Quantum and Experimental Mechanics of Icosahedral Ceramics

QUANTUM AND EXPERIMENTAL MECHANICS OF ICOSAHEDRAL CERAMICS

CODY KUNKA

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2018

To the One who gives life… and life abundantly.

ACKNOWLEDGMENTS

I emphasize that both this dissertation and my professional development have been a group effort. I owe much to my leaders, colleagues, friends, and family so sincerely hope to express my gratitude here and in our daily interactions.

I thank Ghatu Subhash for inviting me into his laboratory when I was only an undergraduate student in need of research experience. He guided me in academics, research, teaching, professional presentations, and job hunting. He is largely the reason

I am well equipped with dual experience in experimentation and modeling and in both mechanical engineering and materials science. In many ways, I plan to model my career after his. If one day, I am honored to guide the development of others, I especially hope to adopt his style of encouragement, discipline, and support in the laboratory, classroom, or wherever else I go.

I thank my fellow labmates for their solidarity, helpfulness, and fun. I enjoyed working in the trenches with them from my first year of undergraduate research through all four years of PhD. I will long remember getting “free” doughnuts in Daytona Beach, jumping off the pier at Lake Wauburg, and celebrating Christmas every year. I especially thank Alison Trachet and Amnaya Awasthi. These two frequently set aside their own time to jumpstart my research career and celebrate our group. They provided my first experience with advanced experimentation and quantum mechanics, respectively.

I thank my friends and family for emotional, financial, intellectual, and spiritual support. I rely on these special people when times feel tough and the people I celebrate with when times feel blessed. My parents’ twenty-six years of dedication to raising me across the United States are so important. My brother and I will be friends for life…

even if we would have the most hilarious loved-ones’ visit. My friends and mentors in

Gator Christian Life have tremendous positive impact on my life. Finally, I thank my beautiful wife, Thandie, for her emotional support and empathetic example. Life is more fun and rewarding together.

Financially, I recognize the National Science Foundation (Graduate Research

Fellowship Program), the University of Florida Department of Mechanical and

Aerospace Engineering, the University of Florida Graduate School, the Department of the Army, and the Army Research Office for providing funding for my research and academics. I am grateful that they entrusted the advancement of their goals to me. I eagerly anticipate the next generation of PhD research at the University of Florida. Go

Gators.

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS ...... 4

LIST OF TABLES ...... 8

LIST OF FIGURES ...... 9

LIST OF ABBREVIATIONS ...... 10

ABSTRACT ...... 11

CHAPTER

1 BACKGROUND AND AIMS ...... 13

Superhard Materials ...... 13 Raman Spectroscopy...... 16 Density Functional Theory ...... 20 Density Functional Perturbation Theory ...... 24 Dissertation Aims ...... 28

2 CRYSTALLOGRAPHIC AND SPECTRAL EQUIVALENE OF BORON- CARBIDE POLYMORPHS ...... 32

Crystallography of B4C ...... 32 Ground States of B4C Polymorphs...... 34 Raman Spectra of B4C Polymorphs ...... 35 Impact of Modified Nomenclature ...... 37

3 EVALUATING BORON-CARBIDE CONSTITUENTS WITH SIMULATED RAMAN SPECTRA ...... 46

Crystallographic Variability in Ceramics ...... 46 Experimental Raman Spectroscopy ...... 47 Simulated Raman Spectroscopy ...... 47 Spectral Superposition ...... 49 Impact of Quantification of Complex Compositions ...... 50

4 NANOTWINNING AND AMORPHIZATION OF BORON SUBOXIDE...... 55

Nanotwinning of Ceramics ...... 55 Mechanisms of Nanotwinning ...... 56 Amorphization of Boron Suboxide ...... 57 Objective ...... 58 Materials Processing...... 59

Indentation, Compression, and Ultrasound ...... 59 Electron Microscopy...... 61 Raman Spectroscopy...... 61 X-Ray Diffraction ...... 62 Biaxial Shear ...... 63 Results ...... 63 Discussion ...... 67 Impact of B6O Characterization ...... 70

5 ICOSAHEDRAL SUPERSTRENGTH AT THE NANOSCALE ...... 77

Modified α-B12 Structures ...... 77 Objective ...... 80 Computational Procedure ...... 81 Results of Ground-State Bonding and Elasticity ...... 84 Results of Shear Deformations ...... 87 Impact of Fundamentals on Icosahedral Bonding ...... 89

6 CONCLUSION ...... 94

Summary of Dissertation ...... 94 Future Work: Nanotwinning Mechanisms for Ceramics ...... 94 Future Work: Systematic Processing of Boron Suboxide ...... 99

APPENDIX

A SIMULATED RAMAN SPECTRA OF ALL B4C POLYMORPHS ...... 100

B CRYSTALLOGRAPHY AND ATOMIC BONDING OF B6O ...... 107

LIST OF REFERENCES ...... 117

BIOGRAPHICAL SKETCH ...... 132

LIST OF TABLES

Table page

2-1 Lattice parameters, mass densities, and static energies of B4C polymorphs...... 39

2-2 Natural frequencies and normalized intensities for dominant Raman peaks of ten equivalent groups of B4C polymorphs...... 40

3-1 Static energies and dominant Raman peaks for all stable B4C polymorphs and for α-B12 ...... 52

4-1 Experimental microhardness values, elastic moduli, and mass densities of HP B6O, SPS B6O, and SPS B4C ...... 71

5-1 Nanotwinning energies, lattice parameters, elastic moduli, microhardness values, shear strengths, and toughness values of hexaborides and α-B12 ...... 90

LIST OF FIGURES

Figure page

1-1 Experimental Raman scans of virgin and indented B4C ...... 31

1-2 HR-TEM of amorphous bands in B4C ...... 31

2-1 Unit cell and connectivities for B4C color-coded by site family ...... 41

2-2 Quantum mechanical bonding in the most popular B4C polymorph ...... 42

2-3 Representative crystal structures from ten equivalent groups of B4C ...... 43

2-4 Simulated Raman spectra of ten equivalent groups of B4C ...... 44

2-5 Simulated Raman spectra that agree with experimental scans and show need for new B4C nomenclature ...... 45

3-1 Prediction of relative abundances of B4C polymorphs based on comparison of experimental and simulated Raman spectra...... 54

4-1 Crystal structures of (B11Cp)CBC, (b) α-B6O, (c) τ-B6O, and (d) 2τ-B6O ...... 72

4-2 Fracture surfaces of HP B6O and SPS B6O ...... 72

4-3 Representative HR-TEM of virgin B6O showing grain size and nanotwinning .... 73

4-4 Representative HR-TEM of indented B6O showing amorphization ...... 73

4-5 Raman spectra of virgin HP B6O and SPS B6O ...... 74

4-6 XRD of experimental B6O and of simulated B6O with nanotwinning ...... 75

4-7 Simulated stress-strain curves for biaxial shear of nanotwinning B6O ...... 76

5-1 Quantum-mechanical bonding of α-B12, B6O, and nanotwinned B6O ...... 91

5-2 Critical slip systems for pure shear of α-B12 and B6O ...... 91

5-3 Electronic densities and localizations for α-B12 and B6i ...... 92

5-4 Simulated stress-strain curves for pure shear of p-block B6i ...... 93

LIST OF ABBREVIATIONS

α-B12 α-Boron

B4C Boron Carbide

B6O Boron Suboxide

B6i Hexaboride

DRP Dominant Raman Peak

C Central

DFT Density Functional Theory

DFPT Density Functional Perturbation Theory

E Equatorial

EB Equatorial Bond

ELF Electron Localization Function

DEN Spatial Distribution of Electronic Density

GGA Generalized Gradient Approximation

HP Hot-Pressed

HR-TEM High-Resolution Transmission Electron Microscopy

I Icosahedral

LDA Local Density Approximation mαB Modified α-Boron Structure

P Polar

PB Polar Bond

SEM Scanning Electron Microscopy

SPS Spark-Plasma-Sintered

XRD X-Ray Diffraction

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

QUANTUM AND EXPERIMENTAL MECHANICS OF ICOSAHEDRAL CERAMICS

Cody Kunka

December 2018

Chair: Ghatu Subhash Major: Mechanical Engineering

Many boron-rich, icosahedral solids command extraordinary mechanical, thermal, electronic, and chemical properties. Applications cover many industries (e.g., defense, nuclear power, automotive, and computing), but structural applications dominate due to the unparalleled combination of superhardness and low mass density.

Unfortunately, uncertainty about crystallography precludes comprehensive failure models and limits both performance and tailoring. Hence, this dissertation probes the nanostructures and physics of the two salient icosahedral systems: boron carbide (B4C) and boron suboxide (B6O).

B4C enjoys widespread investigation and adoption yet suffers solid-state amorphization, which is poorly understood largely because of uncertainty concerning crystallography. Neither traditional nomenclature nor typical characterization methods can distinguish the nanoscale mixture of crystal structures that constitute fabricated samples. Hence, this dissertation revises the B4C nomenclature and quantifies this distribution by fitting experimental Raman scans with quantum-mechanical simulations of over fifty unique B4C crystal structures. This superposition scheme not only reveals

overlooked polymorphs and processing dependency but also could be applied to other complex systems and spectroscopies.

In B4C, nanoscale complexity could help tailoring but also obscures and potentially hastens mechanical failure. Additionally, this heterogeneous structure may not accommodate the highly ordered nanotwinning responsible for recent record mechanical properties in both metals and ceramics. Hence, this dissertation now turns to novel B6O, which has a nanotwinned ground state and does not suffer polymorphism.

Through mechanical, structural, and spectral characterizations, this work demonstrates the superiority and uniqueness of this icosahedral solid. For example, quasistatic compression reveals a strength over 5 GPa, which is both a record for B6O and superior to that of B4C. HR-TEM, XRD, and Raman spectroscopy manifest unique nanotwinning and amorphization. Quantum mechanical simulations of XRD and shear deformation yield tools for tracking and optimizing nanotwinning. Finally, simulations of p-block hexaborides (B6i) demonstrate that polar and equatorial bonding can be used as metrics for elasticity, superstrength, and nanotwinning susceptibility.

Overall, this work probes many structural nuances: atomic-scale complexity, processing dependency, amorphization, and nanotwinning. Especially by facilitating the first understanding of nanotwinning in ceramics, this work may guide development of the next predominant structural material.

CHAPTER 1 BACKGROUND AND AIMS

Superhard Materials

Alongside a high index of refraction and strong thermal conductivity, diamond holds the record for hardness (i.e., resistance to penetration). This naturally occurring ceramic owes its hardness to a three-dimensional network of short, highly covalent bonds [1]. Shorter bonds are typically stronger than longer bonds. Diamond is considered the model system for pure covalent bonding as all the carbon atoms have the same electronegativity. These purely covalent bonds are highly directional so strongly limit the allowed deformation directions. In contrast, ionic and metallic bonds freely change and break along multiple directions. Further, the covalent bonds in diamond form a three-dimensional network, so there is no considerably weaker direction. In contrast, graphite has stronger inplane bonds than diamond but van-der-

Waals forces among planes. Hence, graphite fails by delamination and is significantly weaker than diamond. Finally, note that these covalent bonds resist both elastic and plastic deformation. Therefore, hardness is often empirically related to stiffness, especially shear modulus [2].

Despite its superhardness and other outstanding properties, diamond suffers chemical reaction with iron. Plus, researchers speculate that undiscovered synthetic materials may eventually outperform natural diamond. Hence, significant effort has been spent on discovering new superhard (i.e., microhardness above 40 GPa) and ultrahard

(i.e., microhardness above 80 GPa) materials with unique sets of properties [3-6].

These synthetic materials typically fall into two categories [1]. The first category has low-atomic-number elements (e.g., B, C, N, O) arranged in three-dimensional networks

of short, covalent bonds as in diamond. For example, cubic boron nitride (c-BN) [7] is isoelectronic with and structurally similar to diamond but has each pair of carbon atoms substituted for one boron atom and one nitrogen atom. Despite being difficult to produce and having lower covalency and consequently lower hardness than diamond, c-BN offers superhardness without chemical reactivity with iron. Hence, c-BN is popular for wear and machining applications.

A large number of boron-rich, icosahedral solids also adopt this paradigm of favoring short, covalent, three-dimensional bonds. The most popular material in this class is unquestionably boron carbide (B4C) [8]. This superhard material is softer than c-

BN but has extremely low mass density, high chemical resistance, and strong thermal stability. Combining high hardness with low mass density well serves many applications, such as armor. Hence, improving the hardness of B4C has long been a goal for materials scientists. The main obstacle remains countering the dominant failure mechanism: solid-state amorphization, which is the shear-induced collapse of crystal structure in nanoscale bands [9]. To complicate matters, much debate surrounds the identities and distribution of the crystal structures that comprise fabricated samples of

B4C. Ignorance of the starting structure hurts the effort to counter amorphization.

Hence, one of the major contributions of the current dissertation is demystifying the structural makeup of B4C. Additionally, interest in other icosahedral solids is expanding rapidly. For example, aluminum magnesium boride (BAM) has already demonstrated interesting frictional properties [10,11]. Perhaps most impressively, boron suboxide

(B6O) has already shown exceptional theoretical promise and experimental superhardness, even with non-optimized processing [12-14]. Because of these initial

successes and a crystal structure similar to that of B4C, this dissertation also devotes both experimental and theoretical testing to B6O.

As opposed to exclusively incorporating low-mass elements connected by short, covalent bonds, the other major category of superhard materials combines heavy transition metals with a high bulk modulus and low hardness with the light elements of the former category. Instead of focusing on bond strength alone, this category essentially seeks to increase mass density such that atoms have no space to displace

(i.e., high incompressibility). Initial attempts have achieved high incompressibility but suffered low hardness due to ionic bonds (e.g., RuO2, WC, and Co6WC) [1]. However, incorporating boron seems to increase the covalent bonding and consequently the hardness (e.g., WB4 and WB2) [1,4]. Although this dissertation focuses on the boron-rich icosahedral structures, the findings on the complex bonding of boron may prove useful for exploring this second category of superhard materials as well.

In addition to exploring the elemental composition, significant research is devoted to improving hardness through optimization of features on higher length scales. For example, decreasing grain size (i.e., nanograining) increases hardness according to the

Hall-Petch relationship until a critical grain size is reached [4,5]. Typically, this critical grain size is described as a switch of deformation mechanism from failure through the grains to movement of the grains. For ceramics, other features, such as quantum confinement and resistance to dislocation motion, complicate rationalization of critical grain size. Regardless, experimental evidence strongly supports nanograining as a means to improve hardness. More recently, the cutting edge has become nanotwinning

(i.e., periodic twins separated by nanotwin spacing). Unlike regular twins, which

concentrate stress and lower mechanical performance, nanotwins have recently doubled the record microhardness in diamond to nearly 200 GPa [15], increased the fracture toughness of boron nitride by 140% [16,17], and increased the yield strength of copper by an order of magnitude [18,19]. Because of the success and recency of these advances, this dissertation also explores both the experimental and simulated effects of nanotwinning.

Raman Spectroscopy

Raman Spectroscopy nondestructively probes the characteristic vibrational modes (i.e., phonons) of solids, liquids, and gases by quantifying light inelastically scattered from the sample [20-25]. This change in light frequency was first predicted by

Adolf Smekal in 1923 and first observed by C. V. Raman, K. S. Krishnan, Grigory

Landsberg, and Leonid Mandelstam in 1928. Raman spectroscopy soon provided the first assessments of molecular frequencies. In the 1940’s, sensitivity and cost limitations shifted focus to infrared spectroscopy, which relies on adsorption rather than scattering.

After all, only around 1 in 106 to 108 scattered photons are inelastically scattered [20].

However, the advent of lasers as the light source in the 1960’s, advances in detectors, and new instrumentation have since renewed interest in Raman spectroscopy [26]. For example, Coherent anti-Stokes Raman spectroscopy (CARS) employs two lasers to increase counts through resonance. Angle-resolved Raman spectroscopy relates the angle between the incident laser and sample to phonon dispersion. Spontaneous

Raman spectroscopy (SRS) incorporates the temperature dependence of scattering.

Stimulated Raman spectroscopy uses polarized pulses to observe the Raman spectra of gases or molecular clusters. Surface plasmon polariton enhanced Raman scattering

(SPPERS) separates the focal plane from the excitation plane to remove noise.

