Investigation of Titanium α Plates by EBSD Analysis

Transcription

1 Investigation of Titanium α Plates by EBSD Analysis Master Thesis Arjen Kamp June 2007 Materials Science and Engineering Faculty 3mE Supervisors: Stefan van Bohemen Jilt Sietsma Roumen Petrov

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4 Investigation of Titanium α Plates by EBSD Analysis Summary When titanium Low Cost Beta alloy (Ti-LCB) is cooled from the β phase region, two different types of α plates can be found, which are denoted as primary and secondary α plates. The primary α plates form at relative high temperatures and secondary α plates form at lower temperatures. Many characteristics of the two types of α plates were found to differ, such as the morphology, the growth kinetics and the appearance of surface relief effects. Since the first description of the two types of plates in 1958 there has been several research on the nucleation and growth mechanism of the α plates. In this thesis, both the diffusional primary and the displacive secondary α plates are analysed by Electron Backscatter Diffraction (EBSD). The objective of this study is to enhance the understanding of the influence of the microtexture on nucleation and growth of the two different α plates. In total three different microstructures are analysed in this thesis; two microstructures in which only secondary α plates are found and one in which both the primary and the secondary α plates are found. In total 16 EBSD map scans are performed and 2 sets of point scans. The collected data is subsequently analysed with TexSEM Laboratories Orientation Imaging Microscopy (TSL OIM). The first analysis considers the orientation relationship (OR) between the parent β grains and the α plates. In literature, two important orientation relations are proposed that describe the β-α transformation in titanium alloys, which are the Burgers OR and the Potter OR. In order to compare the orientation relations with the measured misorientations, the numbers of misorientations that fall within defined spreads around the predicted orientation relations are calculated. It is found that for both primary and secondary α plates the Burgers OR is the best description. The influence of the grain boundary misorientation on the nucleation rate of the α plates is also determined. This is done by dividing the measured misorientation angles into four different ranges, each with a width of 15. The nucleation appearance of the α plates is categorized in 6 different types, ranging from grain boundaries without any nucleation to grain boundaries with a high degree of plate nucleation. It was found that the α plate nucleation increases for larger grain boundary misorientations. For misorientations above 30 no change in nucleation is found. The analysis of the possible misorientations between the α plates, shows that one particular axis/angle pair between two plates has a relatively high occurrence. This grain boundary occurred more often than what would be expected in a random situation. Several causes for this occurrence of this type of grain boundaries were proposed and discussed. From the results it is concluded that the high occurrence of this type of grain boundaries is caused by a lower interface energy compared to the other 4 possible grain boundaries between two α plates from one parent β grain. It is conjectured that the high occurrence of grain boundaries with low interface energy is related to autocatalytic nucleation. There is a high variant selection of the α plates that have nucleated at the grain boundaries, both for primary and secondary α plates. This variant selection was not observed for intragranular secondary α plates, the variant selection is therefore considered to be dependent on the β-β grain boundaries. At about 50% of the measured β-β grain boundaries, the variant selection causes the (0001) plane of the α plates to be parallel to one of the two closest (110) planes of the two β grains. When the angle between the two (110) planes increases, the tendency for the α plates to have their (0001) plane to one of the two particular (110) planes decreases. At the lowest angle range between the (110) planes the percentage of (0001) planes i

5 Summary that is parallel to one of the (110) planes is about 70% and the percentage decreases to approximately 30% for the largest minimum angles between the planes. The observation that also for small (110) misorientations the percentage of boundaries that does not follow this rule is still approximately 30%, leads to the conclusion that other factors also influence the variant selection. One proposed factor is the influence of the orientation of the grain boundary plane relative to the possible orientations of the α plates. Also the preferred α-α grain boundaries influence the variant selection by minimizing the grain boundary energy of the α plates with the α plates that have nucleated at the other side of the grain boundaries. ii

6 Investigation of Titanium α Plates by EBSD Analysis Table of contents 1. Introduction General characteristics of titanium Titanium production Mechanical properties Crystal structure Deformation mechanisms Characteristics of α plates in titanium alloys Introduction Morphology Growth directions of α plates Growth kinetics Phase compositions Surface relief effects Texture Introduction Orientation relationships in Ti alloys α-α misorientations Grain boundary theory Grain boundary structure Low-angle grain boundaries High-angle grain boundaries Grain boundary energies Interfacial energy of low-angle grain boundaries Interface energy of high-angle grain boundaries Interfacial energy at triple points Nucleation theory Introduction Classical nucleation Activation energy for nucleation Nucleation rate Non-classical nucleation Introduction Historical background Athermal nucleation Isothermal nucleation Autocatalytic nucleation Variant selection Microtexture analysis EBSD Installation Orientation determination Kikuchi diffraction patterns Pole figures Orientation representation iii

7 Table of contents Orientation matrices Rodrigues-Frank space Experimental Composition and heat treatments Sample preparation EBSD measurement procedure EBSD data analysis Results Optical microscopy EBSD maps Grain boundaries β-β grain boundaries α-β grain boundaries α-α grain boundaries Pole figures Orientation relations Discussion Orientation relations Grain boundary misorientation Misorientation distribution Grain boundary nucleation α-α grain boundaries Variant selection Conclusions Recommendations References Appendix A. EBSD maps A.1. Inverse pole figures A.2. Grain boundary maps Appendix B. Pole figures Appendix C. Matlab scripts C.1. β-β grain boundary misorientation calculation C.2. Orientation relation calculation iv

8 Investigation of Titanium α Plates by EBSD Analysis List of figures Figure 2.1. Effects of the alloying elements on the equilibrium Ti phase diagram... 4 Figure 3.1. Micrograph of primary α plates, reacted for 15 days at 650 C in a Ti-7.22 at.% Cr alloy, magnification ~500 (a), and reacted for 3 days at 650 C, magnification ~1000 (b)... 7 Figure 3.2. Micrograph of secondary α plates, reacted for 3 minute at 600 C in a Ti-7.22at% Cr alloy, magnification ~1000 (a) and secondary α plates reacted for 1 min at 575 C, magnification ~500 (b) Figure 3.3. Micrograph of intragranular sheaves of secondary α plates in a Ti-7.22at%Cr alloy, reacted for 3 min at 600 C, magnification ~500 (a), magnification ~1000 (b) and intragranular secondary α plates reacted for 10 sec at 575 C, magnification ~ Figure 3.4. Interfacial structure with the invariant line of the α-β grain boundary... 8 Figure 3.5. Optical micrographs showing secondary α plates formed at 615 C after 1080 s (a) and primary α plates formed at 735 C after 100,000 s (b) Figure 3.6. The Fe, Mo and Al concentration profiles across several secondary α plates formed at 615 C (a) and across one primary α plate formed at 735 C (b) Figure 3.7. The possible planes and directions for the Burgers OR in a bcc crystal which are parallel to the (0001) plane and [1120] direction of the hcp crystal Figure 3.8. Occurrences of the misorientation angles between all 12 possible α variants calculated from the Burgers OR where the total number of α-α misorientations is Figure 4.1. The orientation between two crystals in a tilt grain boundary (a) and a twist grain boundary (b) Figure 4.2. The random distribution of the misorientation of grain boundaries in cubic-cubic systems by MacKenzie Figure 4.3. Definitions of the angle of misorientation θ and the grain boundary orientation angle φ Figure 4.4. Example of a semicoherent interface between phase α and phase β. d α and d β are the unstressed interplanar spaces of the two phases Figure 4.5. A low-angle tilt grain boundary with misorientation angle θ and dislocation spacing D=b/θ Figure 4.6. Schematic dependence of the misorientation on the grain boundary energy for low-angle and highangle grain boundaries Figure 4.7. A coincidence site lattice of two crystals with a misorientation angle of 22, indicated by the black circles Figure 4.8. The variation of grain boundary energy with orientation. The solid line represents experimental determined values of the grain boundary energies, the dashed curve represents the calculated energy according to equation Figure 4.9. Energy difference for different rotations around the c-axis in hexagonal systems Figure Energy difference for different rotations around the a-axis in hexagonal systems Figure Representation of three intersecting grain boundaries Figure 5.1. The Gibbs free energy as a function of the nucleus radius r Figure 5.2. Schematic representation of the athermal mode (a), the isothermal mode (b) Figure 5.3. Block diagram showing martensitic transformations as a multi level system Figure 5.4. A transformation without variant selection (a) and with variant selection (b) Figure 6.1. A schematic illustration of an EBSD installation Figure 6.2. Schematic illustration describing the relation between the specimen axes, the microscope axes and the phosphor screen axes (a) and the relation between the specimen axes and the acquisition axes (b) Figure 6.3. Example of a Kikuchi pattern measured during EBSD analysis Figure 6.4. Kikuchi pattern (a) and a detail of an IPF (b) of a measurement with pseudosymmetry Figure 6.5. Illustration describing the azimuthal angle α and rotation angle β for plotting a (0001) pole figure Figure 6.6. Illustration of the Euler angles φ 1, Φ and φ 2 and the rotations required describing the relation between the crystal orientation and the reference orientation Figure 6.7.The definition of the crystal orientation angles relative to the specimen reference axes Figure 6.8. The fundamental zone of the R-F space for cubic symmetry Figure 6.9. The fundamental zone of the R-F space for a cubic-hexagonal system Figure 7.1. Schematic illustration of a dilatometer Figure 8.1. Micrograph of the Ti-4.5Fe-6.8Mo-1.5Al alloy isothermally held at 615 C for 750 seconds and subsequently quenched Figure 8.2. Micrograph of the Ti-4.5Fe-6.8Mo-1.5Al alloy isothermally held at 730 C for 100 minutes followed by isothermal holding at 630 C for 12 minutes and subsequently quenched Figure 8.3. Micrograph of the Ti-4.5Fe-6.8Mo-1.5Al alloy isothermally held at 615 C for 960 seconds and subsequently quenched v

9 List of figures Figure 8.4. IPF maps of the Ti-LCB alloy before (top) and after (bottom) cleanup procedures Figure 8.5. Phase map of EBSD scan I-1, in which the hcp α phase is coloured red and the bcc β phase is coloured green. Grain boundaries are indicated by a black line Figure 8.6. Image quality map of scan I Figure 8.7. IPF map of scan I-1with the IPF color legend Figure 8.8. Kernel average misorientation plot of scan I-1with the colour code legend. The minimum and maximum angular deviations are indicated for each colour Figure 8.9. SEM image of a Ti-4.5Fe-6.8Mo-1.5Al alloy isothermally held at 615 for 18 minutes, with the locations of the point scans indicated Figure The measured β-β misorientation distribution compared with the MacKenzie misorientation distribution, indicated by the blue line Figure The measured normalised occurrences of a particular α nucleation class divided over four β-β grain boundary misorientation ranges and for all measured grain boundaries Figure Grain boundary map of scan I-1in which the blue lines indicate the Burgers OR. All other grain boundaries are indicated by a black line Figure Grain boundary map of scan III-3. The 5 possible α-α grain boundaries are indicated by different colors Figure The relative length of the different types of α-α grain boundaries of all EBSD scans and the occurrence in a random situation (section 3.3) Figure The difference in occurrence from a random situation for the 5 types of α-α grain boundaries of all EBSD scans Figure Pole figures according to the Burgers OR of the coloured microstructure shown on the left, the orientations of the crystal lattices are also indicated on the grain map on the left by the unit cells Figure The percentage of grain boundaries at which the (0001) planes of the α plates are parallel to one of the two (110) planes of the two adjacent β grains that have the smallest angle with each other Figure Plot of the misorientation between α plates and the parent β grain of scan I-4, compared with the Burgers ( ), Potter ( ) and Pitsch-Schrader OR ( ) Figure Plot of the variants that have nucleated in a β grain in the R-F space with variant selection (a) of I-4 in which four different variants are found and a plot without a clear sign of variant selection (b) of scan III-1 in which 10 different variants are found Figure 9.1. The fractions of misorientations for different types of α plates that fall within a certain misorientation with the Burgers and Potter OR. The error bars have a length of 2σ Figure 9.2. The fractions of misorientations within ranger for the two orientation relations for the intragranular secondary α plates of two different EBSD scans Figure 9.3. The deviation values for each histogram bin Figure 9.4. The average weights of the grain boundaries for the different misorientation ranges..71 Figure 9.5. Detail of scan I-6 in which the α plates seem to branch into new crystal orientations to increase the α plate volume fraction in the parent grain Figure 9.6. The normalised frequencies of the different α-α grain boundaries and the calculated frequency from the measured orientations of EBSD scan III-1 compared with the random situation Figure 9.7. The normalised frequencies of the different α-α grain boundaries and the calculated frequency from the measured orientations of EBSD scan I-6 compared with the random situation Figure 9.8. The deviations from random for the grain boundaries measured by OIM and the orientation distribution and the OIM measurement compared to the distribution data of scan III Figure 9.9. The deviations from random for the grain boundaries measured by OIM and the orientation distribution and the OIM measurement compared to the distribution data of scan I Figure Pole figures of scan I-4. The α plates have nucleated at a grain boundary with the (0001) pole parallel to the common (110) pole of the β grains. The poles of the closest (110) planes and the parallel (0001) plane are indicated by the circles. The minimum angle between the (110) planes is Figure Pole figures of scan I-4 in which the α plates do not have their (0001) plane parallel to the common (110) planes (yellow and red). The poles of the closest (110) planes are indicated by the circle and it can be seen that no (0001) plane is parallel to any of both (110) planes. The minimum angle between the (110) planes is vi

10 Investigation of Titanium α Plates by EBSD Analysis List of tables Table 2.1. Important properties of titanium compared with steel Table 2.2. Slip systems in α phase titanium... 4 Table 2.3. Twinning systems in α phase titanium... 5 Table 3.1. The 12 variants of the Burgers orientation relationship between the β matrix and the product α phase 14 Table 3.2. Possible axis/angle pairs between two α plates formed in the same β grain according to the Burgers OR Table 4.1. CSL values up to 35b represented in axis/angle pairs and Euler angles for cubic-cubic systems Table 6.1. The 24 symmetric matrix operators S i for a cubic system Table 6.2. The 12 symmetric matrix operators S i for a hexagonal system Table 7.1. Chemical composition of the Ti-LCB alloy Table 7.2. Heat treatments Table 7.3. Grinding steps used during sample preparations Table 7.4. Experimental details of the EBSD scans Table 8.1. The observed occurrences of α nucleations for the different misorientation angle ranges and for all grain boundaries combined Table 8.2. Details of the CSL grain boundaries that were found Table 8.3. Percentages of measured misorientations that fall within a specified maximum misorientation with the Burgers OR Table 8.4. Percentages of measured misorientations that fall within a specified maximum misorientation with the Potter OR Table 8.5. Percentages of measured misorientations that fall within a specified maximum misorientation with the Pitsch-Schrader OR Table 8.6. Smallest misorientation angles between two variants of the orientation relations Table 8.7. Rodrigues vectors and misorientation angle of the four Burgers R-F variants Table 8.8. Rodrigues vectors and misorientation angle of the four Potters R-F variants Table 8.9. Rodrigues vectors and misorientation angle of the two Pitsch-Schrader R-F variants Table 9.1. Comparison of the measured misorientation spread to the MacKenzie distribution Table 9.2. The weight factors for the different grain boundary nucleation types vii

11 List of tables viii

12 Investigation of Titanium α Plates by EBSD Analysis 1. Introduction Titanium is a relatively new construction material that has only been produced on an industrial scale for approximately 60 years. Titanium and its alloys find many applications due to the high strength to density ratio and it excellent corrosion resistance. However, it is relatively expensive to produce titanium from its ore, so the use of titanium is mainly limited to high-tech industries or luxury articles. Titanium Low Cost Beta (Ti-LCB) is a titanium alloy that is used for several different applications, for example in the automotive industry. In general, the mechanical properties of metals and alloys are to a large extent influenced by the microstructure and the texture, which are sometimes combined in the term microtexture. Research on the microtexture enables us to understand and describe the mechanisms that play a role during the phase transformations. With this knowledge, it will be possible to develop microtextures that results in improved characteristics of the metal. The microtexture of two types of α phase plates are investigated during this project by Electron Backscatter Diffraction (EBSD). Primary α plates that are formed during diffusion controlled transformations and secondary α plates that are formed by a displacive mechanism. There has already been much research on both types of α plates on various aspects. It was found that the formation mechanism of secondary α has some analogy with bainite in steel. The general characteristics of titanium are given in chapter 2, and the differences between primary and secondary α plates are described in chapter 3. The theory of grain boundaries is given in chapter 4, in which both the grain boundary structure and the grain boundary energy are described. In chapter 5, some different nucleation theories are given, which are divided in classical nucleation theory and in non-classical nucleation theory. Variant selection is described at the end of the chapter. The theoretical background of microtexture analysis is given in chapter 6. In this chapter the EBSD technique is described and several data processing methods are discussed. In chapter 7 the experimental procedures are given. Various aspects of the nucleation of primary and secondary α plates are analysed with EBSD analysing techniques, of which the results are shown in chapter 8, and are discussed in chapter 9. The results of the EBSD measurements on the α plates are compared to three different orientation relations given in chapter 3. The next analysis considers the influence of β grain boundary misorientation on the α plate nucleation. Also the characteristics of the possible α-α grain boundaries are analysed and compared with random situations. Finally, the variant selection during α plate nucleation is described and analysed according to theories from literature. Finally the conclusions and the recommendations are given in chapter 10 and 11. 1

13 General characteristics of titanium 2. General characteristics of titanium Titanium is the fourth most abundant structural metal in nature, only exceeded by aluminium, iron and magnesium and it ranks ahead of copper, lead and zinc. Of all elements in the earth s crust, titanium is the ninth most available; about 0.57% of the earth s crust consists of titanium. Titanium has atomic number 22, with an atomic mass of 47.9 and is a transition metal element, belonging to group IVa. The melting point of titanium is 1668 C. In this chapter, some of the important general characteristics and possible applications of titanium alloys will be described Titanium production Titanium is chemically very reactive and forms strong ionic bonds. This makes it very difficult to isolate pure titanium from the oxides in the ore. The first who developed a method to produce highly pure titanium on an industrial scale, was Justin Kroll in 1932 in Luxemburg. After many years of research and improving his method, he was the first to introduce commercial titanium on the market in His process, the Kroll-process, is still the most used process in the world to commercially produce titanium. This process is however very labour intensive and titanium can only be produced in batches of around 10 tons. It takes several days for one batch. In this process, titanium is reduced from rutile (TiO 2 ) to titanium in the following two steps [1]: TiO 2 + 2Cl 2 + C TiCl 4 + CO 2 (chlorination) (eq. 2.1) TiCl 4 + 2Mg Ti + 2MgCl 2 (reduction) (eq. 2.2) After the reduction, the titanium has to be cleaned from salts, which is done in a vacuum distillation process. Then, the titanium is further processed to produce titanium sponge. As stated before, titanium has a high tendency to form ionic bonds. In most environments, a TiO 2 surface film is very quickly formed. This film is a passive protective layer that prevents the metal for further oxidising. This passive film forms in many oxidising environments, such as salt or nitric solutions. In some environments, however, the passive layer can dissolve, which occurs in reducing environments like sulphuric, hydrochloric and phosphoric acids. Good corrosion resistance combined with very good mechanical properties, makes titanium very useful in products that are used in extreme environments. Titanium finds its use in the aerospace industry, the chemical industry, but also in luxury objects like golf clubs and frames for glasses [1] Mechanical properties Titanium is often used for its high strength combined with its low weight and the good behaviour at high temperatures. Depending on the alloy and the microstructure, strengths can be obtained that are comparable to that of most construction steels. On the other hand, the density of titanium is about 56% of that of steel. A comparison of some important properties of titanium and steel are given in table 2.1. Of all construction metals, titanium has the highest strength to density ratio. Commercially pure titanium can have a tensile strength around 550 MPa, whereas high strength titanium alloys can reach tensile strengths of 1200 MPa [1]. 2

