INVESTIGATION OF SINGLE CRYSTAL AND BI -CRYSTAL DEFORMATION IN BODY-CENTERED CUBIC TANTALUM USING INDENTATION By Bret Elliott Dunlap A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degre e of Materials Science and Engineering Ð Doctor of Philosophy 2018 ABSTRACT INVESTIGATION OF SINGLE CRYSTAL AND BI -CRYSTAL DEFORMATION IN BODY-CENTERED CUBIC TANTALUM USING INDENTATION By Bret Elliott Dunlap To understand how polycrystalline tantalum (Ta) deforms and develops damage that can lead to fracture, it is necessary to have an understanding of the single crystal deformation as well as deformation at grain boundaries. Metallic crystals generally deform from the motion of dislocations on crysta llographic planes and this motion is dependent on the orientation of the individual crystals in relation to the imposed deformation. In order to study the effects that crystal orientation and grain boundaries have on deformation, as well as quantify and characterize the dislocations involved, three primary experiments were carried out. The first set of experiments involved single crystal microindentation, single crystal nanoindentation, and bi -crystal nanoindentation. The topographies developed were mapp ed using confocal microscopy for microindentations and atomic force microscopy (AFM) for nanoindentations. The single crystal indents revealed the effect crystal orientation has on topography and the bi -crystal nanoindentations reveal the effect that grai n boundaries have on topography. In this work, three grain boundaries were targeted for analysis, with one indent made on each side of the grain boundary. Analysis shows that deformation across the grain boundary is dependent on the side from which defor mation is approaching. The single crystal and bi -crystal nanoindentations were coupled with crystal plasticity finite element modeling (CPFEM) simulations in order to compare the experiments with predictive models. The second set of experiments characte rized and quantified the dislocations involved in the underlying plastic deformation of single crystal and bi -crystal nanoindentations using !electron channeling contrast imaging (ECCI) and cross -correlation electron backscattered diffraction (CC -EBSD). EC CI directly images and characterizes dislocations using contrast analysis. On the other hand, CC -EBSD calculates the geometrically necessary dislocation (GND) density from the subtle shifts in the EBSD patterns. CC -EBSD can also split the GND density onto the specific slip systems. The effectiveness of these two techniques to quantify and characterize dislocations were compared and the advantages and d isadvantages of both outlined. In the third experiment, the sub -surface deformation of a single crystal wedge indentation was analyzed using ECCI and EBSD. The wedge indentation was specifically aligned to the crystal orientation so that when the sample was cut in half, all deformation was manifested as plane -strain in the analyzed surface plane. ECCI and EBSD were carried out on a small area underneath the area where the indenter tip was in contact with the sample. Backscattered electron (BSE) imaging and EBSD mapping reveal thin needle like bands that resemble that of twinning. The crystal orientation alternates between thin bands of rotated crystal orientation and thick bands of the original crystal orientation. ECCI shows high dislocation densities within the bands and at the boundaries between bands. Due to the fact that the alternating banding lea ves no residual lattice curvature, it is concluded that all dislocations within a given band are statistically stored dislocations ( SSDs ). This work reveals that the surface topography and the dislocation distributions that result from indentation reflect the symmetry of the indented crystal orientation. Also, deformation at grain boundaries is dependent on the direction from which deformation is approaching. Even further, t his work shows that the combination of indentation, AFM, ECCI, and CC -EBSD is a great method for investigating deformation but the limitations of each of these tools need to be understood in order to be fully u tilized Copyright b y BRET ELLIOTT DUNLAP 2018 !v!For all of my family . Thank you for encouraging me to do the things I thought w ere impossible. !vi!ACKNOWLEDGEMENTS First, I would like to thank my advisor, Dr. Martin Crimp, for his guidance throughout my PhD work. His influence has greatly enhanced my growth as a researcher, writer, and as an overall person. I have thoroughly appre ciated our discussions that have covered research and non-research related topics. I would also like to thank my committee members, Dr. Philip Eisenlohr, Dr. Thomas Bieler, and Dr. Rebecca Anthony, for their guidance and their flexibility that allowed the m to be easily accessible to discuss ongoing research. I also want to thank Dr. Carl Boehlert for insightful feedback in group meeting discussions. I would like to thank Dr. Per Askeland as well as former and current graduate students in the Òmetals grou pÓ for helping me learn how to use scanning electron microscopes and other research equipment. Most of all, I want to thank my wife Michelle for all of her loving support throughout my undergraduate and graduate work. !vii !TABLE OF CONTENTS LIST OF TABLES ................................................................................................................... ix LIST OF FIGURES .................................................................................................................. x KEY TO ABBREVIATIONS AND SYMBOLS ................................................................... xiv 1 Introduction ....................................................................................................................... 1 1.1 Generalized slip ..................................................................................................................... 2 1.1.1 Predictive Parameters o f Slip ........................................................................................................... 3 1.2 Deformation Mechanisms of bcc Metals .............................................................................. 7 2 Background of Analytical Techniques ........................................................................... 11 2.1 Nanoindentation .................................................................................................................. 11 2.2 Crystal Plasticity Finite Element Method (CPFEM) ......................................................... 22 2.2.1 CPFEM of Nano indentation ........................................................................................................... 24 2.2.2 CPFEM Construction for Nanoindentation ..................................................................................... 27 2.3 Electron Backscattered Diffraction .................................................................................... 27 2.4 Characterization and Mapping of Dislocations .................................................................. 30 2.4.1 Selected Area Channeling Patterns (Obtaining Imaging Conditions for ECCI) ................................ 31 2.4.2 Electron Channeling Contrast Imaging ........................................................................................... 34 2.4.3 Cross -Correlation Electron Backscattered Diffraction .................................................................... 38 2.4.4 Previous Comparisons Between ECCI and CC -EBSD GND Mapping ............................................ 39 3 Experimental Methods .................................................................................................... 41 3.1 Sample Preparation ............................................................................................................ 41 3.2 Single Crystal Microindentation ......................................................................................... 42 3.3 Single Crystal Nanoindentation .......................................................................................... 42 3.3.1 CPFEM of Single Crystal Nanoindenation ..................................................................................... 43 3.4 Bi-crystal/Grain Boundary Nanoindentation ..................................................................... 43 3.4.1 CPFEM of Grain Boundary Nanoindentation ................................................................................. 44 3.4.2 AFM Topography Subtractions ...................................................................................................... 47 3.5 ECCI vs. CC -EBSD ............................................................................................................. 49 3.5.1 ECCI of Nanoindents ..................................................................................................................... 50 3.5.2 CC-EBSD of Nanoindents ............................................................................................................. 50 3.6 Analysis of Wed ge Indent Cross -Section ............................................................................ 53 4 Results .............................................................................................................................. 55 4.1 Single Crystal Microindentation ......................................................................................... 55 4.2 Single Crystal Nanoindentation .......................................................................................... 59 4.2.1 CPFEM of Single Crystal Nanoindentation .................................................................................... 61 4.3 Bi-crystal/Grain Boun dary Nanoindentation ..................................................................... 62 4.3.1 CPFEM of Grain Boundary Nanoindentation ................................................................................. 65 4.4 ECCI vs. CC -EBSD for Single Crystal Indentation ........................................................... 67 4.4.1 Dislocation Distributions ............................................................................................................... 67 4.4.2 Dislocation Density Comparison .................................................................................................... 70 4.4.3 Dislocation Characterization Using ECCI ...................................................................................... 72 4.4.4 Dislocation Characterization Using CC -EBSD ............................................................................... 76 !viii !4.5 ECCI vs. CC -EBSD for Grain Boundary Indentation ....................................................... 79 4.5.1 Dislocations Distributions .............................................................................................................. 79 4.5.2 Dislocation Characterization with ECCI ......................................................................................... 82 4.5.3 Coarser CC -EBSD Over Four Nanoindents .................................................................................... 86 4.6 Analysis of Wedge Indent Cross -Section ............................................................................ 88 5 Discussion ........................................................................................................................ 98 5.1 Single Crystal Microindentation and Nanoindentation ..................................................... 98 5.1.1 CPFEM of Single Crysta l Nanoindentation .................................................................................. 100 5.2 Grain Boundary Nanoindentation .................................................................................... 100 5.2.1 Dislocation Pile -ups at Grain Boundaries ..................................................................................... 102 5.2.2 CPFEM of Grain Boundary Nanoindentation ............................................................................... 103 5.3 ECCI vs. CC -EBSD ........................................................................................................... 104 5.3.1 Dislocation Density Comparison .................................................................................................. 104 5.3.2 Dislocation Characterization Using ECCI .................................................................................... 110 5.3.3 Dislocation Characterization Using C C-EBSD ............................................................................. 111 5.3.4 A Balance of CC -EBSD Noise to SEM Drift ................................................................................ 112 5.3.5 Advantages/Disadvantages of ECCI ............................................................................................. 113 5.3.6 Advantages/Disadvantages of CC -EBSD ..................................................................................... 113 5.4 ECCI/CC -EBSD Compared to AFM ................................................................................ 114 5.5 Analysis of Wedge Indent Cross -Section .......................................................................... 115 6 Conclusions .................................................................................................................... 117 6.1 Suggestions for Future Research ...................................................................................... 118 REFERENCES ..................................................................................................................... 120 !ix!LIST OF TABLES Table 2.1: Comparison of the occurrence of a pop -in event and mÕ in niobium [40]. ................. 19 Table 2.2: Comparison of m' and M [16]. ................................................................................. 19 Table 4.1: Statistics of lobe heights of three microindents, one for each of the primary orientations. ................................................................................................................... 58 Table 4.2: Statistics of lobe heights of three experimental single crystal nanoindents, one for each of the primary orientations. .................................................................................... 60 Table 5.1: Comparison of CC -EBSD GND densities and ECCI dislocation densities for the 5 regions shown in Figure 5.2. ........................................................................................ 108!!x!LIST OF FIGURES Fig ure 1.1: Illustration of both m' and M where the m' equation uses the angles ! and " and M uses the angles # and " [20]. ............................................................................................ 6 Figure 1.2: The relaxed ! <111> screw dislocation core of molybdenu m. The different shades of circles represent consecutive (111) planes and the arrows representing the magnitude of out of plane atom displacements [29]. ....................................................................... 10 Figure 1.3: a) Structure of a scr ew dislocation core after an applied shear that is perpendicular to the Burgers vector in the positive sense and the b) negative sense [12]. ......................... 10 Figure 2.1: This load -displacement curve shows a displacement jump that marks the beginning of plasticity [33]. ........................................................................................................... 13 Figure 2.2: Experimental data for the loads at which the initial pop -ins occur compared to a) an homogenous dislocation n ucleation model and b) a combined model of homogenous dislocation nucleation and the activation of pre -existing dislocations. ............................ 15 Figure 2.3: a) Illustration of a generic curve of pop -ins [15 ]. b) Curves showing a bulk sample indent in Fe -Si and an indent near a grain boundary that exhibits a secondary, or grain boundary, pop -in [16]. ................................................................................................... 17 Figure 2.4: Load at which the pop -ins oc curred versus distance from the indent to the grain boundary [40]. ............................................................................................................... 21 Figure 2.5: Illustration of a) experimental and b) simulated topography evolution as a function of grain orientation [3 2]. ................................................................................................ 26 Figure 2.6: A pictorial example of Kossel -cones formed from backscattered electrons being diffracted at the Bragg angle by lattice planes as they exit the crystal. These Kossel -cones project as lines on the phosphor screen [60]. ........................................................ 29 Figure 2.7: SACP obtained using a Tescan Mira 3 FEG -SEM. ................................................. 33 Figure 2.8: a) Pictorial representation of the backscattered electron yield with respect to the angle at which electrons hit a crystal. b) An imitation SACP corresponding to a condition of high backscattered electron yield, large $. c) An imitation SACP corresponding t o a condition of low backscattered electron yield, small $. The black dots in b) and c) represent the optic axis of the electron beam (amended from Crimp [74] and Joy et al. [72]). ............................................................................................................................. 