Computational Studies of Deformation in HCP Metals and Defects in a Lead-Free Ferroelectric Ceramic

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Computational Studies of Deformation in HCP Metals and Defects in a Lead-Free Ferroelectric Ceramic
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Kim,Dong Hyun
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Doctorate ( Ph.D.)
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University of Florida
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Materials Science and Engineering
Committee Chair:
Phillpot, Simon R
Committee Members:
Sinnott, Susan B
Myers, Michele Viola
Jones, Jacob L.
Tulenko, James S

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Subjects / Keywords:
dft -- dislocation -- hcp -- magnesium -- md -- nanocrystalline -- slip -- sodium -- titanium -- twinning
Materials Science and Engineering -- Dissertations, Academic -- UF
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Materials Science and Engineering thesis, Ph.D.
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Abstract:
Plastic deformation of nanocrystalline Mg and Ti is examined using molecular dynamics (MD) simulation. Slip, twinning, and GB processes are observed in textured (2D) columnar and random (3D) microstructures. The deformation simulations of Mg reproduce various twinning modes: tensile {10(1)overbar2} <10(1)overbar1> twins, and compressive {10(1)overbar1} <10(1)overbar2>, {11(22)overbar} <11(23)overbar> , and {10(1)overbar3} <30(3)overbar2> twins. Two pyramidal <c+a> slip modes are manifested in strained structures: first-order {10(1)overbar1} <(11)overbar23> and second-order {11(2)overbar2} <(11)overbar23> . Nucleation processes and mechanisms of dislocations and twins are identified (e.g. slip-assisted twin nucleation mechanism of {10(1)overbar2} <10(1)overbar1> twin, initiation of {10(1)overbar2} <10(1)overbar1> twins by migration at GB). The crossover of initiation process between slip and twinning are found in 2D textures. The strongest grain size in 3D fully dense Mg occurs at 24nm. Single prismatic <a> dislocations and their interactions directly result in formation of <c+a> and <c> dislocations. Among the empirical potentials for Ti, the Henning MEAM potential displays slip and twinning processes most consistent with experiments. The cation arrangements of Na and Bi in Na0.5 Bi0.5TiO3 are investigated using density functional theory (DFT). The structure with alternative stacking of Na and Bi layers in the perovskite axis has the lowest energy of the cation arrangements. The R3c structure, known tp be the room temperature phase of Na0.5 Bi0.5TiO3 has a higher energies than structures with random cation arrangements. The cation-layered structure is revealed to have a P1 phase by distortion of octahedra in its perovskite. To analyze the structure distortion, elements (Ti-Bi and Ti-La) causing the second-order Jahn?Teller effect are chosen and, compared in structure and charge density. The combination of lone pairs and the d0 transition metal can deform octahedra of a perovskite cell more severely than that of d0 transition metals. The cation-layered structure of Na0.5 Bi0.5TiO3 may be influenced by the Jahn-Teller effect, thereby having the lowest energy.
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by Dong Hyun Kim.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
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Adviser: Phillpot, Simon R.

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1 C OMPUTATIONAL STUDIES OF DEFORMATION IN HCP METALS AND DEFECTS IN A LEAD FREE FERROELECTRIC CERAMIC By DONG HYUN KIM A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011

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2 2011 DONG HYUN KIM

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3 To my family with love

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4 ACKNOWLEDGMENTS First, I would like to thank Prof. Simon Phillpot for his support and guidance thro u ghout my Ph. D course I am really fortunat e and happy to have been one of his students I would also like to thank Prof. Susan Sinnott Her passion and manag ement for work and research are impressive to me Although D r. Ebrahimi ha s passed away, her leg acy will be left in my research I also appreciate Prof. Michele Manuel for serving on my committee and for their helpful suggestions for my research. P rof. Kwangho Kim, my master course advisor, made me first realize what research is and how to do it. H e should be also appreciated here Many people have helped to make my time at the University of Florida enjoyable, and particularly I will never forget SINPOT (Sinnott +Phillpot) group member s I must also thank my family for their constant support, even wi th a thousand miles between us. Without them, none of this would have been possible. Finally, I offer my deepest gratitude to my wife H yunjung to whom this dissertation is dedicated. Her unconditional love and encouragement have been instrumental in my success

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 8 LIST OF FIGURES ................................ ................................ ................................ .......... 9 ABSTRACT ................................ ................................ ................................ ................... 15 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 17 1 1. Motivation ................................ ................................ ................................ ........ 17 1 2. Part I : Nanocrystalline Hexagonal Close P ack ed ( HCP ) Metal ........................ 18 1 3 Part II: Na 0.5 Bi 0.5 TiO 3 ................................ ................................ ........................ 19 2 BACKGROUND: MD SIMULATIONS OF NANOCRYSTALLINE HCP METALS .... 21 2 1 Slip in HCP M etals ................................ ................................ ........................... 23 2 2 Twinning in HCP M etals ................................ ................................ .................. 25 2 3 Nanocrystalline M etals and M olecular D ynamics S imulation ........................... 29 2 4 Molecular Dynamics S imulation ................................ ................................ ....... 31 2 4 1 Pressure Control ................................ ................................ .................... 32 2 4 2 Temperature Control ................................ ................................ .............. 32 2 5 Interatomic Potentials ................................ ................................ ...................... 33 2 5 1. Embedded Atom Me thod ( EAM ) P otential ................................ ............. 35 2 5 2. Modi fied Embedded Atom Me thod ( MEAM ) P otential ............................ 35 2 6 Computational Details ................................ ................................ ...................... 38 2 6 1 Generation of S tructure ................................ ................................ .......... 38 2 6 2 Simulation of Mechanical T est ................................ ............................... 42 2 6 3 Analysis Method ................................ ................................ ..................... 43 2 7 Summary ................................ ................................ ................................ ......... 43 3 OVERALL MECHANICAL RESPONSE OF TEXTURED NANOCRYSTALLINE MG ................................ ................................ ................................ .......................... 44 3 1 Stress D ependence of Mechanical R esponse ................................ ................. 44 3 2. Sign atures of Dislocations and Twins ................................ .............................. 45 3 3 Microstructure Evolution ................................ ................................ .................. 50 3 4. Stress A nalysis of D islocation A ctivation ................................ ......................... 59 3 5. Transitions in Dislocation M ode ................................ ................................ ....... 60 3 6. Competition between S lip and T winning ................................ .......................... 61 3 7. Summary ................................ ................................ ................................ ......... 62

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6 4 OVERALL MECHANICAL RESPONSE OF 2D NANOCRYSTALLINE TI METALS ................................ ................................ ................................ ................. 64 4 1 Potentials and Mechanical R esponse ................................ .............................. 65 4 2 Stress dependence of Mechanical Re sponse in Ti ................................ .......... 69 4 3 Comparison of Mg and Ti ................................ ................................ ................ 70 4 4 Summary ................................ ................................ ................................ ......... 75 5 PYRAMIDAL SLIP IN COLUMNAR NC MG ................................ ............... 76 5 1 Occurrence of Slip ................................ ................................ ................ 78 5 2 Structures of Dislocations ................................ ................................ ..... 80 5 3 Activation Process of Dislocations ................................ ........................ 91 5 4 Comparison of Different Potentials ................................ ................................ .. 95 5 5 Role of Pyramidal Dislocations in Plastic Deformation ......................... 96 5 6 Summar y ................................ ................................ ................................ ......... 99 6 TWINNING IN 2D NANOCRYSTALLINE MG ................................ ....................... 101 6 1 Nucleation of Twins ................................ ................................ ....................... 101 6 2 Nucleation Mechanism of Compressive Twinning from Grain Boundaries .... 106 6 3 Nucleation Mechanism of Tensile Twinning from GB ................................ .... 107 6 3 1 Nucleation P rocess in a L arge G rain ................................ .................... 107 6 3 2 Nucleation process in S mall G rains ................................ ..................... 117 6 4 Summary ................................ ................................ ................................ ....... 118 7 OVERALL MECHANICAL RESPONSE OF RANDOMLY ORIENTED HCP METALS ................................ ................................ ................................ ............... 120 7 1 Strain S tress R esponse ................................ ................................ ................. 120 7 2 Hall Petch R elation ................................ ................................ ........................ 121 7 3 Microstructure Evolution ................................ ................................ ................ 124 7 4 Individual D efect P rocess ................................ ................................ .............. 129 7 4 1. S lip ................................ ................................ ................................ 129 7 4 2 < c+a > and S lip ................................ ................................ .............. 130 7 4 3 Twinning ................................ ................................ ............................... 142 7 5 Grain S ize E ffect on P lasticity ................................ ................................ ........ 144 7 6 Summary ................................ ................................ ................................ ....... 149 8 CATION ORDERI NG IN SODIUM BISMUTH TITANATE ................................ ..... 151 8 1 Background ................................ ................................ ................................ ... 155 8 1 1 First Principles Calculations ................................ ................................ 155 8 1 2. Density Functional Theory (DFT) ................................ ......................... 156 8 1 3. Exchange Correlation Functional ................................ ......................... 158 8 2 Computation al D etails ................................ ................................ .................... 159 8 3 Analysis of S tructure C hange ................................ ................................ ........ 162 8 4 Analysis of L ayered NBT ................................ ................................ ............... 164

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7 8 5 Distortion of Octahedra ................................ ................................ .................. 169 8 6 Summary ................................ ................................ ................................ ....... 173 REFERENCE LIST ................................ ................................ ................................ ...... 175 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 185

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8 LIST OF TABLES Table page 1 1 Classification of d efects found in materials [4] ................................ ................... 18 2 1 Physical properties of Mg [24, 25] and Ti[20, 26]. Add density and specific strength ................................ ................................ ................................ .............. 25 2 2 Twinning planes, directions, and shears in p ure z irconium [28]. ......................... 27 2 3 Slip and twinning modes in Mg and Ti [8, 20, 31] ................................ ................ 29 2 4 Models of plane strain and stress for two dimensional system. ........................... 32 4 1 c/a ratio and slip modes of various HCP metals [8]. ................................ ............. 64 4 2 Lattice constants and stacking fault energies of various HCP metals obtained by First principle calculation.[107]. ................................ ................................ .... 65 4 3 Lattice constants of strained 18nm textured Ti at 0.1K. ............................ 66 5 1 Stacking fault energies in magnesium ................................ ................................ .. 94 6 1 Nucleation conditions for for and primary twinning. ................................ ................................ ... 105 7 1 Hall Petch slopes of pure Cu and Mg from MD simulations and experiments. .. 123 7 2 TS 0.1% SF atom ratio TS and 0.1% SF atom ratio of nc Mg at grain sizes of 9, 18, and 36nm ................................ ................................ ................................ ......... 147 8 1 Three different phases and crystallographic data of NBT [175] ......................... 153 8 2 Comparison of ferroelectric properties of PZT[9] and NBT [179] ....................... 155 8 3 Arrangements of Na and Bi in 2 X 2 X 2 super cell. ................................ .......... 160 8 4 Structural parameters and energies of each ordering type. .............................. 162 8 5 Structural information of the layered structure, T5. ................................ ........... 166 8 6 Radius and electronic structures of Bi and La [198 200]. ................................ .. 171

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9 LIST OF FIGURES Figure page 2 1 Comparison of FCC (right) and HCP (left) lattices. ................................ ............ 22 2 2 Simulation methods at different time and length scales. QM, DFT, and MD denotes quantum mechanics, density functional theory, and molecular dynamics, respectively. ................................ ................................ ..................... 22 2 3 Slip planes and directions of (a) HCP [20]and (b) FCC [21] ............................... 24 2 4 Schematics of original and twinned textures K 1 and K 2 denote t he planes of twinning and the conjugate (or reciprocal) twinni ng. 1 and indicate the directions of twinning and conjugate twinning, respectively [27] ........................ 26 2 5 Twining shears as a function of the c/a ration in HCP metals. Filled circles indicate active twin modes. ................................ ................................ ................. 28 2 6 Schematics of tensile twinning [30] ................................ ............. 29 2 7 Schematic representation of the variation of fl w stress as a function of grain size in metals and alloys [53] . ................................ ................................ ............ 31 2 8 Schematic representation of three major atomic bonding ty pes. ..................... 34 2 9 Schematic of 3d periodic simulation cell with four hexagonal grains. In each axis of HCP unit cell and the x axis of the simulation cell. ................................ ................................ ................... 40 2 10 Schematic of 3D fully dense structure of nc HCP metals Each color indicates a different grain. ................................ ................................ ................................ .. 41 3 1 Strain vs. time plots for nanocrystalline structures with 18nm grain size at various external stresses. ................................ ................................ .................. 45 3 2 Snapshot of 3.5% strained structure with grain size of 18nm. Here, and in subsequent figures, gray, black and brown denote normal (HCP), disordered (non HCP or FCC), and stac king fault (FCC), respectively. ............................. 46 3 3 Burgers vectors and circuits of SF/RH type for the dislocations and stacking faults (SF) found in Fig. 3 2 ................................ ................................ ................ 48 3 4 Snapshot and schematic of an ato mic structure for the extended < a > dislocation, as observed for grain size of 40nm under 1.3GPa. ......................... 4 9 3 5 Snapshots of tensile test for 18n textured structure at 1.18 GPa for four times and strains, increasing from 9(a) to (d). ................................ ............ 53

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10 3 6 Activity of tensile twinning in each grain of 18nm textured structure at 1.18 GPa. ................................ ................................ ........... 54 3.7 Snapshots of strained textured structures with grain size of 60nm at 1.3GPa. (a) and (b): total strains are 4.73 and 7.25% at 1.3GPa, including elastic strain of ~3%. ................................ ................................ ......................... 55 3 8 Snapshots of strained textured structures with grain sizes from 6 to 40nm at 1.0 or 1.18GPa. ................................ ................................ ................... 56 3 9 Snapshots of strained textured structures with 40nm grain size under various external stresses from 1.25 to 1.5GPa. ................................ ................. 59 3 10 CNA image and shear stress map of strained textured structures with 18 nm grain size under 1.1GPa. ................................ ................................ .......... 60 4 1 Strain Time curves of 18nm textured Ti as a function of external stress. ................................ ................................ ................................ ................. 67 4 2 PE and CN image of strained textured structures with 18nm grain size under 3.0GPa. ................................ .... 68 4 3 Snapshots of strained textured structures with 18nm grain siz e under 2 75 GPa. There is no change in CN. Two textures have the same phase. ........ 69 4 4 Creep curves of textured Ti having a grain size of 18nm. ....................... 70 4 5 Snapshots of nano structured Ti ( left ) and Mg ( right ) plastically deformed at 3.25 and 1.2GPa, respectively. ................................ ................................ .......... 71 4 6 Formation of tensile twins during cooling th textured Ti from 700 to100K. ................................ ................................ ........................... 73 4 7 Snapshots of shear strain and central symmetry of the 6% strained textured Ti at 3GPa. ................................ ................................ .......................... 74 5 1 Snapshots of pyramidal < c + a > slip activated in and textured structures. Gray, black and brown denote normal (HCP), disordered (non HCP or FCC), and stacking fault (FCC) atoms respectively. ............................ 79 5 2 Common neighbor analysis (CNA) and p otential energy (PE) map of the partial pyramidal dislocation in the denote the cores of the edge dislocations in (b). ................................ ................ 81 5 3 CNA and PE map of the extended pyramidal dislocation observed in the textured structure. ................................ ................................ .................. 82

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11 5 4 Atomic structures of a (a) leading and (b) trailing of the extended < c + a > dislocation including CNA. (c) and (d): corresponding atomic displacement on the 1 st order pyramidal slip plane, ................................ ........................... 83 5 5 Burgers vectors of the 1 st order pyramidal extended dislocation shown in the HCP unit cell. ................................ ................................ ................................ ...... 84 5 6 (a) CNA and (b) PE map of the pyramidal < c + a > dislocation activated in the textured structure. ................................ ................................ .................... 85 5 7 CNA visualization and PE mapping of t he extended < c + a > pyramidal dislocation projected onto the and planes. ................................ ... 88 5 8 Layer structures of the extended pyramidal < c + a > dislocation in th e texture by CNA. ................................ ................................ ................................ 89 5 9 Potential energy distributions (snapshots) and strain energy curves within cylinders of radius R having a center of an edge pyramidal < c + a > dislocation line. ................................ ................................ ................................ ..................... 93 5 10 Pyramidal < c + a > dislocations simulated by Sun potential. (a ) CNA iamge of texture at 1.45 GPa ................................ ................................ ................. 96 5 11 Creep curves and snapshots of strained columnar Mg with 18nm grain size. Cyan, brown and black in (b) and (c) mean a twinned r egion, stacking fault (FCC), disordered (non HCP or FCC), respectively. ................................ ........ 98 6 1 Three types of twins found during creep, from a simulation of a grain size of 40nm under 1.25GPa. ................................ ................................ ..................... 103 6 2 Snapshots of nucleation process of primary twinning at the grain 1 of textured stru cture with 40nm grain size deformed under 1.25GPa. ................................ ................................ ................................ .......... 104 6 3 twinning activated in the 18nm textured Mg. a and c indicate lattice constants of Mg in (e) and (f) ................................ ................................ ........... 106 6 4 Nucleation process of twinning from GB at 293K. ..................... 108 6 5 Potential Well Overlap (PWO) m odel of t win n ucleatio n at GB. ( a ) S implified 2 D GB structure having a twin nucleation site from Fig. 6 2 ............................. 112 6 6 D ifference between E PWO and E Thermal at scales of temperature (T) and length (R) from Mg potentials. RT, T m and R a denote room temperature (=293K), melting temperature and normal atomic distance at 0K. ................................ 113

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12 6 7 c/a ratios and melting temperatures as functions of cohesive energies of various HCP metals. ................................ ................................ ........................ 116 6 8 Nucleation of twin in 18nm textured Mg. (a) An initial GB structure. ................................ ................................ ................................ ... 119 7 1 Strain stres s curves of nanocrystalline Mg with fully dense 3D structure. Each simulation of tensile test was conducted at 293K under uniaxial stress. A strain rate was 1.5 10 9 s 1 during tensile test . ................................ ............. 121 7 2 T he Hall Petch graph in nc Mg. The flow stress is calculated by averaging strain values between 8 and 11% in st rain stress curves of Fig. 7 1. ............... 123 7 3 Snapshot of 11% strained structure of 18nm grain size at 293K with constant strain rate of 1.5 x 10 9 s 1 a and b represent structures befor e and after straining, respectively. ................................ ................................ ..................... 125 7 4 Shear strain map of nc Mg with 36nm grain size. Normal HCP atoms are not shown. . ................................ ................................ ................................ .......... 127 7 5 Shear strain map of nc Mg with 36nm grain size. Normal HCP atoms are not shown. ................................ ................................ ................................ ............ 128 7 6 < a > slip process in a 3D Mg structure with 24nm g r a in size. ( a ) V arious < a > slip vectors and planes in a HCP unit cell ................................ ........................ 130 7 7 Prismatic < a > slip process in a 3D Mg stru cture with 36nm grain size. I I I and V denote prismatic < a > dislocations. II I and IV denote compressive twins. ................................ ................................ .......................... 132 7 8 Dislocation processes which occur from prismatic < a > slip. (a) < a > dislocations shown in Fig. 7 7c are gliding. basal and prismatic < a > . ................................ ................................ ................ 133 7 9 Evolution of the dislocation source in a < c+a > pyramidal slip [23] (a) cross slip of a dislocat ion (b) formation of junction for < c+a > dislocation, and (c) cross slip of < c+a > dislocation. ................................ ................................ ........ 135 7 10 Activation process of pyramidal < c+a > slip from a single prismatic < a > dislo cation. ................................ ................................ ................................ ........ 136 7 11 Activation process of < c > dislocation between two approaching prismatic
dislocations. Green dot line denote a part of the prismatic dislocation gliding on a different plane. ................................ ................................ .............. 136 7 12 Bowing of a prismatic < a > dislocation and activating of second order pyramidal < c+a > slip. ................................ ................................ .................... 137

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13 7 1 3 Energy difference as a function of bending angle. L denotes bending length of a straight dislocation. ................................ .............. 138 7 1 4 Relation between misorientation angle and dislocation energy per unit length (E/L) of screw < a > and pyramidal < c+a > dislocations at dislocation density ( ) of 10 8 cm 2 . ................................ ................................ ................................ ....... 140 7 1 5 Relation between misorientation angle and dislo cation energy per unit length (E/L) of screw < a > and pyramidal < c+a > dislocations at dislocation density 12 cm 2 . ................................ ................................ ................................ 141 7 1 5 Twin nucleation process of the compressive mode (a ~ d) and tensile mode (e ~ h). ................................ ...................... 143 7 1 6 Potential energy map of SFs in 11.1% strained Mg with 36nm grain size. Normal HCP and disordered atoms like GB are not shown here. .................... 145 7 17 Str ain and evolution of SFs with increasing strain at different grain sizes (9, 18, and 36nm). ................................ ................................ .............................. 146 7 1 8 Stress and evolution of SFs with increasing strain at different grain sizes ( a ) 36nm, ( b ) 18nm, and ( c ) 9nm ................................ ................................ ........... 148 7 1 9 SF evolutions of 11.1% strained samples with three different grain sizes. a. 9nm, b. 18nm, and c. 36nm. ................................ ................................ ............ 149 8 1 R3c structure of 2 X 2 X 1 unit cell sizes. White, green, and red indicate titanium, Na or Bi, and oxygen, respectively. ................................ ................... 152 8 2 Pseudo cubic cells of perovskite. (a) The projection of the rhombohedral cell down [001] o pen circles denotes N a/Bi sites [175] (b) The ideal cubic perovskite of ABX 3 (A,B =cation, X= anion) [180]. ................................ ............ 154 8 3 Cation arrangements adopted in the present work. ................................ ......... 161 8 4 Cation ordering in T4 with 5 X 4 X 4 psuedo cubic size. ................................ .. 162 8 5 Energy and corr e lation between structural parameters. (a) 2a 2b. (b) Energy. ................................ ................................ ................................ .............. 164 8 6 Final optimized structures of T2 (left) and T5 (right) in the perovskite axis (a p b p and c p ). ................................ ................................ ................................ ...... 165 8 7 Change of force and energy in T2 and T5 as a function of an ionic iteration step. E and F denote energy and force. ................................ .......................... 166 8 8 A unit cell of optimized T5 (a and b) and projections of simplified Pr 0.5 Sr 0.5 MnO 3 perovskite structure with symmetry of F4/mmc [188] (c). ......... 167

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14 8 9 Charge density images of P1 phase. a. blue, orange, red, and green denote bismuth, oxygen, titanium, and sodium, respectively. ................................ ...... 168 8 1 0 Optimized unit cell of Na 0.5 La 0.5 TiO 3 Compare with Fig. 8 8 (a) and (b) for NBT. ................................ ................................ ................................ ................. 172 8 11 Contour map of charge density of Na 0.5 Bi 0.5 TiO 3 and Na 0.5 La 0.5 TiO 3 on different planes of pseudo cubic type cell. ................................ ........................ 173

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15 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COMPUTATIONAL STUDIES OF DEFORMATION IN HCP METALS AND DEFECTS IN A LEAD FREE FERROE LECTRIC CERAMIC By Dong Hyun Kim August 2011 Chair: Simon R. Phillpot Major: Materials Science and Engineering P lastic deformation of nanocrystalline Mg and Ti is examined using molecular dynamics (MD) simulation. Slip, t winning, and GB processes are observed in textured (2D) columnar and random (3D) microstructures. The deformation simulations of Mg reproduce various twinning modes : tensile twins and compressive and twins. T wo pyramidal < c + a > slip modes are manifested in strained structures : first order and second order Nucleation processes and mechanisms of dislocations and twins are identified (e.g. slip assisted twi n nucleation mechanism of twin, initiation of twin s by migration at GB). The crossover of initiation process between slip and twinning are found in 2D textures. The strongest grain size in 3D fully dense Mg occurs at 24nm S ingle prismatic
dislocations and their interactions directly result in formation of and dislocations. Among the empirical potentials for Ti, the Henning MEAM potential displays slip and twinning processes most consistent with experiments.