Raman theory is largely attributed to George Placzek for his work by 1934 and has since been expounded [26-28]. In an unperturbed state, phonons cause the positions of the atoms (푄) to be displaced (푑푄) about their equilibria according to the associated natural frequencies (휈푣푖푏):

푑푄 = 푄0 cos(2휋푣푣푖푏푡) (1-1)

The maximum displacements of the atoms (푄0) due to phonons are typically less than

10% of bond lengths, so the effect of these displacements on the polarizability (훼) (i.e., the susceptibility to induced dipoles) can be modeled as a Taylor series:

휕훼 훼 = 훼 + 푑푄 (1-2) 0 휕푄

Therefore, polarizability deviates from that of the equilibrium position (훼0) according to the locations of the atoms (푄). As an electromagnetic wave, light can also disturb the atomic locations and induce a temporary, oscillating dipole. The strength of this dipole

(P) depends on the material polarizability (훼) and the strength of the applied electric field (퐸0):

푃 = 훼퐸0 cos(2휋휈0푡) (1-3)

By combining Eqs. 1-3 and applying a trigonometric identity, the induced dipole moment can be rewritten as:

푄 퐸 휕훼 푃 = 훼 퐸 cos(2휋휈 푡) + ( 0 0) ( ) {cos(2휋[휈 − 휈 ]) + cos(2휋[휈 + 휈 ])} (1-4) 0 0 0 2 휕푄 0 푣푖푏 0 푣푖푏

Therefore, by interacting with the electronic structure of the material, the incident electromagnetic wave can be scattered at three frequencies (i.e., 휈0, 휈0 − 휈푣푖푏, and 휈0 +

휈푣푖푏).

Experimentally, elastic scattering (i.e., Rayleigh scattering) dominates, so the vast majority of scattered photons have the same frequency and energy as the incident

light (i.e., maintains 휈0 and 퐸0). However, if the change in polarizability with respect to atomic positions (i.e., 휕훼⁄휕푄) is nonzero, inelastic scattering (i.e., from 휈0 to 휈0 ± 휈푣푖푏 and from 퐸0 to 퐸0 ± 퐸푣푖푏) can occur. Hence, only the vibrational modes that induce a change in polarizability will be “Raman-active” (i.e., produce non-zero intensities). This

Raman scattering only accounts for a very small fraction of the total photons. Physically, if the frequency is downshifted (i.e., 휈0 − 휈푣푖푏), the scattered light has lost energy to induce characteristic vibration(s) in the sample (i.e., Stokes scattering). Alternatively, if the frequency is upshifted (i.e., 휈0 + 휈푣푖푏), the scattered light has gained energy due to the annihilation of characteristic vibration(s) in the sample (i.e., anti-Stokes scattering).

Note that the Raman effect is also often represented as a change in vibrational energy state through a temporary virtual energy state.

Figure 1-1 provides typical Raman spectra of both virgin and indented regions of

B4C. As usual, this plot represents the Stokes inelastic scattering because induced vibrations are much easier to detect than annihilated vibrations. Stokes and anti-Stokes scattering would produce the same wavenumbers but at different intensities before normalization. The horizontal axis represents the difference between the inverse of the wavelength of the incident light and the inverse of the wavelength of the scattered light

−1 −1 (i.e., 휆푖 − 휆푠 ), Hence, one can think of Raman shift as either a shift in frequency or energy. Also, because the shift reflects natural frequencies (see Eq. 1-4), one can also think of this axis as a list of Raman-active, characteristic vibrational modes or phonons.

Alternatively, the vertical axis provides normalized Raman intensities. Before normalization, this axis can be reported in counts (i.e., the number of inelastic scattering

events). However, these counts are arbitrarily dependent on experimental parameters like scanning time so are frequently normalized.

By revealing the relative strengths of characteristic vibrational modes, Raman spectroscopy essentially reveals the chemical bonding of a material. Also, Raman spectroscopy has significantly higher spatial resolution than other rapid, non-destructive characterization techniques, such as x-ray diffraction. Therefore, Raman spectroscopy is particularly valuable for identifying material composition and structure. For example, forensic scientists can identify fibers, dyes, explosives, drugs, and biological fluids.

Biologists have observed how cells handle toxins, drugs, diseases, death, and differentiation. They have even identified the proteins in DNA and RNA. Because water weakly inelastically scatters, Raman spectroscopy has been particularly useful in non- destructive, automated quality control for pharmaceuticals [24,26].

Perhaps the most frequent application of Raman spectroscopy is materials science from semiconductors to polymers to ceramics to nanotechnology [26]. Materials scientists harnesses not only the ability to identify crystal structures but also their distributions and stresses. For example, a three-phase ceramic of SiC, B4C, and Si was mapped at the micron scale to reveal not only the phase distribution but also the stresses in the weak silicon phase [29]. The resulting maps revealed needle-like silicon- carbide fibers and a non-homogeneous stress distribution contrary to the traditional assumption of homogeneous stresses in microscale composites.

As a second example, Raman spectroscopy is arguably the most valuable tool in tracking the amorphization of B4C, one of the most popular and impressive structural materials [8,9,30]. In this deleterious phenomenon, high shear stresses induce the

formation of nanoscale bands (see Figure 1-2) that concentrate stress, nucleate cracks, soften the material, and ultimately cause catastrophic failure. This solid-state mechanism is the leading issue for the mechanical response of not only B4C but also

B6O and similar systems [31-33]. High-resolution transmission electron microscopy (HR-

TEM) can reveal these amorphous bands but is inefficient and can only sample small volumes. On the other hand, Raman scans can quickly and nondestructively differentiate virgin B4C from amorphized B4C through the intensities of the Raman shifts around 1340, 1520, and 1810 cm-1 [9]. The origins of these bands are frequently debated, especially because Raman spectra reflect vibrations, which require the bonding absent in amorphized regions [8,32]. Perhaps the changes are due to phase transformations or stress. Regardless, Raman spectroscopy remains a powerful tool for tracking the complex failure of B4C.

Density Functional Theory

The most fundamental explanation of all material behavior is from the bottom-up modeling of fundamental particles. Therefore the various flavors of quantum mechanics are often called “first principles” or “ab initio” (“from the beginning”) methods [34-38].

The idea is that every material property from color to strength can be explained simply through the locations of nuclei and electrons. By particle-wave duality, the locations and properties of these particles can be conveniently modeled by a wavefunction. Through the Born-Oppenheimer approximation, this wavefunction can be broken into nuclear and electronic portions:

Ψ = Ψ푒푙푒푐푡푟표푛푖푐 × Ψ푛푢푐푙푒푎푟 (1-5)

Essentially, this approximation independently handles the motion of the nuclei and the motion of the electrons. If negligible relativistic effects, a wavefunction can be determined from the time-dependent Schrödinger Equation:

휕 퐻̂[Ψ(풓, 푡)] = 푖ħ Ψ(풓, 푡) (1-6) 휕푡

On the left side, the Hamiltonian (퐻̂) operator describes the total energy (i.e., kinetic and potential energies) from the wavefunction. This linear, partial differential equation therefore relates total energy to the time derivative of the wavefunction. For a single, non-relativistic particle, this equation adopts a familiar form:

−ħ2 휕 [ ∇2 + 푉(풓, 푡)] Ψ(풓, 푡) = 푖ħ Ψ(풓, 푡) (1-7) 2푚 휕푡

Here, the kinetic energy is computed from the Laplacian, Plank’s constant (ħ), and the reduced mass (푚). Potential energy is given by the potential (푉). Assuming the solution is separable:

푖퐸푡 Ψ(풓, 푡) = Ψ(풓) ∙ 푓(푡) = ∑ 푐 Ψ (풓) exp (− ) (1-8) 푖 푖 푖 ħ

Materials scientists often are interested in the atomic/molecular orbitals (i.e., stationary states) so focus on the time-independent form of Schrödinger’s Equation:

−ħ2 [ ∇2 + 푉(풓)] Ψ(풓) = 퐸Ψ(풓) (1-9) 2푚

In terms of linear algebra, this equation can be considered an eigenvalue problem with the wavefunction as the eigenfunction and the energy of state (E) as the eigenvalue.

To apply to a multi-body system (e.g., multiple electrons), the form of the

Hamiltonian and wavefunction are modified. For example, the crystal potential can be written as a sum of the effects of the N particles’ potentials.

1 푁 2 푁 푍 푁−1 푁 −1 [− ∑푖=1 ∇푖 − ∑푖=1 + ∑푖=1 ∑푗=푖+1|푟푖 − 푟푗| ] Ψ(푟1 … 푟푁) = 퐸Ψ(푟1 … 푟푁) (1-10) 2 푟푖

By assuming that the electrons move independently of each other, the many-body wavefunction can be rewritten as a product of the single-particle orbitals:

푁 Ψ(푟1 … 푟푁) = ∏푖=1 Ψ푖 (1-11)

This equation is the fundamental assumption of the Hartree method. To improve upon this assumption, the Hartree-Fock method mandates asymmetry and adds correlation to the wavefunction. In fact, Hartree-Fock generally well predicts trends in atomic properties but can have large errors (e.g., in energy).

Even with the electron-independence simplification (Eq. 1-7), each wavefunction has three times the number of variables as the number of particles (i.e., three spatial coordinates per particle). The resulting computational expense motivated development of the most popular flavor of quantum mechanics for physicists: density functional theory (DFT). The fundamental notion is that the entire system can be modeled as a function of the spatial distribution of electrons (i.e., 3 instead of 3N spatial variables).

This Nobel-prize-winning idea eventually produced the Hohenberg-Kohn Theorems.

The first theorem states that the electronic density of the ground state dictates the wave function and energy at all states, including excited ones. The second theorem states that minimum energy reveals the ground-state density. These theorems ultimately led to the development of the Kohn-Sham Equations:

1 [− ∇2 + 푉 (푟) + 푉 (푟) + 푉 (푟)] Ψ (풓) = 휖 Ψ (풓) (1-12) 2 퐻 푒푥푡 푋퐶 푖 푖 푖

Interestingly, these equations appear nearly identical to Schrödinger’s Equation for a single, non-relativistic particle. Again, a Hamiltonian operates upon the wavefunction to describe total energy. The Laplacian captures the kinetic energy. The Hartree functional

(푉퐻) captures the potential energy from the Coulombic interaction of electrons. The

external functional captures the potential energy from the Coulomb interaction of atomic nuclei and any other applied potential. Finally, the exchange-correlation functional (푉푋퐶) captures the potential energy related to the swapping and correlation of electrons.

The error of DFT modeling comes exclusively from the exchange-correlation term. Some common approximations for this exchange-correlation functional include the local density approximation (LDA), which is based on a uniform electron gas from Monte

Carlo simulations, and the generalized gradient approximation (GGA), which introduces density gradients to preserve of the exact functional. In practice, all of the potential- energy terms are approximated by a single effective potential, called a pseudopotential.

For example, this dissertation mainly employs the Trollier-Martins norm-conserving pseudopotentials.

Finally, note that each DFT computation requires many choices contingent on the material system and desired properties. First, the atomic structure and its boundary conditions are chosen. For example, modeling the Raman spectrum of a crystal requires periodic boundary conditions and enough atoms to capture all important vibrational modes (i.e., degrees of freedom). The modes associated with higher degrees of freedom are less dominant than the ones with lower degrees of freedom, so an experimental Raman spectrum of a non-deformed material can usually be simulated with the smallest possible unit cell. In contrast, modeling deformation typically requires more atoms in the repeating unit. Next, the basis set (e.g., plane waves) must be chosen for modeling the wavefunction. The pseudopotentials and exchange-correlation functional are chosen to simplify modeling the interaction of atoms (i.e., potentials).

These simplifications are necessary due to computational limitations.

In addition to choosing several parameters through knowledge of the system, some parameters must be chosen through a convergence analysis. Most notably, the cutoff energy essentially dictates the number of atoms that each atom interacts with for computing energy. Also, the mesh must be chosen to determine the points at which properties are computed. Using either a cutoff energy that is too low or a mesh that is too coarse can result in meaningless, non-physical results without any indication of a problem. Hence, a DFT simulation always starts with a convergence analysis.

Essentially, different parameters, like cutoff energy, are manually varied to see the effect on the simulated property (usually total static energy). As described in the next section, the current work requires a particularly strong convergence for meaningful results.

Density Functional Perturbation Theory

After modeling and relaxing the crystal structure with DFT, density functional perturbation theory (DFPT) [39-42] can be employed to compute several material properties, such as Raman spectra and elastic constants. DFPT begins by modeling the energy of the structure as a Taylor series that is a function of small perturbations (λ):

2 3 (0) 훿퐸 1 훿 퐸 1 훿 퐸 퐸[휆] = 퐸 + ∑푖 휆푖 + ∑푖,푗 휆푖휆푗 + ∑푖,푗,푘 휆푖휆푗휆푘 + ⋯ (1-13) ∇휆푖 2 ∇휆푖∇휆푗 6 ∇휆푖∇휆푗∇휆푘

These perturbations can be a variety of things, including atomic displacements, lattice strains, and electric fields. Unlike the wavefunction, each term of this Taylor series can be a physical property. For example, the zeroth derivative of the energy is the ground- state energy. This is the energy that is minimized to find the ground state of a structure in DFT. Likewise, the first derivative of energy with respect to atomic displacement is

force (e.g., like a spring’s energy). These are the forces that should be around zero for a relaxed structure.

To compute the isotropic elastic constants, the second derivatives with respect to both strain and atomic displacement and strain are computed. Taking derivatives with respect to two strains (휂푖 and 휂푗) without allowing the atoms to relax yields the clamped- ion elastic constants as a function of cell volume (Ω):

1 푑퐸2 퐶푖푗̅ = (1-14) Ω 푑휂푖 푑휂푗

However, the material would naturally relax after strain, so this clamped-ion elastic tensor would overestimate stiffness. To correct this issue, the relaxed-ion elastic constants can be obtained as a function of the force constants, which are the differentiations with respect to atomic displacements (푟푚 and 푟푛):

2 2 ̅ 1 푑 퐸 푑푟푚 푑푟푛 푑 퐸 퐶푖푗 = 퐶푖푗 − ( ) ( 2 ) ( ) (1-15) Ω 푑푟푚 푑휂푖 푑 퐸 푑푟푛 푑휂푗

The elastic constants can be used for a variety of calculations. This dissertation mainly uses them to check elastic stability and to compute isotropic elastic moduli for polycrystals. These stability criteria have many forms depending on the material system.

For rhombohedral systems the following four criteria must be met [43]:

퐶11 − |퐶12| > 0 (1-16)

퐶44 > 0 (1-17)

−1 2 2 퐶33(퐶11 + 퐶12) − 퐶13 > 0 (1-18)

−1 2 2 2 2 2 퐶44(퐶11 − 퐶12) − 퐶14 − 퐶15 = 퐶44퐶66 − 퐶14 − 퐶15 > 0 (1-19)

Polycrystalline materials can be modeled as a collection of randomly oriented single crystals. Hence, isotropic elastic constants can be computed from an average of elastic

constants (or the compliance constants) along different directions [44]. The Reuss approximation assumes uniform stress and provides the lower bound of the elastic moduli. For example, for a hexagonal material, the bulk (B) and shear (G) moduli can be computed in terms of the relaxed-ion elastic constants (퐶푖푗):

2 −1 퐵푅 = [(퐶11 + 퐶12)퐶33 − 2퐶13][퐶11 + 퐶12 + 2퐶33 − 4퐶13] (1-20)

퐺푅 = {15푐(퐶11 − 퐶12)퐶44퐶66}{2(퐶11 − 퐶12)[2(퐶11 + 퐶12) + 4퐶13 + 퐶33]퐶44퐶66 +

−1 3푐[2퐶44퐶66 + (퐶11 − 퐶12)(퐶44 + 2퐶66)]} (1-21)

2 푐 = 퐶33(퐶11 + 퐶12) − 2퐶13 (1-22)

Alternatively, the Voigt approximation assumes uniform strain and provides the upper bound for the elastic moduli. For a hexagonal material:

−1 퐵푉 = 9 (2퐶11 + 2퐶12퐶33 + 4퐶13) (1-23)

−1 퐺푉 = 15 (2퐶11 + 퐶33 − 퐶12 − 2퐶13 + 6퐶44 + 3퐶66) (1-24)

Because the Reuss and Voight approximations provide the lower and upper bounds, respectively, it is common practice to compute Hill’s average:

−1 퐵 = 2 (퐵푉 + 퐵퐺) (1-25)

−1 퐺 = 2 (퐺푉 + 퐺퐺) (1-26)

The elastic modulus can then be computed as:

퐸 = (9퐵퐺)(3퐵 + 퐺)−1 (1-27)

In addition to stability and elasticity, this dissertation also employs DFPT to for

Raman spectra. Recall from Chapter 2 that only the subset of characteristic vibrations that induce a change in polarizability are Raman-active. Therefore, instead of simulating the interaction of a laser with a material, DFPT will be employed to model the characteristic vibrations and their corresponding effects on polarizability [39,40,45].

