Challenges of Engineering Grain Boundaries in ...

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DOI: 10.1007/s11837-016-1856-7 Ó 2016 The Minerals, Metals & Materials Society (outside the U.S.)

Challenges of Engineering Grain Boundaries in Boron-Based Armor Ceramics SHAWN P. COLEMAN ,1,3 EFRAIN HERNANDEZ-RIVERA,1 KRISTOPHER D. BEHLER,1,2 JENNIFER SYNOWCZYNSKI-DUNN,1 and MARK A. TSCHOPP 1,4 1.—U.S. Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA. 2.—TKC Global, Herndon, VA 20171, USA. 3.—e-mail: [email protected]. 4.—e-mail: [email protected]

Boron-based ceramics are appealing for lightweight applications in both vehicle and personnel protection, stemming from their combination of high hardness, high elastic modulus, and low density as compared to other ceramics and metal alloys. However, the performance of these ceramics and ceramic composites is lacking because of their inherent low fracture toughness and reduced strength under high-velocity threats. The objective of the present article is to briefly discuss both the challenges and the state of the art in experimental and computational approaches for engineering grain boundaries in boron-based armor ceramics, focusing mainly on boron carbide (B4C) and boron suboxide (B6O). The experimental challenges involve processing these ceramics at full density while trying to promote microstructure features such as intergranular films to improve toughness during shock. Many of the computational challenges for boron-based ceramics stem from their complex crystal structure which has hitherto complicated the exploration of grain boundaries and interfaces. However, bridging the gaps between experimental and computational studies at multiple scales to engineer grain boundaries in these boron-based ceramics may hold the key to maturing these material systems for lightweight defense applications.

INTRODUCTION Engineering the next generation of boron-based armor ceramics will require both experimental and modeling efforts to understand the role of chemistry, processing, microstructure, and design on the system’s performance across multiple scales, i.e., the integrated computational materials engineering (ICME) framework.1,2 However, there are still many challenges in doing so, one of which is how to engineer grain boundaries to optimize the performance of these materials. In terms of boron-based ceramics as armor materials, it is well known that the ballistic performance of these ceramics, such as boron carbide (B4C) and boron suboxide (B6O), are heavily influenced by their underlying microstructure that includes porosity, secondary phases, and grain size distribution.3–7 Grain boundaries and interfaces between grains, different phases, and at pores alter the underlying failure mechanisms,

which lie at the core of their ballistic performance. For this reason, detailed understanding of the processing–structure–property relationships that focuses on grain boundaries and interfaces is critical to advancing the state of the art in boron-based armor ceramics. Boron-based ceramics are appealing for lightweight applications in both vehicle and personnel protection, stemming from their low density (B4C 2.52 g cm 3 and B6O 2.56 g cm 3) as compared to other ceramic and metal alloys. The ballistic protection gained from armor ceramics is achieved through the dissipation of a projectile’s kinetic energy through ideally controlled fracture. Both B4C and B6O can dissipate extreme impact energies due to their high hardness (Vickers hardness of B4C 2632–3450 kg/mm2 (Refs. 8, 9) and B6O 3820– 4587 kg/mm2) (Refs. 8, 10), high elastic modulus (B4C 457–470 GPa11,12 and B6O 472 GPa13), and high Hugoniot elastic limit (HEL) (HEL B4C 15–20

Coleman, Hernandez-Rivera, Behler, Synowczynski-Dunn, and Tschopp

Fig. 1. The atomic structure of the B12-CCC polymorph in (a) shows the complexity of the boron-based ceramics. Icosahedra reside in the lattice vertices of a rhombohedral unit cell with a 3-atom inter-icosahedral chain along the [111] direction. Each of the icosahedra has polar (blue) and equatorial (red) sites shown in (b) where carbon can reside as a substitutional atom. The HRTEM image in (c), reproduced from Ref. 3, shows B4C amorphization that occurs during failure, which can further complicate structural characterization in these systems (Color figure online).

