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Atomistic modeling of the mechanical properties of metallic glasses, nanoglasses, and their composites

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Atomistic modeling of the mechanical properties of metallic glasses, nanoglasses, and their composites

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Content ATOMISTIC MODELING OF THE MECHANICAL PROPERTIES OF METALLIC GLASSES, NANOGLASSES, AND THEIR COMPOSITES by Suyue Yuan A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY MATERIALS SCIENCE May 2023 ii Acknowledgements I would like to express my deepest gratitude to my advisor, Prof. Paulo S. Branicio, for his invaluable guidance, encouragement, and support throughout my six-year PhD journey. I feel incredibly fortunate to have had such a wonderful advisor. His expertise in research taught me essential skills to be a scientist, while his kind-heartedness and resilience taught me how to be an excellent collaborator and persist in the face of challenges. I will always cherish the countless hours Paulo spent with me, tackling tedious problems such as compiling software on HPC, fixing code bugs, revising manuscripts line by line, and submitting papers to online journal systems. Though seemingly small, these problems were time-consuming and required a lot of patience. He also took every opportunity to teach me proper academic etiquette when we attended conferences and met with other researchers. I cannot thank him enough for his tremendous support on these trivial things. These experiences were priceless and played a crucial role in both my academic and personal development. Their significance cannot be quantified, but their impact can certainly be felt in the depths of my heart. Paulo is also a caring elder friend, as when we celebrated successes like publishing new work, winning awards, or just passing exams, by going out for drinks or food, he always ensured our safe return home. Even when he himself was exhausted, he would drive us home whenever possible. I still cherish the time he invited us to his home for an authentic Brazilian BBQ and introduced us to his lovely family and extraordinary dog, Snoopy. Paulo's modest and caring personality has taught me humility and responsibility in my personal and professional life. I admire his hard work, dedication, and passion in research and will always regard him as a role model. Once again, I would like to thank Prof. Branicio for his exceptional support, guidance, and kindness throughout my PhD journey. iii My heartfelt appreciation goes to my collaborators, Chang Liu and Aoyan Liang, for their invaluable support and contributions that significantly enhanced my research experience and contributed to the success of my work. I am particularly grateful to Chang for his selfless help in both my course work and research. It has been a privilege to work with such a dedicated and cooperative colleague. To my junior colleagues, I wish you all the best in your future endeavors. I extend my gratitude to the esteemed members of my thesis committee: Prof. Priya Vashishta, Prof. Aiichiro Nakano, and Prof. Shaama Sharada. Their guidance and feedback were invaluable throughout the development of my work. Their insightful critiques and constructive comments played a vital role in refining my research and presenting it in the best possible light. I am especially grateful to Prof. Nakano, who not only provided exceptional support during the classes I took with him but also throughout my postdoctoral application process. In addition, I would like to express my appreciation to many professors and colleagues at VHE, including Prof. Branicio, Prof. Vashishta, Prof. Sharada, Prof. Ravichandran, Dr. Nomura, Prof. Kalia, Patricia Wong, and other graduate students. Their friendly demeanor and warm smiles never failed to brighten my days at MFD, and their kindness is truly appreciated. Finally, I would like to thank my families, they have been my rock, always encouraging me to pursue my dreams. My high-school teachers and undergraduate professors deserve my deepest appreciation for shaping my academic interests, which I will carry with me throughout my life. I would also like to express my gratitude to my friends who stood by me during the ups and downs of my academic journey. Once again, thank you all from the bottom of my heart. I hope that you all enjoy good health and success in your own endeavors. iv Table of Contents Acknowledgements ......................................................................................................................... ii List of Figures ............................................................................................................................... vii Abstract .......................................................................................................................................... xi Chapter 1: Introduction ................................................................................................................... 1 1.1 Metallic Glasses .................................................................................................................... 1 1.1.1 Bulk Metallic Glasses .................................................................................................... 1 1.1.2 Nanoglasses .................................................................................................................... 3 1.1.3 Metallic Glass Composites ............................................................................................ 4 Chapter 2: Methodology ................................................................................................................. 6 2.1 Molecular Dynamics Simulations ......................................................................................... 6 2.1.1 Classical Large-Scale Molecular Dynamics Simulations .............................................. 6 2.1.2 Embedded Atom Method ............................................................................................... 8 2.1.3 Statistical Ensembles ..................................................................................................... 8 2.1.4 Periodic Boundary Conditions ..................................................................................... 10 2.1.5 Large-scale Atomic/Molecular Massively Parallel Simulator ..................................... 10 2.2 Visualization and Analysis Tools ....................................................................................... 10 2.3 Machine Learning in Materials Science ............................................................................. 11 Chapter 3: Gradient Nanoglasses .................................................................................................. 13 3.1 Introduction ......................................................................................................................... 13 3.2 Simulation Details ............................................................................................................... 14 3.2.1 Tensile Loading of Metallic Glass with Thin-film Geometry ..................................... 14 3.2.2 Nanoindentation of Metallic Glass with Cubic Geometry ........................................... 16 3.3 Simulation Results .............................................................................................................. 19 3.3.1 Tensile Loading Measurements ................................................................................... 19 3.3.2 Deformation Profiles During Tensile loading .............................................................. 20 3.3.3 Grain Sizes, Atomic Volumes, and Local Plasticity Distributions .............................. 23 3.3.4 Nanoindentation Measurements ................................................................................... 25 3.3.5 Local Stress and Strain Profiles under Nanoindentation ............................................. 27 3.3.6 Quantifications of the Deformation Mechanism .......................................................... 31 3.3.7 Analysis of Local Plastic Events .................................................................................. 33 3.4 Discussion ........................................................................................................................... 36 3.4.1 Comparison between Gradient Nanoglass and Other Heterogeneous Metallic Glass . 36 3.4.2 Comparison between Gradient Nanoglass and Gradient Nanocrystals ....................... 38 3.4.3 Effects of Grain Size and Gradient Microstructure in NGs and GNGs ....................... 39 3.4.4 Deformation Mechanisms in NG and GNG structures ................................................ 40 3.4.5 Experimental Synthesis of Gradient Nanoglasses and Comparisons with Simulations ............................................................................................................................................... 42 3.4.6 Potential Applications of Gradient Nanoglasses .......................................................... 44 3.5 Conclusion .......................................................................................................................... 44 v Chapter 4: Nanoglass-Metallic Glass Composites ........................................................................ 46 4.1 Introduction ......................................................................................................................... 46 4.2 Simulation Details ............................................................................................................... 47 4.3 Simulation Results .............................................................................................................. 48 4.3.1 Engineering Stress-strain Curves ................................................................................. 48 4.3.2 Deformation Profiles of Different Metallic Glass-Nanoglass Composites .................. 49 4.3.3 Local Strain Evolutions ................................................................................................ 50 4.4 Discussion ........................................................................................................................... 52 4.4.1 Comparisons between Metallic Glass-Nanoglass Composites and other Metallic Glass Composites ............................................................................................................................ 52 4.4.2 Potential Applications of Metallic Glass-Nanoglass Composites ............................... 54 4.5 Conclusion .......................................................................................................................... 54 Chapter 5: Shape Memory Alloy-Metallic Glass Composites ...................................................... 56 5.1 Introduction ......................................................................................................................... 56 5.2 Simulation Details ............................................................................................................... 57 5.3 Simulation Results .............................................................................................................. 58 5.3.1 Deformation Profiles .................................................................................................... 58 5.3.2 Stress-strain Curves of the Composites and Corresponding Phases ............................ 60 5.3.3 Local Stress Analysis of the Composites during Deformation .................................... 60 5.3.4 Phase Transformations for the Shape Memory Phase ................................................. 61 5.4 Discussion ........................................................................................................................... 64 5.4.1 Comparisons between the Stacked and Staggered Bulk Metallic Glass Composites .. 64 5.4.2 Comparisons among Bulk Metallic Glass Composites with Different Microstructures ............................................................................................................................................... 64 5.5 Conclusion .......................................................................................................................... 65 Chapter 6: Uncovering Metallic Glasses Hidden Vacancy-like Motifs ........................................ 67 6.1 Introduction ......................................................................................................................... 67 6.2 Simulations and Methods .................................................................................................... 69 6.2.1 Molecular Dynamics Simulations ................................................................................ 69 6.2.2 Voronoi Tessellation .................................................................................................... 70 6.2.3 Motif Naming ............................................................................................................... 70 6.2.4 Machine Learning Modeling ........................................................................................ 71 6.3 Simulation Results .............................................................................................................. 73 6.3.1 Identification of Vacancy-like Motifs .......................................................................... 73 6.3.2 Thermal Response of T5 and Q7 motifs ...................................................................... 75 6.3.3 Mechanical Response of T5 and Q7 Motifs under Loading ........................................ 79 6.4 Discussion ........................................................................................................................... 82 6.4.1 Rationales for Considering Q7 as Vacancy-like Defects ............................................. 82 6.4.2 Energetic Implication of T5/Q7 Formation ................................................................. 83 6.4.3 Limitations of This Work ............................................................................................. 85 vi 6.5 Conclusions ......................................................................................................................... 86 Chapter 7: The Effect of Heat Treatment Paths on the Aging and Rejuvenation of Metallic Glasses .......................................................................................................................................... 87 7.1 Introduction ......................................................................................................................... 87 7.2 Simulation Details ............................................................................................................... 88 7.3 Results and Discussion ....................................................................................................... 89 7.3.1 Effects of Thermal Histories on Aging and Rejuvenation ........................................... 89 7.3.2 Structural Evolution of Metallic Glasses Treated with Different Heating and Cooling Rates ...................................................................................................................................... 91 7.3.3 Flow Units Activation and Annihilation along the Heat Treatment Paths .................. 94 7.4 Conclusion .......................................................................................................................... 98 Chapter 8: Conclusion ................................................................................................................... 99 References ................................................................................................................................... 100 vii List of Figures Fig. 3.1 Illustrations of the atomic models for metallic glasses (MG) and gradient nanoglasses (GNG). (a) Homogeneous Cu64Zr36 MG, (b) GNG for tensile loading parallel to the gradient direction (p-GNG), (c) hard-core GNG for tensile loading perpendicular to the gradient direction (hc-GNG), (d) soft-core GNG for tensile loading perpendicular to the gradient direction (sc- GNG). Colors in (a) indicate atomic species while colors in (b-d) indicate different grains. Tensile loading is applied along the indicated z-direction. _____________________________ 16 Fig. 3.2 Nanoindentation simulation models. (a) Spherical indenter with a radius of 10 nm; (b) homogeneous Cu64Zr36 metallic glass (MG); (c) nanoglass (NG) with an average grain size of 7 nm (7nm-NG); (d) NG with an average grain size of 3 nm (3nm-NG); (e) Gradient NG (GNG) with a gradient grain size from 7 nm at the top to 3 nm at the bottom (h-GNG); (f) GNG with a gradient grain size from 3 nm at the top to 7 nm at the bottom (s-GNG). Colors in (b) indicate atomic species while colors in (c-f) indicate different grains. Nanoindentation is applied along the indicated z-direction. _______________________________________________________ 18 Fig. 3.3 Engineering stress-strain curves for the MG, homogeneous NG, and GNG samples during tensile loading test. Open symbols correspond to the snapshots of the deformed GNG samples depicted in Fig. 3.4. ___________________________________________________ 20 Fig. 3.4 Snapshots of the deformation process for the three investigated GNG samples. (a) p- GNG under tensile loading parallel to the gradient direction. (b, c) hc- and sc-GNG under tensile loading perpendicular to the gradient direction. Atoms colors indicate the von Mises shear strain, eM, within the range of values defined in the color bar. For clarity, only atoms with eM > 0.2 are displayed. Tensile engineering strain values for each one of the snapshots in (a to c) are indicated in Fig. 3.2. __________________________________________________________________ 22 Fig. 3.5 Grain size gradient and local Voronoi atomic volume profiles and distribution of local atomic shear strain during deformation. (a) Grain size averages along the gradient direction in p- and sc-GNG samples, in black and red points, respectively, and black dashed guided line. Also, for the hc-GNG sample, in blue points and blue dashed line. (b) Configuration of the hc-GNG before deformation with atom colors indicating the local Voronoi atomic volume. (c) Fraction of atoms with eM > 0.2 along the gradient direction at two strain values with data points color following (a). Guided lines are provided for each curve. (d) Normalized fraction of atoms with eM > 0.2 along the gradient direction at two strain values, following the color choices of (c). _ 23 Fig. 3.6 Mechanical response of the MG, NG, and GNG models under nanoindentation. (a) Load-displacement curves; (b) contact pressure-displacement curves. Calculated values of reduced elastic modulus, E*, and hardness, H, are shown for each model. ________________ 27 viii Fig. 3.7 Local atomic von Mises shear strain for the MG, NG, and GNG models at different indentation depths, d. (a-e) Local shear strain profiles at d = 2 nm; (f-j) local shear strain profiles at d = 10 nm. The views are taken from a thin slab of 1 nm thickness centered in the indentation region of each sample. ________________________________________________________ 28 Fig. 3.8 Local atomic stress for the MG, NG, and GNG models at different indentation depths, d. Local stress profiles at d = 2 nm, (a-e) and d = 10 nm (f-j). The views are taken as in Fig. 3.7. ___________________________________________________________________________ 30 Fig. 3.9 Strain localization and plastic deformation during nanoindentation of the samples. (a) degree of strain localization for the MG, NG, and GNG models at different indentation depths, d; (b) local strain statistics for different models at d = 10 nm. The inset shows the statistics at large local strain values. _______________________________________________________ 32 Fig. 3.10 Non-affine squared displacement (NASD) for the MG, NG, and GNG models at different indentation depths, d. Local stress profiles at d = 2 nm, (a-e) and d = 10 nm (f-j). The views are taken as in Fig. 3.7. ___________________________________________________ 34 Fig. 3.11 Complementary cumulative distribution of cluster sizes. (a) distribution at d = 2 nm and (b) at d = 10 nm. The arrows indicate the gap in cluster sizes, Nc, when a relatively small cluster transits into a large cluster. _______________________________________________ 35 Fig. 4.1 Illustrations of the atomic models for NG and MG-NG composites. (a) Cu64Zr36 NG with 3 nm grain size (3nm-NG). (b) MG-NG composite with Cu64Zr36 MG bricks arranged in a stacked way (stk-C). (c) MG-NG composite with MG bricks arranged in a staggered way (stg- C). Colors indicate different grains in the NG matrix. In (b) and (c) the MG second phase is colored in dark blue. __________________________________________________________ 47 Fig. 4.2 Tensile loading engineering stress-strain curves for the 3nm-NG and MG-NG composite models with different fractions of the MG second phase. The left inset highlights the improvement in the strength of the composites with 30% MG phase. The right inset highlights the difference between the stk-C and stg-C curves at 35% MG phase. ___________________ 49 Fig. 4.3 Deformation profiles of the MG-NG composite models during tensile loading. (a)-(c) Profiles for the MG-NG composites in the stk-C design at different fractions of the MG second phase; (d)-(f) corresponding profiles for the composites with the stg-C design. Atoms are colored according to their von Mises local strain, eM, values. Atoms with eM < 0.2 are not displayed for clarity. _____________________________________________________________________ 50 Fig. 4.4 Evolution of the local strain statistics for different composite designs. ____________ 51 Fig. 5.1 Deformation profiles of the bulk metallic glass composite (BMGC) models during tensile loading. The stacked BMGC (stk-BMGC) model is displayed in (a), and the staggered ix BMGC (stg-BMGC) model is displayed in (b). Atoms are colored based on their von Mises local strain value. _________________________________________________________________ 58 Fig. 5.2 Tensile loading stress-strain curves. (a) Stress-strain curves for the pure MG and B2 phases. (b) Stress-strain curves for the stk-BMGC and the stg-BMGC models. Curves for the subset of atoms in the crystalline bricks and glassy matrix are displayed in dash lines. ______ 59 Fig. 5.3 Local stress distribution of the BMGC models during tensile loading. The microstructure evolution of the stk-BMGC model is displayed in (a), and of the stg-BMGC model is displayed in (b). Atoms are colored based on the local stress value. _____________________________ 62 Fig. 5.4 Local stress distribution and phase transformation for the shape memory phase in the BMGC models during tensile loading. Only one brick in each model is illustrated here. The microstructure evolution of the stk-BMGC model is displayed in (a), and of the stg-BMGC model is displayed in (b). Atoms are colored based on the value of the local stress in the first row and by the corresponding crystalline structure in the second row. ___________________ 63 Fig. 6.1 Machine learning results and illustrations of vacancy-like motifs candidates. The left panel shows the R 2 score for each motif, Sn. The right panel shows the schematics of Voronoi polyhedrons with five triangular faces (T5) and seven quadrangular faces (Q7), which are the motifs with the top two R 2 scores. The schematic of each motif includes three examples with corresponding Voronoi cells, atomic configurations, and Voronoi indices. Green and brown spheres represent Zr and Cu atoms, respectively. The Voronoi indices are denoted with <n3, n4, n5, n6, …> up to the last non-zero ni term, where ni is the number of i-edged faces of the polyhedron. _________________________________________________________________ 73 Fig. 6.2 (a) Potential energy, (b) FI concentration, (c) T5 concentration, and (d) Q7 concentration of CuxZr100-x samples during quenching. The colors indicate the Cu composition of the system. The black dashed lines in (c) and (d) are guides to the eye. ____________________________ 74 Fig. 6.3 Temperature dependence of FI, T5, and Q7 concentrations, as well as the potential energy in a Cu64Zr36 MG during heating. The blue dashed curve is a guide to the eye. The dark purple/black dashed curves are fitted to T5/Q7 data points for T < 1050 K using an Arrhenius function. The light purple/grey dashed lines are fitted to T5/Q7 data points for T > 1050 K using a linear function. _____________________________________________________________ 76 Fig. 6.4 (a)-(c) Local entropy distribution of Cu64Zr36 MG atoms in the whole system, in the FI configuration, in the T5 configuration, and in the Q7 configuration at 310 K, 810 K, and 1630 K (liquid state), respectively. (d)-(f) V oronoi atomic volume distribution and (g)-(i) V oronoi atomic pressure distribution of atoms in the above-mentioned configurations at corresponding temperatures, respectively. _____________________________________________________ 78 x Fig. 6.5 (a)-(c) Stress-strain curves and polyhedral concentration evolutions of Cu64Zr36 MG during uniaxial tensile loading at 50 K, and (d)-(f) compressive loading at 50 