Rongpei Shi-Variant Selection during Alpha Precipitation in Titanium Alloys-Thesis

Variant Selection during Alpha Precipitation in Titanium Alloys

A Simulation Study

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

in the Graduate School of The Ohio State University

By

Rongpei Shi

Graduate Program in Materials Science and Engineering

The Ohio State University

2014

Dissertation Committee:

Yunzhi Wang, Adviosr

Suliman Dregia

Hamish Fraser

Copyright by

Rongpei Shi

2014

ii

Abstract

Variant selection of alpha phase during its precipitation from beta matrix plays a key role

in determining transformation texture and final mechanical properties of ⁄ and

titanium alloys. In this study we develop a three-dimensional quantitative phase field

model (PFM) to predict variant selection and microstructure evolution during beta to

alpha transformation in polycrystalline Ti-6Al-4V under the influence of different

processing variables. The model links its inputs directly to thermodynamic and mobility

databases, and incorporates crystallography of BCC to HCP transformation, elastic

anisotropy, defects within semi-coherent alpha/beta interfaces and elastic

inhomogeneities among different beta grains. In particular, microstructure and

transformation texture evolution are treated simultaneously via orientation distribution

function (ODF) modeling of alpha/beta two-phase microstructure in beta polycrystalline

obtained by PFM. It is found that, for a given undercooling, the development of

transformation texture of the alpha phase due to variant selection during precipitation

depends on both externally applied stress or strain, initial texture state of parent beta

sample and internal stress generated by the precipitation reaction itself. Moreover, the

growth of pre-existing widmanstatten alpha precipitates is accompanied by selective

nucleation and growth of secondary alpha plates of preferred variants.

We further develop a crystallographic model based on the ideal Burgers orientation

relationship (BOR) between GB and one of the two adjacent beta grains to investigate

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how a prior beta grain boundary contributes to variant selection of grain boundary

allotriomorph (GB ). The model is able to predict all possible special beta grain

boundaries where GB is able to maintain BOR with two neighboring grain. In

particular, the model has been used to evaluate the validity of all current empirical variant

selection rules to obtain more insight of how all grain boundary parameters

(misorientation and grain boundary plane inclination) contribute to variant selection

behavior titanium alloys. This work could shed light on how to control processing

conditions to reduce microtexture at both the individual grain level and the overall

polycrystalline sample level.

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Dedication

This document is dedicated to my family.

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Acknowledgments

I can't believe this day has finally come. Firstly, I would like to express my deepest and

sincere gratitude to my advisor, Prof. Yunzhi Wang for offering the opportunity to do my

Ph.D. study in the US that I had never imagined when I was in China. I joined the group

with little background of phase transformation in solid state. Thanks for his incredible

patience and constant support that allows me to survive after a long incubation time in

my learning curve in this field.

Thanks also go to the committee members, Prof. Suliman Dregia. Many fruitful

discussions with him contribute a lot to the work done in the thesis. In particular, his

enthusiasm and humor when discussing about scientific research also lead me to enjoy

doing research.

I must thank Prof. Hamish Fraser for giving me a really big picture about the physical

metallurgy of titanium alloys. Through working with Dr. Yufeng Zheng and Dr. Vikas

Dixit in his group, I find scientific problems remaining unsolved in the field of Ti-alloys

that I, as a modeler, can take over and have some contributions. This is the way I have

been doing most of work described in the thesis.

Thanks to Prof. Xingjun Liu and Prof. Cuiping Wang, my advisor at Xiamen University

in China. The couple gave me the best training in computation thermodynamics, and the

incredible flexibility for doing my Ph.D. study in the US.

vi

Thanks to Prof. Wenzheng Zhang at Tsinghua University for teaching me O-lattice

theory during Gordon Research Conference in 2009, and her continuing help to improve

my understanding of the theory that offers me a complete new insight to think about

phase transformations in solid.

Thanks to Dr. Chen Shen, Dr. Ning Ma and Dr. Ning Zhou. They showed me the beauty

of microstructure modeling that is the one of the most important reasons that I still have

enthusiasm to build new models, do coding, debugging and post-processing data for

publications, though most of them are only between 0 and 1, during midnight.

Special thanks to Dr. Chen Shen. He introduced me to do summer intern in General

Electric, Global Research Center. During that time, he taught me how to work with

industry people within a team and, more importantly, let me realize that how the

knowledge I learn in the college can be direct applied to the R&D of turbine engine.

Thanks to Dr. Ning Ma for the solid basis that he had built for the Ti-research in the

group.

Thanks to Dr. Ning Zhou for his support and help during the hard initial time when I

joined the group. I will always miss our coffee time during his stay in the group.

Thanks go to my group members, Dr. Yipeng Gao, Dr. Dong Wang, Pengyang Zhao,

Xiaoqin Ke. I benefit a lot from many useful discussions with them in phase field

modeling, martensitic transformation, physics, and inter-diffusion.

Thanks to my friends, Lin Li, HongQing Sun, Fan Yang, Yufeng Zheng, Liu Cao, Weiqi

Luo, Xiaoji Li, Huang Lin, Di Qiu, who have encouraged, entertained, and supported me

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through the dark times, celebrated with me through the good, I take this opportunity to

thank you.

A special gratitude and love goes to my family for their unfailing support. Deeply

appreciate my parent’s incredible patience and constant support throughout my Ph.D.

study that allow me to stay in the college until 30 years old without making big money. I

will never truly be able to express my sincere appreciation to the both of you. Without the

great help from my parents-in-law during his stay with us, I would not be able to start

work on the thesis.

Thanks to my adorable daughter, Ellen. A smile from her is able to refresh my mind

much better than cups of coffee. Spending time with her is not a consolation prize, it is

the prize. But, an apology to her, to whom, I should have spent more time with her as a

father.

Finally, I want to express my deepest love and thanks to my wife, Pingting Bai, for her

incredible understanding and support, making amazing food everyday throughout my

Ph.D. study, and taking care of our daughter during the most difficult time of thesis

writing.

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Vita

July 2004 ........................................................B.S. Fuzhou University, Fuzhou, China

Sep 2004- July 2008.......................................Xiamen University, Xiamen, China

Sep 2008 to present .......................................Graduate Research Associate, Department

of Materials Science and Engineering, The

Ohio State University

Publications

[1] Shi R, Vikas D, Fraser H. L. and Wang Y. Crystallographic Studies for Variant

Selection of Grain Boundary Alpha in Titanium Alloys. Acta Materialia ; Under Review

[2] Shi R, Wang Y. Variant Selection during Alpha Precipitation in Ti-6Al-4V under the

Influence of Local Stress - A Simulation Study. Acta Materialia 2013; 61:6006..

[3] Shi R, Wang C, Wheeler D, Liu X, Wang Y. Formation mechanisms of self-organized

core/shell and core/shell/corona microstructures in liquid droplets of immiscible alloys.

Acta Materialia 2012;60:4172.

[4] Shi R, Ma N, Wang Y. Predicting equilibrium shape of precipitates as function of

coherency state. Acta Materialia 2012;60:4172.

[5] Boyne A, Wang D, Shi R, Zheng Y, Behera A, Nag S, Tiley J, Fraser H, Banerjee R,

Wang Y. Pseudospinodal mechanism for fine α/β microstructures in β-Ti alloys. Acta

ix

Materialia 2014;64:188.

[6] Lu Y, Wang C, Gao Y, Shi R, Liu X, Wang Y. Microstructure Map for Self-Organized

Phase Separation during Film Deposition. Physical Review Letters 2012;109:086101.

[7] Li Y, Shi R, Wang C, Liu X, Wang Y. Phase-field simulation of thermally induced

spinodal decomposition in polymer blends. Modelling and Simulation in Materials

Science and Engineering 2012;20:075002.

[8] Gao Y, Liu H, Shi R, Zhou N, Xu Z, Zhu Y, Nie J, Wang Y. Simulation study of

precipitation in an Mg–Y–Nd alloy. Acta Materialia 2012;60:4819.

[9] Shi R, Wang Y, Wang C, Liu X. Self-organization of core-shell and core-shell-corona

structures in small liquid droplets. Applied Physics Letters 2011;98:204106.

[10] Li Y, Shi R, Wang C, Liu X, Wang Y. Predicting microstructures in polymer blends

under two-step quench in two-dimensional space. Physical Review E 2011;83:041502.

Fields of Study

Major Field: Materials Science and Engineering

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Table of Contents

Abstract ............................................................................................................................... ii

Dedication .......................................................................................................................... iv

Acknowledgments............................................................................................................... v

Vita ................................................................................................................................... viii

List of Tables ................................................................................................................. xviii

List of Figures ................................................................................................................. xxii

CHAPTER 1 Introduction................................................................................................... 1

1.1 Motivations................................................................................................................ 1

1.2 Organization of the thesis .......................................................................................... 6

1.3. Reference: ............................................................................................................... 10

CHAPTER 2 Literature Review ....................................................................................... 15

Abstract ......................................................................................................................... 15

2.1 Introduction ............................................................................................................. 16

2.2. precipitation in titanium alloys ........................................................................... 17

2.2.1 Two-phase titanium alloys ......................................................................... 17

2.2.2 Microstructure development during precipitation ......................................... 19

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2.2.3 Orientation relationship between and phases ............................................ 21

2.2.4 Determination of the number of variants ..................................................... 22

2.2.5 The nature of interface between precipitate and matrix ............................ 23

2.2.6 Relationship between microstructure and mechanical properties .................... 24

2.3. Variant selection during precipitation ................................................................ 26

2.3.1 Variant selection of GB .................................................................................. 27

2.3.2 Variant selection of secondary side plates by GB ..................................... 29

2.3.3 Variant selection in basketweave microstructures ............................................ 30

2.3.4 Variant selection due to dislocations ................................................................ 33

2.4. Unresolved issues ................................................................................................... 35

2.4.1 Grain boundary nucleation ............................................................................ 35

2.4.2 Correlations between precipitates with different variants in the basketweave

microstructure ............................................................................................................ 36

2.4.3 The effect of dislocation on variant selection ................................................... 37

2.4.4 Microstructure evolution with variant selection ............................................... 37

2.5. References: ............................................................................................................. 51

CHAPTER 3 Predicting Equilibrium Shape of Precipitates as Function of Coherency

State................................................................................................................................... 59

Abstract: ........................................................................................................................ 59

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3.1. Introduction ............................................................................................................ 60

3.2. Elastic Strain Energy of Coherent and Semi-Coherent Precipitates ...................... 62

3.2.1. Stress-free transformation strain for coherent precipitates .............................. 65

3.2.2. Deformation gradient matrix due to defects at hetero-phase interfaces .......... 66

3.3. Estimation of Interfacial Energy for Semi-Coherent Interfaces ............................. 69

3.4. Worked Examples .................................................................................................. 71

3.4.1. Derivation of effective SFTS for the semi-coherent precipitates ................ 72

3.4.2. Strain energy density and habit plane orientation of semi-coherent

precipitates ................................................................................................................. 77

3.4.3. Interfacial energy anisotropy of semi-coherent precipitates ........................ 78

3.4.4. Equilibrium shape of -precipitates in different cases .................................... 79

3.4.5. Coherency lost ................................................................................................. 81

3.5. Discussions ............................................................................................................. 82

3.6. Summary ................................................................................................................ 88

3.7. Reference ................................................................................................................ 90

CHAPTER 4 Variant Selection during Precipitation in Ti-6Al-4V under the Influence

of Local Stress ................................................................................................................. 106

Abstract: ...................................................................................................................... 106

4.1. Introduction .......................................................................................................... 107

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4.2. Method ................................................................................................................. 111

4.2.1. Determination of number of variants of a low symmetry precipitate phase . 111

4.2.2. Free energy formulation ................................................................................ 113

4.2.2.1. Chemical free energy .................................................................................. 114

4.2.2.2. Elastic strain energy.................................................................................... 116

4.2.3. Stress-free transformation strain for coherent and semi-coherent precipitates

................................................................................................................................. 117

4.2.4. Effect of misfit dislocation on interfacial energy .......................................... 121

4.2.5. Kinetic equations ........................................................................................... 123

4.2.6. Model inputs and parameters ......................................................................... 124

4.3. Results .................................................................................................................. 124

4.3.1. Growth behavior of a single plate .............................................................. 124

4.3.2. Effect of pre-strain on variant selection ........................................................ 126

4.3.2.1. Pre-strain due to compressive stress along [010] ..................................... 127

4.3.2.2. Pre-strain due to tensile stress along [010] .............................................. 128

4.3.3. Variant selection due to pre-existing plates ............................................... 129

4.4. Discussion ............................................................................................................ 131

4.4.1. Lengthening and thickening kinetics of plate ............................................ 131

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4.4.2. Elastic interaction between pre-strain and transformation strain of variants

................................................................................................................................. 132

4.4.3. Competition between pre-strain and evolving microstructure ...................... 135

4.4.4. Variant selection due to pre-existing microstructure ..................................... 138

4.5. Summary .............................................................................................................. 145

4.6. References ............................................................................................................ 164

CHAPTER 5 Evolution of Microstructure and Transformation Texture during Alpha

Precipitation in Polycrystalline Titanium alloys .................................................... 174

Abstract: ...................................................................................................................... 174

5.1. Introduction .......................................................................................................... 175

5.2. Model Formulation ............................................................................................... 180

5.2.1. Polycrystalline sample ................................................................................ 180

5.2.2 Phase Field Model for precipitation in an elastically and structurally

inhomogeneous polycrystalline sample................................................................ 180

5.2.2.1 Chemical free energy for polycrystalline system ........................................ 181

5.2.2.2. Strain energy of an elastically and structurally inhomogeneous system .... 183

5.2.2.3 Kinetic equations ......................................................................................... 187

5.2.3 Orientation Distribution Function modeling of microstructure in

polycrystalline sample ............................................................................................. 189

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5.3. Results .................................................................................................................. 192

5.3.1. Starting polycrystalline and texture ........................................................ 192

5.3.2. Evolution of microstructure and texture during precipitation ......... 193

5.3.3. Effect of pre-strain on variant selection ........................................................ 194

5.3.4. Effect of starting texture on variant selection ............................................ 195

5.3.5. Quantifying the degree of variant selection ................................................... 197

5.3.6. Effect of boundary constraint on variant selection ........................................ 198

5.4. Discussions ........................................................................................................... 198

5.5. Summary .............................................................................................................. 205

5.6. References: ........................................................................................................... 226

CHAPTER 6 Variant Selection of Grain Boundary by Special Prior Grain

Boundaries in Titanium Alloys ....................................................................................... 233

Abstract ....................................................................................................................... 233

6.1. Introduction .......................................................................................................... 234

6.2. Model formulation and Experimental procedures ................................................ 237

6.2.1. Crystallographic model .................................................................................. 237

6.2.2. Experimental procedures ............................................................................... 240

6.3. Results .................................................................................................................. 241

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6.3.1. Special grain boundaries where GB maintainsBOR with both adjacent

grains ....................................................................................................................... 242

6.3.2. Violation of variant selection rule derived from closeness between poles

................................................................................................................................. 244

6.4. Discussion ............................................................................................................ 245

6.5. Conclusions .......................................................................................................... 251

6.6 References: ............................................................................................................ 262

CHAPTER 7 Effects of Grain Boundary Parameters on Variant Selection of Grain

Boundary in Titanium Alloys ...................................................................................... 266

Abstract ....................................................................................................................... 266

7.1. Introduction .......................................................................................................... 267

7.2. Experimental procedure ....................................................................................... 274

7.3. Results .................................................................................................................. 276

7.3.1. Overall Characteristics of variant selection of GB ..................................... 276

7.3.2. Variant selection of GB when different rules are dominant ....................... 278

7.3.2.1. Rule I is dominant....................................................................................... 279

7.3.2.2. Rule II is dominant ..................................................................................... 280

7.3.2.3. Rule III is dominant .................................................................................... 281

7.3.3. Abnormal cases.............................................................................................. 282

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7.3.3.1 Abnormal variant selection when the minimum ....................... 282

7.3.3.2 Abnormal variant selection when .............................................. 284

7.4. Discussions ........................................................................................................... 284

7.5. Summary .............................................................................................................. 293

7.6. Reference .............................................................................................................. 321

CHAPTER 8 Conclusions and Future Works ................................................................. 324

8.1 Conclusions ........................................................................................................... 324

8.2 Direction for future research Conclusions ............................................................ 329

Reference ........................................................................................................................ 332

Appendix A: Determination of the number of variants of precipitate phase .............. 361

Appendix B: Stress free transformation strain for all 12 variants ............................... 363

B.1. Coherent nuclei .................................................................................................... 363

B.2. Fully-grown plates ............................................................................................... 364

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List of Tables

Table 2.1 All 12 variants of the Burgers orientation relation between precipitate sand

matrix ................................................................................................................................ 48

Table 2.2 Axis/angle pairs for all 6 possible boundaries in a single grain [22, 27] 49

Table 2.3 Qualitative correlation between colony and lamellae (single plate) size and

mechanical properties for titanium alloys ................................................................. 50

Table 3.1 Effect of different types of line defects in inter-phase interface on coherency

strain energy and habit plane orientation Lattice parameter of the two phases 3.196a Å,

2.943a Å and 4.680c Å and I is unit tensor ........................................................... 103

Table 4.1 All 12 Burgers orientation variants and symmetry operations associated with

them................................................................................................................................. 161

Table 4.2 Various model parameters and materials properties used in the simulations . 163

Table 6.1 All special misorientations (by angle/axis pairs) between two adjacent

grains, by which GB is able to maintain BOR with both grains ............................... 260

Table 6.2 Orientations of two grains shows Type II misorientation in variant selection

of GB Predicted orientation of GB (GB ) and its misorientation from the measured

one (GB ) ..................................................................................................................... 260

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Table 6.3 Orientations of two grains shows Type III misorientation in variant selection


one (GB ) ..................................................................................................................... 260

Table 6.4 Summary of relationships among misorientaion angle between two closest

poles of two adjacent grains, variant of GB selected, and deviation of the OR

between the GBand the non-Burgers grain from the Burgers orientation relationship

described by ................................................................................................ 261

Table 7.1 Details of grain boundary parameters (misorientation and grain boundary plane

inclination) corresponding to different GB s. Orientation of grain boundary plane with

respect to both crystal reference frame of Burgers grain and Burgers orientation reference

frame associated with selected variant are presented. .................................................... 307

Table 7.2 Orientations of two grains and their misorientation in variant selection of

GB Predicted orientation of GB (GB ) and its misorientation from the measured

one (GB ) ..................................................................................................................... 309

Table 7.3 Details about the effect of all grain boundary parameter in the variant selection

of GB 16. For each variant, associated with , inclination angles

between corresponding , , , and GBP, i.e. , , and

, are presented for Burgers grain and , respectively ........................... 310


of GB 28. ....................................................................................................................... 311

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one (GB ) ..................................................................................................................... 312



one (GB ) ..................................................................................................................... 313



one (GB ) ..................................................................................................................... 314


of GB 7. ......................................................................................................................... 315


of GB 8. ......................................................................................................................... 316



one (GB ) ..................................................................................................................... 317

Table 7.11 Details about the effect of all grain boundary parameter in the variant

selection of GB 32. ........................................................................................................ 318

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one (GB ) ..................................................................................................................... 319


selection of GB 32. ........................................................................................................ 320

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List of Figures

Figure 2.1 Schematic representation of three types of titanium alloys: alloy, alloy,

and alloy in a pseudo-binary section through a isomorphous phase diagram [2] ...... 39

Figure 2.2 Typical microstructures in Titanium alloys: (a) Grain boundary GB (b)

Colony ; (c) Basketweave and (d) Secondary microstructure ......................... 40

Figure 2.3 Schematic illustration of the Burgers orientation relationship, by looking down

[101] // [0001] (pointing into the plane of paper) ........................................................ 41

Figure 2.4 Schematic illustration of the interface and misfit dislocation configuration

........................................................................................................................................... 42

Figure 2.5 stereographic projection shows that GB maintains Burger OR with

grain and exhibits a small deviation from Burger OR with respect to adjacent grain

(G. B. P. indicates grain boundary plane) ......................................................................... 42

Figure 2.6 Schematic illustration of the variant selection rule by the grain boundary plane

(G.B.P.)-conjugate direction tends to parallel to G.B.P. .......................... 43

Figure 2.7 Schematic illustration of GB of different variants formed at a grain boundary

with a slight variation in its boundary plane ..................................................................... 43

Figure 2.8 (a) a prior grain boundary with the colony microstructure in one of the

grain ( grain 1) and the basketweave microstructure in the adjoining grain 2 (b)

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Orientation Image Microscopy (OIM) map of the same region as shown in (a). Regions

with the same color represent the same orientation variant .............................................. 44

Figure 2.9 (a) OIM map of three different laths sharing a common direction in a

basketweave microstructure selected from grain 2; (b) Superimposed pole figures of

{110} poles in matrix with the {0001} poles of the clustering laths; (c)

Superimposed pole figures o matrix with the poles of the

clustering laths [26] ....................................................................................................... 45

Figure 2.10 (a) OIM map of a cluster of three different laths in the basketweave

microstructure; (b) and (c) superimposed pole figures indicate that lath 1 and 2 share a

common basal plane; (d) and (e) superimposed pole figures indicate that lath 2

and 3 share a common .............................................................................. 46

Figure 2.11 (a) precipitates of single variant showed same morphology within slip band

in the matrix [15]; (b) Schematic illustration of the variant selection of on the slip

band [20] ........................................................................................................................... 47

Figure.4.1 Growth behavior of an plate. (a) Thickening kinetic of an infinite plate.

Results by phase field (symbol) and DICTRA (solid line) simulations are compared. (b)

Lengthening and (c) Thickening kinetics of a single finite plate embedded in a

supersaturated matrix. Error bars represent uncertainty in the determination of interface

position ............................................................................................................................ 148

Figure. 4.2 (a) Morphology of an isolated plate visualized by a constant contour of Al

concentration. The transparent light yellow plane denotes the experimentally observed

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habit plane . (b) A cross-section of the matrix phase surrounding the

plate showing variations in Al concentration in the matrix up to the precipitate/matrix

interface. The color bar indicates the relative value of Al concentration. ...................... 149

Figure. 4.3 Variant selection and microstructure development under a pre-stain obtained

via a compressive stress (50Mpa) along [010] . (a) 2D cross-sections showing

microstructure evolution (color online with phase shown in red and phase shown in

blue). Arrows indicate regions with transformation texture. (b) 3D microstructure

obtained at t = 10s. (c) Volume fraction of each variant as function of time. ................ 150


via a tensile stress (50Mpa) along [010] . (a) 2D cross-sections showing microstructure

evolution (color online with phase shown in red and phase shown in blue). Arrows

indicate regions with transformation texture. (b) 3D microstructure at t=10s. (c) Volume

fraction of each variant as a function of time ................................................................. 151

Figure. 4.5 Variant selection of secondary by a pre-existing plate. (a) Pre-existing

plate of variant 1 (V1). (b)-(d) Formation of secondary laths on the broad face of the

pre-existing plate. Different types of secondary are visualized through different

colors (see online version). (e) Volume fraction analysis of each secondary (f) - (h)

Formation of secondary on the other side of broad face of pre-existing plate from a

different view direction. (g) shows the relative locations between secondary (at t = 2s)

and pre-existing plate (at t = 0s). ................................................................................. 152

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Figure.4.6 Interaction energy density between pre-strain and each variant under both

coherent and semi-coherent conditions. The pre-strain is obtained by applying a 50MPa

tensile stress along (a) and (b) , and a 50Mpa compressive stress along (c)

and (d) ...................................................................................................... 153

Figure.4.7 Variant selection caused by a pre-stain obtained via uni-axial tension or

compression (50Mpa) along . Volume fraction of each variant as function of time

under tension (a) and compression (b). 3D microstructure (at t = 10s) under tension (c)

and compression (d). ....................................................................................................... 154

Figure.4.8 Chemical driving force for nucleation around a growing pre-existing plate

(Variant 1). The contour line indicates the chemical driving force in the supersaturated

matrix far away from pre-existing plate .......................................................................... 155

Figure . 4.9 Elastic interaction energy associated with all 12 variants of coherent nuclei

around a pre-existing semi-coherent plate (Variant 1). The contour lines indicate that

the elastic interaction energy is equal to the chemical driving force for nucleation in the

supersaturated matrix far away from the growing pre-existing plate shown in Fig. 4.8.

......................................................................................................................................... 156

Figure. 4.10 Elastic interaction energy associated with all 12 variants of semi-coherent

laths around a pre-existing semi-coherent plate (Variant 1). The contour lines indicate

vanishing elastic interaction energy. ............................................................................... 158

Figure 4.10 (continued) ................................................................................................... 159

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Fig. 4.11 (a) Elastic interaction energy between an nuclei (Variant 5) and a pre-existing

semi-coherent plate (Variant 1). (b) 1D structure order parameter profile (Blue) and

interaction energy (Red) along z-direction across interface. It shows that the

maximum negative values of the elastic interaction energy are located right at the

interface. .................................................................................................................... 160

Figure.5.1 (a) Polycrystalline matrix with different strength of starting texture, i.e.,

(b) a random-textured sample and (c) a strong-textured sample, according to the

maxima intensity in the pole figures ..................................................................... 207

Figure.5.2 (a)-(c) Microstructure evolution due to precipitation in random-texture

sample without any pre-strain, and (a′)-(c′) corresponding texture evolution

represented by pole figures ................................................................................. 208

Figure.5.3 (a)-(d) Microstructure evolution due to precipitation in random-texture

sample under the pre-strain, and (a′)-(d′) corresponding texture evolution

represented by pole figures. The pre-strain is obtained by applying a 50Mpa

compressive stress along x-axis of the system ................................................................ 209

Figure.5.4 (a)-(d) Final microstructure in random-textured sample under different

pre-strains, and (a′)-(d′) corresponding final texture ............................................. 210

Figure.5.5 (a)-(d) Final microstructure in strong-textured sample under different

pre-strains, and (a′)-(d′) corresponding final texture ............................................. 211

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Figure.5.6 (a) Maximum intensity in pole figures as a function of time in random-

textured sample under different pre-strain, (b) Maximum intensity in pole

figures as a function of time in strong-textured sample under different pre-strain, (c)

Maximum intensities in pole figures of final texture in both random-texture and

strong textured samples under different pre-strain ...................................................... 212

Figure.5.7 (a) and (b) pole figures for random-textured and strong-textured

sample; (c) and (d) corresponding pole figures of final texture in random-

textured and strong-textured sample without variant selection ...................................... 213

Figure.5.8 Degree of variant selection in both random-texture and strong textured

samples under different pre-strain .................................................................................. 214

Figure.5.9 (a) Degree of variant selection in random-textured sample under

different boundary constraint, (b) Degree of variant selection in random-textured

sample under different boundary constraint ................................................................... 215

Figure.5.10 (a)-(b) microstructure in the 2nd

and 5th

grain in random-textured

sample under x-tensil pre-strain, respectively; (c)-(d) volume fraction of each variant as a

function of time in the two grains; (e)-(f) local stress state in the two grains; (g)-

(h) interaction energy density between the external loading and each α variant under both

coherent and semi-coherent conditions within these two grains .................................... 216


and 5th

grain in strong-textured



xxviii


coherent and semi-coherent conditions within these two grains .................................... 218


and 5th


sample under x-tensil external loading (Free-end), respectively; (c)-(d) volume fraction of

each variant as a function of time in the two grains; ................................................... 220

Figure.5.13 (a) all possible misorientation between pairs of variants. Misorientation

axes are expressed in a strand triangle for HCP structure; (b) uncorrelated misorientation

analysis for both phase field simulated microstructure and the one without variant

selection; (c) the maximum degree of variant selection within individual grain where a

single variant percolates the whole grain .................................................................... 221

Figure.5.14 (a) degree of variant selection within the largest and the smallest grain in

random-texture sample under different pre-strains and boundary constraint, (b)

corresponding overall degree of variant selection .......................................................... 222

Figure. 5.15 (a) and (b) degree of variant selection within the largest in random-texture

sample under Z-Comp pre-strain and X-Comp external loading (X-Comp-Free),

respectively, (c) and (d) pole figures for final textue under Z-Comp pre-strain

and X-Comp external loading (X-Comp-Free), respectively ......................................... 223

Figure. 5.16 (a) Macro-texture of random-textured sample represented by three different

pole figures, , and poles, respectively; (b) Macro-texture of final

phase without occurrence of variant selection represented by corresponding three

different pole figures, , , and , respectively;(c) Macro-texture of

xxix

final phase with occurrence of variant selection represented by corresponding three

different pole figures, , , and , respectively .................................. 224

Figure. 5.17 Examples showing the pseudo variant selection due to 2D sampling effect.

EBSD scan is performed along at different layers of the sample ................................... 225

Figure 6.1. Illustrations of all special crystallographic orientation relationships between

GB (Red) and two adjacent grains (Blue and Green) that are able to hold the Burgers

Orientation Relationship with the GB (a) Type I - 10.52 º/<110>, (b) Type II- 49.48

º/<110>, (c) Type III- 60º/<110> and Type IV- 60º/<111>. ....................................... 253

Figure 6.2. Experimental observations of a Type II special grain boundary where

GB maintains BOR with two adjacent grainsaOIM image of the Type II

boundary; (b) superimposed pole figures of the poles of the two grains and the

pole of the GB (c) Superimposed pole figures among the poles of the

two grains and the pole of the GB ........................................................... 254

Figure 6.3. Experimental observations of a Type III special grain boundary where

GB maintains BOR with two adjacent grainsaOIM image of the Type III



two grains and the pole of the GB ........................................................... 255

Figure 6.4. OIM images ((a) and (b)) and superimposed pole figures of GB and

pole figures of the two grains with different angular deviation between two

xxx

closest poles ((a) and (c): ; (b) and (d):

). ........................................................................................... 256

Figure 6.6. (a) OIM image for two grains with GB 9 and GB on different locations

of the grain boundary with different inclinations; (b) Disorientation angles associate

with for all 12 variants; (c) Superimposed pole figures among the

poles of the two grains and the pole of GB ; (d) Superimposed pole figures

among the poles of the two grains and the pole of GB (e)

Superimposed pole figures among the poles of the two grains and the

pole of GB ; (d) Superimposed pole figures among the poles of the two grains

and the pole of GB ...................................................................................... 258

Figure 7.1 Overall characteristic of grain boundary alpha (GB ) precipitation shown by

OIM. Presence of GB only occurs at certain grain boundaries .................................... 295

Figure 7.2 Standard stereographic triangle projection shows the orientation of grain

boundary (GB) planes (red solid circles) relative to the crystal reference frame in Burgers

grain ................................................................................................................................ 295

Figure 7.3 (a) Stereographic projection shows the orientation of GB planes relative to the

Burgers reference frame of selected variant, i.e. - - ; (b) and (c) the

frequency of occurrence of variant selection as a function of the inclination angle

between GBP and direction and between GBP and planes

, respectively .................................................................................................. 296

xxxi


Burgers reference frame of selected variant in the case of ; (b) and (c) the

frequency of occurrence of variant selection as a function of and ,

respectively ..................................................................................................................... 297

Figure.7.5 (a) Stereographic projection shows the orientation of GB planes relative to the


frequency of occurrence of variant selection as a function of and ,

respectively ..................................................................................................................... 298

Figure 7.6 Experimental observations of variant selection of GB 16aOIM image; (b)

superimposed pole figures among the poles of the two grains and the

pole of the GB (c) Superimposed pole figures among the poles of the two

grains and the pole of the GB (d) superimposed pole figures of the

poles of the two grains and the pole of the GB (e) Disorientation angles

associate with and (f) for all 12 variants with respect to

different Burgers grain; grain boundary plane orientation is also superimposed in (b)-(d).

......................................................................................................................................... 299



pole of the GB(c) Superimposed pole figures among the poles of the two



xxxii


different Burgers grain .................................................................................................... 300

Figure 7.7 (Continued) .................................................................................................... 301

Figure 7.8 Experimental observations of variant selection of GB 7 and GB 8aOIM

image; (b) Disorientation angles associate with for all 12 variants with

respect to different Burgers grain; (c) and (f) superimposed pole figures among the

poles of the two grains and the pole of the GB and GB 8(d) and (g)


pole of the GB and GB 8(e) and (h) superimposed pole figures of the poles of

the two grains and the pole of the GB and GB 8; (i) and (j) for

all 12 variants with respect to different Burgers grain .................................................... 302

Figure 7.8 (Continued) .................................................................................................... 303








Figure 7.10 Experimental observations of variant selection of GB 26aOIM image;

(b) superimposed pole figures among the poles of the two grains and the

xxxiii






Figure 7.11 A scenario for nucleation of a grain boundary on a prior grain boundary

between and . The nuclei maintain Burgers orientation with , and the low energy

facets and develop into Burgers grain . The zone axis between two facets

is assumed to included in the grain boundary ....................................................... 306

1

CHAPTER 1 Introduction

1.1 Motivations

Titan, the Giant divine being in Greek mythology, a son of Uranos (Father Heaven) and

Gaia (Mother Earth), had lost several wars against the Olympic Gods that resulted in his

being confined in the underground dark world. The element, titanium, confined within

rutile ores, was first discovered by a German chemist Martin Heinrich Klaproth. It was

then confirmed as a new element and, in 1795, was named for the Latin word for Earth

(also the name for the Titan of Greek myth).

Because of their lightweight, high strength-to-weight ratio, low modulus of elasticity, and

excellent corrosion resistance, titanium-based materials (both unalloyed and alloyed)

have been finding increasingly widespread application in many industries for the

production of a wide variety of components and work pieces since the early 1950s. It was

very hard to predict, at that time, that titanium materials would currently receive their

attention, interest and importance not only for industrial applications but also equally for

dental and medical applications. It is believed that the expansion of titanium alloys usage

will continue for the forthcoming years.

The mechanical properties of titanium alloys, such as ductility, strength, creep resistance,

crack propagation resistance and fracture toughness, depend, to a large degree, on the

microstructure, which is formed during the thermomechanical processing (TMP) and

2

thermal treatment procedures. According to the application, a specific properties (or

combination of properties) can be obtained through microstructure fabrication or

modification. Microstructure evolution and control in titanium alloys rely heavily on the

allotropic transformation from a body-centered cubic crystal structure (denoted as beta

phase) at high temperatures to a hexagonal close-packed (HCP) crystal structure (referred

to as alpha phase) found at low temperatures.

The defining characteristic of the transformation is the Burgers orientation relationship

(BOR) [1] between the two phases, i.e. { } and ⟨ ⟩ [ ] . Owing

to the symmetry of the parent and product phases and the BOR between them [2], there

are twelve possible crystallographically equivalent orientation variants of the phase

within a single parentgrain. It is typically the case that only a small subset of the 12

possible variants is formed preferentially within each beta grain under different TMPs,

i.e., variant selection occurs frequently during TMP.

During thermo-mechanical processing, many factors could lead to the occurrence of

variant selection during the transformation and hence formation of microtexture.

For both and processing routes, the transformation starts from prior

grain boundaries that have strong preference to select certain variants for allotriomorphic

GB Colony , i.e., cluster of parallel plates belonging to a single variant

(the same variant as the GB ) could then develop into the grain that holds a BOR [1]

with the GB . The development of colony structures on the other adjacent

3

grain is also subjected to the influence of GB . Defects such as dislocations

and stacking faults generated during TMP in either or phase region act

frequently as preferred nucleation sites for specific subset of variants. Upon further

cooling or aging at a lower temperature within the two-phase region, a specific set

of variants for secondary plates will be further selected to nucleate and fill the retained

matrix between the primary plates or around the primary globular particles.

Besides dislocations, there exists a rich variety of other sources that are able to result in

local stresses and lead to variant selection within sample during TMP. For instance,

owing to the anisotropy of thermal expansion coefficient of the phase (which is 20%

larger than in the ⟨ ⟩ than in the ⟨ ⟩ directions), substantial residual stresses are

common in Ti alloys even after a stress relief annealing treatment [10-12]. Moreover,

local stress fields will also be generated by precipitation and autocatalysis has been

shown frequently to cause variant selection [13, 14]. Furthermore, for polycrystalline

materials under an external stress or strain field, local stress state within the sample will

vary significantly from grain to grain because of the elastic anisotropy in each grain that

leads to elastic inhomogeneity in the sample [15]. Apparently, local stress state, due to a

rich variety of sources, is a key factor in controlling variant selection and hence the final

transformation texture during precipitation in Ti alloys. To sum up, frequent

occurrence of variant selection due to a rich variety of factors during TMP results in a

relative hierarchical and relatively coarse microstructure at the scale of individual

4

grain, across prior grain boundaries, within the overall polycrystalline sample, and

also significant transformation texture (i.e., appearance of large regions of plates

consisting of the same crystallographic orientation variant or different variants but with a

common crystallographic feature such as common basal pole; these regions

within individual grains or across grain boundaries are often referred to as “macro-

zones” or micro-textured regions).

Therefore, in order to control the microstructure, to understand processing-

microstructure-properties relationships, and thus to tailor manufacturing conditions to

obtain specific mechanical properties through TMP, it is of significant importance to

develop a quantitative understanding/prediction of variant selection mechanisms for its

occurrence at different scales and further investigate both microstructure and micro-

texture development of phase due to variant selection. However, variant selections

depend on a wide variety of interacting parameters and thus are very complex. Owing to

this complexity, the mechanisms of variant selection are very difficult to determine

experimentally. For example, the main challenges to study the effect of external

loading/pre-strain variant selection during transformations in polycrystalline

sample under the influence of stress are three-folds: first, one needs to determine stress

distribution in an elastically anisotropic and inhomogeneous polycrystalline matrix

under a given applied stress/strain condition; and second, one needs to describe

interactions of local stress with precipitation of coherent and semi-coherent precipitates,

i.e., to describe interactions of local stress with an evolving microstructures. During

5

early stages of a phase transformation, precipitates or structural non-uniformities tend to

be fully coherent to minimize the interfacial energy. However, they may lose coherency

during continued growth when the elastic strain energy becomes dominant. Defect

structure, including misfit dislocations and structural ledges, at the interfaces will

alter not only the coherency elastic strain energy associated with the precipitation, but

also the interfacial energy and its anisotropy. It could introduce growth anisotropy as

well. These anisotropies, together with the high volume fraction and multi-variants of the

precipitate phase and long-range elastic interactions between the precipitates and local

stress, and among different variants of precipitates themselves, lead to highly non-

random spatial distribution of precipitates with different variants. Third, in order to

provide new insight into materials processing- microstructure- properties relationship,

microstructure and texture needs to be considered together. In other words, variant

selection behavior at the scale of individual parent grains and scale of the whole

polycrystalline sample, and their influence on the microstructure evolution and final

transformation texture need to be considered simultaneously. In sum, variant selections

depend on a wide variety of interaction parameters and thus are very complex.

Based on gradient thermodynamics [16-18] and microelasticity theory [19-23], the phase

field approach [24-30] (also called the diffuse-interface approach) offers an ideal

framework to deal rigorously and realistically with these difficult challenges. As will be

demonstrated in the current study of the transformation in Ti-6Al-4V (in wt%)

[31, 32], in the framework of phase field model , , the formulation of the total free energy

6

functional, which consists of the bulk chemical free energy, elastic strain energy and

interfacial energy, has accounted for the following: (a) a reliable thermodynamic data for

the bulk chemical free energy for Ti-6Al-4V system [32, 33]; (b) crystallography of the

crystal lattice rearrangement, including orientation relationship, i.e. BOR, and

lattice correspondence (LC, i.e. atomic site correspondence for diffusional

transformation) as functions of the lattice parameters of the precipitate and parent phases

(i.e., the effect of alloy chemistry); (c) accommodation of the transformation strain; (d)

development of defect structures (misfit dislocations and structural ledges) at ⁄

interfaces as precipitates grow in size; (e) elastic interaction of nucleating particles with

existing chemical and structural non-uniformities and other stress-carrying defects such

as dislocations [34]. In particular, in combination with orientation distribution function

(ODF) modeling [35] of the simulated ⁄ microstructures, the phase field model allows

for a treatment of both micro- and macro-texture evolution accompanying the ⁄

microstructure evolution during different thermo-mechanical treatments.

1.2 Organization of the thesis

The objective of the current work is to investigate variant selection behavior at the scale

of individual parent grains, on the prior grain boundaries, and scale of the whole

polycrystalline sample, and the influence of occurrence of variant selection at different

scales on the microstructure evolution and final transformation texture. For the

purpose of illustrating this point, a brief literature review about physical metallurgy of

7

titanium alloys based on phase transformation and a variety of factors that would

result in the occurrence of variant selection and transformation texture at different length

scales will be made in Chapter2.