GPa14–16). In fact, the extremely high hardness in boron-rich ceramics comes from the a-B12 icosahedral structural unit (see Fig. 1a and b), which is exceeded only by diamond and boron-based diamond structures (e.g., cubic boron nitride [cBN]). Despite these impressive properties, the performance of both B4C and B6O ballistic armor is limited in the high velocity regime due to their inherent low fracture toughness (B4C 2.7–2.9 MPa m1/2 (Ref. 9) and B6O 2–4.5 MPa m1/2) (Ref. 10). This low fracture toughness combined with the brittle nature in which boron carbide fails and its susceptibility to stress-induced solid-state amorphization3,6 and shear localization currently limit boron carbide as an armor material.17,18 For instance, Fig. 1c shows a high resolution electron microscopy image of an amorphous band taken from a post-mortem fragment after a high rate impact experiment. These mechanisms and possible melting are believed to play a role in the loss of shear strength above the HEL of B4C and affect its poor performance against certain ballistic threats.19,20 The basic properties of B4C and B6O are derived from their complex chemistry and atomic structures, an example of which is shown in Fig. 1a and b. This is especially true of B4C, which can exists as a combination of several different polytypes and stoichiometries as discussed in the ‘‘Experimental Characterization’’ section. In a single crystal, the properties of these boron ceramics are controlled by the strong, directional covalent bonding of the atoms. The structure of these boron ceramics consists of distorted 12atom close-packed icosahedra units that are composed primarily of boron atoms with chains aligned along the [111] direction containing varying configurations carbon and/or boron (B4C) and oxygen (B6O), e.g., the blue atoms in Fig. 1a show the 3-atom carbon chain in B4C (also referred to as B12C3). The icosahedra units are situated on the vertices of ˚ , a = 65.18° rhombohedral unit cells (B4C a = 5.60 A ˚ , a = 62.9°) that form the R3m and B6O a = 5.15 A space group symmetry. While the directional

covalent bonding found in these systems increases compression strength, it also exposes independent slip systems that make these ceramics inherently more brittle. The complex chemistry and structure of these boron-based ceramics also increases the complexity of the structure and interactions at their grain boundaries and interfaces, which makes design optimization difficult. There have been multiple approaches to improving the ballistic performance of armor ceramics. Because of the complexity surrounding the structure and interactions of grain boundaries and interfaces in armor ceramics, one approach to optimize their ballistic performance is to reduce their presence altogether.21 Past research has had limited success reducing the contributions of grain boundaries and interfaces and improving the properties of armor ceramics by producing highly sinterable powders. More recently, however, there has been a renewed focus on understanding and controlling the microstructural effects in armor ceramics to alter their failure mechanisms and improve their ballistic properties. For example, a grain boundary engineering approach has improved fracture toughness in self-reinforced Si3N4 ceramics by shielding crack growth during brittle failure.22–25 Furthermore, engineering intergranular films using Al2O3 additives in B6O has resulted in improved fracture toughness with minimal reduction in hardness.26 Figure 2 shows an example of (a) a clean interface in B6O as well as (b) a transition region to an intergranular glassy film. However, producing bulk samples with intergranular films still has processing challenges. Although, these successes suggest that engineering the grain boundary chemistry and structure is a more promising path forward than reducing grain boundaries altogether. The objective of this article is to discuss some of the challenges (and opportunities) for engineering grain boundary chemistry and structure in boron-based ceramics as a means to optimize their ballistic properties. In short, the path forward is to integrate

Challenges of Engineering Grain Boundaries in Boron-Based Armor Ceramics

Fig. 2. HRTEM images of B6O interfaces, reproduced from Ref. 26, show (a) clean structures without sintering aid dopants and (b) partial wetting of the interface structure due to the usage of Al2O3 dopants.