K. The black dashed lines in (a) and (d) imply the linear elastic response stage. The plumb grey dashed lines underline the yield point and the purple ones underline the failure point of the sample during deformation. ________________________________________________________________ 80 Fig. 6.6 (a) and (b) V oronoi atomic volume distribution of atoms in the whole system, in the FI configuration, in the T5 configuration, and in the Q7 configuration at ε = 0.06 tensile loading and at ε = 0.06 compressive loading, respectively. (c) and (d) V on Mises shear strain distribution of atoms in the above-mentioned configurations at ε = 0.06 tensile loading and at ε = 0.06 compressive loading, respectively. _______________________________________________ 81 Fig. 7.1 Box plots of (a)-(c) Potential energy variation (ΔPE) and (d)-(f) full icosahedra concentration variation (ΔFI) in MGs generated with different quenching rates, grouped by heating time, annealing time, and cooling time, respectively. __________________________ 90 Fig. 7.2 Atomic displacement statistics in MGs (a) heated at different rates and annealed at 850 K for different time periods, (b) cooled to 600 K at different rates after heated to 850 K at a rate of 10 12 K/s. _________________________________________________________________ 92 Fig. 7.3 Activation energy statistics of the as-cast MG and samples subjected to a heat treatment involving reheating to 850 K at a rate of 10 12 K/s, annealing for 100 ps, and subsequent cooling at different rates down to 1 K. __________________________________________________ 94 Fig. 7.4 (a)-(b) Potential energy (PE) and (c)&(d) FI concentration evolution in MGs along different heat treatment paths. (e) Illustration of the associated volumes of flow units involved in activation and annihilation processes along the heat treatment paths shown in (a)-(b). ____ 96 xi Abstract Metallic glasses (MGs) have been widely studied due to their remarkable properties, such as high yield strength, hardness, and wear resistance, making them attractive materials for various engineering applications. However, their tendency to shear localization and premature failure often limits their macroscopic ductility. To overcome this issue, we propose three novel approaches to achieve high strength and ductility in MGs by manipulating their microstructures. We use large- scale molecular dynamics (MD) simulations for heat treatment, tensile loading, compressive loading, and nanoindentation simulations. In addition, we employ machine learning (ML) methods to uncover hidden structure-property relationships in MGs at an atomic level. In the first approach, we introduce seamless gradient microstructures into Cu64Zr36 MGs by incorporating glassy nano grains with various sizes, referred to as gradient nanoglasses (GNGs). To evaluate the mechanical properties of these GNGs, we conduct uniaxial tensile loading and nanoindentation simulations on the prepared samples. The results of the tensile loading simulation demonstrate that plasticity in GNGs follows a grain size gradient, with local plasticity increasing from larger to smaller grain size regions. When loaded parallel to the gradient direction, the samples exhibit intensive necking followed by catastrophic failure. In contrast, when loaded perpendicular to the gradient direction, the samples exhibit diffused shear band propagation from large to small grain size regions, further delocalizing the plastic deformation and delaying the development of a dominant shear band. The nanoindentation results indicate that the deformation mechanisms in GNGs are rooted in the activation of shear transformation zones (STZs) at soft sites and their subsequent evolution into shear band embryos. Furthermore, an increase in grain size leads to enhanced elastic modulus and hardness but reduced abrasion resistance. Although the average grain size at the indentation surface plays a crucial role in the deformation behavior of xii GNG models during nanoindentation, the activation of shear transformation zones far away from the indentation surface has a notable impact on the materials' overall performance. These findings suggest new possibilities for applying heterogeneous gradient designs to MGs to achieve desired combinations of different mechanical properties and provide insights into the study of heterogeneous degradation in functionally graded materials. In the second approach, we propose MG composites that combine a nanoglass (NG) matrix and an MG second phase. The samples are prepared with a brick-and-mortar architecture, and phase fractions are varied. The results of the tensile loading simulation reveal that the global strength of the matrix is significantly improved by the second phase without sacrificing the large ductility of the NG phase. The mechanical synergy of the two phases is further optimized by arranging MG bricks in a staggered way, effectively delocalizing the plastic deformation, hindering the buildup of local stress hot spots, and the generation of critical shear bands. In the third approach, we design and characterize the tensile deformation and failure of shape memory alloy-bulk MG composites (BMGCs) with a brick-and-mortar design. We identify contrasting behaviors when arranging two phases in different ways. The results suggest that composites with crystalline bricks arranged in a staggered way exhibit significantly improved mechanical properties compared to cases where they are arranged in a stacked way. This composite architecture synergizes the mechanical properties of the glassy MG matrix and the crystalline second phase to a great extent by introducing an exceptional strain hardening stage through efficiently delocalizing the local stress and local strain. In addition to studying the unique mechanical behavior of MGs in various architectural designs, we also investigate the structure-property relationships in MGs at an atomic scale. In this work, we employ machine learning (ML) to reveal a previously unknown vacancy-like structural motif xiii in MGs, named Q7, which refers to atoms enclosed in atomic Voronoi polyhedrons with seven quadrangular faces. The Q7 motif is found to significantly contribute to the short-range structural disorder in MGs, with its concentration following an Arrhenius-like relationship with temperature, thereby providing a precise indicator of the glass transition temperature point. The population of Q7-centered atoms shows strong correlations with the yield and failure of MGs during mechanical deformation. Moreover, Q7-centered atoms exhibit larger local entropy, atomic volume, and local tension, indicating their association with vacancy-like configurations. These results offer new insights into the interpretation of local disorder and unfavorable topological descriptors in MG atomic structures and point out a promising avenue for constructing structure-property relationships in MGs through combining MD simulations and ML. Lastly, we investigate the structural dynamics of MGs under various heat treatment paths. In our study, we prepared MGs with different initial structures and subject them to different thermal histories. We find that the heating and cooling rates have contrasting effects in the structural relaxation of MGs. When MGs are annealed for longer durations, they can either age or rejuvenate, depending on the thermal history. Moreover, we observed that the structural relaxation of MGs can be explained by considering the fast dynamics and memory effect of flow units in MGs. These flow units are responsible for structural evolutions in MGs and their behavior can be altered by thermal history. During the relaxation, these flow units reorganize themselves and can lead to aging or rejuvenation. Our work suggests that by carefully controlling the thermal history, it is possible to tailor the properties of MGs to meet specific requirements in various applications. As computing power continues to increase, simulation packages become more sophisticated, and ML techniques are applied more wisely, we anticipate a promising future for gaining further understanding of MG behaviors and designing them with optimized properties. 1 Chapter 1: Introduction 1.1 Metallic Glasses 1.1.1 Bulk Metallic Glasses New discoveries and novel applications of glass continue to emerge despite it being one of the oldest artificial material families. Traditional transparent silicates were commonly used for glass applications before 1960, however, MG, also known as amorphous alloys, is a recent addition. In 1959, Klement et al. performed pioneering work on the vitrification of an AuSi alloy[1]. Turnbull et al. later discovered that MGs exhibit glass transition similar to conventional glasses[2–4]. Since then, interest in understanding this new glass has rapidly increased, leading to the development of diverse MG systems and fabrication techniques[5–15]. MGs have unique and intriguing mechanical, chemical, and physical properties due to their amorphous disordered atomic structure, unlike crystalline metals and alloys that have short and long-range orders[12,16–19]. Most MGs exhibit strengths and elastic limits much higher than those of conventional crystalline alloys and other engineering materials[20–22]. Additionally, they have good wear and corrosion resistance at room temperature due to the absence of dislocation defects in their disordered-lattice structure[13,23–25]. The development of glass-forming theory and solid-state amorphization techniques enabled successful commercial production of MGs in various forms such as bulk, thin-film, powder, and ribbon[6,11–13,15,26,27]. Since then, MGs have gained significant research interest and been widely applied in various fields such as soft magnetic materials and catalytic materials for Fe-based MGs[9,28], solder for Ni-based MGs[29– 31], and magnetic sensors for Co-based MGs[32]. MGs also have potentials for green energy applications, retention, and purification of dangerous pollutants, and in the nuclear industry[13,33– 35]. 2 The first bulk MG (BMG) was prepared by Chen in 1974 from a Pd-Cu-Si ternary system[12,33,35]. BMGs are known for their exceptional mechanical properties. For example, their strength is approximately twice that of conventional steels, and their elastic limit is about ten times higher[13,15,36]. Due to stable viscous flows in their wide supercooled-liquid temperature regions, e.g., up to 120 K for Zr-based BMGs, they can be processed into near-net shapes with minimal shrinkage during the liquid-to-glass transition[37]. Consequently, BMGs are injection- modulable, lack grain boundaries, and exhibit greater strength than conventional alloys, allowing them to be formed into features 100 times smaller than conventional metals[37,38]. BMGs samples and parts also have smooth, shiny surfaces with atomic-level roughness and high corrosion resistance [13,15,36]. These features make them suitable for a wide range of engineering applications, such as sporting goods (golf drivers, tennis rackets, bows), biomedical implants and devices (knee replacement, scalpel blades)[23,39], defense, aerospace, and energy applications (gears, armor coating, neutron shielding)[10,40–42]. However, a major drawback of BMGs is their tendency to fail catastrophically due to the generation and propagation of localized shear bands (SBs) during plastic deformation at ambient temperatures, resulting in relatively lower ductility and plasticity compared to their crystalline counterparts [15,33,35,36]. Despite the remarkable properties and successful applications of MGs, their understanding remains incomplete. This is because, unlike crystalline materials, MGs lack a well-defined long- range order, making it challenging to correlate their macroscopic properties with microscopic structures. Thus, one of the primary challenges is to uncover a discernible atomic packing feature in MGs. While long-range order is absent in MGs, distinct local atomic configurations at the level of nearest neighbors, known as short-range order (SRO), have been widely recognized[43–48]. Previous research has indicated that both favorable SRO packings and geometrically unfavored 3 motifs (GUMs) in MGs are crucial for their response to external stimuli, such as thermal agitation, heterogeneous elastic deformation, and the onset of plastic events[49–51]. These GUMs can be seen as the equivalent of defects in crystalline structures, however, a well-defined and ubiquitous disordered-structure descriptor for MGs is yet to be found. 1.1.2 Nanoglasses As motivated by improving the mechanical properties of MGs, the concept of nanoglass (NG) was first brought up by Gleiter and collaborators in 1989[52]. In the hope of introducing planar defects in MGs akin to the grain boundaries in nanocrystalline materials, a dense solid with glass- glass interfaces referred as NG can be obtained by consolidating nanometer-sized MG particles under high pressure. In 1991, Gleiter synthesized the first NG with spherical glassy nanoparticles of sizes ranging from 1 nm to 10 nm, which is achieved through inert gas condensation followed by a compaction at high pressures of up to 2 GPa[53]. The resulting NGs display microstructures and mechanical behaviors distinct from their MGs parents and have received increasing attention due to their great potential for functional applications. The unique properties of NGs mainly come from their grain boundary-like interfaces between bulk glassy regions These glass-glass interfaces are characterized by excess Voronoi atomic volume and a lack of short-range order, which have been proved to be preferred channels for plastic flow, hindering the development of localized SBs[54–56]. As a result, NGs usually display enhanced ductility and plasticity relative to monolithic MGs. In addition, the ductility of NGs has been reported to increase with decreasing grain size, correspondingly sacrificing the material strength. Different techniques are employed in order to reach a compromise between strength and ductility in an NG. One such approach is to use a microstructure with bimodal grain size distribution, combining glassy grains of 5 nm and 15 nm[57]. Another strategy is to synthesize NG 4 nanolaminate composites, which combine layers of NG with various grain sizes and BMG. Such a composite architecture results in a mild improvement in overall strength while maintaining the large ductility observed in NGs with small grain sizes[58]. Another novel design that may synergize the properties of NG of different length scales involves a microstructure with a seamless gradient variation of grain sizes. Compared to employing a random mix of different grain sizes, the use of gradient grain size microstructures in crystalline metals has been demonstrated to improve materials properties remarkably, achieving an outstanding combination of strength, hardness, and ductility that defies the expected trade-off between strength and ductility with variations in the average grain sizes[59]. 1.1.3 Metallic Glass Composites Bulk metallic glass composites (BMGCs) featuring a ductile second phase have been widely considered as alternatives to monolithic MGs because of their ability to compromise the high strength and ductility of difference phases[60]. These materials usually combine the unique properties of MGs with those of other materials, such as ceramics, crystalline alloy, or polymers. The resulting composite can have properties that are superior to those of the individual components. The design and characterization of BMGCs have been a topic of research in materials science and engineering. Researchers have investigated various fabrication techniques, such as melt infiltration[60–62], powder metallurgy[5,63], and electrodeposition[29], etc., to synthesize these composites. They have also studied their microstructure and mechanical properties to understand their behavior under different conditions. Recent studies have reported BMGCs composed of an MG matrix and a second phase, such as crystalline counterparts of the parent MG[64–67], other crystalline materials[68–71], and other amorphous materials with different compositions[72,73]. Besides the choice of the second phase, 5 microstructure design is another critical factor affecting the mechanical performance of BMGCs. With the development of the freeze-casting technique[74], the second phase may take various forms, such as spherical particles, rectangular inclusions, and dendritic precipitates developed naturally[60,61,63]. Meanwhile, the second phase may be distributed randomly or regularly in the matrix[69,70,75–77]. BMGCs may yet be produced as multilayered or nanolaminate films[78–80], and sandwich structures[81,82]. By tuning the architecture parameters such as the volume fraction and size of the second phase, the mechanical properties of the composites are expected to be further optimized. The development of BMGCs offers a promising path for the design of new advanced materials with tailored properties for specific applications. With ongoing research in this field, BMGCs are expected to find widespread use in various industries in the near future [41,76,77,83–87]. 6 Chapter 2: Methodology 2.1 Molecular Dynamics Simulations 2.1.1 Classical Large-Scale Molecular Dynamics Simulations Due to the immense number of particles that typically make up molecular systems, it is impossible to determine their properties analytically. To surmount this obstacle, molecular dynamics (MD) simulations utilizing numerical methods were developed in the early 1950s and have been demonstrated to be a powerful tool in the application of materials science, condensed matter physics, chemical physics, and biological research. It shows remarkable ability of predicting material properties with significantly reduced computational complexity[88–95]. In the meanwhile, MD simulations can be mathematically ill-conditioned and prone to accumulate errors in numerical integration, hence, it is essential to select algorithms and parameters properly to diminish these errors. Nonetheless, with growing experimental data available and advanced computing power, the accuracy of MD predictions has been continuously improved. The general mechanism of MD is governed by Newton’s equation under a given interatomic force field, the motion of atoms and molecules in the simulation can be predicted, in the meanwhile, various macroscopic and microscopic information can be extracted, serving as an effective tool to evaluate equilibrium and non-equilibrium materials properties[88,93,96]. The time evolution of classical MD simulations is usually computed by discretely integrating the equations of motion using a standard timestep of a few femtoseconds[89]. A typical algorithm for MD simulations of an equilibrium state in the microcanonical ensemble can be simplified as the following steps: 1. Define the initial positions 𝑟 ⃗ ! =(𝑥 " ,𝑦 " ,𝑧 " ,…,𝑥 ! ,𝑦 ! ,𝑧 ! ) and velocities 𝑣 ⃗ ! = (𝑣 #" ,𝑣 $" ,𝑣 %" ,…,𝑣 #! ,𝑣 $! ,𝑣 %! ) of a system consisting of N particles. 7 2. Calculate the force 𝐹 & on each particle from the total interatomic potential energy 𝑉 = 𝑉(𝑟 " ,𝑟 ' ,…,𝑟 ! ), 𝐹 ( / / ⃗ =− )*(, ! ,, " ,…,, # ) ), $ . 3. Calculate the acceleration on each particle at each time step, 𝑎 ( / / / ⃗(𝑡)= 1 % 222⃗(4) 5 $ , where 𝑚 & is the mass of i th particle. 4. Calculate the updated position and velocity of each particle after a given time t based on the calculated acceleration. The velocity-Verlet integration method is commonly used in the integration of Newton’s equation[97], 𝑎 ( / / / ⃗(𝑡) = 6 % 222⃗(47∆4/'):6 % 222⃗(4:∆4/') ∆4 , 𝑣 ( / / / ⃗(𝑡)= , % 222⃗(47∆4):6 % 222⃗(4) ∆4 , where 𝛥𝑡 is a small time interval. 5. Repeat the previous steps, one can update the positions and velocities of a system of N particles until thermodynamic equilibrium is reached in order to calculate thermodynamic average quantities. For systems that obey the ergodic hypothesis, the evolution of one MD simulation may be used to determine macroscopic thermodynamic properties of the system: the time averages of an ergodic system correspond to microcanonical ensemble averages. MD has also been termed "statistical mechanics by numbers" and "Laplace's vision of Newtonian mechanics" of predicting the future by animating nature's forces and allowing insight into molecular motion on an atomic scale. When designing MD simulations, there are a variety of factors to consider. For example, the timestep must be chosen small enough to avoid discretization errors; the simulation duration should match the kinetics of the natural process; the system size should be large enough for drawing statistically valid conclusions; and the computing should be archivable within a reasonable timeframe with available computing power. 8 2.1.2 Embedded Atom Method Though the interatomic forces can be calculated accurately using ab initio calculations, such as using the Density Functional Theory[98–100], simulations involving large number of atoms or long dynamic processes become computationally intractable requiring approximate empirical force fields[101–103]. To model metals, the large computational time can be saved by the application of semi-empirical many-body potentials, such as the embedded atom method (EAM)[103–105]. The EAM method was first developed by Daw and Baskes[104,105]. The basic idea is that each atom in a dynamic system can be viewed as an impurity embedded in a host consisting of all other atoms[106]. The total energy of a system can then be written as 𝐸 4;4<= = ∑ 𝐹 & 7𝜌 >,& 9+ ∑ @ $& A, $& B &'$ ' & , where 𝑟 &C is the distance between atoms i and j. 𝜙 &C 7𝑟 &C 9 is the pair potential accounting for the two-body interaction. 𝜌 >,& is the local density of the host at the site of atom i. 𝐹 & 7𝜌 >,& 9 is the embedding energy of atom i. These potential parameters are then fitted based on a set of experimental data such as atomic sizes, lattice parameters of different structures, cohesive energy, melting temperature, latent heat of melting, etc. Shortly after the basic EAM method was developed, Daw derived a similar expression for the cohesive energy from the local density functional, providing a theoretical basis for the EAM in semi-empirical applications, i.e., for an alloy system, the electron density at any location can be taken as a linear superposition of the atomic electron density and the embedding energy is assumed to be independent of the source of the electron density. The EAM method has been widely applied in studying grain boundaries, fracture and defects, and other material processes in metals and metallic alloys[55,107,108]. 2.1.3 Statistical Ensembles An ensemble is a collection of the possible microstates with identical macroscopic properties of the thermodynamic system. Each statistical ensemble is characterized by a set of conserved 9 thermodynamic quantities such as volume 𝑉, energy 𝐸, temperature 𝑇, and pressure 𝑃. The choice of the ensemble in an MD simulation is based on the experimental conditions that one aims to reproduce[109–111]. The microcanonical ensemble (𝑁𝑉𝐸) is the most commonly used statistical ensemble in MD simulations because of its convenience and simplicity. In this ensemble, the system of 𝑁 atoms is considered to be isolated from the surrounding environment and the total energy is conserved. Other thermodynamic quantities such as 𝑃, 𝑇, and chemical potential 𝜇, fluctuate around their average values at equilibrium. In the thermodynamic limit as 𝑁→∞, the relative fluctuation goes to zero as they are proportional to " √! , and the microcanonical ensemble becomes equivalent to the canonical (𝑁𝑉𝑇), grand canonical (𝜇𝑉𝑇), isothermal-isobaric (𝑁𝑃𝑇), and other ensembles. In MD simulations, the microcanonical ensemble is usually not desirable since most experiments are conducted at constant temperature and/or pressure. In fact, by choosing appropriate methods, the canonical ensemble and isothermal-isobaric ensemble can be readily used in MD simulations. There are several approaches to conduct MD simulations at a constant temperature. The most popular is the one introduced by Nose and Hoover[112], where the constant 𝑇 is achieved by coupling the momenta of the atoms to an external heat bath. In the canonical ensemble, 𝑁, 𝑉, and 𝑇 are conserved. In this ensemble, the system is thermally coupled with a heat reservoir to maintain constant 𝑇 while 𝐸 is allowed to fluctuate. In the isothermal-isobaric ensemble, a mechanical coupling is introduced to the system, which allows the system to maintain N, 𝑃, and 𝑇. A popular method for the isothermal-isobaric ensemble is that proposed by Anderson[90] for the case of hydrostatic pressure and that of Parrinello and Rahman for the general case of the stress tensor[88]. 