14 Investigation of Titanium α Plates by EBSD Analysis Table 2.1. Important properties of titanium compared with steel. Property Titanium Steel Density 4.5 g/cm g/cm 3 Modulus 116 GPa 230 GPa Strength 550 MPa 760 MPa Corrosion Resistance High Low-High The mechanical properties of titanium alloys can be adjusted by adding different amounts of alloying elements. The alloying elements can be divided into two groups: the interstitially soluble elements and the substitutional elements. Increasing the amount of interstitially soluble elements like carbon, oxygen, hydrogen and nitrogen increases the strength of the alloys and will ultimately have a strong embrittling effect on the titanium. The adding of oxygen and nitrogen causes significantly more hardening of the titanium than adding the same amount of carbon to the alloy [2]. However, by adding interstitial elements like carbon, oxygen and hydrogen, the ductility will decrease with increased concentration. Therefore, in most titanium alloys, the content of interstitial elements is kept as low as possible. For hydrogen for example, concentrations of more than 200 ppm are not tolerated. To increase the strength without significantly decreasing the ductility, substitutional elements can be added to the titanium. Important alloying elements in this class are manganese, aluminium, chromium, tin, iron, vanadium, and molybdenum. These alloying elements increase the strength of the titanium, with only a small loss in ductility and toughness. The alloying elements also affect the crystal structure. Some elements tend to stabilize the α phase, others the β phase and some elements are neutral Crystal structure Titanium can exist in several distinct crystal structures, of which the most important are the α phase and the β phase. Under normal conditions at room temperature, commercial pure titanium consists fully of α phase. The crystal structure of the α phase is a hexagonal closedpacked (hcp) structure, which transforms to the β phase when the pure titanium is heated to temperatures above C. Several experiments at room temperature on different specimens resulted in the following values of the lattice parameters of the α phase unit cell a=2.95 Å c=4.68 Å c/a=1.59 The β phase has a body-centred cubic (bcc) crystal structure. The lattice parameter of β phase titanium at 900 C was found to be 3.29 Å. The densities of the two titanium phases are 4.50 g/cm 3 for the α phase at room temperature and 4.35 g/cm 3 for the β phase at 900 C [2]. The α phase is closed-packed, whereas the β phase is not. This difference in density makes the β to α transformation detectable by dilatometry. By adding β stabilizers to the titanium, a dual phase microstructure of both α and β phase can be obtained at room temperature. These alloys are called α+β alloys. Commonly used β stabilizers are Cr, Fe, Mo and V. Common α stabilizers are Al, O, N and C. The β stabilizers can be divided into two classes, the β isomorphous elements and the β eutectoid elements (Figure 2.1). 3

15 General characteristics of titanium α stabilizer (Al,O,N,C) β eutectoid (Fe,Mn,Cr,Co,Cu,Si,H) β isomorphous (V,Mo,Nb,Ta) Figure 2.1. Effects of the alloying elements on the equilibrium Ti phase diagram [3]. Normally, β to α transitions can occur in two different ways. When the cooling is fast, displacive transformations occur and when the cooling is sufficiently slow, nucleation and diffusional growth occurs. The displacive transformation involves a cooperative movement of atoms, resulting in a transformation of the bcc into the hcp lattice. The quenching of β phase titanium can result in two different martensitic crystal structures: the hexagonal martensite α and the orthorhombic martensite α. The most common type is the α martensite, which can occur in two morphologies: massive or lath-shaped martensite and acicular martensite. The latter occurs as a mixture of individual α plates, each having a different variant of the Burgers orientation relation (Chapter 3) [1] Deformation mechanisms At room temperature, α titanium deforms both by slip and twinning. In the α phase, several slip modes are present. The slip systems of the hcp α are given in table 2.2. The predominant 0002 both with an a- slip mode for α titanium is { 10 10} 1120, followed by { } and ( ) type Burgers vector. The a-type Burgers vector corresponds with an edge dislocation and the c-type with a screw dislocation. The slip systems that are observed for β titanium are {112} and {123}, all with the Burgers vector of ½<111>. Table 2.2. Slip systems in α phase titanium [1]. Slip system Burgers vector type Slip direction Slip plane Number of slip systems total 1 a 1120 { 0002 } a 1120 { } a 1120 { } c + a 1123 { } 12 5 independent 4

16 Investigation of Titanium α Plates by EBSD Analysis There are many twinning modes observed in titanium. Two twin systems of α phase titanium are given in table 2.3. Twinning was also observed on the high order pyramidal planes 1124, but only in single crystal flakes where the direction of the applied stress { 1123} and { } was parallel to the principal crystal axis [2]. Twinning can be suppressed by increasing the amount of soluble elements like oxygen and aluminium. Due to this suppression, twinning plays only a role as a deformation mode in highly pure titanium [1]. Table 2.3. Twinning systems in α phase titanium [1]. Twinning plane (First undeformed plane) (K 1 ) Twinning shear direction Second undeformed plane (K 2 ) Direction of intersection of plane of shear with K 2 { 1121 } 1126 { 0001 } 1120 { } { 1122 } 1123 { 1124 } 2243 { } Plane of shear perpendicular to K 1 and K 2 Magnitude of twinning shear

17 Characteristics of α plates in titanium alloys 3. Characteristics of α plates in titanium alloys 3.1. Introduction In several investigations on titanium alloys [4, 5], it was found that when some Ti alloys were aged at different temperatures and times, two types of α phase plates were formed. The α phase that forms at relatively high temperatures is designated in this report as the primary α plates. The other α phase that was observed has a plate like shape, which differs significantly from the primary α phase crystals. These plates are called secondary α plates. The darker etched, thinner secondary plates are formed at lower temperatures (T<600 C for Ti-10.94Cr alloys) and the white etched, thicker primary plates are formed at higher temperatures, therefore the names primary and secondary are applied referring to their order of appearance during cooling. Both types of plates can also form together in the same specimen, when annealed at intermediate temperatures [6]. Primary α plates are also sometimes referred to as normal α plates, type 2 α plates or Widmannstätten α plates [4, 6-8]. Selected area diffraction (SAD) patterns of different specimens have confirmed that the two types of α phase both have an hcp crystal structure [4]. It is useful to summarise some differences in experimental observations for the two types of α phase before discussing the possible origin and mechanism of formation of the two constituents. Much research was done to investigate the two different α phases, which was done with several techniques focusing on the various differences between the primary α plates and the more perfectly shaped secondary α plates. Differences were found in properties such as morphology, etching appearance, growth kinetics, composition and surface relief effects, which are described in the following sections Morphology The first observation of a difference between the two types of α plates is their appearance in optical and electron microscopy [6]. The morphological identification of the two types of α plates is very similar in the various studies. All papers describe the differences in etching appearance, thickness and morphological perfection as the most important differences between the two types of plates. An obvious difference between the primary α phase plates and the secondary α plates is the higher degree of perfection in the secondary plates. It was also observed in several titanium alloys that the secondary plates have a smaller thickness to length ratio compared to the primary α plates as is seen in figures 3.1 to 3.3, which shows primary α plates, secondary α plates at a β-β grain boundary and intragranular secondary α plates. At annealing temperatures in the range of 400 C to 650 C, the dark etched slender α plates are formed at grain boundaries and intragranularly in the α phase. These plates appear dark and are therefore sometimes called black plates [4, 8]. At higher aging temperatures or longer annealing times, the white etched, more irregular primary α plates are formed [4, 8-10]. It is also observed that after longer aging times at the lower temperatures, the secondary α plates slowly whiten and are transformed into by primary α plates [8]. It has been proposed that the secondary plates are actually the same as the primary α plates, but are so thin that they appear as secondary α plates [11]. It is noted that the appearance of the secondary plates in titanium varies little with the alloy that is used. In ternary Ti-14Mo- 6Al, the α phase forms as flat platelets [7]. Initially, the primary α plates are only slightly broader than the secondary plates, but after longer aging times, the primary α plates have 6

18 Investigation of Titanium α Plates by EBSD Analysis grown much wider, whereas the secondary plates remain slender [5]. Investigations of Ti-Mo alloys report that primary α plates grow in a Widmanstätenstructure-like manner into the β grains out of the β grain boundaries and dislocations at β subgrain boundaries [9]. Another difference in the morphology of the plates is that the average spacing between the growth ledges on the broad faces of secondary plates is remarkably greater than that on the broad faces of primary α plates [10]. Figure 3.1. Micrograph of primary α plates, reacted for 15 days at 650 C in a Ti-7.22 at.% Cr alloy, magnification ~500 (a), and reacted for 3 days at 650 C, magnification ~1000 (b) [6]. Figure 3.2. Micrograph of secondary α plates, reacted for 3 minute at 600 C in a Ti-7.22at% Cr alloy, magnification ~1000 (a) and secondary α plates reacted for 1 min at 575 C, magnification ~500 (b) [6]. Figure 3.3. Micrograph of intragranular sheaves of secondary α plates in a Ti-7.22at%Cr alloy, reacted for 3 min at 600 C, magnification ~500 (a), magnification ~1000 (b) and intragranular secondary α plates reacted for 10 sec at 575 C, magnification ~500 [6]. 7

19 Characteristics of α plates in titanium alloys 3.3. Growth directions of α plates Several investigations were done to both experimentally observe and theoretically calculate the growth directions of α plates in titanium alloys [4, 12, 13]. It was found that the strain energy is dependent on the orientation of precipitates in the parent matrix due to the effect of shape strains [14]. This results in preferred growth directions of the product phase crystals during phase transformations. Early observations reported that the habit plane for α plates in the bcc matrix was found to be (334) [15]. Shibata and Ono [12] calculated the habit plane considering the transformation strain energy using Eshelby s theory [16], the habit plane for martensitic α plates in titanium alloys is also approximately (334). The strain energy in the system minimises when hcp precipitates are orientated along this habit plane. The calculated habit planes were found to be almost identical to the observed ones. The authors also considered other habit planes that were less frequently observed, but found that all observed habit planes only deviated a few degrees from the calculated low energy regions [12]. Observations on the black plates showed that there is a difference of the habit plane between the normal α phase plates, which have a (334) habit plane and the black plate α phase. Menon and Aaronson [4] have found experimentally that the secondary α plates have a habit plane of (110). Further investigations on the growth mechanism of secondary α plates showed that the interface of the secondary α plates consists of a ledge structure. These ledges were predicted to be formed because of a lowering of the interface energy between the bcc matrix and the hcp plates. The formation of ledges along the interface results in a higher coherency between the α plates and the β matrix [4]. During observations of the interface structure, two different types of ledges were found: structural ledges and misfit ledges. The structural ledges reduce the interfacial energy between the two faces by causing more coherent areas at the interface, whereas the misfit ledges compensate the misfit of the α plates by creating extra half planes perpendicular to the (0001) plane [17]. The invariant line for the α plates is deduced by connecting the corners of the (10 10) //(112) ledges (Figure 3.4). It was found that this invariant line is [335], which is the growth direction of titanium α plates [17, 18]. The lattices of the two phases have the best fit along this invariant line. Figure 3.4. Interfacial structure with the invariant line of the α-β grain boundary [18]. 8

20 Investigation of Titanium α Plates by EBSD Analysis 3.4. Growth kinetics Investigations on the lengthening behaviour of the two different types of α plates can give some information about the transformation mechanism. The growth rate of primary α plates and secondary α plates was measured as a function of isothermal reaction temperature in Ti6.62at.% Cr alloys [4] with optical microscopy. This was done by measuring the length of the longest plate. For each temperature a constant growth rate was found for both primary and secondary α plates. When the growth rate is plotted as a function of reaction temperature, it can be clearly seen that there is a discontinuity between the growth rate at lower aging temperatures (575 C-600 C) and at higher aging temperatures (650 C-725 C). This growth rate discontinuity in the temperature range of 600 C-650 C is consistent with the observed morphological transition temperature of approximately 617 C between the secondary α plates and primary α plates in this alloy. Equivalent growth rate discontinuities have also been observed in other Ti alloys [4]. Other research has also confirmed that secondary α plates lengthen more rapidly than primary α plates [17]. Three explanations have been proposed to explain the difference between the growth rates of the primary and secondary α plates [4, 5, 17]. The first explanation, in 1986, was that the two different α plates are the result of the existence of a miscibility gap in the phase diagram of binary titanium alloys [4]. The primary α plates are a Ti rich α phase, in which secondary α plates are of Ti lean α phase. This difference between the two α phases will result in a difference in the driving force and hence the growth rate. Furuhara, Howe and Aaronson, [17] attributed the large difference in growth rate between the two α plates to a difference in ledge structure rather than the broad faces of the plates. Transmission Electron Microscopy (TEM) analysis on a Ti-6.62at%Cr alloy has shown that there are some differences between the two α plates in the boundary structure [17]. Both types of plates contain irregular and non-uniformly spaced growth ledges, and uniformly and finely spaced ledges on the growth terraces. The authors assumed that the growth rate difference is caused by the fact that the inter-growth ledge spacing of primary α plates is larger than for secondary α plates. However, this hypothesis was not experimentally confirmed, since it was not possible to examine the ledges of the secondary α plates by TEM [17]. This explanation was done by the same author as the previous one and proposed this ledge mechanism as an addition to his previous work. A recent paper attributes the difference in growth rates to different transformations types for the primary and secondary α plates [5]. The primary α plates are formed by a diffusional transformation process and the secondary α plates by a displacive process. The formation of secondary α plates is not controlled by diffusion, but the transformation is governed by shear stresses. It was argued that some partitioning of alloying elements takes place after the formation of the α plates Phase compositions A key factor in the controversy between the diffusional and displacive formation mechanism of secondary α plates is the observation of the chemical composition of the phases at different stages of the transformations. According to the displacive formalism, supersaturated secondary α plates are formed inheriting the composition of the β phase. The other view on the formation of secondary α plates, is that the transformation is diffusion controlled, at which long-range diffusion takes place during the transformation. Measurements on the phase compositions can provide information about the diffusion of alloying elements during the 9

21 Characteristics of α plates in titanium alloys transformation. Only few investigations [5, 19] have been done on the composition of primary and secondary α plates in titanium. Van Bohemen et al. have measured the local chemical composition of both α plates in a Ti-4.5Fe-6.8Mo-1.5Al alloy with Electron Probe X-ray Micro-Analysis (EPMA) [5]. The partitioning of the alloying elements Fe, Mo and Al was measured by a line scan across a primary α plate after 100,000 s at 735 C and a secondary α plate after 1080 s at 615 C. The paths of the scans are indicated in figure 3.5. This paper denoted the secondary α plates as bainitic α plates. This link to bainite in steel is made because of the many similarities in the formation mechanism and morphology between the two microstructures. The concentration of alloying elements in the primary plate differed strongly from the surrounding phase (Figure 3.6b). In the secondary plates, only a small drop of the iron concentration was found in the plate. The composition of the secondary plate was found to differ only a little from the surrounding phase (Figure 3.6a). It was concluded that the primary α plates are formed by a partitioning process, whereas the secondary α plates are formed by a partitionless process followed by partitioning. After the relatively short time of transformation the concentration of the alloying elements are far from equilibrium after the formation of the plates (Figure 3.6a). (a) Figure 3.5. Optical micrographs showing secondary α plates formed at 615 C after 1080 s (a) and primary α plates formed at 735 C after 100,000 s (b) [5]. Figure 3.6. The Fe, Mo and Al concentration profiles across several secondary α plates formed at 615 C (a) and across one primary α plate formed at 735 C (b)[5]. 10

22 Investigation of Titanium α Plates by EBSD Analysis Enomoto and Fujita investigated the composition of the two types of α plates formed in 5.1wt%Fe and 5.4wt%V titanium alloys [19]. The authors had estimated the T0 values from the binary phase diagrams calculated by Murray [20]. The authors of the report described in their introduction the differences between the two types of α plates that can be found in titanium. However, it is unclear which type of plates they have done research on. For the Ti- Fe alloy, they gave a TEM micrograph and the annealing characteristics, and for the Ti-V they only gave the annealing characteristics without further explanation. The distribution of alloying elements was measured in a Scanning Transmission Electron Microscope (STEM) equipped with an Energy Dispersive X-ray (EDX) analyser. The Ti-Fe alloy was heated for 3 minutes at 730 C after homogenisation at 950 C. The Fe concentration was measured along two different paths. The Fe concentrations were found to be uniform within the primary α plates, but two points along the measured path were supersaturated with Fe. One point was actually on the plate tip and the other was approximately 0.3 µm behind the tip. However, half of the contamination spot was seen to be out of the plate and therefore much of the X-ray counts are likely to come from the matrix phase. The Ti-V alloy was treated at 680 C for 30 s after homogenisation. It was seen that during the early reaction stage, the concentration of V was uniform in the secondary α plate and that it was high in the α matrix near the plate, but dropped to values near the bulk concentration at distances further from the plate. When the concentration of the alloying elements was isothermally measured for different times, it was seen that the Fe and V concentration stayed approximately constant in the α plates and that the concentration in the matrix increased with time [19]. The authors observed that in the Ti-5.4V alloy, at 500 C no solute partitioning did take place. After the formation, no retained β phase was observed and it is likely that the whole matrix transformed at once by a martensitic mechanism. The undercooling was apparently large enough for the whole parent phase to transform displacively. The authors concluded that the growth of the plates was diffusion controlled with two different arguments. They observed that no spot in the α plates had a composition that was halfway to the equilibrium compositions of α and β; the compositions of the plates were in equilibrium immediately after formation, and high concentrations of alloying elements were found. Those at the plate boundaries could be related to count from the matrix. The other argument was that the formation of the plates was well above T0 (200 C to 400 C), and therefore the α plates had to be formed by a partitioning mechanism Surface relief effects Many investigations have been done on the appearance of surface relief effects of the two types of α plates in titanium [21-24]. The surface relief effects that were observed in titanium are of the invariant plane strain (IPS) type, the tent-shaped type or the inverse tent-shaped type. If only one crystal lattice at the interface moves, IPS surface relief effects will form, but IPS relieves can also form when the lattices move at both sides. Investigations on primary and secondary α plates in a Ti-7.15wt%Cr alloy showed that approximately 80% of all surface relieves were tent-shaped. The remainder were of the IPS type. The primary or normal α plates were formed by holding the alloy at 687 C, the secondary plates were formed by holding the alloy at 613 C. The surface relief effects were observed by optical microscopy and measured by differential interference. Research on the surface relief of α plates in a Ti- 9.07wt%Mo alloy gave different results [21]. Primary α plates formed at 750 C, were found to have also tent-shaped relief effects, but with an angle of 11 to 13, which is much higher than that found in the Ti-Cr alloy. However, also an inverse tent-shaped relief was observed for 11

23 Characteristics of α plates in titanium alloys primary α plates formed at the same temperature. During the research on Ti-Mo alloys, IPS surface relieves were hardly observed to occur with the formation of primary α plates [21], in which the surface relief effects of the α secondary plates in the same alloy were mostly of the IPS type [23]. Investigations of surface relief effects of primary α plates in a Ti-9.07Mo alloy showed that many tilt angles of the tent-shaped relief effects were larger than the maximum tilt angle observed with the IPS relief of secondary α plates. It is also observed that the tilt angles of inverse tent-shaped relieves are smaller than those of tent-shaped relieves. This is probably because the constraint to the shape deformation is more severe in the inverse tentshaped relief [23]. Hirth et al. [24] proposed that the formation of surface relief effects is due to the movements of structural ledges at the interface of the plates. When the α plates grow, structural ledges move towards the surface on both the top and bottom of the plate. Calculations on this mechanism predict tilt angles of for tent-shaped relieves and tilt angles of 6 for IPS relieves. The tilt angles can differ from the calculations due to various effects like the inclination of terrace planes, relaxations of strains in the structural ledges can influence the measured tilt angle of the surface relief effects Texture Introduction The research on orientation relationships (OR) is an essential part of the metallurgical science. With the knowledge of the preferred orientations of a product crystal relative to the parent matrix more insight on the nucleation and growth kinetics can be obtained, also the shape of the nucleus can be predicted with the knowledge of the orientation relationships. The orientation relation can be linked to the most favourable growth directions of the crystals. Research on orientation relationships showed that usually the orientations of formed crystals are not random. This indicates that there is a preferred orientation of the formed crystal that probably fits best in the matrix crystal. This will reduce the energy of the product phase, which will cause lower nucleation activation energy. Therefore, when the nucleation in alloys has to be investigated, it is useful to find the orientation relations and the preferred variants. Knowledge of the orientation relationships needs to be incorporated in order to explain observed microstructures [25]. The orientation relationships can provide information about the transformation mechanism, but it is not possible to distinguish between diffusional and shear growth purely on the basis of orientation relationships and habit plane information, because the orientation relation can also be affected by other characteristics [17] such as stresses in the matrix phase or deformations applied to the samples Orientation relationships in Ti alloys In addition to the research on the morphology of the two types of α plates, many investigations were done to distinguish the two types of plates in terms of orientation relationships between the parent β phase and the formed α phase. Several papers have described the orientation relationship between the parent β phase and the product α phase in titanium alloys, in which it was found that the Burgers OR is the predominant OR [7, 10, 17]. There are three other orientation relationships for bcc-hcp transformations that are sometimes described in literature [9]. These are the Potter OR, the Pitsch-Schrader OR and the Rong- 12