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16 The cation arrangements of Na and Bi in Na 0.5 Bi 0.5 TiO 3 are investig a ted using d ensity functional theory (DFT) The structure with al ternative stacking of Na and Bi layers in the perovskite axis has the lowest energy of the cation arrangements The R3c s tructure, known tp be the room temperature phase of Na 0.5 Bi 0.5 TiO 3 has a higher energies than structures with random cation arrangements The cation layered structure is revealed to have a P1 phase by distortion of octahedra in its perovskite. To analyze the structure distortion, elements (Ti Bi and Ti La) causing the second order effect are chosen and, compared in structure and charge density. The combination of lone pair s and the d 0 transition metal can deform octahedra of a perovskite cell more severely than that of d 0 transition metal s. The cation layere d structure of Na 0.5 Bi 0.5 TiO 3 may be influenced by the Jahn Teller effect, thereby having the lowest energy.

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17 CHAPTER 1 INTRODUCTION 1 1. Motivation A wide variety of materials are used as a componen t of devices and machines. Bot h natural materia ls and engineered materials usu ally contain defects. W e control the properties of the materials by removing multiplying or manipulating the se defects. For examp le, the numbers of vacancies or dislocations should be minimized in a silicon wafer to be used as a substrate of a high quality for electron devices [1] A tra nsistor is typically designed with a n n or p type semiconductor producing excess electrons or h ole s. Dopants are essential to make semiconductors perform consistently and can be consi dered as defects in the base materials e.g. Si or Ge [2] In metals, a h i gh density of dislocation s or impurities can enhance their strength by preventing slip process [3] It is a class ic and still significant issue in m aterials science and engineering to control various defects and use their interactions properly D efects appear with various shapes inside materials f rom point defects such as vacancies interstitial or substitutional atoms to one dimension al defects like dislocations and stacking faults and higher dimensional defects including GBs, voids and cracks [4] (see Table 1 1) Microscopic defects are big enough to be experimentally examined without any significant d ifficulty. However, t he atomistic defects and their mechanisms of creation and interaction are still hard to probe in spite of the development of analy tic instruments. In the present dissertation, a computational study is thus carried out to reveal such defect process in two representative materials : one a metal the other a ceramic.

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18 Table 1 1 Classification of d ef ects found in materials [4] Dimension N ame s Point ( 0 D ) vacancy, interstitial, Schott k y, Frenkel, antisite Line ( 1 D ) D islocation stacking fault I nterfacial ( 2D ) G rain boundary, inter phase boundary free surface Bulk ( 3D ) cavit y gas bubble crack, 1 2. Part I : Nanocrystalline HCP Metal Among HCP ( h exagonal c lose p ack ed ) metals, t he importance of m agnesium and titanium alloys in the automotive and aerospace ( jet engines missiles and spacecraft ) industry has greatly increas ed in recent years due to their high specific strengths and light weight [5, 6] Titanium is strong, lustrous, and is corrosion resistant Titanium can be used in its elemental form or can be alloyed with iron aluminum vanadium and molybdenum However in case of Mg, its alloys are favored over pure element due to its susceptibility to oxidation and corrosion. T he most widely us ed magnesium alloys adopt the Mg Al system [6] Their applications are normally limited to temperatures of up to 120 C due to decreasing of strength [6] Further improvement in the high temperature mechanical properties of magnesium alloys will greatly expand their industrial applications. As engineering material s the mechanical properties of Mg alloys need to be full y understood Plasticity i n the metals is normally achieved by a number of deformation events most particularly slip arising from dislo cation s and twi n ning The mechanical response can be changed significantly by promoting or preventing t h e se deformation processes Reducing grain size is one simple method to enhance strength Although nanocrystalline metals exhibit higher strength than coarse grained ones [7] softening

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19 (or inverse Hall Petch behavior) is also found in ultra small grain size of < ~20nm (see Fig. 2.5) Not only is the collective response of nanocrystalline samples analyzed in this dissertation, but also the d efect u nit proces s es The m echanical response s of Mg and Ti are explored using molecular dynamics ( MD ) simulations. The c/a ratio largely determine s the nature of the deformation slip mode s that the metal manifests [8] The anisotropy in HCP metals is revisited with regard to slip and twinning in Chapter 2. In Chapter s 3 through 7 the deform ation behavior of pol y crystal s with columnar texture is invest igated Such samples are selected to understand fundamental properties of plastic deformation The basic phenomena found in columnar nc Mg are verified in fully dense 3D structure in Chapter 8 The MD simulations of 3D Mg elucidate the r elation ship among deformation mechanism s, grain size, and strength T he fully dense 3D Mg polycrystals more closely represent typical experimental samples, and thus should more accurately reproduce experimental observations. 1 3 Part II: Na 0.5 Bi 0.5 TiO 3 With applications in electronic devices, PZT( Pb [ Zr x Ti 1 x ] O 3 ) show s excellent ferroe lectric and piezoelectric properties [9 12] The r e are strong environmental dr ives to replace lead containing PZT with a lead free alternative NBT(Na 0.5 Bi 0.5 TiO 3 ) is one of the promising substitute s for PZT [13 15] The ferroelectric and piezoelectric properties of ferroelectrics depend strongly on the on crystallography and defects e.g. impurit ies and vacanc ies Na and Bi, which occupy the A site in a perovskite structure of ABO, are known to usually be randomly distributed in NBT having R3C symmetry. The fundamental issues are defined in Chapter 9.

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20 In Chapter 10 the energies of various orderings of Na and Bi are determined using D ensity Functional Theory ( DFT ) calculation Different arrangements of Na 1 + and Bi 3 + m ay cause different local distortion s in bond lengths and electron distribution s in the NBT structure due to their very different charges The defect structure determined from the DFT calculations are compared with those determined from experiment.

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21 CHAPTER 2 BACKGROUND: MD SIMUL ATIONS OF NANOCRYSTA LLINE HCP METALS Before beginning MD simulations of n c HCP metal s, the fundamental bac k ground is reviewed in t his chapter T o help understanding plastic deformation in nc HCP metals c rystal structure s deformation modes and the basics o f nanocrystalline metals are discussed. In addition the theor y and practice of molecular dynamics simulations are discussed The crystal structure of a metals largely determined the deform ation mechanisms that control plasticity M etal s normally shows FCC ( face centered cubic ) BCC ( b ody centered cubic ) or HCP ( hexagonal close packed ) lattice structure FCC and HCP are similar structure s as shown in Fig. 2 1. This similarity means that that a stacking fault in FCC corresponds to a region of HCP, a nd vice versa. The FCC lattice ha s a stacking order of ABCABC, while the HCP lat tice ha s a stacking order of ABAB AB as shown in F ig. 2.1 In HCP crystals, the a tomic distance and unit length in the stacking direction are noted as a and c respectively. The c / a ratio known as the anisotropy, is different in different HCP systems and is determined by details of the electronic structure. As discussed in Section s 2 1 and 2 this anisotropy is related with determining the deformation modes in HCP metals M etals, compared to ceramic, h ave good plasticity at even room temperature because of the occurrence of deformation mechanisms such as slip and twinning. Normal engineering metals have polycrystalline structure s and their grain boundaries (GBs) increase strength and decrease plasticity by preventing conventional deformation process of slip and twinning. However, the GBs activate another mechanism in

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22 nanocrystalline metals. As discussed in Section 2 3 these GB processes can increase t he plastici ty and decrease the strength Figure 2 1. Comparison of FCC (right) and HCP (left) lattice s The circle s and red line s indicate atom s and the Brava is lattice s respectively. The letters denote which equivalent layers Figure 2 2 Simulation methods at different time and length scales. QM, DFT, and MD denotes quantum mechanics, density functional theory and molecular dynamics, respec tively. MD and QM are atomistic simulations. N umber s of atoms indicate minimum and maximum numbers of atoms normally calculated at simulation of QM and MD

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23 A c omputational method effective for n anocrystalline HCP metals is a lso explained in the Sec tion 2 3 C omputational methods can be categorized in terms of the scales of length and time as show in in Fig. 2 2 MD simulation operates at the appropriate length scale for nanocrystalline materials The MD simulation method is discussed in Section 2 4 A number of deformation studies of increasing complexity are described in this dissertation. 2 1 Slip in HCP M etals The plane on which slip takes place is different in different HCP systems. However, the three dominant plane s are the (0001) basal plane, the three prismatic planes and the six pyramidal planes as shown in Fig. 2 3 (a) In all cases, however, the s lip direction is one of the three or < a > close packed directions [16, 17] While p rismatic < a > and basal < a > slip is quite common, t he activation of the py ramidal < a > slip system is less common H owever it can take place in polycrystalline aggregates and has been shown to occur primarily due to the large stresses generated in the grain boundary regions arising from the incompatibility of textures between neighboring grains [16] If atoms were ideal hard spheres the c/a lattice parameter ratio of all HCP metals would be 1.633 [17] The c/a ratio of Mg ( 1.623 ) is almost ideal, and s lip in Mg takes place domin antly on the basal plane The preference of an HCP metal for basal or prismatic slip depends on its c/a ratio and on the electronic structure associated with its d electrons [18] The p rimary slip mode is prismatic in Ti, Zr, and Hf, which all have 2 d electrons; Co which has 7 d electrons, s how s predominately basal slip [19]

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24 Fig ure 2 3 Slip planes and directions of (a) HCP [20] and (b) FCC [21] T he slip directions on the basal plane, known to be activated most easily, are perpendicular to the <0001> c axis Such slip does not produce any elongation or contraction parallel to the c axis [16] This indicates that
type dislocations alone cannot produce homogeneous plastic def ormation There are four independent glide systems: two basal (a 1 a 2 ) and two prismatic (a 1 a 2 ) components in Fig. 2 3 [20, 21] The third basal component (a 3 ) the third prismatic component (a 3 ) and the pyramidal can all be constructed as linear combinations of the four independent glide systems. Given von Mises criterio n that five independent slip systems are necessary for a polycrystalline material to undergo general homogeneous deformation, slip or twin sys tems with < c + a > slip/twin directions must also be operative [20] for homogeneous deformation to take place. Neverthele ss particularly in pure Mg, because the prismatic < a> type dislocations can generally only be activated at elevated temperature [22] fewer slip systems are typically operat ive during plastic deformation [20, 23]

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25 T his insuf ficiency in the range of dislocation process es for homogeneous plastic deformation is partially compensated by twinning [17, 19, 20, 23] Twinning modes in HCP structures are particularly significan t for plastic deformation and ductility at low temperatures if the stress axis is parallel to the c axis and if the dislocations with basal plane Burgers vectors cannot move [20] As for the slip processes, the operative twinning systems are strongly correlated with the c / a ratio [18, 19] As shown in Table. 2 1 the c/a ratios of Mg and Ti are quite different from each other and are on opposite ends of the range manifested by HCP metals. Table 2 1 Physical properties of Mg [24, 25] and Ti [20, 26] Add density and specific strength Properties Mg Ti L attice constant a 3.209 2.951 c 5.210 4.679 c/a 1.624 1.586 Melting point /K 923 1943 Bulk modulus/GPa 35.2 109.7 Shear modulus/GPa 16.5 42 C 11 63.5 176.1 C 1 2 26.0 86.9 Elastic constants C 1 3 21.7 68.3 C 33 66.5 190.5 C 44 18.4 50.8 2 2 Twinning in HCP M et als D eformation twinning is classically defined as re orientat ion of the original lattice by atom displacements corresponding to a simple shear of the lattice points. The invariant plane and direction of this shear is called K 1 and 1 respectively. Similar ly, the second undistorted plane ( K 2 ) and its shear direction ( 2 ) can be defined. T he shear pl ane normal to K 1 and K 2 is denoted by P in Fig. 2 4.

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26 Fig ure 2 4 Schematics of original and twinned textures K 1 and K 2 denote t he p lanes of twinning and the conjugate (or reciprocal) t winni ng 1 and i ndicate the directions of twinning and conjugate twinning respectively [27] The reorientation of original lattice essentially results in stress by a lattice mismatch between the original and twinned textures. The lattice mismatch is normally defined by twinning shear. Figure 2 5 shows how the twinning shear appears at real twin modes. T winning shear is the difference of K 2 before and after twinning. Calculated magnitudes of shear and the other twinning elements are shown in Table 2 2

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27 Fig ure 2 5 Twinning shears at three important twin modes of zirconium [28] Table 2 2 Twinning planes, directions, and shears in p ure z irconium [28] Twinning or First u ndistorted plane, K1 Twinning shear d irection, 1 Second u ndistorted plane, K2 Twinning shear d irection, 2 Magnitude of s hear 0.167 0.63 0.225

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28 Twinning shear mentioned above is for zirconium having c/a ratio of 1.598. Since each HCP metal has a different c/a ratio the twinning shear depends on the c/a ratio. However, each twin shear can be described by the same mathematical expression by the c/a ratio (= ). For instance, the twin has common shear equation of in all HCP metals [27] Yoo [29] has reported a relation ship between the dominant shear mode and the c/a ratio of HCP metals as s hown in Fig. 2 5. Fig ure 2 5 Twining shears as a function of the c/a ration in HCP metals. F illed circles indicate active twin modes. There are t wo common twinning modes in Mg : the ten sile twin a nd the compressive twin [19] Fig ure 2 6 de picts the structure and loading conditions of the more common tensile twinning [30] with the orientations of the crystals on the two sides of the twin differing by 86.3 This t winning is favored by tensile stress along [0001] and compressive stress along as shown n Fig. 2 6 (b). Fundamental slip and twinning properties of Mg and Ti are also compared i n Table 2 3

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29 Fig ure 2 6 S chematics of tensile twinning [30] : (a) The twinned structure has mirror symmetry at an angle of 86.3 in Mg ; (b) ap plied loading in the favorable direction (solid arrow) can most easily cause the tensile twin n i ng. Table 2 3 Slip and twinning modes in Mg and Ti [8, 20, 31] Deformation modes Mg Ti
slip Predominant Basal, [0002] Prismatic, Secondary Prismatic, Basal, [0002] slip Predominant 2 n d o rder, 1/3 1 st order, Secondary 1 st order, 2 n d order, 1/3 Twinning Predomina nt Secondary 2 3 Nanocrystalline M etals and M olecular D ynamics S imulation T he mechanical response and the deformation process es of conventional coarse grained polycrystalline metals such as Fe, Ni, Al, Ti and Mg, have been elucidated in some detail [32, 33] The study of ultrafine grained (250~1000nm) and nanocrystalline

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30 (< 100 nm) metal s which have been produced since the 1980s, has provided fundamental understanding of the physical and chemical phenomena associated with small length scale s [34 37] In comparison to conven tional polycryst alline metals, n anocrystalline m etals have more interesting properties such as increased strength and hardness reduced elastic moduli and ductility and enhanced diffusivity [ 37] Computational methods, in particular molecular dynamics (MD) simulation have provided important insights into the structure and properties of nanocrystalline metals [38 50] D islocation process es [40, 49, 50] and g rain boundary ( GB ) phenomena [41, 44 48] have been characteriz e d in nano grains of various FCC metals In particular, t he existenc e of twinning in n anocrystalline Al was identified by simulation [36, 38] before it was observed in experiment [51] In addition considerable attention has been paid to the question of the existence of a strongest grain size (maximum yield point) at the crossover from normal (dislocation process dominated) to inverse (grain boundary process dominated) Hall Petch behavior [36, 37, 42, 43] The t ypical Hall Petch graph of ultra fine and nanocrystalline metals and their alloys is shown in Fig. 2 7 Previous MD studies have mainly focused on FCC metals such as Al [49, 50] Cu [42, 43] and Ni [41, 44] Relatively little attention has been paid to nanocrystalline HCP metals such as Mg, Ti, Co and Zr despite their industrial importance One MD study of 3D nanocrystalline HCP Co identified both an extended and a partial dislocation [52] In addition it was found that mechanical twinning seldom took place during deformation, even at high stress levels This was somewhat surp rising because twinning is known to be a significant deformation mechanism in coarse grained HCP metals T hese findings

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31 indicate that plastic behavior of nanocrystalline HCP metals m ay be substantially different from nanocrystalline FCC metals or coarse gr ained HCP metals. Fig ure 2 7 Schematic representation of the variation of flow stress as a function of grain size in metals and alloys [53] T h e flow stress is adopted instead of yield stress in the Hall Petch analysis 2 4 Molecular Dynamics S imulation Molecular dynamics (MD) simulation [54] is a deterministic approach, in which the motion of atoms in a system is predicted by solving Newton s Equation of motion. According to Newton s second law (2 1 ) W here, F i is the force on an a tom i; m i and a i are its mass and acceleration The force on each atom can be expressed in terms of the gradient of the potential energy with respect to position. V (2 2 )

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32 W here, V is the potential energy. This is calculated through the interatomic potential that describes the interaction of the atoms in the simulation system 2 4 1 Pressure Control There are va rious algorithms developed to maintain an average pre ssure in the system. The Anders e n scheme uses hydrostatic approach to maintain the pressure [54, 55] This method allows the simulation system to expand or contract the same amount in each direction. For a more complex scheme, which can allow different expansions or contractions in different directions, Parrinello and Rahman [56, 57] extended the Andersen scheme to allow change in both shape and size of the simulation system. Stress analysis can be simplifi ed when the physical dimensions and the distribution of loads allow the structure to be treated as one or two dimensional. For a two dimensional analysis a plane stress or a plane strain condition is typically assumed [58] A plain stress condition is said to exist when stress in the z direction is zero but strain i n the z direction is not zero. Also plain strain condition exists when the strain in z direction is zero. Properties of p lane strain and stress models are explained in Table 2 4 Table 2 4 Models of p lane strain and stress for two dimensional system. Plane Stress Models Plane Strain Models No loading or stresses n ormal to the plane z = 0, xz = 0, yz = 0 z 0 Strain occurs only in the xy plane z = 0 xz = yz = 0 z 0 2 4 2 Temperature Control In order to maintain the temperature during simulation, thermostat s can be implemented. Similar to the constant pressure scheme, the instantaneous temperatu re of the system is calculated during simulation.

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33 (2 3 ) W here N is the total number of atoms in the system, m is the mass of each atom, v is the velocity k B is the Boltzmann s constant, and N df is the number of internal degrees of freedom of the system. Theref ore, the average instantaneous temperature at any time can be expressed as (2 4 ) The instantaneous tempera ture is compared with the target temperature (T 0 ) to adjust the temperature. There are various constant temperature schemes available, including velocity rescalin g and Berendsen [59] Langevin [60 62] and Nos Hoover [63, 64] methods The simplest approach to maint ain the temperature is simple velocity rescaling Since the temperature of the system is related to the velocity of each atom present in the system, the velocity of the atoms can be adjusted in order to control the system temperature. This is given by (2 5 ) W here is the rescaled velocity and is the velocity prior to rescaling. Most of the MD simulation s discussed in this dissertation are performed with velo c ity rescaling to maintain the temperature of the system. 2 5 Interatomic Potentials Interatomic potentials form th e basis for all classical molecular dynamics (MD) simulations. They determine the forces that the atoms experience in a simulation, and it

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34 is through them that material properties are specified. There are various potentials describing interactions between atoms with different mathematical expression. It is important that the expression strongly depends on property of interatomic bonding. P ure Al having metallic bonding is usually simulated with an EAM ( Embedded Atom Method ) potential H o wever, if the Al forms ionic bonding with other element, the Al EAM potential cannot produce realistic results. Nowadays potentials for multifunctional materials are being developed, and used in many simulation fields. Figure 2 8 shows m any of the most challenging and impo rtant applications of materials involve interfaces between disparate bonding environments [65] In this study, EAM and MEAM (Modified Embedded Atom Method ) potentials effectively describing metallic bonding are chosen for mechanical properties simulation Fig ure 2 8 Schematic representation of three major atomic bon d ing types For metallic, covalent, and ionic bonding Mg EAM [24] Si COMB [66] and MgO Buckingham type potential [67] with long range electrostatic interaction were used in this figure [65]

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35 2 5 1. Embedded Atom Me thod ( EAM ) P otential The original Embedded Atom Me thod (EAM) potential is an empirical, many atom description of the total energy of a metallic system. The EAM potential has been extensively applied to various systems of interest and has proved to that compare well with experimental findings The EAM potentials, first proposed by Daw and Baskes [68] has been applied to a wide range of prope rties of metals, including point defects, dislocation s and surface s In the embedded atom formalism the total energy of a system is expressed as (2 6 ) Here is the pair interaction energy between atoms i and j at positions and and is the embedding energy of atom i in Eq. 2 6 is the host el ectron density at site i induced by all other atoms in the system. (2 7 ) i s the spherically averaged atomic electron density of atom j 2 5 2. Modified Embedded Atom Method ( MEAM ) P otential The Modified Embedded Atom Method (MEAM) is an extension to the EAM [68] The extension lies in the fact that angular forces and therefore the effects of directional bonding ar e included. The total potential energy ( E tot ) in MEAM potential has the same form as in the EAM potential. The essential difference between the EAM and the MEAM is in the way in which the electron density n i is calculated, In MEAM it is written as: (2 8 )

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36 (2 9 ) where the are corrections t o that include angular dependence and the are weighting factors. Although other for ms for have been proposed [69] the form shown in Eq. 2 8 and 9 is widely used due to its high accuracy The are written as: (2 1 0 ) (2 1 1 ) (2 1 2 ) (2 13 ) where v and denote the components of the vector between atoms i and j The has spherical symmetry S etting to zero would retrieve the EAM expression for with being the spherically averaged atomic electron density of Eq. 2 1 0 The partial atomic electron densities are assumed to be given by a simple exponential for m: (2 14 ) where k = 0 3, are parameters to be determined and where r 0 denotes the equilibrium nearest neighbor separation in a reference structure The embedding function ( F ) i s given by

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37 (2 15 ) where A is a scaling parameter to be determined, E 0 is the sublimation energy ( the negative of the cohesive energy E coh ), and N 0 is the number of nearest neighbors in the reference stru cture ( N 0 = 12 for HCP metals ). Using Eq. 2 6 and 7 the equat ion for the energy per atom in the reference structure ( E u ) as proposed by Rose et al. [70] can be written as : (2 16 ) w here n 0 = N 0 denotes the effective coordin ation number of an atom The universal energy function [70] ( E u ) i s given by: (2 17 ) w ith (2 18 ) w here d is an adjustable parameter. The e xponential decay factor ( ) is rela ted to the bulk modulus and the atomic volume by : (2 19 ) Now we can write the energy per atom for any configuration of single type atoms at r = r 0 as (2 20 ) where N denotes the numbe r of nearest neighbors for a specific configuration.

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38 2 6 Computational D etails In this section, a methodology for MD simulations of nc HCP metals is presented 2D and 3D structures of nc metals are prepared in Section 2 6 1. For the 2D colum nar textures, the specific crystallographic direction is chosen so as to activate the dominant slip and twinning in a given HCP metal. Two different tests (creep and tensile tests) are carried out in Section 2 6 2. Analysis methods of strained s amples are also discussed in Section 3 3. 2 6 1 Generation of S tructure A 2D texture d structure is prepared for the simulation of the mechanical properties of nanocrystalline HCP metals. S ince nanocrystalline metals can be easily grown using th in film processes, the columnar structure has been a popular choice for basic experimental studies and comparisons to simulation results [37, 71, 72] Compared to a 3D structure with a random orientation in each gr ain, a columnar structure allows the simulation of a larger grain size for the same number of atoms. It also makes visualization of deformation processes easier. The simulated polycrystalline structure contains four grains and has a columnar texture. Thus, each grain in the structure has the same crystallog raphic orientation along the z axis and the grains are separated from each other by tilt grain boundaries As noted in previous studies of FCC metals [73] it is necessary to carefully select the crystallographic orientation of the columnar structure so as to allow dislocation process es In addition, for plastic deformat ion of HCP metals twinning has to be taken into account in determining the optimal crystallographic orientation In the simulation cell shown in Fig. 2 9 t he directions is normal to the

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39 texture direction (the z axis), thus promotin g both basal slip parallel to the x y plane and allowing twinning (see Fig 2 3 ). The HCP structure has two fold rotational symmetry about the tilt axis; thus it would be natural to choose misorientation angles of 0, 30, 60, and 90, as in the work for [011] Al [38, 49, 50, 73] However to avoid twinning in the original structure at an angle of 86.3 misorientation angles of 11.25, 33.75, 56.25 , and 78.75 wer e used, as illustrated in Fig. 2 3 Each grain has the direction as the z axis in Fig. 2 9 To compare deformation properties to the t exture, the texture is also tested. T he resulting boundaries are high angle tilt GBs, with highly disordered atomic structure s. The thickness of the simulation cell in the z direction (parallel to the texture) is ~5.0 a 0 ( a 0 = 0.3206 nm), determined by the cutoff radius of the interatomic pote ntial used for the simulations [24] For a tensile load along the x direction in Fig. 2 9 two of the three 1/3 slip d irec tions ( a1 and a3 ) on the basal plane can be activated; the third slip plane, ( a2 ) lies no r mal to the load axis. The x y dimensions range from 1 3 11 nm (for d = 6 nm, giving a total of 9,595 atoms) up to 11 2 97 nm (for d = 60 nm, and a total of 743 ,705 atom s). The Schmid factor s for basal < a > slip for grain s 1, 2, 3, and 4 are 0.46, 0.19, 0.19 and 0.46, respectively The plasticity behaviors obtained from 2D textures are investigated in fully dense 3D structure s As seen in Fig. 2 10 c ube type simulation cel ls containing 16 randomly oriented Voronoi grains. The grain size varies from 6 to 36 nm. The largest number of atoms for a grain size of 36n m, is 16.1 million.