In Chapter 3, a general approach is proposed to predict equilibrium shapes of precipitates

in crystalline solids as function of size and coherency state. The model incorporates

effects of interfacial defects such as misfit dislocations and structural ledges on strain

energy anisotropy and on interfacial energy anisotropy. Using precipitation in

titanium alloys as an example, how the interfacial defects relax the coherency elastic

strain energy and affect the habit plane orientation are analyzed in detail by incorporating

the effect of the defects into the stress-free transformation strain. How the interfacial

defects affect the interfacial energy anisotropy and the final equilibrium shape of

precipitates is also investigated. Various possible equilibrium shapes of precipitates

having different defect contents at interfaces are obtained by phase field simulations.

Determination of habit plane orientation of precipitate due to interplay between the

strain energy minimization and interfacial energy anisotropy will be investigated. In

combination with crystallographic theories of interfaces such as O-lattice theory and

experimental characterization of habit plane of finite precipitates, this approach has the

ability to predict the coherency state (i.e., defect structures at interfaces) and equilibrium

shape of finite precipitates.

In Chapter 4, we develop a three-dimensional (3D) quantitative phase field model to

8

predict variant selection and microstructure evolution during transformation in Ti-

6Al-4V (wt.%) at the scale of a single grain under the influence of both external and

internal stress fields such as those associated with, but not limited to, pre-straining and

pre-existing precipitates. The model links its inputs directly to thermodynamic and

mobility databases, and incorporates the crystallography (Burgers lattice correspondence

and orientation relationship) of BCC to HCP transformation, elastic anisotropy, and

defects within semi-coherent / interfaces in its total free energy formulation.

In Chapter 5, the three-dimensional quantitative phase field model (PFM) formulated in

Chapter 4 is further extended to predict variant selection and microstructure evolution

during transformation in polycrystalline Ti-6Al-4V sample under the influence of

different processing conditions such as pre-strain and boundary constraint. The model

updates local stress state according to the interactions among external loading, elastic

inhomogeneity and structural inhomgeneity due to evolving precipitation using an

iterative solver. In particular, texture evolution is coupled simultaneously with

microstructure evolution through orientation distribution function (ODF) modeling of

two-phase microstructure in polycrystalline obtained by the PFM. Under different

processing routes, degrees of variant selection at the scale of individual parent grains and

scale of the whole polycrystalline sample, and their effects on the final macro-texture of

phase under the influences of different processing variables and starting texture have

been investigated. The effect of non-uniform stress state, due to elastic inhomogeneity

under pre-strain, on the variant selection behavior within individual grain has been

9

investigated. The connection between variant selection within individual grain and the

overall polycrystalline sample will be made.

It has been observed frequently that GB prefers its ⟨ ⟩ pole to be parallel to a

common ⟨ ⟩ pole of the two adjacent grains and results in a micro-textured region

across the grain boundary (GB) and, as a consequence, slip transmission may take place

more easily across that GB. In order to investigate how such a special prior GB

contributes to variant selection of GB, in Chapter 6, we develop a crystallographic

model based on the Burgers orientation relationship (BOR) between GB and one of the

two grains. The model predicts all possible special grain boundaries at which GB is

able to maintain BOR with both grains. A new measure for variant selection of GB,

, i.e. a measure of the deviation of the actual OR between the GB and the

non-Burgers grain from the BOR, is proposed. The validity of the specific variant

selection rule based on the closeness between two closet { } poles between two

grains widely used in literature will be analyzed using the new parameter, .

For variant selection of GBon prior grain boundary, several empirical rules have

been proposed to explain how grain boundary parameters, misorientation and grain

boundary plane inclination, contribute to the selection of GB. However, there is no a

general rule that is able to explain all variant selection behavior of GB. In Chapter 7,

based on the new parameter formulated in Chapter 6, the applicability of all current

10

empirical variant selection rules with respect to grain boundary parameters such as

misorientation and inclination on VS of GBα has been assessed systematically in Ti-

5553. Violations of different variant selection rules will be investigated.

The final conclusions and discussions on some future directions that would extend the

current work are presented in Chapter 8.

1.3. Reference:

[1] Burgers WG. On the process of transition of the cubic-body-centered

modification into the hexagonal-close-packed modification of zirconium. Physica

1934;1:561.

[2] Cahn JW, Kalonji GM. Symmetry in Solid-Solid Transformation Morphologies.

PROCEEDINGS OF an Interantional Conference On Solid-Solid Phase Transformations

1981:3.

[3] Banerjee D, Williams JC. Perspectives on Titanium Science and Technology.


[4] Lutjering G, Williams JC. Titanium (Engineering Materials and Processes).

Berlin: Springer, 2007.

[5] Bhattacharyya D, Viswanathan GB, Denkenberger R, Furrer D, Fraser HL. The

role of crystallographic and geometrical relationships between alpha and beta phases in

an alpha/beta titanium alloy. Acta Materialia 2003;51:4679.

11

[6] Bhattacharyya D, Viswanathan GB, Fraser HL. Crystallographic and

morphological relationships between beta phase and the Widmanstatten and

allotriomorphic alpha phase at special beta grain boundaries in an alpha/beta titanium

alloy. Acta Materialia 2007;55:6765.

[7] Stanford N, Bate PS. Crystallographic variant selection in Ti-6Al-4V. Acta

Materialia 2004;52:5215.

[8] van Bohemen SMC, Kamp A, Petrov RH, Kestens LAI, Sietsma J. Nucleation

and variant selection of secondary alpha plates in a beta Ti alloy. Acta Materialia

2008;56:5907.

[9] Shi R, Dixit V, Fraser HL, Wang Y. Variant Selection of Grain Boundary Alpha

by Special Prior Beta Grain Boundaries in Titanium Alloys. Submitted to Acta Materialia

2014.

[10] Sargent GA, Kinsel KT, Pilchak AL, Salem AA, Semiatin SL. Variant Selection

During Cooling after Beta Annealing of Ti-6Al-4V Ingot Material. Metallurgical and

Materials Transactions a-Physical Metallurgy and Materials Science 2012;43A:3570.

[11] Winholtz RA. Residual Stresses: Macro and Micro Stresses. In: Buschow KHJ,

Robert WC, Merton CF, Bernard I, Edward JK, Subhash M, Patrick V, editors.

Encyclopedia of Materials: Science and Technology. Oxford: Elsevier, 2001. p.8148.

[12] Zeng L, Bieler TR. Effects of working, heat treatment, and aging on

microstructural evolution and crystallographic texture of [alpha], [alpha]', [alpha]'' and

[beta] phases in Ti-6Al-4V wire. Materials Science and Engineering: A 2005;392:403.

12

[13] Kar S, Banerjee R, Lee E, Fraser HL. Influence of crystallography varaiant

selection on microstructure evolution in titanium alloys. In: Howe JM, Laughlin DE, Lee

JK, Dahmen U, Soffa WA, editors. Solid-Solid Phase Transformation in Inorganic

Materials 2005, vol. 1: TMS, 2005.

[14] Lee E, Banerjee R, Kar S, Bhattacharyya D, Fraser HL. Selection of alpha

variants during microstructural evolution in alpha/beta titanium alloy. Philosophical

Magazine 2007;87:3615.

[15] Wang YU, Jin YM, Khachaturyan AG. Three-dimensional phase field

microelasticity theory of a complex elastically inhomogeneous solid. Applied Physics

Letters 2002;80:4513.

[16] Cahn JW, Hilliard JE. Free energy of a nonuniform system. I. Interfacial free

energy. The Journal of Chemical Physics 1958;28:258.

[17] Landau LD, Lifshitz E. On the theory of the dispersion of magnetic permeability

in ferromagnetic bodies. Phys. Z. Sowjetunion 1935;8:101.

[18] Rowlinson JS. Translation of J. D. van der Waals' “The thermodynamik theory of

capillarity under the hypothesis of a continuous variation of density”. Journal of

Statistical Physics 1979;20:197.

[19] Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion, and

related problems. Proceedings of the Royal Society of London. Series A 1957;241.

[20] Eshelby JD. The Elastic Field Outside an Ellipsoidal Inclusion. Proceedings of the

Royal Society A 1959;252:561.

13

[21] Khachaturyan A. Some questions concerning the theory of phase transformations

in solids. Soviet Phys. Solid State 1967;8:2163.

[22] Khachaturyan AG. Theory of Structural Transformations in Solids. New York:

John Wiley & Sons, 1983.

[23] Khachaturyan AG, Shatalov GA. Elastic interaction potential of defects in a

crystal. Sov. Phys. Solid State 1969;11:118.

[24] Boettinger WJ, Warren JA, Beckermann C, Karma A. Phase-field simulation of

solidification. Annual Review of Materials Research 2002;32:163.

[25] Chen L-Q. PHASE-FIELD MODELS FOR MICROSTRUCTURE

EVOLUTION. Annual Review of Materials Research 2002;32:113.

[26] Emmerich H. The diffuse interface approach in materials science: thermodynamic

concepts and applications of phase-field models: Springer, 2003.

[27] Karma A. Phase Field Methods. In: Buschow KHJ, Cahn RW, Flemings MC,

Ilschner B, Kramer EJ, Mahajan S, Veyssière P, editors. Encyclopedia of Materials:

Science and Technology (Second Edition). Oxford: Elsevier, 2001. p.6873.

[28] Shen C, Wang Y. Coherent precipitation - phase field method. In: Yip S, editor.

Handbook of Materials Modeling, vol. B: Models. Springer, 2005. p.2117.

[29] Wang Y, Chen LQ, Zhou N. Simulating Microstructural Evolution using the

Phase Field Method. Characterization of Materials. John Wiley & Sons, Inc., 2012.

[30] Wang YU, Jin YM, Khachaturyan AG. Dislocation Dynamics—Phase Field.

Handbook of Materials Modeling. Springer, 2005. p.2287.

14



[32] Wang Y, Ma N, Chen Q, Zhang F, Chen SL, Chang YA. Predicting phase

equilibrium, phase transformation, and microstructure evolution in titanium alloys. JOM

Journal of the Minerals Metals and Materials Society 2005;57:32.

[33] Chen Q, Ma N, Wu KS, Wang YZ. Quantitative phase field modeling of

diffusion-controlled precipitate growth and dissolution in Ti-Al-V. Scripta Materialia

2004;50:471.

[34] Shi R, Wang Y. Variant selection during α precipitation in Ti–6Al–4V under the

influence of local stress – A simulation study. Acta Materialia 2013;61:6006.

[35] Bunge HJ. Texture Analysis in Materials Science- Mathematical Methods.

London, 1982.

15

CHAPTER 2 Literature Review

Abstract

The ⁄ titanium alloys have been widely used as advanced structural materials in the

aerospace industry. Their mechanical properties mainly depend on the volume fraction,

size, morphology and spatial distribution of precipitates, which form through the

diffusional transformation. According to the symmetry of the

parent (BCC)and product (HCP)phases and their Burgers orientation relationship,

there are twelve possible orientation variants of precipitates within a single prior

grain. However, quite often, some variants appear more frequently than others, a

phenomenon referred to as variant selection. Variant selection during precipitation

generally governs the microstructure evolution and the final mechanical properties of

⁄ titanium alloys. It was found that variant selection is closely related to the

heterogeneous nucleation of phase on pre-existing defects such as grain boundaries

and dislocations. Coupling between plates with different variants also contributes to

variant selection. A full understanding of the mechanism of variant selection can provide

16

important insight into the engineering of microstructure in ⁄ titanium alloys in order

to achieve the desirable mechanical properties.

2.1 Introduction

Titanium alloys have been widely used in industrial and medical applications, ranging

from aircraft jet engine components, to bicycle frames [1-4] and medical implants [5-7]

because the alloys have fascinating combinations of high strength-to-density ratio, high

fracture toughness and high corrosion resistance [6, 8]. Among these alloys, /titanium

alloys are the most widely used because their microstructure and properties can be

manipulated widely by appropriate heat treatments and/or mechanical processing [9].

The basis for the manipulation of microstructure in / alloys relies heavily on the →

+ transformation during cooling, in which, the phase precipitates from the matrix

in the form of laths or plates. It is well known that precipitates usually exhibit a

specific orientation relationship (OR) with the matrix, referred to as Burgers OR [10].

According to the symmetry of the parent and product phases and their Burgers

relationship, there are twelve possible orientation variants of the precipitates in a single

prior grain. Variant selection of phase (some variants grow preferentially over others)

always accompanies with the processing of titanium alloys. Since the HCP phase is

17

highly anisotropic in nature [11], the morphology, distribution and arrangement of it in

the final microstructure influenced by variant selection significantly govern the properties

of the alloys [12-14].

There are many factors that could contribute to the variant selection of phase [15-29].

An understanding of the mechanism of variant selection and its effect on microstructure

development in / titanium alloys will benefit us in terms of better manipulation of

microstructure to obtain desirable mechanical properties.

2.2. precipitation in titanium alloys

2.2.1 Two-phase ⁄ titanium alloys

Titanium and its alloys exist in two allotropic forms: the hexagonal close-packed (HCP)

phase and the body-centered cubic (BCC) phase. Pure titanium exists as phase

below 882 °C (1620 °F). The HCP structure can be defined by placing two atoms at (0, 0,

0) and (2/3, 1/3, 1/2) positions in its unit cell. The space group for the phase is

P63mmc. The predominant slip mode in phase is the 1010 1120 , which is

consistent with the fact that the c/a ratio in pure titanium and its alloys (about 1.587) is

less than the ideal one of 1.633 [1, 2, 30]. The secondary slip systems are 0001

1120 and 1011 1120 . Above 882 °C, pure titanium transforms allotropically

from to phase. The BCC structure can be defined by placing two atoms at (0, 0, 0)

18

and (1/2, 1/2, 1/2) positions in its unit cell. The space group for the phase is Im3m .

The slip systems generally observed in the phase are: 011 111 , 112 111 and

123 111 [1, 2, 30].

In titanium alloys, the to transformation temperature ( transus) strongly depends on

the type and amount of alloying elements [1, 2, 31]. According to their effect on the

transus temperature, alloying elements in titanium alloys can be classified into and

stabilizers. stabilizers, such as Al, C, O and N elements, raise the transus temperature

and thus stabilize the phase. On the other hand, stabilizers, such as V, Cr, Mo, Nb,

etc., stabilize the phase by lowering the transus temperature. The addition of alloying

elements serves one or more of the following functions [2, 6, 31]: to control the

constitution of the alloy, to control the transformation kinetics and to solid-solution

strengthen one or more of the constituent phases [6].

Depending on the phases present and the relative proportions of the constituent phases,

titanium alloys can be classified broadly into three categories: (a) alloys, (b) /alloys

and (c) alloys, as schematically shown in an isomorphous pseudo-binary phase diagram

in Fig. 1 [2].

The/ alloys are in a phase region from the and + phase boundary up to the

intersection of the Ms-line (Martensite starting temperature) with the room temperature.

19

The / alloys have a mixture of and phases at low temperature and contain both

and stabilizing elements. In the most commercially used Ti-6Al-4V (wt. %) alloys, for

example, the Al element partitions selectively to the -phase offering solid-solution

strengthening of phase. The V element, however, is rejected from the -phase due to its

low solubility in this phase and is thus concentrated in the -phase, therefore solid-

solution strengthening the -phase [6]. The phase offers precipitation strengthening for

/ alloys.

2.2.2 Microstructure development during precipitation

For titanium alloys, upon cooling from the single phase region into the +two-

phase phase region, the phase decomposes by nucleation of phase at prior grain

boundaries and subsequently by diffusion controlled growth into the retained matrix.

There are three types of precipitates formed during the to diffusional

transformation in terms of where their nucleation sites: grain boundary allotriomorphic-

(GB), inter-granular and intra-granular [32].

During the cooling from above the transus, a layer of GB heterogeneously nucleates

at and grows preferentially along the grain boundaries (Fig. 2(a)). On further cooling, a

20

set of parallel inter-granular side plates develop either by nucleating directly from the

prior grain boundaries or by branching out from GB(Fig. 2(a)), and shooting into the

interior of prior grains. The plates which nucleate directly from grain boundaries are

referred to as primary side plates while those created by branching of GB are designated

as secondary side plates [33, 34]. Intra-granular plates nucleate and grow within the

interior of the prior -grains. Inter-granular and intra-granular plates comprise the so-

called the widmanstätten microstructure [32]. Depending on the cooling rate, the

widmanstätten plates may group together either in the form of colony attaching to the

prior grain boundaries or in the form of basketweave microstructure within the prior

grains [1, 2]. The Colony microstructure, i.e., clusters of parallel plates belonging to a

single crystallographic variant, forms during slow cooling from the phase field, as

clearly shown in Fig. 2(b). While the basketweave microstructure, i.e., multiple

crystallographic variants of plates clustering in the same region, develops upon higher

cooling rate, as shown in Fig. 2(c). A mixture of the colony and basketweave

microstructures forms at intermediate cooling rates [26].

Depending on their size-scale and sequence of nucleation and growth, widmanstätten

structures can also be subdivided into primary and secondary plates. Secondary

plates nucleate and fill the retained matrix between primary plates upon further

cooling or aging at a lower temperature within two-phase region, as shown in Fig.

2(d).

21

2.2.3 Orientation relationship between and phases

The orientation relationship between the precipitate and matrix during the BCC to

HCP phase transformation has been a subject of intensive research [10, 16, 35-40].

The major orientation relationship between the and phases is the Burgers orientation

relationship [10], described by:

Equation Chapter 2 Section 1

101 // 0001

; 111 // 1120

; 121 // 1100

(2.1)

Recently, more accurate measurements in titanium alloys have showed that the actual

orientation relationship between and phases is deviated slightly from the ideal

Burgers relationship [38], i.e., near Burgers OR. For example, the misorientation angles

between 101 // 0001

and 111 // 1120

are 0.78º and 0.56º [38], respectively.

There is another OR, though less frequently observed than the Burgers OR, in titanium

alloys, i.e., the Pitsch-Schrader OR [41] given by

101 // 0001

; 010 // 1120

; 101 // 1100

(2.2)

Note that a 5.26º crystallographic rotation along 101 // 0001

converts the Pitsch-

Schrader OR to ideal Burgers OR.

22

The precipitate is different from the matrix in both composition and crystal structure.

With the lattice parameters [38] abcc= 0.3196 nm, ahcp=0.2943 nm, and chcp = 0.4680 nm,

the transformation strain ij for an precipitate in matrix with the OR given in Eq.

(2.1) can be described by:

0.0759

0.13620.0356

ij

(2.3)

when referred to 010 // 1120

; 101 // 1100

; 101 // 0001

.

2.2.4 Determination of the number of variants

According to the Burgers OR, the single close-packed basal plane {0001} in the HCP

phase is parallel to one of the six close-packed planes 110

in the BCC phase. In

addition, one of the three close-packed 1120 directions in the basal plane 0001

is

parallel to one of the two close-packed directions 111 lying within the specific

close-packed plane 110

in the phase. However, there are only two distinguishable

combinations of parallel directions 1120 // 111 on 0001

plane due to the

presence of 6 fold rotation symmetry along 0001

. Since there are six possible 101

planes in the phase, there are twelve (6×2) possible equivalent orientation variants of

precipitate allowed by the Burgers OR. Accordingly, the decomposition of a prior grain

23

will give rise to one or more of twelve possible variant of phase, each with its own

distinct orientation with respect to the matrix. All twelve variants are listed in the Table

1. The number of variants can also be derived by group theory [42, 43].

Note that combinations of any two of the 12 variants result in the formation of 6 distinct

types of /grain boundaries in a single grain. These /boundariesare referred to as

Types 1 to 6 boundaries and the reduced axis/angle pairs for each type are listed in Table

2 [22]. Based on a random distribution of each variant, the probability of occurrence of

each type of / boundary, randomP , can be calculated.

2.2.5 The nature of interface between precipitate and matrix

The nature of interface between the precipitate and matrix is crucial to

understand the microstructure evolution during the processing [21, 25] and the properties

of the plastic deformation [44]. Therefore, it has been studied in great details [36-38, 44-

46] including its crystallographic orientation, habit planes and dislocation structures. The

precipitate usually appears in the form of plate and can be characterized as having a

broad face, a side face and an edge face as shown in the Fig 4.

Obviously, the interfaces are semi-coherent with misfit dislocations. Based on the

detailed experimental characterization by Mills et al. [38], the board face is wrapped by a

single set of parallel c-type misfit dislocations with a Burgers vector of 1 2 0001

and

24

the side face is looped by a-type misfit dislocations with a Burgers vector of 1 3 2110

. The dislocations on the broad face and the side face loop around the plate and form a

dislocation network on the edge face. Moreover, the broad face is comprised of structural

ledges [47] or steps that enable the interface to be stepped down along the lattice

invariant line direction 353

1[48], which is also the growth direction (major axis) for

the plate. The terrace plane of step is parallel to 121 // 1100

. The macroscopic

surface of the broad face is generally an irrational plane close to 11 13 11

[36-38, 45].

2.2.6 Relationship between microstructure and mechanical properties

The microstructure of titanium alloys are primarily described by the size, volume

fraction, morphology and spatial distribution of the phase, which in turn has a

substantial influence on the final tensile strength, ductility and fatigue properties of the

alloys.

The size of colonies is the most influential microstructural parameter on the mechanical

properties because it determines the effective slip length [9]. Slip length and colony size

are equal to the width of individual plate. The colony boundaries are major barriers

1 Specific indices are assigned according to the particular variant with OR in Eq.(1)

25

to slip, while the plates do not serve as major deformation barriers because slip transfer

is relatively easy due to the Burgers relationship between the and phases [49].

In general, small colony size and small plate seems to promote better mechanical

properties such as yield stress 0.2, ductility F, high cycle fatigue strength (HCF), except

macrocrack propagation resistance [1, 2, 9], as shown in Table 3.

For example, the yield stress and tensile ductility increase with decreasing colony size

due to the reduction in effective slip length. The dependence of HCF strength on

colony size is qualitatively similar to that of yield stress because the HCF strength

(resistance to crack nucleation) depends primarily on the resistance to dislocation motion.

For the propagation rate of the small, self-initiated cracks (microcracks) , it has been

shown that the microcracks propagate much faster in the coarse colony microstructure as

compared to the fine colony microstructure [9]. In addition, the fatigue cracks usually

nucleate at and propagate through the longest and widest plates due to the preferred slip

band activity within these coarse plates. It will be detrimental to mechanical properties if

a single variant of plate percolates the whole matrix.

It is thus thought that the introduction of more variants for phase leads to an

improvement of fatigue properties because a fine-scale microstructure of phase within

matrix can be obtained through the growth of randomly distributed nuclei of phase

with different variants. In addition, since all variants are distinct in their spatial

26

orientation, the introduction of more variants of phase would increase the tortuosity of

crack paths and thus impede the crack propagation. If there are coarse plates

percolating the whole matrix due to the variant selection of a specific variant, it will

favor the nucleation and propagation of microcracks. Thus, it is important to study the

mechanism of variant selection.

2.3. Variant selection during precipitation

The term “variant selection” will be discussed in detail since it has a significant effect on

the formation of transformation texture and the final mechanical properties [21, 24, 25,

27]. According to the Burgers OR, there are twelve possible orientation variants of

phase that can form within a single prior grain. The chemical driving force, as a

function of only temperature and composition, is the same for all variants, during the to

transformation. It is thus generally thought that all variants should appear with equal

statistical probability. However, some variants can appear more frequently than others

due to certain physical reasons. This phenomenon is referred to as variant selection [21,

22, 24, 25, 27].

There are many factors that could result in the variant selection during precipitation

such as heterogeneous nucleation of phase on the pre-existing defects in parent

27

phases such as grain boundary [27] and dislocations [15, 20]. The coupling between

plates with different variants also induces variant selection [26, 50].

2.3.1 Variant selection of GB

When GB nucleates and develops within the prior grain boundary, selection of one or

multiple specific variants (from the possible 12) are made according to certain rules

[21, 23, 25-27].

It has been commonly observed that Burgers OR exists between GB and one of the two

adjacent prior grains [16, 20, 21, 25, 27] (referred to as , whereas the GB

generally manipulates itself an orientation that has a small deviation from the Burgers

relathiship with respect to the other grain (referred to as , as shown in Fig. 5 [16].

It has been shown by Furuhara et al. [16] that morphologically indistinguishable -

precipitates formed along a relatively straight prior -grain boundary belong to a single

crystallographic variant. Furthermore, the selection of variant from the grain

boundary is made in such a manner that the variant has the minimum possible angle

between the matching direction 1120 // 111 and the grain boundary plane [20,

21], i.e., the matching direction 1120 // 111 of the selected variant tends to

parallel to the grain boundary plane. The variant selection rule seems to be consistent

28

with the proposition [51, 52] that two low energy facets such as 0001 // 110

and

1100 // 112

of precipitates generally make the smallest possible inclination angle

with the grain boundary plane in order to reduce the activation energy for the formation

of critical nucleus. Therefore, the critical nucleus formed at a given grain boundary

tends to elongate along the intersection of these two facets, i.e., 1120 // 111 in

the Burgers related side, as schematically shown in Fig.6.

Notice that, there still remain 3 possible variants sharing a common 111 which

satisfy the smallest angle requirement between 1120 // 111 and the grain

boundary plane, therefore suggesting there are other factors restricting the variant

selected by the grain boundary. It has been shown that the GB also tends to maintain a

minimum possible misorientation from the Burgers relation with respect to the adjoining

grain by selecting a specific variant for GB . Thus the selected precipitate

tends to keep maximum coherency with respect to both of the adjacent grains.

In general, the orientation of grain boundary plane is arbitrary. It was also observed that

[16, 23], for example [16], for a given boundary with a slight variation in its boundary

plane orientation, the grain boundary was decorated by GB precipitates belonging to

two different crystallographic variants (and2), as schematically shown in Fig.7.

29

Note that, the GB 1 is present in a somewhat discontinuous form while the GB 2

exists as a continuous layer.

Both 1 and 2 maintain the Burgers relationship with respect to only one grain.

Though they have the smallest angle between 1120 // 111 and their

corresponding grain boundary plane, neither 1 or 2 holds near Burgers relationship

with respect to the other adjoining grain. This result indicates that the parallelism of

1120 // 111 is the predominant variant selection rule, while maximum

coherency with respect to the opposite grain only plays a secondary role in the variant

selection of GB . However, one critical question remains, i.e., from the three possible

variants which meet the requirement between 1120 // 111 and grain boundary

plane, which one will be selected?

2.3.2 Variant selection of secondary side plates by GB

The GB precipitates also have a pronounced effect on the selection of secondary side

plate formed during the further cooling or aging [24-27].

The side plates growing into the 1 grain choose the same variant as the GB as

shown in Fig. 8(b).It was observed that the side plates usually exhibit a single growth

direction corresponding to the invariant line direction of the operating variant. In contrast,

30

the GB can not grow into the 2 grain with which it does not have a Burgers

relationship. A set ofside plates with Burger relation with 2 matrix develops into the

adjoining 2 grain near the surface of the GB This suggests that the formation of side

plates in 2 grain results from the nucleation on the interface between the GB and 2

matrix [34].

The variant selection of the secondary side plates due to the GB has a pronounced

effect on the microstructure evolution. As has been observed by Lee et al.[26], the colony

microstructure tends to develop in the 1 grain, while the basketweave microstructure

tends to develop in the adjacent 2 grain, as shown in Fig. 8. Based on the interface

instability mechanism [53], the development of colony microstructure from the GB

has been successfully simulated by Wang et al. [54] using phase field approach.

2.3.3 Variant selection in basketweave microstructures

The so-called basketweave microstructure is characterized by multiple crystallographic

variants of laths forming together within a prior grain. Its formation is usually

associated with selective growth of specific multiple variants of laths within the

matrix, i.e., the coupling between variants in the basketweave microstructure is not

random [22, 26, 27, 29].

31

Two types of coupling between variants in basketweave microstructure have been

commonly observed. In one case, laths belonging to three distinct variants tend to

cluster in a same region, as shown in Fig. 9(a). The three distinct variants share a

common 1120

direction that is parallel to the 111

direction of the matrix with

their basal poles 0001

rotated by 60º, as shown in Fig. 9 (b) and Fig. 9 (c). In other

words, three laths with Type 2 / grain boundary tend to cluster in the same region.

In the other case as shown in Fig.10, laths 1 and 2 in the cluster share a common

0001

basal plane with their 1120 directions being rotated by about 10.53º along

the common [0001] direction (Fig. 10 (b) and (c)), i.e., laths 1 and 2 have Type 6

boundary, whereas laths 2 and 3 have Type 2 boundary between them (Fig. 10 (d) and

(e)).

According to the experimental results from Lee [26] and Bohemen [27] , both Type 2 and

Type 6 / grain boundaries occur more frequently than expected on the basis a random

distribution of variants (Table 2), which indicates that variant selection occurs during the

formation of basketweave microstructure too.

It was suggested by Bohemen et al. that the preferential occurrence of Type 2 /

boundaries can not be explained by a favored orientation of the plates during individual

32

nucleation [27]. The clustering of specific multiple variants of laths with Type 2

boundary might be explained based on the principle of self-accommodation [22].

The self-accommodation is a process by which the transformation induced shear strain is

reduced by specific combinations of multiple variants [55]. By working on martensitic

→ transformation in pure titanium, Wang et al. [22] calculated shape strain for each of

12 variants as well as the average shape strain resulting from a cluster of three

variants in different combinations. According to their analysis [22], the lowest shear

strains by combinations of three variants with Type 2 ( 1120 60 )

and Type 4 ( 10 5 5 3 63.26

) boundaries resulted in a relatively high occurrence of

clustering with these two boundaries. The former one, also frequently identified in →

diffusional transformation, is in accordance with their experiment results. However, the

relatively high occurrence of clusters with Type 4 boundary has not been observed yet.

Moreover, the relatively high occurrence of Type 6 boundary in conjunction with Type 2

boundary in the cluster (Fig. 10) can not by itself be explained by self-accommodation

mechanism as well. Kar et al. [26, 56] postulated that the formation of clustering variants

with Type 6 boundary was associated with the heterogeneous nucleation and growth of

new lath near pre-existing laths that already hold Type 2 boundaries. As already

pointed out by Williams et al.[2], in order to minimize the overall elastic strain, the pre-

existing plates have a strong impact on selection of the variant of new plates that can

nucleate and grow near them. For example, the new plates, which nucleate by “point”

33

contact on the broad face (habit plane) of an existing plate, tend to grow nearly

perpendicular to it. As has been observed by Bhattacharyya et al. [21], the growth

directions of two plates with Type 6 boundary are nearly perpendicular, ~80.5 º or 99.5

º, to each other.

In summary, there is a strong coupling between the precipitates, which result in the

formation of the basketweave microstructure. The coupling may be induced by the

accommodation of the strain energy.

2.3.4 Variant selection due to dislocations

It is well known that pre-existing dislocations in the parent phase frequently act as

preferential nucleation sites for precipitates [57, 58]. Furuhara et al. investigated the

influence of dislocations on the selection of variants [15, 20]. In their experiments,

coarse planar slip bands, corresponding to 112 111

slip systems, were introduced by

cold rolling in the matrix at room temperature and precipitates nucleated

preferentially on the dislocations in these slip bands during the subsequent aging [15].

The precipitates in the slip band were of the same morphology and selected a single

variant (V4 in Table 2), as shown in Fig. 11 (a).

The authors tried to explain why only a single variant was selected in terms of the

effective accommodation of transformation stain by the stress field around dislocations.

They derived the maximum misfit strain direction bmax associated with 12 variants of

precipitate by Frank-Bilby equation (FBE) [59, 60]. According to the FBE, the misfit

34

across the interface between the precipitate and matrix can be described in terms of

the net Burgers vector tb crossing a vector p in the interface, as given by:

1

t

Ib A p (2.4)

where A is the homogeneous transformation matrix from lattice to and I is identity

matrix. The calculation results showed that three variants V4, V8 and V12 (in Table 1),

which share a common 111 // 1120

direction, equally gave the closet bmax to the

Burgers vector 2 111a

of the dislocation.

Therefore, according to the effective accommodation of the maximum misfit strain, the

variants of precipitate that are most preferred to nucleate on the 2 111a

dislocation

are limited to three variants V4, V8 and V12. However, only variant V4 was observed in

the slip band. Therefore the criterion of the maximum misfit accommodation is not

sufficient to explain the variant selection rule in this case. Indeed, the other components

of the transformation strain may have further restriction on the selection of the specific

variant of the phase. In fact, the author argued that the exclusive selection of variant 4

can be reasonably explained by the slip plane of dislocation because the slip plane

112

is parallel to the 1100

of variant 4, as shown in Fig. 11(b). As suggested by

Burgers [10], the to transformation begins with shear movements of atoms on 112

35

planes in 111 directions. Thus, prior activity of 112

111 specific systems may

favor the formation of related alpha variants.

2.4. Unresolved issues

The objective of this review on variant selection of phaseduring its precipitation is to

pave the way to better understand the effect of different factors and their interplay on the

variant selection and thus microstructure development in / titanium alloys. According

to the review, there are several unresolved problems:

2.4.1 Grain boundary nucleation

As mentioned above, there are quite stringent restrictions on the possible -variants that

can be precipitated at a given -grain boundaries. The GB is selected from three

possible variants which meet the minimum angle requirement between

1120 // 111 and grain boundary plane. Such a restriction probably serves to

reduce the nucleation barrier for GB . In addition, the selected GB maintains Burgers

OR with respect to one of the two adjacent grains. However, the exact mechanism for

such a variant selection rule is not well understood yet. The relationship between the

morphology of GB and the interface orientation of prior grain boundary is still not

clear. In order to develop a fundament understanding of the variant selection of GB, the

following two problems will be addressed:

36

a) Activation energy and critical nucleus configuration (size, shape, spatial orientation

and OR with the grains) for each of the 12 variants to nucleate at a given grain

boundary between two grains

b) Under what conditions (misorientation and inclination of grain boundary,

undercooling) discontinuous is preferred over a continuous layer of

2.4.2 Correlations between precipitates with different variants in the basketweave

microstructure

As mentioned above, specific variants of plates prefer to cluster together in the

basketweave microstructure. Variants with Type 2 and Type 6 boundary are non-

randomly selected to comprise the basketweave microstructure. However, it is not clear

which of the following two situations occurs: (1) specific variants of appear in group

during their nucleation [61, 62] and then form the cluster; (2) specific variants of new

plates selected by pre-existing plate nucleate and grow near it and thus form the cluster.

It is well known that during nucleation and in early stages of growth, precipitates tend to

be coherent. The observed semi-coherent / interface indicates that precipitates will

lose coherency during its continued growth. Thus the nucleation of coherent plates will

induce large lattice distortion and hence the coherency strain energy will play an

important role during the nucleation process. How the coherency strain energy affects the

nucleation process (the size and configuration of critical nuclei) need to be critically

37

evaluated. In addition, in order to understand the selectivity of new precipitates due to

arbitrary pre-existing plates, anisotropic elastic interaction of a nucleating precipitate

for each variant with the pre-existing plates is also required to analyze.

2.4.3 The effect of dislocation on variant selection

The variant selection due to dislocation is achieved by the heterogeneous nucleation of

phase with specific variant on the dislocation.

However, the maximum misfit accommodation alone fails to explain the variant induced

by dislocation. In order to better understand the variant selection due to dislocation, it still

requires analyzing in details the elastic interaction between the strain field of the coherent

nucleus for each variant and the stress field generated by dislocation in the matrix.

2.4.4 Microstructure evolution with variant selection

As has been demonstrated in this review, variant selection affects the microstructure

evolution in / alloys to a large degree, including the formation of GB, colony and

basketweave microstructure during precipitation. In order to describe and predict the

microstructure evolution and hence to establish a robust microstructure-property

38

relationship, integration of variant selection mechanisms inherent in the precipitation

process is also required in any modeling attempt.

39

Figures

Figure 2.1 Schematic representation of three types of titanium alloys: alloy, ⁄ alloy,

and alloy in a pseudo-binary section through a isomorphous phase diagram [2]

40

Figure 2.2 Typical microstructures in ⁄ Titanium alloys: (a) Grain boundary

GB (b) Colony ; (c) Basketweave and (d) Secondary microstructure

41

Figure 2.3 Schematic illustration of the Burgers orientation relationship, by looking down

[101] // [0001] (pointing into the plane of paper) [10]

42

Figure 2.4 Schematic illustration of the ⁄ interface and misfit dislocation

configuration [38]

Figure 2.5 [ ] stereographic projection shows that GB maintains Burger OR with

grain and exhibits a small deviation from Burger OR with respect to adjacent grain

(G. B. P. indicates grain boundary plane) [16]

43

Figure 2.6 Schematic illustration of the variant selection rule by the grain boundary plane

(G.B.P.)-conjugate ⟨ ⟩ ⟨ ⟩ direction tends to parallel to G.B.P. [16]

Figure 2.7 Schematic illustration of GB of different variants formed at a grain boundary

with a slight variation in its boundary plane [16]

44

Figure 2.8 (a) a prior grain boundary with the colony microstructure in one of the

grain ( grain 1) and the basketweave microstructure in the adjoining grain 2 (b)

Orientation Image Microscopy (OIM) map of the same region as shown in (a). Regions

with the same color represent the same orientation variant [26]

45

Figure 2.9 (a) OIM map of three different laths sharing a common ⟨ ⟩ direction in

a basketweave microstructure selected from grain 2; (b) Superimposed pole figures of

{110} poles in matrix with the {0001} poles of the clustering laths; (c)

Superimposed pole figures of {111} poles in matrix with the { } poles of the

clustering laths [26]

46

Figure 2.10 (a) OIM map of a cluster of three different laths in the basketweave

microstructure; (b) and (c) superimposed pole figures indicate that lath 1 and 2 share a

common basal plane; (d) and (e) superimposed pole figures indicate that lath 2

and 3 share a common ⟨ ⟩ ⟨ ⟩ [64]

47

Figure 2.11 (a) precipitates of single variant showed same morphology within slip band

in the matrix [15]; (b) Schematic illustration of the variant selection of on the slip

band [20]

48

Tables

Table 2.1 All 12 variants of the Burgers orientation relation between precipitate sand

matrix [10, 16]

Variants Orientation Relationship

V1 110 0001//

111 1120//

112 1100//

V2 111 1120//

112 1100//

V3 110 0001//

111 1120//

112 1100//

V4 111 1120//

112 1100//

V5 011 0001//

111 1120//

211 1100//

V6 111 1120//

211 1100//

V7 011 0001//

111 1120//

211 1100//

V8 111 1120//

211 1100//

V9 101 0001//

111 1120//

121 1100//

V10 111 1120//

121 1100//

V11 101 0001//

111 1120//

121 1100

//

V12 111 1120//

121 1100

//

49

Table 2.2 Axis/angle pairs for all 6 possible ⁄ boundaries in a single grain [22, 27]

ype Axis/angle pairs Prandom [%]

1 I (Identity) -

2 1120 60 18.2

3 10 7 17 3 60.83

36.4

4 10 5 5 3 63.26

18.2

5 7 17 10 0 90

18.2

6 0001 10.53 9.1

50

Table 2.3 Qualitative correlation between colony and lamellae (single plate) size and

mechanical properties for titanium alloys [2, 9]

0.2 F HCF Micro-cracks

ΔKth

Macrocracks

KIC

Small

Colonies

Lamellae

+ + + + -

+ (positive), -(negative) to specific mechanical property

51

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Bernard I, Edward JK, Subhash M, Patrick V, editors. Encyclopedia of Materials:


[33] Aaronson HI, Medalist RFM. Atomic mechanisms of diffusional nucleation and

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1993;24A:241.

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[34] Aaronson HI, Spanos G, Masamura RA, Vardiman RG, Moon DW, Menon ESK,

Hall MG. Sympathetic Nucleation - an Overview. Materials Science and Engineering B-

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[alpha]-vanadium. Journal of the Less Common Metals 1973;31:299.

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intragranular proeutectoid [alpha] plates in a hypoeutectoid Ti---Cr alloy. Acta Metall

Mater 1991;39:2873.

[37] Furuhara T, Ogawa T, Maki T. Atomic-Structure of Interphase Boundary of an

Alpha-Precipitate Plate in a Beta-Ti-Cr Alloy. Philos Mag Lett 1995;72:175.