Recall that the horizontal axis for a Raman spectrum essentially represents the

Raman-active natural frequencies. Therefore, the first step is to calculate all of the natural frequencies. First, a matrix of force constants is computed by taking the second derivative of energy with respect to atomic displacements. Diagonalizing this matrix leads to the natural frequencies (휔푖) and phonon eigenvectors (τ). These phonons include the Raman, infrared, and silent vibrational modes. Also, checking to ensure the natural frequencies are positive offers another stability check [46,47]

To determine the Raman activity of all characteristic vibrations, the Raman intensities are computed next. Recall that these intensities are the vertical axis of a typical Raman spectrum and that they result from a change in polarizability (Eq. 1-4).

Hence, the dielectric tensor for an xy-oriented crystal (휒푥푦) is computed from the second-order derivative of the energy with respect to two electric fields. A dielectric tensor reveals the susceptibility of a material to polarization due to an applied electric field. Next, the Raman tensors for an xy-oriented crystal (푅푥푦) are computed from the derivatives of this dielectric tensor with respect to atomic displacements along the phonon eigenvectors (푟휏):

휕휒푥푦 푅푥푦 = √Ω ∑휏 (1-28) 휕푟휏

Hence, the change in polarizability for an xy-oriented crystal is captured by this Raman tensor. To remove dependence on crystallographic orientation, rotational invariants

(퐺(푖)) are calculated:

(0) −1 2 퐺 = ∑푖푖,푖=푥,푦,푧[3 (푅푖푖) ] (1-29)

(1) −1 2 퐺 = ∑푖푗,푖,푗=푥,푦,푧 [2 (푅푖푗 − 푅푗푖) ] (1-30)

(2) −1 2 −1 2 퐺 = ∑푖푗,푖,푗=푥,푦,푧 [2 (푅푖푗 − 훼푗푖) + 3 (푅푖푖 − 푅푗푗) ] (1-31)

Finally, the Raman intensities (퐼) are modeled to match the experimental spectrum that would result from a specific laser wavelength (휔퐿) and the ambient temperature (푇):

(휔 −휔 ) ℎ휔 −1 퐼 = 퐿 푖 [1 − exp (− 푖 )] [10 퐺(0) + 5퐺(1) + 7퐺(2)] (1-32) 30 휔푖 2휋푘푇

Because the Raman tensor results from three derivatives of the energy, these Raman intensities are particularly sensitive to the equilibration step in DFT. Therefore, the computational parameters, especially the cutoff energy and number of relaxation steps, must be carefully chosen. Also, see that this analysis predicts the locations (휔푖) and intensities (퐼) of peaks. Peak widths require higher-order calculations for anharmonic effects but are not necessary for identification of crystal structures, stress, etc.

Dissertation Aims

Boron-rich icosahedral solids often command extraordinary mechanical, thermal, and chemical properties. Therefore, these ceramics are often utilized in applications requiring extreme resistance to impact (e.g., armor), heat (e.g., machining), and chemical reaction (e.g., nuclear reactors). In order to improve performance in these and new applications, this dissertation seeks to understand the underlying physics that rationalize and limit these icosahedral solids.

Phase I (i.e., Chapters 2 and 3) covers B4C, which has long dominated research and industry because of its unparalleled combination of high strength and low mass density. This work attacks one of the most pressing and confounding issues: deciphering the complex composition. Numerous polymorphs of B4C with different atomic arrangements and stoichiometries have been proposed. Unfortunately, the traditional nomenclature for B4C cannot uniquely represent all of these unique

crystallographic forms. Therefore, this phase first redefines the nomenclature by adding full specificity for carbon locations in the unit cell. To confirm this revised nomenclature captures all variability in properties, the static energies, lattice parameters, and Raman spectra of B4C (i.e., 20 at.% C) are modeled with quantum mechanics. The next complication is that almost all polymorphs of B4C were found to have similar formation energies. Therefore, fabricated samples may contain a collection of crystal structures mixed at the nanoscale. Identifying and quantifying this mixture is essential to understanding the fundamental limits of B4C and for establishing models for its mechanical failure (i.e., amorphization). However, the scattering properties of the polymorphs are too similar for distinction by traditional characterization techniques.

Therefore, this phase also develops a new tool that fits a deconvoluted experimental

Raman spectrum with the corresponding spectra generated through quantum mechanics. By using the revised nomenclature, relative abundances, including those of previously overlooked constituents, of fabricated samples are quantified.

Phase II (i.e., Chapter 4) covers B6O, a material at the cutting edge of research.

As with B4C, the principal issue is resolving variability in crystal structure. However, the variability of B6O is not in polymorphism but in nanotwinning (i.e., the incorporation of periodic twins separated by a nanoscale distance). In fact, the lack of polymorphism drives crystallographic homogeneity that likely increases susceptibility to nanotwinning.

This susceptibility is particularly promising because nanotwinning was previously shown to increase the yield strength of copper tenfold and nearly double the record hardness of diamond. Because B6O is relatively unstudied, this dissertation presents results from a wide range of techniques: hot pressing, spark plasma sintering, dynamic/quasistatic

indentation, compression, transmission/scanning electron microscopy, x-ray diffraction,

Raman spectroscopy, density functional theory, and density functional perturbation theory. The HR-TEM reveals that the fabricated samples suffer from heterogeneous nanotwin spacing. Even in this non-optimized crystal structure, however, B6O is found to outperform B4C in hardness, strength, and elasticity. To guide future studies into producing superhard B6O, this work uses quantum mechanics to discover the material- specific critical twin spacing that maximizes strength. Also, these calculations reveal an x-ray-diffraction peak that could be used to track the volume of nanotwinning.

Phase III (i.e., Chapter 5) expands beyond the promising structure of B6O and quantum-mechanically models all p-block hexaborides. By assessing the ground states and shear deformation of this representative subclass of icosahedral solids, fundamental metrics of both elasticity and superstrength are established. These metrics comprise both the strength of polar bonding (i.e., icosahedral separation) and the strength of equatorial bonding (i.e., localization of electrons at equatorial sites).

Identifying these metrics helps rationalize icosahedral superstrength in general and may facilitate tailoring and discovery of next-generation materials. Next, this work confirms that nanotwinning minimally affects elasticity so suggests that mechanism investigations should focus on inelastic deformation. Consistent with the experimental work of Phase

II, the simulated static energies suggest that hexaborides, especially B6O, are susceptible to nanotwinning. Finally, this phase demonstrates that number of valence electrons available at equatorial sites correlates with toughness.

Virgin B4C Indented B4C Normalized Raman Intensity Normalized

450 650 850 1050 1250 1450 1650 1850 -1 Raman Shift in Wavenumbers (cm )

Figure 1-1. Experimental Raman scans of virgin and indented regions of HP B4C. Intensities are normalized to unity.

Figure 1-2. HR-TEM shows the amorphous bands that can induce dislocations, lattice rotation, cracking, and catastrophic failure in B4C. Reproduced from [9].

CHAPTER 2 CRYSTALLOGRAPHIC AND SPECTRAL EQUIVALENE OF BORON-CARBIDE POLYMORPHS

Crystallography of B4C

For B4C, polymorphism affords large variability in both carbon percentage (i.e., from 8.0 to 20 at%) and atomic arrangement to strongly influence physical, mechanical, and electronic properties [8,48-57]. Further, closeness of formation energies of these crystal structures induces many polymorphs to form in the same fabricated sample [56].

Unfortunately, limitations in experimental techniques and naming conventions obscure the identity and distribution of these crystal structures. Hence, this chapter introduces and evaluates a highly specific nomenclature through quantum mechanical simulations of ground states and Raman spectra.

The simplest unit cell for each boron-carbide polymorph comprises a three-atom chain and a twelve-atom icosahedron [see Figure 2-1(a)]. Within each icosahedron, six polar (p) atoms connect multiple icosahedra directly, and six equatorial (e) atoms connect multiple icosahedra through the chains [see Figure 2-1(b) and Figure 2-1(c)].

Traditionally [8], researchers differentiate boron-carbide crystal structures with the following notation: (BmCnfo)ijk where (BmCnfo) represents the icosahedron (sites 1 -- 12), and ijk represents the chain (sites i, j, and k) [see Figure 2-1(a)]. Subscripts “m” and “n” indicate the number of boron atoms and carbon atoms, respectively, in the icosahedron.

If n = 0, Cnfo is excluded. Subscript “f” marks the family of the icosahedral carbons as either polar (p) or equatorial (e). If n = 2 and f = p, “o” is often included to describe relative orientation of polar carbons [8]. The simplest example of this traditional nomenclature is (B12)CCC, which has a CCC chain and a twelve-boron icosahedron.

Here, the six equatorial boron atoms represent one 6h Wyckoff site of space group 166

(푅3̅푚), and the six polar boron atoms represent another [58,59]. Thus, there is no need to differentiate within polar sites or within equatorial sites for (B12)CCC. However, an icosahedron with a carbon atom [i.e., (B11C1f)ijk] no longer exhibits the aforementioned symmetry. Depending on chain composition, location of this icosahedral carbon may need to be explicitly specified in the nomenclature to differentiate unique crystal structures. Adding a second icosahedral carbon atom poses similar issues. Consider

(B10C2p,antipodal)BCB, which has one carbon atom in polar site 1, 2, or 3 and another carbon atom in the polar site 4, 5, or 6. Because the distance between pairs of antipodal atoms varies, additional specificity in the nomenclature is required to describe variability in properties.

Comprehensive understanding of material performance and full tailoring of properties mandate differentiation of all unique polymorphs. Unfortunately, the commonly adopted nomenclature does not always differentiate polymorphs with the same stoichiometry. Further, while fabricated B4C may exhibit 푅3̅푚 symmetry on average, constituent polymorphs may not. For these reasons, this chapter modifies nomenclature from (BmCnfo)ijk to (BmCfz)ijk. Here, “z” specifies location(s) of icosahedral carbon atom(s) in the unit cell according to Figure 2-1(a). Reduced coordinates may slightly change with position and/or number of carbon atoms. The “Cfz” is excluded if no icosahedral carbons are present [e.g., (B12)CCC], and site numbers are separated by commas if multiple icosahedral carbons are present [e.g., (B10Cp2,5)BCB]. Subscript “f” is retained to avoid confusing “z” (site number) for “n” (number of atoms). Unlike traditional nomenclature, this modified scheme can model differences due to icosahedral carbon position within polar sites and within equatorial sites [e.g., (B11Cp1)BCC vs.

(B11Cp4)BCC] and can also differentiate a chain from its mirror [e.g., (B11Cp1)BCC vs.

(B11Cp1)CCB]. Hence, this new nomenclature lays a more robust foundation for understanding polymorphism and the corresponding variation in properties.

Ground States of B4C Polymorphs

The current study employs ABINIT [60,61], a popular atomistic solver for density functional theory (DFT), to model the fifty-two variations of B4C commonly proposed in both experimental and theoretical investigations on B4C [8,54,56,62-64] except those with chain vacancies [65-68] or without Raman activity [57,69]. Single unit cells [Figure

2-1(a)] with periodic-boundary conditions, a Monkhorst k-point mesh, and norm- conserving pseudopotentials with the Troullier-Martins scheme and Pulay mixing are utilized. The exchange-correlation functional is evaluated by both the Teter-Pade local- density approximation (LDA) [70] and the Perdew-Burke-Ernzerhof generalized-gradient approximation (GGA) [71]. From this energy-minimization process (accurate up to 10-14 eV), lattice parameters, mass densities, and relative energies per atom are calculated

(see Table 2-1). Structures with the same properties are considered an “equivalent- lattice group,” assigned a Roman numeral, and ordered by energy relative that of group i. All calculated lattice parameters and energies from the seven B4C polymorphs modeled in a previous DFT study [55] have been replicated within 0.3% error in this investigation. For visualization purpose, Figure 2-2 shows a quantum-mechanical simulation of bonding (i.e., black isosurfaces) of group i, and Figure 2-3 provides schematics of all ten equivalent-lattice groups modeled in this chapter.

Two competing effects seem to drive the ordering of the polymorphs by energy in

Table 2-1. First, small distances between carbon atoms generally were found to correlate with large energies. Carbon has higher electronegativity than boron, so

spatially concentrating carbon atoms concentrates charge and raises potential energy.

Hence, groups i and ii (CBC chain) have lower energy than groups iii – vii (CCC, BCC, or CCB chain). Groups i (0 meV) and iii (166 meV) have lower energy than groups ii (36 meV) and iv (174 meV), respectively, because the icosahedral carbon of the former two groups are not directly bound to chain carbons as in the latter two groups. Likewise, group viii (304 meV) has lower energy than group x (346 meV) because the icosahedral carbons are bonded to icosahedral borons in the former but to icosahedral carbons in the later [see Figure 2-1(b)]. Second, the number of icosahedral carbon atoms was found to correlate with energy. That (B12) is the lowest-energy icosahedron is consistent with the fact that boron icosahedra can form without chains [72,73]. Energy increases from group iii (zero icosahedral carbons) to groups iv – vii (one icosahedral carbon) to groups viii – x (two icosahedral carbons). Polymorphs with a CBC chain and a single icosahedral carbon atom (groups i and ii) have lower energy than (B12)CCC (group iii) simply as a result of the competition of these two effects. In group iii, the rise in energy due to direct bonding of the carbons in the CCC chains supersedes the low energy of the (B12) icosahedron. In groups i and ii, however, the chain carbons are favorably separated by a boron atom.

Raman Spectra of B4C Polymorphs

After fully relaxing lattice parameters and atomic positions, density functional perturbation theory (DFPT) in ABINIT [74] is employed to calculate natural frequencies, phonon eigenvectors, and dielectric tensors. Intensities of the Raman-active natural frequencies can then be computed from a well-established post-processing method

[39]. Because this chapter seeks to establish the equivalencies of polymorphs, Raman- peak widths, which result from anharmonic effects [40], are beyond the scope of this

study and are therefore not included. With a temperature of 300 K and an excitation wavelength of 532 nm, this analysis provides the first high-fidelity simulations for numerous boron-carbide polymorphs. Each row of Table 2-2 lists the top-five Raman peaks (natural frequency; normalized intensity) for polymorphs that exhibit identical

Raman spectra. Most importantly, these “equivalent-Raman groups” are found identical to the equivalent-lattice groups and are therefore labeled in the same fashion as in

Table 2-1. Agreement in these equivalencies is due to the fact that both energy and

Raman activity originate from local bonding environment (see previous rationalization of energy trends and [75]). This agreement suggests that these equivalent groups may share other properties too.

Because LDA and GGA produced similar values and identical trends (see Tables

2-1, Table 2-2, and Figure 2-4), Figure 2-5 plots representative LDA Raman spectra to illustrate salient points. Figure 2-5(a) compares the simulated Raman peaks from group i with an experimental spectrum from HP B4C. For the first time, there is strong agreement of most peak locations and intensities between simulated and experimental spectra. This agreement suggests that group-i polymorphs represent the majority of fabricated B4C. This finding is consistent with many prior studies [54,56,62,63] but offers new information on the vibrational properties of this popular crystal structure. HP B4C contains several components (e.g., free carbon, boron clusters, and boron-carbide polymorphs of possibly different stoichiometries), so peaks not represented by the group-i spectrum are likely due to other species. For example, the relatively large experimental peaks around 980 cm-1 may be due to group viii or group x [see Table 2-2 and Figure 2-5(b)]. This finding is consistent with prior literature [54,63] that implicated

the polymorphs with (B10C2p,antipodal) icosahedra in minority constituents. Figure 2-5(b) and Figure 2-5(c) show variability in Raman spectrum for polymorphs in the same traditional families. The former highlights the need to differentiate antipodal configurations, and the latter highlights the need to differentiate chains from their mirrors

(i.e., ijk vs. kji). The revised nomenclature enables these distinctions.

Impact of Modified Nomenclature

By modeling all members of traditional families, this section offers the first comprehensive description of the B4C crystal structures and their properties. For example, Goddard [76-78] modeled amorphization of B4C based on slip systems, energies, and twins of a single polymorph from the (B11Cp)CBC traditional family. The current investigation found all six polymorphs in this traditional family identical (see group i in Table 2-1 and Table 2-2), so the assumption of those works is valid. On the other hand, Fanchini [56] and Aydin [55] modeled one polymorph from each traditional family with a BCC chain to help explain the kinetics of amorphization and the variability in hardness, respectively. The current investigation has shown that these traditional families exhibit significant variation in energy and Raman spectrum (see groups iv – vii in Tables 2-1 and 2-2), so the results of those studies may be incomplete.

Understanding amorphization is paramount for improving the performance of B4C, so the current results may prove especially useful.

By introducing a new nomenclature and by simulating lattice parameters, energy, and Raman spectra, this chapter has improved the understanding of boron-carbide crystallography. Most importantly, these simulations have enabled novel classifications of polymorphs with differences polar/equatorial occupancies or with differences in chain permutation. These distinctions were not previously possible in the old nomenclature

and serve to highlight the importance of understanding polymorphism-induced variability in local bonding environment. Comprehensive understanding of the structure, properties, and promise of this advanced ceramic mandates these improvements.

