band and points in (x,y) space. (b) Locus of points from (a) in (ρ,ϕ) space. (c) Final Hough transform. (Adapted from Randle [25].)

When the Kikuchi pattern reconstruction is finished, identification is done by comparing measured interplanar angles (angles between bands) and interplanar spacings (band thicknesses) with theoretical values in look-up tables.

The automation of pattern indexing lead to the foundation of orientation imaging microscopy (OIM™), also known as orientation mapping [26,27]. The electron beam is moved over a predefined area with constant stepsize. At every step an EBSD pattern is captured and indexed. The final result is a map of the sample indicating the microstructure, including morphology of the crystals,

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defects and texture. A map indicating the phases in the sample is simply called a phase map and a map indicating the crystal orientation of measurement points with respect to a specific axis is called an inverse pole figure map.

After fully automated EBSD became possible in the 90’s an interest emerged to perform in-situ experiments with the technique. EBSD is combined with a tensile stage or heating stage to observe microstructural changes and phase transformations in the process.

2.2 Phase transformations

A phase transformation occurs when the final state of a system is more stable than the initial state.

At constant temperature and pressure this relative stability is given by the Gibbs free energy (G) and the transformation is driven by a reduction of this energy. The Gibbs free energy is defined as

G = H − T S (2.2)

where H is the enthalpy, T the temperature and S the entropy. The enthalpy H is given by

H = U + P V (2.3)

where U is the energy, P the pressure and V the volume of the system. In solid state phase transformations the second term is negligible with respect to the energy term. The Helmholtz free energy (F )

F = U − T S (2.4)

is then equal to the Gibbs free energy and the terms are often used interchangeably in literature.

When in its most stable state a system is said to be in equilibrium. In a closed system this means it has the lowest possible Gibbs free energy. From equation 2.2 it is seen this occurs for a system with low enthalpy and high entropy. When imagining the Gibbs free energy as a function of atomic arrangement it forms a curve. Next to one stable equilibrium there will always be other local minima. These minima are called metastable and satisfy dG=0, but do not have the lowest value of G. All possible phase transformations are given by the condition

∆G = G2− G1< 0 (2.5)

where G2and G1are the free energies of the final and initial states, respectively. The deviation of G from its minimum causes the driving force for the phase transformation and tries to get the system in equilibrium. So this is not actually a force, but has units of Joule per unit volume, or pressure (N/m²). In polymorphous phase transformations of a parent phase (α) to a product phase (β) the driving force then is

∆Gαβ= ∆Uαβ− T ∆Sαβ (2.6)

A convenient way to represent phases and their stability is by phase diagrams. These diagrams are a graphical representation of the phase structure when the externally controlled parameters of temperature, pressure and composition are plotted in various combinations. The most common of these diagrams is a plot of temperature versus composition, where the pressure is held constant.

Part of the Fe-C phase diagram is shown in figure 2.3. From this figure it is seen that pure iron experiences two phase transformations upon heating, before melting. First, there is a transition from α (BCC) to γ (FCC) at 912 °C. At 1394 °C there is a second solid state phase transformation to δ (BCC). The part of the phase diagram shown in figure 2.3 is considered the iron rich part.

After 6.70 wt% C, the interstitial compound cementite is formed and therefore this part of the phase diagram is omitted. In low carbon steels (up to 0.3 wt%C) there is a temperature region where α and γ coexist and the final transformation temperature to complete austenite is lowered.

2.3 Grain boundaries

Grain boundaries are lattice defects that separate two regions of identical crystal structure having different orientation. Their inherent complex structure makes them difficult to describe and for a three dimensional structure eight parameters are needed to describe them mathematically.

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2.3. GRAIN BOUNDARIES CHAPTER 2. THEORY

Figure 2.3: Iron carbon phase diagram. (Adapted from Massalski et al. [28].)

Three terms are needed for the orientation relationship (for example Euler angles φ1, φ2, Φ), two parameters for the spatial orientation of the boundary through the normal of the grain boundary plane n = (n1, n₂, n₃), with respect to one of the adjacent crystals and three components of the translation vector t = (t1, t₂, t₃). All the properties are essentially a function of these eight parameters.

The transformation of one crystal structure into another has an orientation relationship, which is pure rotation if there is a common origin. A twist boundary has its grain boundary plane perpendicular to the rotation axis. A tilt boundary, however, has its grain boundary plane parallel to the rotation axis. Therefore, a tilt boundary can be both symmetric and asymmetric. The grain boundary is then defined by this orientation relationship and the orientation of the grain boundary plane. The misorientation angle is then used to describe the orientation relationship by a rotation around a specific axis. Boundaries in real polycrystals are not mathematical objects and the atomic structure at the boundary becomes important. There are relaxation processes at the grain boundary which make atoms occupy sites incompatible with the mathematical description.

Using this atomic picture of grain boundaries, it is possible to distinguish between low-angle and high-angle grain boundaries.