experiments, modeling, and theory in these material systems using an ICME approach. This article is laid out as follows. The ‘‘Experimental Methodologies’’ section outlines some of the experimental challenges in both processing and characterizing boron-based ceramics and will highlight some of the recent progress that has been made. The ‘‘Computational Methodologies’’ section outlines some of the modeling challenges of boron-based ceramics and will highlight some of the recent progress in these areas. The ‘‘Future Opportunities’’ section discusses future opportunities for optimizing boron-based ceramic armors using ICME. The overall goal of this article is to review and discuss the challenges and state of the art in experimental and modeling methodologies, analysis tools and techniques, microstructure features, mechanical properties, and recent research in boron-based ceramic systems, with a focus towards development of the next generation of bulk high performance armor ceramics. EXPERIMENTAL METHODOLOGIES One of the primary experimental challenges for boron-based armor ceramics is the ability to process powders and then consolidate them to full density. The difficulty with consolidation is associated with the fact that boron-based materials, and more specifically B4C and B6O, exhibit very high melting points (up to 2450°C and 2640°C, respectively27) and have strong covalent bonding, thus making solid-state sintering difficult. There have been various techniques employed to produce dense B4C and B6O, such as pressureless sintering, field-assisted sintering (FAST) and spark plasma sintering (SPS), and hot-pressing. Additionally, the use of additives to lower the sintering temperature is also an oftenapplied strategy to aid in consolidation. Both the techniques and additives play an important role in densification and in altering the chemistry and structure of grain boundaries and interfaces within the material. In practice, sintering and consolidation is a balance of reducing the porosity (increasing density) while minimizing grain growth.

Powder Consolidation Techniques Densification techniques for powders often require elevated temperatures in excess of 50% of the melting temperature, subjecting powders to kinetic or thermodynamic regimes where the microstructure is altered (e.g., grain growth, precipitation, segregation). During pressure-assisted techniques, there is a trade-off with pressure and temperature: as the sintering load (pressure) is increased, the temperatures required for full densification are decreased. Incorporating electric fields, as in the FAST and SPS techniques, can also enhance densification at lower pressure–temperature combinations. First, pressureless sintering techniques generally require elevated temperatures with a loose powder compact to consolidate the materials. This technique can minimize density variations in the final compact, as opposed to more traditional hot-pressing methods. This technique typically requires temperatures of greater than 2250°C for B4C, a temperature regime where abnormal grain growth is a concern. Far less work has been shown on pressureless sintering of B6O as this has resulted in highly porous materials well below full density due to sintering temperatures above the decomposition temperature of boron suboxide (1760°C). Hence, decreasing the sintering temperature to produce fully dense B4C and B6O typically requires the use of pressure-assisted or field-assisted densification techniques. Second, hot-pressing techniques (i.e., pressureassisted sintering) are commonly used to densify B4C and B6O.28–30 Hot pressing is a convenient, commercially viable technology used to produce armor ceramics at lower temperatures than pressureless methods by applying a load to the ceramics during sintering, thereby reducing the temperature needed for densification and the driving force for grain growth.31 As an example for B6O, full densification has required temperatures of 1850°C with 50 MPa load. A main disadvantage for hot pressing


