Variant Selection during Alpha Precipitation in Titanium Alloys A Simulation Study DISSERTATION

Transcription

1 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

2 Copyright by Rongpei Shi 2014

3 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 ii

4 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. iii

5 Dedication This document is dedicated to my family. iv

6 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. v

7 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 vi

8 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. vii

9 Vita July B.S. Fuzhou University, Fuzhou, China Sep July 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: [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 viii

10 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: [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: [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: [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: Fields of Study Major Field: Materials Science and Engineering ix

11 Table of Contents Abstract... ii Dedication... iv Acknowledgments... v Vita... viii List of Tables... xviii List of Figures... xxii CHAPTER 1 Introduction Motivations Organization of the thesis Reference: CHAPTER 2 Literature Review Abstract Introduction precipitation in titanium alloys Two-phase titanium alloys Microstructure development during precipitation x

12 2.2.3 Orientation relationship between and phases Determination of the number of variants The nature of interface between precipitate and matrix Relationship between microstructure and mechanical properties Variant selection during precipitation Variant selection of GB Variant selection of secondary side plates by GB Variant selection in basketweave microstructures Variant selection due to dislocations Unresolved issues Grain boundary nucleation Correlations between precipitates with different variants in the basketweave microstructure The effect of dislocation on variant selection Microstructure evolution with variant selection References: CHAPTER 3 Predicting Equilibrium Shape of Precipitates as Function of Coherency State Abstract: xi

13 3.1. Introduction Elastic Strain Energy of Coherent and Semi-Coherent Precipitates Stress-free transformation strain for coherent precipitates Deformation gradient matrix due to defects at hetero-phase interfaces Estimation of Interfacial Energy for Semi-Coherent Interfaces Worked Examples Derivation of effective SFTS for the semi-coherent precipitates Strain energy density and habit plane orientation of semi-coherent precipitates Interfacial energy anisotropy of semi-coherent precipitates Equilibrium shape of -precipitates in different cases Coherency lost Discussions Summary Reference CHAPTER 4 Variant Selection during Precipitation in Ti-6Al-4V under the Influence of Local Stress Abstract: Introduction xii

14 4.2. Method Determination of number of variants of a low symmetry precipitate phase Free energy formulation Chemical free energy Elastic strain energy Stress-free transformation strain for coherent and semi-coherent precipitates Effect of misfit dislocation on interfacial energy Kinetic equations Model inputs and parameters Results Growth behavior of a single plate Effect of pre-strain on variant selection Pre-strain due to compressive stress along [010] Pre-strain due to tensile stress along [010] Variant selection due to pre-existing plates Discussion Lengthening and thickening kinetics of plate xiii

15 Elastic interaction between pre-strain and transformation strain of variants Competition between pre-strain and evolving microstructure Variant selection due to pre-existing microstructure Summary References CHAPTER 5 Evolution of Microstructure and Transformation Texture during Alpha Precipitation in Polycrystalline Titanium alloys Abstract: Introduction Model Formulation Polycrystalline sample Phase Field Model for precipitation in an elastically and structurally inhomogeneous polycrystalline sample Chemical free energy for polycrystalline system Strain energy of an elastically and structurally inhomogeneous system Kinetic equations Orientation Distribution Function modeling of microstructure in polycrystalline sample xiv

16 5.3. Results Starting polycrystalline and texture Evolution of microstructure and texture during precipitation Effect of pre-strain on variant selection Effect of starting texture on variant selection Quantifying the degree of variant selection Effect of boundary constraint on variant selection Discussions Summary References: CHAPTER 6 Variant Selection of Grain Boundary by Special Prior Grain Boundaries in Titanium Alloys Abstract Introduction Model formulation and Experimental procedures Crystallographic model Experimental procedures Results xv

17 Special grain boundaries where GB maintainsbor with both adjacent grains Violation of variant selection rule derived from closeness between poles Discussion Conclusions References: CHAPTER 7 Effects of Grain Boundary Parameters on Variant Selection of Grain Boundary in Titanium Alloys Abstract Introduction Experimental procedure Results Overall Characteristics of variant selection of GB Variant selection of GB when different rules are dominant Rule I is dominant Rule II is dominant Rule III is dominant Abnormal cases xvi

18 Abnormal variant selection when the minimum Abnormal variant selection when Discussions Summary Reference CHAPTER 8 Conclusions and Future Works Conclusions Direction for future research Conclusions Reference Appendix A: Determination of the number of variants of precipitate phase Appendix B: Stress free transformation strain for all 12 variants B.1. Coherent nuclei B.2. Fully-grown plates xvii

19 List of Tables Table 2.1 All 12 variants of the Burgers orientation relation between precipitate sand matrix 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 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 a Å, a Å and c Å and I is unit tensor Table 4.1 All 12 Burgers orientation variants and symmetry operations associated with them 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 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 ) xviii

20 Table 6.3 Orientations of two grains shows Type III misorientation in variant selection of GB Predicted orientation of GB (GB ) and its misorientation from the measured one (GB ) 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 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 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 ) 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 Table 7.4 Details about the effect of all grain boundary parameter in the variant selection of GB xix

21 Table 7.5 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 ) Table 7.6 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 ) Table 7.7 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 ) Table 7.8 Details about the effect of all grain boundary parameter in the variant selection of GB Table 7.9 Details about the effect of all grain boundary parameter in the variant selection of GB Table 7.10 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 ) Table 7.11 Details about the effect of all grain boundary parameter in the variant selection of GB xx

22 Table 7.12 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 ) Table 7.13 Details about the effect of all grain boundary parameter in the variant selection of GB xxi

23 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] Figure 2.2 Typical microstructures in Titanium alloys: (a) Grain boundary GB (b) Colony ; (c) Basketweave and (d) Secondary microstructure Figure 2.3 Schematic illustration of the Burgers orientation relationship, by looking down [101] // [0001] (pointing into the plane of paper) Figure 2.4 Schematic illustration of the interface and misfit dislocation configuration 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) 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 Figure 2.7 Schematic illustration of GB of different variants formed at a grain boundary with a slight variation in its boundary plane 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) xxii

24 Orientation Image Microscopy (OIM) map of the same region as shown in (a). Regions with the same color represent the same orientation variant 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] 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 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] 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 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 xxiii

25 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 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 Figure. 4.4 Variant selection and microstructure development under a pre-stain obtained 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 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) xxiv

26 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) 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) 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 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 Figure 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 Figure 4.10 (continued) xxv

27 Fig (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 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 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 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 Figure.5.4 (a)-(d) Final microstructure in random-textured sample under different pre-strains, and (a )-(d ) corresponding final texture Figure.5.5 (a)-(d) Final microstructure in strong-textured sample under different pre-strains, and (a )-(d ) corresponding final texture xxvi

28 Figure.5.6 (a) Maximum intensity in pole figures as a function of time in randomtextured 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 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 randomtextured and strong-textured sample without variant selection Figure.5.8 Degree of variant selection in both random-texture and strong textured samples under different pre-strain 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 Figure.5.10 (a)-(b) microstructure in the 2 nd and 5 th 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 Figure.5.11 (a)-(b) microstructure in the 2 nd and 5 th grain in strong-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)- xxvii

29 (h) interaction energy density between the external loading and each α variant under both coherent and semi-coherent conditions within these two grains Figure.5.12 (a)-(b) microstructure in the 2 nd and 5 th grain in random-textured 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; 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 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 Figure (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 Figure (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 xxviii

30 final phase with occurrence of variant selection represented by corresponding three different pole figures,,, and, respectively Figure Examples showing the pseudo variant selection due to 2D sampling effect. EBSD scan is performed along at different layers of the sample 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 º/<110>, (b) Type II º/<110>, (c) Type III- 60º/<110> and Type IV- 60º/<111> 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 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 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 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 xxix

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

32 Figure 7.4 (a) Stereographic projection shows the orientation of GB planes relative to the 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 Figure.7.5 (a) Stereographic projection shows the orientation of GB planes relative to the 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 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) Figure 7.7 Experimental observations of variant selection of GB 28aOIM 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 xxxi

33 associate with and (f) for all 12 variants with respect to different Burgers grain Figure 7.7 (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 Figure 7.8 (Continued) Figure 7.9 Experimental observations of variant selection of GB 31aOIM 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 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 xxxii

34 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 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 xxxiii

35 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 1

36 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 2

37 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 3

38 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 macrozones or micro-textured regions). Therefore, in order to control the microstructure, to understand processingmicrostructure-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 microtexture 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 4 microstructures. During

39 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 alter not only the coherency elastic strain energy associated with the interfaces will 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 nonrandom 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 5

40 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 6

41 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 7

42 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 8

43 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 9

44 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 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 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. Acta Materialia 2013;61:844. [4] Lutjering G, Williams JC. Titanium (Engineering Materials and Processes). Berlin: Springer, [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:

45 [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 [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, p [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:

46 [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, [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:

47 [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, [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, [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, p [28] Shen C, Wang Y. Coherent precipitation - phase field method. In: Yip S, editor. Handbook of Materials Modeling, vol. B: Models. Springer, p [29] Wang Y, Chen LQ, Zhou N. Simulating Microstructural Evolution using the Phase Field Method. Characterization of Materials. John Wiley & Sons, Inc., [30] Wang YU, Jin YM, Khachaturyan AG. Dislocation Dynamics Phase Field. Handbook of Materials Modeling. Springer, p

48 [31] Shi R, Ma N, Wang Y. Predicting equilibrium shape of precipitates as function of coherency state. Acta Materialia 2012;60:4172. [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,

49 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 15

50 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 16

51 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 precipitation in titanium alloys 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 P6 3 mmc. The predominant slip mode in phase is the , 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, 2, 30]. The secondary slip systems are and Above 882 C, pure titanium transforms allotropically from to phase. The BCC structure can be defined by placing two atoms at (0, 0, 0) 17

52 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: [1, 2, 30]., and 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. 18

53 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 solidsolution strengthening the -phase [6]. The phase offers precipitation strengthening for / alloys Microstructure development during precipitation For titanium alloys, upon cooling from the single phase region into the +twophase 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 19

54 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 socalled 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). 20

55 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 // 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 0.78ºand 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 Note that a 5.26ºcrystallographic rotation along 101 // 0001 ; 101 // 1100 (2.2) converts the Pitsch- Schrader OR to ideal Burgers OR. 21

56 The precipitate is different from the matrix in both composition and crystal structure. With the lattice parameters [38] a bcc = nm, a hcp = nm, and c hcp = nm, the transformation strain ij for an precipitate in matrix with the OR given in Eq. (2.1) can be described by: ij (2.3) when referred to 010 // 1120 ; 101 // 1100 ; 101 // 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 on combinations of parallel directions 1120 // 111 presence of 6 fold rotation symmetry along 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 plane due to the Since there are six possible 101

57 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, P random, can be calculated 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 and 23

58 the side face is looped by a-type misfit dislocations with a Burgers vector of 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 [48], which is also the growth direction (major axis) for the plate. The terrace plane of step is parallel to 121 // The macroscopic surface of the broad face is generally an irrational plane close to [36-38, 45] 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) 24

59 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 25

60 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 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 26