2.3.1 Low-angle grain boundaries

As the name suggests, these boundaries have small misorientation angles. Low-angle grain boundaries are formed by a periodic arrangement of crystallographic defects, called dislocations. In the case of symmetric tilt boundaries, there is a single set of dislocations. The angle is determined by the spacing d of these dislocations and is given by

b

d= 2 sinφ

2 ≈ φ (2.7)

where b is the boundary Burgers vector and φ the angle. This means a smaller dislocation spacing gives a larger angle. An example of a low-angle tilt boundary of a simple cubic structure is shown in figure 2.4 on the left.

At least two sets of dislocations are needed for an asymmetric boundary. These sets must have perpendicular Burgers vectors. The deviation from the symmetric boundary increases with increasing fraction of dislocations in the second set. For twist boundaries at least two sets of screw dislocations are needed. It is possible to have mixed boundaries by a combination of dislocation networks of three Burgers vectors. An analytical solution of the free energy of a low-angle grain

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boundary can be found [29]. Assuming there is an infinite lattice and the stress field has a size on the order of the dislocation spacing d, the energy per unit length then becomes

Ed= µb²

4π(1 − ν) ln_r^d

0 + Ec

(2.8) where µ is the shear modulus, ν de Poisson ratio, r0≈ bthe radius of the dislocation core and E_c the energy of the dislocation core. For a symmetric tilt boundary the number of dislocations per unit length is n = 1/d ≈ φ/b, and hence the grain boundary energy per unit area is

γ = φ b

µb²

4π(1 − ν)ln d r₀ + Ec

= φ(A − B ln φ) (2.9)

where A = Ec/band B = µb/4π(1 − ν). Experimental work has shown this equation is quite a good fit for φ < 15° [30].

Figure 2.4: Low-angle grain boundary (left) and high-angle grain boundary (right) in simple cubic structure, according to a dislocation model. (Adapted from Gleiter [31].)

2.3.2 High-angle grain boundaries

The dislocation model fails to hold for rotation angles larger than 15°. Measurements do not show the energy decrease expected for larger angles. The reason the models fails is that dislocation cores tend to overlap and dislocations can not be identified as individual lattice defects any longer.

The boundaries have a relatively open structure due to large areas of poor atomic fit. Many bonds are broken or distorted because of this fit, leading to a relatively high potential energy.

Boundaries with a misorientation larger than 15° are therefore called high-angle grain boundaries, to distinguish them from the more structured low-angle grain boundaries. See figure 2.4 on the right for the dislocation model of a high-angle grain boundary.

A concept to distinguish special high-angle grain boundaries is the coincident site lattice or CSL [32]. For certain orientation relationships the atomic positions at the grain boundary coincide with both adjacent lattices. These coincidence sites form a periodic lattice of its own, since both adjacent lattices are periodic. This lattice is known as the coincident site lattice. The density of coincidence sites or the size of the CSL unit cell is defined by the parameter Σ, given by

Σ = volume unit cell CSL

volume unit cell crystal lattice (2.10)

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2.4. PHASE BOUNDARIES CHAPTER 2. THEORY

This parameter is always larger or equal to one, as the unit cell of the CSL is larger than the unit cell of the crystal itself. It can also be defined as the ratio of lattice points in the unit cell of the CSL to the number of lattice points in the original lattice. Grain boundaries which have a high density of coincidence sites are simply called CSL boundaries. A smaller Σ value corresponds to more ordering in the grain boundary. Therefore, low-angle grain boundaries are also seen as being Σ = 1, because almost all lattice points are coincident, except for the dislocation cores. There are special low-energy high-angle grain boundaries called twin boundaries. The two crystals are then each others mirror image and the boundary is the mirror plane. This means all lattice sites at the boundary are coincident sites. Twin orientation relationships are Σ3, as every third plane parallel to the twin boundary is in perfect coincidence.

The problem for real grain boundaries is now that coincidence only occurs for special cases and Σdoes not change continually. What happens for small deviations of these cases is that the crystal tries to compensate by introducing dislocations. These dislocations need to have a Burgers vector which will conserve the CSL. Only the energy is determined by the density of coincident sites and not their location. Therefore, it is possible to conserve the CSL without conserving the coincident site locations, by displacement by small vectors. The vectors that satisfy this define another lattice, called the displacement shift complete or DSC. The DSC is the coarsest sub-grid that has all the lattice sites of both crystals in it, though most of its lattice points will be unoccupied. Displacing one of the crystals of a CSL grain boundary by a vector of the corresponding DSC lattice thus preserves the symmetries of the CSL, but shifts the entire CSL. The translation vectors of the DSC lattice are the possible Burgers vectors for the grain boundary dislocations, often called secondary grain boundary dislocations or SGBDs. The SGBDs compensate orientation differences to CSL relationships, while preserving the CSL.

It should be noted that although small Σn tend to have lower energy than average, there is no simple correlation between energy and Σn. There are some values with low energy but most are not significantly different from random orientations.