is that the geometries may be limited for some components (without the use of expensive machining from bulk compacts) due to die restrictions and furnace size. Third, the field- assisted techniques (FAST and SPS) have recently garnered interest as a new technology for rapid densification using electric currents to further enhance pressure-assisted techniques,32 turning densification times normally in the range of hours to minutes. Boron carbide has shown densities of 80–100% of the theoretical density (TD) using temperatures in the range of 1600–2000°C. However, studies have observed abnormal grain growth at higher temperatures of 2200°C.33–35 The main disadvantage for field-assisted techniques is its current limitation in scalability for commercial usage. Sintering Additives Elemental or compound additives are often utilized in combination with the aforementioned techniques to lower the densification temperature. For instance, to lower the temperature for B4C, additives such as C, Al, Al2O3, SiC, Ti, TiC, TiB2 and many more have been employed between 2050°C and 2200°C, resulting in samples with densities >93% TD.31,36–45 In some cases, even low-volume fractions of oxide or metals can reduce the densification temperature to less than 2000°C, minimizing grain growth while achieving high to full density.46–49 These effects are regardless of the densification technique. However, additives not only affect the sintering behavior but can also have significant ramifications on the ceramic microstructure, e.g., forming new phases, segregating to grain boundaries, forming intergranular films, wetting grain boundaries and/or triple junctions, etc. As an example, the SiC system is an appropriate system to show the effects that sintering aids have had in reinforcing materials with intergranular films. Kim et al. have shown intergranular films (consisting of Al-Y-Si-O-C) when using Al2O3-Y2O3 as a sintering additive. This not only produced a core-rim structure as a result of the solution-reprecipitation grain-growth mechanism but was also associated with an increase in grain size, in part due to higher liquid content during sintering.50 Even without evidence of intergranular films in B4C and B6O, sintering additives play an important role in the densification and microstructure of these materials. Sintering aids have been effectively employed in B4C and have shown reduction in the porosity and affect the morphology and grain size.31 Alumina, for example, has been added as a sintering aid to B4C for a number of years and to B6O more recently, yet no film has so far been observed.51–54 Other work on adding rare earth oxides to B6O has expanded upon work shown in other systems but no intergranular films have been observed.55

Sintering additives significantly affect the macroscopic properties of boron-based armor ceramics by modifying the grain boundary behavior. Boron icosahedral materials inherently exhibit clean grain boundaries, which lead to failure through the grains of densified material (i.e., low fracture toughness) as opposed to a more tortuous path around grains through the grain boundaries (i.e., high fracture toughness). Clean boundaries in B4C limit many of the toughening mechanism observed in other ceramics such as grain boundary crack deflection and microcracking. One method that has been shown to impart higher fracture resistance is incorporating intergranular films with the grain boundaries.56 Grain boundary engineering in other ceramics such as silicon carbide (SiC), silicon nitride (Si3N4), zinc oxide (ZnO), and alumina (Al2O3), to name a few, have shown that a higher fracture toughness and resistance to shear localization can be obtained.57–64 However, one of the continuing challenges is that it is often not known beforehand how these additives will affect the processing and properties of these ceramic systems. In this respect, having models that can help elucidate the role of grain boundary chemistry and structure on fundamental mechanisms and properties of boron-rich ceramics may help to accelerate the design optimization of grain boundaries in these boron-based systems. Experimental Characterization Experiments have helped to understand the structure of various boron ceramic polytypes. Neutron and x-ray diffraction have determined that the crystal (Space symmetry of both B4C65,66 and B6O67 is R3m group 166) with primary structural units that consist of cubic closed packing of 12-atom icosahedra at the verticies of the rhombohedral unit cell and either a three-atom carbon–boron chain (for B4C) or a twoatom oxygen chain (for B6O) along the [111] unit cell diagonal. Within the boron icosahedra, there are two chemically distinct atomic positions: (1) boron or carbon around the diameter of the icosahedra (equatorial sites—Be, red atoms in Fig. 1b) which form bonds with boron, carbon or oxygen sites from the inter-icosahedral chains; and (2) boron or carbon near the apex (polar site—Bp, blue atoms in Fig. 1b) of the icosahedra which form inter-icosahedral bonds with Bp from neighboring icosahedra. Experimentally derived B–C phase diagrams indicate that the solid solubility of carbon in boron carbide B12+xC3 x (0.06 < x < 1.7) can range from 8 to 20 at.% carbon without phase separation or interstitial carbon occupancy.29,68 However, the exact atomic occupancy of sites within the icosahedra and along the intericosahedral chains has been difficult to determine experimentally because the carbon, boron, and oxygen atoms have similar atomic numbers, electron densities, nuclear cross-sections, scattering lengths, and form factors.


Fig. 3. High-resolution TEM images of asymmetric twin interfaces in B4C, reproduced from Ref. 76, show the mismatch along the boundary (a, b), the interface orientation (c), and strain map (d) parallel to the interface plane.