61 phases such as grain boundary [27] and dislocations [15, 20]. The coupling between plates with different variants also induces variant selection [26, 50] 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 27

62 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. 28

63 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? 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, 29

64 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 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]. 30

65 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 10.53ºalong 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 31

66 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 ( ) and Type 4 ( ) 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 preexisting 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 32

67 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 ºor 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 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 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 b max associated with 12 variants of precipitate by Frank-Bilby equation (FBE) [59, 60]. According to the FBE, the misfit 33

68 across the interface between the precipitate and matrix can be described in terms of the net Burgers vector b t crossing a vector p in the interface, as given by: 1 b t I 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 b max to the Burgers vector a 2111 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 a 2111 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

69 planes in 111 directions. Thus, prior activity of specific systems may favor the formation of related alpha variants 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: 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: 35

70 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 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 nonrandomly 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 36

71 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 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 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 37

72 relationship, integration of variant selection mechanisms inherent in the precipitation process is also required in any modeling attempt. 38

73 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

74 Figure 2.2 Typical microstructures in Titanium alloys: (a) Grain boundary GB (b) Colony ; (c) Basketweave and (d) Secondary microstructure 40

75 Figure 2.3 Schematic illustration of the Burgers orientation relationship, by looking down [101] // [0001] (pointing into the plane of paper) [10] 41

76 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] 42

77 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] 43

78 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] 44

79 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] 45

80 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] 46

81 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

82 Tables Table 2.1 All 12 variants of the Burgers orientation relation between precipitate sand matrix [10, 16] Variants Orientation Relationship V1 110 // // 1120 V2 111 // // // 1100 V // 11 1 // // 1100 V4 V5 011 // // // 1120 V6 111 // // // // 1100 V // 111 // // 1100 V8 V9 101 // // // 1120 V // // // // 1100 V // 111 // // 1100 V // //

83 Table 2.2 Axis/angle pairs for all 6 possible boundaries in a single grain [22, 27] ype Axis/angle pairs P random [%] 1 I (Identity)

84 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 Macrocracks ΔK th K IC Small Colonies Lamellae + (positive), -(negative) to specific mechanical property 50

85 2.5. References: [1] Leyens C, Peters M, editors. Titanium and Titanium Alloys-Fundamentals and Applications. Weinheim: WILEY-VCH, [2] Lutjering G, Williams JC. Titanium (Engineering Materials and Processes). Berlin: Springer, [3] Boyer RR. Attributes, Characteristics, and Applications of Titanium and Its Alloys. JOM 2010;62:21. [4] Boyer RR. An overview on the use of titanium in the aerospace industry. Materials Science and Engineering a-structural Materials Properties Microstructure and Processing 1996;213:103. [5] Freese HL, Volas MG, Wood JR, Textor M. Titanium and its Alloys in Biomedical Engineering. In: Buschow KHJ, Robert WC, Merton CF, Bernard I, Edward JK, Subhash M, Patrick V, editors. Encyclopedia of Materials: Science and Technology. Oxford: Elsevier, p [6] Froes FH. Titanium Alloys: Properties and Applications. In: Buschow KHJ, Robert WC, Merton CF, Bernard I, Edward JK, Subhash M, Patrick V, editors. Encyclopedia of Materials: Science and Technology. Oxford: Elsevier, p [7] Geetha M, Singh AK, Asokamani R, Gogia AK. Ti based biomaterials, the ultimate choice for orthopaedic implants - A review. Prog Mater Sci 2009;54:397. [8] Froes FH. Titanium Alloys: Corrosion. In: Buschow KHJ, Robert WC, Merton CF, Bernard I, Edward JK, Subhash M, Patrick V, editors. Encyclopedia of Materials: Science and Technology. Oxford: Elsevier, p

86 [9] Lütjering G. Influence of processing on microstructure and mechanical properties of ([alpha]+[beta]) titanium alloys. "Mater Sci Eng, A " 1998;243:32. [10] 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. [11] Wenk HR, Houtte PV. Texture and anisotropy. Rep Prog Phys 2004;67:1367. [12] Peters M, Gysler A, LÜtjering G. Influence of texture on fatigue properties of Ti- 6Al-4V. Metallurgical and Materials Transactions A 1984;15:1597. [13] Bache MR, Evans WJ. Impact of texture on mechanical properties in an advanced titanium alloy. "Mater Sci Eng, A " 2001; :409. [14] Bache MR, Evans WJ, Suddell B, Herrouin FRM. The effects of texture in titanium alloys for engineering components under fatigue. Int J Fatigue 2001;23:153. [15] Furuhara T, Nakamori H, Maki T. Crystallography of alpha phase precipitated on dislocations and deformation twin boundaries in a beta titanium alloy. Materials Transactions JIM 1992;33:585. [16] Furuhara T, Takagi S, Watanabe H, Maki T. Crystallography of grain boundary α precipitates in a β titanium alloy. Metallurgical and Materials Transactions A 1996;27:1635. [17] Gey N, Humbert M, Philippe MJ, Combres Y. Modeling the transformation texture of Ti-64 sheets after rolling in the [beta]-field. "Mater Sci Eng, A " 1997;230:68. [18] Moustahfid H, Humbert M, Philippe MJ. Modeling of the texture transformation in a Ti-64 sheet after hot compression. Acta Mater 1997;45:

87 [19] Divinski SV, Dnieprenko VN, Ivasishin OM. Effect of phase transformation on texture formation in Ti-base alloys. "Mater Sci Eng, A " 1998;243:201. [20] Furuhara T, Maki T. Variant selection in heterogeneous nucleation on defects in diffusional phase transformation and precipitation. Materials Science and Engineering a- Structural Materials Properties Microstructure and Processing 2001;312:145. [21] 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 Mater 2003;51:4679. [22] Wang SC, Aindow M, Starink MJ. Effect of self-accommodation on alpha/alpha boundary populations in pure titanium. Acta Mater 2003;51:2485. [23] Banerjee R, Bhattacharyya D, Collins PC, Viswanathan GB, Fraser HL. Precipitation of grain boundary alpha in a laser deposited compositionally graded Ti-8AlxV alloy - an orientation microscopy study. Acta Mater 2004;52:377. [24] Stanford N, Bate PS. Crystallographic variant selection in Ti-6Al-4V. Acta Mater 2004;52:5215. [25] 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 Mater 2007;55:6765. [26] Lee E, Banerjee R, Kar S, Bhattacharyya D, Fraser HL. Selection of alpha variants during microstructural evolution in alpha/beta titanium alloy. Philos Mag 2007;87:

88 [27] 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 Mater 2008;56:5907. [28] 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, [29] Whittaker R, Fox K, Walker A. Texture variations in titanium alloys for aeroengine applications. Mater Sci Technol 2010;26:676. [30] Mills MJ, Neeraj T. Dislocations in Metals and Metallic Alloys. In: Buschow KHJ, Robert WC, Merton CF, Bernard I, Edward JK, Subhash M, Patrick V, editors. Encyclopedia of Materials: Science and Technology. Oxford: Elsevier, p [31] Froes FH. Titanium: Alloying. In: Buschow KHJ, Robert WC, Merton CF, Bernard I, Edward JK, Subhash M, Patrick V, editors. Encyclopedia of Materials: Science and Technology. Oxford: Elsevier, p [32] Purdy GR. Widmanstätten Structures. In: Buschow KHJ, Robert WC, Merton CF, Bernard I, Edward JK, Subhash M, Patrick V, editors. Encyclopedia of Materials: Science and Technology. Oxford: Elsevier, p [33] Aaronson HI, Medalist RFM. Atomic mechanisms of diffusional nucleation and growth and comparisons with their counterparts in shear transformations. Metall Trans A 1993;24A:

89 [34] Aaronson HI, Spanos G, Masamura RA, Vardiman RG, Moon DW, Menon ESK, Hall MG. Sympathetic Nucleation - an Overview. Materials Science and Engineering B- Solid State Materials for Advanced Technology 1995;32:107. [35] Potter DI. The structure, morphology and orientation relationship of V3N in [alpha]-vanadium. Journal of the Less Common Metals 1973;31:299. [36] Furuhara T, Howe JM, Aaronson HI. Interphase boundary structures of 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, p.295. [39] Miyano N, Ameyama K. Three dimensional near-coincidence site lattice analysis of orientation relationship and interface structure in two phase alloys. J Jpn Inst Met 2000;64:42. [40] Miyano N, Fujiwara H, Ameyama K, Weatherly GC. Preferred orientation relationship of intra- and inter-granular precipitates in titanium alloys. Materials Science and Engineering a-structural Materials Properties Microstructure and Processing 2002;333:85. [41] Pitsch W, Schrader A. Arch. Eisenhutt Wes 1958:

90 [42] Cahn JW, Kalonji GM. Symmetry in Solid-Solid Transformation Morphologies. PROCEEDINGS OF an Interantional Conference On Solid-Solid Phase Transformations 1981:3. [43] Dahmen U. Phase Transformations, Crystallographic Aspects. In: Robert AM, editor. Encyclopedia of Physical Science and Technology. New York: Academic Press, p.821. [44] Suri S, Viswanathan GB, Neeraj T, Hou DH, Mills MJ. Room temperature deformation and mechanisms of slip transmission in oriented single-colony crystals of an alpha/beta titanium alloy. Acta Mater 1999;47:1019. [45] Ye F, Zhang WZ, Qiu D. A TEM study of the habit plane structure of intragrainular proeutectoid [alpha] precipitates in a Ti-7.26 wt%cr alloy. Acta Mater 2004;52:2449. [46] Ye F, Zhang WZ. Dislocation structure of non-habit plane of [alpha] precipitates in a Ti-7.26 wt.% Cr alloy. Acta Mater 2006;54:871. [47] Aaronson H, Plichta M, Franti G, Russell K. Precipitation at interphase 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:

91 [50] 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. [51] Lee JK, Aaronson HI. Influence of faceting upon the equilibrium shape of nuclei at grain boundaries--i. Two-dimensions. Acta Metall 1975;23:799. [52] Lee JK, Aaronson HI. Influence of faceting upon the equilibrium shape of nuclei at grain boundaries--ii. Three-dimensions. Acta Metall 1975;23:809. [53] Mullins WW, Sekerka RF. Morphological Stability of a Particle Growing by Diffusion or Heat Flow. J Appl Phys 1963;34:323. [54] Wang Y, Ma N, Chen Q, Zhang F, Chen S, Chang Y. Predicting phase equilibrium, phase transformation, and microstructure evolution in titanium alloys. JOM Journal of the Minerals, Metals and Materials Society 2005;57:32. [55] Otsuka K, Ren X. Physical metallurgy of Ti-Ni-based shape memory alloys. Prog Mater Sci 2005;50:511. [56] Kar S, Banerjee R, Lee E, Fraser HL. Influence of crystallographic variant selection on microstructure evolution in titanium alloys. Warrendale: Minerals, Metals & Materials Soc, [57] Porter D, Easterling K. Phase transformations in metals and alloys: CRC, [58] Robert WB, Allen SM, Carter WC. Kinetics of materials. Hoboken: John Wiley & Sons, [59] Christian JW. The Theory of Transformations in Metals and Alloys. Oxford: Pergamon,