Equations for the grain boundary energy have been derived for special cases, which show the lower energy of specific CSL boundaries. The energy of a boundary composed of three sets of screw dislocations can be given, which is a good approximation for a (111) twist boundary [33]. The boundary energy is then given by

γ = γ0+ 2Ec

b + γel (2.11)

where Ec is again the dislocation core energy, γ0 the energy of the reference state and γel is the elastic energy per unit area. The elastic energy is derived to be

γ_el= µb√ 3 6π

2πr₀ 3b φ + ln

3b 4πr0

− ln φ

φ (2.12)

Equation 2.11 is a good representation of both low and high-angle grain boundaries of the (111) twist type and the energy shows cusps at φ values of CSL boundaries.

2.4 Phase boundaries

The interfaces between crystals of different lattice structure or composition are called phase boundaries. These interfaces are characterized by their atomic structure in a similar way as interfaces between identical crystals. The phase boundaries occur in three forms: coherent, semicoherent and incoherent.

When the two phases fit perfectly at the boundary, it is called a coherent interface. The lattices are then continuous across the boundary plane. This is only possible if the interface has the same atomic configuration in both phases and if both crystals have a specific orientation. When there is a change in composition across the boundary, atoms bond partially to wrong neighbours. This results in an increase in energy of the atoms at the interface and adds a chemical contribution to the energy. Coherent boundaries are possible even when the lattice spacing is not identical. At least one of the lattices then becomes strained to create a fit.

The total energy is increased by the strains of such a coherent interface. If the misfit is large enough it becomes energetically more favourable to form a semicoherent interface. In this interface, periodic dislocations are present to compensate for the disregistry. The disregistry is defined as the misfit between the lattices and is given by

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δ = dβ− dα

dα (2.13)

where dαand dβ are the lattice spacings when there is no stress. For large dislocation spacing, the coherency strain is not fully compensated and long-range strain fields remain. The interfacial energy has two components for a semicoherent boundary. The first is the same as for the coherent boundary, but there is also the structure term, which is the additional energy from the structural distortions caused by the dislocations.

In some phase boundaries good fit across the interface is not possible. Either because there is a large difference in atomic configuration of the interface in the lattices or because the atomic spacings differ by more than 25%. These are called incoherent boundaries and are typically formed when two randomly oriented crystals coalesce across some interfacial plane. Due to the complex atomic structure only little is known about incoherent boundaries, but there are many similarities to high-angle grain boundaries.

There is a close relation between nucleation and the type of phase boundary which is formed (see section 2.6). When a new phase nucleates from the parent phase a distinct orientation relationship is almost always found. As the assumption can be made that a nucleus will form with a minimal interfacial energy, the two crystal structures will have a good atomic fit. The rate of nucleation would otherwise be drastically reduced by the increased free-energy barrier for nucleation. As explained in section 2.6, the nucleus will have a shape which has a minimum free-energy. Because this precipitate is a three-dimensional object, the interface will have all possible orientations around and therefore have parts with poor atomic alignment and higher interfacial energy.

In the case of heterogeneous nucleation on a grain boundary the new β grain will have a low energy coherent or semicoherent interface with only one of the α grains. There will be a high energy incoherent interface with the other grain [34]. This can be explained by the fact that when a new precipitate has a good atomic matching with one α grain, it is almost impossible to have any atomic fit with the other.

For the α-γ phase transformation this means a low energy semicoherent interface is formed between the closest-packed planes of the phases, i.e. (111)f cc and (110)bcc. The orientation relationships that are then created are called the Nishiyama-Wasserman (N-W) OR [35,36]

(110)_bcc//(111)_{f cc} & [001]_bcc//[¯101]_{f cc} and the Kurdjumov-Sachs (K-S) OR [37]

(110)bcc//(111)f cc & [1¯11]bcc//[0¯11]f cc.

These special orientation relationships are not only important in the case of nucleation (see section 2.6), but also for growth (see section 2.7).

For grain triple junctions and corners the constraints are more complex. In the case of preferred triple junction nucleation the precipitate is seen to have a near K-S OR to two parent grains [13].

2.5 Diffusion

Most of the phase transformations in the solid state happen through thermally activated motion of atoms, called diffusion. Diffusion is always driven by a decrease in Gibbs free energy. On the microscopic scale diffusion is seen as jumping of atoms between sites, which can occur by several mechanisms. This jumping of atoms has a somewhat random nature and does not necessarily prefer a specific direction. An atom experiences a thermal vibration and collides with vibrating neighbours, which causes fluctuations in energy. These fluctuations are sporadically large enough for an atom to overcome the energy barrier of jumping to another site. Because of the random nature of the path an atom follows, the effect of a large number of atoms needs to be considered to give a description of net diffusional flow. There must be a driving force for a net flow, which is typically due to temperature. There is still diffusion of atoms when there is no driving force, but overall there will be no net flow.

Diffusion in the solid state can happen through several mechanisms: an exchange mechanism, substitutional diffusion and interstitial diffusion. The simplest form of diffusion would be a direct

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2.5. DIFFUSION CHAPTER 2. THEORY

exchange of two atoms in the lattice, see figure 2.5. This direct exchange is highly unlikely because the necessary lattice deformation during movement is energetically very unfavourable. It is therefore believed to contribute only little to diffusion.

Figure 2.5: Direct exchange between atoms. (Adapted from Mittemeijer [38].)