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40 Fi gure 2 9 Schematic of 3d periodic simulation cell with four hexagonal grains. In each g rain, the angle axis of HCP unit cell and the x axis of the simulation cell.

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41 Fi gure 2 10 Schematic of 3D fully dense structure of nc HCP metals Each color indicate s a different grain. Due to their anisotropy, the interatomic interactions in HCP metals are typically described by a modified embedded atom method (MEAM) potential [74, 75] However, because the anisotropy in Mg is weak (the c/a ratio is close to ideal) it is possible to apply the simpl er EAM formalism [24] The Mg potential used here [24] has been employed in simulation s of mechanical behavior (e.g. twinning [76] and dislocation [77] processes ) and crystal melting [78] for both sing le component systems and alloys. The s tacking fault energy (SFE s ) as deter mined for this potential, of the intrinsic I1 (ABABACAC) and in trinsic I2 ( ABA B CA C A) type SFs are 27 and 54mJ/m 2 respectively. The latter is consistent with e xperimental values for the I2 type stacking fault energy ( SFE ), variously reported as <50 [79] 60 [80] and 78 mJ/m 2 [81] D ue to its high anisotropy, a MEAM potential is chosen to describe Ti i nstead of an EAM potential In the present study three different MEAM potentials are examined for mechanical simulation of nc Ti metals [26, 75, 82 85] Nanocrystalline structures were constructed in a manner similar to that previously used for FCC structures [73, 86] In particular, a fter removing the small number of atoms at the GBs that have unphysically high energies th e structures we re successively annealed at high, low and room temperatures. Such high temperature (400K) a nnealing enabled a significant amount of atomic rearrangement and diffusion at the GB; the low temperature (200K) and room temperature (293K) anneals equilibrated the structure for the room temperature deformation simulations.

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42 2 6 2 Simulation of Mechanical T est Mechanical simulation s were conducted at 293K and with 3d periodic boundary conditions applied to the simulation cell The time step wa s t 0.4 fs Both creep test s and tensile test s were performed For a creep test a uniaxial tensile load wa s applied along the x direction with magnitudes ranging from 0. 6 to 1.8 GPa f or Mg and from 2.0 to 3.5 GPa for Ti Such creep tests allow the determination of the correlation between deformation mechanisms (GB process, dislocation, and twinning) and the magnitude of the applied stress [50, 7 3] The activation of specific dislocation s at specific values of applied stress can also be determined from creep test s P lasticity of nc metals is also examined by tensile test at strain rate of 1.5 x 10 9 s 1 Flow stress obtained by the tensile test sh ows Hall Petch behaviors between grain size and strength [42, 43] As previously seen in Fig. 2 2, MD simulation s address es atomistic phenomena at s cale of the short time ( ~ nsec ) considering max imum physical time of one MD step and normal total MD time of less than a few months. Such a short time scale requires a high strain rate ( > 10 7 s 1 ) [36] In nc materials simulated with a strain rate rang e of 1x 10 7 ~ 1 x 10 10 s 1 their deformation process es ha ve been consistent with experiments [49, 51] Sch i tz [87] suggest ed the speed of soun d ( 4940 m/s for Mg, 5090 m/s for Ti ) is considered for a maximum strain rate The end s of simulation str u cture have to move below the speed of sound. For example, i f a sample with length of 70 nm is strained with a strain rate of 1 x 10 9 s 1 the strain speed is 70 m/s. S train rates are controlled to be 1x 10 7 ~ 1 .5 x 10 9 s 1 in both creep and tensile tests of this work

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43 2 6 3 Analysis M ethod For the visualization of defects, s trained structures are rapidly cooled down to 0.01K to minimize thermal energy followed by c ommon neighbor analysis ( CN A) [88, 89] The common neighboring analysis ( CN A) [88, 89] was used to distinguish atoms in (f.c.c. like) stacking faults from h c p c oordinated atoms. The AtomEye software package was used for visualization of the MD results [90] Images of coordination number ( CN ) p otenti a l energy ( PE ) common neighboring analysis ( CN A) and shear strain are displayed using the AtomEye in this dissertation 2 7 Summary For MD simulations of n c HCP metal s, the fundamental background w as reviewed in this chapter. Slip and twinning of HCP metals are influenced by t he c/a ratio known as the anisotropy The c/a ratio s of Mg and Ti are 1.624 and 1.586. The domina nt slip mode s of Mg and Ti are basal and prismatic
, respectively The most c ommon twin modes in both Ti and Mg is Computational methods, in particular molecular dynamics (MD) simulation have provided important insights into the structure and properties of nanocrystalline metals In this study, EAM and MEAM (Modified Embedded Atom Method ) potentials effectively describing metallic bonding are chosen for mechanical simulation T he and texture s as well as fully dense 3D structure s are chosen for the m echanical properties simulation s B oth creep and tensile tests are carried out and s train rates are controlled to be 1 x 10 7 ~ 1.5 x 10 9 s 1 in this work.

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44 CHAPTER 3 OVERALL MECHANICAL R ESPONSE OF TEXTURED NANOCRYSTALLINE MG MD simulation of nc metals is first carried out using 2D texture d microstructures The r esponse of nc Mg to external stress is investigated, followed by analysis of individual defects in the strained structures in Section 3 1 and 2 The m icrostructure evolution and interaction between slip and twinning are also analyzed in this c h apter This chapter focus es on verifying that results from MD simulation of this work are consistent with those of exp eriment s. The s tress response of nc Mg is also compared to that of nc FCC metals. Slip and twinning in co arse grained Mg are referred to understand defect process in the deformed Mg samples. Twinning and < c + a > slip, typical and important deformation pr ocesses in HCP metals, will be analyzed in detail later. 3 1 Stress D ependence of Mechanical R esponse A creep test of coarse grained material is conventionally carried out at elevated temperature s Nanocrystalline metals can, however, exhi bit similar creep behavior at room temperature and low external stress b ecause of the enhanced contribution of GB mediated process es [73] The time evolution of the mechanical responses of textured nano crystalline Mg structure s of various grain sizes were determined under a range of tensile stress at 293K Figure 3 1 shows strain vs. time curv es for an 18 nm grain size for stresses ranging from 0. 9 t o 1. 2 GPa. As will be discussed below, GB mediated processes such as grain boundary diffusion and sliding were do minant in plastic deformation below 0 9 GPa, for which the strain rate was ~ 1. 0 x 10 7 s 1 (slope A) A n increase in strain rate is observed at 1.0GPa by an initiation of slip process as denoted as S in Fig. 3 1 These large plastic strain s for stresses in excess of 1.0GPa

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45 arose from slip and twinning. Consequently, 1.0GPa is considered to b e the flow stress ( f ) for this particular system and grain size W ith incre asing applied stress, the strain rates ( slopes B D ) increased gradually and slip process are initiated at shorter and shorter times Fig ure 3 1 Strain vs time plots for nanoc rystalline structure s with 18nm grain size at various external stresses. A to D and the associated triangles, denote increasing strain rate s (A: steady state creep by GB process, and B D: tertiary creep by twinning and slip process). S indicates the initi ation point of slip. 3 2 Signatures of D islocations and T wins Before turning to a detailed analysis of the sequence of plastic deform ation processes it is useful to identify the characteristic signatures of the various kinds of defects that are o bserved in the simulated mechanical tests. Fig ure 3 2 displays a number of dislocations and stacking faults that have been produced in a particular deformation simulation To generate this image the atomic positions at a single instant

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46 of time during an MD system were recorded. The strained structure was then quenched to 0.1K Atoms with non twelve fold coordination (i.e., neither HCP nor FCC ) were identified and are shown in black in Fig. 3 2 Common neighboring analysis ( CN A) [88, 89] was performed to distinguish FCC from HCP environments: FCC atoms are shown in brown HCP atoms are shown in grey. Fig ure 3 2 S napshot of 3.5% strained structure with grain size of 18nm. Here, and in subsequent figures, g ray, black and brown denote normal ( HCP ), disordered (non HCP or FCC ), and stacking fault ( FCC ), respectively. The grains are labeled 1, 2, 3, and 4. Figure 3 3 (a) shows that the s ingle brown lines (labeled 1 in Fig. 3 2 ) are intrinsic I1 type SF (ABABA CACA) stacking faults while Fig. 4 3 (b) shows that the double brown lines (labeled 2 in Fig. 3 2 ) are in trinsic I2 type SF s (ABABCACA) (All dislocations and stacking faults are characterized in the conventional manner with a right handed B urgers circuits in Fig 3 3 [21] ) Th e two single SF lines in Fig. 3 3 (a)

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47 pro pagate with their cores connected, essentially acting as a single dislocation. The two SFs are formed by < a > partial slip of 1 /3 or 2 /3 A or D on the basal plane, as shown in Fig. 3 3 (d). The < a > partial edge dislocation s given in Figs. 3 3 (a) and (b) have a direction and magnitude of 2 /3 The non ba sal dislocation (labeled 3) in Fig. 3 2 is a < c + a > partial edge dislocation whose Burgers circuit is 1 / 6 with a slip vector of BA 0 see Fig. 3 3 (d) In contrast with < a > partial slip, no SF is produced by the < c + a > partial dislocat ion of 1 / 6 ; instead, just an extra half plane of atoms is produced the layer of red A in Fig. 3 3 (c). All of the dislocations are edge type. Thus th e dislocation lines are parallel to the z axis of In a ddition to the partial dislocations, < a > and < c + a > extended dislocation s are also observed A s napshot and a schematic of the atomic st ructure of the extended < a > dislocation are shown in Fig s 3 4 (a) and (b) taken from the same simulation. Layer s A, B a nd C at or near the extended < a > dislocation are parallel to basal plane in Fig. 3 4 (a). T he C layer has slipped in a manner similar to the basal < a > partial dislocation under the shear, thereby forming the extended < a > dislocation. C onsistent with previou s studies [21, 91] i t can be seen from Fig. 3 4 (b) that the extended < a > dislocation consists of two partial dislocations, 1/3 and 1/3 joined by a stacking fault As previously noted dislocation lines are normally parallel to (z directi on). The 1/3 partial slip is at an angle of 90 to the dislocation line, and is thus pure edge type. By contrast the 1/3 partial lies at an angle of 60 to the dislocation line. In Fig. 3 3 (d), the two partial di slocations of 1/3 and 1/3 A and B can be expressed as a complete 1/3 dislocation, AB.

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48 Fig ure 3 3 Burgers vector s and circuits of SF/RH type for the dislocations and stackin g faults (S F) found in Fig. 3 2 : (a) I 1 type SF with < a > partial dislocation of 2 /3 (b) I2 type SF with < a > partial dislocation of 2 /3 (c) < c+a > partial dislocation of 1 / 6 and (d )Burgers vectors in H CP unit cell: A D and BA 0 are 1 /3 2 /3 and 1 / 6 partial dislocations, respectively [21]

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49 Fig ure 3 4 Snapshot and schematic of an atomic structure for the extended < a > dislocation, as observed for grain size of 40nm under 1.3GPa (a) Atomic arrangement on (b ) Three pertinent atomic layers (A: circle, B: dot, and C: triangle ) in (a) on (0001) The extended < a > dislocation with two partial cores, 1/3 and 1/3 m oves from right to left. The dislo cation line s are parallel to (z direction), r ed arrow s indicat es the slip vector of each partial dislocation In some simulations < c + a > extended dislocations were produced T hey sometimes developed from the 1 / 6 partial dislocation, previously seen in Fig. 3 3 (c) showing a complete pyramidal 1 / 3 d islocation It is known from a previous

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50 MD study of HCP metals that the pyramidal dislocation of 1 / 3 can be split into two [92, 93] or three [94] partials. 3 3 Microstructure Evolution In this section, the nucleation and development of slip and twinning are characterized As noted in Fig. 3 1 an accele rated deformation process for the textured structure with grain size of 18nm was found at 1.0GPa ; thus, to observe all of the salient deformation processes, a sample w as strained at 1.18GPa. Figure 3 5 illustrates the evolution of the defects, wit h HCP coordinated atoms not being shown; t winned region s are solid color The twins, indicated as A in Fig. 3 5 (a), are nucleated at grain boundaries. In particular, t he twinning is initiated at the right side of grain 2 grow ing continu ously with time in Fig s 3 5 (b) and (c). This is consistent with the experimental observation that the twins typically appear in the early stage of deformation [95] Grain 2 displays much more t w inning activity than the other grains because its crystallographic orientation is comparatively close to the fav ored stress direction for twinning ( parallel to c axis see Fig s 2 6 and 2 9 ) : the angle between them being only . In Fig 4 5 (d), compressive twins, and are found, having sizes of a few nanometer s In contrast to s mall elastic strain s ( ~ 0.2% ) of coarse grained metals the elastic strain in MD is typically large (2~3 % of total strain) due to the high strain rate, small grain size, and the absence of pre existing dislocations If an elastic strain of 3% is excluded from the tot al strain of 10% shown in Fig. 3 5 (d), then the pure ly plastic strain at saturation of twinning process is about 6%, c onsistent with earlier simulation [96] and experimenta l [97, 98] results that twinning occurs dominantly for 6 ~ 8% strain for Mg and its alloys.

PAGE 51

51 Normalized twinned areas, obtained by dividing total number of atoms by numbers of twinned atoms in each grain, ar e shown a s a function of strain in Fig. 3 6. The twinned area in grain 2 of Fig. 3 6 increase s monotonically from initial step of plastic deformation. This is a typica l process of tensile twin nucleated from GB in this simulation T he t otal twinned area decrease s slightly from 6% plastic deformation after saturation. This implies that twins h ave a limited role in the whole deformation process because they have a mechanism of growing not gliding. Analyzing the situation from the point of view of a basal slip process, the < a > partial dislocations in grain s 2 a nd 4 of Fig. 3 5 (b) were mostly activated from twinned sites. This is consistent with experimental reports that the dislocation emission relieves the long range stress field from an incoherent tip of an advancing twin [21] The < a > partial slip is also observed inside the twinned region. With t winning progressing in grain 2, the < a > par tial slip produced by the growth of the twin is a n I1 type stacking fault having no dislocation core. It has been experimentally noted that dislocations in twinned regions can aid the propagation of twinning in HCP metals due to a local deformation of twin boundary by slip [99 102] This is also agreed with the present simulation results. G rain 2, after being completely twin n ed, can support < a > slip process es more easily due to a change of crystallographic orientation from low Schmid factor (0.19) to a high er Schmid factor (0.26). Indeed, after the twin ning process is completed < a > partial slip cores are found to be ubiquitous in the twin n ed region of Fig. 3 5 (d). In addition to the initiation of intra grain slip process after twinning, the density of dislocations increased substantially in grains 1 and 4, which both a have Schmid factor of 0.46 for basal < a > partial slip Fig. 3 5 (d ). I1 type SFs a lso begin to be formed in grain 4 out of a twinned

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52 area by interaction between < a > partial slips. Thus, activ ation of < a > slip process in our simulation structures is promoted at both twinned and neighboring grains for plastic strains greater than 5 ~ 7 %. Th e majority of < a > dislocations are partial at early and middle stage of deformation ; very few extended < a > dislocations were observed in Fig. 3 5 (d). A pyramidal < c + a > partial dislocation of 1 / 6 noted as C in grain 4 of Fig. 3 5 (a), was form ed from the GB and has a length of a few nanometers ; however i t did not propagate through the grain Given its high Schmid factor, 0.46, basal < a > partial dislocations should be able to propagate through g rain 4 Considering Mg or its alloys, the pyramid al < c + a > partial slip process having a high critical resolved shear stress (CRSS) value in the grain 4, is thus limited by activation of a basal < a > partial slip having a low CRSS value [17, 96] As previo u sly seen in Fig. 3 2 the < c + a > dislocation in grain 2 does propagate. The s uppression of the basal < a > partial slip in grain 2 allows the pyramidal < c + a > partial be activated.

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53 Fig ure 3 5 S napshots of tensile test for 18nm textured structure at 1.18 G P a for four times and strains, increasing from 9 ( a) to (d) Note mic rostructural figures: A tensile twin ; B basal < a > partial dislocations of 1 /3 or 2 /3 ; C pyramidal < c + a > partial dislocations of 1/6 ; D the extended dislocation with basal slip ; and E: and compressive

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54 Fig ure 3 6 Activity of tensile twinning in each grain of 18nm textured structure at 1.18 G P a D islocation processes in a larger grain size sample are shown in Fig. 3 7 A t ensile stress of 1.3GPa was applied to the same initial structure with 60nm grain size. Basically, the deformation behavior is similar to that of the 18nm grain structure shown in Fig 3 6 As exhibited in Fig. 3 7 (a), deformation process took place by tensile twinning, basal < a > and pyramidal < c + a > partial slip. Basal extended dislocations (marked as A in Fig. 3 7 (a)) initiated from a GB w ere also produced Moreover, the number of extended dislocati ons i ncreased markedly at la r ger strain as shown in Fig. 3 7 (b). A pyramidal < c + a > dislocation was also generated in grain 2 Fig. 3 7 (a). Cores of full dislocations were observed, marked as B in Fig. 3 7 (b). Comparing Figs. 3 6 and 3 7 we can identif y a grain size effect on the dislocation mode

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55 for this textured structure During deformation test s of structures with grain size ranging from 6nm to 60nm, the fraction of the total number of dislocations that were extended or full dislocations c onsistently increased a s the grain size increased This tendency is consistent with a transition from partial to the extended slip with increasing a grain size in nanocrystalline Al [73] Fig ure 3 .7 S napshots of strained textured structure s with grain size of 60nm at 1.3GPa. (a) and (b): total strains are 4.73 and 7.25% at 1.3GPa, including elastic stra in of ~3%. Note microstructural features: A extended dislocation, B full dislocation.

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56 Fig ure 3 8 S napshots of strained textured structure s with grain sizes from 6 to 40nm at 1.0 or 1.18GPa. (a): structure with 6nm grain shows both basal < a > slip and tensile twinning at 1.0GPa. Comparing this structure to ones with lager grains (>18nm), the twinning process is however less active than slip through whole plastic deformation There is also no completed twinned gra in over 10% plastic strain. (b) and (c): the crossover of initiation process from twinning to slip using the same initial structure of 18nm grain size by increasing an external stress from 1.0GPa to 1.18Gpa. (d): Plastic behaviors of 40nm grain structure a t 1.0GPa. Circled A and B indicate two different basal slip sources, twinning dislocation and GBs (including a triple point). Cyan, brown and black mean a twinned region, stacking fault ( FCC ), disordered (non HCP or FCC ), respectively. It is well known that in nanocrystalline FCC metals GB mediated process es domina te the deformation mechanism as the grain size decrease s and that a maximum strength exists at a critical size ( typically 10 ~ 20nm) above which conventional slip process es dominate [42, 44, 50] As shown in Fig. 3 8 the grain size dependence (6,

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57 18, and 40nm) of the deformation processes were investigated for loads of 1.0 and 1.18GPa. As shown in Fig. 3 8 (a), at small grain size ( 6nm ) twins and basal < a > partial dislocations were produced This indicates that plastic deformation at extremely small grain size can be promoted by both slip and twinning processes, not only by GB mediated processes. In Fig s 3 8 (b), (c), a nd (d), we compar e the initiation process of slip and twinning at an early stage of deformation First, two structures with grain size of 18nm were deformed from the same initial structure as in Figs. 3 8 (b) and (c) to the same plastic stain, but under dif ferent stresses For the case given in Fig. 3 8 (b), twinning developed prior to activation of slip process. This result is in apparent conflict with experimental findings that s ome dislocations are introduced in polycrystalline Mg before twins are observed [95, 96, 103, 104] Most importantly, Koike [95] reported that t he twins are formed in the early stage of deformation in order to accommodate concentrated stresses due to dislocation slip To investigate the origin of this discrepancy, the initiati on process es of twinning and slip w ere also examined at larger grain size s Consistent with the experimental trend, for 40nm grain size system shown in Fig. 3 8 (d ) slip was activated prior to twinning. Comparing Figs. 3 8 (b) and (d) for a similar strain a nd similar degree of twin growth, it is clear that the strained cell with 40nm grain size follows the known defect activation sequence better than the 18nm system. Analogously, the effect of stress on activation of slip and twinning can be seen by compari ng in Figs. 3 8 (b) and (c). Even though the tensile loads of 1.0 and 1.18GPa are not much differen t the higher stress also promote s slip over twinning.

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58 In order to understand the dependence of defect activation on stress more clearly deformation under even higher external stress was investigated as shown in F ig. 3 9 The same initial structures having a grain size of 40nm were strained at 1.25, 1.32, 1.4, and 1.5GPa. Figures 3 9 (a) to ( d ) show their structures for essentially the same values of strain: these total strains of 9 .6 3 ~ 9 .8 3 %, corresponding to plastic strains of 5.52 ~ 6.63%. Figure 3 9 (a) shows that grain 2 is nearly completely twinn ed The pyramidal < c + a > dislocation s, indicated as A in Fig 4 9 (a), cannot propagate at 1.25GPa because th ey are blocked by the twinning in grain 2 and the basal < a > slip in grain 4. As Fig. 3 9 (b) shows, t he se pyramidal < c + a > dislocations do however propagate at higher stress (1.32GPa) ; this has the effect of preventing the progress of the twin across all of grain 2. The activity of pyramidal < c + a > slip increases continuously with increasing external stress ; at 1. 32 GPa extended < c + a > dislocation s first appear As shown in Fig. 3 9 (d), a high density of pyramidal < c + a > dislocation s is observed at 1.5GPa almost completely suppressing the twinning of grain 2. In addition, grain 4 now manifests basal < a > slip as a dominant deformation mode. Considering the series of twinned regions in Fig s 3 9 (a) (d), we can conclude that the size of the twinned regions decreas e s with increasing external stress. Th is comp eti tion between twinning and pyramidal < c + a > slip has been observed experimentally: Lou et al. [98] suggested that in the AZ31B Mg alloy the pyramidal < c + a > dislocation s act as barriers to twin nucleation and propagation. Proust et al. [96] also reported that the more < c + a > dislocations there are in AZ31 magnesium alloys, the more difficult it becomes to produce tensile twinning.