35 !xi!Figure 2.9: a) Pictorial representation of the rastering electron beam (dotted blue lines) moving over a dislocation. A pictorial representation of the channeling condition for the bulk of the crystal lattice is shown in c) where the optic axis of the electron beam is hitting the crystal planes at the Bragg angle. The lattice distortions from the dislocation change the local channeling condition so that the optic axis hits the crystal planes at angle % (smaller than the Bragg angle) on one side and at angle & (larger than the Bragg angle) on the other side. The channeling conditions for these lattice distortions are shown in b) and d) (figure courtesy Dr. Martin A. Crimp). .......................................................................... 37 Figure 3.1: a) A sn apshot of the mesh generation GUI in STABIX and b) and example of a FIB cross -section on a grain boundary. ................................................................................. 46 Figure 3.2: Example of topography subtraction of AFM measurements for a grain bo undary indent and the corresponding single crystal indents. ....................................................... 48 Figure 3.3: Box plots showing GND density distributions for effective step sizes between 25 nm and 400 nm. ................................................................................................................... 52 Figure 3.4: CC -EBSD derived GND map of a wedge indent that was cut in half and polished. EBSD patterns were collected at a step size of 2.5 µm and the area mapped is 1 x 1 mm. The units are log m/m 3 [80]. The area analyzed in the present study is boxed. .............. 54 Figure 4.1: Indentations of single crystal microindents plotted on a portion of a bcc stereographic projection in order to visualize inde nt topography as a function of orientation. .................................................................................................................... 57 Figure 4.2: Top) Experimental topographies for the nanoindents made in grains of orientations close to [001], [101], and [111], shown left to r ight respectively. Bottom) Corresponding CPFEM simulations for three orientations. .................................................................... 60 Figure 4.3: Left) BSE image of indents along three grain boundaries. Middle -left) AFM measurements of the grain boundary nanoindents. Middle -right) Single crystal nanoindents that correspond to each of the grain boundary nanoindents. Right) Subtraction of the grain boundary AFM measurement with corresponding single crystal indents. .......................................................................................................................... 63 Figure 4.4: Left) AFM measured topographies of experimental nanoindents near grain boundaries. Middle) CPFEM topographies of the same grain boundary experimental indents. Right) Subtraction result of CPFE M minus AFM topographies. ...................... 66 Figure 4.5: a) Multiple ECC images stitched together showing dislocations generated from a single crystal nanoindentation in a grain of approximately [011] orien tation. b) CC -EBSD GND map of the same area, collected with an EBSD scan step size of 100 nm and effective step size of 200 nm, showing dislocation distributions similar to that in the ECC image. ........................................................................................................................... 68 !xii !Figure 4.6: AFM measurement of the same single crystal indent that was imaged and GND mapped in Figure 4.5. .................................................................................................... 69 Figure 4.7: a) ECC image of dislocations from the upper -left of an ind ented area. b) CC -EBSD generated GND density map of the same area showing similar dislocation distributions, using a step size of 50 nm and an effective step size of 200 nm. c) ECC image gridded to the same size as the EBSD step size. d) Dislocation den sity map calculated by counting dislocations in each grid square of gridded the ECC image. ........................................... 71 Figure 4.8: ECC images for the channeling conditions used for contrast analysis, with g indicate d by the white arrows and the black to white contrast indicated by the white dashed arrows. ............................................................................................................... 73 Figure 4.9: Stereographic projections a) corresponding to Figure 4.8a and b) tilted 11 ¡ along t he g = ( -21-1) with each ÒxÓ being a line direction for the four possible screw dislocations. c) ECC image with the same sample tilt as in b), showing a projection of the dislocation line directions. ............................................................................................................... 75 Figure 4.10: Dislocation density of each dislocation type determined using CC -EBSD. ........... 77 Figure 4.11: Dislocation density of the screw dislocations with a Burgers vector of [111] determined by CC -EBSD. ............................................................................................. 78 Figure 4.12: a) BSE image and b) AFM map of an indent located at a grain boundary triple junction. The analysis will be carried out at the grain boundary on the ri ght. ................ 80 Figure 4.13: a) ECC image of the lower right of a grain boundary indent showing dislocations generated on the opposite side of the grain boundary. An CC -EBSD generated GND density m ap of the same area with a step size of 25 nm and an effective step size of 200 nm. ................................................................................................................................ 81 Figure 4.14: ECC images used to characterize three distinct sets of dislocations generated on the opposite side of a grain boundary. The four channeling conditions used are in shown in a) to d). .......................................................................................................................... 83 Figure 4.15: a) Stereographic projection that corresponds to the grain orientation of the ECC image in b). The line directions for the three sets of dislocations are overlaid on the stereographic projection with their respective colors. ..................................................... 85 Figure 4.16: a) BSE image of four indents, three near a grain boundary and one far enough away from the grain boundary to be considered a single crystal indent. b) GND map of the same four indents. ......................................................................................................... 87 Figure 4.17: a) CC -EBSD derived GND map from Ruggles et al. [80] and b) a BSE image of the area analyzed in this work. ............................................................................................ 89 !xiii !Figure 4.18: a) BSE image of the analyzed area and b) an ECC image of one of the boxes/cells formed by the intersection of the bands seen in the BSE image. ..................................... 91 Figure 4.19: Many of the line directions in the ECC image, shown in b), have line directions that are generally parallel to the sur face. Assuming that these dislocations are screw dislocations, overlaying the line directions onto the stereographic projection, shown in a), reveal that the Burgers vector for these dislocations are [ -11-1] ..................................... 92 Figure 4.20: a) Inverse pole figure map for the revealing no change in lattice orientation due to the fact that all lattice rotation is around the surface normal. b) By rotating all crystal lattice points 90 ¡ around the x -axis, the o rientation changes can be more readily discerned. ...................................................................................................................... 94 Figure 4.21: Left Column) Pole figures that represent the shift caused by the left thin bands, Middle Column) pole figures that represent shifts caused by all thin band shifts, and Right Column) are pole figures that represent the shifts caused by the right thin bands. . 95 Figure 4.22: Inverse pole figure EBSD map overlaid wit h the unit cells for the alternating bands. ...................................................................................................................................... 97 Figure 5.1: a) AFM measurement a single crystal nanoindent in a grain of [011] orientation. b) Stereographic projection of the same grain show ing two Burgers vectors lying in the surface plane of the sample and two Burgers vectors pointing to either side of the indent. ...................................................................................................................................... 99 Figure 5.2: Duplicate of Figure 4.7 overlaid with 5 regions for comparison between b) CC -EBSD calculated GND density map and d) a dislocation density map that was derived from the ECC image in a) and c). The oval in a) shows an example of dipole dislocations. ................................................................................................................ 106!!xiv !KEY TO ABBREVIATI ONS AND SYMBOLS AFM atomic force microscopy bcc body-centered cubic crystal structure BSE backscattered electron CC-EBSD cross -correlation electron backscattered diffraction CPFEM crystal plasticity finite element modeling Cr chromium CRSS cri tical resolved shear stress EBSD electron backscattered diffraction ECCI electron channeling contrast imaging ECP electron channeling pattern EDM electronic discharge machining fcc face -centered cubic crystal structure Fe-Si iron -silicon alloy FEG -SEM field emission gun scanning electron microscope FEM finite element methodology FIB focused ion beam GNDs geometrically necessary dislocations GUI graphical user interface hcp hexagonal close -packed crystal structure Mo molybdenum MPIE Max -Planck -Institut f Eisenforschung GmbH !xv!Nb niobium OIM Orientation Imaging Microscopy ROIs regions of interest within an EBSD pattern SACP selected area channeling pattern SEM secondary electron microscopy SiC silicon carbide SSDs statis tically stored dislocations Ta tantalum TEM transmission electron microscopy VTK visualization toolkit file & electron diffraction angle larger than the Bragg angle !"# Nye tensor b Burgers vector "-brass body-centered cubic structure o f brass C elastic tensor C44 elastic modulus $ describes electron backscattered yield F deformation gradient Fe elastic deformation gradient Fp plastic deformation gradient g vector that describes the electron imaging/channeling condition g1, g2 unit vectors along the slip directions $% shear rate !xvi !$&% reference shear rate h!" hardening matrix " angle between slip directions L1, L2 unit vectors along the intersection lines slip planes make with the grain boundary ' angle between slip direction and direction of uniaxial stre ss Lp sum of shear rates on all possible slip systems M predictive parameter of slip transfer using # and " '( rate sensitivity of slip m Schmid factor m slip direction of a dislocation m' predictive parameter of slip transfer using " and # n slip plane % electron diffraction angle smaller than the Bragg angle ( angle between slip plane normal and direction of uniaxial stress q!" latent hardening matrix S stress # angle between slip plane normals ) macroscopic stress )ys yield str ess # resolve shear stress $c critical resolved shear stress $s saturation value of the shear stress #B Bragg angle !xvii !# angle between L1 and L2 u dislocation line direction !1!1!Introduction Due to the body -centered cubic (bcc) structure of tantalum (T a), the deformation mechanisms are complicated , causing it not the follow the widely accepted SchmidÕs law [1Ð3]. In order to fully utilize the properties of Ta in its various applications [4, 5] , an understanding of its deformation mechanisms is necessary; particularly, the transfer of strain across grain boundaries. The purpose of this work is to quantify and characterize the deformation that occurs within individual grains and at grain boundaries from nano indentation induced deformation in bcc Ta. By and large, metals are comprised of many grains, in which each grain has its own crystal orientation. These crystals deform from dislocation motion on crystallographic planes and the deformation of each crystal is dependent upon the stress state placed on it. To understand how polycrystals deform and develop damage that leads to fract ure, it is necessa ry to characterize the disloca tions involved in the underlying pla stic deformation. This disloca tion content is made up of both the statistically stored dislocations (SSD s), consisting of the portion of the overall dislocation density th at effectively cancels itself out, i.e. due to dislocation dipoles, and the geometrically necessary dislocations (GNDs) that are associated with the crystal elastic strain gradients that develop through plastic deformation. In order to accommodate large sc ale deformation, interactions occur between the dislocations and grain boundaries. Ultimately, these interactions will lead to strain transfer across the grain boundary or create a potential sight for damage nucleation, which can lead to fracture. There are four general mechanisms [6] that represent th e interactions between disloca tions and grain boundaries. They include: (1) transmission of dislocations across the grain boundary; (2) absorption of dislocations into the grain boundary; (3) absorption, followed !2!by emission into the neighboring grain; and (4) reflection off the grain boundary back into the parent grain. Gene rally speaking, each of these will cause the energy of the grain boundary to increase by some degree depending on the specifics of the interaction, i.e. the residual Burgers vector stored in the boundary following the interaction. In order to quantify and characterize the deformation that occurs within individual grains and at grain boundaries from indentation induced deformation in bcc Ta, three primary experiments were carried out. The first involved single crystal microindentation, single crystal nanoin dentation, and bi -crystal nanoindentation in order to view the effect s that crystal orientation and grain boundaries have on the topographical lobe s formed from indentation. The single crystal and bi -crystal nanoindentation experiments were coupled with crystal plasticity finite element modeling (CPFEM) simulations of the nanoindentation process . The second set of experiment s characterized and quantified the dislocations formed from nanoindentation in single crystals and near grain boundaries. The charac terization and quantification was carried out using electron channeling contrast imaging (ECCI) and cross -correlation electron backscattered diffraction (CC -EBSD). The efficacy of these two techniques were compared and the advantages and disadvantages of both outlined. The third experiment involved characterizing the sub -surface deformation of single crystal Ta deformed from wedge indentation. 1.1 !Generalized slip Crystals commonly deform by dislocations moving on the crystallographic planes that have the hi ghest planar atomic density with the atomic displacements , caused by dislocations , in the directions that have the highest linear atomic density. These planes are referred to as the slip plane s and the direction of atomic displacements are known as the Bu rgers vectors . The !3!combinations of individual slip planes and the individual Burgers vectors that lie in those planes make up the slip systems of the various crystal systems [7] . Knowing the orientation of a given crystal, along with its possible slip systems, slip trace analysis can be performed on a deformed sample. Slip trace analysis is a method by which the active slip syst ems can be identified from slip lines that appear on the surface following deformation. This was first done with optical microscopy [8] , but it can also be performed using electron microscopy [9Ð11]. Slip trace an alysis has been well demonstrated in the crystal structures of hexagonal close -packed (hcp) and face -centered cubic (fcc) and the slip systems that were found to be active in these crystal structures predominately follow SchmidÕs law [1, 12] . 