24 Investigation of Titanium α Plates by EBSD Analysis Dunlop OR. The parallel planes and directions of the bcc and hcp phase for the different orientation relationships are: Burgers OR: ( ) ( ) , Potter OR: hcp bcc hcp bcc ( ) ( ) ( ) ( ) from 011, , hcp bcc bcc Pitsch-Schrader OR: ( 0001) ( 011 ), 1120 [ 100] hcp bcc hcp bcc Rong-Dunlop OR: hcp hcp bcc ( 0001) ( 021 ),( 1 100) ( 0 12 ), 1120 [ 100] hcp bcc hcp bcc hcp bcc During aging experiments with Ti-14Mo-6Al and Ti-11.6Mo alloys at temperatures in the range of 400 C to 650 C, it was found that secondary α plates have a slightly different orientation than primary α plates [7]. TEM investigations on the alloys confirmed that the secondary α plates are consistent with the Burgers orientation relationship. In these experiments, intense (10 11) reflections were found that occur between the (110) and (200) reflections, which indicated that there is a deviation from a few degrees from the Potter OR and the misorientation is close to the Burgers OR. The plates with these orientations were referred as type 1 α phase, which is a synonym of secondary α. In the other α phase with different etching appearance, the (10 11) reflections were absent between the particular β reflections during X-ray diffraction analysis. The α phase with this orientation was consequently called type 2 α phase, which is a synonym of primary α. It was found that the type 2 or primary α phases do not follow the Burgers relationship, but are in a twin orientation with respect to the Burgers OR. Inspection of all Burgers oriented α phase reflections showed that despite some similarities between the diffraction patterns, there are significant differences between primary and secondary α plates. Rhodes attributes this difference to the presence of internal striations in the secondary α phase. The X-ray diffraction results have shown that the shift in the diffraction peaks between primary and secondary α plates cannot be attributed to a change in composition of the α or β phase. There was no diffraction shift detected between Ti- 0.1wt%Mo and Ti-0.6wt%Mo alloys that could be caused by a change in lattice parameter [7]. Further investigations with electron microscopy and X-ray diffraction on titanium alloys confirmed that the orientation relations between the secondary α plates and the parent β phase follow the Burgers orientation relationship, but have a misfit of a few degrees [9]. This can indicate that the transformation from the β phase matrix to the secondary α phase plates is accompanied with some strains. Investigation of Ti-40wt%Mo alloys gave somewhat different results. When the alloy was aged at a temperature of 500 C, secondary α plates formed in the parent β matrix. The orientation relation of the α phase with the β matrix was investigated and compared to several theoretical orientation relationships. The relationship found with TEM showed that the (01 11) and (011) were inclined to each other by 13. This angle was then 13

25 Characteristics of α plates in titanium alloys compared to the calculated angle for the Burgers relationship of 9.3, the Pitsch-Schrader of 4 and the Potter relationship of 11 [9]. This indicates that secondary α plates in this alloy follow the Potter orientation relationship better than other calculated orientation relations to the β matrix. The small difference of the orientations of the α plates can be explained by extra strains that occur in the microstructure due to the higher volume fraction of the β stabilizing element Mo. The main conclusion from the various studies of the orientation relationships of the two types of α plates is that the secondary α plates are more closely oriented to the β phase regarding to the Burgers orientation relationship [8, 9, 27]. However, Aaronson et al. [3] conclude that in Ti-Cr alloys both α plates obey the Burgers orientation relationship with a comparable misfit. This is consistent with the work of Furuhara et al. [17], which describes the transformations of all α phases as a Burgers orientation relationship transformations. Due to the preferred orientation relations during phase transformations, only a fixed number of relative orientations between the parent and the product phase can be formed. For the Burgers OR, there are 12 different variants possible that relate the β phase to the product α phase. The possible directions and planes for the Burgers OR in a bcc crystal are given in table 3.1 and figure 3.7. Table 3.1. The 12 variants of the Burgers orientation relationship between the β matrix and the product α phase [14]. Variants Orientation Relationship A (1 10) //(0001), [111]//[1120] B (10 1) //(0001), [111]//[1120] C (01 1) //(0001), [111]//[1120] D (110) //(0001), [111]//[1120] E (101) //(0001), [111]//[1120] F (01 1) //(0001), [111]//[1120] G (110) //(0001), [1 11]//[1120] H (10 1) //(0001), [1 11]//[1120] I (011) //(0001), [1 11]//[1120] J (1 10) //(0001), [11 1]//[1120] K (101) //(0001), [11 1]//[1120] L (011) //(0001), [11 1]//[1120] 14

26 Investigation of Titanium α Plates by EBSD Analysis Figure 3.7. The possible planes and directions for the Burgers OR in a bcc crystal which are parallel to the (0001) plane and [1120] direction of the hcp crystal [26] α-α misorientations When the possible crystal orientations of the α plates within a single β grain that obey the Burgers OR are compared with each other, 6 different axis/angle pairs are possible that describe the relative orientation of the crystal. The theory of axis/angle pairs will be described in chapter 6. The axis/angle pairs and their frequencies in a random situation are listed by Wang et al.[14] and are also confirmed by calculations (table 3.2). Some axis/angle pairs occur more often than others when all α orientations are present. The occurrence of the different α-α grain boundaries were calculated and plotted in figure 3.8. In a random situation in which the misorientations between all 12 possible α are calculated, 12 misorientations are of type 1, 24 of type 2, 48 of type 3, 24 of type 4, 24 of type 5 and 12 of type 6, from a total of 144 misorientations. It can be seen that most grain boundaries are high-angle grain boundaries, only the grain boundary between two α plates that share a common [0001] pole are low-angle grain boundaries. 15

27 Characteristics of α plates in titanium alloys Table 3.2. Possible axis/angle pairs between two α plates formed in the same β grain according to the Burgers OR [14]. α-α grain Misorientation angle Axis of rotation boundary type Occurence Misorientation (degrees) Figure 3.8. Occurrences of the misorientation angles between all 12 possible α variants calculated from the Burgers OR where the total number of α-α misorientations is

28 Investigation of Titanium α Plates by EBSD Analysis 4. Grain boundary theory 4.1. Grain boundary structure When phase transformations are considered, it is found that grain boundaries play an important role in the nucleation rate and also in the crystal orientations of the nuclei. In this chapter the orientation of grain boundaries and the grain boundary energy will be described. At the boundary between two adjacent grains, there will always be some form of misorientation between the lattices of the grains. The grain boundaries can be divided into two classes: low-angle grain boundaries and high-angle grain boundaries. Grain boundaries are usually called high-angle grain boundaries when the two lattices have a misorientation angle θ larger than 10 to 15 and grain boundaries with a smaller misfit are therefore called low-angle grain boundaries. The atoms at high-angle grain boundaries are considered to have a high mobility, whereas the atoms at low-angle boundaries have a low mobility. Furthermore, there are two special types of grain boundaries: the tilt grain boundary and the twist grain boundary. In a tilt grain boundary, the axis of rotation of the crystal orientations is parallel to the plane of the boundary and in twist grain boundaries the rotation axis is perpendicular to the boundary plane (Figure 4.1) [28]. In a sample, usually there are several different values of the misorientation angle between all grains. MacKenzie has formulated the random distribution function for the misorientation for cubic-cubic systems, which is given in figure 4.2 [29]. Figure 4.1. The orientation between two crystals in a tilt grain boundary (a) and a twist grain boundary (b) [28]. 17

29 Grain boundary theory Figure 4.2. The random distribution of the misorientation of grain boundaries in cubic-cubic systems by MacKenzie[29]. For a proper understanding of the grain boundary structure, it is important to get a full description of the grain boundary. A grain boundary between two crystals can be determined by five degrees of freedom: the rotation axis, given by the direction of the unit vector ˆn, the orientation angle φ and the orientation of the boundary plane in either of the two crystals, consisting of a set of three perpendicular axes (Figure 4.3) [3,4]. This means that the interfacial energy of grain boundaries is also dependent on these parameters. Calculating the interfacial energy considering all five parameters is very complex. For simplification usually the rotation axis and the inclination of the boundary plane are held constant, which results in an interfacial energy that is only dependent on the misorientation of the grains. However, it is found in this thesis that this simplified description is not sufficient. The inclination of the boundary plane is difficult to determine experimentally, which complicates the analysis of EBSD results as will be discussed later. Figure 4.3. Definitions of the angle of misorientation θ and the grain boundary orientation angle φ [30]. There are three classes in which interfaces of grain boundaries between solids can be divided: coherent, semicoherent and incoherent interfaces. An interface is called coherent when the crystals that are joined match perfectly at the interface. In coherent interfaces, the crystal lattices are continuous across the interface. A coherent interface is mostly achieved when the interfacial plane has the same configuration for both phases. It is also possible to get a 18

30 Investigation of Titanium α Plates by EBSD Analysis coherent interface when the atomic spacing is different for the two phases, i.e. when there is a misfit at the interface. In this case, the lattice will be distorted around the interface. These distortion strains are called coherency strains [28]. When the coherency strains increase, the interface energy will also increase. Therefore, when the misfit is sufficiently large, it will be energetically favourable for semicoherent interfaces to be formed. In semicoherent interfaces the difference in atomic spacing is corrected by misfit dislocations, see figure 4.4, in which the spacing between the dislocations is given by D. When the misfit between the crystals increases, the regions of poor fit overlap and the interface is called incoherent. An interface is called incoherent when the patterns of atoms at either side of the interface are very different, or when the interatomic spacings differ by more than 25% when the atom patterns are similar at the interface [28]. During phase transformations, the incoherent interfaces are highly mobile and the coherent interfaces are immobile. The product phase structures will therefore grow along the coherent interface and the incoherent interface is then the transformation front. Figure 4.4. Example of a semicoherent interface between phase α and phase β. d α and d β are the unstressed interplanar spaces of the two phases [28] Low-angle grain boundaries Low-angle grain boundaries can be easily described with two-dimensional dislocation networks; which is called the dislocation model of a grain boundary. An example of a lowangle grain boundary structure for a tilt grain boundary is given in figure 4.5. In low-angle tilt boundaries, the boundary is considered as an array of parallel edge dislocations, whereas a low-angle twist grain boundary is considered as a set of screw dislocations [28]. The assumption that a low-angle grain boundary is either a tilt or a twist grain boundary is a simplified case. In general, grain boundaries consist of a mixture of tilt and twist grain boundaries and therefore contain sets of both edge and screw dislocations. For low-angle grain boundaries, the grain boundary energy can be calculated by simply adding up the energies of the dislocations in the grain boundary, which depends on the spacing of the dislocations in the boundary. In the case of figure 4.5, the spacing between dislocations, D, can be calculated by 19

31 Grain boundary theory b D = (eq. 4.1) sinθ in which b is the Burgers vector of the dislocations and θ is the misorientation angle between the two crystals. Equation 4.1 can be simplified to D b/θ when the misorientation θ is small [28]. When the misfit is low, the spacing between the dislocations at the grain boundary is large enough for the dislocations to be considered separately. However, at large misfits, the calculated spacing between the dislocations will eventually become smaller, and the dislocation model can no longer be applied. It is observed that in this region with large misorientation angles, the grain boundary energy is almost independent of the misorientation (Figure 4.6). Figure 4.5. A low-angle tilt grain boundary with misorientation angle θ and dislocation spacing D=b/θ[28]. Figure 4.6. Schematic dependence of the misorientation on the grain boundary energy for low-angle and high-angle grain boundaries [28]. 20

32 Investigation of Titanium α Plates by EBSD Analysis 4.3. High-angle grain boundaries A commonly used approach to describe the structure of a high-angle grain boundary is the coincidence site lattice (CSL) model. The CSL describes grain boundaries to have an ordered structure when the orientation of the two adjacent grains is such that the grain boundary is a plane with a high density of coincident sites [31]. The CSL approach has already been used for several decades and it has proven itself to be useful in grain boundary research. A CSL is produced as follows: two crystals are considered to fill up all space. At some relative orientations several lattice points can be found that belong to both crystals and these common lattice points together form a lattice that can be considered as the coincidence site lattice. An example of a CSL is given in figure 4.7 in which the white circles indicate the two adjacent grains and the CSL is indicated by the black circles. The fraction of atoms that are part of the CSL is given by 1/Σ. The parameter Σ is the reciprocal coincidence site lattice density. The simplest CSL grain boundary is a twin boundary, which has a reciprocal coincidence site lattice density Σ of 3 [25]. Figure 4.7. A coincidence site lattice of two crystals with a misorientation angle of 22, indicated by the black circles [32]. The relative number of coincidence sites is given by the parameter Σ. This is simply the group of atom positions that periodically coincide to that of the crystal lattice. The value of Σ in figure 4.7 is 7. The value of Σ can be calculated with the following equation, derived by Ranganathan [31] 2 2 Σ = x + Ny (eq. 4.2) in which N=h 2 +k 2 +l 2 and x and y are integers and [hkl] is rotation axis. When the value of Σ is even, it must be divided by two until an uneven value is acquired. In practice, only relatively low values of Σ are of practical interest, because when the value of Σ is large, the number of coincidence atoms compared to the number of atoms in the lattice is negligible. The angle of misorientation θ can then be calculated by 21

33 Grain boundary theory 1 y θ = 2 tan N (eq. 4.3) x In order to compare the misorientation between measured grain boundaries with CSL misorientations, it is convenient to describe the coincidence boundaries by Euler angles instead of axis/angle pairs. The definition of the Euler angles will be described in chapter 6. The Euler angles together with the axis/angle pairs for the CSL values of Σ up to 35 are given in table 4.1. The maximum misorientation θ between the theoretical axis/angle pairs of a CSL boundary and the axis/angle pair of a measured grain boundary is given by Brandon s criterion by using a constant θ 0 which has a value of approximately 15 for cubic crystals. For hexagonal crystals the value of θ 0 is often chosen to be 10 [33]. The maximum deviation from the theoretical CSL, θ is then given by θ 0 θ = Σ (eq. 4.4) Table 4.1. CSL values up to 35b represented in axis/angle pairs and Euler angles for cubic-cubic systems [34]. Σ θ( ) uvw φ 1 Φ φ a b a b a b a b a b a b a b a b a b c a b

34 Investigation of Titanium α Plates by EBSD Analysis 4.4. Grain boundary energies Interfacial energy of low-angle grain boundaries In 1948, Read and Shockley derived an equation to describe the interfacial energy of lowangle grain boundaries of a simple cubic structure. They described a grain boundary consisting of an array of lattice dislocations. The interfacial energy per unit area of grain boundary for cubic-cubic systems is described by the Read-Shockley formula in terms of the interfacial area A, the misorientation angle θ, the orientation angle of the grain boundary φ, the shear modulus G, the Poisson ratio ν and the lattice constant a [30]. This formula is given by E = E0θ ( A lnθ ) (eq. 4.6) in which E 0 depends only on φ and material constants and is calculated by Ga E0 = cosϕ + sinϕ (eq. 4.7) 4π 1 ν ( ) ( ) The term A depends on φ and the energy of the atoms at the dislocation, in which they do not have the normal number of nearest neighbours. The assumption was made that the spacing of dislocations on the grain boundary is uniform for all values of θ. Read and Shockley derived an equation to determine A from the orientation angle and the radius r 0 of a small circle around the dislocation core determined by the local energy of misfits, which has to be calculated on an atomic basis. A can be calculated with sin 2ϕ sinϕ ln ( sinϕ ) + cosϕ ln ( cosϕ ) A = A0 (eq. 4.8) 2 sinϕ + cosϕ a A0 = 1+ ln 2 π r0 (eq. 4.9) However, the methods that are available to calculate r 0 are not very accurate, so some estimates were made for the calculation of A [30] Interface energy of high-angle grain boundaries The equations derived by Read and Shockley are only applicable for low-angle grain boundaries (Figure 4.6). Therefore several papers [35] have described variations on the Read- Shockley equations for the calculation of the interfacial energy of high-angle grain boundaries. Sutton and Balluffi [36] have described five different situations for high-angle low energy grain boundaries. These are 1) low reciprocal volume density of CSL sites, Σ 2) high planar CSL site density Γ 3) high Γ at constant interplanar spacing, d 4) large interplanar spacing, d 5) high density of locked-in rows of atoms, f Criterion 1 is developed after experimental observations of (111)(111) twins in fcc metals. Fullmann and Fisher [37] observed that these twin grain boundaries have the lowest energy of all high-angle grain boundaries in many fcc metals. This led to the conclusion that low values 23

35 Grain boundary theory of the reciprocal volume density of coincidence lattice sites, Σ, caused low energy grain boundaries. Grain boundaries with a low value of Σ were called special grain boundaries and all other grain boundaries were called random or general grain boundaries, but no limit of Σ was given for this classification [36]. Criterion 1 was not sufficient; it is possible for any coincidence system to find a boundary with a low value of Σ that satisfies criterion 1. Therefore the second classification was proposed by Brandon [38]. He formulated a more reliable value Γ than Σ for the classification of low energy grain boundaries. Γ is the planar density of coincidence sites in the grain boundary. There are two arguments that support criterion 2. There are more atoms at shared sites at grain boundaries with a high value of Γ, which causes lower grain boundary energy, and a high value of Γ causes a lower strain field energy, which is caused by lower periodicity of boundary structure [36]. It is found that the calculation of high-angle grain boundary energies is more complex than that of low-angle grain boundaries. Weins [39] has derived calculations for the grain boundary energy E CSL of high-angle grain boundaries with a CSL with a low value of Σ. In this calculation, the nearest and second-nearest neighbours are taken into account. The grain boundary energy is then given by E CSL =½Σ i Σ j Ψ(r ij ) (eq. 4.10) in which Ψ(r ij ) is the interaction potential between the ith and jth atoms. i is summed over all atoms, and j only over the atoms that are at a separation less than second nearest neighbours. The double counting of the interaction is corrected by the factor ½. The interaction potential Ψ(r ij ) can be calculated by several different equations, which can also calculate the atomic positions associated with the lowest configurational energy. Three of the equations are the Morse potential and the Lennart-Jones (4-7) and (6-12) potentials [39]: { α 0 α 0 } Ψ ( rij ) = B exp 2 ( rij r ) 2exp ( rij r ) 4.0 E r 2.0 E r Ψ ( rij ) = + C4 r C7 r 4 7 s 0 s ij ij 4.0 E r 2.0 E r Ψ ( rij ) = + C6 r C12 r 6 12 s 0 s ij ij (Morse) (eq. 4.11) (Lennart-Jones 4-7) (eq. 4.12) (Lennart-Jones 6-12) (eq. 4.13) in which r ij is the separation between atoms i and j, B and α are constants, r 0 is the equilibrium atomic distance, E s is the sublimation energy and C4, C7, C6 and C12 are crystal constants. The experimental determined interface free energy showed gusps at certain misorientations. The gusps in the interface energy were found to occur at certain values of Σ [40]. This indicated that the grain boundary energy was lowered when the two grains were orientated in such a way that a CSL was formed. These minima in grain boundary energies were found to correspond to that calculated with equation 4.6. This means that at certain high-angle grain boundaries, the low-angle grain boundary energy equation of Read and Shockley is still applicable. An example of the influence of CSL on the grain boundary energy is given in figure