Substitutional diffusion happens when vacancies are present in the material. One of the adjacent atoms can jump to the site of the vacancy, creating a vacancy on its initial site. It can therefore be seen as a direct exchange between an atom and a vacancy. The energy barrier for such a direct exchange is much lower than for the direct exchange of atoms. This energy barrier is given by the difference between the energy before the jump and the energy halfway in the jump, where the atom displaces adjacent atoms to move to the vacancy, as shown in figure 2.6.

Figure 2.6: Substitutional diffusion in lattice and corresponding energy barrier. (Adapted from Mittemeijer [38].)

Smaller solute atoms, such as carbon and nitrogen, occupy interstitial lattice sites in the metal lattice. These atoms diffuse through interstitial diffusion, where the available sites for jumping form a sublattice. An example is the octahedral sublattice in the BCC lattice of ferrite. The diffusing atom can jump to any of the sites in this sublattice, provided it is not occupied, as shown in figure 2.7. Due to the small solubility, the number of available sites is huge. The displacement of the lattice to make room for the interstitial to jump is of the same order as for the substitutional mechanism. Therefore, the diffusion for interstitial atoms is much larger, due to the higher number of available vacancies.

Figure 2.7: Interstitial diffusion. (Adapted from Mittemeijer [38].)

There is also a continuum approach to diffusion, as opposed to the atomic approach. Fick’s law is often used, which was originally developed for diffusion in liquids [39]. Although this is not

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strictly valid for diffusion in metals and alloys, experimental results are often represented in the form of diffusion coefficients, in reference to it. In its original form, Fick’s first law states that the diffusional flux is proportional to the concentration gradient of the diffusing component and is given by

J = −Ddc

dx (2.14)

where J is the flux and D the diffusion coefficient [m²/s]. In a more fundamental sense the law can be written in terms of energy. Since systems prefer a state of minimal energy to be in equilibrium, there will be a driving force to move material to this state of lower energy if the distribution of components does not correspond to this equilibrium. It is then possible to give a law which states that the local flux is proportional to the gradient of the energy, leading to

J = −CdE

dx (2.15)

where C is a constant and E represents an energy term. Through thermodynamics these two forms can be shown to be equivalent, but the concentration form is still more in use.

Combining Fick’s first law together with mass balance it is possible to arrive at another form, called Fick’s second law, which states

dc dt = d

dx

Ddc dx

(2.16) This is a more convenient form when in non steady-state.

2.6 Nucleation

In diffusional phase transformations a new phase with a different composition and or structure is formed by thermally activated atom transfer across an interface. This transformation starts with a process called nucleation and there are two ways for a new phase to nucleate in the parent phase:

homogeneous and heterogeneous nucleation. In the case of homogeneous nucleation, the new phase has equal probability to nucleate anywhere in the material. In heterogeneous nucleation, on the other hand, the process starts at preferred sites; such as defects, impurities and grain boundaries.

Both these types are generally described using classical nucleation theory (CNT). In CNT the equation for free energy change of homogeneous nucleation process is given by

∆G = −V ∆G_V + A_αβγ_αβ+ V ∆G_S (2.17)

where the first term is the volume free energy reduction, the second term the increase in free energy due to creation of surface area between the phases and the last term the misfit strain energy of the new phase. In homogeneous nucleation the nucleus is considered to be spherical with a radius r. If we also assume the surface free energy to be constant, then the equation 2.17 becomes

∆G = −4

3πr³(∆GV − ∆GS) + 4πr²γαβ (2.18) In figure 2.8 the three energy contributions and total free energy change are shown schematically.

Due to the different contributions of the free energy change there is a value of r for which ∆G reaches a maximum. This value is found by differentiation of equation 2.18, which yields

r^∗= 2γαβ

(∆GV − ∆GS) (2.19)

and

∆G^∗= 16πγ_αβ³

3(∆G_V − ∆G_S)² (2.20)

The activation energy barrier is then ∆G^∗ and r^∗ is the critical radius, or the minimum size for a nucleus to be stable and start growing.

Heterogeneous nucleation is, however, much more likely to occur. This is because the preferred site of nucleation has an increased free energy of its own, which is released when a new phase

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2.6. NUCLEATION CHAPTER 2. THEORY

Figure 2.8: Free energy change and energy contributions versus radius. The activation energy barrier is given as ∆G^∗. (Adapted from Porter, Easterling and Sherif [40].)

nucleates here. The release of free energy reduces or even removes the activation energy barrier, making this type of nucleation more likely. Keeping the discussion as general as possible, follow- ing the approach by Clemm and Fisher [41], assume that the boundary area that is lost during nucleation is

Aαα= ar² (2.21)

where r is the radius of curvature of the new surface boundary. The new phase boundary area is

A_αβ= br² (2.22)

and the volume of the new phase nucleus is

V = cr³ (2.23)

The coefficients a, b, c are functions of the α-α and α-β energies. The free energy then becomes

∆G = −V (∆GV − ∆GS) + γαβAαβ− γααAαα (2.24) which can be rewritten in terms of equations 2.21, 2.22 and 2.23 as