10 2.1.4 Periodic Boundary Conditions Edges effects in MD simulation can be removed with the use of periodic boundary conditions (PBC)[109,113]. After a given system size is defined to represent the significant features of interest in the system, the system is enclosed in a fully periodic cell. The rationale of using PBC is that it allows the accurate simulation of bulk properties using a relatively small system. It avoids having to deal directly with surfaces and having to define a system finite dimension explicitly[114,115]. An MD simulation cell with PBC is repeated in all three independent directions in order to properly account for force calculation and atoms migration. As a result, when an atom crosses an MD cell boundary, an exact duplicate of that atom enters the cell through the opposite face, i.e., PBC considers no physical walls around the central computational cell boundary, leading to a constant number of atoms in the entire system over time. 2.1.5 Large-scale Atomic/Molecular Massively Parallel Simulator In our work, MD simulations are performed with the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code[116] using high-performance computing clusters. LAMMPS is a classical MD simulation code designed to run efficiently on parallel computers. It was developed at Sandia National Laboratories, a US Department of Energy facility. It is an open- source code, distributed freely under the terms of the GNU Public License. It can model an ensemble of particles in a liquid, solid, or gaseous state. And it can model atomic, polymeric, biological, metallic, granular, and coarse-grained systems using a variety of force fields and boundary conditions. 2.2 Visualization and Analysis Tools Visualization and analysis of MD simulation results is a crucial step in interpreting and understanding atomic mechanisms. By translating atomic coordinates from simulations into a 11 meaningful graphical representation, researchers can gain deeper insights into the behavior of complex systems at the molecular level. There are several software packages designed specifically for the visualization of atomistic structures generated by MD simulations, including Atomeye[117], VMD[118], and Paraview[119]. In our work, we primarily utilize OVITO (Pro)[120] for all atomistic visualizations. The software provides professional analysis tools that allow us to easily handle large-scale atomistic databases and create clear and detailed images of our simulations, as it offers a range of productive visualization tools that are implemented with high-quality image rendering functions. Through visualization tools, we are able to conduct thorough and comprehensive analyses of our MD simulation results, which in turn allows us to gain a deeper understanding of the complex molecular behaviors we are studying. 2.3 Machine Learning in Materials Science Machine learning has rapidly emerged as a powerful tool in materials science research, with the potential to revolutionize the way materials are designed, synthesized, and characterized. At its core, machine learning involves the use of statistical and computational techniques to identify patterns and relationships within datasets, and to make predictions based on those patterns[121]. In the field of materials science, machine learning is being applied to analyze large datasets from experiments, simulations, and other sources to identify the key factors influencing the properties of materials, to design new materials with targeted properties, and to optimize manufacturing processes[122–127]. With the increasing availability of high-throughput experimental techniques and large-scale computational simulations, the use of machine learning in materials science is poised to become increasingly widespread in the coming years. A variety of machine learning algorithms have been successfully applied in materials science research. Examples include Random Forest, Convolutional Neural Networks (CNNs), Support 12 Vector Machines (SVMs), Gaussian Process Regression (GPR), and Deep Learning techniques. Random Forest is often used for classification tasks in materials science, such as predicting the properties of different types of materials based on their chemical composition or crystal structure[128]. CNNs have been used for image analysis in materials science, particularly for analyzing microscopy images of materials and identifying defects or structural features[129,130]. SVMs have been used for predicting the properties of materials based on their electronic structure, as well as for analyzing data from MD simulations[131,132]. GPR can be used for materials design and optimization, by predicting the properties of hypothetical materials based on their chemical composition and structure[133,134]. Deep Learning techniques, such as artificial neural networks, have been used in materials science for a variety of tasks, including predicting the properties of materials, analyzing X-ray diffraction patterns, and identifying new materials with desired properties[135–138]. 13 Chapter 3: Gradient Nanoglasses 3.1 Introduction To enhance the limited plasticity of MG caused by the tendency of catastrophic failure by shear banding under deformation, various strategies have been proposed to delocalize the plastic flow in MGs. One of the effective approaches is based on the use of the NG architecture[55,139–143]. The unique grain boundary-like interfaces in NGs are reported as preferred channels for plastic flow[55]. Recently, there has been an increasing interest in the study of the mechanical properties of NGs through both experiments and modeling[144,145]. Results indicate an enhancement of ductility because of the delocalization of SBs at glass-glass interfaces. However, the gains in ductility come at the price of lower strength. A promising heterogeneous strategy for the compromise of strength and ductility in crystalline materials is based on a seamless gradient distribution of grain sizes[146–149]. The success of such a strategy in nanocrystalline metals suggests that a gradient NG (GNG) architecture, i.e., an NG designed by combining various grain sizes in a seamless functionally graded style, might be a viable approach as well to synergize strength and ductility in NGs. In this work, large-scale MD simulations are used to investigate the mechanical behavior of Cu64Zr36 GNGs under tensile loading and nanoindentation tests. First, based on the well- investigated Cu64Zr36 NG model[108], GNG samples with thin-film geometry are constructed with seamlessly graded grain sizes from 3 to 15 nm. Three types of grain size distribution patterns are considered: grain size gradient parallel to the tensile loading direction, which is named parallel GNG (p-GNG); grain size gradient perpendicular to the tensile loading direction, with smaller grains at the core, which is named soft-core GNG (sc-GNG) as regions with smaller grains are expected to be mechanically softer; and grain size gradient perpendicular to the tensile loading 14 direction with larger grains at the core, which is named hard-core GNG (hc-GNG) as regions with larger grains are expected to be mechanically harder. After the three types of GNG samples are prepared and relaxed, uniaxial tensile loading is performed. Second, samples with cubic geometry including Cu64Zr36 homogeneous NGs with grain sizes of 3 nm and 7 nm (3nm-NG and 7nm-NG) and GNGs with a harder surface (h-GNG), i.e., surface with larger grains, and softer surface (s- GNG), i.e., surface with smaller grains, are prepared for nanoindentation. The samples are prepared based on the well-investigated Cu64Zr36 NG and GNG models[108,150]. 3.2 Simulation Details 3.2.1 Tensile Loading of Metallic Glass with Thin-film Geometry Large-scale MD simulations are carried out with the LAMMPS package [116] using an EAM potential [151] fitted to Cu64Zr36. The p-GNG sample of size ~5.4 (x) × 80 (y) × 242 (z) nm 3 contains ~ 6,440,000 atoms. Both the hc- and sc-GNG samples have a size of ~5.4 (x) × 242 (y) × 484 (z) nm 3 and contain ~38,600,000 atoms. Before constructing the GNG samples, MGs with the same geometries are prepared by melt and quench. In order to speed up the quenching simulations, a small cubic MG sample (~13,000 atoms) with a side length of ~5.4 nm is first created and subsequently used to generate larger samples by replication. Periodic boundary conditions are applied along all three directions. An integration time step of 2 fs is applied during melting, thermalization, quenching, and relaxation of the system. The small cube is heated up, melted, and equilibrated at 2,000 K for 0.2 ns. Subsequently, it is quenched to 50 K at a rate of 10 10 K/s under zero external pressure. Temperature and pressure are controlled by the Nose-Hoover thermostat[112] using the isobaric-isothermal ensemble. After replicating the small cube according to desired sizes, the GNG samples are then constructed from the corresponding MG samples, as displayed in Fig. 3.1. Nanograins in the GNG samples are defined by the Voronoi 15 tessellation method from randomly distributed points along the y- and z-directions. following a well-defined density designed to produce the aimed gradient in grain size. The density of points is defined by assigning a minimum distance between any two points, following the function c (z − z 2 ) + b for gradient along the z-direction, or c (y − y 2 ) + b for gradient along the y-direction. Where b and c are constants, z and y are the scaled positions along the z- and y-directions, i.e., 0 < (z, y) < 1. The GNG samples are columnar, i.e., there is only one grain along the thickness direction (x- direction). The method we use to produce the GNG models generates flat glass-glass interfaces with constant width, which represents more closely NG microstructures generated by deposition methods[29,145,152] than those generated by inert gas condensation and cold compression. Grain sizes in the three GNG samples vary from 3 to 15 nm in a seamless way, following an approximately linear distribution. These samples are generated, consolidated, and relaxed through similar procedures adopted previously for homogeneous NGs[153]. For the p-GNG, as displayed in Fig. 3.1 (b), the grain sizes along the loading direction (z-direction) at the model extremities are at the range maximum (15 nm) while those in the middle are at the minimum (3 nm). For the hc- and sc-GNGs, as displayed in Figs. 3.1 (c) and (d), the grain sizes at the surfaces are at the range minimum and maximum respectively, while those in the core are at the maximum and minimum respectively. Uniaxial tensile tests are carried out at a constant strain rate of 4×10 7 s −1 along the z- direction. PBCs are imposed in the x- and z-directions. Accordingly, there are free surfaces along the y-direction to allow surface effects and shear offset. A time step of 5 fs is applied during loading. The tensile tests are performed at T = 50 K with zero pressure along the x-direction. Engineering stress is calculated as the Virial stress along the loading direction. 16 Fig. 3.1 Illustrations of the atomic models for metallic glasses (MG) and gradient nanoglasses (GNG). (a) Homogeneous Cu64Zr36 MG, (b) GNG for tensile loading parallel to the gradient direction (p-GNG), (c) hard-core GNG for tensile loading perpendicular to the gradient direction (hc-GNG), (d) soft-core GNG for tensile loading perpendicular to the gradient direction (sc-GNG). Colors in (a) indicate atomic species while colors in (b-d) indicate different grains. Tensile loading is applied along the indicated z-direction. 3.2.2 Nanoindentation of Metallic Glass with Cubic Geometry Similarly, before constructing the NG samples, a Cu64Zr36 MG counterpart with the same geometries, i.e., a ~ 53.7 nm 3 cube containing ~ 9,826,000 atoms, is prepared by replicating a small cube (~ 5.37 nm 3 ) that has been generated from melt and quench. The small cube is heated up, melted, and equilibrated at 2,000 K for 0.2 ns. Subsequently, it is quenched to 50 K at a rate of 10 10 K/s under zero external pressure. Temperature and pressure are controlled by the Nose- Hoover thermostat using the NPT ensemble. An integration time step of 2 fs is applied during the melting, thermalization, quenching, and relaxation processes. Periodic boundary conditions are applied along all three directions. After constructing the MG reference system, Fig. 3.1 (b), grains in NG/GNG samples are defined through Voronoi tessellation based on a given distribution of points in the simulation box. Each grain is filled up with a corresponding volume of material obtained from the reference MG sample. To produce interfaces in the NG/GNG models, the atoms 17 in the fully periodic reference MG sample are translated uniformly by 1 nm in a randomly chosen direction after each grain in the NG/GNG is filled up. To avoid overlapping of atoms at glass-glass interfaces, grains are filled up to 1 Å from the Voronoi tessellation defined interfaces. In addition, atoms are removed to ensure that no pair of atoms is closer than 2.2 Å, as the average nearest neighbor distances for Cu-Cu, Cu-Zr, and Zr-Zr are 2.7 Å, 3.0 Å, and 3.1 Å, respectively. A homogeneous NG with a desired average grain size of 3 nm or 7 nm can be achieved by tuning a density of randomly distributed points in the simulation box, which are used to define the grains in the NG sample, Figs. 3.2 (c) and (d). To produce the GNG sample, a gradient distribution of grain sizes is achieved by using a distribution of points in the simulation box and assigning a minimum distance dmin between any two points. As dmin increases or decreases from the bottom to the top of the simulation box along the z-direction, the resulting grain sizes vary seamlessly between 3 and 7 nm, resulting in an h-GNG or s-GNG structure, Figs. 3.2 (e) and (f). To be noted, the final NG and GNG samples have flat glass-glass interfaces with constant width, which represents closely NG microstructures generated by deposition methods[29,145]. In order to relax the atomic structure at interfaces and densify the samples, the NG and GNG samples are consolidated at 50 K and 3 GPa hydrostatic pressure for 0.04 ns, followed by relaxation at 50 K and 0 GPa for another 0.04 ns. Nanoindentation tests are carried out along the z-direction using a spherical indenter of radius R = 10 nm, Fig. 3.2 (a). On each atom within its interaction range, the rigid indenter exerts a repulsive force of magnitude: 𝐹(𝑟) =𝐾(𝑟−𝑅) ' (3.1) where F is the load, K is the stiffness constant, set as 10 eV/Å 3 , and r is the distance from the atom to the center of the indenter. The indenter is inserted into the sample at a loading rate of 5 m/s until 18 reaching a maximum indentation depth of 10 nm. Then the indenter is held at that position for 0.4 ns and unloaded from the sample at a loading rate of 5 m/s. During the indentation, an NVE ensemble is applied and PBCs are imposed along the x- and y-directions. A free surface is applied along the z-direction, while atoms within a 2 nm thick slab at the bottom of the system are fixed to avoid the translation of the center of mass of the sample. OVITO[120] is used for visualization and analysis. Fig. 3.2 Nanoindentation simulation models. (a) Spherical indenter with a radius of 10 nm; (b) homogeneous Cu64Zr36 metallic glass (MG); (c) nanoglass (NG) with an average grain size of 7 nm (7nm-NG); (d) NG with an average grain size of 3 nm (3nm-NG); (e) Gradient NG (GNG) 19 with a gradient grain size from 7 nm at the top to 3 nm at the bottom (h-GNG); (f) GNG with a gradient grain size from 3 nm at the top to 7 nm at the bottom (s-GNG). Colors in (b) indicate atomic species while colors in (c-f) indicate different grains. Nanoindentation is applied along the indicated z-direction. 3.3 Simulation Results 3.3.1 Tensile Loading Measurements In the investigation of the mechanical behavior of Cu64Zr36 NGs with a homogeneous grain size of 3 nm (3-NG), superplasticity under tensile loading has been reported[153,154]. In contrast, NGs with a homogeneous grain size of 15 nm (15-NG) display a sharp stress drop after yielding, indicating a brittle behavior, similar to what has been observed for MGs[153,154]. This previous knowledge serves as our reference to investigate and understand the relationship between the structure and mechanical behavior of GNGs. The stress-strain curves of the three GNG samples together with those for BMG, 3-NG, and 15-NG under tensile loading are displayed in Fig. 3.3. According to the curves, the ultimate tensile strength (UTS) of all GNG samples lie between the UTS for the 3-NG and 15-NG. In addition, the stress drops from the UTS displayed by the three GNGs are all smoother than that of the 15-NG, suggesting the delayed generation and propagation of a critical SB. Among the GNG samples, the hc- and sc-GNGs display UTS (~2.4 GPa) higher than that of the p-GNG (~2.3 GPa), while their yield strengths (~1.3 GPa, calculated from 0.2% deviation from linearity) are nearly the same. One interesting point to be noted is that the stress-strain curve of the p-GNG contrasts with the curves of the hc- and sc-GNGs. The former does not display a well-defined inflection point beyond the UTS, while the latter display one at ε = ~0.088 (hc-GNG) and the other at ε = ~0.1 (sc-GNG). In fact, the loading curves for the hc- and sc-GNGs are almost identical until the inflection point. Generally, such an inflection point indicates the formation of a critical SB in the MG or NG system. 20 Fig. 3.3 Engineering stress-strain curves for the MG, homogeneous NG, and GNG samples during tensile loading test. Open symbols correspond to the snapshots of the deformed GNG samples depicted in Fig. 3.4. 3.3.2 Deformation Profiles During Tensile loading To further understand the mechanisms behind the different deformation and failure modes of the GNG samples, we characterize the local plastic deformation by calculating the von Mises local shear strain, εM. We illustrate the observed evolution of local plasticity in three GNG samples in a sequence of snapshots in Fig. 3.4. For each GNG sample, six snapshots are displayed at specific strain points, as indicated in Fig. 3.3 by open symbols, highlighting the crucial deformation events. For all cases, plastic deformation is first triggered by local shear at glass-glass interfaces, similar to the deformation of homogeneous NGs reported in previous work[153,154]. Hence, at the embryonic stage of deformation, small grain size regions, i.e., regions with a larger fraction of interfaces, display greater plastic deformation than large grain size regions. That generates a gradient shear transformation zone (STZ) density along the elongation direction of the p-GNG. One can note from the snapshots in Fig. 3.4 (a) (from ε = 0.052 to 0.08) that the gradient in plasticity results in the necking of the system. Beyond ε = 0.08, the p-GNG system fails by the 21 propagation of a combination of multiple critical SBs. The hc- and sc-GNGs display similar mechanical behavior compared with that of the p-GNG. In both cases, a gradient in plasticity is generated following the gradient in grain sizes. As the gradient in the hc- and sc-GNGs is perpendicular to the elongation direction, the produced gradient in plasticity is aligned at 180° compared to that of the p-GNG. In addition, the hc- and sc-GNGs display no signs of necking. Embryonic SBs are generated at large grain size regions for both the hc- and sc-GNGs, indicating they reached their UTS points. These SBs then propagate into small grain size regions along maximally resolved shear stress directions, i.e., oriented at ~45° to the loading direction. As prominent SBs propagate to small grain size regions, they become diffused and connect an increasing density of STZs. The snapshots of Figs. 3.4 (b) and (c) suggest that the propagation of SBs is delayed as they become diffused, allowing the generation of additional embryonic SBs. Eventually, with an increase in strain, critical SBs are generated. It should be noted that the failure of the hc- and sc-GNGs is driven by a combination of multiple critical SBs, in contrast with that of MGs and homogeneous NGs, where the failure is dictated by the propagation of a single dominant SB. In particular, for the hc-GNG, two parallel critical SBs develop simultaneously, while a third critical SB, oriented perpendicular to those two SBs, also contributes to the delocalization of the deformation and failure of the system. As a result of the combination of multiple critical SBs in the hc-GNG, a rather smoother stress drop in the curve shows up compared to that of the sc-GNG, see Fig. 3.3. 22 Fig. 3.4 Snapshots of the deformation process for the three investigated GNG samples. (a) p-GNG under tensile loading parallel to the gradient direction. (b, c) hc- and sc-GNG under tensile loading perpendicular to the gradient direction. Atoms colors indicate the von Mises shear strain, eM, within the range of values defined in the color bar. For clarity, only atoms with eM > 0.2 are 23 displayed. Tensile engineering strain values for each one of the snapshots in (a to c) are indicated in Fig. 3.2. 3.3.3 Grain Sizes, Atomic Volumes, and Local Plasticity Distributions Fig. 3.5 Grain size gradient and local Voronoi atomic volume profiles and distribution of local atomic shear strain during deformation. (a) Grain size averages along the gradient direction in p- and sc-GNG samples, in black and red points, respectively, and black dashed guided line. Also, for the hc-GNG sample, in blue points and blue dashed line. (b) Configuration of the hc-GNG before deformation with atom colors indicating the local Voronoi atomic volume. (c) Fraction of atoms with eM > 0.2 along the gradient direction at two strain values with data points color following (a). Guided lines are provided for each curve. (d) Normalized fraction of atoms with eM > 0.2 along the gradient direction at two strain values, following the color choices of (c). To better understand the deformation profiles displayed in Fig. 3.4, we turn to the distribution of grain sizes, Voronoi atomic volume, and plastic deformation in the systems and possible links between them. The average grain sizes along the gradient direction in the GNG samples are given in Fig. 3.5 (a). Sinusoidal lines are drawn to guide the eyes and highlight the seamless gradient in 24 average grain size present in the nanostructures. To further quantify the microstructure, we show in Fig. 3.5 (b) the distribution of Voronoi atomic volume in the hc-GNG before the deformation, as a representative for all GNG cases. The local volumes are calculated at the position of each atom by averaging over neighbor atoms within a spherical volume of 25 Å radius. Both Cu and Zr atoms have a larger atomic volume at the glass-glass interfaces than the grain interiors, consistent with previous reports. Additionally, the average atomic volume increases from large grain size regions to small grain size regions, as the interface density of the former is larger than that of the latter. To understand the structure-property relationship, we quantify the development of local plasticity across the system during the tensile loading simulations. For that purpose, we calculate the fraction of the atoms that have participated in plastic deformation, i.e., atoms with εM > 0.2. We estimate the fraction across the system by spatially dividing it into 50 bins. The local plasticity is calculated at two different engineering strain levels for each sample, ε = 0.04 and either ε = 0.06 (p-GNG) or ε = 0.08 (sc- and hc-GNGs). The resulting data is displayed in Fig. 3.5 (c) with lines to guide the eye. The distributions quantify the plasticity indicated by the profiles of deformation displayed in Fig. 3.4. The dotted lines corresponding to the initial plastic regime at ε = 0.04 indicate the formation of well-defined seamless gradient plasticity in all cases. Regions with small grain sizes display larger fractions of deformed atoms, at ~5%, while regions with large grain sizes display rather smaller fractions at ~1%. It is worth noting that the p- and sc-GNG curves are nearly indistinguishable at this stage. The solid curves in Fig. 3.5 (c), illustrating the fraction of the atoms with εM > 0.2 at ε = 0.06 (p-GNG) and ε = 0.08 (sc- and hc-GNGs), indicate different patterns of plasticity evolution followed by the systems. While the overall fraction of the atoms with εM > 0.2 increases in all three systems, the curves indicate a startling contrast between the p-GNG and the 25 hc- and sc-GNGs. For the p-GNG, the plasticity level remains constant at large grain size regions while developing strongly at the small grain size region. That is a clear indication of the localization of the plastic deformation, associated with the necking behavior as illustrated in Fig. 3.4 (a). In contrast, the hc- and sc-GNGs display a significant increase in plasticity in all regions. It is worth noting that the deformation behavior of the hc- and sc-GNG systems are rather similar and imply almost symmetrically inversed levels of plasticity at small and large grain size regions. To emphasize the evolution of the degree of strain localization, we illustrate in Fig. 3.5 (d) a histogram of values along the gradient direction, where each point represents the number of atoms with εM > 0.2, normalized by the total number of atoms with εM > 0.2 in the system. As the engineering strain increases from ε = 0.04 to 0.06/0.08, the localization of the plastic deformation in small grain size regions is amplified in the p-GNG, i.e., the curvature of the distribution curve is increased, while the plastic deformation is delocalized in the hc- and sc-GNGs, i.e., the plastic deformation curves become flatter. Such a conclusion on the strain localization/delocalization is in good agreement with the necking behavior of the p-GNG displayed in Fig. 3.4 (a) and the rather homogeneous plastic deformation of the hc- and sc-GNGs displayed in Fig. 3.4 (b) and (c). This contrasting behavior is clear evidence of the synergy between the deformation of small and large grain size regions in the sc- and hc-GNG designs. 