1.1.1 !Predictive Parameters of Slip Schmid Õs law can be used to predict when a given crystal will begin to deform and on which slip systems deformation will occur in single crystals. Schmid Õs law states that a macroscopic stress is resolved onto the specific slip systems and w hen the stress is uniaxial it can be represented by: )*+,- (1.1) -*./01 2/01 3 (1.2) where * is the resolved shear stress , ) is the macroscopic stress , and m is referred to as the Schmid factor . ( and ' are the angles the plane normal and slip direction make with the direction of uniaxial stress, respectfully. According to SchmidÕs law, p lastic flow will occur on a given slip system when the resolved stress reaches a critical value, known as the critical resolved shear stress (CRSS). It is important to realize that the CRSS is dependent on the m aterial and !4!varies for the different slip systems within that material. In part , t he CRSS values are differ ent for the various slip planes due to the Peierls stress . Analogous to the Schmid factor predicting slip in single crystals, predictive parameter s of slip transfer at grain boundaries have been developed. Lee et al. [13] suggested that three conditions are needed to determine the active slip systems in slip transfer; the geometric condition, the resolved shear stress condition, and the residual grain -boundary dislocation condition. The geo metric condition represents the angle between incoming and outgoing slip planes at the grain bound ary, which should be minimized. The resolved shear stress condition says that the resolved shear stress acting on the outgoing plane by th e pile up of dislocations needs to be maximiz ed. Finally, the residual grain -boundary dislocation condition says that the difference between incoming and outgoing Burgers vect ors needs to be minimized [13] . One slip transfer param eter proposed by Luster and Morris [14] is referred to as m$: -4*/01 !5./01 " (1.3) where ! is the angle between the closest slip planes in neighboring grains and " is the angle between the closest slip directi ons that lie within the slip planes. Deviating from one of the conditions proposed by Lee et a l., mÕ does not account for the resolved shear stress condition [15, 16] , m eaning the closest slip planes between in the neighboring grains that are used to calculate mÕ are not necessarily the activated slip planes. One way around this is to use the Schmid factor to predict which slip system is active in each grain given a particular stress [17, 18] . Another factor that is not accounted for in mÕ is the orientation of the grain boundary plane [16] . The ori entation of the grain boundary plane is taken into account by Shen et al. [19] using the parameter M: !5!6*78'.5.89:7;'.5.;9: (1.4) where L1 and L2 are unit vectors along the intersection lines of the slip planes with the grain boundary plane and g1 and g2 are unit vectors along the slip directions . The parameters M and mÕ are illustrated in Figure 1.1 [20] . The angle between the intersection lines of the slip planes with the grain boundary plane is % , the angle between the slip directions is & , and the angle between the slip plane normal s is #. Experimentally, if the orientations of the grains is known, & and # are readily calculated, but unless the subsurface orientation of the boundary is known, % cannot be calculated . Changing the rotation of the grain boundary plane with respect to the sample surface plane normal and /or changing the inclination of grain boundary plane will change the angle of %. The rotation alignment of the grain boundary plane with respect to the crystal orientation of the two grains can easily be determined by imaging the sample surface to view the grain boundary. On the other hand, the inclination of the grain boundary plane cannot be as easily determined. One method involves using a focused ion beam (FIB) to make a cut across the grain boundary to view the inclinat ion. This may not be desirable as it is destructive. !6! Figure 1.1: Illustration of both m' and M where the m' equation uses the angles ! and " and M uses the angles # and " [20] . !7!1.2 !Deformation Mech anisms of bcc Metals The accepted slip systems for bcc metals are composed of the ! <111> Burgers vect ors with the {110} and {112} slip planes [21 Ð23]. Some researchers also consider the {123} slip planes [23, 24] . The activity of the various slip planes is predominately determined by the temperature at which deformation occurs. As the temperature increases, so does the ac tivity of different slip planes, beginning with the {110} being the most active at low temperatures followed by the {112} and then the {123} becoming more active at higher temperatures [23] . This al lows for a total of 48 slip systems; 12 from {110} planes, 12 from {112}, and 24 from {123} plane. Due to the nature of the screw dislocation, the fact that the Burgers vector and line direction a re the same, they do not define a slip plane and are therefo re non -planar. More specifically, in bcc metals, looking down the ! <111> dislocation lin e direction of a screw dislocation, there are three {110}, three {112}, and six {123} planes intersect ing the dislocation core . Screw dislocations can readily cross -slip amongst these various plan es, resulting in what is known as wavy slip [7, 23, 25, 26] . Wavy slip makes slip trace analysis in bcc metals difficult because the slip lin es do not always line up with a predicted slip trace. On the other hand, if the wavy slip lines do line up with a predicted slip trace, it is possible that the slip system for that predicted trace was never activated , but the summation of other cross -slip ping planes makes it appear that way. As a result of this variable slip plane, u nlike the crystal structures of fcc and hcp, bcc materials do not necessarily follow Schmid Õs law. The observation of non -Schmid behavior of bcc metals dates back to the time Schmid Õs law was developed. By way of tension and compression tests on single -crystals of iron and "-brass, Taylor [27] found that deformation slip !8!in one direction, is not the same as deformation slip in the opposite direction, known as asymmetry of slip [12, 27, 28] . Schmid Õs law only accounts for the calcu lated resolved shear stress on specific slip plane s in direction s of slip (known as Schmid Õs Stress), while other projections of the stress tensor are said to have no effect [12] . This does not seem to affec t the ability to predict slip in fcc and hcp materials; but for bcc metals the other projections of the stress tensor do matter because of the screw dislocation core structure [3, 27, 28] . The effect of non-Schmid projections of the stress tensor has been demonstrated in atomistic studies of screw dislocations [3, 12, 29 Ð31]. From atomistic studies [3, 12, 29 Ð31] , it was found that the relaxed screw dislocation core in Mo spreads onto the {110} planes , as shown in Figure 1.2. The dislocation core was relaxed using bond order potential [29] . Dislocations move in the direction their cores spread and d epending on the stress applied to the dislocation, the core will favor spreading on some planes over others [29] . This is demonstrated from an atomistic simulation where a shear stress was applied perpendicular to the Burgers vector in both a positive and negative sense, shown in Figure 1.3a and Figure 1.3b, respectfully [12, 29]. The coordinate system used in this simulation had the z-axis aligned with the <111> direction and the y-axis perpendicular to the ( -101) plane , where the stress t ensor used is given by [29] : <*=>)???)????@A (1.5) The magnitude of * is 0.05 the elastic modulus, C44, which means the core spreading is purely elastic and reverts back when the stress is removed . When the shear stress is applied in the positive sense, the core spreads more on ( -101), but when the shear stress is applied in the negative sense, the core spreads onto the (0 -11) and ( -110) planes. The screw dislocation core !9!spread ing onto intersecting planes is the main reason for a high Peierls stress needed to activate crystal glide [12, 29] . In contrast, edge segments are not found to spread onto other planes, th us maintaining a planar configuration. Therefore, screw dislocations have lower mobility than edge dislocations causing the deformation of bcc materials to be governed, or limited, by the motion of screw dislocations [23] . !10! Figure 1.2: The relaxed ! <111> screw dislocation core of molybdenum. The different shades of circles represent consecutive (111) planes and the arrows representing the magnitud e of out of plane atom displacements [29] . a) b) Figure 1.3: a) St ructure of a screw dislocation core after an applied shear that is perpendicular to the Burgers ve ctor in the positive sense and the b) negative sense [12] . !11!2!Background of Analytical Techniques Thi s chapter gives a literature review of the analytical techniques used for this work. This includ es nanoindentation, crystal plas ticity finite element modeling of nanoindentation s, electron backscattered diffraction (EBSD), cross -correlation EBSD (CC -EBSD) , electron channeling contrast imaging (ECCI), and the collection of selected area channeling patterns (SACPs). 2.1 !Nanoindentation Nanoindentation is a technique that allows for probing a materialÕs properties from specific locations, such as near or away fro m grain boundaries, to mimic single -cry stal or bi -crystal experiments. Using nanoindentation, variables such as the state of stress and indentation methodology can be modified in order to obtain desired measurements. This is done by simply changing the i ndenter tip type and adjusting the mode the nanoindenter opera tes in. The versatility of nano indentation makes it a great technique for understanding dislocation slip. Numerous studies of single -crystal nanoindentation have been perform ed to understand sli p in metals [32 Ð36] . Biener et al . [33] did nanoinden tation in single -crystal bcc Ta to understand dislocation nucleatio n. T hey found dislocatio n nucleation to occur as a pop -in event, also known as a displacement jump in a load -displacement curve, as shown in Figure 2.1. T he displacement jump occurs after a buildup in the elastic strain energy , which has an initial curve that fo llows Hertzian contact theory [37] , and is then released to begin plastic deformation. T his pop-in event (from hereafter will be referred to as an initial pop -in, as there can be a secondary pop-in) occurs in most materials, yet varies in the displacement jump depth, an d marks the point of incipient plastic ity, or where plasticity begins [16, 33] . In Ta , this initial pop -in event occurs !12!at varying loads , but the loads at which they occur can be shifted by changing the loading rates [38]. !13! Figure 2.1: This load -displacement curve shows a displacement jump that marks the beginning of plasticity [33] . !14!Wu et al. [36] studied the effect of tip radius on the stress at which a n initial pop -in occur s in chromium (Cr). They found that as the tip radius decreases, the stress at wh ich initial pop-ins occur increases . Figure 2.2 shows the experimental data compared to a homogenous dislocation nucleation model, shown as green line in Figure 2.2a, and to a model combining homogenous dislocation nucleation and the activation of pre -existing dislocations, shown as a red line in Figure 2.2b. For a large tip radius , there is a significant deviation from the homogen ous dislocation model at low loads, indicating the activation of pre -existing dislocations. As to why this occurs with large tip radii, Wu et al. points out that as the stressed volume of material increases, the chances of activating pre -existing dislocat ions increases [36] . !15!a) b) Figure 2.2: Experimental data for the loads at which the initial pop -ins occur compared to a) an homogenous dislocation nucleation model and b) a combined model of homogenous dislocation nucleation and the activation of pre -existing dislocations. !16!Bi-crystal nanoindentation studies have been performed to understand slip t ransfer across grain boundaries [15, 16, 39 Ð42] . T hese studies have cited a secondary pop-in phenomenon , as shown in Figure 2.3 [15, 16] . In addition to the previously discussed initial pop -in, a second pop-in can be observed that is associated with grain boundary slip transfer. W hile the first pop -in happens in almost all mat erials and marks the beginning of plasticity, the second pop -in only occurs for indents near grain boundaries. T he proposed reason fo r this pop -in is a build up of dislocations at the grain boundary , which causes the hardness (or the slope of the load -disp lacement curve ) to be greater than in th e bulk portion of the crystal. When the energy from the build up in the dislocation pile -ups reaches a critical level, it causes dislocations to nucleate in the neighboring grain , resulting in strain relaxation and a pop-in event . After the pop -in, the hardness at the grain boundary is then comparable to the hardness of the bulk of the crystal [15, 16, 39 Ð41] . An example of load -displacement curve of a secondary pop -in is shown for Fe -Si in Figure 2.3b [16] . !17! a) b) Figure 2.3: a) I llustration of a g eneric curve of pop -ins [15] . b) Curves showing a bulk sample indent in Fe -Si and an indent near a grai n boundary that exhibits a secondary, or grain boundary, pop-in [16] . !18!Secondary pop -ins do not always occur when an indent is made near a grain boundary. Studies [16, 40] have tried to use parameters such as mÕ and M to try to predict when or a reason why secondary pop -ins occur at some grain boundaries and not others. Wang and Ngan [40] showed there is a correlation between secondary pop -ins and the mÕ parameter in niobium. This is s ummarized in Table 2.1 [40] . It must be noted, however, that Wang and Ngan only used the mÕ parameter with relation to the closest slip planes and closest slip directions and made n o approximation for what the activated slip planes and directions are . On the other hand, Soer et al. [16] found M to be a better parameter than mÕ for predicting secondary pop -ins. For comparison, they used a molybdenum (Mo) bi -crystal that had a coincident site lattice (CSL) <111> tilt boundary. Using mÕ in relation to the closest plan es and directions would give an mÕ value of 1 because of the perfectly aligned slip systems between the two grains [16] . For the values of M, Soer et al. assumed uniaxial compression in relation to faces of the Berkovich indenter tip and used the Schmid factor to approximate which slip system would have the maximum resolved shear stress. Their comparison for mÕ to M is summarized in Table 2.2 [16] . !19!Table 2.1: Comparison of the occurrence of a pop-in event and mÕ in niobium [40] . Grain Boundary Number CSL + ! " mÕ Grain Boundary Pop -in 1 0.9970 0.9948 0.9918 Yes 2 0.9953 0.9913 0.9866 Yes 3 0.9946 0.9617 0.9665 Yes 4 41c 0.9953 0.9404 0.9360 Yes 5 41b 0.9976 0.8879 0.8858 No 6 9 0.9972 0.8858 0.8833 No 7 29b 0.9972 0.8339 0.8316 No 8 0.9966 0.8143 0.8115 No Table 2.2: Comparison of m' and M [16] . Material mÕ M Grain Boundary Yielding Observed Mo 1.0 ( +3) 0.78 ( +3) No 0.99 ( +11) 0.25 ( +11) No Fe-Si 0.93 0.82 Yes, depending on indenter orientation Nb 0.90 Ð 0.99 - Yes, regardless of indenter orientation !20!There is a problem with the comparison made by Soer et al. between mÕ and M because they do not compare the parameters at all, but rather only changed what inputs go into the equation s. They consi dered all slip systems for calculating mÕ and only the activated slip systems for calculating M. It is possible that mÕ could be as good as M at predicting pop -ins if the possible slip systems were determined for the mÕ parameter using the Schmid factor approximation as well. Therefore, it would be more thorough to use the same slip plane selection criteria when comparing mÕ and M. Wang and Ngan [40] found a correlation between the load and the distance from the grain boundary the ind ent was made. This correlation is shown in Figure 2.4. Wang and Ngan further quantify this correlation with c/d ratios t o describe pop -in occurrences. T he variable c is the radius of the elasto -plastic boundary and d is the distance from the grain b oundary the indent was made. The v ariable c is estimated with Eq. (1.14 ), which was developed by Zielinski et al . [34] from Johnson Õs model [43, 44] . P represents the load at which the pop-in occurred and )ys is t he yield stress of the material. B*.CDE9FGHI (2.1 ) The results from Wang and Ngan [40] show that pop -ins occur between c/d ratios of 1.5 and 5 but mostly around 2. T hey also found that the c/d ratio is the same for a given grain boundary segm ent. !21! Figure 2.4: Load at which the pop -ins occurred versus distance from the indent to the grain boundary [40] . !22!2.2 !Crystal Plasticity Finite Element Method ( CPFEM) The f ini te element methodology (FEM) is a tool used to solve the no n-linear differential equations of physical systems such as stress, strain, temperature gradients, fluid flow, etc. This can be done for a body that c an have an overall complicated geometry by app lying a mesh composed of many smaller bodies, called elements, which have a simple geometry (i .e. a square , rectangle, or triangle ). Because FEM is only a solver, a material model must be constructed in order to use FEM for specific applications [45, 46] . The crystal plasticity finite element method (CPFEM) i s a specific model used to sim ulate plastic defor mation using FE methodology [46] . The CPFE method is built on the computational techniques of continuum mechani cs and the knowledge gained from experimen tal results. Peirce et al. were the first to introd uce the CPFE method in 1982 [47] . Since then, the developed constitutive models have been applied to a plethora of mechanical problems and used to si mulate experimental conditions. This includes, but is not limited to, nanoindentation, tensile tests, grain boundary deformation, and texture evolution [32, 48, 49] . The kinematic equations for finite strain CPFEM are: J*JKJL (2.2 ) M*'9.NOJKPJK>QR (2.3 ) JL%*.8LJL (2.4 ) 8L*.<$S%TSUVS (2.5 ) where F is the deformation gradient that is composed of both an elastic, Fe, and plastic, Fp, component s. The stress, S, is a function of the 4th order elastic tensor, C, and Fe. The plastic deformation evolves as a function of the sum of the shear rates on all possible slip systems, Lp, !23!which is d etermined by the shear rate, $%, the slip direction, m, and the slip plane , n, of slip system s ! = 1,É , N. There are many constitutive models that can be used depending on the specifics of what is being modeled, but this work will primarily focus on the phenomenological constitutive model, first introduced by Peirce et al. [47] : )S*M.57TWUVS: (2.6 ) $S%*.