36 Investigation of Titanium α Plates by EBSD Analysis Figure 4.8. The variation of grain boundary energy with orientation. The solid line represents experimental determined values of the grain boundary energies, the dashed curve represents the calculated energy according to equation 4.6 [40]. The influence of CSL boundaries on the grain boundary energy was also calculated for hexagonal systems (zinc crystals) by Faraoun et al. [33] using the calculation method derived by Sutton and Chen [41]. The energy difference E=E 0 -E θ for varying misorientations around the c-axis and the a-axis are given in figures 4.9 and The values written inside the diagrams are the misorientation angles for the local minima. Faraoun has also calculated the Σ values for these minima. For example the rotation of 22 at the c-axis has a value of Σ=7. It must be noted that the energy differences for rotations around the a-axis are dependent on the c/a ratio, whereas figure 4.10 can be applied for any value of c/a[33]. Figure 4.9. Energy difference for different rotations around the c-axis in hexagonal systems [33]. 25

37 Grain boundary theory Figure Energy difference for different rotations around the a-axis in hexagonal systems [33] Interfacial energy at triple points The interfacial energy of a three-grain boundary system can be calculated with a relationship derived by Herring [30]. In this situation the dependence of the energy on the grain orientation is taken into account. A schematic representation of a three grain boundary system is given in figure 4.11, in which three grain boundaries with energy E 1, E 2 and E 3 intersect at point O. The angles between the grain boundaries are ψ 1, ψ 2, and ψ 3 and the average orientation of the grain boundaries relative to the specified crystal axes of the two adjacent grains are φ 1, φ 2 and φ 3. The relation for the interfacial energy of three-grain boundaries is given by E 3 E 2 E1 + E2 cosψ 3 + E3 cosψ 2 + sinψ 2 sinψ 3 = 0 (eq. 4.14) ϕ3 ϕ2 The partial derivatives with respect to the boundary orientation are measured counterclockwise about the intersection [30]. Figure Representation of three intersecting grain boundaries [30]. 26

38 Investigation of Titanium α Plates by EBSD Analysis 5. Nucleation theory 5.1. Introduction When phase transformations are considered, it is very important to study nucleation theory. In chapter 3, the characteristics of two types of α plates in titanium were described. It was stated that the secondary α plates are formed by a displacive mechanism [5]. This means that longrange diffusion of substitutional alloying elements does not occur during the nucleation and growth of the α plates. For a full analysis of the process that takes place during the formation of the secondary α plates at the β grain boundaries, it is important to describe the various nucleation theories that are known. The nucleation theories can be divided into two main theories: the classical nucleation theory, which describes the nucleation of diffusionally growing phases and the non-classical nucleation theory, which describes the nucleation of displacive products. The most developed one is the classical nucleation theory that considers the nucleation as a process stimulated by the random movement of atoms in the lattice. The description of diffusionless nucleation is shown to be a bit more complicated and there are several descriptions of this process. Both nucleation theories are described in this chapter, to give a complete overview of the processes that take plate during the nucleation of new phases Classical nucleation Activation energy for nucleation The classical nucleation theory has been formulated for various types of transformations, like liquid-solid transformations and solid-solid transformations. In this section, only the solidsolid case will be described. In the classical nucleation theory, nucleation is considered as the appearing and disappearing of small regions of the product phase due to lattice vibrations. When these small regions become stable, they are critical clusters of the new phase. The formation of stable nuclei can be determined by the total Gibbs free energy change due to the formation of the new phase [28] G = V ( g g ) + A γ G (eq. 5.1) i v s i d i in which V is the volume of the nucleus in m 3, g v is the difference in chemical free energy between of the parent phase and the new phase in Jm -3, g s is the misfit strain energy per unit volume that results from the fact that the nucleus does not perfectly fit in the parent lattice, A i is the created area of interface i, γ i is the free energy per unit area of interface i and G d is the free energy change in J due to the consumption or creation of defects in the lattice. G is dependent on the size and geometry of the nucleus. When the assumption is made that the nucleus is a sphere with radius r, the critical radius r * can be calculated for the formation of a stable nucleus. The Gibbs free energy change can be differentiated with respect to r so that it can be predicted whether the nucleus is stable and grows, or is instable (disappears). d ( G) The nucleus grows when < 0, and alternatively, the nucleus decreases in size when d r this term is positive. ( ) The misfit strain energy, V g s, and the creation of an interface between the new phase and the parent phase, A αβ γ αβ, are the terms that will work against the nucleation, while the chemical 27

39 Nucleation theory driving force, g v, the removal of parent phase-parent phase grain interfaces, A ββ γ ββ, and the removal of dislocations g d stimulate the formation of the nucleus when the temperature is below the transformation temperature. When the nucleus is relatively small, the interface term is the strong factor, but when the nucleus reaches a certain critical size r *, the volume term becomes more important and makes the total free energy change become negative. When this critical size is overcome, the nucleus can only lower its free energy by becoming larger. A typical graph of the Gibbs free energy against the nucleus radius is given in figure 5.1. The critical nucleus size can be determined by introducing two new factors, z i A and z V in which the nucleus is no longer assumed to be spherical shaped. z i A is the geometrical factor for the interfacial surface and z V is the factor for the volume of the nucleus. The critical nucleus radius r * and the nucleation energy barrier are then given by [42] r * = 2 3 i z V z γ i A i g V (eq. 5.2) G * = 4 27 z z γ i i A i 2 2 V gv (eq. 5.3) In the formulation of the change in Gibbs free energy, it was stated that for small particles of the product phase, the interface part is important. Therefore, this term is important for the description of nucleation. The term γ can be described by several parameters like the misorientation angle between the two grains, the orientation of the grain boundary plane, the temperature and the pressure. In many situations, the desctiption of the interface free energy is simplified, by assuming that this term is only dependent on the misorientation angle θ. The value of γ is thus dependent on the misorientation of the interfaces and can be low for coherent interfaces and high for incoherent interfaces [43]. Figure 5.1. The Gibbs free energy as a function of the nucleus radius r [28]. 28

40 Investigation of Titanium α Plates by EBSD Analysis Nucleation rate In the classical nucleation theory the nucleation rate J * is described by * G * * j τ j * τ j J j = N jβ j Z j exp exp = J s exp (eq. 5.4) kbt t t in which N j is the number density of viable nucleation sites of type j, β * is the frequency factor, which describes the rate at which single atoms are added to the critical nucleus, Z is the Zeldovich non-equilibrium factor that corrects the equilibrium rate for sub-critical nuclei that grow beyond the critical size, k B is the Boltzmann constant, T is the reaction temperature, τ is the incubation time and t is the isothermal reaction time. On the right hand side of equation 5.4 the steady state nucleation rate J * s is given, which is the nucleation rate for t>> τ. The equation given above is only valid for one single type of nucleus j, for example only nucleation at grain boundaries. When more types of nucleation sites are taken into account like grain corners or grain edges, the nucleation rate is the sum of the individual nucleation rates of different types and the total nucleation rate is then given by * * J = J (eq. 5.5) tot j j The frequency factor β * can be described in two different ways. The first way describes β * with the use of the activation energy Q for diffusion of atoms in the parent phase. The chance that an individual atom joins the forming phase is determined by the chance that the atom has an energy higher or equal to the activation energy of the transformation. In this approach, the frequency factor is given by [44] * kbt Q β = exp (eq.5.6) h kbt in which h is the Planck constant. The second method describes β * by calculating the possibility that an atom diffuses to a position in the transformed lattice. By using the number of atoms within a single jump distance form the nucleus lattice and the atomic jump frequency, the following description of β * can be found[45] * * 6DS β (eq. 5.7) 2 2 a αd D is the diffusion coefficient, S * is the area of the surface of the nucleus, a is the lattice parameter and α d is the jump distance between the atom and the nucleus. Equation 5.7 can be simplified when α d is approximately the lattice parameter a. β * will then be 6DS a The diffusion coefficient D can be written as exp Q D = D0 kbt * β (eq.5.8) 4 (eq. 5.9) 29

41 Nucleation theory Equations 5.8 and 5.9 can be combined to * 6D0S Q β = exp 4 a kbt (eq. 5.10) 5.3. Non-classical nucleation Introduction The theory of non-classical nucleation involves the nucleation of new phases due to thermodynamic instability of the crystal overcome by a shear process rather than nucleation caused by the random movement of atoms. Transformations that are non-diffusional are also called displacive or shear-like transformations and are characterized by a cooperative movement of atoms in the parent phase that results in the formation of the product phase. No long-range diffusion is taking place during this process. For a displacive transformation to be thermodynamically possible, the temperature of the alloy must be below the temperature at which the parent and the product phase have the same free energy corresponding to the composition of the alloy (T 0 ) during the cooling process. However, often transformations do not take place immediately when the temperature is below T 0, a certain undercooling must be applied for the total free energy of the nucleation to become negative. Martensitic transformations are well known displacive transformations and are often accompanied with the creation of surface relief effects due to the internal stresses that are created with martensitic transformations [40, 46]. It is often found that displacive transformations occur with heterogeneous nucleation, i.e. the nucleation takes place at preferred sites like impurities or dislocations. This preferred nucleation at stress concentrators has the result that the heterogeneous nucleation is not a random process, so when the parent phase contains fewer effective sites, less nucleation will take place when the same undercooling is applied [46, 47]. The theory of displacive nucleation is still a subject that is not fully understood. The reason for this is that displacive transformations occur at very high rates, which makes it difficult to perform measurements during nucleation and growth of the product phase. There are two main kinetic modes for displacive transformations: the athermal mode and the isothermal mode. In the athermal mode, the fraction of transformed phase is only a function of temperature. In the isothermal mode, the fraction of formed phase increases with time when the specimen is held at a constant temperature. [46]. A schematic representation of the athermal and isothermal is given in figure 5.2 will be described in detail in sections and Figure 5.2. Schematic representation of the athermal mode (a), the isothermal mode (b) [48]. 30

42 Investigation of Titanium α Plates by EBSD Analysis Olson [49] has created a schematic representation of the different structural stages and dynamics during displacive transformations (Figure 5.3). This systematic approach of displacive transformations is represented as blocks, divided in two groups. The group of blocks on the left describe the levels of structure during the displacive transformation, while the blocks on the right describe the dynamical phenomena that interact with the different structures during the transformation. The initial nucleation is described by the number density of initial heterogeneous nucleation sites. The second block on the right describes the unit growth of the particle and influences the substructure and the initial average particle volumev o. The behaviour of the growth causes the elastic or plastic strain fields, which influence the number density of nucleation sites N A and the potency of developing the autocatalytic effect. This in turn influences the initial geometry of the product phase, which determines the eventual average particle volume V ( f ). This is only a schematic representation of the hierarchy of displacive transformations. The actual transformation is obviously much more complex and the dynamics stated in the blocks are also dependent of other processes not written in the model [49]. It is also stated that the average volume of the displacive structures changes during the transformations, whereas in other descriptions, this volume is taken to be constant [50]. Figure 5.3. Block diagram showing martensitic transformations as a multi level system [49] Historical background One of the first theories about displacive nucleation was proposed by Kaufman and Cohen in the 1950 s [46]. They assumed an existence of pre-active nuclei to be present at temperatures above the T 0 temperature. These pre-active nuclei, also called embryos, are described as nonequilibrium lattice imperfections, internal surfaces and local strains due to crystal growth and plastic deformation. The embryos will become active when the temperature is lowered at a temperature below T 0, but are able to survive heat treatments above T 0. However, some heat treatments may remove or rearrange the pre-active nuclei [46]. When the temperature is lowered, the Gibbs free energy will become negative and the embryo gets activated when the total free energy change for the transformation drops below zero. Before the active nucleus is created, a dislocation loop in the parent phase is considered in which the nucleus with the shape of a thin disk is created by this loop [46]. The nucleation is in this situation a product of the coupling of defects, which will result in small regions of the product phase. According to 31

43 Nucleation theory this theory, the nucleation already takes place at temperatures above T 0, but the nuclei only become active below this temperature. However, this theory is not verified and the presence of the pre-active embryos is not proven by experiments. Easterling [52] also described the creation of a martensitic nucleus with the coupling of a nucleus with surrounding dislocations. When the strain field caused by the dislocations can interact with the possible strain field caused by the transformation strain, this nucleus can become stable. This interaction of the dislocations leads to the following expression for the total energy in the system with a martensitic nucleus E tot =E surf +E strain +E chem +E int (eq. 5.11) E surf is the interfacial energy between the product phase and the parent phase, E strain is the strain associated with the lattice misfit between the product phase and the parent phase, E chem is the free energy difference between the two phases and E int is the interaction energy between the nucleus and the dislocations. The interaction energy of a disk-like nucleus with semithickness c and radius a with the dislocation can be expressed as 2 2 µ sbπ a 2 v c Eint = (eq. 5.12) 4 1 v a µ is the shear modulus of the parent phase, s is the transformation shear, b is the Burgers vector of the dislocation and v is Poisson ratio. When v=1/3, equation 5.12 can be simplified to Eint = 2µ sπ abc (eq. 5.13) This calculation shows that it is most likely that diffusionless nucleation takes place at regions with a high dislocation density. It must also be noted that the nucleation energy is dependent on the transformation strain s, which is dependent of whether the nucleus is twinned or not. Easterling [52] used these equations to calculate the nucleation energy in different situations. The results showed that a diffusionless nucleus is not stable without interaction with dislocations, unless the nucleus is very large and twinned. Also the surface energy can be of importance, which depends on the coherence between the phases. When all terms are fully worked out, equation 5.11 can be written as 2 2 π 2 π 2 µ 2 π 2 v 16 s 4 sb a 2 v c Etot = 2π a γ + µ ac g a c v a (eq. 5.14) in which γ is the specific interfacial free energy and g v is the free energy difference per unit volume between the parent and the product phase. The coherency term in this equation is the first term of the right part. This term is dependent on the principal strains of the transormation deformation. Using Eshelby s expression [16, 46], the minimum energy can be calculated depending on the orientation of the nucleus relative to the parent phase. The total minimum energy, shear and dilatational, can be plotted in a 2-D stereogram. The minimum energy is not attributed to a single orientation, but to a range of orientations that is called a minimum energy through. In the classical nucleation theory, an embryo is active when its size passes a critical value. For the non-classical theory, Cohen [46] considered the activation of an embryo by the lattice configuration. The path from an inactive embryo to an active embryo is described by the fluctuation of the lattice configuration. Also the interfacial and strain energy must be considered that determine the minimal free energy for the embryo to get activated. Cohen considered such embryos as strain centres consisting of arrays of dislocations. These 32

44 Investigation of Titanium α Plates by EBSD Analysis dislocation arrays are local regions of high free energy. There are two possibilities stated to reduce the high energy. The first is to transform the stressed matrix of the parent phase into the transformed phase. The second is to eliminate the strain fields around the dislocations by strain relaxation, so activation of nuclei is considered to be influenced by strains and not by a critical size that is mainly influenced by the interfacial energy [46] Athermal nucleation In the athermal mode the transformation starts at a well-defined temperature and the amount of transformed phase increases with lowering the temperature. The transformed volume fraction f can be roughly described with the Koistinen and Marburger equation [53] 1 f = exp β M T (eq. 5.15) { ( s )} in which β is the transformation rate parameter, which has a value of approximately K -1. Knapp and Dehlinger [54] described that the initiation of athermal nucleation is possible when the free energy change of nucleation W=0. For athermal nucleation, the nucleation free energy change is described by W= G+ G strain (eq. 5.16) In which G is the change in chemical and interfacial free energy due to the transformation, which is dependent on temperature, and G strain is the nucleation strain energy. Nucleation is in this situation not dependent on the density of nucleation sites, but transformation takes place when the nucleation free energy becomes negative. It is observed that during athermal martensitic transformations in steel, the martensitic plates grow very rapidly until the growth front collides with a barrier at which the plates stop growing. When the samples are further cooled, the plates will not become longer and other plates start to nucleate at new locations [46, 51] Isothermal nucleation A description of the nucleation rate for isothermal martensite has been made by Pati and Cohen [50]. It was assumed that the time dependence of the isothermal transformations is caused by the time dependence of the number of nucleation sites n t. The isothermal transformation rate df dnv ( ) dt is given by df f Q = V = Rn + f 1 f pvν exp (eq. 5.17) dt dt pv RT in which V is the average volume of the product platesphase, R n is the ratio of initial to induced nucleation sites, f is the volume fraction of the formed phase, p is a factor that describes the effect of autocatalysis, ν is the lattice vibration frequency (approximately per second) and Q is the activation energy. For isothermal nucleation the transformation rate is initially low, but the transformation rate increases after some percent of the new phase is formed, which is caused by the autocatalytic effect. Close to the equilibrium transformation fraction, the transformation rate decreases. Isothermal behaviour is less commonly found in displacive transformations than the athermal behaviour. It was found that the isothermal transformation propagates by the nucleation of new plates rather than by the growth of the existing plates. Therefore, the isothermal 33

45 Nucleation theory transformation is characterized by a nucleation rate dependence [46, 51]. Kaufman and Cohen also described the activation energy for isothermal nucleation to be linearly related to the transformation free energy change. For a Fe-29.2%Ni alloy, the activation energy Q is related to the chemical driving force G by Q = G (cal mol -1 ) (eq. 5.18) Autocatalytic nucleation The autocatalytic effect can be observed during isothermal transformations and in extreme form during burst transformations. During autocatalysis, the transformation rate is initially low, but after a few percent has transformed, the transformation rate increases with a factor 10 to 50. This increase can be explained by an increase in nucleation rate, that is stimulated by the already formed phase. This phenomenon is considered to be caused by the increase of internal stresses due to the transformation. A first explanation for the increase of nucleation rate is that the induced strains cause more active nucleation sites. The induced strains in the parent phase can activate non-critical nucleation sites that will cause an increase in nucleation rate. Stresses can also increase the number of nucleation sites when the level of the stress is of such magnitude that plastic deformation occurs, which can induce the creation of new active nucleation sites. The other explanation for the autocatalytic effect is the lowering of the activation energy for the transformation due to the induced stresses in the matrix. The autocatalytic effect can be identified by a large density of the formed phase at certain locations [51]. External and internal stresses and strains can have a great influence on the behaviour of displacive transformations. Kaufman and Cohen have already described the influence of stresses on the martensite start temperature in the late 1950 s [46]. The transformation temperature can be lowered or raised when the stress respectively opposes or aids the chemical driving force G. An effect that is related to the autocatalytic effect is sympathetic nucleation. Sympathetic nucleation is the nucleation of a phase at the interface of the same phase with the parent phase, and is often associated with diffusional transformations. Sympathetic nucleation is a heterogeneous type of nucleation, and can be classified further into three configurations. These are face-to-face sympathetic nucleation, which will result in sheaves, edge-to-face nucleation, which will result in a branched-like structure, and edge-to-edge nucleation which will result in long plate-like structures. The new nucleus will tend to have an as small as possible interface energy with the product phase, so often twin boundaries, CSL lattices or small-angle grain boundaries are found at the sympathetic grain boundaries when the variants make this possible. For edge-to-face sympathetic nucleation, it has been found difficult to distinguish the structures from being sympathetic nucleation or impingement. Therefore, this type of sympathetic nucleation is often analysed when the product phases are widely separated inside the parent grains. An example of this structure are the Widmanstatten stars [55] Variant selection As described earlier, in chapter 3.3, it is possible to describe the relationship between the lattices of the product and the parent phase of transformations with an orientation relationship. For each orientation relationship, there are several possibilities for the product phase to be formed to obey the orientation relationship. Each possibility is called a variant. For example, 34

