PHYSICAL REVIEW B 99, 134105 (2019) Strong anisotropy in strength and toughness in defective hexagonal boron nitride Tousif Ahmed, Allison Procak, Tengyuan Hao, and Zubaer M. Hossain * Laboratory of Mechanics & Physics of Heterogeneous Materials, Department of Mechanical Engineering, University of Delaware, Newark, Delaware 19716, USA (Received 9 September 2018; revised manuscript received 21 March 2019; published 17 April 2019) Using a combination of density functional theory calculations and molecular dynamics simulations we show that the strength and toughness of hexagonal boron nitride (hBN) containing isolated vacancy defects are strongly anisotropic, regardless of the size of the defect core. The degree of anisotropy is preserved for a number of defect structures including monovacancy, tetravacancy, tridecavacancy, triheptacontavacancy, or heptatriacontavacancy defects. The chirality-dependent effects are strongly nonlinear and well characterized by close-form mathematical equations indicating pronounced strength and toughness along the zigzag direction compared to the strength and toughness along the armchair direction. Also, the size dependence of the strength and toughness of the defective lattice shows an inverse relationship with the effective diameter of the defect core. An atomistic analysis of the deformation fields reveals that nonuniformity in bond length, bond strain, and force distribution in the nonlinear regime of mechanical deformation surrounding the defect cores forms the physical basis for the observed anisotropy. The anisotropic character of the lattice is governed primarily by the nearest-neighbor covalent interactions (dominated by the first-nearest neighbors). Consequently, as soon as a set of bonds rupture at the defect core, the entire lattice undergoes catastrophic failure underscoring the brittle nature of the fracture state in hBN. Results also suggest that chirality-dependent elastic effects are dominated by the third-order elastic modulus which stiffens the lattice at higher chiral angles, whereas the second- and fourth-order elastic moduli soften the lattice affecting the strength and toughness of the lattice in an intricate manner. DOI: 10.1103/PhysRevB.99.134105 I. INTRODUCTION Hexagonal boron nitride (hBN) attracts special attention due to its inertness in chemical environments, its thermal stability at higher temperatures, its electrical insulating prop- erties, and more importantly its extraordinary compatibility with other two-dimensional materials (such as graphene) [1–5]. In single-layer hBN, observations of holes or vacancies of discrete sizes and shapes through high-resolution trans- mission electron microscopy have been widely reported in the literature [6–10]. Not only is it inevitable from available scalable synthesis and growth processes that vacancy defects will appear in hBN sheets, they can also presumably be controlled in the future to design and engineer the effective behavior of hBN in a variety of applications. There is a wide body of literature that shows the formation mechanisms of these defects as well as their effects on electronic, optical, and chemical properties in hBN [11–15]. The mechanical behavior of pristine hBN has also been a subject matter of active research in the recent past [16–21]. Nonetheless the implications of vacancy defects on extreme mechanical prop- erties (such as strength, toughness, and higher-order elastic moduli) along different chiral directions that regulate the directional mechanical and thermal response of a solid at finite deformation remain mostly unexplored for hBN. As far as the linearly elastic behavior is concerned, it is now well known that hBN behaves as an isotropic media [18]. It is * zubaer@udel.edu however unclear if the isotropic character is preserved in the presence of defects or if the stiffness is improved or degraded by the vacancy defects due to the suppression of structural fluctuations (as seen in graphene, which is phenomenologi- cally similar to hBN in terms of the mechanical behavior). Likewise there is a substantial lack of information on higher- order elastic moduli in hBN, and their dependence on material defects continues to puzzle scientists and engineers. Relevant to the extreme mechanical behavior of defective hBN, a number of polycrystalline or grain boundary struc- tures and domains with initial cracks have been investigated [22–30]. It has been revealed from first-principles simulations that the misorientation angle at the grain boundary adversely affects the ideal strength of the lattice [31]. Moreover, from atomistic simulations, grain size has been found to reduce the strength of the material substantially [22,23], and temperature and extended defects have been attributed to causing the alter- ation of strength in polycrystalline hBN [24]. In the context of fracture, it is revealed that the criterion for propagation of an existing crack in hBN is direction dependent [25–30], albeit the information about the condition for nucleation of the crack and its atomistic basis for loading along an arbitrary direction are yet to emerge. Apart from the extended defects, spatially confined smaller defects such as the vacancy defects (which are created by irradiation damage during the fabrication of the boron nitride monolayer [6,10,32]) have been studied with a focus on understanding their formation energies and migration path- ways [12,33–35]. Confirmation of monovacancy as one of the stable defects in hBN motivates a detailed exploration of its 2469-9950/2019/99(13)/134105(17) 134105-1 ©2019 American Physical Society