92 [60] Sutton AP, Balluffi RW. Interfaces in Crystalline Materials. Oxford: Oxford University Press, [61] Shen C, Li J, Wang YZ. Finding critical nucleus in solid-state transformations. Metallurgical and Materials Transactions a-physical Metallurgy and Materials Science 2008;39A:976. [62] Wang Y, Khachaturyan AG. Three-dimensional field model and computer 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,

93 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. 59

94 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 interphase interfaces. They are also referred to as transformation dislocations or disconnections to distinguish themselves from defects without the step character in the 60

95 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], 61

96 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 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 62

97 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]: Equation Chapter 3 Section 1 N 0 00 ij ij p p p1 ( x) ( ) ( x ) (3.1) 63

98 which is a linear superposition of all N types of non-uniformities with ( x ) being the p phase fields characterizing the p-th type non-uniformity and 00 ( p) (i,j=1,2,3) being the ij corresponding SFTS measured from a given reference state. Note that 00 ( p) depends on the lattice correspondence (LC) between the precipitate and parent phases. ij 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]. el 1 d g * E C ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 ijklij p kl q ni ij p jk kl q n l p q 2 (2 ) n g g pq, 1 d g 2 (2 ) pq, 3 B * pq( n) p( g) q ( g) (3.2) where g is a vector in the reciprocal space and n g / g, ( p) C ( p) and ij ijkl kl 1 0 [ ( )] ik C ijkl n j n l n is the inverse of the Green s function in the reciprocal space. ( g ) p is the Fourier transform of ( x ). The asterisk denotes complex conjugate. represents p a principle value of the integral that excludes a small volume in the reciprocal space 3 (2 ) /V at g 0, where V is the total volume of the system. The function B ( 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 pq 64

99 ( ) p x that is equivalent to shape functions of the precipitates. We shall introduce a unit vector n 0 such that the function B ( n ) reaches its minimum at n=n 0 [22]. The physical pq meaning of n 0 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 n 0. 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] Stress-free transformation strain for coherent precipitates The calculation of the SFTS, 00 ( p), is an important step towards calculating the ij 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] 65

100 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 F 0 is the transformation matrix or deformation gradient matrix describing the uniform lattice distortion from the parent phase to the p th variant of the product phase following a given LC, then [22, 39] T 00 p 0 0 I F F (3.3) 2 where I is the unit tensor and the symbol ''T'' denotes a transpose operation on the associated matrix. The transformation matrix, F 0, relates the parent crystal lattice site vector, r, to the product crystal lattice site vector, r ', by r' F0r. Generally, three pairs of non-coplanar vectors, r and The crystal lattice site displacements, by ur F 0 I 1 2 r ', related by the LC are selected to construct F 0. ur, associate with the transformation are given r. The strain tensor obtained in Eq. (3.3) is identical to T ur ur, where u r characterizes the displacement gradient 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 66

101 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 b y relaxes the misfit normal to the terrace plane while the dislocation content b x 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, F dis, 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: F dis 0 0 I b D (3.4) where b is the length of the Burgers vector of the misfit dislocations. Following the same approach, the deformation gradient matrix, F, due to the presence of steps with step spacing S and height h (as shown in Fig. 3.1(b) ) can be formulated as: 67

102 bx S Fstep by h I 0 (3.5) with the coordinate system defined on the terrace plane as shown in Fig The total deformation gradient matrix, including contributions from the uniform lattice distortion due to the phase transformation (F 0 ) and contributions from misfit dislocations (F dis ) and structural ledges (F step ), can be written as: and the effective SFTS tensor becomes Ftot Fstep Fdis F (3.6) 0 F F I (3.7) 2 T eff tot tot ij p The effective SFTS tensor can be readily utilized to predict the habit plane normal, n 0, and calculate the minimum coherency strain energy density, B 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. pq 0 68

103 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: 1 ln s m m m E (3.8) where E m 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 ~ bd. 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 69

104 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). 70

105 Therefore, both theory and experimental characterization are required to obtain all the information about the interfacial defects for a finite precipitate 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 (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 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 , which are associated with the 0110 and 110 planes with extra half planes in the phase and spaced by about 9 to atomic plane spacing. On the broad face, in addition to the c-

106 dislocations, the atomic structure of the / interface consists of structural ledges that enable the interface to step down along the invariant line direction 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, and 101. According to the nearest neighbor principle [21], the corresponding three non-coplanar vectors in the phase lattice after the to transformation are , and 0001 [37], respectively. Thus, the lattice correspondence between the BCC and HCP lattices can be described as, ; ; (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.,, 3 // 101 // 0001 x 2 // 101 // 1010 x, as shown in Fig. 3.3., x 1 // 010 // 1210 In the reference frame N1, the transformation matrix to deform the lattice homogeneously to the lattice can be described as: 72

107 a a 0 0 F0 0 3a 2a c 2a (3.11) where the lattice parameters of the two phases are a Å, a Å and c 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 F 0 [37]. However, since shuffling does not change the shape and size of the unit cells of the two crystal lattices, F 0 does represent the actual transformation matrix. In addition to F 0, a rigid body rotation, F, by 5.26º along axis 101 // 0001 R is required to realize the exact Burgers OR shown in Eq. (3.9) and Fig. 3.3(c) F R cos 5.26 sin 5.26 sin 5.26 cos (3.12) Thus, the overall transformation matrix for a coherent precipitate can be expressed as F F F, which expresses the coherent transformation matrix of a specific Burgers R 0 variant with OR defined by Eq. (3.9). 73

108 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 F c I 1 (3.13) Similarly, the deformation gradient matrix due to the a-dislocations on the side face can be described as F a I 0 (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 b bx; b y;0, in a new reference coordinate N2 associated with the terrace plane: 74

109 ; ' 2 // 121 // 0110 x ' 1 // 111 // 2110 x ; ' // 101 // x (3.15) Dislocation b x lies in the terrace plane and has a Burgers vector of associated with the riser of the structural ledges [48, 49]. The component b x 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 λ S (i.e. about 17b x ) 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 b. Thus, the Burgers vector of y b is given by 6 3a 3 2a y 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 75

110 F s bx 17bx bx λs 6 3a 3 2a by h I= I 63a 0 0 (3.16) Note that F s is expressed in the reference frame N2. To express F s in coordinate N1, the deformation gradient matrix ' F due to the structural ledges becomes: S F ' S QF Q', s 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: F F F F F F (3.17) ' tot S a c R 0 eff and the effective SFTS tensor p can be derived using Eq.(3.3). ij 76

111 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, n 0, can thus be predicted by finding the minimum of B ( ) pq n 0. 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 along010, 12.3% expansion along 101 and 3.5% expansion along 101 (Table 1). The minimum strain energy density 77 B n was found to be J/m 3 with the habit plane normal n 0 being [-11; -9.85; 8.07], which deviates about 8ºfrom 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 J/m 3 down to be J/m 3. In addition, the habit plane normal n 0 as predicted by minimizing B n was found to be [-11; ; 11], which deviates pq 0 about 2ºfrom the observed habit plane normal. The introduction of a-type misfit dislocations on the side face further reduces the coherency strain energy down to pq 0

112 J/m 3. However, the habit plane normal n 0 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 0.8ºdeviation. The calculated coherency strain energy density, Bpq 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 Bpq n is obtained where n 0 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, Bpq n reaches its minimum at 0 n corresponding to [- 11; ; 11] as indicated by the dotted arrow, which deviates only by 0.8ºfrom the experimentally observed habit plane [-11; -13; 11] 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 and , respectively. The equivalent misorientation angle of the edge face is 78

113 assumed to be 3 m. According to Eq.(3.8), the structural components, s, of the broad, side and edge faces are 0.39E m, 0.72E m and E m, respectively. E m is assumed to be 250 mj/m 2 [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/m 2, which is reasonable for fully coherent interfaces. Therefore, the interfacial energies of the broad, side and end faces are 150, 230 and 300 mj/m 2, 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] 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/m 2 is shown in Fig. 3.5(a)-1. Also, the projections of the three-dimensional equilibrium shape along010, 101 and 101 are shown in Figs. 5(a)-2-5(a)-4, respectively. The total system size is 16 nm. It 79

114 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/m 2 ), 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 80

115 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) 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 precipitatedislocation 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, r crit, 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 The coherent precipitate has a lower interfacial 81

116 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, r crit, of ~27 nm when s is 50 mj/m 2 and 22 nm when s = 25 mj/m 2. Thus, the critical size r crit 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 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 82

117 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 ton 0, [-11; ; 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 83

118 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 84

119 associated with the riser of the structural ledges is 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 J/m 3 to 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 85

120 pq 0 optimum shape with its broad face corresponding to the minimum of B 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]: E B n 2V E (3.18) pq 0 edge The first term in Eq. (3.18) on the right hand side describes the strain energy of an infinite plate of infinitesimal thickness (D p 0) and the second term can be regarded as energy correction associated with a finite plate thickness. The value of E edge can be described as 2 E ~ D P [22], where, 0 and P are the elastic modulus, edge 0 P transformation strain and plate perimeter P, respectively. The energy E is edge 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 additional a-type dislocation in side face of the precipitate. 86 E exceeds a critical value just as the edge

121 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 B n and also increase the interfacial energy due to its structural contribution. On the other hand, the total strain pq 0 energy of a finite precipitate includes two parts according to Eq. (3.18): E edge and Bpq n V. In our analysis of coherency lost presented in Section 4.5, the contribution 0 2 of E edge is ignored. When the precipitate size is smaller than critical size, however, the E edge part could be dominant and the reduction in strain energy via modification of pq 0 B 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 B n would be dominant and sufficient to pq compensate the increase in interfacial energy. Therefore, the total energy of a semi- 87 0

122 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, r crit, 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 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 88

123 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. 89

124 3.7. Reference [1] Brown LM, Ham RK. In: Kelly A, B. NR, editors. Strengthening methods in crystals. London: Elsevier, [2] Nie JF, Muddle BC, Polmear IJ. The effect of precipitate shape and orientation on dispersion strengthening in high strength aluminium alloys. Mater. Sci. Forum 1996; :1257. [3] Lutjering G, Williams JC. Titanium (Engineering Materials and Processes). Berlin: Springer, [4] Polmear IJ. Light alloys. Metallurgy of the light metals. London, [5] Nie JF. Effects of precipitate shape and orientation on dispersion strengthening in magnesium alloys. Scr. Mater. 2003;48:1009. [6] Nie J, Muddle B. Microstructural design of high-strength aluminum alloys. Journal of Phase Equilibria 1998;19:543. [7] Kovarik L, Unocic RR, Li J, Sarosi P, Shen C, Wang Y, Mills MJ. Microtwinning and other shearing mechanisms at intermediate temperatures in Ni-based superalloys. Prog. Mater Sci. 2009;54:839. [8] Aaronson HI, Enomoto M, Lee JK. Mechanisms of Diffusional Phase Transformations in Metals and Alloys. New York: CRC Press, [9] Porter D, Easterling K. Phase transformations in metals and alloys: CRC, [10] Hall MG, Aaronson HI, Kinsma KR. The structure of nearly coherent fcc: bcc boundaries in a Cu---Cr alloy. Surf. Sci. 1972;31:

125 [11] Pond RC, Celotto S, Hirth JP. A comparison of the phenomenological theory of martensitic transformations with a model based on interfacial defects. Acta Mater. 2003;51:5385. [12] Pond RC, Ma X, Chai YW, Hirth JP. Topological Modelling of Martensitic Transformations. Dislocations in Solids 2007;13:225. [13] Shiflet G, Merwe J. The role of structural ledges as misfit- compensating defects: fcc-bcc interphase boundaries. Metallurgical and Materials Transactions A 1994;25:1895. [14] Zhou N, Shen C, Wagner MFX, Eggeler G, Mills MJ, Wang Y. Effect of Ni4Ti3 precipitation on martensitic transformation in Ti-Ni. Acta Mater. 2010;58:6685. [15] Gao Y, Zhou N, Yang F, Cui Y, Kovarik L, Hatcher N, Noebe R, Mills MJ, Y. W. P-phase precipitation and its effect on martensitic transformation in (Ni,Pt)Ti shape memory alloys. Acta Mater. 2011;In press. [16] Dahmen U. The Role of the Invariant Line in the Search for an Optimum Interphase Boundary by O-Lattice Theory. Scr. Metall. 1981;15:77. [17] Dahmen U. Orientation relationships in precipitation systems. Acta Metall. 1982;30:63. [18] Rigsbee JM, Aaronson HI. The interfacial structure of the broad faces of ferrite plates. Acta Metall. 1979;27:365. [19] Zhang M-X, Kelly PM. Crystallographic features of phase transformations in solids. Prog. Mater Sci. 2009;54:1101. [20] Zhang WZ, Weatherly GC. On the crystallography of precipitation. Prog. Mater Sci. 2005;50:

126 [21] Bollmann W. Crystal defects and crystalline interfaces. Berlin: Springer, [22] Khachaturyan AG. Theory of Structural Transformations in Solids. New York: John Wiley & Sons, [23] Wang Y, Chen LQ, Khachaturyan AG. Kinetics of Strain-Induced Morphological Transformation in Cubic Alloys with a Miscibility Gap. Acta Metall. Mater. 1993;41:279. [24] Johnson WC, Voorhees PW. Elastically-induced precipitate shape transition in coherent solids. Solid State Phenomena 1992;23-24:87. [25] Shen C, Wang Y. Coherent precipitation - phase field method. In: Yip S, editor. Handbook of Materials Modeling, vol. B: Models. Springer, p [26] Wang YZ, Khachaturyan A. Microstructural Evolution during the Precipitation of Ordered Intermetallics in Multiparticle Coherent Systems. Philosophical Magazine a- Physics of Condensed Matter Structure Defects and Mechanical Properties 1995;72:1161. [27] 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. [28] Eshelby JD. The Elastic Field Outside an Ellipsoidal Inclusion. Proceedings of the Royal Society A 1959;252:561. [29] Khachaturyan AG. Some questions concerning the theory of phase transformations in solids. Sov. Phys. Solid State 1967;8:2163. [30] Khachaturyan AG, Shatalov GA. Elastic interaction potential of defects in a crystal. Sov. Phys. Solid State 1969;11:

127 [31] Boettinger WJ, Warren JA, Beckermann C, Karma A. Phase-field simulation of solidification. Annual Review of Materials Research 2002;32:163. [32] Chen L-Q. PHASE-FIELD MODELS FOR MICROSTRUCTURE EVOLUTION. Annual Review of Materials Research 2002;32:113. [33] Wang Y, Li J. Phase field modeling of defects and deformation. Acta Mater. 2010;58:1212. [34] Wang YU, Jin YMM, Khachaturyan AG. Phase field microelasticity theory and modeling of elastically and structurally inhomogeneous solid. J. Appl. Phys. 2002;92:1351. [35] Wang YU, Jin YMM, Khachaturyan AG. Phase field microelasticity modeling of dislocation dynamics near free surface and in heteroepitaxial thin films. Acta Mater. 2003;51:4209. [36] Bain EC, Dunkirk N. The nature of martensite. trans. AIME 1924;70:25. [37] 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. [38] Wayman CM. Introduction to the crystallography of martensitic transformations: Macmillan, [39] Mura T. Micromechanics of Defects in Solids,. Dordrecht: Martinus Nijhoff, [40] Howe JM. Interfaces in Materials. New York: Wiley,

128 [41] Read WT, Shockley W. Dislocation Models of Crystal Grain Boundaries. Physical Review 1950;78:275. [42] Read WT. Dislocations in crystals. New York: Jr. McGraw-Hill, [43] Ye F, Zhang WZ, Qiu D. A TEM study of the habit plane structure of intragrainular proeutectoid alpha precipitates in a Ti-7.26wt%Cr alloy. Acta Mater. 2004;52:2449. [44] Liang Q, Reynolds WT. Determining interphase boundary orientations from nearcoincidence sites. Metall. Mater. Trans. A-Phys. Metall. Mater. Sci. 1998;29:2059. [45] Qiu D, Zhang WZ. An extended near-coincidence-sites method and the interfacial structure of austenite precipitates in a duplex stainless steel. Acta Mater. 2008;56:2003. [46] Zhang WZ, Qiu D, Yang XP, Ye F. Structures in irrational singular interfaces. Metall. Mater. Trans. A-Phys. Metall. Mater. Sci. 2006;37A:911. [47] Ye F, Zhang WZ, Qiu D. Near-coincidence-sites modeling of the edge facet dislocation structures of alpha precipitates in a Ti-7.26 wt.% Cr alloy. Acta Mater. 2006;54:5377. [48] Mou Y, Aaronson HI. O-lattice modeling of ledged, partially coherent b.c.c.:h.c.p. boundaries. Acta Metall. Mater. 1994;42:2133. [49] Furuhara T, Aaronson HI. Computer Modeling of Partially Coherent Bcc - Hcp Boundaries. Acta Metall. Mater. 1991;39:2857. [50] 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. 94

129 Clark, King AH, editors. Boundaries and Interfaces in Materials: The David A. Smith Symposium: The Minerals, Metals & Materials Society, 1998, p.295. [51] Gottstein G, Shvindlerman LS. Grain boundary migration in metals: thermodynamics, kinetics, applications. New York: CRC Press, [52] Cahn JW, Hilliard JE. Free energy of a nonuniform system. I. Interfacial free energy. The Journal of Chemical Physics 1958;28:258. [53] Wang Y, Ma N, Chen Q, Zhang F, Chen SL, Chang YA. Predicting phase equilibrium, phase transformation, and microstructure evolution in titanium alloys. JOM 2005;57:32. [54] Sutton AP, Balluffi RW. Interfaces in Crystalline Materials. Oxford: Oxford University Press,

130 Figures Figure (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). 96

131 Figure Schematic illustration of the interface defect structure in near- titanium alloy Ti-5Al-2.5Sn-0.5Fe (wt.%), following Mills et al.[50] 97

132 Figure 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). 98

133 Figure 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). 99

134 (a)-1 [10-1] (a)-2 (a)-3 (a)-2 (a)-4 [101] [010] [010] 16 [10-1] nm [101] (b)-1 (b)-2 (b)-3 (b)-4 (c) [10-1] [010] Continued Figure 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 100

135 Figure. 3.5 continued (d) (d)-2 (d)-3 (d)-4 (e) (e)-2 (e)-3 (e)-4 (f) (f)-2 (f)-3 (f)-4 101

136 Figure 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/m 2 and the critical size r crit is about 27 nm (along the minor axis on the broad face); (d) =25 mj/m 2 and the critical size r crit 22 nm. 102

137 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 a Å, a Å and c Å and I is unit tensor 103

138 Table 3.1 Defects Deformation gradient F Transformation strain T F F 2-I Case I Defects-free (fully coherent) a 0 0 a cos5.26 sin a RF sin cos a c a Case II c-type misfit dislocation on broad face (habit plane) F c I Case III a-type dislocation on side face F a I Case IV Structural ledge on broad face bx F s λ S b h I y Continued

139 Table 3.1 countinued Minimum Bpq(n) [J/m 3 ] Coherency strain energy density n 0 (habit plane orientation) Deviation from experimental observation ([-11;-13;11]) [ o ] Case I [-11;-9.85;8.07] o 105 Case II [-11;-14.05;11] o Case III [-1; ;1] o Case IV [-11;-12.63;11] 0.79 o 105

140 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 106

141 certain variants that may not be the ones preferred by the initially applied stress. Possible measures to reduce transformation texture are discussed 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 107

142 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., and 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, microcracks 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 108

143 improved fatigue properties. Therefore, the control of variant selection during thermomechanical 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 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]. 109

144 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) physicsbased 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 preexisting precipitates on variant selection in a single grain of. Effects of elastic inhomogeneity and grain boundaries in a polycrystalline sample and effects from other 110

145 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 Method 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 111

146 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, G m, and the product group, G p, under a given OR between them [8], i.e., H G G. Note that G m and G p are the point groups rather than space groups since m p 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 G m, i.e. Equation Chapter 4 Section 1 order of G order of H n (4.1) m 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 2m given that the Burgers OR is maintained between and phases, as described by: ; ; (4.2) 112

147 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 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 nonuniformities [35, 36]. For example, for an + two-phase microstructure, twelve nonconserved order parameters, r 1,2,..., 12 p p N, are needed to describe the structural non-uniformities associated with the twelve OVs and two conserved order parameters, X r k Al, V k, 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, r p 13 p, is introduced to describe the spatial distribution of the matrix phase. For example, r and 1 (r)= 2 (r)= = 12 (r)=0 within the -matrix

148 and 1 p r and qp (r) = 0 inside the p-th OV of precipitates under the constraint that N 1 p r 1 [37, 38]. p 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: κ (4.3) N chem T c F Gm T, X k, p X V k p p p dv V m 2 kal, V 2 p1 κ where κ c and κ p 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]. G m 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]: 114

149 G T X h G T X h G T X N N N N 1,,,, 1,, (4.4) m k p p m Al V p m Al V pq p q p1 p1 p1 q p 3 2 in the current study for simplicity. In the above equation, hp p 6 p 15 p 10 is an interpolation function used to connect the free energy curves (as function of concentrations) of the and phases. The N N1 pq pq term introduces a hump on p1 qp 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 pq pq over the commonly used form, pq, is that it creates an energy cusp at the 2 2 p q 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). G m T, X Al, V and G, m T X Al, V are the equilibrium molar free energies of and phases as function of temperature and individual phase concentration X Al, V and X, Al V, respectively. In the current study, these equilibrium free energies are formulated based on a pseudo-ternary thermodynamic database developed for Ti64 [46]. 115

150 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]: el 1 dg V E B n g g C drc r (4.5) N * 0 0 T ( ) ( ) ( ) 3 2 pq p q ijkl ij kl ij ijkl kl pq, 1 (2 ) 2 where B ( n ) describes the elastic strain energy density of a thin precipitate plates with pq n being its habit plane normal. The detailed form of B ( n ) varies with external * boundary conditions. p( g ) denotes the Fourier transformation of p() r and ( ) p g stands for the complex conjugate of ( g ). g is a vector in the reciprocal space. p pq 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 B ( n ) reads pq B pq Cijklij ( p) kl ( q), g=0 ( n) Cijkl ij ( p) kl ( q) ni ij ( p) jk ( n) kl ( q) nl, g 0 (4.6) 116