$&%.XAYA/YXZ[.1\] .7)S) (2.7 ) )/S%*.^S_`$_%` (2.8 ) ^S_*.aS_b^&cd>A/eA1fgh (2.9 ) In these equations, # is the resolved shear stre ss with $c being the critical resolved shear stress and $s is its saturation value . $&%and m are material parameters of the reference shear rate and rate sen sitivity of slip, respectively. The hardening matrix, h!", gives the effect any slip system " has on the ha rdening behavior of slip system !. q!" represents latent hardening with h0, a, and #s being the structure evolution parameters [46, 47] . With the different constitutive models, there needs to be a material implementation code in order to simulat e crystallographic deformation. Researchers at Max -Planck -Institut f r Eisen forschung GmbH (MPIE) have developed a few of these framew ork codes. One in partic ular is the D sseldorf Advanced Material Simulation Kit (DAMASK ) [50] . This code is able to account for non -Schmid eff ects following the constitutive equation proposed by Koester et al. [51] . !24!The equation for non -Schmid effects proposed by Koester et al. [51] is an extended version of the flow rule developed by G rıger et al. [29, 30, 52] from atomistic studies. The flow rule takes into account the non -glide projections of the stress tensor [51] : )S*.ijTWUVSk.l'ijTWUVSk.l9ij7VSmTW:UVS...................................k.lDij7VnWmTW:UVnWk.loijVSUVS................7pqd?:.........k.lrij7VSmTW:U7VSmTW:..............................................................k.lsijTWUTW In this equation, ! represents the possible slip systems and ! is the stress t enor. VnW includes an angle of -60¡ with reference to n'. a1 through a6 represent specific material constants that can be fitted using atomistic studies. The first term of the equation is SchmidÕs law, the second term accounts for the slip asymmetry, and the third and fourth terms account for the shear stresses that are p erpendicular to the slip directions. The fifth term takes into account the load in the (11 direction and the sixth term takes into account the load in the (22 direction. The last term was added to make the equation independent of hydrostatic stresses [51] . 2.2.1 !CPFEM of Nanoindentation Zambaldi et al . [32] carried out single -crystal nano indentat ion s in hexagonal '-titanium and measured the subsequent topography by AFM mapping . They also cou pled these experimental indents with CPFEM and performed a non -linear optimization to determine the crystal plasticity structure evolution parameters. I n general, e ach optimization process compares resulting CPFEM topographies to experimental topographies, generates a new set of structure evolution parameters based off the comparison of results, and re -runs the process until the parameters are optimized to give the b est CPFEM to experimental match. T his process was carried out for a group of indentations with d ifferent crystal orientations . Figure 2.5 shows two !25!portions of the hexagonal stereographic projection with indent t opographies placed on the projections to represent the different orientations indented. Figure 2.5a shows experimental indents and Figure 2.5b shows simulated indents foll owing the optimization of the structure evolution parameters. The projections show a good match between the experimental and simulated indents and b oth projections illustrate that topography evolution is a function of grain orientation [32] . !26!a) b) Figure 2.5: Illustration of a) experimental and b) simulated topography evolution as a function of grain orientation [32] . !27!2.2.2 !CPFEM Construction for Nanoinden tation Single crystal and bi-crystal simulations of nanoindents can be easily set up using the Matlab [53] toolbox STABiX [54, 55] . This toolbox uses graphical user interfaces (GUIs) to allow for the easy inclusion of experimental conditions, such as crystal type (fcc, bcc, or hcp), slip systems, crystal orientation, indentation tip ty pe and size, and indentation depth. For bi -crystal indents, the GUIs also allow for the inclusion grain boundary inclination and the distance from the indenter tip to the grain boundary. Users can tailor the size/number of elements used to build the fini te element mesh depe nding on their specific needs. Once the needed experimental conditions are incorporated using GUIs and the mesh size decided , a Python [56] file is exported in order to build the m esh in Marc [57] or Abaqus [58] , depending to the finite element solver that is used. STABiX is not only used for setting up nanoindentation simulations , but also helps to chara cterize grain boundaries . One of the GUIs allows for EBSD data to be imported and then calculates the mÕ based off the alignment of the slip system plane normal s and Burgers vectors. The GUI show s all grain boundaries colored based off their mÕ value to assess the grain boundaries relative susceptibility to deformation. Not only can this be done for mÕ, but also for other parameters such as misorientation across the grain boundary. 2.3 !Electron B ackscattered Diffraction When a focused beam of electrons hit the surface of a material, there are a number of scattering events that can occur in all directions [59, 60] . Some electrons are backscattered within the sample and as they exit the sample they are diffracted by the lattice planes at the Bragg angle. As this occur s in all directions, the diffracting lattice planes form Kossel -con es. One set of lattice planes produces two cones. By placing a phosphor screen near the sample, the !28!Kossel -cones appear as lines , known as Kikuchi lines, that are generally parallel to each other . They are only generally parallel as the lines are actua lly hyperbolas that are parallel in the center on the phosphor but bend away from each other at the edges of the phosphor screen. This is due to the fact that the outer edge of the phosphor is further from the source of diffraction than the middle. An exa mple of this is shown for one set of lattice planes in Figure 2.6 [60] . As more than one set of crystallographic planes will form Kossel -cones at once, multiple sets of Kikuchi lines will appear on the phosphor screen , forming a complete electron ba ckscattered diffraction pattern (EBSP ), also known as an EBSD pattern [59 Ð61]. To maximize the yield of electrons that make it to the phosphor screen/EBSD detecto r, samples are generally tilted to ,70¡ [59, 60, 62] . From these EBSD patterns, the orientation of the crystal can be determined. Currently, the process of determining the orientation of crystal in aut omated using the Hough transform [59, 60, 63] . The Hough transform essentially detects the Kikuchi bands of the EBSD pattern and indexes the pattern to related crystal orientation. Early use of the Hough transform for band detection started around 1992 and required ,2 s of computer time to index one pattern [59] . The current ra tes of pattern index ing are well over 1000 points per second [64, 65] . The rate of indexing is primarily affected by EBSD camera sensitivity and the computer processing time for indexing [66] . While the primary use of EBSD is measuring crystal orien tation, the technique can used to measure many mor e things than that. A few examples include: grain size, texture, lattice misorientation, grain boundary disorientation , and even combined with energy dispersive spectroscopy (EDS) for phase identification . !29! Figure 2.6: A pictorial example of Kossel -cones formed from backscattered electrons being diffracted at the Bragg angle by lattice planes as they exit the crystal. These Kossel -cones project as lines on the phosphor screen [60] . !30!Users of EBSD need to be concerned with spatial and angular resolution . Spatial resolution of EBSD is related to the interaction volume of the beam. Due to how EBSD is set up, the electron beam hits the surface as an ellipse with the long portion in the direction of the sample tilt. Beyond this, the interaction volume of the beam is a function of the material, accelerating voltage, the beam current, and the type of filame nt used [67] . Spatial resolution can be between 30 nm and 60 nm [60, 67, 68] . Angular resolution is the ability at which EBSD can determine the relative differences in the orientations of neighboring data points [67] . The angular resolution can be impacted by a number of factor s, including, speed at which EBSD is carried out, the binning of EBSD patterns, and the accuracy of the pattern center calibration for EBSD [60, 63, 66, 67] . The angular resolutions of EBSD is in the range of 0.5 ¡ and 2.0 ¡ [60, 63, 67] . This diminishes the ability of EBSD acc urately determine elastic strains and GNDs. 2.4 !Characterization and Mapping of Dislocations Traditionally, dislocation structures have been characterized using transmission electron microscopy (TEM) [69, 70] ; however, TEM is plagued by a number of l imitations associated with the requisite thin foils. These can include difficult sample preparation, the potential for this sample preparation to affect the apparent dislocation distributions, and limited observation volumes can lead to poor statistical re presentation of the bulk. Two significantly different techniques, electron channeling contrast imaging (ECCI) [71 Ð76] and cross -correlation electron backscattered diffraction ( CC-EBSD) [77 Ð80] , are alternative scanning electron microscopy (SEM) based approaches for characterizing dislocation structures. Both of these techniques involve the examination of the near surface region of bulk samples and require careful preparation of this surface region to be examined; nevertheless, this !31!approach eliminate s many of the limitations imposed by TEM thin foils. Surface preparation may be carried out either before or after the imposed deformation. 2.4.1 !Selected Area Channeling Patterns (Obtaining Imaging Conditions fo r ECCI) In many respects, ECCI is carried out in the same manner as diffraction contrast TEM; that is, imaging is achieved by setting up specific diffraction/channeling conditions. Instead of using electron diffraction patterns to e stablish Ò2 -beamÓ condi tions as with TEM, ECCI relies on either EBSD patterns, electron channeling patterns (ECPs), or selected ar ea channeling patterns (SACPs) to establish electron channeling conditions [68, 72, 76] . EBSD is advantageous to use for setting up ECCI i maging conditions because it can quickly obtain crystal orientations with a high spatial resolution of ,30 nm (spatial resolution is dependent on the specific beam conditions ) [60, 68, 81] . On the other hand, the accuracy of EBSD is limited to 0.5 -2.0 ¡ [63, 67, 81] . Due to this, it can be difficult to k now the exact imaging condition , i.e. the g vector, used to image dislocations. Therefore, ECCI studies that are set up using EBSD patterns are more qualitative and less quantitative in terms of characterizing dislocations. ECPs are better than EBSD patterns in that they give a clear understanding of the specific imaging conditions. ECPs can give an absolute accuracy in crystal orientation of less th an 1 ¡ [72] . An ECP is formed when a single crystal (or large grain from a polycrystal) is viewed at low magnification. If the magnification is low enough, the beam will have a large enough scan angle to view the diffraction contrast of backscattered electrons resulting from the crystal planes being at the Bragg angle relative to the sweeping beam . This contrast makes up the channeling pattern [72] . The utilizatio n of ECPs are limited by that fact it has a spatial resolution of ,1 mm !32![68] and therefore they cannot be achieved for polycrystalline samples with small to moderate grain sizes. SACPs overcome the limitations of ECPs because rather than the beam sweeping over the crystal, the beam is rocked on a Òsingle pointÓ on the crystal surface to achieve the same effect . Single point is written in quotes due to the fact it is a point with an associated area. The user manual for the Tescan Mira 3 field emi ssion gun scanning electron microscope (FEG -SEM) used in this work suggests grains should have a minimum grain size of 100 -150 µm to achieve SACPs. An example of a SACP obtained using the Tescan Mira 3 FEG -SEM is shown in Figure 2.7, with the square ma rking in the center of the SACP, which is the optic axis of the electron beam . The maximum angular range of this SACP is approximately 20¡. It is seen that long the outer edge of the SACP the main channeling pattern is not continuo us. This is because the outside portion of the channeling pattern is formed from the channeling effects of neighboring grains. As the grain size from which a n SACP is obtained gets smaller, the observed channeling of neighboring grains is increased and c hanneling pattern of the grain of interest gets smaller. This demonstrates that spatial resolution goes down as the angular range goes up [68] . !33! Figure 2.7: SACP obtained using a Tescan Mira 3 FEG -SEM. !34!In this work, SACPs were successfully obtained from grain sizes around 40 µm in diameter . There were , however, significant effects from neighboring grains along the outer edge of the channeling pattern image, leading to a limited view of the channeling pattern of interest. This limited view becomes worse with higher tilts as the projection of the grain becomes smaller. Due to the limited spatial resolution of the Tescan Mira 3 FEG -SEM, dislocations cannot be characterized in small grains. Guyon et al. [68] were able to obtain SACPs with a spatial resolution of 500 nm and a rocking beam angle of 4.2 ¡ using a Crossbeam FEG -SEM Zeiss Auriga equipped with a Gemini column. This was done by calibrating the beam shifts in order to accurat ely adjust the rocking beam position. This allows for SACPs to be obtained from samples with small grain sizes. 2.4.2 !Electron Channeling Contrast Imaging In order to view dislocations using ECCI, it is necessary to understand how the channeling conditions rela te to electron interactions within the crystal. Depending on the angle at which an electron beam hits a crystal, there can be either high backscattered electron yield, large $, or a small backscattered electron yield, small $ [72] . A pictorial example of this is shown in Figure 2.8a. Artificial SACPs for high backscattered electron yield and low backscattered electron yield are shown in Figure 2.8b and Figure 2.8c, respectfully. The black dot i n the SACPs shows the optic/beam axis that corresponds to the two SACPs for the different amounts of backscattered electron yield. !35! Figure 2.8: a)Pictorial representation of the backscattered electron yield with respect to the angle at whic h electrons hit a crystal. b) An imitation SACP corresponding to a condition of high backscattered electron yield, large $. c) An imitation SACP correspon ding to a condition of low backscattered electron yield, small $. The black dots in b) and c) represent the optic axis of the electron beam (amended from Crimp [74] and Joy et al. [72] ). !36! A crystal sample is rot ated/tilted so that the optic axis in on the edge of a channeling band, shown in Figure 2.9c. At this point the el ectron beam is hitting the bulk of the crystal at the Bragg angle, #B, for the crystal planes associated with that channeling band. As the rastering electron beam passes over a dislocation , shown in Figure 2.9a, one side of the d islocation distorts the lattice so the electron beam is interacting with the crystal at angle &, which is larger than the Bragg angle. This locally distorts the channeling condition so the optic axis is now off the channeling band, shown in Figure 2.9d. The other side of the dislocation will distort the lattice in the opposite way so that the electron beam is interacting with the crystal at angle %, which is smaller than the Bragg angle. This locally distorts the channeling cond ition so the optic axis is now on the channeling band, shown in Figure 2.9b. These distortions cause one side of the dislocation to be dark and the other side to be bright. Tilting/rotating the sample so the that the opposite g vector is used (the other side of the same channeling band) will cause the bright and dark sides of the dislocation to switch. !37! Figure 2.9: a) Pictorial representation of the rastering electron beam (dott ed blue lines) moving over a dislocation. A pictorial representation of the channeling condition for the bulk of the crystal lattice is shown in c) where the optic axis of the electron beam is hitting the crystal planes at the Bragg angle. The lattice di stortions from the dislocation change the local channeling condition so that the optic axis hits the crystal planes at angle % (smaller than the Bragg angle) on one side and at angle & (larger than the Bragg angle) on the other side. The channeling condit ions for these lattice d istortions are shown in b) and d ) (figure courtesy Dr. Martin A. Crimp) . !38! Knowledge of the channeling conditions allows dislocations to be characterized in te rms of their Burgers vectors, b, and line directions, u, using the well established g ¥ b = 0 and g ¥ b x u= 0 invisibility criterion, where g describes the channeling condition [71 Ð73, 82] . In this respect, characterization of dislocations using ECCI is just like characterizing dislocations using TEM. The dislocation line widths resolved by ECCI are similar to that offered by diffraction contrast bright field imaging TEM, in the r ange of 10 to 12 nm [76] ; however, TEM has the advantage of weak beam microscopy , which decreases the dislocation line width, allowing TEM to image areas with hig h dislocation densities [83, 84] . Conversely, ECCI has the advantage of necessitating only one free s urface, so that image force [7] effects will not be as severe as in TEM thin foils. Also due to the one free surface with ECCI, a sense of the disl ocation line inclination can be easily determined while determining this with TEM is more difficult. 