46 Investigation of Titanium α Plates by EBSD Analysis the Burgers orientation relationship has twelve different variants. There are six possible {110} planes in the β phase that can transform into the (0001) plane of the α phase and for each (110) plane in the β phase, there are two possible <111> directions that are parallel to the <11 2 0> direction in the α phase. This results in the twelve different variants that can be formed according to the Burgers OR. Variant selection describes the selectivity of the formation of one or more variants of the orientation relationship. For nucleation in a perfect crystal, often the formation of the variants occurs random and neighbouring nuclei are usually of different variants [56]. A schematic illustration of transformations without (a) and with variant selection (b) is given in figure 5.4 in which it is shown that variant selection can eventually result in a coarse-grained microstructure. In steels, variant selection is often associated with displacive transformations, but in titanium and zirconium alloys, variant selection is also observed in diffusional transformations [57]. In general, the variant is chosen that minimises the strain energy for the nucleus to be formed. There are several characteristics that can influence the variant selection. Examples are an anisotropic matrix, the asymmetric strains that occur during the transformation and the presence of grain boundaries [56, 58]. To characterise non-diffusional transformations, a tensor is applied on the parent phase lattice to transform it into the product phase lattice. This tensor does not have a homogeneous strain distribution and is dependent on the OR variant. The dependence of the strain field for each variant with an inhomogeneous matrix results in the formation of certain variants to reduce the total elastic energy of the system. Figure 5.4. A transformation without variant selection (a) and with variant selection (b) [56]. During the phase transformation, the formation of an embryo is governed by transformation strains, which increase the elastic energy of the material. When an infinite crystal is considered with homogeneous elastic properties, the strain field of the formation of an embryo will be equal for all different variants to be formed. However, in practice this is almost never the case. When nucleation is taking place at grain boundaries, the presence of other parent 35

47 Nucleation theory phase crystals will result in inhomogeneous elastic properties. The elastic properties of the grain at the other side of the grain boundary will differ from that of the grain in which the nucleation takes place. The local stress and strain can be different for embryos at a grain boundary compared to that of similar embryos in the centre of grains. It was found that variant selection of titanium α occurred at β grain boundaries due to the influence of the adjacent grain on the stress field that controls the formation of a nucleus [59]. 36

48 Investigation of Titanium α Plates by EBSD Analysis 6. Microtexture analysis In microtexture analysis the characteristics of the microstructure and the texture are combined. During microtexture analysis the crystal orientations as well as the influence on the microstructure are analysed. An important texture analysis technique is electron backscatter diffraction (EBSD) EBSD Installation The EBSD technique is based on the use of a scanning electron microscope (SEM). Furthermore, an EBSD installation consists of a video camera coupled to a phosphor screen, a camera control and diffraction processor unit and a computer with data processing and analysis software [60]. A schematic illustration of an EBSD installation is given in figure 6.1. During EBSD measurements, an electron beam is directed on the specimen, which diffracts the electrons. The specimen is tilted; usually over 70, to increase the number of diffracted electrons. The diffracted electrons hit the phosphor screen and form a Kikuchi diffraction pattern, which is captured by the camera. The diffraction data is subsequently processed by the computer for further analysis. Figure 6.1. A schematic illustration of an EBSD installation [61]. There are several different coordinate systems that have to be described when using EBSD. These coordinate systems are the microscope axes, the phosphor screen axes, the specimen axes, the acquisition axes and the crystal axes, see figure 6.2. The microscope axes, X m,y m,z m, refer to the microscope stage when the stage is horizontal for normal SEM imaging, in which Z m is usually taken normal to the stage surface. The phosphor screen axes, X p,y p,z p, are also called the diffraction pattern axes. The orientation of these axes is dependent on the camera position in the microscope chamber. Z p is normal to the phosphor screen surface and X p or Y p is chosen to lie horizontal in the phosphor screen surface. The specimen axes, X s,y s,z s, refer to the natural features, the sectioning surface or the processing history of the sample. Often the rolling direction (RD), the normal direction to the rolling plane (ND) and the transverse direction (TD) are chosen to be the specimen axes for rolled products. The specimen axes are also oriented relative to the microscope axes. X s is aligned parallel to X m and Z s is usually 37

49 Microtexture analysis aligned normal to the acquisition surface. The acquisition axes, X a,y a,z a, describe how the specimen is placed in the SEM for data collection. Z a is oriented normal to the specimen surface, which will be investigated with EBSD. When the specimen has a straight edge, this edge is usually referred as X a or Y a and is aligned parallel with the microscope tilt axis. The crystal axes, X c,y c,z c, refer to the crystallographic orientation. For orthogonal symmetry, the [100], [010]and [001] directions refer to the X c, Y c and Z c axis respectively. For hexagonal and trigonal symmetry these axes are defined by either the[10 10], [12 10] and [0001] directions or the[2 1 10], [01 10] and [0001] directions respectively. PC refers to the pattern centre on the phosphor screen, SP is the pattern source point of the specimen and SSD is the sample to screen distance [62]. Figure 6.2. Schematic illustration describing the relation between the specimen axes, the microscope axes and the phosphor screen axes (a) and the relation between the specimen axes and the acquisition axes (b) [60]. 38

50 Investigation of Titanium α Plates by EBSD Analysis 6.2. Orientation determination Kikuchi diffraction patterns Most microtexture analysis with TEM and SEM is based on Kikuchi diffraction patterns, which is also the case for EBSD measurements. A Kikuchi pattern consists of a set of different Kikuchi lines that forms bands, that corresponds to a distinct crystallographic lattice plane. The intersections of the Kikuchi bands refer to a zone axis (pole) and the major poles are recognisable by the intersection of several bands. Thus a Kikuchi pattern represents the angular relationship in a crystal and hence contains the crystal symmetry. An example of a Kikuchi diffraction pattern is given in figure 6.3. Kikuchi patterns are produced when an electron beam is directed on the surface of the specimen and the electrons are diffracted. Electron diffraction occurs when Bragg s law is satisfied, given by 2d sinθ = nλ (eq. 6.1) d is the interplanar spacing between the planes in which diffraction is occurring, θ is the specific angle at which the diffraction occurs, n is the order of reflection and λ is the electron beam wavelength, which is approximately m. If the diffracted electron beams are recorded on the phosphor screen, a pair of conic sections are projected, which appear to be parallel lines. These lines are the Kikuchi lines and their spacing has an angular distance of 2θ B, which is related to the interplanar spacing by equation 6.1. Figure 6.3. Example of a Kikuchi pattern measured during EBSD analysis. After the detection and capture of the Kikuchi bands by the phosphor screen and the coupled TV camera, the patterns must be identified. The identification is usually done by comparing the measured data with stored patterns in a database. This is done by comparing the interplanar angles between the low Miller indices planes present in the crystal structure including every individual plane in a family of planes. The intersection angles between three Kikuchi bands are measured and compared with the data with a tolerance of about 1-2. After the identification of the planes, the zone axes are indexed from cross-product calculations. Besides this indexing, some other specifications can be used to aid the determination of the orientation. For example, it is possible to use interzonal angle information, bandwidth information or the intensities of the Kikuchi bands, when it is possible to calculate the 39

51 Microtexture analysis structure factors of the specimen. In most cases, the automatic determination and indexing is a proper and reliable method of determining the structure and the orientation, but when low symmetry phases are investigated, it may be useful to work interactively with the program and to correct the solutions given by the database to identify the correct phases and orientations. In this case, the user manually marks a number of bands or axes in the pattern which are transferred to the computer, which indexes the specific points. During EBSD measuring, some practical difficulties may have to be overcome for a good analysis. Sometimes the microstructure of the sample contains a lot of dislocations and stresses. These microstructures are difficult to identify by EBSD, which is a common difficulty when analysing deformed materials. The number of successfully indexed points decreases with the amount of previous plastic deformation. For example, the indexing ratio is typically only 30% for an 80% cold rolled titanium sheet [63]. For some orientations, the Kikuchi patterns are difficult to distinguish from each other within the resolution of the imaging system. This is called pseudosymmetry (Figure 6.4). The problem of pseudosymmetry can by solved by orientation imaging microscopy (OIM), which is and EBSD analysis software package. When pseudosymmetry occurs, a special cleanup procedure can be applied, in which the relation between the two possible orientations must be defined. The most prevalent orientation will then be given to the grain with the pseudosymmetry. However, when two adjacent grains also have the defined pseudosymmetry relation, the smaller grain will be given the same orientation as the largest grain. Pseudosymmetry can also be avoided by using more Kikuchi bands for orientation determination. Another difficulty during orientation determination occurs when the electron beam is diffracted by two different crystal lattices. This occurs when the electron beam is positioned on a grain boundary, thus two crystal orientations fall within the beam spot. Such a double diffraction can also occur when a different grain lays beneath the surface and within the penetration depth of the electron beam. The SEM image will show the microstructure of the surface, but other grains can be measured within the penetration depth of the beam. This problem can be solved for example by decreasing the penetration depth by lowering the beam current. Figure 6.4. Kikuchi pattern (a) and a detail of an IPF (b) of a measurement with pseudosymmetry. a b 40

52 Investigation of Titanium α Plates by EBSD Analysis Pole figures During measurements of the orientations of crystals, often pole figures or inverse-pole figures are used to describe the orientations in a single figure. The direction of any 3-dimensional vector in a crystal can be described as a point on a unit reference sphere. The unit reference sphere is a sphere with radius 1 notionally residing the crystal. The point in which the normal of a plane intersects this sphere is called the pole of this plane. The distributions of poles on the reference sphere is thus a description of the orientations of the planes. The positions of the poles on the sphere can be described by two angles, α and β. α described the azimuth of the pole; α=0 denotes the north pole of the unit sphere. β is the rotation of the pole around the polar axis, in which β=0 is chosen from a reference direction. For the characterisation of the sample, the pole angles have to be determined with respect to the coordinate system of the sample. Often ND is chosen to be in the north pole of the reference sphere in which α=0, and RD is chosen as the reference point in which β=0. A pole figure can be created by projecting the poles on the surface of the reference sphere on a 2-dimensional field, usually the equator plane. A pole figure is a reference system that consists of the coordinates given by the specimen axes with the coordinates of the crystal axes projected into this reference system. An illustration of how a pole figure is created for a hexagonal crystal is given in figure 6.5. To present the complete orientation of a crystal, different poles have to be constructed in a pole figure. Depending on the symmetry of the crystal, at least three poles are necessary to describe the orientation matrix of a crystal. Usually different poles of the same family (hkl) are used to characterise the crystal orientation. Instead of projecting the crystal axes in a reference system of the specimen axes system, visa versa is also possible, resulting in an inverse pole figure. This means that the reference sphere is constructed by the crystal coordinate axes and that the orientation of the crystal is defined by the construction of the positions of the axes system of the specimen. Instead of the angles α and β, the angles γ and δ are equivalently introduced. Inverse pole figures find their use in the determination of axial symmetric specimens in which only one axis is prescribed. Figure 6.5. Illustration describing the azimuthal angle α and rotation angle β for plotting a (0001) pole figure [62]. 41

53 Microtexture analysis 6.3. Orientation representation Orientation matrices There are several methods of describing the orientation and misorientations of crystal lattices. A common used description is to transfer the crystal coordinate system with the use of the Euler angles, φ 1, Φ and φ 2 [62]. The Euler angles are based on the crystal orientation relative to the sample orientation. The definitions of the Euler angles are given in figure 6.6. To transform the specimen coordinate system into the crystal coordinate system, the following rotations must be performed: 1. Rotating φ 1 about Z s 2. Rotating Φ about X s in its new position 3. Rotating φ 2 about Z s in its new position Figure 6.6. Illustration of the Euler angles φ 1, Φ and φ 2 and the rotations required describing the relation between the crystal orientation and the reference orientation [62]. Orientation calculations are usually performed by matrix algebra. The matrix that is used to describe the crystal orientation is the rotation matrix g. Although the rotation matrix is usually not the main output parameter in orientation imaging software, it is used to calculate the spherical coordinates for the plotting of pole figures and inverse pole figures and misorientation parameters. The rotation matrix expresses the crystal direction in terms of the reference coordinate system, which is usually the specimen reference system. The first row of the matrix is then given by the direction cosines of the angles between the crystal axis X c and the three specimen axes X s, Y s, and Z s. An illustration of the three angles α 1, β 1 and γ 1 is given in figure 6.7. The second and the third row are given by the cosines of the angles between Y c and Z c and X s, Y s, and Z s respectively. The full rotation matrix g is then given by 42

54 Investigation of Titanium α Plates by EBSD Analysis cosα1 cos β1 cosγ 1 g11 g12 g13 g = cosα cos β cosγ = g g g (eq. 6.2) cosα3 cos β3 cosγ 3 g31 g32 g 33 The relation with the Euler angles is given by g11=cosϕ cosϕ -sinϕ sinϕ cosφ g12=sinϕ cos ϕ +cosϕ sinϕ cosφ g13=sinϕ sinφ g21=-cosϕ sinϕ -sinϕ cosϕ cosφ g22=-sinϕ sin ϕ +cosϕ cosϕ cosφ g23=cosϕ sinφ g31=sinϕ sin Φ; g32=-cosϕ sinφ g33=cosφ 1 (eq. 6.3) Figure 6.7.The definition of the crystal orientation angles relative to the specimen reference axes [62]. For materials that do not have an orthogonal symmetry, the reference system must be converted to an orthonormal frame. This can be done by normalising the reference system. This is done by the transformation matrix L. The transformation matrix is given by l11 l12 l13 L = l21 l22 l23 (eq. 6.4) l31 l32 l 33 43

55 Microtexture analysis l l l l l l l l = a = b cosγ = c cos β = 0 = bsinγ ( ) = c cosα cos β cos γ / sinγ = 0 = 0 ( ) (eq. 6.5) l 2 33 = c 1+ 2 cosα cos β cosγ cos α cos β cos γ / sinγ in which a, b, and c are the lattice parameters and α, β and γ are the crystal orientation angles. This description can be simplified for hexagonal systems to the transformation matrix [60] a - a / 2 0 L = 0 ( a 3) / 2 0 (eq. 6.6) 0 0 c To transfer the orthonormal coordinates back to the crystal reference system, the orthonormal set must be multiplied by the inverse of the transformation matrix L -1. With the rotation matrices of two crystals, g 1 and g 2, the misorientation between crystal 1 and crystal 2 can be calculated. In this case, a misorientation matrix g can be formed by 1 g = g1g 2 (eq. 6.7) Due to crystal symmetries, there are several misorientation matrices that describe equal misorientations between two crystals. For cubic crystals, there are 24 different ways to describe the crystal orientation. For hexagonal crystals there are 12 different symmetrical solutions. These symmetrical orientations can be described by symmetry matrices which are the symmetry operations of the crystal structure [62]. The 24 symmetric matrices for cubic crystals and the 12 matrices for hexagonal crystals are given in the tables 6.1 and

56 Investigation of Titanium α Plates by EBSD Analysis Table 6.1. The 24 symmetric matrix operators S i for a cubic system [62] Table 6.2. The 12 symmetric matrix operators S i for a hexagonal system [62] The symmetric variant of a misorientation can then be calculated with [64] ( ) 1 g = S g S g ij i 1 j 2 = S g S g i i j 2 = S gs j (eq. 6.8) in which g ij is a symmetric member of the misorientation matrix g and S i and S j are one of the n symmetry operation matrices given in tables 6.1 and 6.2 for cubic and hexagonal systems. Therefore, for a cubic-cubic system, there are 576 possible symmetric variants and for a hexagonal-cubic system there are 288 possibilities. A more convenient method of describing a crystal misorientation is the axis/angle pair. The axis/angle pair method describes the misorientation of two crystals as a rotation about a certain axis to transform the coordinate system of one crystal, X 1, Y 1,Z 1, into the coordinate system of the other crystal, X 2,Y 2,Z 2. The misorientation angle θ and the axis [r 1 r 2 r 3 ] are calculated from the misorientation matrix g by 45

57 Microtexture analysis cos θ = ( g + g + g 1) / 2 (eq. 6.9) r = g g r = g g (eq. 6.10) r3 = g12 g21 Due to the crystal symmetry, there are various possible axis/angle pairs for symmetrical misorientations. Generally the axis/angle pair with the lowest misorientation angle is used Rodrigues-Frank space A convenient method to visualise the misorientations between crystals in a 3-dimensional diagram is the use of the Rodrigues-Frank (R-F) space. In the R-F space, the orientation of a single crystal is expressed by a vector, the Rodrigues vector. The length of this vector is a measure of the misorientation angle and the direction of the vector is a representation of the rotation axis of the misorientation. The Rodrigues vector R is thus a function of the orientation angle θ and the orientation axis r=r 1 +r 2 +r 3 relative to the reference coordinate system. The length and direction of the vector are given by [62] θ R = tan r (eq. 6.11) 2 It is also possible to calculate the Rodrigues vector directly from the Euler angles φ 1, Φ and φ 2 (Bunge) [64], according to Φ R1 = tan sin {( ϕ1 ϕ2 ) / 2 }/ cos {( ϕ1 + ϕ2 ) / 2} 2 Φ R2 = tan cos {( ϕ1 ϕ2 ) / 2 }/ cos {( ϕ1 + ϕ2 ) / 2} (eq. 6.12) 2 R {( ϕ + ϕ ) } = tan / This representation is very useful for visualising the variety of misorientations in samples in a Cartesian coordinate system in which the axes of the coordinate system can be chosen to be reference axes of the sample or the crystal. For example, the vectors R 1, R 2 and R 3 correspond to the [100], [010] and [001] axis respectively. As described in the previous sections, one single misorientation can be described by various axis/angle pairs. Since the misorientation angle influences the length of the Rodrigues vector, thus the distance from the origin, usually the smallest rotation angle and the accompanying axis are chosen to be used in the calculation of the Rodrigues vectors. The space that covers all possible lowest angle misorientations between any crystal and the reference system is the R-F fundamental zone. Any Rodrigues vector that lies outside this zone can be represented by another symmetrical vector that has a lower misorientation angle. The shape of this zone is thus dependent on the crystal symmetry, and has been derived by Heinz and Neumann for several crystal lattices combinations [64]. For cubic symmetry, the fundamental zone is a cube with truncated corners (Figure 6.8) and for a cubic-hexagonal system, the fundamental zone is given in figure 6.9. The maximum length of a Rodrigues vector in a cubic-cubic system corresponds to a rotation angle of and that for the cubic-hexagonal system corresponds to a misorientation angle of The maximum lengths in these system will therefore be 0.61 and 0.54 respectively [64]. 46

58 Investigation of Titanium α Plates by EBSD Analysis Dagggdfgggsdasd tertrt Figure 6.8. The fundamental zone of the R-F space for cubic symmetry [64]. Figure 6.9. The fundamental zone of the R-F space for a cubic-hexagonal system [64].. 47

59 Experimental 7. Experimental 7.1. Composition and heat treatments The titanium alloy that is investigated in this work is a commercial Ti-4.5Fe-6.8Mo-1.5Al alloy. This alloy is commercially designated as Titanium Low-Cost Beta (Ti-LCB). The alloy was received as a rod with a diameter of 15 mm and the chemical composition is given in table 7.1; the concentrations of Mo, Fe and Al were measured with X-ray fluorescence spectroscopy (XRF). Table 7.1. Chemical composition of the Ti-LCB alloy. Mo Fe Al Ti wt. % balance at. % balance All samples were solution treated for 10 minutes at a temperature of 900 C to get microstructure that consists of only β grains. Subsequently the samples were cooled to the holding temperature at a pressure of less then mbar at a cooling rate of approximately 20 C s -1. The heat treatments were applied to the samples as given in table 7.2. At heat treatment II the sample was first held at 750 C for 6000 seconds, subsequently quenched to 630 C and held at this temperature for 720 seconds. The heat treatments were performed with a Bähr 805A/D dilatometer and the samples were cylindrically shaped with a length of 10 mm and a diameter of 5 mm. A schematic illustration of the dilatometer is given in figure 7.1. The length change in the dilatometer is measured with the Linear Variable Displacement Transducer (LVDT). Table 7.2. Heat treatments. Heat treatment Annealing step 1 Annealing step 2 T ( C) t (s) T ( C) t (s) I II III Figure 7.1. Schematic illustration of a dilatometer [65]. 48