3.3.4 Nanoindentation Measurements To quantify the nanoindentation response of the prepared samples, we show their load- displacement curves and contact pressure-displacement curves in Fig. 3.6. Notable differences can be seen in the results for MG and NGs with different grain sizes. Beyond the elastic regime, the load curve for the MG sample displays the largest values at any indentation depth, while the corresponding curve for the 3nm-NG sample displays the lowest values, as shown in Fig. 3.6 (a). 26 Expectedly, the load curve for the 7nm-NG sample lies in between the MG and 3nm-NG cases. Moreover, the load curves for the h-GNG and s-GNG systems mostly overlap with the curves corresponding to the 7nm-NG and 3nm-NG systems, respectively. We then calculate the reduced elastic modulus, 𝐸 ∗ , by fitting the load-displacement curves in the elastic regime according to the relation[155] 𝐹 = 4 3 𝐸 ∗ D 𝑅𝑑 F (3.2) where F is the load, R is the indenter radius, and d is the indentation depth. A guideline fitted to the MG load curve is provided in Fig. 3.6 (a) along with the calculated 𝐸 ∗ values for all the samples. Similar to the relative load trends as a function of indentation depth the 𝐸 ∗ values decrease with grain sizes. In Fig. 3.6 (b), we show the dependence of the contact pressure 𝑃 G with the indentation depth. 𝑃 G is defined as 𝑃 G =𝐹∙𝐴 G :" (3.3) where 𝐴 G is the contact area, which for a spherical indenter is defined as 𝐴 G =𝜋(2𝑅𝑑−𝑑 ' ) (3.4) As 𝑃 G increases and saturates at d = ~ 5 nm, the hardness of the sample, H, can then be evaluated. Here, we calculate H by averaging over the values of 𝑃 G between d = 5 nm and 10 nm. The calculated values for each curve are shown in Fig. 3.6 (b). As expected, the hardness decreases with grain sizes. Meanwhile, the h-GNG (s-GNG) samples show hardness values close to the 7nm- NG (3nm-NG), respectively. To better characterize the abrasion resistance of the material, defined here as the resistance of the material to plastic penetration, we calculate the ratio between H and 𝐸 ∗ ' , as it carries more physical meaning than a single parameter[156]. The results indicate that the 7nm-NG has the 27 lowest resistance, 𝐻/𝐸 ∗ ' = 1.6×10 :F , followed by the h-GNG and MG, 1.8×10 :F , and the 3nm-NG and s-GNG display the largest resistances, 2.6×10 :F . Fig. 3.6 Mechanical response of the MG, NG, and GNG models under nanoindentation. (a) Load- displacement curves; (b) contact pressure-displacement curves. Calculated values of reduced elastic modulus, E*, and hardness, H, are shown for each model. 3.3.5 Local Stress and Strain Profiles under Nanoindentation To further understand the deformation mechanisms in the different systems we evaluate the local atomic von Mises shear strain at different indentation depths, as shown in Fig. 3.7. The deformation evolution can be characterized by comparing the deformation profiles at d = 2 nm and 28 Fig. 3.7 Local atomic von Mises shear strain for the MG, NG, and GNG models at different indentation depths, d. (a-e) Local shear strain profiles at d = 2 nm; (f-j) local shear strain profiles at d = 10 nm. The views are taken from a thin slab of 1 nm thickness centered in the indentation region of each sample. 29 10 nm. In the early indentation stage, i.e., d = 2 nm, STZs in the MG sample are localized around the indenter. Here, we define STZs as regions with von Mises shear strain greater than 0.2. In contrast, all NG/GNG samples show STZs distributed far from the indenter surface in the form of dispersed clusters. Particularly, the density of STZs found in the 3nm-NG sample is higher than that found in the 7nm-NG sample. The density of STZs in the h-GNG and s-GNG is similar. Nonetheless, the distribution of STZs around the indentation center rather follows preferred paths in the h-GNG while it is rather homogeneous in the s-GNG sample. At the maximum indentation depth, i.e., d = 10 nm, severely deformed local regions, i.e., regions with local strain greater than 0.5, are localized around the indenter and a prominent pileup can be identified in the samples, in particular in the MG sample. The transition from severely to mildly deformed regions is considerably smoother in the 3nm-NG/s-GNG compared to the MG sample, resulting in a rather subtle pileup. Specifically, the pileup has a height of ~ 5 nm for the MG and ~ 2 nm for the 3nm- NG, which is a consequence of the localized deformation in the MG sample. The 7nm-NG and h- GNG samples shown in Fig. 3.7 show a similar intermediate pileup deformation behavior compared to the MG and 3nm samples, while the s-GNG sample displays a similar mild pileup as that of the 3nm-NG sample. To further highlight the deformation mechanisms, we show in Fig. 3.8 the evolution of the local stress. In the early stage of the indentation, i.e., d = 2 nm, large local stresses (above 10 GPa) are concentrated around the indenter within a small area in the MG. In comparison, NG/GNG samples show a nearly uniform distribution of large stresses, except for a small region around the indenter. Interestingly, the s-GNG sample shows a notably narrower region around the indenter, see Fig. 3.8 (e) that is under a large local stress level compared to the 3nm-NG, see Fig. 3.8 (c). At the maximum indentation depth, i.e., d = 10 nm, local stresses above 14 GPa, can be found in 30 Fig. 3.8 Local atomic stress for the MG, NG, and GNG models at different indentation depths, d. Local stress profiles at d = 2 nm, (a-e) and d = 10 nm (f-j). The views are taken as in Fig. 3.7. 31 the MG sample see the dark red region in Fig. 3.8 (f). Large local stress in the 3nm-NG and s- GNG gradually radiates outward from the indentation center, while its distribution in the 3nm-NG is slightly smoother than that in the s-GNG. The areas around the indenter under large local stress states in the 7nm-NG and h-GNG are comparable to that in the MG, yet they show a smaller average stress value. 3.3.6 Quantifications of the Deformation Mechanism Quantitative analyses of strain localization are performed to support the examination of the different deformation mechanisms in the models. A useful metric in the evaluation of strain localization propensity is the strain localization parameter, 𝜓 = T " ! ∑ (𝜂 & H&IJI −𝜂 <6J H&IJI ) ' ! &K" , where 𝜂 & H&IJI is the von Mises strain of atom i, 𝜂 <6J H&IJI is the average of the von Mises strain of all atoms, and N is the total number of atoms in the system[22]. In Fig. 3.9 (a), the values of strain localization parameters of different models are plotted as a function of indentation depth. A well- defined crossover between the curve for the MG model and those of other models can be identified at d = 2~3 nm. In the early stage of indentation, the 3nm-NG displays the highest degree of strain localization, whereas the MG displays the lowest. With the indenter further penetrating each sample, the MG develops the highest degree of strain localization, followed by the h-GNG, 7nm- NG, 3nm-NG, and s-GNG. To further quantify the plastic deformation in the systems, in particular in regions with severe plastic deformation, we show in Fig. 3.9 (b) the statistics of von Mises shear strains of each model at an indentation depth of 10 nm on a logarithmic scale. As the calculation only considers atoms participating in plastic deformation in the system, i.e., 0.2 < von Mises strain < 2, we start by dividing this strain range into 100 bins. All the atoms in the system undergoing plastic deformation are then used to fill up the bins according to their von Mises strain values. The bins are then normalized by the total number of atoms in the system. The statistics for the MG 32 sample can again be distinguished from the other models, as it shows a higher fraction of atoms in the moderate deformation range, i.e., 0.2 < von Mises strain < 0.7, whereas a lower fraction of atoms in the severe deformation range, i.e., 0.2 < von Mises strain < 2. Though the difference between NG and GNG models is not significant, one may still find a trend when zooming in on the data at large von Mises strain values. The largest fraction of atoms undergoing severe deformation is found in the h-GNG, followed by the 7nm-NG, 3nm-NG, and s-GNG. Fig. 3.9 Strain localization and plastic deformation during nanoindentation of the samples. (a) degree of strain localization for the MG, NG, and GNG models at different indentation depths, d; (b) local strain statistics for different models at d = 10 nm. The inset shows the statistics at large local strain values. 33 3.3.7 Analysis of Local Plastic Events The plastic events in each sample during the indentation process can be further characterized by evaluating the non-affine squared displacement (NASD)[155], i.e., the local strain at the site of each atom is compared to its homogeneously strained neighborhood[157]. The NASD during the time interval [𝑡−∆𝑡,𝑡] is calculated by minimizing the sum of the squared difference between the actual displacements of the neighboring atoms with respect to the central one and the relative displacements that they would have gone through if they were in a region of uniform strain. 𝐷 5&L ' =𝑚𝑖𝑛 {[ [ (𝑟 L & (𝑡)−𝑟 M & (𝑡)−[ 7𝛿 &C +𝜀 &C 9×[𝑟 L C (𝑡−∆𝑡)−𝑟 M C (𝑡−∆𝑡)] C ) ' & L } (3.5) where n runs over the neighbor atoms within a cutoff range, i and j run over all spatial coordinates, 𝑟 L & (𝑡) is the i th component of the position of the n th atom at time t, 𝛿 &C is the Kronecker delta, 𝜀 &C is the local strain that minimizes the squared difference[157]. Figure 3.10 shows the distribution of NASD in each model at different indentation stages. The identification of groups of atoms with high NASD provides a reliable way for the localization of irreversible shear transformations, i.e., STZs, and incipient shear bands. In the analysis here, we consider regions with averaged NASD greater than 10 Å 2 as STZs. Consistent with the deformation profiles in Fig. 3.7, at d = 2 nm, STZs in the MG are confined within a small region around the indentation center. In comparison, the distribution of STZs in the 7nm-NG and h-GNG around the indentation center is less uniform, revealing preferable shear transformation paths. Among all samples, the 3nm-NG displays the largest number of isolated clusters with high NASD throughout the sample. At the maximum indentation depth, the MG sample shows an STZ layer of ~ 5 nm thickness surrounding the indenter, which is significantly thinner than that in the 3nm-NG and s-GNG, which reaches ~ 10 nm thickness. In the 7nm-NG and h-GNG samples, preferable shear transformation paths can be 34 Fig. 3.10 Non-affine squared displacement (NASD) for the MG, NG, and GNG models at different indentation depths, d. Local stress profiles at d = 2 nm, (a-e) and d = 10 nm (f-j). The views are taken as in Fig. 3.7. 35 identified from the indentation center to the rest of the sample, which are influenced by the distribution of glass-glass interfaces. To quantify the plastic events in each model, we perform cluster analysis based on the calculated local NASD distribution. First, we select atoms with calculated NASD values above 20 Å 2 , which indicates intense local plasticity. To be noted, this NASD threshold value is twice the maximum value shown on the color bar in Fig. 3.10. With atoms selected we define clusters by grouping atoms that are within 2.45 Å. The cluster size is represented by the number of atoms it contains, Nc. Alternatively, Nc can be used to directly quantify the size of STZs. Fig. 3.11 Complementary cumulative distribution of cluster sizes. (a) distribution at d = 2 nm and (b) at d = 10 nm. The arrows indicate the gap in cluster sizes, Nc, when a relatively small cluster transits into a large cluster. 36 In Fig. 3.11, we plot the statistics of Nc in each model at different indentation depths. In the log-log plot, the complementary cumulative, C(Nc), is obtained by calculating the probability of finding a cluster of size Nc and above. Despite the deviation at the tail of the distribution, a power- law relationship can be identified in both Figs. 3.11 (a) and (b), consistent with previously reported observations[155]. At d = 2 nm, one can notice a shift in the distribution among different models. The 3nm-NG shows the highest power-law exponent (-1.80), followed by the s-GNG (-1.71), h- GNG (-1.57), 7nm-NG (-1.51), and MG (-1.45). A large cluster size point with Nc > 10 3 atoms, deviating from the power-law behavior, can be identified in the MG model and indicates the formation of an incipient shear band. At d = 10 nm, the calculated power-law exponents for the systems become comparable, increasing for the NG and GNG models and slightly decreasing for the MG model, i.e., s-GNG (-1.48), h-GNG (-1.47), 3nm-NG (-1.51), 7nm-NG (-1.47), and MG (- 1.47). The arrows and dashed areas in Figs. 3.11 (a) and (b) point out the appearance of a gap in the transition of cluster size that was found in the MG at d = 2 nm and in the 7nm-NG and h-GNG at d = 10 nm. In contrast, continuous data points in a wide range of large Nc can be found at the tail of the 3nm-NG and s-GNG curves. 3.4 Discussion 3.4.1 Comparison between Gradient Nanoglass and Other Heterogeneous Metallic Glass It is worth comparing the results of this work with related studies of heterogeneous MGs and NGs. In our work, we consider a radically different heterogeneous design compared to previous investigations, i.e., a seamless gradient in grain sizes. However, our results resonate with the conclusions of many earlier reports, i.e., heterogeneous structures in MGs are able to compromise, at different efficiencies one may note, the properties of dissimilar phases. That was consistently displayed by NG coating employed to impart surface flaw tolerance to MG[158]; by a bimodal 37 grain size NG designed to impart high ductility to a large and strong grain size NG[57]; by making nanolaminates of MG layers to improve significantly the strength while preserving the ductility of an NG[159]; and in this work, by introducing a GNG where regions with dissimilar properties are connected in a seamless fashion and act to synergize the deformation mechanisms both delaying SB propagation, delocalizing plastic deformation, and strengthening small grain size regions. The strengthening of small grain size regions is implied by the UTS of each GNG, which is superior to what one would predict from the rule of mixtures. Comparable with what was observed in the hc- GNG, Chen et al. also reported a unique pattern of multiple parallel SBs developed in a Zr-based BMG tested under in situ four-point bending[160]. In that case, the mechanism leading to the formation of multiple SBs was linked to the dependence of the energy dissipation for SB nucleation and SB growth with the sample size, which favors incipient SB generation over SB accumulation or propagation in thin samples. Composite designs are often explored as an efficient way to compromise the strength and ductility in MGs. In such composites, the best strength-ductility combination is usually associated with a critical feature size of the composite, e.g., layer spacing in MG nanolaminates[161]. A critical correlation length in heterogeneous MGs involved in the transition of SB formation mechanisms has also been reported by Wang et al.[21]. Here, instead of proposing a homogeneously distributed heterogeneity, we generate a heterogeneity gradient, which combines size effects and spatial correlation. However, the mechanisms involved in the formation and evolution of SBs in composite MGs are distinct from those in GNGs. For example, considering the nanocrystalline MG composite[41,60], STZs initially nucleate at soft amorphous-crystalline interfaces and accumulate there as their development into a critical SB is hindered by hard crystallites. In contrast, in the hc- or sc-GNGs, incipient SBs initialize at glass-glass interfaces at 38 large grain size regions and then propagate into small grain size regions. The delay in the generation of a critical SB is essentially a result of the competition among multiple growing SBs. As it is clear from the mechanical behavior of unimodal NGs, all glass-glass interfaces are activated at the yield point and become effective STZs. One may think of that as a dense network of thin SBs homogeneously distributed throughout the sample. However, the effect of interfaces in the mechanical behavior of unimodal NGs varies with the different grain sizes, i.e., it varies with the fraction of interfacial material[108,162]. In relatively large grain-sized (>10 nm) samples, glass-glass interfaces tend to form paths for the generation and propagation of a critical SB. In a relatively small grain-sized (~3 nm) sample glass-glass interfaces tend to act akin to a soft matrix where grains flow. The latter occurs as a result of an increasing fraction of interfaces, which turn thicken once start to deform. Here, we focus on the combined effect provided by a specific polydisperse NG, i.e., a seamless GNG. In such a structure, we expect both behaviors in a single sample, with large grain-sized NG providing strength to the material and small grain-sized NG providing excellent plasticity. The results indicate that GNGs are in fact able to provide a synergistic combination of mechanical properties, i.e., critical SBs from large grain size regions become diffused and arrested in small grain size regions. Concomitantly, the gradient structure offers support and constrains the uncontrolled flow of small grain size regions. That synergy is particularly clear in the hc-GNG as highlighted in Fig. 3.3 (b) and Figs 3.4 (c) and (d). 3.4.2 Comparison between Gradient Nanoglass and Gradient Nanocrystals Lu et al. experiments with gradient nanocrystalline Cu suggested an unprecedented strength- ductility synergy in a nanocrystalline material[148]. Cheng et al. related the reported synergy to a high density of geometrically necessary dislocations inside grain interiors, which was induced to accommodate the generated plastic strain gradient[146]. Surprisingly, albeit these studies were 39 conducted on crystalline metals, and the mechanism behind the plastic deformation of crystalline and amorphous materials are intrinsically different, they still share some common features in their mechanical behaviors. For both gradient nano-grained metals and GNGs, the graded structure results in a unique patterning of either dislocation or shear banding, which tunes their deformation and failure modes[163]. 3.4.3 Effects of Grain Size and Gradient Microstructure in NGs and GNGs From the quantitative and qualitative analyses of all models during nanoindentation, the effects of grain size and gradient microstructure can be revealed. For the homogeneous NGs, both reduced elastic modulus and hardness increase with grain sizes. Accordingly, the abrasion resistance decreases with increasing grain sizes. For the GNG models, the average grain size at the indentation surface plays a crucial role in their deformation behavior during nanoindentation. The s-GNG with an average grain size of 3 nm at the surface displays a similar mechanical behavior as the 3nm-NG, while the h-GNG with an average grain size of 7 nm at the surface displays a similar mechanical behavior as the 7nm-NG. We attribute such a result to a more delocalized distribution of STZs in the NG with smaller grain sizes, which effectively relieves the build-up of local atomic stress and delays incipient shear banding. As shown in Figs. 3.7, 3.8, and 3.10, a minor load at d = 2 nm can activate “dimples” of soft sites throughout the NG and GNG models. These soft sites are to a large extent contributed by glass-glass interfaces in the models. NGs or regions in GNGs with smaller grain sizes are expected to possess a higher density of glass-glass interfaces, and as a result, display a denser distribution of dimples. One can infer from the difference in the distribution of points shown in Fig. 3.10 (a), that the scattered “dimples” in NGs and GNGs can effectively reduce the concentration of clusters undergoing inelastic shear deformation compared with the MG model. 40 As the indenter penetrates deeper into the samples, see Fig. 3.10 (b), the C(Nc) power-law exponents for NG and GNG models increase to a similar level as that for the MG model, indicating saturation of soft sites, which have been activated as STZs. Meanwhile, the gap in cluster sizes, visually indicated by the arrows, suggests the absence of intermediate-sized clusters in the MG, 7nm-NG, and h-GNG, in comparison with the continuous data points at the tail of the 3nm-NG and s-GNG curves. We relate this observation for the MG, 7nm-NG, and h-GNG samples to a deformation mechanism dominated by stress-dictated nucleation, whereas the observation for the 3nm-NG and s-GNG samples is linked to a deformation mechanism dominated by the structure- dictated strain percolation. These two types of deformation mechanisms have been reported previously by Wang et al.[21]. In their work, they identified a critical spatial correlation length of elastic heterogeneity, which when exceeded would trigger a transition of the shear banding mechanism from stress-dictated nucleation to structure-dictated strain percolation. Such change in deformation mechanisms is similar to our findings that the stress-dictated mechanism can be characterized by the abrupt emergence of large STZs, whereas the structure-dictated mechanism can be characterized by the activation of multiple small STZs which later gradually evolve into shear band embryos. 3.4.4 Deformation Mechanisms in NG and GNG structures In this work, nanoindentation tests quantify the resistance to localized surface deformation in NGs and GNGs, as well as provide new insights into the relationship between the mechanisms involved in the generation and development of STZs and shear bands. It is worth noting that though the average grain size at the indentation surface plays a crucial role in the displayed deformation behavior of GNGs, e.g., the hardness and elastic modulus measurement values of GNGs are close to their homogenous NG counterparts, the corresponding physical interpretation requires a careful 41 investigation of the deformation mechanisms involved. The results shown in Fig. 3.11 depart from a common hypothesis, that statistics of the h-GNG and s-GNG models are expected to be between those of the 7nm-NG and 3nm-NG. The h-GNG displays a higher degree of strain localization and plastic deformation than the 7nm-NG, whereas the s-GNG displays a lower degree of strain localization and plastic deformation than the 3nm-NG, even though the average grain size is larger in the 7nm-NG than the h-GNG and smaller in the 3nm-NG than in the s-GNG. One may also notice a more homogeneous local stress distribution throughout the system in the s-GNG, as shown in Fig. 3.8 (e), than that in the 3nm-NG, as shown in Fig. 3.8 (c). As a result, the 3nm-NG shows higher local stress around the indenter at d = 2nm, which is at the same time reflected by slightly larger contact pressure in the 3nm-NG than that in the s-GNG, see Fig. 3.6 (b). Contrasts in the deformation mechanisms of homogeneous nanograined structures and gradient nanograined structures have also been reported for crystalline Cu[164]. As here the size- dependent strength in the heterogeneous samples induced gradient local stress in the gradient structures promoting the generation of gradient local plasticity. Nonetheless, the stress-strain synergy in gradient nanograined Cu is mainly rooted at partial dislocation nucleation, stacking fault generation, and twinning activated in the gradient stress field, which contrasts with the shear banding mechanism in MGs, NGs, and GNGs. Such a phenomenon can be attributed to a softer bottom (regions with small grain sizes) in the h-GNG and a harder bottom (regions with large grain sizes) in the s-GNG. A soft bottom may serve as a buffer for surface deformation yet contribute less to the stress delocalization compared to a hard bottom. This interesting finding can also be noticed in Figs. 3.7, 3.8, and 3.10. “Dimples” of soft sites in NG and GNG models can reach as far as the bottom of the model, i.e., ~ 50 nm away from the indenter. That implies that the concentration of stress at the surface, with resulting surface damage, 42 on an NG or GNG can be alleviated by the activation of soft sites in the sample, up to a depth of 50 nm. Larger models may be considered in future research to evaluate the extent of the indentation stress release by activatable soft sites in the sample at larger depths. 