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59 Fig ure 3 9 S napshots of strained textured structure s with 40nm grain size under various external stresses from 1.25 to 1.5GPa. (a): total strain 9.67% (plastic 6.63%) at 1.25GPa, (b): total str ain 9.71% (plastic 6.45%) at 1.32GPa, (c): total strain 9.63% (plastic 5.92%) at 1.4GPa, and (d): total strain 9.83% (plastic 5.52%) at 1.5GPa. Cyan, brown and black mean a twinned region, stacking fault ( FCC ), disordered (non HCP or FCC ), respectively. 3 4 Stress A nalysis of D islocation A ctivation The activation of dislocation s depends on str ess. A stress map can give information about si tes where slip occurs. Figure 3 10 shows a shear stress map of 18nm textur ed Mg under constant y s tress of 1.1GPa To observe activation of basal < a > slip, basal plane and of grain 4 are chosen as a s h ear plane and direction, respectively. F orce of each atom on the shear plane to the shear direction is p lotted on the 2D plane of F ig. 3 10. The snapshot was captured during basal < a > slip is activating

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60 at grain 4 : The dislocation size is ~ 0.5nm The partial basal < a > dislocation is mark ed with a blue circle in Fig. 3 10 (a) High shear stress (> 0.7GPa) of a large area (more than 2~3 atoms) is mainly found at GB. This is consistent with classical dislocation theory that a critical concentration of shear stress at GB results in activating slip process. Fig ure 3 10 C N A ima ge and shear stress map of strained textured structure s with 18 nm grain size under 1.1GPa. 3 5 Transitions in Dislocation M ode Typically, slip process in FCC metals involve partial dislocations for grain size s below ~ 20nm [42, 73] with a transition to extended or full dislocations at larger grain size s [73, 105] The cores of the extended dislocations are split into two Shockley partials conn ected by stacking faults. T he splitting distance of two partial cores (the stacking fault distance r ) is a function of the resolved shear stress ( ) [73] Whether a dislocation is partial or extended depends on the splitting distance relative to the grain size For large enough grain sizes extended dislocation s are normally stable, though

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61 partial dislocations are pr esent at very high stresses. However, this behavior is not see n in the present simulations Dislocations in the textured structure consist of basal < a > and pyramidal < c + a > types. Considering the basal < a > dislocation, a partial dislo cation process is frequently seen even for the large grain size of 60nm. By contrast, e xtended dislocations appear in 10nm grain structures when strained over 10%. As the grain size or the external stress increase in the textured str ucture, extended dislocations become more prevalent than partial dislocations A similar tendency is seen for the pyramidal < c + a > slip in the textured structure Because various slip modes basal, prismatic and pyramidal can be ma nifested, the transition of slip mode in the polycrystalline HCP metals can not be explained by a simple equation as it was for the case of FCC metals [73] 3 6 Competition between S lip and T winning T he interaction s between slip and twinning can be viewed as process es of competition and coop eration The first competition takes place between pyramidal < c + a > slip and t winning As already exhibited in Fig. 3 9 (b), the growth of the twinning region can be physically blocked by the pyramidal < c + a > dislocation. Moreover, increasing the stress accelerates this phenomenon by promoting slip rather than twinning. The same trend is also found between basal < a > slip and twinning. At low stress the twinning process is more dominant than the basal slip. However, twinning activity decreases compar ed to the basal slip as the stress increase s

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62 By contrast there are also several cooperati ve processes between slip and twinning with one deformation mechanism help ing the activation of the other. For example, as discussed later, a tensile twin o riginat ing at GBs can initiate a basal < a > slip through dislocation emission Conversely a high density of dislocations leads to compressive twinning A second cooperati ve process is observed in the slip behavior in a twinned area. Whe n a n I1 type SF in twinned area accommodates the propagation of twinning, the Schmid factor for slip increases promot ing a new slip process. 3 7 Summary This chapter has attempted to probe the plastic behavior of a nanocry stalline HCP metal using MD simulation of texture d Mg. We observed typical creep curves for a nanocrystalline metal in which GB mediated process are dominant Further, there was a transition in the nature of the dislocations and a cr ossover in the initiation process of twinning and slip at the nanoscale. In addition, other deformation mechanisms typical of coarse grained HCP metal were observed. Al though twinning behaviors are normally suppressed at small grain sizes, high twinning ac tivity in this structure enabled us to examine various interactions between twinning and slip The twinning process in this simulation was found to be prevalent for ~ 6% plastic strain as has previously been see for coarse grained HCP metal. M ost of the e xperimentally known slip and twinning systems were reproduced in our simulations de spite the limitations of a textured microstructure The GBs of the nanocrystalline structure act ed as a source of twinning as well as dislocation s for slip. The t ransition from partial to extended dislocation we re found for both basal < a > and

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63 pyramidal < c + a > slip. The extended basal < a > dislocation is similar to the corresponding dislocation in FCC metals However its transition from partial to extended cannot be as easily e xplained due to the complex slip mode s available to HCP crystal s In the columnar structure, three types of twinning are formed by GB and dislocation assisted processes tensile twinning is initiated from GBs at low stress, while compressive and twinning a re nucleated from an interaction of dislocations at high stress. A crossover in initiation process from slip to twinning at nanoscale was found with decreasing a grain size. It wa s also found that the transition of deformation mechanism from tensile twinning to pyramidal < c + a > slip occurs with increasing stress. Thus, the p lastic behavior is embodied by competing and cooperati ng processes between slip and twinning

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64 CHAPTER 4 OVERALL MECHANI CAL RESPONSE OF 2D N ANOCRYSTALL I NE TI METALS Although both Ti and Mg have the HCP crystal structu re, they show quite different mechanical properties as previously shown in Table 2 2. Although the c/a ratio is usually considered as a factor to explain different deformation properties in each HCP metals, it is not enough. I n Table 4 1, Zn and Cd havi ng similar c/a ratio show the same slip modes. This is similarly applied to Ti and Zr. However, if we see principle slip system according to deviation change from Cd to Be, Be violates the trend. T here has been an attempt to understand the dependence of deformation modes in HCP metals in terms of the d electrons [106] Table 4 1 c/a ratio and slip modes of various HCP metals [8] The S tacking faul t energ y (SFE) is an important factor in de termining the behavior of dislocations and on twinning The SFEs are shown in Table 4 2 A l ow SFE easily activates partial slip. It is well known that the twinning behavior of fcc metals is strongly affected by the stacking fault energy (SFE). Defor mation twinning is easier in FCC

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65 crystals with very low SFE value, but becomes more difficul t in the fcc crystals with medium to high SFE s As a twin boundary (TB) appears as a SF in FCC metals, low SFE can make the TB more stable. In Table 4 2, the SFEs are obtained from SFs occurring parallel to the basal plane. Hence, behaviors of the basal slip are ex plained with the SFEs in Table 4 2 To relate t he SFEs with twinning in HCP metals the SFEs of TBs should be considered, as will be discussed later. Table 4 2 Lattice constants and stacking fault energies of various HCP metals obta ined by First principle calculation. [107] Parameters Zr Be Ti Mg Zn SFE(mJ/m2) 223 557 287 21 102 stable SFEs on the (0001) plane 4 1 Potentials and Mechanical R esponse For simulation s of the mechanical properties of Ti, e mpirical potential s of the EAM and ME AM types are considered in this study. The EAM potential for Ti is known to give a wrong c/a ratio close to the ideal value of 1.633 instead of 1.58 ; this is due to lack of anisotropy in the EAM formalism Hence, using the EAM potential a domin ant slip mode appear s as a basal < a > slip, not prismatic slip as has been experimentally reported [83 85] The f irst MEAM potential reported by Baskes does not reproduce the experimental lattice parameter or HCP as the lowest energy structure [108, 109] Th e MEAM potential fitted by Kim et al. [26] is chosen for our first MD simulation of Ti. As shown in Fig. 4 1 t he creep test s of 18nm textured Ti ind icate s that plasticity of the nc Ti texture occurs at stresses in excess of 3GPa The structure of the strained samples is analyzed in Fig. 4 2 to determine the deformation mechanism s After an extern al stress of 10GPa is applied t o the y direction the texture is cooling down at 0.01K. A s napshot of potential energy of

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66 Fig. 4 2 (a) exhibits high and low energy boundaries ( A and B ) inside the grains ; these boundaries are produced by the deformation process. It is found that t he e ach region in the deformed texture has different lattice constants as summarized in Table 4 3 Moreover, a cha nge of CN in the intermediate region appears in F ig.4 2(b). While normal atoms show the 12 characteristic of an HCP metal, the atoms between A and B have the CN of 14 After examining the atomic structure, this region is identified as having the BCC structure. Although BCC has 8 nearest neighbors (at 0.866 a), it also has 6 second ne ighbors (at 1.0a), which is within the cutoff used in the CN analysis Under the external stress a phase transition to BCC, instead of dislocation s or SFs of FCC results from higher energy difference between BCC and HCP than that between BCC and HCP [26 ] for this potential This potential is thus not suitable for deformation simulations As seen in Fig. 4 3, two textures near a formed boundary have the same phase and lattice parameter (~ 2.94 ) at 0.01K. The boundary is created by twinning, not a phase transition Hence, t he present research of Ti uses the Ti MEAM potential of Hennig et al. [82] does not have such a problem F urther simulation of Ti is performed with the Hennig potential. Table 4 3 Lattice constants of strained 18nm textured Ti at 0.1K. Normal atoms Region A Intermediate region (between A and B) Region B ~ 2.94 2.72~2.81 ~ 2.80 2.86~2.92

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67 Fig ure 4 1 Strain Time curves of 18nm textured Ti as a function of external stress

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68 Fig ure 4 2 PE and CN image of strained textured structure s with 18 nm grain size under 3.0 G Pa. Kim et. al. s MEAM potential was used. a Cyan region A : boundary with higher energy : lattice constant decreased, and atoms were compresse d. Deep blue region B: boundary with lower energy: lattice constant increased, and atoms were pulled out. In the r egion between A and B, a BCC phase is found.

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69 Fig ure 4 3 Snapshots of strained textured structure s with 18 nm grain size under 2 75 GPa. There is no change in CN. Two textures have the same phase. 4 2 Stress dependence of Mechanical Re s ponse in Ti As Fig. 4 4 sho w ed, the yield stress for Ti, described with the Hennig potential, lies between 2.50 and 2.75GPa at which tertiary creep appears We r e cal l that our earl ier simulations showed that tertiary creep of Mg of the same grain size occurs at 0.9GPa ; the stress in Ti is three times lager. This is similar to a ratio of the cohesive energ ies of Mg and Ti ( E coh Mg : 1.6eV, E coh Ti : 4.7eV) ; thus the occurrence of this creep curve at 2.75GPa seems physically quite reasonable At < 2.5GPa, s train rates are in a range of ~ 10 7 s 1 noted as a typical strain rate by GB Process. As external stress increase s over 2.75GPa, the strain rate jump s from ~10 7 to ~10 9 s 1 In s train stress curve s of Mg and Al [73] displaying a tertiary creep f rom 50 and 100 ps the strain rates are le ss tha n 7 x 10 8 s 1 The c omparati ve high strain indicates that slip and twinning actively takes place at the tertiary creep The stained texture of Ti is discussed in the following section.

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70 Fig ure 4 4 Creep curves of textured Ti having a grain size of 18nm. 4 3 Comparison of Mg and Ti Figure 4 5 (a) and (b) shows snapshots of 10% strained Ti and Mg samples with 18nm grain size under 3.25GPa and 1.2GPa stresses In our previous results, flow stresses of textured Ti and Mg were 2.75 and 1. 0GPa. Th us these specific external stresses were homologous in the sense of being 120% of the flow stress for each material The deformation strain rates were 1.4X10 9 and 1.0X10 9 s 1 for Ti and Mg respectively. As shown in Fig. 4 5 (a) and (b), various slip and twinning processes are observed in these deformed textures.

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71 Fig ure 4 5 Snapshot s of nano structured Ti ( left ) and Mg ( right ) plastically deformed at 3.25 and 1.2GPa, respectively. Their samples have the same grain size (18nm) and crystallographic texture ( ).Cyan, b rown and black mean a twinned region, stacking fault ( FCC ), disordered (non HCP or FCC ), respectively. In the untwined regi on s lip process are actively in Mg (left) than in Ti (right) The d is location density of Mg is also larger tha n that of Ti in Fig. 4 5 This is explained by the ten times high er SFE of Ti, as shown in Table 4 2 D ue to the small grain size b oth Ti and Mg textures exhibit most l y partial basal slip (C) having two FCC lines The different basal SFs (D) having one SF line are also found in the deformed textures. The first order pyramidal < c + a > slip (E) on the plane is often found in t he Mg texture, but rarely occurs in our textured Ti : E glide in Ti image. The twinning mode (A) is manifested in both Ti and Mg of Fig. 4 5 In our textured Mg, the most common thre e twinning modes, as already shown in Fig. 3 5 are observed. The stress condition and texture to activate those modes in Mg can be satisfied in the texture. However, the two specific modes, and of Ti are difficult to nucleate because in the texture. Thus, the next probable twinning mode (B), is f ound in Ti. In Fig 4 4 E E E Ti Mg E E

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72 (a), the twinning, noted as B apparently looks like FCC stacking. The FCC region is, however, believed to be creation of continuous intrinsic stacking fault (D) in twinned area. During preparing the textured Ti, twin nuclei were found near GBs and inside grains. The twins in the strained cell are noted by a circle in Fig. 4 5 ( a ) A texture of pyramidal direction, noted by B C in the HCP unit cell of Fig. 3 3, appears on the plane surrounded with dark atoms of twin boundary in Fig. 4 5 ( b ) Cross section of Fig. 4 5 (b) is seen in Fig. 4 5 ( c ) Untwined and twinned lattices display basal and planes, respectivel y. A red dash dot line on hexagon representing the basal plane is rotation axis, normal to in Fig. 2 6 ( a ) for tensile twinning in Fig. 4 5( c ). The angle is identified to be ~86 of tensile twinning. Thus, the twins whi ch occurs during cooling the textured Ti are the mode. This texture was designed for and because those t wins choose the direction, thickness of the texture as a twin rotation axis [20] However, tensile twins forms just by thermal stress ( < 1 GPa in Fig. 4 6 ). Moreover, some twins are homogeneously nucleated inside grains. The homogeneous nucleation without any external stress is normally very difficult to occur: it is simila r with creation of dislocations without the Frank Read source inside grains not at GBs The Hennig potential particularly describe s martensitic phase transformations between titanium phases of , and [82] Such a m artensitic phase transformation may be related with the homogeneous twin nucleation in this simulation

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73 Fig ure 4 6 Formation of tensile twins during cooling the textured Ti from 7 00 to100K. (a) Light and dark grays denote 12 CN and non 12 CN atoms, respectively. a. twins are indicated by circles. ( b ) Blue square indicates plane. B C, a and c denote pyramidal direction and lattice constants in the HCP uni t cell, respectively. ( c ) B lue hexagon and squares are HCP unit cell. Dash dot line denotes rotation axis for twinning. After obtaining twin free initial textured Ti a creep test was carried out under 3GPa. The 6% str ained sample is seen in Fig. 4 7 (a) and (b). Each snapshot of displays shear strain (a) and central symmetry (b) parameters [110] Figure 4 7 (a) clearly shows two twins, ( ) and ( ). Bright dots at grain interiors are cores of prismatic
dislocations. In this deformed texture, motion ( ) and reorganization to twin boundary ( ) of GBs are also observed SFs of partial dislocations are seen as dark blue in F ig. 4 7 (b). A small central symmetry parameter means a more symmetrical and ordered structure. G rains I and I I I having enough slip

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74 or twinning process are low symmetr y state s due to their high elastic ene rgy We indeed find that prisma tic < a > slip and twinning known as a dominant mode in titanium appear in this simulation Fig ure 4 7 Snapshots of shear strain and central symmetry of t he 6% strained textured Ti a t 3GPa. , , and denote twin, twin, GB moving, reorganizing of GB, and SF on the prismatic
plane, respectively. I ~ IV correspond to each grain.

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75 4 4 Summary M ost EAM and MEAM pote ntials are un able to capture the large anisotropy of Ti ; they thus fail to capture the fundamental deformation processes. However, the Ti MEAM potential reported by Hennig et al. has the properties most consistent with experiment correctly reproduc ing plasticity of Ti by deformation modes of slip and twinning indentified experimentally. The Hen nig potential designed to reproduce phase transitions of Ti seems to results in some distortion of nucleation process of the twin. From t ime strain curves, an acceleration of plastic deformation by successive slip and twinning occu rs at 2.7GPa. Yielding behaviors in nc Ti is achieved by only GB process for stresses less than 2.7GPa. In the present work, o bserved s l ip mode s are prismatic and basal
and twinn i ng also appear at strained textures

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76 CHAPTER 5 PYRAMIDAL SLIP IN COLUMNAR NC MG As discussed in Chapter 2 h exagonal close packed (HCP) metals manifest either prismatic
or basal slip as their princip al slip mode [21] While the p rimary slip mode is prismatic in Ti, Zr, and Hf, basal slip dominates in Co, Zn, Be and Mg [111, 112] Usually, if the primary slip is basal, then the secondary slip mode is prismatic, and vice versa These two slip modes can, however, only produce strain along the direction s Hence, t hese t wo slip systems alone are not enough for homogeneous plastic deformation [111] Activation of pyramidal slip enhances the homogeneity in plastic deformation of HCP metals by providing a strain compo nent along the direction. The pyramidal slip in HCP crystals has two modes : a first order mode and a second order mode. Su ch pyramidal < c+a > slip plays a key role in the ductility of polycrystal line HCP metals [113] The < c+a > dislocations ha ve been studied in exper iment [113 115] and by theory and simulation [112, 116 124] In the experiments, deformed polycrystalline samples were examined using an optical micr oscope ( OM ) or transmission electron microscope ( TEM), in order to identify the < c+a > dislocations It was found in Mg that pyramidal dislocations are strongly bound on the basal planes [113] Tonda and Ando [115] reported that < c+a > edge dislocations are immobilized by their dissociation in to a sessile and a glissile dislocation in Cd, Zn and Mg. S imulation studies have focused on the structures and stability of the < c+a > dislocation. A second order pyramidal edge dislo cation was shown by Minonishi et al. [120, 121] and Liang and Bacon [118] to have two types of cores; undissociated and dissociated. Yoo et al. [112] found the edge

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77 dislocation core with a Shockley partial connected by a basal plane stacking fault to a sessile partial. They also suggested from a calculation using anisotropic elasticity that < c+a > screw dislocations can cross slip from a prism atic plane i nto a pyramidal plane [112] A nalysis [11 6] of simulations using a Lennard Jones potential revealed a temperature dependence of the core structure in < c+a > edge and screw dislocations and double cross slip of < c+a > screw dislocation on and In contrast to studies focusing on second order pyramidal slip of first order pyramidal slip has been addressed in Ti [114] and Mg [11 7] In particular, Numakura et al. [123, 124] suggested three different core structure s for dislocations one planar and two non planar. Li and Ma [117] reported first order pyramidal slip, in Mg In s imulation studies, pyramidal dislocation s have been created i n single crystal s either by manually moving some atoms [118 124] or by applying external stress to an artificial cavity [117] It is not possible to investigate the initiation of < c+a > dislocations when they are constructed by hand In creating a < c+a > dislocation from a cavity the onl y source of the pyramidal dislocation is the cavity itself. I t is thus hard to see the various activation processes of < c+a > slip that might be active in polycrystalline materials, in which grain boundar ies (GBs) can be a source of dislocations [125] This study focuses on the heterogeneous nucleation of < c+a > dislocations from GBs in polycrystall ine Mg. In particular, the structure and properties of b oth first and second order pyramidal < c+a > dislocations in Mg are compared.

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78 5 1 Occurrence of Slip As reported in an earlier paper pyramidal < c + a > slip process es are common in the Mg columnar texture To activate pyramidal slip in this work, the textu re is strained under uniaxial stress of 1.18 GPa After activation of pyramidal dislocations the structure is quenched at 0.01K. The structure is characterized with CNA to distinguish FCC from HCP environments. Gray, brown and black denote atoms in HCP en vironments, FCC environments (in stacking faults) and non twelve fold coordinated toms i.e., non HCP or FCC (in dislocations and grain boundaries) atoms In the textured structure, only partial dislocations are generated from GBs u nder low applied stresses. As discussed previously [5, 126] the structure labeled in Fig. 5 1 (a) is a 1 / 6 pyramidal < c + a > dislocation. At higher applied stresses, a transition of the partial < c + a > dislocation into two different structures takes place Figures 5 1 (b) and (c) show the structure s for a stress of 1.4 5 GPa. One of the regions of the < c + a > partials is of the extended type, as seen in Fig. 5 1 (b). The extended < c + a The other high stress region is denoted in Fig. 5 1 (c). It seems not to be an extended dislocation, but rather the result of the connection of long and short partial pyramidal dislocations. In these simulations the < c + a > slip is also found in a simulation of the textured microstructure at an applied stress of 1.2 4 GPa. In Fig 5 1 (d), and indicate partial and extended pyramidal < c + a > dislocations, respectively. However, < c + a > slip nucleation appears to be less frequent in the texture than in the texture. Moreover, there is a difference between

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79 < c + a > partials in the and textured polycrystals. The < c + a > partials can glide through a grain in the texture, via a stacking fault mechanism. However, after being activated in the text ure, the partial denoted as in Fig. 5 2 (d) is unstable, and hence immediately transforms in to an extended < c + a > dislocation. Fig ure 5 1 Snapshots of pyramidal < c + a > slip activated in and textured structure s Gray, black and brown denote normal ( HCP ), disordered (non HCP or FCC), and stacking fault (FCC) atoms respectively. Arrows denote the directions in which the < c + a > dislocations move (a) The partial < c + a > locatio n s shown as brown lines, are nucleated at 1.18GPa. (b) Extended pyramidal dislocation, is activated at 1.4 5 GPa. Its shape changes during glide. (c) Two connected partial dislocations, appear at 1. 4 5GPa. (d ) Pyramidal < c + a > dislocations w ith partial ( ) and extended types ( ) are found during deformation in textured Mg.