60 Investigation of Titanium α Plates by EBSD Analysis 7.2. Sample preparation Before the samples were suitable for optical microscopy and EBSD analysis, the surface of the samples had to be prepared. The grinding procedures are given in table 7.3. For optical microscopy the samples were mechanically polished: steps 1 to 8 were performed and subsequently the samples were polished for 480 seconds with OPS, followed by etching for 10 to 20 seconds with Krolls reagent. The composition of this etching solution is 100 ml distilled water, 2 ml HF and 5 ml HNO ³. For EBSD, the samples were grinded up to step 5. After the grinding, the samples were electrolytically polished with A2 electrolyte (90 ml distilled water, 730 ml ethanol, 100 ml butylcellosolve and 78 ml perchloric acid), at 50 V for 35 s at an electrolyte temperature of 10 C. During all grinding steps, the samples were grinded and polished in opposite direction of the rotation of the grinding disk. One EBSD sample was prepared by mechanical polishing only. After the grinding procedure, the sample was polished with 3 and 1 µm diamond paste followed by electrolytic polishing. Table 7.3. Grinding steps used during sample preparations. Step Grid Cycles (min -1 ) Time(s) Pressure (N) EBSD measurement procedure In total 18 EBSD scans were performed. Most of the EBSD measurements were performed at samples that were heat treated according to heat treatment I. Four scans were done on samples treated according to treatment II and one scan III-1 was done on a sample treated by heat treatment III (see table 7.4). The EBSD scans are numbered according to the heat treatment, the point scans are also indicated by ps. EBSD scans I-1 to I-7 were done with a FEI microscope of type Environmental Scanning Electron Microscope (ESEM) XL30 with a Kimble LaB 6 filament at the Ghent University. The scans were preformed with an acceleration voltage of 25 kv and a beam current of 60 na. The working distances varied, scan I-1 had a working distance of 18 mm, scan I-2 of 15 mm, scan I-3 had a working distance of 10 mm and scan I-4 to I-7 had working distances of 20 mm. All other measurements were preformed at the Delft University of Technology with a JEOL 6500F hot Field Emission Gun Scanning Electron Microscope (FEG SEM) installation equipped with a HKL channel 5 EBSD acquisition software. The measurements in Delft were done with an acceleration voltage of 20 kv, a beam current of 14 na and a working distance of 25 mm. The step size varied for the different scans; the smallest step size was 0.03 µm for scan I-6 and scan II-4. The largest step size used was 1.2 µm for scan 10. Most scans had a step size of 0.08 to 0.1 µm. During every scan, the orientation calculation characteristics like the minimum and maximum number of detected bands were adjusted to get the highest possible number of correctly indexed points. The data collection for scan I-1 to I-3 was done with TSL OIM data collection, the data of all other scans was collected with Channel 5 from HKL. The data collection speed varied from approximately 40 patterns per second to 60 patterns per second, 49

61 Experimental depending on the quality of the patterns. Besides the EBSD map scans, 2 point scans were performed. During a point scan several series of point scans were performed at different areas of the sample to make it possible to analyse a larger number of β grains. At scan I-ps1, 6 different series of point scans were measured and at scan I-ps2 another 2 series were measured. Table 7.4. Experimental details of the EBSD scans. Scan number Heat treatment Step size (µm) Scanned area (µm 2 ) I-1 I I-2 I I-3 I I-4 I I-5 I I-6 I I-7 I I-8 I I-9 I I-10 I I-11 I II-1 II II-2 II II-3 II II-4 II III-1 III I-ps1 I - - (Point scan series 1) I-ps2 (Point scan series 2) I EBSD data analysis All collected data were analysed with TSL Orientation Imaging Microscopy version 4.6. This is a software package for the analysis of the orientation and misorientation of the crystals in the sample. To use the data collected by Channel 5 with OIM, the dataset had to be converted to a format that is compatible with OIM. This was done with the program HKLtoANG.exe, which converts a.ctf file into a.ang file which is compatible with TSL OIM software. Before the raw data could be analysed, the data had to be cleaned of misindexations, which is done by the cleanup tool of OIM. The cleanup procedures were performed by the iterative process of grain dilatation with a grain tolerance angle of 2 degrees and a minimum grain size of 2 pixels. Subsequently a grain confidence index standardisation cleanup procedure was performed with equal grain tolerance angles and grain CI standardisations. It is also possible to remove misindexed points due to pseudosymmetry (section 6.2.1), by a special cleanup procedure. During the cleanup, one has to be careful not to delete useful information. Therefore, the cleaned map has to be checked whether no artificial structures are created. After the cleanup procedures, the EBSD data is more suitable for further analysis. The first analysis was done by creating image quality maps, phase maps and inverse pole figure (IPF) maps with OIM. These three map types are very useful for the first analysis of the samples, 50

62 Investigation of Titanium α Plates by EBSD Analysis and make it possible to get an indication of the quality of the EBSD measurement, phase transformation and possible variant selection of the product phase in the sample. The grain boundaries were analysed by calculating the angular misorientation between the two β grains at each side of the grain boundary. Subsequently, the relationships between the formed secondary α plates and the parent β phases were analysed by plotting grain boundary maps, in which special grain boundaries were indexed with colours. This is done for various orientation relations and for different α-α grain boundary variants. The lengths of the different specified grain boundaries are automatically calculated by OIM. It is also possible to analyse separate selections of the scans. This is done by making partitions of the desired grains to analyse. This has the advantage that wrongly indexed grains are excluded and that it is more easy to analyse parent-product relationships. Another analysis that is performed with OIM is the creation of pole figures. OIM makes it possible to create pole figures of selections for desired planes and directions. Pole figures were created for the [1120] direction and the (0001) plane for the α phase and the [111] direction and (110) plane for the β phase, which corresponds to the Burgers OR. Much analysis was done with OIM, but the point scans and the misorientations between individual pixels had to be analysed with matlab. During the point measurements, the phase and the Euler angles were stored from which the grain boundary misorientations were calculated from the Euler angles by using equation 6.3. A similar method is used to calculate the misorientations between parent β grains and product α plates. To extract the correct misorientation data, partitions were made to exclude the α plates with the parent grain. From the partitions the orientation data were extracted and analysed with matlab. The calculated misorientations between the α and β phase were compared with the orientation relations using the Euler angles that define that orientation relation. This was done by comparing the phase index of two horizontally adjacent pixels. When the pixels had a different phase index, the misorientation is calculated using the Euler angles. This method made it possible to calculate a high amount of misorientations between the parent and product phase. For a proper analysis of the ORs, several angular distances between the ORs were created, and the corresponding percentages of misorientations that fall within the range were calculated. The calculations were done for three different types of α plates, namely secondary α plates from grain boundaries, intragranular secondary α plates and primary α phase. In order to improve statistical results the data off all possible partitions from all EBSD scans are combined. Also several orientations are calculated and represented in the R-F space. This is done for the observed misorientations between the parent β phase and the product α phase, in which also the three described orientation relations are represented in the R-F. Also for several EBSD measurements, the orientations of the α plates from one specified β grain are represented in the R-F space. The matlab scripts for processing the point scans and the OR determination are given in appendix C. 51

63 Results 8. Results 8.1. Optical microscopy After the heat treatments described in the chapter 7 were performed, the resulting microstructures were first analysed with optical microscopy. The three different heat treatments resulted in three distinct microstructures. The first heat treatment resulted in a microstructure consisting of β phase grains with secondary α plates predominantly formed at the grain boundaries (Figure 8.1). At some grain boundaries a large density of secondary α plates was found at both sides of the boundary, while at other grain boundaries the α plates are found at only one side or at neither side of the β-β grain boundary. The second heat treatment resulted in a mixed microstructure: at most β-β grain boundaries primary α phase was formed and secondary α plates were found mainly inside the β grains (Figure 8.2). The third treatment resulted in a similar microstructure as for treatment 1, but due to the longer holding time, more secondary α plates were nucleated, especially intragranularly (Figure 8.3). Some important characteristics of both primary and secondary α plates that were not determined during this project are described in chapter 3. Figure 8.1. Micrograph of the Ti-4.5Fe-6.8Mo-1.5Al alloy isothermally held at 615 C for 750 seconds and subsequently quenched. Figure 8.2. Micrograph of the Ti-4.5Fe-6.8Mo-1.5Al alloy isothermally held at 730 C for 100 minutes followed by isothermal holding at 630 C for 12 minutes and subsequently quenched. 52

64 Investigation of Titanium α Plates by EBSD Analysis Figure 8.3. Micrograph of the Ti-4.5Fe-6.8Mo-1.5Al alloy isothermally held at 615 C for 960 seconds and subsequently quenched EBSD maps In figure 8.4 two inverse pole figures are given. In the upper part, and IPF before cleaning is given, and the IPF after cleaning is given in the lower part. A great variety of orientations is measured at the grain boundaries, but after the cleanup procedure, many indexed points are given the same orientation as neighbouring pixels that have a higher confidence index, see section 7.4. Figure 8.4. IPF maps of the Ti-LCB alloy before (top) and after (bottom) cleanup procedures. The basic analysis with OIM after cleaning the data was the construction of phase maps, image quality (IQ) maps and IPF maps. These maps give an indication of the microtexture of the sample and the quality of the measurement. A phase map is a map of the EBSD scan in which the different crystal phases are indexed with different colours. An example of a phase 53

65 Results map, in which it is clearly visible that the platelike structures are hcp phase, is shown in figure 8.5 in which the α phase is coloured red and the β phase is coloured green. The IQ map is a mapping in greyscale of the quality of the Kikuchi patterns obtained during the EBSD measurement, in which spots with a low image quality are indicated with a darker greyscale than spots with a higher quality. IQ maps can also be used to display the microstructure of the sample. Examples of an IQ and IPF map of secondary α plates at β grain boundaries are given in figure 8.6 and 8.7. The cleaned IPF maps of the other scans are given in appendix A.1. In the IPF maps each measured point is given a colour corresponding to the crystal orientation, indicated by the legend of figure 8.7. Crystal orientations that are very close to each other are indicated with colours that are close in the colour scheme as can be seen in figure 8.7, but equal colours do not necessary imply that the orientations are equal, since only one reference pole is given by the colour scheme. For a full description of the crystal orientations by IPF maps, two IPF maps with a different reference system have to be constructed. All IPF maps given in this thesis have the same reference system and the colours are therefore all described by the legend of figure 8.7. Comparing figures 8.6 and 8.7 shows that it was not possible to index every point correctly during the EBSD. In the IQ map it is seen that inside the bottom left β grain, many α plates have grown that are not identified in the IPF map. Figure 8.5. Phase map of EBSD scan I-1, in which the hcp α phase is coloured red and the bcc β phase is coloured green. Grain boundaries are indicated by a black line. Figure 8.6. Image quality map of scan I-1. 54

66 Investigation of Titanium α Plates by EBSD Analysis Figure 8.7. IPF map of scan I-1with the IPF color legend. The induced stresses due to the β to α transformation can be qualitatively visualised by creating kernel misorientation maps. In the kernel maps the degree of misorientation correlated to the neighbouring data points are indicated by a colour scheme. A kernel map and the associated colour legend are shown in figure 8.8, giving the misorientation in degrees. Red colours indicate a high degree of misorientation compared to the surrounding data points; blue colours indicate a low misorientation and thus low strains. Figure 8.8. Kernel average misorientation plot of scan I-1with the colour code legend. The minimum and maximum angular deviations are indicated for each colour. 55

67 Results 8.3. Grain boundaries β-β grain boundaries The point scans were performed in order to obtain a larger amount of misorientation data between adjacent β grains than can be obtained by automated EBSD mapping. They were done at manually chosen locations in the β phase to include a maximum number of β grains. An example of a SEM image with the locations of the point scans and the β grain identification is given in figure 8.9. The point scans were labelled with the point scan number and the different β grains were indexed with a letter and the number boxes belonging to the same β grain were given an equal colour. Scanned points that appeared to be wrongly indexed were not included in the analysis and are manually coloured red in the SEM images. With the obtained β orientations, the misorientations between the β grains were calculated. Combined with the grain boundaries from the EBSD maps, a total of 162 different β-β grain boundary misorientations were measured between 105 β grains. A histogram of the measured grain boundary misorientations compared with the MacKenzie distribution function is given in figure Figure 8.9. SEM image of a Ti-4.5Fe-6.8Mo-1.5Al alloy isothermally held at 615 for 18 minutes, with the locations of the point scans indicated. 56

68 Investigation of Titanium α Plates by EBSD Analysis Frequency Misorientation ( ) Figure The measured β-β misorientation distribution compared with the MacKenzie misorientation distribution, indicated by the blue line [29]. For each analysed grain boundary the degree of α plate nucleation was determined. The degree of α plate nucleation was classified in 6 different types: A. No α plates at the grain boundary. B. α plates at one side of the grain boundary, none at the other side. C. Few α plates at both sides of the grain boundary. D. Many α plates at one side of the grain boundary, and few α plates at the other side. E. Many α plates at both sides of the grain boundary. F. Undefined, i.e. bad SEM picture or very short grain boundary. This classification is used to obtain a simple overview of the influence of the grain boundary misorientation on the α nucleation rate. The influence of the grain boundary misorientation on the α nucleation are given in table 8.1 and figure Type F grain boundaries were not included in figure The statistical probability f that one type of β-β grain boundary, low or high-angle, belongs to a particular α nucleation class is calculated according to the distribution function described by Spoormaker [66] m + 1 f = (eq. 8.1) N + 2 and m + 1 m + 2 m + 1 σ f = (eq. 8.2) N + 2 N + 3 N + 2 where N is the total number of data points, which was162 for all boundaries. m is the number of data points that fall within the specified class and σ f is the standard deviation. 57

69 Results Also the occurrence of CSL boundaries was calculated. This was done by using the Euler angles of the various CSL misorientations up to a value of Σ of 35 from table 4.1. It is also possible to determine CSL boundaries with OIM. The maximum determinable value of Σ by OIM is 49. The results of the found CSL grain boundaries and the corresponding grain boundary nucleation classes are given in table 8.2. Table 8.1. The observed occurrences of α nucleations for the different misorientation angle ranges and for all grain boundaries combined. Grain Observed number of grain boundaries (m) boundary All grain boundaries type A B C D E F Total Statistical probability Type A Type B Type C Type D Type E all grain boundaries Grain boundary misorientation range Figure The measured normalised occurrences of a particular α nucleation class divided over four β-β grain boundary misorientation ranges and for all measured grain boundaries. 58

70 Investigation of Titanium α Plates by EBSD Analysis Table 8.2. Details of the CSL grain boundaries that were found. Σ Scan Deviation from CSL boundary Grain boundary misorientation Grain boundary type E A E C 39b C 41b C 49c C α-β grain boundaries The α-β grain boundaries were analysed by colouring specified axis/angle pairs between neighbouring α and β crystals. In the EBSD maps, the α-β grain boundaries that obeyed the ) with an orientation spread of 5 were indicated with a blue line Burgers OR (45.3 at [ ] and all other grain boundaries were indicated by a black line. An example of the grain boundary map is given in figure The blue lines can be used to determine which β grain is the parent of the specific α plate. This was used for a more detailed analysis of the relation between the parent β grains and the product α plates. For small plates at grain boundaries, it was sometimes not directly clear in which β grain the α plates was growing. Figure Grain boundary map of scan I-1in which the blue lines indicate the Burgers OR. All other grain boundaries are indicated by a black line. 59

71 Results α-α grain boundaries In section 3.3, the possible misorientations between α plates within one single parent grain according to the Burgers OR are given. There are 6 possibilities between α plates from one single parent grain, each with a different expectancy. The lengths of the different α-α grain boundaries that were found during EBSD analysis were determined by OIM with an orientation spread of 2. This range is small enough to distinguish between the various grain boundaries, since the minimum difference in angle between any grain boundary variant is calculated to be An example of a grain boundary map in which the different types of α- α are indexed is given in figure The axis/angle pairs and the total length of the grain boundaries are given in the legend. The other grain boundary maps are given in appendix A.2. The total lengths from all EBSD scans were used to create a histogram of the frequencies of the observed grain boundary types, which is shown in figure The measured grain boundary lengths are compared with the random occurrences in figure It is not possible to measure type 1 boundaries, and therefore they are not included. It is important to notice that the grain boundary frequencies given in figure 8.15 are compared to the random situation in which type 1 α-α grain boundaries do not exist. For example, the frequency of type 2 grain boundaries is compared with a random occurrence of 24 out of 132 possible grain boundaries, because the 12 possible type 1 grain boundaries are excluded from the calculations. Figure Grain boundary map of scan III-3. The 5 possible α-α grain boundaries are indicated by different colors. 60

72 Investigation of Titanium α Plates by EBSD Analysis Frequency α α grain boundary type Measured Random Figure The relative length of the different types of α-α grain boundaries of all EBSD scans and the occurrence in a random situation (section 3.3) Relative frequency Measured relative to random α α grain boundary type Figure The difference in occurrence from a random situation for the 5 types of α-α grain boundaries of all EBSD scans measured with OIM Pole figures An important EBSD analysis tool is the construction of pole figures. For an analysis of the Burgers OR and the influence of parent crystal orientations on variant selection, the pole figures of the (0001) and (110) planes and [1120] and [111] directions were constructed. An example of these pole figures is given in figure The symmetries of the crystal lattices are also indicated in the pole figures. For a single crystal orientation there is only one {0001}reflection and there are three different < 1120 > symmetries, with a misorientation of 60 around the [0001] axis. The colours in the IPF in figure 8.16 correspond to the colours of the pole figure reflections. When a (0001) plane of an α plate is parallel to a (110) plane of a 61

73 Results β, the reflections will be on the same location in the pole figures. A complete overview of the pole figures of α plates at β-β grain boundaries is given in appendix B in which the colours in the IPF correspond to the colours of the poles. From all pole figures, the angles of the closest (110) planes between two adjacent β grains were calculated and compared with the orientation of the (0001) plane of the α plates that nucleated at the grain boundaries. This is done because some literature refer the common (110) poles between two adjacent β grains to be an important cause of variant selection [18, 57]. Figure 8.17 shows a histogram in which the frequency of β-β grain boundaries at which the (0001) planes are parallel to the closest (110) planes is given for different ranges of the smallest angles between those closest (110) planes of the two adjacent β grains θ (110). Figure Pole figures according to the Burgers OR of the coloured microstructure shown on the left, the orientations of the crystal lattices are also indicated on the grain map on the left by the unit cells. 62

74 Investigation of Titanium α Plates by EBSD Analysis Percentage all grain boundaries Smallest angle between (110) planes Figure The percentage of grain boundaries at which the (0001) planes of the α plates are parallel to one of the two (110) planes of the two adjacent β grains that have the smallest angle with each other Orientation relations In the chapter 3 several orientation relationships for bcc-hcp transformations are described. The most commonly described orientation relations for titanium alloys are the Burgers OR and the Potter OR. Also the Pitsch-Schrader OR is sometimes given as an approximation of the other two orientation relationships. To test the validity of the three possible orientation relations, the percentages of measured bcc to hcp misorientations that fall within eight specified ranges around the three predicted orientation relations are calculated. The method of calculating the misorientations has been described in the chapter 7. The percentages of the angular deviations from the orientation relations are given in tables 8.3 to 8.5 for different misorientation ranges with increasing angle of deviation. The α plates have been divided in three types, primary α plates, secondary α plates and intragranular secondary α plates. The primary α plates are the α plates measured in scan II-1 to II-4, the secondary α plates are the α plates at the grain boundary in scan I-1 to I-11 and the intragranular α plates are the plates in scan III-1 and the intragranular α plates at the top of scan I-3. The smallest possible misorientations between each pair of the three orientation relations and between two variants of one single orientation relation has been calculated and are given in table