3.4.5 Experimental Synthesis of Gradient Nanoglasses and Comparisons with Simulations One can envisage different ways that GNGs can be generated. MGs with a gradient microstructure composed of a gradient density of SBs have been reported previously[165,166]. If one considers that glass-glass interfaces in NGs behave effectively as SBs, one can think of a GNG as an MG with a gradient density of SBs, which can be synthesized by surface treatment methods such as surface mechanical attrition processing. Regardless, one can envisage four other different paths for the generation of true gradient NGs. I) Glassy nanoparticles generated by inert gas condensation can have their size controlled by tuning condensation parameters[167,168]. Thus, a GNG could be generated by cold compression of a combination of different glassy nanoparticles arranged in a gradient range of sizes. II) NGs can also be generated by magnetron sputtering[145,152]. The size of NGs, as in the case of inert gas condensation, can be tuned by adjusting deposition parameters. Thus, sputtering offers another path to potentially generate a size gradient in the NG sample by continuously changing deposition parameters. III) The third alternative to generate a GNG, following a similar procedure as for sputtering, is by tuning electrodeposition parameters[29,169]. IV) A fourth possibility, for the generation of GNG, is by turning to the chemical generation of amorphous nanoparticles. Such metallic amorphous nanoparticles can be generated by colloidal chemistry[170,171]. The size of such nanoparticles can be controlled, and a gradient of sizes can be generated by combining different sets of nanoparticles and then applying cold compression. We hope that the results of this work and such suggestions motivate further experimental work in this area. 43 Due to the lack of experimental counterparts to compare our simulation results with, here we address the possible artifacts that may arise from such computational modeling. In this work, we prepared the reference BMG sample with a cooling rate of 10 10 K/s, which is orders of magnitude higher than those typically applied in experiments. Cu–Zr-based BMGs generated with higher cooling rates usually exhibit higher structural heterogeneity, i.e., tend to form a lower fraction of full icosahedra[107,172]. However, the difference in short-to-medium range structural order between simulation and experimental results is minor[51] and can be sufficiently reduced when the MD quenching rate is reduced to 10 10 K/s[173]. We perform tensile loading at a temperature of 50 K to highlight the effect of stress rather than the temperature on the mechanical response of GNGs. Temperature effects were believed to significantly change the mechanical behavior of BMGs[174]. Nonetheless, the mechanical behavior of BMGs at 50 K in simulations is representative of their behavior in experiments at room temperature[175]. To be noted, temperature effects in NGs or GNGs could result in a complex scenario as higher temperatures will additionally promote glass-glass interfacial relaxation[176]. Strain rate is another important parameter that can affect the mechanical behavior of BMGs, NGs, GNGs, and other materials. Typically, the strain rates used in MD simulations, 10 7 to 10 10 s −1 , are orders of magnitude higher than those in experiments. Interestingly, MD simulation results using such high strain rates are able to reproduce the critical features of the mechanical behavior of BMGs reported in experiments, e.g., formation of critical SBs and brittle failure under tension/compression loading[177–180]. Arguably, the effects coming from the use of much higher strain rates in MD simulations are offset by the use of much higher quench rates in the generation of the BMG structure[181]. As highlighted in the previous paragraph, there are currently no experimental results on the properties of GNGs. This lack of results prevents us from validating our simulation predictions. However, it 44 is worth mentioning that the simulation results on gradient nanocrystalline systems were instrumental to understand the complex deformation behavior in these materials reported in experiments[149,182]. 3.4.6 Potential Applications of Gradient Nanoglasses Before we summarize our work, we should mention that GNGs may broaden the traditional range of applications of MGs. MGs have been widely explored in biomedical applications, e.g., implants and stents[39,183,184]. Besides biocompatibility and strength often such applications require a degree of ductility beyond what is commonly found in MGs. Our results indicate that the ability of GNGs to generate gradient plasticity and synergize strength and ductility may offer a competitive edge enabling further applications in this field. Gradient amorphous structures, such as the GNG, have also been proposed as effective coating materials[166]. The gradients in the GNG models considered here are relatively steep since the simulation model lengths along the gradient direction are relatively small at ~0.25 μm (p-GNG) and ~0.5 μm (hc/sc-GNG). These gradients would correspond for instance to a thin surface coating. However, the predictions of this work are expected to be valid in other length scales as well. Besides structural applications, GNGs offer a framework to investigate novel magnetic designs. Recent reports on FeSc NGs demonstrate that glass-glass interfaces display ferromagnetic properties in contrast with the paramagnetic glass particles[141,185]. The gradient in the density of glass-glass interfaces present in the structure of the GNG may be used to generate a gradient ferromagnetic profile. Such structures could be used to tune magnetic properties in unprecedented ways. 3.5 Conclusion In summary, we studied the mechanical behavior of Cu64Zr36 GNGs under deformation using MD simulations. Results suggest distinct deformation mechanisms for GNG samples with 45 different gradient grain size designs. For the tensile loading test, initial gradient plasticity is generated in all systems following the gradient in grain sizes, i.e., larger plasticity for smaller grain size regions. For the p-GNG case, that is followed by necking and generation of delocalized SBs in the necking region, resulting in the failure of the system. In contrast, critical SBs are generated in large grain size regions for the sc- and hc-GNGs. Their propagation is then delayed as they become diffused traversing small grain size regions. As a result, significant delocalization of the plastic deformation is developed until failure occurs by a combination of critical SBs. While fundamentally consistent behavior is observed for the sc- and hc-GNGs, the best strength-ductility compromise is achieved by the latter. Additionally, the results from nanoindentation test show that the deformation mechanisms in both NGs and GNGs with small grain sizes are rooted in the activation of STZs at soft sites and their evolution into shear band embryos. An increase in grain sizes results in the enhancement of elastic modulus and hardness and a reduction in abrasion resistance. These results demonstrate the subtle synergistic effects triggered by heterogeneous gradient designs in the mechanical behavior of MGs and point out exciting new possibilities. 46 Chapter 4: Nanoglass-Metallic Glass Composites 4.1 Introduction To compromise the high MG strength and NG ductility, one can turn to the design of bulk MG composites (BMGCs) as the heterogeneous architectures are expected to synergize the mechanical properties of the two phases, e.g., constraining the premature strain localization while preserving the strength. A composite composed of MG and NG (MGNG-C) is expected to combine the strength of the MG phase and the ductility of the NG phase. Previously, MGNG-Cs with a sandwich or nanolaminate architecture have been investigated using MD simulations[158,186]. A transition in failure mode from localized shear banding to superplastic flow was found for the increasing volume fraction of the NG phase in MGNG-Cs. It was demonstrated that with an effective nanolaminate heterostructure, the composite was able to preserve superplasticity while producing a peak strength of 15% higher than that of the NG counterpart[186]. Those early works on MGNG-Cs motivate us to further investigate the synergy of the two phases and to search for alternative architectures that may optimize the displayed mechanical properties of the composite. Generally, many structure parameters can affect the mechanical property of a composite, such as the size, volume fraction, and arrangement of the second phase[41,60,63,76,80,187–191]. In this work, we consider MGNG-Cs with a brick-and-mortar design, where the matrix is composed of a 3 nm grain-sized NG (3nm-NG) and the second phase is composed of a homogeneous MG. Three second phase volume fractions (30%, 35%, 40%) and two brick-and-mortar arrangements (stacked MG bricks, named stk-C, and staggered MG bricks, named stg-C) are considered. 47 4.2 Simulation Details Large-scale MD simulations are carried out using validated CuZr interatomic potential functions[173]. A 2-fs time step is used during the sample preparation. To construct the composites with dimensions of ~4.9 ´ 156 ´ 314 nm 3 containing ~15 million atoms, we first prepare the bulk MG and 3nm-NG counterparts respectively. First, a small Cu64Zr36 sample with ~10 thousand atoms is thermalized at 2000 K for 0.2 ns, then quenched to 50 K at 10 10 K/s, and finally relaxed at 50 K for 0.4 ns to generate the amorphous phase. The bulk MG is obtained by replicating the small sample to the desired size and annealing it at 800 K for 0.5 ns. Subsequently, the 3nm-NG, as displayed in Fig. 4.1 (a), is constructed from the MG sample by defining its nanograins using the Voronoi tessellation method[108,192]. Then in the 3nm-NG, following the desired MGNG-C design, we replace the required brick regions with the MG phase, as displayed in Figs. 4.1 (b) and (c). Here, both MG and NG phases are columnar along the x-direction. Samples with 30%, 35%, and 40% MG phase have bricks of sizes of 15.6 ´ 82.2 nm 2 , 17.7 ´ 84.4 nm 2 , and 19.7 ´ 86.4 nm 2 . Fig. 4.1 Illustrations of the atomic models for NG and MG-NG composites. (a) Cu64Zr36 NG with 3 nm grain size (3nm-NG). (b) MG-NG composite with Cu64Zr36 MG bricks arranged in a stacked way (stk-C). (c) MG-NG composite with MG bricks arranged in a staggered way (stg-C). Colors indicate different grains in the NG matrix. In (b) and (c) the MG second phase is colored in dark blue. 48 The corresponding mortar regions have a thickness of 24.4 nm, 22.3 nm, and 20.2 nm. The six MGNG-C samples (30%/35%/40%MG-stk/stg-C) are later sintered at 50 K and 3 GPa hydrostatic pressure for 0.4 ns and then relaxed at 50 K and 0 GPa for 0.4 ns. Uniaxial tensile loading with a 5-fs time step is applied along the z-direction at 50 K, at a strain rate of 4´10 7 s -1 . 4.3 Simulation Results 4.3.1 Engineering Stress-strain Curves To demonstrate the overall deformation behavior of the Cu64Zr36 MGNG-Cs, we plot their engineering stress-strain curves in Fig. 4.2. The corresponding 3nm-NG curve is also given as a reference. Consistent with previous results[108], the 3nm-NG exhibits nearly superplastic behavior, i.e., the stress beyond the UTS point decreases smoothly to a flow value within the simulation time. As the volume fraction of the MG phase increases from 30% to 40%, the UTS is increased in both the stk-C and stg-C designed composites. Interestingly, the 30%MG-stk-C sample reveals an enhanced UTS (~2.0 GPa) compared with the 3nm-NG (~1.8 GPa) while preserving superplasticity (left inset in Fig. 4.2). The composites with higher MG volume fractions sacrifice their ductility for strength, displaying a steeper stress drop at ε > 0.1. Moreover, it is indicated that the stk-C composites, for all cases, display slightly higher overall strength compared to the stg-C composites. Whereas the stress drops in the stk-C curves are smoother than those in the stg-C curves. This is particularly notable in the 35%MG-stk/stg-C case (right inset in Fig. 4.2). While the two composites have almost identical UTS, ~2.01/2.05 GPa, the 35%MG-stg-C displays a much larger toughness than the 35%MG-stk-C, which implies the ability of the stg-C design to improve the mechanical properties of the composite. 49 Fig. 4.2 Tensile loading engineering stress-strain curves for the 3nm-NG and MG-NG composite models with different fractions of the MG second phase. The left inset highlights the improvement in the strength of the composites with 30% MG phase. The right inset highlights the difference between the stk-C and stg-C curves at 35% MG phase. 4.3.2 Deformation Profiles of Different Metallic Glass-Nanoglass Composites To further understand the deformation mechanisms of different MGNG-C models, we visualize the evolution of local strain distribution in each model, as displayed in Fig. 4.3. Colors indicate the level of von Mises local strain, eM. Only atoms under plastic deformation (eM < 0.2) are displayed. In all cases, most plastic deformation occurs in the relatively softer NG matrix compared to the relatively stronger MG bricks. Among the models with different volume fractions of the second phase, one can observe significant contrasts. While both the 35% and 40%MG- stk/stg-Cs develop a critical SB under large deformation (e > 0.15), the 30%MG-stk/stg-C illustrates symmetrical crisscross shear flow paths, which prevent the development of a critical SB. Such unique deformation profiles of the MGNG-Cs originate from the essential difference in the mechanical response of the two phases and can be affected by their arrangements. As displayed in Figs. 4.3 (a)-(c), the stk-C designed composites develop noticeable shear flows across MG bricks under large loading (e > 0.15). However, in the stg-C designed composites, SBs propagate solely 50 Fig. 4.3 Deformation profiles of the MG-NG composite models during tensile loading. (a)-(c) Profiles for the MG-NG composites in the stk-C design at different fractions of the MG second phase; (d)-(f) corresponding profiles for the composites with the stg-C design. Atoms are colored according to their von Mises local strain, eM, values. Atoms with eM < 0.2 are not displayed for clarity. in the NG matrix and circumvent the MG phase, as displayed in Figs. 4.3 (d)-(f). It is worth noting that the degree of distinction in the stk/stg deformation profiles also depends significantly on the phase fraction. When the MG volume fraction is at 35%, SB propagates within a sharp well- defined path at ~49° to the tensile axis in the stk-C, see Fig. 4.3 (b). Nonetheless, it is rather diffused in the corresponding stg-C, deviating significantly from the maximum resolved shear stress angle of 45°, and rather following the connected path of the deformed NG matrix at 58°, as displayed in Fig. 4.3 (e). 4.3.3 Local Strain Evolutions Considering the distinctive local strain distribution in the 35%MG-stk-C and 35%MG-stg-C, it is instructive to make a quantitative evaluation of the difference in their degree of strain localization. In Fig. 4.4, we illustrate the statistics of the composite atomic eM at two loading values, e = 0.10 and 0.15. As the calculations only consider atoms undergoing plastic deformation (0.2 < 51 eM < 2) in the system, we start by dividing this strain range into 200 bins. All the atoms in the system with 0.2 < eM < 2 are then used to fill up the bins according to their eM value. The statistics are then calculated by normalizing the number of atoms in each bin by the total number of atoms in all bins. To better depict the difference, we plot the curves on a logarithmic scale. At e = 0.10, the two composites display nearly identical statistics. As the engineering strain increases to e = 0.15, statistics of both composites suggest a significant increase in the fraction of atoms with eM > 0.5. However, the 35%MG-stg-C curve displays higher values for the moderate deformation (0.3 < eM < 0.8) and lower values for the severe deformation (0.8 < eM < 2.0), in comparison with the 35%MG-stk-C. That quantifies the higher efficiency of the former design at delocalizing local strain deformations and constraining severe shear flows than the latter. Fig. 4.4 Evolution of the local strain statistics for different composite designs. 52 4.4 Discussion 4.4.1 Comparisons between Metallic Glass-Nanoglass Composites and other Metallic Glass Composites It is helpful to compare the MD predicted results for MGNG-Cs in this work with previously reported results on other MG composites. For instance, a particularly interesting nanolaminate Cu50Zr50 MG composite with an NG second phase was investigated[186]. In agreement with our predictions, the composite indicates a significant synergy of the mechanical behavior of the two phases. In both works, by modifying the volume fraction of the two phases, the strength and ductility of the composite can be tuned, i.e., a higher volume fraction of the MG phase yields higher strength, while a higher fraction of the NG phase yields higher ductility. However, the synergy mechanism in this brick-and-mortar model is different from that in nanolaminates. In nanolaminate models, in order to prevent the failure by SB propagation, incipient SBs must be confined within NG layers by carefully choosing the thickness of the MG layers, e.g., the thickness of MG layers has to be comparable with the grain size of NG layers. That constrains the maximum volume fraction that can be used as the second phase and hence the enhancement of the composite strength. By comparison, the grid layout of the two phases in the 30%MG-stk/stg-C prompts symmetrical crisscross paths for shear flows in the composite. These paths effectively delocalize the local strain across the system as a whole, preventing the formation of critical SBs. The mechanical behavior of nanopillars constructed with nanolaminate MGNG-Cs was the topic of yet another work[159], where the conclusion was similar to that of the bulk nanolaminate[186]. The enhanced ductility has also been reported in a sandwich structured MGNG-C[24,193]. It was indicated that the softer NG layers can provide effective protection and shielding to the MG core, absorbing impact energy. Compared with the other designs mentioned above, the sandwich 53 architecture can only delay the formation of incipient SBs but not their propagation. It is worth mentioning that NGs with a bimodal grain size distribution may also be viewed as MGNG-Cs, where large grains are analogous to the randomly arranged MG second phase and small grains are analogous to the NG matrix[57]. Therefore, the deformation mechanism of the bimodal NG is rather similar to the brick and mortar designed MGNG-Cs. As presented in this work, the MG phase in brick and mortar designed MGNG-Cs can be arranged in two different ways, stacked or staggered. It is worthwhile to highlight the differences in the synergy of the two phases for these two arrangements. According to the deformation profiles in Fig. 4.3, when the volume fraction of the second phase is ~35%, the second phase arrangement makes a more noticeable difference in the phase synergy. However, the mechanical behavior of MGNG-Cs is less dependent on the phase arrangement for higher (> 40%) or lower (< 30%) volume fractions of the second phase, i.e., the former always fails by critical SB propagation and the latter always overwhelms SB localization. While Fig. 4.3 suggests that the threshold volume fraction of the transition in failure mode is ~35% for both designs, the deformation profiles in Figs. 4.3 (b) and (e) indicate that the stg-C design prompts a rather diffused SB propagation path, indicating a distinctly stronger synergy effect in the composite. The effectiveness of the stg-C design in synergizing the two phases in MGNG-Cs can be further verified by the local strain statistics in Fig. 4.4. As the overall strain increases to e = 0.15, the stg-C curve displays a higher fraction of atomic local strain in the moderate deformation range compared with the stk-C curve. Consequently, the fraction of atomic local strain in the severe deformation range is limited in the stg-C case. 54 4.4.2 Potential Applications of Metallic Glass-Nanoglass Composites MGs have been widely explored in biomedical applications, e.g., implants and stents, due to their high strength and biocompatibility[39,174]. Our results indicate that MGNG-Cs may broaden investigation in this area. Firstly, the MGNG-C considered here is composed of a bulk NG matrix and an MG second phase of the same composition, which implies that the biocompatibility of the composite should be as outstanding as its MG counterpart. Secondly, to fulfill a specific application requirement, composites with desired mechanical properties, e.g., strength-ductility compromise, could be achieved by tuning the second phase volume fraction and architecture. It is worth noting that when it comes to real applications, more complex mechanical tests such as multiaxial loading and cycling loading are usually carried out, where other mechanical properties of the composite such as fatigue, plays an important role in the failure mechanism[194]. In future work, the effects of aspect ratio and size in the composites should be clarified. As has been reported in previous experimental work, the aspect ratio strongly affects the compressive ductility of the bulk MG[195,196], and decreasing sample size tends to hinder shear banding[197–199]. Such effects could involve more complicated mechanisms in MGNG-Cs, as the size and aspect ratio should be considered not only for the overall sample but also for each phase constituent. 4.5 Conclusion In summary, we studied the mechanical behavior of MGNG-Cs under uniaxial tensile loading by performing MD simulations. Samples were constructed from a 3 nm-grain-size Cu64Zr36 NG matrix and an MG second phase with a brick-and-mortar design. By tuning the volume fraction and arrangement of the second phase, distinct deformation mechanisms are observed. With 30% MG second phase, the composites display significantly enhanced overall strength compared with the NG counterpart, while preserving the superplasticity of the matrix. Higher MG volume 55 fractions render higher strength while sacrificing ductility. At the same phase fractions, an optimized mechanical synergy of the two phases can be achieved by arranging the MG phase in a staggered way. Such an architecture is able to effectively delocalize local plastic deformation and hence delay the generation of critical SBs. These results point out new possibilities in the field of MG composites. 56 Chapter 5: Shape Memory Alloy-Metallic Glass Composites 5.1 Introduction As been mentioned in the previous chapter, the use of BMGCs is an effective approach to overcome the poor ductility of monolithic MGs. Such a conclusion has been widely reported in previous work of BMG matrix-based composites with crystalline inclusions or precipitates second phase[56,60,66,200–202]. Recently, BMGCs with a shape memory second phase have drawn attention due to their superior tensile ductility and work-hardenability[60,61,67,76,191,203,204]. The unique tensile behavior of these BMGCs was suggested to be a result of a deformation-induced martensitic transformation[61,63,205,206]. While recent reports indicate that shape memory BMGCs can enhance significantly the mechanical properties of BMGs[31,61,191,207–209] it is still to be determined the microstructure that maximizes the synergy of the composite phases. Commonly, the crystalline phase takes the form of dendrites or irregular precipitates[60,66]. The development of freeze-casting[74] suggests alternative microstructure designs such as nanolaminates[67], nanoparticles[190,210], and rectangular inclusions [77,211] which can optimize the mechanical properties by tuning the second phase parameters, e.g., volume fraction and size. Alternatively, the mechanical properties can be further optimized by following nature- inspired designs[212–214]. In this work, we considered the design of shape memory BMGCs, which as discussed are expected to constrain the formation of critical shear bands in the MG matrix. When the failure mechanism shifts from MG matrix shear banding to crystalline inclusion fracture different strategies are required to further improve the mechanical performance of BMGCs. We generate BMGCs with brick-and-mortar designs inspired by nacre[215] and investigate their mechanical 57 properties. We perform tensile loading with MD simulations on two designs: one where crystalline ‘bricks’ are stacked, named stk-BMGC, and one where bricks are staggered, named stg-BMGC. 5.2 Simulation Details Large-scale MD simulations are carried out using a validated CuZr force field[216] and PBCs. To be noted, the description of the instability of the CuZr B2 phase at low temperature and the related transformation into a stable structure is a challenging task for the available interatomic potentials for CuZr. In this work, we first consider the interatomic potential functions of Cheng et al.[107] and Mendelev et al.[216] and then decided to use the latter because it provides an overall robust description of the two concerned phases, i.e., it describes well both the mechanical properties of the Cu64Zr36 MG and the instability of the CuZr B2 phase at low temperatures. In addition, the use of Mendelev et al.'s potential allows us to complement and compare directly our results with previous simulation studies. The stk- and stg-BMGC samples are generated with dimensions of 80 ´ 40 ´ 5.4 nm 3 and contain ~1.04 million atoms. A 2-fs time step is used during the sample preparation stage. Initially, a Cu64Zr36 liquid is thermalized at 2000 K for 0.2 ns, then quenched to 50 K at 10 10 K/s, and finally relaxed at 50 K for 0.4 ns at zero pressure to produce a BMG[108]. The sample is then cut according to the composite design and filled with Cu50Zr50 crystalline bricks, as illustrated in the top left corners in Figs. 5.1 (a) and (b). The stk- and stg- BMGCs have bricks with dimensions of 21.1 ´ 4.4 nm 2 and mortars with 5.5 nm thickness. Bricks are columnar along the z-direction. This construction yields a 0.35/0.65 volume fraction of crystalline/glassy phases, respectively. The BMGCs are sintered at 50 K and 3 GPa external hydrostatic pressure for 0.4 ns and finally relaxed at 50 K for 0.4 ns at zero pressure. Uniaxial tensile loading is applied at 50 K with a 5-fs time step, using a strain rate of 4´10 7 s -1 along the 58 longest system dimension (B2 phase in <100> direction). OVITO is used for analysis and visualizations[120]. 