2.4.3 !Cross -Correlation Electron Backscattered Diffraction One of the biggest limitat ions of EBSD is its angular resolution, which has a range of 0.5 ¡-2.0 ¡ [60, 63, 67] . Cross -correlation electron backscattered diffraction (CC -EBSD), also referred to as high resolution or high angular resolution EBSD [77, 78, 85, 86] , is able to obtain an angular resolution of 2 x 10-4 rad, which corresponds to 0.01 ¡ [77] . The cross -correlation method selects regions of interest (ROIs) within a given EBSD pattern an d uses computer software to cross -correlate the ROIs across all EBSD patterns in order to detect the subtle shift s in the EBSD patterns . From this, the entire elastic strain tensor can be determined [77, 78] . The acquisition of EBSD patterns for CC -EBSD analysis is the same as traditional EBSD, however, longer exposure times are used to ensure adequate pattern quality and the patterns are save d for cross -correlation analysis to be perfor med offline. Due to the !39!longer exposure times and the offline analysis, CC -EBSD is significantly slower than traditional EBSD. CC-EBSD can be used to map the GND content deduced from the Nye tensor [87] : !"#* directions that proje ct from the middle of the indent . Indents in t he [001] oriented grains show four lobes equally separated from one another . I ndents in t he [101] oriented grains show two pairs of lobes on opposite sides of the indents . Indents in the [111] oriented grain s show three lobes equally arranged at 120¡ around the indent. As orientati ons gradually change from these primary orientations, the topography change is gradual as well. The max height of each lobe for three microindent s, one indent for each of the prima ry orientations , were measured and summarized in Table 4.1. The [100] indent has the highest lobes heights with an average of 467 nm, then [101] with 335 nm, and the [111] indent has the lowest average lobes heights with 288 nm. Since [100] had the highest average lobe height, the !56![101] and [111] heights were calculated as a percentage of [100] height in order to compare later with nanoindentations. !57! Figure 4.1: Indentations of single crystal microindents plotted on a portion of a bcc stereographic projection in order to visualize indent topography as a function of orientation. !58!Table 4.1: Statistics of lobe heights of three microi ndent s, one for each of the primary orientations. Primary Orientations [100] [101] [111] Lobe # Max Lobe Heights (nm) 1 455 381 300 2 550 287 305 3 408 347 260 4 456 326 - Average 467 335 288 St. Dev 60 39 25 % of [100] Height 100% 72% 62% !59!4.2 !Sing le Crystal Nanoindentation AFM measurements of nanoindents place d in the middle of grains with [001], [101], and [111] orientations are shown in the top of Figure 4.2. Similar to the microindents shown above, thes e nanoindents ref lect four -fold symme try for the [001] orientation, two -fold symmetry for [101 ] orientation, and three -fold symmetry for [111] orientation. They also agree with the fact that the lobes are generally at the highest heights along the <111> d irections that project out from each indent. For indents the [001] and [111] , there is an agreement on the number of topographical lobes that are observed with microindentation and nanoindentation. But for the indents in [101] oriented grains, there is o nly two broad lobes on either side of the nanoindent , while there are four lobes on the microindent. This is evidence of a size effect in the indentation process. The max lobe height s of the nanoindents are summarized in Table 4.2. The [100] orientation had the highest average height 56.8 nm, then the [101] with 26.6 nm, and [111] had the lowest with 24.9 nm. The average heights of the [101] and [111], however, are statistically equivalent due to the fact that each of the stand ard deviations encompass the average of the other. !60! Figure 4.2: Top) Experimental topographies for the nanoindents made in grains of orientations close to [001], [101], and [111] , shown left to right resp ectively . Bottom) Corresponding CPFEM simulations for three orientations. Table 4.2: Statistics of lobe heights of three experimental single crystal nanoindents, one for each of the primary orientations. Primary Orientations [100] [101] [111] Lobe # Max Lobe Heights (nm) 1 62.4 24.7 31.3 2 54.9 28.5 30.1 3 51.7 - 13.4 4 58.2 - - Average 56.8 26.6 24.9 St. Dev 4.6 2.7 10.0 % of [100] Height 100% 47% 44% !61!4.2.1 !CPFEM of Single Crystal Nanoindentation To test whet her the CPFE mode l of Ta correctly predicts the topography of single crystal nanoindentation, c orresponding CPFEM simulations of the three experimental nanoindents are shown in the bottom row of Figure 4.2. The [001] and [101] nanoindents appear to have reasonable agreement, particularly in terms of symmetry, but disagree in the details. The [111] has the largest disagreement between the experiment and the simulation. The agreement/disagreement between t he experimental and simulated nanoindents are described below. The [001] simulation appears to have the closest match to the experimental AFM measurem ent and it also shows the same four -fold symmetry. Nevertheless , there is disagreement between the [001] experiment and simulation in regards to the topographical lobe height and spread of the lobes from the middle of the nanoindent. The lobe height is higher in the experimental measurement and the lobes in the simulation extend further out from the middle of the indent. For both the experimental and simulated nanoindents, there are four corners along the edge of the indent rim between the four lobes. These corners are more exaggerated in the simulated indent. The [ 101] experimental and simulated indents also show agreement in terms of the 2 -fold symmetry of the lobes. The experimental indent appears to have the same lobe height as the simulation near the edge of the indent but, going further out from the middle of the indent, the lobe height drops off more quickly than in the simulated indent. Along the indent rim of the simulated indent there are two distinct corners between the two lobes on the top -left and bottom -right of the indent, similar to the corners found in the [001] indent. The experimental in dent only shows one distinct corner on the top -left , but there is still separation between the lobes on the !62!bottom -right. Also, the lobes of the experiment al indent appear to surround mor e of the indent center , while the simulated indent has a larger gap with no topography between the lobes, at the top -left and bottom right of the indent center. While the experimental [111] nanoindent has 3 -fold symmetry, the simulated nanoindent is close r to 6 -fold symmetry. In the places where the experimental nanoinden t shows the highest topography, the simulated nanoindent shows zero topography. Also, the three lobes in the experimental nanoindent come to a point further out from the indent while the simulated indents appears to have six lobes that fan out moving furt her from the indent. 4.3 !Bi-crystal/Grain Boundary Nanoindentation To characterize the manner in which grain boundaries influence strain transfer, over 200 nanoindents were placed near grain boundaries. Even with this many nanoindents, t he secondary pop-ins, or grain boundary pop -ins, that are discussed in the literature were not observed in Ta. This indicates that there is no conclusive evidence that secondary pop -ins occur in Ta. Three grains boundaries were selected for more in -depth analysis of deformatio n transfer with each grain boun dary having a nanoindent on either side of the grain boundary, shown in Figure 4.3. In the left column are BSE images of the three grain boundaries analyzed with each grain boundary indent identified by a number at the lower right of the indent and the mÕ value for the grain boundary at the top of the image. The mÕ value is based on the closest aligned slip systems since the activated slip systems are unknown. The middle -left column show s the AFM measurements of each of the grain boundary indents. The middle -right column shows the AFM measurements of the single crystal indents that correspond to the grains that each grain boundary indent was made in. The right column shows the resu lts of the subtraction procedure outlined in section 3.4.2 . !63! Figure 4.3: Left) BSE image of indents along three grain boundaries. Middle -left) AFM measurements of the grain bo undary nanoindents. Middle -right) Single crystal nanoindents that correspond to each of the grain boundary nanoindents. Right) Subtraction of the grain boundary AFM measurement with corresponding single crystal indents. !64!From the AFM grain boundary mea surements in the middle -left column in Figure 4.3, strain transfer is considered to have occurred if topography appears across the grain boundary from the indent and not to occur if there is no topography across the boundary . All three grain boundaries show that strain transfer is more significan t from one indent and less prevalent , or non-existent, from the indent on the opposite side of the grain boundary . This suggests that grain boundary strain transfer is dependent on the di rection th e shear approaches the boundary. This also shows that even though all grain boundaries have reasonably high mÕ, strain transfer should not be predicted solely on slip system alignment. For grain boundary 33, the AFM grain boundary maps show sign ificant strain transfer for indent number 3 , but little strain transfer for indent number 4. Regardless of amount of strain transfer between the two grain boundary indents, the su btraction results for both show significant strain transfer resistance. Thi s is exhibited by red (positive) topography in the parent grains of the subtraction result. On grain boundary 42, the AFM grain boundary measurements shows significant strain transfer for indent 11 and very little to no strain transfer for indent 12. The subtraction result for indent 11 shows a slightly positive topography at the top right of the indent indicating a small amount of resistance to strain transfer. Indent 12 also shows a small amount of resistance to strain transfer due to positive topograph y at the top left of the indent. For grain boundary 43, the AFM g rain boundary measurements show no strain transfer for indent 14 and significant strain transfer for indent 15. The subtraction result of indent 14 suggests very little resistance to strain transfer due to the slightly positive topography above the indent. On the other hand, the subtraction result for indent 15 suggests strong resistance to strain transfer due to the positive topography, indicated by the bright red, to the right of the inden t. !65!4.3.1 !CPFEM of Grain Boundary Nanoindentation A comparison of the AFM measure d topographies to CPFEM topograph ies are shown in Figure 4.4. The AFM measurements are shown in the left column, CPFEM results are shown i n the middle column , and the subtraction results (CPFEM minus AFM) are shown in the right column. The simulation for grain boundary indent 11 at grain boundary 42 did not converge to a solution , so this indent is ignored for the comparison between AFM and CPFEM. CPFEM simulations of grain boundary indents agree with AFM measurements in that they also indicate that grain boundaries can have an effect on indent topography /strain . Upon closer comparison to the AFM measurements , however, the e ffect that grain boundaries have on topography is only reasonably similar to experimental indents 3 and 4 for grain boundary 33 , in that the experimental and simulated indents at grain boundary 33 have the best topographical alignment on both sides of the grain boundary . Comparison of CPFEM results and AFM measurements for other indents show a clear difference in topography. The CPFEM results indicate topography transfer across the grain boundary for every grain boundary indent, disagreeing with the AFM results of indents 12 and 14. As for indents 4 and 15 , CPFEM results agree with the AFM measurements that topography transfer occurs , but the CPFEM significantly over -estimates the amount topography transfer. This is indicated by the red topography in the subtraction resu lt of the neighboring grain. Indent 3 has the best match between CPFEM and AFM for the overall indent topography in the parent grain and the a mount of topography transferred into the neighboring grain . For all other indents the CPFEM overestimates the top ography of the parent grain and the topography transferred into the neighboring grain . !66! Figure 4.4: Left) AFM measured topographies of experimental nanoindents near grain boundaries. Middle) CPFEM topo graphies of the same grain boundary experimental indents. Right) Subtraction result of CPFEM minus AFM topographies. !67!4.4 !ECCI vs. CC -EBSD for Single Crystal Indentation 4.4.1 !Dislocation Distributions The dislocation distribution around a single crystal indent in a grain oriented near [011] was analyzed. Figure 4.5a, which was produced by taking multiple ECC images at the g = ( -21-1) channeling condition and stitching them together, shows the general deformation fields ar ound the indent. The strong intensity near the edge of the indentation can be attributed to the nominally tear shaped backscattered electron interaction volume escaping from the interior surface of the indent when the electron beam is scanned close to the edge of the indent. This effect is most likely complicated by the extensive deformation and localized rotations expected near the indent. Furthermore, the bright region appears asymmetrical due to the sample being tilted. Moving away from the high inte nsity region, dislocation fields extend from the indent in a number of directions. Most of the dislocations in these fields appear as black/white dots, representing dislocations roughly normal to the surface (examples shown in the dashed circle in Figure 4.5a), but some appear more extended due to their lines being more parallel to the surface (example show n in dashed rectangle in Figure 4.5a). More detailed images showing individual dislo cations are illustrated in subsequent figures. The corresponding CC -EBSD calculated GND map (tot al GND density), shown in Figure 4.5b, displays dislocation distributions comparable to those in the ECC image. The p ixels that correspond to EBSD patterns that have a confidence index less than 0.15 are whited -out. The topography of the same indent was measured with AFM and is shown in Figure 4.6. The stitched ECC image and the GND map from Figure 4.5, as well as the AFM measurement , reflect two -fold symmetry about the [011] axis. !68!a)! b)!!Figure 4.5: a) Multiple ECC images stitched together showing dislocations generated from a single crystal nanoinden tation in a grain of approximately [011] orientation. b) CC -EBSD GND map of the same area, collected wit h an EBSD scan step size of 100 nm and effectiv e step size of 200 nm, showing dislocation distributions similar to that in the ECC image. !69! Figure 4.6: AFM measurement of the same single crystal indent that was imaged and GND mapped in Figure 4.5. !70!4.4.2 !Dislocation Density Comparison A more detailed comparison between the ECCI and CC -EBSD re sults, carried out on a neighboring indent within the same grain, is shown in Figure 4.7. Here the ECCI, Figure 4.7a, shows a broad band of dislocations extending to the upper left of the indent and a fainter band near the right hand edge of the image, which curves to the left moving up in the image. Individual dislocations can be readily discerned, with the majority of the dislocations appearing close to end -on in the image. As before, there are also smaller numbers of dislocations with line directions m ore parallel to the sample sur face. A comparison of this image with the corresponding GND map from C C-EBSD, Figure 4.7b, again shows good agreement with the approximate locations of the dislocations. Nevertheless, there is not an exact one -to-one correlation between ECCI and the CC -EBSD images for reasons which will be discussed below. In order to facilitate a robust comparison, the ECC image was gridded to the same step size as the EBSD scan, shown in Figure 4.7c. The number of dislocations in each grid square were cou nted and divided by the area of the grid square to give an effective dislocation density. This result is shown in Figure 4.7d and is plotted with the same color scale as the CC -EBSD derived GND map . Regions of th e ECC image where dislocations could not be reliably imaged, i.e. the indent rim and inside the indent, are whited -out, seen in the lower right Figure 4.7d. !71! Figure 4.7: a) ECC image of dislocations from the upper -left of an indented area. b) CC -EBSD generated GND density map of the same area showing similar dislocation distributions, using a step size of 50 nm and an ef fective step size of 200 