75 Results Table 8.3. Percentages of measured misorientations that fall within a specified maximum misorientation with the Burgers OR. Types of α Maximum angular deviation plates Primary Secondary Intragranular Table 8.4. Percentages of measured misorientations that fall within a specified maximum misorientation with the Potter OR. Types of α Maximum angular deviation plates Primary Secondary Intragranular Table 8.5. Percentages of measured misorientations that fall within a specified maximum misorientation with the Pitsch-Schrader OR. Types of α Maximum angular deviation plates Primary Secondary Intragranular Table 8.6. Smallest misorientation angles between two variants of the orientation relations. OR Burgers Potter Pitsch-Schrader Burgers Potter Pitsch-Schrader 30 Plotting the misorientation distribution in the R-F space is an excellent method of visualising orientation relations and the occurrence of variant selection. The general method used to calculate the Rodrigues vectors for the R-F space was described in chapter 7. The normalised Rodrigues vectors and the rotation angles θ for the three described orientation relations are given in tables 8.7 to 8.9. Due to symmetry, there are only four different vectors for the Burgers and Potter OR and two for the Pitsch-Schrader OR. An example of a R-F plot of the misorientations between parent β grains and product α plates is given in figure 8.18 in which the misorientations are coloured red. The orientation of the parent β grain is chosen to be the reference system. The three orientation relationships are indicated by a circle (Burgers), a triangle (Potter) and a diamond (Pitsch-Schrader). This figure also gives an indication of the measured orientation spread relative to the angular differences between the ORs. In figure 8.19a and b, the orientations of the different variants within one β grain were also plotted in the R-F space. In this R-F space, the reference system is taken to be the sample axes. This was done to visualise in a convenient way the number of variants that are present in a specific area. In the figure the measured orientations of the α phase are plotted to distinguish each single variant. Figure 8.19a is constructed from data of scan I-6 and figure 8.19b represents the orientations of all α plates of scan III-1. 64

76 Investigation of Titanium α Plates by EBSD Analysis Table 8.7. Rodrigues vectors and misorientation angle of the four Burgers R-F variants. R-F Variant R 1 R 2 R 3 θ Table 8.8. Rodrigues vectors and misorientation angle of the four Potters R-F variants. R-F Variant R 1 R 2 R 3 θ Table 8.9. Rodrigues vectors and misorientation angle of the two Pitsch-Schrader R-F variants. R-F Variant R 1 R 2 R 3 θ Figure Plot of the misorientation between α plates and the parent β grain of scan I-4, compared with the Burgers ( ), Potter ( ) and Pitsch-Schrader OR ( ). 65

77 Results a Figure Plot of the variants that have nucleated in a β grain in the R-F space with variant selection (a) of I-4 in which four different variants are found and a plot without a clear sign of variant selection (b) of scan III-1 in which 10 different variants are found. b 66

78 Investigation of Titanium α Plates by EBSD Analysis 9. Discussion 9.1. Orientation relations The analysis aimed to investigate the strictness of the three different ORs, shows that both the Burgers and the Potter OR describe the relation between the α and the β phase correctly for approximately 75% of the measured points within a spread of 3. The Pitsch-Schrader OR does not describe the relation between the α and β phase correctly. Most of the calculated misorientations differ more than 5 with the Pitsch-Schrader OR (Table 8.5). Approximately 92% of the misorientations with the secondary α plates from the grain boundary fall within 5 from the Burgers and Potter OR, and 78% of the primary α plates fall within this range. For all plates it is found that the percentages are stable between 5 and 10 for the Burgers and Potter OR and the calculated percentages all not reach 100%. This means that the remaining α plates are misindexed α grains or misorientations that are calculated between an α plate and a β grain that was not the parent grain. For a proper comparison between the orientation relations and the different types of α plates, the fractions are therefore normalized according to the ending percentages, which are taken to be the percentages at 10. The normalized fractions for the Potter and Burgers OR are plotted for each misorientation range and different α plate types, see figure Fraction Burgers Primary Burgers Secondary Burgers Intragranular Potter Primary Potter Secondary Potter Intragranular Misorientation range ( ) Figure 9.1. The fractions of misorientations for different types of α plates that fall within a certain misorientation with the Burgers and Potter OR. The error bars have a length of 2σ. In figure 9.1 it is seen that the Burgers OR fits better for all types of α plates compared to the Potter OR, especially for the intragranularly nucleated secondary α plates. Only at the range of 2, the Potter OR has a higher frequency compared to the Burgers OR for the primary α plates. However, at this range the borders of 2 around both relationships overlap since the angular distance between the two is only 1.7 and the difference at 2 is much smaller compared to the differences at 0.8 and 1. When the frequencies of the three α plate types according to the different Burgers ranges are compared, it is seen that the intragranular plates have a higher 67

79 Discussion fraction compared to the other grain boundary α plates at every misorientation range. The intragranular secondary α plates were measured from scan I-3 and scan III-1, at which scan III-1 accounts for more than 3 times as many calculated misorientations. When the intragranular plates from both measurements are compared, it is seen that the intragranular plates of scan I-3 show the same strictness to the orientation relations as the primary and secondary α plates at the grain boundaries, see figure 9.2. The difference between the grain boundary α and the intragranular α is caused by the very good fit of the plates measured in scan III-1. The intragranularly nucleated plates of scan I-3 showed the same result as the grain boundary secondary α plates. The better Burgers OR fit is probably a result of the longer holding time for scan III-1 compared to the other secondary α plates which are treated according to treatment I, or due to differences that occurred during the sample preparation Fraction Burgers OR scan 3 Burgers OR scan Misorientation range ( ) Figure 9.2. The fractions of misorientations within ranger for the two orientation relations for the intragranular secondary α plates of two different EBSD scans. It was expected that there would be a difference in the strictness of the Burgers OR between secondary α plates and primary α plates as was previously described by Rhodes and Williams [7]. There is only a slight difference observed of the orientation relations at the spreads of 1.5 to 3. The small difference is probably caused by the stresses between the secondary α plates Grain boundary misorientation Misorientation distribution In chapter 4, the theory of grain boundary misorientations was described. According to the Read and Shockley theory [30], low-angle grain boundaries have a lower grain boundary energy than high-angle grain boundaries and heterogeneous nucleation is therefore preferred at high-angle grain boundaries. The influence of the β-β grain boundary misorientation on the nucleation of secondary α plates was analysed by determining the orientations of β grains followed by calculating the misorientation of β-β grain boundaries. The measured misorientations are compared with the random misorientation of the MacKenzie distribution 68

80 Investigation of Titanium α Plates by EBSD Analysis [29] to get an indication of the spread of measured grain boundary misorientations. A strong deviation from a random distribution can imply that factors like preferred texture in the initial structure influences the α nucleation. Figure 8.10 shows the measured misorientation distribution compared to the MacKenzie distribution. The measured misorientation distribution is compared with MacKenzie according to the χ 2 goodness of fit method [67] in which the misorientations have to be displayed in a histogram for comparison with the random distribution. For a set of n data, the number of bins k can be calculated with [68] k = log n (eq. 9.1) which results in the use of 8 bins. The general function for the χ 2 goodness of fit test is [67] χ ( O E ) 2 k 2 i i = (eq.9.2) i= 1 Ei O i is the observed frequency for bin i and E i is the expected frequency form literature, in this case the MacKenzie distribution [29]. The expected frequency is calculated by E = F( Y ) F( Y ) (eq. 9.3) i u l F(Y u ) is the cumulative distribution function of the upper limit of the bin and F(Y l ) is the cumulative distribution function of the lower limit of bin i. The probability that a random distribution has an equal goodness of fit can be found by comparing the found value of χ 2 with the tables from literature [67]. The value of χ 2 for the measured data compared to the MacKenzie distribution is 9.84 and given in table 9.1. The corresponding value of P is 0.2, whereas the value of P will be 1 if the measured dataset corresponds exactly to the expected values. The value of 0.2 means that the measured misorientation distribution differs significantly from a random distribution. The single values of χ 2 for each bin are plotted in figure 9.3 to show in which misorientation angle region the deviation from a random situation is the largest. Negative values in figure 9.3 correspond to a misorientation range with less grain boundaries than what is expected; positive values correspond to more observed grain boundaries compared to the random expectation. It can be seen that for the first bin, the deviation value is much higher compared with the other bins. This bins accounts for more than 40% of the total value of χ 2. The other deviation values are each less than half of the value of that of the first bin. This means that in the range 0 to 8 a strong deviation from a random distribution is present. Table 9.1. Comparison of the measured misorientation spread to the MacKenzie distribution. Bin number (i) Misorientation range O i MacKenzie number E i Deviation value fraction Total

81 Discussion 5 4 Deviation value Bin (i) Figure 9.3. The deviation values for each histogram bin. The results of the χ 2 goodness of fit test show that the grain boundary distribution is not random, there are more low-angle grain boundary observed than what would be expected. However, the large χ 2 is caused by measuring two extra grain boundaries in the first misorientation region, the observed number of grain boundaries (O i ) is 3, whereas the expected number (E i ) is 1. Thus only a few extra grain boundaries cause a large value of χ 2. Therefore it is assumed that the measured data set is not large enough and that the deviation from random is not influenced by the texture of the β phase. It was consequently not necessary to perform more measurements to observe a random misorientation distribution Grain boundary nucleation When the α plate nucleation is compared with the CSL values of the β grain boundaries, it is found that there is no relationship between the α plate nucleation and the Σ value. At the Σ=3 grain boundary many α plates were found, but at the Σ=5 grain boundary no plates were observed. However, the number of observed CSL boundaries is small and also the angular deviation has to be taken into account. Therefore the CSL data are insufficient to draw any conclusions about the relation between the CSL value and the α plate nucleation. The measured grain boundaries were divided into four misorientation ranges of 15 wide to analyse the influence of the grain boundary misorientation on α plate nucleation. The frequencies of the different α plate nucleation types have been plotted for the four misorientation ranges and for all measured grain boundaries combined (Figure 8.11). To analyse the influence of the grain boundary misorientation on the α plate nucleation grade, different weight factors are attributed to the different nucleation types. Sides of grain boundaries with no plates are given the weight of 0, sides with few plates are given the weight of 1 and sides with many plates have a weight of 2. The resulting weight factors are given in table 9.2. Type B grain boundaries are given a factor 1.5 because both grain boundaries with many plates at one side as boundaries with only few plates at one side are included. The weight factors are multiplied by the number of grain boundaries of the particular type and then divided by the total number of determined grain boundaries in the particular grain boundary 70

82 Investigation of Titanium α Plates by EBSD Analysis misorientation range. The resulting values are an indication of the α plate nucleation grade for the different misorientation ranges and are given in figure 9.4. Table 9.2. The weight factors for the different grain boundary nucleation types. Grain boundary Weight factor nucleation type A 0 B 1.5 C 2 D 3 E 4 F Not included Grain boundary average weight all grain boundaries Grain boundary misorientation range Figure 9.4. The average weights of the grain boundaries for the different misorientation ranges. There is a significant difference between the lower two (<30 ) and higher two (>30 ) grain boundary misorientation ranges for the nucleation grade. At approximately 50% of the grain boundaries with the highest misorientation, many α plates nucleate at both sides. For low misorientation, it is found that approximately 25% show this type of α nucleation (Figure 8.11). Also the frequencies of type A grain boundaries decrease significantly for larger misorientation angles and become stable at the misorientations above 30. Also the occurrence of grain boundary types B and C tend to decrease with increasing grain boundary misorientation. Only few type D grain boundaries are found for all misorientation ranges. These results in figure 9.4 show that the α plate nucleation is influenced by the grain boundary misorientation; the amount of α plate nucleation increases for increasing grain boundary misorientation angles. This has some analogy with the theory of Read and Shockley, that describes the influence of the misorientation angle on the grain boundary energy [30], which 71

83 Discussion results in different heterogeneous nucleation rates. However, they describe a clear distinction between low-angle grain boundaries that have a misorientation below 10 to 15 and highangle grain boundaries. When the data of figure 9.4 are analysed it is seen that no clear distinction between this classification of low and high-angle grain boundaries is found. The nucleation grade continues to increase after the range. Therefore, the amount of nucleation at the grain boundaries cannot be explained by dividing the grain boundaries in low and high-angle grain boundaries. The classification between grain boundary misorientation does also not explain why many α plates are found at one side of a grain boundary while none are found at the other side of the boundary. The results indicate that the classical grain boundary energy as derived by Read and Shockley is not primarily determining the α plate nucleation rate. Probably the inclination of the grain boundary and the relative orientation of the (110) β are also influence the amount of α plate nucleation α-α grain boundaries It is found that the secondary α plates sometimes tend to branch into one or more sideplates with a different crystal orientation (Figure 8.1). Often it seems that a change of crystal orientation is required before the plates are able to grow further into the β grain. This is illustrated in figure 9.5 which is a detail of scan I-6. In chapter 3, the possible misorientations between α plates that obey the Burgers OR from one single β grain are described. There are six different misorientations possible when the misorientations are calculated between any of the twelve variants. The lengths of each grain boundary type were determined with OIM for each EBSD scan. When the total lengths of the different types of α grain boundaries are compared to the theoretical predictions, it is seen that grain boundary type 2 is found much more often than the other grain boundaries (Figure 8.15). Type 2 grain boundaries are found 2.5 times more than what would be expected in random situations. Figure 9.5. Detail of scan I-6 in which the α plates seem to branch into new crystal orientations to increase the α plate volume fraction in the parent grain. The high occurrence of type 2 grain boundaries was investigated by Wang et al. [14]. They also measured a frequency of 2.5 times the random situation for type 2 grain boundaries. This is in accordance with the results shown in figure However, the report also shows a frequency of type 4 grain boundaries that is 1.8 times the random situation, which was not observed during the present research, in which type 4 grain boundaries were found less frequent than what is expected in a random situation. It was also stated [14] that type 6 grain boundaries have an expectancy of 0.036, which is much lower than observed at the measurements done on Ti-LCB. Wang attributes the high frequencies of type 2 and 4 grain 72

84 Investigation of Titanium α Plates by EBSD Analysis boundaries to self-accommodation. Self-accommodation is the effect that the transformation induced shear strain is lowered by the combination of two or more variants of the product phase. Wang found that the lowest shear strains for three variants result in variants that form type 2 grain boundaries and the second lowest shear strains result in type 4 boundaries [14]. Two other explanations can be proposed that cause the high occurrence of type 2 boundaries. The first is that this boundary type has a low grain boundary energy compared to the other grain boundaries. The other is that preferred crystal orientations of the α plates cause this type of grain boundaries when they collide. Therefore further analysis was needed to find out which factor causes this grain boundary selection. This was done by calculating the frequencies of the different boundaries by comparing the possible misorientations between each measured α pixel, to include the distribution of crystal orientations. In contrast, the measured grain boundaries by OIM are based on neighbouring α pixels. It was chosen to perform this analysis on the intragranular α plates of scan III-1 and on α plates near a β-β boundary (scan I-6). The results are given in figures 9.6 to 9.9. In figure 9.6 and 9.7, the misorientation occurrences calculated from the orientation distributions are compared with the measured lengths from OIM and the random distribution. The values of the distribution data are much closer to the random distribution compared to the values measured with OIM. The measured values are compared with the random occurrence and with each other in figures 9.8 and 9.9. In complete random situations, the last three bars in figures 9.8 and 9.8 will have a height of 1. In scan III-1, the measured data from the orientation distribution are close to a random distribution, whereas the measured grain boundary frequency by OIM deviates strongly from a random situation, indicated by the yellow bars (figure 9.8). This difference indicates that the occurrence of the type 2 boundary is not caused by favoured orientation of the α plates during individual nucleation, but that this grain boundary is highly favoured compared to the other possible grain boundaries. A similar deviation between the grain boundary frequencies from OIM and the frequencies from the orientation distribution for type 2 grain is found for scan I-6 in figure 9.9, indicated by the height of the purple bar. The conclusion can be drawn that the type 2 has a lower grain boundary energy compared to the other α-α grain boundary types. 1 Frequency Measured w ith OIM Measured from orientation distribution α α grain boundary type Random Figure 9.6. The normalised frequencies of the different α-α grain boundaries and the calculated frequency from the measured orientations of EBSD scan III-1 compared with the random situation. 73

85 Discussion Frequency α α grain boundary type Figure 9.7. The normalised frequencies of the different α-α grain boundaries and the calculated frequency from the measured orientations of EBSD scan I-6 compared with the random situation. 6 Relative frequency α α grain boundary type OIM data relative to random Distribution data relative to random OIM data relative to distribution data Figure 9.8. The deviations from random for the grain boundaries measured by OIM and the orientation distribution and the OIM measurement compared to the distribution data of scan III-1. 74

86 Investigation of Titanium α Plates by EBSD Analysis 6 5 Relative frequency α α grain boundary type Figure 9.9. The deviations from random for the grain boundaries measured by OIM and the orientation distribution and the OIM measurement compared to the distribution data of scan I-6. The measurements done during this project did not show any sign of self-accommodation, which indicates that other factors cause the high occurrence of the type 2 grain boundaries. It is claimed that the high frequencies of type 2 grain boundaries are a result of autocatalysis. The results of the preferred grain boundary show that the side plates have nucleated at the grain boundaries of already existing plates. The autocatalysis is observed for both grain boundary and intragranular secondary α plates and was not found at primary α plates. At the grain boundary plates, it seems that the plates have nucleated to enable the plates to increase the fraction α plates in the β grain, because α plates from the same grain boundaries often have similar growth directions. The grain boundary energies for several different grain boundaries between hexagonal crystals are described in chapter 4, but the description of hcphcp grain boundaries is complicated due to the deviation of the c/a ratio for different alloys Variant selection In section 8.3 it is seen that there is a high degree of variant selection for the secondary α plates at the grain boundaries. At each side of the grain boundary, often only a few of the possible 12 variants are present, in contrast with scan III-1 where ten out of twelve variants are found. These plates have nucleated at a distance from any grain boundary or nucleated in an autocatalytic manner with the preferred type 2 α-α boundary. This implies that the variant selection is caused by the influence of grain boundaries on the nucleation. There are several possible explanations for the observed variant selection at the grain boundaries in previous literature [18, 56, 57, 69, 70]. The first explanation includes the surface energy minimization between the product phase and the grain boundary. Bhattacharyya [18] and Stanford and Bate [57] found that the α crystals that grow at β-β grain boundaries often have an orientation of the (0001) planes that is parallel to the (110) plane of the parent β grain that has the smallest angular deviation regarding to any 75

87 Discussion (110) plane of the β grain at the other side of the grain boundary. This was found for diffusional transformations and is in accordance with the pillbox nucleation shape model where the nucleus is considered to have coherent and semicoherent interfaces [71]. Stanford and Bate reported a maximum angle of approximately 10 between the (110) planes of the two β grains involved for which this rule of variant selection applies. This selection of the (0001) plane parallel to close (110) planes would reduce the interfacial energy of the grain boundary nucleus, according to the Burgers OR. The results of the measurements done during this project show that this rule applies at approximately 50% of the grain boundaries. For angles below 5 between the two (110) planes, there is a higher tendency to select the variant with the (0001) plane parallel to the common (110) planes, which decreases for higher angles between the planes (Figure 8.17). The pole figures of a situation in which the (0001) plane is parallel with the common (110) planes is given in figure In some cases there seems to be no connection between the orientations of the (110) planes of both β grains and the orientation of the α plates at the grain boundary, even though the angle between two (110) planes is in the range of 3 to 7. An example is given in figure 9.11 in which the angle between the (110) planes is 6.6. The pole figures describing the variant selection at the other grain boundaries are given in appendix B in which the angles between the most common (110) planes (θ (110) ) are also indicated. At more than 50% of the determined grain boundaries there is a tendency of variant selection with the (0001) plane parallel to the closest (110) planes. When this value is compared to a probability of 1/6 in random situations, it is seen that the frequency of the variants is more than 3 times random. When the described variant selection rule applies, there is a high occurrence of α plates with the preferred (0001) plane. For one (0001) orientation, there are two different crystal orientations possible of the α phase. These two crystal orientations will result in type 6 α-α grain boundaries when they collide. This is shown in figure 9.9 in which there is a high tendency of type 6 grain boundaries. It shows that the high occurrence of type 6 boundaries is caused by preferred orientations of the α plates which is illustrated by the high value of the light blue bin. Also figure 8.15 in which all EBSD scans are combined shows a high occurrence of type 6 boundaries compared to random, which can be explained by the variant selection rule of Bhattacharyya [18] and Stanford and Bate [57]. 76