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80 5 2 Structures of Dislocations We first examine the partial and extended dislocations appearing in the text ured structure. To understand the mechanism of partial slip, the structure of the leading core connected to a staking fault is illustrated in Fig. 5 2 ; the < c + a > dislocation is seen as black atoms among the atoms in HCP environments ( gray ). B lue dots denot e the positions of the atoms before slip The HCP stacking sequence in a basal direction layers Green arrows denote directions of the shear s tress that causes the slip. In the non basal SF, t he atomic displacement indicated by the yellow arrows is parallel to and normal to These two directions are on the plane between two < c + a > pyramidal slip planes previously illustrated in Fig 2 1 (a); h ence, th e depicted dislocation is a first order pyramidal partial The observed dislocation has a partial core and staking fault (SF), as shown in Fig. 5 2 (b). The pyramidal < c + a > dislocation line exists in the direction, indicated as in Fig. 5 2 (b). The potential energy (PE) of atoms in dislocation structure is also shown in Fig. 5 2 (b). Although the atom positions show some diso r der at the core of the partial dislocation, a stacking fault is formed by coordinated atomic motion. The t wo layers of the stacking fault colored in black move in opposite directions: one along 1/12 the other along 1/12 These opposite at omic displacements combine to pro duc e 1 / 6 partial slip a s shown previous ly in [23] An extended pyramidal dislocation in the texture is more complicated than the partial dislocation. CNA results and PE s of the extended dislocation are shown in Figs. 5 3 (a) and (b), respectively. The e xtended dislocation can be divided into leading

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81 and t railing partials connected with a SF see Fig. 5 3 (a) The regions with high PE in Fig. 5 3 (b) correspond to the leading and trailing partial cores. Extra half planes are found near the cores. There is a boundary of layers, denoted by the dotted (blue) line, at the upper side of the extended dislocation. The nature of this boundary will be discussed below in connection with sequential slip that leads to the formation of an extended dislocation. Fig ure 5 2 Common neighbor analysis (CNA) and potential energy (PE) map of the partial pyramidal dislocation in the textured denote the cores of the edge dislocations in (b)

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82 Fig ure 5 3 CNA and P E m ap of the extended pyramidal dislocation observed in the textured denotes the extended < c + a > dislocation. Blue dots indicate a boundary of different layers: A and B denote different layer in the direction of (b) T dislocations. The structures of the leading and t r ail ing partials are characterized in detail in Fig. 5 4 Among the three stacking fault lines (layers I, II, and III) of the t railin g core, layers II and III move in opposite directions, as indicated by the yellow arrows in Fig. 5 4 (a). To quantitatively determine their displacements, the atomic structure of layers I, II, and III are illustrated in Fig. 5 4 (c). The slip of layer s II an d III is 1/12 and 1/12 They thus form a total of 1 / 6 partial slip, in the same manner as shown earlier in Fig. 5 2 The atomic structure of the dislocation core in the leading partial is a lso exhibited in Fig. 5 4( b ) Each atom moves in the direction denoted by the yellow

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83 arrows. S ome atomic displacements are not parallel to the glid e direction of the extended pyramidal dislocation in the leading core of Fig. 5 4 (d) In t he leading core all the atomic displacements collectively make two components of partial slip: 1/18 and 1/9 Figure 5 5 shows the Burgers vectors of the extended pyramidal dislocation in the HCP unit cell The three pa rtial s o bserved in the leading and trailing parts produce perfect pyramidal slip : Fig ure 5 4 Atomic structures of a (a) leading and (b) t r ail ing of the extended < c + a > di slocation including CN A. (c) and (d): corresponding a tomic displacement on the 1 st order pyramidal slip plane

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84 0 + B 0 B 1/9 + 1/18 + 1 / 6 (5 1) Fig ure 5 5 Burgers vectors of the 1 st order pyramidal extended dislocation shown in the HCP unit cell. In the above, it has been demonstrated that the < c + a > dislocations in the textured structure lie on the first order pyramidal plane, appearing either as partials or as fully extended dislocation s Now we focus on the pyramidal dislocations found in the texture. In order to characterize their structure, t he atomic level details of the < c + a > dislocation found in the texture of Fig. 5 1 are shown in Fig. 5 6 The CNA analysis in Fig. 5 6 (a) shows that the dislocation has an extra half plane parallel to the basal direction. The di slocation seems to be divided into two different parts, as seen in the potential energy map of Fig. 5 6 (b). The head partial which contains a series of high energy atoms, lies parallel to the z direction, The left image of Fig. 5 6 (b)

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85 shows that the head core extends to the basal plane. This extended plane connects to the tail part, seen as an angular plane. This shape is quite different from the extended pyramidal dislocation formed in the textured structure. Because the structures of the head and tail partials are key s to understanding a formation mechanism of the extended < c + a > dislocation they ar e displayed in further detail in Fig. 5 7 Fig ure 5 6 (a) CNA and (b) PE map of the pyramidal < c + a > dislocation activated in the textured structure. Figure 5 7 shows CNA images projected on both the and planes in the two left columns, while the potential energy (PE) on is mapped in the right column. The first and second order pyramidal < c + a > slip planes are seen as semi transparent red and blue. Blue dots and y ellow arrows denote the sites of atoms before slip and the atomic displacement, respectively. The s tacking sequence of HCP in the < c + a > plane can be seen in the CNA results of Fig. 5 7 (a). Referring to the oriented CNA figure, the yellow arrows indicate that the ato ms do not displace in the same direction as in secondary pyramidal slip. The difference in the slip behavior at each atomic layer

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86 parallel to the slip plane is due to the motion of zonal dislocations known to typically occur in pyramidal slip [21] The zonal dislocations appear as z displacements. They are also known to make displacement of ~1/4B 'C ( ) see Fig. 5 5 [21] Most of the atomic displacements shown in the vicinity of secondary pyramidal slip plane, of Fig. 5 7 are also ~1/4B 'C In additio n, the PEs of atoms on the slip plane are higher than those of bulk HCP atoms. The n ext basal layer of the cross sect ion of Fig. 5 7 (a) along is shown in Fig. 5 7 (b). The secondary pyramidal slip has progressed more in Fig. 5 7 (b), as evidenced by atomi c displacements and PEs on the slip plane of Fig. 5 7 (b). As shown in the oriented CNA result of Fig. 5 7 (a), the atomic displacement takes place on one layer parallel to the secondary < c + a > slip plane of ; yel low arrows form a monolayer in the microstructure Two layers move however by individual displacement of atoms adjacent to the slip plane, as shown by the two yellow arrows in the oriented CNA result of Fig. 5 7 (b). As the secondary pyramidal slip progress, a region of atoms with high PEs appears between the secondary pyramidal slip plane and an angular boundary in the oriented PE map of Fig. 5 7 (b). This imp l ies that additional displace ments occur on a basal plane by the motions of zonal dislocations on a second pyramidal plane. The stress on the basal plane thus activates slip on the first order pyramidal plane of as exhibited in Fig. 5 7 (c). The extended pyramid al dislocation is thus revealed to contain both first and second order pyramidal slip. As already observed in Fig. 5 6 (a), the secondary pyramidal slip is edge type, as evidenced by an extra half plane of

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87 atoms. However, because the first order < c + a > part half plane, it appears to be a screw dislocation Figure 5 8 exhibits the tail partials of the extended < c + a > dislocation structure in the texture The semi transparent red and blue sheets denote t he first and second order pyramidal slip planes, respectively. The atomic displacement s associated with slip are represented by green arrows. The edge dislocation line of the secondary < c + a > slip is denoted in Figs. 5 8 (a) and (b). The first order pyramidal slip is initiated in Fig 5 8 (b) A s shown in the oriented CNA result of Fig. 5 7 (c), the atomic motion s occur along However, as Fig. 5 8 (c) shows, the atomic motion s change to the py ramidal < c + a > direction. The presence of a screw dislocation can be seen in Figs. 5 8 (g) and (h). Although the same plane is seen in both, in Fig. 5 8 (g) line 5 8 pyramidal dislocation at the t r ail partial is screw type, having a disl ocation line, In the above analyses, both first order [114, 117, 118, 123, 124] and second order [113, 115, 116, 119 122] pyramidal dislocation stru ctures have been considered. For the first order pyramidal plane, Liang and Bacon [118] studied the pyramidal slip on the plane using a Lennard Jones potential. They identified a dissociati on reaction: 1 /4 + 1 /12 + 1 /6 (5 2)

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88 Fig ure 5 7 CNA visualization and PE mapping of the extended < c + a > pyramida l dislocation projected onto the and planes. Atomic structures where one and two layers of top basal surface are removed at (a) are shown at (b) and (c), respectively. represent t he stacking sequence of HCP in the basal direction The s hear direction i s marked by the green arrow.

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89 Fig ure 5 8 Layer structures of the extended pyramidal < c + a > dislocation in the texture by CNA. The semi transparent red and blue sheets denote the first and second order pyramidal slip plane s respectively. represent t he stacking sequence of HCP in the basal direction The direction of local slip is seen as a green arrow. and i ndicate a disl ocation line of screw and edge, respectively. and denote discontinuous atomic layers in the basal direction. H o wever, each atom has les s level difference than atomic radius (1.6 ) in continuous lines of or Li and Ma suggested a dissociation process of the first order < c + a > dislocation in Mg: 1 /6 + 1 /6 + 1 /6 (5 3) A lthough their Mg EAM potential [127] is the same as that used in th is work, the reaction in Eq. 5 3 is quite different from that seen in our simulations: 1/9 + 1/18 + 1 / 6 (5 4) This mechanism was previously rep orted by Jones and Hutchinson [114] Using a hard sphere model for a titanium alloy ( Ti 6Al 4V ) they characterized the first order pyramidal slip as dissociated. Thus, at this point, there is a qu estion as to why the same dissociation process takes place in both Mg and Ti. While Mg has an essentially

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90 spatially isotropic electronic structure of s and p orbitals [126] the 3d 2 orbitals of Ti make it strongly anisotropic [126] Such an anisotropy is responsible for the non ideality in the c/a ratio and differences in the dominant slip mode: Mg and Ti have basal and prismatic slip as a dominant slip mode, respectively [128] It is important to note that the work by Jones and Hutchinson [114] was performed using an intrinsically isotrop ic hard sphere model. Hence, it is no t surpris ing that the partial dissociation s they observe d are the same as those we see in Mg. The secondary < c + a > edge dislocation is known to display three variants [120, 122] Type I is a perfect dislocation, while Type II consists of two 1/2 < c + a > partials. The Type I transforms to the Type II on heating from 0K to 293K [116] The dissociation reaction of Type II is: 1 / 6 + 1 / 6 (5 5) Morris et al. [122] suggested a Type III variant, in which the dissociatio n is different: 1/3 + 1/3 (5 6) Partial edge core s of Type II are found in the textured structure in our simulations. In contrast to these reactions, p yramidal dislocations in the textured structure have both edge and screw components. Considering a screw dislocation, there are two issues to be compared to previous studies. First, the screw dislocation is likely to cross slip: it h as been reported that a < c + a > screw dislocation spreads on two first order pyramidal planes at 0K [116] Yoo et al. [112] concluded that a < c + a > screw dislocation may split on multiple slip planes, e.g. and In the

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91 textured structure the pyramidal < c + a > dislocation takes place by cross slip on the first and second order slip planes mediating a basal plane (see Fig s 5 7 and 8 ). The screw core also appears as a junction of two p lanes in this simulation. Second, the secondary < c + a > edge dislocation becomes immobilized: the secondary pyramidal edge dislocations of Type II is known to be sessile at > 30K because of the exten ded core along the (0001) [116] T ype III is also sessile with a basal stacking fault, as seen in Eq. 5 6 [122] This spreading of the second order pyramidal dislocation on the basal plane was alread y observed in Figs. 5 6 7 and 8 This non plana r core spreading in HCP causes dislocations to be sessile [128] The immobilization of secondary < c + a > edge dislocations may be one of the reason s why it is difficult to form long partial < c + a > dislocations in the texture, a s previously mentioned in Sec tion 5 1 5 3 Activation Process of Dislocations The simulations described above show that < c + a > pyramidal slip processes are more easily observed in the texture than in the texture. Hence, first order pyramidal slip in the texture takes place more easily than second order pyramidal slip in the textu re This is in disagreement with the experimental findings that secondary pyramidal slip on 1/3 dom inates in Mg. To address this discrepancy w e compare the atomic motion potential energ ies, and stacking fault energies of pyramidal edge dislocations in the two textures. In the texture the first order partial pyramidal disloca tion, 1 / 6 arises from partial motion of two adjacent layers, ~ 1/12 and 1/12 as previously shown in Fig. 5 2 By contrast t he core of secondary pyramidal slip with edge type is

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92 obtained b y full motion of only one layer with zonal behavior in Fig. 5 7 For the purposes of an energy analysis of the 1 st and 2 nd order pyramidal dislocations, the distributions of total strain energy per Burgers vector are represented in Fig. 5 9 as a function o f distance from the centers of dislocation cores. The total strain energy (E total ), expressed by adding core and elastic strain energy (E core + E elastic ) [129] is useful for analyz ing the size and energy of a dislocation core. P yramidal disloca tions are partial or extended types in this work: the 1 st and 2 nd pyramidal dislocations are connected with a SF and with screw dislocation consisting of two planes, respectively. In order to focus on the ed ge co res of pyramidal slip, we examine the total strain energy of core regions (E core ) rather than that of the elastic region (E elastic ). The magnitude of Burger s ve c tor ( b ) for both partial pyramidal slip is 3.2 ; the total strain energy is plotted to 3.75 b a s shown in Fig. 5 9 Taking the dislocation core size to be 3 b [129 132] it is possible to estimate the core energy from Fig. 5 9 The second order pyramidal dislocation core in the texture is found to have much higher strain energy than the first order pyramidal slip in the texture In addition, analysis of stacking faults energies (SFEs ) in Mg from various simulation and experimental methods are compared in Table 5 1 The S FEs in the present work were determined from SF regions > 5nm apart from a core, created in polycrystalline by partial slip. Since there has been little analysis of SFEs on the < c + a > plane, we tried to obtain consistency of our results through comparing well known SFEs on the basal plane. The calculated e nergies of basal intrinsic stacking fault 1 (ISF 1) and 2 (ISF 2) are less than those which were reported by Liu [127] T his discrepancy may be caused by imperfect relaxation of our polycrystalline structure due to the

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93 presence of residual stress a fter partial slip process ; by contrast single crystals having no other defects would allow complete stress relaxation. However, our SFEs are still in the range of values experimentally reported. As shown in Table 5 1 the SFE on the second order < c + a > slip plane, 253 mJ/m 2 is twice that of the SFE on the first order < c + a > slip plane, 122mJ/m 2 The second order pyramidal < c + a > dislocations d id not show long SFs connecting their partial cores, whereas long SFs are easily produced in the first order < c + a > disl ocations. Presumably, this is a result of the differences in their SFEs. Fig ure 5 9 Potential energy distributions (snapshots) and strain energy curves within cylinders of radius R having a center of an edge pyramidal < c + a > dislocation line. In summary, the atomic motions and partial core energies explain the more frequent occurrence of the first order pyramidal dislocation Although the second order < c + a > edge dislocation s nucleate GBs, their non plana r structure prevent the m from

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94 gliding inside a grain. As local stress is concentrated at these se ssile dislocations, however, the nucleated partials become mobile. Due to their high SFE s they normally appear as extended dislocations rather than partial cores separated by l ong SFs Moreover, the high SFE of second pyramidal < c + a > slip indicate s that t he second order < c + a > slip are likely to be a screw type or dissociat ed ( < c + a > sessile < c > + glissile < a >) process es rather than pure edge type, as reported in previous studies [113] Tabl e 5 1 Stacking fau lt energies in magnesium Stacking Fault Energy (E sf mJ/m 2 ) Method Basal
, ( 0001 ) 1 st order slip plane, 2 nd order slip plane, ISF 1 ISF 2 This work 24 48 122 253 MD (poly), Liu EAM Nogaret [1 33] 27 54 121, 225 198 MD (single), Liu EAM MD (single), Sun EAM 44 118, 240 180 236 Ab initio Morris [134] 173 Chetty [135] 44 Jelinek [136] 18 37 Yasi [137] 34 Wang [138] 21 Sastry [139] 78 Experiment Devlin [140] 60 Fleischer [141] >90 Couret [142] <50 It is instructive to compare these simulation results with experimental findings. Tonda and Ando [115] reported a difference in yield shear stress between first and second order pyramidal slip of only ~ 20%. This difference is small compar able to the critical resolves shear stress ( CRSS ) ratio for non basal to ba sal slip This indicates that first order pyramidal slip of 1/3 should also be manifested for appropriate textures and applied stresses ; activation of deformation modes in HCP metals, such as Mg, strongly depends on texture and applied stress [143] Indeed, Li and Ma [117] also

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95 suggested that first order pyramidal slip takes place in MD simulation of single crystalline Mg and identified the associated stacking faults in TEM [4] 5 4 Comparison of Different Potentials The properties of a material as determined in a simulation can depend strongly on the description of the interatomic interactions. There are two literature EAM potentials for Mg : one developed by Liu et al. [127] the other developed by Sun et al. [ 144] To this point, w e have adopted the Liu potential to simulate plastic deformation of nanocrystalline Mg [5 126] The EAM potential fitted by Sun et al. has been found to manifest more active basal slip behavior than the Liu potential a result of the lower Pei e rls stress [133] We have compared the deformation behavior as given by the Sun potential with the results above for the Liu potential [126] We find that there are far fewer nucleation events for twinning and < c + a > slip for the Sun potential [126] In addition, intergranular cracks are frequently seen even at small plastic strain s (< 3 %) during plastic deformation process of polycrysta lline sample, as shown in Fig 5 10 (b) presumably a consequence of the reduced twinning and < c + a > slip activity Such crack s are common in simulation of 3D polycrystalline Mg using Sun potential As described by the Liu potential, the 1 st order pyramidal < c + a > partial slip in the texture shows partia l dislocations joined by long SFs ; moreover the extended < c + a > dislocations glide easily In contrast, Sun potential produces comparatively short non basal SFs often connected with a SF of FCC on the bas al plane, as exhibited in Fig. 5 10 (a), and less mobi le extended < c + a > dislocations. Comparing the structure of < c + a > dislocations using Liu and Sun potentials in and textur es of Mg,

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96 there was no significant difference (Fig. 5 10 ), consistent with the results of Nogaret et al. [133] Fig ure 5 10 Pyramidal < c + a > dislocations simulated by Sun potentia l (a ) CN A iamge of texture at 1.45 GPa; b lack, brown and cyan denote disordered (non HCP or FCC ), stacking fault (FCC) atoms and twinned area 1 is connected with FCC stacking faults, and 2 shows two < c + a > dislocations joined on two different slip planes (b) Potential energy analysis of texture at 1.32 GPa in (a). Arrow indicates the directions in which the < c + a > dislocat ions move and denote pyramidal < c + a > and basal < a > dislocations. has the same structure as the extended < c + a > dislocation illustrated in Fig. 5 6 An intergranular crack appears between grain s 2 and 4. 5 5 Role of Pyramidal Dislocations in Plastic Deformation The pyramidal < c + a > dislocations are fo und to play a number of roles First, homogeneous plastic deformation requires pyramidal < c + a > slip [34, 35] A basal or prismatic mode ap pears as dominant slip in HCP metals. However, if som e grains require high CRSS to activate the basal or prismatic mode, the secondary slip mode, pyramidal < c + a > should be activated for plastic deformation A ctivation of this secondary mode has a direct effect on mechanical response of the material at meso and macro level s Single crystalline textures of Mg show n oticeable differences in stress strain curve according to the directions in which stress is applied [36] The

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97 activa tion of secondary < c + a > slip is one of the important reasons for these differing deformation responses Second the interacti on between < c + a > slip and twi nning results in hardening The twinning and < c + a > slip compete in grains in which basal slip is rare Figure 5 11 shows hardening from the interaction between the < c + a > dislocati on and the twinning boundary. I niti al ly the simulated textures are defect free except for the GB s The interaction can be thus examined between dislocations and twinning created under the external stress. To maintain the specific st ress value needed for the < c + a > slip twinning interaction, a creep test is more suitable than a tensile test. The initial textures are strained at 1.035 and 1.18GPa, showing a < c + a > dislocation twinning interaction and only twinning in Fig 5 11 (b) and (c) respectively While the 1.18GPa curve shows the strain rate ( A ) of 1.5 10 9 s 1 a significant decrease ( B = 1.4 10 8 s 1 ) of stain rate appears in 1.035GPa curve of Fig 5 11 (a). As shown in Fig. 5 11 (b) and (c) We attribute this hardening to growt h of the tensile twinning blocked by the < c + a > dislocations of order In addition it was experimentally reported that under compres sive stress the < c + a > slip has a harmful effect on room temperature ductil ity of Mg : it produces severe strain hardening by dissociation to mobile < a > and immobile < c > dislocations [4, 37].

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98 Fig ure 5 11 C reep curves and snapshots of strained columnar Mg with 18nm grain size. Cyan, brown and black in (b) and (c) mean a twinned region, stacking fault ( FCC ), disordered (non HCP or FCC ), respectively. (a) Time strain curve s at 1.035 and 1.18GPa. (b) Snapshot at of 1.18Gpa curve. (c) Snapshot at of 1.035Gpa curve. A red circle indicates a region where an interaction between twinning and < c + a > dislocation s take place.

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99 5 6 Summary Pyramidal < c + a > dislocations were created at GBs in a nd textured Mg using MD simulation. It wa s found that both the first and second order pyramidal slip can be activated in Mg. The first order < c + a > dislocations in the texture manifested various forms, depending on the external stress. At low stress they are 1 / 6 partials. Two phases, extended dislocation and connected partials, appear at high stress. The extended type of the first order pyramidal slip takes place through a dissociation reacti on of 1/9 +1/18 + 1 / 6 While all of the dislocations in the texture are of edge type, the extended secondary pyramidal dislocation in the texture exhibits both types of edge and screw. The edge partial appears as a leading core of the secondary < c + a > slip, formed by typical zonal dislocation in HCP metals. The partial core, due to spreading on the basal plane and high SFE, r emain sess ile. During deformation, stress concentration which occur s at the sessile dislocation finally causes an activation of screw dislocation on the basal plane connected with the edge core. Hence, the complex dislocation is possible to glide in the columnar textures without leaving SF having high energy. Comparing the two different EAM potentials, the atomistic structure s of pyramidal < c + a > dislocations are found to be very similar though the Liu potential shows more ac tive < c + a > slip than Sun potential. The textured columnar nano struct ures considered here are in many ways ideal for fundamental studies; however, they are optimized for edge dislocation. As a next step, large scale randomly oriented, non texture d Mg polycrystals will be simulated; these will

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100 allow for a greater diversity of plastic process and allow more direct comparisons with experiment.

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101 CHAPTER 6 TWINNING IN 2D NANOCRYSTALLINE MG Three types of twinn ing were manifested in the deformed magnesium of 40nm grain size under a tensile load of 1.25 GPa Fig ure 6 1 shows tensile twinning (A) identical to the ideal case shown in Fig. 2 6 This twin was the most preval ent The other twins (B) and (C) c ompressi ve twinning rarely appeared. T he most probable twinning modes in Mg are , and [145] as our simulations indeed display Having illustrated the signatures of all of the relevant microstructural features in th is chapter we address the issue of how they are created and how they evolve under applied stresses 6 1 Nucleation of Twin s The deformation simulations of textured Mg manifested all three expected types of twinning : tensile twins, and compressive and twins. With regard s to t winning nucleation, it is known that e xtended defect structures such as GBs and cross slip can act a heterogeneous nucleation source of twinning [27] The p olycrystalline model used in this study provide s various extended defect structures as potential nucleation sites for twinning that are not available in single crystals [76] or bi crystal s [146] In a single crystalline structure of pure Mg, the ratio CRSS twin /CRSS basal range s from 2.4 to 4.4 [98] indicat ing that activation of slip i s easier than activation of twinning. However, the simulations of the polycrystalline structure show that tensile twinning take s place at the GBs in an early stage of a plastic deformation process Moreover, this GB mediate d

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102 process operates for all grain sizes studied at low stress es ( 0.6 to 0.9GPa ) where there is no slip activity. The GBs in the polycrystalline structure thus appear to act as heterogeneous nucleation source s that decreas e CRSS twin for tensile twinning. As the external stress gradually increases the dislocation process es are ac tivated in the grain interior s with the high density of dislocations providing another nucleation site for the and compressive twin s A n example of twin formation is illustrated in Figs. 6 2 (a) (d). The nucleation of the twin takes place at dislocation interactions, which are located in the center of a sequence of stacking faults. This nucleation of twinning by Shockley dislocations has been observed in simulation work of FCC Cu where it occurred under the high local compressive stress arising from dislocation s [147] It is similarly noted that t winning may be nucleated from the partial dislocations that are on parallel and neighboring glide planes [148]

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103 Fig ure 6 1 Three types of twins found during creep from a simulation of a grain size of 40nm under 1.25 GPa. (a) tensile twinning at the boundary of grain 2 and 4, (b) compressive twinning at the triple junction among grain 1,3, and 4. (c) compressive twinning a t the boundary of grain 3 and 4.

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104 Fi gure 6 2 S napshots o f nucleation process of primary twinning at the grain 1 of textured structure with 40nm grain size deformed under 1.25GPa. Here in contrast to to the tensile twin the and compre ssive twins are formed by the interaction between dislocations or between a GB and dislocations, possibly contributed by either a basal < a > or a pyramidal < c + a > dislocation. T h us, the compressive twins require higher stress for their nucleation There is e vidence to support our finding that the CRSS (76 to 153MPa) of the f twinning is much larger than the CRSS (2 to 3MPa) of twinning in Mg alloys [95] In the simulations, the tensile twin

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105 is nucleated only at GB s; however, the f and compressive twins are nucleated both at GBs and in the interior of grain s The two compressive twins were not found for structure s with grains smaller than ~18nm. As previously mentioned, due to the crossover of deformati on mechanism from slip to GB mediated process es dislocation activity in nanocrystalline materials decreases at grain size below 10 ~ 20nm. It can be deduced from the simulation results that the restriction of the compressive twin nucleation is caused by t he decreas e in slip involved in their nucleation at grain size s below ~18nm. Twinning normally consists of primary and secondary (or double) types [27] After primary twinning occurs se condary twinning can take place within the reoriented primary twinned area [27, 149] Although and double twins are frequently found e xperimentally [31, 149, 150] no such secondary twinning is seen here. Further study is required about the double twin The n ucleation conditions of and prim ary twins from 6 to 60nm grain size are summarized in Table 6 1. Table 6 1 Nucleation conditions for and primary twinning. Twinning Places Mechanism External Stress Strain O nly GB Separation of twinning dislocation from GB Low ( < f ) < 0.5% plastic GB or Grain interior Interaction between disl o cation s or GB High ( f ) > ~ 3% P lastic GB or Grain interior Interaction between disl o cation s or GB High ( f ) > ~ 3% P lastic

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106 6 2 Nucleati on Mechanism of Compressive Twinning from Grain Boundaries The t hree twinning modes shown in Table 6 1 are found in textured Mg. A nother twin mode is found in a creep te s t of the texture, and displayed in Fig. 6 3. Figure 6 3. twinning activated in the 18nm textured Mg. a and c indicate lattice constants of Mg in (e) and (f).