∆G = −cr³(∆GV − ∆GS) + γαβbr²− γααar² (2.25) The energy barrier then becomes

∆G^∗ = 4 27

(γ_αβb − γ_ααa)³

c²(∆GV − ∆GS)² (2.26)

and the grain boundary nucleation problem then simplifies to finding the coefficients a, b and c. To see how this works for a grain boundary see figure 2.9. The critical nucleus will have a shape which minimizes the total interfacial free energy, two connected spherical caps in this case. The angle θ is determined by the equilibrium between interfacial energies and hence is given by

cos θ = γαα

2γαβ (2.27)

The coefficients a, b and c can then be calculated, see figure 2.10. The area Aααbecomes

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Figure 2.9: Nucleation of new phase β at grain boundary between two α phase grains. The energies of the interfaces γααand γαβare shown together with the dihedral angle θ. (Adapted from Clemm and Fisher [41].)

Aαα= ar²

= π sin²θr² and so

a = π sin²θ.

The new phase boundary area is

Aαβ= br²= 2(2πr)h

= 4π(1 − cos θ)r² thus

b = 4π(1 − cos θ).

The value of c is found from volume of the two spherical caps

V = cr³=2πh²

3 (3r − h)

=2π

3 (2 − 3 cos θ + cos³θ)r³ hence

c = 2π

3 (2 − 3 cos θ + cos³θ)

The ratio between the activation energy barriers for homogeneous and heterogeneous nucleation is then given by

∆G^∗_Het

∆G^∗_Hom = V_Het^∗

V_Hom^∗ = S(θ) (2.28)

which is called the shape factor and equals

S(θ) = 1

2(2 + cos θ)(1 − cos θ)² (2.29)

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2.6. NUCLEATION CHAPTER 2. THEORY

Figure 2.10: Figure to determine the coeffiecients a, b and c for nucleation on a grain boundary.

(Adapted from Clemm and Fisher [41].)

So the efficiency of a grain boundary as a nucleation site depends on cos θ or the relation γ_αα/2γαβ. In a similar way the coefficients can be calculated for grain edges and grain corners, see figure 2.11 for a schematic of triple junction and corner nucleation. The critical volume and energy barrier become smaller for nucleation on grain edges or grain corners. Figure 2.12 shows the ratio ∆G^∗Het/∆G^∗_Hom dependency on the cosine of θ. It is seen that the activation energy is decreasing with increasing number of grains meeting in the junction.

Figure 2.11: Nucleation of new phase at triple junction (left) and grain corner. (Adapted from Clemm and Fisher [41].)

In the case the parent and product phases are similar enough, it becomes possible to reduce V^∗ and ∆GHet further by the creation of low energy interfaces, see figure 2.13. The new grain then has a special orientation relationship with one or more of the parent phase grains (see section 2.4).

Because of the smaller nucleation barrier these faceted nuclei are expected to occur frequently.

The rate laws for nucleation follow from the equation for the concentration of critical nuclei

C^∗= C₀exp(−∆G^∗/kT ) (2.30)

where C0 is the number of atoms per unit volume in the phase. Assuming every nucleus can be made supercritical at a rate f per second, the homogeneous nucleation rate then is

N_Hom= f C^∗ (2.31)

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Figure 2.12: Energy barrier ratio as a function of cosine of angle for heterogeneous nucleation.

(Adapted from Cahn [42].)

Figure 2.13: Formation of low energy interface during nucleation on grain boundary. (Adapted from Porter, Easterling and Sherif [40].)

this factor f depends on frequency with which the nucleus can receive atoms from the parent phase, which depends on the surface area of the nucleus and the rate of diffusion. It can be rewritten as ω exp(−∆G_m/kT ), where ∆Gm is the activation energy for diffusion per atom and ω is a factor that includes vibration frequency of atoms and the area of the critical nucleus and so the nucleation rate is

NHom= ωC0exp

−∆Gm

kT

exp

−∆G^∗ kT

(2.32)

which has units nuclei m⁻³s⁻¹. In a similar way, a rate equation for heterogeneous nucleation is given by

NHet= ωC1exp

−∆G_m kT

exp

−∆G^∗ kT

(2.33)

where C1is the concentration of heterogeneous nucleation sites per unit volume. The ratios of nucleation rates do not necessarily increase the same as the ratios of activation energy decrease, because the density of sites also decreases as the type of nucleation changes. The ratio of nucleation rates is given by

NHet

N_Hom = C1

C₀exp∆G^∗_Hom− ∆G^∗_Het kT

(2.34)

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2.7. GROWTH CHAPTER 2. THEORY

where differences in ω and ∆Gm are ignored, due to importance. The ∆G^∗ is always smallest for heterogeneous nucleation and hence the exponential is a large quantity, favouring high heterogeneous nucleation rate. But the factor C1/C0 must also be considered, which is the ratio of heterogeneous nucleation sites compared to the total number of atoms per unit volume. If we assume the effective grain boundary thickness to be δ and the mean grain boundary diameter to be D, then the ratio can be approximated to be

C₁ C0

= δ

D (2.35)

For grain edges and grain corners this factor reduces further to (δ/D)²and (δ/D)³, respectively.