5.3 Simulation Results 5.3.1 Deformation Profiles To obtain an overview of the deformation process, we illustrate the evolution of the BMGCs local strain in Fig. 5.1. Colors indicate the level of von Mises shear strain, eM. Results reveal a sharp contrast between the deformation and failure modes in the two models. In the stk-BMGC, see Fig. 5.1 (a), plastic deformation initially occurs in the MG matrix, wrapping around the top and bottom sides of bricks. Its microstructure is preserved well at further deformation until a nanocavity shows up in a crystalline brick. Concurrently, the local strain distribution becomes asymmetrical and “dendritic” paths for shear flow develop in the MG matrix. In contrast, the Fig. 5.1 Deformation profiles of the bulk metallic glass composite (BMGC) models during tensile loading. The stacked BMGC (stk-BMGC) model is displayed in (a), and the staggered BMGC (stg-BMGC) model is displayed in (b). Atoms are colored based on their von Mises local strain value. 59 stg-BMGC develops a noticeable crisscross plastic deformation pattern, see Fig. 5.1 (b), which helps to homogeneously release the stress. With further loading, the bricks shift from each other, and larger diffused shear transformation zones emerge, whereas no critical shear band forms. Fig. 5.2 Tensile loading stress-strain curves. (a) Stress-strain curves for the pure MG and B2 phases. (b) Stress-strain curves for the stk-BMGC and the stg-BMGC models. Curves for the subset of atoms in the crystalline bricks and glassy matrix are displayed in dash lines. 60 5.3.2 Stress-strain Curves of the Composites and Corresponding Phases To gain further insights into the deformation process, we plot tensile loading stress-strain curves for the composites in Fig. 5.2 (a) and their pure phases in Fig. 5.2 (b). The B2 phase curve displays peaks corresponding to the two martensitic transitions[217]. The plateau at the very beginning of loading is a result of the detwinning of the initially twinned B2 structure, which is spontaneously developed after relaxation as it is unstable under low temperature[218,219]. The MG curve displays a typical response with a large elastic region followed by a sharp stress drop at the yield point. Considering the significant contrast in the response of the composite phases, we calculate and display their stress-strain curves separately. The stk-BMGC curve suggests common features as that of the pure B2 phase. Both composite phases curves follow a similar trend, albeit at different stress levels. A sharp stress drop at e = ~0.4 followed by a long-lasting flow indicates the failure of the system. By comparison, the stg-BMGC curve indicates a tensile behavior dramatically different than those displayed by the stk-BMGC. The curve unexpectedly yields in a two-step manner, i.e., the first one at e = ~0.05, 𝜎 = ~2.5 GPa and the second one at e = ~0.9, 𝜎 = ~7.5 GPa, separated by a strain-hardening regime. Although the following peaks on the overall curve are hard to distinguish, the ones for the crystalline phase can be roughly recognized at e = ~0.9, ~1.2, and ~1.4. The stress drops in a stepped way after the third peak into a flow stress regime, pointing out the failure of the system. The amorphous phase displays a smooth curve with a constant stress level suggesting a homogeneous deformation. 5.3.3 Local Stress Analysis of the Composites during Deformation To further highlight the deformation mechanisms of the glassy matrix, we illustrate in Fig. 5.3 the evolution of the local stress. The local stress is calculated based on the local atomic virial stress as following, 61 𝜎 "" = − " N ( ∑ ( " ' ∑ 𝑓 &C # 𝑟 &C # +𝑚 & 𝑣 & # ' ! CK", CO& ) ! &K", &∈N ( , (5.1) where, 𝜎 "" is the component of the virial stress along the x loading direction and averaged over N neighboring atoms within a spherical volume of radius 8 Å. 𝛺 G is the total Voronoi atomic volume of the atoms within the spherical volume. One can see bold contrast in the local stress between the glassy and crystalline phases. The deformation in the regions where crystalline and glassy phases coexist is heterogeneous and displays a net Poisson ratio different from the glassy region at the joints. Consequently, the local stress in the amorphous layer parallel to the bricks becomes low and even negative. In Fig. 5.3 (a), the local stress forms aligned bands in the stk-BMGC matrix during the early stage of deformation. At e = 0.4, stress distribution distortions accompanied by bending and necking of bricks can be observed. In Fig. 5.3 (b), the stg-BMGC displays initially localized stress at the two phases joints and develops a zigzag pattern with further loading. The design generates a rather delocalized stress distribution in the matrix and peak local stress levels are comparatively lower than those displayed in Fig. 5.3 (a) for the stk-BMGC at similar strain values. Remarkably, in the whole range of deformation before failure, the stress distribution in the MG matrix remains symmetrical and crystalline bricks maintain their integrity. 5.3.4 Phase Transformations for the Shape Memory Phase We choose a single representative brick in each model to illustrate their mechanical response. Local stress distribution in the crystalline bricks is displayed in Fig. 5.4 upper panel while the corresponding crystalline structure is displayed in the lower panel. To be noted, here we extend the strain range to depict a full picture of the stress localization and stress release process. Stress initially localized at the core of the bricks radiates outwards as the BMGCs further deform. For the chosen bricks an overall stress release is found at ε = 0.7 for the stk-BMGC and ε = 2.0 for the stg-BMGC. In Fig. 5.4 lower panel, atoms are colored according to their crystalline structure at 62 Fig. 5.3 Local stress distribution of the BMGC models during tensile loading. The microstructure evolution of the stk-BMGC model is displayed in (a), and of the stg-BMGC model is displayed in (b). Atoms are colored based on the local stress value. different strain levels in order to characterize the strain-induced martensitic transformation in the second phase. Results indicate a multi-step phase transition in both models: B2 to intermediate R, then to L10 phase. For reference, crystal structures of the B2, R, and L10 phases are illustrated in the lower part of Fig. 5.4. Once the stress is released at the brick failure, the L10 phase partially reverts to the B2 phase. To be noted, it is challenging to accurately distinguish the R phase from other irregular crystal structures such as the boundaries between the MG matrix and the crystalline bricks. Hence, all these regions are colored pink. It is worth mentioning that many previous studies report the B19’ rather than the L10 phase as the crystal structure after martensitic transformation of the B2 phase[61], but accurate first-principles calculations of the energetics of CuZr in nine candidate crystal structures, e.g. B2, B11, L10, B33, B19, B27, B19’, p, µ, indicates that the difference in energy among all the stable phases is no more than 0.6%[220]. Such a claim of the L10 phase was also reported elsewhere[221]. 63 Fig. 5.4 Local stress distribution and phase transformation for the shape memory phase in the BMGC models during tensile loading. Only one brick in each model is illustrated here. The microstructure evolution of the stk-BMGC model is displayed in (a), and of the stg-BMGC model is displayed in (b). Atoms are colored based on the value of the local stress in the first row and by the corresponding crystalline structure in the second row. The results indicate a strong synergy of deformation mechanisms in the stg-BMGC. While maintaining a high average stress level comparable with that of the stk-BMGC, the stg-BMGC exhibits a much higher failure strain, implying both higher ductility and toughness. It is natural to ask how the ductility of the stg-BMGC can far surpass that of the stk-BMGC? A key point is that the two designs generate very different stress distributions. The induced strain distribution in the stk-BMGC is effective for delaying the formation of shear bands. For the stg-BMGC, though there are induced visible “concentrated strain bands”, the local stress distribution indicates that they do not form a favorable path for the formation of a runaway shear band either. While the stress is 64 concentrated in the stk-BMGC matrix at the joints between B2 bricks, the stress is continuously released and delocalized in the stg-BMGC matrix as the B2 bricks flow for increasing straining. As a consequence, voids emerge in crystalline bricks in the stk-BMGC earlier than the ones in the stg-BMGC and lead to failure. Once the brick fails, it no longer helps to delay the formation of a runaway shear band. Hence, the stg-BMGC turns out to be more effective. 5.4 Discussion 5.4.1 Comparisons between the Stacked and Staggered Bulk Metallic Glass Composites As mentioned above, the remarkable enhancement in ductility of the stg-BMGC comes with a surprisingly modest sacrifice of the ultimate tensile strength. In our work, we demonstrated that brick and mortar designed BMGC with a large 35% volume fraction of the second phase is able to prevent the propagation of shear bands, shifting the failure mode from shear bands propagation to deformation and failure of the crystalline phase. One can note that the MG matrix plays a major role in defining the strain profile, as it is able to release stress by delocalized plastic deformation, while the crystalline bricks dominate the stress profile as they are able to absorb a large amount of elastic energy. The synergy of deformation between crystalline and amorphous phases in the two BMGC models is ultimately reflected in their stress-strain curves. It can be seen that both the stk and stg-BMGC curves combine a strain-hardening regime, indicating a strong synergy of the stress. Consequently, stk- and stg-BMGCs display similar strengths. 5.4.2 Comparisons among Bulk Metallic Glass Composites with Different Microstructures It is instructive to compare the predictions of this work with related recent reports. There are a bulk of akin investigations focused on crystalline nanoprecipitates-based composites, which indicates enhanced global plasticity [66,71,222]. In general, those investigations indicate that nanocrystals are able to effectively constrain the catastrophic propagation of shear bands. The 65 same general mechanism applies to shape memory BMGCs, even though work hardening caused by the deformation-induced martensitic transformation of the B2 phase is also of key importance to describe the deformation profile[61,63,206]. Optimization of BMGC microstructures in order to improve their mechanical properties is an active research field[77,190]. Simulations of Sopu et al. on BMGCs indicate that high crystalline volume fractions lead to a constraint of critical shear bands[77]. These results corroborate the modeling results of Shete et al. that SBs are forced to circumvent crystalline inclusions delaying strain localization[209] and conclusions from Guo et al. for porous NiTi shape memory alloy BMGC[191]. In this work, we first consider BMGCs with the second phase volume fraction and crystal spaces purposely chosen, to prevent the failure by the development of critical shear bands. Then a brick-and-mortar microstructure with staggered crystalline bricks is considered to further improve their mechanical properties. Results demonstrate such a design can effectively delay the deformation of crystalline inclusions, which further enhances the ductility of shape memory BMGCs. Advances in the synthesis of BMGCs, based on flash annealing suggest that further tuning of heat-treatment procedures may pave the way to the synthesis of shape memory BMGCs with such brick and mortar designs[204,223,224]. 5.5 Conclusion In this work, we use MD simulations to characterize the tensile deformation and failure of shape memory BMGCs. When the failure of BMGCs is transited from a critical shear band to the second phase for given structural parameters, we seek different strategies to further improve their mechanical properties. Results indicate that a brick-and-mortar design with staggered crystalline bricks can better synergize the mechanical properties of the glassy and crystalline phases by 66 effectively distributing the local stress and preventing the generation of critical shear bands. Hence, it displays enhanced ductility while preserving strength. 67 Chapter 6: Uncovering Metallic Glasses Hidden Vacancy-like Motifs 6.1 Introduction In contrast with crystalline materials, where physical properties are largely predictable by their well-defined long-range order, the relationship between atomic arrangement and material properties in MGs is not well understood. The first challenge is to uncover structural differences in MGs that show seemingly no discernible atomic packing patterns. Once that is complete, the next step is to bridge the MG’s structural description to its macroscopic properties. Though long- range order is absent in MGs, distinct local atomic configurations at the level of nearest neighbors, i.e., short-range order (SRO), have been widely recognized[19,51,225]. Many structural characterization techniques have been used to discover key features of the atomic-level structure of MGs, such as pair distribution function, structure factor, common neighbor analysis, bond angle analysis, etc.[107,226,227]. The atomic Voronoi tessellation is another effective tool that can be used to extract information about the atomic packing topology[228,229], i.e., the atomic Voronoi polyhedron, where the arrangement of the nearest-neighbor atoms can be implied by the Voronoi index. Many studies have revealed that the prevalent atomic Voronoi motifs found in MGs depend on the chemical composition and play an essential role in determining the material properties[16,226–228]. For example, the Cu-centered clusters with the Voronoi index <0, 0, 12, 0>, named Cu-centered full icosahedra (Cu-FI), are the predominant atomic packing in Cu-rich Cu-Zr MGs[50]. This icosahedron motif is linked to the tendency to form five-fold bonds, allowing an efficient polytetrahedral packing. In contrast, the prism-type SRO is the featured polyhedron motif in Mg-based MGs due to bond shortening caused by charge transfer[230]. While favorable SRO motifs in MGs have been intensely discussed, less attention has been given to unfavorable local atomic packings. Nonetheless, unfavorable motifs, such as those with 68 liquid-like features, may play a crucial role in the response to external stimuli, leading to thermal agitation, heterogeneous elastic deformation, and the onset of plastic events. Previous research on configurational disorder has highlighted the importance of geometrically unfavored motifs (GUMs)[21,50,231,232]. Local regions rich in GUMs tend to possess larger excess free volume and corresponding lower energy barriers for thermal- or stress-induced relaxation. This points out an opportunity for designing MGs with desired properties by tuning the population of favorable as well as unfavorable SRO motifs. However, the lack of a proper interpretation of the GUMs topological descriptions precludes further quantitative predictions of structure-sensitive properties. One can see the role of GUMs for MGs as the equivalent of defects in crystalline structures. Nonetheless, it remains to be found a well-defined and ubiquitous disordered-structure descriptor for MGs as defects are for crystals. As the simplest type of defect in crystalline metals, vacancies denote missing atoms from lattice sites. These defects play an important role in atomic diffusion, relaxation, and deformation processes, affecting material properties such as the bulk and Young’s modulus, melting temperature, thermal and electrical conductivity, etc.[233,234]. As the vacancy-like formation process gives rise to MG specimen expansion after thermal treatments, the corresponding formation energy can be estimated by differential scanning dilatometry[235] and the change in radial atomic distances can be observed in synchrotron X-ray diffraction experiments[236]. However, identifying a specific topological signature representing a vacancy like defect is challenging due to the absence of a reference lattice in the amorphous structure[237,238]. Noticeably, some local features, e.g., higher local tensile stress, larger excess volume, and lower atomic packing, strongly suggest the existence of vacancy-like motifs in MGs[43,239,240]. 69 In this work, we describe the SRO of MGs by calculating the atomic Voronoi tessellation, which provides data that is used in ML models to mine eligible candidates for a topological signature that resembles vacancy defects. The data used in ML are collected from MD simulations of quenching Cu-Zr samples. The results reveal that the T5 (polyhedron with five triangular faces) and Q7 (polyhedron with seven quadrangular faces) are the two most likely motifs with vacancy- like features among all the atomic polyhedrons considered. The preliminary conclusion from the ML analysis is supported by additional simulations of a Cu64Zr36 MG sample with heat treatment and mechanical loading, showing that concentrations of the T5 and Q7 motifs follow an Arrhenius- like relationship with temperature and exhibit a strong correlation with the elastic and plastic behavior of the sample during mechanical deformation. Both the T5- and Q7-centered atoms display larger local entropy compared with the system average. However, larger atomic volume and local atomic tension are consistently displayed by Q7-centered atoms. Hence, we further recognize the Q7 as the most representative vacancy-like defect motif in MGs. These results provide new insights into the interpretation of unfavorable topological descriptors and a novel path for the construction of structure-property relationships in MGs. 6.2 Simulations and Methods 6.2.1 Molecular Dynamics Simulations Large-scale MD simulations are carried out with the LAMMPS package[241], using an EAM potential developed for the CuZr systems[173]. For the quenching process, cubic MG samples containing ~ 21,300 atoms are created by melting their parent CuxZr100-x alloy (x = 0, 20, 36, 40, 44, 50, 56, 60, 64, 70, 80, 100) and equilibrating at T = 2000 K for 0.2 ns, followed by cooling down to T = 50 K at a rate of 1010 K/s. To investigate the MG under heat treatment, we select the specific composition, Cu64Zr36, as an example. A larger Cu64Zr36 MG sample containing ~2.1 70 million atoms is generated by replicating the small MG model generated through the quenching process mentioned above. The sample is then heated from T = 50 K to 1800 K at a rate of 5×1011 K/s. Periodic boundary conditions are applied in all directions. The NPT ensemble algorithm with zero external pressure and an integration time step of 2 fs are employed throughout the quenching and heating simulations. The temperature is changed following a linear ramp by setting start and end temperature values. For the mechanical loading simulations, we generate another Cu64Zr36 MG sample of size ~5.4 (x) × 54 (y) × 140 (z) nm 3 containing ~2.5 million atoms. Uniaxial tensile and compressive loading is carried out at a constant strain rate of 4×10 7 s −1 along the z-direction with a time step of 5 fs. The temperature is kept at T = 50 K and PBCs are applied in all directions. The engineering stress is extracted by calculating the Virial stress along the loading direction. 6.2.2 Voronoi Tessellation The atomic packings in each MG sample are monitored using the Voronoi tessellation method to yield detailed information about the packing topology. Planes are drawn to bisect each line connecting the central atom and all neighboring atoms. Then the region enclosed by these inner planes is defined as the Voronoi cell of the central atom[242]. The computational efficiency of our in-house code is optimized by constructing Delaunay tetrahedra[243]. OVITO is used for Voronoi polyhedral visualization[120]. 6.2.3 Motif Naming Considering the intrinsic flexibility in atomic arrangements in amorphous structures, we broaden the range of possible topological descriptors to represent the vacancy-like local environment in MGs. Commonly, atomic Voronoi polyhedrons are analyzed considering the number of each type of face in a Voronoi polyhedron, i.e., denoting it with Voronoi index <n3, n4, n5, n6, …>, where ni denotes the number of i-edged faces of a polyhedron. A vacancy-like defect, 71 representing a local region of lower density and short-range order, is not a well-defined point topology represented by a corresponding Voronoi index. Here, to better identify vacancy-like motifs, we adopt an approach. We decouple the Voronoi index as individual structural descriptors representing a polyhedron as Sn, where S indicates the shape of the face, i.e., triangular (T), quadrangular (Q), pentagonal (P), or hexagonal (H), and n is the number of times that face is represented in the polyhedron. One can expect substantial overlaps in motifs represented by such notations. For instance, an atom characterized by the Voronoi index <0, 2, 8, 2> is denoted by motifs named either T0, Q2, P8, and H2, since its polyhedron has zero triangular, two quadrangular, eight pentagonal, and two hexagonal faces, simultaneously. 6.2.4 Machine Learning Modeling The goal of the ML analysis is to identify atomic packings that meet the necessary criteria for being considered vacancy-like motifs. The training criterion assumes that vacancy-like motifs in MGs should follow an expected vacancy physical behavior, i.e., their populations should follow a well-defined correlation with temperature, regardless of composition. If an atomic packing motif is recognized as a vacancy-like defect, one should be able to predict its population at a certain temperature without knowing any information about the MG composition. It is worth mentioning that this assumption is not in contradiction with the high likelihood that the activation energy of these motifs is dependent on the MG composition. Following these considerations, we construct a dataset using the results collected from MD quenching simulations, i.e., populations of thirty-eight different structural descriptors, Sn (T0-T5, Q0-Q7, P0-P12, H0-H10), in twelve CuxZr100-x systems (x = 0, 20, 36, 40, 44, 50, 56, 60, 64, 70, 80, 100) sampled at an interval of ~20 K from T = ~2000 K to 50 K. Then we design an ML regression model to evaluate the likelihood of predicting the population of each Sn by providing a temperature value. The greater the prediction success, the 72 more likely an Sn represents vacancy-like motifs. To avoid heavy influence from data outliers on the regression model, while not completely ignoring their effect, we employ a Huber regressor[244], as implemented in the Scikit-learn package[245], for training and testing the dataset. The loss function in the Huber regressor is formulated as 𝑚𝑖𝑛 Q,R ∑ d𝜎+𝐻 S d T $ Q:$ $ R e𝜎e+𝛼‖𝑤‖ ' ' L &K" , (6.1) 𝐻 S (𝑧) =i 𝑧 ' , 𝑖𝑓 |𝑧| < 𝜖 2𝜖|𝑧|−𝜖 ' , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 , (6.2) where w is the model coefficients to be optimized, 𝜎 is the scaling factor to be optimized, 𝛼 is the penalty of the squared L2 regularization, and 𝜖 determines the outliers, i.e., the smaller the 𝜖, the more robust to outlier data points. Here, we set 𝛼 = 10 -4 and 𝜖 = 1.35. The target value 𝑦 & is the normalized data of the population of Sn and the input 𝑋 & is the temperature record in polynomial combinations with a degree up to five. The dataset is randomly split with 70% samples for training and 30% samples for testing. We use the coefficient of determination, denoted R2 score, see Eq. (6.3), to evaluate the performance of the regression model and interpret it as a successful criterion in predicting the population of Sn by temperature. 𝑅 ' =1− UVV WVV , (6.3) where, 𝑅𝑆𝑆 is the residual sum of squares, 𝑅𝑆𝑆 =∑(𝑦 s & −𝑦 & ) ' & , where 𝑦 s & is the predicted value from the model and 𝑦 & the actual value and 𝑇𝑆𝑆 is the total sum of squares, 𝑇𝑆𝑆 = ∑(𝑦 & −𝑦 ( t) ' & , where 𝑦 ( t is the average value. 