Experimentally resolving and characterizing grain boundary (and single crystal) structure and the presence of intergranular films often requires a host of techniques: x-ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy (TEM), x-ray energy dispersive spectroscopy (EDS) and electron energy loss spectroscopy (EELS). For instance, XRD characterizes the crystal structure, which results in peaks of crystalline phases that can be fit using techniques such as Rietveld analysis to determine the ratios of different crystalline polytypes. SEM allows for the microstructure such as grain size, morphology, porosity, and observation of secondary phases in the grain boundary junctions, trapped in grains, or along the grain boundaries. The use of backscattered electrons (BSE) allows for Z-contrast to help identify the different phases. If enough of the grain boundary phase is present, SEM imaging can show evidence of wetting along the grain boundaries. For films less than a few nanometers in thickness, TEM imaging further resolves details of the crystal structure of grains and grain boundaries. EDS and EELS help analyze the chemical species present

and, combined with elemental mapping, can provide a distribution of the different chemistries in a spot, line, or area. Observing the nanometer scale films in TEM is of particular interest as (1) amorphous and crystalline phases are distinguishable, (2) the size of the films can accurately be measured, and (3) the composition can be further analyzed. Advances in microscopy have helped with the identification and understanding of atomic structure at boundaries. For instance, Fig. 3a–d shows HREM images of an asymmetric twin interface in B4C as well as the associated electron diffraction pattern and strain map. Improved microscopy resolution, more developed techniques, and coupling with computational approaches will enable the understanding and engineering of grain boundaries (including twins) and intergranular films. COMPUTATIONAL METHODOLOGIES Integrated computational materials engineering (ICME) relies on understanding the underlying relationships between alloy composition, processing, structure, properties, and performance. This


approach is driven by modeling and simulation that are validated by experiments. Some of the main challenges associated with boron-rich ceramics is how additives affect grain boundaries during processing as well as the precise characterization of their effects on microstructure. ICME methods can be used to address ongoing experimental processing and characterization challenges while moving towards a design optimization that improves the fracture toughness and ballistic performance in polycrystalline boron-based armor ceramics. An ICME approach for improving boron-based armor ceramics presents several computational modeling challenges. These include:

Accurate thermodynamic databases: Wide compositional space in the B-C-N-O ceramic systems as well as the additional chemistry of sintering aids. Single crystal structure–property relationships: Complex atomic structures and atomic bonding within each individual phase. Grain boundary structure–property relationships: Vast degrees of freedom effecting interface and grain boundary character, e.g. boundary planes, misorientation, stoichiometry, reconstructions, interfacial terminations, etc. Grain boundary segregation: How solute, impurity, and point defects affect the structure of all grain boundary types. Grain boundary construction methods: Geometrical constraints limiting bicrystal simulations with periodic boundary conditions. Grain boundary mobility: Limited methods for systemic modeling the migration of flat ceramic boundaries. Experimental validation of computational results: The need for methods to validate subcontinuum simulation results with experiments (structure, mechanical properties, etc.). Scale bridging and multiscale modeling: Connecting models across spatiotemporal scales.

A number of these challenges require further development of the methods for exploring ceramic systems, not just the application of existing techniques to boron-based ceramics. However, it is illustrative to recap some of the work that has been done in boron-based ceramics as a way to point out the future opportunities within these systems. The following sections address the current status in addressing these challenges as well as point towards opportunities for continued research. First Principles Atomistic Simulations First principles-based atomistic simulations have been used both to (1) understand the structure, vibrational and electronic properties of single crystal boron-based ceramics and isolated defects, as well as (2) understand the structure and mechanical properties of boron-based ceramic grain boundaries.