151 where 00 C is the elastic modulus tensor of the matrix phase and p 0 ijkl is the stress-free ij transformation strain (SFTS) or inelastic strain of the p th orientation variant of the - phase. ( p) C ( p) 1 and n ij ijkl ij n C jk i ijkl l n is the inverse of the Green s function in the reciprocal space, and n g g with ni being its i th component. In Eq.(4.5), the macroscopic homogenous strain, ij, is equal to the pre-strain, appl ij by the initial load applied to the system that has a volume V in the real space. which is established The last term in Eq. (4.5) represents the coupling between the pre-strain and the transformation strain induced by the precipitates, T r. T ij ij r represents the spatial distribution of the SFTS field associated with structural non-uniformities and can be T 00 expressed by r ( p) non-uniformities N ij ij p p1 r, which is a linear superposition of all N types of The total strain energy of the system can be easily obtained and incorporated into the total free energy of the system, i.e., chem el F F E Stress-free transformation strain for coherent and semi-coherent precipitates 00 Derivation of SFTS, ij, associated with a phase transformation is one of the key steps towards formulating the coherency elastic strain energy in the KS microelasticity theory 117

152 [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 118

153 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 dis F F F F (4.7) where step F and dis F represent the deformation gradient matrices due to structural ledges 00 and dislocations, respectively. Then, the transformation strain 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: e, and the total strain is related to T ij ij ij the displacement through ij ui, j uj, i 1 where ui is the ith component of the total 2 displacement field u and u i, j represents it gradient. F u or ij ij i, j F I 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) : 119

154 00 F tot T 2 F tot 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 semicoherent 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 B pq (n 0 ) (from J/m 3 to J/m 3 ) obtained when using coherent and semi-coherent SFTSs. In addition, the minimum of B pq (n 0 ) is reached respectively at n 0 =(-11,-9.85,8.07) and n 0 =(-11,-12.63,11) for coherent and semicoherent precipitates, which deviate respectively 8ºand 0.8 ºfrom the experimentally observed habit plane orientation (-11,-13,11). Therefore, the introduction of defects 120

155 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. 00 T 00 p p S S (4.9) p where the symmetry operations S p 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 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 121

156 broad, side and edge faces are 0.39E m, 0.72E m and E m, respectively, where E is m assumed to be 250 mj/m 2. 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/m 2, 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/m 2, 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, x : 353 1, x 2 : and x : 101. In the global coordinate system where the SFTSs have been derived,, the 3 gradient energy coefficient tensor for the structural order parameters, has a form 1 R 2 3 R T, (4.10) where i 1 3 denotes the eigenvalues of the gradient energy coefficients i 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 122

157 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 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 timedependent 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: 2 m 1 M T, X, r, t V t X t n1 Xk r, t F kj i j k j1 jr, (4.11) chem chem el p r, t 1 F F E L p t pq p, t q, t r r p r, t r, t (4.12) where,, kj i M T X is the chemical mobility [65], L is the mobility of the longrange 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 123

158 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., N 1 p 1. p1 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 diffuseinterface region is treated as a homogeneous mixture of the precipitate and matrix phases with different compositions but equal diffusion potentials [67] Model inputs and parameters All the parameters for the model and materials properties used in the simulations are listed in Table Results 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 124

159 chosen to be 0.25 m and its composition is assumed to be at equilibrium (for Ti-64): at.% Al and 2.38 at.% V. The initial composition of the supersaturated matrix is 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, multiplatform 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 125

160 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 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 prestrain 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 126

161 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 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: appl x ; appl y and appl 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 127

162 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 79.8ºor 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) 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 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. 128

163 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 stepquenched 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 129

164 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, , 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 130

165 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 Discussion 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 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, G L, 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 131

166 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 G L. 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] 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 and can be rationalized by the interaction between the pre-strain and the SFTS of each variant characterized through the interaction energy density, int appl 0 E C 00 ( p). In general, the SFTS of each OV, p ij ijkl kl 132

167 00 ( p), is different from each other when defined in a common reference frame and thus ij 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 133

168 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 prestrain is obtained via a compressive stress (50MPa) along z direction, its principal value is: appl x 3 appl ; y and appl 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 semicoherent 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. 134

169 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) 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 135

170 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. e. Since the system T ij ij ij 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) T is equal to the pre-strain. As the microstructure and thus inelastic strain, 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 = 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 136

171 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 selfaccommodation. 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 137

172 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 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 138

173 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 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, calculated as: GV, can be G nucleus G G c nucleus nucleus Al, cv c c G cal, cv (4.13) V i i ial,v ci c 139

174 where c i is the average composition of the matrix phase, nucleus ci 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 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 The minimum self-elastic strain energy density of a nucleus is 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 140

175 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, c 0.2. c In our simulation, however, = = 150 mj/m 2. 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-toface 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 141

176 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 It is obvious that the interaction is highly anisotropic around the pre-existing plate. Again, negative values of int E 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 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 142

177 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 143

178 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 and ) 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 and 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 angle/axis pair and that for V1, V9 and V11 is described by 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, selfaccommodating plate) will be published in a follow-up paper. 144

179 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 selfaccommodation 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 microtexture 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 145

180 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). 146

181 6) Secondary plates having a misorientation of 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. 147

182 Figures 0.25 m (a) (b) (c) 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

183 (a) (b) 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. 149

184 (a) x y z (b) t = 2 s t = 4 s t =10 s t = 6 s (c) t = 8 s 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

185 (a) x y z (b) t = 2 s t = 4 s t =10 s t = 6 s (c) t = 8 s Figure. 4.4 Variant selection and microstructure development under a pre-stain obtained 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

186 (a) (b) (c) (d) v1 v4 v6 v4 v6 v8 v9 v11 t = 0s t = 1s t = 2s t = 3s (e) (f) (g) (h) t =3s v8 v9 v11 t = 1s t = 2s t =3s 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

187 (a) (b) (c) (d) 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

188 (a) (b) (c) (d) t =10 s t =10 s 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

189 G V =-0.5 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

190 E int = 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

191 Figure 4.9 continued

192 E int = Continued Figure 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

193 Figure 4.10 continued

194 Order parameter Interaction Energy E int Maximum E int Interaction V1-V5 Fig (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

195 Tables Table 4.1 All 12 Burgers orientation variants and symmetry operations associated with them Variants Orientation Relationship Symmetry operation S i // // I // // // // // // // // // // Continued 161

196 Table 4.1 continued Variants Orientation Relationship Symmetry operation S i // // // // // // // // // // // // n uvw denotes a n-fold rotation around axis uvw and superscript of of n uvw -1 n uvw indicates inverse 162

197 Table 4.2 Various model parameters and materials properties used in the simulations Physical properties Symbol Value Unit Temperature T 1023 K Grid size l m Interface width 5 l m Interfacial energy 150,230,300 mj/m 2 Broad-, Side-, End-facet Gradient Coefficients [78] 0.038, 0.089, J m 2 /mol Hump height 192 J/mol Interface mobility L J/m 3 /s Molar volume V m 10-5 m 3 /mol Elastic constants of phase [79] C11, C12, C 97.7,82.7,37.5 GPa 44 Lattice parameter of and a, a, c 3.196,2.943,4.680 Å phase [59] Coherent SFTS[50] Semi-coherent SFTS eff

198 4.6. References [1] Lutjering G, Williams JC. Titanium (Engineering Materials and Processes). Berlin: Springer, [2] Geetha M, Singh AK, Asokamani R, Gogia AK. Ti based biomaterials, the ultimate choice for orthopaedic implants - A review. Prog Mater Sci 2009;54:397. [3] Froes FH. Titanium Alloys: Properties and Applications. In: Buschow KHJ, Robert WC, Merton CF, Bernard I, Edward JK, Subhash M, Patrick V, editors. Encyclopedia of Materials: Science and Technology. Oxford: Elsevier, p [4] Purdy GR. Widmanstätten Structures. In: Editors-in-Chief: KHJB, Robert WC, Merton CF, Bernard I, Edward JK, Subhash M, Patrick V, editors. Encyclopedia of Materials: Science and Technology (Second Edition). Oxford: Elsevier, p [5] Leyens C, Peters M, editors. Titanium and Titanium Alloys-Fundamentals and Applications. Weinheim: WILEY-VCH, [6] Ahmed T, Rack HJ. Phase transformations during cooling in alpha+beta titanium alloys. Mater Sci Eng, A 1998;243:206. [7] 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. [8] Cahn JW, Kalonji GM. Symmetry in Solid-Solid Transformation Morphologies. PROCEEDINGS OF an Interantional Conference On Solid-Solid Phase Transformations 1981:3. 164

199 [9] Peters M, Gysler A, LÜtjering G. Influence of texture on fatigue properties of Ti- 6Al-4V. Metall Mater Trans A 1984;15:1597. [10] Bache MR, Evans WJ. Impact of texture on mechanical properties in an advanced titanium alloy. Mater Sci Eng, A 2001; :409. [11] Bache MR, Evans WJ, Suddell B, Herrouin FRM. The effects of texture in titanium alloys for engineering components under fatigue. Int J Fatigue 2001;23:153. [12] Whittaker MT, Evans WJ, Lancaster R, Harrison W, Webster PS. The effect of microstructure and texture on mechanical properties of Ti6-4. Int J Fatigue 2009;31:2022. [13] Whittaker R, Fox K, Walker A. Texture variations in titanium alloys for aeroengine applications. Mater Sci Technol 2010;26:676. [14] Gey N, Humbert M, Philippe MJ, Combres Y. Investigation of the alpha- and beta-texture evolution of hot rolled Ti-64 products. Mater Sci Eng, A 1996;219:80. [15] Glavicic MG, Goetz RL, Barker DR, Shen G, Furrer D, Woodfield A, Semiatin SL. Modeling of texture evolution during hot forging of alpha/beta titanium alloys. Metall Mater Trans A 2008;39A:887. [16] Bate P, Hutchinson B. The effect of elastic interactions between displacive transformations on textures in steels. Acta Mater 2000;48:3183. [17] 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 Mater 2008;56:5907. [18] Stanford N, Bate PS. Crystallographic variant selection in Ti-6Al-4V. Acta Mater 2004;52:

200 [19] 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 Mater 2003;51:4679. [20] 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 Mater 2007;55:6765. [21] Furuhara T, Takagi S, Watanabe H, Maki T. Crystallography of grain boundary α precipitates in a β titanium alloy. Metall Mater Trans A 1996;27:1635. [22] Lee E, Banerjee R, Kar S, Bhattacharyya D, Fraser HL. Selection of alpha variants during microstructural evolution in alpha/beta titanium alloy. Philos Mag 2007;87:3615. [23] Moustahfid H, Humbert M, Philippe MJ. Modeling of the texture transformation in a Ti-64 sheet after hot compression. Acta Mater 1997;45:3785. [24] Gey N, Humbert M, Philippe MJ, Combres Y. Modeling the transformation texture of Ti-64 sheets after rolling in the [beta]-field. Mater Sci Eng, A 1997;230:68. [25] 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, [26] Wang SC, Aindow M, Starink MJ. Effect of self-accommodation on alpha/alpha boundary populations in pure titanium. Acta Mater 2003;51:

201 [27] Furuhara T, Maki T. Variant selection in heterogeneous nucleation on defects in diffusional phase transformation and precipitation. Mater Sci Eng, A 2001;312:145. [28] 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. Mater Sci Eng, A 2005;392:403. [29] 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, p [30] Bate PS, Hutchinson WB. Imposed stress and variant selection: the role of symmetry and initial texture. J Appl Crystallogr 2008;41:210. [31] Li DY. Morphological evolution of coherent Ti11Ni14 precipitates under inhomogeneous stresses. Philos Mag A 1999;79:2603. [32] Guo W, Steinbach I, Somsen C, Eggeler G. On the effect of superimposed external stresses on the nucleation and growth of Ni4Ti3 particles: A parametric phase field study. Acta Mater 2011;59:3287. [33] Wen YH, Wang Y, Chen LQ. Influence of an applied strain field on microstructural evolution during the [alpha]2 --> O-phase transformation in Ti-Al-Nb system. Acta Mater 2001;49:13. [34] Zhou N, Shen C, Wagner MFX, Eggeler G, Mills MJ, Wang Y. Effect of Ni4Ti3 precipitation on martensitic transformation in Ti-Ni. Acta Mater 2010;58:6685. [35] Wang Y, Li J. Phase field modeling of defects and deformation. Acta Mater 2010;58:

202 [36] Chen L-Q. PHASE-FIELD MODELS FOR MICROSTRUCTURE EVOLUTION. Annual Review of Materials Research 2002;32:113. [37] Steinbach I, Pezzolla F, Nestler B, Seeßelberg M, Prieler R, Schmitz GJ, Rezende JLL. A phase field concept for multiphase systems. Physica D 1996;94:135. [38] Steinbach I, Pezzolla F. A generalized field method for multiphase transformations using interface fields. Physica D 1999;134:385. [39] Cahn JW, Hilliard JE. Free energy of a nonuniform system. I. Interfacial free energy. The Journal of Chemical Physics 1958;28:258. [40] Wheeler AA, McFadden GB. On the notion of a xi-vector and a stress tensor for a general class of anisotropic diffuse interface models. P Roy Soc Lond a Mat 1997;453:1611. [41] Wang Y, Wang H-Y, Chen L-Q, Khachaturyan AG. Microstructural Development of Coherent Tetragonal Precipitates in Magnesium-Partially-Stabilized Zirconia: A Computer Simulation. J Am Ceram Soc 1995;78:657. [42] Wang YZ, Khachaturyan A. Microstructural Evolution during the Precipitation of Ordered Intermetallics in Multiparticle Coherent Systems. Philos Mag A 1995;72:1161. [43] Wu K, Chang YA, Wang Y. Simulating interdiffusion microstructures in Ni Al Cr diffusion couples: a phase field approach coupled with CALPHAD database. Scr Mater 2004;50:1145. [44] Warren JA, Boettinger WJ. Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method. Acta Metall Mater 1995;43:

203 [45] Chen Q, Ma N, Wu KS, Wang YZ. Quantitative phase field modeling of diffusion-controlled precipitate growth and dissolution in Ti-Al-V. Scr Mater 2004;50:471. [46] Wang Y, Ma N, Chen Q, Zhang F, Chen SL, Chang YA. Predicting phase equilibrium, phase transformation, and microstructure evolution in titanium alloys. Jom- Us 2005;57:32. [47] Khachaturyan AG. Some questions concerning the theory of phase transformations in solids. Sov. Phys. Solid State 1967;8:2163. [48] Khachaturyan AG. Theory of Structural Transformations in Solids. New York: John Wiley & Sons, [49] Khachaturyan AG, Shatalov GA. Elastic interaction potential of defects in a crystal. Sov. Phys. Solid State 1969;11:118. [50] 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. [51] Bower AF. Applied Mechanics of Solids: CRC Press, [52] Bollmann W. Crystal defects and crystalline interfaces. Berlin: Springer, [53] Hall MG, Aaronson HI, Kinsma KR. The structure of nearly coherent fcc: bcc boundaries in a Cu---Cr alloy. Surf Sci 1972;31:257. [54] Pond RC, Ma X, Chai YW, Hirth JP. Topological Modelling of Martensitic Transformations. Dislocations in Solids 2007;13:

204 [55] Pond RC, Celotto S, Hirth JP. A comparison of the phenomenological theory of martensitic transformations with a model based on interfacial defects. Acta Mater 2003;51:5385. [56] Furuhara T, Howe JM, Aaronson HI. Interphase boundary structures of intragranular proeutectoid [alpha] plates in a hypoeutectoid Ti---Cr alloy. Acta Metall Mater 1991;39:2873. [57] 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, p.295. [58] Ye F, Zhang WZ, Qiu D. A TEM study of the habit plane structure of intragrainular proeutectoid alpha precipitates in a Ti-7.26wt%Cr alloy. Acta Mater 2004;52:2449. [59] Shi R, Ma N, Wang Y. Predicting equilibrium shape of precipitates as function of coherency state. Acta Mater 2012;60:4172. [60] Mura T. Micromechanics of Defects in Solids,. Dordrecht: Martinus Nijhoff, [61] Read WT. Dislocations in crystals. New York: Jr. McGraw-Hill, [62] Read WT, Shockley W. Dislocation Models of Crystal Grain Boundaries. Phys Rev 1950;78:275. [63] Cahn JW. On spinodal decomposition. Acta Metall 1961;9:

205 [64] Gunton J. D., Miguel M. S., P.S. S. The dynamics of first-order phase transitions. In: Domb C, L. LJ, editors. Phase transitions and critical phenomena, vol. 8, vol. 8. New York: Academic Press, [65] Andersson J-O, Agren J. Models for numerical treatment of multicomponent diffusion in simple phases. J Appl Phys 1992;72:1350. [66] Kim SG, Kim WT, Suzuki T. Phase-field model with a reduced interface diffuseness. J Cryst Growth 2004;263:620. [67] Kim SG, Kim WT, Suzuki T. Phase-field model for binary alloys. Phys Rev E 1999;60:7186. [68] [69] Zener C. Trans. AIME 1946;167:950. [70] Hillert M. Jernkontorets Annaler 1957;141. [71] Semiatin SL, Kinsel KT, Pilchak AL, Sargent GA. Effect of Process Variables on Transformation-Texture Development in Ti-6Al-4V Sheet Following Beta Heat Treatment. Metall Mater Trans A 2013; online at 17 April 2013 [72] Aaronson HI, Enomoto M, Lee JK. Mechanisms of diffusional phase transformations in metals and alloys: CRC Press New York, [73] Enomoto M. Computer modeling of the growth kinetics of ledged interphase boundaries I. Single step and infinite train of steps. Acta Metall 1987;35:935. [74] Enomoto M. Computer modeling of the growth kinetics of ledged interphase boundaries II. Finite train of steps. Acta Metall 1987;35:

206 [75] Madangopal K, Singh JB, Banerjee S. The nature of self-accommodation in Ni Ti shape memory alloys. Scripta Metallurgica et Materialia 1993;29:725. [76] Menon ESK, Aaronson HI. Overview no. 57 Morphology, crystallography and kinetics of sympathetic nucleation. Acta Metall 1987;35:549. [77] Aaronson HI, Spanos G, Masamura RA, Vardiman RG, Moon DW, Menon ESK, Hall MG. Sympathetic nucleation: an overview. Mater Sci Eng, B 1995;32:107. [78] Spanos G, Fang HS, Aaronson HI. A mechanism for the formation of lower bainite. Metall Trans A 1990;21:1381. [79] Russell KC. Nucleation in solids. Phase transformations. Metals Park, OH: ASM, p.219. [80] Shen C, Simmons JP, Wang Y. Effect of elastic interaction on nucleation: I. Calculation of the strain energy of nucleus formation in an elastically anisotropic crystal of arbitrary microstructure. Acta Mater 2006;54:5617. [81] Shen C, Simmons JP, Wang Y. Effect of elastic interaction on nucleation: II. Implementation of strain energy of nucleus formation in the phase field method. Acta Mater 2007;55:1457. [82] Gao Y, Zhou N, Yang F, Cui Y, Kovarik L, Hatcher N, Noebe R, Mills MJ, Wang Y. P-phase precipitation and its effect on martensitic transformation in (Ni,Pt)Ti shape memory alloys. Acta Mater 2012;60:1514. [83] Wang Y, Khachaturyan AG. Three-dimensional field model and computer modeling of martensitic transformations. Acta Mater 1997;45:

207 [84] Teng CY, Zhou N, Wang Y, Xu DS, Du A, Wen YH, Yang R. Phase-field simulation of twin boundary fractions in fully lamellar TiAl alloys. Acta Mater 2012;60:

208 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. 174

209 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 175

210 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 microtextured 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 176

211 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 elastic strain energy associated with the interfaces will alter not only the coherency precipitation, but also the interfacial energy 177

212 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 178

213 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

214 5.2. Model Formulation 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 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 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 180

215 transformation in polycrystalline Ti 6Al 4V (wt.%) sample under the influence of pre-strain or external stress 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, X r k Al, V k, are required to describe the chemical nonuniformities of components Al and V in the system. At the scale of individual grain, twelve non-conserved order parameters,, r 1,2,..., 12 p p N, are needed to describe the structural non-uniformities associated with the total N OVs. One more dependent order parameter,, r p 13 p, 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 N 1 p1 p, r 1[39, 40]. For a polycrystalline consisting of grain, the number of nonconserved 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 181

216 can be simply extended to polycrystalline system from its counterpart for single crystal as follows: Equation Chapter 5 Section 1 M N chem 1 κ 2 1 T c F Gm T, X k, p X k p p V p dv V m 2 kal, V 2 κ 1 p1 (5.1) κ c where and κ are the gradient energy coefficient and gradient energy coefficient p tensors characterizing contributions from non-uniformities in concentration and structure within polycrystalline system, respectively. A specific set of eigenvalues for the κ 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, c T Q Q p p κ is obtained via κ κ where Q is a rotation matrix that describes the orientation of grain. p The bulk chemical free energy density in Eq.(5.1) is expressed as: 182

217 M N M N G T, X, h G T, X 1 h G T, X,, m k p p m Al V p m Al V 1 p1 1 p1 M N N 1 p1 qp pq p q (5.2) where, 3 2 p p p p h is an interpolation function connecting the free energy surface (as function of concentration and temperature) of and phase. N N1 The term p1 qp pq p q introduces a hump on the free energy surface between either variant and q, or variants and matrix, and hump height is proportional to. pq Gm T, X Al, V and G, m T X Al, V are the equilibrium molar free energies of and phases as function of temperature and individual phase concentration X and Al, V X, Al, V respectively 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, 183 T ij r, (transformation strain due to precipitation), and an inhomogeneous distribution of elastic modulus C () r due to of ijkl