Table 2-1. Lattice parameters, mass densities, and relative static energies per icosahedron of B4C (20 at%) polymorphs for the LDA (GGA). Structures with the same energy and combination of lattice parameters are listed on the same row and assigned a Roman numeral. Equivalent Lattice Mass Rel. Group Polymorphs Parameters Density Energy 3 -- zijk of (BmCfz)ijk Å g/cm meV i 1CBC = 2CBC = 3CBC = {5.00, 5.16, 5.16} 2.60 0

4CBC = 5CBC = 6CBC ({5.04,5.19,5.19}) (2.56) (0) ii 7CBC = 8CBC = 9CBC = {5.11, 5.15, 5.15} 2.59 36

10CBC = 11CBC = 12CBC ({5.15,5.18,5.18}) (2.55) (36) iii 0CCC {5.14, 5.14, 5.14} 2.55 77

({5.17,5.17,5.17}) (2.50) (71) iv 7BCC = 10CCB = 8BCC = {5.05, 5.16, 5.16} 2.55 166

11CCB = 9BCC = 12CCB ({5.10,5.19,5.19}) (2.50) (161) v 4BCC = 1CCB = 5BCC = {4.98, 5.17, 5.17} 2.54 169

2CCB = 6BCC = 3CCB ({5.02,5.20,5.20}) (2.50) (164) vi 1BCC = 4CCB = 2BCC = {4.96, 5.17, 5.17} 2.55 174

5CCB = 3BCC = 6CCB ({5.00,5.20,5.20}) (2.50) (169) vii 10BCC = 7CCB = 11BCC = {5.10, 5.16, 5.16} 2.53 225

8CCB = 12BCC = 9CCB ({5.13,5.19,5.19}) (2.49) (219) 3,5BCB = 2,6BCB = viii {5.01, 5.01, 5.21} 2.54 304 3,4BCB = 1,6BCB = 1,5BCB = ({5.05,5.05,5.24}) (2.59) (299) 2,4BCB 5,6BCB = 2,3BCB = ix {5.01, 5.01, 5.23} 2.52 328 4,6BCB = 1,3BCB = 4,5BCB = ({5.05,5.05,5.25}) (2.47) (322) 1,2BCB 1,4BCB = 2,5BCB = x {4.81, 5.22, 5.22} 2.55 346 3,6BCB ({4.85,5.25,5.25}) (2.50) (342)

Table 2-2. Natural frequencies (ω) and normalized intensities (I) for the dominant Raman peaks (DRPs) of each equivalent group of B4C (20 at% C) polymorphs for the LDA (GGA). Group DRP #1 DRP #2 DRP #3 DRP #4 DRP #5 -- ω in cm-1; I ω in cm-1; I ω in cm-1; I ω in cm-1; I ω in cm-1; I i 1099; 1.00 1058; 0.72 486; 0.69 1076; 0.25 724; 0.19 (1019; 1.00) (1074; 0.97) (480; 0.85) (1044; 0.26) (1116; 0.24) ii 1082; 1.00 1118; 0.60 512; 0.50 747; 0.30 703; 0.26 (1054; 1.00) (1080; 0.58) (504; 0.54) (730; 0.34) (688; 0.29) iii 1160; 1.00 758; 0.87 1098; 0.53 999; 0.37 707; 0.08 (1154; 1.00) (743; 0.97) (1059; 0.59) (968; 0.54) (683; 0.12) iv 1067; 1.0 724; 0.52 355; 0.38 776; 0.38 498; 0.37 (1046; 1.00) (952; 0.61) (1153; 0.58) (754; 0.56) (1039; 0.55) v 1062; 1.00 1160; 1.00 377; 0.79 460; 0.61 1076; 0.42 (1021; 1.00) (1147; 0.73) (455; 0.60) (383; 0.55) (1039; 0.34) vi 354; 1.00 1071; 0.89 1685; 0.46 460; 0.38 1164; 0.36 (1030; 1.00) (369; 0.80) (453; 0.47) (1144; 0.43) (1673; 0.39) vii 394; 1.00 1668; 0.58 1140; 0.45 723; 0.30 457; 0.27 (398; 1.00) (1653; 0.66) (737; 0.45) (1126; 0.41) (453; 0.36) viii 975; 1.00 1015; 0.76 454; 0.62 1035; 0.61 1080; 0.56 (978; 1.00) (948; 0.94) (448; 0.65) (431; 0.60) (1045; 0.59) ix 264; 1.00 409; 1.00 348; 0.96 1016; 0.91 971; 0.75 (267; 1.00) (979; 0.54) (394; 0.53) (345; 0.38) (944; 0.31) x 968; 1.00 502; 0.77 453; 0.70 1028; 0.69 1180; 0.69 (993; 1.00) (942; 0.71) (446; 0.66) (498; 0.62) (1151; 0.53)

Figure 2-1. (a) Simplest boron-carbide unit cell color-coded by site family: polar (red, sites 1 -- 6), equatorial (blue, sites 7 -- 12), and chain (black, sites i, j, and k). (b) Connectivity of chain atoms. (c) Nearest neighbors of icosahedral atoms. Reproduced from [79].

Figure 2-2. Two views of a quantum mechanical simulation of bonding (i.e., black surfaces) in 1CBC. Purple spheres represent boron nuclei, and green spheres represent carbon nuclei.

Figure 2-3. Representative crystal structures from ten equivalent groups of boron- carbide polymorphs.

Figure 2-4. Simulated Raman spectra of ten equivalent groups of B4C polymorphs had similar features in the LDA (blue diamonds) and GGA methods (red x’s). These symbols represent the wavenumbers and intensities of the peak centers. For comparison, an experimental scan of B4C is shown by a black curve in each subfigure.

Figure 2-5. (a) Strong agreement of Raman spectra from group-i simulations (LDA) and experimentally (EXP) HP B4C. (b,c) Variation within (B10C2p,antipodal)BCB, (B11Ce)BCC and (B11Ce)CCB and traditional families. Reproduced from [79].

CHAPTER 3

EVALUATING BORON-CARBIDE CONSTITUENTS WITH SIMULATED RAMAN SPECTRA

Crystallographic Variability in Ceramics

Ceramics often exhibit a high propensity for variation in their crystal structures

(e.g., polymorphs, polytypes, allotropes, isomorphs, and isomers). For example, silicon carbide has over 250 polytypes [80], and B4C is speculated to have hundreds of polymorphs [8]. Identification of these structural variants typically comprises quantum- mechanical calculations of Gibbs free energy (i.e., the sum of static/lattice, internal, phonon, and pressure-volume energies [81]). Because this calculation requires significant computational expense, many studies exclusively report static energy [82].

However, this approximation forgoes kinetics considerations (e.g., temperature- dependent polymorphic transformations [83]) and may incorrectly predict fabricated structures [84]. Unfortunately, even including all contributors to Gibbs free energy does not guarantee correct prediction for systems that have energetically similar structural variants [85]. For those cases, researchers often rely on a comparison of experimental and theoretical properties (e.g., TEM diffraction and spectroscopies). For decades, comparison methodology has been employed to analyze the structure of B4C [8].

However, success has been greatly limited by the large number of proposed polymorphs and by the tendency for fabricated samples to contain several crystal structures mixed at the nanoscale (i.e., below experimental resolution) [56]. Hence, this chapter approximates components of both HP and SPS B4C by fitting experimental

Raman scans with simulated Raman spectra.

Experimental Raman Spectroscopy

Raman scans of both hot-pressed (HP) and spark-plasma-sintered (SPS) B4C are performed with 532-nm wavelength, 250-mW energy, 1800-l/mm grating, 1-s exposure, and 1-μm spatial resolution. This green wavelength is the most common for measuring the Raman spectra of B4C [8]. Also, each scan (i.e., HP and SPS) can be considered representative of its processing method because the experimental Raman spectrum of B4C was shown to be independent of stoichiometry [86]. Next, these two experimental Raman scans are normalized and deconvoluted with the Renishaw

WiRE® 4.3 software [see “EXP” curves in Figure 3-1]. These deconvolutions recreate the experimental spectra within 0.01% error. Note that the locations (i.e., wavenumbers) of the dominant Raman peaks are nearly identical for HP and SPS samples. These shared, dominant peaks are labeled “A” through “K” in Fig 3-1. The major differences between these two spectra are only the relative intensities of several peaks (especially

266, 320, 480, 533, and 825 cm-1).

Simulated Raman Spectroscopy

To evaluate the fabricated constituents, Raman spectra are now simulated to match the aforementioned deconvoluted peaks. For this study, stoichiometry of simulated polymorphs is limited to B4C (i.e., 20 at% C), which offers high theoretical stability [49,50,53,54,87], Raman activity [88], and consistency between experimental and simulated twinning behaviors [78]. Also, α-boron (i.e., α-B12) is included because prior studies speculated that this crystal structure may be formed in B4C [54,89]. To offer comprehensive and efficient simulation of boron-carbide polymorphs, this section employs the recently modified nomenclature: zijk [79]. Here, z specifies the site number(s) of icosahedral carbon(s), and ijk specifies chain permutation. For example,

this nomenclature can differentiate 3,5BCB [i.e., (B10Cp3,p5)BCC] from 1,4BCB [i.e.,

(B10Cp1,p4)BCB], but the traditional nomenclature considers both to be

(B10C2p,antipodal)BCB. Hence, by using this revised nomenclature, all variations of the B4C unit cell that have twelve-atom icosahedra and three-atom linear chains can be modeled.

Ground-state electronic configurations and static energies are calculated with density functional theory (DFT) in ABINIT [61] with periodic unit cells, 4x4x4 Monkhorst k-point meshes, 30-Ha (816-eV) cutoff energy, Troullier-Martins norm-conserving pseudopotentials, and the Teter-Pade local-density approximation (LDA) [70]. This exchange-correlation functional was chosen over the generalized-gradient approximation (GGA) because of a prior experimental comparison [79]. As validation, static energies of six boron-carbide polymorphs reported in another study [55] were reproduced within 0.5% error by using the current parameters. After relaxation, elastic tensors and natural frequencies are calculated with density functional perturbation theory (DFPT) [74] under the harmonic approximation. Except for 1,2,3BBB and

1,2,9BBB, all fifty-eight unique B4C structures (i.e., “equivalent-lattice groups” [79]) are found to be both elastically stable (i.e., having positive values for all four elastic-stability criteria [90]) and dynamically stable (i.e., lacking negative natural frequencies). Table 3-

1 ranks all stable crystal structures by their static energies (Estatic), which are listed relative that of 1CBC, the most abundant polymorph [8].

For the stable crystal structures, Raman intensities are calculated with the post- processing deployed by the WURM This WURM method is independent of crystallographic orientation and explicitly incorporates both excitation frequency (i.e.,

532 nm) and temperature (i.e., 300 K). As a validation of the Raman-simulation parameters, the peak positions of multiple Raman spectra in the WURM database [39] and of both experimental and simulated Raman spectra of α-B12 [91,92] were reproduced within 0.5% error. project [39,93]. For conciseness, Table 3-1 reports the top three dominant peaks for each structure (except for 1CBC). Peaks within 11 cm-1 of a deconvoluted experimental (EXP) peak are labeled with superscripts corresponding to the peak locations of Figure 3-1 (i.e., “A” through “K”). This buffer should account for deviation due to the harmonic approximation [40] and ensures that the most abundant polymorph (i.e., 1CBC) matches the most dominant experimental peaks. Full spectra for all equivalent groups of B4C polymorphs are provided in Appendix A.

Spectral Superposition

An experimental Raman spectrum represents the combination of characteristic scattering events from all constituents in the interaction volume, and the relative contribution of each constituent is directly proportional to its relative abundance. Hence, the relative abundance of a constituent is determined by the scaling factor required for fitting that constituent’s simulated spectrum to a dominant peak in the experimental spectrum. The harmonic approximation and pseudopotentials only induce small errors, so simulated polymorphs with strong dominant Raman peaks in regions far from experimental peaks (e.g., 1160 cm-1 of 0CCC) are disqualified as constituents. As suggested by experimental peak splitting [91], nearby peaks can superimpose if peak separation and widths are appropriate. For example, the simulated peaks of 1CBC at

1059 cm-1 and 1099 cm-1 likely superimpose to form experimental peak K (1089 cm-1).

The high scaling factor required for fitting this peak translates to 65 at% for HP B4C and

45 at% for SPS B4C [see Figure 3-1] and is consistent with prior work [8]. Also, 1CBC

can account for peaks C (479 cm-1), D (532 cm-1), and F (725 cm-1). Peak G (825 cm-1) can be attributed to α-B12 and note that no structure with B4C stoichiometry could account for this peak. This selection of α-B12 is consistent with the fact that boron clusters are sometimes considered minority constituents [56,92] and proposed to form during solid-state amorphization [9,72,73,94]. Peaks A (265 cm-1), B (319 cm-1), and H

(961 cm-1) can be attributed to polymorphs with multiple icosahedral carbons (e.g.,

7,8,10BBB, 1,8BCB, and 1,2,6BBB). Many of these structures were previously overlooked.

To establish the origin of peak I (998 cm-1), future simulations of other boron phases[89,95], defected/twinned structures [78], non-B4C stoichiometries (e.g., polymorphs without Raman-activity [8] and/or with two/four-atom chains [51,69,87,96]), or heterogeneous supercells [97] are recommended. To identify non-Raman-active constituents, other spectroscopies would prove useful. To more easily distinguish crystal structures that produce similar Raman peaks and to determine when an experimental peak is a superposition of nearby peaks, peak widths could be included with anharmonic simulations [40]. To include the interactions of dissimilar constituents, supercells could be simulated if computational expense decreases. However, although these potential improvements would further generalize our method, the current form still provides a powerful prediction.

Impact of Quantification of Complex Compositions

This chapter has presented a novel and efficient method for predicting relative abundances of constituents in complex materials through comparison of experimental and quantum-mechanically simulated spectra. This method has been applied to quantify the constituents produced in HP and SPS B4C. Based on the similarities of the

dominant Raman-active natural frequencies, it was argued that these two processing methods yield the same crystal structures but at different abundances for B4C. The identification of the majority constituent (i.e., 1CBC) was consistent with multiple prior investigations, and several minority constituents have been identified for the first time. In particular, the association of the relative abundance of α-B12 with the Raman peak near

-1 825 cm could be particularly helpful for understanding the amorphization of B4C [9].

Table 3-1. Static energies (Estatic) and dominant Raman peaks for all stable B4C polymorphs and for α-B12. B4C static energies are relative 1CBC. Superscripts indicate agreement of simulated wavenumbers with the twelve experimental (EXP) peaks shown in Figure 3-1.