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107 First pyramidal < c+a > slip appears from the GB in the 18nm textured Mg sample strained at 1.2 GPa. The < c+a > slip occurs on t he second order pyramidal plane, in Fig. 6 3 ( a ) After growing to ~ 5nm, the < c+a > edge dislocation is changed to twinning (see Fig. 6 3(a) (d)). In the previous chapter, we observed second order < c+a > dislocation cross slipp ing to first order pyramidal plane in textured Mg. Thus, the second order pyramidal slip can choose either a transformation to the other < c+a > slip or twinning. The twin mode has not been experimentally found in Mg [27] It indicates that the twin can take place under particular conditions, e.g. in the columnar texture. Indeed, some MD simulations reported activation of the twin [151, 152] 6 3 Nucleation Mechanism of Tensile Twinning from GB 6 3 1 Nucleat ion P rocess in a L arge G rain The nucleation of the twin is first manifested at a boundary of a grain (A in Fig. 6 4 ) with the c the external stress, exhibited as a series of the nucleati on process in panels a to e The twin is initiated from atoms in a disordered inter grain region, marked as a red circle gradually grain A, along th e GB. In Fig 6 4( b ) a twin nucleus noted as a red area surrounded with a yellow solid line below the disordered atoms, is formed to thickness of a fe w (0001) layers. The atoms in the disordered region move to a neighboring lattice site on the next basal plane, thereby growing the twin texture to the c axis of the grain A, as shown in Fig. 6 4( c ) Such a process also causes thickening of the twin to tilted direction from an original texture

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108 in Fig. 6 4( d ) Figure 6 4 ( f ) and ( g ) are reproduced from Fig. 6 4( a ) and ( e ) to understand an original and twined texture separated by a twin boundary. Finally, a well developed twin is observed in Fig. 6 4( g ) having a width of ~ 2nm. In summary, the nucleation process of twin is found to consist of stepping forward to the c axis of an original texture, [0001] and thicken the c axis. The thickening of the twin is clearly conducted by a zonal twin dislocation, as discussed extensively in the literature. [153 156] Figure 6 4 Nucleation process o f twinning from GB at 293K. Atoms having a c oordination number of 9, 10, 11, 12 and 13 are shown as magenta, green, violet, yellow and blue, respectively ( a ) initial GB structure. A crystallographic direction is indicated by a unit cell of HCP, having t wo different lattice constants of a ( is seen due to z direction of = 2 a cos 30 ) and c in grain A and B. Solid and dot straight lines note alternative stacking (ABAB) of two atomic layers in c direction of grains A (gray) and B (green). Miscoordinated area, GB, between grains is surrounded by small black dot line. ( b ) ~ ( e ) duri ng a tensile test, snapshots from b to e were captured at 18.8 (0.9), 21.2 (1.3), 22.3 (1.4), and 23.5psec (1.6% removing an elastic strain in a total strain), respectively. Cyan area surrounded by a dot line indicates an initial GB. Red circle denote a cluster of disorder ed atoms. T ransparent red inside yellow line indicates twinned atoms. ( f ) and ( g ) initial and twin nucleated GB (23.5psec) are shown as larger atoms than in ( a )~( e ) The most interesting points are the nucleation site and propagation o f the twin achieved by the disordered atoms. The disordered atoms successively move through a

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109 path of 1/6 between the two closest lattice sites. On the basis of classical dislocation theory, that direction is sessile [21] with a low Schmid factor of < 0.1. Accordingly, the simulations strongly suggest that the propagation of the twin occurs by migration of individual disordered atoms. Figure 6 5 show s a pr oposed mechanism for the propagation of a twin in a 2 dimensional vi ew. First, the GB structure shown in Fig. 6 4 is shown schematically in Fig. 6 5( a ) If t here is an atom at site III (atom III), a very high repulsive would occur between two atoms (atom II and III) due to their small interatomic separation This is essen tially unlikely to occur in nature. Hence, site III should be vacant as shown in Fig 6 5( a ) Considering the atoms in Fig. 6 5( a ) there is an overlap of the potential curve s between the atom II and vacancy III in Fig. 6 5( b ) defined as an intergrain PWO (Potenti al Well Overlap) site. In Fig. 6 5( c ) the atomic energy should contain a T ) by thermal vibration at temperature T. The atom at site II, thus, has its increased energy to E RT, no stress from E 0K, no stress at 0K. As soon as an external stress is applied to the x direction, a t om II has E Atom 0 at 0K in the absence of external stress, being affected by attractive force, F B 0 ), arising from the potential field of vacancy shown as a blue dotted line Because the potential is the interac tion between atoms, vacancy doesn t have such a n interaction. If, however, an atom approach es the vacancy, a f orce is generated between the atom and lattice atoms surrounding the vacancy. The effective potential field of vacancy thus indicates the interact ion which the lattice atoms surrounding the vacancy act on. 0 can be expressed in Eq. 6 1. (6 1)

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110 As discussed above, by thermal activation and applied stress, the energy of atom II is increased from E 0K, no stress to E RT, stress ; the atoms also move 0 2 The 0 2 increase s the attractive force acting on the atom II from the potential curve of grain B, from F B 0 ) to F B 2 ). The total energy of the atom II TOT ) is thus express ed T ), interatomic elastic energy E A ) arising from the attractive force (F B ) of potential of grain B. TOT = T + E + A (6 2) PWO the atom can move to III. The specific energy condition is (6 3) where, k b and T are the B oltzman n constant and temperature, and c is th e intersection point between the blue dot ted line and the r axis. F A and F B are derived from dU A (r)/dr and dU B (r)/dr. After that the atomic postions and potentials are changed in Fig. 6 5( d ) t here are two unoccu p ied sites, II and V: a vacancy caus ed by the atom which moved to III and an empty lattice point in a new potential produced by the moved atom,. Hence, atom mo tion is possible from I to V or from III to II A driving force f or the sucessive migration can be identified by considering an applie d stress acting on two textu res of grain A and B In Fig. 6 5( a ) an external stress x y z =0) is reso l v Ax x A Bx x B ) at the c axis of A B Ax Bx Each interatomic strain produced by t he resolved stress es Ax Bx at sites of I and III can be expressed as : (6 4)

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111 (6 5) Because Ax Bx Eqs. 7.4 and 5 can be combined to give: (6 6) This result, due to = means 2 is closer to a vacancy site than 2 The total energy of atoms I and III are then given by: (6 7) (6 8) The energy difference (E TOT, I E TOT, I II ), obtained from Eqs. 7.7 and 7. 8 of atoms I and III may indicate which process between I V and III II take s place. Considering U A ( 1 ) = U B ( 1 ) = U A (c) = U B (c)=0, the energy difference can be summarizes as : E TOT, I E TOT, I II = [ U A ( 2 ) U B ( 2 )] + [U A ( 2 ) U B ( 2 ) ] (6 9) Because 2 is closer to a vacancy site than 2 the following conditions are obtained from Fig. 6 5( d ) (6 10) (6 11) (6 12) Therefore, (6 13) (6 14)

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112 Figure 6 5. Potential Well Overlap (PWO) m odel of t win n ucleation at GB. ( a ) S implified 2 D GB structure having a twin nucleation site from Fig. 6 2 Circles of violet and red rots denote Mg atoms and vacancy. In real HCP stacking, each basal plane is stacked in a manner of ABAB to the c axi s A GB between grain A and B appears as a gray dash line. x Ax and Bx are resolved atomistic stresses at each texture of grain A and B, respectively. ( b ) O verlap of potential curves at I IV sites of the GB. U(r) is a potential energy as a function of atomic position, r. Circl es of violet and orange surrounded by red dots denote atom and vacancy. r a and r PWO indicate the atomic distances at lattice and PWO site s PWO is the energy barrier for atom moving from site II to III. ( c ) and ( d ) t he first and second atom moving at PWO site. T E indicate therm al and interatomic elastic en erg y E 0K, no stress E RT, no stress and E RT, stress indicate atomic energy of unstrained sample at 0 and 293K, and of strained one at 293K, respectively. 0 1 and 2 ( 0 2 and 0 2 ) deno te atom position at E 0K no stress E 293K no stress and E 293K stress respectively. Red and blue dash curve s show pathways of an atom with changing an energy state of atom. Equation 6 1 4 indicates the atom at site I, having a higher energy state than the one at site II, can more easily jump over E PWO T he atom moving from I to V is thus

PAGE 113

113 favored. This indicates that successive atom mo tion is possible in this model, having selectivity and consistency of its direction. Furthermore, this m odel explains why the twin experimentally nucleates and propagates into one grain with the c axis more parallel to an external stress direction at two neighboring grains. Fig ure 6 6 a difference between E PWO and E T hermal at scales of temperature (T) and length (R) fro m Mg potentials. RT, T m and R a denote room temperature (=293K), melting temperature and normal atomic distance at 0K. The calculation is conducted using Mg EAM potential [24] In our 2D sim ulation, the density of PWO sites in a GB having misorientation angle ; this is for twin nucleation in coarse grained metals. Although the discussion above was in terms of a PWO site consisting of only single

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114 atom and vacan cy, twin nucleation at PWO sites with multi atoms and vacancy is also observed in this simulation In the PWO model, temperat ure is one of the important factors. Figure 6 6 shows the difference between E PWO and E Thermal as a function of temperature and length. If R is 0.4Ra at RT, E PWO is of the same magnitude as E Thermal This means atom mo tion can occur at intergrain PWO sites of Mg having a separation of < 0.4Ra by thermal energy alone even without any external stress. In addition, a high temperatu re is not necessary for such a migration The case of low temperature will be discussed later. We now extend the discussion of the nucleation mechanism of twinning obtained in Mg to all HCP metals by addressing the critical issues and validating the results against experimental findings. First twinning shear doesn t fully explain the occurrence of the twinning. T here is a relation between twinning shear (TS) and nucleation of twinning that has been established for a long time [157] In spite of its important physical meaning, the theory has raised two key questions : (rhenium) show the other dominant twin modes, not the mode having low TS? why does twinning occur dominantly in Be although the or mode has a lower TS? PWO depends on a potential well depth and thus strongly i nfluences the bonding strength cohesive energy, melting temperature etc. The c/a ratios and melting point s of HCP metals are illustrated in Fig 6 7 as a function of their cohesive energie s (CE) Interestingly, their CEs appear to be related to the c/a ratio. The nucleation of

PAGE 115

115 twinning depends systematically on the CEs and MPs of various HCP metals Since the CEs increase PWO for a twin nucleation, regime s I, II, and III indicate g roup s of elements for which twin ning is dominant subordinate and rare, respectively Filled circles indicate elements in which twinning modes have been reported from experiments. The boundaries of each regime are assigned by the twinning modes reported from experiments. All fil l ed circle s (Cd [158] Zn [158] Mg [158] Be [158, 159] and Co [160] ) denote elements showing the mode as a dominant twin in the regime I. The crossover of the dominant twin mode from to the other twin ning mode (e.g. twinning at Ti [161] and at Zr [162] and Hf [162, 163] ) first occurs at Ti. Hence, Ti is identified as the b oundary between regime s I and II. In regime II, the twins are still active enough to have an effect on plasticity of each element although the other twin mode is dominant. This is noted by some papers which found both and the other twin modes frequently take place at those metals. In contrast to Ti, Zr, and Hf, the twin is very inactive in Re [164] Hence, Re needs to be separated from the regime II, and regime III includes Re and Os having similar CE with Re.

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116 Figure 6 7 c/a ratios and melting temperatures as functions of cohesive energ ies of various HCP metals For the c/a ratio, [c/a ratio] = 12.985 exp ( [CE] / 0.315) + 1.586, R 2 =0.871. For CE and MP fi lle d and unfil l ed circle s denote element s whose twin mode is known and those where it is not, respectively. In regime I, II, and II each element show s the twin dominantly, subordinately and rarely respectively. Second, there is a lso a sensitive dependence of twin nucleation activity on grain size. Twinning becomes more inactive than slip with decreasing grain size, thus leading to a crossover in the dominant deformation mode from twinning to slip in coarse grained metals. It is found that the areal density of the PWO sites depends on the intergrain there are more PWO s in lar ger area s control the rate of twin nucleation is the area of GB per grain Assuming a spher ical

PAGE 117

117 2 t he decrease of r causes a rapid drop proportional to r 2 of twin nucleation activity [165] For over a half century, twinni ng shear theory and classical dislocation theory have been applied to understand the twinning in HCP metals [27] It is, however, been argued here that the twinning of polycrystalline HCP metals occurs by directional migration at intergrain PWO sites. Surprisingly, nucleation sites for the twin already exist at GBs in contrast to slip which can take place at any place of high stress concentration. Moreover, t he external stress fi eld not shear stress, to caus e enough interatomic strain is required to nucleate the twin at PWO sties Furthermore, our theory may be a key to explain ing GB phenomena such as stress induced grain growth [166] 6 3 2 Nucleation process in S mall G rain s In C ha pter 4, the crossover of initiation deformation mechanism s between basal < a > slip and twinning was observed in textured Mg. As discussed above, the twin nucleation mechanism by interatomic strain at PWO sties is explained for grai n size s of > 40nm. To examine whether the crossover is caused by a different twinning mechanism of at small grain size, the twin nucleation in 18nm textured Mg is analyzed in Fig. 6 8. A n initial GB structure is seen i n Fig. 6 8( a) The w hite zigzag lines indicate the same z position along the thickness direction of After the tensile stress is applied to the texture, disordering of atomic structure having coordination number (CN) of 11 is found on the right side of the G B in Fig. 6 8(b). The white zigzag lines show the initial atomic positions mismatch with the lattice of grain 2. A displacement vector between from the initial positions at grain 2

PAGE 118

118 indicates shear parallel to the GB. Such a shear component is not found at the twin nucleation at large grain size s (>20nm). We thus need to consider the possibility that GB processes, such as GB sliding or diffusion are enhanced in nano size grain s GB sliding may lead to twin nucleation for grain size s of < 18 nm. As already shown in the time strain curves, the GB processes begin before the slip process in dislocation free nanocrystalline metals. The change into the twin nucleation mode mediated by the GB sliding is a reason for the crossover of initiation defor mation modes from slip to twinning in the texture. 6 4 Summary The deformation simulations of textured Mg reproduce all three expected types of twinning : tensile twins, and compressive and twins. T he and compressive twins are formed by the interaction between dislocations or by the interaction between a GB and dislocations. T he twinning is created from a first order < c+a > edge disloca tion It is revealed that the twinning of polycrystalline HCP metals occurs by directional migration at int ergrain PWO sites. twin sites already exist at G Bs in contrast to slip whi ch can take place at any places of high s tress concentration.

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119 Figure 6 8 Nucleation of twin in 18nm textured Mg (a) An initial GB structure. White zigzag lines indicate atoms on the same atomic plane normal to the thickness direction. (b) A plastically straine d GB structure at 27.8psec. (c) An obvious twin nucl ei of is found at 29.8psec.

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120 CHAPTER 8 OVERALL MECHANICAL R ESPONSE OF RANDOMLY ORIENTED HCP METALS In previous chapters, the plasticity of HCP metals was studied in 2D texture d materials As is known from studies of FCC metals, the 2D textures successfully reproduce the fundamental processes of GB process, slip and twinning [49, 73, 86] However, the activations and interac tions of deformation modes occur somewhat independently in the 2D system For example, although basal < a > slip can appear in textured Mg, prismatic < a > slip is not observed at any value of external stress in the same texture This is because a GB in the 2D columnar texture does not produce a shear stress in the thickness direction. To examine interactions bet w een basal and prismatic, 3D structure s having a real istic grain shape, not a columnar texture, are thus required This chapter examines the comparative eas e of activation of slip and twinning processes. Furthermore, not only the Hall Petch behavior but also GB processes are addressed here. 7 1 Str ain S tress R esponse Strain stress curves of fully dense 3D Mg are shown in Fig 7 1 Grain sizes range from 36nm to 6nm. The polycrystalline structure already shown in Fig. 2 10 was strained at constant rate of 1.5 10 9 s 1 Young s modulus and a tensile strength decrease with decreasing the grain size in Fig. 7 1 (a) Such behavior in a n anocrystalline metal is already well known [44, 87] One of the important phenomena in nanocr y stalline is an inverse Hall Petch response S trength is normally enhanced, as the grain size decrease s However, when th e grain size becomes less than a critical value, the strength begins to decreas e due to GB processes [42, 43] To find such a critical grain size in nanocrystalline Mg, a Hall Petch plot is constructed in terms of the

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121 flow stress (average stress at strain range from 8 % to 10%) of Fig. 7 1 (b) as shown in Fig. 7 2. Fig ure 7 1 S train str ess curves of nanocrystalline Mg with fully dense 3D structure. Each simulation of tensile test was conducted at 293K under uniaxial stress. A strain rate was 1.5 10 9 s 1 during tensile test a. stress strain curves of full scale. x and y axis indicate st rain and stress. b. stresses at strain between 0.06 and 0.11. 7 2 Hall Petch R elation As expressed in Eq. 7 1, the s trength of materials normally increases with decreasing grain size according to Hall Petch behavior [3] ; however, the trend becomes the opposite (i.e., decreasing with decreasing grain size) b elow a critical grain size. This phenomenon is well known as inverse Hall Petch (H P) behavior. (7 1)

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122 The crossover from normal to inverse H P behavior at a critical grain size of typically 20 ~ 30 nm is thought to be due to a change in the dominant deformation mode, from one based on dislocations to o ne mediated by GB processes [167] Figure 7 2 shows the flow stress calculated from 8 11% strain of Fig. 7 1 as a function of grain size in nc 3D Mg. The p olycrystalline structure already shown in Fig. 2 10 was strained. The H P relation is obtained from the relation between the yield strength and the grain size. However, the flow stress is often adopted for the H P graph because it is hard to identify the yield strength in nc metals [42, 43] The crossover in the H P slope in Mg appears at grain size o f 24nm, which is similar to previous MD results for FCC metals such as Cu (~15nm) [42, 43] and A l (~20nm) [36] The n ormal and inverse H P slopes are 1.2 and 0.9 MPa mm 0.5 respectively. In general, the normal H P slop e obtained in MD simulation of nc metals, as previously shown in the general H P graph of Fig. 2 7, is smaller than that in coarse grained metals due to reduced activity of slip and twinning [43, 87] Neverthel ess, it is instructive to compare simulations and experiments, as this can help to understand the difference of plasticity in HCP and FCC metals. Table 7 1 represents the normal H P slop e s of pure Mg and Cu as determined from MD simulations and experiments The slop e of simulated nc Mg is much smaller than that of simulated nc Cu or experimental Mg. By contrast, the simulated nc Cu shows a similar value of H P slop with experimental one. The small slop e in the simulated nc Mg is due to the restrained twinni ng activity at this small length scale. As the grain size r decrease s the twin activity rapidly decrease proportionally to r 2 [165] If grain size is not small enough (~ a few ten nanometers) to yield significant GB process es in HCP metals, then only basal slip is

PAGE 123

123 active, and this cannot properly produce plasticity without a help of twinn ing. However, slip process es in nc Cu are still dominant at grain size of 20 ~ 100 nm because slip activity decrease proportionally to r 0.5 in coarse grains and r 1 for nano grains [50] Fig ure 7 2 the Hall Petch graph in nc Mg. The flow str ess is calculated by averaging strain values between 8 and 11% in strain stress curves of Fig. 7 1. Table 7 1 Hall Petch slopes of pure Cu and Mg from MD simulations and experiments. Mg Cu K y / MPamm 0.5 G rain size K y / MPamm 0.5 G rain size MD simu lation 1.2 24 ~ 36 nm 5.9 [43] 24 ~ 49 nm Experiment 10 [168] 43 ~ 172 m 4.7 [169, 170] 15~120 m From the present simulation

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124 7 3 Microstructure Evolution In this section, the deformation mechanism in nc Mg is characterized by analyzing the st r ained structure and the deformatio n modes. From Fig. 7 2 w e f ind that the critical grain size in the Hall Petch graph of Mg is 24nm. The normal and inverse Hall Petch behaviors separated at the critical size need to be related to the def ormation process es To examine the relation between the Hall Petch behavior and microstructure evolution, the comparative prevalence of individual deformation mechanisms (slip, twinning and GB process) is examined in samples of three different grain sizes (9, 18, and 36nm ) Figure 7 3 shows an 18nm gra in size polycrystal strained by 11% at 293K at a constant strain rate of 1.5 x 10 9 s 1 Initial and deformed structures are seen in Fig. 7 3 (a) and (b), respectively. After cooling down to 0.01K, the sample was analyzed with CN A. FCC SFs pyramidal disloca tions and twins appear as deformation structure s. The deformation modes are very similar to those manifested in the 2D columnar micro structure. S lip and twinning processes occur in different grains. This means that as previously revealed in columnar textu re s there is complementar ity between slip and twinning in order to promote plasticity under the external stress.

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125 Fig ure 7 3 Snapshot of 11% strained structure of 18nm grain size at 293K with constant strain rate of 1.5 x 10 9 s 1 a and b represent structures before and after straining, respectively. G ray, black and red denote normal (HCP), disordered (non HCP or FCC), and stacking fault (FCC), respectively. Twinned area is shown transparent orange. Figure s 7 4 and 5 show a plastic deformation pro cess during a tensile test of the 36nm Mg structure. B asal < a > slip first occurs, as denoted by in Fig. 7 4(b), at a total strain of 4.81%. Twinning is not observed for this relatively low strain. From the yield point to the beginning point of slip, GB p rocess es may thus contribute to the plasticity of nc Mg. The strain range ( GB ) of almost 3% is large when compared to GB of < ~ 0.25% in the columnar structures. This obvious difference in GB can be explained by less active GB processes caused from smal ler GB area and the columnar textures with a hexagonal grain shape In Fig. 7 4 (c) the activated < a > dislocation glides with an extended type, and new < a > slip is activated at another grain. The basal < a > dislocations normally have a low shear strain, sho wn a s a uniform blue color ; however it

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126 is possible for the stress to locally have a high value shown in orange, near the GB. The prismatic dislocation and twinning appear for the first time in Fig 7 4 ( d ) The directions with which the prismatic d islocation propagates does not have any stable sites for partial slip. Therefore, in contrast to basal < a > slip, the prismatic < a > slip is not manifested with a wide SF. The twin boundaries are normally seen as parallel orange SFs owing to their high shea r strain. The tensile strength (TS) point of Mg with 36nm grain size appear s at ~ 5.5% strain in the Fig. 7 .4. Full scale plasticity a chie ved by twinning and slip leads to softening after the TS point. As the strain to the loading direction keeps increas ing, the density of dislocations, SFs, and twins rapidly increase in Fig 7 4 ( d ) to 5. The stress drop after the TS point is closely related with the activation of slip and twinning, as will be discuss ed later. In the following section the specific deforma tion mechanisms and their properties are addressed.

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127 Fig ure 7 4 Shear strain map of nc Mg with 36nm grain size. Normal HCP atoms are not shown. , and denote a basal < a > dislocation, pyramidal < c+a > dislocation and twin, respectively ( a) 27.4psec, 4.19% strain, (b) 31.3psec,4.81%, (c) 35.2psec, 5.43%, (d) 39.2psec, 6.05%, (e) 43.1psec, 6.68%, and (f) 47.0psec, 7.3%.

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128 Fig ure 7 5 Shear st rain map of nc Mg with 36nm grain size. Normal HCP atoms are not shown. , and denote a basal < a > dislocation, pyramidal < c+a > dislocation and twin, respectively (g) 50.9psec, 7.94% strain, (h) 54.8psec, 8.57%, (i) 58.7psec, 9.21%, (j) 62.6psec, 9.86%, (k) 66.6psec, 10.5.%, and (l) 70.5psec, 11.1%

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129 7 4 Individual D efect P rocess 7 4 1
slip The < a > slip mode is the dominant deformation process in Mg. As illustrated in Fig. 7 6, various types of slip can take place during plastic deforma tion. In nc Mg, due to the absence of Frank Read source s, each < a > slip mode can only be produced by direct activation from GB s and from cross slip from the other < a > slip. I n grain s in which < a > slip is the first process, it is almost always the sole pro cess, e.g. the grain noted by a red circle in Fig s 7 4 and 5. Because the basal < a > slip mode is easier to generate than the prismatic or pyramidal modes, the basal < a > mode does not cross slip to the other higher energy modes. The basal < a > slip mode h as a tendency to appear as a full, a partial, or an extended dislocation. The prismatic < a > slip mode is shown in Fig. 7 6 ( b ) Initially two prismatic < a > edge dislocations are activated from one and the o ther si de of a GB, as noted by A an B. Th e prismatic < a > dislocation, denoted by A is changed into screw type at both ends of the edge type, and one of the screw dislocation cross slips to a basal < a > dislocation. The other prismatic < a > edge dislocation, denoted by B is similarly altered t o screw and extended basal < a > dislocations. In conclusion, even if the prismatic < a > slip is initiated with an edge type, it is energetically favor ed to become screw type and to cross slip to the basal < a > slip.