[38] M. J. Mills, D. H. Hou, S. Suri, Viswanathan GB. Orientation relationship and

structure of alpha/beta interface in conventional titanium alloys. In: R. C. Pond, W. A. T.

Clark, King AH, editors. Boundaries and Interfaces in Materials: The David A. Smith

Symposium: The Minerals, Metals & Materials Society, 1998, 1998. p.295.

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of orientation relationship and interface structure in two phase alloys. J Jpn Inst Met

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relationship of intra- and inter-granular precipitates in titanium alloys. Materials Science

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deformation and mechanisms of slip transmission in oriented single-colony crystals of an

alpha/beta titanium alloy. Acta Mater 1999;47:1019.

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intragrainular proeutectoid [alpha] precipitates in a Ti-7.26 wt%Cr alloy. Acta Mater

2004;52:2449.

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in a Ti-7.26 wt.% Cr alloy. Acta Mater 2006;54:871.

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boundaries. Metallurgical and Materials Transactions A 1978;9:363.

[48] Dahmen U. Orientation relationships in precipitation systems. Acta Metall

1982;30:63.

[49] Lin F, Starke E, Chakrabortty S, Gysler A. The effect of microstructure on the

deformation modes and mechanical properties of Ti-6Al-2Nb-1Ta-0.8Mo: Part I.

Widmanstätten structures. Metallurgical and Materials Transactions A 1984;15:1229.

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[51] Lee JK, Aaronson HI. Influence of faceting upon the equilibrium shape of nuclei

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[54] Wang Y, Ma N, Chen Q, Zhang F, Chen S, Chang Y. Predicting phase


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selection on microstructure evolution in titanium alloys. Warrendale: Minerals, Metals &

Materials Soc, 2005.

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[58] Robert WB, Allen SM, Carter WC. Kinetics of materials. Hoboken: John Wiley &

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[59] Christian JW. The Theory of Transformations in Metals and Alloys. Oxford:

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[60] Sutton AP, Balluffi RW. Interfaces in Crystalline Materials. Oxford: Oxford

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modeling of martensitic transformations. Acta Mater 1997;45:759.

[63] Courtesy of H.L. Fraser. Courtesy of H.L. Fraser in The Ohio state university.

[64] Lee E. Microstructure evolution and microstructure/mechanical properties

relationships in alpha+beta titanium alloys. vol. Ph.D. dissertation: The Ohio State

University, 2004.

59

CHAPTER 3 Predicting Equilibrium Shape of Precipitates as

Function of Coherency State

Abstract:

A general approach is proposed to predict equilibrium shapes of precipitates in crystalline

solids as function of size and coherency state. The model incorporates effects of

interfacial defects such as misfit dislocations and structural ledges on transformation

strain and on interfacial energy. Using precipitation in titanium alloys as an

example, various possible equilibrium shapes of precipitates having different defect

contents at interfaces are obtained by phase field simulations. The simulation results

agree with experimental observations in terms of both precipitate habit plane orientation

and defect content at the interface. In combination with crystallographic theories of

interfaces and experimental characterization of habit plane of finite precipitates, this

approach has the ability to predict the coherency state (i.e., defect structures at interfaces)

and equilibrium shape of finite precipitates.

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3.1. Introduction

Most engineering alloys are strengthened by second-phase particles and their quantity,

size, shape, orientation, coherency state and spatial distribution determine the

deformation mechanism and mechanical behavior of the alloys [1, 2]. Classical examples

include Al- , Ti- and Mg-based light alloys [3-6] and high-temperature Ni-base

superalloys [7], to name a few. To assist in alloy design, it is essential to develop

modeling capabilities to predict these key microstructural features. However these

features are determined by the interplay between interfacial and elastic strain energy

minimization during precipitation, which is difficult to quantify theoretically or by

experiment.

New phases formed during precipitation reactions in solids usually have different

compositions and structures from those of the parent phase. During nucleation and in the

early stages of growth, precipitates tend to be coherent with the matrix, which minimizes

the interfacial energy [8, 9]. They may lose coherency during continued growth when the

elastic strain energy contribution to the total free energy of the system becomes

dominant. Formation of line defects such as misfit dislocations within the interface

relieves misfit stress at the expanse of increasing interfacial energy. In addition to misfit

dislocations, another type of line defects, structure ledges [10], which exhibit step

character as well as dislocation properties [11, 12], are also frequently observed at inter-

phase interfaces. They are also referred to as transformation dislocations or

disconnections to distinguish themselves from defects without the step character in the

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topological model for structural phase transformations [11]. In contrast to misfit

dislocations, it is well recognized that the existence of structure ledges increases the

degree of coherency of a hetero-phase interface and hence lowers the interfacial energy

[13]. Examples of these line defects at a BCC-HCP interface are shown schematically in

Fig. 3.1(a).

Since misfit dislocations and structure ledges not only alter the coherency stress and

change the interfacial energy and its anisotropy, but also introduce growth anisotropy,

they impact all the key microstructural features mentioned above. In addition, the

structural defects at interfaces may alter the nature of precipitate-dislocation interactions

and change the deformation mechanisms (e.g., cutting vs. looping), as well as the nature

of precipitate-martensite interactions and change the transformation paths [14, 15].

Therefore, in order to predict the key microstructural features of precipitates and how

they interact with dislocations and other types of precipitates, the interfacial defect

structure as a function of precipitate size has to be determined first. With the advances in

high resolution electron microscopy, defect structures at many hetero-phase interfaces

have been characterized. However, it is difficult to determine how the defect structure

changes when particle size changes. Models accounting for misfit dislocations and

structural ledges in an integrated manner, in terms of their effect on coherency elastic

strain energy, interfacial energy and final equilibrium shape of finite precipitates, are still

lacking. Existing crystallographic theories, such as the invariant line model [16, 17],

structure ledge model [18], edge-to-edge matching model [19], O-lattice model [20, 21],

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and topological model [11, 12], have been successful in predicting some of the major

crystallographic features of hetero-phase interfaces in infinite systems, including

orientation relationship (OR), habit plane orientation and defect structure within

interfaces. Nevertheless, it is difficult to predict the shape and interfacial defect structure

of a finite precipitate, which is a typical variational problem where the sum of the

interfacial and elastic strain energies as a functional of interfacial defect structure is

minimized.

Most of the existing models for microstructural evolution during precipitation consider

either coherent [22-24] or incoherent precipitates and ignore interfacial defects. In this

chapter, we propose a general approach that incorporates interfacial defects in a phase

field model. Using precipitation reaction in a near titanium alloy as an example, we

show how different types of interfacial defects relieve the coherency elastic strain,

change the interfacial energy and its anisotropy, and affect the habit plane orientation and

equilibrium shape of precipitates. We also discuss how to predict interfacial defect

structures and the critical information required.

3.2. Elastic Strain Energy of Coherent and Semi-Coherent Precipitates

As aforementioned, a precipitate phase is usually different from the matrix in terms of

composition, crystal structure and orientation, which results in lattice misfit across the

precipitate-matrix interface. The elastic deformation that accommodates the misfit in the

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crystal lattices of adjoining phases to form coherent or semi-coherent inter-phase

boundaries, known as coherency strain, usually plays a significant role in solid-state

phase transformations [22-26]. Being both volume- and morphology-dependent, the

coherency elastic strain energy affects precipitate shape, spatial arrangement, as well as

the overall driving force for the transformation. In addition, as a nucleating phase may

possibly adopt a metastable structure with low-energy coherent interfaces with the parent

matrix, the final transition to the stable phase structure is controlled by the coherency

strain energy and its interplay with the interfacial energy.

Coherency strain energy of an arbitrary coherent or semi-coherent multi-phase mixture

can be treated in the framework of Eshelby [27, 28] using the general theory of phase

field microelasticity by Khachaturyan and Shaltov (KS theory) [22, 29, 30], formulated

upon spatial distribution of the stress-free transformation strain (SFTS). The SFTS

associated with arbitrary compositional and structural non-uniformities in an arbitrary

multi-phase mixture can be expressed in terms of a set of conserved (e.g., concentration)

and non-conserved (e.g., long-range order parameter, inelastic displacement, etc.) order

parameter fields (also called phase fields) [22, 25, 31-33]:


0 00

1

( ) ( ) ( )N

ij ij p

p

p

x x (3.1)

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which is a linear superposition of all N types of non-uniformities with ( )p x being the

phase fields characterizing the p-th type non-uniformity and 00 ( )ij p (i ,j=1,2,3) being the

corresponding SFTS measured from a given reference state. Note that 00 ( )ij p depends on

the lattice correspondence (LC) between the precipitate and parent phases.

Using the above SFTS fields as input the total coherency strain energy of the system at

mechanical equilibrium can be readily obtained by following the Eshelby procedure [27]

in the KS theory [22].

0 00 00 00 00 *

3,

*

3,

1 d g( ) ( ) ( ) ( ) ( ) ( ) ( )

2 (2 )

1 d g ( ) ( ) ( )

2 (2 )

el

ijkl ij kl i ij jk kl l p q

p q

pq p q

p q

E C p q n p q n

B

n g g

n g g

(3.2)

where g is a vector in the reciprocal space and /n g g , 00 0 00( ) ( )ij ijkl klp C p and

1 0[ ( )]ik ijkl j lC n n n is the inverse of the Green’s function in the reciprocal space. ( )p g

is the Fourier transform of ( )p x . The asterisk denotes complex conjugate. represents

a principle value of the integral that excludes a small volume in the reciprocal space

3(2 ) /V at 0g , where V is the total volume of the system. The function ( )pqB n

characterizes the density of coherency elastic strain energy and carries all the information

about the crystallography of the phase transformation and the elastic properties of the

system [22, 25], while information on the shape and volume of precipitates is included in

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( )p x that is equivalent to shape functions of the precipitates. We shall introduce a unit

vector n0 such that the function ( )pqB n reaches its minimum at n=n0 [22]. The physical

meaning of n0 is that it represents the normal to the habit of a precipitate in the real space.

In other words, the minimum strain energy for a given precipitate volume is obtained if

the precipitate develops into an infinite platelet with infinitesimal thickness whose habit

is normal to n0.

In the current study, precipitates are assumed to have the same elastic modulus as that of

the matrix (i.e. the homogeneous modulus case), which simplifies significantly the strain

energy analysis. However, the analysis is valid in cases of inhomogeneous modulus as

well. As a matter of fact, the KS microelasticity theory has been extended to

inhomogeneous modulus systems [34, 35].

3.2.1. Stress-free transformation strain for coherent precipitates

The calculation of the SFTS, 00 ( )ij p , is an important step towards calculating the

coherency elastic strain energy. For fully coherent precipitates, the SFTS can be

calculated directly from the LC between the precipitate and matrix phases, which could

be obtained according to Bollman’s nearest neighbor principle [21]. There are a number

of choices for relating the lattices of the parent and product phases by uniform lattice

deformation. The one with the minimum energy barrier is, in general, the one that

involves the minimum lattice distortion and rotation. Both the Bain correspondence [36]

http://en.wikipedia.org/wiki/Normal_%28mathematics%29

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for the BCC FCC transformation and the Burgers correspondence [37] for the BCC

HCP are such LCs. The calculation of SFTS for a given LC is then straightforward

[22, 38]. For example, if F0 is the transformation matrix or deformation gradient matrix

describing the uniform lattice distortion from the parent phase to the pth

variant of the

product phase following a given LC, then [22, 39]

T

00 0 0 I2

F Fp

(3.3)

where I is the unit tensor and the symbol ''T'' denotes a transpose operation on the

associated matrix. The transformation matrix, 0F , relates the parent crystal lattice site

vector , r , to the product crystal lattice site vector, 'r , by 0' Fr r . Generally, three

pairs of non-coplanar vectors, r and 'r , related by the LC are selected to construct 0F .

The crystal lattice site displacements, u r , associate with the transformation are given

by 0 IF u r r . The strain tensor obtained in Eq. (3.3) is identical to

T1

2 u r u r , where u r characterizes the displacement gradient.

3.2.2. Deformation gradient matrix due to defects at hetero-phase interfaces

For semi-coherent precipitates, the effect of misfit dislocations and structural ledges on

misfit strain can be considered by superposition of their eigenstrains with the SFTS

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calculated for fully coherent precipitates, which can be achieved by treating the

interfacial defects as successive deformations, following the uniform lattice deformation,

applied to the precipitate phase. For structural ledges, their eigenstrains are determined by

the requirements [11, 12] that the misfits in and normal to the terrace plane are cancelled

by dislocation characters associated with the ledges, as shown in Fig. 3.1(b). The

dislocation content by relaxes the misfit normal to the terrace plane while the dislocation

content bx compensates the misfit in the terrace plane. The step character of the structural

ledges causes the habit plane to become inclined by an angel from the terrace plane, as

depicted schematically in Fig. 3.1(b). Knowing the dislocation characters of the structural

ledges, one could treat them as regular dislocations when deriving their eigenstrains.

The deformation gradient matrix, Fdis, due to the presence of a set of periodical misfit

dislocations with spacing D and its Burgers vector being parallel to the z-direction (Fig.

3.1(a)) can be expressed as the following:

dis

D

0

0 IF

b

(3.4)

where b is the length of the Burgers vector of the misfit dislocations. Following the same

approach, the deformation gradient matrix, stepF , due to the presence of steps with

spacing S and height h (as shown in Fig. 3.1(b) ) can be formulated as:

68

x S

step y I

0

b

F b h

(3.5)

with the coordinate system defined on the terrace plane as shown in Fig. 3.1.

The total deformation gradient matrix, including contributions from the uniform lattice

distortion due to the phase transformation (F0) and contributions from misfit dislocations

(Fdis) and structural ledges (Fstep), can be written as:

tot step dis 0F F F F (3.6)

and the effective SFTS tensor becomes

T

eff tot totij I

2

F Fp

(3.7)

The effective SFTS tensor can be readily utilized to predict the habit plane normal, n0,

and calculate the minimum coherency strain energy density, 0pqB n , associated with it,

which can be used conveniently to evaluate the effect of individual interfacial defect on

habit plane orientation and degree of accommodation of coherency elastic strain.

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3.3. Estimation of Interfacial Energy for Semi-Coherent Interfaces

The energy of a semi-coherent hetero-phase interface sc consists of both chemical c

and structural contributions s [9, 40]. The contributions from the misfit dislocations to

the interfacial energy could be determined by using the microscopic phase field model

(MPF) of dislocations [33] or approximated simply by the Read–Shockley formula [41,

42]. With the information of Burgers vector and spacing between misfit dislocations at

the interface, one could estimate the contribution from misfit dislocations to the structural

part of the interfacial energy in different facets of the precipitates according to the Reed-

Shockley formula [42] as following:

s m m m1 lnE (3.8)

where mE represents the energy of a general high misorientation boundaries, is the

misorientaton angle and m (~ 10 - 20º) is a constant determined by the structures of the

two joining crystals. If the magnitude of the Burgers vector and the spacing between

misfit dislocations are b and D, respectively, the misorientation angle can be determined

by ~ b D .

By knowing the effect of interfacial defects on interfacial energy and SFTS, the

equilibrium shape of a finite precipitate can be predicted using phase field simulations

[31-33] that minimize the total free energy of the system, i.e., the sum of bulk chemical

70

free energy, interfacial energy and elastic strain energy. The interfacial defects of an

infinite planar interface could be predicted by specific crystallography theories. For

example, the dislocation content (Burgers vector and spacing) has been well predicted for

precipitate in Ti-7.26 wt. % Cr system [43]. Besides the habit plane or broad face, a

finite precipitate is also surrounded by other non-habit facets. When the O-lattice model

[20] in 3D is not solvable for the OR that permits only one set of periodical dislocation

on the habit plane, the possible orientations and dislocation structures of non-habit facets

(out of habit plane) could be predicted by an extended near-coincidence-sites (NCS)

methods [44, 45], which combines the analysis of fit/misfit distribution in three

dimensions (3D) at a given OR by NCS model [44], with analysis of properties of Moiré

planes [46] developed based on the O-lattice theory [21]. The approach has been applied

successfully to predict orientation of non-habit facets, the Burgers vector and the spacing

of the misfit dislocations in non-habit facets of precipitates in FCC/BCC [45] and

BCC/HCP [47] systems. When structure ledges are also present on irrational habit planes,

the dislocation content, ledge height and inter-ledge spacing associated with the structural

ledges can be derived by the O-lattice theory [48], computer-aid graphical techniques [10,

49], and the topological model [12].

Note that the outputs from these geometrical methods in terms of Burgers vector and

spacing of misfit dislocation are not always unique [43, 47]. Thus, additional constraints

such as the condition of maximum dislocation spacing are required to refine the results,

which rely on advanced experimental characterization (such as TEM or HRTEM).

71

Therefore, both theory and experimental characterization are required to obtain all the

information about the interfacial defects for a finite precipitate.

3.4. Worked Examples

In Ti-alloys, the interfaces between the (HCP) precipitates and the (BCC) matrix

are typical examples of semi-coherent interfaces that contain both misfit dislocations [43,

50] and structural ledges [49]. The precipitates usually exhibit a specific orientation

relationship with the matrix, referred to as the Burgers OR [37], i.e.,

101 // 0001

, 111 // 2110

(3.9)

The interface structures in a near titanium alloy (Ti-5Al-2.5Sn-0.05Fe)

characterized by TEM [50] are shown schematically in Figure 3.2. The broad and edge

faces are wrapped with a single set of c-type misfit dislocations with a Burgers vector of

0001 2

(specific indices are assigned in accordance with the particular variant of OR

described in Eq. (3.9)). The spacing between these dislocations is about 7 nm, which

corresponds to ~30 atomic planes of 0002

or 101

, and their line direction is close

to the invariant line direction 353

. The side and end faces are wrapped with a set of

a-type misfit dislocations with a Burgers vector of 2110 3

, which are associated

with the 0110

and 110

planes with extra half planes in the phase and spaced by

about 9 to 13 0110

atomic plane spacing. On the broad face, in addition to the c-

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dislocations, the atomic structure of the / interface consists of structural ledges that

enable the interface to step down along the invariant line direction.

3.4.1. Derivation of effective SFTS for the semi-coherent precipitates

To formulate the SFTS, it is required to know the lattice correspondence between the

parent and product phases. Figure 3.3 shows the Burgers orientation relationships

between the two phases in both three-dimension Fig.3.3 (a)-(c) and two-dimension

Fig.3.3 (d)-(f). Three non-coplanar vectors in lattice are selected, i.e., 010

,

111 2

and 101

. According to the nearest neighbor principle [21], the

corresponding three non-coplanar vectors in the phase lattice after the to

transformation are 1210 3

, 2110 3

and 0001

[37], respectively. Thus, the

lattice correspondence between the BCC and HCP lattices can be described as,

010 1210 3

; 111 2 2110 3

; 101 0001 (3.10)

as shown in the Fig. 3.3 (a)-(b). Three non-coplanar vectors are chosen as the axes of a

new orthogonal reference coordinate N1, i.e., 1 // 010 // 1210x

,

2 // 101 // 1010x

, 3 // 101 // 0001x

, as shown in Fig. 3.3.

In the reference frame N1, the transformation matrix to deform the lattice

homogeneously to the lattice can be described as:

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0

0 0

0 3 2 0

0 0 2

a a

F a a

c a

(3.11)

where the lattice parameters of the two phases are 3.196a Å, 2.943a Å and

4.680c Å [50]. Note that, as shown in Fig. 3.3(b), during the actual BCC to HCP

lattice transformation, atomic shuffling on every other 0002

plane is required in

addition to the homogeneous deformation 0F [37]. However, since shuffling does not

change the shape and size of the unit cells of the two crystal lattices, 0F does represent

the actual transformation matrix.

In addition to 0F , a rigid body rotation, RF , by 5.26º along axis 101 // 0001

is

required to realize the exact Burgers OR shown in Eq. (3.9) and Fig. 3.3(c)

cos 5.26 sin 5.26

sin 5.26 cos 5.26

1

RF

(3.12)

Thus, the overall transformation matrix for a coherent precipitate can be expressed as

0RF F F , which expresses the coherent transformation matrix of a specific Burgers

variant with OR defined by Eq. (3.9).

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The c-dislocations accommodate the misfit between the 0002

and 101

planes.

According to Eq. (3.4), the deformation gradient matrix due to the c-dislocations on the

broad face can be described as

01

0 I30

1

cF

(3.13)

Similarly, the deformation gradient matrix due to the a-dislocations on the side face can

be described as

01

1 I11

0

aF

(3.14)

In addition to the misfit dislocations, structure ledges also exhibit dislocation characters.

As shown in Fig 1(a), the Burgers vector associated with the structure ledges can be

expressed as x y; ;0 b b b , in a new reference coordinate N2 associated with the terrace

plane:

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'

1 // 111 // 2110x

; '

2 // 121 // 0110x

; '

3 // 101 // 0001x

(3.15)

Dislocation xb lies in the terrace plane and has a Burgers vector of 1 12 111

associated with the riser of the structural ledges [48, 49]. The component xb

compensates the misfit along 111

direction on the terrace plane. Thus, the existence

of the structural ledges can eliminate one set of the a-type (i.e. b= 111

) misfit

dislocations on the terrace plane. The structural ledges have also been shown to have

Burgers vectors and inter-ledge spacing Sλ (i.e. about 17 xb ) both one sixth of their misfit

dislocation counterparts [48]. Moreover, the height of risers or steps, h, is found to be 2

atomic layer of the terrace plane, i.e., 121

or 0110

[34, 35]. On the other hand,

structural ledges step down along the invariant line direction in order to accommodate

simultaneously the misfit normal to the terrace plane [48], which can be well represented

by the dislocation yb . Thus, the Burgers vector of

yb is given by 6 3 3 2a a

according to the O-lattice calculation [48].

Thus, the deformation gradient matrix due to the structural ledges on the terrace face can

be determined as

76

x x

x S

y

b 17bb λ

6 3 3 2b h I= I

6 30

0

s

a aF

a

(3.16)

Note that sF is expressed in the reference frame N2. To express sF in coordinate N1, the

deformation gradient matrix '

SF due to the structural ledges becomes: ' 'S sF QF Q ,

where Q is the transformation matrix between coordinate N1 and N2.

Thus, the total deformation gradient matrix, including contributions from the uniform

lattice distortion due to the phase transformation and from misfit dislocations and

structural ledges, can be formulated as:

'

tot 0S a c RF F F F F F (3.17)

and the effective SFTS tensor eff

ij p can be derived using Eq.(3.3).

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3.4.2. Strain energy density and habit plane orientation of semi-coherent

precipitates

The details of the deformation gradient matrix and transformation strain associated with

the coherent to lattice deformation, misfit dislocations and structural ledges are

summarized in Table 1. With the information of the effective transformation strain, the

habit plane orientation, 0n , can thus be predicted by finding the minimum of 0( )pqB n .

For fully coherent precipitates, on substituting the values for the lattice parameters and

ignoring the shearing components, the magnitudes of the principal strains are as follows:

8.3% contraction along 010

, 12.3% expansion along 101

and 3.5% expansion

along 101

(Table 1). The minimum strain energy density 0pqB n was found to be

6.4026×107 J/m

3 with the habit plane normal 0n being [-11; -9.85; 8.07], which deviates

about 8º from the observed [50] habit plane normal [-11; -13; 11]. By introducing a set

of c-type misfit dislocations on the broad face, the 3.5% expansion along 101

was

eliminated, which results in a significant coherency strain energy reduction, from

6.4026×107 J/m

3 down to be 5.1232×10

3 J/m

3. In addition, the habit plane normal 0n as

predicted by minimizing 0pqB n was found to be [-11; -14.05; 11], which deviates

about 2º from the observed habit plane normal. The introduction of a-type misfit

dislocations on the side face further reduces the coherency strain energy down to

78

4.7269×103 J/m

3. However, the habit plane normal 0n deviates further from the

experimental observation (see Table I). Finally, the structural ledges on the broad face

further relax the coherency strain energy, though not by much. However, due to the

presence of the structural ledges, the habit plane rotated towards the experimentally

observed orientation with only about 0.8º deviation.

The calculated coherency strain energy density, pqB n , as a function of habit plane

normal is projected onto the basal plane, as shown in Fig. 4. The solid circles represent

the case without considering defects at the interfaces. The minimum of pqB n is

obtained where 0n is indicated by the solid arrow. For comparison, the result obtained for

the case where interfacial defects are considered is shown in Fig. 4 by the open circles. It

can be readily seen that the defects on the interface relax considerably the coherency

elastic strain energy. In addition, pqB n reaches its minimum at 0n corresponding to [-

11; -12.63; 11]as indicated by the dotted arrow, which deviates only by 0.8º from the

experimentally observed habit plane [-11; -13; 11]

3.4.3. Interfacial energy anisotropy of semi-coherent precipitates

On substituting the dislocation spacing and the corresponding Burgers vector, the

equivalent misorientation angles for the broad and side faces are 1 1 30 1.9 and

3 1 11 5.2 , respectively. The equivalent misorientation angle of the edge face is

79

assumed to be 3 m . According to Eq.(3.8), the structural components, s , of the

broad, side and edge faces are m0.39E , m0.72E and mE , respectively. mE is assumed to

be 250 mJ/m2 [51]. In addition, the chemical components of the interfacial energy for the

three faces are assumed all equal for simplicity and have a value of 50 mJ/m2, which is

reasonable for fully coherent interfaces. Therefore, the interfacial energies of the broad,

side and end faces are 150, 230 and 300 mJ/m2, respectively. The results are incorporated

in the gradient energy coefficient characterizing structural non-uniformities in the phase

field free energy formulated based on the gradient thermodynamics [52].

3.4.4. Equilibrium shape of -precipitates in different cases

The equilibrium shapes of an isolated precipitate determined by the interplay between

the interfacial energy and strain energy under different cases are obtained by phase field

simulations [14] and are presented in Fig. 3.5 with the coordinates being indicated in Fig.

3.5(a)-1. In all cases, a specific initial composition is selected along the tie-line of Ti-

6Al-4V (wt. %) [53] to obtain an equilibrium volume fraction of 5% for the precipitate.

The three-dimensional equilibrium shape of a fully coherent precipitate obtained under

the assumption of isotropic interfacial energy of 50 mJ/m2

is shown in Fig. 3.5(a)-1. Also,

the projections of the three-dimensional equilibrium shape along 010

, 101

and

101

are shown in Figs. 5(a)-2 - 5(a)-4, respectively. The total system size is 16 nm. It

80

is readily seen that the particle has a disk-like shape with a well-defined habit plane.

However, the orientation of the habit plane obviously deviated from the experimentally

observed one, which is indicated by a transparent light yellow plane across the center of

the simulation cell. The deviation manifests itself via the shaded top end of plate since it

lies below the light yellow plane, as shown in Fig. 3.5(a)-1 and 5(a)-3. Note that the

minor axis of the disk in the habit plane is about 7 nm, which is commensurate with the

spacing of the c-dislocations on the broad face. Therefore, particles of such a size are

most likely to be coherent.

The equilibrium shape for a semi-coherent precipitate with the consideration of c- type

misfit dislocations on the broad face is presented in Fig. 3.5(b). It can be easily found in

this case that the habit plane is almost parallel to the light yellow plane which indicates

the experimental observed habit plane. Superposition of Figs. 5(a)-4 and 5(b)-4 indicates

that the c-dislocations change the habit normal to nearly parallel to the experimentally

observed one, as shown in Fig. 3.5(c). When considering all the defects, including misfit

dislocations and structural ledges, the equilibrium shape of the precipitate, in the case

of isotropic interfacial energy (200 mJ/m2), is shown in Fig. 3.5(d). The habit plane

normal almost coincides with the experimentally observed one. In the case of interfacial

energy anisotropy alone, the equilibrium shape of the precipitate is shown in Fig.

3.5(e), which becomes an ellipsoid with no obvious habit plane. Finally Fig. 3.5(f)

presents the equilibrium shape of the semi-coherent particle obtained when the

anisotropy in interfacial energy is considered. Obviously, it has a well developed plate

81

shape with a habit plane normal almost parallel to the experimentally determined habit

plane normal. However, compared to Fig. 3.5(d)-4, it is more elongated along the

invariant line direction and contracted along z or [101] direction as shown in Fig. 3.5(f)-

4.

3.4.5. Coherency lost

A coherent precipitate may lose its coherency during its continued growth. Although the

dislocation spacing on the semi-coherent interface can be well characterized by high

resolution TEM or predicted by O-line theory, it is still desirable to estimate a critical size

beyond which the coherent precipitate changes to a semi-coherent one. As mentioned

earlier, the coherency state of precipitates may change the nature of precipitate-

dislocation interactions and hence the deformation mechanisms. There is no doubt that

the critical size must be larger than the inter-dislocation spacing. However, this critical

size depends on the difference between the total energy (interfacial energy and strain

energy) of the coherent precipitate and its semi-coherent counterpart of a given size [9].

In other words, beyond a critical size, rcrit, it becomes energetically favorable for a

coherent particle to lose coherency. Since the optimum shapes of the precipitate in

different cases have been obtained for both coherent and semi-coherent precipitate (with

c-dislocations only since their contribution to the strain energy reduction is dominant as

shown in Table 3.1) (Fig. 3.5(a)-(b)), it is possible to evaluate its interfacial energy [52]

and strain energy [22] and hence the total energy of a given size by the phase field

method. The results are shown in Fig. 3.6. The coherent precipitate has a lower interfacial

82

energy than the semi-coherent one (Fig. 3.6(a)) due to extra structural contribution to the

interfacial energy s. On the other hand, as shown in Fig. 3.6 (b), the semi-coherent

precipitate has lower strain energy than the coherent one since misfit dislocation release

part of the coherency strain energy. Therefore, there exists a crossover point between the

total energy of the coherent (open circle) and semi-coherent (solid) precipitates as

function of their volumes, as show in Fig. 3.6(c), which yields a critical size, rcrit, of ~27

nm when s is 50 mJ/m2 and 22 nm when s = 25 mJ/m

2. Thus, the critical size rcrit scales

with the structural part of the interfacial energy due to misfit dislocations, which agrees

with the analysis by Porter and Easterling [9]. Note that the predicted critical sizes are

about 3-4 times of the c-dislocation spacing on the broad face.

3.5. Discussions

A plane remains undistorted under the action of a homogeneous lattice strain if and only

if one of the principal strains is zero and the other two are of opposite signs [38]. Due to

the introduction of c-type misfit dislocations on the broad face, an interesting feature of

the BCC to HCP transformation is that the principal strain ( 3 ) along the 101

direction becomes very small (-0.03%) and the other two are of opposite signs, i.e., 8.3%

contraction along 010

, and 12.3% expansion along 101

. It is thus not

unreasonable to treat the transformation with the approximation that 3 is zero.

Therefore, the lattice deformation would have left a plane undistorted when considering

83

the effect of the c-type dislocations. As shown in Table 3.1, the introduction of the c-type

misfit dislocations relax most of the coherency strain energy if the precipitate develops

into a thin plate whose habit is normal to 0n , [-11; -14.05; 11]

Based on the assumption that the long-range misfit strain in the interface is completely

accommodated by a single set of periodical array of misfit dislocations on a singular

interface [54], Zhang et al. [43] predicted the habit plane structure of precipitates in Ti-

7.26wt%Cr alloy using the O-line model developed in the frame work of the O-lattice

theory. The Burgers vector, line direction and spacing of dislocations on the habit plane

from their predictions agree well with their experimental observations [43]. However, the

assumption that there is no long-range strain may not be truly satisfied. Otherwise, the

set of a-type misfit dislocations would not develop on the side face of the precipice. The

O-lattice theory, in essence, is a geometrical method that considers two infinite crystals.

In practice, precipitates are finite and the strain energy of a plate-shaped precipitate

having a finite thickness is finite. Accumulation of the residual strain will induce misfit

dislocations on the other facets of interfaces, such as the a-type dislocations on the side

face of the precipitate. This additional set of misfit dislocations further relaxes the

coherency strain energy (see the calculation results in Table 3.1). However, due to the

introduction of the a-type misfit dislocations on the side face, the predicted habit plane

obviously deviates from the experimentally observed one. When further considering the

structural ledges on the habit plane, the orientation of the precipitated is rotated back to

84

the experimentally observed one. Therefore, the a-type misfit dislocations and the

structural ledges seem to be coupled and they have to be considered simultaneously.

It is worth mentioning that in predicting dislocation structures on non-habit facets, the

OR was fixed to the one determined by the O-line model when applied to predict

dislocation structures on the habit plane. This suggests that the dislocation structures on

the non-habit planes would not change the OR any more. This may not be the case for

finite precipitate. As suggested by Aaronson et al. [8], for example, the interface itself

may rotate, allowing the Burgers vector of the dislocations to lie in the interface and

hence be fully utilized to compensate the lattice misfit.

The structural ledges are present at hetero-phase interfaces as a means to increase the

level of coherency between the BCC and HCP lattice. In addition, structural ledges have

Burgers vector associated with their risers and also lying in the terrace plane. In other

words, structural ledges play a role in compensating interface misfit alternative to misfit

dislocations. It has been showed that [48, 49] on the Burgers-related 0110 // 121

terrace planes, the misfit along the 0001 // 101

direction is relaxed by a single set of

parallel c-type misfit dislocation, while the misfit along 2110 // 111

direction is

compensated by the dislocation content associated with the structural ledges. However, in

comparison with the misfit dislocations, the Burgers vector associated with the structural

ledges is relatively small. For example, in the BCC/HCP system, the Burgers vector

85

associated with the riser of the structural ledges is 1 12 111

. Thus the introduction of

structural ledges would not effectively relax the coherency strain energy. According to the

calculation results, the structural ledges only reduce the coherency strain energy from

4.73×103 J/m

3 to 4.71×10

3 J/m

3.

It should be emphasized that the structural ledges in the BCC/HCP example considered

are different from the defect structures of habit planes of an internally twinned or slipped

martensitic plate embedded in austenite, where the twinning or slip (called lattice

invariance deformation) is required to provide an invariant plane strain (IPS) and the

defect structures at the invariant plane habit are “by-products” of the twinning or slip.

These defects (facets or steps) do not contribute to the long-range elastic strain energy,

though locally contribute to the interfacial energy. In contrast, the structural ledges on the

broad face of an plate are required to relax the long-range elastic strain. As can be seen

from Table 3.1, the structural ledges compensate the lattice misfit in a manner similar to

misfit dislocation.

As has been mentioned earlier, the interplay between interfacial energy and strain energy

minimization determines the final equilibrium shape of a precipitate, i.e., the equilibrium

shape of a precipitate is determined by the condition that the sum of elastic and interface

energies reaches minimum at a given precipitate volume. In the cases of isotropic

interfacial energy (e.g. Fig. 3.5(a), (b) and (d)), the precipitate tends to develop an

86

optimum shape with its broad face corresponding to the minimum of 0pqB n

to

minimize the strain energy. As the coherent particle continues to grow, the

precipitate/matrix interface can no longer maintain coherency and misfit dislocations will

be generated to relieve the coherency stress. As discussed above, the c-dislocations on the

broad face reduce greatly the density of the strain energy. The variation of the habit plane

orientation from coherent to semi-coherent particle is well illustrated by Fig. 3.5(c).

The strain energy of a finite plate-like inclusion, E, can be described as [22]:

0 edge2pqE B V E n (3.18)

The first term in Eq. (3.18) on the right hand side describes the strain energy of an

infinite plate of infinitesimal thickness (Dp0) and the second term can be regarded as

energy correction associated with a finite plate thickness. The value of edgeE can be

described as 2

edge 0 P~E D P [22], where , 0 and P are the elastic modulus,

transformation strain and plate perimeter P, respectively. The energy edgeE is

proportional to the perimeter length since it is associated with the lattice mismatch

between the precipitate and matrix along the edges of the plate-like precipitate [22]. It is

quite possible that misfit dislocations would appear at the edges (side face) to further

relieve the strain energy of the precipitate when edgeE exceeds a critical value just as the

additional a-type dislocation in side face of the precipitate.

87

If considering interfacial energy alone, one would expect a plate-like shape if there is a

strong cusp in the -plot. From the equilibrium shape in Fig. 3.5(e), it is clear that the

precipitate is more like an ellipsoid than a plate, which suggests that the interfacial

energy anisotropy in this system (at least for the parameters used in the phase field

simulations) is not strong enough to generate a plate-like precipitate. However, the

interfacial energy anisotropy does cause elongation of the particle along invariant line

direction, as shown in Fig. 3.5(f).

The predicted critical size is clearly much larger than the inter-dislocation spacing. On

one hand, the introduction of a single c-type misfit dislocation on the broad face will

relax part of the coherency strain energy via modification of 0pqB n and also increase

the interfacial energy due to its structural contribution. On the other hand, the total strain

energy of a finite precipitate includes two parts according to Eq. (3.18): edgeE and

0 2pqB Vn . In our analysis of coherency lost presented in Section 4.5, the contribution

of edgeE is ignored. When the precipitate size is smaller than critical size, however, the

edgeE part could be dominant and the reduction in strain energy via modification of

0pqB n through the introduction of c-dislocations to the broad face may not be able to

compensate the increase in interfacial energy. Beyond the critical size, the relaxation of

the strain energy via modification of 0pqB n would be dominant and sufficient to

compensate the increase in interfacial energy. Therefore, the total energy of a semi-

88

coherent precipitate would become lower than that of a fully coherent one when its size is

about 3-4 times of the dislocation spacing. It should be pointed out here that the analysis

does not consider the difficulties in dislocation acquisition to convert a coherent interface

into an incoherent one. Therefore, the actual transition from coherent to semi-coherent

interfaces may occur at even larger precipitate sizes [9].

The equilibrium shape of a finite precipitate is determined by the interplay between the

interfacial energy and the strain energy. So the final results on equilibrium shape

predicted in different cases depend critically on the accuracy of the evaluation of the

interfacial energy and its anisotropy. In the current paper, however, how the structure

ledges increase the coherency on the broad face and thus reduce the interfacial energy has

not been considered. In addition, the prediction of the critical size, rcrit , also relies on the

accuracy of the evaluation of the structural part of the interfacial energy. Despite these

limitations, the current approach provides a general method to study the relationship

among the coherency state, equilibrium shape and, critical size of a finite precipitate.

3.6. Summary

In summary, we have formulated a general method to derive effective transformation

strain during precipitation that considers the effect of interfacial defects including misfit

dislocations and structural ledges. How the interfacial defects relax the coherency elastic

strain energy and affect the habit plane orientation are analyzed in detail by incorporating

89

the effect of the defects into the stress-free transformation strain. How the interfacial

defects affect the interfacial energy anisotropy and the final equilibrium shape of

precipitates is also investigated. The equilibrium shapes of an isolated precipitate

generated by the interplay between interfacial energy and elastic strain energy are

obtained from phase field simulations. The habit plane orientation is found to be

dominated by the strain energy minimization, while interfacial energy anisotropy

contributes to the in-plane shape (ratio of the two major axes). The present work may

build a bridge between the O-line theory of precipitate habit planes and interfacial

dislocation structures based on pure geometrical consideration and the theory of optimum

shapes of precipitate based on the consideration of strain energy that depends on

precipitate size, coherency state, shape and orientations. However, both theoretical

analysis and experimental characterization are required to obtain all the information

about the interfacial defect structures for a finite precipitate.

90

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96

Figures

Figure. 3.1. (a) Schematic illustration of an inter-phase interface between BCC and HCP,

exhibiting both structural ledges (disconnections) [12] and misfit dislocation arrays. The

interface is decorated by arrays of structural ledges (b, h) with height h and spacing

and misfit dislocations with spacing . The terrace coordinate frames, the line

direction of the ledges, , and dislocations, , are also shown; (b) The dislocation

properties associated with structure ledges (disconnections) with Burgers vector resolved

in the terrace plane. The terrace plane (bold) is inclined at an angle to the habit plane

(dashed).

97

Figure. 3.2. Schematic illustration of the ⁄ interface defect structure in near-

titanium alloy Ti-5Al-2.5Sn-0.5Fe (wt.%), following Mills et al.[50]

98

Figure. 3.3. Schematic lattice correspondence between the BCC -phase and the HCP -

phase during to transformation maintaining Burgers OR in both three-dimension (a)-

(c) and two-dimension (e)-(f).