Structure Estatic Dominant Raman Peaks zijk meV EXP Peak(cm-1; a.u.) 1CBC 0 K(1099; 1.00) (1058; 0.72) C(486; 0.69) F (1076; 0.25) (724; 0.19) (1151; 0.17) D (772; 0.13) (534; 0.12) (740; 0.12) 7CBC 35 K(1082; 1.00) (1118; 0.60) (512; 0.50) 0CCC 77 (1160; 1.00) (758; 0.87) K(1098; 0.53) 7,8BBC 151 J(1045; 1.00) J(1036; 0.95) (1120; 0.64) 1,10CBB 156 (1068; 1.00) (492; 0.92) J(1031; 0.75) 7BCC 166 (1067; 1.00) F(724; 0.52) (355; 0.38) 4BCC 169 (1062; 1.00) (1160; 1.00) (377; 0.79) 1BCC 174 (354; 1.00) (1071; 0.89) (1685; 0.46) 1,8BBC 176 (1056; 1.00) C(477; 0.81) (1113; 0.64) 1,7BBC 193 (307; 1.00) C(471; 0.69) (439; 0.50) 7,10BBC 199 (1019; 1.00) (1061; 0.65) D(535; 0.58) 1,11CBB 201 (347; 1.00) (431; 0.74) J(1033; 0.63) 1,5BBC 202 (344; 1.00) (1074; 0.88) (1024; 0.65) 1,2BBC 221 (341; 1.00) (1019; 0.56) (378; 0.35) 10BCC 225 (394; 1.00) (1668; 0.58) (1140; 0.45) 1,2CBB 229 B(324; 1.00) (362; 0.99) (420; 0.43) 1,10BBC 228 (334; 1.00) (461; 0.71) J(1046; 0.37) 1,8CBB 232 (360; 1.00) K(1100; 0.68) J(1043; 0.64) 7,11BBC 246 J(1036; 1.00) (1167; 0.50) (510; 0.49) 1,4BBC 248 B(309; 1.00) (457; 0.93) J(1034; 0.84) 1,7CBB 260 (404; 1.00) (455; 0.70) (1074; 0.66) 7,10BCB 265 (1063; 1.00) (689; 0.68) F(723; 0.68) 1,11BBC 271 (352; 1.00) (307; 0.24) J(1040; 0.17) 1,10BCB 272 A(257; 1.00) (1059; 0.41) (678; 0.28) 7,8CBB 283 (364; 1.00) (412; 0.66) J(1050; 0.50) 7,8BCB 287 (347; 1.00) (1604; 0.56) (417; 0.32) 1,8BCB 289 J(1029;1.00) B(322; 0.98) E(702; 0.83) 1,5BCB* 304 (975; 1.00) (1015; 0.76) (454; 0.62)

Table 3-1. Continued

Structure Estatic Dominant Raman Peaks zijk meV EXP Peak(cm-1; a.u.) 1,7BCB 305 (214; 1.00) (1053; 0.44) C(470; 0.42) 7,11BCB 314 (345; 1.00) (453; 0.68) (672; 0.67) 1,11BCB 316 (281; 1.00) D(363; 0.42) (1625; 0.40) 1,2BCB 328 A(264; 1.00) A(264; 1.00) (348; 0.96) 7,8,9BBB 337 B(325; 1.00) H(966; 0.13) (853; 0.08) 1,4BCB 346 H(968; 1.00) (502; 0.77) (453; 0.70) 1,7,10BBB 352 (424; 1.00) A(263; 0.61) (974; 0.54) 1,2,4BBB 356 A(268; 1.00) (304; 0.65) (393; 0.46) 7,8,10BBB 358 H(966; 1.00) (348; 0.49) (291; 0.40) 1,8,9BBB 359 (296; 1.00) (981; 0.16) D(522; 0.10) 1,5,8BBB 373 (932; 1.00) (452; 0.93) (462; 0.87) 1,10,11BBB 375 (350; 1.00) (293; 0.68) (985; 0.27) 1,8,10BBB 375 (379; 1.00) (977; 0.91) A(258; 0.79) 1,8,11BBB 379 (331; 1.00) (975; 0.66) (1164; 0.56) 1,7,8BBB 384 (379; 1.00) (415; 0.43) I(1000; 0.43) 7,8,12BBB 405 (337; 1.00) (988; 0.64) (415; 0.47) 1,5,9BBB 414 A(267; 1.00) B(315; 0.94) (1021; 0.24) 1,11,12BBB 415 B(319; 1.00) B(324; 0.20) (978; 0.13) 1,8,12BBB 423 (351; 1.00) (304; 0.75) I(992; 0.65) 1,2,10BBB 430 A(265; 1.00) (398; 0.42) B(327; 0.35) 1,2,7BBB 432 (378; 1.00) A(266; 0.49) (459; 0.27) 1,7,11BBB 433 (250; 1.00) (307; 0.90) (362; 0.33) 1,4,7BBB 440 (394; 1.00) (580; 0.41) (212; 0.40) 1,2,12BBB 442 (245; 1.00) (351; 0.80) (429; 0.48) 1,2,6BBB 442 A(266; 1.00) A(262; 0.39) (934; 0.25) 1,5,7BBB 443 (404; 1.00) A(254; 0.63) (373; 0.61) 1,4,8BBB 455 (300; 1.00) A(263; 0.63) (226; 0.31) G H α-B12 -- (818; 1.00) (951; 0.46) (791; 0.24)

Figure 3-1. Fitting of deconvoluted experimental Raman spectra (dotted curves) with quantum mechanical simulations. Differences between (a) HP and (b) SPS B4C seem to be merely relative abundances. Reproduced from [98].

CHAPTER 4 NANOTWINNING AND AMORPHIZATION OF BORON SUBOXIDE

Nanotwinning of Ceramics

Twins are crystallographic defects represented by a reflection of a parent lattice.

They can occur during nucleation, growth, phase transformation, recrystallization, annealing, or deformation. Regardless of origin, twins generally induce stress concentrations that promote crack nucleation and lower mechanical performance [99].

However, recent investigations showed that increasing twin density (i.e., reducing twin spacing, λ) at the nanoscale can dramatically increase mechanical properties even beyond those of nanograined structures [4,5]. Nanotwinned copper (nt-Cu) produced by pulsed electrodeposition exhibited an order-of-magnitude increase in yield strength over regular copper (i.e., 900 vs. 70 MPa) without lowering electrical conductivity [18,19]. A later study showed nt-Cu had up to 85% increased fatigue strength (i.e., 370 vs. 80

MPa) [100]. Nanotwinned cubic boron nitride (nt-c-BN, 휆 ≈ 3.8 푛푚) produced from onion nanoparticles exhibited a 40% increase in microhardness (> 100 GPa) and a 140% increase in fracture toughness (>12 MPa·m1/2) [16,17]. Also produced from onion nanoparticles, nanotwinned diamond (nt-D, 휆 ≈ 5 푛푚) reached a 200-GPa record hardness and showed improved thermal stability [15].

The latest proposals for nanotwinning are two ceramics: boron carbide (nt-B4C)

[78,101,102] and boron suboxide (nt-B6O or β-B6O) [103-105]. Experimentally, both B4C and B6O have exhibited superhardness (i.e., microhardness above 40 GPa), low mass

3 3 density (2.52 g/cm for B4C and 2.60 g/cm for B6O), and moderate fracture toughness

1/2 1/2 (3.4 MPa·m for B4C and 4.2 MPa·m for B6O) [8,12-14,106]. Crystallographically, these materials share boron-rich icosahedra but have important differences [see Figure

4-1(a) and Figure 4-1(b)]. B4C has three-atom chains that connect icosahedra, but the oxygen atoms that bond icosahedra in B6O do not form chains. Icosahedral carbons can break the sixfold-symmetry in B4C but not in B6O. Also, B4C is highly susceptible to structural heterogeneity due to polymorphism [8,98], while B6O is not [12]. These differences in crystal structure affect the orientation of planar defects, such as twins. In

B6O, the {100}푟 family of planes is preferred for twinning [104,105,107] while many factors, such as stoichiometry and processing, strongly affect the preferred planes in

B4C [78,108,109]. As explained later, these differences make B6O more structurally homogeneous and therefore a better candidate for nanotwinning. Hence, this chapter focuses on B6O but compares to B4C when warranted.

Mechanisms of Nanotwinning

Just as nanograining has a critical grain size that maximizes mechanical strength

(i.e., Hall-Petch relationship), nanotwinning has a critical twin spacing (휆푐푟). Both experimental [19,110] and theoretical [111-113] investigations on nanotwinned metals suggest that twin boundaries beneficially act as barriers to slip (i.e., dislocation motion) but deleteriously act as nucleation sites for partial dislocations. The balance of the effects of slip barriers and dislocation-nucleation sites is material-dependent and ultimately dictates 휆푐푟. Importantly, recognize that this mechanism was developed for metals. Ceramics, on the other hand, have strong chemical bonds that largely counter dislocation motion [114]. Regardless, some still argue that nanotwinning benefits ceramics by resisting slip [16,114]. Others argue that nanotwinning beneficially increases quantum confinement of bandgap energy of ceramics [16,115]. Few

investigations of nanotwinned ceramics are available, so the correct mechanism may not have been speculated yet.

Determining 휆푐푟 for B6O represents a first step in demystifying the mechanism of nanotwinning in ceramics and in potentially achieving record properties B6O. To begin, a prior work established a new nomenclature for B6O [104]. They retained the name of

α-B6O for the non-twinned structure but discarded β-B6O and nt-B6O for the nanotwinned structure. Instead, they proposed iτ-B6O where i represents the number of layers of icosahedra between twin boundaries. For example, τ-B6O has twin boundaries separated by a single layer of icosahedra and a λ of approximately 0.44 nm [Figure 4-

1(c)]. Likewise, 2τ-B6O has twin boundaries separated by two layers of icosahedra and a λ of approximately 0.89 nm [Figure 4-1(d)]. That study also found that the DFT ground-state static energies of α-B6O, τ-B6O, 2τ-B6O, 3τ-B6O, and 4τ-B6O were essentially equivalent. Therefore, experimentally varying twin spacing is likely feasible, so a fabricated sample could contain multiple regions with dissimilar twin spacings.

Amorphization of Boron Suboxide

Apart from nanotwinning at critical spacing, countering solid-state amorphization represents the principal concern for maximizing the mechanical strength of many boron- rich icosahedral solids, including B6O. In this deleterious mechanism, high pressures induce disordered bands at the nanoscale to concentrate stress and promote microcracking, post-yield softening, and catastrophic failure [8]. Before the present work, only two studies have experimentally observed amorphization of B6O. They showed that nanoindentation induced amorphous bands with a width of 2-3 nm and a length of 200-300 nm mostly along {01̅11} and {1̅012} [31,105]. Although no other

experimental works covered the amorphization of B6O, similarly sized amorphous bands were frequently found in B4C [8,9,30,116]. In contrast to amorphization of B6O, however, the amorphous bands in B4C lay along numerous planes: (112̅3) and (21̅1̅3) for ballistic impact [30], (213̅5) and (22̅01) for laser shock [117], (314̅0) for hydrostatic depressurization [118], and many planes for indentation [9,119]. This variety of preferred directions in B4C is likely due to its aforementioned polymorphism-driven heterogeneity. Therefore, there should be less variety in the preferred directions for

B6O, which is much more homogeneous crystallographically. Regardless, the origins of amorphization, especially for B6O, remain unclear.

Objective

The objective of this chapter is to characterize and rationalize the mechanical response of nanotwinned B6O. First, the results of pulse-echo ultrasound, quasistatic/dynamic indentation, and quasistatic compression for both HP SPS samples are presented. This is the first dynamic testing of B6O. With scanning electron microscopy (SEM) and high-resolution transmission electron microscopy (HR-TEM), the trends in properties are explained through measurements of porosity, grain size, nanotwinning, and amorphization. The twins and amorphous bands in B6O are shown to heavily prefer certain crystallographic planes. Raman spectroscopy is also employed to investigate the uniqueness of the amorphization of B6O. Experimental and quantum mechanical x-ray diffraction (XRD) are employed to reveal a potential tool for quantifying the volume fraction of nanotwinning, which is likely beneficial to mechanical response. Finally, biaxial shear of nanotwinned B6O is employed to present a new mechanism by which nanotwinning may benefit ceramics.

Materials Processing

This study covers the two most popular high-pressure-high-temperature (HPHT) processing techniques for polycrystalline ceramics: hot pressing (HP) and spark plasma sintering (SPS). While both techniques mechanically apply uniaxial pressure, HP and

SPS apply temperature through radiation and conduction, respectively. This difference in heating mechanism makes SPS significantly faster than HP and therefore can produce microstructural differences. The HP sample had been produced with a peak temperature of 1850 °C, peak pressure of 50 MPa, and total time of 2 hours. The SPS sample had been produced with a peak temperature of 1600 °C, peak pressure of 55

MPa, and total time of 15 minutes. Starting powder for both samples had been obtained from the sole provider: Fraunhofer Institute for Ceramic Technologies and Systems.

After processing, both samples were ground and cut into rectangular prisms measuring

3.4 x 3.4 x 5.0 mm. Because processing methods for B6O are still unrefined, the grinding and cutting revealed severe cracking in both materials. Hence, only a few intact samples were extracted for mechanical testing.

Indentation, Compression, and Ultrasound

Both HP and SPS samples were polished to a 1-μm surface finish with lapping under maximum pressures of 40 N for 50 minutes. Quasistatic indentation (i.e., 휖̇ ≈ 10−3

푠−1) was performed with a Wilson Instruments Instron Tukon® 2100B hardness tester with a Vickers indenter and loads ranging from 0.98 to 14.7 N. Dynamic indentation (i.e.,

휖̇ ≈ 10+3 푠−1) was performed with a Vickers indenter and a custom dynamic hardness tester that uses a momentum trap to ensure single indentations within a hundred microseconds [120]. For these dynamic tests, load was stochastically varied from 2.40 to 17.3 N. To facilitate direct comparison of the quasistatic and dynamic tests and to

standardize measurements across investigations, slope hardness (퐻푉퐺푃푎) [106,121]

(i.e., the slope of Eq. 4-1) is computed. Here, 푃푁 is the indentation load in N, and 푑푚푚 is the average indentation diagonal in mm.

2 0.0018544 푃푁 = (퐻푉퐺푃푎)푑푚푚 (4-1)

Limited sample size precluded fracture-toughness testing using Chevron-notch specimens. Instead, fracture toughness was approximated from indentation-induced radial cracks. This technique may suffer error up to 25% [122,123] but provides a common, semi-quantative comparison of materials. At low loads, radial cracks were either non-existent or unmeasurable in scanning electron microscopy. At high loads, the diversion of some energy from radial cracking to lateral cracking induced error in the toughness calculation. Hence, the tip-to-tip lengths of the radial cracks (2c) and diagonals (2a) from 9.8-N indents were measured and substituted into Eq. 4-2, which is one of several empirical estimates of fracture toughness (KIC) [124].

−1.5 1/2 퐾퐼퐶 = 0.16(푐⁄푎) (퐻푉푎 ) (4-2)

Due to limited number of samples, quasistatic compression was performed on a single sample of HP B6O with a TestResources® Model 314-150 with a 222-kN (50-klbf) load cell. To avoid indentation of the small ceramic sample into the metal platens, tungsten carbide (WC) were inserted between the sample and loading column.

The mass densities of the HP and SPS samples were obtained by performing

Archimedes’ method on several rectangular specimens. Elastic moduli were measured through pulse-echo ultrasound with an Olympus 5072PR pulser/receiver, Olympus longitudinal/shear piezoelectric transducers, and an InfiniiVision® MSO-X 2012A Mixed-

Signal Oscilloscope.

Electron Microscopy

Microstructural observations of porosity, phase composition, and fracture patterns were made on a FEI Nova NanoSEM 430. Because B6O is highly resistant to thermal and chemical etching, grain size was approximated from micrographs of fracture surfaces induced by 300-N indentations. Because observation of nanotwinning, amorphization, and other crystallographic defects required higher magnification, HR-

TEM was performed on a JEOL 2010F with a 200-kV accelerating voltage and a Gatan

Orius SC200B camera. TEM samples were prepared from both virgin and indented

(0.98 N, Vickers) regions on an FEI Helios Nanolab 600 with an Omniprobe Autoprobe

200. Because of the hexagonal crystal structures, ±[112̅0] was chosen for the zone axis for TEM imaging.

Raman Spectroscopy

For B4C, several studies suggested that volume of amorphized material could be correlated to the intensities of the Raman peaks at 1340 (i.e., “D” peak), 1580 (i.e., “G” peak), and 1820 cm-1 [8,9,30,116]. In particular, tracking the 1340-cm-1 peak produced by a 532-nm laser is popular. Because Raman-active vibrational modes require bonding, considerable debate questions why new Raman peaks (i.e., new bonding characteristics) accompany amorphization (i.e., a loss of bonding). Regardless, to investigate if this quick and non-destructive technique could track amorphization in B6O as well, Raman spectroscopy was performed on virgin and indented regions of HP and

SPS B6O with a Renishaw InVia® Raman spectrometer. A prior experimental investigation [125] indicated that red and green lasers induce florescence in B6O, so both 325-nm (i.e., ultraviolet) and 532-nm (i.e., green) lasers were used.

X-Ray Diffraction

As nanotwinning was previously shown to minimally affect the static energy of

B6O [104], experimental samples likely contain a mixture of α-B6O and nt-B6O. This coexistence of α-B6O and nt-B6O is supported by the HR-TEM images presented in the results section. Therefore, an efficient means of quantifying the volume of nanotwinned material would be useful for processing investigations that seek to optimize nanostructure for mechanical response. HR-TEM can detect nanotwinning when appropriately aligned but is time-consuming and only probes a small volume. Therefore, x-ray diffraction (XRD) was performed on a PANalytical X’Pert3® Powder XRD system.

This bulk measurement should be able to nondestructively detect and perhaps quantify the twin boundaries in a sample of B6O.

For comparison to the experimental XRD scans, quantum mechanical simulations of α-B6O, τ-B6O, and 2τ-B6O were made. First, the relaxed atomic positions and lattice parameters were determined with density functional theory (DFT). For the smaller systems (i.e., α-B6O and τ-B6O), the ABINIT® software [60,61,70] with plane- wave-basis sets, periodic boundaries, Troullier-Martins norm-conserving pseudopotentials, the Teter-Pade local-density approximation (LDA) of the exchange- correlation functional [70], and a 1900-eV cutoff energy were used. These parameters had previously been used to successfully calculate static energies and relative abundances of B4C polymorphs in HP and SPS samples [79,98]. The large-atom system (i.e., 2τ-B6O) was modeled with the Vienna Ab initio Simulation Package (VASP)

[126-129] with plane-wave-basis sets, periodic boundaries, the projector-augmented- wave (PAW) method, the Perdew−Burke−Ernzerhof (PBE) exchange-correlation functional, and a 600-eV cutoff energy. After computing ground-state configurations

(i.e., the relaxed atomic positions and lattice parameters) for all three structures, XRD spectra were simulated with the CrystalDiffract® software.