Typically, the sites with the highest volume nucleation rates depend on the driving force ∆GV. When the driving force is small, the activation energy barriers are high and the highest nucleation rates will be at grain corners. As the driving force increases, grain edges and grain boundaries start to dominate behavior. At high driving forces it will become possible for the C1/C₀term to dominate and homogeneous nucleation has the highest rate. To express these conditions quantitatively take the variable R = kT ln(D/δ), the conditions for the different dominant types of nucleation then become [43]

Dominant Nucleation rate

N_Hom R > ∆G_Hom− ∆G^B_Het

N_Het^B ∆G_Hom− ∆G^B_Het> R > ∆G^B_Het− ∆GÊ_Het N_HetÊ ∆G^B_Het− ∆GÊ_Het> R > ∆GÊ_Het− ∆G^C_Het N_Het^C ∆GÊ_Het− ∆G^C_Het> R

It needs to be noted that the above equations are for the isothermal case, if there is a changing temperature, the driving force will change with time. The initial stages will then be dominated by the nucleation sites which produce measurable nucleation volume first.

In the kinetics of transformation two types of behavior are shown that relate to the nucleation.

These two types of behavior are called normal and abnormal, as observed and classified by Liu et al. [20,44]. In the abnormal behavior multiple maxima are observed in the ferrite formation rate as a function of transformed fraction, as opposed to one in the normal behavior. This abnormal behavior is due to further occurring nucleation processes ahead of the phase boundary, which is called autocatalytic nucleation [44]. This autocatalytic nucleation is a consequence of the misfit between parent and product phase. In the γ/α transformation there is a reasonable amount of volume misfit strain energy. The austenite matrix is deformed by induced strain and defects, caused by growing ferrite grains. Nucleation is easier in such deformed austenite. This ultimately leads to several repeated maxima of nucleation ahead of the interface and abnormal transformation kinetics.

2.7 Growth

Growth in phase transformations, where the phases have different crystal structure, is highly depending on the atomic fit at the interface. Atom transfer across an incoherent interface is expected to be much higher by thermally activated jumps of atoms. This process is analogous to high angle grain boundaries moving during recrystallization. In the case of a good atomic fit across the boundary, it is much harder for atoms to find atomic sites to jump to. This creates a growth barrier for interfaces with good atomic fit. This difference in growth barrier leads to different growth speeds, which in turn leads to anisotropic growth in precipitates nucleated on grain boundaries, corners or edges.

The type of growth in phase changes provides an easy way to divide the nucleation and growth processes. This type of growth is characterized by the type of interface, which is either glissile or non-glissile. The difference in these two interfaces lies in the requirement for thermal activation.

A glissile boundary does not need thermal activation and can move even at low temperatures, it is therefore also known as athermal. These interfaces move by a shearing of the parent lattice into the product caused by dislocation glide. Non-glissile boundaries, on the other hand, do need a thermal activation. Migration here is caused by thermally activated random jumping of atoms across the interphase boundary. If the growth during a transformation is by motion of glissile boundaries the process is called military and if it is by migration of non-glissile it is known as civilian [43].

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In a military transformation there is no compositional change between the parent and product phase and the nearest neighbours are more or less unchanged. This means this is also a diffusionless transformation. In this regime, movement occurs by coordinated, simultaneous migration of atoms.

A well-known example is the martensitic transformation. These military transformations are of no further interest in this report and in subsequent mentioning of phase transformations the discussion applies only to the civilian regime.

A civilian transformation can be either diffusionless or diffusional. As in the military regime, a civilian diffusionless transformation has no change in composition between parent and product phase. An example is a massive transformation, in which there is no long-range diffusion and migration is by individual atoms moving across the interface. The limiting factor in the growth of the new phase is in this case the speed of jumping of atoms across the interface. Therefore this type of transformation is interface controlled.

For a change is composition, diffusion of atoms of various sizes towards or away from the interface is needed for boundary migration. There are boundaries that can move slowly even in presence of high driving forces. The speed of the boundary is then determined by this interface and mostly independent on the rate of diffusion and said to be interface controlled. If, however, the boundary has high mobility as compared to the diffusion, the limiting factor for growth will be the diffusion rate and the process is termed diffusion controlled. It is possible that the rates for the interface and diffusion are of equal size, in which case it is called mixed control. In phase changes with a substantial difference in composition it is most likely that the process is diffusion controlled. As an atom needs to jump hundreds to thousands of times within the parent phase, while only having to jump once or twice at the interface.

Both types of growth can conveniently be described by one equation. This equation gives the volume of a particle (Y ) at a time t, that nucleated at a time τ and is given by [45]

Y (τ, t) = g Z t

τ

vdt⁰

!_m^d

(2.36) where g represents the particle geometry factor (m³m^−d), v is the growth velocity of the interface and m is the growth parameter (m = 1 for interface-controlled growth and m = 2 for volume diffusion-controlled growth; assuming there is a parabolic growth law), d is the dimensionality of the growth (d = 1, 2 or 3).