Sony

附注

[-1 1 1]和[-1 -1 1]方向的夹角为70.5度因为点乘=1-1+1=sqrt(3)*sqrt(3)*cos(theta) --- theta=arccos(1/3)=70.5288度将最密排方向[-1 1 1]beta 平行于 [-2 1 1 0]alpha时，需要旋转(70.5288-60)/2=5.2644度；即该2D点阵对应关系图所示。（该图已清晰描述，我仅写出具体理解过程）

99

Figure. 3.4. Density of coherency elastic strain energy, in the case of with (red

open circles) and without (black solid circles) considering defects at the interface

(projected on the plane).

100

Continued

Figure. 3.5. Equilibrium shapes of an isolated -precipitate in different cases. (a)–(d)

isotropic interfacial energy with/without interfacial defects: (a) fully-coherent; (b) c-

dislocations on the broad face; (c) superposition of (a) and (b) showing the difference in

habit plane orientations; (d) all interfacial defects present.(e) Anisotropic interfacial

energy alone. (f) Both anisotropic interfacial energy and coherency elastic strain energy

with all interfacial defects considered. The transparent light yellow plane in each case on

the left column denotes the experimentally observed habit plane

[010][101]

[10-1]

(a)-1

16 nm

(b)-1

(a)-2

[010] [10-1]

(a)-2 (a)-3

(b)-2 (b)-3 (b)-4

[101]

(a)-4

(b)-4

(c)

[010]

[10-1]

101

Figure. 3.5 continued

(d)

(e)

(f)

(e)-2 (e)-3 (e)-4

(d)-2 (d)-3 (d)-4

(f)-2 (f)-3

(d)-4

(e)-4

(f)-4

102

Figure. 3.6. The interfacial energy (a), strain energy (b) and total energy (c) vs.

precipitate volume for coherent and semi-coherent precipitate with its equilibrium shape.

(a)-(c) is obtained with the structural part of interfacial energy =50 mJ/m2 and the

critical size rcrit is about 27 nm (along the minor axis on the broad face); (d) =25

mJ/m2 and the critical size rcrit ≈ 22 nm.

103

Table 3.1 Effect of different types of line defects in inter-phase interface on coherency

strain energy and habit plane orientation Lattice parameter of the two phases 3.196a Å,

2.943a Å and 4.680c Å and I is unit tensor

104

Table 3.1

Defects Deformation gradient F Transformation strain TF F 2-I

Case I

Defects-free (fully coherent)

0

0 0

cos 5.26 sin 5.26 0

3sin 5.26 cos 5.26 0 0 0

20 0 1

0 0 2

a

a

aRF

a

c

a

3

3

0.083 9.486 10

9.486 10 0.123

0.035

Case II

c-type misfit dislocation

on broad face (habit plane)

01

0 I30

1

cF

3

3

4

0.083 9.486 10

9.486 10 0.123

2.998 10

Case III

a-type dislocation

on side face

01

1 I11

0

aF

4

0.083 0.013

0.013 0.021

2.998 10

Case IV

Structural ledge on broad face

x S

y

b λ

b h I

0

sF

3

3

4

0.049 3.1 10

3.1 10 0.067

2.998 10

Continued

104

Sony

附注

sqrt(2)在分母；式(3.11)是对的！ R矩阵应参考Eq.(3.12)，R(1,2)和R(2,1)元素的正负号有误。最终写的Epsilon矩阵是对的

105

Table 3.1 countinued

Minimum Bpq(n) [J/m3]

Coherency strain energy density

n0

(habit plane orientation)

Deviation from experimental

observation ([-11;-13;11]) [o]

Case I 6.4026×107 [-11;-9.85;8.07] 7.997

o

Case II 5.1232×103 [-11;-14.05;11] 2.2048

o

Case III 4.7269×103 [-1;-3.8185;1] 29.79

o

Case IV 4.71×103 [-11;-12.63;11] 0.79

o

105

106

CHAPTER 4 Variant Selection during Precipitation in Ti-

6Al-4V under the Influence of Local Stress

Abstract:

Variant selection of (HCP) phase during its precipitation from (BCC) matrix plays a

key role in determining the microstructural state and mechanical properties of /

titanium alloys. In this work, we develop a three-dimensional (3D) quantitative phase

field model to predict microstructural evolution and variant selection during

transformation in Ti-6Al-4V (wt.%) under the influence of both external and internal

stresses. The model links its inputs directly to thermodynamic and mobility databases,

and incorporates the crystallography of BCC to HCP transformation, elastic anisotropy,

and defects within semi-coherent / interfaces in its total free energy formulation. It is

found that for a given undercooling, the development of a transformation texture (also

called micro-texture) of the phase due to variant selection during precipitation is

determined by the interplay between externally applied stress or strain and internal stress

generated by the precipitation reaction itself. For example, the growth of pre-existing

precipitates is accompanied by selective nucleation and growth of secondary plates of

107

certain variants that may not be the ones preferred by the initially applied stress. Possible

measures to reduce transformation texture are discussed.

4.1. Introduction

Titanium alloys have many advanced applications, ranging from aircraft engine

components [1] to medical implants [2], owing to their excellent combinations of high

strength-to-density ratio and excellent fracture toughness and corrosion resistance [3].

Among these alloys, two-phase /Ti alloys are the most widely used ones because a

rich variety of microstructures and mechanical properties can be obtained simply by

varying thermo-mechanical processing. For example, the -processed microstructure

consisting solely of widmanstätten [4] (HCP) plates in a BCC) matrix shows

superior resistance to creep and fatigue crack growth [1]. On the other hand, the /-

processed microstructure consisting of a combination of globular (or equiaxed) grains

with transformed structure (fine scale widmanstätten laths with adjacent ribs) offers

high ductility as well as good fatigue strength [1].

Microstructure engineering of / titanium alloys through thermo-mechanical

processing is based largely on the → + transformation upon cooling, which involves

both composition and structure changes. Usually precipitates form either as an

allotrimorph along prior -grain boundaries or as Widmanstätten plates nucleating either

108

from the grain boundary or in the interior of grain [1, 5, 6]. It is well known that the

plates usually maintain a specific orientation relationship (OR) with the matrix,

referred to as the Burgers OR [7], i.e., 101 || 0001

and 111 || 2110

.

According to the symmetry of the parent and product phases and the Burgers OR between

them [8], there are twelve crystallographically equivalent orientation variants of the

phase within a single prior grain. Since each of them has different transformation

strains, different variants may have different degree of elastic self-accommodation when

they are in contact with each other and different interfacial energy when they are in

contact with grain boundaries, variant selection (i.e. some variants appear more

frequently than the others) usually accompanies precipitation, leading to the formation

of transformation texture or micro-texture and relatively coarse microstructures.

Since the phase is highly anisotropic in its physical and mechanical properties, the

transformation texture of due to variant selection will determine, to a large extent, the

final mechanical properties of the / Ti alloys [9-12]. For example, the fatigue cracks

usually nucleate at and propagate through the longest and widest plate [5]. Also, micro-

cracks propagate much faster through a highly textured coarse colony microstructure as

compared to a fine colony one of random texture due to the increased tortuosity of crack

propagation paths [13]. Thus, it will be detrimental if, due to variant selection, plates of

a single or a few variants percolate through the whole matrix and it is not unreasonable to

assume that having more variants of plates in an microstructure would lead to

109

improved fatigue properties. Therefore, the control of variant selection during thermo-

mechanical processing is a key to control micro-texturing in the final products and hence

their fatigue properties.

A variety of factors could contribute to variant selection during precipitation,

including external or residual stresses within a polycrystalline matrix [14-16] due to

thermo-mechanical processing, heterogeneous nucleation on pre-existing defects in the

matrix such as grain boundaries [17-22], stacking faults and dislocations [21, 23, 24], and

correlated nucleation due to grain boundary [22] and pre-existing widmanstätten

[25]. For example, the nucleation and growth of coherent plates are accompanied by

significant lattice distortion in the surrounding matrix due to the transformation strain.

Therefore, the surrounding matrix will favor nucleation and growth of specific variants

to accommodate the strain [26]. Pre-existing dislocations in the matrix have been

shown to act frequently as preferential nucleation sites for specific variants. For

instance, dislocations belonging to the 112 111

slip systems would favor the

nucleation and growth of a single variant whose orientation relationship is described by

the components of the specific slip system [21, 23, 27]. Furthermore, stresses may also

arise from anisotropic thermal expansion of grains within a polycrystalline sample [28,

29] and initial texture of grains could enhance variant selection during precipitation

[30].

110

It is clear that the local stress state of an untransformed matrix, due to a variety of

sources, is a key factor in controlling variant selection during precipitation and the final

transformation texture. However, the challenges to study variant selection during

transformations under the influence of stress are two-folds: first, one needs to determine

stress distribution in an elastically anisotropic and inhomogeneous media under a given

applied stress or strain condition; second, one needs to describe interactions of local

stress with coherent and semi-coherent precipitates, i.e. to describe interactions of local

stress with evolving two-phase microstructures. In particular, the coherency state of an

precipitate may change during its growth and defect structures, including misfit

dislocations and structural ledges, at the interfaces may alter the coherency stress.

Because of these complications, limited work exists in literature on modeling variant

selection during transformation in titanium alloys. Moreover, most previous work

about effect of stress on precipitation deals with coherent precipitates only [31-34].

The main objective of this chapter is to develop a three-dimensional (3D) physics-

based phase field model for quantitative prediction of microstructural evolution and

variant selection during transformation in Ti-alloys under the influence of

both external and internal stresses. In the current study, we will focus on effects of a

constant externally applied strain (pre-strain) and internal stresses generated by pre-

existing precipitates on variant selection in a single grain of . Effects of elastic

inhomogeneity and grain boundaries in a polycrystalline sample and effects from other

111

stress-carrying defects such as dislocations and stacking faults on variant selection will be

investigated in a follow-up chapter.

The rest of the chapter is organized as the following. In section 4.2, a 3D quantitative

phase field model is formulated, which incorporates the crystallography of the

transformation and defect structures at interfaces, and links its model inputs directly

to available thermodynamic and mobility databases. All symmetry operations to generate

the 12 orientation variants that have Burgers OR with the matrix phase are derived. In

Section 4.3, the lengthening and thickening kinetics of a single plate are first compared

between simulation predictions and theoretical solutions. Then variant selection of

precipitates within a single grain of due to different applied strains is investigated.

Finally the effect of a growing pre-existing plate on variant selection of secondary

plates is studied. The results are analyzed in Section 4.4 and the main findings are

summarized in Section 4.5.

4.2. Method

4.2.1. Determination of number of variants of a low symmetry precipitate phase

For structural phase transformations, a low-temperature product phase usually has

lower symmetry than its high-temperature parent phase. Such a symmetry reduction (i.e.,

some of the symmetry elements of the parent phase are no longer shared by the product

112

phase) leads to multiple crystallographically equivalent domains of the product phase

referred to as orientation variants (OVs). The set of remaining common symmetry

elements between the parent and product phases is given by the intersection group, H, of

the parent group, Gm, and the product group, Gp, under a given OR between them [8], i.e.,

m pH G G . Note that Gm and Gp are the point groups rather than space groups since

all the common translations are usually destroyed by the transformation. Then the

number of all OVs, n, is given by the index of H in Gm, i.e.


morder of G order of Hn (4.1)

The number of variants produced by the transformation in titanium alloys can

be determined readily by the above equation. For example, the point groups of the parent

(BCC) and product (HCP) phases are m3m and 6 mmm , respectively, and the orders of

m3m and 6 mmm are 48 and 24, respectively. The intersection group is thus

determined to be 2 m given that the Burgers OR is maintained between and phases,

as described by:

101 || 0001

; 111 || 1120

; 121 || 1100

. (4.2)

113

The order of H is 4 and thus the number of variants is n = 48/4=12. All the other OVs

can be obtained readily via symmetry operations on the variant described by Eq. (4.2)

(See Appendix A for details about the derivation of the intersection group H and all 12

symmetry operations associated with each OV). All 12 Burgers orientation variants used

in the current study and their corresponding symmetry operations to derive the other 11

from the one described by Eq.(4.2) are summarized in Table 4.1. The symmetry

operations are quite useful to obtain misorientations between different variant pairs, and

more importantly, to derive the transformation strain of others variants from that of the

variant described by Eq. (4.2), as will be described further latter.

4.2.2. Free energy formulation

Any given microstructure, no matter how complicate it is, can be described by two

types of order parameters that characterize, respectively, structural and chemical non-

uniformities [35, 36]. For example, for an + two-phase microstructure, twelve non-

conserved order parameters, 1,2,..., 12p p N r , are needed to describe the

structural non-uniformities associated with the twelve OVs and two conserved order

parameters, Al, VkX k r , are needed to describe the chemical non-uniformities of

components Al and V in the system. In the multi-phase field model, one more dependent

order parameter, 13p p r , is introduced to describe the spatial distribution of the

matrix phase. For example, 13 1 r and 1(r)=2(r)=…=12(r)=0 within the -matrix

114

and 1p r and qp(r) = 0 inside the p-th OV of precipitates under the constraint

that 1

1

1N

p

p

r [37, 38].

4.2.2.1. Chemical free energy

According to the gradient thermodynamics [39], the total chemical free energy of a

chemically and structurally non-uniform system can be formulated as a functional of the

order parameters introduced above:

2

, 1

1 1, ,

2 2

NT

chem cm k p k p p p

Vk Al V pm

F G T X X dVV

κκ (4.3)

where cκ and pκ are the gradient energy coefficient and gradient energy coefficient

tensors characterizing contributions from non-uniformities in concentration and structure,

respectively. In particular, different choices of eigenvalues for the κ tensors allow for the

consideration of interfacial energy anisotropy [40]. mG is the non-equilibrium local

molar free energy of the system defined in both concentration and structural order

parameter space. Even though it can be formulated by using Landau expansion

polynomials [41-43], it is approximated by [44, 45]:

115

1

, ,

1 1 1

, , , 1 ,N N N N

m k p p m Al V p m Al V pq p q

p p p q p

G T X h G T X h G T X

(4.4)

in the current study for simplicity. In the above equation, 3 26 15 10p p p ph is

an interpolation function used to connect the free energy curves (as function of

concentrations) of the and phases. The 1

1

N N

pq p q

p q p

term introduces a hump on

the free energy surface between two structurally degenerated states, i.e., between variants

p and q, and the hump height is proportional to pq . The advantage of using p q

p q

over the commonly used form, 2 2

p q

p q

, is that it creates an energy cusp at the

equilibrium values of the structural order parameters and hence prevents significant

deviations of the order parameters from 1 (or 0) in the bulk phases (i.e., creates higher

energetic penalty for deviation). ,,m Al VG T X and ,,m Al VG T X are the equilibrium

molar free energies of and phases as function of temperature and individual phase

concentration ,Al VX and ,Al VX

, respectively. In the current study, these equilibrium free

energies are formulated based on a pseudo-ternary thermodynamic database developed

for Ti64 [46].

116

4.2.2.2. Elastic strain energy

The theoretical treatment of elasticity problem associated with phase transformations

was due to Khachaturyan and Shatalov (KS) [47-49] who derived a close form of

coherency elastic strain energy of a system with arbitrary compositional and structural

non-uniformities by following Eshellby’s approach [50]:

* 0 0

3, 1

1( ) ( ) ( )

2 (2 ) 2

Nel T

pq p q ijkl ij kl ij ijkl kl

p q

d VE B C d C

g

n g g r r (4.5)

where ( )pqB n describes the elastic strain energy density of a thin precipitate plates with

n being its habit plane normal. The detailed form of ( )pqB n varies with external

boundary conditions. ( )p g denotes the Fourier transformation of ( )p r and *( )p g

stands for the complex conjugate of ( )p g . g is a vector in the reciprocal space.

In the current study, a clamped boundary condition is employed, where the system’s

boundary is fixed after applying an external load to the system. Then ( )pqB n reads

0 00 00

0 00 00 00 00

( ) ( ), =0( )

( ) ( ) ( ) ( ) ( ) , 0

ijkl ij kl

pq

ijkl ij kl i ij jk kl l

C p qB

C p q n p q n

gn

n g (4.6)

117

where 0

ijklC is the elastic modulus tensor of the matrix phase and 00

ij p is the stress-free

transformation strain (SFTS) or inelastic strain of the pth

orientation variant of the -

phase. 00 0 00( ) ( )ij ijkl ijp C p and 1

jk i ijkl ln C n n is the inverse of the Green’s function in

the reciprocal space, and n g g with in being its ith

component. In Eq.(4.5), the

macroscopic homogenous strain, ij , is equal to the pre-strain, appl

ij which is established

by the initial load applied to the system that has a volume V in the real space.

The last term in Eq. (4.5) represents the coupling between the pre-strain and the

transformation strain induced by the precipitates, T

ij r . T

ij r represents the spatial

distribution of the SFTS field associated with structural non-uniformities and can be

expressed by 00

1

( )N

T

ij ij p

p

p

r r , which is a linear superposition of all N types of

non-uniformities

The total strain energy of the system can be easily obtained and incorporated into the

total free energy of the system, i.e., chem elF F E .

4.2.3. Stress-free transformation strain for coherent and semi-coherent precipitates

Derivation of SFTS, 00

ij , associated with a phase transformation is one of the key steps

towards formulating the coherency elastic strain energy in the KS microelasticity theory

118

[48]. During nucleation and in the early stages of growth, precipitates tend to be coherent

with the matrix, which minimizes the interfacial energy. For coherent precipitates, the

transformation matrix [48] or deformation gradient matrix [51], F , associated with the

phase transformation could be determined by the lattice correspondence (LC) according

to the nearest neighbor principle [52] for a given OR between the precipitate and parent

phases. F is a discrete mapping from a Bravais lattice of the parent phase to a Bravais

lattice of the product phase and describes geometrical change between them under the

given OR via a uniform lattice deformation. Usually, it is expressed in a common

orthogonal coordinate system (usually chosen in the parent phase reference frame).

During their continued growth, coherent precipitates may lose coherency when the

elastic strain energy contribution to the total free energy of the system becomes

dominant. Line defects, such as misfit dislocations and structural ledges [53], are then

introduced within the interface to relieve misfit stress. Structural ledges, which exhibit

step character as well as dislocation properties, are also referred to as transformation

dislocations or disconnections to distinguish themselves from defects without the step

character in the topological model for structural phase transformations [54, 55]. Interface

between fully grown precipitate and matrix has been frequently observed to have

both types of defects within it [56-58]. For semi-coherent precipitates, the effect of misfit

dislocations and structural ledges on the transformation strain can be considered by

superposition of their eigenstrains with the SFTS calculated for fully coherent

precipitates, which can be achieved by treating the interfacial defects as successive

119

deformations, following the uniform lattice deformation, applied to the precipitate phase

[59].

Thus, the total deformation gradient resulting from the phase transformation

and defects on the / interfaces can be formulated as:

tot step disF F F F (4.7)

where stepF and dis

F represent the deformation gradient matrices due to structural ledges

and dislocations, respectively. Then, the transformation strain 00

ij can be derived from

the total transformation matrix. To be consistent with the assumptions made during the

derivation of the micro-elasticity theory [48, 60], the total strain in the system is given by

the sum of the elastic and inelastic strains: T

ij ij ije , and the total strain is related to

the displacement through , ,

1

2ij i j j iu u where iu is the ith component of the total

displacement field u and ,i ju represents it gradient.

,ij ij i jF u or

I F u where ij denotes the Kronecker delta. Thus, the SFTS for the orientation

variant described in Eq. (4.2) is given by (under small strain approximation)

:

120

tot tot

00

2

T

F F

I (4.8)

According to the aforementioned approach, the transformation strains for both coherent

and semi-coherent precipitate have been obtained [49] based on the LC [7] deduced from

the Burgers OR and detailed interface structure. Details of the derivation procedures

can be found in Ref. [59].

It is worth mentioning that the transformation strains should be altered when a

coherent nucleus grows beyond a critical size when the coherency is lost [59]. As a result,

the response of the precipitation process to the applied strain or pre-straining would vary

with the coherency state of precipitates in terms of the sign and magnitude of the

interaction energy density. Therefore, the derivation of SFTS for the coherent and semi-

coherent precipitate is an important first step towards variant selection study.

In fact, it has been shown that [59] the introduction of defects at the / interfaces

relaxes significantly the coherency strain energy according to the minimum elastic strain

energy density Bpq(n0) (from 6.426×107 J/m

3 to 4.71×10

3 J/m

3) obtained when using

coherent and semi-coherent SFTSs. In addition, the minimum of Bpq(n0) is reached

respectively at n0=(-11,-9.85,8.07) and n0=(-11,-12.63,11) for coherent and semi-

coherent precipitates, which deviate respectively 8º and 0.8 º from the experimentally

observed habit plane orientation (-11,-13,11). Therefore, the introduction of defects

121

relaxes the coherency strain and alters the habit plane orientation at the expense of

increasing interfacial energy.

The transformation strain for all the other orientation variants of precipitate can be

obtained by symmetry operations, i.e.

T

00 00 p pp S S (4.9)

where the symmetry operations 1..12pS p are listed in Table 4.1. The SFTSs for all

12 variants of nuclei and semi-coherent plates are presented in Appendix B1 and

Appendix B2, respectively.

4.2.4. Effect of misfit dislocation on interfacial energy

Defects at the precipitate/matrix interfaces alter not only the elastic strain energy but

also interfacial energy and its anisotropy, and growth anisotropy as well. Contributions

from misfit dislocations at different facets of the precipitates to the structural part of the

interfacial energy s could be evaluated according to the Read-Shockley equation [61,

62]. Based on the misfit dislocation structure at different facets [57], the structural

components, s , of the interface energy due to the presence of misfit dislocations at the

122

broad, side and edge faces are m0.39E , m0.72E and mE , respectively, where mE is

assumed to be 250 mJ/m2. In addition, the chemical component c of the semi-coherent

interfacial energy for the three facets are assumed equal for simplicity and have a value

of 50 mJ/m2, which is reasonable for a fully coherent interface. Therefore, the interfacial

energies of the broad, side and end faces are 150, 230 and 300 mJ/m2, respectively. The

results are then incorporated in the gradient energy coefficients characterizing chemical

and structural non-uniformities in the phase field free energy formulated based on the

gradient thermodynamics (Eq.(4.3)). Neglecting the details of -plot, we further assume

that the interfacial energy is a quadratic function of direction vector in the local reference

frame attached to lath, which is referred to as N3, 1 : 353x

, 2 : 11 13 11x

and

3 : 101x

. In the global coordinate system where the SFTSs have been derived, , the

gradient energy coefficient tensor for the structural order parameters, has a form

1

2

3

R R

T , (4.10)

where i 1 3i denotes the eigenvalues of the gradient energy coefficients

associated with broad, edge and side faces, respectively, whose values (all listed in Table

4.2) are so chosen to account for the anisotropy in interfacial energy mentioned above. R

123

is the rotation operation from local to laboratory (global) coordinate system. There are

total twelve rotation matrices corresponding to the 12 orientation variants of the phase.

4.2.5. Kinetic equations

The temporal and spatial evolutions of concentration and structural order parameters

are governed by the Cahn-Hilliard generalized diffusion equations [63] and the time-

dependent Ginzburg-Landau equation [64], respectively. In particular, considering the

multi-variant of phase, we employ the multi-phase field method [37, 38], which was

developed to treat multi-phase and multi-component material systems:

1

21

,1, , ,

,

nk

kj i j k

jm j

X t FM T X t

V t X t

rr

r (4.11)

, 1,

, , ,

chem chem elp

p

p q p q p

t F F EL t

t t t t

r

rr r r

(4.12)

where 1...12, ,kj iM T X is the chemical mobility [65], L is the mobility of the long-

range order parameters characterizing interface kinetics, and k and p are the Langevin

noise terms for composition and long-range order parameter, respectively. If the interface

motion is diffusion-controlled, L can be determined at a vanishing kinetic coefficient

124

condition [66]. In Eq.(4.12), is the number of phases that co-exist locally, not the

number of all phases which is N+1. Note that similar to Eq. (4.11), Eq. (4.12) was also

derived [38] in a variational framework with the use of Lagrange multiplier to account for

the local constraint among p , i.e., 1

1

1N

p

p

.

In order to remove the length scale limit of the conventional phase field model,

the Kim-Kim-Suzuki-Steinbach model [37, 67] is implemented, where the diffuse-

interface region is treated as a homogeneous mixture of the precipitate and matrix phases

with different compositions but equal diffusion potentials [67].

4.2.6. Model inputs and parameters

All the parameters for the model and materials properties used in the simulations are

listed in Table 4.2.

4.3. Results

4.3.1. Growth behavior of a single plate

In order to demonstrate the quantitative nature of the model, the growth (thickening)

behavior of an precipitate (an infinite plate) in a supersaturated matrix at 1023K is

investigated and compared to DICTRA simulations. The initial thickness of the plate is

125

chosen to be 0.25 m and its composition is assumed to be at equilibrium (for Ti-64):

10.48 at.% Al and 2.38 at.% V. The initial composition of the supersaturated matrix is

10.19 at.% Al and 3.60 at.% V. The total system size is 10 m. The phase field

simulation results are compared with DICTRA simulations in Fig. 4.1(a), which shows a

good quantitative agreement. In addition, the thickening kinetics of an infinite plate is

found to follow a parabolic law, which confirms that the transformation is diffusion

controlled.

Then, the growth behavior of a single finite plate embedded in a supersaturated

matrix with anisotropy in both elastic strain energy and interfacial energy is investigated.

A super-critical nucleus (a spherical particle having a radius of 37.5 nm) is placed at the

center of the computation cell. During growth, the precipitate develops into a disk with

its broad face parallel to 11,13,11

(as indicated by the shaded plane) (Fig. 4.2(a)). The

kinetics of both lengthening along the invariant line 3;5;3

and thickening normal to

the habit plane 11,13,11

are investigated. The thickness and length of the disk are

measured using Ruler in Source Toolbar of ParaView [68], an open-source, multi-

platform data analysis and visualization software application, at the = 0.5 contour line.

The results are shown in Fig. 4.1(b)-(c). Note that in measuring the interfacial position

there are uncertainties in determining the exact position of = 0.5 and the exact position

of the broad face because it is not exact flat. However, the uncertainties should be smaller

126

than the interface width for the broad and edge faces. Therefore, the interface width is

used as a measure of the error bar shown in the plots and the fitting process has taken the

uncertainty into account. Also note that the amplitude of the error bar in Fig. 4.1(c) at the

first time step is larger than that of the others since the broad face of the plate at that

moment has not yet developed due to its relative small size.

The lengthening kinetic is linear with time, while the thickening follows a parabolic

law, which agrees with the growth kinetic of a widmanstatten plate [69, 70]. And the

lengthening is about 10 times faster than the thickening.

The shapes of the disk outlined by a constant contour of Al concentration is shown

in Fig. 2(b). The color bar indicates the relative value of Al concentration.

4.3.2. Effect of pre-strain on variant selection

To simulate the type of constraint and stress conditions that might be experienced by

a component during processing [71], we carry out simulations under an uniform pre-

strain that is realized by applying either an uniaxial tensile or compressive stress along

the x (i.e. [010]direction of the reference frame and then clamping the system. Such a

boundary condition parallels to the recent experimental study on variant selection in

and titanium alloys [71]. During the nucleation stage, the SFTS calculated for coherent

127

precipitates (Section 2.3) is used. When enough nuclei are generated, the Langevin

noise terms are removed from Eq. (4.12) and the SFTS calculated for semi-coherent

precipitates is used during the growth and coarsening stages. The interfacial energy

anisotropy associated with misfit dislocations is also introduced for each variant at this

moment.

4.3.2.1. Pre-strain due to compressive stress along [010]

The macroscopic strain appl

ij is introduced by a compressive stress that is applied

along the x-axis (i.e., [010] direction) before the system’s boundary is clamped. The

magnitude of the applied stress is -50MPa and the resulting principal pre-strain is:

32.13 10appl

x ;

31.02 10appl

y and

31.02 10appl

z . Figure 3(a) shows

microstructure evolution during (Red online) precipitation from matrix (Blue online)

viewed at x - [010], y - [-101] and z - [101] cross-sections, respectively. It is obvious

that only limited numbers of orientation variants are formed under this specific pre-strain

condition and a strong transformation texture develops with time (see the arrows in the

last row of Fig. 4.3(a)). The whole entire precipitate microstructure in the parent -grain

consists of a few colonies of parallel array of plates.

A 3D plot of the microstructure developed at time t=10s is shown in Fig. 4.3(b) and

the volume fraction of each variant as function of time is shown in Fig. 4.3(c). It is

128

readily seen that only 4 variants, i.e. V1, V2, V7 and V8 have finite volume fractions.

Note that V1 and V7 have a common basal plane (101)//(0001) and their habit plane are

11,13,11

and 11,13,11

, respectively. The angle between the two habit planes is

79.8º or 100.2 º. Hence, the two plates are nearly perpendicular to each other, so do the

two plates of V2 and V8, as shown in Figs. 3(a) and 3(b).

4.3.2.2. Pre-strain due to tensile stress along [010]

In case where the pre-strain is generated by a 50 MPa tensile stress applied along

[010], the results are shown in Fig. 4.4. Different from the previous case, there are more

variants present, which suggests that the variant selection is sensitive to the initial

stress state and has a tension-compression asymmetry. Nucleation and growth of phase

seems to be enhanced at early stages when more specific variants are selected

simultaneously. For example, the volume fraction of each selected variant is much larger

than that obtained under compressive stress at t=2s, as shown in Fig.3(c) and Fig.4(c).

However, the volume fraction of each favored variant at t=10s is almost the same in the

two cases, i.e. around 8%. Also, some V-shaped patterns of plates are observed at very

beginning in this case, as indicated by the arrows in Fig. 4(a). These patterns have been

observed in experiment [25]. The variants in the V-shape pattern are found to have a

misorientation of 60◦ rotation around their common [111] or [11-20] axes.

129

4.3.3. Variant selection due to pre-existing plates

For -processed Ti-alloys, the specimen is usually step quenched to a specific

temperature within the + phase region after solution heat treatment above transus.

After isothermal holding for certain period of time, the sample is aged at another lower

temperature within the + phase region or cooled continuously at different rates. The

plates formed at the higher aging temperature are referred to as primary and the ones

formed at the lower aging temperature or during continuous cooling are referred to as

secondary . Since the secondary plates are formed in the untransformed matrix in

between the primary plates, their variant selection process is affected strongly by the

present of the primary . Grain boundary may have a similar effect on variant selection

of the primary plates. It has been suggested by a recent experimental study on step-

quenched Ti-550 alloys [25] that the development of a basket-weave structure could be

associated with nucleation and growth of different variants of secondary from primary

one. However, the mechanism is still not well understood.

In order to investigate the effect of existing plates on variant selection of new

plates, a phase field simulation study was designed as the following: a single variant of

plate (V1) is allowed to grow at 1123K for certain period of time and then the system is

quenched to 1073K (Fig. 4.5(a)). The random noise terms are introduced to simulate the

nucleation process of the secondary . The amplitudes of the random noises are

specifically chosen (by trial-and-error) to avoid homogeneous nucleation (i.e., nucleation

130

uncorrelated to the presence of the primary ). The results are shown in Fig.5, where it is

readily seen that two secondary particles of variants V4 and V6 nucleate and grow on

the broad face and near the edge of a growing primary disk (V1) (Fig. 4.5(b) and 5(f)

show two different views of the same microstructure). Note that the shape of the

secondary particles is different from that of the primary particle; they have lath-like

shapes rather than the disk-like shape. It should also be noted that the orientation

relationship or the misorientation between variants V1, V4 and V6 is such that when

described by the axis/angle pair, 60 1120

, three of them share a common

111 // 1120

direction according to the Burgers OR.

As the primary and secondary precipitates continue to grow, new particles of the

same variant 4 and 6 are induced in an arrangement of edge-to-edge adjacent to the

previously formed secondary laths as shown in Fig. 5(c). The relative location between

the newly formed secondary laths and the initial primary plate or disk (Fig. 4.5(a)) is

shown in Fig. 4.5(g). At even later stages, new laths of variants 8, 9 and 11 are formed

near the interface between the primary and secondary of variant 4 and 6 as shown in

Fig. 4.5(d) and Fig. 4.5(h) (viewed from the other side of broad face as compared to Fig.

4.5(d)) at t=3s. It appears that the growth of a secondary plate formed at an early

moment can lead to the growth of new secondary alpha around them. The volume

fraction analysis of each variant at t=3s (Fig. 4.5(e)) suggests that the nucleation and

131

growth of secondary of variants V4, V6, V9, V11 and V8 would be favored by the

primary alpha of variant V1, and V4 and V6 are the most favored ones.

4.4. Discussion

4.4.1. Lengthening and thickening kinetics of plate

It is clear that the growth of an isolated plate is highly anisotropic since its

lengthening is much faster than its thickening. In our simulations, the growth anisotropy

results from a non-uniform distribution of solute depletion zone surrounding the growth

plate, as shown clearly in Fig. 4.2. The plate-like anisotropic shape is determined by

the anisotropy in interfacial energy and elastic strain energy. Quantitative studies of both

lengthening and thickening kinetics of a widmanstatten plate have been carried out in

literature. For example, the mechanism of lengthening (edgewise growth) has been

explored by Zener [69] and later by Hillert [70] in steels and linear growth kinetics was

derived. Our simulation results (Fig. 4.1(b)) also show that the plate lengthens at a

constant rate, GL, which agrees with the Zener and Hillert’s analysis. Since Al is an

stabilizer and V is a stabilizer, the growth of plates is controlled by partitioning of Al

and V between and phases. The plate shape ensures that the pilling-up of V or

depletion of Al by a growing plate takes place at the sides of the plate. Therefore, the

132

Al and V concentration profile ahead of the tips of an plate remains constant as it

lengthens. As a result, unlike the thickening process, the plate lengthens at a constant rate

GL.

The thickening (sidewise growth) of a plate is, however, less understood than its

lengthening. The current phase field simulation result obtained for the growth of an

isolated plate shows a parabolic thickening kinetics, which is self-consistence with the

diffusion-controlled growth condition assumed in the simulations. It should be

mentioned, however, that the broad face of an plate is comprised of terrace of good

atomic fit with steps at the atomic scale, which is beyond the resolution of the current

phase field method. According to Aaronson’s analysis [72], the more coherent broad face

could only grow perpendicular to itself by nucleation and migration of ledges. Enomoto

studies the migration of an array of steps using finite difference scheme [73, 74] and

showed that the thickening kinetics follows nearly a parabolic law at long reaction time.

More detailed discussion on ledge growth can be found in Refs. [72-74].

4.4.2. Elastic interaction between pre-strain and transformation strain of variants

In this Section we demonstrate that the simulation results on selective nucleation and

growth of specific variants presented in Section 3.2.1 and 3.2.2 can be rationalized by

the interaction between the pre-strain and the SFTS of each variant characterized through

the interaction energy density, int 0 00 ( )appl

p ij ijkl klE C p . In general, the SFTS of each OV,

133

00 ( )ij p , is different from each other when defined in a common reference frame and thus

the interaction energy density for each OV will be different. In other words, the

interaction is crystallographically anisotropic in nature. As a result, the nucleation and

growth of some variants will be favored over others by the applied strain, appl

ij . To be

specific, a variant whose growth reduces the strain energy (i.e., the interaction energy

density is negative) will be favored to nucleate and grow, while the variant whose growth

results in a positive interaction energy density will be suppressed. The resulted uneven

distribution of the different orientation variants of precipitate then gives rise to

transformation texture.

Since a coherent precipitate will lose its coherency when its size exceeds a critical

value [59], its interaction with the pre-strain will be size-dependent because of the change

in the SFTSs (see Section 2.3). In other words, the interaction energy also depends on the

coherency state of precipitate. When the pre-strain is produced by [010] tension or

compression, the coherency state alters only the magnitude but not the sign of the

interaction energy (see, e.g., Fig. 4.6 (a) and 6 (c)). In the case of tension along [010],

for example, variants V3-V6 and V9-V12 have the same negative interaction energy

while variants V1, V2, V7 and V8 have the same positive interaction energy for both

coherent and semi-coherent precipitates. In the case of compression, the interaction

energies reverse their signs, i.e., variants V1, V2, V7 and V8 have the same negative

interaction energy while variants V3-V6 and V9-V12 have the same positive interaction

134

energy. These interaction energy calculations agree well with the phase field simulation

results (Fig. 4.3 and Fig. 4.4), which means that such simple interaction energy

calculation could be used to predict variant selection behavior in this case.

Nevertheless, when a tensile or compressive stress is applied along the z-[101]

direction to obtain the pre-strain, the situation is much more complicated. When the pre-

strain is obtained via a compressive stress (50MPa) along z direction, its principal value

is: 31.02 10appl

x ;

30.28 10appl

y and

30.93 10appl

z . It can be readily seen

that the coherency state alters both the magnitude as well as the sign of the interaction

energy (see, e.g., Fig. 4.6 (b) and 6 (d)). In the case of tension (Fig. 4.6(b)), variants V2

and V8 have the most negative interaction energy for both coherent and semi-coherent

precipitates, but variants 9-12 have positive interaction energy if the precipitates are

semi-coherent and negative interaction energy if the precipitates are fully coherent.

Variants 1 and 7 have large negative interaction energy at the coherent stage and nearly

vanishing interaction energy when they lose coherency. If a compressive stress is applied

along the [101] (Fig. 4.6(d)), the variants favored during both coherent and semi-

coherent stages are only V3-V6. Variants V9-V12 are unfavored when they are coherent

but become favored when they are semi-coherent by the pre-strain. These complicated

interactions may alter the nucleation rate as well as the growth rate at different stages of a

precipitation process, making it impossible to predict transformation texture by simple

interaction energy density calculations.

135

From the results of interaction energy calculations shown in Fig. 4.6(b), one would

expect that V2 and V8 would be the dominant ones during precipitation. This is,

however, not the case according to the phase field simulation results. As can been seen

from Fig. 4.7(a), the most dominant two variants are V1 and V7, which have a much

smaller interaction energy than that of variant V2 and V8. The cause of this discrepancy

will be discussed in the following Section.

In case of compression, both interaction energy calculations and phase field

simulations predict that variants V3-V6 are the most dominant variants. In addition,

phase field simulations show that variants V1 and V7 become more favored than variants

V9-V12 by the evolving microstructure during their growth at semi-coherent stage even

though the interaction energy calculations show that they are less favored than variants

V9-12 (see Fig. 4.6(d)).

The 3D microstructures obtained in the two cases are shown in Figs. 4.7(c) and

4.7(d), respectively. There are many V-shaped precipitate configurations as well as a few

enclosed triangle configurations as indicated by the arrows in the Fig. 4.7(d).

4.4.3. Competition between pre-strain and evolving microstructure

From the examples discussed above, one can see that the interaction energy

calculations, though simple and fast, cannot predict the overall variant selection behavior

136

in all cases. This is because of the fact that the interaction energy calculations do not

consider how an evolving microstructure alters the local stress state. The total strain of a

system is the sum of elastic strain and inelastic strain, i.e. T

ij ij ije . Since the system

considered in this study is under a clamed boundary condition with a specific pre-strain

determined by an initially applied stress, the average total strain (over the whole system)

is equal to the pre-strain. As the microstructure and thus inelastic strain, T

ij , evolve, the

elastic strain fields of different variants evolve as well and nuclei of different variants

would become favored by the elastic interactions. For example, as have been shown

when the pre-strain is obtained by applying a 50MPa tensile stress along [101] (Fig.

4.7(a)), the most favored variants at later stages do not agree with the interaction energy

calculation. This situation would not occur if the boundary is stress-controlled since the

system is free to change its shape and volume.

In order to confirm these arguments, a microstructure is obtained by only switching

the boundary condition to be stress-controlled while keeping all the other simulation

parameters the same. In this case, the volume fraction of V2 and V8 will increase sharply

to 30% at t=1s. Assume that this microstructure can also exist under the clamped

boundary condition and the internal stress is calculated. The principle stress components

obtained are: 11=-102.4 MPa, 22=27.9MPa and 33=44.3MPa. It is obvious that the

magnitude of some of the internal stress components exceed 50MPa that is used to

generate the pre-strain. Based on the interaction energy calculation, such stress state will

favor two pair of variants, i.e., V1/V7 and V2/V8. Therefore, the precipitation of V1 and

137

V7 is favored by both the pre-strain and the internal stress. Moreover, the combination of

V1 + V2, and V7 + V8 could reduce the overall transformation strain by self-

accommodation. Thus, within a clamped system, internal stress generated by an evolving

microstructure could exceed the initially applied stress that generated the pre-strain,

inducing the formation of other variants and hence reducing the degree of transformation

texture. In this regard, a clamped boundary could be a way of preventing the development

of transformation texture and, in particular, preventing percolation of a single or a few

variant through a whole entire grain. It seems that the experimental results reported

in a recent study on effects of different processing variables on transformation texture

development (due to variant selection) in Ti-64 sheet during -processing by Semiatin et

al [71] support this simulation finding. On the other hand, the prediction from the

interaction energy calculation is valid only when the internal stress generated by the

evolving microstructure is significantly smaller that created by the pre-strain under a

clamped boundary condition.