Biaxial Shear

To help determine the critical twin spacing of B6O, collaboration with another research group provided biaxial shear deformations parallel to the twin boundary [i.e.,

(001)/<100> in the rhombohedral system] on structures with one [i.e., τ-B6O, λ ≈ 0.44 nm, Figure 4-1(c)], two [i.e., 2τ-B6O, λ ≈ 0.89 nm, Figure 4-1(d)], and four (i.e., 4τ-B6O, λ

≈ 1.75 nm) layers of icosahedra between twin boundaries [33]. Biaxial shear approximates the complex stress state under indentation experiments [104,130]. To mimic a Vickers indenter, 휎푧푧 = 휎푧푥 tan(68°) was forced, and stresses were relaxed in the other directions [104,130]. For this biaxial shear, the Vienna Ab initio Simulation

Package (VASP) [126-129] with the aforementioned computational parameters were employed.

Results

Table 4-1 compares the results of the indentation, compression, and ultrasonic testing of HP B6O and SPS B6O to those from a prior study on SPS B4C [106]. First, see that all three materials exhibited strain-rate hardening (i.e., the dynamic properties superseded the quasistatic ones for a given material). Most importantly, see that for most mechanical properties, SPS B6O outperformed the other two materials. At both quasistatic and dynamic strain rates, the hardness of SPS B6O superseded those of

SPS B4C and HP B6O by significant margins. For approximate fracture toughness, which may have large uncertainty outside the standard error reported in Table 4-1 (e.g., from differences in measurement method), SPS B6O surpassed its HP variant by 10% but lost to B4C by 8%. For quasistatic strength, HP B6O dominated SPS B4C by almost

40%. This extraordinary 5-GPa strength is the highest ever reported for B6O. Limitation of number of SPS-B6O specimens precluded measurement of the quasistatic strength of

SPS B6O, but its superior hardness and fracture toughness suggest a strength even higher than that of HP B6O. Finally, all elastic moduli were significantly higher for SPS

B6O than the other two materials. Hence, our mechanical testing demonstrated not only the promise of B6O but also the advantage of SPS.

The SEM micrographs of the fracture surfaces (Figure 4-2) show that both samples benefited from phase uniformity (i.e., no major secondary phases) but that HP produced more porosity than SPS. This finding is consistent with the relative densities of the HP and SPS samples in Table 4-1. Assuming a theoretical density of 2.68 g/cm3

(i.e., the value from our ground-state DFT simulations), HP and SPS B6O achieved 96% and 99% mass density, respectively. This porosity disparity could contribute to the differences in elastic moduli, yield strength, hardness, and fracture toughness. For example, see that a 3% difference in porosity (i.e., 96% vs. 99% mass density) correlated with a 13% drop in Young’s modulus (i.e., 484 vs. 427 GPa). This observation is consistent with a prior study that revealed a 5% drop in Young’s modulus for approximately each percent of porosity in zirconia, another hard ceramic [131].

Finally, the SEM images reveal that the two B6O samples had the same grain size as the previously studied SPS B4C (i.e., 300 nm) [106]. Hence, differences in Table 4-1 cannot be attributed to grain size.

Figure 4-3 presents representative HR-TEM images of virgin B6O. Consistent with the SEM images, Figure 4-3(a) indicates a grain size of approximately 300 nm. As shown by Figure 4-3(b) and Figure 4-3(c), both samples exhibited heterogeneous

nanotwinning. For regions with the densest twinning (see figure inserts), HP B6O had double the twin spacing of SPS B6O (i.e., 2 vs. 1 nm). As proposed in the Discussion section, nanotwin spacing could affect hardness, toughness, and strength (see Table 4-

1). However, nanotwinning generally does not affect elasticity (see Introduction) so cannot account for the differences in moduli. Regardless of processing method, twins and faults were found along (01̅11) as in previous studies [31,105] and in contrast to

B4C, which had many more preferred directions (see Introduction).

Fig 4(a) shows that indentation induced amorphous bands within a depth of two microns from the indented surface. These bands were only a few nanometers in width but extended over several hundred nanometers in length. At this low magnification, the growth of these bands seems randomly oriented as in B4C. However, higher magnification shows that the nucleation of these bands was preferentially along (011̅2).

Figure 4(b) and Figure 4(c) show that these amorphous bands can shear apart twins along [01̅11] by up to a few nanometers (see dotted lines). This deformation highlights the shear-driven nature of amorphization.

Although the HR-TEM clearly indicated the presence and distribution of amorphous bands, tremendous effort is required to sample a large region. To evaluate if

Raman spectroscopy can track amorphization, Figure 4-5 presents the Raman spectra of virgin and indented samples of B6O probed by a 325-nm laser (note: the 532-nm

Raman scans had similar features but excessive fluorescence). The scans of the virgin

-1 HP and SPS B6O differ mostly by intensities at 330, 790 and 1290 cm [see Figure 4-

5(a)]. This observation parallels the fact that HP and SPS B4C differ by the Raman intensities around 265, 320, 480, 533, and 825 cm-1 [98]. However, this similarity of the

trends in Raman spectra of B4C and B6O did not extend to amorphization. While three new Raman peaks appear in amorphized B4C [8], the Raman spectra of virgin and indented B6O are nearly identical [see Figure 4-5(b) and Figure 4-5(c)]. Hence, Raman spectroscopy can efficiently track amorphization in B4C but not in B6O.

Unlike Raman spectroscopy, XRD cannot easily differentiate HP and SPS B6O

[see Figure 6(a)]. To assess if XRD can quantify volume of nanotwinned material,

Figure 4-6(b) – Fig 4-6(d) present the simulated x-ray diffractions of α-B6O, τ-B6O, and

2τ-B6O. See that the simulated XRD pattern of α-B6O [Figure 4-6(b)] captures all of the features of the experimental scan [Figure 4-6(a)]. This similarity suggests that the heterogeneous nanotwinning revealed by the HR-TEM [see Figure 4-3(b) and Figure 4-

3(c)] occupies a negligible volume fraction of the fabricated samples. This finding is consistent with the fact that the experimental hardness values of B6O were not excessively larger than those of B4C [see Table 4-1]. Interestingly, see that the simulated XRD spectra of τ-B6O and 2τ-B6O have only minor differences, which may be due to differences in software (i.e., ABINIT vs. VASP) and parameters (e.g., exchange- correlation functional). Most importantly, a strong peak at 2ϴ ≈ 37° exists for both nanotwinned structures but not for α-B6O. Hence, this peak could be used to track the volume fraction of nanotwinning in fabricated B6O.

From the stress-strain curves for the biaxial shear [Figure 4-7(a)], the critical (i.e., maximum) stress of 2τ-B6O (i.e., 37.8 GPa) exceeded than that of τ-B6O (i.e., 36.2 GPa) and 4τ-B6O (i.e., 36.3 GPa). Likewise, the energy (i.e., area under the stress-strain curve up to critical stress) is highest for 2τ-B6O by over 15%. Hence, the deformation of

2τ-B6O is most difficult, and the ideal twin spacing (λcr) for B6O is likely around 0.89 nm.

This ideal twin spacing was achieved in the heavily nanotwinned regions of the fabricated SPS B6O [see Figure 4-1(c)]. Although other factors may be at play, especially heterogeneous nanotwinning and porosity, note that the ideal twin spacing coincides with the superior properties in Table 4-1. Understanding why this nanotwin spacing may be ideal is complicated by the absence of a consensus on the mechanisms by which nanotwinning affects ceramics and by potential limitations of the computational modeling. Hence, new arguments for this strengthening mechanism are provided in the Discussion section.

Discussion

Recall from the Introduction that for metals, nanotwins are thought to beneficially resist slip but deleteriously promote formation of partial dislocations. Although slip is typically far more difficult in ceramics than in metals, some still argue that these materials may share the same mechanisms. For example, see that the twin boundaries

(i.e., sites susceptible to partial dislocations) are most numerous in τ-B6O and are close together at critical stress [see Figure 4-7(b)]. Perhaps proximity of these dislocation- nucleation sites promotes the formation of enough dislocations to counter the benefits of the increased number of barriers to slip. Alternatively, 2τ-B6O has an entire layer of icosahedra between the twin boundaries to avoid interaction of the partial-dislocation sites but still benefits from barriers to slip [see Figure 4-7(c)]. Adding another two layers of icosahedra between twin boundaries in 4τ-B6O may merely reduce the number of barriers to slip without significantly further reducing the interaction of partial-dislocation sites [see Figure 4-7(d)].

In light of the fact that dislocations typically initiate cracks rather than slip in ceramics, perhaps the twin boundaries beneficially resist the growth of cracks and/or

amorphous bands instead of slip. After all, nanotwinning and nanograining share characteristics, such as a lack of influence on elastic properties. For nanograining, the variation of fracture toughness with grain size is often explained with crack deflection.

Maybe nanotwins work in a similar way to arrest/divert growth of cracks or even the amorphous bands that are thought to initiate cracks. The fact that amorphous bands were exclusively found in non-twinned or lightly twinned regions in the HR-TEM [e.g.,

Figure 4-4] supports this idea.

Testing this theory that twin boundaries resist the growth of cracks and/or amorphous bands, however, requires expensive computational modeling different from the high-periodicity models of the current investigation. Cracks and amorphous bands relieve stress locally so would likely be spread out. Hence, development of molecular- dynamics potentials to enable simulation of larger cells may prove useful. Also, a fundamental model for the amorphization of B6O would need to be established to properly simulate the effect of twin boundaries on the growth of amorphous bands. Most predict that amorphization is the dominant failure mechanism for many icosahedral solids [9,30-32], but the specifics of the mechanism are frequently debated. The amorphization of B4C is most studied [9], but the results of the current investigation seem to suggest that the amorphization of B6O is unique. The HR-TEM revealed that the amorphous bands formed along much fewer number of directions in B6O than in

B4C. This finding suggests that the deformation of B6O is much more ordered than that of B4C. Also, amorphization altered the Raman spectrum of B4C [8] but not of B6O (see

Figure 4-5). This finding suggests that the bonding in B6O changes differently than in

B4C when deformed.

These amorphization differences between B4C and B6O can be attributed to crystal structure. In the most abundant polymorph of B4C, the serially bonded, C-B-C chain [see Figure 4-1(a)] is susceptible to bending that has been theorized to initiate amorphization [76,77]. On the contrary, the two oxygen atoms that bond icosahedra in

B6O are not bonded serially, and there is no easily displaceable central chain atom

[Figure 4-1(b)]. Further, B4C suffers from structural heterogeneity due to polymorphism

[98] while B6O does not. Polymorphism-induced differences in adjacent cell volumes have been theorized to affect stability [65]. Overall, these structural differences likely allow the amorphization in B6O to be different from that of B4C. Therefore, future studies on experimentally tracking and theoretically modeling the unique amorphization of B6O are recommend.

Regardless of modeling difficulty, B6O is highly susceptible to nanotwinning. The highly ordered nature of nanotwinning likely requires a crystal structure with little crystallographic variability. B6O is largely immune to polymorphism and substitutional disorder, and the current HR-TEM showed twins and amorphous bands along few directions. Consistent with prior modeling of the static energies of α-B6O and nt-B6O

[102-104], the current HR-TEM also experimentally showed nanotwinning in B6O.

Unfortunately, the properties of our experimental samples (see Table 4-1) were likely limited by porosity (see Figure 4-2) and non-critical twin spacing (see Figure 4-3).

Therefore, future work focused on the careful control of processing kinetics (e.g., selection of starting powder and processing parameters) is suggested. After all, nt-c-BN

[16] and nt-D [15] were only fabricated after starting from onion-like nanoparticles. To

guide processing of nt-B6O, the aforementioned tracking of the XRD peak at 37° is recommended.

Impact of B6O Characterization

Through mechanical, structural, and spectral characterizations, the promise and uniqueness of B6O, especially with respect to B4C, was demonstrated. For example, quasistatic compression revealed a strength of over 5 GPa, which is both a record for

B6O and superior to that of B4C. The HR-TEM demonstrated heterogeneous nanotwinning at non-ideal twin spacing and amorphous bands along few directions.

Because nanotwinning has set mechanical records for other ceramics, the TEM suggests high potential of B6O as a structural ceramic. Raman spectroscopy was also performed to demonstrate the uniqueness of B6O’s amorphization as opposed to that of

B4C. To guide future investigations in exploiting nanotwinning in B6O, quantum mechanical simulations were performed to predict the critical twin spacing (i.e., 휆푐 =

0.89 nm) and develop a tool for tracking volume fraction of twinning (i.e., x-ray diffraction at 2ϴ ≈ 37°). Overall, these findings support the uniqueness and promise of

B6O as a next-generation superhard material. They also significantly advance the field toward the ultimate goal of determining the exact mechanism by which nanotwinning influences structural ceramics.

Table 4-1. Experimental microhardness values, elastic moduli, and mass densities reflect the superiority of SPS B6O over HP B6O and SPS B4C. Reported hardness values are independent from the indentation load through the slope- hardness calculation.

Experimental Property HP B6O SPS B6O SPS B4C [106]

Quasistatic Hardness (HVQS, GPa) 27.0±0.3 29.9±0.4 29.3±0.3

Dynamic Hardness (HVD, GPa) 29.5±0.5 31.8±0.5 29.9±0.7

1/2 Quasistatic Toughness (KIC, MPa·m ) 2.77±0.04 3.15±0.05 3.4±0.1

Quasistatic Strength (σQS, GPa) 5.0 N/A 3.6±0.2 Bulk Modulus (K, GPa) 203 243 229 Shear Modulus (G, GPa) 186 207 186 Elastic Modulus (E, GPa) 427 484 460 Mass Density (ρ, g/cm3) 2.57 2.64 2.50

Figure 4-1. The crystal structures of (a) (B11Cp)CBC, (b) α-B6O, (c) τ-B6O, and (d) 2τ- B6O share icosahedral fundamental units. Spheres are colored by element (i.e., green for boron, grey for carbon, and red for oxygen) and are sized according to covalent radius. Bends in the dashed yellow lines indicate twin planes. Reproduced from [33].

Figure 4-2. Fracture surfaces of (a) HP B6O and (b) SPS B6O reveal porosity and a 300-nm grain size. Reproduced from [33].

Figure 4-3. Representative HR-TEM of virgin B6O show (a) 300-nm grain size, (b-c) heterogeneous nanotwin spacing, and (b-c) a preferred twinning plane of (01̅11). Minimum twin spacing is larger in (b) HP B6O than in (c) SPS B6O. Reproduced from [33].

Figure 4-4. Representative HR-TEM of indented B6O show (a) an indentation profile with cracking and (b-c) amorphous bands primarily along (011̅2). These bands can shear apart the twins (see dotted lines) and were only found in regions with large twin spacing. Reproduced from [33].

Figure 4-5. (a) Raman spectra (325-nm laser) of virgin HP and SPS differ mostly by intensities of the peaks at 330, 790, and 1290 cm-1. Indentation has little effect on Raman spectrum for both (b) HP B6O and (c) SPS B6O. Reproduced from [33].

Figure 4-6. X-ray diffraction (XRD) of (a) experimental B6O, (b) simulated α-B6O, (c) simulated τ-B6O (i.e., β-B6O), and (c) simulated 2τ-B6O suggest limited nanotwinning in experimental samples. The unique peak at 2ϴ = 37° in τ-B6O and 2τ-B6O correlates with nanotwinning regardless of twin spacing. Reproduced from [33].

Figure 4-7. (a) Stress-strain curves for biaxial shear of τ-B6O (σcr=36.2 GPa), 2τ-B6O (σcr=37.8 GPa), and 4τ-B6O (σcr=36.3 GPa) suggest 2τ-B6O has λ_c. Structures at critical stress are shown for (b) τ-B6O, (c) 2τ-B6O, and (d) 4τ- B6O. Dotted lines indicate twin planes. Reproduced from [33].

CHAPTER 5 ICOSAHEDRAL SUPERSTRENGTH AT THE NANOSCALE

Modified α-B12 Structures

Many materials that exhibit extreme hardness, low mass density, and high thermal/chemical stability can be viewed as simple modifications to α-B12 [4,5,89].

Hence, this manuscript refers to these boron-rich, icosahedral ceramics as a modified α- boron structures (mαB’s). The main structural units of most mαB’s are fully boron icosahedra directly bonded at all six polar sites of each icosahedron [i.e., (B12)]. Without modification, this α-B12 had an experimental Vickers microhardness (HV) of 34 GPa, which is lower than a theoretical prediction of 39 GPa[132]. However, note that this phase of boron has been difficult to produce [89].