While limited in terms of the simulation supercell size, which limits the complexity of grain boundaries among other things, first principles are ideal for exploring the role of various elements and point defects on equilibrium and non-equilibrium polytypes. Single Crystal Density functional theory (DFT) has been successfully exploited to determine the crystal structure, phase stability, and site occupancy in chemically complex, compositionally disordered B4C and B6O polytypes. For example, it was originally argued that as B4C became boron-rich, the carbon-based chains disappeared based on the disappearance of two peaks at 481 cm 1 and 534 cm 1 in the Raman spectra.69 However, using DFT perturbation theory to calculate the infrared and Raman spectra for three B4C polytypes, Lazzari et al.70 showed that these peaks are associated with the rotation of the inter-icosahedral chain about an axis perpendicular to the [111] direction and a vibrational mode of the icosahedra. Furthermore, Saal et al.71 extended this work by considering the effect of vacancies on the structure of carbon and boron-rich boron carbide polytypes. Their results show that, in the carbon-rich regime, the lowest energy structure that most closely agrees with experimental lattice parameters is B11Cp-CBC, in which one C atom occupies a polar site in the icosahedra and a B lies in the center of the intericosahedral chain (predicted DHf = 10.78 kJ/mol atom versus experimental DHf = 11.5 ± 2.3 kJ/mol atom72). Similarly, at 13.33 at.% C, the predicted structure is B12-CBC ( 7.89 kJ/mol atom) and in the boron-rich regime, it is B12-BVaC (6.43 kJ/mol atom). From these results, Saal et al.71 proposed that, as the carbon concentration decreases, boron preferentially replaces carbon in the icosahedra until all icosahedral carbon has been replaced around 13.33 at.% C. As the system becomes further boron rich, the boron begins replacing carbon from the chain ends and creates vacancies at the chain center. Raman spectra predicted for the above mechanism matches the experimentally measured shift of the highest frequency mode to lower frequency as a function of carbon concentration. DFT has also been utilized to determine the effect of stoichiometry on elastic moduli and strain response of the different boron carbide polytypes by Taylor et al.73,74 They found that the shear modulus was reduced by 5% (234 GPa to 222 GPa) as carbon replaced boron to create a CCC intericosahedral chain, thereby reducing the boron carbide stiffness. At low carbon content (6.6 at.% C), the minimum energy, elastically stable B12-CCB structure has a nonlinear 3-atom chain that demonstrated a considerable reduction in stiffness compared to other systems. Their model suggests that stress-induced amorphization can be initiated by a


Fig. 4. High-resolution TEM images and DFT model of asymmetrical (a–d) and symmetric (e–h) twin boundaries in B4C, reproduced from Ref. 75. Results showed that asymmetrical twins are the result of a phase boundary from B11Cp-CBC to B11Ce-CBC and that boron-rich B12-CBC can only form symmetric twins.

discontinuity in the pressure–volume curve caused by bending of the 3-atom chain at a critical pressure. The buckling of the 3-atom chain allowed chain atoms to form new bonds with the equatorial B from neighboring icosahedra, leading to a pressure softening of the C44 modulus with increasing hydrostatic load. Grain Boundaries Recently, DFT models for grain boundaries in B4C have focused on special boundaries such as symmetric and asymmetrical twins, which have been well characterized experimentally and are known to activate deformation processes, thereby affecting mechanical performance. A combined TEM and DFT study conducted by Xie et al.75 found that B4C had approximately 30% asymmetric and 70% symmetric twins. Experimentally, TEM determined that the angles between (100) and (010) in the symmetric twin were 73.8° ± 0.3°, whereas in the asymmetric twins these angles were a = 73.8° ± 0.3° on one side of the boundary and a = 72.0° ± 0.4° on the other (see Fig. 4a–d; also observed by Fujita76). Interestingly, Fig. 4e–h shows that when the boron carbide was boron-rich (B13C2), asymmetric twins were absent. Xie et al.75 used DFT to explain the effect of stoichiometry on the formation of asymmetric and symmetric twins, examining the shear translation in a B11Cp-CBC and a B12-CBC twin boundary. After applying shear in the B11Cp-CBC cell (e.g., Fig. 4c), the carbon atom, which was previously in a polar site (Cp),