218 elastic anisotropy and different orientations of individual grains, i.e., both C () r and T ij r are functions of spatial coordinate r. Elastic strain energy of such an system ijkl under external loading or pre-strain, el T appl E C ( r), r,, is obtained using the ijkl ij ij iterative method developed by Wang et al. [17, 41]. By introducing a virtual strain field 0 r and a reference modulus C 0 () ij ijkl, 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 () r is an energy minimizer of the total strain energy functional ij that determines the equilibrium state of the elastically and structurally inhomogeneous system. In practice, it can be obtained numerically through a solution of the timedependent Ginzburg-Landau (TDGL) type equation [41], 0 el ij ( r, ) E Lijkl ( r, ) 0 kl (5.3) where elastic strain energy of the system, el E, is given by: 184

219 el T 0 T E ( ) ( ) 3 2 V CijmnSmnpq r Cpqkl C ijkl ij r ij r kl r kl r d r V 0 V Cijklij ( ) kl ( ) d ij V Cijkl kl ( ) d Cijklij kl 2 r r r r r d g n ( ) ( ) ( )* 3 iij g jk n kl g nl (2 ) (5.4) where L is the kinetic coefficient tensor (a convenient choice is ijkl 1 L L C [42]) ijkl 0 ijkl and the parameter describes the elastic relaxation process. In Eq.(5.4), 1 0 S r ijkl Cijkl C r ijkl.in order to ensure the convergence [43], 1 0 L C ijkl has been chosen to be 0 1 ijnm Cmnkl r during the iteration of Eq (5.3). Elastic inhomogeneity L C ijkl r C defined in the sample reference is then described by T T T T 0 C r Q r Q r Q r Q r C, where stands for the transpose operator. ijkl ip jq km nl pqmn Einstein s summation convention for repeated indices is assumed throughout. ( ) 0 0 g C g, g is the Fourier transform of r, n ij ijkl kl ij ij C 1 0 ij ijkl k l n n, and the superscript asterisk denotes the complex conjugate. The last integral in Eq. (5.4)(?) excludes a volume of 2 3 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. In this case, the macroscopic shape is obtained by allowing the body to ij 185

220 relax at fixed appl to minimize the strain energy with respect to ij. The energy ij 0 0 appl minimization is obtained when minimizer : = + S, ij ij ijkl ij 1 () d V r r ij V kl 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: M N T ij ij p 1 p1 000, p, r r (5.5) as a linear combination of individual phase field order parameter within each b grain, 000, r., p p is the SFTS tensor of variant in grain expressed in the ij 000 T T 00 sample reference frame. It can be obtained through, p Q Q p 00,where p kl ij ik jl kl 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 ( r) in Eq.. Then, the total strain in the system is: ij ij r 1 ij 2 d 3 g n n n e igr i jk n jik n kl g l (5.6) 186

221 And the stress distribution () r in the polycrystalline system can be obtained through 0 0 C r ij ijkl r kl r kl. ij The elastically and structurally inhomogeneous polycrystalline sample during 0 precipitation is then equivalent to an elastic homogeneous system with modulus C with ijkl the equilibrium internal stress distribution described by Eq. (5.3) 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. 2 m 1 M T, X, r, t V t X t n1 Xk r, t F kj i j k j1 jr, (5.7) There are total M N 1 order parameters for precipitate in the polycrystalline system and the governing equation is given as [38]: 187

222 el p, r, t 1 F F E, rl p r, t t pq p,, t q,, t r r p, r, t (5.8) where, r defines the shape of individual grain that is equal to unity inside the th grain and vanishes outside of it.,, M T X is the chemical mobility, L is the kj mobility of the long-range order parameters characterizing interface kinetics, and k and are the Langevin noise terms for composition and long-range order parameter, p respectively. In Eq. (5.8), is the number of phases that co-exist locally. i 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 Cijkl r is then treated as a constant in the calculation of the functional variation of the strain energy with respect to order parameter field, r. p 188

223 el E, r p T r ( r), r V Cijmn Smnpq C pqkl C ijkl ij ij p 0 T 3 1 kl ( r) kl p, rd r 2, r p (5.9) All the parameters for the model and materials properties can be referred to Ref. [36] 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 189

224 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 gdg V (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: 190

225 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. 191

226 5.3. Results 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. 192

227 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 193

228 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 to 10.45, due to coarsening 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 ( random) at 1.5 s to 23.2 ( random) at 9.0 s. While 194

229 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. Prestrains 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 ) 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 strongtextured sample (Fig. 5.1(c)) with the same grain geometry as that of the randomtextured sample. Pre-strains are also generated by applying a 50Mpa tensile/compressive 195

230 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 prestrains, 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). 196

231 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. 197

232 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 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 198

233 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 to zoom into individual texture and pre-strain/external loading. Thus, it is necessary 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 2 nd and 5 th 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 2 nd grain and, in contrast, only two variants (V8 and V12) survive, though with a relative small volume fractions. Microstructures in 2 nd and 5 th 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 199 grains are

234 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 2 nd grain, the favored variants would be variants V3-V6, and V9-V10 in nucleation and growth stage, while in the 5 th grain, the most favored ones are variants V2, V6, V8 and V12. From the volume fraction analysis, within 2 nd grain, only variants V12, V5, V9, V3, and V11 have been preferentially selected from the above-mentioned most favored ones. Within the 5 th 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 nonuniform 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 200

235 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 2 nd and 5 th 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 (g) and Fig. 5.10(h). It is clearly that, in the 2 nd 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 5 th 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 201

236 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: 202

237 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 almost identical. To be specific, in the case of Z-Comp pre-strain, is 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. 203

238 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 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 204

239 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 crosssections 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 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 205

240 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. 206

241 Figures: (a) Random-Textured (b) Strong-Textured (c) 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

242 (a) (b) (c) t=1.0s t=3.0s t=5.0s (a ) (b ) (c ) 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

243 (a) (b) (c) (d) t=1.5s t=3.0s t=6.0s t=9.0s (a ) (b ) (c ) (d ) 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

244 X - Comp X - Tensil Z - Comp Z - Tensil t=10.0s (a) (b) (c) (d) (a ) (b ) (c ) (d ) Figure.5.4 (a)-(d) Final microstructure in random-textured sample under different pre-strains, and (a )-(d ) corresponding final texture 210

245 X - Comp X - Tensil Z - Comp Z - Tensil (a) (b) (c) (d) (a ) (b ) (c ) (d ) Figure.5.5 (a)-(d) Final microstructure in strong-textured sample under different pre-strains, and (a )-(d ) corresponding final texture 211

246 Random-Textured Strong-Textured (a) (b) (c) 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 randomtexture and strong textured samples under different pre-strain 212

247 Random-Textured Strong-Textured (a) (b) (c) (d) 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 randomtextured and strong-textured sample without variant selection 213

248 Figure.5.8 Degree of variant selection in both random-texture and strong textured samples under different pre-strain 214

249 Random -Textured Strong -Textured (a) (b) 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

250 (a) (c) (b) (d) Continued Figure.5.10 (a)-(b) microstructure in the 2 nd and 5 th 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

251 Figure 5.10 continued (e) (f) (g) (h) 217

252 (a) (c) (b) (d) Continued Figure.5.11 (a)-(b) microstructure in the 2 nd and 5 th grain in strong-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 218

253 Figure 5.11 continued (e) (f) (g) (h) 219

254 (a) (c) (b) (d) Figure.5.12 (a)-(b) microstructure in the 2 nd and 5 th grain in random-textured 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

255 (a) (b) (c) 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

256 (a) (b) 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

257 (a) (b) (c) (d) Figure (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

258 (a) (b) (c) Figure (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 224

259 (a) (b) (c) (a ) (b ) (c ) Figure Examples showing the pseudo variant selection due to 2D sampling effect. EBSD scan is performed along at different layers of the sample 225

260 5.6. References: [1] Banerjee D, Williams JC. Perspectives on Titanium Science and Technology. Acta Materialia 2013;61:844. [2] TITANIUM ALLOYS TOWARDS ACHIEVING ENHANCED PROPERTIES FOR DIVERSIFIED APPLICATIONS. Croatia: InTech, [3] Semiatin SL, Furrer DU. Modeling of Microstructure Evolution during the Thermomechanical Processing of Titanium Alloys. In: L. SS, U. FD, editors. ASM Handbook Volume 22A: Fundamentals of Modeling for Metals Processing [4] 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. [5] Cahn JW, Kalonji GM. Symmetry in Solid-Solid Transformation Morphologies. PROCEEDINGS OF an Interantional Conference On Solid-Solid Phase Transformations 1981:3. [6] Lutjering G, Williams JC. Titanium (Engineering Materials and Processes). Berlin: Springer, [7] Whittaker R, Fox K, Walker A. Texture variations in titanium alloys for aeroengine applications. Materials Science and Technology 2010;26:676. [8] 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:

261 [9] 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, p [10] 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. [11] Humbert M, Germain L, Gey N, Bocher P, Jahazi M. Study of the variant selection in sharp textured regions of bimodal IMI 834 billet. Materials Science and Engineering: A 2006;430:157. [12] Moustahfid H, Humbert M, Philippe MJ. Modeling of the texture transformation in a Ti-64 sheet after hot compression. Acta Materialia 1997;45:3785. [13] Gey N, Humbert M, Philippe MJ, Combres Y. Modeling the transformation texture of Ti-64 sheets after rolling in the [beta]-field. Materials Science and Engineering A 1997;230:68. [14] Gey N, Humbert M, Philippe MJ, Combres Y. Investigation of the alpha- and beta-texture evolution of hot rolled Ti-64 products. Materials Science and Engineering a- Structural Materials Properties Microstructure and Processing 1996;219:80. [15] 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,

262 [16] 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. [17] Wang YU, Jin YM, Khachaturyan AG. Three-dimensional phase field microelasticity theory of a complex elastically inhomogeneous solid. Applied Physics Letters 2002;80:4513. [18] Cahn JW, Hilliard JE. Free energy of a nonuniform system. I. Interfacial free energy. The Journal of Chemical Physics 1958;28:258. [19] Landau LD, Lifshitz E. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion 1935;8:101. [20] 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. [21] 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. [22] Eshelby JD. The Elastic Field Outside an Ellipsoidal Inclusion. Proceedings of the Royal Society A 1959;252:561. [23] Khachaturyan A. Some questions concerning the theory of phase transformations in solids. Soviet Phys. Solid State 1967;8:2163. [24] Khachaturyan AG. Theory of Structural Transformations in Solids. New York: John Wiley & Sons,

263 [25] Khachaturyan AG, Shatalov GA. Elastic interaction potential of defects in a crystal. Sov. Phys. Solid State 1969;11:118. [26] Boettinger WJ, Warren JA, Beckermann C, Karma A. Phase-field simulation of solidification. Annual Review of Materials Research 2002;32:163. [27] Chen L-Q. PHASE-FIELD MODELS FOR MICROSTRUCTURE EVOLUTION. Annual Review of Materials Research 2002;32:113. [28] Emmerich H. The diffuse interface approach in materials science: thermodynamic concepts and applications of phase-field models: Springer, [29] 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, p [30] Shen C, Wang Y. Coherent precipitation - phase field method. In: Yip S, editor. Handbook of Materials Modeling, vol. B: Models. Springer, p [31] Wang Y, Chen LQ, Zhou N. Simulating Microstructural Evolution using the Phase Field Method. Characterization of Materials. John Wiley & Sons, Inc., [32] Wang YU, Jin YM, Khachaturyan AG. Dislocation Dynamics Phase Field. Handbook of Materials Modeling. Springer, p [33] Shi R, Ma N, Wang Y. Predicting equilibrium shape of precipitates as function of coherency state. Acta Materialia 2012;60:4172. [34] 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:

264 [35] 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. [36] 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. [37] Bunge HJ. Texture Analysis in Materials Science- Mathematical Methods. London, [38] Artemev A, Jin YM, Khachaturyan AG. Three-dimensional phase field model and simulation of cubic -> tetragonal martensitic transformation in polycrystals. Philosophical Magazine a-physics of Condensed Matter Structure Defects and Mechanical Properties 2002;82:1249. [39] Steinbach I, Pezzolla F, Nestler B, Seeßelberg M, Prieler R, Schmitz GJ, Rezende JLL. A phase field concept for multiphase systems. Physica D: Nonlinear Phenomena 1996;94:135. [40] Steinbach I, Pezzolla F. A generalized field method for multiphase transformations using interface fields. Physica D: Nonlinear Phenomena 1999;134:385. [41] Wang YU, Jin YMM, Khachaturyan AG. Phase field microelasticity theory and modeling of elastically and structurally inhomogeneous solid. Journal of Applied Physics 2002;92:1351. [42] Wang YU, Jin YMM, Khachaturyan AG. Mesoscale modelling of mobile crystal defects - dislocations, cracks and surface roughening: phase tield microelasticity approach. Philosophical Magazine 2005;85:

265 [43] Shen Y, Li Y, Li Z, Wan H, Nie P. An improvement on the three-dimensional phase-field microelasticity theory for elastically and structurally inhomogeneous solids. Scripta Materialia 2009;60:901. [44] Engler O, Randle V. Introduction to texture analysis: macrotexture, microtexture, and orientation mapping: CRC press, [45] Hielscher R, Schaeben H, Siemes H. Orientation Distribution Within a Single Hematite Crystal. Mathematical Geosciences 2010;42:359. [46] Bachmann F, Hielscher R, Schaeben H. Texture Analysis with MTEX- Free and Open Source Software Toolbox. Solid State Phenomena 2010;160:63. [47] Rycroft CH. VORO plus plus : A three-dimensional Voronoi cell library in C plus. Chaos 2009;19. [48] Gruber J, Ma N, Wang Y, Rollett AD, Rohrer GS. Sparse data structure and algorithm for the phase field method. Modelling and Simulation in Materials Science and Engineering 2006;14:1189. [49] Shi R, Wang Y. Evolution of Microstructure and Transformation Texture due to Variant Selection during Alpha Precipitation in Polycrystalline Titanium alloys- A Simulation Study. To be submitted Acta Materialia [50] Semiatin SL, Kinsel KT, Pilchak AL, Sargent GA. Effect of Process Variables on Transformation-Texture Development in Ti-6Al-4V Sheet Following Beta Heat Treatment,. Metallurgical and Materials Transactions a-physical Metallurgy and Materials Science 2013;under review. 231

266 [51] Ivasishin OM, Shevchenko SV, Vasiliev NL, Semiatin SL. 3D Monte-Carlo simulation of texture-controlled grain growth. Acta Materialia 2003;51:1019. [52] Li DY, Chen LQ. Selective variant growth of coherent Ti11Ni14 precipitate in a TiNi alloy under applied stresses. Acta Materialia 1997;45:471. [53] Wang SC, Aindow M, Starink MJ. Effect of self-accommodation on alpha/alpha boundary populations in pure titanium. Acta Materialia 2003;51:2485. [54] Gourgues-Lorenzon AF. Application of electron backscatter diffraction to the study of phase transformations. International Materials Reviews 2007;52:

267 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. 233

268 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 234

269 formation of large regions of colonies in multiple grains with a common crystallographic feature, which are also referred to as macro-zones [2] or microtextured regions [3] that would have detrimental effect on the mechanical behavior of Tialloys 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 235

270 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. 236

271 6.2. Model formulation and Experimental procedures 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 [ ] [ ], Equation Chapter 6 Section 1 J reads: J BOR (6.1) BOR 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 i S i on 1 BOR J, i.e. 237

272 i 1 BOR BOR J J S i i symmetry operations, S i i. All the 12 orientation variants and the, 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 i th 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: i 2 BOR J J J (6.2) where J represents the misorientation between the two grains and 1 2 i J is the inverse of J i BOR 2 2 BOR. Similar to calculating misorientation 238

273 between two adjacent grains, how J deviates from the BOR can be evaluated 1 quantitatively by defining a misorientation matrix [17, 18] as the following: BOR j BOR1 j 1..6 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 i j 1 ΔJ 1 1 J 2 2 JBOR JBOR J 2 Sj JBOR Si JBOR (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 r1, r2, r 3 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 239

274 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 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 240

275 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 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. 241

276 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: Sj Si J J BOR J. (6.5) BOR The results are summarized in Table 6.1 and illustrated in Fig 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 242

277 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 (GB M ) 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. 243

278 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 (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

279 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 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 245

280 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 246

281 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 247

282 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 248

283 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 249

284 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 250

285 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 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 Such variant selection of GB at special grain 251

286 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 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. 252

287 Figures [111] β [111] β [111] β [111] β [111] β [001] β2 [111] β [001] β1 [001] [001] β1 β1 [001] β2 [001] β2 (a) Type I º/<110> (b) Type II º/<110> (c) Type III- 60º/<110> [111] β [111] β /[ 111] [001] β1 [100] β2 [110] [011] β1 β2 [010] β2 [10 1] (d) Type IV - 60º/<111> β2 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 º/<110>, (b) Type II º/<110>, (c) Type III- 60º/<110> and Type IV- 60º/<111>. 253

288 (a) 1 2 (b) (c) 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

289 1 (a) 2 (b) (c) 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 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 255

290 (a) (b) (c) (d) 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): ). 256

291 (a) (b) 1 2 (c) (d) 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 257

292 (a) (b) 1 2 (c) (d) 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) 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

293 Figure 6.6 continued (b) (110) pole (e) (f) 259

294 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 º/<110> II º/<110> º/<211> III 60º/<110> 60.8 º/< > IV 60º/<111> 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 ) Orientation / Misorientation r 1 r 2 r GB GB Table 6.3 Orientations of two grains shows Type III misorientation in variant selection of GB Predicted orientation of GB (GB ) and its misorientation from the measured one (GB ) Orientation / 260 Misorientation r 1 r 2 r GB GB

295 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 [ ] 261

296 6.6 References: [1] Bache MR. Processing titanium alloys for optimum fatigue performance. International Journal of Fatigue 1999;21:S105. [2] Whittaker R, Fox K, Walker A. Texture variations in titanium alloys for aeroengine applications. Materials Science and Technology 2010;26:676. [3] Banerjee D, Williams JC. Perspectives on Titanium Science and Technology. Acta Mater 2013;61:844. [4] 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. [5] Furuhara T, Takagi S, Watanabe H, Maki T. Crystallography of grain boundary α precipitates in a β titanium alloy. Metallurgical and Materials Transactions A 1996;27:1635. [6] Stanford N, Bate PS. Crystallographic variant selection in Ti-6Al-4V. Acta Mater 2004;52:5215. [7] 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 Mater 2008;56:5907. [8] 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:

297 [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 : TMS (The Minerals, Metals & Materials Society), [10] Lutjering G, Williams JC. Titanium (Engineering Materials and Processes). Berlin: Springer, [11] 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 Mater 2003;51:4679. [12] 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 Mater 2007;55:6765. [13] Semiatin SL, Kinsel KT, Pilchak AL, Sargent GA. Effect of Process Variables on Transformation-Texture Development in Ti-6Al-4V Sheet Following Beta Heat 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,

298 [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, [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. [23] Shi R, Ma N, Wang Y. Predicting equilibrium shape of precipitates as function of 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:

299 [26] Ye F, Zhang WZ, Qiu D. A TEM study of the habit plane structure of intragrainular proeutectoid alpha precipitates in a Ti-7.26wt%Cr alloy. Acta Materialia 2004;52:2449. [27] Lee JK, Aaronson HI. Influence of faceting upon the equilibrium shape of nuclei at grain boundaries II. Three-dimensions. Acta Metallurgica 1975;23:809. [28] Lee JK, Aaronson HI. Influence of faceting upon the equilibrium shape of nuclei at grain boundaries I. Two-dimensions. Acta Metallurgica 1975;23:

300 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 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 266

301 barrier for each variant. The violation could be associated with the characteristic of grain boundary plane orientation population 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 Tialloy 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 267

302 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 fivedimensional 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 268

303 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. 269

304 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 270

305 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 271

306 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. 272

307 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

308 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 274

309 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 focusedion beam (FIB) in FEI NOVA Dual Beam TM (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 275

310 measurements to determine local orientation of GBP can be found in referred to Ref. [17] Results 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 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. 276

311 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 277

312 frequently followed. In contrast, as can be found in Figs , 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 [ ] 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 278

313 GB, which would help to clarify how grain boundary parameters contribute to select one or two variants of GBfrom 24 candidates 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 279

314 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 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.,, 280

315 [ ] [ ], 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 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 281

316 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 grain) nor V8 (for GB 8 with 2 as Burgers grain) has the minimum as Burgers 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 Abnormal cases 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

317 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 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. 283

318 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 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 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 ( 284 )) nucleates on the

319 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, ( ) 285

320 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 certain critical value of, will most likely result in a rise in these energy terms up to. This is akin to the Read-Shockley relationship between grain boundary energy and misorientation angle [21]. The critical angle of be about 15 for the alloy Ti-5553 considered in this study. When seems to, 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 286

321 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 and develop low-energy facets into, while the other one could maintain BOR with. 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 with a smaller is still not clear. For example, the VS of GB shows that the one 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 287 always

322 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 288

323 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 closedpacked 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 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 289

324 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 under most circumstances, it could be independent of increases rapidly with 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 290

325 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 291

326 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. 292

327 7.5. Summary Variant selection of grain boundary (GB) by prior grain boundaries (GBs) in Ti under small undercooling is investigated to understand the effects of grain boundary structure characterized by misorientation and inclination. All existing empirical variantselection (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 293

328 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. 294

329 Figure (a) (b) (b) 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 295

330 (a) (b) (c) 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

331 (a) (b) (c) Figure 7.4 (a) Stereographic projection shows the orientation of GB planes relative to the 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

332 (a) (b) (c) Figure.7.5 (a) Stereographic projection shows the orientation of GB planes relative to the 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 298

333 (a) (b) (c) (d) 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

334 (a) (b) (c) (d) Continued Figure 7.7 Experimental observations of variant selection of GB 28aOIM 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 300

335 Figure 7.7 continued (e) (f) 301

336 (a) (b) 1 2 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 302

337 Figure 7.8 continued (c) (d) (e) (f) (g) (h) (i) (j) 303

338 (a) (b) (c) 1 2 (d) (e) (f) Figure 7.9 Experimental observations of variant selection of GB 31aOIM 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 304

339 (a) (b) (c) (d) (e) (f) 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 305