To assess whether a combination of a group of plates of all the most and equally

favored variants is able to reduce significantly the overall transformation strain as

compared to an isolated one, we calculate the average transformation strain of all of the

most favored variants based on the interaction energy calculations. The results show that

when the pre-straining is obtained by an applied stress along the [010] axis (irrespective

tensile or compressive), the average transformation strain of a group of plates of the

most favored variants is reduced significantly as compared to the transformation strain of

138

a single plate. Nevertheless, when the pre-straining is obtained via tensile stress along

[101], the average transformation strain of the most favored variants V2 and V8 is

almost identical to that of a single variant in both coherent and semi-coherent state.

Thus, such a variant selection of only two variants sharing a common basal plane (V2 and

V8) cannot be driven by the pre-straining. In contrast, the average strain of variants V3-

V6 is much smaller than that of a single variant when the pre-straining is achieved via a

compressive stress along [101]. As a result, the variant selection behavior can be

predicted by the interaction energy calculation. This approach is similar to the method for

an approximate estimation of the effectiveness of self-accommodation among different

groups of martensitic plates, developed by Madangopal et al. [75]. It should be

mentioned that such an analysis is based on the assumption that the volume fractions of

the most favored variants are identical and the nature of the interfaces between the

variants has been ignored. In addition, the elastic interaction is assumed to play a more

important role than that by supersaturation. Thus, the analysis may not apply to systems

under relative large undercooling.

4.4.4. Variant selection due to pre-existing microstructure

It is obvious that the nucleation and growth of secondary laths occur at the interphase

boundary between the primary plate and the matrix. Such nucleation of secondary

laths off primary plates (same phase, but different orientation) is usually referred to

139

as sympathetic nucleation (SN) [76, 77]. Depending on the morphological configurations,

SN can be classified further into: a) face-to-face nucleation, leading to a structure known

as sheaves; b) edge-to-face nucleation, resulting in the formation of a branched structure;

and c) edge-to-edge nucleation, which end up with a larger plate-like structure with a

small angle grain boundary in it. According to this classification, the SN of V4 and V6

shown in Fig. 4.5 belongs to the edge-to-face one (with respect to the existing primary

plate), while the formation of the second-generation of V4 and V6 take places in a

manner of the edge-to-edge SN (with respect to the first-generation of V4 and V6). The

SN of V9 and V11 occurs in a face-to-face manner (with respect to the primary plate).

The orientation relationship between V1, V9 and V11 can be described by the angle/axis

pair of 63.26 10 5 5 3

.

To analyze the energetics of the nucleation processes observed in our simulations,

below we calculate contributions from the chemical free energy and self-elastic energy to

the driving force for nucleation. From the Gibbs free energy database employed in the

current study [46], the chemical driving force for nucleation of from , VG , can be

calculated as:

nucleus nucleus nucleus

Al V Al V

Al,V

, ,V i i

i i c

GG G c c c c G c c

c

(4.13)

140

where ic is the average composition of the matrix phase, nucleus

ic is the composition of the

nucleus, G and G

are the molar Gibbs free energy of and phase respectively from

the thermodynamic database. The results are shown in Fig. 4.8. It is obvious that the

nucleation driving force near the / interface has been consumed partially by the growth

of the primary plate. There is only 390J/mol left at the / interface as compared to

500J/mol in the bulk away from the interface as indicated by the white contour line in

Fig. 4.8. The minimum self-elastic strain energy density of a nucleus is 640.26 J/mol

[59]. Thus the formation of an nucleus at an existing / interface is

thermodynamically impossible without the consideration of contributions from interfacial

energy or elastic interaction energy.

Aaronson et al. [76, 77] have conducted a detailed analysis, from both energetics and

kinetics points of view, to evaluate the feasibility of SN as an alternative to grain

boundary (heterogeneous) or homogeneous nucleation. Using a pillbox (circular disc with

its height much less than its radius) to represent the critical nuclei configuration, they

showed that the nucleation activation barriers *G associated with SN are comparable

provided that the SN replace the matrix/precipitate (i.e. /) boundary with a

precipitate/precipitate (i.e. /) grain boundary of relatively low energy, which could be

the case of the edge-to-edge SN or a coincidence lattice type precipitate/precipitate

boundary in the cases of edge-to-face and face to face SN. In particular, SN can still

occur even when the driving force for SN is less than half of that for heterogeneous or

141

homogeneous nucleation when the interfacial energy of an / boundary, , is much

smaller than that of the coherent broad faces of the pillbox, c

, for instance,

0.2c

.

In our simulation, however, c

= = 150 mJ/m2. Thus, the replacement of an /

boundary by an / would not contribute to the reduction of *G for SN. Therefore, the

SN mechanisms discussed above cannot explain the correlated nucleation phenomena

observed in our simulations. Furthermore, based on the analysis made by Aaronson et al.

[76, 77], the edge-to-edge SN will have the lowest activation barrier *G according to

both supersaturation and interfacial energy considerations, followed by face-to-face and

then edge-to-face SNs. But this analysis cannot explain the replacement of the face-to-

face SN by the edge-to-face SN with increasing supersaturation observed in their

experiment [78]. On the other hand, our simulation result (Fig. 4.5(e)) does show that the

edge-to-face SN is more favorable than the face-to-face one.

It has been shown, by both experimental observations [79] and computer simulations

[34, 80-84] that contributions from elastic interactions between the strain fields

associated with a nucleus and a pre-existing microstructure to the nucleation process

could be significant and make certain locations preferred nucleation sites. Even though

the self-elastic strain energy associated with a nucleus has been considered, the analysis

by Aaronson et al. [76, 77] does not consider the long-range elastic interactions between

142

nucleating particles and pre-existing semi-coherent precipitates. In order to understand

the contributions from elastic interactions between a nucleating particle and an existing

microstructure, nucleation of phase particles in the stress field created by a pre-existing

precipitate is investigated through the calculation of the elastic interaction energy [80,

81]. The results are shown in Fig. 4.9. It is obvious that the interaction is highly

anisotropic around the pre-existing plate. Again, negative values of intE promote

nucleation while positive ones suppress it. One key finding is that for all variants the

maximum negative values of the elastic interaction energy (located right at the /

interface as shown in Fig. 4.11 are far more negative than the chemical driving force for

nucleation as indicated by the white contour line in each of the plots. Therefore, the “SN”

phenomena observed in our simulations are actually stress-induced nucleation caused by

the elastic interactions between the nucleating particles and the pre-existing one. The

most negative values are located at the edge of the plate. On the other hand, the

positive values of the interaction energy for all the 12 variants are located on top of the

broad face of the primary plate.

Despite the fact that nuclei with the most negative interaction energy belong to

variant 3, 5, 9 and 11, these variants are not the most frequently selected ones by the

primary plate according to the phase field simulation results (Fig. 4.5(d)). Note that the

growth of pre-existing plates (i.e., the primary plates) and nucleation of secondary

plates occur simultaneously in our simulations. If a secondary precipitate is formed by

an edge-to-edge SN process, it may simply be absorbed (via coarsening) by the

143

continuously growing primary . The nucleation rate is determined by both the

abundance of available nucleation sites and their nucleation barrier. Based on the

interaction energy calculations, the most likely SN nucleation sites available for

secondary to nucleate and grow are at certain locations on the broad face away from

the tips of a growing primary plate.

When stable nuclei of different variants of secondary start to grow, they will lose

their coherency at a critical size [59] and become semi-coherent. Their growth behavior

will still be affected by the continuously growing primary plate. As has been shown by

the interaction energy calculations presented in Fig. 4.6, a secondary selected by the

primary may not be favored anymore after it loses its coherency. The interaction

energies between semi-coherent particles of all 12 variants and a pre-existing primary

plate are shown in Fig. 10. One feature is found quite different from the coherent ones,

i.e., for some variants the negative values of the interaction energy are located on the

top of the broad face of the primary plate such as V2 (V8), V4 (V10), and V6 (V12), as

indicated by arrows. The continued growth of secondary of V4 and V6 on the broad

face of the primary plate (V1) will lead to the formation of a closed “triangle”, which

has been observed frequently in experiments.

When different plates of different orientations within a single grain meet with

each other, 5 distinct morphological patterns have been identified. The axis/angle pairs to

144

represent the misorientation between different plates and their occurrence frequencies

in a random situation are presented by Wang et al. [26]. However, some axis/angle pairs

(such as 60 1120

and 63.26 10 5 5 3

) occur far more frequently than their

counterparts in a random situation according to the misorientation distribution analysis

obtained using OIM for both pure Ti [26] and Ti-alloys system [17]. By treating the

as martensitic transformation, Wang et al. calculated the shape strain (assuming fully

coherent) for each of the 12 variants as well as the average shape strain resulting from a

cluster of three variants in different combinations. According to their analysis, the

largest degree of self-accommodation can be achieved by a combination of three specific

variants with 60 1120

and 63.26 10 5 5 3

misorientation between them. Self-

accommodation mechanism is thus believed to account for the relatively high frequency

of these two types of misorientations. According to the volume fraction analysis in our

simulations, the most favored misorientation occurs among plates of variant V1, V4

and V6, followed by V1, V9 and V11. The misorientation among V1, V4 and V6 is

described by 60 1120

angle/axis pair and that for V1, V9 and V11 is described by

63.26 10 5 5 3

angle/axis pair. Therefore, the current simulation results and

interaction energy analysis are consistent qualitatively with their analysis. The study

about whether or not the coupling between specific variants occurs during the nucleation

stage via collective or correlated nucleation [83], or during growth via variant selection

(i.e., the growth of an existing plate induces the nucleation and growth of a new, self-

accommodating plate) will be published in a follow-up paper.

145

4.5. Summary

A quantitative three-dimensional phase field model is developed to investigate

variant selection during precipitation from matrixin Ti-6Al-4V under the influence

of both external and internal stress fields such as those associated with, but not limited to,

pre-straining and pre-existing precipitates considered in this paper. The model

incorporates the crystallography of BCC to HCP transformation, elastic anisotropy and

interface defect structures in its total free energy formulations. Model inputs are

linked directly to thermodynamic and mobility databases. The main findings are:

1) Under a given undercooling, there is a competition between internal stress associated

with an evolving microstructure and external applied stress or pre-strain on the

development of micro-texture. If the transformation strain or internal stress produced

by variants selected by a specific external stress or pre-strain during early stages of

precipitation cannot be accommodated among themselves, the internal stress would

prevent further development of such a transformation texture and induce the

formation of other variants to achieve self-accommodation. Since self-

accommodation can be achieved only by multiple variants (minimum two variants not

sharing a common basal plane), any constraints on macroscopic shape change of a

sample (e.g., by clamping) will prevent effectively the development of strong micro-

texture or transformation texture.

2) The development of micro-texture is sensitive to the loading axis of an external stress

or strain. From the elastic interaction energy calculations, we have learned that when

146

an external stress or pre-strain is applied in certain directions multiple variants of

phase could be favored simultaneously with the same interaction energy. Therefore, if

a polycrystalline sample has a strong macro-texture of the grains, control of

external load (if any) orientation could prevent strong micro-texture of plates from

percolating through the sample leading to poor fatigue properties.

3) There exists an obvious tension/compression asymmetry in variant selection behavior,

i.e., the types and numbers of variants produced under tensile and compressive

stresses are different. For example, pre-straining obtained via uniaxial tensile and

compressive stress along [010] will result in the selection of 8 and 4 out of 12

variants, respectively.

4) The interaction energy calculations, though simple and fast, cannot predict the overall

variant selection behavior at all cases. In addition, the prediction is valid only when

the internal stresses generated by an evolving microstructure are significantly smaller

than the externally applied stress.

5) Although nucleation of specific variants of secondary plates on interfaces between

primary plates and matrix observed in Ti-alloys could be classified as

sympathetic nucleation (SN), the elastic interaction analysis in this study suggests that

such nucleation phenomenon observed in our simulations is coherency stress induced

correlated nucleation (i.e., auto-catalytic effect) rather than the conventional SN

discussed in literature (which is caused by the relatively low grain boundary energy

between the secondary and primary particles).

147

6) Secondary plates having a misorientation of 60 1120

with the pre-existing (i.e.,

primary) ones (i.e. sharing a common 111 / / 1120

) tend to nucleate and grow on

the broad faces of the pre-existing plates, which could serve as an auto-catalytic

mechanism underlying the formation of basket-weave microstructures.

7) The stress-free transformation strain (SFTS) of precipitate varies with its coherency

state and variant selection rules (in terms of the sign and magnitude of elastic

interaction energy) are found different for coherent and semi-coherent precipitates.

When considering effect of primary plates whose sizes are usually above the

critical size for coherency (around 20 nm [59]), the SFTS for semi-coherent

precipitate should be employed for the primary precipitate while the SFTS for

coherent precipitate should be used for the nucleating secondary precipitates.

Applications of the model to study effects from other stress-carry defects such as

dislocations, stacking faults, grain boundaries, as well as effects from thermal stress on

variant selection during precipitation in polycrystalline samples are straight forwards

and corresponding work is underway.

148

Figures

Figure.4.1 Growth behavior of an plate. (a) Thickening kinetic of an infinite plate.

Results by phase field (symbol) and DICTRA (solid line) simulations are compared. (b)

Lengthening and (c) Thickening kinetics of a single finite plate embedded in a

supersaturated matrix. Error bars represent uncertainty in the determination of interface

position.

0.25 m

(a) (b) (c)

149

Figure. 4.2 (a) Morphology of an isolated plate visualized by a constant contour of Al

concentration. The transparent light yellow plane denotes the experimentally observed

habit plane . (b) A cross-section of the matrix phase surrounding the

plate showing variations in Al concentration in the matrix up to the precipitate/matrix

interface. The color bar indicates the relative value of Al concentration.

(a) (b)

150


via a compressive stress (50Mpa) along [010] . (a) 2D cross-sections showing

microstructure evolution (color online with phase shown in red and phase shown in

blue). Arrows indicate regions with transformation texture. (b) 3D microstructure

obtained at t = 10s. (c) Volume fraction of each variant as function of time.

x y z (b)(a)

t =10 s

(c)

t = 2 s

t = 4 s

t = 6 s

t = 8 s

151


via a tensile stress (50Mpa) along [010] . (a) 2D cross-sections showing microstructure

evolution (color online with phase shown in red and phase shown in blue). Arrows

indicate regions with transformation texture. (b) 3D microstructure at t=10s. (c) Volume

fraction of each variant as a function of time

(b)(a) x y z

(c)

t =10 s

t = 2 s

t = 4 s

t = 6 s

t = 8 s

152

Figure. 4.5 Variant selection of secondary by a pre-existing plate. (a) Pre-existing

plate of variant 1 (V1). (b)-(d) Formation of secondary laths on the broad face of the

pre-existing plate. Different types of secondary are visualized through different

colors (see online version). (e) Volume fraction analysis of each secondary (f) - (h)

Formation of secondary on the other side of broad face of pre-existing plate from a

different view direction. (g) shows the relative locations between secondary (at t = 2s)

and pre-existing plate (at t = 0s).

(a) (c) (d)

(h)(e)

v11

v4

v6 v11

v8

v1

t = 0s t = 2s t = 3s

t =3s

t =3s

v4

v6

(b)

t = 1s

(g)

t = 2s

(f)

t = 1s

v9

v8

v9

153

Figure.4.6 Interaction energy density between pre-strain and each variant under both

coherent and semi-coherent conditions. The pre-strain is obtained by applying a 50MPa

tensile stress along (a) [ ] and (b) [ ] , and a 50Mpa compressive stress along (c)

[ ] and (d) [ ]

(a) (b)

(c) (d)

154

Figure.4.7 Variant selection caused by a pre-stain obtained via uni-axial tension or

compression (50Mpa) along [ ] . Volume fraction of each variant as function of time

under tension (a) and compression (b). 3D microstructure (at t = 10s) under tension (c)

and compression (d).

t =10 s t =10 s

(a) (b)

(c) (d)

155

Figure.4.8 Chemical driving force for nucleation around a growing pre-existing plate

(Variant 1). The contour line indicates the chemical driving force in the supersaturated

matrix far away from pre-existing plate

∆GV=-0.5

156

Continued

Figure . 4.9 Elastic interaction energy associated with all 12 variants of coherent nuclei

around a pre-existing semi-coherent plate (Variant 1). The contour lines indicate that

the elastic interaction energy is equal to the chemical driving force for nucleation in the

supersaturated matrix far away from the growing pre-existing plate shown in Fig. 4.8.

Eint=-0.5

1-1

1-3

1-5 1-6

1-4

1-2

157

Figure 4.9 continued

1-7 1-8

1-9

1-11 1-12

1-10

158

Continued

Figure. 4.10 Elastic interaction energy associated with all 12 variants of semi-coherent

laths around a pre-existing semi-coherent plate (Variant 1). The contour lines indicate

vanishing elastic interaction energy.

1-1 1-2

1-41-3

1-5 1-6

Eint=-0.0

159


1-7

1-9

1-11 1-12

1-10

1-8

160

Fig. 4.11 (a) Elastic interaction energy between an nuclei (Variant 5) and a pre-existing

semi-coherent plate (Variant 1). (b) 1D structure order parameter profile (Blue) and

interaction energy (Red) along z-direction across ⁄ interface. It shows that the

maximum negative values of the elastic interaction energy are located right at the

⁄ interface.

Interaction V1-V5

Order parameter

Interaction Energy Eint

Maximum Eint

161

Tables

Table 4.1 All 12 Burgers orientation variants and symmetry operations associated with

them

Variants Orientation Relationship Symmetry operation iS

1 101 // 0001

111 // 1120

1

1

1

I

2 101 // 0001

111 // 1120

100

1

2 1

1

3 011 // 0001

111 // 1120

111

-1

1

3 1

1

4 011 // 0001

111 // 1120

111

-1

1

3 1

1

5 110 // 0001

111 // 1120

111

1

3 1

1

6 110 // 0001

111 // 1120

111

1

3 1

1

Continued

162

Table 4.1 continued

Variants Orientation Relationship Symmetry operation iS

7 101 // 0001

111 // 1120

010

1

2 1

1

8 101 // 0001

111 // 1120

001

1

2 1

1

9 011 // 0001

111 // 1120

111

-1

1

3 1

1

10 011 // 0001

111 // 1120

111

-1

1

3 1

1

11 110 // 0001

111 // 1120

111

1

3 1

1

12 110 // 0001

111 // 1120

111

1

3 1

1

uvwn denotes a n-fold rotation around axis uvw and superscript of -1

uvwn indicates inverse

of uvwn

163

Table 4.2 Various model parameters and materials properties used in the simulations

Physical properties Symbol Value Unit

Temperature T 1023 K

Grid size l0 0.0125 m

Interface width 5 l0 0.0625 m

Interfacial energy

Broad-, Side-, End-facet

150,230,300 mJ/m2

Gradient Coefficients [78] 0.038, 0.089, 0.152 J·m2/mol

Hump height 192 J/mol

Interface mobility L 6.0×10-8

J/m3/s

Molar volume Vm 10-5

m3/mol

Elastic constants of phase [79] 11 12 44, ,C C C 97.7,82.7,37.5 GPa

Lattice parameter of and

phase [59]

a , a , c 3.196,2.943,4.680 Å

Coherent SFTS[50] 00 0.083 0.0095

0.0095 0.123

0.035

-

Semi-coherent SFTS eff 0.049 0.0031

0.0031 0.067

0.0003

-

164

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174

CHAPTER 5 Evolution of Microstructure and Transformation

Texture during Alpha Precipitation in Polycrystalline Titanium alloys

Abstract:

A previously developed three-dimensional phase field model of transformation in

single crystal Ti-6Al-4V is extended to polycrystals to study variant selection and

microstructure evolution under the influence of different processing conditions such as

pre-straining and boundary constraint. Effect of starting texture is also investigated.

Degrees of variant selection at both the individual grain and the whole polycrystalline

sample levels and their effects on the final macro-texture of precipitates are analyzed.

In particular, the effect of non-uniform local stress state, arising from elastic

inhomogeneity of a polycrystalline sample under a uniform external strain (thermal or

applied), on the variant selection behavior within individual grains is addressed. It is

found that when subjected to certain pre-strains, a sample having strong starting texture

could end up with a relatively small degree of micro-texture when local stresses

associated with the pre-strains promote multiple variants simultaneously within the

whole polycrystalline sample. The results could shed light on how to control processing

conditions to reduce transformation texture at both the individual grain and the overall

polycrystalline sample levels.

175

5.1. Introduction

Titanium and its alloys are currently finding increasingly wide application in the

aerospace, shipbuilding, automotive, sports, chemical and food processing industries due

to their desirable and versatile combination of good mechanical and chemical properties

such as extreme lightness, high specific strength and good corrosion resistance [1, 2].

Depending on the application, a specific property (or combination of properties) can be

obtained through tailoring / microstructure whose evolution and control depends

heavily on the allotropic transformation from phase (BCC structure) at high

temperature to phase (HCP structure) found at low temperatures [3]. The defining

characteristic of the transformation is the Burgers orientation relationship (BOR) [4]

between the two phases, i.e. { } and ⟨ ⟩ [ ] . Owing to the

symmetry of the parent and product phases and the BOR between them [5], there are

twelve possible crystallographically equivalent orientation variants of the phase within

a single parentgrain. If all 12 variants are able to form within each grain, the

resulting microstructure would be relatively fine, with large amount of boundaries,

and transformation texture would be relatively weak since matrix and its orientation

density needs to be partitioned by all 12 variants of phase. However, it is typically the

case that only a small subset of the 12 possible variants is formed preferentially within

each beta grain under different thermo-mechanical processing (TMP), i.e. combination of

working and heat treatment. In other words, variant selection occurs frequently, resulting

176

in a relative coarse microstructure with final texture of phase with various

strengths.

The statistical and spatial distribution of each orientation variants then determines the

texture state of alpha phase during the transformation. Since the sources of strength in

titanium is the barrier to dislocation movement represented by hetero-phase

interface, the density and orientation of hetero-phase interfaces, and their spatial

uniformity determine the deformation mechanisms and mechanical behavior of the alloys

[1, 6]. Variant selection, due to a wide variety of factors during TMP, would lead to the

formation of large regions of phase with a common crystallographic feature (such as

common basal pole, or common orientation), i.e. “macro-zone” or micro-

textured region, within individual grain or across grain boundaries, and thus would

result in a significant reduction in fatigue life of Ti-component that is undesirable in a

safety critical operating environment [7]. Consequently, understanding and thus control

of alpha phase size, morphology, and distribution including that of its orientation

variants, i.e. microstructure and texture state, under the influence of variant selection, are

of fundamental importance in control and tailoring the properties of titanium alloys[1, 6].

During TMP, there exists a rich variety of sources that are able to result in local stresses

and lead to variant selection within a sample during TMP. In other words, the

development of local stress could not be avoided during TMP. For instance, owing to the

anisotropy of thermal expansion coefficient of the phase (which is 20% larger than in

177

the ⟨ ⟩ than in the ⟨ ⟩ directions), substantial residual stresses are common in Ti

alloys even after a stress relief annealing treatment [8-10]. Moreover, defects such as

dislocations and stacking faults generated during TMP in either or phase region act

frequently as preferred nucleation sites for specific subset of variants[11-14]. Local

stress fields will also be generated by precipitation and autocatalysis has been shown

frequently to result in variant selection [15, 16]. Furthermore, for polycrystalline

materials under an external stress or strain field, local stress state within the sample will

vary significantly from grain to grain because of the elastic anisotropy in each grain that

leads to elastic inhomogeneity in the sample [17]. It is clear that the local stress state, due

to a rich variety of sources, is a key factor in controlling variant selection and hence the

final transformation texture during precipitation in Ti alloys.

Nevertheless, the main challenges to study variant selection during

transformations in polycrystalline sample under the influence of stress are three-folds:

first, one needs to determine stress distribution in an elastically anisotropic and

inhomogeneous polycrystalline matrix under a given applied stress/strain condition;

and second, one needs to describe interactions of local stress with precipitation of

coherent and semi-coherent precipitates, i.e., to describe interactions of local stress with

an evolving microstructures. In particular, defects structure, including misfit

dislocations and structural ledges, at the interfaces will alter not only the coherency

elastic strain energy associated with the precipitation, but also the interfacial energy

178

and its anisotropy. It could introduce growth anisotropy as well. These anisotropies,

together with the high volume fraction and multi-variants of the precipitate phase and

long-range elastic interactions between the precipitates and local stress, and among

precipitates themselves, lead to highly non-random spatial

distribution of precipitates with different variants. Third, in order to provide new insight

into materials processing- microstructure- properties relationship, microstructure and

texture needs to be considered together. In other words, variant selection behavior at the

scale of individual parent grains and scale of the whole polycrystalline sample, and their

influence on the microstructure evolution and final transformation texture need to

be considered simultaneously. In sum, variant selections depend on a wide variety of

interaction parameters and thus are very complex. Owing to this complexity, the

mechanisms of variant selection are very difficult to determine experimentally. And,

existing modeling attempts have been only taking care of one or two parameters.

Based on gradient thermodynamics [18-20] and microelasticity theory [21-25], the phase

field approach [26-32] (also called the diffuse-interface approach) offers an ideal

framework to deal rigorously and realistically with these difficult challenges. As has been

demonstrated in a recent phase field simulation study of the transformation in Ti-

6Al-4V (in wt%) [33, 34], the formulation of the total free energy functional, which

consists of the bulk chemical free energy, elastic strain energy and interfacial energy, has

accounted for the following: (a) a reliable thermodynamic data for the bulk chemical free

energy for Ti-6Al-4V system [34, 35]; (b) crystallography of the crystal lattice

179

rearrangement, including orientation relationship, i.e. BOR, and lattice correspondence

(LC) as functions of the lattice parameters of the precipitate and parent phases (i.e., the

effect of alloy chemistry); (c) accommodation of the transformation strain; (d)

development of defect structures (misfit dislocations and structural ledges) at

interfaces as precipitates grow in size; (e) elastic interaction of nucleating particles with

existing chemical and structural non-uniformities and other stress-carrying defects such

as dislocations [36]. In particular, in combination with orientation distribution function

(ODF) modeling [37] of the simulated microstructures, the phase field model allows

for a treatment of both micro- and macro-texture evolution accompanying the

microstructure evolution during different thermo-mechanical treatments, as we shall

discussed in greater details in the following sections.

The main objective is to explore the effect of different processing route on both

microstructure and transformation texture evolution. The paper is organized as the

following. In section 2, we first make an extension of a 3D quantitative phase field model

[33, 36] formulated by the authors to investigate both microstructure and texture

evolution in during precipitation in polycrystalline Ti-alloys. In Section 3, the model is

employed investigate variant selection behavior (i.e. degree of variant selection) under

the influence of different pre-strain, starting texture, and the type of boundary

constraint of the sample, and its effect on the evolution both microstructure and final

alpha texture in polycrystalline sample. The results will be analyzed in Section 4 via

considering variant selection behavior at the scale of both individual b grain and

polycrystalline sample. Main findings will be summarized in Section 5.

180

5.2. Model Formulation

5.2.1. Polycrystalline sample

In the current work, we consider a polycrystalline matrix that is assumed to be formed

by a periodical repetition of M grains that occupy a computation cell of 128×128×128.

Different grains have different orientations. An orientation, often given the letter , of

grain or crystal in sample reference frame can be described by the rotation matrix

between crystal and sample co-ordinates. In practice, it is convenient to describe the

rotation by a triplet of Euler angles, e.g. [ ] by Bunge [37]. A rotation

matrix field is then introduced to describe the polycrystalline structure [38],

where assumes a constant value but different within each grain depending on

its orientation.

5.2.2 Phase Field Model for precipitation in an elastically and structurally

inhomogeneous polycrystalline sample

In this section, we extend a three-dimensional quantitative phase field model developed

by current authors to predict variant selection and microstructure evolution during

181

transformation in polycrystalline Ti–6Al–4V (wt.%) sample under the influence

of pre-strain or external stress .

5.2.2.1 Chemical free energy for polycrystalline system

Arbitrary two-phase microstructure in polycrystalline parent sample includes both

structural and chemical non-uniformities. For Ti-Al-V ternary system, two conserved

phase field parameters, Al, VkX k r , are required to describe the chemical non-

uniformities of components Al and V in the system. At the scale of individual grain,

twelve non-conserved order parameters, , 1,2,..., 12p p N r , are needed to

describe the structural non-uniformities associated with the total N OVs. One more

dependent order parameter, , 13p p r , is also introduced to describe the spatial

distribution of the matrix phase with grain. In the frame work of multi-phase field

model, such all non-conserved order parameters are subjected to the constraint that

1

1

, 1N

p

p

r [39, 40]. For a polycrystalline consisting of grain, the number of non-

conserved order parameters to describe the spatial distribution of phase is .

The total free energy of such a system having an arbitrary coherent or semi-coherent

two-phase microstructure, including both chemical and structural non-uniformity, is

formulated on the basis of the gradient thermodynamics [18]. The chemical free energy

182

can be simply extended to polycrystalline system from its counterpart for single crystal as

follows:


2

, 1 1

1 1, ,

2 2

M NT

chem cm k p k p p p

Vk Al V pm

F G T X X dVV

κκ

(5.1)

where and p κ are the gradient energy coefficient and gradient energy coefficient

tensors characterizing contributions from non-uniformities in concentration and structure

within polycrystalline system, respectively. A specific set of eigenvalues for the

tensors has been employed to describe the interfacial energy anisotropy by considering

contributions from misfit dislocations at different facets of interface to the structural

part of the interfacial energy [36]. The gradient energy coefficient tensors for all 12 OVs,

have also been derived according to symmetry operation associated with

each variant. In the sample reference frame, p κ is obtained via

Tc

p pQ Q κ κ where Q is a rotation matrix that describes the orientation of

grain.

The bulk chemical free energy density in Eq.(5.1) is expressed as:

cκ

κ

183

, ,

1 1 1 1

1

1

, , , 1 ,

M N M N

m k p p m Al V p m Al V

p p

M N N

pq p q

p q p

G T X h G T X h G T X

(5.2)

where, 3 26 15 10p p p ph is an interpolation function connecting

the free energy surface (as function of concentration and temperature) of and phase.

The term 1

1

N N

pq p q

p q p

introduces a hump on the free energy surface between

either variant and 𝑞, or variants and matrix, and hump height is proportional to

pq . ,,m Al VG T X and ,,m Al VG T X are the equilibrium molar free energies of and

phases as function of temperature and individual phase concentration ,Al VX and

,Al VX ,

respectively.

5.2.2.2. Strain energy of an elastically and structurally inhomogeneous system

Polycrystalline sample with two-phase microstructure is a typical elastically and

structurally inhomogeneous system. The system is characterized by an arbitrary

distribution of the crystalline lattice misfit strain, T

ij r , (transformation strain due to

precipitation), and an inhomogeneous distribution of elastic modulus ( )ijklC r due to of

184

elastic anisotropy and different orientations of individual grains, i.e., both ( )ijklC r and

T

ij r are functions of spatial coordinate r . Elastic strain energy of such an system

under external loading or pre-strain, ( ), , el T appl

ijkl ij ijE C r r , is obtained using the

iterative method developed by Wang et al. [17, 41]. By introducing a virtual strain field

0 ( )ij r and a reference modulus 0

ijklC , the exact elastic equilibrium, including total strain

and stress distributions, of an elastically and structural inhomogeneous system are

obtained [17, 41] by solving the elasticity problem in an equivalent elastically

homogeneous system. 0 ( )ij r is an energy minimizer of the total strain energy functional

that determines the equilibrium state of the elastically and structurally inhomogeneous

system. In practice, it can be obtained numerically through a solution of the time-

dependent Ginzburg-Landau (TDGL) type equation [41],

0

0

( , )

( , )

elij

ijkl

kl

EL

r

r (5.3)

where elastic strain energy of the system, elE , is given by:

185

0 0 0 0 0 3

0 0 0 3 0 0 3 0

1( ) ( )

2

1( ) ( ) ( )

2 2

1

2

el T T

V ijmn mnpq pqkl ijkl ij ij kl kl

V ijkl ij kl ij V ijkl kl ijkl ij kl

E C S C C d

VC d C d C

r r r r r r

r r r r r

3

0 0

3( ) ( ) ( )*

(2 )i ij jk kl l

dn n

gg n g

(5.4)

where ijklL is the kinetic coefficient tensor (a convenient choice is

10

ijkl ijklL L C

[42])

and the parameter describes the elastic relaxation process. In Eq.(5.4),

1

0

ijkl ijkl ijklS C C

r r .In order to ensure the convergence [43], 1

0

ijklL C

has been

chosen to be 1

0

ijnm mnklL C C

r during the iteration of Eq (5.3). Elastic inhomogeneity

rijklC defined in the sample reference is then described by

0T T T T

ijkl ip jq km nl pqmnC Q Q Q Q Cr r r r r , where stands for the transpose operator.

Einstein’s summation convention for repeated indices is assumed throughout.

0 0 0( )ij ijkl klC g g , 0

ij g is the Fourier transform of 0

ij r , 1 0

ij ijkl k lC n n n , and the

superscript asterisk denotes the complex conjugate. The last integral in Eq. (5.4)(?)

excludes a volume of 3

2 V around the point at g=0 . is the th component

of a unit vector , ⁄ ,in the reciprocal space. The strain energy in the form in

Eq. (5.4) is convenient when the body is fully clamped and thus its macroscopic

deformation is specified by the pre-strain appl

ij . The strain energy in Eq.(5.4) needs to be

modified if the macroscopic deformation of the body is controlled by the applied external

stress, appl

ij . In this case, the macroscopic shape is obtained by allowing the body to

186

relax at fixed appl

ij to minimize the strain energy with respect to ij . The energy

minimization is obtained when minimizer : 0 0 appl

ij ij ijkl ijS = + , 0 0 31

( )ij klV

dV

r r .

From Eq.(5.4), it is obvious that the stress-free transformation strain T

ij r from

precipitates in a whole system is a critical input in formulating the strain energy for such

an inhomgeneous system. In the sample reference frame, T

ij r is formulated upon

spatial distribution of the transformation strain field:

000

1 1

, ,M N

T

ij ij p

p

p

r r (5.5)

as a linear combination of individual phase field order parameter within each b grain,

,p r . 000 ,ij p is the SFTS tensor of variant in grain expressed in the

sample reference frame. It can be obtained through 000 00, T T

ij ik jl klp Q Q p

,where 00

kl p denotes SFTS tensor of variant in the crystal reference frame.

Thus, the elastic equilibrium is obtained through the converged value of the virtual misfit

strain 0 ( )rij in Eq.. Then, the total strain in the system is:

1

2ij ij r

30

32

i

i jk j ik kl l

dn n n e

g rg

n n g (5.6)

187

And the stress distribution ( )ij r in the polycrystalline system can be obtained through

0 0

ij ijkl kl klC r r r .

The elastically and structurally inhomogeneous polycrystalline sample during

precipitation is then equivalent to an elastic homogeneous system with modulus 0

ijklC with

the equilibrium internal stress distribution described by Eq. (5.3).

5.2.2.3 Kinetic equations

The temporal and spatial evolutions of both concentration and structural order

parameters, i.e. microstructure evolution, are governed by the Cahn-Hilliard equation and

the time-dependent Ginzburg-Landau equation, respectively. For simplicity, the

diffusion along grain boundaries and within bulk is treated as the same. The diffusion

equation is then the same as that within a single grain [36], i.e.

1

21

,1, , ,

,

nk

kj i j k

jm j

X t FM T X t

V t X t

rr

r (5.7)

There are total 1M N order parameters for precipitate in the polycrystalline

system and the governing equation is given as [38]:

188

, , 1, ,

, , , , , ,

elp

p

p q p q p

t F F EL t

t t t t

r

r rr r r

(5.8)

where , r defines the shape of individual grain that is equal to unity inside the th

grain and vanishes outside of it. 1...12, ,kj iM T X is the chemical mobility, L is the

mobility of the long-range order parameters characterizing interface kinetics, and k and

p are the Langevin noise terms for composition and long-range order parameter,

respectively. In Eq. (5.8), is the number of phases that co-exist locally.

It needs to be mentioned that the variation of the strain energy with respect to order

parameter is calculated based on the assumption that elastic relaxation occurs much faster

than precipitation. That is, the time-dependent Ginzburg-Landau equation Eq. (5.3) is

solved first to obtain a steady-state solution for the virtual strain 0

ij r with a clamped

order parameter and concentration field. In addition, it is also assumed that there would

be no grain growth within the polycrystalline sample. The effective strain and

inhomogeneous elastic modulus ijklC r is then treated as a constant in the calculation of

the functional variation of the strain energy with respect to order parameter field ,p r .

189

el

0 0 0 0

0 3

,

( ) ,

( ) , r1

2 ,

p

T

V ijmn mnpq pqkl ijkl ij ij p

T

kl kl p

p

E

C S C C

d

r

r r r

r r

r

(5.9)

All the parameters for the model and materials properties can be referred to Ref. [36].

5.2.3 Orientation Distribution Function modeling of microstructure in

polycrystalline sample

It has been realized that when characterizing titanium in general, texture should not be

ignored since its influence on mechanical properties can be significantly strong in

polycrystalline Ti-alloys due to low symmetry α phase with strong anisotropy in

properties. Therefore, microstructure and texture needs to linked together to obtain new

insight to materials processing.

Orientation of crystals in a polycrystalline is measured by individual orientation

measurements using a virtual electron back-scattered diffraction EBSD. A total of

individual orientations are measured. The orientation density of individual orientations of

grain in a polycrystalline sample describes starting texture. Similarly, the orientation

density of individual orientation variants represents its texture. To permit a quantitative

190

evaluation of textures for both phases, it is necessary to describe the orientation density

of each phase in a polycrystalline in an appropriate 3-D representation, that is, in terms of

its orientation density function (ODF) [44]. The ODF is defined as a probability density

function of orientation that models the relative frequencies of crystal orientations

within the specimen by volume [37]. Mathematically, the ODF is defined by the

following relationship:

dV

f g dgV

(5.10)

where is the sample volume and is the volume of all crystalline with the orientation

in the angular element . To be specific, ⁄ is the volume

of the region of integration in orientation space. Choice of such that ∫ to

normalize the ODF implies that the uniform random ODF, , which is then

consistent with the custom of expressing in terms of multiples of the uniform

random ODF (M.R.D.).

The ODF estimation from individual EBSD data in MTEX is implemented using the

function . The underlying statistical method is called kernel density estimation,

which can be interpreted as a generalized histogram. To be more precisely, assume be

a radially symmetric, unimodal model ODF [45]. Then the kernel density estimator for

the individual orientation data is defined as:

191

∑

The choice of the model ODF and in particular its half width has a great impacting in

the resulting ODF.

Experimentally, pole figures are frequently used to represent textures. The pole density

function (PDF) of a specimen models the relative frequencies of specific lattice plane

orientations , i.e. the relative frequencies of normal vectors of specified lattice planes,

within the specimen volume. For example, a pole that is defined by the direction in a

given 2-D pole figure, , corresponds to a region in the 3D-ODF that contains

all possible rotations with angle about this direction in the pole figure.

∫

where { } is the position of a given pole on the reference sphere.

The angle describes the azimuth of the pole and the angle characterizes the rotation

of the pole around the polar axis[44]. When ODF is obtained, pole figure for a set

specific plane can be readily obtained. For convenience, the freely available MTEX

Matlab Toolbox for Quantitative Texture Analysis [46] is utilized to make all pole

figures. As will be shown latter, a comparison of the alpha- and beta-phase textures,

represented respectively by and { } pole figures, is able to quantify

preferential variant selection during transformation due to BOR between two

phases.