73 6.3 Simulation Results 6.3.1 Identification of Vacancy-like Motifs In Fig. 6.1 we show the results from the ML analysis (left panel) and illustrations of the top two candidate vacancy-like motifs, T5 and Q7 (right panel). The R2 score, also known as the coefficient of determination, measures how well the population of the Voronoi polyhedron motif, Sn, can be predicted solely based on temperature. It should be noted that this ML process is used as a screening tool for identifying vacancy-like motifs, and a high R2 score is necessary yet not sufficient for a motif to be considered a vacancy-like motif. The R2 score maximum value is 1.0, indicating perfect prediction, while the value can be negative, indicating a lack of predictability. Accordingly, we conclude that Q7 and T5, which yield an R2 score greater than 0.9, are the top Fig. 6.1 Machine learning results and illustrations of vacancy-like motifs candidates. The left panel shows the R 2 score for each motif, Sn. The right panel shows the schematics of Voronoi polyhedrons with five triangular faces (T5) and seven quadrangular faces (Q7), which are the motifs with the top two R 2 scores. The schematic of each motif includes three examples with corresponding Voronoi cells, atomic configurations, and Voronoi indices. Green and brown spheres represent Zr and Cu atoms, respectively. The Voronoi indices are denoted with <n3, n4, n5, n6, …> up to the last non-zero ni term, where ni is the number of i-edged faces of the polyhedron. 74 two candidates representing vacancy-like motifs. The R2 scores for the T4, T3, and P1 motifs are relatively lower, but still above 0.7, suggesting they may also be potentially recognized as vacancy- like motifs. Nevertheless, for the aim of this work, here we focus on the top two motifs and leave further analysis of the other candidate motifs for future study. To illustrate the variety of possible atomic packings of each candidate motif, we select several Voronoi polyhedrons characterized as T5 or Q7, see Fig. 6.1 right panel. Significant variations in the topology can be noticed within the T5 and Q7 motifs. These are attributed to the inherent tendency of vacancy-like motifs to develop diverse local structural distortions. Fig. 6.2 (a) Potential energy, (b) FI concentration, (c) T5 concentration, and (d) Q7 concentration of CuxZr100-x samples during quenching. The colors indicate the Cu composition of the system. The black dashed lines in (c) and (d) are guides to the eye. 75 Figure 6.2 shows the variation of the T5/Q7 population with temperature, as these data are extracted from the quenching process of CuxZr100-x samples simulated by MD and collected for ML analysis. The potential energy and full icosahedra (FI, with the Voronoi index <0, 0, 12, 0>) concentration are also evaluated for references. It is worth mentioning that the Cu composition of the system ranges from 0% to 100%, which can cause crystallization after quenching, particularly for pure Cu and pure Zr. However, we include these systems in our analysis to increase the robustness of our ML model and further analyses. The concentration of FI varies greatly at low temperatures due to differences in composition. However, a universal correlation between the T5/Q7 concentration and temperature can be observed, which is also evidenced by our previous ML results, i.e., an R2 score above 0.9 for Q7 and T5. Interestingly, when we conduct the same ML evaluation on FI, the model yields a negative R2 score, -0.22, indicating a poor correlation and predict ability of the concentration of FI based on temperature. Both FI and T5/Q7 display a distinct transition point as the temperature changes, which is a result of the liquid-to-glass/crystal transition during quenching. This transition causes a significant increase in the concentration of FI at a lower temperature for most systems, reaching a maximum of up to 18%. However, the FI concentration can also drop to zero due to the crystallization of pure Cu and Zr. In contrast, the concentration of T5/Q7 decreases continuously with temperature, reaching a maximum of only 0.5% at the lowest temperatures measured, regardless of composition. 6.3.2 Thermal Response of T5 and Q7 motifs With T5 and Q7 being identified as vacancy-like motif candidates based on results from the quenching process, we conduct further heating simulations on a larger MG sample with a fixed composition to detailly analyze their characteristics of thermal response. The temperature dependence of T5/Q7 and FI concentrations of a Cu64Zr36 MG during heating is plotted in Fig. 6.3, 76 with potential energy provided as a reference to indicate the glass transition point between the brittle glassy state and the viscous liquid state. The results indicate that as the temperature increases, there is a decrease in the concentration of FI and an increase in the concentrations of T5 and Q7 motifs. A significant change in the concentrations of FI, T5, and Q7 aligns well with the glass transition at ~950-1050 K. It is worth noting that this temperature is higher than the transition point observed in Fig. 6.2, as they depict different thermal processes. The changes in T5 and Q7 concentrations after the transition point are well described by a linear fit, represented by light purple and grey dashed lines. On the other hand, an Arrhenius-like function fit, represented by dark purple and black dashed lines, describes well the data before the transition, 𝑐 WX,YZ [ =𝛼∙𝑒𝑥𝑝d :\ )*,, - ] . W e, (6.4) where 𝛼 is a scaling factor, 𝐴 WX,YZ is the activation energy for T5/Q7 motif formation, 𝑘 ^ is the Boltzmann constant, 𝑇 is the temperature, and 𝑐 WX,YZ [ is the rescaled concentration of T5/Q7, Fig. 6.3 Temperature dependence of FI, T5, and Q7 concentrations, as well as the potential energy in a Cu64Zr36 MG during heating. The blue dashed curve is a guide to the eye. The dark purple/black dashed curves are fitted to T5/Q7 data points for T < 1050 K using an Arrhenius function. The light purple/grey dashed lines are fitted to T5/Q7 data points for T > 1050 K using a linear function. 77 obtained by subtracting a residual concentration, 𝑐 WX,YZ , , from the calculated concentration, 𝑐 WX,YZ , 𝑐 WX,YZ [ =𝑐 WX,YZ −𝑐 WX,YZ , . (6.5) The fitting yields the activation energies 𝐴 WX =~0.793 𝑒𝑉 and 𝐴 YZ =~1.157 𝑒𝑉 as well as the residual concentration 𝑐 WX , =0.033%, 𝑐 YZ , =0.009%. To understand the intrinsic difference between the T5 and Q7, as well as their contributions to the MG behavior, we scrutinize the correlations between the T5/Q7 motifs and other atomic-level properties. First, we employ the simulation results from the above-mentioned Cu64Zr36 MG sample during heating and calculate the local entropy[246], atomic volume, and atomic level pressure statistics of atoms centered in the T5, Q7, or FI polyhedrons, as well as the whole system statistics. The results obtained at 310 K, 810 K, and 1630 K are shown in Fig. 6.4, where each of the 50 points curves represents bin average atomic fractions. The FI and T5/Q7 local entropy statistics contrast at each of the three temperatures considered, as shown in Figs. 6.4 (a)-(c). While the curve for the FI motif has a left shift from the whole system average curve, the T5/Q7 curves display a right shift in their distributions. Their distribution curves become less distinct as temperature increases and tend to converge with that of the whole system at 1630 K, as shown in Fig. 6.4 (c). Differences are also found among the T5, Q7, and FI statistics of Voronoi atomic volume compared to that of the whole system at a lower temperature, see Figs. 6.4 (d) and (c). The distribution curve of the whole system has double peaks corresponding to Cu and Zr atomic volumes, respectively. In contrast, the Q7 and T5 corresponding curves display only shoulders, which are more prominent in the Q7 distribution. Meanwhile, the single peak of the FI curve is a consequence of the fact that FI-centered atoms are mostly Cu, whereas T5/Q7-centered atoms show no atom-type bias as expected. Among the three motifs, the FI shows the lowest average atomic volume, while the T5 shows a value close to that of the whole system, and the Q7 clearly 78 Fig. 6.4 (a)-(c) Local entropy distribution of Cu64Zr36 MG atoms in the whole system, in the FI configuration, in the T5 configuration, and in the Q7 configuration at 310 K, 810 K, and 1630 K (liquid state), respectively. (d)-(f) V oronoi atomic volume distribution and (g)-(i) V oronoi atomic pressure distribution of atoms in the above-mentioned configurations at corresponding temperatures, respectively. shows a value exceeding the system average at both 310 K and 810 K. As the temperature increases beyond the transition point, see Fig. 6.4 (f), all distribution curves shift to larger atomic volumes and diminish the difference between the corresponding peaks for Cu and Zr atoms. Meanwhile, the contrast between the T5 and Q7 curves disappears while the sharp contrast between the FI and T5/Q7 curves remains. The average atomic volume of the FI is below the system average, whereas those for the T5/Q7 are visibly above. 79 To evaluate atomic level pressure, we calculate the virial per-atom stress tensor [247] and take the negative sum of the diagonal components as the per-atom pressure, which is then used in averages. The T5 and Q7 centered atoms in the pressure statistics curves show a distinct difference, with the FI curve aligning well with the curve for the entire system, see Figs. 6.4 (g)-(i). The Q7 curve shows a negative shift compared to the reference whole system, indicating a tendency towards a tensile environment. In contrast, the T5 curve shows a positive shift compared to the whole system, indicating a tendency towards a compressive environment. As temperature increases, the T5 curve tends to align with the FI and whole system curves, while the Q7 curve’s negative shift remains, indicating the persistence of a tensile environment. Based on these results, we conclude that the Q7 motif is rather vacancy-like than the T5 motif and is therefore the most suitable representative motif for MG vacancy-like defects. 6.3.3 Mechanical Response of T5 and Q7 Motifs under Loading Vacancies and vacancy-like defects can significantly affect the deformation and failure of materials. To evaluate the behavior of T5/Q7 motifs in the deformation of Cu64Zr36, we generate a sample with thin-film geometry and quantify the T5/Q7 statistics during uniaxial tensile and compressive loading. The results are illustrated in Fig. 6.5. During tensile loading (Figs. 6.5 (a)- (c)), both the yield and failure points of the MG sample can be distinctly inferred from the inflection point of T5 and Q7 concentrations at engineering strain 𝜀 = ~0.03 and ~0.07, respectively, as indicated by the grey and purple dashed lines. In contrast, such transitions during yielding cannot be identified in the evolution of the FI concentration, while the failure point can be identified in the FI concentration curve as its value halts the dropping and starts fluctuating around a given value. During compressive loading (Figs. 6.5 (d)-(f)), the yield and failure points of the sample (ε = ~0.032 and ~0.063) can be accurately predicted by the inflection point of the 80 Fig. 6.5 (a)-(c) Stress-strain curves and polyhedral concentration evolutions of Cu64Zr36 MG during uniaxial tensile loading at 50 K, and (d)-(f) compressive loading at 50 K. The black dashed lines in (a) and (d) imply the linear elastic response stage. The plumb grey dashed lines underline the yield point and the purple ones underline the failure point of the sample during deformation. Q7 concentration, whereas the inflection points of T5 and FI curves do not correlate well with the yield and failure points. To further understand the behavior of T5/Q7 motifs during mechanical loading, we compare the Voronoi atomic volume and local von Mises shear strain 𝜀 H statistics of atoms centered in FI, T5, and Q7 polyhedrons in the sample mentioned above. The calculations are done at 𝜀 = 0.06 for tensile and 𝜀 = 0.06 for compressive strain, corresponding to plastically deformed states before the 81 Fig. 6.6 (a) and (b) V oronoi atomic volume distribution of atoms in the whole system, in the FI configuration, in the T5 configuration, and in the Q7 configuration at 𝜀 = 0.06 tensile loading and at 𝜀 = 0.06 compressive loading, respectively. (c) and (d) V on Mises shear strain distribution of atoms in the above-mentioned configurations at 𝜀 = 0.06 tensile loading and at 𝜀 = 0.06 compressive loading, respectively. sample failure at 50 K. For both tensile and compressively deformed states, as shown in Figs. 6.6 (a) and (b), the FI motif has the lowest average atomic volume, the T5 displays a value close to the whole system average, and the Q7 has the largest value. There are striking contrasts in the von Mises shear strain distributions of the FI and T5/Q7, as shown in Figs. 6.6 (c) and (d). Compared to atoms centered in FI, atoms centered in T5 and Q7 participate less in the elastically deformed regime, i.e., 𝜀 H < ~0.2. Thereafter, as implied in the graph, a relatively larger fraction of T5 and Q7 participates in the plastically deformed regime, i.e., 𝜀 H > ~0.2. The difference between the T5 and Q7 local strain distribution curves is prominent under tension yet nearly non-existent under compression, suggesting their inherent motif topologies display distinct stress-strain asymmetries. 82 6.4 Discussion 6.4.1 Rationales for Considering Q7 as Vacancy-like Defects The results suggest that the T5 and Q7 polyhedrons are unique motifs that strongly correlate with the local structural disorder in MGs. Furthermore, the Q7 displays a rather striking similarity to crystalline vacancy defects. The rationale for considering the Q7 a vacancy-like motif in MGs is summarized as follows: (1) The Q7 concentration displays a strong correlation with temperature, as indicated by an R2 score above 0.9, regardless of the material phase change between brittle glassy state and viscous liquid state, or variations in the composition of the system, CuxZr100-x with x from 0 to 100. (2) At temperatures close to the transition point, the magnitude of the Q7 concentration in the simulated samples is ~10 -4 -10 -3 , which is similar to the typical vacancy concentration found in crystalline metals[233]. It is worth noting that, the measurement of Q7 concentration extends beyond the transition temperature. While it may not be realistic to define vacancies in ideal liquids, we use the term "vacancy-like" defects as they possess a unique defect topology in the viscous liquid state with a concentration below 10 -3 . (3) The Q7 concentration in a Cu64Zr36 MG during heating can be well-fitted into an Arrhenius-like relationship with temperature before transforming into a viscous liquid state, similar to the dependence of vacancy concentration in crystals obtained from theoretical analysis of the Gibbs free energy associated with single lattice vacancy formation[248]. (4) The Q7-centered atoms tend to display larger local entropy, atomic volume, and local atomic tension in comparison with the system average, indicating more disordered/liquid-like packing and increased excess free volume, similar to the characteristics of vacancy-like defects. Specifically, larger local entropy indicates more disordered/liquid-like packing[249], larger Voronoi atomic volume implies increased excess free volume[250], and vacancy-like defects can be defined as localized regions of lower density under tension[238,251]. 83 (5) The Q7 motif can be induced by plastic deformation. Although its concentration saturates at a magnitude as low as 10 -4 upon tensile/compressive loading, its variation from the initial value, 10- 5, can well predict the elastic-plastic transition and the failure point. Similar trends have been reported for vacancies in nanometals, i.e., increasing plastic deformation results in a higher concentration of vacancies or vacancy agglomerates[252]. Interestingly, Castellero et al. reported that decreasing the population of “defects” leads to a longer structural relaxation time and contribute to the formation of shear transformation zones[253]. For the reasons outlined above, we conclude that Q7 motifs embody vacancy-like motifs in MGs based on their correlation with local geometric frustration. 6.4.2 Energetic Implication of T5/Q7 Formation By fitting the T5 and Q7 data curves in Fig. 6.3 with Eq. (6.4) and Eq. (6.5), we obtain the activation energy values 𝐴 WX =~0.793 𝑒𝑉 and 𝐴 YZ =~1.157 𝑒𝑉 . Here, we interpret the physical meaning of 𝐴 WX and 𝐴 YZ as the formation energy of the T5 and Q7 motifs. The magnitude of 𝐴 WX and 𝐴 YZ are similar to the vacancy formation energy of pure metals predicted by empirical relations, as seen in Eq. (6.6): 𝑊 ≈10𝑘 ^ 𝑇 5 (6.6) where W is the vacancy formation energy, 𝑘 ^ is the Boltzmann constant, and 𝑇 5 is the melting point. Here, we substitute 𝑇 5 with the upper glass transition temperature, 1050 𝐾, measured in the Cu64Zr36 MG sample and estimate the vacancy formation energy with Eq. (6.6), obtaining 𝑊 = ~0.9 𝑒𝑉, within the magnitude of 𝐴 WX and 𝐴 YZ . To be noted, the Gibbs free energy of vacancy formation for pure metals is typically a function of temperature, whereas a universal description of their relationship is still not conclusive[254]. Nonetheless, the agreement of results suggests the existence of a fundamental link between the T5/Q7 motifs and local geometric frustration in MGs. 84 Alternatively, one may regard the formation of the T5/Q7 motifs as a structural relaxation process and associate it with thermally activated dynamics in glassy materials, e.g., the 𝛼-, 𝛽-, and other faster relaxation modes[37,255–257]. The primary 𝛼-relaxation refers to a low-frequency mode and is responsible for large cooperative atomic rearrangements above the glass transition temperature 𝑇 _ . The secondary 𝛽-relaxation usually initiates at a temperature well below 𝑇 _ and affects the mechanical properties of the glassy material via structural rearrangements on a smaller scale. The 𝛽 [ -relaxation (or 𝛾-relaxation) activated at an even lower temperature, is related to the fast-caged dynamics and reversible stress inhomogeneities. Though the connection between structural rearrangements leading to the formation of motifs and the different relaxation modes in MGs is uncertain, we propose that the formation of T5/Q7 motif is mostly correlated with 𝛽 [ and β relaxation modes, as their activation energies can be approximated[37] as: E ` / ab` ≈nRT c , (6.7) where R is the molar gas constant, n is the scaling number, i.e., n = 26 for β-relaxation and n = 12 for β [ -relaxation. Fitting the function with the lower glass transition temperature measured in the Cu64Zr36 MG sample, i.e., 950 K, we can calculate E ` / =~0.98 eV, E ` =~2.13 eV . The obtained A dZ =~1.157 eV is of a magnitude between the activation energies for β [ - and β- relaxation, hence we consider a strong correlation between the formation of Q7 motif and β [ , β- relaxation. In comparison, the A eX =~0.793 eV is slightly lower than the activation energies of β [ -relaxation, which indicates the T5 motif is likely to associate with fast-caged dynamics. In addition, the non-zero residual concentrations of the T5 and Q7 motifs, c eX b =0.033% and c dZ b = 0.009%, are compatible with the inherent amorphous structure of an MG, i.e., atoms are inherently arranged in a state of higher energy, compared with a reference crystalline structure, with an associated larger configurational entropy[258]. 85 6.4.3 Limitations of This Work Despite evidence that indicates a strong correlation between the T5/Q7 motifs and local structural disorder in MGs, and further, between the Q7 and vacancy-like defects, it is important to address some possible limitations of this work. First, all atomic configuration data for analysis are generated from atomistic MD simulations, which might not represent realistically experimental samples due to rather limited sizes and complete elimination of surfaces. The accuracy of the energetic property evaluation is heavily dependent on the empirical force field[259]. Second, we adopt a scheme of motifs that only considers faces with 3-6 edges in Voronoi polyhedrons, i.e., T, Q, P, and H, whereas the possible face order in an MG motif can surpass six at times. This can be seen from the T5/Q7 polyhedron examples illustrated in Fig. 6.1, which show maximum face order ranging between seven to ten. Also, we truncate Sn according to the maximum number of each i- edged face that can be found in a polyhedron that is statistically significant at a system level, i.e., T5, Q7, P12, and H10. This implies that the vacancy-like Q7 motifs are not restricted to polyhedrons with exactly seven quadrangular faces, but also include those with more than seven quadrangular faces. Though the latter shows no statistical significance alone due to their minuscule quantity, we make this extrapolation based on the structural implication of the Q7 motif, i.e., the more quadrangular faces in a polyhedron, the more vacancy-like its local environments is expected to be. In fact, the mathematical calculation of Voronoi polyhedrons is sensitive to small perturbations of the atomic positions, which might result in very small edges and faces. Nevertheless, we do not assign a threshold for face area or edge length to avoid introducing arbitrary parameters that could affect the results and analysis. These limitations are acknowledged, and the results and conclusions of this work are expected to provide a framework that motivates and supports further investigations of MGs. More 86 underlying dynamics and physical properties of T5 and Q7 centered atoms are left to be characterized, such as cluster formation[107] and percolation[50], vibrational density of states[239], atomic mobility[16], and flexibility volume[250,260]. Additionally, it remains to be determined the dependence of Q7 activation energy in CuxZr100-x as a function of composition. The formation and evolution of the Q7 motif may also be useful in understanding plastic deformation in MGs driven by the activation of shear transformation zones. 6.5 Conclusions In summary, the present work is focused on uncovering underlying structural patterns in CuZr MG atomic configurations by examining atomic Voronoi statistics using ML. Results show that the T5/Q7 polyhedrons are found to be MG motifs strongly correlated with an enhanced local structural disorder. The Q7 is further identified as a vacancy-like motif, which is revealed by intrinsic correlations between its formation and the material microscopic properties, e.g., local entropy, atomic volume, and local atomic tension, as well as macroscopic properties, e.g., glass transition and mechanical deformation. These findings suggest exciting potential applications of ML to understand microscopic polyhedral motifs, which are expected to promote better understanding and prediction of amorphous materials properties, such as the glass forming ability, shear and bulk modulus, and plastic behavior. 87 Chapter 7: The Effect of Heat Treatment Paths on the Aging and Rejuvenation of Metallic Glasses 7.1 Introduction Metallic glasses display attractive properties over conventional crystalline metals such as high strength, high elastic limit, and corrosion resistance[13,20,42,56]. They are also highly malleable and can be molded into complex shapes, making them promising candidates for various engineering applications[261]. However, MGs' properties are strongly affected by their thermal histories and heat treatments, resulting in aging and rejuvenation, due to their inherent non- equilibrium state[37,262–264]. Predicting these processes is challenging because of the nonlinear and non-exponential nature of MGs, which result from complex relaxation modes, including α- relaxation, β-relaxation, β'-relaxation, fast cage dynamics, and boson peak[37,265–269]. Each relaxation mode corresponds to different characteristic times, frequencies, and activated temperatures, which are strongly correlated with intrinsic dynamics and structural heterogeneities in MGs. Molecular dynamics simulations have successfully investigated the structure-property relationships in MGs[16,77,173,257]. However, simulated MG samples are typically generated by fast-cooling processes using extremely high quenching rates due to computational constraints. A demonstrated approach to obtain well-relaxed glassy structures is to heat treat quenched samples with systematically controlled annealing temperatures and times[270]. Nevertheless, the extant understanding of the relaxation behavior, in particular the fast dynamics and memory effect of MG structures during heat treatment, remains elusive and incomplete, hindering the development of optimal thermal processing strategies for tailoring the structure of MGs. In this work, we employ MD simulations to investigate the structural dynamics of MGs under various heat treatment paths. We prepare MGs with different initial structures and subject them to 88 different thermal histories. Our findings show that heating and cooling rates have contrasting effects in the structural relaxation of MGs. Increasing the annealing duration can lead to either aging or rejuvenation, contingent upon the thermal history. These observations are elucidated by considering the fast dynamics and memory effect of flow units in MGs. 7.2 Simulation Details We simulate the MG prototype CuZr alloy system using the LAMMPS package[241] with interactions described by an EAM potential[271]. To create the initial MG structures, we melt a Cu64Zr36 alloy sample containing ~16,000 atoms and equilibrate it at 2000 K for 0.2 ns. The thermalized melt configuration is then used to generate different amorphous samples by bring it to 50 K using quenching rates of 10 9 , 10 10 , and 10 11 K/s. The quenching simulations are performed applying zero external pressure. Periodic boundary conditions along all directions and a 2-fs timestep are employed throughout the quenching and heat treatment simulations. The calculated glass transition temperature (Tg) of the MGs produced at 10 9 , 10 10 , and 10 11 K/s is 960, 1000, and 1050 K, respectively. Subsequently, the “as-cast” MG samples are subjected to sub-Tg annealing treatment, which involves reheating the samples from 50 K to 850 K at a rate of 10 11 , 10 12 , or 10 13 K/s, equilibrating at this temperature for 10, 100, 1000, or 10,000 ps, and then cooling back down to 1 K at a rate of 10 11 , 10 12 , or 10 13 K/s. The thermal history is then determined by the combination of different heating rates (rh), annealing times, and cooling rates (rc). We use ART nouveau[272,273] to study the potential energy landscape (PEL) of the MG structure by examining its local excitations[274,275]. To introduce initial perturbations, an atom and its nearest neighbors are randomly moved by 0.5 Å in a random direction. The saddle point is considered found when the overall force in the system is less than 0.1 eV/Å. The activation energy is then calculated by 89 subtracting the energy at the saddle point from the initial state energy. We identify ~2,500 unique activations for each sample after removing failed searches. 