Figure 2.14: Energy change of system upon jump of atom from parent phase to product phase.

(Adapted from Mittemeijer [38].)

The interface velocity can be found by considering figure (2.14). When an atom of the parent phase jumps to the product phase, there is an accompanying energy change ∆G = Gβ− Gα. Note that ∆G is negative here, indicating a positive driving force. An activation energy, ∆G^a, needs to be overcome to reach the state of lower energy. The quantity of atoms jumping the interface per unit time is then given by

v⁰exp −∆G^a RT

!

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2.8. JMAK EQUATION CHAPTER 2. THEORY

where v⁰ is an atomic vibration frequency. In a similar way the quantity of atoms crossing the interface in opposite direction is found, where the barrier is now given by (∆G^a− ∆G) and hence

v⁰exp −∆G^a− ∆G RT

!

The difference between these quantities then leads to an expression for the interface velocity [43]

v(T (t)) = v₀exp − ∆G^a RT (t)

!

1 − exp ∆G RT (t)

!!

(2.37) where v0 is a pre-exponential factor, that incorporates the atomic jump distance and ∆G^a is the activation energy for the transfer of atoms through the phase interface and ∆G is the difference in energy between the parent phase and the new phase.

During growth in the α-γ phase transformation there generally is formation of annealing twins.

It is proposed these annealing twins form resulting from a decrease in interfacial free energy of the phase boundaries, which would not be possible without these twins [46]. In this phase transformation the process is believed to be caused by a two dimensional nucleation process on the {111}

planes of the growing grain [47].

In a few rare cases growth speeds have been presented in literature for phase transformations in pure iron and steels. From dilatometric data an average speed of 3.05 µm/s was found in the case of the γ/α transformation in pure Fe for cooling rates of 5 - 15 K/min [20]. In similar experiments an average speed of 1.6 µm/s was found for the γ/α interface of a Fe-0.01 at.% C at 20 K/min cooling, as compared to 3.9 µm/s for pure iron at the same cooling rate [21]. In the same paper a transition from diffusion-controlled growth to interface-controlled growth was discussed.

Diffusion-controlled growth was observed in the initial stages of transformations for cooling rates under 15 K/min. Speeds about 5 times smaller (∼10⁻⁷ m/s) than for massive interface-controlled growth were found. In optical microscopy experiments the α/γ transformation of Fe-4.2 at.% Cr with 5 K/min heating showed two stage transformation behaviour with 0.33 - 0.75 µm/s and 3.7 - 7.6 µm/s stages, respectively [1]. The two stage behaviour was believed to be due to a change in driving force related to the phase transformation from the α/γ coexisting region to the γ single state region of the Fe-Cr alloy.

2.8 JMAK equation

Several models have been developed to describe the kinetics of phase transformations. The most well-known of these models is the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation [48–51].

The model represents the kinetics of a solid state phase transformation as a process of nucleation and growth of a product phase at the expense of the parent phase. The original approach is only valid for random homogeneous nucleation and isotropic growth, but was later shown to be applicable for continuous cooling [52] and heterogeneous nucleation [42], when modified. The most important concept in these models is that of extended volume, or the volume of new phase that would have been formed if there is no interference from other nuclei. This means the regions that are already transformed are counted and regions continue to grow irrespective of other regions. In its most general form this extended volume, V^e, is

V^e= Z t

0

V ˙N (T (τ ))Y (t, τ )dτ (2.38)

where ˙N is the rate of nucleation per unit volume, V the total volume and Y the volume of a new phase grain nucleated at time τ. Equations for the fraction of transformed material have been derived for various modes of nucleation, growth and impingement [53]. In the case of random distribution and isotropic growth the fraction becomes

f = 1 − exp(−xe) (2.39)

where xe= Ve/V is the fraction of extended volume over total volume. For anisotropic growth there is more chance for the nuclei to impinge and the equation becomes

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df dxe

= (1 − f )^ζ (2.40)

where ζ is a term that indicates if growth is anisotropic (ζ > 1) or isotropic (ζ = 1). For ζ = 1 equation 2.40 reduces to equation 2.39. When ζ > 1 equation 2.40 can be integrated to give

f = 1 − [1 + (ζ − 1)xe]^{−1/(ζ−1)} (2.41) Nucleation of preferred sites, such as grain boundaries and triple points, is more likely. In this case the equation is given in its general form

df dxe

= 1 − f^ε (2.42)

where ε indicates a non-random distribution of nuclei (ε > 1) or a random distribution (ε = 1).

For ε = 1 equation 2.42 reduces to equation 2.39 An analytical equation for the fraction f is only possible for specific values of ε.

Independent of the type of nucleation, growth and impingement, an analytic formulation of the extended volume can be derived [54]. For an isothermal transformation it is given by

xe= K0(t)^n(t)t^n(t)exp −n(t)Q(t) RT

!

(2.43) where K0is a kinetic parameter, n the growth exponent and Q the effective activation energy.