192

5.3. Results

5.3.1. Starting polycrystalline and texture

A polycrystalline sample of Ti-6Al-4V (Fig. 5.1(a)) is first created by the Voronoi

algorithm [47] and, further relaxed by a phase field grain growth code [48] to obtain

equilibrium junctions among grains. The orientation of each beta grain with respect to

the sample reference is specified by a set of Euler angler [ ] using the Bunge

notation [37]. The estimation of orientation distribution functions (ODFs) of texture is

then made from sampled individual orientation data using MTEX Quantitative Texture

Analysis Software [46]. The chosen kernel is a de la Vallée Poussin kernel with a

smoothing half-width of 5 deg. The beta texture is then represented by { } pole

figures as shown in Fig. 1(b), whose intensity contours are represented in times-random

units. Since the starting texture of the grains may have a strong influence on the

transformation texture of the phase because of the BOR between the two phases, two

sets of initial texture are considered in the current study. One is a relatively random

texture referred to as “random-textured” sample and the other one has a relatively strong

texture and is referred to as “strong-textured” sample. Their textures are represented by

{ } pole figures shown in Fig. 5.1(b) and 5.1(c), respectively. As can be seen from

the pole figures, the strong-textured sample has a relatively larger maximum pole

intensity of the { } pole than that of the random-textured one.

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5.3.2. Evolution of microstructure and texture during precipitation

Evolution of microstructure during precipitation is obtained by solving Eq. (5.7)

and Eq. (5.8) simultaneously. In case of clamped boundary without any pre-strain

(referred to as fix-end) in “random-textured” sample, microstructure evolution during

precipitation is shown Fig. 2 through (a)-(c). The red color represents phase, while blue

color denotes matrix phase. In particular, the matrix is set be transparent to make the

precipitation visible within the bulk. The white lines indicate the locations of grain

boundaries. It is clearly that nucleation of a phase occurs not only near grain boundaries

but also within the bulk. phase gradually fills the whole polycrystalline sample as

shown if Figs 2(b)-(c). Formation of closed-triangle pattern is also observed, as pointed

out by an arrow in Fig. 2(c).

In order to quantify the texture evolution during precipitation, a virtual EBSD scan is

preformed through the sample to read in orientation information of

individualprecipitates according to the index of the variants [36] and orientation of

its matrix grain based on BOR. The ODFs for phase are obtained using the same

approach as that used in describing texture [49]. The final textures are represented

by the pole figures as shown in Fig. 2(a′)-(c′). By comparing with { } pole

of starting texture (Fig. 5.1(b)), pole figures had similar locations of intensity

maxima, confirming the validity of the Burgers relationship during the decomposition of

the phase. The strength of the transformed texture is simply represented by the

194

texture-component maxima in these pole figures. It is found that maximum pole intensity

in pole figures larger than that in { } pole figure representing starting

texture, and the maxima is increasing, from 10.04 to 10.45, due to coarsening.

5.3.3. Effect of pre-strain on variant selection

In “random-textured” sample, when a pre-strain is introduced by a compressive stress,

50MPa, that is applied along the x-axis of the sample before the system’s boundary is

clamped, as we can seen clearly, nucleation of phase in different grains tend to occur at

grain boundaries as shown in Fig. 3(a). Formation of closed-triangle pattern is also

observed, as pointed out by an arrow in Figs.3(c)-(d). Precipitation behavior varies

significantly from one to another grain. For example, in some grains, multiple

variants are favored simutanously; while in other grains, only a limited number of

variants survive. The corresponding texture evolution of phase represented by

pole figures is shown through Figs. 3(a′)-(d′). When compared with { } pole of

starting texture (Fig. 5.1(b)), pole figures have much less number of intensity

maxima. In particular, there are clear quantitative differences in the intensities at specific

locations, which suggest orientation densities of parent phase are shared by only

limited numbers of variants in different grains. The strength of the transformation

texture is found to decrease with precipitation. The maxima of intensity in

pole figures decrease from 39.05 ( random) at 1.5 s to 23.2 ( random) at 9.0 s. While

195

in the mean time, several new texture components within the basal poles appear with

precipitation as pointed out by red arrows from Figs. 3(b)-(d). The evolution of texture

suggests that more variants come out during precipitation.

For “random-textured” sample, different pre-strains have been introduced to the sample

to investigate their influences on both microstructures and final texture. Pre-

strains are generated by applying a 50Mpa tensile/compressive along x, and z direction of

the sample that are referred to as x-comp, x-tensil, z-comp, and z-tensil, respectively. As

can been seen clearly from Fig. 4(a)-(d), when subjected to different pre-strain, final

microstructure at t =10.0 s are quite different in the sample in terms of numbers and types

of orientation variants within individual grains. So do the final textures in terms of

numbers of maxima, intensities and locations of maxima, Fig. 4(a′)-(d′).

5.3.4. Effect of starting texture on variant selection

Both grain-boundary geometry and parent texture would change due to grain growth or

hot deformation in phase region [8, 50]. Such grain growth may result in the weakening

of initial-stronger texture component and also the strengthening of initially-weaker

texture [51]. Thus, it has been noticed that preferential variant selection needs to be

interpreted according to the specific starting texture right prior to decomposition. In

order to focus to the effect of starting texture on variant selection, we consider a strong-

textured sample (Fig. 5.1(c)) with the same grain geometry as that of the random-

textured sample. Pre-strains are also generated by applying a 50Mpa tensile/compressive

196

along x, and z direction of the sample that are also referred to as x-comp, x-tensil, z-comp,

and z-tensil, respectively. When compared with random-textured sample, a similar trend

for variant selection behavior and microstructure can be easily found as well at the

scale of individual grain and overall polycrystalline sample in strong-textured sample,

as shown in Fig. 5(a′)-(d′). The final textures in terms of number of maxima,

intensities and locations of maxima are also sensitive the type of pre-strain, as shown in

Fig. 5(a′)-(d′). The evolution of texture strength represented by the maxima intensity in

pole figures in both random-textured and strong-textured sample is shown in

Fig. 6(a) and Fig. 6(b), respectively. In addition, the maxima pole intensity for

both random-textured and strong-textured sample under different pre-strain is

represented by red and green bars in Fig. 6(c). It is found that, when subjected to pre-

strains, the strength of texture will gradually decrease with precipitation process,

with different rates depending on both initial texture and type of pre-strain. Moreover,

final texture will be always stronger if there is a concurrent pre-strain during the

decomposition irrespective of starting texture. Since the phase acquires a specific

texture, from texture during phase transformation through Burgers orientation, it

is generally believed that the stronger is the starting texture, the stronger is the final

texture, which is also true in most of the cases considered in our simulations. However,

when the pre-strain is generated via a 50 MPa tensile stress, the final maximum intensity

of the basal pole in the strong-textured sample is smaller than that in the random-textured

sample, as shown in Fig. 6(c).

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5.3.5. Quantifying the degree of variant selection

The differences in both microstructure and texture reflect the occurrence of

different variant selection behaviors resulting from the combined effect of the starting

texture and specific pre-strains. The degree of variant selection could be quantified using

the ratio of the maxima pole intensities in the and { } pole figures [50].

As shown in Fig.7, irrespective of the starting texture is, if there is no variant selection,

i.e. the orientation density of individual grain is distributed equally to all 12 variants

within it, the corresponding { } and pole figures will be identical (See

comparison between Fig. 7(a) and 7(c), and Fig. 7(b) and 7(d) ), and thus .

However, if variant selection occurs [50]. Thus, magnitude of is able to

characterize the overall degree of variant selection within sample. The degrees variant

selection associated with Fig. 6(a) is shown in Fig. 8. It can be seen from Fig. 8 that

in the strong-textured sample is smaller than that in the random-textured sample except

the one when the pre-strain is associated with a compression along the x direction. Thus,

when subjected to a certain pre-strain, the strong-textured sample could promote more

variants simultaneously within the whole polycrystalline sample and thus have a

relatively small degree of micro-texture. To sum up, the degree of variant selection and

the resulting strength of the transformed texture depend heavily on both the degree of

preferred orientation in the parent phase and type of pre-strain.

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5.3.6. Effect of boundary constraint on variant selection

All the above simulations are performed in which the system’s boundary is fixed after

applying an external force to the system, followed by the precipitation process. Such a

boundary constraint has been referred to as strain-constraint [52], or Fix-end [50] that is

used in the current study. In practice, in order to prevent an axis stress/strain due to

thermal contraction, another type of constraint is also employed, in which the system is

subjected to a constant external force without fixing the system’s boundary. Such a

constraint is then referred to as stress-constraint [52] or Free-end [50] in the current

study. Type of boundary constraint has been found to have significant influence on the

degree of variant selection. For instance, in random-textured sample, when a 50Mpa

external stress is applied along the x-axis of the sample, the strength of final texture

will be larger in the case of Fix-end boundary irrespective of compressive or tensile

loading, as shown in Fig. 9(a). However, in strong-textured sample, when a 50Mpa

external stress is applied along the z-axis of the sample, the strength of final texture

will be much smaller in the case of Fix-end boundary irrespective of compressive or

tensile loading, as shown in Fig. 9(b). Thus, the effect of boundary constraint on the final

texture depends on the strength of starting texture and type of pre-strain.

5.4. Discussions

It has been demonstrated that the overall degree of variant selection and strength of final

texture depend sensitively on processing variables such as type of pre-strain, boundary

199

constraint and starting texture. The strength of final texture results from the

occurrence of different degree of variant selection within individual grains under the

influence of both starting texture and pre-strain/external loading. Thus, it is necessary

to zoom into individual grain to see how variant selection occurs at the scale of

individual grains, and, more importantly, how it contributes to the overall variant

selection behavior in the whole polycrystalline matrix.

In the case of random-textured sample under x-tensile pre-strain, microstructures in 2nd

and 5th

grains are shown in Figs. 10(a) and 10(b), respectively. The volume fraction of

each variant as a function of time within two grains is shown in Fig. 4.10(c) and 10(d),

respectively. It is clearly that there are more than 4 (V12, V5, V9, V3, and V11, in order

of their volume fractions) variants selected in the 2nd

grain and, in contrast, only two

variants (V8 and V12) survive, though with a relative small volume fractions.

Microstructures in 2nd

and 5th

grains, in the case of strong-textured sample also under x-

tensil pre-strain, are shown in Figs. 11(a) and 11(b), respectively. The volume fraction of

each variant as a function of time within two grains is also shown in Figs. 11(c) and

11(d), respectively. In both two grains, there are 4 different variants selected. Thus,

when subjected to a specific pre-strain, the strong-textured sample could promote more

variants simultaneously within the whole polycrystalline sample, which thus results in a

relatively weak micro-texture. The variant selection behaviors relate closely to the local

stress state within individual grain. For random-textured sample, the initial local stress

field ( expressed in the sample reference frame) within two grains are calculated

and shown in Figs. 10(e) and 10(f). Firstly, local stress states within two grains are

200

non-uniform as indicated by the inset color bar, and deviate dramatically from the

external loading ( ) that generates the pre-strain. In order to quantify effect

of external loading on selective nucleation and growth of specific variants, interaction

energy density between the external loading and each α variant under both coherent and

semi-coherent conditions within these two grains are calculated within both grains and

represented in Figs. 5.10(g) and 10(h), respectively. From interaction energy analysis, the

initial external loading would result in significant different variant selection behavior

within the two grains, as confirmed by the volume fraction analysis. In the 2nd

grain,

the favored variants would be variants V3-V6, and V9-V10 in nucleation and growth

stage, while in the 5th

grain, the most favored ones are variants V2, V6, V8 and V12.

From the volume fraction analysis, within 2nd

grain, only variants V12, V5, V9, V3,

and V11 have been preferentially selected from the above-mentioned most favored ones.

Within the 5th

grain, only variants V8 and V12 have been selected from all possible

favored variants. It is noticed that, within both grain, only a fraction of all favored

variants by the external loading has been selected that could be ascribed to the non-

uniform stress state with significant deviation from initial external stress. Another reason

could be associated with evolving local stress state due to evolving precipitation. In the

case of strong-textured sample under x-tensil pre-strain, local stress state within these two

parent grains (Figs. 5.11(e) and 11(f)) are also non-uniform with deviation from initial

external loading, but both of them are under tension. According to interaction energy

analysis as presented in Figs. 5.11(g) and 5.11(h), within both grains, variants V3-V6

and V9-V12 would be favored in nucleation and growth stage by the initial external

201

loading, while with different magnitude for each favored one. Again, not all favored

variants have been selected. Therefore, compared with random-textured sample, there

would be a relative uniform stress state that is possible to promote relative more variants

simultaneous within relative more grain in the strong-textured sample when subjected

to a specific pre-strain.

It has been found that boundary constraint of polycrystalline sample also have a

significant influence on the strength of final texture. For instance, in random-textured

sample, when a 50Mpa tensile stress is applied along the x-axis of the sample, the

strength of final texture will be larger in the case of fix-end boundary as shown in Fig.

5.9(a). Microstructures in 2nd

and 5th

grains are shown in Figs. 5.12(a) and 5.12(b),

respectively. The volume fraction of each variant as a function of time within two

grains is shown in Fig. 5.12(c) and 5.12(d), respectively. The interaction energy results

will be identical to those in Fig. 5.10 (g) and Fig. 5.10(h). It is clearly that, in the 2nd

grain, there are 7 variants (V12, V9, V5, V3, V11, V6, V10) have been selected out of all

8 possible favored variants, having volume fractions much larger than those in the fix-end

case. In the 5th

grain, there are still only two variants (V8 and V12) survived but,

almost percolating the whole grain according to their volume fraction. In the case of

the fix-end boundary constraint, there will be a competition between pre-strain and

evolving microstructure on variant selection [36]. In particular, it is possible that internal

stress generated by an evolving microstructure exceeds the initially applied stress that

generates the pre-strain. In the case of free-end boundary condition, the system would be

202

free to change its shape and volume to relax the internal stress due to evolving

microstructure.

In order to make connection between variant selection behaviors within individual

grain and the overall polycrystalline sample, it is necessary to quantify degree of

variant selection within a single parent grain. It is known that among all 12 variants,

there are only 6 type possible misorientation between any two variants [53]. As, shown in

Fig. 13(a) when represented as misorientation/axis pair, all 6 type are, , [ ]⁄ ,

[ ]⁄ , [ ] ⁄ , [ ] ⁄ ,

[ ] ⁄ referred to as Type I to Type VI misorientations, respectively.

In particular, two variants with type II misorientation share a common basal

plane.

In the case of no variant selection, occurrence frequencies are 12, 12,

24, 48, 24 and 24 for each type out of all 144 possible combinations within one single

parent grain when doing uncorrelated misorientation analysis, i.e., two variants are not

necessary in contact with each other. The measured uncorrelated misorientation

distribution can then be compared with the random occurrences, as shown

in Fig. 5.13(b) using red and green bars, respectively. The degree of deviation from the

random case can be quantified by the summation of the deviation of each type of

misorientation from the random occurrences. Thus, the degree of variant selection within

is defined as:

∑|

|

203

The physical meaning can be interpreted as: if there is no variant selection, there will be

no deviation from random occurrence frequencies for each type of misorientation, i.e.

; if a single variant is able to percolate through a whole single grain, maximum

degree of variant selection is reached with , as shown in Fig. 13(c). Thus,

the value of is able to characterize qutitatively the degree of variant selection within

a single grain. The computation of an uncorrelated misorientation distribution from the

EBSD data of all individual phase in a parent grain is performed using MTEX. For

random-textured sample under different processing conditions such as pre-strain and

boundary constraints, degree of variant selection within individual grains, is

calculated within the largest and the smallest grain represented respectively by red and

green bars in Fig. 5.14(a). The corresponding overall degree of variant selection is also

presented in Fig. 5.14(b). It can be found that, in most case, the larger the is, the

larger the overall degree of variant selection for the polycrystalline sample will be.

However, it is also noticed that with the largest and smallest grains under Z-Comp

pre-strain are larger than those under X-Comp-Free external loading, but the final is

almost identical. To be specific, in the case of Z-Comp pre-strain, in the

largest grain, while, in the case of X-Comp-Free, , as shown in Fig. 5.15(a)

and Fig. 5.15(b), respectively. Compared with the difference in (maximum value of

), the difference in corresponding maximum intensity in poles, and

thus is relatively small.

204

It has been stated that when there is no variant selection, and when variant

selection occurs, . It is then believed that the larger is, the larger the overall

degree of variant selection will be. However, there is a special case where even if variant

selection already occurs, one still has . For instance, the macro-texture of

random-textured sample is represented using three different pole figures, { } , { }

and { } poles, i.e. three component in describing BOR, as shown in Fig. 5.16. The

macro-texture of final phase without occurrence of variant selection is represented

using corresponding three different pole figures, , { } , and { }

according to BOR. As has been stated that { } pole is identical to { } one.

Another set of ODF for the final texture from random textured sample is obtained by

assuming a special variant selection occurring in a single grain that multiple random

distributions of six variants are 3 times of those of other 6 variants sharing common

pole. In other words, in the grain, variant selection occurs due to the bias

between any two variants with misorientation of [ ] ⁄ or having common

basal . While in all the other grains, their orientation density is shared equally

by all 12 variants within them. The resulting macro-texture of final phase is

represented using three different pole figures, , { } , and { } as well. It

is readily seen that { } pole is still identical to { } one even if variant selection

occurs. The occurrence of variant selection can be noticed by comparing the difference

between { } or { } poles in the case of with/without occurrence of variant

selection. Thus, when using to evaluate the influence of different set of processing

variables on the overall degree of variant selection, it is better to double check with

205

degree of variant selection within individual grain. As a matter of fact, as a 2-D

projection of the 3-D orientation distribution, pole figures may bear some losses in

information. Thus, the evaluation of degree of variant selection by comparing two sets of

pole figures could be inadequate.

Alpha precipitate has a strong anisotropy in shape that appears as a lath. This anisotropy

in shape may lead, even in the absence of variant selection, to an inhomogeneity in

variant distribution due to the morphological orientation of phase with respect to the

sample surface plane. For example, as shown in Fig. 17, three different cross-sections

have been made along x, y and z surface layer. It is obvious those microstructures are

significantly different among these three sections. Variant selection is then studied, from

a statistical point of view, using the average texture obtained by a virtual EBSD scan

through these sections. Apparently, the final texture varies significantly with cross-

sections in terms of maximum pole intensity and distribution with each basal pole figure.

This is due purely to the 2D stereology sampling artifact that has recently been referred to

as “pseudo variant selection” [54]. Thus, when analyzing experimental data in literature

one has to bear in mind this possible 2D sampling effect.

5.5. Summary

A three-dimensional quantitative phase field model (PFM) has been developed to study

the variant selection process during transformation in polycrystalline sample

under the influence of different processing variables such as pre-strains. The effect of

206

elastic and structural inhomogeneities on the local stress state and its interaction with

evolving microstructure is also considered in the model. In particular, microstructure and

transformation texture evolution are treated simultaneously via orientation distribution

function (ODF) modeling of ⁄ two-phase microstructure in polycrystalline systems

obtained by PFM. The variant selection behavior at the scale of individual grain and

the overall polycrystalline sample, and the resulting final texture are found to be

heavily dependent on type of pre-strain, boundary constraint of the sample, and strating

texture. It is found that, when subjected to a certain pre-strain, the sample with strong

texture component could promote more variants simultaneously within the whole

polycrystalline sample and thus lead to a relatively small degree of microtexture. The

results could shed light on how to control processing conditions to reduce the strength

micro-texture at both the individual grain level and the overall polycrystalline sample

level according to its starting texture.

207

Figures:

Figure.5.1 (a) Polycrystalline matrix with different strength of starting texture, i.e.,

(b) a random-textured sample and (c) a strong-textured sample, according to the

maxima intensity in the { } pole figures

(a)

(b) (c)

“Random-Textured” “Strong-Textured”

208

Figure.5.2 (a)-(c) Microstructure evolution due to precipitation in random-texture

sample without any pre-strain, and (a′)-(c′) corresponding texture evolution represented

by { } pole figures

t=1.0s t=3.0s t=5.0s

(a) (b) (c)

(a′) (b′) (c′)

209

Figure.5.3 (a)-(d) Microstructure evolution due to precipitation in random-texture

sample under the pre-strain, and (a′)-(d′) corresponding texture evolution represented

by { } pole figures. The pre-strain is obtained by applying a 50Mpa compressive

stress along x-axis of the system

t=1.5s

(a) (b) (c) (d)

t=3.0s t=6.0s t=9.0s

(a’) (b’) (c’) (d’)

210

Figure.5.4 (a)-(d) Final ⁄ microstructure in random-textured sample under different

pre-strains, and (a′)-(d′) corresponding final texture

X - Comp X - Tensil Z - Comp Z - Tensil

(a) (b) (c) (d)

(a′) (b′) (c′) (d′)

t=10.0s

211

Figure.5.5 (a)-(d) Final ⁄ microstructure in strong-textured sample under different

pre-strains, and (a′)-(d′) corresponding final texture

X - Comp X - Tensil Z - Comp Z - Tensil

(a) (b) (c) (d)

(a′) (b′) (c′) (d′)

212

Figure.5.6 (a) Maximum intensity in pole figures as a function of time in

random-textured sample under different pre-strain, (b) Maximum intensity in

pole figures as a function of time in strong-textured sample under different pre-strain,

(c) Maximum intensities in pole figures of final texture in both random-

texture and strong textured samples under different pre-strain


(a) (b)

(c)

213

Figure.5.7 (a) and (b) { } pole figures for random-textured and strong-textured

sample; (c) and (d) corresponding pole figures of final texture in random-

textured and strong-textured sample without variant selection


(a) (b)

(c) (d)

214

Figure.5.8 Degree of variant selection in both random-texture and strong textured

samples under different pre-strain

215

Figure.5.9 (a) Degree of variant selection in random-textured sample under

different boundary constraint, (b) Degree of variant selection in random-textured

sample under different boundary constraint

“Random”-Textured “Strong”-Textured

(a) (b)

216

Continued

Figure.5.10 (a)-(b) ⁄ microstructure in the 2nd

and 5th





coherent and semi-coherent conditions within these two grains

(a) (b)

(c) (d)

217


(e) (f)

(g) (h)

218

Continued


and 5th

grain in strong-textured




coherent and semi-coherent conditions within these two grains

(a) (b)

(c) (d)

219


(e) (f)

(g) (h)

220


and 5th


sample under x-tensil external loading (Free-end), respectively; (c)-(d) volume fraction of

each variant as a function of time in the two grains;

(a) (b)

(c) (d)

221

Figure.5.13 (a) all possible misorientation between ⁄ pairs of variants. Misorientation

axes are expressed in a strand triangle for HCP structure; (b) uncorrelated misorientation

analysis for both phase field simulated ⁄ microstructure and the one without variant

selection; (c) the maximum degree of variant selection within individual grain where a

single variant percolates the whole grain

(a)

(b) (c)

222

Figure.5.14 (a) degree of variant selection within the largest and the smallest grain in

random-texture sample under different pre-strains and boundary constraint, (b)

corresponding overall degree of variant selection

(a) (b)

223

Figure. 5.15 (a) and (b) degree of variant selection within the largest in random-texture

sample under Z-Comp pre-strain and X-Comp external loading (X-Comp-Free),

respectively, (c) and (d) pole figures for final textue under Z-Comp pre-strain

and X-Comp external loading (X-Comp-Free), respectively

(a) (b)

(c) (d)

224

Figure. 5.16 (a) Macro-texture of random-textured sample represented by three different

pole figures, { } , { } and { } poles, respectively; (b) Macro-texture of final

phase without occurrence of variant selection represented by corresponding three

different pole figures, , { } , and { } , respectively;(c) Macro-texture

of final phase with occurrence of variant selection represented by corresponding three

different pole figures, , { } , and { } , respectively

(a)

(b)

(c)

225

Figure. 5.17 Examples showing the pseudo variant selection due to 2D sampling effect.

EBSD scan is performed along at different layers of the sample

(a)

(a′)

(b) (c)

(b′) (c′)

226

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233

CHAPTER 6 Variant Selection of Grain Boundary by

Special Prior Grain Boundaries in Titanium Alloys

Abstract

During -processing of / and titanium alloys, variant selection (VS) of grain

boundary GB is one of the key factors in determining the final transformation

texture and mechanical properties. It has been observed frequently that GB prefers its

⟨ ⟩ pole to be parallel to a common ⟨ ⟩ pole of the two adjacent grains and

results in a micro-textured region across the grain boundary (GB) and, as a consequence,

slip transmission may take place more easily across that GB. In order to investigate how

such a special prior GB contributes to VS of GB, we develop a crystallographic

model based on the Burgers orientation relationship (BOR) between GB and one of the

two grains. The model predicts all possible special grain boundaries at which GB is

able to maintain BOR with both grains. A new measure for VS of GB, ,

i.e. a measure of the deviation of the actual OR between the GB and the non-Burgers

grain from the BOR, is proposed. For the particular alloy chosen for experimental

observations, Ti-5553, it is found that when the misorientation angle of

(instead of the closeness between two closet { } poles between two grains widely

used in literature) is less than , misorientation between the two grains dominates the

VS of GB and, in particular, the variant with minimum is always selected for GB.

A possible effect due to grain boundary plane inclination on VS is also discussed.

234

6.1. Introduction

Similar to steels, the influence of grain boundary (GB) on subsequent intragranular

microstructure development and microstructure-properties relationships in Ti-alloys has

been an active area of research for decades [1-3]. In most cases, GB has the Burgers

orientation relationship (BOR) [4], i.e. and [ ] [ ] , with

one of the two adjacent grains that form the prior grain boundary (GB). The grain that

maintains BOR with GB is then referred to as the Burgers grain and the other as the

non-Burgers grain. There are 12 crystallographically equivalent orientation variants

available for the GBif it develops BOR with only one of the grains. In most cases,

however, only limited number of variants of GBare observed on most GBs [3, 5-7].

Such variant selection of GB not only has a direct influence on the overall

microstructure and transformation texture evolution [8, 9], but also has a significant

impact on the mechanical properties of the alloy [10]. For example, it has been

demonstrated frequently that different colonies (e.g. different in both orientation and

growth directions) growing from GB into the grains share common crystallographic

features such as a common basal plane orientation (though with their { } pole

being misoriented by a ~10.5˚ rotation about their common [ ] axis) provided that

the two grains have special misorientations [6, 7, 11]. An even more special case has

also been discovered recently [12] where colonies growing into two adjacent grains

have exactly the same orientation as that of the GB [3, 5-7]. This results in the

235

formation of large regions of colonies in multiple grains with a common

crystallographic feature, which are also referred to as “macro-zones” [2] or micro-

textured regions [3] that would have detrimental effect on the mechanical behavior of Ti-

alloys such as cold-dwell crack initiation and propagation [2]. Ample experimental

observations have shown that the formation of such macro-zones is closely related to

crystallographic characters of the prior grain boundaries [2, 6, 7, 11, 12]. For instance,

it has been reported repeatedly that if two adjacent grains share a common { } pole

with angular deviation within 10°, the preferred variant for GB always has its { }

pole parallel to one of the common { } poles [6, 7, 11]. In particular, when two

adjacent grains are misorientated by a 10.5˚ rotation about a common ⟨ ⟩ axis or a

60˚ rotation about a common ⟨ ⟩ axis [12], the selected GBis able to maintain BOR

with both adjacent grains. In this case, colonies in both grains will have the same

crystallographic orientation as that of the GB.

A statistical analysis [6, 7] of EBSD data has shown that variant selection for GB

usually obeys the following rule: when there exist two nearly common { } poles of

the two adjacent grains (with angular deviation within 10°), the normal of

plane of GB will be parallel to the common { } pole. Even though it was

mentioned in the literature [7, 11] that selection of specific variant for GB by such a

rule might minimize the interfacial energy between the selected GB and the two

grains and thus lower GB nucleation barrier, no fundamental argument or analysis has

236

been provided. In consequence, it is still not understood why the rule is not always

followed. For example, it has been observed that the { } pole of GB is parallel to

one of the { } pole that does not belong to one of the two closet { } poles though

still with an angular deviation less than 10˚ in between [6, 7]. Apparently, the parameter

describing angular deviation between two closest { } poles is not always a primary

factor in determining variant selection of GB. Thus, the application of the rule to

predict the overall transformation texture of phase would not necessarily lead to a

satisfactory result [13]. As a matter of fact, the single parameter may not be sufficient

considering the fact that it requires at least two independent parallelisms to describe an

orientation relationship, for example ⟨ ⟩ ⟨ ⟩ besides { } { } for the

BOR.

In this paper, we first formulate a crystallographic model which allows us to evaluate

quantitatively deviation of the OR between GB and grains from BOR. Then we

characterize experimentally ORs of GB with two adjacent grains at different prior

GBs using electron-backscatter diffraction (EBSD) and analyze the experimental results

against the specific rule. Cases that follow and do not follow the rule are identified and

analyzed, and new criteria of variant selection of GB by prior grain boundaries are

provided.

237

6.2. Model formulation and Experimental procedures

6.2.1. Crystallographic model

Any GB will always have a relative orientation with respect to its mother

grains in contact with it. Here, is a 3×3 matrix representing a co-ordinate

transformation that transforms the components of vectors defined in the basis of the

parent phase to those defined in the basis of the product phase. The notation, which

was due to Bowles and Mackenzie [14, 15], is particularly convenient in avoiding

confusion about different bases [16]. When GB maintains BOR with the Burgers grain,

is referred to as . For the specific orientation variant described by

, [ ] [ ] and [ ] [ ] , BOR J reads:


1

BOR

1 1 1

3 3 3

1 1 2

6 6 6

1 10

2 2

J . (6.1)

See Supplementary Materials for more details about calculation of the orientation matrix

in Eq.(6.1). The orientation matrix for all other variants can be obtained readily by

applying symmetryoperations 1..12i i S on 1

BOR J , i.e.

238

1

BOR BOR i

i

J J S 1..12i . All the 12 orientation variants and the

symmetry operations, 1..12i i S , associated with them can be found in

Supplementary Materials Table 6.1. It should be emphasized that the model is able to

offer orientations of grain boundary for all 12 variants provided that the orientation of

the Burgers grain is known. To be consistent with the experimental measurement of

orientation of GB, a right-handed Cartesian coordinate system is assigned for a

hexagonal crystal using the same convention as that employed in the TSL-OIM software,

i.e., x-axis // [ ] , y-axis // [ ] and z-axis // [ ] . Thus, by comparing the

predicted orientations of GB with the measured ones, it would be straightforward to

know which variant has been selected.

Suppose that the ith

variant of GB forms at a grain boundary between two neighboring

grains, 1 and 2, and maintains BOR with 2. Then the misorientation between the GB

and the non-Burgers 1 grain follows that:

1 1 2 2 BOR i J J J (6.2)

where 1 2 J represents the misorientation between the two grains and

2 BOR i J is the inverse of BOR 2 i J . Similar to calculating misorientation

239

between two adjacent grains, how 1 J deviates from the BOR can be evaluated

quantitatively by defining a “misorientation” matrix [17, 18] as the following:

BOR

1 1 1 BOR 1 1..6j j ΔJ J J (6.3)

where the superscript j in the right hand side of Eq. (6.3) describes the order of all 6

possible orientations for 1 if BOR is maintained between itself and GBsince there are

only 6 possible orientation variants of if BOR exists between the parent and product

phases during transformation. Combining Eqs. (6.2) and (6.3), we have

BOR

1 1 1 2 2 BOR BOR 1

11 1

1 2 BOR BOR

i j

j iS S

ΔJ J J J

J J J (6.4)

Thus, one only needs misorientation between two adjacent grains, symmetry operations

of and phases, and an orientation matrix describing BOR to evaluate the deviation of

the OR between GB and non-Burgers grain from the BOR for all 12 variants. Note that

symmetry operations of both and phase will result in multiple equivalent

misorientation matrices associated with . Thus, the misorientation matrix

such as or , is also described by the angle/axis (r pair with

misorientation angle and axis 1 2 3, , r r r being calculated from the misorientation

matrix [19], and the angle/axis pair that has the minimum misorientation angle, i.e., the

disorientation, is selected to represent the misorientation. The magnitude of the

240

disorientation angle characterizes the closeness between two orientations. The

misorientation between the measured and predicted orientations for GB is evaluated

through the misorientation matrix , where and represent the measured

(by Experiment) and predicted (by Model) orientation of GB, respectively. The matrix

is then also expressed by the disorientation angle/axis pairs.

6.2.2. Experimental procedures

Twenty-eight examples of GB at different prior grain boundaries are observed in the

experiment. They belong to only one or two specific variants. The material used is a

forged -Ti alloy, Ti-5553, i.e., Ti-5Al-5Mo-5V-3Cr-0.5Fe (wt. %). The as-received

alloy was sectioned to small samples (~20mm×20mm×40mm) for heat treatment. The

samples were initially -annealed at 1000˚C for 15 minutes using a conventional tube

furnace in an inert Argon atmosphere. The samples were wrapped in titanium foil to

further reduce the possibility of oxygen ingress. The samples were then cooled in the

furnace to 825˚C at a controlled rate of 5˚C/min, and were soaked for 2 hours to allow for

the phase transformation to complete. Finally, the samples were water-quenched to room

temperature. Since the -transus for this alloy is close to 850˚C, such a heat treatment

procedure would allow for only limited numbers of variants of GBform on a given

planar grain boundary if the precipitation would occur while the colony structure has

not developed from GB and thus allow us to avoid the influence of relative large under

241

cooling on the variant selection behavior. In other word, under such a small undercooling,

grain boundary characters would dominate the variant selection of GB. For subsequent

characterization, all specimens are further sectioned in the middle and the exposed

surfaces were subjected to mechanical polishing using standard metallographic

techniques. In the final step, the material is kept in a vibratory polisher in a suspension of

0.05 m silica particles for a number of hours to achieve a mirror-finish. The

crystallographic orientation of GB and the parent grains are determined using EBSD

data collection in Philips XL30 ESEM FEG SEM at 20kV, with a spot size of 4 and a

working distance of 20 mm. Suitable step-size (1.7m) in OIM-TSL software was

selected to allow for the collection of data from a large area (~1 mm × 1 mm) in a

reasonable amount of time. The reliability of this data collection was verified on a ⟨ ⟩

silicon sample under similar conditions.

6.3. Results

The orientations of two adjacent grains determined in the experiment are used as model

inputs to predict orientations of GB by evaluating for all 12 variants. By

comparing orientations of GB between the measured and predicted values, we then

further investigate variant selection of GB by a special grain boundary having a

nearly common { } pole between two adjacent grains.

242

6.3.1. Special grain boundaries where GB maintainsBOR with both adjacent

grains

The experimental observations have demonstrated that GB is able to hold BOR with

both adjacent grains. In this case, the misorientation matrix , i.e.,

J with respect to non-Burgers grain is coincident with BOR and then both

grains are Burgers grains. All possible special misorientations between the two Burgers

grains can be determined from Eq. (6.4) as follows:

BOR BOR

11 1

1 2 j iS S

J J J . (6.5)

The results are summarized in Table 6.1 and illustrated in Fig. 6.1. There are 4 types of

special misorientations between the two adjacent grains in total, among them Type I

and Type IV have been observed previously by Bhattacharyya et al. [12] and Type II and

Type III are observed in the current study as shown in Figs. 6.2 and 6.3, respectively. The

OIM images in Fig. 6.2(a) and Fig. 6.3(a) show two adjacent grains and GB in

between in different colors according to their orientations (i.e., Euler angles: Bunge

notation, [, , ]) for Type II and Type III special GBs, respectively. Orientations of

and misorientations between grains for these two cases are summarized in Tables 6.2

and III, respectively. The corresponding superimposed pole figures of the GB and two

adjacent grains for Fig. 6.2(a) are shown in Figs. 6.2(b) and 6.2(c) for {110}/{0001}

and {111} /{ } , respectively. From the pole-figures, it can be seen that the GB

243

appears to have its basal (0001) pole coincident with the nearly common (110) pole of

the two grains, i.e. and

, as indicated by an arrow in Fig. 6.2(b), while

its { } poles are parallel to different { } poles in the two grains,

i.e. and

with an angular spread of 61.8˚ in between

the two given { } poles. This suggests that the ORs of GB could be different with

respect to different grains (i.e. , [ ]

[ ] and

, [ ]

[ ] ) and also shows how GB maintains BOR with

both adjacent grains with such a misorientation. The same is true for the case of Type

III, whose pole figures are shown in Figs 6.3(b) and 6.3(c). Predicted orientations of GB

(GBM

) on these two types of special grain boundaries are also presented in Tables 6.2

and 6.3, respectively. The disorientation angles associated with the misorientation matrix

are only 2.01˚ and 1.53˚ in Type II and Type III, respectively. Such variant

selection of GB at special grain boundaries will result in the development of large

colony structures from the GB into two adjacent grains with identical orientation as

that of the GB.

244

6.3.2. Violation of variant selection rule derived from closeness between

poles

In the current work, we show two examples in Fig. 6.4 that demonstrates the violation of

the variant selection rule mentioned earlier, i.e., when there exist two nearly parallel

{ } poles between the two adjacent grains (with angular deviation within 10°), the

normal of plane of GB will be parallel to the common { } pole. The OIM

images for these two examples are shown in Figs. 6.4(a) and 4(b), respectively. As shown

in Fig. 6.4(c), the GB has its pole to be parallel to that is neither one

of the two nearly common { } poles, i.e., or

, though the angular

deviation between them is only 8.96° as indicated by the arrow. The results in Fig. 6.4(d),

in contrast, show that the pole is still parallel to one of two closet { } poles,

though the angular deviation between them is 12.40° (larger than 10°) as indicated by the

arrow. The relationships among the misorientation angle between two nearly common

{ } pole of two adjacent grains (with angular deviation up to 18°), variant of GB

selected, and deviation of the OR between the GB and the non-Burgers grain from the

Burgers orientation relationship described by are summarized in Table 6.4.

All the Euler angle sets for the two adjacent grains and the GB are presented in

Supplementary Materials Table 6.2.

245

It is clear that when the rule is followed, the misorientation angle, , associated with

is always larger than that between the two closet { } poles since it also

takes into account the deviation of other two poles, i.e. ⟨ ⟩ and ⟨ ⟩ , from the two

grains. Moreover, it is found that { } of the GB would be parallel to one of the

two closet { } poles only when of associated with such a GB is

less than 15°, as shown via the selection of GB1-6 and GB12 in Table 6.4. In contrast,

for those cases where the rule is violated, of associated with the selected

GBis always larger than 15°, no matter how close the two { } poles are, as

confirmed by the variant selection of GBandGB8. In fact, a minimum of

for all 12 possible variants in these cases is always larger than 15°. While

for those cases where the rule is followed, the minimum of is always

less than 15°, and the variant with the smallest is also the one selected for

GBCompared with the closeness of poles between two adjacent grains, the

parameter , that describes the deviation of the OR between GB and the non-

Burgers grain from the BOR, should be a more general criterion to serve as GBvariant

selection rule.

6.4. Discussion

The variant selected among all 12 possible variants by a prior GB during nucleation

should arrange itself to have the minimum interfacial energy and elastic strain energy

246

with the two contacting grains. Here, we refer to the interfacial energy between GB

and the non-Burgers grain and that between GB and the Burgers grain, as and

, respectively. In general, the nature of an interface between two phases with

different Bravais lattices depends on the composition, crystal structure and lattice

parameters of each phase, OR between the two crystals, and interface plane orientation

(inclination). For all 12 variants of GB, though the first three factors are identical for

each variant, there is still a preferred set of variants selected as observed in the current

study. It is thus concluded that it is the differences of the crystallographic orientation of

GB relative to the two adjacent grains and their interface inclinations among all 12

possible variants that have more significant influence on specific variant selection.

The minimum interfacial energy occurs, as has been demonstrated by Shiflet and Van der

Merwe [20], Nie [21], and Zhang et al [22], when rows of close-packed atoms in the two

phases match at the interface (habit plane), which may most likely provide the minimum

elastic strain energy as well [23]. The frequently observed BOR between GB

precipitates and Burgers grains results from the atom row matching between

⟨ ⟩ ⟨ ⟩ at the interface [22]. Analogous to the CSL introduced in the study of

grain boundaries between two grains with a special misorientation, when BCC and HCP

lattices penetrate into each other under BOR, there will be a reasonable fit among atomic

sites from the two crystals at the interface and thus the interfacial energy as well as the

elastic strain energy will be reduced. Following the assumption that BOR offers a

relatively low interfacial energy of as well as a low elastic strain energy, any

247

deviation from it, measured by associated with , will most likely result in

a rise in these energy terms up to certain critical value of . This is akin to the

relationship between grain boundary energy and misorientation angle. The critical angle

of seems to be about 15˚ for the alloy Ti-5553 considered in this study. The variant

with minimum of would have the lowest interfacial as well as the

elastic strain, energy and, thus, the lowest nucleation barrier among all 12 variants. It is

reasonable to assume that it would be selected for GB variant by the prior GB when

. This may explain why the variant with the smallest is always

selected as GB, such as GB1-6 and GB12 shown in Table 6.2 and, of course, the

special cases where GB maintains BOR with both grains, e.g., GB1 and 5.