88 Investigation of Titanium α Plates by EBSD Analysis Figure Pole figures of scan I-4. The α plates have nucleated at a grain boundary with the (0001) pole parallel to the common (110) pole of the β grains. The poles of the closest (110) planes and the parallel (0001) plane are indicated by the circles. The minimum angle between the (110) planes is 2.5. Figure Pole figures of scan I-4 in which the α plates do not have their (0001) plane parallel to the common (110) planes (yellow and red). The poles of the closest (110) planes are indicated by the circle and it can be seen that no (0001) plane is parallel to any of both (110) planes. The minimum angle between the (110) planes is 6.6. The observation of variant selection that differs from the criterion described by Bhattacharyya [18], and Stanford and Bate [57] implies that there are more characteristics that can cause variant selection. In some publications [56, 69] the variant selection of nucleation at grain boundaries is described as a result of selecting the variant that has the closed packed direction as parallel as possible to the grain boundary normal direction. These measurements are done 77

89 Discussion on grain boundary α crystals in a Ti-15.3 alloy. It was found that the low energy facets, which are the closed-packed planes, are nearly parallel to the grain boundary planes, which results in low energy interfaces between the nucleus and the grain boundary. During this project it was however not possible to determine the grain boundary planes. Therefore it was not possible to investigate this variant selection description in detail. Humbert et al. [70] described variant selection in titanium alloys as a product of anisotropic strain fields and a preferred texture of the β grains. It was chosen not to analyse this description further because no preferred texture was found within the β grains. It is also rather difficult to investigate the possible anisotropic strain fields and their influence on the nucleation energy. The observation of grain boundaries at which at one side many α plate have nucleated and none at the other can be explained as an extreme form of variant selection. At the side where no α plates are found, no nucleus variant will probably have a low enough nucleation energy. The in this thesis described variant selection rule cannot explain this form of extreme variant selection. Both the influence of the grain boundary misorientation and the angle between the most common (110) planes will result in an equal form of variant selection at both sides of the grain boundary. This is however not observed. These situations will be probably be caused by the influence of the orientation of the grain boundary plane on the interface energy or due to the presence of anisotropic strain fields. This indicates that the nucleation of displacive structures is dependent on many characteristics, which is also described in chapter 5 in which the complexity of displacive nucleations is described. 78

90 Investigation of Titanium α Plates by EBSD Analysis 10. Conclusions The investigation of both the primary and secondary α plates in Ti-LCB by EBSD analysis led to several different conclusions. In this chapter, the most important conclusions are given, concerning the orientation relations, the influence of grain boundary misorientation on the α plate nucleation, the α-α grain boundary selection and the variant selection. The analysis of the orientation relationships shows that the Burgers OR describes the relation between the parent β and the product α phase better than the Potter OR. There is no significant difference observed of the strictness of the Burgers OR between primary or secondary α plates at the grain boundaries, although they are formed by a different mechanism. The intragranular secondary α plates are stricter according to the Burgers OR, but this result is only measured in one of the two scans of which the intragranular α plates were analysed. In the other scan the results were similar as to the α plates at the grain boundaries. This difference between the intragranular plates and all other plates may be due the difference in heat treatment or due to an effect of the sample preparation. The grain boundary misorientation between the β grains influences the α plate nucleation; there is a significant difference between low-angle and high-angle grain boundaries of the degree of α plate nucleation. It is found that the grain boundary energy calculated by Read and Shockley does not give a correct relation between the grain boundary misorientation and the α plate nucleation. The theory does not explain why at some grain boundaries many plates have nucleated at one side, but none at the other side. When the misorientation between two β grains increases, the amount of α plate nucleation also increases, and becomes stable at misorientations above 30. When a specimen consists of many low-angle β-β grain boundaries, it is expected that less α plates are formed compared to specimens with many high-angle grain boundaries. During the analysis of the grain boundaries between the different variants of α plates, one particular α-α grain boundary is found much more often compared to a random distribution. Analysis on the orientations of the different α plates show that these particular grain boundaries were formed due to autocatalysis where the new α plates are formed at α-β interfaces. These α-α grain boundaries with an axis angle pair of [1120] /60 are considered to have a lower grain boundary energy than the other possible α-α grain boundaries. The type 2 grain boundaries will therefore probably have a high coherency. At almost all β-β grain boundaries, a high degree of variant selection of the α orientations is found for secondary α plates. When the orientations of the plates at the grain boundaries are compared with the orientations of the intragranular α plates, it is found that the variant selection is caused by influences of the β-β grain boundary. Analysis of the closest (110) planes of the two β grains at grain boundaries shows that there is some tendency of choosing the α variant with the (0001) plane parallel to this most common (110) plane for angles between the planes up to 10. However, there are also some situations observed where a high degree of variant selection is found, for which the (0001) pole is not parallel to close (110) planes. This indicates that also other factors cause the variant selection. One possibility is that the orientation of the grain boundary plane influences the interfacial energy between nuclei and the grain boundary, which will cause the unexplained variant selection. It was not possible to investigate the 3-dimensional structure of the grain boundary planes during this project. Also the α-α grain boundary selection as described above leads to variant selection during the branching of the α plates. When one crystal orientation of the α phase is formed at 79

91 Conclusions the grain boundary, only two different crystal orientations are possible that will create a type 2 α boundary. The type 2 α-α grain boundaries are only possible between sets of three different α orientations. The tendency to form the type 2 α-α grain boundaries, will therefore also result in some kind of variant selection. The results from the analysis on the secondary α plates provided more insight in the characteristics that play a role during displacive transformations mechanisms. In chapter 5 it is stated that displacive transformations are still not fully understood. The results from this thesis provide information about the dependence of the parent structure on the nucleation of the secondary α plates and the difference between displacive transformations at grain boundaries and intragranular displacive transformations. 80

92 Investigation of Titanium α Plates by EBSD Analysis 11. Recommendations With EBSD analysis it is possible to perform many different investigations regarding to the orientation of the crystals. It is therefore not possible to perform a complete analysis of the α plates during the time of this project. In this chapter several recommendations are given for further analysis on the α plates, not only with EBSD but also with other analysing techniques. An important aspect of this thesis is the analysis of the grain boundaries between two α plates. The results show that the α-α grain boundaries are a result of autocatalysis in which the new secondary α plates are formed at the sides of the already existing plates. The autocatalysis makes it possible for the β to α transformation to occur in all directions and not only at the tip of the α plates, to increase the volume fraction of the α phase. There are many descriptions of grain boundaries of cubic-cubic systems, but the hexagonal-hexagonal system has not been described in full detail. One of the problems is that the grain boundary energy often is dependent on the c/a ratio, as described in chapter 4. To the author s knowledge, there is little research published about autocatalysis of displacive plates in titanium structures and the influence of the grain boundary energy of the hexagonal system. It is recommended to perform calculations on the nature of the type 2 α-α grain boundaries using the lattice parameters of the α and β crystals in the alloy used during this project. Also analysis with TEM of 3-D EBSD with a focussed ion-beam (FIB) on the structure of the α plates may explain the tendency of autocatalysis of the α plates and why it is in many cases favourable to form a branched structure instead on lengthening or thickening of the present plates. It may be assumed that the α plates do not nucleate exactly at the same time at both sides of the grain boundary. It is very likely that the orientation of the α plates that nucleate at a later stage will have an orientation that will be as close as possible to the orientation of the plates that already have nucleated at the grain boundary, due to the generally high volume fraction of α plates at the grain boundaries. It is likely that special grain boundaries as twins or CSL boundaries are formed, although the chance that one of the 12 variants will result in such a grain boundary is rather small. Investigation of the characteristics of the α-α grain boundaries, will possibly make it possible to analyse at which side of the β-β grain boundary the α plates have nucleated first. Another factor of interest in this thesis is the analysis of the variant selection during α plate nucleation. Displacive transformations are one of the most interesting open fields of metals science. There has already been much research on displacive transformations at grain boundaries of various metals and alloys, but there are still many parts not fully understand. It is therefore not possible to get a complete view of the mechanisms that play a role during the nucleation and growth of the α plates that were investigated in this report. In this thesis several theories from literature are verified. The conclusions drawn in this report show that there are many characteristics of the grain boundaries that influence the variant selection during nucleation. It has been described in chapter 4 that grain boundaries have more dimensions than could be determined by the EBSD analysis. The results show that a 2- dimensional analysis is not sufficient to describe the full process of grain boundary displacive nucleations. Therefore it is necessary to perform further analysis on the grain boundaries by TEM or 3-D EBSD to get an enhanced view of the grain boundary structure. A detailed analysis of the grain boundaries will also give more insight on the coherency between both the primary and secondary α plates and the non Burgers OR β grain at the other side of the β-β grain boundary. This information can be used for developing nucleus shape models and nucleation rate theories. 81

93 Recommendations It is found that for equal crystal orientations, the growth directions are not always the same. For one crystal orientation of the α plates, at least two growth directions are possible, the growth direction of the α plates is not determined by the crystal orientation alone. The observation that α plates with an equal orientation grow in a different direction when they nucleate at different locations suggests that the local stress field surrounding the nucleus has an influence on the growth behaviour of the α plates. Analysis of α plates with equal crystal orientations but different growth directions can provide further information of displacive growth kinetics. The last recommendation that will be made is the analysis of the structures by EPMA. In chapter 3 some results of EMPA analysis on the alloying elements are shown and described. However, it is known that T 0 is dependent on the concentration of interstitial elements. It is possible that the concentration of interstitial elements was not homogeneous in the sample during the transformations, which will have an effect on the nucleation of the α plates. A difference of these elements in the β grains can therefore explain why at some grain boundaries the α plate nucleation is much larger than for other grain boundaries with the same grain boundary characteristics. During this project is was not possible to perform measurements of the interstitial elements, it is foreseen that the results of the recommended experiments will increase our understanding of the mechanism of α plate nucleation in Ti alloys. 82

94 Investigation of Titanium α Plates by EBSD Analysis References [1] G. Lütjering, J. C. Williams, Titanium, Springer, Berlin [2] A. D. McQuillan, M. K. McQuillan, Titanium, [3] The Selection and Use of Titanium, A Design Guide, M. I. Service [4] E. S. K. Menon, H. I. Aaronson, Acta Metallurgica 1986, 34, [5] S. M. C. van Bohemen, et al., Physical Review B 2006, 74. [6] H. I. Aaronson, et al., Transactions of the American Institute of Mining and Metallurgical Engineers 1957, 209, [7] C. G. Rhodes, J. C. Williams, Metallurgical Transactions a-physical Metallurgy and Materials Science 1975, 6, [8] R. F. Bunshah, Transactions of the American Institute of Mining and Metallurgical Engineers 1958, 212, 424. [9] T. Furuhara, et al., Materials Transactions Jim 1998, 39, 31. [10] E. S. K. Menon, H. I. Aaronson, Acta Metallurgica 1986, 34, [11] T. Furuhara, et al., Metallurgical Transactions a-physical Metallurgy and Materials Science 1990, 21, [12] M. Shibata, K. Ono, Acta Metallurgica 1975, 23, 587. [13] M. Shibata, K. Ono, Acta Metallurgica 1977, 25, 35. [14] S. C. Wang, et al., Acta Materiala 2003, 51, [15] Y. C. Liu, Trans. AIME 1957, 206, [16] J. D. Eshelby, Proceedings of the Royal Society of London Series a-mathematical and Physical Sciences 1957, 241, 376. [17] T. Furuhara, et al., Acta Metallurgica Et Materialia 1991, 39, [18] D. Bhattacharyya, et al., Acta Materialia 2003, 51, [19] M. Enomoto, M. Fujita, Metallurgical Transactions a-physical Metallurgy and Materials Science 1990, 21, [20] J. L. Murray, Phase Diagrams of Binary Titanium Alloys, OH [21] H. Guo, et al., Scripta Materialia 2000, 43, 899. [22] H. J. Lee, H. I. Aaronson, Acta Metallurgica 1988, 36, 787. [23] H. Guo, M. Enomoto, Acta Materialia 2002, 50, 929. [24] J. P. Hirth, et al., Acta Materialia 1998, 46,

95 References [25] H. K. D. H. Bhadeshia, Worked examples in the geometry of crystals, Institute of Metals, London [26] T. Karthikeyan, et al., Scripta Materialia 2006, 55, 771. [27] Y. Ohmori, et al., Materials Transactions Jim 1998, 39, 49. [28] D. A. Porter, K. E. Easterling, Phase Transformations in Metals and Alloys, Stanley Thornes Publishers, Cheltenham [29] J. K. Mackenzie, Biometrika 1958, 45, 229. [30] W. T. Read, W. Shockley, Physical Review 1950, 78, 275. [31] P. H. Pumphrey, K. M. Bowkett, Scripta Metallurgica 1971, 5, 365. [32] H. Gleiter, Materials Science and Engineering 1982, 52, 91. [33] H. Faraoun, et al., Scripta Materialia 2006, 54, 865. [34] A. D. Rollet, P. Kalu, in Advanced Characterisation & Microstructural Analysis, MRSEC, Carnegie Mellon [35] G. J. Wang, V. Vitek, Acta Metallurgica 1986, 34, 951. [36] A. P. Sutton, R. W. Balluffi, Acta Metallurgica 1987, 35, [37] R. L. Fullman, J. C. Fisher, Journal of applied Physics 1951, 22. [38] D. Brandon, Acta Metallurgica 1966, 14. [39] M. J. Weins, Surface Science 1972, 31, 138. [40] J. W. Christian, The Theory of Transformations in Metals and Alloys, Vol. Part 2, Pergamon, Amsterdam [41] A. P. Sutton, J. Chen, Philosophical Magazine Letters 1990, 61, 139. [42] S. E. Offerman, Vol. Ph.D., Delft University of Technology, [43] P. J. Goodhew, ASM Materials Science Seminar , 155. [44] H. Eyring, Journal of Chemical Physics 1935, 3, 107. [45] K. C. Russel, Acta Metallurgica 1969, 17, [46] L. Kaufman, M. Cohen, Progress in Metal Physics 1958, 7, 165. [47] H. Y. Yu, et al., Materials Science and Engineering 1995, B32, 153. [48] N. N. Thadhani, M. A. Meyers, Progress in Materials Science 1986, 30, 1. [49] G. B. Olsen, Materials Science and Engineering 1999, A , 11. [50] S. R. Pati, M. Cohen, Acta Metallurgica 1969, 17, 189. [51] G. B. Olsen, W. S. Owen, Martensite, ASM International, [52] K. E. Easterling, A. R. Thölen, Acta Metallurgica 1976, 23,

96 Investigation of Titanium α Plates by EBSD Analysis [53] D. P. Koistinen, R. E. Marburger, Acta Metallurgica 1959, 7, 59. [54] H. Knapp, U. Dehlinger, Acta Metallurgica 1956, 4. [55] H. I. Aaronson, et al., Materials Science and Engineering B-Solid State Materials for Advanced Technology 1995, 32, 107. [56] T. Furuhara, T. Maki, Materials Science and Engineering A 2001, 312, 145. [57] N. Stanford, P. S. Bate, Acta Materialia 2004, 52, [58] M. Kato, et al., Science and Technology of Advanced Materials , 375. [59] M. Humbert, et al., Materials Science and Engineering 2006, A 430, 157. [60] V. Randle, Microtexture Determination and its applications, The Institute of Materials, Minerals and Mining, London [61] [62] V. Randle, O. Engler, Introduction to Texture Analysis, CRC Press, [63] F. Wagner, et al., Solid State Phenomena 2005, 105, 15. [64] A. Heinz, P. Neumann, Acta Crystallographica Section A 1991, 47, 780. [65] T. A. Kop, et al., Journal of Materials Science 2001, 36, 519. [66] J. L. Spoormaker, Ontwerpen voor bedrijfszekerheid, Delft University of Technology Februari [67] in Dataplot Vol. 1. Auxiliary Chapter, National Institute of Standards and Technology. [68] H. A. Sturges, J. Amer. Statist. As , 21. [69] T. Furuhara, et al., Metall. Mater. Trans. 1996, A27. [70] M. Humbert, et al., in Icotom 14: Textures of Materials, Pts 1and 2, Vol , 2005, [71] W. F. Lange III, et al., Metallurgical Transactions 6A

97 Appendix A. EBSD maps Appendix A. EBSD maps A.1. Inverse pole figures IPF Legend 86

98 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-1 EBSD scan I-2 87

99 Appendix A. EBSD maps EBSD scan I-3 88

100 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-4 EBSD scan I-5 89

101 Appendix A. EBSD maps EBSD scan I-6 EBSD scan I-7 90

102 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-8 EBSD scan I-9 91

103 Appendix A. EBSD maps EBSD scan I-10 EBSD scan I-11 92

104 Investigation of Titanium α Plates by EBSD Analysis EBSD scan II-1 EBSD scan II-2 93

105 Appendix A. EBSD maps EBSD scan II-3 EBSD scan II-4 94

106 Investigation of Titanium α Plates by EBSD Analysis EBSD scan III-1 95

107 Appendix A. EBSD maps A.2. Grain boundary maps Grain boundary legend. Grain boundary type Rotation angle Rotation axis Colour index Burgers OR (β-α) 45.3 [ ] β 2 (α α) 60 3 (α α) (α α) (α α) 90 6 (α α) α α α α α Scan I-1 Scan I-2 96

108 Investigation of Titanium α Plates by EBSD Analysis Scan I-3 97

109 Appendix A. EBSD maps Scan I-4 Scan I-5 Scan I-6 98

110 Investigation of Titanium α Plates by EBSD Analysis Scan I-7 Scan I-8 Scan I-9 99

111 Appendix A. EBSD maps Scan I-10 Scan I-11 Scan II-1 Scan II-2 100

112 Investigation of Titanium α Plates by EBSD Analysis Scan II-3 Scan II-4 Scan III-1 101

113 Appendix B. Pole figures Appendix B. Pole figures EBSD scan I-1, θ (110) =

114 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-1, θ (110) =

115 Appendix B. Pole figures EBSD scan I-1, θ (110) =

116 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-2, θ (110) =

117 Appendix B. Pole figures EBSD scan I-2, θ (110) =

118 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-2, θ (110) =

119 Appendix B. Pole figures EBSD scan I-2, θ (110) =

120 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-3, θ (110) =

121 Appendix B. Pole figures EBSD scan I-3, θ (110) =

122 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-3, θ (110) =

123 Appendix B. Pole figures EBSD scan I-3, θ (110) =

124 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-4, θ (110) =

125 Appendix B. Pole figures EBSD scan I-4, θ (110) =

126 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-4, θ (110) =

127 Appendix B. Pole figures EBSD scan I-4, θ (110) =

128 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-4, θ (110) =

129 Appendix B. Pole figures EBSD scan I-5, θ (110) =

130 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-5, θ (110) =

131 Appendix B. Pole figures EBSD scan I-5, θ (110) =

132 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-6, θ (110) =

133 Appendix B. Pole figures EBSD scan I-6, θ (110) =

134 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-6, θ (110) =

135 Appendix B. Pole figures EBSD scan I-8, θ (110) =

136 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-9, θ (110) =

137 Appendix B. Pole figures EBSD scan I-10, θ (110) =

138 Investigation of Titanium α Plates by EBSD Analysis EBSD scan I-11, θ (110) =

139 Appendix B. Pole figures EBSD scan II-1, θ (110) =

140 Investigation of Titanium α Plates by EBSD Analysis EBSD scan II-2, θ (110) =

141 Appendix B. Pole figures EBSD scan II-2, θ (110) =

142 Investigation of Titanium α Plates by EBSD Analysis EBSD scan II-2, θ (110) =

143 Appendix B. Pole figures EBSD scan II-2, θ (110) =

144 Investigation of Titanium α Plates by EBSD Analysis EBSD scan II-2, θ (110) =

145 Appendix B. Pole figures EBSD scan II-3, θ (110) =

146 Investigation of Titanium α Plates by EBSD Analysis EBSD scan II-3, θ (110) =