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130 (a) (b) Fig ure 7 6 < a > slip process in a 3D Mg structure with 24nm gr a in size. ( a ) V arious < a > slip vectors and planes in a HCP unit cell: Basal, prismatic, and pyramidal slip plane are and A (or B) and A are 1 / 3 and 2/3 partial dislocations, respectively (b) Potential energy map of dislocations and GBs. 7 4 2 < c+a > and slip In contrast to active < a > slip in a deformed sample, no independent < c+a > process is found. However, some prismatic < a > dislocations are identified as creating < c+a > and < c > dislocations. Activation processes of < c > and < c+a > from prismatic < a > dislocations are exhibited in Fig. 7 7. The snapshots show a grain being compressed in the c d irection. Normal HCP atoms are not shown, and each color denotes the boundar ies between different grains. In Fig. 7 7 ( a ) two prismatic < a > dislocations (I and II) activated from a right corner of the grain are gliding on the plane in the direction. The dislocation ( I ) keeps gliding with curvature by bowing in Fig 7 7 ( b ) The curvatures are different at the upper and lower parts of the pure edge component. S imply considering the relation between stress and radius of the curvature by ten sion, [4] bowing stresses ( ) resulting in bowing of the dislocation

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131 are due to The gliding behavior of prismatic < a > dislocation s, II and V, is a little different from that with I. III and V which denote compressive twin n ing have a role in preventing one of end of II and V from moving, respectively. Significant changes appear in the ends of dislocation s I, as seen in Fig. 7 7 (c) and ( d ) Th e prismatic < a > dislocation terminated as a screw type is comparatively easy to move up or down on the prismatic plane. The upper end of I cross slip s to a basal < a > extended edge dislocation. Hence, the edge part of basal < a > cannot follow the vertical mo vement of I to the c axis, finally separated from I The separated edge basal < a > is seen as I in Fig. 7 7 ( d ) At the lower end of I, the curvature ( ) become s smaller in Fig. 7 7 ( c ) The dislocation I is transformed into the hook in Fig. 7 7 ( d ) T he tail of the hook ( 1) is separated in Fig. 7 7 ( e ) As dislocation I glides, dislocation segments of 2 and 3 are left in Fig. 7 7 ( f ) On the other hand, two prismatic < a > dislocations, II and V, gliding to the opposing directions exhibit two reactions. After two dislocation s encounters, the upper parts keep moving while the lower parts dissociate to 1 and 1, as shown in Fig. 7 7 ( d ) Formation of < c > and < c+a > dislocations from prismatic < a > slip process is described in Fig. 7 8. T hree prismatic < a > d islocations activated from a GB are captured in Fig. 7 8 ( a ) In contrast to the pure prismatic < a >dislocation of I, dislocation II consists of pyramidal as well as prismatic and basal < a > components on planes of , and While I include s many jogs, II has one jog at its middle. The jog is normally known to move on a non slip plane with the help of a vacancy [21] Although a few vacancies denoted by O, are found, they are not enough to explain the motion of the jogs. In this simulation, a ledge on the prismatic < a > dislocations is called as a jog due to its protrusion out of slip plane. It is, how ever, necessary to understand

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132 the jog on the other slip plane, N on vacancy assisted process may be possible to move the jog. The jog at II is the direct reason for the complex process es observed between II and V. II and V glide away without any col lision above the jog due to some distance between their slip planes in Fig. 7 8 ( a ) But, below the jog, the II and V dislocations collide on the exact same slip plane. After the collision, the upper parts of the dislocation II and V are connected at the jo g, the other ends at the GB keep gliding to the opposite directions. The l ower parts become dissociated, leaving some dislocation segments denoted 1 and 2 in Fig. 7 7 ( d ) Fig ure 7 7 Prismatic < a > slip process in a 3D Mg structure with 36nm grain size. I and II, and V denote prismatic < a > dislocations. II and IV denote compressive twins O indicates a vacancy. I an d II are dislocation segments separated from I and II. 1 and 2 are segments produced by interaction between II and V. 1, 2, and 3 indicates < a > dislocations. ( a ) ( b ) ( c ) ( d ) ( e ) and ( f ) are captured at 43.1psec ( total strain: 6.68%), 47.0psec (7.3%), 50.9psec (7.94%), 54.8psec (8.57%), 58.7psec (9.21%), and 62.6psec (9.86%) respectively.

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133 Fig ure 7 8 Dislocation processes which occur from prismatic < a > slip. (a) < a > dislocations shown in Fig. 7 7c are gliding. and denote slip planes of basal and prismatic < a > J denotes jog. ( b) Creation of dislocation segments in Fig. 7 7(d). (c) A screw < c+a > dislocation cross slipped from V. (d) < c > dislocation segment produced from interaction between V and II. (e) Initial stage of < c+a > slip. Re d line denotes second order pyramidal slip plane, (f) Separation of the < c+a > dislocation

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134 Figure 7 8 ( b ) shows a potential energy map for the dislocations in Fig. 7 7 ( d ) Different energy structures with normal prismatic < a > dislocation are seen at ( c ) and ( d ) : analysis of the energies of the structures shows that the energy of the dislocation cores is ~ 0.2eV higher than that of normal prismatic < a >. It is evident that the segments with the high energy are not typical prismatic < a > dislocations. In order to identify the dislocations, they are magnified in Fig. 7 8 ( c ) and ( d ) Leading prismatic < a > and the trailing part with high energy are aligned along the c axis in Fig. 7 8 ( c ) The basal planes in the trailing part are lifted, thereby being between the original A and B layers. It means that the slip vector of the trailing segment contains the < c > component. The dislocation segment is revealed to have pure < c > component, as atomic displacements to the are denoted by red arrows in Fig 7 8 ( d ) The formation of the slip vector component to the c axis from prismatic < a > slip is further investigated in Fig 7 8 ( e ) and ( f ) Initiation of slip process is observed from screw < a > on the second order pyramidal < c+a > slip plane in Fig 7 8 ( e ) Even thoug h the < c+a > slip develop s a screw dislocation, the < c+a > screw is separated due to the different direction and magnitude of slip vector with the prismatic < a > dislocation body, as shown in Fig 7 8 ( f ) The separation of the screw < c+a > dislocation is imper fect, thus a part of it is left. The left part on the screw < a > dislocation acts as a source for new screw < c+a > dislocations. 1, 2 and 3 were < c+a > screws, generated from the source in Fig. 7 7 ( e ) and ( f ) However, because they are not independently stable, the < c+a > dislocation s are rapidly changed into pure < a > dislocation after separated. The slip vector of < c+a > in Fig. 7 8 ( c ) corresponds to 1/18 shown as B 0 in Fig. 6 5, and a pure < c > component is very small. How the

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135 < c+a > can be changed to the < a > without any other interaction may be due to such a small < c > component in the slip vector of the < c+a > In the present simulation, t h e < c+a > slip is directly activated from one prismatic < a > dislocation without the help of < c > dislocations. This is in contradiction to an earlier model, reported by Yoo et al. [23] of < c > dislocation assisted activation for < c+a > dislocation in Fig. 7 9. The activation process of < c +a > and < c > dislocations in this study are summarized in Fig 7 10 and 11. Fig ure 7 9 Evolution of the dislocation source in a < c+a > pyramidal slip [23] (a) cross slip of a dislocation (b) formation of junction for < c+a > dislocation and (c) cross slip of < c+a > dislocation

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136 Fig ure 7 10 Activation proces s of pyramidal < c+a > slip from a single prismatic < a > dislocation. Fig ure 7 11 Activation process of < c > dislocation between two approaching prismatic
dislocations. Green dot line denote a part of the prismatic dislocation gliding on a differe nt plane.

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137 The competition between activation of the < c+a > slip and bowing is now examine d Figure 7 12 ( a ) shows the prismatic < a > dislocation in the shape of a hook before the < c+a > dislocation is separated. First the bowing of pure edge part (II) is required to make screw < a > component. After fully bowing, a screw < a > component forms. Although < c+a > slip is observed in the present simulation, there appear to be two competing processes depending on direction of shear stress. The two cases are illust rated in Fig. 7 12 ( b ) and ( c ) If the shear stress in a grain is less than the CRSS for second order pyramidal < c+a > the screw < a > dislocation would bow, as shown in Fig. 7 12 ( b ) For the purposes of analysis of bowing of II and IV, the bowing is treated as arising from bending According to the elastic theory of dislocations, the bending energy depends on the type of dislocation (e.g. edge or screw) and the bending length and angle [21] Fig ure 7 12 Bowing of a prismatic < a > dislocation and activating of second order pyramidal < c+a > slip. I, II, III and IV denote dislocations of pure edge prismatic < a > bowed (mixe d) < a > pure screw < a > and bowed (mixed) < a > respectively. indicates dislocation line. B and p denote the bowing angle of screw < a > dislocation and the angle between basal and second order pyramidal slip planes, respectively and indicates shear stress causing the < a > dislocation t o glide on the prismatic plane, resolved shear stresses to the directions of x, z, and the second order pyramidal slip.

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138 Figure 7 13 is obtained the stand a rd equations for the energies of edge and screw types [21] : (7 2) (7 3) Fig ure 7 1 3 Energy difference as a function of bending angle. L denotes b ending length of a straight dislocation. Bowing may be considered as a series of b e nding processes at very small distance s In Fig. 7 13, the difference of bending energy between screw and edge

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139 dislocations per unit length, (W screw W edge )/L, is always positive number at small angles (<15 ). Thus, the screw dislocation is harder to bend than the edge dislocation. This tendency increase s with increas ing bending length, L. Compared to the prismatic edge < a > dislocations, the bending or bowing of the scr ew dislocations, IV would be rare. This is consistent with the present simulation results and general theory of elasticity [21] The bending energy mentioned above can be expressed as the total energy by combining with the strain energy : E T otal, screw = E Strain Energy, s crew + E Bending Energy, s crew (7 4) where, E T otal, screw E Strain Energy, s crew and E Bending Energy, s cre w denotes total, strain, and bending energy of screw dislocation, respectively. The stain energy is normally noted : (7 5) w here, b R and r 0 denotes bulk modulus, B urg er s vector, the range of the strain field, and dislocation core size, respectively. In real materials, R is considered as a half of the distance between neighboring dislocations. R can be replaced by dislocation d ensity : (7 6) In addition r 0 is normally taken to be ~ 3b [129, 171, 172] Thus, substituting r 0 and R in Eq. 7 5 by 3b and Eq. 7 6, yields

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140 (7 7) T he to tal energy of < a > and < c+a > screw dislocation s in Fig. 7 12(b) and (c) can be determined by combining Eqs. 7 3 and 7 : (7 8) Fig ure 7 1 4 Relation between misorientation angle and dislocation energy per unit length (E/L) of screw < a > an d pyramidal < c+a > dislocation s at dislocation density ( ) of 10 8 cm 2 L denotes bending length.

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141 Fig ure 7 1 5 Relation between misorientation angle and dislocation energy per unit length (E/L) of screw < a > and pyramidal < c+a > dislocation s at dislocation density ( ) of 10 12 cm 2 L denotes bending length. Ac cording to Eq. 7 8, screw < a > and < c+a > dislocations are energetically compared in Fig s. 7 1 4 and 1 5. It is here necessary to understand the meaning of misorientation angle First, determines the bowing of < a > and activati on of < c+a > As the direction of the shear stress ( ) is close to the second pyramidal slip plane, in Fig. 7 12 ( a ) and ( c ) the < c+a > slip occurs more easily due to the higher RSS to the direction. Low may allow only bending of < a > not activation of < c+a > to take place. If the < c+a > slip once occurs, Figs. 7 14 and 15 are obtained.

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142 = 0 and = 90 indicat ing the < a > axis of and the < c > axis, respectively. If the screw < a > dislocation glides along is zero, and there is no bending energy. If the dislocation bends to some angle the total energy increases by contributing to its bending energy. On the other hand, the < c+a > screw dislocation is most stable on the pyramidal plane tilted 58.8 from the direction in the basal plane. The < c+a > dislocation can also bend like the < a > dislocation But, bending of the < a > and < c+a > dislocation occurs on different planes : the and respectively. As displayed in Fig. 7 14, at low dislocation density ( =10 8 cm 2 ) the < c+a > dislocatio n always has a lower energy than the < a > dislocation regardless of the bending length The reason of this result is much smaller burgers vector of < c+a > dislocation: the magnitude of b < c+a > is equal to that of ~1/3 b < a > However, in Fig. 7 15 the energy dif ference becomes small at the high dislocation densit ies ( =10 12 cm 2 ) which appear in severely deformed metals. As the dislocation density increase s bending of the < a > dislocations may be thus more frequent. In additio n, the energy of the < a > dislocation increase s more rapidly than that of < c+a > dislocation wi th increasing bending length. 7 4 3 Twinning As observed in columnar structures, tensile twins and or compressive twins are nucleated with very different mechanism and conditions. A similar trend is a lso seen in 3D structures. Figure 7 15 shows nucleation processes of and The compressive twin is initiated from a triple junction in Fig. 7 15 ( a ) propagating with a sharp shape normal to GB in Fig. 7 15 ( b ) After fully penetrating the grain the twin begins to thicken. As shown in Fig. 7 15 ( d ) new twins sometimes occur inside the

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143 twin which has sufficiently grown. On the other hand, the boundary of the twin slowly expands parallel to the GB in the initial stage of nucleation. The tensile twin is verified to follow the mechanism revealed in a 2D columnar texture (see chapter 7). Hence, the compressive twin is further examined. Fig ure 7 1 5 Twin nucleation process o f the compressive mode (a ~ d) and tensile mode (e ~ h). G ray, black and red denote normal (HCP), disordered (non HCP or FCC), and stacking fault (FCC), respectively. The twin propagates much faster than the twin. The twin is seen through its potential energy in Fig 7 16. The b lue plane (~ 1.51eV) indicates a n FCC SF shown as the red line in the 2D snapshot of Fig. 7 15. The energy structure of compressive twin bounda ry (TB) is quite differ ent with that of the FCC SF. The twin boundary (TB) has alternati ng lines of atom s with high (~ 1.45 eV) and low (~ 1.55eV) energy. This energy and structure are the same as those of the non basal SF found in partial first order pyramidal < c+a > dislocation previously shown in

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144 Fig. 6 2 ( b ) Hence, the TB is a non basal SF, and it is highly probable that the compressive twin propagate s by dislocation process. The mechanism of twin nucleation by slip has been frequently reported in FC C metals [49, 147, 173] In addition, it is the reason why t he twin propagates much faster than the twin mentioned in Fig. 7 15: both twins have different propagation mechanisms 7 5 Grain S ize E ffect on P lasticity The maximum strength of polycrystalline Mg was noted to occur at a grain size of 24nm from Fig. 7 2. The reason for the maximum strength is the crossover of deformation mechanisms from slip and twinning to GB process es with decreasing grain size [36, 42, 43] Basal < a > slip, pyramidal < c+a > slip, and compressive twin boundar ies were successively identified as SFs in Fig. 7 16. R emoving the GBs from the images enables us to cle arly see a series of deformation process es Figure 7 16 was obtained using CN A twice. First CN A identifies HCP, FCC, and disordered atoms. After removing HCP atoms from the list of total atoms, the coordination number s (CNs) of FCC SFs become 6 or 9, and CN of non Basal SFs appears to be 5. A s econd CN A process is carried out to distinguish pure SFs. Both GB s and n on basal SF are identified as disordered structure after the first CN A. However, since one atom has the same coordination as neighboring atoms in the SF, the atoms in the SF can be distinguished from those in GB s Some clusters consisting of a few atoms exist because some atoms can sometimes have the same CN. The clusters are removed using a filtering condition that atoms having CN of smaller tha n 3 or larger than 11 are not processed.

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145 Now the crossover of deformation mechanisms at nc Mg is examined coupling a strain stress curve and evolution of the SFs. Initiation point of slip and twinning is examined at three grain sizes in Fig. 7 17. F ig ure 7 1 6. Potential energy map of SFs in 11.1% strained Mg with 36nm grain size. Normal HCP and disordered atoms like GB are not shown here. R emoving method of GB is explained at next page.

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146 Fig ure 7 1 7. Str ain and evolution of SFs with increasing strain at different grain sizes (9, 18, and 36nm). The strains of 0.1% SF atoms per total atoms in a increases with decreasing grain size. In a strained structure, the large number of SF atoms means high activity of conventional deformation process: sl ip and twinning. Evolution of SFs in Fig. 7 17(a) begins more lately with decreasing grain size. Comparing SF atoms of three grain sizes

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147 at same strain, small grain size shows a small number of SFs. Hence, at small grain size s GB processes support insuff icient slip and twinning. The SF atom ratio of 0.1% is chosen to remove noise and to accelerate slip and twinning processes in Fig. 7 17(a). The strains at SF atom ratio of 0.1% ( 0.1% SF atoms ) are shown as a function of grain size in Fig. 7 18(b). The st rain point which slip and twinning begins in nc Mg has the following relation with grain size, r (7 9) G raphs of SF evolution and stress strain at each grain size are seen in Fig. 7 19. In the strain stress curve of coarse grained material the elastic limit depends on how much stress is needed to make a dislocation This process occurs by pre existing dislocations inside the material. In Fig. 7 19, SFs are not observed at the yield point. This indicates that the yielding p rocess begins by GB processes in nc Mg. N ormally the tensile limit at which stress decreases first in the strain stress curve appears by acceleration of slip or twinning process in coarse grain materials. Initiation of the SFs before the tensile strength (TS) point in Fig. 7 19 is consistent with that in the coarse grain materials. Thus, the SFs initiate yielding, but are involved with the TS. To deal with a tensile point and initiation of SFs, a str ain offset is defined in Eq. 7 10 S train offset = TS 0. 1% SF atom ratio (7 10 ) The strain offset, TS and 0.1% SF atom ratio are summarized in Table 7 2 Table. 7 2 TS 0. 1% SF atom ratio TS and 0. 1% SF atom ratio of nc Mg at grain sizes of 9, 18, and 36nm Grain size TS 0. 1% SF atom ratio TS /GPa 0. 1% SF atom ratio /GPa 9nm 0.02375 0.732 0.601 18nm 0.01025 0.918 0.898 36nm 0.00375 1.022 1.021

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148 Fig ure 7 1 8 Stress and evolution of SFs with increasing strain at different grain sizes ( a ) 36nm, ( b ) 18nm, and ( c ) 9nm

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149 S napshots of 11.1% strain ed Mg with 9, 18, and 36nm are displayed in Fig. 7 19. The maximum strength of Mg appears at grain size of 24nm. There are fewer SFs at grain sizes smaller than 24nm than that at the 36nm grain size. The larger number of SFs is observed at 36nm grain size, thereby causing strength hardening in nc Mg. The much l ower density of SFs would be insufficient for such hardening. This explains the inverse Hall Petch behavior at small grain size. The increase in the nucleation stress for dislocations at small grain size s reduces the evolution of SFs [50] In nanocrystalline materials, large GB area per unit volume and the high nucleation stress of dislocation allow s a GB process to significantly contribute to plasticity. Fig ur e 7 1 9 SF evolutions of 11.1% strained samples with three different grain sizes. a. 9nm, b. 18nm, and c. 36nm. GB is not shown in these snapshots. Gray and black denotes twin boundary or dislocation cores and FCC SFs, respectively. 7 6 Summary U sing molecular dynamics simulations, physical phenomena in nc 3D Mg were reproduced and analyzed in this chapter The crossover of the Hall Petch slope in Mg

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150 appears at grain size of 24nm < a > slip including basal, prismatic, and pyramidal modes is the d ominant deformation process in Mg. There is some direct activation of < c+a > and < c > dislocations from the GB s The < c+a > and < c > dislocations are nucleated f ro m the prismatic < a > slip process. W hile the < c > dislocation appears by the interaction between t wo prismatic < a > dislocations gliding in opposite directions, t h e < c+a > slip is directly activated from one prismatic < a > dislocation without the help of < c > dislocations. The activation of < c+a > slip can be considered as competing with the bending of a sc rew < a > dislocation according to the shear stress direction. As the dislocation density increase s bending of the < a > dislocations may be more frequent than the activation of < c+a > dislocations. In additio n, the energy of an < a > dislocation increase s more rapidly than that of a < c+a > dislocation with increasing bending length. In contrast to boundary of tensile twin, compressive TB is mainly a non basal SF formed by dislocation process es Considering the evolution of SFs, slip and twinni ng are main deformation mechanisms in normal Hall Petch regime (grain size: > 24nm). The increase in the nucleation stress for dislocations at small grain size reduces the evolution of SFs. Hence, the overall conclusion is that GB processes enhanced at sma ll grain size produce plasticity in nc Mg.

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151 CHAPTER 8 CATION ORDERING IN S ODIUM BISMUTH TIT ANATE Piezoelectric behavior in Na 0.5 Bi 0.5 TiO 3 (NBT), an A site complex perovskite compound, was first identified by Smolenskii et al. in 1960 [174] NBT displays three different phases as the temperature increa ses : a rhombohedral phase (< 533K), a tetragonal phase (from 533K to 783 813K), and a cubic phase (>~ 800K) phases [175] The c rystallograph ic structure of NBT at room temperature has R3 c symmetry as Figure 8 1 shows. Four R3c rhomboh edral unit cells ( in a 2X2 configuration ) distinguished by yellow lines are combined in the a b plane of Fig. 8 1(a). The R3c structure has one unit cell length in the z direction of Fig. 8 1(b). The R3c structure can also be viewed as a hexagonal structur e as delineated by black line on the xy plane. The Ti and the other cations (Na and Bi) are alternatively arranged along the c axis. Six o xygen s surrounding a titanium atom form an octahedr on, as noted by transparent blue structure in Fig. 8 1(b). Two ver tices of the octahedra are corner connected with other octahedra in diagonal direction of R3c cell seen in Fig. 8 1(b). Figure 8 1(c) the R3c structure in pseudo cubic axis (x p ,y p and z p ) The R3c phase of NBT can be expressed with a a a tilt system which appears in rhombohedral perovskites arising from oxygen octahedra along each axis This results in cell doubling of all three pseudo cubic axes [175] In this case, the cH setting of hexagonal R3c structure corresponds to [111] p of pseudo cubic axis. In the rhombohedral system a rhombohedrally centered hexagonal lattice projected down the threefold axis be considered as hexagonal axes, whose lattice parameters a H and c H

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152 are specified in relation to the double pseudo cubic cell ( 2a p x 2b p x 2c p ) by the following matrix [175] (9.1) Figure 8 1 R3c structure of 2 X 2 X 1 unit cell siz es. White, green, and red indicate titanium, Na or Bi, and oxygen, respectively. a, b, and c denotes hexagonal axis, and x p y p and z p represents pseudo cubic axis.