formed a bond with a carbon atom from the CBC chain; the equatorial Be formed an inter-icosahedral bond with a polar Bp from the neighboring icosahedra, thereby changing the local stoichiometry from B11Cp-CBC to B11Ce-CBC. Upon fully geometric relaxation of the twin boundary, the angles between the (100) and (010) was 73.8° for the B11-CpCBC twin boundary region and 72.2° for the B11Ce-CBC (see Fig. 4d), and the calculated interfacial energy associated with symmetric and asymmetric twins in B11Cp-CBC was 83.2 mJ/m2 and 189.2 mJ/m2, respectively. The DFT simulations showed that the observed asymmetric twins in B11Cp-CBC are actually phase boundaries between two different boron carbide polytypes. Moreover, DFT showed that shear applied to boron-rich B12-CBC does not produce a change in the atomic occupancy of the twin boundary. Therefore, without the presence of C in a polar site within the B12 icosahedra, it is not possible to create an asymmetrical twin; thus, all twins observed in B12-CBC are symmetric twins, in agreement with experimental findings. Classical Atomistic Simulations and Interatomic Potentials Molecular statics and molecular dynamics simulations of boron-based ceramics have focused on the properties and mechanisms of bulk material and have explored only a select few nanostructures (e.g., boron nitride nanotubes).77–80 Although interfaces are important in nanotubes and other two-dimensional structures, these are not representative of


grain boundaries in bulk ceramic materials that would enhance mechanical (and ballistic) properties. Additionally, classical atomistic studies of grain boundaries and interfaces in boron-based ceramics have been limited by the accuracy of the interatomic potentials when modeling these very complex structures and interactions. Traditional pair and multibody empirical potentials fall short of capturing the correct atomic structure and properties. However, more recent reactive potentials (i.e., ReaxFF81) are showing promising results in capturing the correct thermodynamic structure and defect phenomena in select boron ceramic systems. Historically, the majority of classical atomistic studies in boron ceramics have focused on boron nitride (BN). The thermodynamically stable layered hexagonal-BN (h-BN) structure and the extremely hard cubic-BN (c-BN) structure have been modeled using several parameterizations of Tersoff-like potentials.82–84 These simulation results are often limited to modeling h-BN, c-BN and BN nanotubes due to their focused parameterization and the restrictive form of the potential. A BN ReaxFF potential parameterized by Han et al.85 was developed later using DFT-calculated bond and reaction energies, crystalline structural parameters, and relative energetic stability. The more detailed bond order description of the atomic interactions and dynamic optimization of elementary charge86,87 enable ReaxFF to successfully model the complex effects of hydrogen on BN nanotubes.88 In terms of B–C and B–O systems, very few potentials have been developed for bulk ceramic properties in these systems. Recently, a ReaxFF potential was parameterized for the B4C structure by An et al.89 to study the mechanisms of shear amorphization within the bulk; this potential agrees well with quantum mechanical and experimental results. Other interatomic potential parameterizations with B and C-O have been developed for boron glasses in complex multi-component systems where B acts more as an additive. For instance, Maranas et al.90 developed a B–O–H interatomic potential, using pair interactions driven by Coulombic forces, to study polarization and boroxol rings in B-O. The optimized potential incorporated bond ordering for chemical bonding between B and O. Overall, there has been a lack of potential development that aims to capture the structural complexity present in B6O, but some of these potential formulations have shown promise. Mesoscale Modeling and Beyond: Microstructure Evolution Many of the computational efforts at the mesoscale and above have focused on continuum-scale methods due to the limited understanding of microstructure evolution from atomic-scale simulations. At present, many of the applications of interest and experimental studies have centered on the macroscopic performance