nm. c) ECC image g ridded to the same size as the EBSD step size. d ) Dislocation density map calculated by counting dislocations in each grid square of gridded the ECC image . !72!4.4.3 !Dislocation Characterization Using ECCI The dislocations imaged using E CCI were characterized usi ng channeling contrast criteria supplemented with the approximate line directions [73 Ð76]. This analysi s is focused on the region out lined by white da shes in the upper lef t portion in Figure 4.8a. This im age, collected using the g = ( -21-1) channeling condition, shows what appears to be 64 dislocations in the circled region, (in a few cases the contrast is complicated and may represent more than one dislocation). Careful examination of these dislocations reveals that many of them have their characteristic black/white contrast in the same orientation, while others d isplay reversed or rotated con trast. These differences in cont rast can indicate different Burgers vectors and/or edge or sc rew type dislocations [69, 98, 99] . Overall, 39 of the dislocations reveal t he same contrast orientation, wi th four having reversed contrast. An addit ional 21 display different con trast orientation or are difficult to categorize due to weak contrast. The six different channeling conditions used for the analysis shown in Figure 4.8 were established b y rotating and tilting the sam ple in conjunction with SACPs. The majority of the dislocations do not go out of contrast with any of the channeling conditions, but the orientation of the black/white contrast varies with each channeling condition. The fact that the dislocations do not go out of contrast suggests that these are screw dislocations that are generally perpendicular to the surface. That is, despite the fact that g ¥ b = 0 for all of the g vectors perpendicular to the screw line direction, the surface relaxation caus es them to always be visible [70] . The white dashed arrows in Figure 4.8 shows that the direction of the black to white contras t is roughly perpendicular to g, consistent with the c ontrast expected from screw dislocations generally perpendicular to the surface [69, 98, 99] . !73! Figure 4.8: ECC images for the channeling conditions used for contrast analysis, with g indicated by the white arrows and the black to white contrast indicated by the white dashed arr ows. !74!The four possible <111> screw dislocation line directions in this region are each shown as an ÒxÓ on the stereographic proj ection with respect to the back scatter detector, shown in Figure 4.9a. Two of these line directions, the [1-11] and [-1-11] are nearly parallel and can be eliminated as potential Burgers vectors/line directions of the dislocations that are close to perpendicular. To distinguish between the two remaining possibilities, [111] and [-111] (which are 40 ¡ and 31¡ from perpendicular to the beam axis, respectively), the sa mple was tilted 11 ¡ along g = ( -21-1), with the resulting orientation shown in the st ereographic projection in Figure 4.9b. This tilt would cause [111] screw dislocat ions to become more parallel to the detector (48 ¡ from the beam axis) while [-111] screw dislocations would become more perpendicular to the detector (27 ¡ from the beam axis). The ECC image corresponding to this tilt, Figure 4.9c, shows the dislocations now projecting as lines that project (fade) towards the bottom of the image, indicating the majority of the dislocations have line directions close to [111]. Combined with the sense of contrast discussed above, it is reasonable to conclude that these most common dislocations are a/2 [111] screw dislocations. It is worth noting that the other dislocations that display different black/white contrast do not project in the same direction as the a/2 [111] screws, suggesting they have d ifferent line directions and Burgers vectors. !75! Figure 4.9: Stereogr aphic projections a) correspondi ng to Figure 4.8a and b) tilted 11 ¡ along the g = (-21-1) with each ÒxÓ being a line direction for the fou r possible screw dislocations. c) ECC image w ith the same sample tilt as in b), showing a projection of the dislocation line directions. !76!4.4.4 !Dislocation Characterization Using CC -EBSD In addition to the total dislocation density sho wn in previous sections, the Nye tensor determined from CC -EBSD analysis may also be used to characterize the Burgers vector and edge/screw character of the local dislocation density, as well as the slip plane of the edge dislocations (the slip plane of sc rew dislocations is not determinable because it has no effect on the Nye tensor) via the Nye -Krıner method. The GND densities were determined using the line length minimization approach outlined by Ruggles et al. [88] . For this analysis, the smallest available effective step size of 25 nm was employed to maximize the spatial resolution of the method. The dislocation densities of each screw an d edge dislocation possi bility are shown in Figure 4.10. The disloca tion densities we re locally av eraged to better show trends. In the highly deformed region near the indent, the Nye -Krıner method identifies the Burgers vector of dislocation content where ECCI was incapable of resolving dis locations. In the region further from the indent , where individual dislocations were discernible via ECCI, CC -EBSD also characterized the dislocation content as being composed of s crew dislocations with a [111] Burgers vector. To highlight agreement with the two methods, the dis location density for the [111] screw dislocation determined via CC -EBSD is shown in greater detail in Figure 4.11. !77! Figure 4.10: Dislocation density of each dislocation type determined using CC -EBSD. !78! Figure 4.11: Dislocation density of the screw dislocations with a Burgers vector of [111] determined by CC -EBSD. !79!4.5 !ECCI vs. CC -EBSD for Grain Boundary Indentation 4.5.1 !Dislocations Distribution s An indent locat ed at a grain boundary triple junction was chosen for analysis of dislocations generated across a grain boundary. An overview BSE image and AFM map of the indent in is shown in Figure 4.12. The analysis was carried out at the gr ain boundary on the right of the indent. The AFM map, Figure 4.12b, shows topography transfer extending across the grain boundary to the right and up. The BSE image shown Figure 4.12a shows an area of b right contrast extending across the grain boundary to the right and down , away from the topography . This bright area is dislocation s generated on the opposite side of the grain boundary. A higher magnification ECC image of these dislocations is show in Figure 4.13a. !80! Figure 4.12: a) BSE image and b) AFM map of an indent located at a grain boundary triple junction. The analysis will be carried out at the grain boundary on the ri ght. !81! Figure 4.13: a) ECC image of the lower right of a grain boundary indent showing dislocations generated on the opposite side of the grain boundary. An CC -EBSD generated GND density map of the same a rea with a step size of 25 nm and an effective step size of 200 nm. !82!Figure 4.13a shows an ECC image of the dislocations generated across a grain boundary and Figure 4.13b shows the corresponding CC -EBSD derived GND density map. The EBSD for this map was carried out with a step size of 25 nm and an effective step size of 200 nm. In the ECC image, dislocations immediately across the grain boundary cannot be individually resolved because the concentration of dislocations is too high. Individual dislocations become discernable further away from the grain boundary. Most of the dislocations appear as lines, with a few appearing as dots, indicating most dislocations line directions are close to parallel with the sample surface. The CC -EBSD derived GND density map in Figure 4.13b shows good agreement with the dislocation distributions of the ECC image. Just as the ECC image shows a dislocation concentration gradient moving away from the grain boundary , the GND map also shows a high density and then decreases moving away from the grain boundary. This agreement if further exhibited by looking at the areas of low dislocation density that are located amid the areas of high dislocation de nsity in both the ECC image and GND den sity map. 4.5.2 !Dislocation Characterization with ECCI The dislocations generated on the opposite side of the grain boundary were characterized using g ¥ b = 0 analysis by way of the four channeling conditions shown in Figure 4.14. In this analysis, three distinct sets of dislocations were identified. Groupings of these sets are outlined in red, green, and orange. The channeling condition, g = (1 -12), in Figure 4.14a sho ws all three sets of dislocation. For all other channeling conditions, one set of dislocations becomes invisible in each channeling condition. !83! Figure 4.14: ECC images used to characterize three distin ct sets of dislocations generated on the opposite side of a grain boundary. The four channeling conditions used are in shown in a) to d). !84!It is important to note that the plane normal is approximately [ -111] and all the channeling conditions used inters ect the [ -111] zone axis. Ther efore, edge dislocations with a [-111] Burgers vector (which would have line direction parallel to the sample surface) will always be in an invisibility condition. On the other hand, screw dislocations of the [ -111] Burgers vector would always be visible due to surface relaxation due the line directions being perpen dicular to the sample surface. For these reasons, the sets of dislocation outlined are not expected to have a Burgers vector of [ -111] The dislocation s outlined in red become invisible at the channeling condition of g = ( -1-10), indicating a Burgers vector of [1 -11]. The dislocations outlined in green become invisible at the channeling condition of g = (01 -1), indicating a Burgers vector of [111]. The dislocati ons outlined in orange become invisible at the channeling condition g = ( -10-1), indicating a Burgers vector of [11 -1]. By combining this g ¥ b = 0 analysis with an analysis of the line direction with respect to the grain orientation (shown in Figure 4.15), a sense of the dislocation type s is obtained. Figure 4.15a shows the stereographic projection of the grain orientation for the ECC image that is again shown in Figure 4.15b. The line directions of the three sets of dislocations are overlaid onto the stereographic projection with their respective colors. The line directions of each set of dislocations lies close to the Burgers vectors, indicating that these dislocations are predominately screw type. It was attempted to characterize these dislocations using CC -EBSD but the scan quality was too low to resolve the GND density onto specific slip systems. Possible reasons for low quality will be discussed later. !85! Figure 4.15: a) Stereographic projection that corresponds to the grain orientation of the ECC image in b). The line directions for the three sets of dislocations are overlaid on the stereographic projection with their res pective colors . !86!4.5.3 !Coarser CC -EBSD Over Four Nanoindents To demonstrate the efficiency/effectiveness of CC -EBSD to map dislocation distributions , CC -EBSD was performed over a broa der area. EBSD was carried out at a step size of 200 nm and the cross -correla tion calculation was carried out with an effective step size of 200 nm. This is shown in Figure 4.16. Figure 4.16a is a BSE image of the four indents with a grain boundary running from the upper left of the image to the bottom right of the image. Three indents are near the grain boundary with two indents on the right and one on the left. The fourth far enough away from the grain boundary to be considered a single crystal indent. The top left indent is closer to the grain boundary than the bottom right indent. Below and to the left of the top left indent, the BSE images shows distortion of the grain boundary. The bottom right indent has the same effect on the grain boundary but not as much. Figure 4.16b shows the corresponding CC -EBSD calculated GND map. It show s GND build up on the opposite side of the grain boundary as the top left and bottom right indents. This GND build up is in the same areas as that of the distortions at the grain boundaries of the BSE images and with the same relative amounts between the two indents. However, the GND density level is not much higher than the noise and can barely be seen at the grain boundary for the bottom right indent. As done is the previous GND maps, pixels that correspond to an EBSD confidence index of less than 0.15 were whited -out. The grain boundary in the GND map is dotted with these white pixels. !87! Figure 4.16: a) BSE image o f four indents, three near a grain boundary and one far enough away from the grain boundary to be considered a single crystal indent. b) GND map of the same four indents. !88!4.6 !Analysis of Wedge Indent Cross -Section Figure 4.17a show s the CC -EBSD derived GND density map from Ruggle et al. [80] along with a BSE image , Figure 4.17b, of the area analyzed in this work. The channeling contrast in the BSE image shows thin band s that become pointed from the outside of the image towards the center in a downward direction. These bands are seen from left to right and from right to left, however, they are more faint fro m the right to left direction. If a line was drawn from the tip of the indent, it would go down the middle of the BSE image, where t he bands from both sides intersect and the intersections form parallelogram shaped boxes/cells of dark contrast. !89! Figure 4.17: a) CC -EBSD derived GND map from Ruggles et al. [80] and b) a BSE image of the area analyzed in this work. !90!A magnified ECC image of one these boxes/cells reveals dislocations, shown in Figure 4.18b. Many of the dislocations seen in the middle box /cell are generally parallel to the sample surface. Contrast from dislocations can also be seen within the bands as well as the boxes /cells on the other sides of the thin bands. There is also significant contrast from a large number of dislocations on the boundaries between the boxes/cells and the bands. The dislocations cannot be characterized using contrast analysis because the g vector could not be determined using SACPs. This is because the boxes/cells are too small to obtain SACP s. For other portio ns of this work, SACPs were only obtain ed for grains as small as 40 µm in diameter , but the box/cell in Figure 4.18b is only about 4 µm across. Because screw dislocations have a lower mobility than edge dislocations in bcc Ta, it can be assumed that the majority of dislocations in the interior of the box/cell in Figure 4.18b are of the screw type . Therefore , the Burgers vectors of the dislocations that have line directions parallel to the sample surface c an be determined using the stereographic projection , which was calculated from the data collected with EBSD. Figure 4.19a shows the line directions of the dislocations overlaid on the stereographic projection , revealing that thes e dislocations are likely [-11-1] screw dislocations. !91! Figure 4.18: a) BSE image of the analyzed area and b) an ECC image of one of the boxes/cells formed by the intersection of the bands seen in the BSE im age. !92! Figure 4.19: Many of the line directions in the ECC image, shown in b), have line directions that are generally parallel to the surface. Assuming that these dislocations are screw dislocations, overlaying the line directions onto the stereographic projection, shown in a), reveal that the Burgers vector for these dislocations are [ -11-1] !93! EBSD was carried out with a 60 nm step size over the area of interest shown in Figure 4.17b and Figure 4.18a. The EBSD map is shown in Figure 4.20a with the points that had a confidence index lower than 0.15 being discarded . Due to the fact that this is plane strain in the surface plane (xy plane) , all lattice rotation is around the surface normal. For this reason, the original inverse pole figure color of green (the color of the {101} plane normal for the inverse pole figure legend) will remain green for all lattice rotations. Therefore , strictly for visualization purposes, the crystal lattice for all EBSD points were rotated 90 ¡ about the x -axis in order to visualize the changing crystal orientation, shown in Figure 4.20b. In this view, the thin -pointed bands seen in the BSE image in Figure 4.17b and Figure 4.18a are pink and the original/bulk orientation is now red. All the thick bands on both sides generally have the same orientation (the original bulk crys tal orientation) . The thin bands on the left rotate the crystal lattice an average of 10.2 ¡ counter clockwise (maximum being 17.7 ¡ and the minimum being 6.1 ¡) and the thin bands on the right rotate the crystal clockwis e an average of 7.2 ¡ (maximum being 1 1.8 ¡ and the minimum being 4.4 ¡). This is further revealed in the pole figure maps shown in Figure 4.21. The left column of pole figures show the shifts caused by the thin bands on left side of the EBSD map cause, the middle col umn shows the shift caused by all thin bands, and the right column shows the shifts cause d by the thin bands on the right side of the EBSD map. In all of the pole figures the shifts are seen as streaks. The differences between the shifts for the left thi n bands and right thin bands is seen by the rotational shifts in the left and right columns of Figure 4.21. There are, however, minor effects of the right bands in the left band