Therefore, it suggests that when , the misorientation between two grains plays

a dominant role over grain boundary plane (GBP) inclination in determining GBvariant

selected.

For example, it can be seen from the OIM image shown in Fig. 6.5(a) that the two GB

precipitates with the same color (i.e., same orientation) appear at two different locations

with different GBP inclinations of the GB between 1 and 2. Superimposed pole figures

among poles of two adjacent grains and pole of GBfor GB6 is

shown in Fig. 6.5(c), while superimposed pole figures among poles of two

adjacent grains and pole of GBfor the same GB precipitate are shown in

Fig. 6.5(d). It is readily seen that is parallel to one of the two closest as

248

indicated by the arrow in Fig. 6.5(c). Moreover, GB6 maintains BOR with the 2 grain.

Disorientation angles associated with for all 12 variants are provided in

Fig. 6.5(b) when 1 or 2 servers as the Burgers grain. In the former,

measures the deviation of GB from the BOR with the non-Burgers grain 2, while in

the latter measures the deviation of GB from the BOR with the non-

Burgers grain 1. It is readily seen that variant V1 (See Supplementary Table 6.1 for

details) that has the minimum (less than 15°) has been selected for GB and

maintained BOR with Though the selection of GB near grain triple junction (upper

left of Fig. 6.5(a)) may be related to a third grain, a variation of GBP inclination (as

indicated by an arrow) does not changes the results of variant selection of GB.

When , however, the GBP inclination may play a dominant role over the

misorientation between two grains in determining GBvariant selected. As also shown

in Table 6.4, selection of a variant with its being parallel to a common

pole would not necessarily result in a with its . It should also be

noted that when the smallest associated with is greater than , such a

variant selection rule will never be valid. Still making the assumption that the interfacial

energy between GB and the non-Burgers grain becomes approximately independent of

misorientation when [24], then if of for all 12

variants are larger than the critical value of 15 , the difference in may not result in

significant differences in anymore (here the contribution of misorientation axis to

249

is also ignored since the system temperature is close to transus and hence the

degree of interfacial energy anisotropy is small and could be neglected as well). In such

cases, the other factor in quantifying nucleation barrier, i.e., inclination of interface

for a given variant with respect to the GBP, needs to be considered to determine variant

selection rules. In other words, the inclination of GBP would play a more important role

than that by misorientation between two adjacent grains when .

Variant selections of GB9 and GB10 observed in the experiments also support the

above analysis, as shown in Figure 6.6. It can be seen from the OIM image in Fig. 6.6(a)

that two GBs precipitate with different colors (i.e., different orientations) form at the

GB between 1 and 2, located at two different places having different GBP inclinations.

Superimposed pole figures among poles of the two grains and pole of

the GBfor GB9 and GB10 are shown in Figs. 6.6(c) and 6.6(e), respectively, while

superimposed pole figures among poles of the two grains and pole of

the GB are shown in Figs. 6(d) and 6(f), respectively. It is clear that is parallel

to neither one of the two closest as indicated by arrows in Fig. 6.6(c) and Fig.

6.6(e), respectively. Moreover, GB9 maintains the BOR with 1 grain, while GB10

keeps the BOR with 2 grain. Disorientation angles associate with for

all 12 variants are provided in Fig. 6.6(b) when either 2 or 1 serves as Burgers grain.

Obviously all 24 values are larger than and, more importantly, neither GB9 nor

GB10 listed in Table 6.4 is the one with minimum . All the above facts suggest that

250

inclination of GBP determines which variant will be selected and which grain will be

the Burgers one, though we are not clear about how without considering the information

of GBP inclination.

An unique interface inclination at a fixed OR that contains matching atom rows is

believed to allow a minimum energy state to be realized [22, 25]. The / interface

between precipitate and matrix has been characterized to have a broad facet, a side

facet and an edge facet [5, 11, 26]. The broad face is made of structural ledges with their

terrace plane parallel to and the habit plane of side facet is near

. Both of these facets probably are low energy interface portions of an

interface. In the Burgers grain , for different variants, the nucleation barrier would

vary with the inclination of low energy facets with respect to the grain boundary plane.

Lee and Aaronson have shown that a grain boundary precipitate should arrange its low

energy facet to be parallel as much as possible to the grain boundary plane in order to

minimize the nucleation barrier [27, 28]. Based on this argument, Furuhara et al. [5]

concluded that variant selection of GB is made in such a manner that the variant of

GB has its ⟨ ⟩ ⟨ ⟩ direction nearly parallel to the grain boundary plane by

arranging these low energy (broad and side) facets to eliminate as much as possible grain

boundary area.

It should also be mentioned that the broad face of / interface consists of structural

ledges (steps) [5, 11, 26] with their terrace plane parallel to . Under

251

such a microscopic configuration, the macroscopic broad face is generally an irrational

plane close to { } , e.g., that is also the habit plane that minimizes the

elastic strain energy [23]. Therefore, the macroscopic habit plane should also be parallel

to the grain boundary plane in order to reduce the elastic energy contribution to the

nucleation energy barrier. The relative contributions from alignment of the low energy

facets { } or the habit plane { } with the grain boundary plane will depend

on the size of the critical nucleus, i.e., whether it exceeds the spacing of structural ledges

or not. It thus suggests that both misorientation and inclination of a grain boundary plane

play a role in the selection of GB. A comprehensive study about how all grain boundary

parameters contribute to variant selection of GB on a general grain boundary will be

presented in a separate paper.

6.5. Conclusions

A crystallographic model based on the Burgers orientation relationship between GB and

one of two grains has been developed to study how variant selection occurs on prior

grain boundary in / and titanium alloys. In particular, a new parameter,

that describes quantitatively the deviation of OR between a GB and the non-Burgers

grain from BOR, is identified and a new GB selection rule is proposed. All possible

special misorientations between two grains that make GB in the Burgers orientation

relationship (BOR) with both grains have been predicted and confirmed by experimental

observations made for Ti-5553. Such variant selection of GB at special grain

252

boundaries will result in the development of large colony structures from the GB into

two adjacent grains with identical orientation as that of the GB. Through the analysis of

the experiment observations of GBin Ti-5553 using the model, it is found that when the

disorientation angle associated with is less than 15º, the variant with the

smallest of is always selected for GB, and the selected GBwill have

its ⟨ ⟩ pole parallel to a common ⟨ ⟩ pole of the two adjacent grains. When

, grain boundary plane inclination may play more important role for GB

variant selection in Ti-5553. Theoretical arguments why the parameter, , is a

better measure than the closeness between two closest { } from two grains widely

used in literature in analyzing GB variant selection are provided. It would be

straightforward to extend the model and approach to study variant selection of grain

boundary precipitate in other alloys.

253

Figures

Figure 6.1. Illustrations of all special crystallographic orientation relationships between

GB (Red) and two adjacent grains (Blue and Green) that are able to hold the Burgers

Orientation Relationship with the GB (a) Type I - 10.52 º/<110>, (b) Type II- 49.48

º/<110>, (c) Type III- 60º/<110> and Type IV- 60º/<111>.

β2[001]

β1[001]

(a) Type I - 10.52 º/<110> (b) Type II - 49.48 º/<110> (c) Type III- 60º/<110>

β1[111] || 2110

β2[111] || 1120

β2[111] || 1210

β1[001] β1[001]

β2[001]

β2[001]

β1[111] || 2110

β1[111] || 2110

β2[111] || 1210

β1[001]

β1[111] || 2110

β2

β2

[100]

[0 10]

β2[111] || 2110

β1[110] β2

β2

[011]

[101]

(d) Type IV - 60º/<111>

60 / [ ]111

254

Figure 6.2. Experimental observations of a Type II special grain boundary where

GB maintains BOR with two adjacent grainsaOIM image of the Type II



two grains and the pole of the GB

1

2

(a)

(b) (c)

255

Figure 6.3. Experimental observations of a Type III special grain boundary where

GB maintains BOR with two adjacent grainsaOIM image of the Type III



two grains and the pole of the GB

1

2

(a)

(b) (c)

256

Figure 6.4. OIM images ((a) and (b)) and superimposed pole figures of

GB and pole figures of the two grains with different angular deviation

between two closest { } poles ((a) and (c):

; (b) and (d):

).

1

2

1

2

(a) (b)

(c) (d)

257

Figure 6.5. (a)

grain boundary with different inclinations; (b) Disorientation angles associate with

for all 12 variants; (c) Superimposed pole figures among the poles

of two adjacent grains and the pole of the GB

figures among the pole of

the GB

1

2

(a)

(c) (d)

(b)

258

Continued

Figure 6.6. (a) OIM image for two grains with GB 9 and GB on different locations

of the grain boundary with different inclinations; (b) Disorientation angles associate

with for all 12 variants; (c) Superimposed pole figures among the

poles of the two grains and the pole of GB ; (d) Superimposed pole figures

among the poles of the two grains and the pole of GB (e)


pole of GB ; (d) Superimposed pole figures among the poles of the two

grains and the pole of GB

1

2

(a) (b)

(c) (d)

259


(b) (110) pole

(f)(e)

260

Tables

Table 6.1 All special misorientations (by angle/axis pairs) between two adjacent

grains, by which GB is able to maintain BOR with both grains

Type disorientation Equivalent misorientation

I 10.52 º/<110>

II 49.48 º/<110> 63.26 º/<211>

III 60º/<110> 60.8 º/<0.568 0.392 0.392>

IV 60º/<111>

Table 6.2 Orientations of two grains shows Type II misorientation in variant selection


one (GB )

Orientation / ˚ Misorientation

˚ r1 r2 r3

2 99.4 9.6 301.7 47.9 0.3735 0.4214 0.0321 1 305.9 45.7 30.3

GB 265.8 78.7 86.9 2.01 -0.003 0.031 -0.002

GB

Table 6.3 Orientations of two grains shows Type III misorientation in variant selection


one (GB )

Orientation / ˚ Misorientation

˚ r1 r2 r3

2 119.2 42.7 227.1 59.3 -0.242 -0.241 -0.003

1 188.3 66.9 182.3

GB 303.2 46.0 82.1 1.53 0.011 0.024 0

GB

261

Table 6.4 Summary of relationships among misorientaion angle between two closest

{ } poles of two adjacent grains, variant of GB selected, and deviation of the OR

between the GBand the non-Burgers grain from the Burgers orientation relationship

described by

Example Two Closest { } Poles Selected Variant

for GB

GB1

[ ]

GB2

[ ]

GB3

[ ]

GB

[ ]

GB

[ ]

GB

[ ]

GB

[ ]

GB

[ ]

GB

[ ]

GB

[ ]

GB

[ ]

GB

[ ]

GB

[ ]

GB

[ ]

262

6.6 References:

[1] Bache MR. Processing titanium alloys for optimum fatigue performance.

International Journal of Fatigue 1999;21:S105.


aeroengine applications. Materials Science and Technology 2010;26:676.


Acta Mater 2013;61:844.



1934;1:561.


precipitates in a β titanium alloy. Metallurgical and Materials Transactions A

1996;27:1635.


2004;52:5215.



2008;56:5907.



Magazine 2007;87:3615.

263

[9] Kar S, Banerjee R, Lee E, Fraser HL. Influence of Crystallography Variant

Selection on Microstructure Evolution in Titnaium Alloys. In: Howe JM, Laughlin DE,

Dahmen U, Soffa WA, editors. Solid-to-Solid Phase Transformation in Inorganic

Materials, vol. 133-138: TMS (The Minerals, Metals & Materials Society), 2005.












Treatment. Metallurgical and Materials Transactions A 2013;44:3852.

[14] Bowles JS, Mackenzie JK. The crystallography of martensite transformations I.

Acta Metallurgica 1954;2:129.

[15] Mackenzie JK, Bowles JS. The crystallography of martensite transformations II.

Acta Metallurgica 1954;2:138.

[16] Bhadeshia HKDH. Worked examples in the geometry of crystals, 2001.

264

[17] Kim D, Suh D-W, Qin RS, Bhadeshia HKDH. Dual orientation and variant

selection during diffusional transformation of austenite to allotriomorphic ferrite. Journal

of Materials Science 2010;45:4126.

[18] Kim DW, Qin RS, Bhadeshia HKDH. Transformation texture of allotriomorphic

ferrite in steel. Materials Science and Technology 2009;25:892.

[19] Engler O, Randle V. Introduction to texture analysis: macrotexture, microtexture,

and orientation mapping: CRC PressI Llc, 2010.

[20] Shiflet GJ, Merwe JH. The role of structural ledges as misfit- compensating

defects: fcc-bcc interphase boundaries. Metallurgical and Materials Transactions A

1994;25:1895.

[21] Nie JF. Crystallography and migration mechanisms of planar interphase

boundaries. Acta Mater 2004;52:795.

[22] Zhang M-X, Kelly PM. Crystallographic features of phase transformations in

solids. Progress in Materials Science 2009;54:1101.


coherency state. Acta Mater 2012;60:4172.

[24] Read WT, Shockley W. Dislocation Models of Crystal Grain Boundaries.

Physical Review 1950;78:275.

[25] Zhang WZ, Weatherly GC. On the crystallography of precipitation. Progress in

Materials Science 2005;50:181.

265


intragrainular proeutectoid alpha precipitates in a Ti-7.26wt%Cr alloy. Acta Materialia

2004;52:2449.


at grain boundaries—II. Three-dimensions. Acta Metallurgica 1975;23:809.


at grain boundaries—I. Two-dimensions. Acta Metallurgica 1975;23:799.

266

CHAPTER 7 Effects of Grain Boundary Parameters on

Variant Selection of Grain Boundary in Titanium Alloys

Abstract

In titanium alloys, variant selection (VS) of grain boundary (GB ) by prior grain

boundaries during precipitation has a significant influence on the transformation

pathway and transformation texture of phase and thus on the final mechanical

properties. In this paper, the applicability of all current empirical VS rules with respect to

grain boundary parameters such as misorientation and inclination on VS of GB has been

assessed systematically using experimental characterizations of Ti-5553. It is found that

when the minimum misorientation angle associated with , a measure of

deviation of the orientation relationship between the GB and the non-Burgers grain

from the Burgers orientation relationship, is less than , misorientation plays a

dominant role in VS of GB , and when grain boundary plane inclination plays

a more important role. The violations of the empirical VS rules related to inclination may

be attributed to the interplay among grain boundary energy and interfacial energies

between GB and the Burgers and non-Burgers grains in determining the nucleation

267

barrier for each variant. The violation could be associated with the characteristic of grain

boundary plane orientation population.

7.1. Introduction

Titanium and its alloys (Ti-alloys) are currently finding increasingly widespread use in

many applications, ranging from structural components in aircrafts, automobiles and

ships to bio-implants [1]. The effectiveness to tailor the microstructures of a given Ti-

alloy through relatively simple thermo-mechanical processing to meet selected

engineering requirements has contributed to their current dominance. Microstructure

evolution in most Ti-alloys during heat treatments is dominated by the (BCC) to

(HCP) transformation upon cooling. For both and + processing route,

microstructure evolution initiates from the formation of allotriomorphic on the prior

grain boundaries (GBs) that is referred to as grain boundary (GB). Similar to ferric

alloys, the influence of GB on the subsequent microstructure development and

microstructure-properties relationships in Ti-alloys has been an active area of research for

decades [1-3]. In most cases, GB has the Burgers orientation relationship (BOR) [4], i.e.

; [ ] [ ] , with one of two adjacent grains. The grain that

maintains BOR with the GB is referred to as Burgers grain and the other grain is

referred to as non-Burgers one. The BOR would result in 12 crystallographically

equivalent orientation variants available for GBwith respect to Burgers grain. However,

a trend of strong variant selection (VS) of GBhas been demonstrated frequently by

268

experimental observations that, instead of 12, only limited numbers of variants seem to

be dominant on most of GBs. VS of GB has [5] a direct influence on the overall

microstructure and transformation texture evolution [6, 7], and thus has a significant

impact on the mechanical properties [5]. For instance, when two adjacent grains are

specially misorientated [8-10], colonies developed from GB into the two grains

would have common crystallographic features such as common basal plane orientations

(though with their [ ] pole being misorientated by a ~10.5˚ rotation about their

common [ ] axis) and, even more specifically [11], may have exactly the same

orientation as that of GB. Large regions of phase with a common crystallographic

feature are also referred to as “macro-zones” [3] or micro-textured regions that would

result in a significant reduction in fatigue life of Ti-component [2, 12] for a given

operation stress. Thus the formation of macro-zone across grain boundaries due to VS of

GB is undesirable in a safety critical operating environment. It is of great importance to

investigate how a given grain boundary select a variant for GBfrom the 12 possible

candidates.

VS of GB by a specific grain boundary is determined by the structure of that grain

boundary, which depends on both misorientation and (GBP) inclination, defined in a five-

dimensional space. In terms of how the five parameters of a grain boundary contribute to

VS of GB, several empirical rules have been proposed. For example, it has been

commonly observed that GB maintains an OR with the non-Burgers grain that has only

a small deviation from the BOR [9, 10, 13]. In other words, deviation of the OR between

269

the GBand the non-Burgers grain from the BOR should be as small as possible. In

special cases, GBcould have BOR with both grains [11], which should be the most

preferred configuration. It is thus referred to as Rule I in the current study that variant

selection of GB is made to maintain BOR with both adjacent grains as much as

possible. Such an arrangement in orientations among a GB and the two grains may

lead to low energy interfaces with respect to both grains and thus result in a low

nucleation barrier for the GB. A new parameter, , that is a measure of

deviation of the OR between the GB and the non-Burgers grain from the BOR, has been

proposed [14]. The disorientation angle associated with the deviation matrix is able to

quantify the deviation, akin to the dependence of grain boundary energy on

misorientation angle. For the particular alloy, Ti-5553, considered in [14], it has been

found that when the misorientation angle (instead of the closeness between two closet

{ } poles between two grains widely used in literature) is less than ,

misorientation between the two grains dominates VS of GB and, in particular, the

variant with minimum is always selected for GB. For each variant,

depends on misorientation and BOR matrix associated with the variant. Thus, Rule I

accounts only for the effect of misorientation on VS of GB.

As for the influence of GBP inclination, two additional rules have been proposed so far.

It has been accepted in general that there exists a pronounced tendency for the low energy

facet or habit plane developed between GB and the matrix grains to be parallel to GBP.

270

As have been demonstrated by Lee and Aaronson [15, 16], the inclination angle, ,

between GBP and the low energy facet of an precipitate has a significant effect on the

nucleation barrier, . For two ratios between the grain boundary energy and the

interfacial energy (assumed to be the same for the two interfaces), 1.07 and 1.57,

the minimum of always occurs when for different ratios between the

interfacial energy of the low energy facet and , no matter whether the GBP is

planar or puckered. In other words, in order to reduce nucleation barrier, the low energy

facet needs to be parallel to the GBP as much as possible as such to maximize the area of

grain boundary eliminated by such GB nucleation. According to the interface

structure when BOR is maintained in between, the low energy facets have been

characterized to be { } { } (terrace plane orientation) and { } { }

(side facet orientation). Thus, we refer the above criterion to as Rule II, i.e., the major low

energy facet { } { } of the selected variant should have the minimum deviation

from GBP.

Nevertheless, in a crystallographic study of GB formed in a titanium alloy (Ti-15V-

3Cr-3Sn-3Al in wt.%), Furuhara et al [13] concluded that selection of a variant for GB

is made in a different manner that the matching direction (i.e. ⟨ ⟩ ⟨ ⟩ ) of the

selected variant makes the smallest deviation angle from the GBP, i.e., ⟨ ⟩

⟨ ⟩ tends to be parallel to the GBP. It is referred to as Rule III in the present work.

As has been argued by Furuhara et. al. [13], Rule III results from the requirement that the

271

two low energy facets, i.e., { } { } and { } { } developed into the

Burgers grain, make the smallest inclination angle with respect to the GBP to minimize

. Accordingly, the critical nucleus formed at a given grain boundary tends to

elongate along the intersection of these two facets, i.e., ⟨ ⟩ ⟨ ⟩ zone axis of

these two facets. According to reference [13], Rule III is a modified version of Rule II for

lath- or needle-shaped precipitates [17], in which the most effective way to eliminate

grain boundary area by grain boundary nucleation is making the growth direction of a

lath nuclei parallel to the GBP. Note that since [ ] is the zone axis of and

planes, even if [ ] is included in the grain boundary, there are still many

ways to arrange two planes with respect to the grain boundary. Thus, Rule III accounts

for how to arrange two different low energy facets developed in the Burgers grain with

respect to the GBP to reduce the nucleation barrier of GB, while keeping the zone axis

of the two facets included in the GBP.

Up to now, there is no critical assessment of the general applicability of these rules and,

hence, their predictive powers are limited. For example, for a given GB there will be

three variants that share a common ⟨ ⟩ and satisfy Rule III. Which one will be

finally selected will be determined by factors beyond what are considered in Rule III. As

has been suggested by Furuhara et al, Rule I and Rule III result in the selection of a single

variant of GB on a planar grain boundary [13]. Moreover, Rule III could be

predominant over Rule I when the minimum associated with is more

than since Rule III is more frequently (though not always) followed than Rule I in

272

variant selection of GB, while Rule I is more important when the minimum is less

than because in this case the variant with minimum is always selected as the

GB. It should be noted that Rule I is not able to determine the Burgers grain. In fact,

there are always two variants having the same minimum , e.g., when assuming

different adjacent grain to be the Burgers one during the prediction, corresponding to

with respect to the Burgers grain and

with respect to the

Burgers grain , respectively. Nevertheless, when the minimum , it is found

that GBP inclination is able to determine the Burgers grain, though the manner is still not

clear without considering the information of GBP inclination. In terms of predicting

capability, Rule II would predict a single variant to be selected by the Burgers grain. For

Rule III, there exist three variants with common ⟨ ⟩ meeting the requirement. For

Rule I, it would predict, at least, two variants with identical minimum of

but having BOR with respect to different grains. In other words, Rule I is not able to

predict the Burgers grains. In particular, as will be shown in the current study, there are

some cases where variant selection behavior follow none of three rules. There is no doubt

that all grain boundary parameters play their own roles in VS of GB, but how they

contribute to VS of GB on a given grain boundary has not yet been completely

understood. Therefore, further validation and consideration of these three criteria,

especially the physical mechanisms behind them, requires further investigation.

Apparently, the work first requires a complete description of grain boundary characters.

273

In order to make a precise prediction of VS of GB for a given set of grain boundary

parameters, the applicability and limitation of each rule needs to be clarified. In

particular, as will be shown in the current study, all VS rules could expire on a given

grain boundary. It is thus crucial to investigate why a rule is violated and also whether the

physic mechanism lying behind each rule is reasonable or not. Therefore, the objective of

the current study is to elucidate that, on a prior grain boundary, how to use all current

VS rules to make a right prediction of grain boundary nucleation, validities and

limitations of all rules, the reasons why a single or multiple rules are violated.

The chapter is organized as the follows. In Section 7.2, we first design an experiment that

highlights the effect of grain boundary parameters on the grain boundary nucleation.

Individual orientations of two adjacent grains and GB are measured using electron

backscattered diffraction (EBSD) technique, and GBP orientation is determined using a

three-dimensional two-surface trace approach. In Section 7.3, variant selection of GB at

different prior GBs are analyzed according to three aforementioned variant selection

rules. The analysis is conducted by comparing the measured orientation of GB and the

predict one by a crystallographic model developed by the present authors to investigate

how three rules are followed or violated. The effect of grain boundary parameters on

variant selection of GB and possible physical insights of different rules are then

discussed in Section 7.4. Finally, major findings are summarized in Section 7.5.

274

7.2. Experimental procedure

In current study, the material used is a forged -Ti alloy, Ti-5553, i.e., Ti-5Al-5Mo-5V-

3Cr-0.5Fe (wt. %). A -titanium alloy is selected in order to avoid the martensitic

transformation and thus retain phase during a thermal quench. The as-received alloy

was sectioned to small samples (~20mm×20mm×40mm) for heat treatment. The samples

were initially -annealed at 1000˚C for 15 minutes using a conventional tube furnace in

an inert Argon atmosphere. The samples were wrapped in titanium foil to further reduce

the possibility of oxygen ingress. The samples were then cooled in the furnace to 825˚C

at a controlled rate of 5˚C/min, and were soaked for 2 hours to allow for the phase

transformation to complete. Finally, the samples were water-quenched to room

temperature. Since the -transus for this alloy is close to 850˚C, such a heat treatment

procedure would allow for only limited one or two variants of GBform on a given

planar grain boundary if the precipitation would occur while colony structure has not

develop from GB and thus allow us to avoid the influence of relative large under

cooling on the variant selection behavior, and to determine the orientations of two grain

and GB . In other word, under such a small undercooling, grain boundary characters

would play a dominant role in the determination of variant selection of GB. For

subsequent characterization, all specimens are further sectioned in the middle and the

exposed surfaces were subjected to mechanical polishing using standard metallographic

techniques. In the final step, material is kept in a vibratory polisher in the suspension of

0.05 m silica particles for a number of hours to achieve a mirror-finish. The

275

crystallographic orientation of GB and the parent grains are measured at various

locations using EBSD data collection in Philips XL30 ESEM FEG SEM at 20kV, with a

spot size of 4 and a working distance of 20 mm. Suitable step-size (1.7m) in OIM-TSL

software is selected to allow for the collection of data from a large area (~1 mm × 1 mm)

in a reasonable amount of time. The reliability of this data collection was verified on a

⟨ ⟩ silicon sample under similar conditions.

The local orientations of GBP are determined by producing a site specific section into the

sample to expose trace of a GB on two mutually perpendicular surfaces, i.e. sample

surface and a trenched one. A combination of secondary electron imaging and focused-

ion beam (FIB) in FEI NOVA Dual BeamTM

(SEM/FIB) microscope is used for this

purpose. In other words, trenched sections are produced nearly perpendicular to the grain

boundary traces present on the sample surface using FIB. The crystallographic orientation

of the grain boundary relative to the adjacent grain has been determined by combining

their geometry with the crystallographic information provided by the EBSD data. The

accuracy of the method has been validated and analyzed using the knowledge of

crystallographic characteristics of twins present in both cubic (IN-100 Ni-based

superalloy) and hexagonal systems (commercially pure (CP)-titanium). A certain degree

of reorientation of grain boundary planes occurs as a result of the GBprecipitation. In

order to take this change into account, FIB sections have been produced nearly normal to

the site specific projection of trace of GB. Details on the experimental techniques and

276

measurements to determine local orientation of GBP can be found in referred to Ref.

[17].

7.3. Results

7.3.1. Overall Characteristics of variant selection of GB

In the current study, thirty five examples of GB at different prior grain boundaries are

observed and analyzed. For all grain boundaries, there is only one or two variants are

selected, as shown in Fig. 7.1. Details about all 5 grain boundary parameters of different

grain boundaries corresponding to selection of all GB in the current study are presented

in Table 7.1. In particular, grain boundary plane orientations are expressed in both crystal

reference frame of Burgers grain, i.e. [ ] ; [ ] ; and [ ] , and

Burgers reference frame associated with the selected variant in the Burgers grain, i.e.

[ ] [ ] ; [ ] [ ] ; and [ ] [ ] .

Results about 35 orientations of different GBs (i.e. 70 GB surfaces) are displayed in Fig.

7.2 in the form of solid circles in the standard stereographic triangle for the BCC

structure. The locations of circles indicate the orientation of surface normal of GB planes

with respect to the crystal reference frame in Burgers grain. It can be found that grain

boundary planes with the orientation have a relative large population areas, while

there is only one example where grain boundary plane orientation is close to .

277

Figure 7.3 shows orientations of all grain boundary planes displayed in a pole figure with

respect to the Burgers reference frame. Such a plot is really convenient to illustrated the

relationship among GBP orientation, ⟨ ⟩ axis and { } plane of the selected

variant, which are key parameters described by different empirical VS rules. For

example, the frequency of occurrence of variant selection as a function of the inclination

angle between GBP and { } planes ( ), and between GBP and ⟨ ⟩ direction

( ) are presented in Fig. 3(b) and Fig. 3(c), respectively. Inclinations angles within

[ ] are divided into six groups with interval of 15° for each. It is readily seen that

the distribution is quite randomly scattered. In particular, variant selection occurs most

frequently when falls in the group of [ ] , or when

is located within

[30-45]°. It should also be noted that there is only one case where is within [

] . In contrast, there are 6 examples where is within [ ] . On the basis of

this statistic, it seems that Rule III plays a more important role in the variant selection of

GB than Rule II. However, even more frequently, VS of GB does not follow either

Rule II or Rule III.

It has been suggested that when the minimum is less than 15°, misorientation between

two grain would be dominant over GBP inclination in the VS of GB, Thus, the same

analysis as that in Fig. 7.3, is furthered refined into two groups, one is for , and

the other one is for . The results are shown in Fig. 4 and Fig. 5, respectively. It

is readily seen that the similar trend as that in Fig. 7.3 is still followed. In the current

study on VS of GB in Ti5553 alloy, it seems that neither Rule II nor Rule III is

278

frequently followed. In contrast, as can be found in Figs. 7.4-7.5, there are some

examples that violate both Rule II and Rule III no matter is larger or less than 15°. For

example, there are 3 examples that is within [ ] , and 4 examples that

is within [ ] .

7.3.2. Variant selection of GB when different rules are dominant

It is worth mentioning that the above analysis only considers the relationship

among ⟨ ⟩ , { } of selected variant, and GBP. However, according to

aforementioned empirical rules, it should be the differences in , and

among all 24 possible variants that determine the VS of GB. Therefore, in this section,

several examples are analyzed individually to evaluate how these different parameters

contribute to VS of GB. Examples are divided into 4 groups, a) Rule I is dominant; b)

Rule II is dominant; c) Rule III is dominant; d) Abnormal cases where none of three rules

is followed. For a GB in a given grain boundary, the measured orientations of two

adjacent grains are also used as model inputs to predict all 24 possible orientations of

GBby assuming one of two adjacent grains to be the Burgers one alternatively. For

each variant, associated with , inclination angles between [ ] ,

, , and GBP, i.e. ,

, and , are all evaluated. By comparing

orientations of GB between the measured and predicted values, we would then have a

picture about how a variant follows or violates these three rules to make it selected for

279

GB, which would help to clarify how grain boundary parameters contribute to select

one or two variants of GBfrom 24 candidates.

7.3.2.1. Rule I is dominant

The OIM image in Fig. 7.6(a) shows two adjacent grains and a GB (Example 16 in

Table 7.1, or GB16) in between in different colors according to their orientations (i.e.,

Euler angles: Bunge notation, [, , ]). The corresponding superimposed pole figures

of the GB and two adjacent grains (with respect to sample reference frame) for Fig.

6(a) are shown in Figs.7.6(b)-7.6(d) for { } /{ } , { } / { } and

{ } /{ } ,respectively. In particular, orientation of grain boundary plane is also

superimposed in Fig. 7.6(c) and (d). In Fig. 7.6(b), trace of grain boundary plane is

superimposed. From the superimposed pole figures, it seems that GB exist BOR with

both grains, i.e. , [ ]

[ ] (V5) and ,

[ ] [ ] (V12). When comparing the predicted orientations for GB and the

measured ones, it is found that the variant V12 that has been selected for GB and

maintains BOR with 1, as shown in Table 7.2. Disorientation angles associated with

for all 12 variants are provided in Fig. 7.6(e) when 1 or 2 servers as the

Burgers grain, respectively. In the former, measures the deviation of GB

from the BOR with respect to the non-Burgers grain 2, while in the latter

280

measures the deviation of GB from the BOR with respect to the non-Burgers grain 1.

It is readily seen that the variant V5 (2 as Burgers grain) and the variant V12 (1 as

Burgers grain) have the same minimum (less than 15°). By comparing for V5

and V12 as shown in Fig. 7.6(f), it is found that V12 has a smaller value and thus meets

the requirement of Rule II. Note that neither V5 nor V12 has the minimum . The

variant V8 has the minium among 24 possible variants. however, it is not selected.

All the above facts seem to suggest that in this case, Rule I is dominant over Rule II while

Rule II may contribute to the determination of Burgers grain. Details about the effect of

all grain boundary parameters in the variant selection of GB 16 are referred to Table 7.3.

7.3.2.2. Rule II is dominant

The OIM image in Fig. 7.7(a) shows two adjacent grains and a GBGB28 in Table

7.1) in between. The corresponding superimposed pole figures of the GB and two

adjacent grains (with respect to sample reference frame) for Fig. 7.7(a) are shown in

Figs. 7.7(b)-(d) for { } /{ } , { } / { } and { } /{ } ,

respectively. For pole figure in Fig. 7.7(b), there are three ⟨ ⟩ poles , i.e. [ ]

and [ ] , that are very close to trace of GBP. However, only [ ]

pole is selected.

Details about , ,

, and

are referred to Table 7.4. Variant V9 has

been selected as GB and maintains BOR with 2, i.e., ,

281

[ ] [ ] , which is also confirmed by the comparison between predicted and

measured orientations for GB as presented in Table 7.5. As shown in Fig. 7.7(e), all 24

values are larger than 15°, the selected variant (V9) is not the one having the

minimum . However, the selected variant has its plane closet to GBP among

all 24 candidates, as indicate by an arrow in the Fig. 7.7(e). The analysis suggests that

Rule II is decisive in terms of selection of Burgers grain and variant for GB in the

example.

7.3.2.3. Rule III is dominant

The OIM image in Fig. 7.8(a) shows two adjacent grains and two GB s in different

locations with different GBP orientations in between. precipitate in the upper right is

GB7, and the one in the bottom left is GB8. The corresponding superimposed pole

figures of the GB and two adjacent grains for Fig. 7.8(a) are shown in Figs. 7.8(b)-

8(d) for { } /{ } , { } / { } and { } /{ } , respectively.

Superimposed pole figures for GBare shown in Figs. 7.8(e)-8(g).From the pole

figures and comparisons between predicted and measured orientations for GBTable

7.6), variant V7 is selected and holds BOR with 1, i.e. ,

[ ] [ ] ; for GB8 Table 7.7),, variant V8 is selected and maintains BOR

with 2, , [ ]

[ ] . Similar to the example in Sec 3.2.2, all

282

24 values are larger than 15°, neither one of two selected variants is the one with the

minimum . It suggests that, in this case, GBP orientation also determines VS of GB.

However, Rule II does not work. Note that neither V7 (for GB 7 with as Burgers

grain) nor V8 (for GB 8 with 2 as Burgers grain) has the minimum among all

24 candidates. But both two selected variants have their corresponding close to 90˚.

Details about the effect of all grain boundary parameter in the variant selection of GB7

and GB8 are referred to Table 7.8 and Table 7.9, respectively. It seems that Rule III

plays a leading role in the determination of VS of GB 7 and GB 8than Rule II. It

should be mentioned that Rule III would offer three candidate variants sharing a

common⟨ ⟩ . But how the Rule III results in the selection of a single variant on a

given planar GB with fixed GBP orientation is still not clear.

7.3.3. Abnormal cases

7.3.3.1 Abnormal variant selection when the minimum

An example is presented in this section to show how both Rule II and Rule III expires in

the determination of Burgers grain when the minimum associated with is

less or equal to 15°. For example, it can be seen from the OIM image shown in Fig.

7.9(a) that the two GB precipitates with the same color (i.e., same orientation) appear at

two different locations with different GBP inclinations of the GB between 1 and 2.

283

The corresponding superimposed pole figures of the GB and two adjacent grains

(with respect to sample reference frame) for Fig. 7.9(a) are shown in Figs. 7.9(b)-7.9(d)

for { } /{ } , { } / { } and { } /{ } , respectively. From the pole

figures and comparisons between predicted and measured orientations (Table 7.10), GB

selects variant V1 and maintains BOR with 2, i.e. ,

[ ] [ ] (V1). The variation in GBP orientation does not change the result of

VS that thus suggests that misorientation between two grains determines the variant

selection. The argument is supported by the fact that Variant (V5 with 1 as Burgers

grain) that has the minimum has not been selected for GB. Moreover, none of

variants having[ ] , [ ]

and [ ] that are closet to the GBP trace (

) has been selected. Variant V1 is one of the two candidates that have the

minimum associated with and, in particular, minimum . When

compared with variant V3 that maintains BOR with , ,

[ ] [ ] , the variant V1 has a relative lager

and smaller . Details

about the effect of all grain boundary parameter in the variant selection of GB 32 are

referred to Table 7.11. The above analysis confirms that when misorientation is dominant

in VS, Rule II and Rule III are not always able to further determine the Burgers grain.

284

7.3.3.2 Abnormal variant selection when

The example presented in this section is to show how both Rule II and Rule III are

violated in the determination of Burgers grain even when the minimum associated

with is larger than 15°. The OIM image for such an example is shown in

Fig. 7.10(a). The corresponding superimposed pole figures of the GB and two adjacent

grains for Fig. 7.10(a) are shown in Figs. 7.10(b)-10(d) for { } /{ } , { } /

{ } and { } /{ } , respectively. From the pole figures and comparisons

between predicted and measured orientations (Table 7.12), GB selects variant V7 and

maintains BOR with 2, i.e. , [ ]

[ ] . Details about the

effect of all grain boundary parameters in the variant selection of GB 26 are referred to

Table 7.13. Again, all 24 values are larger than 15°, the selected variant is not the one

with the minimum . Nevertheless, the selected variant does have neither the largest

nor the smallest

. In the case, all empirical VS rules, would mislead

predictions towards t VS of GB from all 24 candidacies.

7.4. Discussions

Existing studies suggest that heterogeneous nucleation of at prior grain boundaries

occurs as the following: for a given undercooling, an embryo/nucleus of phase

maintaining BOR with one of the grains (i.e., the Burgers grain ( )) nucleates on the

285

grain boundary between and . Low energy facets such as { } and { }

develop into the Burgers grain to minimize the interface energy between the GB and the

Burgers grain. A unique interface inclination at a fixed OR has been shown to have

the minimum energy state [19], i.e., singular or vicinal [20] interface in the 5-dimensional

space (misorientation and interface inclination between phases). On the non-Burgers

grain side, the nucleus adopts the shape of a spherical cap to minimize the interfacial

energy. Thus the nucleation barrier depends on the chemical driving force for nucleation

(determined by undercooling), grain boundary energy , interfacial energies between

the GB and the non-Burgers and Burgers (non-facet portion) grains, and

,

respectively, and interfacial energies of the low energy facets of the GB in the Burgers

grain,

and

. Among all these parameters, is variant sensitive since each

variant has different value of that links directly to the interfacial energy.

If nucleation occurs in the bulk of the Burgers grain, ,

and

and thus the

activation energy for nucleation will be identical for all 12 variants. However, for grain

boundary nucleation, low energy facets of different variants will have different

inclination angles, and

, with respect to the grain boundary plane (GBP). As a

result, the 2 interface will consist of different areas of a non-facet portion having

energy and facets portions having energy

and

[15, 16]. Thus, the

activation energy for each variant will depend on and

as well. Thus we have,

(

)

286

where is the activation energy for GB nucleation. This is the origin that leads to the

occurrence of VS of GBon prior GBs. In order to make a precise prediction of VS of

GB, therefore, a rule in general needs to take all these parameters into account and,

more importantly, demonstrate quantitatively how these parameters contribute to .

Nevertheless, none of the current VS rules discussed earlier has taken all these

parameters into consideration.

Through the analysis presented in the Results sections, the empirical rules of VS of

GBare valid only in limited cases. For example, Rule I considers only the differences in

misoriention for all possible variants and thus accounts only for .