While some mαB’s have icosahedral substitutions [e.g., (B11C)CBC], the main differences among mαB’s are interstitial atoms placed around the six equatorial sites of each icosahedron. The most important interstitial sites are the centers of each triplet of icosahedra, so many mαB’s have two interstitial atoms per icosahedron [e.g., (B12)OO

[12] and (B12)PP [133]]. Alternatively, some mαB’s have one [e.g., (B12)S [134]], three

[e.g., (B12)CBC [135] and (B12)NBN [136]], or four [e.g., (B12)BBBB [69]] interstitial atom(s) per icosahedron. Especially for binary compounds, the interstitial atoms in the mαB’s typically come from Groups IV-A (carbon group), V-A (pnictogen group), and VI-

A (chalcogen group) of the p-block of the periodic table [89,137].

Group IV-A provides the most prevalent icosahedral solid: boron carbide (B4C), which has been researched for over fifty years and is employed in numerous fields [8].

Depending on grain size, Vickers microhardness (HV) has varied between 28 and 36

GPa [106,138]. This range is high but lower than the 45-GPa theoretical prediction [139]

likely because of nanoscale complexity from extensive polymorphism, defects, and variable stoichiometry (i.e., 8 – 20 at% C) [8]. Most agree that (B12)CBC [135] and

(B11C)CBC [140] are the majority constituents, but minority constituents are frequently debated [69,98,141]. Additionally, these B-C mαB’s also accept metals to form ternary compounds. For example, Mgx(B12)4(CBC)2(C2)2, which had a HV of 32 GPa, is essentially a modification to (B12)CBC [142]. Likewise, Li has been included to form

LiB12PC [143] (27-GPa HV), LiB13C2 [144], and Li2B12C2 [144].

Amorphization, which is shear-induced, localized loss in crystallinity in nanoscale bands, is the dominant failure mechanism of many icosahedral systems, especially B4C

[8,9,30-32]. To counter this deleterious phenomenon, recent studies have attempted to dope B4C with Si (Group IV-A) to increase ductility [145]. The goal is to mitigate bending of the CBC chains by locally introducing structures, such as (B11C)SiSi [146] or

(B10Si2)SiSi [147]. While these Si-containing mαB’s have mainly been the subject of theoretical investigations, Li2B12Si2 [148] and Mg2B12Si2 [149] have been produced experimentally. Note that the incorporation of Mg into mαB’s has received particular attention due to the BAM (boron aluminum magnesium) materials (e.g., MgAlB14), which have excellent frictional properties [10,11].

When boron subnitride (Group V-A) was first discovered, researchers assumed a structure of (B12)NN due to similarity of the experimental x-ray diffraction (XRD) with that of another hexaboride [150]. Later analysis of XRD [136,151,152] and phase stability

[153,154] instead suggested (B12)NBN, which resembles (B12)CBC. However, quantum- mechanical modeling of Gibbs free energy [155] and enthalpy [156] suggested that stability requires a 2:1 mixture of (B12)NBN and (B12)NN. Neither structure is stable

alone. Regardless, experimental samples exhibited a bulk modulus of 200±15 GPa

[157,158] and a HV of 41 GPa [159]. This hardness is close to the 40-GPa theoretical prediction, which assumed a structure of (B12)NBN alone. Finally, theoretical work on ternary, N-containing mαB’s predicted that B12N2Zn, B12N2Cd, and B12N2Be are stable, superhard, and semiconducting mαB’s [132]. The theoretical HV values for these materials were predicted to be 46, 42, and 49 GPa, respectively. That last value is one of the highest of any icosahedral solid.

Unlike boron subnitride, boron subphosphide (also Group V-A) only has one structure: (B12)PP [133,160-163]. This lack of a three-atom chain [i.e., no (B12)PBP] is likely responsible for the theoretical ability to recover from much more shear deformation than B4C [e.g., (B12)CBC] [164,165]. Experimentally, (B12)PP processed from micropowders and nanopowders exhibited HV of 28 [166] and 35 GPa [167], respectively. The latter value is close to the theoretical predictions of 35 [132] and 37

GPa [139]. This variability due to processing is likely responsible for different bulk moduli in other studies: 179±1 [168] and 192±12 GPa [169].

Like boron subphosphide, boron suboxide (B6O) (Group VI-A) only has one form

[(B12)OO] [12,170,171] and the ability to recover from significantly more shear deformation than B4C [164]. Recent experimentation revealed a bulk modulus of 243

GPa [33] and HV between 30 and 38 GPa [12,33,172]. The latter value well matches the theoretical prediction of 38 GPa [139] but is lower than another theoretical prediction of 47 GPa [132]. While these values are already extraordinary, B6O is arguably the most promising icosahedral solid because of its theoretical [103,104] and experimental [33] susceptibility to nanotwinning (i.e., periodic twins separated by a nanoscale spacing).

Whereas regular twins concentrate stress and decrease mechanical performance, nanotwins have set impressive records for copper (i.e., 1200% increase in yield strength

[18,19] and 85% increase in fatigue strength [100]), cubic boron nitride (i.e., 40% increase in microhardness and 140% increase in fracture toughness [16]), and diamond

(i.e., 100% increase in microhardness [15]). Overall, B6O is arguably the most promising mαB.

Other mαB’s with p-block interstitial elements have been experimentally observed but not characterized much beyond XRD. Arsenic (Group V-A) can form boron subarsenide [(B12)AsAs [173,174]] and a composite with boron subphosphide [(B12)(As1- xPx)2 [175]]. The relatively low 33-GPa theoretical hardness of (B12)AsAs [132] suggests these materials will probably receive little attention. Group VI-A provides (B12)S

[134,176], B6S0.6 [177], and (B12)SeB [134,178]. As indicated by the results of the current investigation, elements larger than As and Se are likely too large to act as interstitials in a network of polar-bonded, boron-rich icosahedra (i.e., mαB’s).

Objective

As indicated by the introduction, mαB’s have exhibited high potential for structural applications. Interestingly, because many of these compounds are formed with nearby elements, mαB’s often form a “continuous series of closely related crystals” in terms of lattice parameters [137]. By quantum-mechanically modeling the ground states and shear deformations of α-B12 and binary mαB’s with two interstitial atoms (i) per icosahedron [(B12)ii], this investigation shows that these mαB’s have fundamental trends in mechanical properties as well. By testing all i atoms in the p-block of the periodic table, the simulation space includes some of the salient mαB’s [i.e., (B12)OO

and (B12)PP] and can well represent some of the chief physics of more complicated systems (e.g., ternary systems and compounds with three-atom chains).

In addition to α-B12 and these p-block hexaborides, this work also simulates the corresponding nanotwinned variations with minimum twin spacing [i.e., τ-(B12) and τ-

(B12)ii]. Here, τ indicates that the nanotwins are separated by a single layer of icosahedra. This nanotwin spacing is likely close enough to the critical spacing to capture fundamental physics [104]. Recall from the Introduction that nanotwinning in many ways represents the cutting edge of mechanical performance for structural ceramics. Overall, the goal is to establish physics-based metrics of nanoscale bonding responsible for the mechanical performance and nanotwinning susceptibility of icosahedral solids.

Computational Procedure

To establish these fundamental trends in mechanical properties, this investigation first employs density functional theory (DFT) to model the ground-states of

(B12), (B12)ii, τ-(B12), and τ-(B12)ii. Although hexagonal and rhombohedral representations are interchangeable, the latter is used throughout this investigation for computational efficiency, easy association with the nomenclature, and consistency with the α-B12. Figure 5-1 shows examples of the quantum-mechanical bonding of these structures. The α-boron [i.e., (B12) and τ-(B12)], which comprises boron icosahedra connected exclusively by polar bonds (PB’s), is simulated to isolate the influence of these icosahedra from that of the i atoms in (B12)ii and τ-(B12)ii. To examine the influence of the interstitial (i) atoms, which connect the icosahedra through equatorial bonds (EB’s), the identities of these atoms are systematically varied within the entire p- block of the periodic table.

These ground-state simulations are performed with the ABNIT software

[41,42,60,61,74], an open-source solver for DFT. Computational details include plane- wave basis sets, periodic boundaries, Troullier-Martins norm-conserving pseudopotentials, the Teter-Pade local-density approximation (LDA) [70] of the exchange-correlation functional, and a 1900-eV cutoff energy. This scheme already successfully revealed the complex composition [79,98] and amorphization [32] of B4C and the x-ray diffraction [33] of B6O. Also, these parameters ensure convergence down to 0.1 GPa in elastic moduli for B6O in the current work. Visualizations of the ground- state bonding and atomic structure are performed with VESTA 3.2.1 and Crystal Maker

9.0.0, respectively.

After equilibration, density functional perturbation theory (DFPT) is performed under the harmonic approximation to compute the natural frequencies and relaxed-ion elastic tensors. If these frequencies are positive, dynamic stability is assumed, and the elastic tensors (퐶푖푗) are computed to check elastic stability according to following four criteria for rhombohedral systems (Eq. 1-16 to Eq. 1-19). For the stable structures, bulk and shear moduli are computed from Hill’s arithmetic average of the Reuss (i.e., minimum moduli) and Voigt (i.e., maximum moduli) approximations (Eq. 1-20 to Eq. 1-

26).

For a structural ceramic, perhaps the most useful estimate of performance is hardness, which is the resistance to penetration. Unfortunately, hardness is not a material property and can vary with testing method. Hence, comparison of simulation and experiment often requires approximations, such as empirical curve fits. This

investigation estimates Vickers microhardness (HV) through a function specifically designed for hard, polycrystalline materials [179]:

퐻푉 ≈ 2(퐺3퐾−2)0.585 − 3 퐺푃푎 (5-1)

Although this hardness estimate approximates inelastic effects, the above simulations have been purely elastic. Hence, this work incorporates deformations of pure shear from a collaboration with another research group. Pure shear is a homogeneous flattening that stretches in one direction and compresses perpendicular to that direction. This deformation can reveal ideal shear strength (휏푚푎푥) and may facilitate an understanding of amorphization, which is widely attributed to shear deformation [9]. Based on a prior B6O investigation [104], the slip systems for the pure shear of (B12)ii and τ-(B12)ii are chosen to be (001)〈21̅1̅〉 and (010)〈001〉, respectively

[see Figure 5-2(a) and Figure 5-2(b)]. To break the 3c-2e bonds (i.e., three atoms acting as centers of bonding by one electron pair) instead of the stronger 2c-2e bonds, the slip systems for (B12) and τ-(B12) are chosen to be (101)〈1̅20〉 and (010)〈001〉, respectively

[see Figure 5-2(c) and Figure 5-2(d)]. From these simulations, plots of true stress versus true strain are produced. Also, toughness is approximated by the area under the plot of engineering stress versus engineering strain until failure strain. Toughness can viewed as work, which equals force times displacement from the original position, so engineering values are more appropriate. Failure strain is chosen to be the strain after which a significant drop of stress occurs [180].

Because computational efficiency is important for high-strain simulations, these shear deformations are performed with the Vienna Ab initio Simulation Package (VASP)

[126-129]. Computational parameters include plane-wave-basis sets, periodic

boundaries, the Perdew−Burke−Ernzerhof (PBE) exchange-correlation functional, the projector-augmented-wave (PAW) method, and a 600-eV cutoff energy. This scheme was used in prior work to successfully model the stability and deformation of B6O with various nanotwin spacings [104].

Results of Ground-State Bonding and Elasticity

Only the p-block hexaborides with interstitial atoms from groups V-A (i.e., pnictogens) and VI-A (i.e., chalcogens) are stable. The instability of (B12)NN is consistent with the experimental and theoretical work mentioned in the introduction. For each stable (B12)ii, the corresponding nanotwinned structure passes the same stability criteria and has similar static energy (Δ퐸휏) and elastic moduli (퐾 and 퐺) [see Table 5-1].

This Δ퐸휏, which is reported per icosahedron, is the largest component of formation energy [81] so can help predict likelihood of nanotwinning. See that Δ퐸휏 decreases as the covalent radius of the interstitial atoms (푟푖) increases and even slightly favors the nanotwinned structure for B6O by 3 meV. This susceptibility to nanotwinning is consistent with prior investigations that underscores the appeal of B6O [33,103,104]. As compared to the energy differences among B4C polymorphs [98], the small magnitude of this 3-meV Δ퐸휏 suggests that experimental B6O samples may contain both nanotwinned and nontwinned regions. This finding is consistent with the HR-TEM of a prior B6O investigation [33] and highlights the need for systematic processing.

For the structures with dynamic and elastic stability, Table 5-1 also presents the bulk moduli (퐾), shear moduli (퐺), and estimated Vickers-microhardness values (퐻푉).

For B6O, the simulation well matches experimental measurements of bulk modulus (i.e.,

241 vs. 243 GPa [33]) and hardness (i.e., 37 vs. 38 GPa [172]). Likewise, the simulation

well matches recent experimental hardness of B6P (i.e., 37 vs. 35 GPa [167]). However, recall that reported bulk moduli values varied (e.g., 179 [168] and 192 GPa [169]), so the current value of 211 GPa may reveal the theoretical maximum. Overall, the current simulations seem reasonably consistent with experimental evidence of nanotwinning, moduli, and hardness values.

With consistency between simulation and experiment verified, consider the trends in the simulated elastic moduli and estimated hardness values. See that 푟푖 positively correlates with 퐾, G, and 퐻푉. To rationalize this trend, Figure 3 provides visualizations of the spatial distributions of electronic density (DEN) and electron localization function (ELF) for the (111̅) planes of (B12) and all stable (B12)ii. Essentially, these maps bisect icosahedra and i atoms as indicated by the superimposed ball-and- stick model on the DEN. The DEN indicates the probability of locating single electrons and is strongest at the red regions (≥0.7 electrons/Å3) and absent at the blue regions.

Note that the DEN is highly localized so reflects strong covalent bonding. The ELF, on the other hand, indicates the probability of locating pairs of electrons (i.e., bonding) and varies between unitless 1 to 0[181-183]. In Figure 3, the ELF is plotted from 0.8 (red) to

0 (blue). See that the trends in DEN and ELF are similar due to the high covalency of these systems. See that the trends in DEN and ELF are similar due to the high covalency of these systems. As described below, strength of the polar bonds (PB’s) and strength of the equatorial bonds (EB’s) drive the trends in elastic moduli. Recall that

PB’s connect neighboring icosahedra directly while EB’s connect icosahedra through i atoms.

Within the hexaborides, PB strength offers the strongest correlation with elastic moduli. As 푟푖 increases, PB’s lengthen/weaken, moduli and stiffness decrease. For example, see how the PB’s weaken near the icosahedra when moving from (B12)OO to

(B12)SS [see the regions indicated by black arrows regions in Figure 3(b)]. Although

(B12)OO and (B12)SS have the same number of valence electrons, (B12)OO has a 50% higher shear modulus and a 20% higher bulk modulus (see Table I). A similar trend can be seen for (B12)PP and (B12)AsAs [see Figure 3(c)]. Note that each of these short and highly directional PB’s connects two icosahedra, so PB strength is reflected by the center-to-center icosahedral spacing, which is identical to the lattice parameter (푎) [see

Table I]. Equilibrium spacing of icosahedra is dictated by (B12) [see Figure 3(a)]. Adding low-period elements, such as oxygen, minimally affects 푎 [see Figure 3(b)]. However, increasing 푟푖 eventually pushes PB’s too far beyond this highly stable configuration [see

Figure 3(c)]. Hence, structures with large i atoms generally have weaker PB’s than structures with small i atoms. This trend is consistent with the experimental evidence that mαB’s form three-atom chains only with small interstitial atoms [e.g., (B12)NBN and

(B12)CBC] (see Introduction).

Although PB’s are arguably more important, EB’s also contribute directional, covalent bonding. For example, (B12) and (B12)OO have similar PB’s, but the latter has stronger EB’s and a 7% higher bulk modulus [see Figure 5-3 and Table 5-1].

Importantly, EB’s connect icosahedra through these i atoms so should be localized near the icosahedra and not in the central region. See that (B12)OO has higher electronic density near the equatorial regions and less electronic density at the central region as compared to (B12), (B12)SS, (B12)PP and (B12)AsAs (see “E” and “C” regions in Figure 5-

3). The large sizes of the S, P, and As atoms especially force valence electrons away from the E region and into the C region. In other words, increasing 푟푖 lengthens and therefore weakens the EB’s.

These trends in elastic moduli (see Table 5-1) and features in ground-state bonding (see Figure 5-3) are identical for (B12)ii and τ-B(12)ii. Essentially, elastic deformation does not provide enough strain for the twin boundaries to interact, so the local bonding environments in nontwinned and nanotwinned structures are effectively identical. This finding is similar to how nanograining minimally affects elasticity in other ceramics. As expected, if nanotwinning affects the mechanical response of mαB’s, the effects of nanotwinning will manifest during inelastic deformation.