of these armor ceramics, e.g., ballistic impact. It is important to note, though, that many of the failure mechanisms are tied directly to their structure, thermodynamics, and kinetics at the nanoscale which are not specifically modeled nor well understood through simulation. Nonetheless, these continuum studies provide a useful understanding of how realistic microstructures perform under extreme environments. Constitutive Models The majority of computational work on boronbased ceramics has been on the continuum length scales. Many have taken an approach that considers the bulk reaction and performance of these ceramics without consideration of their microstructural detail.91–94 These works often assume that untextured polycrystalline boron ceramics can be described as anisotropic elastic materials,95 which is likely not a valid assumption given that experimental observations confirm that it is highly textured. Moreover, these models do not account for the intrinsic properties of the microstructure (including the grain boundaries and interfaces between grains, different phases, and at pores) which can heavily influence the fracture toughness of the material, as previously discussed. That being said, these continuum models have shed light on our understanding of the failure behavior in boron-based armor systems. For example, Clayton96,97 and collaborators98 have developed nonlinear constitutive elastic and inelastic models to simulate the shock response of B4C. These models have been applied to single96 and polycrystalline96–98 B4C by applying shock compression to simulate ballistic impact. These mesoscale models captured important features observed from DFT calculations, such as the elastic instability due to compressive loading parallel to the three-atom inter-icosahedral chain in B4C. Detailed B-instability studies show that the ‘‘B-criterion’’ is a good predictor of failure in shocked B4C. This agrees with DFT calculations99 that show that the B-instability associated with decreasing shear stiffness (B44 and/ or B55) leads to a vanishing eigenvalue. Therefore, the B-criterion has been employed as a threshold for failure activation due to intrinsic instability of the structure, assuming that failure presents itself as amorphization of the ceramic.97 Failure due to the crystalline-to-amorphous transition was studied by performing shock simulations in two similar microstructures: mixed polymorphs (B11Cp-CBC and B4C), and a pure B11-CpCBC material. Shock impacts were simulated by applying uniform pressure boundary conditions on the top face of the simulation domain. Using intrinsic instability considerations to simulate amorphization, Clayton showed that complete grains become unstable (i.e., amorphous) in the mixed microstructure (see Fig. 5a and b; the red region in Fig. 5b indicates a transition to amorphous region). On the


Fig. 5. Mesoscale models of polycrystalline B4C, reproduced from Ref. 97, show the formation of amorphous regions in mixed-polytype systems (a, b) occurs under less stress than in a purely polar B4C polytope (c, d).

other hand, the purely polar microstructure was observed to transition in areas of concentrated axial stress and along grain boundaries (see Fig. 5c and d). The simulation results suggest that purely polar microstructures have a higher average stiffness and are more tolerant to stress-induced amorphization, which is encouraging as the polar polymorph is believed to be most abundant.100 While these continuum simulations have captured important physical phenomena, there are intrinsic limitations to the model. The most obvious is resolution. The aforementioned constitutive model has a minimum resolution of tens of nanometers, while the amorphous regions (glassy phase bands) observed from experiments are on the order of a few nanometers. Furthermore, this model does not address the thermal effects (e.g., temperature rise during shock loading), which could be important in shear localization. Mesoscale Models To address resolution challenges in continuum models, Clayton incorporated the nonlinear model into a phase field model formulation.101 The phase field model shows that amorphization during quasistatic loading only takes place under a shear strain loading condition (i.e., no failure observed for hydrostatic loading), in agreement with DFT calculations. However, the shear strains required for

amorphization are half of those reported by DFT. In this respect, the phase field model may be more realistic as (brittle) ceramics are unlikely to reach > 10% strains. Moreover, Clayton’s mesoscale model was expanded to account for elastic anisotropy, nonlinear elasticity, thermoelastic coupling, and various inelastic deformation mechanisms (i.e., fracture, pore crushing, bulking, and stress-induced amorphization).98 This extension to the model captures the elastic–fracture–granular flow transitions as outlined by the shock stress versus volume change (Hugoniot) curve. Furthermore, the simulation results of shock along the c-axis show that poled anisotropic microstructures (aligned along the caxis) are more likely to fail than strain localization of axial and/or shear stresses (i.e., conditions causing amorphization). It should be noted that the model requires that porosity during the granular flow regime be kept at very small concentrations (