pole figures (and vice versa) because the bands cro ss and could not be completely separated out. !94! Figure 4.20: a) Inverse pole figure map for the revealing no change in lattice orientation due to the fact that all lattice rotation is around the surface nor mal. b) By rotating all crystal lattice points 90 ¡ around the x -axis, the orientation changes can be more readily discerned . !95! Figure 4.21: Left Column) Pole figures that represent the shift caused by the left thin bands, Middle Column) pole figures that represent shifts caused by all thin band shifts, and Right Column) are pole figures that represent the shift s caused by the right thin bands. !96!As the thick an d thin bands are alternating, it can be seen t hat the orientations are alternating between the rotated and un-rotated version of the crystal lattice . This is shown in Figure 4.22. It is not a gradual rotation across the thick and thin bands , but rather immediate shifts in e ither side of the boundary. The thin bands on the left rotate the original orientation counter clockwise, while the thin bands on the right rotate the original orientation clockwise. The rotations on the right side are smaller and therefore less clear. The alternating rotations are consistent with the alternating contrast of the bands observed in Figure 4.17b. !97! Figure 4.22: Inverse pole figure EBSD map overlaid with the unit cell s for the alternating bands. !98!5!Discussion 5.1 !Single Crystal Microi ndentation and Na noindentation Both micr oindentation and nanoindenation in single crystals show that indentation topography is a function of crystal orientation. There is , however , a size effec t between microindentation and nanoindentation, which is seen in the indentation of crystals with a surface normal of the <101> type. While nanoindentation shows what appears to be one long topographical lobe on either side of the indent, microindenation shows two lobes on either side of the ind ent. By looking at the stereographic projection of a <101> type orientation, it is seen that there are three <111> directions pointing out from either side of the indent. Two of the directions , [1 -11] and [ -1-11], lie in the surface plane of the sample (contributing two <111> directions on either side of the indent) and two of the directions , [111] and [ -111], are inclined at about 35 ¡ (contributing one <111> direction on either side of the indent) , shown in Figure 5.1b. As noted earlier, the lobes generally line up with the <111> directions , therefore, one would expect to see three topographical lobes on either side of the indent. This is observed best in the AFM map shown in Figure 5.1a and the corresponding stereographic projection in Figure 5.1b. Because the three lobes are close together, they can appear as a single lobe. Contrasting with nanoindentation, microindentation shows two lob es on either side of the indent. T his is likely due to the changing stress state imposed by the indenter tip . While nanoinden tation and microindentation use d the same size tip radius of 1 µm, nanoinden tation general stays in the spherical portion of t he nanoindenter and the deformation induced by microinden tation is dominated by the conical portion. This suggests the slip that is activated by the spherical portion of the indenter , which cause s the middle lobe in nanoinden tation , is suppressed by the s tress state of the conical portion of the indenter in microinden tation. !99! Figure 5.1: a) AFM measurement a single crystal nanoindent in a grain of [011] orientation. b) Stereographic projection of the same grain showing two Burgers vectors lying in the surface plane of the sample and two Burgers vectors pointing to either side of the indent. !100! For microindentation, the lobes of the <100> type indents have the highest average maximum height, followed by <101 > and the <111>. For nanoindentation, the lobes of the <100> type also have the highe st average maximum height but the <101> and <111> types are statistically equivalent. In other word s, it seems that as indentation continues, the <101> type has a greate r relative increase in the average maximum height than the <111> type. This could be due to the fact that for nanoindentation , the slip activity is pushing lobes out in six directions but on ce the microindentation scale is reached , all the slip activity i s pushing lobes out in only four of the original six directions, increasing the relative slip activity of the lobes in comparison to the <111> type. 5.1.1 !CPFEM of Single Crystal Nanoindentat ion The comparison of CPFEM to experimental nanoindentation show s the b est match for [001] oriented grain and the worst match for the [111] oriented grain. Even though the [001] orientation has the best match, the re is a difference in lobe height and lobe spread for this orientation . Furthermore, the re is a complete differe nce in symmetry for the [111] orientation . These two major differences suggest that the structure evolution parameters need to be optimized for Ta and the non-Schmid effects need to be correctly implemented into the phenomenological model. 5.2 !Grain Boundary Nanoindentation AFM measurements of grain boundary nano indents in section 4.3 show that there is significant topography transfer from one side of the grain boundary and not from the other , indicating the directionality influenc e of dislocation flow . For indents where there is significant transfer, i.e. indents 3, 11, and 15 from Figure 4.3, the deformation in the neighboring grain !101!follows the symmetry/orientation of the crystal. This is seen by the la ck of significant topography of the neighboring grain in the topography subtraction result. The directionality depen dence of deformation transfer indicates tha t a single mÕ value for a given grain boundary is inadequate for describing whether a gra in bound ary is susceptible to, or at which level it is able to accommodate deformation transfer. Rather , mÕ should be dependent on the activated slip systems in the parent grain. Even further, the resistance to deformation at the grain boundary may depend on the side of the grain boundary from which deformation is approaching. In other words , a given crystal orientation may be able to transmit deformation to a neighboring grain , but is less prone to accommodate deformation coming across the grain boundary. This is be directly related to the anisotropy of the crystal i n that some crystal s are in a ÒhardÓ, or less accommodating to deformation, orientation and some crystal s are in a ÒsoftÓ, or more accommodating to deformation , orientation . From Figure 4.3, resistance to topography transfer across the grain boundary is shown by positive (red) topography in the parent grain of the subtractions results ( i.e. more red topography indicates more resistance). With this, it is seen that the amou nt of topography transfer resistance at a grain boundary is not synonymous with whether or not there is topography transfer. For example, indents 3, 15 , and 11 show significant topography transfer across the grain boundary , however, while indents 3 and 15 show significant resistance, indent 11 shows little to no resistance. In a similar regard, indents 4, 12, and 14 all show no topography transfer, however, while indent 4 shows significant resistance, indents 12 and 14 show little to no resistance. In sum mary, these six indents made along the th ree grain boundaries illustrate four combinations of topography transfer and resistance at the grain boundary. Those combinations include: (1) significant transfer and significant resistance (indents 3 and 15), (2) significant !102!transfer and little to no resistance (indent 11), (3) little to no transfer and significant resista nce (indent 4), or (4) little to no transfer and little to no resistance (indents 12 and 14). The combination of little to no transfer and littl e to no resistance likely means that due to the crystal orientation, the deformation from nanoindentation is not heavily directed towards the grain boundary. On the other hand, the combination of little to no transfer and significant resistance suggests d eformation is directed at the grain boundary and possibly the dislocation s are being reflected back from the grain boundary or slip systems transferring deformat ion towards the boundary lock up to cause an increased activation of other slip systems. The co mbination of significant transfer and significant resistance suggests that dislocations are reflected back from the boundary or slip systems lock up to cause an increased activation of other slip systems but eventually the stress at the grain boundary is h igh enough to activate the slip systems in the neighboring grain. The combination for significant transfer and little resistance suggests that deformation directed at the grain boundary and the neighboring grain is in an orientation the can easily accommo date deformation. 5.2.1 !Dislocation Pile -ups at Grain Boundaries In general, deformation at grain boundaries is thought of as dislocations piling up at the grain boundary and after certain amount of stress develops, dislocation s are generated on the other side o f the grain boundary [6, 7, 15, 19, 39, 40] . Contrasting , CC -EBSD and ECCI measurements shown in Figure 4.13 show a build -up of dislocations in the n eighboring grain rather than a dislocation pile -up in the originating grain. But the indent is quite close to the grain boundary not allowing for any dislocation pile -up to be viewed within the parent grain. This concern is addressed , however , with the grain boundary indents in Figure 4.16. Here the CC -EBSD shows no dislocation build -up in the parent grain but shows dislocation build -up !103!immediately on the other side of the grain boundary. This indicates that dislocations readily leave the parent grain but dislocation propagation into the neighboring grain is restricted . One possibility for the lack of dislocations at the grain boundary in the parent grain is the dislocations coming from the indent are edge dislocations and because of their high mobility they exit the grain quickly going in the neighboring grain. The dislocations propagating into the neighboring grain could be screw dislocat ions, which have a low mobility, restricting their propagation into the neighboring grain. This could be the case for the grain boundary inden t in Figure 4.13 as the dislocations were identified to be screw dislocations. The second possibility is that deformation from nanoindentation is so localized that the motion of dislocations does not reach very far beyond the gra in boundary. Regardless of the reasons for dislocation build -up on either side of the grain boundary, this work illustrates that ECCI and CC -EBSD combined with nanoindentation is a good method for investigating dislocation build -up at grain boundaries. One of the biggest advantages of nanoindentation is that the originating and neighboring grains are easily identified in relation to deformation. With experiments such as tensile test s, however, it is not always intuitive which grain is the ori ginating and which grain is the neighboring grain to deformation. Assuming that the dislocation pile -up occurs in the originating grain can be incorrect assumption according to the results in Figure 4.13 and Figure 4.16. 5.2.2 !CPFEM of Grain Boundary Nanoindentation As mentioned earlier , the CPFEM topography results show that grain boundaries have an effect on topography , but they disagree with the experimental results in that the CPFEM suggests all grain boundaries exhibit strain transfer and CPFEM overestimates the topography height (only indent 3 and 4 are close to the experimental result). Su et al. [42] als o showed that the !104!CPFEM overestimated topography transfer for &-Ti, but to a lesser degree than in this work. Overall, just as the single crystal CPFEM results suggest , the structure evolution parameters need to be optimized for Ta and the non -Schmid effe cts need to be correctly implemented into the phenomenological model in order to achieve a better analysis between experimental and CPFEM nanoindenation. 5.3 !ECCI vs. CC -EBSD 5.3.1 !Dislocation Density Comparis on Qualitatively, there is good agreement between areas o f high dislocations density of the CC-EBSD GND results and the locations of individual dislocations measured from ECCI. Nevertheless, as shown in Figure 4.7, ECCI has superior spatial resolution, which allows for individual dislocations to be detected within a single grid square while data from CC -EBSD is more diffuse and noisy. The diffusivity and noise from CC -EBSD is due to the fact that a dislocation is treated as a continuum based on the strain field in the l attice, causing the limited resolution of CC -EBSD to be controlled by the original step size at which the EBSD data was acquired and the effective step size at which the GND map was calculated. While ECCI has advantages for identifying individ ual dislocat ions at low densities, CC -EBSD is advantageous because it is able to detect large lattice ro tations and observe dislocation strain effects in high deformation regions that are too densely packed for ECCI, i.e. around the rim of the indent. To obtain a more robust quantitative comparison of the measurements presented in Figure 4.7, dislocation densities measured via ECCI and CC -EBSD were averaged for five separate regions , shown in Figure 5.2. In regions 1, 2, and 3, ECCI and CC -EBSD both detected dislocations, in region 4 only ECCI observed distinct dislocations, and in region 5 no dislocations were observed using ECCI. For each of these five regions, an average GND density !105!from CC -EBSD was determined by averaging the GND density associated with each pixel in the region. Dislocation densities from ECCI were determined by counting the number of dislocation intersections with the surface. !106! Figure 5.2: Dup licate of Figure 4.7 overlaid with 5 regions for comparison between b ) CC -EBS D calculated GND density map and d) a dislocation density map that was deri ved from the ECC image in a ) and c) . The oval in a) shows an example of dipol e dislocations. !107!Dislocations were initially assumed to have line directions perpendicular to the surface, but if dislocations are not normal to the counting area, dislocation densities are underestimated [100] . To obtain corrected densities, the dislocation density should be multiplied by 1/cos( #), where # is the angle between the line direction and the beam axis. Most of the d islocations in regions 1 and 4 were identified as [111] screw dislocations with a line direction 40 ¡ to the beam axis when the sample was in the channeling condition for the ECC image in Figure 5.2a and c. The dislocations in reg ions 2 and 3 were not identified and are not all the same dislocation type but many of these dislocations are likewise inclined. Since all line directions are possible in these two regions , an average angle of 58 ¡ has been used for calculating the disloca tion density. This was calculated by averaging the angles that the 22 possible line directions (12 for {110} slip plane systems, 6 for {112} slip plane systems, and 4 for screw disloc ations) make with the beam axis . For all five regions, Table 5.1 presents both the initial and line direction corrected dislocation densities. !108!Table 5.1: Comparison of CC -EBSD GND densities and ECCI dislocation densities for the 5 regions shown in Figure 5.2. Region # CC-EBSD GND Density ECCI Density ECCI (Line Direction Correction) ECCI (Dipole Correction) 1 2.6 x 10 14 m-2 1.6 x 10 14 m-2 2.1 x 10 14 m-2 1.9 x 10 14 m-2 2 2.1 x 10 14 m-2 8.6 x 10 13 m-2 1.6 x 10 14 m-2 No dipoles 3 1.7 x 10 14 m-2 1.1 x 10 14 m-2 2.2 x 10 14 m-2 No dipoles 4 6.4 x 10 13 m-2 6.1 x 10 13 m-2 8.0 x 10 13 m-2 No dipoles 5 5.2 x 10 13 m-2 0 0 0 !109!Due to the spatial resolution limitations of CC -EBSD as compared to ECCI, it is possible that dipole dislocati on pairs will fall within a given CC -EBSD step, canceling the contribution to the dislocation density, i.e. on the local scale ECCI may resolve dipoles while CC -EBSD may not. A few dipole pairs are observed in the ECC images, for examp le in the small oval in Figure 5.2a. ECCI shows 22 dislocations in region 1 with one dislocation displaying reversed contrast (i.e. opposite Burgers vectors). From the CC -EBSD perspective this dislocation will cancel out with another closely spaced dislocation and neither will be accounted for, leaving a net 20 dislocations in the CC -EBSD determined dislocation den sity. This effect is accounted for in the Dipole Correction Column in Table 5.1. Dipoles were observed in reg ion 1, but not observed in regions 2 through 5. The total dislocation density is made up of both GNDs and SSDs. Thus, as ECCI images reveal both the GNDs and SSDs, one would expect that the ECCI measured density would be greater than or equal to that dete rmined by CC -EBSD. However, the results presented here do not reflect this for regions 1 and 2. This may indicate that the comparison here is being carried out in regions where the CC -EBSD GND density measurements are close to their noise floor. Indeed, region 5 is an area where no dislocations were observed using ECCI, but the CC -EBSD indicated a GND density average of 5 x 1013 m-2. This noise floor is near the CC -EBSD GND density noise range suggested by the work of Jiang et al. [101] in which they measured the GND density noise on single crystal Si. This noise is likely due to binning/resolution of the EBSD camera [101] , pattern quality due to EBSD scan rate [102] , and the EBSD step size/effective step size [95, 103] . Errors may also be associated with increased diffusiveness of the EBSD patterns taken from areas with a higher density of dislocations, but it would be expected that this error would b e averaged out over a number of EBSD steps. Nevertheless, if the noise level !110!indicated by region 5 outlines an uncertainty level that is then applied to the measurements in the other regions, the CC -EBSD and ECCI measurements appear quite close. ECCI coul d also result in lower measured dislocation densities simply because some dislocations may be in a zero contrast condition for the particular 2 -beam channeling condition used, i.e. g ¥ b = 0 and/or g ¥ b x u= 0. In this work, however, this was not the cas e as this effect was account ed for by taking images at mul tiple channeling conditions and other dislocations do not appear. CC-EBSD will never have dislocations that are ÒmissedÓ due to this effect and will be able to identify al l of the dislocations that contribute to the GNDs. Another potential limitation of ECCI is that at higher dislocation densities it becomes impossible to resolve the individual dislocations. This appears to be the case for the regions close to the indent that appear very bright. CC -EBSD does in fact identify higher dislocation density pixels in this near -indent region that appear only bright in ECCI. Overall, both CC -EBSD and ECCI have some inherent limitations to determining dislocation densities, and users should be aware of thes e restrictions when using these techniques. 