Results of Shear Deformations

From Table 5-1, note that the 39-GPa HV of (B12) exceeds the 37-GPa HV of

(B12)OO despite smaller elastic moduli and weaker EB’s (see Figure 5-3). Perhaps the

PB’s are much more important than the EB’s, so the ideal icosahedral spacing in (B12) counters the lack of EB’s. However, this notion is inconsistent with the fact that HV favored B6O by 4 GPa experimentally and by 8 GPa theoretically [132,172]. Further, the current investigation has suggested that both PB’s and EB’s can positively contribute to mechanical performance of hexaborides. Most likely, the current hardness prediction based solely on elastic moduli fails to compare significantly different structures [i.e.,

(B12) vs. (B12)ii]. Hence, this investigation now turns to the results of shear deformations.

Figure 5-4 plots true shear stress (휏) versus true shear strain (훾) for the structures with dynamic and elastic stability. In contrast to the hardness comparison, see that the (B12) has significantly lower shear strength than (B12)OO. This finding is consistent with the aforementioned experimental evidence and with the current notion

that EB’s mechanically benefit mαB’s when small interstitial atoms are incorporated.

Ignoring this (B12) exception, the rankings of the simulated structures by 휏푚푎푥 match the rankings by 퐾, 퐺, and 퐻푉 (see Table 5-1). This similarity of elastic and inelastic responses is consistent with the fact that the covalent bonding in ceramics generally resists both elastic and inelastic deformation [1]. In particular, note the superiority of

(B12)OO over all other p-block hexaborides in terms of 휏푚푎푥. Interestingly, see that the chalcogen hexaborides, especially (B12)OO, have higher toughness than the pnictogen hexaborides, such as (B12)PP. Perhaps the extra valance electrons from the chalcogen atoms increase bonding with the boron atoms, which are known to bond in complex ways. Overall, these results are consistent with the elasticity trends within hexaborides and validate the growing focus on B6O [33].

As with the elasticity simulations, the pure-shear deformations seem minimally affected by nanotwinning. Nanotwinning only slightly increases 휏푚푎푥 and decreases 푈

(see Table 5-1 and Figure 5-4]. If nanotwinning benefits icosahedral solids in similar fashion as diamond and cubic boron nitride (see Introduction), a larger difference in

휏푚푎푥 would be reasonable. However, these highly periodic DFT simulations (see simulation cells in Figure 5-2) may not be able to capture important, low-periodicity features present in experimental samples. For example, cracking and amorphization would likely occur at relatively isolated regions of the sample. If these extremely expensive computational features could be simulated, the benefits of nanotwinning would likely be reflected in the simulated 휏푚푎푥. Regardless, these high-periodicity results are still useful in the development of the first model for nanotwinning in ceramics.

Note that the highly directional, covalent bonding of hard ceramics typically strongly

resists the dislocation motion commonly thought to be affected by nanotwinning in metals [19]. Hence, nanotwinning of ceramics may be quite different from that of metals

[16,114].

Impact of Fundamentals on Icosahedral Bonding

Through quantum-mechanical simulations of both ground states and shear deformations, this work provides a generalized, physics-based understanding of elasticity and strength in icosahedral solids. Analysis of ground-state bonding trends reveals two main predictors of elastic moduli: icosahedral separation (i.e., strength of polar bonds) and localization of equatorial bonding (i.e., strength of equatorial bonds).

Consistency of trends in ground-state bonding and shear deformation suggests these metrics may apply to strength as well. As for nanotwinning, this work demonstrates that susceptibility to nanotwinning relies on key bonding traits and that nanotwinning minimally affects elasticity and high-periodicity inelasticity. For ceramics, these findings aid development of the first model of nanotwinning, a cutting-edge mechanism responsible for dramatic records in experimental properties. Alongside a fundamental understanding of bonding physics and processing kinetics, such a model would likely yield the next generation of structural materials.

Table 5-1. Relative nanotwinning energies (Δ퐸휏), lattice parameters (푎), bulk moduli (퐾), shear moduli (퐺), microhardness values (퐻푉), shear strengths (휏푚푎푥), and toughness values (푈) are calculated from DFT simulations. Structures are ordered according to the radii of the interstitial atoms (푟푖). Structure ΔEτ ri a K G HV τmax U -- meV Å Å GPa GPa GPa GPa GJ/m3

(B12) -- -- 5.00 203 145 39 31 8.0

τ-(B12) 459 -- 5.00 200 143 36 26 3.2

(B12)OO -- 0.66 5.10 241 212 37 38 10.4

τ-(B12)OO -3 0.66 5.09 239 214 38 39 10.3

(B12)SS -- 1.05 5.29 225 209 22 28 7.2

τ-(B12)SS 119 1.05 5.28 221 199 22 30 4.7

(B12)PP -- 1.07 5.20 211 196 37 36 8.1

τ-(B12)PP 138 1.07 5.21 211 194 37 37 6.2

(B12)AsAs -- 1.19 5.25 199 180 34 24 8.3

τ-(B12)AsAs 199 1.19 5.27 199 178 33 24 5.8

Figure 5-1. Quantum-mechanical bonding of (a) α-B12 [(B12)], (b) B6O [(B12)OO], and (c) nanotwinned B6O [τ-(B12)OO] are shown along (100). The fundamental units of these structures are boron icosahedra directly bonded by polar bonds (PB’s). In (b) and (c), icosahedra are also bonded by equatorial bonds (EB’s) connecting the icosahedra and interstitial oxygen atoms. In (c), nanotwin boundaries are indicated by bends in the dotted line. Reproduced from [184].

Figure 5-2. The critical slip systems for pure shear of the nontwinned and nanotwinned (τ) variations of α-B12 and B6O (an example hexaboride) are indicated by the dotted lines and yellow planes. Boron and oxygen atoms are represented by green and red spheres, respectively. Reproduced from [184].

Figure 5-3. Spatial distributions of electronic density (DEN) and electron localization function (ELF) along (111̅) of (a) α-B12, (b) chalcogen hexaborides, and (c) pnictogen hexaborides are indicated by gradients of red (≥0.7 electrons/Å3 for DEN and 0.8 for ELF) to blue (0 electrons/Å3 for DEN and 0 for ELF). Example polar (P), equatorial (E), and central (C) regions are indicated in (a). Ball-and-stick models are superimposed on DEN and have atomic sizes corresponding to covalent radii. Modified from [184].

Figure 5-4. Plots of true stress (휏) versus true strain (훾) are shown for highly-periodic simulations of pure shear of nontwinned and nanotwinned variations of p- block hexaborides and α-B12. With the exception of α-B12 (i.e., “α”), curves are labeled according to the interstitial element [e.g., “O” in the nontwinned plot is (B12)OO]. Reproduced from [184].

CHAPTER 6 CONCLUSION

Summary of Dissertation

Overall, this dissertation makes three major contributions to the field of structural ceramics. First, experimental and computational spectroscopy are incorporated into a new method that can decipher and quantify nanoscale mixtures. The power of this method is demonstrated for B4C after revising the nomenclature to enable comprehensive modeling of polymorphism. Understanding the makeup of complex materials is essential for modeling material behavior, tailoring properties, and ultimately discovering new materials. Second, nanotwinning of ceramics is explored through a large experimental effort on B6O. Materials without polymorphism are suggested to be more susceptible to nanotwinning, and influence of processing on and the importance of nanotwin spacing are demonstrated. This work also suggests that B6O may be the next- generation lightweight, structural ceramic. Finally, this dissertation simulated both ground states and shear deformations of a large class of icosahedral solids to identify metrics of both elasticity and strength. These metrics can be used to facilitate a fundamental understanding of superhardness and as a means of discovering materials for new applications.

Future Work: Nanotwinning Mechanisms for Ceramics

Computational Materials Science has long complimented and motivated experimentation in discovering, understanding, and improving advanced materials essential to the interests of Oakridge National Laboratory and to the nation as a whole.

Traditionally, this field massively iterated atomic configurations to discover materials and then assessed them to find appropriate applications. However, recent advances in

physics-based fundamentals and in computational resources have propelled developments in Materials Data Science. Instead of discovering materials and then selecting their applications, the state of the art now invents materials based on a desired set of properties [32,185]. This paradigm has many names and approaches, including Combinatorial Materials Science [186], Materials Informatics [187,188], Big-

Deep-Smart Data [189], Nanotechnology Initiative [190], Integrated Computational

Materials Engineering Initiative [191], and Materials Genome Initiative [192]. This proposal for the Oakridge Distinguished Staff Fellowship adopts this property-driven method of materials discovery to facilitate the development of cutting-edge frameworks in intelligent materials-discovery algorithms.

Because intelligent materials discovery is essentially in nascent stages, much opportunity for innovation exists, especially for advanced materials. In many ways, superhard ceramics are also a frontier of materials science and engineering. No other materials exhibit such extraordinary resistance to impact/deformation, heat, and chemical reaction. These record properties are harnessed by an ever-expanding array of applications in ballistics, wear, nuclear power, electronics, etc. Despite widespread adoption, however, much potential for improvement exists because of the complexity of the ceramic structures at both the microscale and nanoscale. Essentially, complexity acts as degrees of freedom in optimization and tailoring. Arguably, the cutting edge of these ceramic optimizations is nanotwinning (i.e., periodic twins separated by a nanoscale distance). Unlike regular twins, which concentrate stress and lower mechanical performance, nanotwins have recently doubled the record microhardness in diamond to nearly 200 GPa [15], increased the fracture toughness of boron nitride by

140% [16,17], and increased the yield strength of copper by an order of magnitude

[18,19]. These few experimental successes have inspired preliminary theoretical investigations in additional advanced ceramics, like B4C [101,102] and B6O [33,103-

105]. Hence, this future work would target the novel mechanism of nanotwinning in superhard ceramics.

Despite the aforementioned experimental successes with nanotwinning, however, a fundamental understanding of the corresponding mechanisms eludes materials scientists. Both experimental [19,110] and theoretical [111-113] investigations on nanotwinned metals suggested that nanotwin boundaries beneficially act as barriers to slip but deleteriously act as nucleation sites for partial dislocations. To complicate matters, the balance of the effects has been suggested to be material-dependent. Also, this theory was designed for metals so likely cannot rationalize the effects of nanotwinning in ceramics, which have strong chemical bonds that largely counter slip

[114]. While some researchers still assume this metals theory for ceramics

[16,114,130], others have speculated ideas like quantum confinement of bandgap energy [16,115] and resistance to the growth of amorphous bands and cracks [33].

These ideas are largely conjecture and may not even include the true mechanism.

Hence, this future work would employ computational chemistry to develop and validate the first accurate model for nanotwinning in superhard ceramics.

As mentioned in Chapter 1, superhard ceramics typically fall within two structural classifications: those with diamond crystal structure (e.g., diamond and cubic boron nitride) and those with boron-rich icosahedra (e.g., B4C, B6O, and BAM). Because of the complex bonding abilities of boron atoms [89], the icosahedral category is subject to

much investigation into tailoring of properties and discovery of materials. Alternatively, the diamond category has lower potential for tailoring due to lower polymorphic variability but has higher mechanical records. This future work start with the diamond category because the lower variability and higher symmetry would facilitate development of generalized theories. Also, the fact that cubic boron nitride is isoelectric and isostructural with diamond would help in validation. After material selection, highly- periodic simulations of both ground states and deformations would be performed with density functional theory (DFT) with an efficient pseudopotential in the Vienna Ab Initio

Software Package (VASP). Special focus would be devoted to evaluating how the covalent-bonding environment is affected by nanotwinning and deformation. For example, consider that nanotwinned diamond experimentally exhibited double the microhardness of regular diamond but has similar local connectivities in the ground state. Perhaps geometric considerations increase the directionality of bonding or allow bond lengths to remain shorter and stronger for relatively higher strains. Note that different twin spacings and perhaps different twin orientations may be necessary to capture the appropriate physical response.

While nanotwinning may affect fundamental bonding physics in localized, periodic regions at the nanoscale, characteristics at higher length scales may also be affected. For example, perhaps nanotwinning acts as a barrier to the growth or even formation of nanocracks or amorphous bands. These features would require relaxed periodicity constraints. Therefore, understanding how global bonding environment evolves during high-strain deformations may be essential to deciphering the mechanism of nanotwinning for ceramics. If the property difference between regular and

nanotwinned materials in highly periodic simulations does not match that of experimental evidence, new approaches would be taken. First, expensive supercells would be attempted with quantum mechanics. Convergence of such large-atom structures would require particular finesse and careful consideration of modeling assumptions. Alternatively, if supercells do not converge, this work would turn to molecular dynamics with a material-specific potential (e.g. Tersoff bond-order potential for diamond and cubic boron nitride) in the LAMMPs software package. Again, a variety of non-deformed and deformed structures at various twin spacings would be simulated to examine the evolution of bonding with strain.

After simulating the effects of nanotwinning in the diamond category at different length scales, a first draft of the generalized model of nanotwinning in ceramics would be produced. By incorporating physics-based bonding trends, this model would be fundamental rather than empirical. To evaluate and potentially further generalize this model, icosahedral structures would be simulated next. The first one would likely be

B6O because this material system has experimentally shown susceptibility to nanotwinning and has high crystallographic order thought necessary for the highly ordered structure of nanotwinning [33].

This fundamental theory for nanotwinning in ceramics could then be incorporated into intelligent materials-discovery algorithms that are harnessed by collaborations, such as the Materials Genome Initiative, to advance Combinatorial Materials Science, which is the cutting edge and evolution of Computational Materials Science. These collaborations would ignite processing endeavors that would ultimately lead to a revolution in structural materials. Discovering new superhard materials with unique sets

of mechanical, chemical, thermal, electronic, and optical properties would advance existing applications in defense (e.g., armor), energy (e.g., nuclear shielding), electronics (e.g., wide-bandgap semiconductors), refractories, etc. This discovery of new property combinations could even open new opportunities (e.g., functional materials) by adding otherwise unavailable extraordinary strength to traditionally required properties.

Future Work: Systematic Processing of Boron Suboxide

As shown by Chapters 4 and 5, B6O may be the next-generation lightweight, structural ceramic. Unfortunately, the experimental samples in this dissertation had heterogeneous nanotwinning and significant porosity. Future investigations would target the processing of B6O to maximize material performance. The previous future-work section on the mechanics of nanotwinning in ceramics would help identify the target ending structure (e.g., critical twin spacing). Kinetics investigations would be helpful for determining the target starting structure(s) before processing. For example, consider that nanotwinned diamond and nanotwinned cubic boron nitride were only produced after starting with onion-like nanoparticles [15]. Perhaps a similar structure would be useful for B6O.

APPENDIX A SIMULATED RAMAN SPECTRA OF ALL B4C POLYMORPHS

100

101

102

103

104

105

Notes: Simulated Raman peaks (DFT LDA) are marked with red diamonds, and spectra are presented in order of increasing static energy (see Table 3-1). For interpretation of the zijk nomenclature, see Chapter 2. The black curves depict an experimental Raman scan of HP B4C for comparison to these simulations.

106

APPENDIX B CRYSTALLOGRAPHY AND ATOMIC BONDING OF B6O

[111]

107

[111]

108

[111̅]

109

[111̅]

110

[112̅]

111

[112̅]

112

[110]

113

[110]

114

[100]

115

[100]

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BIOGRAPHICAL SKETCH

In 2013, Cody Kunka initiated research in the Laboratory for the Dynamic

Response of Advanced Materials (LDRAM) under the University Scholars Program

(USP) and Professor Ghatu Subhash at the University of Florida (UF). Assessing interacting cracks in silicon carbide, he earned second place in the International Student

Paper Competition (ISPC) by the Society for Experimental Mechanics (SEM), second place in an international poster competition by the American Ceramic Society (ACS), and publication in the Journal of the American Ceramic Society.

In 2014, he graduated Summa Cum Laude as the Four-Year Scholar for the entire UF College of Engineering and began his PhD in LDRAM. By 2015, he contributed to a collaborative processing project with the Department of the Army, earned fellowships from the National Science Foundation (GRFP) and the Department of Defense (NDSEG and SMART), and quickly learned quantum mechanics. With expertise in Mechanical Engineering, Materials Science, modeling, and experiment, he introduced a new nomenclature, quantified complex crystallography, and characterized a promising new ceramic. By 2018, he presented this work at fourteen professional conferences and published in Scripta Materialia (x3), Acta Materialia, and Physical

Review Materials.

In his final months of PhD, he spearheaded an experimental investigation on the oxidation of polycrystalline copper coated by few-layer graphene to fuel a seventh publication. He also accepted a postdoctoral position at the Center for Integrated

Nanotechnology (CINT) of Sandia National Laboratory (SNL) to explore cutting-edge nanomechanics.

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