Following the assumption that the BOR offers a relatively low interfacial energy of

as well as a low coherency elastic strain energy, any deviation from it, measured by

associated with , will most likely result in a rise in these energy terms up to

certain critical value of . This is akin to the Read-Shockley relationship between

grain boundary energy and misorientation angle [21]. The critical angle of seems to

be about 15˚ for the alloy Ti-5553 considered in this study. When , the variant

with minimum of seems to have the lowest interfacial energy, , as

well as the minimum elastic strain energy and, thus, the lowest among all the 12

variants. In this case, Rule I would be valid. In particular, there are four special grain

boundaries in terms of misoriention [14] at which certain GB variant is able to maintain

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BOR with both grains, i.e., . As has been shown through VS of GB and

GB in Fig. 7.6 and Fig. 7.9, Rule I always shortlists the candidate variants from 24 to

2 that have identical minimum associated with . However, Rule I is not

capable of further discriminating against these two variants and thus cannot predict the

Burgers grain. The limitation of the predicting power of Rule I could be ascribed to the

fact that and the coherency elastic strain energy are only two among the many

parameters that determine .

Because of the relatively small undercooling considered in the current study, only one

variant of GB is able to nucleate at most of the GBs. Therefore, the other grain

boundary parameters such as the GBP inclination must be considered to further refine the

predictions offered by Rule I. One candidate variant of GB could maintain BOR with

and develops low-energy facets into , while the other one could maintain BOR with

and develop low-energy facets into . The low-energy facets developed on each side

may have different inclinations with respect to the same GBP, which results in different

that will then further shortlist the two candidates into a finally selected one.

However, the manner of the arrangement of the two low-energy facets that would lead to

a relatively low is still not clear. For example, the VS of GB shows that the one

with a smaller is selected while the VS of GBshows the opposite.

It is noticed from all the tables about the comparisons between experimental measured

and predicted orientations that two candidate variants with the two smallest always

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share a common basal plane for a given Burgers grain. Thus, these two variants would be

able to decorate a grain boundary simultaneously under a relatively large undercooling. It

seems that the experimental observations reported in an orientation microscopy study on

the precipitation of GB in a laser deposited, compositionally graded Ti-8Al-xV by

Banerjee et al. [22] support this statement. It has been observed [22] that GB of such

two variants decorates the GB in a nearly alternating manner.

When , however, the GBP inclination may play a more dominant role over the

misorientation between two grains in determining variant selected for GB. For

example, it has been observed frequently that when , the variant with the

minimum was not selected for GB . Still making the assumption that the interfacial

energy between the GB and the non-Burgers grain becomes approximately independent

of misorientation when , then if of for all the 12

variants are larger than 15 , the difference in may not result in significant differences

in anymore (here the contribution of misorientation axis to

is also ignored

since the system temperature is close to transus and hence the degree of interfacial

energy anisotropy is small and could be neglected as well). In such cases, other factors in

quantifying the activation energy for nucleation such as inclination of interface for a

given variant with respect to the GBP needs to be considered.

Both Rule II and Rule III focus on the influence of GBP inclination with respect to the

low energy facets, i.e., or

, on . Rule II considers the influence of

289

inclination of a single low-energy facet, i.e., { } in the current study with respect to

the GBP. Rule III, on the other hand, addresses the influence of inclination of a closed-

packed direction with respect to the GBP. As stated by Furuhara et al.[13], Rule III is

essentially a modified version of Rule II. As a matter of fact, as the zone axis of { }

and { } facets, the orientation of ⟨ ⟩ relative to the GBP determines the relative

orientations of two low-energy facets with respect to the GBP. Thus, Rule III actually

addresses the problem about how to arrange two low-energy facets relative to the GBP to

reduce the activation energy of nucleation. It states that the two facets should be arranged

in such a manner that ensures ⟨ ⟩ to be included in the GBP, i.e., should be

close to 90°. It should be noticed that even if ⟨ ⟩ is included in the GBP (

), there are still numerous ways to arrange the two low energy facets.

As has been shown by GB28, the selected variant has the minimum among all 24

possible variants. Among all 35 examples of GB , Rule II only works in this case. More

frequently, Rule II and Rule III are violated according to the statistics presented in Figs.

7.2-7.4. The underlying mechanism for both Rule II and Rule III is that the fraction of

grain boundary area eliminated by a GB nucleus will be maximized by selecting a

variant with minimum . Both rules are derived based on Lee and Aaronson’s study

on the influence of faceting on the equilibrium shape of nuclei and activation energy of

nucleation at grain boundaries in both two- (2D) [16] and three-dimension (3D) [15]. The

facet is present only in one grain, i.e., the { } facets develops only in the Burgers

grain. In 2D the equilibrium shapes of critical nuclei at different inclination angles

290

between the low-energy facet and GBP, i.e., , are derived graphically through a new

generalization of the Wulff construction [16]. The dependence of on at

various ratios of

⁄ is then calculated based on the derived equilibrium shapes. It

is found that is significantly smaller at small values of under most

circumstances. Based on the exact equilibrium shapes of critical nuclei derived in 2D,

nucleus shapes and in 3D are also studied under the same conditions. It is also found

that is significantly smaller at small values of under different ratios of

⁄ ,

and increases rapidly with at small . Thus, it is believed that nucleation at a

disordered grain boundary should occur with pronounced preference parallel to only one

of all cyrstallographically equivalent low energy facet or habit plane, i.e. , as

described by Rule II.

However, it should also be mentioned that, though increases rapidly with

under most circumstances, it could be independent of as well [15, 16]. In particular

in the 3D cases studied [15], when

⁄ , first increases sharply with

till a critical value, and then decreases over a range of

, and further increases

up to . In this case, does not increase monotonically with

anymore.

It should be noted that the applicability of both Rule II and Rule III depends on GBP

population that is influenced strongly by GB energy and texture. In a random-textured

microstructure, the grain boundary population is expected to be inversely proportional to

the grain boundary energy. It has been observed in a ferritic steel [23] and also the

291

current study (See Fig. 7.2) that, when misorientation is ignored, GBP with the { }

orientation would have the minimum energy and the largest population area. This may

explain that in most examples VS of GB does not follow Rule II since GBs with { }

orientation is relative rare to meet. Thus, it is likely that the orientation of a grain

boundary relative to all the 12 equivalent low-energy facets is such that is

appreciable for all of them, i.e., within a region where is not increasing

monotonically with or is independent of

. In the former case, Rule II

expires; in the latter case, the orientation of secondary low-energy facets, { }, may

play a role in determining VS of GB that, however, has not been considered yet.

It worth mentioning that the dependence of on is calculated [15, 16] under a

representative condition of relative interfacial energy

and

for different ratios of

⁄ . The grain boundary

energy has been found to have a strong dependence on both misorientation and GBP

inclination [24]. Therefore, it is still not yet clear whether the findings (i.e. the minimum

nucleation occurs when ) by Lee and Aaronson are valid under all combination

among different values of , and

.

In addition, one should also be aware of several assumptions made during the derivation

[15, 16] of the equilibrium shape and thus the calculation of . In both 2D and 3D

cases, shapes of critical nuclei are investigated with an additional constraint, i.e., the

grain boundary plane is constrained to remain planar. As such, the force balance at

292

junctions among two grains and the nucleus could never be achieved. In 3D cases,

the equilibrium shape of a critical nucleus derived in 2D is assumed directly for the

faceted portion of the nucleus in the Burgers side. Though the grain boundary is also

allowed to be displaced, the exact force balance at the triple junction in 3D could not be

obtained. Furthermore, and

are always assumed to be equal to make the same

chemical potential all along the surface of a nucleus since solute redistribution cannot be

considered by a purely geometrical method used in the derivation. According to the

analyses on atomic site matching between two crystals [14], should be, in general,

smaller than . The effect of non-equal diffusion potential between the facetted

portions and non-facetted portions of an interface is then also ignored.

For nucleation of GB on arbitrary prior grain boundaries, both misorientation and

GBP inclination should play their roles in determining VS of GB . It is in general

difficult to predict which factors (e.g., or

for each variant) are dominant.

Therefore, using a rule that considers only the effect of either misorientation or GBP

inclination to predict VS of GB may result in frequently wrong predictions.

293

7.5. Summary

Variant selection of grain boundary (GB) by prior grain boundaries (GBs) in Ti-

5553 under small undercooling is investigated to understand the effects of grain boundary

structure characterized by misorientation and inclination. All existing empirical variant-

selection (VS) rules about the influence of grain boundary parameters and, in particular,

how a single or a combination of different rules contributes to the VS have been analyzed

and evaluated systematically against the experimental observations. It is found that when

the minimum misorientation angle associated with , a measure of

deviation of the orientation relationship (OR) between the GB and the non-Burgers

grain from the Burgers OR, is less than , the value of plays an dominant role in

determining the interfacial energy between the GB and the non-Burgers grain as well

as the coherency elastic strain energy associated with this interface and, thus, the

activation energy of nucleation of different variants of GB. When , the grain

boundary plane (GBP) inclination may play a more important role in determining VS of

GB than the value of . However, the rules commonly accepted, e.g., the variant of

selected at a given -grain boundary is the one that has the minimum possible angle

between one of the matching directions ⟨ ⟩ ⟨ ⟩ and the grain boundary plane,

or the one that has the minimum possible angle between the one of the matching planes

{ } { } and the GBP, are found to be violated frequently by the experimental

observations. The violations of the empirical VS rules could be associated with the fact

that the activation energy of nucleation of GB is determined by a complicated interplay

294

among the five parameters related to misorientation and inclination of a GB or an

interphase interface that define the structure and energy of the GB and interfaces,

while the individual empirical rules account for only a subset of these parameters. In

order to make more accurate predictions of VS of GB a general rule needs to be further

developed that take all the parameters (grain boundary energy, interfacial energies

between GB and two grains, interfacial energies of low-energy facets, and

orientations of the low-energy facets with respect to GBP) playing their roles during the

grain boundary nucleation and, more importantly, demonstrate quantitatively how these

parameters contribute to the activation energy of the nucleation.

295

Figure

Figure 7.1 Overall characteristic of grain boundary alpha (GB ) precipitation shown by

OIM. Presence of GB only occurs at certain grain boundaries

Figure 7.2 Standard stereographic triangle projection shows the orientation of grain

boundary (GB) planes (red solid circles) relative to the crystal reference frame in Burgers

grain

(a) (b) (b)

296


Burgers reference frame of selected variant, i.e. ⟨ ⟩ -⟨ ⟩ -⟨ ⟩ ; (b) and (c) the

frequency of occurrence of variant selection as a function of the inclination angle

between GBP and ⟨ ⟩ direction and between GBP and { } planes

,

respectively

(a)

(b) (c)

297



frequency of occurrence of variant selection as a function of and

, respectively

(a)

(b) (c)

298

Figure.7.5 (a) Stereographic projection shows the orientation of GB planes relative to the


frequency of occurrence of variant selection as a function of and

, respectively

(a)

(b) (c)

299


superimposed pole figures among the [ ] poles of the two grains and the [ ]

pole of the GB (c) Superimposed pole figures among the poles of the two



associate with and (f) for all 12 variants with respect to different

Burgers grain; grain boundary plane orientation is also superimposed in (b)-(d).

(a)

(b) (c) (d)

300

Continued







Burgers grain

(a)

(b) (c) (d)

301


(e) (f)

302

Continued

Figure 7.8 Experimental observations of variant selection of GB 7 and GB 8aOIM

image; (b) Disorientation angles associate with for all 12 variants with

respect to different Burgers grain; (c) and (f) superimposed pole figures among the

[ ] poles of the two grains and the [ ] pole of the GB and GB 8(d) and

(g) Superimposed pole figures among the poles of the two grains and the

pole of the GB and GB 8(e) and (h) superimposed pole figures of the

poles of the two grains and the pole of the GB and GB 8; (i) and

(j) for all 12 variants with respect to different Burgers grain

1

2

(a) (b)

303


(c) (d) (e)

(f) (g) (h)

(i) (j)

304







Burgers grain

1

2

(a) (b) (c)

(e) (f)(d)

305

Figure 7.10 Experimental observations of variant selection of GB 26aOIM image;

(b) superimposed pole figures among the [ ] poles of the two grains and the

[ ] pole of the GB(c) Superimposed pole figures among the poles of the

two grains and the pole of the GB (d) superimposed pole figures of the

poles of the two grains and the pole of the GB (e) Disorientation

angles associate with and (f) for all 12 variants with respect to

different Burgers grain

(a) (b) (c)

(e) (f)(d)

306

Figure 7.11 A scenario for nucleation of a grain boundary on a prior grain boundary

between and . The nuclei maintain Burgers orientation with , and the low energy

facets and develop into Burgers grain . The zone axis between two

facets [ ] is assumed to included in the grain boundary

307

Table 7.1 Details of grain boundary parameters (misorientation and grain boundary plane

inclination) corresponding to different GB s. Orientation of grain boundary plane with

respect to both crystal reference frame of Burgers grain and Burgers orientation reference

frame associated with selected variant are presented.

Crystal reference frame

Burgers-OR reference frame

GB [100] [010] [001] [110] [111] [112] Misorientation

angle Misorientation

axis

1 0.24 -0.32 0.92 0.85 -0.21 -0.48 45.1° [17 -2 -11]

2 0.05 -0.26 0.97 0.44 -0.62 0.65 43.4° [13 -15 18]

3 -0.98 0.13 0.14 0.57 -0.80 0.19 44.4° [0 -16 -13]

4 -0.73 -0.16 0.66 0.90 0.17 0.40 32.8° [12 13 -20]

5 -0.63 -0.20 0.75 0.91 0.28 0.30 30.9° [18 7 6]

6 -0.36 -0.31 0.88 0.12 -0.99 0.03 48° [-3 -7 11]

7 0.71 0.10 -0.70 0.06 -0.90 0.43 25.6° [-5 -2 -14]

8 0.35 0.20 0.91 0.61 0.68 0.40 26.6° [-5 -2 -14]

9 0.17 0.83 0.54 0.69 -0.69 -0.20 27.7° [-13 -15 -19]

10 -0.34 0.93 0.15 0.82 -0.17 -0.55 42.2° [-24 7 13]

11 -0.18 -0.78 0.60 0.00 0.22 0.98 41.6° [-25 12 13]

12 0.29 -0.38 0.88 0.89 0.27 -0.36 27.9° [1-11]

13 0.39 -0.82 0.42 0.94 -0.15 0.30 28.9° [1-11]

14 -0.03 -0.98 -0.20 0.43 -0.90 -0.12 29.9° [1-11]

15 0.40 -0.92 -0.03 0.74 0.04 0.67 30.9° [1-11]

16 -0.53 -0.84 -0.16 0.69 0.69 0.22 51° [-9 8 0]

Continued

308

Table 7.1 continued

Crystal reference frame

Burgers-OR reference

frame

GB [100] [010] [001] [110] [111] [112] Misorientation

angle

Misorientation

axis

17 0.13 0.46 0.88 0.70 -0.65 0.30 35° [12 -7 -3]

18 -0.37 0.87 -0.31 0.54 0.77 -0.35 47.2° [5 -15 -13]

19 -0.65 -0.39 0.65 0.52 -0.42 0.74 51.4° [23 -12 -19]

20 -0.35 -0.93 0.10 -0.28 0.62 0.73 35.6° [25 14 6]

21 0.35 -0.63 0.69 0.97 -0.17 -0.20 39.9° [10 -6 -11]

22 -0.76 -0.38 0.53 0.09 -0.59 -0.80 57.7° [14 10 9]

23 0.51 0.74 0.44 0.13 0.54 0.83 22.3° [18 13 11]

24 -0.43 -0.73 0.53 0.98 0.16 -0.14 37.9° [-13 7 7]

25 -0.78 -0.60 -0.16 -0.20 -0.82 -0.54 34.5° [13 -2 9]

26 0.97 0.08 0.23 0.74 -0.24 0.63 37.1° [10 13 25]

27 -0.17 0.83 0.53 -0.08 -0.26 0.96 38° [-3 -8 10]

28 -0.71 0.52 0.48 0.17 0.99 -0.03 39° [-11 -19 -14]

29 0.26 0.95 -0.15 0.62 0.62 0.49 33.5° [-6 2 5]

30 0.54 0.52 0.66 -0.23 -0.47 0.85 34.5° [-6 2 5]

31 0.99 -0.09 0.09 0.67 -0.37 -0.64 47° [22 17 2]

32 0.84 0.04 0.55 0.77 0.12 -0.62 47° [22 17 2]

33 0.67 0.63 0.39 0.98 0.21 -0.02 48° [14 -16 -1]

34 0.20 0.79 0.59 0.91 -0.08 0.42 48° [14 -16 -1]

35 -0.73 -0.43 -0.52 -0.13 0.55 0.82 59.3° [110]

309



one (GB )

Grain Orientation / ˚ Misorientation

˚ u v w

2 209.4 41.9 150.5 51.12 0.4418 -0.3915 0.017

1 112.3 40.0 262.1

GB 50.1 52.6 311.8

Model Prediction:2 as burgers grain

Variant GBorientation ( ) ( )

Number

5 2.01

7 10.17



Number

12 1.31

9 9.70

310


of GB 16. For each variant, associated with , inclination angles

between corresponding [ ] , , , and GBP, i.e. ,

, and ,

are presented for Burgers grain and , respectively

Variants Burgers Orientation Relationship

No.

2 1

2 1 2 1

V1 29.60 31.65 75.05 72.42 15.77 21.73

V8 26.57 26.48 [ ] 85.08 77.64 15.76 17.15 75.07 78.30

V10 31.40 29.87 45.22 49.20 45.20 43.44

V2 19.71 19.71 66.31 83.30 75.07 78.30

V7 9.44 19.71 [ ] 28.51 13.53 64.91 76.47 77.34 89.99

V9 19.71 9.44 88.72 83.26 61.52 78.32

V3 29.04 31.65 89.71 72.39 15.78 21.73

V6 29.87 29.04 [ ] 74.22 77.68 33.81 49.22 60.96 43.40

V12 26.48 3.37 33.29 17.11 61.53 78.31

V4 29.27 29.60 57.77 65.23 60.95 43.41

V5 3.37 26.57 [ ] 46.16 56.99 46.60 33.01 77.32 89.99

V11 31.40 29.27 81.15 65.19 45.20 43.44

311


of GB 28.


No.

2 1

2 1 2 1

V1 30.10 34.74 26.84 85.10 82.60 53.56

V8 27.22 25.43 [ ] 64.36 36.88 70.46 61.85 33.16 68.23

V10 34.74 31.00 56.10 56.14 44.90 77.11

V2 22.69 22.69 58.70 24.30 33.16 68.22

V7 22.02 22.69 [ ] 80.15 79.74 62.27 39.02 29.72 52.85

V9 22.69 22.02 10.02 82.27 88.18 12.90

V3 23.03 32.48 83.65 38.50 82.59 53.56

V6 31.00 23.03 [ ] 9.78 79.27 86.77 25.08 80.78 67.62

V12 25.43 22.02 80.39 82.93 88.20 12.90

V4 27.91 30.10 24.45 61.48 80.77 67.62

V5 22.02 27.22 [ ] 67.57 37.64 71.56 84.77 29.71 52.85

V11 32.48 27.91 53.56 55.36 44.91 77.10

312



one (GB )


˚ u v w

2 181.2 16.4 151.4 39.04 -0.4344 -0.3344 -0.2762 1 57.5 55.5 319.6

GB 318.5 31.5 11.7



Number

9 22.69 1.15 12 25.43 10.51



Number

2 22.69 23.8

8 25.43 26.48

313



one (GB )


˚ u v w

2 122.3 39.8 210.4 25.6 0.065 0.0270 0.1825 1 328.7 55.0 49.4

GB 239.4 89.8 148.6



Number

4 25.60 29.17

6 25.60 30.29



Number

7 4.36

5 12.39

314



one (GB )


˚ u v w

2 122.3 39.8 210.4 25.6 0.065 0.0270 0.1825 1 328.7 55.0 49.4

GB 193.6 39.6 176.1



Number

8 1.85

2 25.60 11.52



Number

3 18.91 23.00

1 25.60 24.0

315


of GB 7.


No.

2 1

2 1 2 1

V1 25.60 25.60 44.26 35.07 48.35 55.10

V8 24.18 25.60 [ ] 77.69 86.97 21.00 85.05 73.26 5.81

V10 23.17 18.91 77.44 25.21 17.73 65.00

V2 25.60 23.17 83.67 85.49 73.26 5.81

V7 25.60 25.60 [ ] 17.96 86.34 78.80 25.73 76.14 64.57

V9 16.83 25.60 72.27 34.69 87.23 55.56

V3 18.91 24.18 63.92 71.18 48.35 55.10

V6 25.60 25.60 [ ] 52.83 41.06 69.17 48.94 44.51 89.99

V12 25.60 21.50 37.31 70.47 87.22 55.57

V4 25.60 16.83 46.51 60.53 44.52 89.99

V5 21.50 25.60 [ ] 82.30 29.47 15.93 76.10 89.99 64.73

V11 25.60 25.60 74.13 75.42 46.68 65.15

316


of GB 8.

Va Burgers Orientation Relationship

No.

2 1

2 1 2 1

V1 25.60 25.60 79.89 2.84 10.83 87.42

V8 24.18 25.60 [ ] 86.17 88.82 20.22 57.42 70.18 32.61

V10 23.17 18.91 40.28 62.59 49.98 27.44

V2 25.60 23.17 75.53 80.79 70.18 32.61

V7 25.60 25.60 [ ] 24.91 59.03 65.26 49.50 87.27 55.96

V9 16.83 25.60 80.29 35.94 67.30 73.60

V3 18.91 24.18 83.02 69.37 10.82 87.46

V6 25.60 25.60 [ ] 81.77 20.80 37.83 82.12 53.39 70.88

V12 25.60 21.50 24.31 77.57 67.30 73.59

V4 25.60 16.83 66.53 33.32 53.39 70.87

V5 21.50 25.60 [ ] 45.82 63.84 44.31 45.45 87.27 55.95

V11 25.60 25.60 71.50 82.30 49.97 27.44

317



one (GB )


˚ u v w

2 173.1 16.6 196.1 47.03 0.0514 0.0401 0.0048 1 65.1 39.5 302.0

GB 324.4 76.6 43.2



Number

1 2.45

3 14.89 11.08



Number

3 5.84

1 14.89 14.20

318


selection of GB 32.


No.

2 1

2 1 2 1

V1 7.11 14.89 83.04 54.76 51.64 48.02

V8 26.21 24.78 [ ] 39.22 62.05 53.28 29.80 78.13 80.49

V10 36.20 24.78 61.52 73.09 65.47 33.48

V2 26.68 30.89 57.92 81.22 78.15 80.49

V7 30.27 26.68 [ ] 34.71 13.00 84.97 77.33 55.76 87.14

V9 30.89 30.27 63.68 86.17 69.07 77.60

V3 14.89 7.11 39.57 50.64 51.65 48.02

V6 24.78 31.87 [ ] 81.72 67.18 81.26 74.80 12.08 27.91

V12 24.78 30.27 22.66 26.31 69.06 77.61

V4 30.21 33.24 85.11 63.06 12.09 27.91

V5 30.27 26.21 [ ] 78.97 83.23 36.46 7.35 55.76 87.15

V11 36.20 30.21 27.20 57.40 65.47 33.47

319



one (GB )


˚ u v w

2 168 33.3 174.7 37.16 0.1042 0.1339 0.2588 1 57.8 55.4 319.2

GB 31.1 64.5 315.0



Number

7 2.4 5 33.58 12.4



Number

9 28.80 28.60 12 33.58 32.73

320


selection of GB 32.


No.

2 1

2 1 2 1

V1 32.61 22.64 79.85 61.98 42.22 65.21

V8 25.54 32.18 [ ] 49.57 39.03 56.41 53.27 58.48 78.63

V10 22.64 29.01 43.17 82.63 77.41 51.94

V2 25.54 28.80 64.72 57.79 58.48 78.64

V7 28.80 25.54 [ ] 42.46 34.63 76.16 64.07 50.86 68.72

V9 28.80 28.80 48.22 84.50 83.76 55.93

V3 37.16 28.09 56.45 27.25 42.22 65.21

V6 29.01 37.16 [ ] 67.53 79.40 68.60 85.34 31.90 11.60

V12 32.18 33.58 23.41 36.13 83.77 55.93

V4 27.43 32.61 76.19 81.59 31.90 11.60

V5 33.58 25.54 [ ] 61.87 82.07 51.99 22.86 50.86 68.71

V11 28.09 27.43 31.29 39.18 77.42 51.94

321

7.6. Reference


Acta Mater 2013;61:844.

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Fatigue 1999;21:S105.


aeroengine applications. Mater Sci Tech-Lond 2010;26:676.



1934;1:561.




variants during microstructural evolution in alpha/beta titanium alloy. Philos. Mag.

2007;87:3615.

[7] Kar S, Banerjee R, Lee E, Fraser HL. Influence of Crystallography Variant

Selection on Microstructure Evolution in Titnaium Alloys. In: Howe JM, Laughlin DE,

Dahmen U, Soffa WA, editors. Solid-to-Solid Phase Transformation in Inorganic

Materials, vol. 133-138: TMS (The Minerals, Metals & Materials Society), 2005.




322


2004;52:5215.



2008;56:5907.





[12] Wilson RJ, Randle V, Evans WJ. The influence of the Burgers relation on crack

propagation in a near alpha-titanium alloy. Philos Mag A 1997;76:471.


precipitates in a β titanium alloy. MMTA 1996;27:1635.

[14] Shi R, Dixit V, Fraser HL, Wang Y. Variant Selection of Grain Boundary Alpha

by Special Prior Beta Grain Boundaries in Titanium Alloys. Acta Mater 2014:Submitted.


at grain boundaries—II. Three-dimensions. Acta Metall Mater 1975;23:809.


at grain boundaries—I. Two-dimensions. Acta Metall Mater 1975;23:799.

[17] Furuhara T, Kawata H, Morito S, Miyamoto G, Maki T. Variant selection in grain

boundary nucleation of upper bainite. Metall Mater Trans A 2008;39A:1003.

323

[18] Vikas D. Grain-Boundary Parameters Controlled Allotriomorphic Phase

Transformations in Beta-Processed Titanium Alloys. vol. Ph.D.: The Ohio State

University, 2013.

[19] Zhang WZ, Weatherly GC. On the crystallography of precipitation. Progress in


[20] Sutton AP, Balluffi RW. Interfaces in Crystalline Materials. Oxford: Clarendon

Press, 1995.


[22] Banerjee R, Bhattacharyya D, Collins PC, Viswanathan GB, Fraser HL.

Precipitation of grain boundary alpha in a laser deposited compositionally graded Ti-8Al-

xV alloy - an orientation microscopy study. Acta Mater 2004;52:377.

[23] Beladi H, Rohrer GS. The relative grain boundary area and energy distributions in

a ferritic steel determined from three-dimensional electron backscatter diffraction maps.

Acta Mater 2013;61:1404.

[24] Rohrer GS. Grain boundary energy anisotropy: a review. J Mater Sci

2011;46:5881.

324

CHAPTER 8 Conclusions and Future Works

8.1 Conclusions

Mechanisms responsible for the occurrence of variant selection at different length scales

have been studied systematically in the current work as follows:

For a single variant, the phase forms as laths with broad faces that are semi-coherent

and contain structural ledges and dislocations. A general method has been formulated to

derive effective transformation strain during precipitation that considers the effect of

interfacial defects including misfit dislocations and structural ledges, which is an

important first step toward variant selection study since the coupling between

precipitation and external bias such as stress field varies with the coherency state of

precipitate. How the interfacial defects relax the coherency elastic strain energy and

affect the habit plane orientation are analyzed in detail by incorporating the effect of the

defects into the stress-free transformation strain. How the interfacial defects affect the

interfacial energy anisotropy and the final equilibrium shape of precipitates is also

investigated. The equilibrium shapes of an isolated precipitate generated by the

interplay between interfacial energy and elastic strain energy are obtained from phase

field simulations. The habit plane orientation is found to be dominated by the strain

325

energy minimization, while interfacial energy anisotropy contributes to the in-plane shape

(ratio of the two major axes). The present work may build a bridge between the O-line

theory of precipitate habit planes and interfacial dislocation structures based on pure

geometrical consideration and the theory of optimum shapes of precipitate based on the

consideration of strain energy that depends on precipitate size, coherency state, shape and

orientations.

For variant selection within individual grain, a quantitative three-dimensional phase

field model (PFM) is developed to investigate variant selection during precipitation

from matrixin Ti-6Al-4V under the influence of both external and internal stress fields

such as those associated with, but not limited to, pre-straining and pre-existing

precipitates considered in this work. The model incorporates the crystallography of BCC

to HCP transformation, elastic anisotropy and interface defect structures in its total

free energy formulations. Model inputs are linked directly to thermodynamic and

mobility databases. The main findings are:

1) Under a given undercooling, there is a competition between internal stress associated

with an evolving microstructure and external applied stress or pre-strain on the

development of micro-texture. If the transformation strain or internal stress produced

by variants selected by a specific external stress or pre-strain during early stages of

precipitation cannot be accommodated among themselves, the internal stress would

prevent further development of such a transformation texture and induce the

formation of other variants to achieve self-accommodation. Since self-

accommodation can be achieved only by multiple variants (minimum two variants not

sharing a common basal plane), any constraints on macroscopic shape change of a

sample (e.g., by clamping) will prevent effectively the development of strong micro-

texture or transformation texture.

326

2) The development of micro-texture is sensitive to the loading axis of an external stress

or strain. From the elastic interaction energy calculations, we have learned that when

an external stress or pre-strain is applied in certain directions multiple variants of

phase could be favored simultaneously with the same interaction energy. Therefore, if

a polycrystalline sample has a strong macro-texture of the grains, control of

external load (if any) orientation could prevent strong micro-texture of plates from

percolating through the sample leading to poor fatigue properties.

3) There exists an obvious tension/compression asymmetry in variant selection behavior,

i.e., the types and numbers of variants produced under tensile and compressive

stresses are different. For example, pre-straining obtained via uniaxial tensile and

compressive stress along [010] will result in the selection of 8 and 4 out of 12

variants, respectively.

4) The interaction energy calculations, though simple and fast, cannot predict the overall

variant selection behavior at all cases. In addition, the prediction is valid only when

the internal stresses generated by an evolving microstructure are significantly smaller

than the externally applied stress.

5) Although nucleation of specific variants of secondary plates on interfaces between

primary plates and matrix observed in ⁄ Ti-alloys could be classified as

sympathetic nucleation (SN), the elastic interaction analysis in this study suggests that

such nucleation phenomenon observed in our simulations is coherency stress induced

correlated nucleation (i.e., auto-catalytic effect) rather than the conventional SN

discussed in literature (which is caused by the relatively low grain boundary energy

between the secondary and primary particles).

6) Secondary plates having a misorientation of [ ] ⁄ with the pre-existing

(i.e., primary) ones (i.e. sharing a common ⟨ ⟩ or ⟨ ⟩ ) tend to nucleate and

grow on the broad faces of the pre-existing plates, which could serve as an auto-

catalytic mechanism underlying the formation of basket-weave microstructures.

7) The stress-free transformation strain (SFTS) of precipitate varies with its coherency

state and variant selection rules (in terms of the sign and magnitude of elastic

327

interaction energy) are found different for coherent and semi-coherent precipitates.

When considering effect of primary plates whose sizes are usually above the

critical size for coherency (around 20 nm ), the SFTS for semi-coherent precipitate

should be employed for the primary precipitate while the SFTS for coherent

precipitate should be used for the nucleating secondary precipitates.

For variant selection within polycrystalline sample, the three-dimensional quantitative

phase field model has been further developed to study the variant selection process

during transformation under the influence of different processing variables such as

pre-strains. The effect of elastic and structural inhomogeneities on the local stress state

and its interaction with evolving microstructure is also considered in the model. In

particular, microstructure and transformation texture evolution are treated simultaneously

via orientation distribution function (ODF) modeling of ⁄ two-phase microstructure in

polycrystalline systems obtained by PFM. The variant selection behavior at the scale of

individual grain and the overall polycrystalline sample, and the resulting final

texture are found to be heavily dependent on type of pre-strain, boundary constraint of

the sample, and strating texture. It is found that, when subjected to a certain pre-strain,

the sample with strong texture component could promote more variants simultaneously

within the whole polycrystalline sample and thus lead to a relatively small degree of

microtexture. The results could shed light on how to control processing conditions to

reduce the strength micro-texture at both the individual grain level and the overall

polycrystalline sample level according to its starting texture.

For variant selection of GB on prior grain boundaries, a crystallographic model based

on the Burgers orientation relationship between GB and one of two grains has been

328

developed to study how variant selection occurs on prior grain boundary in / and

titanium alloys. In particular, a new parameter, that describes quantitatively

the deviation of OR between a GB and the non-Burgers grain from BOR, is identified

and a new GB selection rule is proposed. All possible special misorientations between

two grains that make GB in the Burgers orientation relationship (BOR) with both

grains have been predicted and confirmed by experimental observations made for Ti-

5553. Such variant selection of GB at special grain boundaries will result in the

development of large colony structures from the GB into two adjacent grains with

identical orientation as that of the GB. Through the analysis of the experiment

observations of GBin Ti-5553 using the model, it is found that when the disorientation

angle associated with is less than 15º, the variant with the smallest of

is always selected for GB, and the selected GBwill have its ⟨ ⟩

pole parallel to a common ⟨ ⟩ pole of the two adjacent grains. When ,

grain boundary plane inclination may play more important role for GB variant selection

in Ti-5553. Theoretical arguments why the parameter, , is a better measure

than the closeness between two closest { } from two grains widely used in literature

in analyzing GB variant selection are provided.

All current empirical variant selection rules about the influence of grain boundary

parameters have been investigated systematically. For the precipitation of GB son

different prior grain boundaries, how a single rule or different rules combined together

329

contributes to the final variant selection are studied. It has been found that when the

minimum misorientation angle associated with is less than ,

misorientation between two adjacent grains plays a leading role in the VS of GB .

When , GBP inclination is more important in the determination of VS of GB

than misorientation. However, the common employed rules that the variant of a selected

at a given -grain boundary is the one that has the minimum possible angle between the

matching direction ⟨ ⟩ ⟨ ⟩ and the grain boundary plane, or the that has the

minimum possible angle between matching planes { } { } and the grain

boundary plane, are found to be frequently violated. The violations of VS rules are

associated with the influence of interplay between different parameters such as grain

boundary energy, interface energy between GB and Burgers grain, interface energy

between GB and non-Burgers grain, on nucleation barrier. The violations of the

empirical VS rules could be associated with the fact that the activation energy of

nucleation of GB is determined by a complicated interplay among the five parameters

related to misorientation and inclination of a GB or an interphase interface that define the

structure and energy of the GB and interfaces, while the individual empirical rules

account for only a subset of these parameters..

8.2 Direction for future research Conclusions

Applications of the developed PFM model to study effects from other stress-carry

defects such as dislocations, stacking faults, grain boundaries, as well as effects from

330

thermal stress on variant selection during precipitation in polycrystalline samples are

straight forwards and corresponding work is underway. During all the simulations for

precipitations in polycrystalline sample by PFM, the effect of cooling rate and variant

selection of GB have been ignored. To mimic real precipitation during TMP,

however, these two factors have to be considered.

The nucleation barrier of GB, , for each variant is a function of multiple

parameters, grain boundary energy , interfacial energy between GB and non-

Bugers grain, , interfacial energy between GB and non-Bugers grain,

,

interfacial energy of low-energy facets,

, and inclinations between low-energy

facets and grain boundary plane,

(

)

Nucleation is to be made easiest for the variant with the minimum . This is the origin

that leads to the occurrence of VS of GB. In order to make more accurate predictions of

VS of GB a general rule needs to be further developed that take all the parameters

(grain boundary energy, interfacial energies between GB and two grains, interfacial

energies of low-energy facets, and orientations of the low-energy facets with respect to

GBP) playing their roles during the grain boundary nucleation and, more importantly,

demonstrate quantitatively how these parameters contribute to the activation energy of

the nucleation. The work is currently on its way.

How to implement these variant selection mechanisms working at different scales

simultaneously to investigate their effects on both microstructure and transformation

331

texture evolution in a representative volume element of sample, also needs to be

addressed.

332

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361

Appendix A: Determination of the number of variants of

precipitate phase

To find the intersection group H, it is necessary to specify the OR between and . In

Ti alloys, the precipitates usually hold the Burgers OR with the matrix, i.e.

101 // 0001

2 m∩ 6 m→ 2 m

111 // 1120

3∩ 2 m→ 1

121 // 1100

1 ∩ 2 m→ 1

The parallelism of 101 // 0001

places the symmetry element of 6 m in on top of the

symmetry element of 2 m in [1]. The other two parallelism relationships place

respectively 2 m on top of 3 and 2 m on top of 1 . The surviving symmetry element is

2 m , i.e. the intersection group H of the matrix Gm = m3m and the precipitate Gp =

6 mmm is 2 m when the Burgers orientation relationship is maintained between the

precipitate and matrix phases.

The 48 symmetry operations in the m3m point group can be divided into two groups:

first 24 proper rotational symmetry operations and second 24 improper rotational ones

that are obtained by a combination of first 24 rotations with the inversion center ( 1 ). The

24 proper rotational symmetry operations include, the identity I (monad); 3 rotations by

362

about the axes 100

(diad-3 2

4C ); 6 rotations by 2 about the axes 100

(tetrad-6

4C ); 6 rotations by about the axes 110

(diad-6 2C ) and 8 rotations by 2 3 about the

diagonals of the cube 111

(triad-8 3C ). From the Burgers OR., it can be readily seen

that 8 3C are missing in the product phase, which, thus, will generate 8 new variants by

operating them on the matrix. The remaining 3 unique symmetry operations for the new

variants are 3 2

4C because 4C can be generated by a combination of 3C about 111

and 2C

about 110

in the cube with m3m point group. Since the 1 is common to product and

matrix phases, the 24 improper symmetry operation will not generate new variants.

363

Appendix B: Stress free transformation strain for all 12

variants

B.1. Coherent nuclei

Table A1 SFTS for all 12 variants of nuclei

0.083 0.0095

V2= 0.035

0.0095 0.123

0.079 0.0359 0.0264

V3 0.0359 0.0047 0.0810

0.0264 0.0810 0.0087

0.079 0.0359 0.0264

V4 0.0359 0.0047 0.0810

0.0264 0.0810 0.0087

0.079 0.0359 0.0264

V5= 0.0359 0.0047 0.0810

0.0264 0.0810 0.0087

0.079 0.0359 0.0264

V6= 0.0359 0.0047 0.0810

0.0264 0.0810 0.0087

0.083 0.0095

V7= 0.0095 0.123

0.035

0.083 0.0095

V8= 0.035

0.0095 0.123

0.079 0.0264 0.0359

V9= 0.0264 0.0087 0.0810

0.0359 0.0810 0.0047

0.079 0.0264 0.0359

V10= 0.0264 0.0087 0.0810

0.0359 0.0810 0.0047

0.079 0.0264 0.0359

V11= 0.0264 0.0087 0.0810

0.0359 0.0810 0.0047

0.079 0.0264 0.0359

V12= 0.0264 0.0087 0.0810

0.0359 0.0810 0.0047

0.083 0.0095

V1 0.0095 0.123

0.035

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B.2. Fully-grown plates

Table A2 SFTS for all 12 variants of fully-grown plates

0.049 0.0031

V1 0.0031 0.067

0.0003

0.049 0.0031

V2= 0.0003

0.0031 0.067

0.0334 0.0222 0.0253

V3= 0.0222 0.010 0.0412

0.0253 0.0412 0.0056

0.0334 0.0222 0.0253

V4= 0.0222 0.010 0.0412

0.0253 0.0412 0.0056

0.0334 0.0222 0.0253

V5= 0.0222 0.010 0.0412

0.0253 0.0412 0.0056

0.0334 0.0222 0.0253

V6= 0.0222 0.010 0.0412

0.0253 0.0412 0.0056

0.049 0.0031

V7= 0.0031 0.067

0.0003

0.049 0.0031

V8= 0.0003

0.0031 0.067

0.0334 0.0253 0.0222

V9= 0.0253 0.0056 0.0412

0.0222 0.0412 0.010

0.0334 0.0253 0.0222

V10= 0.0253 0.0056 0.0412

0.0222 0.0412 0.010

0.0334 0.0253 0.0222

V11= 0.0253 0.0056 0.0412

0.0222 0.0412 0.010

0.0334 0.0253 0.0222

V12= 0.0253 0.0056 0.0412

0.0222 0.0412 0.010

Rongpei Shi-Variant Selection during Alpha Precipitation in Titanium Alloys-Thesis

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