7.3 Results and Discussion 7.3.1 Effects of Thermal Histories on Aging and Rejuvenation MG samples are prepared with 108 unique thermal histories. The statistical results extracted from these samples are presented in Fig. 7.1 in a traditional box-and-whisker graphical representation of the data distribution, which displays the median, quartiles, and range using a rectangular box, and whiskers extending from the box to minimum and maximum data points within 1.5 times the interquartile range, and individual outlier points beyond this threshold. The results are shown as a function of the quench rate used to generate the initial samples. These histories consist of combinations of the quench rates (3), heating rates (3), annealing times (4), and cooling rates (3). Heating and cooling rates are converted to their corresponding times: a heating/cooling rate of 10 11 , 10 12 , and 10 13 K/s corresponds to a heating time of 15000, 1500, and 150 ps or a cooling time of 16000, 1600, and 160 ps, respectively. Figures 7.1 (a)-(c) depict the change in potential energy, ΔPE, which is calculated by comparing the initial potential energy per atom (PE) of the as-cast MG sample with that of the same sample after a full heat treatment process. A negative ΔPE indicates aging, while a positive ΔPE indicates rejuvenation of the MG. The impact of heating time on the ΔPE is strongly dependent on the initial MG structure and displays no monotonic effect, see Fig. 7.1 (a). Surprisingly, increasing the annealing time results in a prominent reduction of the aging effect in MGs with compact initial structures, i.e., MGs prepared at lower quenching rates, see Fig. 7.1 (b). In contrast, for MGs prepared at a higher quenching rate, the annealing time displays an intricate effect on ΔPE. Interestingly, increasing cooling time 90 Fig. 7.1 Box plots of (a)-(c) Potential energy variation (ΔPE) and (d)-(f) full icosahedra concentration variation (ΔFI) in MGs generated with different quenching rates, grouped by heating time, annealing time, and cooling time, respectively. consistently promotes the aging effect in MGs, regardless of varying heating and annealing times, or initial structures, see Fig. 7.1 (c). One of the key features of the short-range order (SRO) in MGs is the presence of full icosahedra (FI) or icosahedral-like atomic clusters[19,50,51]. A higher fraction of FI indicates a compact MG structure and a lower energy state, while a lower fraction of FI indicates a soft, liquid- 91 like structure and a higher energy state. Figures 7.1 (d)-(f) show the changes in FI in the samples caused by different heat treatment paths, with the trend of ΔFI agreeing with the above conclusions about ΔPE. The impact of heating rate on ΔFI is heavily influenced by the initial state of the MG, and longer annealing times result in a lower fraction of FI in MGs with compact initial structures, while longer cooling times generally result in a higher fraction of FI. However, we notice that after heat treatment, particularly for MGs quenched at 10 9 K/s, the samples exhibit simultaneous negative ΔPE and ΔFI, indicating a lower degree of SRO in an aged MG. We suggest that the decoupling of ΔPE and ΔFI can be clarified by the idea that increasing the SRO in the structure requires a considerable decrease in the energy of the system. This can also be viewed from the perspective of the PEL[275], where a specific structural evolution corresponds to the hopping between metabasins or sub-basins confined within a metabasin. Therefore, the negative ΔPE could be attributed to changes in the medium-range order (MRO) of MGs. 7.3.2 Structural Evolution of Metallic Glasses Treated with Different Heating and Cooling Rates To further understand the effects of heating and cooling rates on the structural dynamics of MGs, we measure the atomic displacement statistics during sample relaxation by computing the instantaneous difference between current and initial particle coordinates at the beginning of the annealing. Figure 7.2 (a) shows the results obtained by heating MGs produced at a quenching rate of 10 10 K/s to 850 K using two different heating rates, rh = 10 11 and 10 13 K/s, which is followed by annealing at different periods of time, ranging from 10 ps to 20 ns. When the equilibrating time is less than 100 ps, the displacement magnitude is below the average bond length values (Cu-Cu: 2.6 Å, Cu-Zr: 2.8 Å, Zr-Zr: 3.3 Å) for both heating rates, indicating that the major dynamic mode of atoms is confined within the "cage"[276–278]. However, when the relaxation time reaches 10 ns, 92 the displacement distribution widens and shifts above the average bond length values. A critical relaxation time is identified as 1 ns, during which the difference in atomic displacement between MGs treated with different heating rates is most prominent. Both curves exhibit a second peak at ~2.6 Å, with the MG heated at a lower rate having a greater proportion of atoms at this peak. We use fast-caged dynamics theory to interpret these observations[279,280]. The MG atoms are initially confined to cages formed by neighboring atoms or clusters due to an anharmonic intermolecular potential. Only a limited part of the configuration space is available for the motion of these atoms within a short relaxation time. As the relaxation time increases, fluctuations of Fig. 7.2 Atomic displacement statistics in MGs (a) heated at different rates and annealed at 850 K for different time periods, (b) cooled to 600 K at different rates after heated to 850 K at a rate of 10 12 K/s. 93 atomic movements within the cages leads to dissipation and the escape of cages results in a different relaxation mode[37,281]. We suggest that the cage radius can be determined by identifying the second peak in the displacement magnitude distribution. Moreover, when the MG is subjected to a reduced heating rate, it transitions into an elevated energy state within the PEL. Consequently, this facilitates the liberation of atoms from their constrained local environments, i.e., from their cages. Similarly, Fig. 7.2 (b) presents atomic displacement statistics obtained from MGs that undergo heat treatment using different cooling rates. The MGs quenched at a rate of 10 10 K/s are reheated to 850 K at a rate of 10 12 K/s and then separately cooled to 600 K at rates of 10 11 , 10 12 , and 10 13 K/s. The results are calculated by comparing the difference between particle coordinates at 850 K and at 600 K. The peak of the 10 12 and 10 13 K/s curves drop rapidly to near zero within 2 Å, whereas the 10 11 K/s curve shifts slightly towards larger values and exhibit a second peak at ~2.6 Å, in good accordance with the second peak position shown in Fig. 7.2 (a). This supports our previous suggestion that the second peak is related to the cage dynamics and should be independent of the temperature. A lower cooling rate applied to the MG results in a larger fraction of atoms with a greater displacement due to a longer relaxation time for transitioning into lower energy states from high temperatures, which leads to fewer atoms frozen in their loosely packed configurations. In MGs, the process of structural relaxation is mediated by transitions in atomic configurations where atoms hop between two local energy minima separated by an energy barrier[275]. We now quantify how cooling rates affect energy states after heat treatment. First, the MG samples are produced using a quenching rate of 10 10 K/s. They are then heated to 850 K at a rate of 10 12 K/s, annealed for 100 ps, and cooled to 1 K at three different rates (r c = 10 11 , 10 12 , and 10 13 K/s). Next, 94 Fig. 7.3 Activation energy statistics of the as-cast MG and samples subjected to a heat treatment involving reheating to 850 K at a rate of 10 12 K/s, annealing for 100 ps, and subsequent cooling at different rates down to 1 K. we employ the activation-relaxation technique (ART) to evaluate the activation energy distribution in different MG samples[272,273]. As seen in Fig. 7.3, the peaks for the distribution curves do not differ significantly. However, the proportions of events on the two sides of the curve are noticeably distinct. Compared to the as-cast curve, the curve for rc = 10 11 K/s shows more events with higher activation energies and slightly fewer events with lower activation energies, indicating overall aging of the MG. In contrast, the curve for rc = 10 13 K/s displays a wider dispersion, with both the fractions of events with lower and higher activation energies increasing. This suggests the presence of local structural heterogeneities induced by complex relaxation mechanisms during thermal treatment, causing atomic packing to become more or less compact. 7.3.3 Flow Units Activation and Annihilation along the Heat Treatment Paths To better compare the effects of heating and cooling rates on the structural evolution of MGs during heat treatment, we varied each rate while keeping the total heat treatment time constant. In Figs. 7.4 (a) and (c), the sample quenched at a rate of 10 10 K/s is heated to 850 K at a rate of 10 11 K/s, annealed for 100 ps, and then cooled to 1 K at a rate of 10 13 K/s. Only minor changes are 95 observed in the PE and FI% comparing the initial and final MG structures. However, when the sample is treated with heating and cooling rates of 10 13 K/s and 10 11 K/s, respectively, see Figs. 7.4 (b) and (d), a clear decrease in PE and increase in FI% can be identified after the full thermal treatment. These findings imply that reducing the cooling rate is a more effective way to promote MG aging, in contrast to slowing down the heating rate, provided that the total treatment time remains constant. The dissimilar effects of heating and cooling rates can be elucidated considering the concept of "flow units" and their activation and annihilation during heat treatment. These flow units harbor local structural heterogeneities at the atomic level, wherein atoms exhibit weak bonding and form local packing with diminished density. Rejuvenation correlates with the activation of flow units, whereas aging is linked to their annihilation[268,282]. Building upon prior research on flow units[37,283], we illustrate in Fig. 7.4 (e) the manner in which the heat treatment process influences the dynamics of flow units in MGs. At lower temperatures, flow units remain inactive and in a static state. As temperature increases, these units are activated through atomic reconfigurations, gradually increasing their number and size over time. As temperature is reduced flow units are gradually annihilated in a time-controlled manner where the most unstable “reversible” flow units are rapidly destroyed and those in stable “irreversible” configurations tend to endure. We hypothesize that this time-controlled process, i.e., the memory effect, accounts for the discrepancies in the effects of heating and cooling rates on MGs during heat treatment. Slow heating activates large, irreversible flow units, whereas rapid heating activates small, reversible flow units. Similarly, slow cooling provides sufficient time for the annihilation of activated flow units, while fast cooling preserves their configuration. The associated energy required for flow unit activation can be estimated based on the corresponding state-hopping across energy barriers in the 96 Fig. 7.4 (a)-(b) Potential energy (PE) and (c)&(d) FI concentration evolution in MGs along different heat treatment paths. (e) Illustration of the associated volumes of flow units involved in activation and annihilation processes along the heat treatment paths shown in (a)-(b). 97 PEL. We now discuss the findings in this work by connecting the concepts of relaxation mode, flow unit, and memory effect. Results in Fig. 7.1 (b) show that at a temperature below Tg, longer annealing times raise the energy of MGs with compact initial structures, whereas have a complex effect on MGs with loosely packed structures. This upset conventional ideas about sub-Tg annealing of MGs[263,270,284]. It is commonly believed that longer annealing times result in a more compact structure and lower energy state. To understand this, note that total simulation times in this work range from pico- to nanoseconds. Thus, the simulations here probe β’-relaxation and fast-cage dynamics in the relaxation spectrum (10 5 to 10 10 Hz) as the driving forces behind the competing aging and rejuvenation processes in the MG samples[268,269,285]. A higher quenching rate leads to a less relaxed MG state and a larger fraction of loosely packed atomic regions, i.e., flow units. During the annealing process of such structure, the energy of the MG is lowered, implying that the annihilation of flow units exceeds their activation. Conversely, a lower quenching rate results in a more relaxed MG state and a lower fraction of flow units. During the annealing of such structure the activation of flow units tends to surpass their annihilation as annealing time increases. In contrast, the ability to reverse the activation of flow units after annealing depends on the cooling rate. Slower cooling has associated a longer time frame that allows flow units to gradually deactivate (low memory), while faster cooling prevents the relaxation from higher energy states (high memory). The competing mechanism between aging and rejuvenation, i.e., between the annihilation and activation of flow units, can be inferred by the activation energy distributions shown in Fig. 7.3. Fast cooling (red curve) during heat treatment leads to both an increase in the fraction of events with higher activation energy and particularly the fraction of events with lower 98 activation energy compared to the as-cast MG (highlighted by dish rectangles). This indicates the occurrence of both annihilation and activation of flow units. As depicted in Fig. 7.4 (e), the flow unit perspective provides intuitive insights into the fast dynamics in MGs with distinct thermal histories and the memory effect of their heterogeneous microstructures. Additionally, the results reveal a second peak in the displacement magnitude distribution, as illustrated in Fig. 7.2. We hypothesize that this peak is associated with the cage radius, which is essential in understanding caging effects in MGs. Previous studies have demonstrated that fast-caged dynamics continue to exist across all temperatures without a distinct time scale[37,269,279,286]. Our investigation may offer novel perspectives on short-time dynamics during the relaxation of the glass. 7.4 Conclusion In summary, we investigate the relaxation dynamics of MGs subjected to various thermal histories. The influence of heating rate is considerably contingent upon the MG initial state, while reducing the cooling rate consistently promotes MG aging. The annealing time exhibit a complex interplay in the structural relaxation of MGs, leading to either aging or rejuvenation, contingent upon their specific thermal history. We hypothesize that the memory effect of flow units in MGs can account for the variations in the effects of heating and cooling rates during heat treatment. Our findings hold substantial implications for the design of heat treatments that can effectively fine- tune MG structures. 99 Chapter 8: Conclusion In conclusion, this study first proposes three novel approaches to achieve high strength and ductility in MGs by tuning their microstructures. We employed large-scale MD simulations and visualization tools to evaluate the mechanical properties of MGs with these novel designs. The results suggest that introducing seamless gradient microstructures into MGs, using MG composites that combine a nanoglass matrix and an MG second phase, and designing shape memory alloy- MG composites with a brick-and-mortar design can effectively delocalize the plastic deformation, hindering the buildup of local stress hot spots and the generation of critical shear bands. Moreover, the structure-property relationships in MGs at an atomic scale are investigated with the help of ML tools, revealing new insights into the interpretation of local disorder and unfavorable topological descriptors in MG atomic structures. Lastly, the study investigates the structural dynamics of MGs under various heat treatment paths, showing that controlling the thermal history can tailor the properties of MGs to meet specific requirements in various applications. Overall, this study offers new possibilities for applying heterogeneous microstructure designs and efficient heat treatment strategies to MGs to achieve desired combinations of different mechanical properties. The study also provides insights into understanding local structural disorder in MGs and constructing the structure-property relationships. 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Abstract (if available)

Abstract Metallic glasses (MGs) have been widely studied due to their remarkable properties, such as high yield strength, hardness, and wear resistance, making them attractive materials for various engineering applications. However, their tendency to shear localization and premature failure often limits their macroscopic ductility. To overcome this issue, we propose three novel approaches to achieve high strength and ductility in MGs by manipulating their microstructures. We use large-scale molecular dynamics (MD) simulations for heat treatment, tensile loading, compressive loading, and nanoindentation simulations. In addition, we employ machine learning (ML) methods to uncover hidden structure-property relationships in MGs at an atomic level. In the first approach, we introduce seamless gradient microstructures into Cu64Zr36 MGs by incorporating glassy nano grains with various sizes, referred to as gradient nanoglasses (GNGs). To evaluate the mechanical properties of these GNGs, we conduct uniaxial tensile loading and nanoindentation simulations on the prepared samples. The results of the tensile loading simulation demonstrate that plasticity in GNGs follows a grain size gradient, with local plasticity increasing from larger to smaller grain size regions. When loaded parallel to the gradient direction, the samples exhibit intensive necking followed by catastrophic failure. In contrast, when loaded perpendicular to the gradient direction, the samples exhibit diffused shear band propagation from large to small grain size regions, further delocalizing the plastic deformation and delaying the development of a dominant shear band. The nanoindentation results indicate that the deformation mechanisms in GNGs are rooted in the activation of shear transformation zones (STZs) at soft sites and their subsequent evolution into shear band embryos. Furthermore, an increase in grain size leads to enhanced elastic modulus and hardness but reduced abrasion resistance. Although the average grain size at the indentation surface plays a crucial role in the deformation behavior of GNG models during nanoindentation, the activation of shear transformation zones far away from the indentation surface has a notable impact on the materials' overall performance. These findings suggest new possibilities for applying heterogeneous gradient designs to MGs to achieve desired combinations of different mechanical properties and provide insights into the study of heterogeneous degradation in functionally graded materials. In the second approach, we propose MG composites that combine a nanoglass (NG) matrix and an MG second phase. The samples are prepared with a brick-and-mortar architecture, and phase fractions are varied. The results of the tensile loading simulation reveal that the global strength of the matrix is significantly improved by the second phase without sacrificing the large ductility of the NG phase. The mechanical synergy of the two phases is further optimized by arranging MG bricks in a staggered way, effectively delocalizing the plastic deformation, hindering the buildup of local stress hot spots, and the generation of critical shear bands. In the third approach, we design and characterize the tensile deformation and failure of shape memory alloy-bulk MG composites (BMGCs) with a brick-and-mortar design. We identify contrasting behaviors when arranging two phases in different ways. The results suggest that composites with crystalline bricks arranged in a staggered way exhibit significantly improved mechanical properties compared to cases where they are arranged in a stacked way. This composite architecture synergizes the mechanical properties of the glassy MG matrix and the crystalline second phase to a great extent by introducing an exceptional strain hardening stage through efficiently delocalizing the local stress and local strain. In addition to studying the unique mechanical behavior of MGs in various architectural designs, we also investigate the structure-property relationships in MGs at an atomic scale. In this work, we employ machine learning (ML) to reveal a previously unknown vacancy-like structural motif in MGs, named Q7, which refers to atoms enclosed in atomic Voronoi polyhedrons with seven quadrangular faces. The Q7 motif is found to significantly contribute to the short-range structural disorder in MGs, with its concentration following an Arrhenius-like relationship with temperature, thereby providing a precise indicator of the glass transition temperature point. The population of Q7-centered atoms shows strong correlations with the yield and failure of MGs during mechanical deformation. Moreover, Q7-centered atoms exhibit larger local entropy, atomic volume, and local tension, indicating their association with vacancy-like configurations. These results offer new insights into the interpretation of local disorder and unfavorable topological descriptors in MG atomic structures and point out a promising avenue for constructing structure-property relationships in MGs through combining MD simulations and ML. Lastly, we investigate the structural dynamics of MGs under various heat treatment paths. In our study, we prepared MGs with different initial structures and subject them to different thermal histories. We find that the heating and cooling rates have contrasting effects in the structural relaxation of MGs. When MGs are annealed for longer durations, they can either age or rejuvenate, depending on the thermal history. Moreover, we observed that the structural relaxation of MGs can be explained by considering the fast dynamics and memory effect of flow units in MGs. These flow units are responsible for structural evolutions in MGs and their behavior can be altered by thermal history. During the relaxation, these flow units reorganize themselves and can lead to aging or rejuvenation. Our work suggests that by carefully controlling the thermal history, it is possible to tailor the properties of MGs to meet specific requirements in various applications. As computing power continues to increase, simulation packages become more sophisticated, and ML techniques are applied more wisely, we anticipate a promising future for gaining further understanding of MG behaviors and designing them with optimized properties.

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Creator Yuan, Suyue (author)

Core Title Atomistic modeling of the mechanical properties of metallic glasses, nanoglasses, and their composites

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Materials Science

Degree Conferral Date 2023-05

Publication Date 05/11/2023

Defense Date 05/10/2023

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag mechanical properties,metallic glass,molecular dynamics simulations,nanoglass,OAI-PMH Harvest

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Branicio, Paulo (committee chair), Nakano, Aiichiro (committee member), Sharada, Shaama Mallikarjun (committee member), Vashishta, Priya Darshan (committee member)

Creator Email excellentstudentyuan@gmail.com,suyueyua@usc.edu

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