To give an analytical expression for an isothermal transformation, the concept of nucleation index (a) is introduced [55]. If (a = 0) it means there is no nucleation, if (a = 1) there is constant nucleation and if (a > 1) nucleation accelerates. For an isochronal transformation (a = 1) the extended volume then is

xe= K0(T )^{n(T )} RT² Φ

!n(T )

exp −n(T )Q(T ) RT

!

(2.44) and if (a > 1) it is

xe= K0(T )^{n(T )}(T²)^{n(T )}exp −n(T )Q(T ) RT

!

(2.45) where Φ is the constant heating/cooling rate.

Liu et al. [53] have developed a recipe for determination of the time and/or temperature dependent activation energy Q, which requires no adaption to one of the specific models, and the time and/or temperature dependent growth exponent n, which requires only choice of a impingement model. This determination of Q uses the temperature Tf needed to reach a certain fixed value of phase fraction f, measured for different heating/cooling speeds. A plot of ln(Tf²/Φ)versus 1/Tf

is then made to obtain Q [56]. To determine the growth exponent n, the transformed fraction fT

at a fixed value of T is needed at different heating/cooling speeds. Plotting of ln[− ln(1 − fT)], or ln([(1 − fT)^1−ζ/(ζ − 1)]), or ln arctanh(fT) (depending on the type of impingement) versus ln Φyields n. When a line is drawn between two points of different heating rate, the values for Q(f )and n(T )Φcan only be considered approximations of the time and/or temperature dependent constants.

Although the JMAK approach is still widely in use in the study of solid-state phase transformations, it has its limitations. From a transformation curve, the factors and processes being present in the transformation can never be reconstructed unambiguously.

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Chapter 3

Experimental

The experiments were conducted in a Tescan Lyra FEG/FIB Dual beam microscope, equipped with an OIM system by EDAX and a Hikari super camera (up to 1400 indexed points per second).

The EBSD data were obtained using TSL OIM Data Collection 7. A Kammrath & Weiss 1500

°C heating module was used for in-situ heating of the samples, which is calibrated for optimal performance at high temperatures (i.e. 500 - 1500 °C). It has a maximum heating/cooling speed of 5 °C/s. This heating module is mounted on the SEM stage like a "large sample" and is positioned by moving the stage. The setup inside the Tescan Lyra chamber is shown in figure 3.1. The heating module works with a PID-temperature controller, which regulates the temperature within an error of ±0.1 K. The heating profile was monitored using Kammrath & Weiss DDS software. See figure 3.2 for the graphical interface and an example of a heating profile using DDS.

Figure 3.1: Setup for high temperature in-situ EBSD in Tescan Lyra chamber.

The heating module is designed for sample sizes up to 10 x 10 mm². These samples were secured with a tungsten spring on the top and optionally ceramic plates, to compensate for thickness, on the bottom. A SEM image of the setup is shown in figure 3.3. The sample is surrounded by three overlapping tantalum shields that minimize thermal radiation pollution in the EBSD measurements.

The experiments started with finding a feature on the sample. This feature can be either a particle or a hole and is used to center the image, in order to correct for drift. This centering is necessary to be sure to have the same area for every scan. During heating or cooling, the image is drifting both to the side and up and down. This is due to a combination of creep, thermal expansion and shifting of the beam by changes in applied current. In the case of an isothermal experiment there is only a creep drift, moving the the sample down (or image up). See figure 3.4 for a SEM image indicating the directions of drift and the feature used to correct for drift.

After a feature is found, a scan of a large area is made to find a smaller area within this larger area that is suitable for the experiment. In this smaller area there preferentially are a few complete grains, triple points and grain boundaries. The quality of the image in the area is also

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Figure 3.2: Heating profile in Kammrath & Weiss DDS software.

an important factor. If there are already many misindexed points at room temperature, this will not be a suitable scan area.

To get a quality EBSD scan, a subtraction of the background is needed. During heating the sample becomes so hot it starts to radiate and so the background changes. Therefore, every other couple degrees this background needs to be captured again and subtracted to conserve the quality of the scans being done. Another problem arises due to heating of the phosphor screen in the EBSD detector. The screen gets damaged when its temperature becomes to high. Hence, a thermocouple is placed on top of the EBSD detector that monitors the temperature. When this temperature gets to high, the screen is retracted to cool down, before the experiment is resumed.

The used samples were made of a S355 low carbon steel with a 0.15wt% carbon content. The samples were prepared with a standard grinding and polishing procedure.

The data was processed in TSL OIM Data Analysis 7. First the data was dewrapped or cropped to remove the first row of points on the left side, which are misplaced by the OIM Collection software. Then the data was cleaned with Grain Confidence Index Standardization (tolerance 5.0, size 5, Multi Row 1) and a Neighbour Orientation Correlation (level 4, tolerance 5, minimal Confidence Index 0.10). A partition was made were all data points with a confidence index below 0.10 were dropped. If possible, consecutive datasets were aligned using the average offsets calculated from the red, green and blue channels of the inverse pole figures.

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CHAPTER 3. EXPERIMENTAL

Figure 3.3: SEM image of low carbon steel sample secured by a tungsten spring and shielded with tantalum plates.

Figure 3.4: SEM image of sample showing feature, creep drift direction and other drift causes.