5.3.2 !Dislocation Characterization Using ECCI All the dislocations characterized in this work were identified to be screw dislocations. Considering that edge dislocations have a higher mobility than screw dislocations , it is reasonable to conclude that at the stage of d eformation where these analyses were carried out, the edge dislocations have exited the crystal leaving predominate ly screw dislocations, i.e. as deformation continues the concentration of screw dislocat ions will likely increase because as more dislocation loops are generated, the edge components will exit the crystal quickly leaving behind the screw components. !111!5.3.3 !Dislocation Characterization Using CC -EBSD A few caveats apply when employing the Nye -Krıner m ethod at the limits of its spatial and dislocation density resolution (i.e. when there are countably few dislocations per area resolution). First, all dislocation content is assumed to be a linear superposition of pure edge or pure screw dislocations [88 ]. This means that dislocations of mixed character will be represented by superimposed fields. Additionally, at these low step sizes, noise effects are more dominant [95] . One caveat often mentioned [80, 88, 95, 97] when interpreting dislocation density fields measured via CC -EBSD is not particularly cogent at the extremes of its resolution: the Nye -Krıner method only detects geometrically necessary dislocations. Because the length scale of the scan approaches that of dislocation dipole spacing, virtually all of the dislocations in the scan area may be thought of as geometrically necessary. Despite the challenges of employing CC -EBSD dislocation characterization at a res olution suitable for comparison at the same length scale, t he level of agreement for single crystal indent in section 4.4.3 is strong . The noise effects for the grain boundary indent in section 4.5.2 were too prevalent to resolve dislocations onto specific slip systems. It is possible that this crystal orientation and/or the orientation of the dislocation type causes the noise effects to be too prevalent since the Nye tensor determ ination is sensitive to orientation [80] . It can be seen in Figure 4.16b that noise of the GND calculations is affected by crystal orientation. In the top right grain there is more noise than in the bottom left grain. This is indicated by higher concentration of light blue pixels scattered throughout the upper right grain than the lower le ft grain. !112!5.3.4 !A Balance of CC-EBSD Noise to SEM Drift Mentioned earlier, the three main causes of noise for CC -EBSD is binning/resolution of the EBSD camera [101] , pattern quality due to EBSD scan rate [102] , and the EBSD step size/effective step size [95, 103] . For this work, the EBSD camera had a pixel resolution for 480 x 480 and no binning was employed, therefore, noise due to camera resolution cannot be reduced. Noise related to step size could not be reduced because this work sought to have a step size s mall enough as to not skip dipole dislocations. That leaves pattern quality as the main source for noise reduction. Other than exceptional sample preparation, good pattern quality is primarily achieved by increasing the exposure time of the camera or inc reasing the beam current . However, increasing the beam current will increase the spot size which will increase the interaction volume of the beam. In this work it was desired to maintain the spatial resolution, therefore the beam current was not increase d. The exposure time used in this work to achieve EBSD patterns was 0.1 s . While not all researchers state what exposure was used to collect EBSD patterns, some report 1 s or more [102, 104] . Britton et al. [102] achieved the best pixel intensity r esolution , 0.02 pixels, with an exposure time of 5 s . In comparison, the exposure time used here is extremely fast. However, due to drift inside the SEM, 0.1 s was the longest exposure time that could be used as to not have distortions in the EBSD map. Drift inside an SEM can be caused by a number of factors. A few possibilities include mechanical stage drift from a worn -out stage , beam drift due to a poorly grounded sample or stage, beam drift from charging particles inside the column, and other problem s that can be related to the electromagnetic lenses or column of the beam. Mechanical stage drift is usually enlarged by tilting to 70 ¡ for EBSD because the weight of the stage will cause to it move down !113!in the direction of the tilt. Beam drift is also enhanced for EBSD in the directional component that is perpendicular to the beam and rotational axis for the 70 ¡ tilt. For example, a beam shift of 1 µm shift on a flat surface would obviously be 1 µm. But due to the geometry of EBSD this would cause a shift of 1.58 µm, from an EBSD perspective. 5.3.5 !Advantages/Disadvantages of ECCI ECCI is advantageous because it allows for the direct imaging of defects. In contrast to CC-EBSD , it able to resolve individual defects and quantify low dislocation densities. By taking images at multiple channeling conditions, defects can be characterized using g ¥ b = 0 and/or g ¥ b x u= 0 analysis. On the other hand, taking images at multiple channeling conditions to characterize the defects however, can be time consuming. Y et, even if characterization is not desired and only dislocation density is wanted, ECCI still may not reveal all dislocations in a single image, still requiring multip le images. A nother disadvantage of ECCI technique is that it is difficult to quantify h igh dislocation densities because individual dislocations are difficult to resolve at high densities. Also, few microscope s have a rocking beam function to allow for the collection of SACPs which would make characterizing defects difficult. 5.3.6 !Advantages/Dis advantages of CC -EBSD In comparison to ECCI, CC -EBSD is much faster due to the fact that much of the data collection and analysis is automated. This can allow for scans of large areas where it would not be plausible to use ECCI because ECCI would require many high magnification images to cover the same area . CC-EBSD can also quantify areas of high dislocation density that cannot be reso lved with ECCI. !114!However, the ability of CC -EBSD to accurately quantify the density is related to the noise , which can com e from binning/resolution of the EBSD camera [101] , pattern quality due to EBSD scan rate [102] , and the EBSD step size/effective step size [95, 103] . Using pattern quality to overcome noise has to be balance d with the possibility of sample/beam drift. But the main disadvantage of CC -EBSD relate s to its inability to image/characterize individual defects. For example, it is likely to miss closely spaced dislocation dipoles (ie. GNDs vs. SSDs). Also, the noise may limit its ability to quanti fy low dislocati on densities. A less thought of disadvantage of CC -EBSD is that saving patterns can generate a lot of data. This would require research facilities where CC -EBSD is carried out regularly to maintain large data repositories. As an example of size, some EBS D scans used in this work collected 10,000 patterns. These patterns were 480 x 480 pixels and each image was saved as 16 bit TIFF file in order to maintain good intensity resolution of each pixel, making each image 461 KB. In total, the data size for a 1 0,000 image scan is 4.61 GB. But using an EBSD camera with 1000 x 1000 pixels, 10,000 images would take up 20 GB. With an exposure time of 0.1 s that was used in this work, it only took 16.7 minutes to generate this much data. Extrapolating this for a research facility that does CC -EBSD scan s regularly, it would only take about 14 hours of scan data to use up an entire 1 TB h ard -drive! 5.4 !ECCI/CC -EBSD Compared to AFM ECCI, CC -EBSD, and AFM measurements all reflect the symmetry of the crystal that nanoinde ntation was performed in , but the AFM topography measurements do not identify the location s of dislocations. Dislocations may or may not appear in the topographical lobes and they also appear where there are not any topographical lobes. To that point, it is important to realize that while topography that appears on the opposite side of a grain boundary from !115!nanoindentation is indicative of deformation transfer, the lack of topography does not mean there is a lack of deformation transfer. This is seen in the data from Figure 4.12 and Figure 4.13. The AFM map in Figure 4.12b shows that the majority of topography that is across the grain boundary is to the right of the indent, in an area where this is not a concentration of dislocations. The concentration of dislocations, seen in Figure 4.13a, appears at the lower right of the indent across the grain boundary in an area where there is not much topography. The fact that topography across the grain boundary does not have an exact correlation with deformation transfer brings up two points. First, AFM only detects slip that has components that are out of the sample surface plane. Any deformation that occurs parallel to the surface plane may not be seen with AFM . The topographical lobes are made from many slip steps that are formed from dislocations exiting the sample surface [33] . This leads to the second point , that topography is evidence of dislocations that have already exited the crys tal and ECCI/CC -EBSD assess the dislocations that are still in the crystal. Therefore, both techniques of AFM and ECCI/CC -EBSD should be used to seek an understanding of the past and current deformation processes taking place in a material. 5.5 ! Analysis of W edge Indent Cross -Section The dislocations generated under the wedge indent cannot be characterized due to the inability to obtain quantitative channeling patterns from the small areas. Assuming most are screw dislocations, however, some are able to be ch aracterized using the data from EBSD to form a stereographic projection. Regardless of how many dislocations can be characterized, the amount of GNDs versus SSDs can be reasoned from the EBSD. Due to the fact that the bands keep alternating between the sa me two general lattice or ientations, overall there is no net lattice curvature across a given !116!band. This means , all of the dislocations in a given band can be thought of as SSDs since GNDs would require some net lattice curvature. In other word s, there w ill be an equal amount of positive and negative dislocations in the opposing boundaries on either side of the band In terms of what are the thin pointed bands observed in the BSE image ( Figure 4.17b and Figure 4.18a) and EBSD maps ( Figure 4.20 and Figure 4.22), the analysis from this work is that they are most likely twins. Twinning in bcc metals is said to occur on {112} [7, 105 Ð109], however, the thin band s could not be related to any {112} planes. Regardless, the thin bands resemble twins visually due to their thin needle like structure and quantitatively show significant and immediate change in orientation across a boundary of abo ut 7.3 ¡ and 10.2 ¡. !117!6!Conclusion s The resulting topographies from indentation show that topography is a function of crystal orientation. While CPFEM also shows the same symmetry for indent topography for some indents, it does not for all. For CPFEM to be fully utilized as tool of understanding nanoindentation in bcc Ta the structure evolution parameters need to be optimized and the non -Schmid effects need to be properly implemented into the phenomenological model. This work has demonstrated that individu al grain boundaries have different response to strain depending on the side of the grain boundary from which the strain is ap proaching. To that point, using a single mÕ value to represent the susceptibility of a grain boundary to accommodating deformation is inadequate. With the three different grain boundaries that were targeted , this worked showed four different combinations of deformation response at grain boundaries. The y include: (1) significant transfer and significant resistance, (2) significant transfer and little to no resistance, (3) little to no transfer and significant resistance, or (4) little to no transfer and little to no resistance. ECCI and CC -EBSD reveal very similar dislocation distributions associated with nanoindentation deformation. While there is not a one -to-one correlation between maps from these two techniques, the dislocation densities measured by ECCI are generally similar to those determined by CC -EBSD. The discrepancies between the two techniques may be in part due to infer ior spatial resolution of CC -EBSD, allowing for CC -EBSD to miss dipole arrangements, and the potential for ECCI to miss dislocations that are either under invisibility conditions or are in areas that have too many dislocations to image. Despite these mino r discrepancies, the strong correlation in distributions, densities, and characterization of dislocations determined by the two techniques suggest that CC -EBSD can be used with confidenc e for characterizing GND !118!struc tures with higher dislocation densi ties than those that can be im aged using ECCI. At the other extreme, this work suggests that CC - EBSD has the potential to resolve i ndividual dislocations , but can not do so at this time with high confidence in deformed metallic materials. This work also shows that combination of nanoindentation, AFM, ECC I, and CC -EBSD is a great method for investigating deformation at grain boundaries. Nanoindentation is a good tool for probing specific areas of material, such as grain boundaries, so that the directionality of strain can be easily known . AFM shows evidence of dislocations that have already exited the crystal with an out -of-plane component , while ECCI and CC -EBSD show dislocations that are still in the cry stal. Of course, nanoindentation, AFM, ECCI, and CC -EBS D all have limitations and these limitations need to be understood in order to fully utilize these methods. This work discovered thin needle like bands that developed by way of plane strain under wedge indentation. These thin bands resemble twins both vis ually due to their thin needle like feature s and quantitatively due to their high amount of disorientation across the boundary. Drawing a line from the tip of where the indent was made divides the EBSD data set in half. The thin bands on the left side ro tated the original crystal orientation counter clockwise and the thin bands on the right rotated the crystal clockwise. Due to the fact that the alternating banding leaves no residual lattice curvature, it is concluded all dislocations within a given band are SSDs. 6.1 !Suggestions for Future Research It would be advantageous to know what dislocation are involved in the development of topographical lobes that result from indentation. Is it primarily edge dislocations or screw dislocations? To do this AFM, ECC I, and CC -EBSD can be performed on nanoindents that were made with in the same large grain at varying loads. The first indentation to be analyzed would be an indentation that is stopped right after the initial pop -in, i .e. where plasticity begins. !119!Another worthwhile study would be doing AFM and CC-EBSD on a large number of indentations to see which grain boundaries exhibit deformation and which side of the grain boundary dislocation pile -ups are found. Many places in the literature show that dislocations are found in the parent grain and examples in this work show dislocation pile -ups in the neighboring grain. Future work should be done to identify the dislocations generated between the thin bands formed form wedge indentation. This can be done using the previously mentioned Zeiss Auriga SEM because it has the capabilities to obtain channeling patterns form and area of ,500 nm [68] . 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