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153 T able 8 1 Three different phases and crystallographic data of NBT [175] Crystal system Rhombohedral ** Tetragonal Cubic Temp erature ( K ) < 673 > 673 < 873 > 873 Space group R3c P4bm Pm m a H ( ) 5.4887 c H ( ) 13.5048 a p ( ) 5.5179 3.91368 c p ( ) 3.9073 , ( ) 90, 90, 120 90, 90, 90 90, 90, 90 p ( ) 89.83 Tilt system a a a (three tilt system, antiphase) a 0 a 0 c + (one tilt system, in phase) a 0 a 0 a 0 (zero tilt system) Displacements Parallel along [111] p Antiparallel along [001] None ( ) 8.24 3.0 6 Some parameters are updated. ** Monoclinic phases are being reported instead of the rhombohedral phase with R3c The pseudo cubic cells are shown in Fig. 8.2. The left of Fig. 8.2 represents a view of the rhombohedral structure of double pseudo cu bic cell down [001] showing the tilt system and circle denote a tilting angle of octahedra and Na/Bi atoms, respectively. S ingle pseudo cubic cell with a shape of ABX 3 (A, B=cation, X=anion) perovskite structure is seen in right of Fig. 8.2. Hatched and black circles indicate A and B atoms, respe ctively. PbZr 0.5 Ti 0.5 O 3 (PZT) having a perovskite structure of ABO 3 (A=Pb, B=Zr and Ti) has numerous applications due to its superior ferro electric and piezoelectric properties. However, as previously mentioned in Chapter 1, since PZT contains lead there have been various attempts to find identify alternative compositions [176 178] Na 0.5 Bi 0.5 Ti O 3

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154 (NBT) is one potential alternative for PZT The f erro electric and piezo electric properties of NBT are similar t o those of PZT. A hi gh coercive fi el d and a relative high conductivity are however obstacles to overcome for application s [179] (a) (b) Figure 8 2 Pseudo cubic cells of perovskite. (a) The projection of the rh ombohedral cell down [001] o pen circles denotes Na/Bi sites [175] (b) The ideal cubic perovskite of ABX 3 (A,B = cation, X= anion). The anions are at the vertices of the octahedra. H atched and b lack circles denotes A and B cations [180] To improve the properties of NBT, we need to understand its structure at the atomic level In the structure of NBT, sodium and bismuth occupy the A in the ABO x perovskite structure There ha ve been only a few studies of the arrange ment of the Na and Bi cations Moreover, the results are contradictory. S hort range cation ordering of Na and Bi was found in neutron scattering by Vakhrushev et. al [181] However l ong range ordering between Na + and Bi 3+ cations was observed in XRD analysis using a single c rystal rotation camera [182]

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155 As previously mentioned in Fig. 2 2, a b initio or first principles, calculation are useful for characterization of the atomic an d electronic structures at small scales Since the cation ordering is presumed to involve a relatively small number (less than a few hundred) of Na and Bi atoms, first principles calculations can be used Table 8 2 Comparison of ferroelectric properties of PZT [9] and NBT [179] Pr o perties PZT NBT Remn a nt polarization (Pr) 35 2 38 2 Coercive fi e l d (Ec) 20KV/cm 73KV/cm 8 1 Background 8 1 1 First Principles Calculations First principles calc ulations are based on the interactions of the electrons and nucle i in a system These interactions are described by q uantum mechanics through the Schrdinger Equation. The time in dependent, non rel ativistic Schrdinger Equation [183] is written as: (9 3) where H is the Hamiltonian operator, the e igenvalue (energy). Eq 9 3 can be solved exactly for only a very few specific cases, such as the hydrogen atom. T he Hamiltonian in Eq. 9 3 can expanded with Born Openheimer appoxim ation to: (9 4 ) Where, m is the mass of the electr on and N is the total number of e l ectrons in the system [184] The individual te rms in the Hamiltonian in Eq. 9 4 are from left to right,

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156 the kinetic energy of each electron, the electron nuclei interactions, and the electronic electron interactions, respectively. is a function of the spatial coordinates of the N electrons, or (9 5 ) This full wave function can be approximated as a product of individual wave functions knows as the Hartree product, and exp ressed as: (9 6 ) In the Copenhagen interpretation of quantum mechanics, the electron density at a particular position r can be written in terms of the wave function as: (9 7 ) w here, the asterisk indicates the complex conjugate of the wave funct ion. The prefactor of 2 in Eq. 9 7 takes into account the P auli exclusion principle that each individual electron wave function can be occupied by two electrons with opposite spin. Cons idering N electro ns in the system, this density approach reduces Eq. 9 4 to be a function of 3N coordinates. This electron density is the main motivation towards the development of density functional th e ory (DFT). 8 1 2 Density Functional Theory (DFT) T he Kohn Sham theorem [185] states that energy of the overall functional is the true electron density correspon ding to the full According to this theorem, t he properties of a many electron system are determined by using functionals which are the spatially

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157 dependent electron density T he name of density functional theor y t hus comes from the use of functionals of the electron density Based on the Hohenberg Kohn theorem [184] the simplified energy functional can be written as : (9 8 ) where, (9 9 ) and is the exchange correlation functional which defines all the other inte ractions not included by Eq. 9 9 The terms in Eq. 9 9 are, from left to right, the kinetic energy of electrons, the Coulomb interaction between the electrons and nuclei, the Coulomb interactions between electrons, and the Coulomb interaction s betwee n nuclei. In order to solve for Kohn Sham showed that the correct electron density ca n be expressed as solving a set of Equations involving single electrons only. Therefore, Eq.9 4 can be written in terms of the Kohn Sham Equation as: (9 10 ) W here is the electron ion potential, is the Hartree potential, is the exchange correlation potential, is the Kohn Sham eigenvalue, and is the wave function of state i The Hartree potential is given b y: (9 11 ) The exchange correlation potential is given as:

PAGE 158

158 (9 12 ) w here, is the exchange correlation energy. Even though Eq.9 10 looks very sim ilar to Eq.2 32, there is one very distinct difference : Eq.9 10 does not contain any of the summations that are present in Eq.9 4.This makes replacement of the electron wave function by the electron density much more useful from a computational standpoint. The solution of Eq.9 10 proceeds as follows [184] : a) A n initial guess of t he electron density, n(r) is made. b) T he Kohn Sham equation is solved to find the single particle wave functions c) T he electron density is calculated from the single particle wave functions obtained from step b) using d) The previous value of n(r) and the calculated are compared. If the two densities are the same then the ground state electron density has been identifi ed. If not, then steps b) d) are repeated to convergence 8 1 3 Ex change Correlation Functional The key idea behind the Kohn Sham theory was to make the unknown contribution as small as possible Nevertheless, it is important to choose a good exchange correlation approximation for the system under consideration. One common approximation is the local density approximation (LDA) [184] In this approximation the exchange correlation potential at any position r is assumed to be describable as a homogeneous electron gas having the same density. Since a homogeneous gas approximation is not always physically appropriate in r eal systems, a

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159 more sophisticated description of the functional is given by the generalized gradient approximation (GGA) [186] Th e GGA functional includes both the local electron density and the local gradient in the electron density. One would expect that GGA with a more elaborate description of the exchange correlation functional for individual calculations would match experiment better However, this is not always the case [184] Hence, care must be take n in choosing the exchange correlation functional for individual calculations. 8 2 Computational D etail s To investigate the comparative stability of various arrangements of the Na and Bi cations around the Ti and O octahedr a the Vienna ab initio Si mulation Package (VASP) based on the DFT is used in the present work For these calculations, t he electron exchange correlation energy is described in the generalized gradient approximation (GGA) with pseudopotentials of projector augmented wave (PAW) T he GGA is chosen because GGA generally gives substantially better materials properties than the LDA, particularly for the cohesive properties bonding length and atom ization energies The cuto ff energy for the plane wave basis set is chosen to be 4 5 0 e T h e criteria for considering the calculations as fully converges is that the total force on all of the atoms atoms is < 0.02 eV/. The 2 X 2 X 2 NBT supercell is treated with a 2 X 2 X 1 Monkhorst Pack k point mesh to sample the Brillouin zone. The s imulat ion structure has symmetry of R3c which is the room temperature phase of NBT shown in Table 8 1. In the R3 c hexagonal unit cell of NBT, the A sites are occupied by 3 Na and 3 Bi atoms. Both short and long range ordering of Na and Bi are considered in the p resent work. For charge neutrality the number of Na and Bi in the supercell are set to be equal. If the hexagonal structure is considered as being formed

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160 from three identical rhombohedral unit cells, then the s ix inequivalent cation sites can be occupied by three Na and Bi, result ing in four different arrangements of T ypes I IV as shown in Table 8 3 and Figure 8 3 Type IV appears not to have any long range order ing in the pseudo cubic perovskite structure in contrast to T ype s I, II, and III (see Fig. 8 4 ). By contrast, if all of the cations in a 2 X 2X2 supercell of hexagonal unit cell are treated as being independent, then there are many more possible cation arrangements Types VI IX have twenty four Na and twenty four Bi atoms which occupy the forty eight cation sites in the 2X2X2 cell in various different ways In particular Type V is obtained by expanding the Type I cation ordering of 1X1X1 supercell to the 2X2X2 supercell When a unit cell of Type I is connected with the other one, of the A catio n change s identity. Thus the structure becomes N a N a N a |B i B i B i or B i B i B i |N a N a N a (see [100] and [010] in Table 8 3 ). However, there is no such change in cation ordering in T ype V. When viewed as a pseudo cubic perovskite structure, the layers of Type V consi st of cations of the same species along the [100] and [010] directions and alternati ng cation s along the orthogonal [001] direction. Table 8 3 Arrangements of Na and Bi in 2 X 2 X 2 super cell. Types Cation arrangements Ordering of Na and Bi Method of cation arrangement Hexagonal unitcell Perovskite axis A B C D E F [100] p [010] p [001] p I Duplicating a hexagonal unit cell N B B B N N N N N B B B N N N B B B N B B N B B long range II N B N B N B N B N B N B N B N B N B N B N B N B III B B B N N N N N N B B B N N N B B B N N N B B B IV B N N B N B Not fixed S hort range V Arranging Na an Bi in a 2X2X2 frame of Ti O All N (or B) All N (or B) N B N B N B long range VI IX Not fixed Short range N and B denote Na and Bi, respectively 3. [100] p [010] p and [001] p indicate x, y, and z axis of pseudo cubic.

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161 Figure 8 3 Cation arrangements adopted in th e present work. (a) a unit cell of NBT on the x y plane (basal) (b) a unit cell of NBT in the z axis (c ,d, e, and f ) Type 1, 2, 3, and 4 (T1,T2,T3,T4) (g) a 2X2X2 frame of NBT for T ype 5, 6, 7, 8, and 9 (T5, T6, T7, T8, and T9). Red dot circle denote s c ation sites for Na and Bi. Each blue dot line, indicating one of the perovskite ax e s, has alternat ing cations in T ype 5, as denoted with a chemical element of Na or Bi. From T6 to T9, the red dots are occupied by Na and Bi

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162 Figure 8 4 Cation ordering in T4 with 5 X 4 X 4 psuedo cubic size. 8 3 Analysis of S tructure C hange Structural information and energies of samples from T1 to T9 optimized using the GGA calculations are shown in Table 8 4 Among the vari ous types, the R3C symmetry of by the T1, T2 and T3 structure s To compare each type effectively the lattice constants and angles are plotted in Fig. 8 5 Table 8 4 Structural parameters and energies of each ordering type. Type T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 E /eV 1803.33 1803.65 1803.37 1803.77 1807.27 1804.41 1804.65 1804.61 1804.36 a/ 10.98919 10.9908 10.98999 10.99561 10.88887 10.99918 10.99815 10.99954 10.99877 b/ 10.98917 10.9908 10.99023 11.00228 11.02058 10.99924 10.9 9961 10.99736 11.0081 c/ 26.98994 27.42721 26.99251 27.29441 27.14211 27.36157 27.37067 27.34045 27.2759 / 90.0005 90.0001 89.9993 89.818 90.1559 89.9733 89.9725 89.9253 89.9579 / 89.9996 90 .0000 89.9993 90.1202 89.9998 89.9917 89.9739 90.0723 89.9291 / 120.0003 120.0001 119.9999 119.9271 119.6057 119.9762 119.9931 119.9716 120.0415 V / 3 2822.677 286 9.261 2823.437 2861.688 2831.856 2867.467 2867.759 2864.989 2858.797 T5 sample doesn t satisfy the R3c symmetry due to anistropy in its lattice constants (a and b) and angles ( and ) as shown in Table 8 4 and in Fig. 8 5 (a). In all

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163 samples exept T5, t he lattice parameters, a and b, are essentially identical, a=b Interestingly, the energy of the T5 ( 1807.27eV) having the distorted strcture from R3c is clearly lower than that of the other samples ( 1803.33 1804.65eV) in the 2 X 2 X 2 super cell conta ining 240 atoms in Fig. 8 5(b) The T5 sample will be disscued later. Except T5, samples from T6 to T9 with random arrangement s of cations have low er energy than cation orde r ing manipulated with some rules from T1 to T4. This indicates that if NBT is ident ified from R3c by XRD( X ray D iffraction ) or ND (Nuetron Diffraction) in experiments, random ordering of Na and Bi would be prevalent. This is also coincident with Vakhrushev et. al. [181] s ND results and Park et. al. [182] s XRD results of NBT sample prepared by the flux method. On the other hand, non R 3c NBT at room temperature has been reported. Park et al. [182] identified NBT grown by the Czochralski method as pseudo cubic with a small deviation from ideal cubic. Aksel et al. [187] also found a monoclinic phase of NBT instead of R3c symmetry using XRD analysis. Assuming that the T5 structure c ould be the low symmetry structure found experimentally, structure analysis and formation mechanism of T5 would be important Henc e, in the next section, the T5 sample having the lowest energy is compar ed with T2 sample satisfying R3c sy m metry In addition a symmetrical analysis and probable reason for the distortion of T5 structure are dealt with in detail.

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164 Figure 8 5 Ener gy and correlation between structural parameters. (a) 2a 2b. (b) Energy. (c) 8 4 Analysis of L ayered NBT The f inal optimized structures of T2 (left) and T5 (right) are exhibited in the pseudo cubic perovskite ax e s ( a p b p and c p ) in Fig. 8 6. T2 is a reference structure of R3c symmetry. T5 displays the largest deviation between the initial and final structures, as discussed above. While T2 appears as the pseudo cubic cells of Fig. 8 2, cells of zigzag by distorted oxygen (pink dots) structu re are seen inT5. In contrast to sodium (green dots), bismuth (blue dots) is off center in cells of Ti (white dots) and O in T5 of

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165 Fig. 8 6. Hence, it is revealed that there is a big structur al change in T5, resulting in a large difference in lattice param eters shown in Table 8 4 and Figure 8 5. Figure 8 6 Final optimized structures of T2 (left) and T5 (right) in the perovskite axis (a p b p and c p ). Blue, green, pink, white dots denote bismuth, sodium, titanium, and oxygen, respectively. Blue, red, and green arrows indicate three perovskite axis, a p b p and c p respectively. V ariations of energy and force are compared between T2 and T5 in Fig. 9 7 As shown in Fig. 8 7( a ) the final structure of T2 having the lowest energy ( 1803.6 eV) is obtained at 38 i o nic iteration steps (ITS) .The final optimized structure of T2 is R3c. On the other hand, T5 shows similar trend of energy and force with T2 for the first 55 ionic iteration steps. For the next ~15 iteration steps, the force actually increases e ven though the energy decreases from 1806.2 to 1807.2 eV after 60 ITS. Such changes of force and energy indicate that there is a significant change in the structure of T5. Analyzing the final optimized structure of T5, a new phase is found which does n ot satisfy the R3C symmetry of the initial structure

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166 Figure 8 8 shows the optimized T5 structure As shown in the two schematics, it has lower symmetry than R3C. Along the c axis, the layers of Na and Bi are alternatively stacked, with the Na and B i columns offset from each other Such a distorted structure can be attributed to the charge balance in and between layers of Na and Bi having a charge of +1 and +3, respectively. The s tr uc trual characterization of the layered str uc ture is shown in Table 8 5 Table 8 5 Structural information of the layered structure, T5. Symmetry Lattice constant s Lattice angles P 1 a b c 7.77322 7 .77768 7.82 025 90.0030 89.9993 88.8383 Figure 8 7 Change of force and energy in T2 and T5 as a function of an ionic iteration step. E and F denote energy and force.

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167 Figure 8 8 A unit cell of optimized T5 after structural optimization (a and b) and projections of simplified Pr 0.5 Sr 0.5 MnO 3 perovskite structure with symmetry of F 4/mmc [188] (c). Only manganese (small dot) and oxygen (large dot) are shown

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168 Figure 8 9 Charge density images of P1 phase. a. blue, orange, red, and green denote bismuth, oxygen, titanium, and sodium, respec tively. Pink indicates interior of oxygen atom and bismuth is seen yellow in b and c.

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169 I n Fig. 8 8(a), the network of oxygen and titanium is similar to that of the F4/mmc structure shown in Fig. 8 8(c) [188 ] To display F4/mmc symmetry, a structure has to be tetragonal with a=b and = = =90 However, T5 having alternative layers of Na and Bi is monoclinic with symmetry of P1 due to the deviation s in the positions of Na and Bi sites in the c direction (see Fig. 8 8(a)) As a result, the lattice parameters of T5 are not consistent wit h th ose of a tetragonal system (see Table 8 5). The displacement of the oxygen atoms is much larger than that of titanium in the P1 phase of T5. In order to characterize the bonding of the Na and Bi atoms with neighboring oxygen atoms, an analysis of c harge density is carried out. Figure 9 9 exhibits charge density images for the P1 phase of T5. Neighboring oxygen atoms strongly bond with bismuth atoms, but no obvious bonding is observed between Na and neighboring oxygen atoms. Therefore, the bonding of Bi O is highlighted in Fig. 8 11(a) and (b). The c harge density is captured on two planes to show all oxygen atoms neighboring a Bi in Fig. 8 9(a). There are two kinds of Bi. O ne is coordinated with four oxygen atoms, and the other with two oxygen atoms in Fig. 8 11(a). On the other plane the Bi atom with two neighboring oxygen atoms in Fig. 8 11(a) is coordinated with four oxygen atoms in Fig. 8 11(b). Similarly, the Bi atom with four neighboring oxygen atoms is surrounded with other two oxygen atoms in Fig. 8 11(b). This is consistent with the Bi atom being 6 fold coordinated and forming [BiO 6 ] 9 units [189] 8 5 Distor tion of O ctahedra The structure change from R3c to P1 phase results from th e distortion of octahedra consisting of oxygen atoms. There are three main reasons why the structure of lattice may be distorted [190] :

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170 T he bonds valence and length may not be topological ly equivalent A toms may have an intrinsically anisotrop y in their electronic structure S teric strains may be caused by stretch ing or compress ing of bonds. The first and second reasons are applied for new systems or external stress. In this study, two different phases, R3c and P1, of NBT takes place by change of cation ordering in the same perovskite frame consisting of Ti O bonding. Therefore, the electronic structure is focused on to elucidate the relation between cation ordering and structure change. [191] a system with a degenerate electronic structure in the ground state will distort if such a distortion can remove the degeneracy. Th e distortion in ceramic materials normally results from the following s [190] : T he stereoactive lone pairs associated with main group cations in lower valence states (e.g. S 4+ and As 3+ ) T he octahedrally coordinated transition metals with a d 0 or d 1 configu ration (e.g. V 5+ ) T found around octahedrally coordinated Cu 2+ and Mn 3+ The first and second mechanisms are called (SOJT) [192, 193] Ti 4+ in NBT is considered as a d 0 transition metal owing to its electronic structure of [ Ar ] 4s 2 3d 2 [194] The second mechanism can be applied to

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171 explain the structure distortion. As mentioned in the analysis of the charge density results of NBT, Bi has an important role in the P1 structure by bonding neighboring oxygen atoms Bi has 10 d electrons in Table 8 6. In octahedral complexes, the Jahn Teller effect is most commonly observed when an odd number of electrons occupy the e g orbitals; i.e. in d 9 low spin d 7 or high spin d 4 complexes [195] The only factor to cause the distortion is a lone pair ( 3d 2 ) in Bi. Hence, Ti 4+ and Bi +3 can produce the distortion in NBT. T he primary distortive cause ( intr a octahedral displacement [196] ) can be attributed to SOJT effects w ith both the d 0 transition metals and lone pair cations [194, 197] In thi s case, the primary distortion (intra octahedral displacement) of the d 0 transition metal occurs by the secondary distortion (interactions between the polyhedra having lone pairs and the MO 6 octahedra) [194] Combination of two mechanisms and distortion of a pseudo cubic structure are related to examine Na 0.5 Bi 0.5 TiO 3 and Na 0.5 La 0.5 TiO 3 Na 0.5 Bi 0.5 TiO 3 (N B T) : lo ne pair s (Bi +3 ) + d 0 transition metal (Ti +4 ) Na 0.5 La 0.5 TiO 3 (NLT) : d 0 transition metal (La +3 ) + d 0 transition metal (Ti +4 ) As shown in Table 9 6, La is chosen to compare to Bi owing to its charge, electronic structure, and ionic radius. Table 8 6. Radius and electronic structures of Bi and La [198 200] Element Charge Radius/nm Electronic structure After ionization Ionic Cova lent Atomic Bi +3 103 148 156 [Xe] 4f 14 5d 10 6s 2 6 p 3 6s 2 is le ft: lone pair La +3 103 207 187 [ Xe ] 5d 1 6s 2 [Xe] is left All the Bi atoms are substitu t ed with La in the P1 phase ; the lowest energy structure of Na 0.5 La 0.5 TiO 3 (NLT) is then determined using the GGA ap proximation, see

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172 Fig 9 1 0 The system still has P1 symmetry H o wever, the cell is more symmetrical than that of NBT as characterized by small displacements of the oxygen atoms and the near colinearity of the Na and Bi cations. Hence, the combination of lone pair s and the d 0 transition metal can deform octahedra of a perovskite cell more severely than that of d 0 transition metal s. E xcessive Bi is reported to increases the SOJT effect in Bi 4 Ti 3 O 12 [201] The distortion results in make cations and octahedra a symmetric [194] or off center [196] Figure 8 1 1 represents this. W e cannot see Na a toms i n Fig. 8 1 1 (a) and (b). As shown in Fig. 8 8, the Na and Bi atoms do not sit on the same plane. The La and Na are, however, on the same plane in Fig. 8 1 1 (c) and (d). Moreover, some Na atoms in Fig 9 1 1 (d) form weak bonding with oxygen atoms. Na atoms contribute to determining the structure NLT structure, in contrast to their small effect in NBT. Figure 8 1 0 Optimized unit cell of Na 0.5 La 0.5 TiO 3 Compare with Fig. 8 8 (a) and (b) for NBT.

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173 Figure 8 11 Contour map of charge density of N a 0.5 Bi 0.5 TiO 3 and Na 0.5 La 0.5 TiO 3 on different planes of pseudo cubic type cell. 8 6 Summary Understanding the cation structure of NBT is important for engineering the properties of NBT as a replacement for PZT for ferro electric and piezoelec tric applications Density functional theory calculations at the level of the GGA w ere used to determine the relative energies of various arrangements of Na and Bi cations The T2 structure, having al ternative arrangement of Na and Bi in the perovskite a xis has the lowest energy of the cation arrangements that have R3c structure However, the R3c structures with ordered cations have higher energies than structures with random cation arrangements This indicates th at the random arrangement of cation

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174 can b e expected to be favored The T5 with layers of Na and Bi shows the lowest energy among all of the structures analyzed and revealed to have a P1 phase by distortion of octahedra in its perovskite. To analyze the structure distortion, elements (Ti Bi and T i La) causing the second effect are chosen and, compared in structure and charge density. The combination of lone pair s and the d 0 transition metal can deform octahedra of a perovskite cell more severely than that of d 0 transition metal s. The cation layere d structure of Na 0.5 Bi 0.5 TiO 3 may be influenced by the Jahn Teller effect, thereby having the lowest energy.

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185 BIOGRAPHICAL SKETCH Dong Hyun Kim was born in January 1976 at Masan, Republic of Korea. He graduated d ept. Inorganic Materials Engineering of Pusan National University in 2008 He finished his master course at the same depart ment under the supervision of Prof. Kwangho Kim in 2000 He studied chemical sensors and thin film processes. He has joined the research center of Daeyang Electrics Co ., Ltd as a researcher of the sensor device team in 2001 He worked at the Plasma Displa y Panel division of Samsung SDI as a n assistant manager form 2004 to 2007. He has begun Ph. D course under the supervision of Prof. Simon R. Phillpot in Dept. Materials Scie n ce and Engineering of University of Florida in Aug 2007. His research is oriented towards atomistic simulation of metals and ceramics. Aug 2011, Dong Hyun earned his Doctor of Philosophy in material science and engineering.